Handbook of Measurement in Science and Engineering Volume 2 [2] 978-1-118-38464-0, 978-1-118-38463-3

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HANDBOOK OF MEASUREMENT IN SCIENCE AND ENGINEERING

HANDBOOK OF MEASUREMENT IN SCIENCE AND ENGINEERING Volume 2 Edited by

MYER KUTZ

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per‐copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750‐8400, fax (978) 750‐4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748‐6011, fax (201) 748‐6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762‐2974, outside the United States at (317) 572‐3993 or fax (317) 572‐4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging‐in‐Publication Data: Handbook of Measurement in Science and Engineering / Myer Kutz, editor. volumes cm Includes bibliographical references and index. ISBN 978-0-470-40477-5 (volume 1) – ISBN 978-1-118-38464-0 (volume 2) – ISBN 978-1-118-38463-3 (set) – ISBN 978-1-118-64724-0 (volume 3) 1. Structural analysis (Engineering) 2. Dynamic testing. 3. Fault location (Engineering) 4. Strains and stresses–Measurement. I. Kutz, Myer. TA645.H367 2012 620′.0044–dc23 2012011739

To Jayden, Carlos, Rafael, Irena, and Ari. Watch them grow.

CONTENTS

VOLUME 2

CONTRIBUTORS

xxiii

PREFACE

xxvii

PART IV 31

MATERIALS PROPERTIES AND TESTING

Viscosity Measurement

945 947

Ann M. Anderson, Bradford A. Bruno, and Lilla Safford Smith

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viii

CONTENTS

0DMRUYLVFRVLW\PHDVXUHPHQWPHWKRGV 'UDJW\SHYLVFRPHWHUV %XEEOHWXEH YLVFRPHWHUV 5RWDWLRQDOYLVFRPHWHUV )ORZW\SHYLVFRPHWHUV 2UL¿FHW\SHFXS YLVFRPHWHUV 9LEUDWLRQDOUHVRQDQW YLVFRPHWHUV $670VWDQGDUGVIRUPHDVXULQJYLVFRVLW\ 4XHVWLRQVWRDVNZKHQVHOHFWLQJDYLVFRVLW\PHDVXUHPHQW WHFKQLTXH 5HIHUHQFHV 32 Tribology Measurements Prasanta Sahoo ,QWURGXFWLRQ 0HDVXUHPHQWRIVXUIDFHURXJKQHVV 6XUIDFHSUR¿ORPHWHU 2SWLFDOPLFURVFRS\ $GYDQFHGWHFKQLTXHVIRUVXUIDFHWRSRJUDSK\ HYDOXDWLRQ 0HDVXUHPHQWRIIULFWLRQ ,QFOLQHGSODQHULJ 3LQRQGLVFULJ &RQIRUPDODQGQRQFRQIRUPDOJHRPHWU\ULJ (QYLURQPHQWFRQWURO 7HFKQLTXHVIRUIULFWLRQIRUFHPHDVXUHPHQW 0HDVXUHPHQWRIZHDU 0HDVXUHPHQWRIWHVWHQYLURQPHQW 7HPSHUDWXUHPHDVXUHPHQW 7KHUPRFRXSOHV 7KLQ¿OPVHQVRUV 5DGLDWLRQGHWHFWRUV 0HWDOORJUDSKLFREVHUYDWLRQ /LTXLGFU\VWDOV +XPLGLW\PHDVXUHPHQW 0HDVXUHPHQWRIR[\JHQDQGRWKHUJDJHV 0HDVXUHPHQWRIPDWHULDOFKDUDFWHULVWLFV +DUGQHVV t4>t3…)

(a) A Fluid

(b)

Shear stress t3 t t1 t 2

4

Fluid

t4

t2, t3,t4 Solid

A

Shear stress reset to zero at t4 t1

Shear stress

t1

t5…

Shear stress reset to zero at t4 t5… Solid

FIGURE 31.1 Fluid (viscous) versus Solid (elastic) behavior. (a) Fluid Behavior: A thin layer of fluid held between two parallel plates, the top plate caused to move relative to the bottom plate by application of a shear stress, will continue to deform (experience shear strain) as long as the shear stress is applied. The rate of shear strain is related to the magnitude of the shear stress. When the shear stress is removed the fluid will remain in its deformed state. (b) Solid (Elastic) Behavior: A solid will deform (experience shear strain) through some fixed angle when a shear stress is applied. The amount of shear strain is related to the magnitude of the shear stress. When the shear stress is released the solid will return to its original form.

plane of application. If you slide your open hand over a desk top the friction between your hand and the desk will create a shear stress on your hand.) As illustrated in Figure 31.1, a solid (within its elastic limit) will deform through some limited angle while a shear stress is applied, and will then return to its original configuration when the stress is removed. A fluid on the other hand will deform continuously as long as a shear stress is applied, and will not return to its original shape when the stress is removed. For a fluid the rate of shear deformation is related to the magnitude of the shear stress applied, while for a solid the amount of shear deformation is related to the magnitude of the shear stress applied. This seemingly innocuous mechanical difference in reaction to shear stresses gives rise to the large difference in character between solids and fluids. In fact it gives rise to all of the behaviors that one thinks of as inherently “fluid”: the ability to flow, the ability to fill volumes of arbitrary shape, the ability to spread out and “wet” certain surfaces, etc. It also gives rise to the complex nature of fluid mechanics because it allows for the very large material deformations that in turn give rise to phenomena like turbulence. Viscosity (or more precisely the shear viscosity, defined below) is the material property that defines the quantitative relation between the applied shear stress and the shear deformation rate in a fluid. Qualitatively the viscosity indicates the “thickness” or resistance to flow of a fluid. Since viscosity is the property that controls and quantifies the shear stress/shear rate behavior that is definitional to fluids, it is in many regards the most important physical property of a fluid.

COMMON UNITS OF VISCOSITY

949

Unfortunately, as alluded to above, the term “viscosity” is actually used to denote several related, but different, physical properties. It is important to understand these distinctions in terms from the outset. First, the term “viscosity” is most commonly used in conjunction with effects arising from shear forces and shear deformations in fluids. When used in this context, the most common one, the property is more precisely called the “shear viscosity” or the “first coefficient of viscosity.” However, when used in this sense, it is almost always simply referred to as “viscosity.” This is contrasted with the “bulk viscosity,” associated with volume dilatation. Bulk viscosity is rarely an important parameter and hence is not as well known or understood as the more common shear viscosity. Bulk viscosity is discussed briefly in Section 31.1.3. Second, it should be noted that even the “shear viscosity” described above is often stated in two different forms, the absolute or dynamic viscosity, m, and the kinematic viscosity or momentum diffusivity, n, where n ¼ m/r and r is the fluid’s density. Although the dynamic and kinematic viscosities are clearly related properties, they are dimensionally dissimilar and it is critically important to always distinguish between them. More is said on the distinction between dynamic and kinematic viscosity in the following section on common viscosity units. The remainder of this chapter begins by discussing the units in which viscosity is measured. Then the distinction between the larger field of rheology and its subfield viscometry is made in the context of differentiating between the so-called Newtonian and non-Newtonian fluids. After that the chapter provides a brief theoretical and mathematical overview of viscosity. Finally, the majority of the chapter provides detailed and practical information on methods for measuring viscosity.

31.2 COMMON UNITS OF VISCOSITY There are several systems of units used with viscosity; many of them are archaic and/or closely tied to one specific viscosity measuring technique (e.g., the Saybolt cup and the “Saybolt Universal Second,” and the Krebs unit) or one particular industry (e.g., SAE oil grade and the automotive industry). It is impossible to capture all of these systems in one document, but an attempt is made below to define and relate the most common and standard units associated with viscosity measurement.

31.2.1 Absolute Viscosity, m In terms of the SI (Le Systeme Internationale d’Unites) system of fundamental units the derived units for absolute viscosity, m, are kg=m  s which is equivalent to Pa  s (Pascal-seconds). This grouping of units has not received a name of its own. In the closely related cgs (centimeter, gram, and second) system of units, the derived unit of g=cm  s or dyne  s=cm2 is called a “Poise” (after Poiseuille). More commonly a centipoise, cP ¼ 1/100th of a Poise is used. In the FPS (foot, pound, and second) system of units, the units of absolute viscosity are lbF  s=in2 , which is called the Reyn (after Osbourne Reynolds). Refer to Table 31.1 for a collection of units of absolute viscosity.

950

VISCOSITY MEASUREMENT

TABLE 31.1 Units for Absolute/Dynamic Viscosity, m Units System SI cgs English (FPS)

Derived Viscosity Unit
kg or ms Pa  s g =cm s
dyne  s lbF  s




Unit Name

Equivalence

none

1 Pa  s ¼ 10 Poise ¼ 1000 Centipoise

Poise

100 Centipoise ¼ 1 Poise

Reyn

1 Reyn ¼ 68,948 Poise

or

cm2

in2

31.2.2 Kinematic Viscosity, n Recall that the dynamic (kinematic) viscosity, n, is defined as the absolute viscosity divided by the fluid density, r. In the SI system of fundamental units the units for kinematic viscosity are meter square per second, which is not a named grouping. It should be noted that the units of kinematic viscosity (m2/s) are identical to the units of thermal diffusivity used in heat transfer, and mass (species) diffusivity used in diffusion. This leads to the kinematic viscosity being referred to as the coefficient of momentum diffusivity by analogy. In the cgs system the unit of kinematic viscosity is the centimeter square per second called the “Stokes” (after G.G. Stokes). More commonly the kinematic viscosity is given in centistokes (cSt) where 100 cSt ¼ 1 Stokes. In the FPS system kinematic viscosity would be foot square per second or inch square per second, neither of which is a named unit. 31.2.3 Nonstandard Units Kinematic viscosity is also often given in “Saybolt Universal Seconds” or SUS (also sometimes SSU “Saybolt Seconds Universal” or SUV “Saybolt Universal Viscosity), which is directly related to the Saybolt viscosity cup measuring system (see Section 31.2.2). Of course, the unit of “seconds” is not a dimensionally correct unit for the physical quantity of kinematic viscosity, so this system is problematic. The Saybolt measurement system is based on ASTM method D88 and measurements in SUS can be converted into more standard (dimensionally correct) viscosity units using procedures provided in ASTM 2161. There are countless other such “legacy” scales of viscosity associated with different industries, and unfortunately there is often no standard method for converting these legacy measures into dimensionally correct viscosity units. A number of online viscosity converters exist (see www.coleparmer.com, www.gardco.com, or www.cannon.com, for example) (Table 31.2). 31.2.4 Distinction Between Rheology and Viscometry A simple linear relationship between shear stress and shear strain rate is observed in a wide variety of fluids (Figure 31.2a). The constant slope of the line labeled Newtonian is TABLE 31.2 Units for Kinematic Viscosity, n Units System

Derived Viscosity Unit

SI cgs English (FPS)

m2 =s cm2 =s in2 =s; ft2 =s

Unit Name None Stokes None

Equivalence 1 m2 =s ¼ 10; 000 Stokes 100 Centistokes ¼ 1 Stokes 1 in2 =s ¼ 645:16 Centistokes ¼ 0.00694 ft2 =s

COMMON UNITS OF VISCOSITY

(a) Shear stress

951

(b) Bingham plasc

Shear stress

Constant strain rate applied Rheopecc

Dilatant Newtonain const. slope = µ

Time invariant Thixotropic

Yield stress

Pseudoplasc

Shear strain rate

Time

FIGURE 31.2 Newtonian and non-Newtonian fluid behavior (a) shear stress vs. strain rate (b) shear stress vs. duration of applied strain rate.

the (shear) viscosity of the fluid. Fluids demonstrating such a relationship are known as Newtonian fluids. Many common fluids like air, gases in general, water, or simple oils demonstrate Newtonian behavior meaning constant viscosity with respect to strain rate over a very wide range (many orders of magnitude) of strain rates. The measurement of the shear viscosity of Newtonian fluids is referred to as viscometry and is the focus of this chapter. Fluids with more complicated molecular structures (e.g., polymers) or fluids with other phases suspended in them (e.g., mixtures, slurries, and colloids) often demonstrate more complicated shear stress to strain rate behaviors (see Figure 31.2a). Fluids exhibiting such behaviors are broadly characterized as “non-Newtonian” fluids. Non-Newtonian fluids can be further classified according to how they react to changes in shear deformation rates, to the duration of application of the applied loading, and to whether or not they exhibit a threshold elastic (solid-like) shear resistance prior to deforming like a fluid. Fluids that show increasing apparent viscosity (the apparent viscosity is the local slope of the stress versus strain rate curve, see Figure 31.2a) as the applied strain rate increases are called “shear thickening” or dilatant fluids. The classic example of a shear thickening fluid is a mixture of cornstarch in water. If one attempts to shear this fluid quickly (e.g., hit it with a hammer) the viscosity will rise to such a level that the fluid seems almost solid, the hammer blow will bounce off the surface. Yet at lower shear rates the mixture will act like a “normal” fluid (e.g., a hammer set on its surface would sink right into the fluid). Fluids which show the opposite behavior (decrease in apparent shear viscosity with increasing strain rate) are called “shear thinning” or pseudo-plastic fluids. A common example of a shear thinning fluid would be “no drip” paint, which behaves as a fairly thick (viscous) fluid while adhering to a paintbrush (a low shear rate circumstance), but which spreads easily (i.e., exhibits lower viscosity) when the paintbrush is dragged along a surface thereby increasing the shear strain rate applied to the fluid (Cengel and Cimbala, 2010). Some fluids will “thin” (produce a lower shear stress resisting the motion) or some will “thicken” (produce a higher shear stress resisting the motion) as the duration for which a constant strain rate is applied increases, (see Figure 31.2b). Fluids exhibiting the former behavior are referred to as “thixotropic” the latter as “rheopectic.” Such fluids are also sometimes referred to as “time-thinning” or “time-thickening” fluids. Examples of thixotropic fluids include yogurt and some classes of paint. Rheopectic behavior is much rarer.

952

VISCOSITY MEASUREMENT

Examples include gypsum paste and printers ink. Newtonian fluids exhibit constant strain rate with regard to loading duration for a constant applied shear stress. Newtonian fluids will exhibit constant strain rate to shear stress behavior down to very low (theoretically zero) applied shear stresses. However some fluids, called “Bingham plastic” fluids will initially show “solid-like” behavior until a threshold shear stress (called the “yield stress”) is applied; after which they will show “fluid-like” behavior (continuously deforming while the shear stress is applied) (see Figure 31.2a). A common example of this type of fluid is toothpaste, which will not flow at all until a threshold shear value is exceeded. Broadly this kind of behavior is described as viscoelasticity. Bingham plastic materials can show dilatant, Newtonian, or pseudo-plastic behavior after their yield point. It is also worth noting that these terms are often not consistently applied. The study and measurement of these more complicated, non-Newtonian, shear stress/shear strain rate behaviors is a subset of the larger science referred to as rheology and is largely beyond the scope of this chapter. 31.2.5 Mathematical Formalism In this section we develop the mathematical formulations governing viscosity, and explain the roles and relations between “shear” viscosity and “bulk” viscosity. The discussion begins with the consideration of a very simple one-dimensional (1D) flow situation, and then introduces the more general 3D form of the equations. Shear viscosity is defined mathematically by Newton’s Law of Viscosity. Newton’s law defines viscosity as the physical property that relates the shear stress produced as a reaction to an applied strain deformation rate. If one considers the simple flow shown in Figure 31.1a, a thin layer of fluid confined between two infinite parallel flat plates the upper moving within its own plane relative to the lower, the total shear strain rate on any layer in the fluid is given by @u=@y which is the rate of change of x direction velocity in the perpendicular (y) direction. Newton’s law states that the shear stress experienced on the lower face of such a layer is given by: t ¼ m

du dy

ð31:1Þ

where m is the (shear) viscosity of the fluid at the applied strain rate. If the shear viscosity is constant with regard to strain rate then the fluid is said to be “Newtonian.” If the fluid exhibits a more complex relationship between shear stress and strain rate the fluid is defined as “non-Newtonian.” Distinctions between Newtonian and non-Newtonian behavior is discussed in greater detail in Section 31.1.2. The viscosity picture becomes more complicated if we allow for more complex (3D) motions of the fluid. Any motion that a “particle” of fluid can undertake can be constructed from a superposition of the following four simpler types of motion: pure translation (movement without rotation or deformation), pure rotation (rotation without movement or deformation), pure linear strain (deformation without motion that does not disturb angles within the fluid particle), and pure shear strain (deformation without motion which does change angles within the fluid particle). Figure 31.3 illustrates these types of motions in two dimensions. In a fully general flow any combination of these motions can occur in any or all of the three coordinate directions. In such cases independent shear deformations (Figure 31.3d) can occur on any or all of three orthogonal planes.

COMMON UNITS OF VISCOSITY

(a) Translaon

953

(b) Rotaon

(c) Linear strain

(d) Shear strain

FIGURE 31.3 Types of fluid motion and deformation illustrated in 2-D.

The fluid can also undergo purely extensional deformations (i.e., elongations without shear, Figure 31.3c) in any or all of the three dimensions. These extensional deformations will in some circumstances also contribute to the stress response of the fluid, for example, if they combine in such a way that the volume of the fluid element changes then the “bulk viscosity” described below will also be important. Thus for general 3D deformations it is necessary to use tensors (a branch of mathematics that describes vectors pointing in several directions) to describe the full relation between the stress at a point in the fluid and the resulting strain. The stress response of a Newtonian fluid element in response to a fully general deformation is given in White (2006) as tij ¼ Pdij þ m



@ui @uj þ @xj @xi



VÞ þ dij gðr  ~

ð31:2Þ

Here m is the shear viscosity, g is the “Bulk Viscosity” coefficient and dij is the Kronecker delta function (i.e., d ¼ 1 when i ¼ j, and d ¼ 0 when i 6¼ j), and i and j are indices used to refer to the three orthogonal planes. Those interested in the derivation of this relation are directed to White or any other graduate level fluids text. One important point to note arising from Equation (2) is that the stress response to purely extensional strains is described by the bulk viscosity, g, or through the closely related “second coefficient of viscosity” m’ (the bulk viscosity, g is equal to the second coefficient of viscosity, m’, plus 2/3 m, see Owczarek (1964), for example, for a more thorough discussion). The topic of bulk viscosity is largely beyond the scope of this chapter, but these few comments are made to inform the reader of when it may be safely ignored, and when it may become important. First it is important to note that the bulk viscosity is not as well understood or characterized as the more common shear viscosity. Fortunately, for Newtonian fluids, the bulk viscosity coefficient occurs only in combination with the divergence of the velocity field ðr  ~ V Þ; and for incompressible fluids conservation of mass (i.e., continuity) requires that r  ~ V ¼ 0: Hence the bulk viscosity will play no role in a truly incompressible fluid. Of course no substance is truly 100% incompressible, so if one is concerned with acoustic issues (which are inherently a compressibility phenomenon) or flows at significant Mach number the bulk viscosity may play a role. One way to gain insight on when the bulk viscosity may play a significant role is to examine its role in viscous dissipation.

954

VISCOSITY MEASUREMENT

Ultimately the role of viscosity (shear or bulk) is to irreversibly convert the mechanical energy in a flow into thermal energy (heat). This effect is known as “viscous dissipation.” Hence the magnitude of the dissipation caused by the viscosity can give one indication of the qualitative importance of viscosity in the flow. The dissipation (F) caused by the bulk viscosity is given by Shaughnessy et al. (2005) as: Fbulk

  g dr 2 ¼ 3 r dt

ð31:3Þ

where g is the “bulk viscosity” and r is the density. In order to be significant, either large changes in density are required, or the changes in density must occur over very short time scales. Thus, due to the relative magnitudes of the extensional strains typically involved, the dissipation due to bulk viscosity (and indeed the bulk viscosity itself) may safely be ignored for almost all practical applications except shock waves (which involve large changes in density) and attenuation of high frequency (small dt) sound or “ultrasound”. Henceforth in this chapter the term “viscosity” shall refer to “shear viscosity” only.

31.2.6 Relation of Viscosity to Molecular Theory As discussed above, ultimately the role of viscosity is to dissipate the ordered kinetic energy associated with the macroscopic motions of a flow to disordered, randomly distributed, microscopic molecular energy (i.e., thermal energy). As such it becomes clear that the quantity that we refer to as viscosity is the macroscopic manifestation of molecular level effects; much in the same way that the macroscopic quantity of “pressure” represents the average net force per unit area caused by countless individual molecular collisions on a surface. That is to say that viscosity, like pressure, temperature, and density, is a “continuum property” of a substance. It makes no more sense to talk about the “viscosity” of a single molecule than it does to talk about the “density” of a single atom of a gas. The fact that the viscosity is an emergent property arising only for large collections of molecules places an important limitation on the concept of fluid viscosity, namely the continuum limit. Essentially the (continuum) concept of viscosity breaks down when applied on length scales that are comparable to the mean free path of the molecules in the fluid, or when applied on time scales that are comparable to the mean time period between molecular collisions in the fluid. For ordinary macroscopically sized flows at ordinary pressures the continuum limit is not a concern. However in applications such as nanotechnology (where the length scales of concern become very small) or rarefied gas dynamics (where the mean free path of molecules in very low density gases becomes very large) this limit should be kept in mind. Examining how the macroscopically observed property of viscosity arises from molecular effects can provide insight and physical intuition about viscosity. If we examine how momentum is transported by the thermal (random) motions of molecules within a flow we can come to understand the molecular basis for viscosity. First, let us consider the physics qualitatively. To do so consider again the simple 1D shear flow shown in Figure 31.1. Specifically consider the molecules near a plane parallel to the top plate and half way between the top plate and the bottom plate. All of these molecules will have velocities that are composed of their individual, random (thermal)

COMMON UNITS OF VISCOSITY

955

velocities (which average out to zero bulk velocity), plus a small extra velocity component which depends on the local value of the bulk velocity. Those molecules slightly above the plane will, on average, have slightly larger values of velocity in the x direction and therefore also have slightly higher values of x direction momentum. Similarly, on average, those below the plane will have slightly lower values of x velocity and momentum. All of the molecules (above and below the plane) will have random thermal velocities with components in all three directions. The y component of these random velocities will occasionally carry molecules across the plane in both directions. But because of the asymmetry in velocity above and below the plane (i.e., because of the gradient of velocity perpendicular to the plane) the net effect of these random cross plane exchanges will be to transport x momentum from above the plane to below the plane. That is, there will be a net flux of x momentum in the negative y direction. In fact the quantity we call “viscosity” is precisely defined by this “diffusive” transport (i.e., transport caused by random molecular motions rather than bulk macroscopic fluid motions) of momentum in the direction opposite to a velocity gradient. A simple dimensional analysis will convince the reader that a stress, such as shear stress (measured in Pa ¼ kg/m s2), is dimensionally identical to a flux of momentum (transport of momentum per second per unit area ¼ kg/m s2). Thus, referring back to Equation (1), we can consider viscosity to be the fluid property that relates the diffusive momentum flux (t) to the velocity gradient driving it.

31.2.7 Effect of Pressure and Temperature on Viscosity 31.2.7.1 Low Density Gases We can make this relationship more quantitative by examining the actual molecular interactions occurring in the fluid. The following development parallels that given by Bird et al. (1960), but similar developments can be found in any text covering the kinetic theory of gases. The simplest model of viscosity arises from a consideration of simple kinetic theory of a low density gas where we assume that the gas molecules are rigid spheres with diameter “d” that only interact through collisions (i.e., there are no forces causing “action at a distance” between molecules). Thus molecules will exchange momentum and come into equilibrium with their surroundings only through collisions. Kinetic theory for rigid sphere molecules also provides these other important results that will be used in this development: 

The average random (thermal) velocity of the molecules in the gas will be
¼ V

rffiffiffiffiffiffiffiffiffi 8kT pm

ð31:4Þ

where k is Boltzman’s Constant, m is the mass of the individual molecules and T is the absolute temperature.  The mean free path of a molecule between collisions, l, will be 1 l ¼ pffiffiffi 2 2pd n

where n is the number density (number per unit volume) of the gas molecules.

ð31:5Þ

956 

VISCOSITY MEASUREMENT

The average frequency of collision per unit area (from one side) on any plane in the flow 1
Z ¼ nV 4

ð31:6Þ

If one again considers the situation of a plane between the two plates in Figure 31.1 it can be shown that a molecule will travel an average vertical distance of 2/3 l between collisions. The flux of x momentum transferred across the plane from below is then Zmvx,y-2/3l where vx,y-2/3l is the average excess (nonthermal) x velocity component at a y location “one collision distance” below the plane (i.e., at y-2/3l). Similarly the flux of x momentum transferred across the plane from above is, Zmvx,yþ2/3l. Thus the net x momentum flux is: t ¼ Zmvx;y2=3l  Zmvx;yþ2=3l

ð31:7Þ

And, if the x velocity gradient in the vicinity of y is linear we can replace the vx at locations above and below the plane with a first order Taylor series in terms of the gradient at the plane, yielding:   2 dvx 2 dvx t ¼ Zm ðvx;y  l Þ  ðvx;y þ l Þ 3 dy 3 dy

ð31:8Þ

Simplifying and substituting in the definition of Z gives: t¼

1
dvx nmVl 3 dy

ð31:9Þ

Comparing back to Equation (31.1) we can see that: 1
m ¼ nmVl 3

ð31:10Þ


and l yields: Further substitution for V 2 1 m¼ 3 p3=2

pffiffiffiffiffiffiffiffiffiffi mkT d2

ð31:11Þ

Thus p ifffiffiffithe ffi molecules are considered as perfectly rigid spheres (the simplest model) then m  T . Note also that the viscosity of such a gas is not expected to be a function of pressure based on this very simple molecular model. This is an important result which is largely borne out by experimental observations of real low-density gases; their dynamic viscosity is observed to depend only very weakly on pressure. Of course molecules are not perfectly rigid spheres for that would imply no force whatsoever between molecules until they come into contact (when their center to center distance equals d) and then an infinite repulsion force. Clearly the idea of an infinite repulsion force is unphysical. In fact all molecules will show some “action at a distance” and, more importantly, act as if they have some “give” or flexibility when they “collide”

COMMON UNITS OF VISCOSITY

957

which will eliminate the unphysical infinite forces inherent in the simple rigid sphere model. Typically these interactions are modeled as an intermolecular “potential” function from which the magnitudes of attractive and/or repulsive forces as a function of center to center distance can be calculated; as can an “effective” molecular diameter. The exact nature of these intermolecular potential functions is complicated, and has received extensive study, but is largely beyond the scope of this chapter. The interested reader is directed to Kogan (1969) or Vicenti and Kruger (1965), for example. Here it is sufficient to make a few points. First, as more complex and precise relations for the potential theory are used, the predictions of macroscopic properties such as viscosity improve markedly. Second, the “action at a distance” effects associated with these potentials allow for pressure to have an effect on the viscosity of gases. But, as stated above, this is typically found to be a weak effect for common gases. And finally, the functional relation between the viscosity and the temperature in low-density gases depends critically on the details of this potential function. The next simplest model of intermolecular relations (after the rigid sphere model) is one proposed by Maxwell. In such a model the potential function drops off as 1/s4 where s is the center to center distance between the molecules. With such pffiffiffiaffi model it can be shown that the “effective molecular diameter” will vary as: d 2  1= T and so, referring back to Equation 11, m  T. That is, a gas composed of “Maxwellian” molecules will have a viscosity that will increase linearly with temperature, as opposed to with the square root of temperature predicted from the rigid sphere model. 31.2.8 Correlations of Viscosity with Temperature for Gases In reality, gases typically exhibit behaviors between the two extremes discussed in the section above (rigid and “Maxwellian”) and so their viscosity’s dependence on temperature is commonly correlated with a power-type law, as given by White (2006): m ¼ mo



T To

n

ð31:12Þ

Where the parameters n, To, and mo are specific to the particular gas. Values for “n” for most simple gas molecules fall between 0.5 and 1, as predicted by the above discussion. Some values for n, To, and mo can be found in White. Another common correlation technique for gas viscosities is based on the work of Sutherland (1893) (as covered in Vicenti and Kruger, 1965). Here viscosity is correlated as m ¼ mo



T To

 3  2 To þ S T þS

ð31:13Þ

where S is the so-called Sutherland Parameter and To, and mo are reference values. A completely equivalent form of this equation: 0 1 3 2 C1 T A m¼@ ð31:14Þ T þS

958

VISCOSITY MEASUREMENT

where mo

C1 ¼

3

To 2

!

ðT o þ SÞ

ð31:15Þ

is often used as well. Poling et al. ( 2004) provided a detailed discussions of several much more detailed methods for estimating the variation of viscosity of both pure gases and mixtures of gases at various temperatures. 31.2.9 Correlations of Viscosity with Temperature for Liquids In real (higher density, more complex molecular structure) gases and especially in liquids intermolecular forces (beyond the “collisional” forces discussed previously) play a critically important role. Molecules in such substances can exert significant force (and hence transfer significant momentum) “at a distance” without colliding. Since viscosity as a property arises from transfers of momentum, these “actions at a distance” must be accounted for in any physical model that hopes to adequately predict a material’s viscosity. The nature and magnitude of these non-collisional interactions are so complex and so large in liquids that currently no one general model exists that will adequately predict the viscosity of all liquids. Instead many specialized empirical and semi-empirical relations are available. Some general trends however are observed for liquids, the most important being that the viscosity of liquids falls off strongly with increasing temperatures. One type of curve fit that is recommended for liquid viscosity, recommended by White (2006) is ln

   2 m T0 T0 ffiaþb þc m0 T T

ð31:16Þ

where a, b, and c are curve fit parameters and To and mo are reference values. Another, slightly simpler, empirical correlation often used is “Andrade’s equation” (Munson et al., 2009) B

m ¼ DeT

ð31:17Þ

which is often presented in the alternate form lnm ¼ A þ

B T

ð31:18Þ

Viswanath et al. (2007) provide a lengthy discussion of such correlation methods and coefficients A and B for a wide variety of liquids.

31.2.10 Effect of Pressure on Viscosity The effects of pressure on viscosity are not nearly as significant as the effects of temperature. In many practical circumstances it is entirely sufficient to simply neglect the effect

MAJOR VISCOSITY MEASUREMENT METHODS

959

of pressure on viscosity. This has lead to pressure effects being much less studied, and to data on viscosity at different pressures being much more sparse. For low density gases, the molecular dynamics models discussed above (refer to Equation 31.10) indicate that the absolute viscosity, m, should not depend on pressure at all; due to the competing effects of increasing number density (n) and decreasing mean free path (l) as pressure is increased. Of course the value of the kinematic viscosity, n ¼ m/r, of gases will decrease with increasing pressure due to the increase in density, r, of the gas as the pressure increases. These predictions are largely borne out by experimental data for common low-density (ideal) gases. Poling et al. ( 2004) provide a detailed discussion of several methods for estimating the viscosities of both pure gases and mixtures of gases at higher pressures. Viscosity data for liquids at high pressure is sparse when compared to the quantity of data available at or near atmospheric pressures. In general, as pressure increases, so does the viscosity of most liquids. Both Viswanath et al. (2007) and Poling et al. ( 2004) recommend a method for estimating the effect of pressure on liquid viscosity attributed to Lucas (1981):   DPr A 1þD m 2:118 ð31:19Þ ¼ mSL 1 þ CvDPr where m is the viscosity of the liquid at pressure P; mSL is the viscosity of the liquid at vapor pressure Pvp; Pvp is the the vapor pressure; DPr ¼ ðP  Pvp Þ=Pc ; T r ¼ T=T C ; Pc ; T c are the critical pressure and temperature; v is the acentric factor 4:674 104 A ¼ 0:9991  1:0523T r0:03877  1:0513 



"

D¼

0:3257 1:0039  T r2:573

#

0:2906  0:2086

C ¼ 0:07921 þ 2:1616T r  13:4040T r 2 þ 44:1706T r 3  84:8291T r 4 þ 96:1209T r 5  59:8127T r 6 þ 15:6719T r 7 In spite of the theory and correlation techniques described above, in many practical situations one must resort to simply measuring viscosity. The following section describes the different measurement techniques available.

31.3 MAJOR VISCOSITY MEASUREMENT METHODS Viscometers are designed to make use of the theoretical relationship between shear stress and strain rate to measure viscosity. They do this using simple flows (1D, steady, fully developed) in which both the shear stress and strain rate can be measured. There are three primary types of viscometers: flow, drag, and resonant. The flow-type viscometers measure the rate of flow of the fluid in a tube or through an orifice. The shear stress can be calculated from theory (e.g., capillary tube viscometer) or estimated based on theory (e.g., orifice cup viscometers). Use of these types of viscometers yields values for kinematic viscosity. Design parameters for flow-type viscometers include minimizing entrance and exit effects, maintaining a constant pressure head (which drives the flow), minimizing surface tension effects and mitigating effects of temperature variation.

960

VISCOSITY MEASUREMENT

Drag-type viscometers measure either the force on an object as it moves at a specified rate in the fluid (rotational viscometers) or measure the time it takes for an object to move a specified distance through the fluid (falling object and bubble tube viscometers). Use of these types of viscometers yields values for absolute viscosity (except for the bubble tube which measures kinematic viscosity). Design parameters for drag-type viscometers include minimizing the effects of turbulence and flow separation through the specification of a flow condition (generally a low relevant Reynolds number), controlling for transients, minimizing surface tension effects and mitigating effects of temperature variation. The third type of viscometer is the resonant or vibrational viscometer which is most commonly used in in-line process applications. These are designed so that changes in the viscous damping bring about significant changes in the resonance behavior of the instrument. Use of these viscometers yields values for kinematic viscosity. This section presents information on each of the three major viscometer types. It begins with the drag-type viscometers (falling object, bubble tube, and rotational) followed by the flow-type viscometers (capillary and orifice) and concludes with a discussion of vibrational viscometers. Each section includes information on the theory of operation, a description of the types of viscometers available, a list of available manufacturers, and the capabilities and advantages/limitations. This chapter focuses on the use of laboratory-type viscometers; however some information is included on the use of process viscometers. This list of viscometers is not intended to be exhaustive but includes many of those that are most readily available commercially. There are other specialized methods for measuring viscosity and the reader is referred to Viswanath et al. (2007) for more information. 31.3.1 Drag-Type Viscometers 31.3.1.1 Falling Object Viscometers Theory of Operation Falling object viscometers determine viscosity (m) by measuring the drag force acting on a falling object under specific flow conditions. Use of the falling object viscometer requires a separate measurement of density to calculate kinematic viscosity. Figure 31.4 illustrates the forces acting on a falling object. This case shows a spherical object (a ball), however there are a variety of falling objects that can be used such as needles, and cylinders. There are three forces acting on the object: FB the buoyancy force and FD the drag force act upwards while FG the gravitation force (weight) acts down. The buoyancy force is calculated using Archimedes principle and is equal to the weight of the fluid displaced by the object: F B ¼ 8B  rf  g

ð31:20Þ

where 8B is the volume of the object, rf is the density of the fluid, and g is the gravitational constant. The weight of the object is simply: F G ¼ 8B  rB  g

ð31:21Þ

where rB is the density of the object. If the object is travelling at terminal speed, the acceleration will be zero and application of Newton’s second law yields an equation relating the three forces:

MAJOR VISCOSITY MEASUREMENT METHODS

961

FIGURE 31.4 Schematic showing the forces acting on a falling object.

F D ¼ F G  F B ¼ 8B  ðrB  rf Þ g

ð31:22Þ

The drag force is composed of a shearing force (due to the fluid) and a pressure force (due to flow separation). Falling object viscometers are generally designed to operate in the Stokes (creeping) flow regime which is characterized by a lack of flow separation and occurs for very low Reynolds numbers (Re < 0.1). In this case the drag force is due only to the shearing force. For example, if the object is a sphere then the Reynolds number is calculated as rf VD Re ¼ ð31:23Þ m Where V is the terminal speed of the object, D is the diameter of the sphere, and m is the viscosity. For the low Reynolds number situation, the drag force is related to Reynolds number by Stokes Law: FD ¼

3pmVD Re

ð31:24Þ

Combination of these equations yields an equation for viscosity in terms of the speed, diameter, and density difference: m¼

 gD2

r  rfluid 18V obj

ð31:25Þ

This theory applies to balls moving at low Reynolds number in an infinite media (see Brizard et al. 2005 for development of the theory accounting for more realistic conditions). In many commercial applications, the falling object is placed in a tube of specified diameter (see Figure 31.5). The object, typically a ball, will fall or slide down the tube and

962

VISCOSITY MEASUREMENT

FIGURE 31.5 Falling ball viscometers (a) Gilmont and (b) Hakke type (Courtesy of Brookfield Engineering Labs).

the user measures the time it takes for the ball to travel between two timing lines. The first timing line is placed sufficiently far from the top of the viscometer to allow the ball to reach terminal velocity. The manufacturer supplies a calibration equation of the sort:

 m ¼ K  t robj  rfluid ð31:26Þ

where K is a calibration constant and t is the measured time to fall the specified distance. To obtain the calibration constant the manufacturer measures the fall time of the ball in a series of liquids of known viscosity.

Types of Viscometers/ Options Falling object viscometers use a variety of objects including spheres, needles, and cylinders. Falling object viscometers often come with a set of objects, each with different mass/density which allows one to measure viscosity over a range of values. The most readily available commercial falling object viscometers are the relatively inexpensive Gilmont-type falling ball viscometers and the more expensive, more accurate Haake-type falling ball viscometers as shown in Figure 31.5a and b. The Haake viscometers include a mounting mechanism and an outer chamber that can be used for temperature control of the sample during testing. Falling needle viscometers use thin needle like objects which are designed to minimize wall effects and are more stable as they fall (see Davis and Brenner, 2001). They can be used to measure viscosity of non-Newtonian fluids. Falling cylinder viscometers involve a more complex flow field subject to significant end effects; however, they are useful for measuring viscosity at high pressure (Cristescu et al., 2002). Table 31.3 provides further information about suppliers of falling object viscometers. Summary Although falling object viscometers are relatively inexpensive, the use of one requires some skill and is labor intensive. The tubes must be carefully cleaned before use and when filling the tube with the fluid of interest, care needs to be taken to avoid air

MAJOR VISCOSITY MEASUREMENT METHODS

963

TABLE 31.3 Sampling of Common Falling Ball Viscometer Types Type

Name

Falling ball

Gilmont Falling-ball Viscometers

Falling ball Falling ball Falling needle

HAAKE Falling Ball Viscometer Brookfield Falling Ball Viscometer PDV-100 Portable Field Viscometer

Manufacturer/Vendor Gilmont; Cole Parmer/ Thermo Scientific; Gardco HAAKE/Thermo Scientific Brookfield/Gardco Stony Brook Scientific/Gardco; Cole Parmer

Range (cP)

Price

0.2–200

$ ($200)

0.5–10 (up to 7500) 0.5–70,000

$$$ ($3500) $$$ ($3000)

5–106

$$ ($700)

bubbles. They cannot be used with opaque liquids. After setup, each individual measurement can take 1–2 min to complete. Gilmont-type viscometers must be mounted or held vertically and care needs to be taken when handling them to avoid heating up the fluid in the viscometer. Haake-type viscometers are pre-mounted at a specified angle and the falling ball tube is located inside an outer glass tube that can be easily connected to a circulating water bath for temperature control. They are not automated and require manual timing. However, it is relatively easy to compare the viscosity of different fluids by using multiple viscometers. The primary sources of error that arise in the use of a falling ball viscometer are related to temperature effects, handling and contamination. 31.3.2 Bubble (Tube) Viscometers Theory of Operation Although most drag-type viscometers measure absolute viscosity and require a separate measurement of density to calculate kinematic viscosity, bubble viscometers measure kinematic viscosity (n) by measuring the drag on a rising bubble of air which has low density compared to the fluid. The bubble tube consists of a glass tube (see Figure 31.6a) which is filled with the liquid of interest (leaving space for a bubble to

FIGURE 31.6 (a) Schematic of a bubble tube viscometer and (b) BYK-Gardner Bubble Viscometer Set (Courtesy of BYK Gardner USA).

964

VISCOSITY MEASUREMENT

form). The tube is marked with timing lines and the time for the bubble to rise is measured and compared to times for liquids with known viscosities. The theory in this case is similar to that for a falling object, except that here one measures the time for the bubble to rise a specific distance (see Goldsmith et al., 1962, for more information on the theory). If we ignore the effect of bubble viscosity the same equation for viscosity applies: m¼

 gD2

r  rfluid 18V obj

ð31:27Þ

gD2 ðr Þ 18V fluid

ð31:28Þ

gD2 18V

ð31:29Þ

However in the case of a rising gas bubble one can assume that the object (air) density is much less than that of the fluid and the calibration equation becomes: m¼ Or in terms of kinematic viscosity n¼

m rfluid

¼

In practical applications, bubble viscometers use carefully manufactured precision tubes which are calibrated using known viscosity standards. The kinematic viscosity is then calculated as: n ¼ k t

ð31:30Þ

where k is the calibration constant for the tube and t is the time for the bubble to rise a specific distance. The faster the bubble travels, the lower the viscosity. Types of Viscometers/ Options Bubble tube viscometers are used to measure viscosity using either a direct measurement method or a comparison method. In the direct method, a single tube (like that shown in Figure 31.6a) is used to measure the bubble rise time and that time is converted to viscosity using information from the manufacturer. In the comparison method, the user selects reference tubes of known viscosity (see Figure 31.6b) and directly compares the bubble rise time in the fluid of interest to the rise time in the reference tubes. The fluid of interest is then assumed to have the same viscosity as that of the reference tube with the closest rise time. Table 31.4 includes a list of suppliers for bubble tube viscometers. Summary Bubble tube viscometers are relatively inexpensive. Their use requires some skill and measuring viscosity is labor intensive. The tubes must be carefully cleaned before use. After setup, each individual measurement typically takes 1–2 min to complete. TABLE 31.4 Sampling of Common Bubble Tube Viscometer Types Type Bubble viscometers Bubble viscometers

Name Gardner Standard Bubble Viscometers Bubble Viscometer Kits

Manufacturer/Vendor

Range (St)

Price

Gardner/Gardco

0.5–5.5

$$

Cole Parmer

0.005–0.320

$–$$

MAJOR VISCOSITY MEASUREMENT METHODS

965

The viscometer needs to be held in a vertical position during measurement (some manufacturers supply a stand to hold and flip the tubes) and care must be taken not to heat the liquid in the tube when handling. They are not suitable for use with opaque liquids. They are not automated and require manual timing; however one can easily use multiple tubes to compare the viscosity of different fluids. The primary sources of error that arise in the use of a falling ball viscometer are related to temperature effects (it is not easy to supply external temperature control), handling, and contamination. 31.3.3 Rotational Viscometers Theory of Operation Rotational viscometers determine viscosity by measuring the resistance on a shaft rotating in the liquid of interest. They are designed to make a direct measurement of the absolute viscosity m. A schematic of a rotational viscometer is shown in Figures 31.7a and 31.8a. Figure 31.7b shows a commercially available Brookfield rotational viscometer. The viscometer includes a spindle (cylinder), attached to a rotating shaft. The spindle is placed in the liquid of interest and rotated at constant speed. The torque required to rotate the spindle is measured and then related to the fluid viscosity. Although their setup is somewhat more elaborate than that of the falling ball or capillary viscometers, they can be used to study the behavior of non-Newtonian fluids. The theory of operation of a rotational viscometer is based on the Couette flow model for fully developed, steady, incompressible laminar flow between two surfaces, one of which is moving. If we model a viscometer as a rotating cylinder inside a stationary cylinder as shown in Figure 31.7a, the Couette flow solution for the wall shear t; assuming the gap between the two cylinders is small and the velocity varies linearly, is given as t¼m

du vRi ¼m dy R o  Ri

ð31:27Þ

where v is the rotational speed of the cylinder, Ri is the radius of the rotating cylinder, and Ro is the radius of the outer, stationary cylinder. The torque due to this wall shear is then equal to the shear force times the area over which the shear acts times the moment arm

FIGURE 31.7 (a) Schematic of a cylinder in cylinder rotational viscometer and (b) spindle-type viscometer (Courtesy of Brookfield Engineering Labs).

966

VISCOSITY MEASUREMENT

(in this case the inner radius): T ¼ tARi ¼ m

vRi ð2pRi LÞRi Ro  Ri

ð31:28Þ

where T is the torque and A is the area of the rotating cylinder ð2pRi LÞ. Under steady state conditions the torque due to the viscous forces equals that which is applied to keep the cylinder rotating. Therefore, the viscosity can be measured using the measured torque value and the geometry of the system: Ro  R i  m ¼ T

ð31:29Þ 2pvR3i L In practice, rotational viscometers come with a variety of rotating spindles and calibration information that allows one to relate the torque measurement to viscosity. A typical calibration equation is in a form such as: m¼

T cv

ð31:30Þ

where T is the measured torque at a given rotation rate, w and c is the calibration constant used to account for geometry and end effects. Some viscometers are equipped with a digital readout. Types of Viscometers/Options There are three main types of rotational viscometers: Concentric cylinders, cone and plate, and parallel plate. The concentric cylinder theory is described above for a rotating inner cylinder which is more common. Systems are also designed to have a rotating outer cylinder to minimize the centrifugal forces which lead to the formation of Taylor vortices. A variation on the concentric cylinder-type viscometer uses a rotating spindle in an infinite medium. Cone and Plate viscometers (see Figure 31.8b) are designed to provide a uniform shear rate across the rotating plate. They consist of a conical surface and a flat plate, separated by a small gap that is filled with the liquid of interest. One of the plates is rotated and the torque required to hold the other in place is measured. The theory of operation is similar to that described above for the concentric cylinder viscometer, although the geometry is different. The cone angle in Figure 31.8b is designed to provide the uniform shear rate.

FIGURE 31.8 Geometries typically used in rotational viscometer (a) cylindrical, (b) cone and plate, and (c) parallel plate.

967

MAJOR VISCOSITY MEASUREMENT METHODS

Parallel plate viscometers provide a non-uniform shear rate. They consist of two parallel plates (see Figure 31.8c) separated by a gap filled with the liquid of interest. The shear rate acting on the fluid depends on the radial location. One plate typically rotates and the torque required to hold the other is measured. Again, the theory is similar to that described above; however in this case the shear rate varies with radial distance. Table 31.5 presents a listing of types and manufacturers of rotational viscometers. Summary Rotational viscometers are more expensive than the other drag-type viscometers but they are somewhat easier to use. Measurements can be made rapidly. Rotational speed can be altered to vary shear rates and test non-Newtonian fluids. The spindles, cones and plates must be carefully cleaned before use so the testing of multiple fluids is more complex. The primary sources of error that arise in the use of a rotational viscometer are related to temperature effects, set up errors (the system must be level and care must be taken to avoid end effects and eccentricity) and calibration of the torque measurement system. Some rotational viscometers have built in temperature control. Process versions exist that can be used in-line (see Table 31.5). 31.3.4 Flow-Type Viscometers 31.3.4.1 Capillary Viscometers Theory of Operation Capillary viscometers determine viscosity through measurement of the flow rate of the liquid traveling through a capillary tube. A capillary tube is one with a large length to diameter ratio (i.e., long and skinny). A schematic of a capillary tube

TABLE 31.5 Sampling of Common Rotational Viscometer Types Type Cylindrical Cylindrical

Cylindrical Cylindrical Cylindrical Cylindrical inline Cylindrical inline Cone and plate

Cone and plate

Name

Manufacturer/Vendor

Dial Reading DV-E; DV-I Prime; DVIIþ Pro; DV-IIþ Pro EXTRA HAAKE Rotational Plus Viscometer Thomas Stormer Viscometers KU-2 VTA Pneumatic Viscosel; VTE Electric Viscosel TT Series In-Line PV Process LV, RV, HA and HB Series Viscometers

Brookfield Eng./Gardco Brookfield Eng./Gardco

1–64 M 1–320 M

$$–$$$$

Thermo Scientific; Cole Parmer Thomas Stormer/Gardco Brookfield Eng./Gardco Brookfield Eng.

0.3–4000

$$$$$

332–2000

$

High Shear CAP1000þ, CAP-2000þ

Range (cP)

Price

27–5,274 0–10,000

Brookfield Eng.

2–10,000,000

Wells-Brookfield/ Brookfield Eng.; Gardco Wells-Brookfield/ Brookfield Eng.; Gardco

0.3–7,864,000

20–1,500,000

$$$$$

968

VISCOSITY MEASUREMENT

FIGURE 31.9 Capillary tube viscometer (Ostwald type).

viscometer is shown in Figure 31.9. Capillary viscometers are typically made of glass and consist of a bulb reservoir connected to the capillary tube. The theory of operation for a capillary tube viscometer is based on the Poiseuille model of laminar flow which describes flow through a round pipe. The volume flow rate, Q in a pipe can be derived from the Navier–Stokes equations for steady, laminar, and fully developed, incompressible flow as pR4 dP ð31:31Þ Q¼ 8m dy where R is the pipe radius, m is the viscosity and dP/dy is the pressure gradient (i.e., the change in pressure over the length of the pipe) which is the driving head for the flow. In the case of a vertical tube with both ends open to the ambient, the pressure gradient is caused by the hydrostatic pressure gradient: dP ¼ rg dy

ð31:32Þ

where r is the density and g is the gravitational constant. Combining equations and rearranging m pR4 g ¼y¼ r 8Q

ð31:33Þ

Which allows for calculation of the kinematic viscosity, v, from the measured flow rate and the geometry of the capillary tube. In practical applications, capillary viscometers are carefully calibrated and the manufacturer supplies a calibration constant. The user is

MAJOR VISCOSITY MEASUREMENT METHODS

969

instructed to measure the time for the fluid to travel a specified distance and then the kinematic viscosity is calculated as: y ¼ Kt

ð31:34Þ

where K is the manufacturer supplied calibration constant and can be a function of temperature depending on the viscometer type. Types of Viscometers/Options There are four primary types of capillary tube viscometers. They are the original Ostwald, the Modified Ostwald, the Suspended level (Ubbelohde) and the reverse flow capillary viscometers. Each is described below. The Ostwald Viscometer is one of the simplest capillary tube viscometers. As shown in the schematic (Figure 31.9), the viscometer consists of a bulb connected to a long capillary tube. To use the viscometer one partially fills it and then draws the fluid to the upper mark above the right side bulb (typically using a syringe system). The fluid is released to flow through the capillary tube and the time for the upper bulb to empty (fluid level at upper marks to lower marks) is measured. Some of the problems associated with the use of the Ostwald viscometer include the need to keep the viscometer vertical, the requirement for a specific volume of fluid and the effect of temperature on the viscosity measurement. A number of Modified Ostwald-type viscometers exist. These include the Cannon– Fenske routine viscometer, Pinkevitch viscometer, Zeitfuchs, (see Viswanath et al. 2007). Each is designed to address some of the sources of error found in the Ostwald type. For example, the Cannon–Fenske Routine viscometer is designed to minimize the effect of tilt angle by placing the upper and lower bulbs along the same vertical axis. Suspended Level Viscometers are designed to address loading issues by using a constant pressure gradient (driving head) during measurement of viscosity. They do this by suspending the test liquid above the capillary tube and using a pressure equalization tube. They include the Ubbelohde and Cannon–Ubbelohde-type viscometers. To determine viscosity, the test liquid is loaded into the upper bulb and then released. The liquid flowing through the capillary is separated from the reservoir bulb at the bottom. The third tube which connects the bottom of the capillary tube to the ambient ensures that the only pressure difference between the top of the bulb and the bottom of the capillary is that due to the hydrostatic pressure that is, the weight of the liquid. Reverse Flow Viscometers are used to measure the viscosity of opaque fluids (although they can also be used to measure that of transparent liquids). They measure the flow rate through a “dry” capillary tube so that the leading edge of the opaque fluid can be easily identified. Reverse Flow viscometers must be cleaned between each measurement. There are a number of variations on the standard capillary tube viscometer, including small volume (micro or semi-micro) viscometers requiring 1 mL or less of fluid (useful in the measurement of the viscosity of blood and plasma), dilution viscometers with extra large reservoirs for dilution of the sample and vacuum viscometers for fluids with high viscosities such as asphalt. There also exist more rugged capillary tube viscometers that are used under continuous flow conditions in industrial applications, and disposable capillary models to avoid the labor (and error potential) associated with cleaning. Table 31.6 provides a sampling of the various capillary tube viscometers. Summary Capillary tubes are used to measure viscosity for a wide range of fluids from oils to blood/plasma and even asphalt. They are relatively inexpensive (although generally

970

VISCOSITY MEASUREMENT

TABLE 31.6 Sampling of Common Capillary Viscometer Types Type Cannon-Fenske Routine Ubbelohde Reverse flow

Small volume

Dilution

Vacuum

In-line viscometer

Name

Manufacturer/Vendor

Range (St)

Price

Cannon-Fenske Routine

Cannon

0.5–100,000

$

Cannon-Ubbelohde; Ubbelohde BS/U-Tube; Zeitfuchs Cross-Arm; CannonFenske Opaque Cannon-Manning; Cannon-Ubbelohde SemiMicro Cannon-Ubbelohde FourBulb Shear; CannonUbbelohde Dilution Asphalt Institute; CannonManning; Modified Koppers Vacuum; and Zeitfuchs Cross-Arm KV100 Capillary Viscometer

Cannon

0.5–100,000

$

Cannon

0.5–100,000

$

Cannon

0.5–20,000

$

Cannon

0.5–20,000

$

Cannon

0.036– 5,800,000 P

$

Brookfield Eng.

0–500 cP

more costly than the falling ball and bubble tube viscometers). They require skill to use and can be labor intensive. Measurements take 1–5 min. Use requires manual timing and a method for filling the viscometer (a syringe system is typically used). The glass tubes are fragile. The tubes must be carefully cleaned before use but the viscosity of different fluids is easily measured using different tubes. The primary sources of error that arise in the use of a capillary viscometer are related to temperature effects, set-up errors (the tubes must be mounted vertically) and contamination. The tubes can be mounted in a temperaturecontrolled bath to minimize temperature effects and care must be taken not to affect the fluid temperature when handling. The calibration constant can be sensitive to temperature. Process versions exist that can be using in-line. 31.3.5 Orifice-Type (Cup) Viscometers Theory of Operation Orifice-type viscometers measure kinematic viscosity by comparing the time it takes the fluid to pass through an orifice (the efflux time) to the time that it takes a fluid of known viscosity to pass through the same orifice. They generally consist of a fluid reservoir from which the fluid flows, an orifice and a capturing reservoir (see Figure 31.10). Although they were originally designed using the Poiseuille flow model, it was determined that entrance and exit effects in this type of device are significant. Therefore, cup methods only provide a relative measurement. Absolute values of viscosity cannot be measured using this type of viscometer. In practice, a “cup” or reservoir is filled with a specified quantity of the fluid of interest and allowed to equilibrate thermally. Then a valve is opened at the bottom of the

MAJOR VISCOSITY MEASUREMENT METHODS

971

FIGURE 31.10 (a) Schematic of cup viscometer and (b)image of a Ford cup viscometer (Courtesy of Paul N. Gardner Company, Inc.).

cup and the time for the cup to empty is measured. Viscosity is then calculated using an empirical equation: n ¼ k t ð31:35Þ where k is supplied by the manufacturer. These types of viscometers are commonly used to measure the viscosity of oils, varnishes, and paints. Specifications in certain industries or for certain industrial processes are often closely tied to one particular type of cup measurement, and thus it may be difficult to use or generalize viscosity information found with other viscosity measuring methods to these industrial applications. Types of Viscometers/Options Cup viscometers are commonly used for in field measurements. There are a variety of cup types including the early models used in the petroleum industry such as the Saybolt, Redwood, and Engle Cups. The Ford, Zahn, and Shell viscosity cups (to name just a few) are more commonly used for measuring the viscosity of paints, varnishes, and lacquers. A sampling of cup viscometer types is listed in Table 31.7. Summary Cup viscometers are relatively inexpensive but generally not very accurate. In fact cup “viscometers” do not produce true viscosity measurements, only relative ones. Their primary drawback is that information gained using them is not directly translatable or comparable to information obtained by other methods (although empirical relations making comparisons to other methods abound). In practice this leads to significant “legacy” effects, necessitating the continued use of the same exact cup method if comparisons to historical data are desired. Thus a particular type of cup’s use is typically closely

972

VISCOSITY MEASUREMENT

TABLE 31.7 Sampling of Common Cup Viscometer Types Type

Name

Manufacturer/Vendor

Range (cSt)

Dip viscosity cups

EZ (Equivalent Zahn) Viscosity Cups S90 Zahn Signature Cups

Zahn/Gardco

10–1401

$

Zahn/Gardco; Spectrum Chemical DIN/Gardco

15–1627

$

38–545

$

Ford/Gardco Fisher/Gardco

2–1413 11–1125

$ $

ISO/Gardco

4.6–823

$

Gardco

3.3–400

$

Gardco

29 –1413

$

Gardco/ISO Viscosity Cups

Gardco

4.6–2611

$

Gardco/DIN Standard Viscosity Cups The Parlin Cups Gardco/Fisher Standard Viscosity Cups Gardo/ISO Standard Viscosity Cups and Orifices Stein Hall Viscosity Cup

DIN/Gardco

38–545

$

Gardco Fisher/Gardco

7–15000 11–1125

$ $

ISO/Gardco

4.6–2611

$

Dip viscosity cups Dip viscosity cups Dip viscosity cups Dip viscosity cups Dip viscosity cups Dip viscosity cups Laboratory or “Ring Stand” viscosity cups Laboratory or “Ring Stand” viscosity cups Standard cups Standard cups Standard cups Standard cups Standard cups

Gardco/DIN 4mm Dip Viscosity Cups Ford Dip Viscosity Cups Gardco/Fisher Dip Viscosity Cups Gardco/ISO Mini Dip and Orifices Norcross1 Shell Viscosity Cup Ford Standard Viscosity Cups

Gardco

Price

$

tied to a certain industry, industrial process, or product and the primary benefit of using such an “industry standard cup” is that their measurements can be directly compared to previous industrial knowledge or standards developed using the same method. Cup viscometers also require some skill to use (the cup requires steady holding during measurement) and can be labor intensive. Some models include a ring stand, others have handles, to hold the cup still and improve accuracy. Typical measurements are made in the field and each can take 1–2 min. The primary sources of error that arise in the use of a cup viscometer are related to temperature effects, setup errors, unsteadiness, and contamination. Most cups do not come with temperature control and this can affect the measurement although Saybolt-type cup viscometers do include temperature control. Cups must be properly cleaned because even slight clogging of the orifice can cause significant error. 31.3.6 Vibrational (Resonant) Viscometers Theory of Operation Vibrational viscometers determine viscosity by forcing a resonator to vibrate in the fluid of interest and then measuring the damping that is associated with fluid viscosity. This dampening effect is measured by one of three methods: (a) through

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MAJOR VISCOSITY MEASUREMENT METHODS

TABLE 31.8 Sampling of Common Vibrational Viscometer Types Type

Name

Vibrational Vibrational Vibrational Vibrational Vibrational

Process viscometers Portable viscometers Special viscometers Reactor viscometer SV-A100

Vibrational Vibrational Vibrational

SV-10 SV-100 In-line viscometer

Tuning fork vibrational

SV-A1 tuning fork vibration viscometer

Tuning fork vibrational

SV-A10 tuning fork vibration viscometer

Manufacturer/Vendor Vindum Eng. Vindum Eng. Vindum Eng. Vindum Eng. Gardco; Spectrum Chemical Gardco Gardco Nametre/Galvanic Applied Sciences Gardco; Spectrum Chemical; Cole Parmer Gardco; Spectrum Chemical; Cole Parmer

Range (cP)

Price

1,000–100,000

$$$$$

0.3–10,000 1,000–100,000

$$$$ $$$$

0.3–1,000

$$$$–$$$$$$

0.3–10,000

$$$$$–$$$$$$

measurement of the power required to maintain a constant vibration (the higher the viscosity of the fluid, the higher the power need); (b) measurement of the signal decay time upon halting the vibration (higher viscosity fluids will have a shorter decay time); or (c) looking at the frequency of the resonator as a function of the phase angle between the excitation and response signals. Types of Viscometers/ Options Vibrational viscometers are commonly used in in-line process measurement applications, although lab versions exist. Table 31.8 presents information on a sampling of commercially available vibrational viscometers. The most common vibrational viscometers are configured as a tuning fork, an oscillating sphere, or a vibrating rod. Tuning fork vibrational viscometers have two sensor plates which are immersed in the liquid of interest. The plates (which have the same natural frequency) are made to vibrate at the same amplitude using an electromagnetic force. The viscosity of the liquid is determined by measuring the current required to maintain the constant amplitude motion. Figure 31.11 shows a schematic of one type of tuning fork viscometer. The oscillating sphere viscometers use the dampening associated with the torsional motion of a sphere immersed in the fluid of interest to measure viscosity. Vibrating rod viscometers do the same with a rod shaped sensor. Tuning fork vibrational viscometers are capable of simultaneously measuring density and viscosity; however the oscillating spheres and rod types require separate measurement of density (see Retsina et al., 1987, for more details). Summary Vibrational viscometers are more expensive than the other types of viscometers but they are much easier to use. Once they are installed inline they require minimal skill and measurements can be made quickly. Their fundamental lack of moving parts makes them more robust and resilient, and thus better suited for use in harsh environments. The primary sources of error that arise in the use of a vibrational viscometer are related to calibration and drift errors.

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VISCOSITY MEASUREMENT

FIGURE 31.11 (a) Schematic of a vibrational viscometer and (b) Image of an SV-A Series Sine Wave Vibro Viscometer (Courtesy of A&D Weighing).

31.3.6.1 Viscosity Calibration Fluids Viscosity standards are standard fluids of known viscosity that are used to calibrate a viscometer. Most manufacturers recommend three or more fluids specified by NIST that can be used to calibrate a viscometer. Viscosity standard oils can be found through Sigma–Aldrich, Gardco, Cole–Parmer or other similar companies. 31.4 ASTM STANDARDS FOR MEASURING VISCOSITY There are a variety of published standards available. In this section we focus on the standards put out by the American Society for Testing and Materials (ASTM). Many industries adhere to specific guidelines provided by these or similar standards. The ASTM viscosity standards come in two types: (a) standards focused on methods (method standards) and (b) standards based on materials (material standards). A method standard describes a viscosity measurement method that can be applied in a variety of circumstances. An example of this would be the Standard Test Method for High-Shear Viscosity Using a Cone/Plate Viscometer, D-4287, which specifies a method for using a cone and plate viscometer and in particular applies this to paint. Because such standards specify the method, they can be applied beyond the particular material discussed in the standard. A material standard outlines multiple techniques that can be used to measure the viscosity of one specific material. For example, the Standard Test Methods for Viscosity of Adhesives, D1084, provides four different methods to test free-flowing adhesives with a viscosity range from 50 to 20 k cP. One limitation of the material standards is that they have a narrow focus. Table 31.9 provides a starting point for finding both types of standards. In this table the different methods (viscometer type) are listed by column and the material types are listed by row. For example, if one wishes to use a cup-type viscometer to measure the viscosity of paints ASTM D1200-10, D4212-10 and D5125-10 apply. One can also utilize the ASTM website to search for standards. The bolded standards in Table 31.9 are the recommend starting standards as they are the most general and provide a good list of referenced documents.

TABLE 31.9 ASTM Standards

Material/Method Iron and steel products Construction

Petroleum Products, lubricants and fossil fuels Paints, related coatings and aromatics Plastics Rubber Medical and surgical materials Adhesives High temperature

Rotational Viscometers

Capillary Viscometers

Falling Object Viscometers

Cup-Type Viscometers

Bubble Viscometer

Tapered Plug/Bearing Viscometer

D562; D2196 D4016-08; D7226-06e1; D4402-06; D7496-09; D562; D2196 D2983-09; D7042-04; D6896-03; D2161-05e1; D7279-08 D2196-10; D7394-08; D56210; D4287; D6606; D7395-07 D1824-95 D1992-91; D2196 D2196 D2556-93a; D562; D1084-08 D2669-06; D6267-08

D2171M-10; D2170M10; D4957-08; D4603-03 D5481-10; D446-07; D445-09

D5225-09; D4603-03; D446 D1646-07 D4603

D1823

D7483-08

D5478-09; D660600; D4040-10; D1343-95 D1823

D1545-07

D1200-10; D4212-10; D5125-10;

D1725-04

D6616-07; D4683-09; D4741-06

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VISCOSITY MEASUREMENT

31.5 QUESTIONS TO ASK WHEN SELECTING A VISCOSITY MEASUREMENT TECHNIQUE This section is intended as a guide to the practicing engineer on how to choose the appropriate viscometer. It is structured as a series of nonhierarchical questions. Table 31.10 also provides a summary comparison of the different viscometer types. 1. Do you need an online/process measurement, or an offline measurement? If you need to continuously monitor the viscosity of an industrial process stream online or in situ your choices of available technique will be limited, and the cost of your system will be higher. On the positive side, very little operator labor will be required once the system is installed and there will be no need to remove and handle samples from the process stream. Measurements can be taken at much shorter time intervals, and the viscosity will be known at exactly the conditions (temperature, pressure) associated with the process. The main viscometry techniques available for online measurement are resonant, capillary, and rotational. More information on the measurement range and suppliers of these viscometers can be found in Tables 31.5, 31.6 and 31.8. 2. What is the viscosity range that you will be measuring? Matching the viscometer to the viscosity range to be measured is one of the most important tasks associated with selecting a viscometer. Since many techniques measure the time for a certain amount of fluid to flow (e.g., cup or capillary types), or for an object to fall through the fluid (e.g., falling ball or rising bubble types), if the fluid viscosity is higher than optimal for a given viscometer the measurements may take an exceedingly long time to conduct. On the other hand, if the viscosity is too low key assumptions related to the operation of the viscometer (e.g., the low Reynolds number assumption) may be violated and accuracy will be lost. Rotational-type viscometers simply will not give a reading, and may in fact be damaged, if the fluid being measured is too viscous for the instrument and settings selected. On the other hand if the viscosity is such that the reading is less than 10% of the instrument’s full range (for a given setting) the uncertainty of the measurement will become unacceptably high. Thus good results necessitate careful matching of the viscosity range and the instrument. Operating ranges for many common types of viscometers are listed in Tables 31.3–31.8. 3. What other characteristics of the fluid must be considered? Other physical properties of the fluid beyond viscosity are also important to consider. For example, certain techniques or instruments lend themselves to use with opaque fluids (e.g., the reverse flow viscometer) while others cannot be used because the technique depends on being able to see through the fluid (e.g., falling ball). Likewise, chemical compatibility of the fluid with the exposed viscometer surfaces is important to consider, as are any issues associated with safe handling of the fluid. Also various properties such as how difficult the fluid will be to clean off of the viscometer surfaces, and how likely the fluid is to solidify and clog are critical. Certain specialized fluids, like blood, have specific viscosity instruments designed specifically for use with them. 4. Do you wish to measure kinematic or dynamic viscosity? Each type of viscometer fundamentally measures either the absolute viscosity of a fluid (m) or the kinematic viscosity (n). In order to convert between the two a separate

TABLE 31.10 Comparison of Viscometer Types Type

Falling Object

Bubble Tube

Rotational

Capillary

Cup

Vibrational

0.5–100 k cSt (general), 0.5–20 k cSt (small volume) up to 5.8 MP (vacuum) High Some Some $200–$300

4–2600 cSt

4–2600 cSt

Med No No

Low Yes Yes Expensive

Viscosity range

0.2–70 k cP

0.005–5.5 cSt

0.3–320 M cP

Labor/skill level Automated? In-line? Expense

High No No $100–$200 Gilmont $1000 Haake

High No No ($50–100 each or $400–1000 for a kit)

Med Yes Some Expensive

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VISCOSITY MEASUREMENT

5.

6.

7.

8.

9.

10.

measurement of the fluid’s density is required. Thus care must be exercised in selecting a viscometer if one wishes to avoid the need for this additional measurement. How much and what type of labor is required? The quantity of labor (time) and skill level required to make accurate viscosity measurements varies significantly among different viscometer types. Broadly speaking the lower cost viscosity measurement instruments (falling ball, rising bubble, and capillary tube) are more labor intensive to use and require significantly more skill (operator training) than the more expensive viscometer types (rotating disk, vibrating rod, etc.). The frequency of measurements required, as well as the labor associated with maintenance, calibration, and cleaning of the viscometer should be carefully considered in this context. What is the required accuracy? Another major factor influencing the type of viscometer to chose is the required accuracy. Here some of the simplest models (e.g., capillary and falling ball) compare quite well on the basis of accuracy with more expensive techniques, assuming a sufficiently skilled and careful operator. However the old truism that increased accuracy leads to increased expense does generally hold within a given class of viscometer. As discussed in point 5 above, labor costs may be significant in regards to the cost/accuracy decision. Are there standard methods associated with your industry or fluid? Viscosity measurements for many substances, for many industries, and for the use of many types of viscometers are the subject of government or industrial standards. For example, the ASTM standards governing viscosity measurement are discussed at length in Section 31.3. Likewise many industries or industrial processes are closely tied to particular viscosity measurement techniques, even if these are outmoded. In such cases there may be a considerable body of industrial or process control knowledge that is not readily translatable to more standard viscosity measurements (e.g., the use of the “Stein–Hall Cup” measurement with starch based adhesives used in corrugated box manufacturing). In such cases sticking with the “standard method” may be the most desirable approach. Is temperature control needed? Is viscosity as a function of temperature needed? The viscosities of liquids and gases are a strong function of temperature, and this should always be kept in mind when collecting and reporting viscosity data. Some viscosity measurement techniques lend themselves more readily to maintaining temperature control of the fluid during the measurement (e.g., the Haake falling ball viscometer has a built-in temperature bath capability, for other types that capability must be added ad hoc). Likewise some viscometers (e.g., Brookfield rotational viscometer with small sample adapter) lend themselves to measuring the viscosity of a fluid over a wide range of temperatures. Quantity of fluid needed for the measurement. This is an important factor to consider when dealing with scarce or expensive fluids (e.g., blood). Viscosity measurement techniques can vary by more than an order of magnitude in the amount of liquid required to make a measurement. Specialized needs. If your process involves specialized needs or requirements, that is, measurements at high pressure or measurements at temperature extremes there are viscometers available specifically for those purposes.

REFERENCES

979

REFERENCES Bird RB, Stewart WE, Lightfoot EN. Transport Phenomena. New York: John Wiley and Sons; 1960. Brizard M, et al. Design of a high precision falling-ball viscometer. Review of Scientific Instruments 2005;76:025109. Cengel YA, Cimbala JM. Fluid Mechanics, Fundamentals and Applications. 2nd ed. New York: McGraw Hill; 2010. Cristescu ND, Conrad BP, Tran-Son-Tay R. A closed form solution for falling cylinder viscometers. International Journal of Engineering Science 2002;40.6:605–20. Davis AMJ, Brenner H. The falling-needle viscometer. Physics of Fluids 2001;13:3086. Goldsmith HL, Mason SG. The movement of single large bubbles in closed vertical tubes. Journal of Fluid Mechanics 1962;14.01:42–58. Kogan MN. Rarefied Gas Dynamics. New York: Plenum Press; 1969. Lucas K. Die Druckabh€angigkeit der Viskosit€at von Fl€ ussigkeiten eine einfache Absch€atzung. Chemie Ingenieur Technik 1981;53:959. Munson BR, Young DF, Okiishi TH, Huebsch WW. Fundamentals of Fluid Mechanics. 6th ed. New York: John Wiley and Sons; 2009. Owczarek JA. Fundamentals of Gas Dynamics. Scranton, PA: International Textbook Co.; 1964. Poling BE, Prausnitz JM, O’Connell, JP. The Properties of Gases and Liquids. 5th ed. New York: McGraw Hill; 2004. Retsina T, Richardson SM, Wakeham WA. The theory of a vibrating-rod viscometer. Applied Scientific Research 1987;43:325–346. Shaughnessy JJr., EJ, Katz IM, Schaffer JP. Introduction to Fluid Mechanics. New York: Oxford University Press; 2005. Viswanath DS, Ghosh TK, Prasad DHL, Dutt NVK, Rani K. Viscosity of Liquids. Theory, Estimation, and Data. New York: Springer; 2007. White FM. Viscous Fluid Flow. 3rd ed. New York: McGraw Hill Inc.; 2006. Vicenti WG, Kruger CH. Introduction to physical gas dynamics. New York: Wiley; 1965.

32 TRIBOLOGY MEASUREMENTS PRASANTA SAHOO 32.1 Introduction 32.2 Measurement of surface roughness 32.2.1 Surface profilometer 32.2.2 Optical microscopy 32.2.3 Advanced techniques for surface topography evaluation 32.3 Measurement of friction 32.3.1 Inclined plane rig 32.3.2 Pin-on-disc rig 32.3.3 Conformal and non-conformal geometry rig 32.3.4 Environment control 32.3.5 Techniques for friction force measurement 32.4 Measurement of wear 32.5 Measurement of test environment 32.5.1 Temperature measurement 32.5.2 Thermocouples 32.5.3 Thin film sensors 32.5.4 Radiation detectors 32.5.5 Metallographic observation 32.5.6 Liquid crystals 32.5.7 Humidity measurement 32.5.8 Measurement of oxygen and other gages 32.6 Measurement of material characteristics 32.6.1 Hardness 32.6.2 Young’s modulus and the elasticity limit 32.6.3 Fracture toughness 32.6.4 Residual stresses 32.6.5 Chemical composition of a surface 32.7 Measurement of lubricant characteristics 32.7.1 Analysis of chemical changes 32.7.2 Viscosity measurement 32.7.3 Lubricant oxidation tests

Handbook of Measurement in Science and Engineering. Edited by Myer Kutz. Copyright Ó 2013 John Wiley & Sons, Inc.

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32.8 Wear particle analysis 32.8.1 Chemical analysis of particles in lubricant 32.8.2 Analysis based on separation of wear particles 32.9 Industrial measurements 32.10 Summary

32.1 INTRODUCTION Tribology as a subject is relatively new, but the practice of tribological principles is older than recorded history. Tribology is defined as the science and technology of interacting surfaces in relative motion and of related subjects and practices. It deals with the technology of lubrication, control of friction, and prevention of wear. Successful design of machine elements depends essentially on the understanding of tribological principles. During contact of two nominally flat surfaces, contact occurs at discrete spots due to surface roughness and adhesion occurs due to intimate contact. When one solid body moves over another it experiences the resistance to motion called friction. The surface damage or material removal that takes place in a moving contact is termed as wear. Surface coatings and treatments are provided to monitor friction and to control wear. The most effective way of friction and wear control is by using proper lubricants either liquid, solid or gas. The lubricant properties and the mechanisms of lubrication constitute the essence of optimum performance and reliability of bearings. The recent emergence of proximal probes and high capability computational techniques has stimulated systematic investigations of interfacial problems with high resolution in micro and nano components leading to the development of the new field of micro/nano tribology. Tribology is different from other science branches, for example, fundamental physics where theoretical predictions are made long before the experimental validation of the same. Tribology uses a more empirical methodology based on experimental observation and theoretical concepts follow the findings later. A classical example of experimental observation leading to the development of basic tribological concept of a hydrodynamic pressure field operating in a lubricated bearing is Tower’s friction experiments. The railway axle bearings were fitted with numerous oil holes in order to supply oil to the bearing. But during operations, oil leaked through the holes and even wooden plugs were unable to prevent the leakage. Tower fitted pressure gauge at the oil holes to find that the oil pressure was capable to support the bearing load. Figure 32.1 is a schematic representation of the partial bearing used by Tower. Tower’s results led to the development of hydrodynamic theory of lubrication by Osborne Reynolds. Most tribological phenomena, for example, friction, wear, and frictional heating, are not intrinsic material properties. These depend on a number of competing factors. For example, to evaluate load capacity of a hydrodynamic bearing one needs to consider viscous heating of the lubricant, cavitation and turbulent lubricant flow, elastic deformation of bearing structure and so on. Friction and wear being chaotic processes, these are described in terms of specific experimental findings that are systematically analyzed to get an engineering model of friction, wear, and lubrication. Tribological investigation

MEASUREMENT OF SURFACE ROUGHNESS

Lubrication hole W

983

Partial bronze bearing

Lubricant level N

Journal

FIGURE 32.1 Schematic representation of the partial bearing used by Tower.

may be categorized into two groups: fundamental research for understanding of basic mechanisms of friction and wear; and applied research for resolving specific industrial friction and wear issues. On both counts, tribological measurements mainly include surface roughness, friction, wear, test environment, material characteristics, and lubricant characteristics. Wear particle analysis and industrial tribology form an important part of tribological measurement.

32.2 MEASUREMENT OF SURFACE ROUGHNESS The surfaces of any engineering component contain a vast number of peaks and valleys and it is not possible to measure the height and location of each of the peaks. So what is done is to take measurements from a small and representative sample of the surface so chosen that there is a high probability for the surface lying outside the sample to be statistically similar to that lying within the sample. Over the years different methods have been devised to study the topography of surfaces. A brief outline of some of the methods is presented here. 32.2.1 Surface Profilometer The most common method of studying surface texture features is the stylus profilometer, the essential features of which are illustrated in Figure 32.2. A fine, very lightly loaded stylus is dragged smoothly at a constant speed across the surface under examination. The transducer produces an electrical signal, proportional to displacement of the stylus, which is amplified and fed to a chart recorder that provides a magnified view of the original profile. But this graphical representation differs from the actual surface profile because of difference in magnifications employed in vertical and horizontal directions. Surface slopes appear very steep on profilometric record though they are rarely steeper than 10 in actual cases. The shape of the stylus also plays a vital role in incorporating error in measurement. The finite tip radius (typically 1–2.5 mm for a diamond stylus) and the

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TRIBOLOGY MEASUREMENTS

Motor and gearbox

Transducer

Skid Amplifier

Stylus

Specimen

A–D converter

Chart recorder

Data logger

FIGURE 32.2 Component parts of a typical stylus surface-measuring instrument.

included angle (of about 60 for pyramidal or conical shape) results in preventing the stylus from penetrating fully into deep and narrow valleys of the surface and thus some smoothing of the profile are done. Some error is also introduced by the stylus in terms of distortion or damage of a very delicate surface because of the load applied on it. In such cases noncontacting optical profilometer having optical heads replacing stylus may be used. Reflection of infrared radiation from the surface is recorded by arrays of photodiodes and analysis of the same in a microprocessor result in the determination of the surface topography. Vertical resolution of the order of 0.1 nm is achievable though maximum height of measurement is limited to a few microns. This method is clearly advantageous in case of very fine surface features. 32.2.2 Optical Microscopy In this method, the surface of interest is held to reflect a beam of visible light and then these are collected by the objective of the optical microscope. An image of the surface is produced and is analyzed at very high rates of resolution (up to 0.01 mm) by optical interferometers. Depth of field achievable is up to 5 mm. But success of the method depends on the reflective property of the material, which limits the use of the same. Optical methods may be divided into two groups: geometrical methods and physical methods. Geometrical methods include light-sectioning and taper-sectioning methods. Physical methods include specular reflection; diffuse reflection, speckle pattern, and optical interference. In light-sectioning method, the image of a slit is thrown onto the surface at an incident angle of 45 . The reflected image appears as a straight line if surface is smooth and as an undulating line if the surface is rough. In taper-sectioning method, a section is cut through the surface to be examined at an angle of u, thus effectively magnifying the height variation by a factor of cotu and is subsequently examined by an optical microscope. The surface is supported with an adherent coating that prevents smearing of the contour during

MEASUREMENT OF SURFACE ROUGHNESS

985

the sectioning process. The taper section is lapped, polished, and lightly heat tinted to provide good contrast for optical examination. The process suffers from the disadvantages like destruction of test surface and tedious specimen preparation. In specular reflection method, gloss or specular reflectance that is a surface property of the material and function of reflective index and surface roughness, is measured by gloss meter. Surface roughness scatters the reflected light and affects the specular reflectance. Thus a change in specular reflectance provides a measure for surface roughness. Diffuse reflection method is particularly suitable for on-line roughness measurement during manufacture since it is continuous, fast, non-contacting and non-destructive. This method employs three varieties of approaches. In total integrated scatter (TIS) approach, one measures the total intensity of the diffusely scattered light and the same is used to generate the maps of asperities, defects and particles rather than micro-roughness distribution. The diffuseness of scattered light (DSL) approach measures a parameter that characterizes the diffuseness of the scattered radiation pattern and relates the same to surface roughness. In angular distribution (AD) approach, the scattered light provides roughness height, average wavelength or average slope. With rougher surfaces, this may be useful as a comparator for monitoring both amplitude and wavelength surface properties. In speckle pattern method, surface roughness is related to speckle, which is basically the local intensity variation between neighbouring points in the reflected beam when a surface is illuminated with partially coherent light. The principle of laser speckle roughness measuring system is schematically shown in Figure 32.3. The optical interference technique involves looking at the interference fringes and characterizing the surface with suitable computer analysis. Common interferometers include the Nomarski polarization interferometer and Tolanski multiple beam interferometer. 32.2.3 Advanced Techniques for Surface Topography Evaluation

In re ten fle si ct ty ed of lig ht

A further improvement in the resolution of surface topographic examination is possible by the use of electron microscopes. Two basic types of electron microscopes are available: scanning electron microscopes and transmission electron microscopes. In scanning

Laser Difference in path length creates interference

Speckle pattern formed by a surface roughness

Photo detector array Scattering of reflected laser light

Rough surface Test specimen

FIGURE 32.3 Schematic of the principle of laser speckle roughness measuring system.

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TRIBOLOGY MEASUREMENTS

electron microscopy (SEM) a focused beam of high-energy electrons is incident on the surface at a point resulting in the emission of secondary electrons. These are then collected and fed to an amplifier to send an electric signal to a cathode ray tube (CRT). The electron beam is scanned over the surface to have a complete picture. The CRT screen gives a topographical image of the entire area of interest. Depth of field is up to 1000 mm, which acts as a primary advantage of this method over optical method. The requirement on size of the specimen to be placed within the vacuum chamber of the instrument raises a drawback of the method. This can be overcome by preparing a replica of the surface. In transmission electron microscopy (TEM), the focused beam of high-energy electrons is made to transmit through a very thin specimen. Then the deflection and scattering of the electrons is recorded to analyze the surface topography. Preparation of a specimen thin enough to transmit electrons plays a vital role. Sometimes a replica of the surface retaining all the texture features but of a material having greater electron transparency is produced for the same purpose. Recently, a different type of electron microscopy called scanning tunneling electron microscopy (STM) is in use. It incorporates the electron-tunneling phenomenon through an insulating layer separating two conductors. The sharp pointed tip of a probe forms one electrode and the surface of the specimen the other. The probe is moved by a highly precise positional controller to keep the tunneling current at a steady value. The probe provides an image of the surface under examination. The method is superior to the earlier ones in a sense it does not require any vacuum. Only disadvantage is the proper design of the controller mechanism. The principle of the STM is very simple. Just like in a record player, the instrument uses a sharp needle, referred to as the tip, to investigate the shape of the surface. But the STM tip does not touch the surface. The schematic of the method is shown in Figure 32.4. A voltage is applied between the metallic tip and the specimen, typically between a few millivolts and volts. The tip touching the surface of the specimen results in a current and when the tip is far away from the surface, the current is zero. The STM operates in the regime of extremely small distances between the tip and the surface of only 0.5–1.0 nm, which are typically 2–4 atomic diameters. At these distances, the

Piezo element

A

tip Trajectory

Specimen

Specimen

FIGURE 32.4 Schematic of STM.

MEASUREMENT OF SURFACE ROUGHNESS

987

electrons can jump from the tip to the surface or vice versa. This jumping is necessarily a quantum mechanical process known as “tunneling” and hence the name scanning tunneling microscope. The STMs usually operate at tunneling currents between a few picoAmperes (pA) and a few nano-Amperes (nA). The tunneling current depends critically on the precise distance between the last atom of the tip and the nearest atom or atoms of the underlying specimen. When this distance is increased a little bit, the tunneling current decreases heavily. As a thumb rule, for each extra atom diameter that is added to the distance, the current becomes a factor of 1000 lower. Thus the tunneling current provides a highly sensitive measure of the distance between the tip and the surface. The STM tip is attached to a piezo-electric element, which changes its length a little bit, when it is put under an electrical voltage. The distance between the tip and the surface can be regulated by adjusting the voltage on the piezo element. In most STMs, the voltage on the piezo elements is adjusted in a manner that the tunneling current always has the same value, say 1 nA. Thus the distance between the last atom on the tip and the nearest atoms on the surface is kept constant. Using the so-called electronics; the distance regulation is done automatically. The feedback electronics continually measures the deviation of the tunneling current from the desired value and accordingly adjust the position of the tip. While this feedback system is active, two other parts of the piezo elements are used to move the tip in a plane parallel to the surface to scan over the surface. In the scanning process, every time that the last atom of the tip is precisely over a surface atom, the tip needs to be retracted a little bit, while it has to be brought slightly closer when the tip atom is between the surface atoms. This automatically leads the tip to follow a bumpy trajectory, which replicates the atoms of the surface. Then this information about the trajectory is available in the form of the voltages that have been applied by the feedback electronics are then finally visualized in the form of a collection of individual height lines or in the form of grey scale/color scale representation or in the form of some three dimensional perspective views. More recently AFM (atomic force microscope) is developed to investigate surfaces of both conductors and insulators on an atomic scale. Like the STM, AFM relies on a scanning technique to produce very high-resolution, three-dimensional images of sample surfaces. In AFM, the ultra-small forces (less than 1 nN) present between the AFM tip and sample surface are measured by measuring the motion of a very flexible cantilever beam having an ultra small mass. The AFM combines the principles of the STM and the stylus profiler. The important difference between the AFM and the STM is that in the AFM, the tip gently touches the surfaces. The AFM does not record the tunneling current but the small force between the tip and the surface. The AFM tip is attached to a tiny leaf spring, known as the cantilever, which has a low spring constant. The bending of the cantilever is detected with the use of a laser beam, which is reflected from the cantilever. The AFM thus measures contours of constant attractive or repulsive force. The detection is made very sensitive such that the forces as small as a few pico Newton can be detected. Forces below 1 nN are usually sufficiently low to avoid damage to either the tip or the surface. Since AFM does not rely on the presence of a tunneling current, it can also be used on non-conductive materials. Soon after the introduction of the AFM, it was realized that the same instrument could be used to also measure forces in the direction parallel to the surface, that is the friction forces. When modifications are incorporated for atomic scale and micro scale studies of friction, it is termed as the friction force microscope (FFM) or the lateral force microscope (LFM). The FFM usually detects not only the deflection of the cantilever perpendicular to the surface, but also the torsion of the cantilever, resulting

988

TRIBOLOGY MEASUREMENTS Filter Mirrored prism

Diode laser Lens

AFM signal

T

Mirror

R L FFM signal

B Sample

Split diode photo-detector X-Y-Z PZT tube scanner

Cantilever

FIGURE 32.5 Schematic operation of AFM/FFM.

from one lateral force. Schematic of the AFM/FFM commonly used for measurements of surface roughness, friction, adhesion, wear, scratching, indentation and boundary lubrication from micro to atomic scales is shown in Figure 32.5. In all surface profilometric methods, roughness (small-scale irregularities) and form error (deviation from its intended shape) remain coupled in the recorded data. Form error may be subtracted from the recorded data to provide only the roughness features by different means. The two most common methods used in stylus profilometer are (a) use of datum-generating attachments and (b) use of large radius skids or flat shoes. With these the average local level is used as datum and form error or waviness is not recorded. Other methods include the use of filtering the displacement signal corresponding to waviness. Table 32.1 summarizes the comparison of the different roughness measuring methods.

32.3 MEASUREMENT OF FRICTION Friction is defined as the force of resistance to motion that occurs when a solid body moves tangentially with respect to the surface of another body that it touch. The friction force acts in a direction opposite to that of motion. Even when an attempt is made to initiate the motion, the friction force exists. The friction force required to initiate the sliding is called the static friction force and that required to maintain sliding is called the kinetic friction force the value of which is usually lower than the former for the same combination of material and other parameters. The basic

TABLE 32.1 Summary and Comparison of Roughness Measurement Methods Resolution at Maximum Magnification, nm

Method

Quantitative Data 3-D Data

Horizontal

Vertical

On-Line Measurement Capability

Stylus method

Yes

Yes

15–100

0.1–1

No

Operates along linear track, contact type can damage sample, slow speed in 3-D mapping

Optical methods Light sectioning Taper sectioning Specular reflection Diffuse reflection Speckle pattern Optical interference SEM TEM

Limited Yes No Limited Limited Yes Limited Limited

Yes No No Yes Yes Yes Yes Yes

500 500 105–106 105–106

0.1–1 25 0.1–1 0.1–1

Qualitative Destructive, tedious specimen preparation Semiquantitative Smooth surfaces ( i swap (xi,xk) End If m = n/2 While m  1 And k > m k km m m/ 2 End While k k+m Next i Mmax 1

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SIGNAL PROCESSING

While n > Mmax ic 2Mmax u p/Mmax wp 2 sin2 (u/2) jsin(u) w 1 For m = 1 To Mmax For i = m To n By ic k i + Mmax xtemp xi  wxk xi xi + wxk xk xtemp Next i w w (wp + 1) Next m Mmax = ic End While

49.2 BASIC ANALOG FILTERS Linear filters apply frequency-specific gains to a signal. This is often done to enhance desired portions of the spectrum while attenuating or eliminating other portions. Four common filters are low pass, high pass, bandpass, and band reject. The objective of an ideal low-pass filter is to eliminate a range of undesired high frequencies from a signal and leave the remaining portion undistorted. To this end, an ideal low-pass filter will have a gain of 1 for all frequencies less than some desired cutoff frequency fc and a gain of 0 for all frequencies greater than fc, as seen in Figure 49.1. There are various rational functions that approximate this ideal. But because of the discontinuity in the ideal low-pass response, all realizations of this ideal with be an approximation. The various approximation functions generally trade off between three characteristics: passband ripple,

FIGURE 49.1 Frequency response for an ideal low-pass filter.

BASIC ANALOG FILTERS

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FIGURE 49.2 Typical frequency response for an approximate low-pass filter.

stop-band ripple, and the transition width, shown in Figure 49.2. Four common rational function approximations for low-pass filters are the Butterworth, the Tchebyshev Types I and II, and the elliptical filter. By convention, we use H(s) as the transfer function from which we determine the frequency response, where HðsÞ ¼

V output ðsÞ V input ðsÞ

ð49:6Þ

is the output-to-input gain of a standard two-port system. 49.2.1 Butterworth The Butterworth filter has a smooth passband region (frequencies less than fc hertz) and a smooth stop band (frequencies greater than fc) and a comparatively wide transition region as shown in Figures 49.3 and 49.4. Let Sc be the cutoff frequency; then a low-pass rational function approximation is as follows: jHðsÞj2 ¼

1 1 þ ðs=sc Þ2N

ð49:7Þ

;

where 2pf c ¼ sc . In factored form HðsÞ ¼

1 ; ðs  p0 Þðs  p1 Þ    ðs  pN1 Þ

ð49:8Þ

where pi ¼ ai þ jbi and p  ðN þ 2i þ 1Þ ai ¼ 2pf c cos  2N  p ðN þ 2i þ 1Þ bi ¼ 2pf c sin 2N

i ¼ 0; . . . ; N  1 i ¼ 0; . . . ; N  1

:

ð49:9Þ

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SIGNAL PROCESSING

FIGURE 49.3 Frequency response to a third-order Butterworth filter.

49.2.2 Tchebyshev Unlike the Butterworth rational function, the Tchebyshev (Type I) rational function permits some ripple to occur in the passband (frequencies less than fc in the low-pass filter), in exchange for a sharper transition region compared to a Butterworth filter of equal order N, as seen in Figure 49.5: jHðsÞj2 ¼

1 ; 1 þ e2 T 2N ðs=sc Þ

FIGURE 49.4 Frequency response to Butterworth filters of order 2, 5, and 11.

ð49:10Þ

BASIC ANALOG FILTERS

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FIGURE 49.5 Frequency response to a fifth-order Tchebyshev filter.

where TN is the Tchebyshev polynomial defined as T N ðsÞ ¼ cos½n cos1 ðsÞ:

ð49:11Þ

The pole placements pi ¼ ai þ jbi for the Tchebyshev Type I filter are p  ðN þ 2i þ 1Þ ai ¼ 2pf c sinhðv0 Þcos 2N p  bi ¼ 2pf c coshðv0 Þsin ðN þ 2i þ 1Þ 2N

i ¼ 0; . . . ; N  1

;

i ¼ 0; . . . ; N  1

ð49:12Þ

where   1 1 1 ; v0 ¼ sinh N e sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  1 0 < r < 1; e¼ ð1  rÞ2

ð49:13Þ ð49:14Þ

where r is this amplitude of the ripple in proportion to the gain of the passband. 49.2.3 Inverse Tchebyshev The inverse Tchebyshev (or Tchebyshev Type II) filter has a smooth passband, ripple in the stop band (frequencies less than fc for the low-pass filter), and a sharper transition region compared to the Butterworth function of equal order N, as seen in Figure 49.6: jHðsÞj2 ¼

e2 T 2N ðsc =sÞ : 1 þ e2 T 2N ðsc =sÞ

ð49:15Þ

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SIGNAL PROCESSING

FIGURE 49.6 Frequency response to a fifth-order inverse Tchebyshev filter.

The zero placements zi ¼ ai þ bi for the Tchebyshev Type II filter are zi ¼

1 sinðip=2NÞ

0  i  N  1:

ð49:16Þ

The pole placements pi ¼ ai þ jbi for the Tchebyshev Type II filter are simply the reciprocals of the pole placements computed for the Tchebyshev Type I filter. 49.2.4 Elliptical The elliptical filter has ripple in both the passband and the stop band but, in exchange, has the narrowest transition region for equal filter order N, as seen in Figure 49.7. The

FIGURE 49.7 Frequency response to a fifth-order elliptical filter.

BASIC ANALOG FILTERS

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derivation and implementation for the determination of the poles and zeros involve the Jacobian elliptic function: Zt dx f ðt; kÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð49:17Þ 2 2 1  k sin ðxÞ 0 The method is beyond the scope of this chapter. The interested reader is referred to Parks and Burrus (1987); Vlc9ek and Unbehauen (1989). 49.2.5 Arbitrary Frequency Response Curve Fitting by Method of Least Squares It is possible to design a filter to approximate a desired frequency response F(v) by the method of least squares. Consider a transfer function in factored form as QM

HðsÞ ¼ G QNi¼i

k¼1

Minimize

Z

2

x ðG; z; pÞ ¼

D

s  zi

s  pk

ð49:18Þ

:

½jHðjvÞj  FðvÞ2 dv;

ð49:19Þ

subject to

< N 1 winðiÞ ¼ > : 2  2i N1

N1 0i 2 : N1 i N1 2

ð49:29Þ

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BASIC DIGITAL FILTER

TABLE 49.2 Comparison of Characteristics for Commonly Used Windowing Functions Window Name

Minimum Stop-Band Attenuation (dB) 21 25 44 53 74

Rectangular Bartlett Hanning Hamming Blackman

Hanning: winðiÞ ¼

   1 2pi 1  cos 0  i  N  1: 2 N1

ð49:30Þ

Hamming: winðiÞ ¼ 0:54  0:46 cos



2pi N1



0  i  N  1:

ð49:31Þ

Blackman: 

   2pi 4pi þ 0:08 cos winðiÞ ¼ 0:42  0:5 cos N 1 N1

0  i  N  1:

ð49:32Þ

Table 49.2 gives a list of several common windowing functions together with their characteristics (Oppenheim and Schafer, 1975). Figure 49.10 demonstrates the effect of a Hanning window.

FIGURE 49.10 Comparison of FIR filter with Hanning window and with windowing.

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SIGNAL PROCESSING

49.3.2.2 FIR High-Pass and Bandpass Design The design of a high-pass filter is simply 1  H(z). In the time domain, this is      N N hðiÞ ¼ vc sinc vc i   vc d 2 2

1  i  N;

ð49:33Þ

where dðiÞ ¼



1 0

i¼0 : i 6¼ 0

ð49:34Þ

We construct a bandpass by filtering the data with a high pass-filter, then filtering the output with a low-pass filter. Or we can combine the two filters together into a single filter by convolving the coefficients. 49.3.3 Design of IIR Filters The common strategy in designing IIR filters is as follows: Step 1. Design a rational function in the S-domain in factored form that best approximates the desired frequency response characteristics (using Butterworth, Tchebyshev, elliptical, etc.). Step 2. Transform the poles and zeros into the z-domain. Step 3. Reconstruct the rational function. Step 4. Realize the difference equation by inverse z-transform of the rational polynomials. There are different S- to z-transforms. The two most common are the impulse-invariant and the bilinear transformation. The impulse-invariant transformation is z ¼ esT ;

ð49:35Þ

where T is the sampling period. This method is usually not used because it can cause aliasing. The bilinear transformation avoids this but distorts the mapping in other ways for which we need to compensate. The bilinear transformation is z¼

2=T þ s : 2=T  s

ð49:36Þ

The S- to z-transformation is not exact and always involves various trade-offs. Because of this, the actual placement of the cutoff in a designed digital filter is misplaced and the error increases for cutoff frequencies closer to the Nyquist frequency. To compensate for this effect, we apply Equation (49.37) in the design of our filter: f 0c ¼ This process is called “prewarping.”

1 tanðpf c TÞ: pT

ð49:37Þ

BASIC DIGITAL FILTER

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49.3.3.1 Example IIR Design For a digital system with a sampling rate of 100 Hz, a third-order low-pass Tchebyshev filter with a cutoff frequency at 15 Hz and with 20% ripple in the passband is designed as follows: Step 1. Compute the Nyquist frequency fN ¼ 100 Hz/2 ¼ 50 Hz. Step 2. Compute f 0 using Equation (49.37) to prewarp the cutoff frequency: f 0 ¼ 16:22 Hz: Step 3. Compute v0 using Equations (49.13) and (49.14): e ¼ 0:75

v0 ¼ 0:3662:

Step 4. Determine the poles and zeros of the analog system using Equation (49.12): p1 ¼ 19:0789 þ j94:2364 p2 ¼ 38:1578 p3 ¼ 19:0789  j94:2364: Step 5. Map the poles from the S-domain into the z-domain using the bilinear transformation Equation (49.36) and T ¼ (1/sampling rate) ¼ 0.01 s: pz1 ¼ 0:5407 þ j0:6627 pz2 ¼ 0:6796 pz3 ¼ 0:5407  j0:6627: Step 6. Map the zeros from the S-domain into the z-domain also using Equation (49.36). Although we may be tempted to conclude that there are no zeros in the S-domain, since our numerator is constant, we note that HðsÞ ! 0 as s ! 1. Thus mapping the bilinear transformation for this case yields lim

2=T þ s ¼ 1: s

s!1 2=T

And since this is a third-order system, z1 ¼ 1

z2 ¼ 1

z3 ¼ 1

Step 7. Expand the numerator and denominator polynomials: ðz  z1 Þðz  z2 Þðz  z3 Þ 1 þ 3z1 þ 3z2 þ z3 ¼ : ðz  pz1 Þðz  pz2 Þðz  pz3 Þ 1 þ 1:7611z1  1:466z2 þ 0:4972z3 Step 8. Normalize the filter so that the gain in the passband will be 1. In this case, we know that the gain should be 1 at v ¼ 0. Hence, we evaluate jHðe jv Þj for v ¼ 0: jHðe0 Þj ¼ 38:4. The normalized transfer function is then HðzÞ ¼

0:026 þ 0:078z1 þ 0:078z2 þ 0:026z3 : 1 þ 1:7611z1  1:466z2 þ 0:4972z3

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SIGNAL PROCESSING

Step 9. Realize the difference equation from inverse z-transformation of the derived transfer function: yn ¼ 1:7611yn1  1:466yn2 þ 0:4972yn3 þ 0:026xn þ 0:078xn1 þ 0:078xn2 þ 0:026xn3 :

49.3.4 Design of Various Filters from Low-Pass Prototypes The procedure for designing a high-pass, bandpass, or band-reject IIR filter is as follows: First, design, by pole-zero placement, a low-pass filter (Butterworth, Tchebyshev, etc.) with an arbitrary cutoff frequency (though for practical considerations, it is best to choose a value midway between 0 Hz and Nyquist), transform the poles and zeros according to the following formulas, reconstitute a new transfer function from the transformed poles and zeros, then realize the digital filter by taking the inverse z-transform of the new transfer function. The formulas with an example are given below, where vL and vH are the low- and high-frequency cutoffs, respectively, normalized between 0 and p, that is, vL ¼ 2pf L /sample rate, where fL is the cutoff frequency in hertz; fL is the normalized cutoff frequency of the low-pass prototype (Oppenheim and Schafer, 1975). 49.3.4.1 High Pass z0 ¼ 

1 þ Az ; ZþA

ð49:38Þ

where Z is the pole or zero to be transformed and z0 is the transformed pole or zero and A¼

cos½ðvH þ fL Þ=2 : cos½ðvH  fL Þ=2

ð49:39Þ

49.3.4.2 Bandpass The bandpass filter has two transitions: a rising edge and a falling edge. For this reason, we need twice as many coefficients for the same approximate transition width as the prototype filter. Thus, the order of these polynomials will be twice the order of polynomials in the prototype. Each pole from the prototype will transform into a pair of poles (z01 and z02 ). Likewise, each zero will transform into a pair of zeros: z01 ¼ z02 ¼

ðA þ AZÞ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA þ AZÞ2  4ðZ þ BÞðBZ þ 1Þ

2ðZ þ BÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA þ AZÞ  ðA þ AZÞ2  4ðZ þ BÞðBZ þ 1Þ 2ðZ þ BÞ

A¼ B¼

2CD Dþ1

D1 Dþ1

ð49:40Þ ð49:41Þ ð49:42Þ ð49:43Þ

BASIC DIGITAL FILTER

cos½ðvH þ vL Þ=2 cos½ðvH  vL Þ=2 v  v  f H L D ¼ cot tan L : 2 2 C¼

1677

ð49:44Þ ð49:45Þ

49.3.4.3 Band Reject z01 ¼ z02 ¼

ðAZ  AÞ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAZ  AÞ2  4ðZ  BÞðBZ  1Þ

2ðZ  BÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAZ  AÞ  ðAZ  AÞ2  4ðZ  BÞðBZ  1Þ 2ðZ  BÞ

cos½ðvH þ vL Þ=2 cos½ðvH  vL Þ=2 v  v  f H L D ¼ tan tan L : 2 2 C¼

ð49:46Þ ð49:47Þ ð49:48Þ ð49:49Þ

As an example, design a Tchebyshev (Type I) bandpass filter for a system sampled at 100 Hz with a low cutoff frequency of 20 Hz and a high cutoff frequency of 35 Hz. Step 1. We can use the poles and zeros designed in Section 49.3.3 as the low-pass prototype. Step 2. Convert the low and high cutoff frequencies to values normalized between 0 and p, where p corresponds to the Nyquist frequency: 20 35 vL ¼ 2p  ¼ 1:2566 vH ¼ 2p  ¼ 2:1991: ð49:50Þ 100 100 Step 3. Map the poles from the prototype using Equations (49.40) and (49.41):  0:6095  j0:9105 p1 ! 0:2940 þ j1:0722  0:2302  j1:2527 p2 ! 0:2302 þ j1:2527  0:6095 þ j0:9105 p3 ! : 0:2940  j1:0722 Then map the zeros from the prototype using Equations (49.40) and (49.41):  1 z1 ! 1  1 z2 ! 1  1 z3 ! : 1

ð49:51Þ ð49:52Þ ð49:53Þ

ð49:54Þ ð49:55Þ ð49:56Þ

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SIGNAL PROCESSING

Step 4. Expand the numerator and denominator polynomials: ðz  z1 Þ    ðz  z6 Þ ðz  pz1 Þ    ðz  pz6 Þ 1  3z2 þ 3z4  z6 ¼ ð1 þ 1:091z1 þ 3:632z2 þ 2:616z3 þ 4:642z4 þ 1:982z5 þ 2:407z6 :

ð49:57Þ

Step 5. Normalize the transfer function so that it will have unity gain in the passband. For this, we estimate M ¼ max jHðejv Þj:

ð49:58Þ

1 HðzÞ: M

ð49:59Þ

0vp

Then compute H normalized ðzÞ ¼

For this example, M ¼ 14.45. Step 6. Realize the difference equation from inverse z-transformation of the derived transfer function: yn ¼ 1:091yn1  3:632yn2  2:616yn3  4:642yn4  1:982yn5

: ð49:60Þ 2:407yn6 þ 0:0692  xn  0:2076xn2 þ 0:2076xn4  0:0692xn6

49.3.5 Frequency-Domain Filtering It is possible to filter the data in the frequency domain. The method involves the use of the Fourier transform. We Fourier transform the data, multiply by the desired frequency response, then inverse Fourier transform the data. This is similar to the FIR filters discussed earlier. Deriving the FIR coefficients by performing a discrete cosine transform (DCT) of the desired frequency response and then convolving the coefficients with the data is equivalent to filtering the data in the frequency domain. One difference, however, is that the frequency domain filtering is generally done on blocks of data and not on streaming data, as is done in the time domain, which can be of concern when processing highly nonstationary data with abrupt transients. The inverse Fourier transform is 1 F fFðvÞg ¼ pffiffiffiffiffiffi 2p 1

Z1

1

f ðvÞejvt dv:

ð49:61Þ

49.4 STABILITY AND PHASE ANALYSIS 49.4.1 Stability Analysis Consider a transfer function HðsÞ ¼

1 ; sp

ð49:62Þ

STABILITY AND PHASE ANALYSIS

1679

where the pole p is a complex number a þ jb. The inverse Laplace transform of this function is eat ½cosðbtÞ þ j sinðbtÞ. This function is bounded as t ! 1 if and only if a < 0. From this, we can determine the stability of a function by inspecting the real components of all poles of a given transfer function. The procedure for a rational function (a ratio of polynomials) would be to factor the polynomials in the denominator and inspect to ensure that the real components to all of the poles are less than zero. Suppose

HðsÞ ¼ ¼

a0 þ a1 s þ    þ am s m b0 þ b1 s þ    þ bn s 2

: ðs  z0 Þðs  z1 Þ    ðs  zm Þ ðs  p0 Þðs  p1 Þ    ðs  pn Þ

ð49:63Þ

In a similar way, by inspection of the S-to-z transformation z ¼ es we see that the entire left half of the plane in the S-domain maps inside the unit circle in the z-domain. For this reason, we analyze the stability of systems in the z-domain by inspecting the poles of the transfer function. The system is stable if the norm of all poles is less than 1. 49.4.2 Phase Analysis While processing the data in real time, our filters must act on the signal history. For this reason, there will always be some delay in the output of our process. Worse, certain filters will delay some frequency components by more or less than other frequency components. This results in a phase distortion of the filter. For a certain class of FIR filters, it is possible to design filters that shift each frequency component by a time delay in proportion to the frequency. In this way, all frequency components are shifted by an equal time delay. Although it is possible to design certain nonreal-time, noncausal IIR filters that are phase shift distortionless, in general, IIR filters will produce some phase shift distortion. We can determine the actual phase shift for each frequency component by computing arg Hð jvÞ ¼ ffH real þ jH imag ¼ tan1

H imag : H real

The arctan will produce the principal value of the phase shift, not necessarily the cumulative phase shift, since the arctan function produces principal value p  tan1 ðfÞ  p. It is possible to recover the accumulated phase shift by factoring the rational function into its binomial parts, then expressing this in exponential form as a summation. One can then determine the principal angle on each part and the accumulated phase shift by summing the parts.

49.4.3 Comparison of FIR and IIR Filters There are various factors when deciding on a particular filter for a given application. Table 49.3 summarizes these.

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SIGNAL PROCESSING

TABLE 49.3 Comparison of FIR and IIR Characteristics FIR

IIR

Run time efficiency

Less efficient; requires high-order filter

Stability

Always stable

Phase shift distortion

Can be designed to be phase shift distortionless Simpler design process, usually involving Fourier transforms or solving linear systems

Higher efficiency; usually possible to achieve a desired design specification in fewer computations Stable if all poles are inside the unit circle Generally distorts phase

Ease of design

Design is more complex, involving special functions or solving nonlinear systems

49.5 EXTRACTING SIGNAL FROM NOISE The PSD of white noise is uniformly distributed over all frequencies. Therefore, it is possible to detect the PSD signature of a signal corrupted by white noise by inspecting spectral components that rise above some baseline. From this, we can design a matching filter to optimally extract the signal from the noise. Figures 49.11 and 49.12 illustrate this procedure.

FIGURE 49.11 Use of bandpass filter for discriminating signal from noise.

REFERENCES

1681

FIGURE 49.12 Improved matching filter for better discrimination of signal from noise.

REFERENCES Marple S. Digital Spectral Analysis with Applications. Englewood Cliffs (NJ): Prentice-Hall; 1987. Oppenheim A, Schafer R. Digital Signal Processing. Englewood Cliffs (NJ): Prentice-Hall; 1975. Parks T, Burrus C. Digital Filter Design. New York: Wiley; 1987. Press W, Flannery B, Teukolsky A, Vetterline W. Numerical Recipes in C. Cambridge, UK: Cambridge University Press; 1988. Vlc9ek M, Unbehauen R. Analytical solutions for design of IIR equiripple filters. IEEE Transactions on Acoustics, Speech, and Signal Processing 1989;37(10): 1518–1531.

50 DATA ACQUISITION AND DISPLAY SYSTEMS PHILIP C. MILLIMAN 50.1 Introduction 50.2 Data acquisition 50.3 Process data acquisition 50.3.1 Sampling interval 50.3.2 Accuracy and precision of data 50.3.3 Time-based versus event-driven collection 50.4 Data conditioning 50.4.1 Simple linear fit 50.4.2 Nonlinear relationships 50.4.3 Filtering 50.4.4 Compression techniques 50.4.5 More on sampling and compression 50.5 Data storage 50.5.1 In-memory storage 50.5.2 File storage 50.5.3 Database storage 50.5.4 Using third-party data acquisition systems 50.6 Data display and reporting 50.6.1 Current-value inspection 50.6.2 Display of individual data points 50.6.3 Display of historical data 50.7 Data analysis 50.7.1 Distributed systems 50.7.2 System error analysis 50.8 Data communications 50.8.1 Serial communications 50.8.2 Parallel communications 50.8.3 Networks 50.8.4 OSI standard 50.8.5 OPC standard 50.8.6 Benefits of standard communications

Handbook of Measurement in Science and Engineering. Edited by Myer Kutz. Copyright Ó 2013 John Wiley & Sons, Inc.

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50.9 Other data acquisition and display topics 50.9.1 Data chain 50.9.2 Web programs and interfaces 50.9.3 Configuration versus implementation 50.9.4 Store and forward 50.9.5 Additional communications topics 50.10 Summary References

50.1 INTRODUCTION The industry has changed significantly since this chapter was first written in the months before 1990. The personal computer has become part and parcel of everyday life. Control systems have become increasingly based on standard systems and interfaces; sensors themselves are often based on just smaller versions of the same operating system as large manufacturing systems. This has tended to change the focus from the technology of data acquisition to the software and systems to support data acquisition. The trend has been away from requiring the engineer to understand the science of how sensors work and the lowest levels of data acquisition and more toward the engineer understanding the collection, coordination, storage, access, and manipulation of data. With that in mind, this chapter has been updated to focus more on the latter and less on the former. Other chapters in this book cover details of the electronics, transducers, sampling, and calibration. To control any process or understand what occurs during the life cycle of a process, the system (a human or machine) must have information about what is occurring. In the simplest of control loops, the measured variable must be converted into usable units, comparison in some form to a target occurs, and a response is determined. At the plant level, improvement of plant operation relies upon understanding the relationships between processes within the plant (not only current, but historical), which in turn requires collecting data throughout the plant, characterizing the relationship of the data with other data, storing the data in such a way as to be retrievable in a useful, timely way, and manipulating the data for presentation and hopefully providing an aid to understanding the relationships between processes. In today’s competitive environment, focusing on local control and ignoring the interaction between processes, both internal to the plant and external, can be disastrous. If one is not focused on improvement, one can bet the competitor is. Larger corporations, especially, can bring analytical tools to bear to improve local processes, plantwide processes, and their relationships to external influences, such as the supply chain. On the other hand, today’s computing tools bring very powerful data acquisition and analysis capability within the reach of the average technician with a little bit of motivation. Data acquisition and display systems have changed dramatically. Twenty years ago, terms referring to specialized systems such as SCADA (supervisory control and data acquisition) and data loggers were common terms. Now, with the proliferation and broadening role of computer systems and their intrusion into every aspect of manufacturing,

DATA ACQUISITION

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many of the features that used to be in specialized instruments and systems are now part of the everyday tools available to anyone with a computer. This chapter attempts to cover aspects of data acquisition and manipulation that may help the engineer better understand issues and give a foundation for using and even constructing tools. The organization is as follows: 

The initial sections cover the nature of data and the acquisition and conversion of data to usable units and includes some discussion of useful display techniques. The discussion attempts to identify issues of which the engineer should be aware and give guidelines on how to manage data.  The latter sections cover the coordination, storage, access, and manipulation of data. A discussion of pros and cons of different strategies should help the reader understand the trade-offs in system selection and construction. It is difficult to do this without describing specific technologies and brands, but the author has endeavored to level the discussion in such a way that changes in technology will not change the value of the discussion. Time will tell if the approach is effective. 50.2 DATA ACQUISITION Data acquisition includes the following: (1) acquiring raw data from the process being measured and (2) converting data into usable units. Included in this section are also some topics of data display closely related to the nature of the data being acquired. Other aspects of data display will be covered in later sections. In process industries, much of the data are analog in nature, such as pressure, temperature, and flow rate. The values acquired are sampled representations of process data that have a scale and a range, with various issues around effective range and whether values over a range are linear or more complex. When acquired in a data acquisition system there are a number of issues that must be addressed related to how data is sampled; how it is represented in the computer as a digital value but still able to be manipulated as an analog number; and how a continuously changing value can be stored without exceeding the capacity of storage or computation of the acquisition system. Discrete manufacturing still has a number of analog data sources, but a larger proportion involves discrete data, such as motor stops, starts, and pulses. These have their own issues of acquisition and storage and are often related to attributes of the process. The next section deals primarily with process data, additionally covering some discrete data and issues around data collection, representation, and storage. As businesses begin to broaden the scope of optimization and understand their global processes, the context of the data in terms of product, plant conditions, market conditions, and other environmental aspects has increasingly added discrete data to the set of data to be obtained. The interaction of the process with factors such as which crew is managing the process, which customer’s needs are highest priority, legal controls such as environmental limits impacting allowable process rates, operator decisions, which product is being manufactured, grade achieved, and a large number of other factors become important when a company is competing with other companies that have already achieved excellent local control of processes. These data involve less understanding how to deal with continuous data and more with the coordination of data within and between processes. These can be termed manufacturing attributes to emphasize their importance in

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providing an environment around process data. Later sections deal with manufacturing attribute data and issues around their collection and coordination with process data.

50.3 PROCESS DATA ACQUISITION Most modern data acquisition is via digital systems that may have a lower level analog collection mechanism but is now so removed from the engineer that the engineer is only concerned with the digital portion of the system. The ability to use digital microprocessors as building blocks for data collection, the prevalence of computer tools, and the creation of widely available operating systems that operate on small footprints have virtually eliminated the need for analog instrumentation. While the data may be analog in nature, the technology has been developed to such a degree that the engineer decreasingly needs to pay attention to the analog aspects of the data. A digital-to-analog (D/A) and analog-to-digital (A/D) converter performs the actual processing required to bring analog information from or to the process. While the resulting signal may be digital, it is a representation of a continuous number that has characteristics that, if not understood, can result in erroneous conclusions from data, including missing data, misinterpreting trends, or improperly weighting certain values. The engineer should be aware of several features of analog data to ensure that the data are used properly. An understanding of sampling interval, scaling, and linearization will facilitate the use of data once collected. 50.3.1 Sampling Interval One of the important steps with any data collection process includes the proper choice of sampling interval. As an example of the impact of selection of sampling interval, or frequency, a sine wave with a period of 1 s (Figure 50.1a) is measured with several sampling intervals. Both 0.5-s (Figure 50.1b) and 0.1-s (Figure 50.1c) intervals provide different impressions of what is actually happening. The 1-s sampling rate being in phase with the sine wave yields the impression that we are measuring a nonvarying level. The 0.5-s sampling rate yields several different results depending on what phase shift is encountered. This is known as aliasing (see Johnson, 1984, pp. 122–125). If a 0.1-s period is used, we finally begin to obtain a realistic idea of what the waveform truly looks like. The sampling interval has a different impact when collecting manufacturing attribute data. With manufacturing attribute data, every change in value has importance. They provide a context to the process that assists with the tieback to business interactions. The values are often coded. The sampling must be close enough to the time of occurrence to allow determination of state in relation to other events. The sampling interval must be fast enough to capture any change in state. Sampling at slower than the change rate of the data will mean lost events—potentially critical in a situation where the count of items processed is important. Sampling at a rate slightly faster than the maximum change rate of the manufacturing attribute data assures that no change will be missed. Another important consideration is to know at what time an event occurred. For instance, if a value changes only infrequently but other related manufacturing attribute values are changing at a faster rate, then the scan rate has to be fast enough to match the fastest change rate of all the related manufacturing attribute data. This is sometimes called the master scan rate, meaning that the frequency of scanning must be fast enough to capture faster events and

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FIGURE 50.1 Sampling (a) sine wave form; (b) aliasing of data; (c) sampled every 10 s.

determine the state of other variables relative to those events. Similarly, if there are related analog data, the scan rate may have to be fast enough to even characterize the curve of the analog data (remember Figure 50.1). The capacity of the target system must also be taken into account. Storing large volumes of data is becoming more feasible as systems increase in speed and power, but the retrieval and organization of those data may become a time-consuming, overly complex task with too much data or poorly organized data. Consequently, even though storage itself is less of an issue, other factors impact how much data are retained and how organized for later retrieval. Later sections examine approaches for organizing and retrieving data. 50.3.2 Accuracy and Precision of Data Accuracy and precision are dependent on the sampling interval as well as the resolution of the system (Murrill, 1981; Liptak, 2003, pp. 78–80; and Chapter 1 in this volume). When dealing with the A/D conversion process, the step size or number of bits used is critical when determining the system precision and accuracy (Johnson, 1984, pp. 78– 81). Figure 50.2 illustrates the difference between accuracy and precision of data. Table 50.1 illustrates the effect the number of bits has on the precision.

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Precise but inaccurate measurement

Imprecise but accurate measurement

Actual signal (accurate and precise)

FIGURE 50.2 Difference between accuracy and precision.

This also interacts with range, which will be discussed later, since having an accurate number over a small percent of the desired range would not allow the ability to fully characterize the process. For example, highly accurate readings with 1% moisture resolution over a range from 10 to 20% moisture content would be inadequate if one were attempting to measure moisture over a 5–40% range. When selecting transducers, it is necessary that they have both the accuracy and the range needed for the process being observed. When selecting converters, one should be aware of the settling time (governs how often readings can be obtained), resolution of the converter (affects range and detail of measurements), and accuracy of sensor. Chapter 1 includes some characteristics of transducers, including calibration, and the sampling of data. One should be aware that an event that has been stabilized in a data collection system may be offset in time, resulting in a potential discrepancy between events or values when values are compared from different sources or from multiplexed data. It should be verified that the transducer is collecting the data fast enough to allow one to have relevant times of collection in the data acquisition system. Also, one should assure that the potential relationship between events from different sources and their intended use is understood when considering the speed and accuracy of transducers. 50.3.3 Time-Based Versus Event-Driven Collection There are two major approaches when collecting data with a general-purpose data acquisition system. In one approach, data are collected on a regular frequency based on TABLE 50.1 Relationship Between Number of Bits and Precision Number of Bits 8 10 12 16

Steps

Resolution on 5-V Measurement

Percent Resolution

256 1,024 4,096 65,536

0.01950 0.00488 0.00122 0.00008

0.3900 0.0980 0.0240 0.0015

DATA CONDITIONING

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time, such as once per second. This is easy to institute and it is relatively easy to analyze the data and their relationships after the fact. This approach tends to require more data storage and can make it difficult to identify events or the interactions with manufacturing attribute data. The other approach is event-based acquisition. An event is identified, such as when a package is dropped onto a platform, the time of that event is recorded, and the values of related variables are collected for that time. The sampling rate of the transducers to acquire the other variables may be important, as their values may become irrelevant if too long a time interval has passed after the event has occurred when the related variables are sampled. Batch processes, such as mixing a tankful of chemicals, often have some data collected only at the start and end of the process. Other data may be recorded at fixed time intervals during the batch process. Depending on the needs, the data during the actual reaction may be of great or of little use. The time between sampled events may be several minutes, hours, or even days in length, but the time of the event may be critical, as well as detailed data at the time of the event, resulting in a common tactic of using high-speed scanning to detect the occurrence of an infrequent event. Approaches for combining and analyzing data will be covered in a later section.

50.4 DATA CONDITIONING Often the data obtained from a process are not in the form or units desired. This section describes several methods of transforming data to produce proper units, reduce storage quantity, and reduce noise. There are many reasons why process measurements might need to be transformed in order to be useful. Usually, the signals obtained will be values whose units (e.g., voltage, current) are other than the desired units (e.g., temperature, pressure). For example, the measurement from a pressure transducer may be in the range 4–20 mA. To use this as a pressure measurement in pounds per square inch (PSI), one would need to convert it using some equation. As environmental conditions change, the performance characteristics of many sensors change. A parametric model (equation) can be used to convert between types of units or to correct for changes in the parameters of the model. The parameters for this equation may be derived through a process known as calibration (Chapter 1 covers much of the process of calibration and sampling). This involves determining the parameters of some equation by placing the sensor in known environmental conditions (such as freezing or boiling water) and recording the voltage or other measurable quantity it produces. Some (normally simple) calculations will then produce the parameters desired. (See the following discussion of simple linear fit for the procedure for a simple, twoparameter equation.) The complexity of the model increases when the measured value is not directly proportional to the desired units (nonlinear). Additionally, as sensors get dirty or age, the parameters might need adjusting. There are a variety of control techniques to assist with compensating for changes in the environment around the sensor, including adaptive control. 50.4.1 Simple Linear Fit The simplest formula for converting a measured value to the desired units is a simple linear equation. The form of the equation is y ¼ ax þ b, where x is the measured value, y

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100 90 80 70 60

°C

50 40 30 20 10 0 0

0.2

0.4

0.6

0.8

1

Voltage (V)

Legend Line

Voltage Voltage Equation Equation for 0°C for 100°C for a for b 100 – 0 0 – 100 0 1 1–0 100 – 0 0 – 50(–1) –1 1 1 – (–1) 100 – 0 0 – 250(0.2) 0.2 0.6 0.6 – 0.2

Value Equation for °C for b 0 C =100V 50

C =50V + 50

–50

C =250V – 50

FIGURE 50.3 Relationship between measured values and engineering units.

represents the value in the units desired, and a and b are parameters to adjust the slope and offset, respectively. The procedure for finding a and b is as follows: 1. create a known state for the sensor in the low range. An example would be to put a temperature sensor in ice water; 2. determine the value obtained from the sensor; 3. create a known state for the sensor in the high range. An example would be to immerse the sensor in boiling water; 4. determine the value obtained from the sensor; 5. calculate the values of a and b from these values using the equations. a¼

actual high value measured high value

b ¼ actual low value

actual low value measured low value

ða  measured low valueÞ:

ð50:1Þ ð50:2Þ

Figure 50.3 demonstrates example relationships between measured values and engineering units. 50.4.2 Nonlinear Relationships Often, there is not a simple linear relationship between the engineering units and the measured units (Figure 50.4). Instead, for a constantly rising pressure or temperature, the

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30 25

°C

20 15 10 5 0 –10

–5

0

5

10

Voltage (V) Legend

Y = 10 + 4 sin(1.3X ) (transcendental) Y = 0.5X 2 + 10 Y = 0.9X 2 + 0.6 sin(0.5X 2)

FIGURE 50.4 More complex data relationships.

measured value would form some curve. If possible, we use a portion of the sensor’s range where it is linear, and we can use Equations (50.1) and (50.2). When this is not possible, we have to characterize the sensor by a different equation, which could be a polynomial, a transcendental, or a combination of a series of functions. One can imagine several sensors which are linear in different ranges to be used in conjunction to create a larger range of operational data. This variety of formulas should make one point clear: without an understanding of the basic model of the sensor, one cannot know what type of conversion to use. Many sensors have known differences in output depending on the range of sensed data. Be aware of the effect environmental conditions have on the sensor readings. If the characteristics of a sensor are unknown, then the sensor must be measured under a variety of conditions to determine the basic relationship between the measured values and the engineering units. Some knowledge of the theory of the sensor’s mechanism will help to give an idea of which model to use. The development and evaluation of a model is beyond the scope of this chapter, but other chapters in this volume provide assistance. 50.4.3 Filtering Even after data are converted into the appropriate units, the data may have characteristics that inhibit understanding the important relationships for which one is looking. For instance, the data may have occasional fluctuations caused by factors other than the process or the process may have short-term perturbations, which are not really an indication of the major process factors.

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Filtering is a technique that allows one to retain the essence of the data while minimizing the effects of fluctuations. The data may then appear to be “smoothed.” In fact, the terms “filtering” and “smoothing” are often interchanged. Filtering can occur when the data are still in an analog state (Johnson, 1984, p. 54) or can occur after the data are converted into digital data (digital filtering). Measurement variability comes from a variety of sources. The process itself may undergo fluctuations that result in variation in measurement but that are only temporary and should be ignored. For instance, if the level of an open tank of water were to be measured but waves cause fluctuations in the height of a float, then the exact value at any given time would not be an accurate reflection of the level of the tank. The sensor itself may have fluctuations due to variability in its method for acquiring data. For instance, the proximity of a 60-Hz line may induce a 60-Hz sinusoidal variation in the signal (measured value). Examples of filtering approaches are as follows and are also given in other chapters in this volume (Wright and Edgar, 1983, p. 538): (a) Repeated Sample Average: take a number N of samples at once and average them: 1X ValueðiÞ N (b) Finite-Length Average (Moving): take the average of the last N measurements, averaging them to obtain a current calculated value. (c) Digital Filters: y ¼ ð1

aÞyi

1

þ axi 1 :

The simple average is useful when repeated samples are taken at approximately the same point of time. The more samples, the more random noise is removed. Chapter 1 addresses some of the issues with sampling and the concept of population distribution. The formula for an average is shown in (a) above. However, if the noise appeared for all the samples (as when all the samples are taken at just the time that a wave ripples through a tank), then this average would still have the noise value. A moving average can be taken over time [see (b) above] with the same formula, but each value would be from the same sensor, only displaced in time. A disadvantage of this approach is that one has to keep a list of previous values at least as long as the time span one wishes to average. A simpler approach is the first-order digital filter, where a portion of the current sample is combined with a portion of previous samples. This composite value, since it contains more than one measured value, will tend to discard transitory information and retain information that has existed over more than one scan. The formula for a first-order digital filter is described in (c) above, where a is a factor selected by the user. The more one wants the data filtered, the smaller the choice of a; the less one wants filtered, the larger the choice of a. Alpha must be between 0 and 1 inclusive. The moving average or digital filter can tend to make the appearance of important events to be later than the event occurred in the real world. This can be mitigated somewhat for moving averages by including data centered on the point of time of interest in the moving-average calculation. These filters can be cascaded, that is, the output of a filter can be used as the input to another filter.

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A danger with any filter is that valuable information might be lost. This is related to the concept of compression, which is covered in the next section. When data are not continuous, with peaks or exceptions being important elements to record, simple filters such as moving-average or digital filters are not adequate. Some laboratory instruments such as a gas chromatograph may have profiles that correspond to certain types of data (a peak may correspond to the existence of an element). The data acquisition system can be trained to look for these profiles through a pre-existing set of instructions or the human operator could indicate which profiles correspond to an element and the data acquisition system would build a set of rules. An example is to record the average of a sample of a set of data but also record the minimum and maximum. In situations where moisture or other physical attributes are measured, this is a common practice. Voice recognition systems often operate on a similar set of procedures. The operator speaks some words on request into the computer and it builds an internal profile of how the operator speaks to use later on new words. One area where pattern matching is used is in error-correcting serial data transmission. When serial data are being transmitted, a common practice to reduce errors is to insert known patterns into the data stream before and after the data. What if there is noise on the line? One can then look for a start-of-message pattern and an end-of-message pattern. Any data coming over the line that are not bracketed by these characters would be ignored or flagged as extraneous transmission. 50.4.4 Compression Techniques For high-speed or long-duration data collection sessions there may be massive amounts of data collected. It is a difficult decision to determine how much detail to retain. The tradeoffs are not just in space but also in the time required to store and later retrieve the data. Sampling techniques, also covered in other chapters, provide a way of retaining much of the important features of the data while eliminating the less important noise or redundant data. As an example, 1000 points of data collected each second for 1 year in a database could easily approach 0.5 terabyte when index files and other overhead are taken into account. Even if one has a large disk farm, the time required to get to a specific data element can be prohibitive. Often, systems are indexed by time of collection of the data point. This speeds up retrieval when a specific time frame is desired but is notoriously slow when relationships between data are explored or when events are searched for based on value and not on time. Approaches to reducing data volume include the following: 

Reduce the volume of data stored by various compression tools, such as discovering repeating data, storing one copy, and then indicating how many times the data are repeated. There are many techniques for compressing data, covered elsewhere. Zip files are a common instance of compression to make the data consume less storage and to take less time in transmission from one computer to another. Compression tends to increase the storage and retrieval time slightly. Increasingly, file systems associated with common operating systems include compression as a standard option or feature of mass storage. These systems are quite good at compressing repeated data but are less effective when data vary but have a mathematical relationship, such as a straight line between two points.  Normalize the data. When the developer knows relationships between data, redundancy can be avoiding by normalizing the data–following some basic principles to organize the data in such a way that redundancy is avoided (Date, 1990). C. J. Date

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(1990) describes the levels of normalization of data in a relational database. For instance, if a person has several addresses, then one could store the person’s name once, store each address, and store the links from the person to the address. While very similar to compression, it relies on the developer identifying and taking advantage of the relationships between data to eliminate redundancy and reduce space. This creates significant effort in planning for acquisition and storage of data. It pays off in reduced storage and significantly improved retrieval and analysis times.  Eliminate nonessential data. If one is not interested in the shape of a sinusoidal signal, for instance, but only interested in how many cycles occurred during a given time frame, then sampling techniques can be used to characterize the data without having to store significant data. The engineer or researcher has to make assumptions about how the data will be used and factor those into the acquisition and storage system. A project attempting to discover the relationships between waveforms would require high-frequency sampling and probably time-based storage. A project attempting to record the number of times a boiler went over a certain temperature level might have a high-speed scanning capability but only store those values that were above the temperature limit. An inventory tracking system may have triggers that cause scanning only when some event occurs. Often, a batch or pallet of product may contain a large number of items. The items can be sampled for some process attribute. The customer may want to know summary statistics about the pallet, but the storage of all the data may not be feasible. In this case, statistical results can normally be derived from summary data. Average, standard deviation, total, correlation, maximum, and minimum are easily calculated from summary, accumulated data1: Averages: Keep a running sum of data and count of readings: Average ¼ Sum of values=count of values Totals: Keep a running sum of data. Standard Deviation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Pn 2 P n ni¼1 x2i i¼1 xi nðn 1Þ Correlation: P P P n xi yi ð xi Þð yi Þ r ¼ rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ih P i P P P n x2i ð xi Þ2 n y2i ð yi Þ2

ð50:3Þ

Range: Save largest and smallest values. Median: Find the middle value of a distribution, which requires keeping all values. 1

Standard deviation and correlation from Beyer ( 1976, pp. 473, 477).

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The median (the true center of the data) requires the raw data to be calculated. A compromise for depicting the distribution of data without having to store the full details is to store a distribution of the data. For instance, the range of possible important data can be broken into a series of totals, reflecting the count of items that fit into the particular total. A histogram representing the distribution of data can be created from the totals without requiring the full set of original data. In addition, the median can be approximated using this technique. The distribution can also be used to supply data for statistics based on distribution of data, such as the Taguchi loss function (Crossley, 2000, pp. 397–400). 50.4.5 More on Sampling and Compression Rather than just sample the data, why not save all the changed values of the data, discarding values which are the same or within some limits of the previous reading? This really applies best to continuous processes. Quite significant space reduction can be maintained in processes that are slowly changing and have only occasional large upsets. Variations of this technique can provide additional improvements. For instance, rather than just checking to see if the current value is the same as or within some limits from the previous reading, see if it is on the same line or curve as the previous value. This can result in a great reduction of storage requirements at the loss of a slight amount of accuracy in reconstruction. The more flexible the compression technique, the more work must be done to reconstruct the data later for examination. For instance, if the user of the data acquisition system wants to retrieve a data point within data that has been reduced to a line segment, the user or the system must determine which line segment is wanted using the time stamp for the beginning and ending of the line segment interval and then recalculate the point from the equation. This is referred to as a boxcar algorithm. Values that are close to the line segment can be treated as on the line segment if one can afford to lose some accuracy (http://www.aspentech.com/publication_files/White_Paper_ for_IP_21.pdf). The formula for a boxcar has to take into account the length of the interval (maximum), the height of the box (how much noise is allowed), and how peak or exception values are treated. For instance, Table 50.2 presents a set of data with several types of compression applied for data sampled at a constant interval. In Table 50.2, the simple repeating-value compression will not lose data but will result in little or no compression if the data are changing value frequently, including having any noise. The boxcar compression technique results in much higher compression for slowly changing data with only the loss of fine detail data (depending on the height of the window). For data that are nonlinear or changing frequently, the boxcar compression method results in little compression. Process information systems often use the boxcar compression method. If data are slow moving with occasional bursts of activity, the boxcar and repeating-value methods can result in dramatic reductions in space required. If data changes tend to be linear, then the boxcar algorithm tends to be superior to the repeatedvalue approach. For an extreme example, see Table 50.3. The raw data would have resulted in 631 data points being stored. The boxcar method would result in five data points being stored, less than 1% of the storage required. In the example above, the repeated-value method would have resulted in almost exactly the same compression as the boxcar. However, if there had been a 0.1% slope in data throughout the period, the boxcar would remain the same but the repeated-value method would have resulted in no compression. A deadband (the height of the boxcar) around the repeated value (meaning two values within some small deviation from each other would

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TABLE 50.2 Examples of Compression Techniques Time Stamp and Raw Value

Simple Repeating Value

Boxcar Compression

Time

Value

Time

Value

Count

Start Time

Start Value

12:00 12:01 12:02 12:03 12:04 12:05 12:06 12:07 12:08 12:09 12:10 12:11 12:12 12:13 12:14 12:15 12:16 12:17

1 1 1 2 3 4 5 6 7 5 3 1 5 5 5 5 5 1

12:00 12:03 12:04 12:05 12:06 12:07 12:08 12:09 12:10 12:11 12:12 12:17

1.0 2.0 3.0 4.0 5.0 6.0 7.0 5.0 3.0 1.0 5.0 1.0

3 1 1 1 1 1 1 1 1 1 5 1

12:00 12:03 12:08 12:11 12:12 12:16

1 2 7 1 5 5

Slope 0 1 2 4 0 4

be counted as the same value) would result in very high compression in the above example. A long ramp-up of the value during that time frame would have further differentiated the two compression methods. There are many techniques for compression of data. As aforementioned, many rely on assumptions about the underlying nature of the data, such as being continuous data. TABLE 50.3 Data Compression: Raw Data Versus Boxcar Raw Dataa Start Time 12:00 12:01 ... 14:59 15:00 15:01 15:02 15:03 15:04 ... 18:59 19:00 a

Boxcar Method

Start Value

Start Time

Start Value

1 1

12:00 15:00 15:01 15:02 15:03

1 1 2 5 1

1 1 2 5 1 1 1 1

Gaps represent no change in data.

Slope 0 1 3 4 0

DATA STORAGE

1697

Where data are directly related to events, more traditional compression techniques which look for repeating patterns in the data may be used. These are typically performed by operating systems and database systems and can therefore be taken advantage of with little or no work on the part of the engineer.

50.5 DATA STORAGE In whatever ways data are sampled, collected, filtered, smoothed, and/or compressed, at some point the data must be stored on some media if long-term data are to be analyzed (covered later in this chapter). There are several approaches to data storage that will be discussed in brief here. 50.5.1 In-Memory Storage There are normally limitations on how much data can be stored, particularly when lowfrequency events have high-frequency data surrounding them that are of interest. For example, if scientists are monitoring Mt. St. Helens for seismic data, it would be prohibitive to capture millisecond data for years while waiting for an eruption. It would be of interest to capture data at high density just before, during, and after each eruption but not in the quiet times in the intervening years. Collecting the millisecond data on many sensors would overflow the storage capability of most systems. There are techniques to store subsets of the data that allow high-density data from constrained time intervals to be stored. An approach for collecting and later reporting high-density data around an event of interest is to collect the data continuously using the triggered snapshot method. High-speed data are temporarily retained for a fixed time interval or memory capacity, with the start and end time of the data moving forward with time. Older data are discarded as the time range moves past it. This moving window is useful for creating trends and summary data for that interval. The user can be shown dynamic displays that update over time and reflect characteristics of the moving window of the process. Periodically, a set of the data can be extracted to mass storage, especially triggered by some event of interest. An event is recognized by some means (automatic or user generated) that the engineer has preconfigured to cause the transfer of the current instance of the moving window to permanent storage. The relationship between the trigger and the moving window can be configured several ways, as depicted in Figure 50.5. The handling of the data involves moving values through a data array, adding more recent values at the end and pushing the rest toward the beginning—a queue. The triggered snapshot is particularly useful when knowledge about the sequence of events just before the event of interest can help discover problems. As an example in manufacturing, in sawmills there are often very high-speed sequences of events, such as where a board may come out of one conveyor and is transferred to another conveyor and some event such as the board leaving the conveyor occurs. High-speed video can be always in progress, and the detection of the board leaving the system can be used to trigger the transfer to permanent storage of the video. Events of concern can be safety issues and the triggered snapshot method can be used to help eliminate potential life-threatening situations. The triggered snapshot method is particularly useful for discovering the causes of unusual events.

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Data can be stored around trigger in three ways: Beginning trigger: Window can be stored with trigger occurring at earliest point of stored interval. Benefits: Can aid in detecting what happened after the trigger event. Disadvantages: Cannot tell what led up to the triggered event.

Ending trigger: Window can be stored with trigger occurring at latest point of stored interval. Benefits: Can aid in detecting what happened leading up to the trigger event. Disadvantages: Cannot tell what happened after the triggered event.

Beginning of moving window

End of moving window

Timeline This was the earliest data acquired in the moving window

Centered trigger: Window can be stored with trigger occurring at middle point of stored interval. Benefits: Can aid in detecting what happened before and after the trigger event. Disadvantages: None, other than still have to predetermine what constitutes a trigger.

This was the most recent data acquired in the moving window

FIGURE 50.5 Relationship between trigger and moving window.

This has the advantage of allowing monitoring and analysis of high-speed events and still capturing some data to enable determining some data relationships. This is most useful if the engineer has some idea of what events may yield valuable relationships. It is much less useful when events, triggers, or relationships are unknown or unexpected. Sampling data at slower intervals may serve to allow accidental capture and identification of useful relationships, but the work required to find those relationships is much higher and of questionable probability of success. As an example, in the sawmill many variables are changing state at high speed. For diagnostic purposes, it is valuable to see a high-speed snapshot of states of photo eyes compared to saw drops, gate openings, and grade decisions. However, the volume of data is normally too great for storage and later analysis. There are some events that are of more importance than others, such as when a gate is opening early or a saw is failing to drop consistently. These can often be recognized and the data captured in the window of time before and after the event can be stored, allowing later analysis of what led to the event and what happened shortly afterward. Some characteristic data can be summarized for each time window, stored, and used later for analysis, such as the number of photo eye changes, number of saw drops, and number of gate openings. More complex relationships can be tallied to aid in diagnostics, such as number of gate openings for grade 2. The more complex the relationship, the more difficult the programming task to ensure capturing the incidence to storage. A typical pattern is to collect process variables that may be of interest, often from a programmable logic controller. As a given problem begins to be identified, additional logic can be added to examine relationships between process inputs and sequences of events, creating a new variable

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TABLE 50.4 Example of Storing Multiple Variables in Files Filename Data in file

Process Data 2004-03-12, 14, 23, 05 Timestamp, Temperature A, Temperature B, Temperature C, Rate 1, Rate 2, Rate 3 2004-03-20, 12:34, 15, 14, 15, 12, 13, 13 2004-03-20 12:35, 15, 14, 14, 12, 13, 13 2004-03-20 12:36, 16, 14, 13, 12, 13, 13 2004-03-20 12:37, 15, 14, 12, 12, 13, 13 2004-03-20 12:38, 15, 14, 11, 12, 13, 13 2004-03-20 12:39, 14, 14, 11, 12, 13, 13 2004-03-20 12:40,15, 14, 10, 12, 13, 13 2004-03-20 12:41, 15, 14, 10, 12, 13, 13 2004-03-20 12:42, 16, 14, 10, 12, 13, 13 2004-03-20 12:43, 15, 14, 09, 12, 13, 13

that reflects some attribute of that relationship. The data collection system can store the results of that variable, such as the sum of the number of times it occurred in some larger time interval, allowing it to be low enough in volume to be mass stored for later analysis. 50.5.2 File Storage An easy way to store data is in a file, often a comma-delimited file. This is easy to program and can easily be imported into analysis tools such as spreadsheets. It is not well suited to the compression techniques mentioned earlier because of the complexity of storing and interpreting data. However, for storing records of multiple variables collected at a time interval this can be a very useful technique. An example is shown in Table 50.4. In the above example, the filename was chosen so that it would be unique. The date and time were concatenated together, eliminating every invalid character with an underscore. This helps to prevent files from being overwritten accidentally and facilitates storeand-forward techniques described below. Files can be sorted by date or name. Often, they are stored in a directory structure so the number of files in any one directory does not get too great. This speeds up file search in a given directory but can make programs more complex that search for files across directories. An example directory structure is the following: C: DataDirectory 2003 11–files created during November 2003 are stored here 12 2004 01 02 03

Archiving files to backup media is easy in this file organization because one only needs to reference the particular directory for the time frame desired. Files can fill up mass

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media and so either a manual process to check for file limits, backup old files, and delete them must be instituted or a program to provide the same functions would need to be created. A major deficiency with a file-based system is that when the time range of a search is larger than one file, the analysis can become very difficult. One may be searching for events, specific time frames that go across file boundaries, or relationships between variables that may not be effectively evaluated within the time frame of one file. The analysis task usually consists of importing a number of files into some analysis tool and then using the analysis tool to look for relationships. This means that the importation process has to include the organizing of data and identifying relationships between events, a difficult task at best. A common tactic is to import data via a script or macro for a given time range, so the user only has to specify a beginning and ending time. 50.5.3 Database Storage Database technology has been improving for many years, resulting in database management systems being increasingly the storage tool of choice for data acquisition systems. Database management systems provide organization tools, compression of data, access aids in the form of indexes, and easy access for analysis tools. A special benefit of database management systems is that they allow the combination of discrete data and time-based data collected on different time intervals. Relational databases are now the dominant database management system type. Data are organized in tables. Each table is composed of a set of rows, each row having a fixed set of columns. Indexes are provided to speed access to data. In data acquisition systems, the designer often adds a time stamp column to each row to facilitate retrieval and analysis of data. An example of a simple database is given in Table 50.5. The TimeData table contains data that are sampled every second, whereas the BatchEvent table contains a record for each batch that has occurred. The SQL language is a common language used to examine and extract data in the tables. An example of its power is that a query can be constructed to use the BeginBatchTime and EndBatchTime to extract data from the TimeData table and combine it with related batch events in the BatchEvent table. A sample query to combine event- and time-based data is as follows:

TABLE 50.5 Example Database Structure Time Data Table Timestamp TemperatureA TemperatureB TemperatureC Rate1 Rate2 Rate3

Batch Event Table Datetime Float Float Float Integer Integer Integer

Timestamp BatchNumber MixPercent InputMaterialAQty InputMaterialBQty OutputProductType OutputProductQty BeginBatchTime EndBatchTime

Datetime Integer Float Integer Integer Varchar Integer Datetime Datetime

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Select BatchEvent.Timestamp, Batchevent.BatchNumber, Batchevent.MixPercent, Min(TimeData.TemperatureA),Max(TimeData.TemperatureA) From BatchEvent, TimeData Where BatchEvent.OutputProductType( ’ BENCH’ and TimeData.Timestamp between BatchEvent. BeginBatchTime and BatchEvent. EndBatchTime Group by BatchEvent.BatchNumber Order by BatchEvent.BatchNumber

The above query searches for batches that created a certain output product type (BENCH) and reports the maximum and minimum temperatures from those batches. This can facilitate research, for example, on what conditions lead to the best yield of a particular product. The ease of performing this operation is a particular advantage of relational databases. There are some disadvantages, including overhead due to the access methods, extra space requirements due to the creation of indexes, and costs and complexity associated with the database management system. Indexes may add as much or more than 100% to the size of a database. Old data must be managed and removed as with any other storage system. This typically is via an automated program, since the structure is not as simple as just looking for the file creation date of a file. 50.5.4 Using Third-Party Data Acquisition Systems When data storage is fairly simple, it is not hard to store data in the aforementioned methods, but when one is using sophisticated methods of data compression, it is recommended that systems that have robust implementations of those methods be used rather than attempting to reinvent the wheel. They can be quite complex to implement reliably. Transfer of data to those systems can occur through a variety of methods, including creation of files that are captured by the other systems, insertion of data into standard interfaces such as OPC or message buses, or insertion into database tables which are monitored by the other systems (often ODBC links). Third-party systems often have software development kit (SDK) interfaces that allow the engineer with some programming skills to store data directly into the system. Process historians are optimized for storing time-based data. A technique used to provide some relational capability to the data is the following:     

store-related data at exactly the same time stamp (time stored in the database for when the data elements were collected), treat data stored with the same time stamp as being part of the same record, select a set of these “records” based on a time range, search a variable for some attribute value (e.g., having some value or range of values), provide to the display system the values of some related variable in the same record having the same time stamp as the desired attribute variables.

This is functionally the same as performing a relational database query on a set of records in a table, with criteria based on values in some columns.

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50.6 DATA DISPLAY AND REPORTING There are a variety of ways to reference and display data acquired from a sensor and stored in suitable media. The current value can be inspected, values can be stored for inspection later, values can be trended, alarm conditions can be detected and reported, or some output back to the process can be performed. 50.6.1 Current-Value Inspection Often, one wants to see the data as they are being collected. This can be of critical importance in experiments which are hard or costly to repeat, allowing the researcher to react to situations as they occur. As it is collected, each data item will be called the current value for that sensor. Current data are usually stored in high-speed storage (the computer main memory). As new values are obtained, they replace the value from the last reading. The collection rate can vary widely (Table 50.6). For instance, detecting the profile at 10-mm intervals for a log moving at 100 m/min requires values to be obtained for each sensor 167 times per second. In continuous processes, data may only need to be acquired once per minute, as in monitoring the level of a large vat. It is useful to remember that human reaction time is in terms of tenths of a second, so displaying data at a faster rate would only be useful if it were easier to program the data. Do not waste time and energy attempting to record data at a high frequency if the only reason is for display to an operator, even if the operator must immediately react to an alert. Human–machine interfaces often show data changes at the time the new value arrives from a sensor. They have display elements that are tied to sensing points. Process historians (data acquisition systems architected for the long-term storage of process data) provide tools to extract data and present the data to the analyst. Their display systems normally provide update tools that automatically refresh the user’s display at some display refresh rate (often in terms of seconds, such as 10 s). The data being collected by the historian may be changing faster, and the data may be stored at a faster rate, but the display normally is still refreshed at the standard refresh rate. It is useful to have the time displayed when the value was collected, as there is often a time delay between the collection and the display of data values. This is especially true where the data collection system may be disconnected temporarily from the display system. The data may come back in a rush when the link is reconnected and is useful for the user (and systems performing analysis) to provide a context for what time the data represent. TABLE 50.6 Data Collection Rates: Examples Type of Operation Discrete manufacturing operations Assembly line manufacturing/assembly Video image processing Parts machining Continuous processes Paper machine Boiler Refinery Dissolving operations

Time per Event 0.01 to multiple seconds 0.001 s 0.002–0.02 s 1–60 s Several seconds to several minutes Seconds to minutes 1 s–20 min

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50.6.2 Display of Individual Data Points Display of the data is normally in text or some simple bar graph representation. Other techniques include button or light indicators where color may represent the state of some value or range of values of the current value. Coded values may take the current value and translate it into some form that provides more value to the user such as zero being translated to the string “FALSE” on the display. Often, one can better understand the data being obtained by using an analog representation (Bailey, 1982, pp. 243–254; Tufte, 1983). This involves representing the measured quantity by some other continuously variable quantity such as position, intensity, or rotation. A common example is the traditional wristwatch. The hours and minutes are determined by the position of a line indicator on a circle. A common analog representation for data acquisition is the faceplate. This is a bar graph where the height of the bar corresponds to the value being measured. Often, lines or symbols may be overlaid on the bar to indicate high or low ranges. A frequent indicator is the meter. A needle rotates in a circle with the degrees of movement corresponding to the value obtained from a sensor. Many voltmeters use this technique (Figure 50.6). Increasingly sophisticated calculations can be established to translate a flow rate, for example, into a cost number, providing the user with immediate feedback on the costs being incurred by the current process rate. A common technique for representing trends of current value is to create a simple array and plot it as a trend line on the display. As new values are gathered, the array values are shifted through the array, with old values shifted out at one end of the array while the new values are shifted in at the other. This is a simple technique that provides some of the benefits of data storage without requiring the complexity of actually storing data in mass storage and managing it.

Digital

Coded

17:00

5 P.M.

Analog 12 9

3 6

(a)

50

Red

(b)

FIGURE 50.6 Comparison of digital, coded, and analog data representation: (a) time; (b) temperature,  F.

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50.6.3 Display of Historical Data There are two main issues with display of historical data:  

selection of the data, the representation the data will have.

50.6.3.1 Selection of Historical Data Selection of historical data involves several factors, including time frame and attributes of the data. Identifying a time frame is probably the most common activity in selecting historical data. How the data are updated can be an important consideration when comparing third-party historians. The time frame is often referenced by a span (such as a number of hours) and a starting point which can be absolute time (e.g., 2004-12-14 16:22:03) or relative time (e.g., 4H for starting 4 h in the past). Another option is to provide an absolute start time and an absolute end time. It is common to have the time frame updating (moving forward with time) if the start point is relative (but check your particular vendor’s software for their practice) and to be fixed if the start point is absolute. For example, if the span is 2 h and the start point is 2 h at 15:00, the starting point on a trend line would be 13:00 and the ending point would be 15:00. Ten minutes later the starting point on the trend line would be 13:10 and the ending point would be 15:10. For relational data, there is often a desired time frame as described earlier (to restrict the size of the data to be searched) and some selection criteria based on characteristics of the data itself or of related data, including events. For example, one may wish to find those manufactured units for the past month that were for product X and see how many had quality defects. As described earlier under third-party data acquisition systems, there are techniques for selecting data from process history databases that approach (but do not equal) relational capability. 50.6.3.2 Representation of Historical Data The primary difference between current data and historical data is that there are multiple data points, normally with an order defined by the time they were acquired (for process history data) and/or by their relationship to other variables and events (for relational data). Once the time frame and relationships are selected, the data must have some method of representation on a display. Historical data have a number of potential representation techniques. Multiple data values can be combined into a single number such as average, standard deviation, mode, maximum, minimum, range, variance, and so on. These can then be represented by techniques for individual data elements as described earlier. The equations in Section 50.4.4 are examples of summary statistics formulas. The simplest form to represent historical data is the list. Create a column for each variable of interest. If they are all collected at the same time (the “records” described above), then each time data were collected can be used as the first value on the left in each row. The data for each variable of interest can be placed on the row corresponding to its time stamp, similar to that for files shown in Table 50.4. Where data were collected with different time stamps, a new row can be created whenever a new value is obtained and values only placed in the column–row combinations where there is a corresponding time stamp between the row’s time stamp and the time stamp of the variable in question. While useful, the problem with this is that there are now holes in the data. This list is very useful when viewing a trend line or graphical tool to validate numbers, to verify time stamps, and diagnose problems with collection.

DATA ANALYSIS

1400

50

1000 30

800

20

600 400

10

Second series

1200

40 First series

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First series Second series

200

0

0

X values

FIGURE 50.7 Multiple-axis chart.

Often, one desires to view the relationships between data and time or other variables. Trend plots typically are used to show the relationship between variables and time. The X axis is typically represents the time range, and the Y axis represents the value of the desired variable. The range of data on the Y axis can vary depending on how one wishes to view the data. The maximum and minimum of the data can be used to set the top and bottom of the range of the Y axis, respectively. This can have two undesired effects. It may make small movements in data appear to be very large when the range is small. In cases where there are spikes in the data where a value is disproportionately high or low, representing the Y axis based on the maximum and minimum could make it difficult to view normal variation in the data. One has to understand the potential use of the data to choose the Y-axis scale appropriately. When more than one variable is shown on a trend chart, the selection of scale of the Y axis becomes more complex. If all the variables are representative of the same domain, such as all temperatures, then perhaps the same Y axis can be used for all of them. Often, however, the viewer is attempting to compare relative variations in data, sort of a poor man’s correlation analysis. In this case, it may be useful to have multiple Y axes and select the range of each of them such that they represent the range of one or more of the variables being viewed (Figure 50.7). Another approach to compare variation between two variables is to use one variable for the X value and the other variable for the Y value (an X–Y chart). This is useful when two variables are related by sample time or some other selection technique that results in a paired relationship between the two variables. The correlation function in Section 4.4 represents the mathematical correlation between two variables and can be used to determine the strength of that relationship. Chapter 1 discusses correlation and the calculation of the line through a distribution of data.

50.7 DATA ANALYSIS 50.7.1 Distributed Systems Distributed systems are a powerful approach to data acquisition systems because they combine some of the best of both stand-alone and host-based systems. The data acquisition portion is located on a small processor that has communication capability to a host computer system. The small system collects the data, possibly reducing some to a more

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compact form, and then sends the data to the host systems for analysis. The host system can analyze the data when it has the available time to do so. Only the data acquisition portion needs to be very responsive to the process. If the data acquisition task gets too big for the small system, the cost of expansion is limited to moving the data acquisition software to a new computer or splitting it up over several computers and changes to the host computer portion are not required. The major disadvantage of distributed systems is that they suffer from a more complex overall architecture even though the individual parts are simple. This leads to problems with understanding error sources and increases the potential errors because of more parts. Unless the communications are designed carefully, the messages sent between the small systems and the host system may be inflexible, causing increased effort when one wants to change the type of data being collected. Distributed systems may be expensive because of the number of individual components and the complexity required but often fit well with environments where one already has a host computer. 50.7.2 System Error Analysis The errors that can occur at different stages in the data acquisition process must be analyzed, as they can add up to make the data meaningless. For instance, one may have very accurate sensors, but by the time the data reach the host computer they might have been converted into integer data or from real to integer and back to real again. This can lead to dangerous assumptions about the accuracy of the received numbers, because each conversion can cause rounding or other errors. It is the responsibility of the person setting up the acquisition system and the analyst to examine each source of potential error, discover its magnitude, and reduce it to the point where it will not have a significant impact on the conclusions to be derived from the data. Use of the filtering techniques described earlier under data collection can be of use to eliminate random error. It is not within the scope of this chapter to cover system error analysis, but Chapter 1 gives some foundation.

50.8 DATA COMMUNICATIONS Data communications are involved in many aspects of data acquisition systems. The communications between the sensing and control elements and data acquisition devices, as well as the communications between the data acquisition system and other computer systems, can be carried out in many ways. This section will cover some aspects of communications, especially as they pertain to computer systems. 50.8.1 Serial Communications A serial communication link means that data sent over a communications line is spread out over time on one physical data path. For instance, if a character is sent from a sensor to a computer, each bit making up the character (normally eight bits) will be sent one after the other (Table 50.7). This is often useful for low-cost, low-speed (usually less than 10,000 cps) rates of data transfer.

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TABLE 50.7 Time Sequence of Bits Sent over Serial Communication Line Character “A” is (in bit form) Bit number 6 5 4 3 2 1 0 Bit value 1 0 0 0 0 0 1 The communications are using the RS232C communications standard and sending the ASCII character A Bit Value Start bit 1 (bit 0 of A) 0 (bit 1 of A) 0 (bit 2 of A) 0 (bit 3 of A) 0 (bit 4 of A) 0 (bit 5 of A) 1 (bit 6 of A) Parity bit Stop bit

Time 0 s after start 1/9600 s after start 2/9600 s after start 3/9600 s after start 4/9600 s after start 5/9600 s after start 6/9600 s after start 7/9600 s after start 8/9600 s after start 9/9600 s after start

50.8.2 Parallel Communications A serial communication link may not require very many wires, but the time spent to transfer data can add up. A way to improve the speed of communications is to use parallel communication links. This is done by having a number of wires to carry data. For instance, sending the same “A” over a nine-wire bus would only require one transfer (Table 50.8). TABLE 50.8 Time Sequence of Bits Sent over Parallel Communications Interface Character “A” is (in bit form) Bit number 6 5 4 3 2 1 0 Bit value 1 0 0 0 0 0 1 The communications are using a hypothetical nine-line parallel communications bus sending the ASCII character A Bit Value

Time

Start bit 0 s after start 1 (bit 0 of A) 0/9600 s after start 0 (bit 1 of A) 0/9600 s after start 0 (bit 2 of A) 0/9600 s after start 0 (bit 3 of A) 0/9600 s after start 0 (bit 4 of A) 0/9600 s after start 0 (bit 5 of A) 0/9600 s after start 1 (bit 6 of A) 0/9600 s after start Parity bit 0/9600 s after start If the bus could handle the same rate of change of bits as the serial interface, then the next character could be sent 1/9600 s after the first character (the A)

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50.8.3 Networks Ethernet with Transmission Control Protocol/Internet Protocol (TCP/IP) has become the dominant communications network protocol for data collection. There are proprietary process control and data acquisition networks that serve special purposes, but Ethernet has proven to be versatile for everything from office communications to data collection from smart sensors. Many computers can be connected to the same network segments. The use of switches and routers provides ways to isolate and limit communications to improve performance and security. Firewalls provide filters and protection for entire classes of messages and sources. While Ethernet cannot guarantee delivery (being based on a collision detection and retransmit strategy), it has been shown to provide excellent response to moderate network activity. Communications speeds are regularly being improved to provide an even greater range of applicability. 50.8.4 OSI Standard The International Standards Organization has developed a set of standards for discussing communications between cooperating systems called the Open Systems Interconnect (OSI) model (see Table 50.9). This defines communications protocols in terms of seven layers (American National Standards Institute, 1981). While not providing for specific interface protocols, the OSI model has had a significant impact on communications because it has provided a framework for compartmentalizing aspects of communications to allow the handoff of information from one device to another in a standard way. For instance, the transmission of data from one media type to another (such as copper wire to fiber to satellite to copper wire and then to wireless) is a result of standards enabling the seamless transfer of messages in a way that is transparent to the user. 50.8.5 OPC Standard A recent standard of use in manufacturing is the OPC (OLE for process control) standard, which provides for a standard way of communicating with process equipment. It is sponsored by the OPC Foundation and originated as an extension for process control from the Microsoft OLE functionality (http://www.opcfoundation.org). Functions provided by OPC include ability to browse the variable database of a device and monitor data on demand or when events occur. The capability of OPC has been TABLE 50.9 Open Systems Interconnect Model Layer 7. Application 6. Presentation 5. Session 4. Transport 3. Network 2. Data link 1. Physical

Principle

Example

Application Display, format, edit Establish communications Virtual circuits Route to other networks Correct errors Synchronize communications Electrical interface

Millwide reporting Convert ASCII into EBCDIC Log onto remote computer Make sure all message parts got there in order Talk to Internet Send character downline Send Ack-Nak Wire and voltages

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expanded to work with Web communication methods such as XML and cover complex data such as record structures. The power of OPC is that data from an instrument can be available via a standard network interface so that any data acquisition program that uses the OPC interface can gain access to any OPC device. The need to know the protocol of each device or adhere to the wiring of specialized communications or use custom database access methods is eliminated through the use of a standard protocol. Multiple programs can be simultaneously monitoring the same piece of data, performing different functions at different time intervals, as events occur. The last point is particularly significant, because much of the work of data acquisition systems is spent in polling for changes in data or otherwise attempting to determine when an event has taken place. A program can subscribe to an OPC item and it will be notified when the item changes value, reducing the complexity of monitoring data dramatically. As an example of the power of this approach, consider the following example (Figure 50.8). A device can collect the identification from a unit of material such as unit number, color, and manufacturing date and make it available via OPC. As the unit is processed in a manufacturing center, another device collects defect counts and makes the unit number available via OPC. A human–machine interface program can monitor both sources with the same interface protocol and software and display it live for an operator to see. Simultaneously, another program can collect the defect counts and summarize them into totals. Yet another program can monitor the totals and wait for the unit number to change, triggering a transaction to a database or an email if there was a problem. The power of the OPC interface is that it provides real-time access to the data from each of the sources and multiple programs can monitor the same OPC sources to perform work. Diagnostics can monitor the same data to evaluate system processing, downtime, or quality issues. Other programs can sample and store data to log files or diagnostic files for further analysis. All can be operating in parallel with no need to understand the internals of the other programs, via a standard interface and standard protocols that support asynchronous delivery of data.

Multiple server/clients with OPC Operator monitoring Statistics to history

Unit entry

Defect detection

OPC provides clearing house for process data: • Real time • Event driven • Standard

Diagnostic logging

Unit totalization

Defect alerts

FIGURE 50.8 Example use of OPC communications.

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50.8.6 Benefits of Standard Communications When implementing data acquisition and display systems, the ability to communicate in a standard fashion can play a large role in the cost of the system. This is realized in a variety of ways: 1. Different sensors can be connected to a system without having to buy a whole new system. 2. Data can be sent to other systems as needed for further processing. 3. As technology or need changes, portions of a system can be mixed and matched. 4. Increased competition from vendors tends to bring prices down. 5. A standard will have many people using products based on the standard, resulting in more vendors and greater availability of parts with a greater variety of options.

50.9 OTHER DATA ACQUISITION AND DISPLAY TOPICS 50.9.1 Data Chain As materials and parts move through a manufacturing operation, the data collected are separated by time and type of data. Combining that data together presents a number of challenges. One can consider the first piece of data collected about some object to be the beginning of a chain of data and each successive data acquisition point in the manufacturing process to be a link in that chain until the end of the chain, where the last piece of data about a manufactured item is collected. Often, the steps of the manufacturing process may proceed from raw material to some intermediate work-in-process unit, to some other step that may be time based, then to some other step that may be finished-unit based. Many varieties of the above exist. Each link can represent one of the following:  

Data collected from a start time to an end time. A set of attributes about a particular manufacturing unit with an associated time of processing.

Often referred to as the genealogy, the steps can be linked together through: 

Some assumptions based on time stamp relationship of one link back to the previous one.  Recording of units that entered or left the time-based portion of the process and the beginning and ending of the entry time.  Some assumptions about the mixing of elements of the manufactured item. Using the techniques described above for combination of time-based and event data, a set of data for the whole life cycle of a manufactured item can be created. Where mixing occurs, the data will be less accurate but may provide clues as to the factors that went into the final product, such as proportions of additives. As an example, Table 50.10 shows various queries that can be combined to provide one picture of all data sources for a manufactured unit.

TABLE 50.10 Data Chain Sample Queries State in Process Raw material inventory Intermediate goods processing

Important Data (Typical)

Example Data

Intermediate package creation

Identifiers: raw material batch ID Data: supplier, quality, time in inventory Identifiers: raw material lots consumed; raw material batch ID; start and end time of entry to process; time delay through process Data: piece count, rejects, downgrades, process characteristics such as rate, temperature, modification to materials Identifiers: raw package ID, start/end time of creation Data: piece count, package dimensions

Intermediate inventory

Identifier: intermediate package ID Data: location in inventory

Package PR1 location warehouse 1

Finished goods processing

Identifier: finishing batch ID, intermediate package consumed, start/end time of consumption; time delay through process Data: piece count, rejects, downgrades, process characteristics such as rate, temperature, modification to materials Identifiers: finished package ID; start/end time of creation Data: piece count, grade, package dimensions, customer order

Intermediate packages consumed at breakdown: 5:30–6:00 PR1; 95 pieces; 400 6:00–6:30 PR2; 147 pieces; 390 6:30–6:40 PR3; 25 pieces; 395 6:40–7:00 PR4; 40 pieces; 410 Typical residence time: 15

Finished goods package creation

Simplified Query to Combine Data with Previous Step

Lot 1: 5% rejects; lot 2: 7% rejects 3:00–4:00 raw materials from lot 1; 1000 pieces; 10 rejects; 200 4:00–5:00 raw materials from lot 2; 1045 pieces; 14 rejects; 205 ; 10 residence time in process

Use raw material batch ID to match to raw materials characteristics; material run at 4:00–5:00 had 7% rejects when delivered to plant

Package PR1 created: 3:30–4:10; 100 pieces; package PR2 created: 4:10–5:10; 150 pieces

Use time of manufacture, time lag through process to identify characteristics from intermediate process and raw materials lots; package PR1 was processed at 200 , was created from lot 1 and was from a lot that had 5% rejects detected when delivered to the plant Use ID of intermediate package to match to package at intermediate package creation; package PR1 had 100 pieces Use ID of intermediate package to match to intermediate inventory; in this example, package PR1 came from warehouse 1, lost 5 pieces in consumption

Finished package PF1 created from 6:00 to 6:30; 40 pieces; prime grade; order PO5670

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Use time of manufacture, time lag through process to identify characteristics from breakdown, and which raw packages sourced this finished package. There may be significant mixing. In this example, PR1 and PR2 would be sources for PF1. PF1 was possibly created at 400 , stored in warehouse 1, lost 5 pieces when loading into finished goods process, was probably processed at 200 , and was probably from lot 1

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DATA ACQUISITION AND DISPLAY SYSTEMS

Starting at the finished-good item, the batch ID of the previous step acts as a link into the range of time data in the previous step. If the list of items broken down is retained from the prior step, then those can be used to link back to the previous time frame. Depending on the amount of mixing, the results will be more or less indicative of what actually happened. The smaller the batch sizes, the easier the tracking back to source data will be. Time lags between process steps can dramatically impact the ability to assume when the raw materials for a particular item were processed. 50.9.2 Web Programs and Interfaces Web interfaces have improved to the point that user interfaces can be written in Web browser screens. This eliminates the effort and organization required to deploy code across a company. The application is written for a Web interface. When the user uses a Web browser to access the page, functionality is downloaded to the user’s page or is executed in such a fashion as to obtain the results of a query and transmit a data page to the user. The developer does not have to get involved in the process of manually installing software on the user’s machine. This reduces the demand on the desktop computer and allows the developer to make a change and have it proliferated to all users when they next reference the Web. 50.9.3 Configuration Versus Implementation As a general rule of thumb, third-party data acquisition and storage systems provide configurable tools for acquisition and display. For simple applications which do not require great flexibility in program functions, such as generating alarms, unusual graphics types, control of the process, or integration into larger systems, it is appropriate to use these, often simple, question–answer or menu-type systems. When the system must be very flexible or customized, it may be more appropriate to write a custom program. When considering this approach, be cautious, for the cost of implementing, a program is often much higher than expected. For instance, if one wanted to perform simple data acquisition and storage from a sensor, the cost to write a program would probably be higher than buying a small off-theshelf system and entering the parameters for data collection. Writing a program involves analysis, design, development, debugging the program, and testing of results. The cost of documenting a program is often a large unplanned cost. If the results are intended to be used to make economic or process-related decisions, then the program must be tested carefully. Additionally, maintenance of the program can be quite expensive. Someone must be trained in the technologies used to build the program, the logic of the program, and the installation of the program. Another factor to consider is that costs of improvement of third-party software are borne by many customers and driven by many customers. The net result to the user of this software is that it is normally constantly improving, constantly tested, and maintained by a group of developers whose primary job is software development. One reason to build and maintain software internally is that a company can keep special knowledge within the company and thus maintain competitive advantage. 50.9.4 Store and Forward When data acquisition and data storage are on two separate machines, it is important to provide methods to retain data in case the link between systems is broken. Message buses

REFERENCES

1713

provide automated methods of maintaining a link between data acquisition systems. The developer inserts data into the message bus. If the link between the two systems is broken, the message bus queues up the data messages on the collection machine. When the data storage machine connection is reestablished, the message bus passes on the data to the data storage machine. When a message bus is not available or feasible, a simplified mechanism can be created where a file representing each sample of data is created. If the data collection and data storage system are linked, then the data storage system monitors the directory of the collector for a new data file. If the data storage system detects one or more files on the data collection computer, then it will process them into storage. If the link is broken, then the files build up until the link is reestablished. A related technique is to store data in a database or similar mechanism on the data collection computer and scan it periodically for missing data from the storage computer. This is particularly useful when connection to the data collection computer is unreliable.

50.9.5 Additional Communications Topics When considering transmission media, some points may provide value to the engineer. Fiber-optic cabling is less sensitive to noise than other transmission media. Wireless access points provide increased flexibility in positioning of sensors and greatly reduce wiring costs. Particularly, if one wishes to collect data from sites that may move, such as environmental sampling sites, the costs of wiring and rewiring can be quite significant. Using wireless transmission technology, it eliminates much of the wiring costs and facilitates moving the sensors from one location to another. Wireless transmission has a set of concerns that must be taken into account by the engineer, including security, since other units can monitor signals (still evolving) and ability to be jammed.

50.10 SUMMARY The tremendous change in technology for data acquisition and display systems since this chapter was first written has driven us to take a different approach than with the first edition. The technologies for data acquisition and display have become more standardized. Engineers are increasingly reliant upon and versed in computing technologies. The combination of data from various sources into an integrated view of the process has facilitated process improvement and leads to competitive advantage. This chapter has attempted to provide tools and techniques to aid in the acquisition, storage, and manipulation of process data, expanding from the previous edition into techniques to aid in the manipulation of data for integration and analysis.

REFERENCES American National Standards Institute. Open Systems Interconnection–Basic Reference Model, Draft Proposal 7498 97/16 N719. New York: American National Standards Institute; 1981. Bailey, RW. Human Performance Engineering: A Guide for Systems Designers, Englewood Cliffs (NJ): Prentice-Hall; 1982.

1714

DATA ACQUISITION AND DISPLAY SYSTEMS

Beyer, WH, editor. CRC Standard Mathematical Tables. 24th ed. Boca Raton (FL): CRC; 1976. Crossley, ML. The Desk Reference of Statistical Quality Methods. Milwaukee (WI): ASQ Quality Press; 2000. Date, CJ. An Introduction to Database Systems. 5th ed. Addison-Wesley; 1990. Johnson, CD. Microprocessor-Based Process Control. Englewood Cliffs (NJ): Prentice-Hall; 1984. Liptak, BG. System accuracy. In: Liptak BG, editor. Instrument Engineer’s Handbook. Vol. 1, 4th ed. Boca Raton (FL): Process Measurement and Analysis, CRC; 2003. Murrill, PW. Fundamentals of Process Control Theory, Research Triangle Park (NC): Instrument Society of America; 1981. Tufte, ER. The Visual Display of Quantitative Information, Cheshire, England: Graphics; 1983. Wright, JD, Edgar TF. Digital Computer Control and Signal Processing Algorithms. In: Real-Time Computing. Mellichamp DA, editors. New York: Van Nostrand Reinhold; 1983.

Magazines that Carry Relevant Information Control Engineering International: http://www.controleng.com/. Design Engineering: http://www.designengineering.co.uk/. IEEE Control Systems Magazine: http://www.ieee.org/organizations/pubs/magazines/cs.htm. Industrial Technology: http://www.industrialtechnology.co.uk/. Instrumentation and Automation News: http://www.ianmag.com/. Pollution Engineering Online: http://www.pollutionengineering.com/. Scientific Computing and Instrumentation: http://www.scamag.com/. Sensors Magazine: www.sensorsmag.com.

51 MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES JACK H. WESTBROOK 51.1 Symbols and abbreviations Bibliography for letter symbols Bibliography for graphic symbols 51.2 Mathematical tables 51.3 Statistical tables 51.4 Units and standards 51.4.1 Physical quantities and their relations 51.4.2 Dimensions and dimension systems 51.4.3 Dimension and unit systems 51.4.4 The international system of units 51.4.5 Application of SI prefixes 51.4.6 Other units 51.4.7 Length, mass, and time (English units and standards) 51.4.8 Standard of time 51.4.9 Force, energy, and power 51.4.10 Thermal units and standards temperature 51.4.11 Quantity of heat and some derived quantities 51.4.12 Chemical units and standards 51.4.13 Theoretical, or absolute, electrical units 51.4.14 Internationally adopted electrical units and standards Bibliography for units and measurements 51.5 Tables of conversion factors 51.6 Standard sizes 51.6.1 Preferred numbers 51.6.2 Gages 51.6.3 Paper sizes 51.6.4 Sieve sizes 51.6.5 Standard structural sizes—steel 51.6.6 Standard structural shapes—aluminum 51.7 Standard screws 51.7.1 Nominal and minimum dressed sizes of American Standard Lumber 

This chapter is a revision and extension of Sections 1 and 3 of the third edition, which were written by Mott Souders and Ernst Weber, respectively. Section 51.4.4 is derived principally from ASTM’s Standard for Metric Practice, ASTM E380-82, Philadelphia, 1982 (with permission). Section 51.6.1 is derived from MIS Newsletter, General Electric Co., 1980 (with permission).

Handbook of Measurement in Science and Engineering. Edited by Myer Kutz. Copyright Ó 2013 John Wiley & Sons, Inc.

1715

1716

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

51.1 SYMBOLS AND ABBREVIATIONS TABLE 51.1 Greek Alphabet A B G D E Z

Alpha Beta Gamma Delta Epsilon Zeta

a b g d e z

H Q I K L M

h q u i k l m

Eta Theta Iota Kappa Lambda Mu

N X O P P S

n j ο p r s z

Nu Xi Omicron Pi Rho Sigma

T Y F X C V

t y f x c v

Tau Upsilon Phi Chi Psi Omega

TABLE 51.2 Symbols for Mathematical Operationsa Powers and Roots

Addition and Subtraction a þ b, a plus b a b, a minus b a  b, a plus or minus b a  b, a minus or plus b Multiplication and Division a  b, or a
b, or ab, a times b a a  b, or , or a/b, a divided by b b Symbols of Aggregation () parentheses () parentheses {} braces vinculum Equalities and Inequalities a ¼ b, a equals b a  b, a approximately equals b a 6¼ b, a is not equal to b a > b, a is greater than b a < b, a is less than b a  b, a much larger than b a  b, a much smaller than b a 3 b, a equals or is greater than b a 2 b, a is less than or equals b a  b, a is identical to b a ! b, or a ¼ b, a approaches b as a limit Proportion a/b ¼ c/d, or a: b:: c: d, a is to b as c is to d a / b, a  b, a varies directly as b %, percent

2

a , a squared an, a raised to the nth power pffiffiffi a, square root of a ffiffiffi p 3 a, cube root of a p ffiffiffi n a, or a1/n, nth root of a a n, 1/an 3.14  104 ¼ 31,400 3.14  10 4 ¼ 0.000314 Miscellaneous a, mean value of a a!, ¼ 1
2
3 . . . a, factorial a jaj ¼ absolute value of a Pðn; rÞ ¼ nðn 1Þðn 2Þ


ðn r þ 1Þ   Pðn; rÞ n ¼ binomial Cðn; rÞ ¼ ¼ r r! coefficients pffiffiffiffiffiffiffi i (or j) ¼ 1, imaginary unit p ¼ 3.1416, ratio of the circumference to the diameter of a circle 1, infinity Plane Geometry d 2 Þ tan b þ d sin ax a b2 d 2 bþd 4 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx 1 d þ b sin ax þ d 2 b2 cos ax ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi loge ðd 2 > b2 Þ 112. b þ d sin ax a d 2 b2 b þ d sin ax Z sinða bÞx sinða þ bÞx 113. sin ax sin bx dx ¼ ða2 6¼ b2 Þ 2ða bÞ 2ða þ bÞ Integrals Involving cosnax 114.

Z

1 cos3 ax dx ¼ sin ax a

1 sin3 ax 3a

3 1 1 sin 4ax cos4 ax dx ¼ x þ sin 2ax þ 8 4a 32a Z Z cosn 1 ax sin ax n 1 þ cosn 2 ax dx ðn ¼ positive integerÞ 116. cosn ax dx ¼ na n Z cos ax x sin ax 117. x cos ax dx ¼ þ a2 a  2  Z 2x x 2 sin ax 118. x2 cos ax dx ¼ 2 cos ax þ a a3 a  2   3  Z 3x 6 x 6x 119. x3 cos ax dx ¼ cos ax þ sin ax a2 a a4 a3 Z Z xn sin ax n xn 1 sin ax dx ðn > 0Þ 120. xn cos ax dx ¼ a a 115.

Z

MATHEMATICAL TABLES

1753

TABLE 51.18 (Continued ) Z Z cos ax 1 cos ax a sin ax 121. dx ¼ dx xn n 1 xn 1 n 1 xn 1 Z Z dx 1 sin ax n 2 dx ¼ þ ðn integer > 1Þ 122. cosn ax aðn 1Þ cosn 1 ax n 1 cosn 2 ax Z xdx x 1 123. ¼ tan ax þ 2 loge cos ax cos2 ax a a Z dx 1 ax 124. ¼ tan 1 þ cos ax a 2 Z dx 1 ax ¼ cot 125. 1 cos ax a 2 Z x dx x ax 2 ax ¼ tan þ loge cos 126. 1 þ cos ax a 2 a2 2 Z x dx x ax 2 ax 127. ¼ cot þ loge sin 1 cos ax a 2 a2 2 ! rffiffiffiffiffiffiffiffiffiffiffi Z dx 2 b d ax 128. ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan 1 tan ðb2 > d 2 Þ b þ d cos ax a b2 d 2 bþd 2 129.

Z

130.

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 1 d þ b cos ax þ d 2 b2 sin ax ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi loge b þ d cos ax a d 2 b2 b þ d cos ax cos ax cos bx dx ¼

sinða 2ða

bÞx sinða þ bÞx þ bÞ 2ða þ bÞ

ðd 2 > b2 Þ

ða2 6¼ b2 Þ

Integrals Involving sinnax, cosnax  1 cosða bÞx cosða þ bÞx þ ða2 6¼ b2 Þ 2 a b aþb

131.

Z

sin ax cos bx dx ¼

132.

Z

sin2 ax cos2 ax dx ¼

x 8

133.

Z

sinn ax cos ax dx ¼

1 sinnþ1 ax ðn 6¼ aðn þ 1Þ

1Þ

134.

Z

sin ax cosn ax dx ¼

1 cosnþ1 ax aðn þ 1Þ

1Þ

sin 4ax 32a

ðn 6¼

Z sinn 1 ax cosmþ1 ax n 1 þ sinn 2 ax cosm ax dx ðm; n posÞ aðn þ mÞ nþm Z Z sinn ax sinnþ1 ax n mþ2 sinn ax 136. dx ¼ dx ðm; n pos; m 6¼ 1Þ m m 1 cos ax aðm 1Þ cos ax m 1 cosm 2 ax Z Z cosm ax cosmþ1 ax n m 2 cosm ax dx ¼ 137. þ dx ðm; n pos; n 6¼ 1Þ n 1 sinn ax n 1 aðn 1Þsin ax sinn 2 ax Z dx 1 ¼ loge tan ax 138. sin ax cos ax a 135.

Z

sinn ax cosm ax dx ¼

(continued)

1754

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.18 (Continued )  Z dx 1 1 139. ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi loge tan ax þ tan b sin ax þ d cos ax a b2 þ d 2 2 Z sin ax 1 dx ¼ loge ðb þ d cos axÞ 140. b þ d cos ax ad Z cos ax 1 dx ¼ loge ðb þ d sin axÞ 141. b þ d sin ax ad

142.

Z

143.

Z

144.

Z

145.

Z

146.

Z

147.

Z

148.

Z

151.

Z

Z

d b



Integrals Involving tannax, cotnax, secnax, cscnax Z 1 tann ax dx ¼ tann 2 ax dx ðn integer > 1Þ tann 1 ax aðn 1Þ Z 1 cotn 1 ax cotn 2 ax dx ðn integer > 1Þ cotn ax dx ¼ aðn 1Þ Z 1 sin ax n 2 þ secn ax dx ¼ secn 2 ax dx ðn integer > 1Þ aðn 1Þ cosn 1 ax n 1 Z 1 cos ax n 2 cscn ax dx ¼ þ cscn 2 ax dx ðn integer > 1Þ aðn 1Þ sinn 1 ax n 1  dx 1 d ¼ 2 log ðb cos ax þ d sin axÞ bx þ e b þ d tan ax b þ d 2 a "rffiffiffiffiffiffiffiffiffiffiffi # dx 1 b d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffi sin 1 sin ax ðb pos; b2 > d 2 Þ b b þ d tan2 ax a b d tan ax sec ax dx ¼

Z 149. tann ax sec2 ax dx ¼ 150.

1

1 sec ax a

1 tannþ1 ax ðn 6¼ aðn þ 1Þ

1Þ

sec2 ax dx 1 ¼ loge tan ax tan ax a cot ax csc ax dx ¼

152.

Z

153.

Z

154.

Z

xbax dx ¼

xb a loge b

155.

Z

xeax dx ¼

eax ðax a2

1 csc ax a 1 cotnþ1 ax ðn 6¼ aðn þ 1Þ

cotn ax csc2 ax dx ¼ csc2 ax dx ¼ cot ax

1Þ

1 loge cot ax a Integrals Involving bax, eax, sin bx, cos bx

156. 157.

Z

Z

ax

xn bax x b dx ¼ a loge b n ax

1 xn eax dx ¼ xn eax a

bax a2 ðloge bÞ2 1Þ Z n xn 1 bax dx ðn positiveÞ a loge b Z n xn 1 eax dx ðn positiveÞ a

MATHEMATICAL TABLES

1755

TABLE 51.18 (Continued ) Z dx 1 158. ¼ ½ax loge ðb þ deax ފ b þ deax ab Z eax dx 1 loge ðb þ deax Þ ¼ 159. b þ deax ad rffiffiffi! Z dx 1 b 1 ax ðb and d positiveÞ ¼ pffiffiffiffiffiffi tan e 160. ax ax be þ de d a bd Z ax e ðaxÞ2 ðaxÞ3 161. þ þ


dx ¼ loge x þ ax þ 2
2! 3
3! x   Z ax Z e 1 eax eax 162. dx ¼ þa dx ðn integer > 1Þ n 1 xn xn 1 xn 1 Z eax 163. eax sin bx dx ¼ ða sin bx b cos bxÞ 2 a þ b2 Z eax 164. eax cos bx dx ¼ ða cos bx þ b sin bxÞ a2 þ b2 Z xeax ða sin bx b cos bxÞ xeax sin bx dx ¼ 165. 2 a þ b2ax

2  e ða b2 Þ sin bx 2ab cos bx ða2 þ b2 Þ2 Z xeax xeax cos bx dx ¼ ða cos bx þ b sin bxÞ 166. 2 a þ b2ax

2  e ða b2 Þ cos bx þ 2ab sin bx ða2 þ b2 Þ2 Integrals Involving logeax 167. 168.

Z

Z

loge ax dx ¼ x loge ax

x

ðloge axÞn dx ¼ xðloge axÞn 

loge ax nþ1

169.

Z

xn loge ax dx ¼ xnþ1

170.

Z

ðloge axÞn ðloge axÞnþ1 dx ¼ x nþ1

nðloge axÞn 1 dx 

1 ðn þ 1Þ2 ðn 6¼

ðn positiveÞ

ðn 6¼

1Þ

1Þ

dx ¼ loge ðloge xÞ x loge ax " # Z dx 1 ðloge axÞ2 loge ðloge axÞ þ loge ax þ ¼ þ


172. loge ax a 2
2! Z Z n xmþ1 ðloge axÞ n 173. xm ðloge axÞn dx ¼ xm ðloge axÞn 1 dx ðm; n 6¼ 1Þ mþ1 mþ1 Z Z xm dx xmþ1 mþ1 xm dx 174. þ n ¼ n 1 ðloge axÞ n 1 ðn 1Þðloge axÞ ðloge axÞn 1

171.

Z

(continued)

1756

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.18 (Continued ) Some Definite Integrals 1.

Z

2.

Z

a 0 a 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa2 a2 x2 dx ¼ 4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa2 2ax x2 dx ¼ 4

1

dx p ¼ pffiffiffiffiffi ða and b positiveÞ 2 a þ bx 2 ab 0 ffiffiffiffiffi ffi p Z a=b Z 1 dx dx p 4. dx ¼ pffiffiffiffiffiffi a þ bx2 ¼ pffiffiffiffiffi ða and b positiveÞ 2 a þ bx 4 ab 0 a=b p ffiffiffiffiffi ffi Z a=b dx p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi ða and b positiveÞ 5. 2 2 b a bx 0 8p > ðb > 0Þ > > Z 1

x 0 > > : p ðb < 0Þ 2 Z 1 tan x p dx ¼ 7. x 2 0 Z p=2 Z p=2 2
4
6


2n ðn > 0Þ cos2nþ1 x dx ¼ sin2nþ1 x dx ¼ 8. 3
5
7


ð2n þ 1Þ 0 0 Z p=2 Z p=2 1
3
5


ð2n 1Þ p 2n 9.
ðn > 0Þ cos2n x dx ¼ sin x dx ¼ 2
4
6


2n 2 0 0 Z p Z p 10. cos ax cos bx dx ¼ 0 ða 6¼ bÞ sin ax sin bx dx ¼ 3.

Z

0

0

11.

Z

12.

Z

13.

Z

14.

Z

15.

Z

16.

Z

17.

Z

p

sin2 ax dx ¼

0

Z

p 0

cos2 ax dx ¼

p=2

loge cos x dx ¼

0 1

e

ax2

0 1 0

rffiffiffi p a

ax

dx ¼

n! anþ1

1

loge x dx ¼ 1 x

p2 6

1

loge x dx ¼ 1þx

p2 12

1

loge x p2 dx ¼ 2 8 1 x

0

0

0

xn e

1 2

dx ¼

Z

p 2

p=2 0

loge sin x dx ¼

p loge 2 2

ða > 0; n ¼ 1; 2; 3; . . .Þ

MATHEMATICAL TABLES

1757

TABLE 51.19 Haversinesa Value

Log

u

Value

Log

u

Value

Log

u

Value

Log

0 1 2 3 4

0.00000 0.00008 0.00030 0.00069 0.00122

– 0.88168 0.48371 0.83584 0.08564

45 46 47 48 49

0.14645 0.15267 0.15900 0.16543 0.17197

0.16568 0.18376 0.20140 0.21863 0.23545

90 91 92 93 94

0.50000 0.50873 0.51745 0.52617 0.53488

0.69897 0.70648 0.71387 0.72112 0.72825

135 136 137 138 139

0.85355 0.85967 0.86568 0.87157 0.87735

0.93123 0.93433 0.93736 0.94030 0.94318

5 6 7 8 9

0.00190 0.00274 0.00373 0.00487 0.00616

0.27936 0.43760 0.57135 0.68717 0.78929

50 51 52 53 54

0.17861 0.18534 0.19217 0.19909 0.20611

0.25190 0.26797 0.28368 0.29905 0.31409

95 96 97 98 99

0.54358 0.55226 0.56093 0.56959 0.57822

0.73526 0.74215 0.74891 0.75556 0.76209

140 141 142 143 144

0.88302 0.88857 0.89401 0.89932 0.90451

0.94597 0.94869 0.95134 0.95391 0.95641

10 11 12 13 14

0.00760 0.00919 0.01093 0.01281 0.01485

0.88059 0.96315 0.03847 0.10772 0.17179

55 56 57 58 59

0.21321 0.22040 0.22768 0.23504 0.24248

0.32281 0.34322 0.35733 0.37114 0.38468

100 101 102 103 104

0.58682 0.59540 0.60396 0.61248 0.62096

0.76851 0.77481 0.78101 0.78709 0.79306

145 146 147 148 149

0.90958 0.91452 0.91934 0.92402 0.92858

0.95884 0.96119 0.96347 0.96568 0.96782

15 16 17 18 19

0.01704 0.01937 0.02185 0.02447 0.02724

0.23140 0.28711 0.33940 0.38867 0.43522

60 61 62 63 64

0.25000 0.25760 0.26526 0.27300 0.28081

0.39794 0.41094 0.42368 0.43617 0.44842

105 106 107 108 109

0.62941 0.63782 0.64619 0.65451 0.66278

0.79893 0.80470 0.81036 0.81592 0.82137

150 151 152 153 154

0.93301 0.93731 0.94147 0.94550 0.94940

0.96989 0.97188 0.97381 0.97566 0.97745

20 21 22 23 24

0.03015 0.03321 0.03641 0.03975 0.04323

0.47934 0.52127 0.56120 0.59931 0.63576

65 66 67 68 69

0.28869 0.29663 0.30463 0.31270 0.32082

0.46043 0.47222 0.48378 0.49512 0.50625

110 111 112 113 114

0.67101 0.67918 0.68730 0.69537 0.70337

0.82673 0.83199 0.83715 0.84221 0.84718

155 156 157 158 159

0.95315 0.95677 0.96025 0.96359 0.96679

0.97016 0.98081 0.98239 0.98389 0.98533

25 26 27 28 29

0.04685 0.05060 0.05450 0.05853 0.06269

0.67067 0.70418 0.73637 0.76735 0.79720

70 71 72 73 74

0.32899 0.33722 0.34549 0.35381 0.36218

0.51718 0.52791 0.53844 0.54878 0.55893

115 116 117 118 119

0.71131 0.71919 0.72700 0.73474 0.74240

0.85206 0.85684 0.86153 0.86613 0.87064

160 161 162 163 164

0.96985 0.97276 0.97553 0.97815 0.98063

0.98670 0.98801 0.98924 0.99041 0.99151

30 31 32 33 34

0.06699 0.07142 0.07598 0.08066 0.08548

0.82599 0.85380 0.88068 0.90668 0.93187

75 76 77 78 79

0.37059 0.37904 0.38752 0.39604 0.40460

0.56889 0.57868 0.58830 0.59774 0.60702

120 121 122 123 124

0.75000 0.75752 0.76496 0.77232 0.77960

0.87506 0.87939 0.88364 0.88780 0.89187

165 166 167 168 169

0.98296 0.98515 0.98719 0.98907 0.99081

0.99254 0.99350 0.99440 0.99523 0.99599

35 36 37 38 39

0.09042 0.09549 0.10068 0.10599 0.11143

0.95628 0.97996 0.00295 0.02528 0.04699

80 81 82 83 84

0.41318 0.42178 0.43041 0.43907 0.44774

0.61613 0.62509 0.63389 0.64253 0.65102

125 126 127 128 129

0.78679 0.79389 0.80091 0.80783 0.81466

0.89586 0.89976 0.90358 0.90732 0.91098

170 171 172 173 174

0.99240 0.99384 0.99513 0.99627 0.99726

0.99669 0.99732 0.99788 0.99838 0.99881

40 41 42 43 44

0.11698 0.12265 0.12843 0.13432 0.14033

0.06810 0.08865 0.10866 0.12815 0.14715

85 86 87 88 89

0.45642 0.46512 0.47383 0.48255 0.49127

0.65937 0.66757 0.67562 0.68354 0.69132

130 131 132 133 134

0.82139 0.82803 0.83457 0.84100 0.84733

0.91455 0.91805 0.92146 0.92480 0.92805

175 176 177 178 179

0.99810 0.99878 0.99931 0.99970 0.99992

0.99917 0.99947 0.99970 0.99987 0.99997

180

1.00000

0.00000

u

hav u ¼ 12 vers u ¼ 12 ð1 cos uÞ ¼ sin2 12 ua hav( u) ¼ hav u hav(180 u) ¼ hav(180 þ u) ¼ 1 hav u Characteristics of the logarithms are omitted. a

1758

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.20 Complete Elliptic Integralsa sin 1k

K

log K

E

log E



0 1 2 3 4

1.5708 1.5709 1.5713 1.5719 1.5727

0.196120 0.196153 0.196252 0.196418 0.196649

1.5708 1.5707 1.5703 1.5697 1.5689

0.196120 0.196087 0.195988 0.195822 0.195591

5 6 7 8 9

1.5738 1.5751 1.5767 1.5785 1.5805

0.196947 0.197312 0.197743 0.198241 0.198806

1.5678 1.5665 1.5649 1.5632 1.5611

10 11 12 13 14

1.5828 1.5854 1.5882 1.5913 1.5946

0.199438 0.200137 0.200904 0.201740 0.202643

15 16 17 18 19

1.5981 1.6020 1.6061 1.6105 1.6151

20 21 22 23 24

sin 1k

K

log K

E

log E

45 46 47 48 49

1.8541 1.8691 1.8848 1.9011 1.9180

0.268127 0.271644 0.275267 0.279001 0.282848

1.3506 1.3418 1.3329 1.3238 1.3147

0.130541 0.127690 0.124788 0.121836 0.118836

0.195293 0.194930 0.194500 0.194004 0.193442

50 51 52 53 54

1.9356 1.9539 1.9729 1.9927 2.0133

0.286811 0.290895 0.295101 0.299435 0.303901

1.3055 1.2963 1.2870 1.2776 1.2681

0.115790 0.112698 0.109563 0.106386 0.103169

1.5589 1.5564 1.5537 1.5507 1.5476

0.192815 0.192121 0.191362 0.190537 0.189646

55 56 57 58 59

2.0347 2.0571 2.0804 2.1047 2.1300

0.308504 0.313247 0.318138 0.323182 0.328384

1.2587 1.2492 1.2397 1.2301 1.2206

0.099915 0.096626 0.093303 0.089950 0.086569

0.203615 0.204657 0.205768 0.206948 0.208200

1.5442 1.5405 1.5367 1.5326 1.5283

0.188690 0.187668 0.186581 0.185428 0.184210

60 61 62 63 64

2.1565 2.1842 2.2132 2.2435 2.2754

0.333753 0.339295 0.345020 0.350936 0.357053

1.2111 1.2015 1.1920 1.1826 1.1732

0.083164 0.079738 0.076293 0.072834 0.069364

1.6200 1.6252 1.6307 1.6365 1.6426

0.209522 0.210916 0.212382 0.213921 0.215533

1.5238 1.5191 1.5141 1.5090 1.5037

0.182928 0.181580 0.180168 0.178691 0.177150

65 66 67 68 69

2.3088 2.3439 2.3809 2.4198 2.4610

0.363384 0.369940 0.376736 0.383787 0.391112

1.1638 1.1545 1.1453 1.1362 1.1272

0.065889 0.062412 0.058937 0.055472 0.052020

25 26 27 28 29

1.6490 1.6557 1.6627 1.6701 1.6777

0.217219 0.218981 0.220818 0.222732 0.224723

1.4981 1.4924 1.4864 1.4803 1.4740

0.175545 0.173876 0.172144 0.170348 0.168489

70 71 72 73 74

2.5046 2.5507 2.5998 2.6521 2.7081

0.398730 0.406665 0.414943 0.423596 0.432660

1.1184 1.1096 1.1011 1.0927 1.0844

0.048589 0.045183 0.041812 0.038481 0.035200

30 31 32 33 34

1.6858 1.6941 1.7028 1.7119 1.7214

0.226793 0.228943 0.231173 0.233485 0.235880

1.4675 1.4608 1.4539 1.4469 1.4397

0.166567 0.164583 0.162537 0.160429 0.158261

75 76 77 78 79

2.7681 2.8327 2.9026 2.9786 3.0617

0.442176 0.452196 0.462782 0.474008 0.485967

1.0764 1.0686 1.0611 1.0538 1.0468

0.031976 0.028819 0.025740 0.022749 0.019858

35 36 37 38 39

1.7312 1.7415 1.7552 1.7633 1.7748

0.238359 0.240923 0.243575 0.246315 0.249146

1.4323 1.4248 1.4171 1.4092 1.4013

0.156031 0.153742 0.151393 0.148985 0.146519

80 81 82 83 84

3.1534 3.2553 3.3699 3.5004 3.6519

0.498777 0.512591 0.527613 9.544120 0.562514

1.0401 1.0338 1.0278 1.0223 1.0172

0.017081 0.014432 0.011927 0.009584 0.007422

40 41 42 43 44

1.7868 1.7992 1.8122 1.8256 1.8396

0.252068 0.255085 0.258197 0.261406 0.264716

1.3931 1.3849 1.3765 1.3680 1.3594

0.143995 0.141414 0.138778 0.136086 0.133340

85 86 87 88 89 90

3.8317 4.0528 4.3387 4.7427 5.4349 1

0.583396 0.607751 0.637355 0.676027 0.735192 1

1.0127 1.0086 1.0053 1.0026 1.0008 1.0000

0.005465 0.003740 0.002278 0.001121 0.000326 0.000000

MATHEMATICAL TABLES

1759

TABLE 51.20 (Continued) sin 1k

k

K

log K

89 89 89 89 89 89 89 89 89 89

20 22 24 26 28 30 32 34 36 38

5.840 5.891 5.946 6.003 6.063 6.128 6.197 6.271 6.351 6.438

0.76641 0.77019 0.77422 0.77837 0.78269 0.78732 0.79218 0.79734 0.80284 0.80875

sin 1k

k

K

log K

89 89 89 89 89 89 89 89 89 89

40 41 42 43 44 45 46 47 48 49

6.533 6.584 6.639 36.696 6.756 6.821 6.890 6.964 7.044 7.131

0.81511 0.81849 0.82210 0.82582 0.82969 0.83385 0.83822 0.84286 0.84782 0.85315

sin 1k

k

K

log K

89 89 89 89 89 89 89 89 89 89

50 51 52 53 54 55 56 57 58 59

7.226 7.332 7.449 7.583 7.737 7.919 8.143 8.430 8.836 9.529

0.85890 0.86522 0.87210 0.87984 0.88857 0.89867 0.91078 0.92583 0.94626 0.97905

90

0

1

1

TABLE 51.21 Gamma Functionsa n

G(n)

n

G(n)

n

G(n)

n

G(n)

1.00 1.01 1.02 1.03 1.04

1.00000 0.99433 0.98884 0.98355 0.97844

1.25 1.26 1.27 1.28 1.29

0.90640 0.90440 0.90250 0.90072 0.89904

1.50 1.51 1.52 1.53 1.54

0.88623 0.88659 0.88704 0.88757 0.88818

1.75 1.76 1.77 1.78 1.79

0.91906 0.92137 0.92376 0.92623 0.92877

1.05 1.06 1.07 1.08 1.09

0.97350 0.96874 0.96415 0.95973 0.95546

1.30 1.31 1.32 1.33 1.34

0.89747 0.89600 0.89464 0.89338 0.89222

1.55 1.56 1.57 1.58 1.59

0.88887 0.88964 0.89049 0.89142 0.89243

1.80 1.81 1.82 1.83 1.84

0.93138 0.93408 0.93685 0.93969 0.94261

1.10 1.11 1.12 1.13 1.14

0.95135 0.94739 0.94359 0.93993 0.93642

1.35 1.36 1.37 1.38 1.39

0.89115 0.89018 0.88931 0.88854 0.88785

1.60 1.61 1.62 1.63 1.64

0.89352 0.89468 0.89592 0.89724 0.89864

1.85 1.86 1.87 1.88 1.89

0.94561 0.94869 0.95184 0.95507 0.95838

1.15 1.16 1.17 1.18 1.19

0.93304 0.92980 0.92670 0.92373 0.92088

1.40 1.41 1.42 1.43 1.44

0.88726 0.88676 0.88636 0.88604 0.88580

1.65 1.66 1.67 1.68 1.69

0.90012 0.90167 0.90330 0.90500 0.90678

1.90 1.91 1.92 1.93 1.94

0.96177 0.96523 0.96878 0.97240 0.97610

1.20 1.21 1.22 1.23 1.24

0.91817 0.91558 0.91311 0.91075 0.90852

1.45 1.46 1.47 1.48 1.49

0.88565 0.88560 0.88563 0.88575 0.88595

1.70 1.71 1.72 1.73 1.74

0.90864 0.91057 0.91258 0.91466 0.91683

1.95 1.96 1.97 1.98 1.99 2.00

0.97988 0.98374 0.98768 0.99171 0.99581 1.00000

a

Values of GðnÞ ¼

Z

1 0

e x xn 1 dx; Gðn þ 1Þ ¼ nGðnÞ.

For large positive integers, Stirling’s formula gives an approximation in which the relative error decreases as n increases: nn Gðn þ 1Þ ¼ ð2pnÞ1=2 e Source: From CRC Standard Mathematical Tables, Chemical Rubber Publishing Co., 12th ed., 1959. Used by permission.

1760

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.22 Bessel Functions J0(x) and J1(x)a

J0(x) and J1(x)a

J0(x) and J1(x)a

x

J0(x)

J1(x)

x

J0(x)

J1(x)

x

J0(x)

J1(x)

0.0 0.1 0.2 0.3 0.4

1.0000 0.9975 0.9900 0.9776 0.9604

0.0000 0.0499 0.0995 0.1483 0.1960

3.0 3.1 3.2 3.3 3.4

0.2601 0.2921 0.3202 0.3443 0.3643

0.3391 0.3009 0.2613 0.2207 0.1792

6.0 6.1 6.2 6.3 6.4

0.1506 0.1773 0.2017 0.2238 0.2433

0.2767 0.2559 0.2329 0.2081 0.1816

0.5 0.6 0.7 0.8 0.9

0.9385 0.9120 0.8812 0.8463 0.8075

0.2423 0.2867 0.3290 0.3668 0.4059

3.5 3.6 3.7 3.8 3.9

0.3801 0.3918 0.3992 0.4026 0.4018

0.1374 0.0955 0.0538 0.0128 0.0272

6.5 6.6 6.7 6.8 6.9

0.2601 0.2740 0.2851 0.2931 0.2981

0.1538 0.1250 0.0953 0.0652 0.0349

1.0 1.1 1.2 1.3 1.4

0.7652 0.7196 0.6711 0.6201 0.5669

0.4401 0.4709 0.4983 0.5220 0.5419

4.0 4.1 4.2 4.3 4.4

0.3971 0.3887 0.3766 0.3610 0.3423

0.0660 0.1033 0.1386 0.1719 0.2028

7.0 7.1 7.2 7.3 7.4

0.3001 0.2991 0.2951 0.2882 0.2786

0.0047 0.0252 0.0543 0.0826 0.1096

1.5 1.6 1.7 1.8 1.9

0.5118 0.4554 0.3980 0.3400 0.2818

0.5579 0.5699 0.5778 0.5815 0.5812

4.5 4.6 4.7 4.8 4.9

0.3205 0.2961 0.2693 0.2404 0.2097

0.2311 0.2566 0.2791 0.2985 0.3147

7.5 7.6 7.7 7.8 7.9

0.2663 0.2516 0.2346 0.2154 0.1944

0.1352 0.1592 0.1813 0.2014 0.2192

2.0 2.1 2.2 2.3 2.4

0.2239 0.1666 0.1104 0.0555 0.0025

0.5767 0.5683 0.5560 0.5399 0.5202

5.0 5.1 5.2 5.3 5.4

0.1776 0.1443 0.1103 0.0758 0.0412

0.3276 0.3371 0.3432 0.3460 0.3453

8.0 8.1 8.2 8.3 8.4

0.1717 0.1475 0.1222 0.0960 0.0692

0.2346 0.2476 0.2580 0.2657 0.2708

2.5 2.6 2.7 2.8 2.9

0.0484 0.0968 0.1424 0.1850 0.2243

0.4971 0.4708 0.4416 0.4097 0.3754

5.5 5.6 5.7 5.8 5.9

0.0068 0.0270 0.0599 0.0917 0.1220

0.3414 0.3343 0.3241 0.3110 0.2951

8.5 8.6 8.7 8.8 8.9

0.0419 0.0146 0.0125 0.0392 0.0653

0.2731 0.2728 0.2697 0.2641 0.2559

Y0(x) and Y1(x)

Y0(x) and Y1(x)

Y0(x) and Y1(x)

x

Y0(x)

Y1(x)

x

Y0(x)

Y1(x)

x

Y0(x)

Y1(x)

0.0 0.5 1.0 1.5 2.0

( 1) 0.445 0.088 0.382 0.510

( 1) 1.471 0.781 0.412 0.107

2.5 3.0 3.5 4.0 4.5

0.498 0.377 0.189 0.017 0.195

0.146 0.325 0.410 0.398 0.301

5.0 5.5 6.0 6.5 7.0

0.309 0.340 0.288 0.173 0.026

0.148 0.024 0.175 0.274 0.303

a J1(x) ¼ 0 for x ¼ 0, 3.832, 7.016, 10.173, 13.324, . . . J0(x) ¼ 0 for x ¼ 2.405, 5.520, 8.654, 11.792, . . .

STATISTICAL TABLES

1761

51.3 STATISTICAL TABLES1 TABLE 51.23 Binomial Coefficients n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

  n 0

  n 1

  n 2

  n 3

  n 4

  n 5

  n 6

  n 7

  n 8

  n 9

  n 10

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190

1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140

1 5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 3060 3876 4845

1 6 21 56 126 252 462 792 1287 2002 3003 4368 6188 8568 11628 15504

1 7 28 84 210 462 924 1716 3003 5005 8008 12376 18564 27132 38760

1 8 36 120 330 792 1716 3432 6435 11440 19448 31824 50388 77520

1 9 45 165 495 1287 3003 6435 12870 24310 43758 75582 125970

1 10 55 220 715 2002 5005 11440 24310 48620 92378 167960

1 11 66 286 1001 3003 8008 19448 43758 92378 184756

nCm ¼

n m

!

ðp þ qÞn ¼ pn þ

¼ n 1

½ðn !

n! ¼ mÞ!m!Š

pn 1 q þ


þ

n n n s

m

!




n 0

!

¼ 1:

!

ps qt þ


þ qn ; s þ t ¼ n:

Probability Let p be the probability of an event e in one trial and q the probability of failure t times   of e. The probability that, in n trials, the event e will occur exactly n is nt pn t qt . The probability that an event e will happen at least r times in n trials is Pt¼n r  n  n t t Pt¼n  n  n t t q. p q ; at most r times in n trials is t¼0 t¼n r t p t

In a point binomial, (p þ q)n, distribution, the mean number of favorable events is np; pffiffiffiffiffiffiffiffi the mean number of unfavorable events is nq; the standard deviation is s ¼ pqn; and pffiffiffiffiffiffiffiffi a3 ¼ ðp qÞ=s. The mean deviation from the mean MD is s 2=p ¼ 0:7979s; the semiquartile deviation from the mean is 0:6745s ¼ 0:845 MD. The probability that a deviation of an individual measure from the average lies between y ¼ a and y ¼ a is Z y¼a Z x¼b Z y¼a 2 2 1 1 1 2 2 2 pffiffiffi he h y dy ¼ pffiffiffiffiffiffi e x =2 dx e y =2s dy ¼ pffiffiffiffiffiffi p y¼ a 2p x¼ b s 2p y¼ a pffiffiffi pffiffiffi pffiffiffi where x ¼ hy 2, b ¼ ha 2, and s ¼ 1=h 2; h is called the modulus of precision and s the standard (quadratic mean) deviation. 1

Tables 51.23–51.25 from Burington, Handbook of Math Tables and Formulas, published by McGraw-Hill.

1762

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.24 Probability Functions 1 ð1 þ aÞ ¼ 2

a¼

Z

x

þ aÞ

1

FðxÞ dx ¼ area under FðxÞ from 1 FðxÞ ¼ pffiffiffiffiffiffi e 2p

FðxÞdx;

Fð2Þ ðxÞ ¼ ðx2

1Þ

Fð3Þ ðxÞ ¼ ð3x

x3 Þ

F ðxÞ ¼ ðx 1 2 ð1

x

x

ð4Þ

x

Z

F(x)

F(2)(x)

4

x2 =2

1 to x

¼ normal function

FðxÞ ¼ second derivative of FðxÞ

FðxÞ ¼ third derivative of FðxÞ

2

6x þ 3Þ FðxÞ ¼ fourth derivative of FðxÞ F(3)(x)

F(4)(x)

x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

0.00 0.01 0.02 0.03 0.04

0.5000 0.5040 0.5080 0.5120 0.5160

0.3989 0.3989 0.3989 0.3988 0.3986

0.3989 0.3989 0.3987 0.3984 0.3980

0.0000 0.0120 0.0239 0.0359 0.0478

1.1968 1.1965 1.1956 1.1941 1.1920

0.35 0.36 0.37 0.38 0.39

0.6368 0.6406 0.6443 0.6480 0.6517

0.3752 0.3739 0.3726 0.3712 0.3697

0.3293 0.3255 0.3216 0.3176 0.3135

0.3779 0.3864 0.3947 0.4028 0.4107

0.8556 0.8373 0.8186 0.7996 0.7803

0.05 0.06 0.07 0.08 0.09

0.5199 0.5239 0.5279 0.5319 0.5359

0.3984 0.3982 0.3980 0.3977 0.3973

0.3975 0.3968 0.3960 0.3951 0.3941

0.0597 0.0716 0.0834 0.0952 0.1070

1.1894 1.1861 1.1822 1.1778 1.1727

0.40 0.41 0.42 0.43 0.44

0.6554 0.6591 0.6628 0.6664 0.6700

0.3683 0.3668 0.3653 0.3637 0.3621

0.3094 0.3059 0.3008 0.2965 0.2920

0.4184 0.4259 0.4332 0.4403 0.4472

0.7607 0.7408 0.7206 0.7001 0.6793

0.10 0.11 0.12 0.13 0.14

0.5398 0.5438 0.5478 0.5517 0.5557

0.3970 0.3965 0.3961 0.3956 0.3951

0.3930 0.3917 0.3904 0.3889 0.3873

0.1187 0.1303 0.1419 0.1534 0.1648

1.1671 1.1609 1.1541 1.1468 1.1389

0.45 0.46 0.47 0.48 0.49

0.6736 0.6772 0.6808 0.6844 0.6879

0.3605 0.3589 0.3572 0.3555 0.3538

0.2875 0.2830 0.2783 0.2736 0.2689

0.4539 0.4603 0.4666 0.4727 0.4785

0.6583 0.6371 0.6156 0.5940 0.5721

0.15 0.16 0.17 0.18 0.19

0.5596 0.5636 0.5675 0.5714 0.5753

0.3945 0.3939 0.3932 0.3925 0.3918

0.3856 0.3838 0.3819 0.3798 0.3777

0.1762 0.1874 0.1986 0.2097 0.2206

1.1304 1.1214 1.1118 1.1017 1.0911

0.50 0.51 0.52 0.53 0.54

0.6915 0.6950 0.6985 0.7019 0.7054

0.3521 0.3503 0.3485 0.3467 0.3448

0.2641 0.2592 0.2543 0.2493 0.2443

0.4841 0.4895 0.4947 0.4996 0.5043

0.5501 0.5279 0.5056 0.4831 0.4605

0.20 0.21 0.22 0.23 0.24

0.5793 0.5832 0.5871 0.5910 0.5948

0.3910 0.3902 0.3894 0.3885 0.3876

0.3754 0.3730 0.3706 0.3680 0.3653

0.2315 0.2422 0.2529 0.2634 0.2737

1.0799 1.0682 1.0560 1.0434 1.0302

0.55 0.56 0.57 0.58 0.59

0.7088 0.7123 0.7157 0.7190 0.7224

0.3429 0.3410 0.3391 0.3372 0.3352

0.2392 0.2341 0.2289 0.2238 0.2185

0.5088 0.5131 0.5171 0.5209 0.5245

0.4378 0.4150 0.3921 0.3691 0.3461

0.25 0.26 0.27 0.28 0.29

0.5987 0.6026 0.6064 0.6103 0.6141

0.3867 0.3857 0.3847 0.3836 0.3825

0.3625 0.3596 0.3566 0.3535 0.3504

0.2840 0.2941 0.3040 0.3138 0.3235

1.0165 1.0024 0.9878 0.9727 0.9572

0.60 0.61 0.62 0.63 0.64

0.7257 0.7291 0.7324 0.7357 0.7389

0.3332 0.3312 0.3292 0.3271 0.3251

0.2133 0.2080 0.2027 0.1973 0.1919

0.5278 0.5309 0.5338 0.5365 0.5389

0.3231 0.3000 0.2770 0.2539 0.2309

0.30 0.31 0.32 0.33 0.34

0.6179 0.6217 0.6255 0.6293 0.6331

0.3814 0.3802 0.3790 0.3778 0.3765

0.3471 0.3437 0.3402 0.3367 0.3330

0.3330 0.3423 0.3515 0.3605 0.3693

0.9413 0.9250 0.9082 0.8910 0.8735

0.65 0.66 0.67 0.68 0.69

0.7422 0.7454 0.7486 0.7517 0.7549

0.3230 0.3209 0.3187 0.3166 0.3144

0.1865 0.1811 0.1757 0.1702 0.1647

0.5411 0.5431 0.5448 0.5463 0.5476

0.2078 0.1849 0.1620 0.1391 0.1164

STATISTICAL TABLES

1763

TABLE 51.24 (Continued) x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

0.70 0.71 0.72 0.73 0.74

0.7580 0.7611 0.7642 0.7673 0.7704

0.3123 0.3101 0.3079 0.3056 0.3034

0.1593 0.1538 0.1483 0.1428 0.1373

0.5486 0.5495 0.5501 0.5504 0.5506

0.0937 0.0712 0.0487 0.0265 0.0043

1.15 1.16 1.17 1.18 1.19

0.8749 0.8770 0.8790 0.8810 0.8830

0.2059 0.2036 0.2012 0.1989 0.1965

0.0664 0.0704 0.0742 0.0780 0.0818

0.3973 0.3907 0.3840 0.3772 0.3704

0.6561 0.6643 0.6720 0.6792 0.6861

0.75 0.76 0.77 0.78 0.79

0.7734 0.7764 0.7794 0.7823 0.7852

0.3011 0.2989 0.2966 0.2943 0.2920

0.1318 0.1262 0.1207 0.1153 0.1098

0.5505 0.5502 0.5497 0.5490 0.5481

0.0176 0.0394 0.0611 0.0825 0.1037

1.20 1.21 1.22 1.23 1.24

0.8849 0.8869 0.8888 0.8907 0.8925

0.1942 0.1919 0.1919 0.1872 0.1849

0.0854 0.0890 0.0890 0.0960 0.0994

0.3635 0.3566 0.3566 0.3425 0.3354

0.6926 0.6986 0.6986 0.7094 0.7141

0.80 0.81 0.82 0.83 0.84

0.7881 0.7910 0.7939 0.7967 0.7995

0.2897 0.2874 0.2850 0.2827 0.2803

0.1043 0.0988 0.0934 0.0880 0.0825

0.5469 0.5456 0.5440 0.5423 0.5403

0.1247 0.1455 0.1660 0.1862 0.2063

1.25 1.26 1.27 1.28 1.29

0.8944 0.8962 0.8980 0.8997 0.9015

0.1826 0.1804 0.1781 0.1758 0.1736

0.1027 0.1060 0.1092 0.1123 0.1153

0.3282 0.3210 0.3138 0.3065 0.2992

0.7185 0.7224 0.7259 0.7291 0.7318

0.85 0.86 0.87 0.88 0.89

0.8023 0.8051 0.8078 0.8106 0.8133

0.2780 0.2756 0.2732 0.2709 0.2685

0.0771 0.0718 0.0664 0.0611 0.0558

0.5381 0.5358 0.5332 0.5305 0.5276

0.2260 0.2455 0.2646 0.2835 0.3021

1.30 1.31 1.32 1.33 1.34

0.9032 0.9049 0.9066 0.9082 0.9099

0.1714 0.1691 0.1669 0.1647 0.1626

0.1182 0.1211 0.1239 0.1267 0.1293

0.2918 0.2845 0.2771 0.2697 0.2624

0.7341 0.7361 0.7376 0.7388 0.7395

0.90 0.91 0.92 0.93 0.94

0.8159 0.8186 0.8212 0.8238 0.8264

0.2661 0.2637 0.2613 0.2589 0.2565

0.0506 0.0453 0.0401 0.0350 0.0299

0.5245 0.5212 0.5177 0.5140 0.5102

0.3203 0.3383 0.3559 0.3731 0.3901

1.35 1.36 1.37 1.38 1.39

0.9115 0.9131 0.9147 0.9162 0.9177

0.1604 0.1582 0.1561 0.1539 0.1518

0.1319 0.1344 0.1369 0.1392 0.1415

0.2550 0.2476 0.2402 0.2328 0.2254

0.7399 0.7400 0.7396 0.7389 0.7378

0.95 0.96 0.97 0.98 0.99

0.8289 0.8315 0.8340 0.8365 0.8389

0.2541 0.2516 0.2492 0.2468 0.2444

0.0248 0.0197 0.0147 0.0098 0.0049

0.5062 0.521 0.4978 0.4933 0.4887

0.4066 0.4228 0.4387 0.4541 0.4692

1.40 1.41 1.42 1.43 1.44

0.9192 0.9207 0.9222 0.9236 0.9251

0.1497 0.1476 0.1456 0.1435 0.1415

0.1437 0.1459 0.1480 0.1500 0.1519

0.2180 0.2107 0.2033 0.1960 0.1887

0.7364 0.7347 0.7326 0.7301 0.7274

1.00 1.01 1.02 1.03 1.04

0.8413 0.8438 0.8461 0.8485 0.8508

0.2420 0.2396 0.2371 0.2347 0.2323

0.0000 0.0048 0.0096 0.0143 0.0190

0.4839 0.4790 0.4740 0.4688 0.4635

0.4839 0.4983 0.5122 0.5257 0.5389

1.45 1.46 1.47 1.48 1.49

0.9265 0.9279 0.9292 0.9306 0.9319

0.1394 0.1374 0.1354 0.1344 0.1315

0.1537 0.1555 0.1572 0.1588 0.1604

0.1815 0.1742 0.1670 0.1599 0.1528

0.7243 0.7209 0.7172 0.7132 0.7089

1.05 1.06 1.07 1.08 1.09

0.8531 0.8554 0.8577 0.8599 0.8621

0.2299 0.2275 0.2251 0.2227 0.2203

0.0236 0.0281 0.0326 0.0371 0.0414

0.4580 0.4524 0.4467 0.4409 0.4350

0.5516 0.5639 0.5758 0.5873 0.5984

1.50 1.51 1.52 1.53 1.54

0.9332 0.9345 0.9357 0.9370 0.9382

0.1295 0.1276 0.1257 0.1238 0.1219

0.1619 0.1633 0.1647 0.1660 0.1672

0.1457 0.1387 0.1317 0.1248 0.1180

0.7043 0.6994 0.6942 0.6888 0.6831

1.10 1.11 1.12 1.13 1.14

0.8643 0.8665 0.8686 0.8708 0.8729

0.2179 0.2155 0.2131 0.2107 0.2083

0.0458 0.0500 0.0542 0.0583 0.0624

0.4290 0.4228 0.4166 0.4102 0.4038

0.6091 0.6193 0.6292 0.6386 0.6476

1.55 1.56 1.57 1.58 1.59

0.9394 0.9406 0.9418 0.9429 0.9441

0.1200 0.1182 0.1163 0.1145 0.1127

0.1683 0.1694 0.1704 0.1714 0.1722

0.1111 0.1044 0.0977 0.0911 0.0846

0.6772 0.6710 0.6646 0.6580 0.6511

(continued)

1764

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.24 (Continued) x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

1.60 1.61 1.62 1.63 1.64

0.9452 0.9463 0.9474 0.9484 0.9495

0.1109 0.1092 0.1074 0.1057 0.1040

0.1730 0.1738 0.1745 0.1751 0.1757

0.0781 0.0717 0.0654 0.0591 0.0529

0.6441 0.6368 0.6293 0.6216 0.6138

2.05 2.06 2.07 2.08 2.09

0.9798 0.9803 0.9808 0.9812 0.9817

0.0468 0.0478 0.0468 0.0459 0.0449

0.1563 0.1550 0.1538 0.1526 0.1513

0.1203 0.1225 0.1245 0.1265 0.1284

0.2222 0.2129 0.2036 0.1945 0.1854

1.65 1.66 1.67 1.68 1.69

0.9505 0.9515 0.9525 0.9535 0.9545

0.1023 0.1006 0.0989 0.0973 0.0957

0.1762 0.1766 0.1770 0.1773 0.1776

0.0468 0.0408 0.0349 0.0290 0.0233

0.6057 0.5975 0.5891 0.5806 0.5720

2.10 2.11 2.12 2.13 2.14

0.9821 0.9821 0.9830 0.9834 0.9838

0.0440 0.0440 0.0422 0.0413 0.0404

0.1500 0.1500 0.1474 0.1460 0.1446

0.1302 0.1302 0.1336 0.1351 0.1366

0.1765 0.1765 0.1588 0.1502 0.1416

1.70 1.71 1.72 1.73 1.74

0.9554 0.9564 0.9573 0.9582 0.9591

0.0940 0.0925 0.0909 0.0893 0.0878

0.1778 0.1779 0.1780 0.1780 0.1780

0.0176 0.0120 0.0065 0.0011 0.0042

0.5632 0.5542 0.5452 0.5360 0.5267

2.15 2.16 2.17 2.18 2.19

0.9842 0.9846 0.9850 0.9854 0.9857

0.0395 0.0387 0.0379 0.0371 0.0363

0.1433 0.1419 0.1405 0.1391 0.1377

0.1380 0.1393 0.1405 0.1416 0.1426

0.1332 0.1249 0.1167 0.1086 0.1006

1.75 1.76 1.77 1.78 1.79

0.9599 0.9608 0.9616 0.9625 0.9633

0.0863 0.0848 0.0833 0.0818 0.0804

0.1780 0.1778 0.1777 0.1774 0.1772

0.0094 0.0146 0.0196 0.0245 0.0294

0.5173 0.5079 0.4983 0.4887 0.4789

2.20 2.21 2.22 2.23 2.24

0.9861 0.9864 0.9868 0.9871 0.9875

0.0355 0.0347 0.0339 0.0332 0.0325

0.1362 0.1348 0.1333 0.1319 0.1304

0.1436 0.1445 0.1453 0.1460 0.1467

0.0927 0.0850 0.0774 0.0700 0.0626

1.80 1.81 1.82 1.83 1.84

0.9641 0.9649 0.9656 0.9664 0.9671

0.0790 0.0775 0.0761 0.0748 0.0734

0.1769 0.1765 0.1761 0.1756 0.1751

0.0341 0.0388 0.0433 0.0477 0.0521

0.4692 0.4593 0.4494 0.4395 0.4295

2.25 2.26 2.27 2.28 2.29

0.9878 0.9881 0.9884 0.9887 0.9890

0.0317 0.0310 0.0303 0.0297 0.0290

0.1289 0.1275 0.1260 0.1245 0.1230

0.1473 0.1478 0.1483 0.1486 0.1490

0.0554 0.0484 0.0414 0.0346 0.0279

1.85 1.86 1.87 1.88 1.89

0.9678 0.9686 0.9693 0.9699 0.9706

0.0721 0.0707 0.0694 0.0681 0.0669

0.1746 0.1740 0.1734 0.1727 0.1720

0.0563 0.0605 0.0645 0.0685 0.0723

0.4195 0.4095 0.3995 0.3894 0.3793

2.30 2.31 2.32 2.33 2.34

0.9893 0.9896 0.9898 0.9901 0.9904

0.0283 0.0277 0.0270 0.0264 0.0258

0.1215 0.1200 0.1185 0.1170 0.1155

0.1492 0.1494 0.1495 0.1496 0.1496

0.0214 0.0150 0.0088 0.0027 0.0033

1.90 1.91 1.92 1.93 1.94

0.9713 0.9719 0.9726 0.9732 0.9738

0.0656 0.0644 0.0632 0.0620 0.0608

0.1713 0.1705 0.1697 0.1688 0.1679

0.0761 0.0797 0.0832 0.0867 0.0900

0.3693 0.3592 0.3492 0.3392 0.3292

2.35 2.36 2.37 2.38 2.39

0.9906 0.9909 0.9911 0.9913 0.9916

0.0252 0.0246 0.0241 0.0235 0.0229

0.1141 0.1126 0.1111 0.1096 0.1081

0.1495 0.1494 0.1492 0.1490 0.1487

0.0092 0.0149 0.0204 0.0258 0.0311

1.95 1.96 1.97 1.98 1.99

0.9744 0.9750 0.9756 0.9761 0.9767

0.0596 0.0584 0.0573 0.0562 0.0551

0.1670 0.1661 0.1651 0.1641 0.1630

0.0933 0.0964 0.0994 0.1024 0.1052

0.3192 0.3093 0.2994 0.2895 0.2797

2.40 2.41 2.42 2.43 2.44

0.9918 0.9920 0.9922 0.9925 0.9927

0.0224 0.0219 0.0213 0.0208 0.0203

0.1066 0.1051 0.1036 0.1022 0.1007

0.1483 0.1480 0.1475 0.1470 0.1465

0.0362 0.0412 0.0461 0.0508 0.0554

2.00 2.01 2.02 2.03 2.04

0.9772 0.9778 0.9783 0.9788 0.9793

0.0540 0.0529 0.0519 0.0508 0.0498

0.1620 0.1609 0.1598 0.1586 0.1575

0.1080 0.1106 0.1132 0.1157 0.1180

0.2700 0.2603 0.2506 0.2411 0.2316

2.45 2.46 2.47 2.48 2.49

0.9929 0.9931 0.9932 0.9934 0.9936

0.0198 0.0194 0.0189 0.0184 0.0180

0.0992 0.0978 0.0963 0.0949 0.0935

0.1459 0.1453 0.1446 0.1439 0.1432

0.0598 0.0641 0.0683 0.0723 0.0762

STATISTICAL TABLES

1765

TABLE 51.24 (Continued) x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

2.50 2.51 2.52 2.53 2.54

0.9938 0.9940 0.9941 0.9943 0.9945

0.0175 0.0171 0.0167 0.0163 0.0158

0.0920 0.0906 0.0892 0.0878 0.0864

0.1424 0.1416 0.1408 0.1399 0.1389

0.0800 0.0836 0.0871 0.0905 0.0937

2.95 2.96 2.97 2.98 2.99

0.9984 0.9985 0.9985 0.9986 0.9986

0.0051 0.0050 0.0048 0.0047 0.0046

0.0396 0.0388 0.0379 0.0371 0.0363

0.0865 0.0852 0.0838 0.0825 0.0811

0.1364 0.1358 0.1352 0.1345 0.1337

2.55 2.56 2.57 2.58 2.59

0.9946 0.9948 0.9949 0.9951 0.9952

0.0154 0.0151 0.0147 0.0143 0.0319

0.0850 0.0836 0.0823 0.0809 0.0796

0.1380 0.1370 0.1360 0.1350 0.1339

0.0968 0.0998 0.1027 0.1054 0.1080

3.00 3.01 3.02 3.03 3.04

0.9987 0.9987 0.9987 0.9988 0.9988

0.0044 0.0043 0.0042 0.0040 0.0039

0.0355 0.0347 0.0339 0.0331 0.0324

0.0798 0.0785 0.0771 0.0758 0.0745

0.1330 0.1321 0.1313 0.1304 0.1294

2.60 2.60 2.62 2.63 2.64

0.9953 0.9953 0.9956 0.9957 0.9959

0.0136 0.0136 0.0129 0.0126 0.0122

0.0782 0.0782 0.0756 0.0743 0.0730

0.1328 0.1328 0.1305 0.1294 0.1282

0.1105 0.1105 0.1152 0.1173 0.1194

3.05 3.06 3.07 3.08 3.09

0.9989 0.9989 0.9989 0.9990 0.9990

0.0038 0.0037 0.0036 0.0035 0.0034

0.0316 0.0309 0.0302 0.0295 0.0288

0.0732 0.0720 0.0707 0.0694 0.0682

0.1285 0.1275 0.1264 0.1254 0.1243

2.65 2.66 2.67 2.68 2.69

0.9960 0.9961 0.9962 0.9963 0.9964

0.0119 0.0116 0.0113 0.0110 0.0107

0.0717 0.0705 0.0692 0.0680 0.0668

0.1270 0.1258 0.1245 0.1233 0.1220

0.1213 0.1231 0.1248 0.1264 0.1279

3.10 3.11 3.12 3.13 3.14

0.9990 0.9991 0.9991 0.9991 0.9992

0.0033 0.0032 0.0031 0.0030 0.0029

0.0281 0.0275 0.0268 0.0262 0.0256

0.0669 0.0657 0.0645 0.0633 0.0621

0.1231 0.1220 0.1208 0.1196 0.1184

2.70 2.71 2.72 2.73 2.74

0.9965 0.9966 0.9967 0.9968 0.9969

0.0104 0.0101 0.0099 0.0096 0.0093

0.0656 0.0644 0.0632 0.0620 0.0608

0.1207 0.1194 0.1181 0.1168 0.1154

0.1293 0.1306 0.1317 0.1328 0.1338

3.15 3.16 3.17 3.18 3.19

0.9992 0.9992 0.9992 0.9993 0.9993

0.0028 0.0027 0.0026 0.0025 0.0025

0.0249 0.0243 0.0237 0.0232 0.0226

0.0609 0.0598 0.0586 0.0575 0.0564

0.1171 0.1159 0.1146 0.1133 0.1120

2.75 2.76 2.77 2.78 2.79

0.9970 0.9971 0.9972 0.9973 0.9974

0.0091 0.0088 0.0086 0.0084 0.0081

0.0597 0.0585 0.0574 0.0563 0.0552

0.1141 0.1127 0.1114 0.1100 0.1087

0.1347 0.1356 0.1363 0.1369 0.1375

3.20 3.21 3.22 3.23 3.24

0.9993 0.9993 0.9994 0.9994 0.9994

0.0024 0.0023 0.0022 0.0022 0.0021

0.0220 0.0215 0.0210 0.0204 0.0199

0.0552 0.0541 0.0531 0.0520 0.0509

0.1107 0.1093 0.1080 0.1066 0.1053

2.80 2.81 2.82 2.83 2.84

0.9974 0.9975 0.9976 0.9977 0.9977

0.0079 0.0077 0.0075 0.0073 0.0071

0.0541 0.0531 0.0520 0.0510 0.0500

0.1073 0.1059 0.1045 0.1031 0.1017

0.1379 0.1383 0.1386 0.1389 0.1390

3.25 3.26 3.27 3.28 3.29

0.9994 0.9994 0.9995 0.9995 0.9995

0.0020 0.0020 0.0019 0.0018 0.0018

0.0194 0.0189 0.0184 0.0180 0.0175

0.0499 0.0488 0.0478 0.0468 0.0458

0.1039 0.1025 0.1011 0.0997 0.0983

2.85 2.86 2.87 2.88 2.89

0.9978 0.9979 0.9979 0.9980 0.9981

0.0069 0.0067 0.0065 0.0063 0.0061

0.0490 0.0480 0.0470 0.0460 0.0451

0.1003 0.0990 0.0976 0.0962 0.0948

0.1391 0.1391 0.1391 0.1389 0.1388

3.30 3.31 3.32 3.33 3.34

0.9995 0.9995 0.9996 0.9996 0.9996

0.0017 0.0017 0.0016 0.0016 0.0015

0.0170 0.0106 0.0102 0.0157 0.0153

0.0449 0.0439 0.0429 0.0420 0.0411

0.0969 0.0955 0.0941 0.0927 0.0913

2.90 2.91 2.92 2.93 2.94

0.9981 0.9982 0.9982 0.9983 0.9984

0.0060 0.0058 0.0056 0.0055 0.0053

0.0441 0.0432 0.0423 0.0414 0.0405

0.0934 0.0920 0.0906 0.0893 0.0879

0.1385 0.1382 0.1378 0.1374 0.1369

3.35 3.36 3.37 3.38 3.39

0.9996 0.9996 0.9996 0.9996 0.9997

0.0015 0.0014 0.0014 0.0013 0.0013

0.0149 0.0145 0.0141 0.0138 0.0134

0.0402 0.0393 0.0384 0.0376 0.0367

0.0899 0.0885 0.0871 0.0857 0.0843

(continued)

1766

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.24 (Continued) x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

x

1 2 ð1

þ aÞ

F(x)

F(2)(x)

F(3)(x)

F(4)(x)

3.40 3.41 3.42 3.43 3.44

0.9997 0.9997 0.9997 0.9997 0.9997

0.0012 0.0012 0.0012 0.0011 0.0011

0.0130 0.0127 0.0123 0.0120 0.0116

0.0359 0.0350 0.0342 0.0334 0.0327

0.0829 0.0815 0.0801 0.0788 0.0774

3.80 3.81 3.82 3.83 3.84

0.9999 0.9999 0.9999 0.9999 0.9999

0.0003 0.0003 0.0003 0.0003 0.0003

0.0039 0.0038 0.0037 0.0036 0.0034

0.0127 0.0123 0.0120 0.0116 0.0113

0.0365 0.0356 0.0347 0.0339 0.0331

3.45 3.46 3.47 3.48 3.49

0.9997 0.9997 0.9997 0.9998 0.9998

0.0010 0.0010 0.0010 0.0009 0.0009

0.0113 0.0110 0.0107 0.0104 0.0101

0.0319 0.0311 0.0304 0.0297 0.0290

0.0761 0.0747 0.0734 0.0721 0.0707

3.85 3.86 3.87 3.88 3.89

0.9999 0.9999 1.0000 1.0000 1.0000

0.0002 0.0002 0.0002 0.0002 0.0002

0.0033 0.0032 0.0031 0.0030 0.0029

0.0110 0.0107 0.0104 0.0100 0.0098

0.0323 0.0315 0.0307 0.0299 0.0292

3.50 3.51 3.52 3.53 3.54

0.9998 0.9998 0.9998 0.9998 0.9998

0.0009 0.0008 0.0008 0.0008 0.0008

0.0098 0.0095 0.0093 0.0090 0.0087

0.0283 0.0276 0.0269 0.0262 0.0256

0.0694 0.0681 0.0669 0.0656 0.0643

3.90 3.91 3.92 3.93 3.94

1.0000 1.0000 1.0000 1.0000 1.0000

0.0002 0.0002 0.0002 0.0002 0.0002

0.0028 0.0027 0.0026 0.0026 0.0025

0.0095 0.0092 0.0089 0.0086 0.0084

0.0284 0.0277 0.0270 0.0263 0.0256

3.55 3.56 3.57 3.58 3.59

0.9998 0.9998 0.9998 0.9998 0.9998

0.0007 0.0007 0.0007 0.0007 0.0006

0.0085 0.0082 0.0080 0.0078 0.0075

0.0249 0.0243 0.0237 0.0231 0.0225

0.0631 0.0618 0.0606 0.0594 0.0582

3.95 3.96 3.97 3.98 3.99

1.0000 1.0000 1.0000 1.0000 1.0000

0.0002 0.0002 0.0002 0.0001 0.0001

0.0024 0.0023 0.0022 0.0022 0.0021

0.0081 0.0079 0.0076 0.0074 0.0072

0.0250 0.0243 0.0237 0.0230 0.0224

3.60 3.61 3.62 3.63 3.64

0.9998 0.9999 0.9999 0.9999 0.9999

0.0006 0.0006 0.0006 0.0006 0.0005

0.0073 0.0071 0.0069 0.0067 0.0065

0.0219 0.0214 0.0208 0.0203 0.0198

0.0570 0.0559 0.0547 0.0536 0.0524

4.00 4.05 4.10 4.15 4.20

1.0000 1.0000 1.0000 1.0000 1.0000

0.0001 0.0001 0.0001 0.0001 0.0001

0.0020 0.0017 0.0014 0.0012 0.0010

0.0070 0.0059 0.0051 0.0043 0.0036

0.0218 0.0190 0.0165 0.0143 0.0123

3.65 3.66 3.67 3.68 3.69

0.9999 0.9999 0.9999 0.9999 0.9999

0.0005 0.0005 0.0005 0.0005 0.0004

0.0063 0.0061 0.0059 0.0057 0.0056

0.0192 0.0187 0.0182 0.0177 0.0173

0.0513 0.0502 0.0492 0.0481 0.0470

4.25 4.30 4.35 4.40 4.45

1.0000 1.0000 1.0000 1.0000 1.0000

0.0001 0.0000 0.0000 0.0000 0.0000

0.0008 0.0007 0.0006 0.0005 0.0004

0.0031 0.0026 0.0022 0.0018 0.0015

0.0105 0.0090 0.0077 0.0065 0.0055

3.70 3.71 3.72 3.73 3.74

0.9999 0.9999 0.9999 0.9999 0.9999

0.0004 0.0004 0.0004 0.0004 0.0004

0.0054 0.0052 0.0051 0.0049 0.0048

0.0168 0.0164 0.0159 0.0155 0.0150

0.0460 0.0450 0.0440 0.0430 0.0420

4.50 4.55 4.60 4.65 4.70

1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0003 0.0003 0.0002 0.0002 0.0001

0.0012 0.0010 0.0009 0.0007 0.0006

0.0047 0.0039 0.0033 0.0027 0.0023

3.75 3.76 3.77 3.78 3.79

0.9999 0.9999 0.9999 0.9999 0.9999

0.0004 0.0003 0.0003 0.0003 0.0003

0.0046 0.0045 0.0043 0.0042 0.0041

0.0146 0.0142 0.0138 0.0134 0.0131

0.0410 0.0401 0.0392 0.0382 0.0373

4.75 4.80 4.85 4.90 4.95

1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0001 0.0001 0.0001 0.0001 0.0000

0.0005 0.0004 0.0003 0.0003 0.0002

0.0019 0.0016 0.0013 0.0011 0.0009

STATISTICAL TABLES

1767

The sum of those terms of ðp þ qÞn 

n   X n n t t p q t t¼0

pþq¼1

in which t ranges from a to b inclusive, a and b being integers ða 2 t 2 bÞ, is (if n is large enough) approximately Z

x2

fðxÞ dx þ

x1



q

 1 1 f ðxÞ þ 6s 24 s 2 p

2

where x1 ¼ ða 12 qnÞ=s; x2 ¼ ðb þ 12 The sum of the first t þ 1 terms of

 x2 6 ð3Þ f ðxÞ n x1

qnÞ=s.

n   X n n t t ðp þ qÞ  p q t t¼0 n

pþq¼1

is approximately Z

1

x

where x ¼ ðs

1 2

fðxÞ dx þ

npÞ=s; s ¼ n Z

1

fðxÞ dx

x

q

p 6s

2

f ðxÞ

 1 1 24 s 2

 6 ð3Þ f x n

t. The sum of the last s þ 1 terms is approximately q

p 6s

f2 ðxÞ

 1 1 24 s 2

 6 ð3Þ f x n

where x ¼ ðt 12 nqÞ=s. The probable error of a single observation in a series of n measures, t1 ; t2 ; . . . ; tn , the arithmetic mean of which is m, is 0:6745 e ¼ pffiffiffiffiffiffiffiffiffiffiffi n 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm t1 Þ2 þ ðm t2 Þ2 þ


þ ðm tn Þ2

the probable error of the mean is 0:6745 E ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm t1 Þ2 þ ðm t2 Þ2 þ


þ ðm tn Þ2

Approximate values of e and E are Pn i¼1 d i e ¼ 0:8453 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ where

Pn

i¼1

Pn i¼1 d i E ¼ 0:8453 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

d i is the sum of the deviations d i ¼ jti

mj.

1768

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.25 Factors for Computing Probable Errors

2 3 4

1 pffiffiffi n

0.707 107 0.577 350 0.500 000

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

0:6745 pffiffiffiffiffiffiffiffiffiffiffi n 1

0:6745 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

0:8453 pffiffiffiffiffiffiffiffiffiffiffi n n 1

0:8453 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

5 6 7 8 9

0.447 214 0.408 248 0.377 964 0.353 553 0.333 333

0.223 607 0.182 574 0.154 303 0.133 631 0.117 851

0.3372 0.3016 0.2754 0.2549 0.2385

0.1508 0.1231 0.1041 0.0901 0.0795

0.0845 0.0630 0.0493 0.0399 0.0332

0.1890 0.1543 0.1304 0.1130 0.0996

10 11 12 13 14

0.316 228 0.301 511 0.288 675 0.277 350 0.267 261

0.105 409 0.095 346 0.087 039 0.080 064 0.074 125

0.2248 0.2133 0.2034 0.1947 0.1871

0.0711 0.0643 0.0587 0.0540 0.0500

0.0282 0.0243 0.0212 0.0188 0.0167

0.0891 0.0806 0.0736 0.0677 0.0627

15 16 17 18 19

0.258 199 0.250 000 0.242 536 0.235 702 0.229 416

0.069 007 0.064 550 0.060 634 0.057 166 0.054 074

0.1803 0.1742 0.1686 0.1636 0.1590

0.0465 0.0435 0.0409 0.0386 0.0365

0.0151 0.0136 0.0124 0.0114 0.0105

0.0583 0.0546 0.0513 0.0483 0.0457

20 21 22 23 24

0.223 607 0.218 218 0.213 201 0.208 514 0.204 124

0.051 299 0.048 795 0.046 524 0.044 455 0.042 563

0.1547 0.1508 0.1472 0.1438 0.1406

0.0346 0.0329 0.0314 0.0300 0.0287

0.0097 0.0090 0.0084 0.0078 0.0073

0.0434 0.0412 0.0393 0.0376 0.0360

25 26 27 28 29

0.200 000 0.196 116 0.192 450 0.188 982 0.185 695

0.040 825 0.039 223 0.037 743 0.036 370 0.035 093

0.1377 0.1349 0.1323 0.1298 0.1275

0.0275 0.0265 0.0255 0.0245 0.0237

0.0069 0.0065 0.0061 0.0058 0.0055

0.0345 0.0332 0.0319 0.0307 0.0297

30 31 32 33 34

0.182 574 0.179 605 0.176 777 0.174 078 0.171 499

0.033 903 0.032 791 0.031 750 0.030 773 0.029 854

0.1252 0.1231 0.1211 0.1192 0.1174

0.0229 0.0221 0.0214 0.0208 0.0201

0.0052 0.0050 0.0047 0.0045 0.0043

0.0287 0.0277 0.0268 0.0260 0.0252

35 36 37 38 39

0.169 031 0.166 667 0.164 399 0.162 221 0.160 128

0.028 989 0.028 172 0.027 400 0.026 669 0.025 976

0.1157 0.1140 0.1124 0.1109 0.1094

0.0196 0.0190 0.0185 0.0180 0.0175

0.0041 0.0040 0.0038 0.0037 0.0035

0.0245 0.0238 0.0232 0.0225 0.0220

40 41 42 43 44

0.158 114 0.156 174 0.154 303 0.152 499 0.150 756

0.025 318 0.024 693 0.024 098 0.023 531 0.022 990

0.1080 0.1066 0.1053 0.1041 0.1029

0.0171 0.0167 0.0163 0.0159 0.0155

0.0034 0.0033 0.0031 0.0030 0.0029

0.0214 0.0209 0.0204 0.0199 0.0194

n

0.707 107 0.408 248 0.288 675

0.6745 0.4769 0.3894

0.4769 0.2754 0.1947

0.4227 0.1993 0.1220

0.5978 0.3451 0.2440

STATISTICAL TABLES

1769

TABLE 51.25 (Continued )

45 46 47 48 49

1 pffiffiffi n

0.149 071 0.147 442 0.145 865 0.144 338 0.142 857

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

0:6745 pffiffiffiffiffiffiffiffiffiffiffi n 1

0:6745 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

0:8453 pffiffiffiffiffiffiffiffiffiffiffi n n 1

0:8453 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

50 51 52 53 54

0.141 421 0.140 028 0.138 675 0.137 361 0.136 083

0.020 203 0.019 803 0.019 418 0.019 048 0.018 692

0.0964 0.0954 0.0945 0.0935 0.0927

0.0136 0.0134 0.0131 0.0129 0.0126

0.0024 0.0023 0.0023 0.0022 0.0022

0.0171 0.0167 0.0164 0.0161 0.0158

55 56 57 58 59

0.134 840 0.133 631 0.132 453 0.131 306 0.130 189

0.018 349 0.018 019 0.017 700 0.017 392 0.017 095

0.0918 0.0910 0.0901 0.0893 0.0886

0.0124 0.0122 0.0119 0.0117 0.0115

0.0021 0.0020 0.0020 0.0019 0.0019

0.0155 0.0152 0.0150 0.0147 0.0145

60 61 62 63 64

0.129 099 0.128 037 0.127 000 0.125 988 0.125 000

0.016 807 0.016 529 0.016 261 0.016 001 0.015 749

0.0878 0.0871 0.0864 0.0857 0.0850

0.0113 0.0112 0.0110 0.0108 0.0106

0.0018 0.0018 0.0018 0.0017 0.0017

0.0142 0.0140 0.0138 0.0135 0.0133

65 66 67 68 69

0.124 035 0.123 091 0.122 169 0.121 268 0.120 386

0.015 504 0.015 268 0.015 038 0.014 815 0.014 599

0.0843 0.0837 0.0830 0.0824 0.0818

0.0105 0.0103 0.0101 0.0100 0.0099

0.0016 0.0016 0.0016 0.0015 0.0015

0.0131 0.0129 0.0127 0.0125 0.0123

70 71 72 73 74

0.119 523 0.118 678 0.117 851 0.117 041 0.116 248

0.014 389 0.014 185 0.013 986 0.013 793 0.013 606

0.0812 0.0806 0.0801 0.0795 0.0789

0.0097 0.0096 0.0094 0.0093 0.0092

0.0015 0.0014 0.0014 0.0014 0.0013

0.0122 0.0120 0.0118 0.0117 0.0115

75 76 77 78 79

0.115 470 0.114 708 0.113 961 0.113 228 0.112 509

0.013 423 0.013 245 0.013 072 0.012 904 0.012 739

0.0784 0.0779 0.0773 0.0769 0.0764

0.0091 0.0089 0.0088 0.0087 0.0086

0.0013 0.0013 0.0013 0.0012 0.0012

0.0113 0.0112 0.0111 0.0109 0.0108

80 81 82 83 84

0.111 803 0.111 111 0.110 432 0.109 764 0.109 109

0.012 579 0.012 423 0.012 270 0.012 121 0.011 976

0.0759 0.0754 0.0749 0.0745 0.0740

0.0085 0.0084 0.0083 0.0082 0.0081

0.0012 0.0012 0.0012 0.0011 0.0011

0.0106 0.0105 0.0104 0.0103 0.0101

85 86 87 88 89

0.108 465 0.107 833 0.107 211 0.106 600 0.106 000

0.011 835 0.011 696 0.011 561 0.011 429 0.011 300

0.0736 0.0732 0.0727 0.0723 0.0719

0.0080 0.0079 0.0078 0.0077 0.0076

0.0011 0.0011 0.0011 0.0010 0.0010

0.0100 0.0099 0.0098 0.0097 0.0096

n

0.022 473 0.021 979 0.021 507 0.021 054 0.020 620

0.1017 0.1005 0.0994 0.0984 0.0974

0.0152 0.0148 0.0145 0.0142 0.0139

0.0028 0.0027 0.0027 0.0026 0.0025

0.0190 0.0186 0.0182 0.0178 0.0174

(continued)

1770

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.25 (Continued )

90 91 92 93 94

1 pffiffiffi n

0.105 409 0.104 828 0.104 257 0.103 695 0.103 142

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

0:6745 pffiffiffiffiffiffiffiffiffiffiffi n 1

0:6745 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

0:8453 pffiffiffiffiffiffiffiffiffiffiffi n n 1

0:8453 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ

95 96 97 98 99

0.102 598 0.102 062 0.101 535 0.101 015 0.100 504

0.010 582 0.010 471 0.010 363 0.010 257 0.010 152

0.0696 0.0692 0.0688 0.0685 0.0681

0.0071 0.0071 0.0070 0.0069 0.0069

0.0009 0.0009 0.0009 0.0009 0.0009

0.0089 0.0089 0.0088 0.0087 0.0086

100

0.100 000

0.010 050

0.0678

0.0068

0.0008

0.0085

n

0.011 173 0.011 050 0.010 929 0.010 811 0.010 695

0.0715 0.0711 0.0707 0.0703 0.0699

0.0075 0.0075 0.0074 0.0073 0.0072

0.0010 0.0010 0.0010 0.0010 0.0009

0.0094 0.0093 0.0092 0.0091 0.0090

TABLE 51.26 Statistics and Probability Formulas pðxÞ ¼ dPðxÞ=dx PðxÞ ¼

Rx

Differential probability (density) function of random variable x; univariate frequency function Cumulative probability function of random variable x; univariate distribution function Cumulative probability that x is between A and B Probability of simultaneous (joint) occurrence of E and F Probability of occurrence of E or F or both Conditional probability; probability of occurrence of E provided F has occurred Expected value of function of a random variable x

Pðx0 Þ dx0

1

PðA < x < BÞ PðE \ FÞ PðE [ FÞ PðEjFÞ ¼ PðE \ FÞ=PðFÞ E½f ðxފ ¼

R1

f ðxÞpðxÞ dx R11 1 xpðxÞdx

EðxÞ ¼ x ¼ ar ¼ Eðxr Þ mr ¼ Eðx

xÞ

rth moment of random variable x; rth moment about the origin

r

Var x ¼ E½ðx xÞ2 Š ¼ ðx ¼ x2 x2 s ¼ ðVar xÞ1=2 M x ðsÞ ¼ Eðesx Þ cx ðqÞ ¼ Eðejqx Þ

jqgðxÞ

cg ðqÞ ¼ E½e

Expected (mean) value of random variable x

Š

xÞ2

rth moment of random variable x from mean value; r th central moment Variance value of random variable x Standard deviation of random variable x Moment generating function associated with random variable x Characteristic function associated with random variable x Characteristic function of g(x) with random variable x

pðx; yÞ ¼ d 2 Pðx; yÞ=dx dy

Differential probability (density) function of random variables x and y; bivariate frequency function Rx Ry 0 0 0 0 Cumulative probability function of random variables x and y; Pðx; yÞ ¼ 1 1 pðx ; y Þdx dy bivariate distribution function PðA < x < B; C < y < DÞ Cumulative probability that x is between A and B and that also y is between C and D; cumulative joint probability Cov ðx; yÞ ¼ E½ðx xÞðy yފ Covariance value of random variables x and y ¼ ðx xÞðy yÞ Correlation coefficient of random variables x and y rðx; yÞ ¼ Covðx; yÞ=s x s y Source: Giacoletto, Electronic Designers’ Handbook, Copyright # 1977 by McGraw-Hill, pp. 1–8.

UNITS AND STANDARDS

1771

51.4 UNITS AND STANDARDS 51.4.1 Physical Quantities and Their Relations Mathematics is concerned with relations between numerical quantities, either constant or varying in a specified manner over a specified range of values. The numerical values are unique, absolute, and the same all over the world, being the expression of a fundamental perception of the mind. Any mathematical equation defines the values of one numerical quantity, known as the dependent, in terms of constants and one or more other numerical quantities, known as the independent variables, as, for example, 2

z ¼ r þ 3x þ 4

y¼c


Z

0

x

x2 dx cos x

ð51:1Þ

where z and y are dependent variables, r and x independent variables, and c a constant. Physics, comprising the knowledge of inanimate nature and its laws, is concerned fundamentally with the measuring of the various quantities founded or created by definition, as, for example, length, mass, and electric charge. In order to specify a physical quantity it is not sufficient to state merely a number. The value of a physical quantity can be determined only by comparison of the sample with a known amount of the same quantity by the process of measuring. The reference amount is called a unit, and the result of any measurement must be a statement of “how many times the sample was found to contain the reference amount.” Thus a physical quantity Q naturally appears to be the product of a numerical value N and a unit U, Q¼N
U

ð51:2Þ

as, for example: The length of a particular rod is 3.5 ft, or the rod is 3 12 times the length of 1 ft. Obviously, the reproduction of a unit must be possible at any time in order to facilitate correct measurements. This is being done by means of the “standards,” which are simply a set of fundamental unit quantities kept under normalized conditions in order to preserve their values as accurately as facilities permit. Any physical relation must be the result of a more or less obvious measurement, so that equations in physics are not merely numerical relations but express dependencies between physical quantities. Mathematics does not know “standards”; physics cannot be without “standards.” The fact that physics often uses the methods of mathematics must not lead to the identification of the two sciences; it is merely an overlapping in the border regions. 51.4.1.1 Relations Between Units A unit is a particular amount of the physical quantity to be measured defined in terms of a standard. The choice of a unit depends on convenience, facility of reproduction, and easy subdivision so as to obtain smaller units if desired. The value of a physical quantity Q must be independent of the units used, so that for two different units of the same type Q ¼ N1
U1 ¼ N2
U2

ð51:3Þ

The size of the unit and the numerical value of the quantity are inversely related: the larger the unit the smaller the number of units.

1772

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

A unit relation is an equation between two different units of the same type, U 1 ¼ N 12
U 2

ð51:4Þ

and serves to convert from one unit U1 to a different one U2. The conversion is achieved by replacing U1, taken as a factor, by its equivalent according to Eq. (51.4) so that Q ¼ N 1
U 1 ¼ N 1
ðN 12
U 2 Þ ¼ ðN 1
N 12 Þ
U 2

ð51:5Þ

As an example, express the length 3.5 ft in centimeters. The unit relation is 1 ft ¼ 30.5 cm, and therefore l ¼ 3.5 ft ¼ 3.5  (30.5 cm) ¼ 106.75 cm. No error is possible if this rule is followed properly. 51.4.1.2 Physical Equations Relations between physical quantities are usually given in the form of equations. It is always possible, by the proper use of unit relations (see previous paragraph), to express each side in the same units. Since units are to be considered as factors, they may be canceled and a numerical identity must result. This fact always can be used to check the proper numerical relations and the consistency of the units used. There are two fundamental types of physical equations: 1. The mathematical definition of a physical quantity determines a new quantity uniquely in terms of known quantities. An example is Newton’s definition of mass by f ¼ m
a, where f is the force and a the acceleration of a moving body. If f and a are measured, m can be computed as a physical quantity with numerical value N(f)/N(a) and unit U(f)/U(a) ¼ U(m). A definition should be in agreement with all the other known relations in a particular field of science; it can only be of restricted value if it contradicts other relations (see later the “absolute” electric systems). 2. The statement of proportionality defines one physical quantity as linearly depending on a combination of other, known quantities. It is always the result of an experimental investigation. An example is Newton’s law of the gravitational force f ¼ kðm1 m2 =r2 Þ, where m1 and m2 are the two masses, r their center distance, and k the proportionality factor. In the case of a proportionality it is permissible to choose arbitrary units for all measurable physical quantities involved and to use the equation as a definition of the proportionality constant that, in general, will be a physical constant with numerical value and unit. In the example the value of k would be Nðf Þ
Nðr2 Þ Uðf Þ
Uðr2 Þ  ¼ NðkÞ
UðkÞ Nðm1 Þ
Nðm2 Þ Uðm1 Þ
Uðm2 Þ Most of the fundamental laws of physics are statements of proportionalities, leading to universal physical constants, as, for instance, the gravitational constant k, the Planck constant h, the gas constant R, the absolute permeability of free space mv, and the absolute dielectric constant of free space eu. It may be observed that each branch of physics is represented by at least one fundamental proportionality constant. Derived physical quantities are, in general, the result of mathematical definitions. The units of derived quantities are expressed from the combinations of the units used

UNITS AND STANDARDS

1773

in the definition. All proportionality constants are ordinarily considered as derived physical quantities. 51.4.1.3 Fundamental Physical Quantities The physical quantities, arbitrarily chosen to define new quantities or derived quantities, are called fundamental physical quantities. Their number may vary according to needs and convenience. There is no possibility to designate any physical quantity as absolutely fundamental, or a priori fundamental. Quantities that appear to be fundamental in one special field may be derived quantities in some other field. 51.4.2 Dimensions and Dimension Systems 51.4.2.1 Definition of Dimension To choose a unit for a physical quantity one has an infinity of possibilities. The numerous units of length that were in use about 100 years ago present a good practical illustration. Yet all these units have in common the quality of being a distinct length and not, for example, a volume. It is convenient to state this fact by representing with the notation [L] any unit of length whatsoever. The measurement of a physical quantity Q, therefore, leads to the statement Q ¼ N
½QŠ

ð51:6Þ

where N is a numeric denoting the number of general units [Q] that constitute the total quantity Q. According to Fourier, who first introduced this concept into the literature, [Q] is called the “dimension” of the quantity Q. Be it clearly understood that dimension is simply the expression of a general unit and therefore a characteristic peculiarity of physical quantities not occurring in mathematics. Each new physical quantity gives rise to a new “dimension”, as, for instance, time [T], force [F], mass [M], and so on. There are as many dimensions, or general units, as there are kinds of physical quantities. 51.4.2.2 Derived Dimensions Many physical quantities have been introduced by mathematical definition. Velocity, for example, is defined as v ¼ ds/dt, where s is the length of the path measured from a definite origin and t is the time. A possible expression for the dimension of velocity would be [V]. It is customary and convenient, however, to make use of the mathematical definition that is but the rule for the measurement of velocity and to express the dimension in terms of the more familiar dimensions of length and time as a derived dimension [V] ¼ [L]/[T] ¼ [L][T] 1. [Read: velocity is of þ1 dimension in length and 1 dimension in time.] The use of mathematical definitions, leading to derived dimensions of a composite nature, reduces the number of symbols. Thus the measurement of volume, if scientifically conducted, gives [Vol] ¼ [L]3, or in words, “volume is of þ3 dimensions in length [L].” Proportionality constants of physics have, in general, derived dimensions, as they are defined by the corresponding physical equations. 51.4.2.3 Fundamental Dimensions The more familiar dimensions used to express derived dimensions are referred to as fundamental dimensions. It is advantageous to use as few of these fundamental dimensions as possible, not because the physical relations become simpler or clearer, but merely as a matter of economy in symbols. In fact, any dimension can be chosen to be a fundamental dimension in a particular field and a derived

1774

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

dimension in some other field of physics. No fundamental dimension can be made a starting point of natural philosophy. 51.4.2.4 Dimensional Equations Since a physical equation constitutes in fact two equations, one for the units and one for the numerics, one can disregard the numerical factors entirely and write the general units or dimensions only, arriving thus at a dimensional equation. For instance, the law of gravitation would read [F] ¼ [k][M]2[L] 2 using [F], [k], [M], and [L] as dimensions for force, gravitation constant, mass, and length, respectively. From this dimensional equation a derived dimension can be obtained for any quantity involved. Conversely, dimensional equations are used to check the correctness of physical relations if all dimensions can be made to cancel. Finally, the validity of dimensional equations leads to the method of dimensional analysis. A set of fundamental dimensions is any group of fundamental dimensions convenient and useful to express all the physical quantities of a particular field in terms of derived dimensions. The number of fundamental dimensions to make a set may vary according to the field of application. Whether or not a set of fundamental dimensions can be used beyond the field for which it was originally intended will depend upon its suitability as a dimension system. (See next paragraph.) In no case should it be used where it can lead to confusion. A set of fundamental dimensions is incomplete when the number of fundamental dimensions composing it is less than the number required for a dimension system. Incomplete sets of fundamental dimensions should not be used outside the very restricted field for which they are defined; they necessarily would lead to confusing relations. A dimension system is composed of the smallest number of fundamental dimensions that will form a consistent and complete set for a field of science. Since each relation between physical quantities can be split up into one relation of numerics and another one of dimensions (as general units), it is possible to combine all known relations of dimensions. In setting up these relations, all proportionality factors must be taken as physical quantities. If there are m independent relations known, m þ p dimensions may be involved, of which m dimensions can be expressed by any p “fundamental” dimensions chosen arbitrarily. This set of p “fundamental” dimensions is then called a dimension system. From the   p theory of numbers, therefore, it is known that one generally has a choice of m þ p

possible dimension systems. Thus, if p ¼ 3, m ¼ 3, then one has

  6 3

¼ 20 different

possibilities. A necessary condition, however, is that each independent relation involve at least p þ 1 dimensions. If this  is not the case, then the number of possible dimension mþp systems is less, so that indicates the upper limit. p Any dimension system chosen in the described manner is consistent, as well as correct, and never leads to ambiguity with respect to the expression of physical quantities. Complete dimension systems in mechanics must have three, in thermodynamics four, and in electromagnetism four fundamental dimensions. It seems, according to present knowledge, that five fundamental dimensions suffice for the entire range of physics, namely, the three fundamental dimensions of mechanics, an additional one for thermodynamics, and another additional one for electromagnetism. All the known dimension systems use length [L] and time [T] as primary fundamental dimensions, adding various fundamental dimensions from the available physical quantities of the fields of physics. The choice of [L] and [T] reduces at once the maximum  mþp 2 number of possible dimension systems to p 2 .

UNITS AND STANDARDS

1775

51.4.2.5 Why Dimension Systems? Since the proper choice of units is the ultimate goal of any critical analysis of physical quantities, the question may be asked: Why is it necessary to discuss dimension of systems? The answer is that each physical quantity may be measured by an infinite variety of units but has only one dimension within a given dimension system. The process of deciding upon the fundamental dimensions before fixing the units within the scope of the fundamental dimensions is, therefore, essentially a matter of economy and logic. 51.4.3 Dimension and Unit Systems In the past different dimension systems were introduced for various fields of technology (mechanics, heat, electromagnetism) and based on different choices of fundamental dimensions, for example, for mechanics, length and time plus mass or force or energy or gravitational constant gave potentially four different dimension system classes. In turn, for a given dimensional system, a unit system could be developed, choosing for each fundamental dimension a unit desirably related to a fundamental standard or standards. In seeking to define units with appropriate size values, relationships, and so on, many different unit systems, for example, centimeter–gram–second (cgs), meter–kilogram–second (mks), “absolute,” “technical,” and so on, have been introduced over the years. (See O. W. Eshbach and M. Souders, Handbook of Engineering Fundamentals, 3rd ed., Wiley, New York, 1975, for a detailed exposition of the subject.) In recent years a major step toward simplification and standardization has been taken by the increasing adoption of the International System of Units (SI). 51.4.4 The International System of Units The SI system, composed of six fundamental units, has been adopted by the Conference Generale (BIPM Sevres, Paris, 1954 and 1960) to cover the whole range of physics and one in which all international reports are to be expressed. Quantity Length Mass Time Intensity of electric current Thermodynamic temperature Luminous intensity Amount of substance

Unit Fundamental Units Meter Kilogram Second Ampere Degree kelvin Candela Mole

Symbol m kg s A K cd mol

Derived Units with Special Names Area Volume Frequency Density (mass density) Velocity Angular velocity

Square meter Cubic meter Hertz Kilogram per cubic meter Meter per second Radian per second

m2 m3 Hz kg/m3 m/s rad/s (continued)

1776

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Quantity

Unit

Symbol

Acceleration Angular acceleration Force Pressure, stress Kinematic viscosity Dynamic viscosity Work, energy, heat (quantity of heat) Power, radiant flux Plane angle Solid angle Electric charge Electric potential, potential difference, electromotive force Electric field strength Resistance (to direct current) Electric conductance Capacitance Magnetic flux Inductance Magnetic flux density (magnetic induction) Magnetic field strength Magnetomotive force Luminous flux Luminance Illumination Activity (of a radionuclide) Absorbed dose Dose equivalent

Meter per square second Radian per square second Newton Newton per square meter Square meter per second Newton-second per square meter Joule Watt Radian Steradian Coulomb Volt

m/s2 rad/s2 N, kg.m/s2 N/m2 m2/s N.s/m2 J, N
m W, J/s rad sr C, A
s V, W/A

Volt per meter Ohm Siemens Farad Weber Henry Tesla Ampere per meter Ampere Lumen Candela per square meter Lux Becquerel Gray Sievert

V/m V, V/A S, A/V F, A
s/V Wb, V
s H, V
s/A T, Wb/m2 A/m A lm, cd
sr cd/m2 lx, lm/m2 Bq, l/S Gy, J/kg Sv, J/kg

Other Common Derived Units Absorbed dose rate Acceleration Angular acceleration Angular velocity Area Concentration (of amount of substance) Current density Density, mass Electric charge density Electric field strength Electric flux density Energy density Entropy Exposure (X and gamma rays) Heat capacity Heat flux density Irradiance Luminance Magnetic field strength Molar energy Molar entropy

Gray per second Meter per second squared Radian per second squared Radian per second Square meter Mole per cubic meter Ampere per square meter Kilogram per cubic meter Coulomb per cubic meter Volt per meter Coulomb per square meter Joule per cubic meter Joule per kelvin Coulomb per kilogram Joule per kelvin Watt per square meter Watt per square meter Candela per square meter Ampere per meter Joule per mole Joule per mole kelvin

Gy/s m/s2 rad/s2 rad/s m2 mol/m3 A/m2 kg/m3 C/m3 V/m C/m2 J/m3 J/K C/kg J/K W/m2 W/m2 cd/m2 A/m J/mol J/(mol
K)

UNITS AND STANDARDS

1777

Quantity

Unit

Symbol

Molar heat capacity Moment of force Permeability (magnetic) Permittivity Power density Radiance Radiant intensity Specific heat capacity Specific energy Specific entropy Specific volume Surface tension Thermal conductivity Velocity Viscosity, dynamic Viscosity, kinematic Volume Wave number

Joule per mole kelvin Newton meter Henry per meter Farad per meter Watt per square meter Watt per square meter steradian Watt per steradian Joule per kilogram kelvin Joule per kilogram Joule per kilogram kelvin Cubic meter per kilogram Newton per meter Watt per meter kelvin Meter per second Pascal second Square meter per second Cubic meter 1 per meter

J/(mol
K) N
m H/m F/m W/m2 W/(m2
sr) W/sr J/(kg
K) J/kg J/(kg
K) m3/kg N/m W/(m
K) m/s Pa
s m2/s m3 1/m

Definitions of Derived Units of the International System Having Special Names Quantity 1. Absorbed dose

2. Activity 3. Celsius temperature

4. Dose equivalent

5. Electric capacitance 6. Electric conductance 7. Electric inductance

Unit and Definition The gray is the absorbed dose when the energy per unit mass imparted to matter by ionizing radiation is one joule per kilogram. Note: The gray is also used for the ionizing radiation quantities: specific energy imparted, kerma, and absorbed dose index, which have the SI unit joule per kilogram. The becquerel is the activity of a radionuclide decaying at the rate of one spontaneous nuclear transition per second. The degree Celsius is equal to the kelvin and is used in place of the kelvin for expressing Celsius temperature (symbol t) defined by the equation t ¼ T T0, where T is the thermodynamic temperature and T0 ¼ 273.15 K by definition. The sievert is the dose equivalent when the absorbed dose of ionizing radiation multiplied by the dimensionless factors Q (quality factor) and N (product of any other multiplying factors) stipulated by the International Commission on Radiological Protection is one joule per kilogram. The farad is the capacitance of a capacitor between the plates of which there appears a difference of potential of one volt when it is charged by a quantity of electricity equal to one coulomb. The siemens is the electric conductance of a conductor in which a current of one ampere is produced by an electric potential difference of one volt. The henry is the inductance of a closed circuit in which an electromotive force of one volt is produced when the electric current in the circuit varies uniformly at a rate of one ampere per second. (continued)

1778

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Quantity 8. Electric potential difference, electromotive force 9. Electric resistance

10. Energy 11. Force 12. Frequency 13. Illuminance 14. Luminous flux 15. Magnetic flux

16. Magnetic flux density 17. Power 18. Pressure or stress 19. Quantity of electricity

Unit and Definition The volt (unit of electric potential difference and electromotive force) is the difference of electric potential between two points of a conductor carrying a constant current of one ampere when the power dissipated between these points is equal to one watt. The ohm is the electric resistance between two points of a conductor when a constant difference of potential of one volt, applied between these two points, produces in this conductor a current of one ampere, this conductor not being the source of any electromotive force. The joule is the work done when the point of application of a force of one newton is displaced a distance of one meter in the direction of the force. The newton is that force that, when applied to a body having a mass of one kilogram, gives it an acceleration of one meter per second squared. The hertz is the frequency of a periodic phenomenon of which the period is one second. The lux is the illuminance produced by a luminous flux of one lumen uniformly distributed over a surface of one square meter. The lumen is the luminous flux emitted in a solid angle of one steradian by a point source having a uniform intensity of one candela. The weber is the magnetic flux that, linking a circuit of one turn, produces in it an electromotive force of one volt as it is reduced to zero at a uniform rate in one second. The tesla is the magnetic flux density given by a magnetic flux of one weber per square meter. The watt is the power that gives rise to the production of energy at the rate of one joule per second. The pascal is the pressure or stress of one newton per square meter. The coulomb is the quantity of electricity transported in one second by a current of one ampere.

Prefixes. The SI system has adopted the following standard set of prefixes: Multiplication Factor 1 000 000 000 000 000 000 ¼ 1018 1 000 000 000 000 000 ¼ 1015 1 000 000 000 000 ¼ 1012 1 000 000 000 ¼ 109 1 000 000 ¼ 106 1 000 ¼ 103 100 ¼ 102 10 ¼ 101 0.1 ¼ 10 1 0.01 ¼ 10 2 0.001 ¼ 10 3 0.000 001 ¼ 10 6 0.000 000 001 ¼ 10 9 0.000 000 000 001 ¼ 10 12 0.000 000 000 000 001 ¼ 10 15 0.000 000 000 000 000 001 ¼ 10 18

Prefix Exa Peta Tera Giga Mega Kilo Hecto14 Deka14 Deci14 Centi14 Milli Micro Nano Pico Femto Atto

Symbol E P T G M k h da d c m m n p f a

UNITS AND STANDARDS

1779

51.4.5 Application of SI Prefixes General In general the SI prefixes should be used to indicate orders of magnitude, thus eliminating nonsignificant digits and leading zeros in decimal fractions and providing a convenient alternative to the powers-of-10 notation preferred in computation. For example, 12; 300 mm becomes 12:3 m 12:3  103 m becomes 12:3 km 0:00123 mA becomes 1:23 nA Selection When expressing a quantity by a numerical value and a unit, a prefix should preferably be chosen so that the numerical value lies between 0.1 and 1000. To minimize variety, it is recommended that prefixes representing 1000 raised to an integral power be used. However, three factors may justify deviation: 1. In expressing area and volume, the prefixes hecto-, deka-, deci-, and centi- may be required, for example, square hectometer, cubic centimeter. 2. In tables of values of the same quantity or in a discussion of such values within a given context, it is generally preferable to use the same unit multiple throughout. 3. For certain quantities in particular applications, one particular multiple is customarily used. For example, the millimeter is used for linear dimensions in mechanical engineering drawings even when the values lie far outside the range 0.1–1000 mm; the centimeter is often used for body measurements and clothing sizes. Prefixes in Compound Units2 It is recommended that only one prefix be used in forming a multiple of a compound unit. Normally the prefix should be attached to a unit in the numerator. One exception to this is when the kilogram occurs in the denominator. For example, V=m; not mV=mm; and MJ=kg; not kJ=g Compound Prefixes Compound prefixes formed by the juxtaposition of two or more SI prefixes are not to be used. For example, use 1 nm; not 1 mmm 1 pF; not 1 mm F If values are required outside the range covered by the prefixes, they should be expressed by using powers of 10 applied to the base unit. Powers of Units An exponent attached to a symbol containing a prefix indicates that the multiple or sub-multiple of the unit (the unit with its prefix) is raised to the power expressed by the exponent. For example, 1 cm3 ¼ ð10 2 mÞ3 ¼ 10 6 m3 1 ns 1 ¼ ð10 9 sÞ 1 ¼ 109 s 1 1 mm2 =s ¼ ð10 3 mÞ2 =s ¼ 10 6 m2 =s 2

A compound unit is a derived unit that is expressed in terms of two or more units rather than by a single special name.

1780

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Calculations Errors in calculations can be minimized if the base and the coherent derived SI units are used and the resulting numerical values are expressed in powers-of10 notation instead of using prefixes. 51.4.6 Other Units Units from Different Systems To assist in preserving the advantage of SI as a coherent system, it is advisable to minimize the use with it of units from other systems. Such use should be limited to units listed in this section. A following section presents conversion factors to and from SI units. 51.4.7 Length, Mass, and Time (English Units and Standards) 51.4.7.1 Units of Length The foot (ft) is the fundamental unit of length in the foot– pound–second (fps) system. It equals, by definition, one-third of a yard (yd), which is the English legalized standard unit of length. The U.S. yard was defined by Act of Congress, July 28, 1866, as 3600/3937 the length of the meter. (See discussion of metric system for definitions of metric length.) In Great Britain, the Imperial yard is measured by a bronze bar preserved in the Standards Office, Westminster. Its length, in terms of the international prototype meter, is 3600/3937.0113 m. For engineering purposes, the U.S. and British yards may be considered identical. As subunits, the inch (in.) is defined as one-twelfth of one standard foot, and the mil as the one-thousandth part of one inch. The nautical mile (mi) is defined as one minute of arc on the earth’s surface at the equator, whereas the U.S. mile (U.S. mi statute) is exactly 5280 ft and practically identical with the British mile. 51.4.7.2 Unit of Capacity (Dry) The bushel (bu) is the standard unit of dry capacity. The Winchester bushel (U.S. standard) has a volume of 2150.42 in.3 In Great Britain, the Imperial bushel (bu) is defined as the volume of 80 lb of pure water at 62 F weighed against brass weights in air at the same temperature as the water and with the barometer at 30 in. Its volume is approximately 2219.36 in.3 51.4.7.3 Unit of Capacity (Liquid) The gallon (gal) is the standard unit of liquid capacity. The U.S. gallon has a volume of 231 in.3 In Great Britain, the Imperial gallon is defined as the volume of 10 lb of pure water at 62 F weighed against brass weights in air at the same temperature as the water and with the barometer at 30 in. Its volume is approximately 277.420 in.3 The Imperial gallon (liquid measure) equals exactly one-eighth of the Imperial bushel (dry measure). Subunits are the quart (qt), which is one-fourth of the standard gallon, and the pint (pt), which is 12 qt. 51.4.7.4 Units of Mass The pound (avoirdupois) (lb avdp) is the fundamental unit of mass in the fps system.3 It is also the English legalized standard unit of mass. The U.S. 3

The slug of mass, which is extensively used by engineers and physicists, is (in the English system) the mass to which an acceleration of one foot per second per second would be given by the application of a one-pound force. Under any gravity conditions, 1 slug of mass ¼ 32.1739 lb of mass.

UNITS AND STANDARDS

1781

pound (avoirdupois) was defined by Act of Congress, 1866, as 1/2.2046 kg, but since 1895 there has been used, for greater accuracy, a value that agrees with that given by law as far as the latter is given, namely, 453.5924277 g. This value is now used by the Bureau of Standards as an exact definition and is the basis of the customary U.S. weights (Circular 47, Bureau of Standards). In Great Britain, the Imperial pound (avoirdupois) is the mass of a platinum cylinder preserved in the Standards Office, Westminster. Its legal equivalent is 453.59243 g. For engineering purposes, the U.S. and British pounds (avoirdupois) may be considered as identical. 1 Subunits of mass are the grain (gr), defined as 7000 of the standard pound (avoirdupois) 1 and the ounce (avoirdupois) (oz-avdp), which is 16 of the standard pound (avoirdupois). The grain was used as a fundamental unit in the so-called foot–gram–second (fgs) system of units prior to 1873. 51.4.7.5 Weight versus Mass Unfortunately, the word “weight” is used in two different senses, namely, (1) by the layman (as well as loosely by the scientist) to designate a given mass or quantity of matter and (2) by the scientist to designate the pull in standard gravitational force units that is exerted by the earth upon a piece of matter. The result of the commercial act of “weighing” a specific quantity is independent of the local gravitational pull of the earth, since both spring scales and balances are calibrated locally by comparison with standard masses. 51.4.7.6 Auxiliary Fundamental Units Auxiliary Fundamental Units and their principal derived units are defined and discussed under the sections of this handbook pertaining to the topics to which they apply. In general, however, conversion factors are included in the tables of Section 51.5. For an interesting and rather complete history see British Weights and Measures, London, 1910, by Sir C. M. Watson. Metric (or SI) Units and Standards The development of the SI system and the operations of the international bodies (BIPM, CIPM, and CGPM) having cognizance over weights and measures are described in appendices to ASTM’s Standard for Metric Practice (ASTM E380-82, American Society for for Testing and Materials, Philadelphia, 1982). Units of Length The centimeter (cm) is the fundamental unit of length in the cgs system. 1 It equals, by definition, 100 of a meter (m). The meter has been standardized by international agreement as 1,650,763.73 times the wavelength in vacuum of the unperturbed transition (2p10–5d5) of krypton 86. The basic meter for international comparisons is the international prototype meter, which is the distance, at zero degrees Centigrade, between two lines on a platinum–iridium bar located at the International Bureau of Weights and Measures at Sevres, France. This meter is the nearest to a duplicate ever constructed of the original mefter, which was constructed and deposited in the Archives of the French Republic in 1799. The meter is very nearly equal to one ten-millionth of the distance, measured at sea level, from the equator to either pole. An interesting history of the development of the international prototype meter (as well as the international prototype kilogram—see discussion on unit of mass) is given

1782

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

by W. Parry, National Bureau of Standards, in Merriman’s Civil Engineers’ Handbook, as follows: The use of the meter as the basis of geodetic surveys had become so general throughout Europe that a conference was called in Paris, France, in 1870, for the purpose of establishing a central bureau where the standards of the different countries could be compared. As a result of this conference an International Bureau of Weights and Measures was established near Paris in 1875, by the concurrent action of the principal nations of the world. One of the first tasks undertaken by the Bureau was the construction of exact copies of the meter and kilogram deposited in the Archives. Thirty-one standard meters of iridio-platinum and forty kilograms of the same alloy were constructed and carefully compared with the standards of the Archives and with one another. This great work was completed in 1889, and the meter and kilogram which agreed most nearly with the original standards were called international prototypes, and were deposited at the International Bureau, where they are maintained today subject to the authority of the International Committee on Weights and Measures. The remaining meters and kilograms were distributed by lot to the different nations which contributed to the support of the Bureau. The United States secured two copies of the meter and two copies of the kilogram, which are in the custody of the Bureau of Standards at Washington. One of the meters, known as No. 27, and one kilogram, No. 20, were selected as the United States standards, while the other meter and kilogram are used as secondary standards. It was the declared intention of the International Committee that the various national prototypes should be returned to the International Bureau at regular intervals for the purpose of recomparing them with the international standards and with one another. In this way all measurements based upon metric standards throughout the world are ultimately referred to the international meter and kilogram.

Unit of Capacity The liter (L) is the standard unit of capacity. It is defined as the volume of one kilogram of pure water at the temperature of maximum density (4 C) under a pressure of 76 cm of mercury. For all practical purposes, the liter may be regarded as the equivalent of the cubic decimeter, although the former is actually slightly greater, in the amount of less than three parts in one hundred thousand. Unit of Mass The gram (g) is the fundamental unit of mass in the cgs system.4 It equals, 1 by definition, 1000 of a kilogram (kg), which is the standard unit of mass. The basic kilogram for international comparisons is the international prototype kilogram, which is a cylinder of platinum–iridium located at the International Bureau of Weights and Measures at Sevres, France. This mass is the nearest to a duplicate ever constructed of the original kilogram, which was constructed and deposited in the Archives of the French Republic in 1799. The latter was made as nearly as possible equal to the mass of a cube of pure water at 4 C, the sides of the cube being one-tenth the length of the original meter. An interesting history of the development of the international prototype kilogram was given under the discussion on units of length. Weight versus Mass See discussion under this same subheading of the English units and standards. 4

The slug of mass, which is extensively used by engineers and physicists, is (in the metric system) the mass to which an acceleration of one meter per second per second would be given by the application of a one-kilogram force. Under any gravity conditions, 1 slug of mass ¼ 9.80665 kg of mass.

UNITS AND STANDARDS

1783

Auxiliary Fundamental Units Auxiliary fundamental units and their principal derived units are defined and discussed under the sections of this handbook pertaining to the topics to which they apply. In general, however, conversion factors are included in Tables 51.27–51.64. 51.4.8 Standard of Time Unit of Time The second has been standardized by international agreement as 1/31,556,925.9747 of the tropical year at 12 hr, ephemeris time, January 0 for the year 1900.0. (This definition has been retained for the time being as an astronomical time standard—the following atomic standard of time interval is 100 times more precise.) The second has been standardized by international agreement as the time taken for 9,192,631,770.0 vibrations of the unperturbed hyperfine transition 4,0–3,0 for the 2 S1=2 fundamental state of the cesium 133 atom. The 133 Cs standard has been adopted provisionally (see resolution 5 of the 12th General Conference of Weights and Measures, BIPM, Sevres, Paris, Oct. 1964). A more accurate hydrogen maser standard may be available in the near future that is 100 times more accurate than the 133 Cs standard. Measures of Time A solar day is measured by the rotation of the earth about its axis with respect to the sun. In astronomical computations and in nautical time the day commences at noon, and in the former it is counted throughout the 24 hr. In civil computations the day commences at midnight and is divided into two parts of 12 hr each. A solar year is the time in which the earth makes one revolution around the sun. Its average time, called the mean solar year, is 365 days, 5 hr, 48 min, and 45.9747 sec, or nearly 365 14 days. 51.4.9 Force, Energy, and Power 51.4.9.1 Dynamical and Gravitational Units According to the use of two different dimension and unit systems, the dynamical (or physical, or “absolute”) system and the gravitational (or technical) system, two different sets of units of force, energy, power, and derived quantities are defined in both the English and the metric systems. One dynamical unit of force produces an acceleration of unity on unit standard mass. The gravitational unit of force is defined as that force required to give a unit standard mass an acceleration equal to that produced by the gravitational pull of the earth. As the acceleration due to gravity, g, varies with location and altitude,5 the gravitational unit of force is not constant, and, therefore, its relation to the dynamical unit of force will vary. By international agreement, the value g0 ¼ 980.665 cm/ sec sec ¼ 32.1739 ft/ sec sec (British) has been chosen as the standard acceleration of gravity to make invariant the gravitational unit of force. English Units Units of Force The dynamical or physical unit of force is the poundal, defined as the force required to give a mass of one pound an acceleration of one foot per second per second. 5

The variation of g with latitude f and altitude H is given approximately by (f in degrees, H in meters) g ¼ 978.039 (1 þ 0.005295 sin2 f) 0.000307H. See International Critical Tables, Vol. 1, p. 395.

1784

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

The pound-force (or weight of the pound mass) is the gravitational or technical unit of force. It is, by definition, the force required to give a mass of one pound an acceleration of 32.1739 ft/sec sec. If a force is measured by “weighing,” the result in pounds weight must be multiplied by g/g0, the ratio of local to standard acceleration of gravity, in order to obtain the absolute value in pound-force units. For engineering purposes this correction can usually be neglected. Unit of Pressure This is defined as the unit of force acting upon a unit area. The most commonly employed unit is the pound (force) per square inch. Standard atmospheric pressure is defined to be the force exerted by a column of mercury 760 mm (29.92 in.) high at 0 . This corresponds to 0.101325 MPa or 14.695 psi. Reference or fixed points for pressure calibration exist and are analogous to the phase changes used for temperature standards. These pressure references are based on phase changes or resistance jumps in selected materials. Units of Work or Energy The foot-poundal is the physical unit of work or energy and is defined as the work done by a force of one poundal in moving a body through the distance of one foot in the direction of the force. The foot-pound (force) is the technical unit of work or energy and is defined as the work required to raise a mass (or weight) of one pound through a vertical distance of one foot at standard acceleration of gravity g0. If measurements are made in places where the local value of the acceleration of gravity g is different from g0, a correction factor g/g0 must be applied if the exact value of work or energy is desired. The British thermal unit (Btu) is the quantity of heat required to raise the temperature of a one-pound mass of water either at 39 F (at its maximum density) or at 60 F and 1 standard pressure through 1 F. The mean British thermal unit is defined as the 180 part of the heat required to raise the temperature of a one-pound mass of water from 32 to 212 F at standard pressure. It is obvious that the reference temperature must be indicated with the unit used. Units of Power Power is the time rate at which work is done. Its physical unit is the footpoundal per second, its technical units are the foot-pound (force) per second, or the British thermal unit per second. The horsepower (hp or Hp) is defined as 33,000 ft-lb (force) per minute or 550 ft-lb (force) per second. Units of Torque Torque is the effectiveness of a force to produce rotation. It is defined as the product of the force and the perpendicular distance from its line of action to the instantaneous center of rotation. Its physical unit is the poundal-foot, and its technical unit the pound (force)-foot. (Note the reversal of force and length units in the designation of the units of torque as compared with the units of energy or work.) Metric Units Units of Force The dynamical, or physical, unit of force is the dyne, defined as the force required to give a mass of one gram an acceleration of one centimeter per second per second. The newton is the SI unit of force. It is the force required to give a mass of one kilogram an acceleration of one meter per second per second.

UNITS AND STANDARDS

1785

The kilogram force (or weight of the kilogram mass) is the gravitational or technical unit of force. It is, by definition, the force required to give a mass of one kilogram an acceleration of 980.665 cm/sec sec. If a force is measured by “weighing,” the result in kilograms weight must be multiplied by g/g0, the ratio of local to standard acceleration of gravity, in order to obtain the absolute value in kilogram-force units. For engineering purposes this correction can usually be neglected. In the electrotechnical system of units the systematic unit of force is defined as the joule per meter, based on the fundamental definition of the joule. (See discussion on metric units of energy.) Unit of Pressure This is defined as the unit of force acting upon a unit of area. The newton per square meter is the SI unit of pressure and is called the pascal. The kilogram force per square meter is the technical unit of pressure. With respect to correction for local gravity, see discussion on force versus weight. Pressure is measured also by the height in centimeters of the column of water at 4 C, or of the column of mercury at 0 C, which it supports. (See conversion Table 51.41.) The normal atmosphere (at), or the standard atmospheric pressure, is defined as the pressure exerted by a column of 76 cm of mercury at sea level and 0 C at standard acceleration of gravity g0. It is equal to 1.01321 bars or 1.0332 kg/cm2 force and is used extensively in the engineering literature. Some confusion exists since the unit of 1 kg/cm2 is occasionally called 1 practical atmosphere. Units of Work or Energy The joule is the physical or so-called absolute unit of work or energy. It is defined as the work done by a force of one newton acting through the distance of one meter. A larger unit is the theoretical or “absolute” joule defined as 107 ergs; it is a systematic unit in the practical electrical unit systems that is based on the theoretical unit systems. (See discussion on electrical units.) The international joule is defined as the energy expended during one second by an electric current of one international ampere flowing through a resistance of one international ohm. (See discussion on electrical units.) The latest value of the international joule is equal to 1.000165 theoretical joules.6 The kilowatt-hour is the practical unit of energy in electrical metering. It is defined as a theoretical or an international unit (see definition of joule already given) and is equal to 3.6 megajoules. The meter-kilogram force (commonly referred to as the kilogram-meter) is the technical unit of work or energy. It is defined as the work required to raise the mass (or weight) of one kilogram through a vertical distance of one meter at standard acceleration of gravity g0. If measurements are made in places where the local value of the acceleration of gravity g is different from g0, a correction factor g/g0 has to be applied, if the exact value of work or energy is desired. (See discussion on force versus weight.) The gram-calorie or small calorie is the physical unit of heat energy. It is defined as the quantity of heat required to raise the temperature of one gram mass of water either from 14.5 to 15.5 C or from 19.5 to 20.5 C at standard pressure. The two values are designated 1 as 15 and 20 C cal, respectively. The mean gram-calorie is defined as 100 part of the quantity of heat required to raise the temperature of one gram mass of water from 0 to 100 C at

6

Mechanical Engineering, Feb., 1930, pp. 122, 139.

1786

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

standard pressure. The same definitions apply to kilogram-calorie, or large calorie, if the kilogram mass is used as reference standard mass. The Ostwald calorie is the quantity of heat required to raise the temperature of one gram mass from 0 to 100 C. This unit is frequently used by electrochemists and is equal to 100 mean gram-calories. The international kilocalorie or international steamtable calorie (IT cal) is defined as 1 the 860 part of the international kilowatt-hour. This new unit avoids any reference to the thermal properties of water and was recommended for international adoption at the first International Steam Table Conference (1929).7 Its value is very nearly equal to the mean kilocalorie, 1 IT cal ¼ 1.00037 kilogram-calories (mean). Units of Power Power is the time rate at which work is done. Its physical unit is the watt, defined as the power which gives rise to the production of energy at the rate of one joule per second. The international watt is defined as the power expended by an electric current of one international ampere flowing through a resistance of one international ohm. (See discussion on electrical units.) The latest value of the international watt is equal to 1.000165 theoretical watts.8 The electrical horsepower is defined as 746 absolute watts and is commonly used in the United States and in England in rating electrical machinery. The meter-kilogram force per second (commonly referred to as the kilogram-meter per second) is the technical unit of power. The metric horsepower is defined as 75 kg-m/sec and is the most common mechanical unit of power. Units of Torque Torque is the effectiveness of a force to produce rotation. It is defined as the product of the force and the perpendicular distance from its line of action to the instantaneous center of rotation. Its physical unit is the dyne-centimeter, and its technical unit the kilogram force meter. (Note the reversal of force and length units in the designation of the units of torque as compared with the units of energy and work.) 51.4.10 Thermal Units and Standards Temperature 51.4.10.1 Definition of Temperature The temperature of a body may be defined as its thermal state considered from the standpoint of its ability to communicate heat to other bodies. When two bodies are placed in thermal communication, the one that loses heat to the other is said to be at the higher temperature. 51.4.10.2 Standard Temperatures Certain thermal states or “temperatures” may be reproduced and recognized by the fact that definite physical phenomena occur at these temperatures. Such thermal states are called “fixed points,” and they may, quite apart from any temperature scale, be specified by the physical phenomena characteristic of those temperatures. The two fundamental fixed points are the ice point and the steam point.

7

Mechanical Engineering, Nov., 1935, p. 710. Announcement of Changes in Electrical and Photometric Units, Circular of National Bureau of Standards C459, Washington, DC, 1947. 8

UNITS AND STANDARDS

1787

The ice point is defined as the temperature of melting ice, which is realized experimentally as the temperature at which pure finely divided ice is in equilibrium with pure, air-saturated water under standard atmospheric pressure. The effect of increased pressure is to lower the freezing point to the extent of 0.007 C per atmosphere. The steam point is defined as the temperature of condensing water vapor at standard atmospheric pressure, and it is realized experimentally by the use of a hypsometer so constructed as to avoid superheat of the vapor around the thermometer or contamination with air or other impurities. If the desired conditions have been attained, the observed temperature should be independent of the rate of heat supply to the boiler, except as this may affect the pressure within the hypsometer, and of the length of time the hypsometer has been in operation. 51.4.10.3 Definition of Temperature Scale The purpose of establishing a temperature scale is to assign a number to every thermal state or temperature and to provide a means for determining the temperature of any particular body. A temperature scale may be defined by (1) selecting definite numbers for certain fixed points, (2) selecting some physical property of a definite substance that varies with temperature, and (3) selecting a mathematical law expressing temperatures on the scale in question in terms of the selected property of the thermometric substance. For example, on the Centigrade mercury-in-glass scale, the ice and steam points are numbered 0 and 100, respectively, the relative or “apparent” expansion of a volume of mercury enclosed in glass of a definite kind is the property used, and the mathematical relation used to express temperature on this scale is that equal increments of apparent volume of the mercury in this glass correspond to equal increments of temperature. If some other substance is substituted for mercury, or if glass of a different kind is used, another scale is obtained that agrees with it at 0 and 100 but not at other temperatures. Although, in general, a temperature scale depends on the thermometric substance as well as on the expression for the temperature in terms of some property of this substance, Lord Kelvin has shown that, if the property selected is the availability of energy, the scale so defined is wholly independent of the substance and depends only on the mathematical relation chosen. Any scale so defined is known as a thermodynamic scale. 51.4.10.4 Kelvin Temperature Scale The temperature scale finally chosen by Lord Kelvin is the one on which the temperature interval from the ice point to the steam point is 100 and the ratio of the values of any two temperatures is equal to the ratio of the heat taken in to the heat rejected by a reversible thermodynamic engine working with a source and refrigerator at the higher and lower temperatures, respectively. On this scale, which is also known as the absolute thermodynamic scale, the lowest attainable temperature is 0 and the ice point is found experimentally to be 273.16 . The steam point therefore is 373.16 or 100 higher. The degree Kelvin ( K) or degree of absolute temperature is the absolute unit of temperature and is, for practical purposes, identical with the degree Centigrade ( C) of the international temperature scale. 51.4.10.5 Thermodynamic Centigrade Scale This is derived by subtracting from the Kelvin scale a constant number of the proper magnitude to make the ice point 0 . On this

1788

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

scale, therefore, the ice and steam points are 0 and 100 , respectively, and the so-called absolute zero is 273.16 . 51.4.10.6 International Centigrade Scale This is a practical representation of the thermodynamic Centigrade scale to such a degree of accuracy as is possible with present-day apparatus and methods. It was adopted at the General Conference on Weights and Measures at Sevres, France, in 1927 and is subject to revision and amendment as improved and more accurate methods of measurement are evolved. The unit of temperature on the international scale is the degree Centigrade ( , or C int) 1 and is very nearly equal to 100 the difference between the temperature of melting ice and the temperature of condensing water vapor under standard atmospheric pressure. (See discussion on metric units for pressure.) The standard of the international temperature scale between 190 and þ660 C is deduced from the electrical resistance of a standard platinum resistance thermometer by means of a formula connecting the resistance Rt at any temperature t C within the above range with the resistance R0 at 0 C. The purity of the platinum of which the thermometer is made should be such that the ratio Rt/R0 for certain fixed temperatures is within specified limits. See also U.S. Bureau of Standards, Journal of Research, Vol. 1, p. 636, 1928. The degree Centigrade is most widely used in scientific publications and increasingly also in the engineering literature. In many countries in Europe it is the common everyday temperature unit. The subdivision into a hundred degrees of the temperature interval between the ice point and the steam point was first used by Celsius, a German, in 1742; therefore, in the European literature “ C” is read “degree Celsius.” 51.4.10.7 Fahrenheit Temperature Scale This scale subdivides the temperature interval between the ice point and the steam point into 180 parts, one part of which is chosen as the unit of temperature and named degree Fahrenheit ( F). The ice point is assigned the value 32 F, so that the steam point has a temperature of 212 F. The Fahrenheit unit of temperature is in common everyday use in the English-speaking countries. It was first introduced in England about 1665 by the physicist Fahrenheit; the choice of 32 F for the ice point has its explanation in the fact that Fahrenheit chose as zero the lowest temperature attainable by means of a salt–ice mixture. 51.4.10.8 Rankine Absolute Temperature Scale ( R) This is the thermodynamic Fahrenheit scale where absolute zero is 0 R ( 459.69 F). The ice point is assigned the value 491.69 R and the steam point 671.69 R. 51.4.10.9 Relations Between Temperature Scales The following table shows the interrelations between the various temperature scales in the form of equations. Temperature Interrelationships x F ¼ x K ¼ 5/9(t F 32) þ 273.16 x C ¼ 5/9(t F 32) x R ¼ (t F) þ 459.699

9/5(t K

273.16) þ 32

(t K) 273.16 9/5(t K)

9/5(t C) þ 32 (t ) þ 273.16 9/5(t C) þ 491.69

UNITS AND STANDARDS

1789

Here, X indicates the unknown number of chosen temperature units and t the known number of given temperature units: 51.4.11 Quantity of Heat and Some Derived Quantities Units of Quantity of Heat Quantity of heat is defined as the energy transferred from one body to another by a thermal process, that is, by radiation or conduction. The units for the quantity of heat are the British thermal unit and the calorie as specific thermal units and the erg and joule as general physical units (see discussion on units of energy, metric and English system of units). Thermal Capacity or Specific Heat of a Substance This is the quantity of heat required to produce a unit change in temperature in a unit of mass of the substance. The common English unit is the British thermal unit per degree Fahrenheit per pound mass (Btu per  F per lb); the usual metric unit is the gram-calorie per degree Centigrade per gram mass (cal per  C per g); and the general physical unit used in the scientific literature is the erg per degree Centigrade per gram mass (erg per  C per g). In the technical literature thermal capacity of a substance is often expressed in watt-seconds (or joules) per degree Centigrade per kilogram mass (W-sec per  C per kg) on account of the easy comparison with other technical units. Calorimetric or Water Equivalent This is the quantity of heat required to produce a unit change in temperature of a body or system. It is numerically equivalent to the mass of water (in units as involved in the definition of the unit of quantity of heat used) that could be raised a unit temperature by the same total quantity of heat. The thermal capacity is expressed in British thermal units per degree Fahrenheit (Btu per  F), calories per degree Centigrade (cal per  C), or watt-seconds per degree Centigrade (W-sec per  C). Thermal Conductivity This is the time rate of heat transfer through a unit area across a unit thickness per unit difference in temperature between the end surfaces. It is measured in British thermal units per second per degree Fahrenheit per inch thickness per square inch cross section (Btu per sec per  F per in. per in.2), in calories per second per degree Centigrade per centimeter thickness per square centimeter cross section (cal per sec per  C per cm per cm2), or in watts per degree Centigrade per meter thickness per square meter cross section (W per  C per m per m2). Thermal Transmittance The surface coefficient of transfer is the time rate of heat emitted by a unit area for a unit difference in temperature between the surface in question and the surroundings. It is measured in British thermal units per second per degree Fahrenheit per square inch (Btu per sec per  F per in.2), in calories per second per degree Centigrade per square centimeter (cal per sec per  C per cm2), or in watts per degree Centigrade per square meter (Wa per  C per m2). Joule Equivalent The Joule equivalent is defined as the number of foot-pounds of energy per Btu. The numerical values for the various energy units used in the English and metric systems are shown in the table below.

1790

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

1 British thermal unit (Btu) (mean) ¼ 1 gram-calorie (cal) (mean) ¼ 1 international kilocalorie (IT cal) ¼ 1 Ostwald calorie ¼

Joules “Absolute”

Footpounds (force)

Footpoundals

Meterkilogram (force)

Kilowatthour “international”

1055.18 4.1873

778.26 3.0884

25.040 99.366

107.599 0.42699

2.93019  10 1.16279  10

4

4187.3 418.73

3088.4 308.84

99.366 9936.6

426.99 42.699

1.16279  10 1.16279  10

3

6

4

51.4.12 Chemical Units and Standards 51.4.12.1 Atomic Weight The present definition of atomic weights (1961) is based on 12 C, which is the most abundant isotope of carbon and whose atomic weight is defined as exactly 12. 51.4.12.2 Standard Cell Potential A very large class of chemical reactions are characterized by the transfer of protons or electrons. Substances losing electrons in a reaction are said to be oxidized, those gaining electrons are said to be reduced. Many such reactions can be carried out in a galvanic cell that forms a natural basis for the concept of the half-cell, that is, the overall cell is conceptually the sum of two half-cells, one corresponding to each electrode. The half-cell potential measures the tendency of one reaction, for example, oxidation, to proceed at its electrode; the other half-cell of the pair measures the corresponding tendency for reduction to proceed at the other electrode. Measurable cell potentials are the sum of the two half-cell potentials. Standard cell potentials refer to the tendency of reactants in their standard state to form products in their standard states. The standard conditions are 1 M concentration for solutions, 101.325 kPa (1 atm) for gases, and for solids, their most stable form at 25 C. Since half-cell potentials cannot be measured directly, numerical values are obtained by assigning the hydrogen gas–hydrogen ion half reaction the half-cell potential of 0 V. Thus, by a series of comparisons referred directly or indirectly to the standard hydrogen electrode, values for the strength of a number of oxidants or reductants can be obtained, and standard reduction potentials can be calculated from established values. Standard cell potentials are meaningful only when they are calibrated against an electromotive force (emf) scale. To achieve an absolute value of emf, electrical quantities must be referred to the basic metric system of mechanical units. If the current unit ampere and the resistance unit ohm can be defined, then the volt may be defined by Ohm’s law as the voltage drop across a resistor of one standard ohm (V) when passing one standard ampere (A) of current. In the ohm measurement, a resistance is compared to the reactance of an inductor or capacitor at a known frequency. This reactance is calculated from the measured dimensions and can be expressed in terms of the meter and second. The ampere determination measures the force between two interacting coils while they carry the test current. The force between the coils is opposed by the force of gravity acting on a known mass; hence, the ampere can be defined in terms of the meter, kilogram, and second. Such a means of establishing a reference voltage is inconvenient for frequent use and reference is made to a previously calibrated standard cell. Ideally, a standard cell is constructed simply and is characterized by a high constancy of emf, a low temperature coefficient of emf, and an emf close to 1 v. The Weston cell,

UNITS AND STANDARDS

1791

which uses a standard cadmium sulfate electrolyte and electrodes of cadmium amalgam and a paste of mercury and mercurous sulfate, essentially meets these conditions. The voltage of the cell is 1.0183 V at 20 C. The alternating current (ac) Josephson effect, which relates the frequency of a superconducting oscillator to the potential difference between two superconducting components, is used by the National Bureau of Standards to maintain the unit of emf, but the definition of the volt remains the V/A derivation described. 51.4.12.3 Concentration The basic unit of concentration in chemistry is the mole, which is the amount of substance that contains as many entities, for example, atoms, molecules, ions, electrons, protons, and so on, as there are atoms in 12 g of 12 C, that is, Avogadro’s number NA ¼ 6.022045  1023. Solution concentrations are expressed on either a weight or volume basis. Molality is the concentration of a solution in terms of the number of moles of solute per kilogram of solvent. Molarity is the concentration of a solution in terms of the number of moles of solute per liter of solution. A particular concentration measure of acidity of aqueous solutions is pH, which, usually, is regarded as the common logarithm of the reciprocal of the hydrogen ion concentration (qv). More precisely, the potential difference of the hydrogen electrode in normal acid and in normal alkali solution ( 0.828 V at 25 C) is divided into 14 equal parts or pH units; each pH unit is 0.0591 V. Operationally, pH is defined by pH ¼ pH (soln) þ E/K, where E is the emf of the cell: H2 jsolution of unknown pHjjsaturated KCljj solution of known pHjH2 and K ¼ 2.303 RT/F, where R is the gas constant, 8.314 J/(mol/K) [1.987 cal/(mol
K)], T is the absolute temperature, and F is the value of the Faraday, 9.64845  104C/mol. pH usually is equated to the negative logarithm of the hydrogen ion activity, although there are differences between these two quantities outside the pH range 4.0–9.2: log qH þ mH þ ¼



pH þ 0:014 ðpH pH þ 0:009 ð4:0

9:2Þ pHÞ

for pH > 9:2 for pH < 4:0

51.4.13 Theoretical, or Absolute, Electrical Units With the general adoption of SI as the form of metric system that is preferred for all applications, further use of cgs units of electricity and magnetism is deprecated. Nonetheless, for historical reasons as well as for comprehensiveness, a brief review is included in this section and section 4.10. The definitions of the theoretical, or “absolute,” units are based on a particular choice of the numerical value of either ke, the constant in Coulomb’s, electrostatic force law, or km, the constant in Ampere’s electrodynamic force law. The designation absolute units is generally used because of historical tradition; an interesting account of the history can be found in Glazebrook’s Handbook for Applied Physics, Vol. II, “Electricity,” pp. 211 ff., 1922. Because of the theoretical background of the unit definitions, they have also been designated as “theoretical” units, which is in good contradistinction to practical units based on physical standards.

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MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Theoretical Electrostatic Units The theoretical electrostatic units are based on the cgs system of mechanical units and the choice of the numerical value unity for kev in Coulomb’s law. They are frequently referred to as the cgs electrostatic units, but no specific unit names are available. In order to avoid the cumbersome writing, for example, one “theoretical electrostatic unit of charge,” it had been proposed to use the theoretical “practical” unit names and prefix them with either stat or E.S. as, for example, statcoulomb, or E.S. coulomb. The first alternative will be used here. The absolute dielectric constant (permittivity) of free space is the reciprocal of the Coulomb constant kev and is chosen as the fourth fundamental quantity in the theoretical electrostatic system of units. Its numerical value is defined as unity, and it is identical with one statfarad per centimeter if use is made of prefixing the corresponding unit of the “practical” series. The theoretical electrostatic unit of charge, or the statcoulomb, is defined as the quantity of electricity that, when concentrated at a point and placed at one centimeter distance from an equal quantity of electricity similarly concentrated, will experience a mechanical force of one dyne in free space. An alternative definition, based on the concept of field lines, gives the theoretical electrostatic unit of charge as a positive charge from which in free space exactly 4p displacement lines emerge. The theoretical electrostatic unit of displacement flux (dielectric flux) is the “line of displacement flux,” or 14 p of the theoretical electrostatic unit of charge. This definition provides the basis for graphical field mapping insofar as it gives a definite rule for the selection of displacement lines to represent the distribution of the field quantitatively. The theoretical electrostatic unit of displacement, or dielectric flux density, is chosen as one displacement line per square centimeter area perpendicular to the direction of the displacement lines. It can be given also as 14 p statcoulomb per square centimeter (according to Gauss’s law). In isotropic media the displacement has the same direction as the potential gradient, and the surfaces perpendicular to the field lines become the equipotential surfaces; the theoretical electrostatic unit of displacement can then be defined as one displacement line per square centimeter of equipotential surface. The theoretical electrostatic unit of electrostatic potential, or the statvolt, is defined as existing at a point in an electrostatic field, if the work done to bring the theoretical electrostatic unit of charge, or the statcoulomb, from infinity to this point equals one erg. This customary definition implies, however, that the potential vanishes at infinite distances and has, therefore, only restricted validity. As it is fundamentally impossible to give absolute values of potential, the use of potential difference and its unit (see below) should be preferred. The theoretical electrostatic unit of electrical potential difference or voltage, is the statvolt and is defined as existing between two points in space if the work done to bring the theoretical electrostatic unit of charge, or the statcoulomb, from one of these points to the other equals one erg. Potential difference is counted positive in the direction in which a negative quantity of electricity would be moved by the electrostatic field. The theoretical electrostatic unit of capacitance, or the statfarad, is defined as the capacitance that maintains an electrical potential difference of one statvolt between two conductors charged with equal and opposite electrical charges of one statcoulomb. In the older literature, the cgs electrostatic unit of capacitance is identified with the “centimeter”; this was replaced by statfarad to avoid confusion. The theoretical electrostatic unit of electric potential gradient, or field strength (field intensity), is defined to exist at a point in an electric field if the mechanical force exerted

UNITS AND STANDARDS

1793

upon the theoretical electrostatic unit of charge concentrated at this point is equal to one dyne. It is expressed as one statvolt per centimeter. The theoretical electrostatic unit of current, or the statampere, is defined as the time rate of transfer of the theoretical electrostatic unit of charge and is identical with the statcoulomb per second. The theoretical electrostatic unit of electrical resistance, or the statohm, is defined as the resistance of a conductor in which a current of one statampere is produced if a potential difference of one statvolt is applied at its ends. The theoretical electrostatic unit of electromotive force (emf) is defined as equivalent to the theoretical electrostatic unit of potential difference if it produces a current of one statampere in a conductor of one statohm resistance. It is identical with the statvolt but, according to its concept, requires an independent definition. The theoretical electrostatic unit of magnetic intensity is defined as the magnetic intensity at the center of a circle of 4p centimeters diameter in which a current of one statampere is flowing. This unit is equal to 4p statamperes per centimeter but has no name as the factor 4p excludes the possibility of using the prefixed “practical” unit name. The theoretical electrostatic unit of magnetic flux, or the statweben, is defined as the magnetic flux whose time rate of change through a linear conductor loop (linear conductor is used to designate a conductor of infinitely small cross section) produces in this loop an emf of one statvolt. The theoretical electrostatic unit of magnetic flux density, or induction, is defined as the electrostatic unit of magnetic flux per square centimeter area, or the statweber per square centimeter. The absolute magnetic permeability of free space is defined as the ratio of magnetic induction to the magnetic intensity. Its unit is the stathenry per centimeter as a derived unit. The theoretical electrostatic unit of inductance, or the stathenry, is defined as connected with a conductor loop carrying a steady current of one statampere that produces a magnetic flux of one statweber. A more general definition, applicable to varying fields with nonlinear relation between magnetic flux and current, gives the stathenry as connected with a conductor loop in which a time rate of change in the current of one statcoulomb produces a time rate of change in the magnetic flux of one statweber per second. Theoretical Electromagnetic Units The theoretical electromagnetic units are based on the cgs system of mechanical units and Coulomb’s law of mechanical force action between two isolated magnetic quantities m1 and m2 (approximately true for very long bar magnets) that must be written as Fm ¼

km m1 m2 2 r2

ð51:7Þ

where km is the proportionality constant of Ampere’s law for force action between parallel currents that is more basic, and amenable to much more accurate measurement, than (51.7). The factor 12 appears here because of the three-dimensional character of the field distribution around point magnets as compared with the two-dimensional field of two parallel currents. The theoretical electromagnetic units are obtained by defining the numerical value of kmv/2 (for vacuum) as unity; they are frequently referred to as the cgs electromagnetic units. Only a few specific unit names are available. In order to avoid cumbersome writing,

1794

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

for example, one “theoretical electromagnetic unit of charge,” it had been proposed to use the theoretical “practical” unit names and prefix them with either ab-or E.M. as, for example, abcoulomb, or E.M. coulomb. The first alternative will be used here. The absolute magnetic permeability of free space is the value kmv/2 in (51.7) and is chosen as the fourth fundamental quantity in the theoretical electromagnetic system of units. Its numerical value is assumed as unity, and it is identical with one abhenry per centimeter if use is made of prefixing the corresponding unit of the “practical” series. The theoretical electromagnetic unit of magnetic quantity is defined as the magnetic quantity that, when concentrated at a point and placed at one centimeter distance from an equal magnetic quantity similarly concentrated, will experience a mechanical force of one dyne in free space. An alternative definition, based on the concept of magnetic intensity lines, gives the theoretical electromagnetic unit of magnetic quantity as a positive magnetic quantity from which, in free space, exactly 4p magnetic intensity lines emerge. The theoretical electromagnetic unit of magnetic moment is defined as the magnetic moment possessed by a magnet formed by two theoretical electromagnetic units of magnetic quantity of opposite sign, concentrated at two points one centimeter apart. As a vector, its positive direction is defined from the negative to the positive magnetic quantity along the center line. The theoretical electromagnetic unit of magnetic induction (magnetic flux density), or the gauss, is defined to exist at a point in a magnetic field, if the mechanical torque exerted upon a magnet with theoretical electromagnetic unit of magnetic moment and directed perpendicular to the magnetic field is equal to one dyne-centimeter. The lines to which the vector of magnetic induction is tangent at every point are called induction lines or magnetic flux lines; on the basis of this flux concept, magnetic induction is identical with magnetic flux density. The theoretical electromagnetic unit of magnetic flux, or the maxwell, is the “field line” or line of magnetic induction. In free space, the theoretical electromagnetic unit of magnetic quantity issues 4p induction lines; the unit of magnetic flux, or the maxwell, is then 1/4p of the theoretical electromagnetic unit of magnetic quantity times the absolute permeability of free space. The theoretical electromagnetic unit of magnetic intensity (magnetizing force), or the oersted, is defined to exist at a point in a magnetic field in free space where one measures a magnetic induction of one gauss. The theoretical electromagnetic unit of current, or the abampere, is defined as the current that flows in a circle of one centimeter diameter and produces at the center of this circle a magnetic intensity of one oersted. The theoretical electromagnetic unit of inductance, or the abhenry, is defined as connected with a conductor loop in which a time rate of change of one maxwell per second in the magnetic flux produces a time rate of change in the current of one abampere per second. In the older literature, the cgs electromagnetic unit of inductance is identified with the “centimeter”; this should be replaced by a henry to avoid confusion. The theoretical electromagnetic unit of magnetomotive force (mmf) is defined as the magnetic driving force produced by a conductor loop carrying a steady current of 14 p abamperes; it has the name one gilbert. The concept of magnetomotive force as the driving force in a “magnetic circuit” permits an alternative definition of the gilbert as the magnetomotive force that produces a uniform magnetic intensity of one oersted over a length of one centimeter in the magnetic circuit. Obviously, one gilbert equals one oersted-centimeter. The theoretical electromagnetic unit of magneto-static potential is defined as the potential existing at a point in a magnetic field if the work done to bring the theoretical

UNITS AND STANDARDS

1795

electromagnetic unit of magnetic quantity from infinity to this point equals one erg. This customary definition implies, however, that the potential vanishes at infinite distances, and the definition has therefore only restricted validity. The unit, thus defined, is identical with one gilbert. The difference in magnetostatic potential between any two points is usually called magnetomotive force (mmf). The theoretical electromagnetic unit of reluctance is defined as the reluctance of a magnetic circuit in which a magnetomotive force of one gilbert produces a magnetic flux of one maxwell. The theoretical electromagnetic unit of electric charge, or the abcoulomb, is defined as the quantity of electricity that passes through any section of an electric circuit in one second if the current is one abampere. The theoretical electromagnetic unit of displacement flux (dielectric flux) is the “line of displacement flux,” or 14 p of the theoretical electromagnetic unit of electric charge. This definition provides the basis for graphical field mapping insofar as it gives a definite rule for the selection of displacement lines to represent the character of the field. The theoretical electromagnetic unit of displacement, or dielectric flux density, is chosen as one displacement line per square centimeter area perpendicular to the direction of the displacement lines. It can also be given as 14 p abcoulombs per square centimeter (according to Gauss’s law). In isotropic media the theoretical electromagnetic unit of displacement can be defined as one displacement line per square centimeter of equipotential surface. (See discussion on theoretical electrostatic unit of displacement.) The theoretical electromagnetic unit of electrical potential difference, or voltage, is the abvolt and is defined as the potential difference existing between two points in space if the work done in bringing the theoretical electromagnetic unit of charge, or the abcoulomb, from one of these points to the other equals one erg. Potential difference is counted positive in the direction in which a negative quantity of electricity would be moved by the electrostatic field. The theoretical electromagnetic unit of capacitance, or the abfarad, is defined as the capacitance that maintains an electrical potential difference of one abvolt between two conductors charged with equal and opposite electrical quantities of one abcoulomb. The theoretical electromagnetic unit of potential gradient, or field strength (field intensity), is defined to exist at a point in an electric field if the mechanical force exerted upon the theoretical electromagnetic unit of charge concentrated at this point is equal to one dyne. It is expressed as one abvolt per centimeter. The theoretical electromagnetic unit of resistance, or the abohm, is defined as the resistance of a conductor in which a current of one abampere is produced if a potential difference of one abvolt is applied at its ends. The theoretical electromagnetic unit of electromotive force (emf) is defined as the electromotive force acting in an electric circuit in which a current of one abampere is flowing and electrical energy is converted into other kinds of energy at the rate of one erg per second. This unit is identical with the abvolt. The absolute dielectric constant of free space is defined as the ratio of displacement to the electric field intensity. Its unit is the abfarad per centimeter, a derived unit. Theoretical Electrodynamic Units The theoretical electrodynamic units are based on the cgs system of mechanical units and are therefore frequently referred to as the cgs electrodynamic units. In contradistinction to the theoretical electromagnetic units, these units are derived from a significant experimental law, Ampere’s experiment on the mechanical force between two parallel currents.

1796

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

The units as proposed by Ampere and used by W. Weber differ from the electromagnetic units by factors of 2 and multiples thereof. They can be made to coincide with the theoretical electromagnetic units by proper definition of the fundamental unit of current. Some of the important definitions will be given for this latter case only. For the absolute magnetic permeability of free space, see discussion on theoretical electromagnetic units. The theoretical electrodynamic unit of current, or the abampere, is defined as the current flowing in a circuit consisting of two infinitely long parallel wires one centimeter apart when the electrodynamic force of repulsion between the two wires is two dynes per centimeter length in free space. If the more natural choice of one dyne per centimeter length is made, pffiffiffi the original proposal of Ampere is obtained and the unit of current becomes 1= 2 abampere. The theoretical electrodynamic unit of magnetic induction is defined as the magnetic induction inducing an electromotive force of one abvolt in a conductor of one centimeter length and moving with a velocity of one centimeter per second if the conductor, its velocity, and the magnetic induction are mutually perpendicular. The unit thus defined is called one gauss. The theoretical electrodynamic unit of magnetic flux, or the maxwell, is defined as the magnetic flux represented by a uniform magnetic induction of one gauss over an area of one square centimeter perpendicular to the direction of the magnetic induction. The theoretical electrodynamic unit of magnetic intensity, or the oersted, is defined as the magnetic intensity at the center of a circle of 4p centimeters diameter in which a current of one abampere is flowing. All the other unit definitions, which do not pertain to magnetic quantities, are identical with the definitions for the theoretical electromagnetic units. 51.4.14 Internationally Adopted Electrical Units and Standards In October 1946, at Paris, the International Committee on Weights and Measures decided to abandon the so-called international practical units based on physical standards (see below) and to adopt effective January 1, 1948, the so-called absolute practical units for international use. Adopted Absolute Practical Units By a series of international actions, the “absolute” practical electrical units are defined as exact powers of 10 of corresponding theoretical electrodynamic and electromagnetic units because they are based on the choice of the proportionality constant in Ampere’s law for free space as kmv ¼ 2  10 7 H/m. The absolute practical unit of current, or the absolute is defined as the current flowing in a circuit consisting of two very long parallel thin wires spaced 1 m apart in free space if the electrodynamic force action between the wires is 2  10 7 N ¼ 0.02 dyne per meter length. It is 10 1 of the theoretical or absolute electrodynamic or electromagnetic unit of current and was adopted internationally in 1881. The absolute practical unit of electric charge, or the absolute coulomb, is defined as the quantity of electricity that passes through a cross-sectional surface in one second if the current is one absolute ampere. It is 10 1 of the theoretical or absolute electromagnetic unit of electric charge and was adopted internationally in 1881.

UNITS AND STANDARDS

1797

The absolute practical unit of electric potential difference, or the absolute volt, is defined as the potential difference existing between two points in space if the work done in bringing an electric charge of one absolute coulomb from one of these points to another is equal to one absolute joule ¼ 107 ergs. It is 108 of the theoretical or absolute electromagnetic unit of potential difference and was adopted internationally in 1881. The absolute practical unit of resistance, or the absolute ohm, is defined as the resistance of a conductor in which a current of one absolute ampere is produced if a potential difference of one absolute volt is applied at its ends. It is 109 of the theoretical or absolute electromagnetic unit of resistance and was adopted internationally in 1881. The absolute practical unit of magnetic flux, or the absolute weber, is defined to be linked with a closed loop of thin wire of total resistance one absolute ohm if upon removing the wire loop from the magnetic field a total charge of one absolute coulomb is passed through any cross section of the wire. It is 108 of the theoretical or absolute electromagnetic unit of magnetic flux, the maxwell, and was adopted internationally in 1933. The absolute practical unit of inductance, or the absolute henry, is defined as connected with a closed loop of thin wire in which a time rate of change of one absolute weber per second in the magnetic flux produces a time rate of change in the current of one absolute ampere. It is 109 of the theoretical or absolute electromagnetic unit of inductance and was adopted internationally in 1893. The absolute practical unit of capacitance, or the absolute farad, is defined as the capacitance that maintains an electric potential difference of one absolute volt between two conductors charged with equal and opposite electrical quantities of one coulomb. It is 10 9 of the theoretical or absolute electromagnetic unit of capacitance and was adopted internationally in 1881. Abandoned International Pactical Units The International System of electrical and magnetic units is a system for electrical and magnetic quantities that takes as the four fundamental quantities resistance, current, length, and time. The units of resistance and current are defined by physical standards that were originally aimed to be exact replicas of the “absolute” practical units, namely the absolute ampere and the absolute ohm. On account of long-range variations in the physical standards, it proved impossible to rely upon them for international use and they recently have been replaced by the absolute practical units. The international practical standards are defined as follows: The international ohm is the resistance at 0 C of a column of mercury of uniform cross section having a length of 106.300 cm and a mass of 14.4521 g. The international ampere is defined as the current that will deposit silver at the rate of 0.00111800 g/sec. From these fundamental units, all other electrical and magnetic units can be defined in a manner similar to the absolute practical units. Because of the inconvenience of the silver voltameter as a standard, the various national laboratories actually used a volt, defining its value in terms of the other two standards. At its conference in October 1946 in Paris, the International Committee on Weights and Measures accepted as the best relations between the international and the absolute practical units the following: 1 mean international ohm ¼ 1:00049 absolute ohms 1 mean international volt ¼ 1:00034 absolute volts

1798

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

These mean values are the averages of values measured in six different national laboratories. On the basis of these mean values, the specific unit relation for converting international units appearing on certificates of the National Bureau of Standards, Washington, DC, into absolute practical units are as follows: 1 international 1 international 1 international 1 international 1 international 1 international

ampere coulomb henry farad watt joule

¼ ¼ ¼ ¼ ¼ ¼

0:999835 absolute 0:999835 absolute 1:000495 absolute 0:999505 absolute 1:000165 absolute 1:000165 absolute

ampere coulomb henries farad watts joules

BIBLIOGRAPHY FOR UNITS AND MEASUREMENTS Cohen, E. R., and Taylor, B. N., “The 1986 Adjustment of the Fundamental Physical Constants,” Report of the CODATA Task Group on Fundamental Constants, November 1986, CODATA Bulletin No. 63, International Council of Scientific Unions, Committee on Data for Science and Technology, Pergamon, 1986. Hvistendahl, H. S., Engineering Units and Physical Quantities, Macmillan, London, 1964. Jerrard, H. G., and McNeill, D. B., A Dictionary of Scientific Units, 2nd ed., Chapman & Hall, London, 1964. Letter Symbols for Units of Measurement, ANSI/IEEE Std. 260-1978, Institute of Electrical and Electronic Engineers, New York, 1978. Quantities, Units, Symbols, Conversion Factors, and Conversion Tables, ISO Reference 31, 15 sections, International Organization for Standardization Geneva, 1973–1979. Standard for Metric Practice, ASTM E 380-82, American Society for Testing and Materials, Philadelphia, 1982. Young, L., System of Units in Electricity and Magnetism, Oliver and Boyd, Edinburgh, 1969. Young, L., Research Concerning Metrology and Fundamental Constants, National Academy Press, Washington, DC, 1983.

51.5 TABLES OF CONVERSION FACTORS9 J. G. Brainerd (revised and extended by J. H. Westbrook) TABLE 51.27 Temperature Conversion F ¼ ( C  95) þ 32 ¼ ( C þ 40)  95 40 C ¼ ( F 32)  59 ¼ ( F þ 40)  59 40  R ¼  F þ 459.69  K ¼ C þ 273.16 



9

Boldface units in Tables 51.28–51.63 are SI.

TABLE 51.28 Length [L] Multiply Number of ! by !

to Obtain # Centimeters Feet Inces Kilometers Nautical Miles Meters Mils Miles Millimeters Yards

Centimeters 1 3.281  10 0.3937 10 5 0.01 393.7 6.214  10 10 1.094  10

2

6

2

Feet 30.48 1 12 3.048  10 1.645  10 0.3048 1.2  104 1.894  10 304.8 0.3333

Inces

4 4

4

2.540 8.333  10 1 2.540  10 — 2.540  10 1000 1.578  10 25.40 2.778  10

2

5

2

5

2

Kilometers

Nautical Miles

Meters

105 3281 3.937  104 1 0.5396 1000 3.937  107 0.6214 106 1094

1.853  105 6080.27 7.296  104 1.853 1 1853 — 1.1516 — 2027

100 3.281 39.37 0.001 5.396  10 4 1 3.937  104 6.214  10 4 1000 1.094

Mils 2.540  10 8.333  10 0.001 2.540  10 1 — 2.540  10 2.778  10

Miles 3 5

8

2 5

1.609  105 5280 6.336  104 1.609 0.8684 1609 — 1 — 1760

Millimeters 0.1 3.281  10 3.937  10 10 6 — 0.001 39.37 6.214  10 1 1.094  10

3 2

7

3

Yards 91.44 3 36 9.144  10 4.934  10 0.9144 3.6  104 5.682  10 914.4 1

4 4

4

1799

1800

Length Land Measure 7:92 inches ¼ 1 link

25 links ¼ 1 rod ¼ 16:5 feet ¼ 5:5 yards ð1 rod ¼ 1 pole ¼ 1 perchÞ 4 rods ¼ 1 chain ðGunther’sÞ ¼ 66 feet ¼ 22 yards ¼ 100 links

10 chains ¼ 1 furlong ¼ 660 feet ¼ 220 yards ¼ 1000 links ¼ 40 rods

8 furlongs ¼ 1 mile ¼ 5280 feet ¼ 1760 yards ¼ 8000 links ¼ 320 rods ¼ 80 chains Ropes and Cables 2 yards ¼ 1 fathom

120 fathoms ¼ 1 cable length

Nautical Measure 6080:27 feet ¼ 1 nautical mile ¼ 1:15156 statute miles

3 nautical miles ¼ 1 league ðU:S:Þ 3 statute miles ¼ 1 league ðGr: BritainÞ (Note: A nautical mile is the length of a minute of longitude of the earth at the equator at sea level. The British Admiralty uses the round figure of 6080 feet. The word “knot” is used to denote “nautical miles per hour.”) Miscellaneous 3 inches ¼ 1 palm

4 inches ¼ 1 hand

9 inches ¼ 1 span

2 12 feet ¼ 1 military pace

TABLE 51.29 Area [L2]

by

to Obtain #

Multiply Number of ! !

Acres Circular Mils Square Centimeters Square Feet Square Inches Square Kilometers Square Meters Square Miles Square Millimeters Square Yards

Acres

Circular Mils

1 — — 1 — 5.067  10 4.356  104 — 6,272,640 7.854  10 4.047  10 3 — 4047 — — 1.562  10 3 — 5.067  10 4840 —

Square Centimeters

6

7

4

— 1.973  105 1 1.076  10 3 0.1550 10 10 0.0001 3.861  10 11 100 1.196  10 4

Square Feet

Square Inches

Square Kilometers

Square Meters

2.296  10 5 1.833  108 929.0 1 144 9.290  10 8 9.290  10 2 3.587  10 8 9.290  104 0.1111

— 1.273  106 6.452 6.944  10 3 1 6.452  10 10 6.452  10 4 — 645.2 7.716  10 4

247.1 — 1010 1.076  107 1.550  109 1 106 0.3861 1012 1.196  106

2.471  10 4 1.973  109 104 10.76 1550 10 6 1 3.861  10 7 106 1.196

Square Miles

Square Millimeters

640 — — 1973 2.590  1010 0.01 2.788  107 1.076  10 4.015  109 1.550  10 2.590 10 12 6 2.590  10 10 6 1 3.861  10 — 1 3.098  106 1.196  10

5 3

13

6

Square Yards 2.066  10 4 — 8361 9 1296 8.361  10 7 0.8361 3.228  10 7 8.361  105 1

1801

1802

Area Land Measure 30 14 square yards ¼ 1 square rod ¼ 272 14 square feet 2 12

16 square rods ¼ 1 square chain ¼ 484 square yards ¼ 4356 square feet square chains ¼ 1 rood ¼ 40 square rods ¼ 1210 square yards 4 roods ¼ 1 acre ¼ 10 square chains ¼ 160 square rods

640 acres ¼ 1 square mile ¼ 2560 roods ¼ 102; 400 square rods

1 section of land ¼ 1 square mile; 1 quarter section ¼ 160 acres Architect ’ s Measure 100 square feet ¼ 1 square Circular Inch and Circular Mil

A circular inch is the area of a circle 1 inch in diameter ¼ 0.7854 square inch 1square inch ¼ 1:2732circular inches

A circular mil is the area of a circle 1 mil (or 0.001 inch) in diameter ¼ 0.7854 square mil 1 square mil ¼ 1:2732 circular mils

1 circular inch ¼ 106 circular mils ¼ 0:7854  106 square mils 1 square inch ¼ 1:2732  106 circular mils ¼ 106 square mils

TABLE 51.30 Volume [L3] Multiply Number of ! by !

to Obtain #

Bushels (Dry) Cubic Centimeters Cubic Feet Cubic Inches Cubic Meters Cubic Yards Gallons (Liquid) Liters Pints (Liquid) Quarts (Liquid)

Brushels (Dry) 1 3.524  104 1.2445 2150.4 3.524  10 2 — — 35.24 — —

Cubic Centimeters — 1 3.531  10 6.102  10 10 6 1.308  10 2.642  10 0.001 2.113  10 1.057  10

5 2

6 4

3 3

Cubic Feet 0.8036 2.832  104 1 1728 2.832  10 2 3.704  10 2 7.481 28.32 59.84 29.92

Cubic Inches 4.651  10 16.39 5.787  10 1 1.639  10 2.143  10 4.329  10 1.639  10 3.463  10 1.732  10

4

4

5 5 3 2 2 2

Cubic Meters

Cubic Yards

28.38 — 106 7.646  105 35.31 27 6.102  104 46,656 1 0.7646 1.308 1 264.2 202.0 1000 764.6 2113 1616 1057 807.9

Gallons (Liquid) — 3785 0.1337 231 3.785  10 4.951  10 1 3.785 8 4

Pints (Liquid)

Liters

3 3

2.838  10 1000 3.531  10 61.02 0.001 1.308  10 0.2642 1 2.113 1.057

Quarts (Liquid)

2

2

3

473.2 1.671  10 28.87 4.732  10 6.189  10 0.125 0.4732 1 0.5

2

4 4

946.4 3.342  10 57.75 9.464  10 1.238  10 0.25 0.9464 2 1

2

4 3

1803

1804

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Volume Cubic Measure 1 cord of wood ¼ pile cut 4 feet long piled 4 feet high and 8 feet on the ground ¼ 128 cubic feet 1 perch of stone ¼ quantity 1 12 feet thick; 1 foot high; and 16 12 feet long ¼ 24 34 cubic feet (Note: A perch of stone is, however, often computed differently in different localities; thus, in most if not all of the states west of the Mississippi, stonemasons figure rubble by the perch of 16 12 cubic feet. In Philadelphia, 22 cubic feet is called a perch. In Chicago, stone is measured by the cord of 100 cubic feet. Check should be made against local practice.) Board Measure In board measure, boards are assumed to be one inch in thickness. Therefore, feet board measure of a stick of square timber ¼ length in feet  breadth in feet  thickness in inches. Shipping Measure For register tonnage or measurement of the entire internal capacity of a vessel, it is arbitrarily assumed, to facilitate computation, that 100 cubic feet ¼ 1 register ton For the measurement of cargo: 40 cubic feet ¼ ¼ 42 cubic feet ¼ ¼

1 U:S: shipping ton 32:143 U:S: bushels 1 British shipping ton 32:703 Imperial bushels

Dry Measure One U.S. Winchester bushel contains 1.2445 cubic feet or 2150.42 cubic inches. It holds 77.601 pounds distilled water at 62 F. (Note: This is a struck bushel. A heaped bushel in general equals 1 14 struck bushels, although for apples and pears it contains 1.2731 struck bushels ¼ 2737.72 cubic inches.) One U. S. gallon (dry measure) ¼ 18 bushel and contains 268.8 cubic inches. (Note: This is not a legal U.S. dry measure and therefore is given for comparison only.) One British Imperial bushel contains 1.2843 cubic feet or 2219.36 cubic inches. It holds 80 pounds distilled water at 62 F. 1 British Imperial gallon ¼

1 8

Imperial bushel and contains 277:42 cubic inches:

1 Winchester bushel ¼ 0:9694 Imperial bushel 1 Imperial bushel ¼ 1:032 Winchester bushels

TABLES OF CONVERSION FACTORS

1805

Same relations as before maintain for gallons (dry measure). [Note: 1 U.S. gallon (dry) ¼ 1.164 U. S. gallons (liquid)).] U.S. UNITS10 2 pints ¼ 1 quart ¼ 67.2 cubic inches 4 quarts ¼ 1 gallon ¼ 8 pints ¼ 268.8 cubic inches 2 gallons ¼ 1 peck ¼ 16 pints ¼ 8 quarts ¼ 537.6 cubic inches 4 pecks ¼ 1 bushel ¼ 64 pints ¼ 32 quarts ¼ 8 gallons ¼ 2150.42 cubic inches 1 cubic foot contains 6.428 gallons (dry measure) Liquid Measure One U.S. gallon (liquid measure) contains 231 cubic inches. It holds 8.336 pounds distilled water at 62 F. One British Imperial gallon contains 277.42 cubic inches. It holds 10 pounds distilled water at 62 F. 1 U:S: gallon ðliquidÞ ¼ 0:8327 Imperial gallon

1 Imperial gallon ¼ 1:201 U:S: gallons ðliquidÞ

[Note: 1 U.S. gallon (liquid) ¼ 0.8594 U.S. gallon (dry).] U.S. UNITS 4 gills ¼ 1 pint ¼16 fluid ounces 2 pints ¼ 1 quart ¼ 8 gills ¼ 32 fluid ounces 4 quarts ¼ 1 gallon ¼ 32 gills ¼ 8 pints ¼ 128 fluid ounces 1 cubic foot contains 7.4805 gallons (liquid measure) Apothecaries’ Fluid Measure 60 minims ¼ 1 fluid drachm 8 drachms ¼ 1 fluid ounce In the United States a fluid ounce is the 128th part of a U.S. gallon, or 1.805 cubic inches or 29.58 cubic centimeters. It contains 455.8 grains of water at 62 F. In Great Britain the fluid ounce is 1.732 cubic inches and contains 1 ounce avoirdupois (or 437.5 grains) of water at 62 F.

10

The gallon is not a U.S. legal dry measure.

1806 TABLE 51.31 Plane Angle (No Dimensions)

to Obtain #

by

Multiply Number of !

!

Degrees Minutes Quadrants Radiansa Revolutionsa (Circumferences) Seconds a

Degrees 1 60 1.111  10 1.745  10 2.778  10 3600

2p rad ¼ 1 circumference ¼ 360 by definition.

Minutes

2 2 3

1.667  10 1 1.852  10 2.909  10 4.630  10 60

2

4 4 5

Quadrants

Radiansa

Revolutionsa (Circumferences)

90 5400 1 1.571 0.25

57.30 3438 0.6366 1 0.1591

360 2.16  104 4 6.283 1

3.24  105

2.063  105

1.296  106

Seconds 2.778  10 1.667  10 3.087  10 4.848  10 7.716  10 1

4 2 6 6 7

TABLE 51.32 Solid Angle (No Dimensions)

to Obtain #

by

Multiply Number of ! !

Hemispheres Spheresa Spherical Right Angles Steradiansb a b

A sphere is the total solid angle about a point. 4p steradians ¼ 1 sphere by definition.

Hemispheres

Spheresa

Spherical Right Angles

1 0.5 4 6.283

2 1 8 12.57

0.25 0.125 1 1.571

Steradiansb 0.1592 7.958  10 0.6366 1

2

1807

1808 TABLE 51.33 Time [T]

to Obtain #

Multiply Number by of ! !

Days Hours Minutes Months (Average)a Seconds Weeks a

Days 1 24 1440 3.288  10 2 8.64  104 0.1429

Hours 4.167  10 1 60 1.370  10 3600 5.952  10

Months (Average)a

Minutes 2

3

3

6.944  10 1.667  10 1 2.283  10 60 9.921  10

4 2

5

5

1 of a common year. One common year ¼ 365 days; one leap year ¼ 366 days; one average month ¼ 12

30.42 730.0 4.380  10 4 1 2.628  106 4.344

Weeks

Seconds 1.157  10 2.778  10 1.667  10 3.806  10 1 1.654  10

5 4 2 7

6

7 168 1.008  104 0.2302 6.048  105 1

TABLE 51.34 Linear Velocity [LT 1]

by

to Obtain #

Multiply Number of ! !

Centimeters per Second Feet per Minute Feet per Second Kilometers per Hour Kilometers per Minute Knotsa Meters per Minute Meters per Second Miles per Hour Miles per Minute a

Nautical miles per hour.

Centimeters per Second 1 1.969 3.281  10 0.036 0.0006 1.943  10 0.6 0.01 2.237  10 3.728  10

2

2

2 4

Feet per Minute

Feet per Second

0.5080 1 1.667  10 1.829  10 3.048  10 9.868  10 0.3048 5.080  10 1.136  10 1.892  10

30.48 60 1 1.097 1.829  10 0.5921 18.29 0.3048 0.6818 1.136  10

2 2 4 3

3 2 4

Kilometers per Hour

2

2

27.78 54.68 0.9113 1 1.667  10 0.5396 16.67 0.2778 0.6214 1.036  10

Kilometers per Minute

2

2

1667 3281 54.68 60 1 32.38 1000 16.67 37.28 0.6214

Meters per Minute

Knotsa 51.48 101.3 1.689 1.853 3.088  10 1 30.88 0.5148 1.152 1.919  10

2

2

1.667 3.281 5.468  10 0.06 0.001 3.238  10 1 1.667  10 3.728  10 6.214  10

Meters per Second

2

2

2 2 4

100 196.8 3.281 3.6 0.06 1.943 60 1 2.237 3.728  10

2

Miles per Hour 44.70 88 1.467 1.609 2.682  10 0.8684 26.82 0.4470 1 1.667  10

Miles per Minute

2

2

2682 5280 88 96.54 1.609 52.10 1609 26.82 60 1

1809

1810

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Linear Velocity The Miner’s Inch. The miner’s inch is used in measuring flow of water. An act of the California legislature, May 23, 1901, makes the standard miner’s inch 1.5 ft3/ min, measured through any aperture or orifice. The term miner’s inch is more or less indefinite, for the reason that California water companies do not all use the same head above the center of the aperture, and the inch varies from 1.36 to 1.73 ft3/min, but the most common measurement is through an aperture 2 in. high and whatever length is required and through a plank 1 14 in. thick. The lower edge of the aperture should be 2 in. above the bottom of the measuring box and the plank 5 in. high above the aperture, thus making a 6-in. head above the center of the stream. Each square inch of this opening represents a miner’s inch, which is equal to a flow of 1.5 ft3/min. Avoirdupois Weight. Used Commercially. 27:343 grains 16 drachms 16 ounces 28 pounds 4 quarters

¼ ¼ ¼ ¼ ¼ ¼

1 drachm 1 ounceðozÞ ¼ 437:5 grains 1 pound ðlbÞ ¼ 7000 grains 1 quarter ðqrÞ 1 hundredweight ðcwtÞ 112pounds

20 hundredweight ¼ 1 gross or long ton11 200 pounds ¼ 1 net or short ton 14 pounds ¼ 1 stone 100 pounds ¼ 1 quintal Troy Weight. Used in weighing gold or silver. 24 grains ¼ 1 pennyweight ðdwtÞ 20 pennyweights ¼ 1 ounce ðozÞ ¼ 480 grains 12 ounces ¼ 1 pound ðlbÞ ¼ 5760 grains The grain is the same in avoirdupois, troy, and apothecaries’ weights. A carat, for weighing diamonds, ¼ 3.086 grains ¼ 0.200 gram (International Standard, 1913.) 1 pound troy ¼ 0:8229 pound avoirdupois 1 pound avoirdupois ¼ 1:2153 pounds troy Apothecaries’ Weight. Used in compounding medicines. 20 grains 3 scruples 8 drachms 12 ounces

¼ ¼ ¼ ¼

1 scrupleðÞ 1 drachmðÞ ¼ 60 grains 1 ounceðÞ ¼ 480 grains 1 poundðlbÞ ¼ 5760 grains

The grain is the same in avoirdupois, troy, and apothecaries’ weights. 1 pound apothecaries ¼ 0:82286 pound avoirdupois 1 pound avoirdupois ¼ 1:2153 pounds apothecaries 11 The long ton is used by the U.S. custom houses in collecting duties upon foreign goods. It is also used in freighting coal and selling it wholesale.

TABLES OF CONVERSION FACTORS

1811

TABLE 51.35 Angular Velocity [T 1]

by

to Obtain #

Multiply Number of ! !

Degrees per Second

Degrees per Second Radians per Second Revolutions per Minute Revolutions per Second

1 1.745  10 0.1667 2.778  10

Revolutions per Minute

Radians per Second 57.30 1 9.549 0.1592

2

3

6 0.1047 1 1.667  10

Revolutions per Second 360 6.283 60 1

2

TABLE 51.36 Linear Accelerationa [LT 2]

by

Multiply Number of ! !

to Obtain #

Centimeters per Second per Second

Centimeters per Second per Second Feet per Second per Second Kilometers per Hour per Second Meters per Second per Second Miles per Hour per Second

1 3.281  10 0.036 0.01 2.237  10

Feet per Second per Second

Kilometers per Hour per Second

Meters per Second per Second

Miles per Hour per Second

30.48 1 1.097 0.3048 0.6818

27.78 0.9113 1 0.2778 0.6214

100 3.281 3.6 1 2.237

44.70 1.467 1.609 0.4470 1

2

2

a The (standard) acceleration due to gravity (g0) ¼ 980.7 cm/sec sec, ¼ 32.17 ft/sec sec ¼ 35.30 km/hr sec ¼ 9.807 m/sec sec ¼ 21.94 mph/sec.

TABLE 51.37 Angular Acceleration [T 2]

by

Multiply Number of ! !

to Obtain # Radians per Second per Second Revolutions per Minute per Minute Revolutions per Minute per Second Revolutions per Second per Second

Radians per Second per Second 1 573.0 9.549 0.1592

Revolutions per Minute per Minute 1.745  10 1 1.667  10 2.778  10

3

2 4

Revolutions per Minute per Second 0.1047 60 1 1.667  10

2

Revolutions per Second per Second 6.283 3600 60 1

1812 TABLE 51.38 Mass [M] and Weighta

by

to Obtain # Grains Grams Kilograms Milligrams Ouncesb Poundsb Tons (Long) Tons (Metric) Tons (Short) a

Multiply Number of ! !

Grains 1 6.481  10 6.481  10 64.81 2.286  10 1.429  10 — — —

Grams 2 5

3 4

15.43 1 0.001 1000 3.527  10 2.205  10 9.842  10 10 6 1.102  10

Kilograms

2 3 7

6

1.543  104 1000 1 106 35.27 2.205 9.842  10 4 0.001 1.102  10 3

Ouncesb

Milligrams 1.543  10 0.001 10 6 1 3.527  10 2.205  10 9.842  10 10 9 1.102  10

2

5 6 10

9

Poundsb

Tons (Long)

437.5 7000 28.35 453.6 1.016  106 2 2.835  10 0.4536 1016 2.835  104 4.536  105 1.016  109 1 16 3.584  104 2 6.250  10 1 2240 2.790  10 5 4.464  10 4 1 2.835  10 5 4.536  10 4 1.016 3.125  10 5 0.0005 1.120

Tons (Metric)

Tons (Short)

106 1000 109 3.527  104 2205 0.9842 1 1.102

9.072  105 907.2 9.072  108 3.2  104 2000 0.8929 0.9072 1

These same conversion factors apply to the gravitational units of force having the corresponding names. The dimensions of these units when used as gravitational units of force are MLT 2; see Table 51.40. b Avoirdupois pounds and ounces.

TABLE 51.39 Density or Mass per Unit Volume [ML 3]

to Obtain #

by

Multiply Number of ! !

Grams per Cubic Centimeter Kilograms per Cubic Meter Pounds per Cubic Foot Pounds per Cubic Inch Pounds per Mil Foota a

Grams per Cubic Centimeter 1 1000 62.43 3.613  10 3.405  10

2 7

Unit of volume is a volume one foot long and one circular mil in cross-sectional area.

Kilograms per Cubic Meter 0.001 1 6.243  10 3.613  10 3.405  10

2 5 10

Pounds per Cubic Foot 1.602  10 16.02 1 5.787  10 5.456  10

Pounds per Cubic Inch 2

4 9

27.68 2.768  104 1728 1 9.425  10 6

1813

1814 TABLE 51.40 Forcea [MLT 2] or [F]

by

to Obtain #

!

Multiply Number of !

Dynes Grams Joules per Centimeter Newtons, or Joules per Meter Kilograms Pounds Poundals a

Dynes 1 1.020  10 10 7 10 5 1.020  10 2.248  10 7.233  10

Joules per Centimeter

Grams 3

6 6 5

980.7 1 9.807  10 9.807  10 0.001 2.205  10 7.093  10

7

5 3

3 2

10 1.020  104 1 100 10.20 22.48 723.3

Newtons, or Joules per Meter 5

10 102.0 0.01 1 0.1020 0.2248 7.233

Kilograms 5

9.807  10 1000 9.807  10 2 9.807 1 2.205 70.93

Pounds

Poundals 5

4.448  10 453.6 4.448  10 2 4.448 0.4536 1 32.17

Conversion factors between absolute and gravitational units apply only under standard acceleration due to gravity conditions. (See Section 51.4.)

1.383  104 14.10 1.383  10 3 0.1383 1.410  10 2 3.108  10 2 1

TABLE 51.41 Pressure or Force per Unit Area [ML 1T 2] or [FL 2]

by

Multiply Number of ! !

to Obtain # Atmospheresa Baryes or Dynes per Square Centimeter Centimeters of Mercury at 0 Cb Inches of Mercury at 0 Cb Inches of Water at 4 C Kilograms per Square Meterc Pounds per Square Foot Pounds per Square Inch Tons (Short) per Square Foot Pascal

Atmospheresa

Baryes or Dynes per Square Centimeter

Centimeters of Mercury at 0 Cb

1 1.013  106

9.869  10 1

7

76.00

7.501  10

5

29.92 406.8 1.033  104 2117 14.70 1.058 1.013  105

2.953  10 4.015  10 1.020  10 2.089  10 1.450  10 1.044  10 10 1

5 4 2 3 5 6

Inches of Mercury at 0 Cb

Inches of Water at 4 C

1.316  10 2 3.342  10 2 2.458  10 1.333  104 3.386  104 2.491  10 1

2.540

0.3937 1 5.354 13.60 136.0 345.3 27.85 70.73 0.1934 0.4912 1.392  10 2 3.536  10 2 1.333  103 3.386  103

3 3

0.1868 7.355  10 1 25.40 5.204 3.613  10 2.601  10 2.491  10

2

2 3 4

Kilograms per Square Meterc

Pounds per Square Foot

Pounds per Square Inch

9.678  10 98.07

5

4.725  10 478.8

4

7.356  10

3

3.591  10

2

5.171

2.896  10 3.937  10 1 0.2048 1.422  10 1.024  10 9.807

3

1.414  10 0.1922 4.882 1 6.944  10 0.0005 47.88

2

2.036 27.68 703.1 144 1 0.072 6.895  103

2

3 4

3

Tons (Short) per Square Foot

Pascal

6.804  10 2 0.9450 9.869  10 6.895  104 9.576  105 10

6

71.83

7.501  10

4

28.28 384.5 9765 2000 13.89 1 9.576  104

2.953  10 4.015  10 0.1020 2.089  10 1.450  10 1.044  10 1

4 8

2 4 5

a

Definition: One atmosphere (standard) ¼ 76 cm of mercury at 0 C. To convert height h of a column of mercury at t degrees Centigrade to the equivalent height h0 at 0 C use h0 ¼ h{1 (m l)t/(1 þ mt)}, where m ¼ 0.0001818 and l ¼ 18.4  10 if the scale is engraved on brass; l ¼ 8.5  10 6 if on glass. This assumes the scale is correct at 0 C; for other cases (any liquid) see International Critical Tables, Vol. 1, p. 68 c 1 g/cm2 ¼ 10 kg/m2. b

6

1815

1816

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.42 Torque or Moment of Force [ML2T 2] or [FL]a Multiply Number of ! by !

to Obtain #

Dyne-Centimeters Gram-Centimeters Kilogram-Meters Pound-Feet Newton-Meter

DyneCentimeters

GramCentimeters

KilogramMeters

Pound-Feet

1 1.020  10 1.020  10 7.376  10 10 7

980.7 1 10 5 7.233  10 9.807  10

9.807  107 105 1 7.233 9.807

1.356  107 107 1.383  104 1.020  104 0.1383 0.1020 1 0.7376 1.356 1

3 8 8

5 4

NewtonMeter

a

Same dimensions as energy; more properly torque should be expressed as newton-meters per radian to avoid this confusion.

TABLE 51.43 Moment of Inertia [ML2]

by

Multiply Number of ! !

to Obtain # Gram-Centimeters Squared Kilogram-Meters Squared Pound-Inches Squared Pound-Feet Squared Slug-Feet Squared

GramCentimeters Squared 1 10 7 3.4169  10 2.37285  10 7.37507  10

KilogramMeters Squared

4 6 8

Pound-Inches Squared

Pound-Feet Squared

Slug-Feet Squared

2.9266  103 4.21434  105 1.3559  107 107 1 2.9266  10 4 4.21434  10 2 1.3559 3.4169  103 1 144 4.63304  103 23.7285 6.944  10 3 1 32.1739 0.737507 2.15841  10 4 3.10811  10 2 1

TABLE 51.44 Energy, Work and Heata [ML2T 2] or [FL] Multiply Number of ! by !

to Obtain #

British Thermal Unitsb

British Thermal Unitsb 1 Centimeter-Grams 1.076  107 Ergs or Centimeter-Dynes 1.055  1010 Foot-Pounds 778.0 Horsepower-Hours 3.929  10 4 Joules,c or Watt-Seconds 1054.8 0.2520 Kilogram-Caloriesb Kilowatt-Hours 2.930  10 4 Meter-Kilograms 107.6 Watt-Hours 0.2930 a

CentimeterGrams 9.297  10 1 980.7 7.233  10 3.654  10 9.807  10 2.343  10 2.724  10 10 5 2.724  10

8

5 11 5 8 11

8

Ergs or CentimeterDynes 9.480  10 1.020  10 1 7.367  10 3.722  10 10 7 2.389  10 2.778  10 1.020  10 2.778  10

11 3

8 14

11 14 8 11

Foot-Pounds 1.285  10 3 1.383  104 1.356  107 1 5.050  10 7 1.356 3.239  10 4 3.766  10 7 0.1383 3.766  10 4

HorsepowerJoules,c or Hours Watt-Seconds 2545 2.737  1010 2.684  1012 1.98  106 1 2.684  106 641.3 0.7457 2.737  105 745.7

9.480  10 4 1.020  104 107 0.7376 3.722  10 7 1 2.389  10 4 2.778  10 7 0.1020 2.778  10 4

KilogramCaloriesb

KilowattHours

3.969 3413 4.269  107 3.671  1010 4.186  1010 3.6  1013 3087 2.655  106 1.559  10 3 1.341 4186 3.6  106 1 860.0 1.163  10 3 1 426.9 3.671  105 1.163 1000

MeterKilograms

Watt-Hours

9.297  10 3 3.413 105 3.671  107 9.807  107 3.6  1010 7.233 2655 3.653  10 6 1.341  10 3 9.807 3600 2.343  10 3 0.8600 2.724  10 6 0.001 1 367.1 2.724  10 3 1

See note at the bottom of Table 51.45. Mean calorie and Btu used throughout. One gram-calorie ¼ 0.001 kilogram-calorie; one Ostwald calorie ¼ 0.1 kilogram-calorie. The IT cal, 1000 international steam table calories, has been defined as the 1/860th part of the international kilowatthour (see Mechanical Engineering, Nov. 1935, p. 710). Its value is very nearly equal to the mean kilogram-calorie, 1 IT cal-1.00037 kilogram-calories (mean). 1 Btu ¼ 251.996 IT cal. c Absolute joule, defined as 107 ergs. The international joule, based on the international ohm and ampere, equals 1.0003 absolute joules. b

1817

1818

TABLE 51.45 Power or Rate of Doing Worka [ML2T 3] or [FL 1] Multiply Number of ! by !

to Obtain # British Thermal Units per Minute Ergs per Second Foot-Pounds per Minute Foot-Pounds per Second Horsepowera Kilogram-Calories per Minute Kilowatts Metric Horsepower Watts

British Thermal Units per Minute 1

5.689  10

1.758  108 778.0 12.97 2.357  10 2 0.2520

1 4.426  10 7.376  10 1.341  10 1.433  10

2

10 10 1.360  10 10 7

1.758  10 2.390  10 17.58

KilogramFoot-Pounds Foot-Pounds Calories per per Minute per Second Horsepowera Minute

Ergs per Second

2

9

6 8 10 9

10

1.285  10

3

7.712  10

2

42.41

3.969

Kilowatts

Metric Horsepower

56.89

41.83

Watts 5.689  10

2.259  105 1.356  107 7.457  109 6.977  108 1010 7.355  109 107 4 4 4 1 60 3.3  10 3087 4.426  10 3.255  10 44.26 1.667  10 2 1 550 51.44 737.6 542.5 0.7376 3.030  10 5 1.818  10 3 1 9.355  10 2 1.341 0.9863 1.341  10 3.239  10 4 1.943  10 2 10.69 1 14.33 10.54 1.433  10 2.260  10 3.072  10 2.260  10

5 5 2

1.356  10 1.843  10 1.356

3 3

0.7457 1.014 745.7

6.977  10 9.485  10 69.77

2 2

1 1.360 1000

0.7355 1 735.5

10 3 1.360  10 1

2

3 2

3

Note: 1 Cheval-vapeur ¼ 75 kilogram-meters per second

1 Poncelet ¼ 100 kilogram-meters per second

a

The “horsepower” used in these tables is equal to 550 foot-pounds per second by definition. Other definitions are one horsepower equals 746 watts (U.S. and Great Britain) and one horsepower equals 736 watts (continental Europe). Neither of these latter definitions is equivalent to the first; the “horsepowers” defined in these latter definitions are widely used in the rating of electrical machinery.

1819

TABLES OF CONVERSION FACTORS

TABLE 51.46 Quantity of Electricity and Dielectric Flux [Q]

to Obtain #

by

Multiply Number of ! !

Abcoulombs Ampere-Hours Coulombs Faradays Statcoulombs

Abcoulombs

AmpereHours

Coulombs

Faradays

Stat coulombs

1 2.778  10 3 10 1.036  10 4 2.998  1010

360 1 3600 3.731  10 2 1.080  1013

0.1 2.778  10 4 1 1.036  10 5 2.998  109

9649 26.80 9.649  104 1 2.893  1014

3.335  10 9.259  10 3.335  10 3.457  10 1

11 14 10 15

TABLE 51.47 Charge per Unit Area and Electric Flux Density [QL 2]

by

Multiply Number of ! !

Coulombs per Square Centimeter

Coulombs per Square Inch

to Obtain #

Abcoulombs per Square Centimeter

Abcoulombs per Square Centimeter Coulombs per Square Centimeter Coulombs per Square Inch Statcoulombs per Square Centimeter Coulombs per Square Meter

1 0.1 1.550  10 2 10 1 0.1550 64.52 6.452 1 2.998  1010 2.998  109 4.647  108 105 104 1550

Statcoulombs per Square Centimeter 3.335  10 3.335  10 2.151  10 1 3.335  10

11 10 9

6

Coulombs per Square Meter 10 5 10 4 6.452  10 4 2.998  105 1

TABLE 51.48 Electric Current [QT 1]

to Obtain # Abamperes Amperes Statamperes

by

!

Multiply Number of ! Abamperes

Amperes

1 10 2.998  1010

0.1 1 2.998  109

Statamperes 3.335  10 3.335  10 1

11 10

1820

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.49 Current Density [QT 1L 2]

by

Multiply Number of ! !

to Obtain # Abamperes per Square Centimeter Amperes per Square Centimeter Amperes per Square Inch Statamperes per Square Centimeter Amperes per Square Meter

Abamperes per Square Centimeter

Amperes per Square Centimeter

1 10 64.52 2.998  1010 105

0.1 1.550  10 2 3.335  10 1 0.1550 3.335  10 6.452 1 2.151  10 2.998  109 4.647  108 1 104 1550 3.335  10

Amperes per Square Inch

Statamperes per Square Centimeter 11 10 9

6

Amperes per Square Meter 10 5 10 4 6.452  10 4 2.998 105 1

TABLE 51.50 Electric Potential and Electromotive Force [MQ 1L2T 2] or [FQ 1L]

to Obtain # Abvolts Microvolts Millivolts Statvolts Volts

Multiply Number of ! by !

Abvolts 1 0.01 10 5 3.335  10 10 8

Microvolts

11

100 1 0.001 3.335  10 10 6

Millivolts

Statvolts

5

9

10 1000 1 3.335  10 0.001

Volts 10

6

2.998  10 2.998 108 2.998 105 1 299.8

108 106 1000 3.335  10 1

3

TABLE 51.51 Electric Field Intensity and Potential Gradient [MQ 1LT 2] or [FQ 1]

by

to Obtain #

Multiply Number of ! !

Abvolts per Centimeter Microvolts per Meter Millivolts per Meter Statvolts per Centimeter Volts per Centimeter Kilovolts per Centimeter Volts per Inch Volts per Mil Volts per Meter

Abvolts per Centimeter 1 1 0.001 3.335  10 10 8 10 11 2.540  10 2.540  10 10 6

11

8 11

Microvolts per Meter 1 1 0.001 3.335  10 10 8 10 11 2.540  10 2.540  10 10 6

11

8 11

Millivolts per Meter

Statvolts per Centimeter

1000 1000 1 3.335  10 10 5 10 8 2.540  10 2.540  10 10 3

2.998  1010 2.998  1010 2.998  107 1 299.8 0.2998 761.6 0.7616 2.998  104

8

5 8

Volts per Centimeter 108 108 105 3.335  10 1 0.001 2.540 2.540  10 100

3

3

Kilovolts per Centimeter

Volts per Inch

1011 1011 108 3.335 1000 1 2.540 2.540 105

3.937  107 3.937  107 3.937  104 1.313  10 3 0.3937 3.937  10 4 1 0.001 39.37

Volts per Mil

Volts per Meter

3.937  1010 106 10 3.93710 106 3.937  107 1000 1.313 3.335  10 393.7 10 2 0.3937 10 5 1000 2.540  10 1 2.540  10 3.937  104 1

5

2 5

1821

1822

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.52 Electric Resistance [MQ 2L2T 1] or [FQ 2LT]

by to ! Obtain #

Multiply Number of !

Abohms Megohms Microhms Ohms Statohms

Abohms

Megohms

1 10 15 0.001 10 9 1.112  10

1015 1 1012 106 1.112  10

21

Microhms

6

1000 10 12 1 10 6 1.112  10

Note: Electric Conductance [F 1Q2L 1T 1]. 1 Siemens ¼ 1 mho ¼ 1 ohm

18 1

Statohms

Ohms 109 10 6 106 1 1.112  10

¼ 10

6

12

8.988  1020 8.988  105 8.988  1017 8.988  1011 1

megmho ¼ 106 micromho.

TABLE 51.53 Electric Resistivitya [MQ 2L3T 1] or [FQ 2L2T]

by !

to Obtain #

Multiply Number of ! AbohmCentimeters

Abohm-Centimeters Microhm-Centimeters Microhm-Inches Ohms (Mil, Foot) Ohms (Meter, Gram)b Ohm-Meters

1 0.001 3.937  10 6.015  10 10 5d 10 11

4 3

MicrohmCentimeters 1000 1 0.3937 6.015 0.01d 10 8

MicrohmInches

Ohms (Mil, Foot)

Ohms (Meter, Gram)b

OhmMeters

1011 2540 166.2 105/d 2.540 0.1662 100/d 108 2 1 6.545  10 39.37/d 3.937  107 15.28 1 601.5/d 6.015  108 2.540  10 2d 1.662  10 3d 1 10 6d 8 9 6 2.540  10 1.662  10 10 /d 1

a In this table d is density in grams per cubic-centimeters. The following names, corresponding respectively to those at the tops of columns, are sometimes used: abohms per centimeter cube; microhms per centimeter cube; microhms per inch cube; ohms per milfoot; ohms per meter-gram. The first four columns are headed by units of volume resistivity, the last by a unit of mass resistivity. The dimensions of the latter are Q 2L6T 1, not those given in the heading of the table. b One ohm (meter, gram) ¼ 5710 ohms (mile, pound).

TABLE 51.54 Electric Conductivitya [M 1Q2L 3T] or [F 1Q2L 2T 1]

to Obtain #

by

!

Multiply Number of !

Abmhos per Centimeter Mhos (Mil, Foot) Mhos (Meter, Gram) Micromhos per Centimeter Micromhos per Inch Siemens per Meter a

Abmhos per Centimeter 1 166.2 105/d 1000 2540 1011

Mhos (Mil, Foot)

Mhos (Meter, Gram)

6.015  10 3 10 5d 1 1.662  10 3d 601.5/d 1 6.015 0.01d 15.28 2.540  10 2d 8 6.015  10 106d

Micromhos per Centimeter 0.001 0.1662 100/d 1 2.540 108

Micromhos Siemens per per Inch Meter 3.937  10 4 10 11 2 6.524  10 1.662  10 39.37/d 10 6/d 0.3937 10 8 1 2.54  10 3.937  107 1

9

8

See footnote of Table 51.53. Names sometimes used are abmho per centimeter cube, mho per mil-foot, etc. Dimensions of mass conductivity are Q2L 6T.

TABLES OF CONVERSION FACTORS

1823

TABLE 51.55 Capacitance [M 1Q2L 2T2] or [F 1Q2L 1]

to Obtain #

by

!

Multiply Number of !

Abfarads Farads Microfarads Statfarads

Abfarads

Farads

Microfarads

Statfarads

1 109 1015 8.988  1020

10 9 1 106 8.988  1011

10 15 10 6 1 8.988  105

1.112  10 1.112  10 1.112  10 1

21 12 6

TABLE 51.56 Inductance [MQ 2L2] or [FQ 2LT2]

by

to Obtain #

Multiply Number of !

!

Abhenriesa

a

Abhenries Henries Microhenries Millihenries Stathenries a

1 10 9 0.001 10 6 1.112  10

Microhenries

Henries 9

21

10 1 106 1000 1.112  10

12

1000 10 6 1 0.001 1.112  10

Millihenries 6

18

10 0.001 1000 1 1.112  10

15

Stathenries 8.988  1020 8.988  1011 8.988  1017 8.988  1014 1 1

An abhenry is sometimes called a “centimeter.”

TABLE 51.57 Magnetic Flux [MQ 1L2T 1] or [FQ 1LT]

to Obtain #

by

!

Multiply Number of !

Kilolines Maxwells (or Lines) Webers

Kilolines

Maxwells (or Lines)

Webers

1 1000 10 5

0.001 1 10 8

105 108 1

TABLE 51.58 Magnetic Flux Density [MQ 1T 1] or [FQ 1L 1T]

by

to Obtain #

!

Multiply Number of !

Gausses (or Lines per Square Centimeter) Lines per Square Inch Webers per Square Centimeter Webers per Square Inch Tesla (Webers per Square Meter)

Gausses (or Lines per Square Centimeter) 1 6.452 10 8 6.452  10 10 4

8

Lines Webers Webers Tesla (Webers per Square per Square per Square per Square Inch Centimeter Inch Meter) 0.1550 1 1.550  10 10 8 1.550  10

1.550  107 104 108 6.452  108 108 6.452  104 9 1 0.1550 10 4 6.452 1 6.452  10 4 5 104 1550 1

1824

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.59 Magnetic Potential and Magnetomotive Force [QT 1]

to Obtain #

by

Multiply Number of !

!

Abampere-Turns

Ampere-Turns

1 10 12.57

0.1 1 1.257

Abampere-Turns Ampere-Turns Gilberts

Gilberts 7.958  10 0.7958 1

2

TABLE 51.60 Magnetic Field Intensity, Potential Gradient, and Magnetizing Force [QL 1T 1]

by

!

Multiply Number of !

to Obtain #

AbampereTurns per Centimeter

AmpereTurns per Centimeter

1 10 25.40 12.57 103

0.1 1 2.540 1.257 102

Abampere-Turns per Centimeter Ampere-Turns per Centimeter Ampere-Turns per Inch Oersteds (Gilberts per Centimeter) Ampere-Turns per Meter

AmpereTurns per Inch 3.937  10 0.3937 1 0.4950 39.37

Oersteds (Gilberts per Centimeter) 2

7.958  10 0.7958 2.021 1 79.58

2

AmpereTurns per Meter 10 3 10 2 2.54  10 1.257  10 1

2 2

TABLE 51.61 Specific Heat [L2T 2t 1] (t ¼ temperature) To change specific heat in gram-calories per gram per degree Centigrade to the units given in any line of the following table, multiply by the factor in the last column. Unit of Heat or Energy

Unit of Mass

Temperature Scalea

Gram-calories Kilogram-calories British thermal units British thermal units Joules Joules Joules Kilowatt-hours Kilowatt-hours

Gram Kilogram Pound Pound Gram Pound Kilogram Kilogram Pound

Centigrade Centigrade Centigrade Fahrenheit Centigrade Fahrenheit Kelvin Centigrade Fahrenheit

a

Temperature conversion formulas: tc tf tK 1F 1K tc tt tK

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

temperature in Centigrade degrees temperature in Fahrenheit degrees temperature in Kelvin degrees 5 9 C 1 C 5 32Þ 9 ðtf 9 5 tc þ 32 tc þ 273

Factor 1 1 1.800 1.000 4.186 1055 4.187  103 1.163  10 3 2.930  10 4

TABLE 51.62 Thermal Conductivitya [LMT 3t 1]

to Obtain #

Multiply Number of ! by !

Btu
ft/h
ft2
 F Btu
in./h
ft2
 F Btu
in./s
ft2
 F J/m
s
 C kcal/m
h
 C erg/cm
s
 C kcal/m
s
 C cal/cm
s
 C W/ft
 C W/m
K a

Btu
ft/h
ft2
 F 1 12 3.333  10 3 1.731 1.483 1.731  105 4.134  10 4 4.134  10 3 5.276  10 1 1.731

Btu
in./ h
ft2
 F 8.333  10 1 2.778  10 1.442  10 1.240  10 1.442  10 3.445  10 3.445  10 4.395  10 1.442  10

Btu
in./ sec
ft2
 F 2

4 1 1 4 5 4 2 1

3.0  102 3.6  103 1 5.192  102 4.465  102 5.192  107 1.240  10 1 1.240 1.582  102 5.192  102

J/m
s
 C 5.778  10 6.933 1.926  10 1 8.599  10 1.0  105 2.388  10 2.388  10 3.048  10 1.0

kcal/m
h
 C erg/cm
s
 C kcal/m
s
 C cal/cm
s
 C 1

3

1

4 3 1

6.720  10 1 8.064 2.240  10 3 1.163 1 1.163  105 2.778  10 4 2.778  10 3 3.545  10 1 1.163

5.778  10 6.933  10 1.926  10 1.000  10 8.599  10 1 2.388  10 2.388  10 3.048  10 1.00  10

International Table Btu ¼ 1.055056  103 joules and International Table cal ¼ 4.1868 J are used throughout.

6 5 8 5 6

9 8 6 5

2.419  103 2.903  104 8.064 4.187  103 3.6  103 4.187  108 1 10 1.276  103 4.187  103

W/ft
 C

2.419  102 1.895 2.903  103 2.275  101 8.064  10 1 6.319  10 3 4.187  102 3.281 3.6  102 2.821 4.187  107 3.281  105 1.0  10 1 7.835  10 4 1 7.835  10 3 2 1.276  10 1 4.187  102 3.281

W/m
K 5.778  10 6.933 1.926  10 1.0 8.599  10 1.0  105 2.388  10 2.388  10 3.048  10 1

1

3

1

4 3 1

1825

1826

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.63 Photometric Units Common Unit Luminous intensity Luminance

Luminous flux Quantity of light flux Luminous exitancea Illuminanceb

Multiply by

International candle cd/in.2 cd/cm2 Foot
lambert cd
sr Candle power (spher.)

9.81  10 1.550  103 1  104 3.4263 1.0000 12.566

lm Foot candles lmft2 lx Phots

3.103  103 1.0764  10 1.0764  10 1.000 1  104

Luminous efficacy a b

to Get SI Unit

1

cd cd/m2 cd/m2 cd/m2 lm lm lm
lm/m2 cd/m2 lm/m2 lm/m2 lm/m2 lm/m2 Im/W

Luminous emittance. Luminous flux density.

TABLE 51.64 Specific Gravity Conversions Specific Gravity 60 /60 0.600 0.605 0.610 0.615 0.620 0.625 0.630 0.635 0.640 0.645 0.650 0.655 0.660 0.665 0.670 0.675 0.680 0.685 0.690 0.695 0.700 0.705 0.710 0.715 0.720 0.725



Be



API

103.33 104.33 101.40 102.38 99.51 100.47 97.64 98.58 95.81 96.73 94.00 94.90 92.22 93.10 90.47 91.33 88.75 89.59 87.05 87.88 85.38 86.19 83.74 84.53 82.12 82.89 80.53 81.28 78.96 79.69 77.41 78.13 75.88 76.59 74.38 75.07 72.90 73.57 71.44 72.10 70.00 68.58 67.18 65.80 64.44 63.10

70.64 69.21 67.80 66.40 65.03 63.67

lb/ft3 at lb/gal 60 F, wt 60 F, wt in Air in Air 4.9929 5.0346 5.0763 5.1180 5.1597 5.2014 5.2431 5.2848 5.3265 5.3682 5.4098 5.4515 5.4932 5.5349 5.5766 5.6183 5.6600 5.7017 5.7434 5.7851

37.350 37.662 37.973 38.285 38.597 39.910 39.222 39.534 39.845 40.157 40.468 40.780 41.092 41.404 41.716 42.028 42.340 42.652 42.963 43.275

5.8268 5.8685 5.9101 5.9518 5.9935 6.0352

43.587 43.899 44.211 44.523 44.834 45.146

lb/ft3 at lb/gal 60 F, wt 60 F, wt in Air in Air

Specific Gravity 60 /60



0.730 0.735 0.740

61.78 60.48 59.19

62.34 61.02 59.72

6.0769 6.1186 6.1603

45.458 45.770 46.082

0.745 0.750 0.755 0.760 0.765 0.770 0.775 0.780 0.785 0.790 0.795

57.92 56.67 55.43 54.21 53.01 51.82 50.65 49.49 48.34 47.22 46.10

58.43 57.17 55.92 54.68 53.47 52.27 51.08 49.91 48.75 47.61 46.49

6.2020 6.2437 6.2854 6.3271 6.3688 6.4104 6.4521 6.4938 6.5355 6.5772 6.6189

46.394 46.706 47.018 47.330 47.642 47.953 48.265 48.577 48.889 49.201 49.513

0.800 0.805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 0.845 0.850 0.855

45.00 43.91 42.84 41.78 40.73 39.70 38.67 37.66 36.67 35.68 34.71 33.74

45.38 44.28 43.19 42.12 41.06 40.02 38.98 37.96 36.95 35.96 34.97 34.00

6.6606 6.7023 6.7440 6.7857 6.8274 6.8691 6.9108 6.9525 6.9941 7.0358 7.0775 7.1192

49.825 50.137 50.448 50.760 51.072 51.384 51.696 52.008 52.320 52.632 52.943 53.225

Be



API

TABLES OF CONVERSION FACTORS

1827

TABLE 51.64 (Continued) lb/gal lb/ft3 at  60 F, wt 60 F, wt in Air in Air

Specific Gravity 60 /60



0.860 0.865 0.870 0.875 0.880 0.885 0.890 0.895

32.79 31.85 30.92 30.00 29.09 28.19 27.30 26.42

33.03 32.08 31.14 30.21 29.30 28.38 27.49 26.60

7.1609 7.2026 7.2443 7.2860 7.3277 7.3694 7.4111 7.4528

53.567 53.879 54.191 54.503 54.815 55.127 55.438 55.750

0.900 0.905 0.910 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995

25.76 24.70 23.85 23.01 22.17 21.35 20.54 19.73 18.94 18.15 17.37 16.60 15.83 15.08 14.33 13.59 12.86 12.13 11.41 10.70

25.72 24.85 23.99 23.14 22.30 21.47 20.65 19.84 19.03 18.24 17.45 16.67 15.90 15.13 14.38 13.63 12.89 12.15 11.43 10.71

7.4944 7.5361 7.5777 7.6194 7.6612 7.7029 7.7446 7.7863 7.8280 7.8697 7.9114 7.9531 7.9947 8.0364 8.0780 8.1197 8.1615 8.2032 8.2449 8.2866

56.062 56.374 56.685 56.997 57.410 57.622 57.934 58.246 58.557 58.869 59.181 59.493 59.805 60.117 60.428 60.740 61.052 61.364 61.676 61.988

Be

Specific Gravity 60 /60



1.000 1.005 1.010 1.015 1.020 1.025 1.030 1.035 1.040 1.045 1.050 1.055 1.060 1.065 1.070

10.00 0.72 1.44 2.14 2.84 3.54 4.22 4.90 5.58 6.24 6.91 7.56 8.21 8.85 9.49

Be





API

TW

10.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14

lb/ft3 at lb/gal  60 F, wt 60 F, wt in Air in Air 8.3283 8.3700 8.4117 8.4534 8.4950 8.5367 8.5784 8.6201 8.6618 8.7035 8.7452 8.7869 8.8286 8.8703 8.9120

62.300 62.612 62.924 63.236 63.547 63.859 64.171 64.483 64.795 65.107 65.419 65.731 66.042 66.354 66.666

lb/gal lb/ft3 at  60 F, wt 60 F, wt in Air in Air

Specific Gravity 60 /60



1.075 1.080 1.085 1.090 1.095

10.12 10.74 11.36 11.97 12.58

15 16 17 18 19

8.9537 8.9954 9.0371 9.0787 9.1204

66.978 67.290 67.602 67.914 68.226

1.100 1.105 1.110 1.115 1.120 1.125 1.130 1.135 1.140 1.145 1.150 1.155 1.160 1.165 1.170 1.175 1.180 1.185 1.190 1.195

13.18 13.78 14.37 14.96 15.54 16.11 16.68 17.25 17.81 18.36 18.91 19.46 20.00 20.54 21.07 21.60 22.12 22.64 23.15 23.66

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

9.1621 9.2038 9.2455 9.2872 9.3289 9.3706 9.4123 9.4540 9.4957 9.5374 9.5790 9.6207 9.6624 9.7041 9.7458 9.7875 9.8292 9.8709 9.9126 9.9543

68.537 68.849 69.161 69.473 69.785 70.097 70.409 70.721 71.032 71.344 71.656 71.968 72.280 72.592 72.904 73.216 73.528 73.840 74.151 74.463

1.200 1.205 1.210 1.215 1.220 1.225 1.230 1.235 1.240 1.245 1.250 1.255 1.260 1.265 1.270 1.275 1.280 1.285 1.290 1.295 1.300

24.17 24.67 25.17 25.66 26.15 26.63 27.11 27.59 28.06 28.53 29.00 29.46 29.92 30.38 30.83 31.27 31.72 32.16 32.60 33.03 33.46

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

9.9960 10.0377 10.0793 10.1210 10.1627 10.2044 10.2461 10.2878 10.3295 10.3712 10.4129 10.4546 10.4963 10.5380 10.5797 10.6214 10.6630 10.7047 10.7464 10.7881 10.8298

74.775 75.087 75.399 75.711 76.022 76.334 76.646 76.958 77.270 77.582 77.894 78.206 78.518 78.830 79.141 79.453 79.765 80.077 80.389 80.701 81.013

Be



TW API

1828

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.64 (Continued)

API

lb/gal lb/ft3 at  60 F, wt 60 F, wt in Air in Air

Specific Gravity 60 /60



33.89 34.31 34.73 35.15 35.57 35.98 36.39 36.79 37.19 37.59 37.99 38.38 38.77 39.16 39.55 39.93 40.31 40.68 41.06

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

10.8715 10.9132 10.9549 10.9966 11.0383 11.0800 11.1217 11.1634 11.2051 11.2467 11.2884 11.3301 11.3718 11.4135 11.4552 11.4969 11.5386 11.5803 11.6220

81.325 81.636 81.948 82.260 82.572 82.884 83.196 83.508 83.820 84.131 84.443 84.755 85.067 85.379 85.691 86.003 86.315 86.626 86.938

1.56 1.57 1.58 1.59

1.400 1.405 1.410 1.415 1.420 1.425 1.430 1.435 1.440 1.445 1.450 1.455 1.460 1.465 1.470 1.475 1.480 1.485 1.490

41.43 41.80 42.16 42.53 42.89 43.25 43.60 43.95 44.31 44.65 45.00 45.34 45.68 46.02 46.36 46.69 47.03 47.36 47.68

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

11.6637 11.7054 11.7471 11.7888 11.8304 11.8721 11.9138 11.9555 11.9972 12.0389 12.0806 12.1223 12.1640 12.2057 12.2473 12.2890 12.3307 12.3724 12.4141

87.250 87.562 87.874 88.186 88.498 88.810 89.121 89.433 89.745 90.057 90.369 90.681 90.993 91.305 91.616 91.928 92.240 92.552 92.864

1.495 1.500 1.51 1.52 1.53 1.54 1.55

48.01 48.33 48.97 49.61 50.23 50.84 51.45

99 100 102 104 106 108 110

12.4558 12.4975 12.581 12.644 12.748 12.831 12.914

93.176 93.488 94.11 94.79 95.36 95.98 96.61

Specific Gravity 60 /60



Be

 TW

1.305 1.310 1.315 1.320 1.325 1.330 1.335 1.340 1.345 1.350 1.355 1.360 1.365 1.370 1.375 1.380 1.385 1.390 1.395



lb/gal lb/ft3 at  60 F, wt 60 F, wt in Air in Air

52.05 52.64 53.23 53.81

112 114 116 118

12.998 13.081 13.165 13.248

97.23 97.85 98.48 99.10

1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79

54.38 54.94 55.49 56.04 56.59 57.12 57.65 58.17 58.69 59.20 59.71 60.20 60.70 61.18 61.67 62.14 62.61 63.08 63.54 63.99

120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158

13.331 13.415 13.498 13.582 13.665 13.748 13.832 13.915 13.998 14.082 14.165 14.249 14.332 14.415 14.499 14.582 14.665 14.749 14.832 14.916

99.73 100.35 100.97 101.60 102.22 102.84 103.47 104.09 104.72 105.34 105.96 106.59 107.21 107.83 108.46 109.08 109.71 110.32 110.95 111.58

1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89

64.44 64.89 65.33 65.77 66.20 66.62 67.04 67.46 67.87 68.28

160 162 164 166 168 170 172 174 176 178

14.999 15.082 15.166 15.249 15.333 15.416 15.499 15.583 15.666 15.750

112.20 112.82 113.45 114.07 114.70 115.31 115.94 116.56 117.19 117.81

1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00

68.68 69.08 69.48 69.87 70.26 70.64 71.02 71.40 71.77 72.14 72.50

180 182 184 186 188 190 192 194 196 198 200

15.832 15.916 16.000 16.083 16.166 16.250 16.333 16.417 16.500 16.583 16.667

118.43 119.06 119.68 120.31 120.93 121.56 122.18 122.80 123.43 124.05 124.68

Be

TW API

STANDARD SIZES

1829

51.6 STANDARD SIZES 51.6.1 Preferred Numbers Selection of standard sizes or ratings of many diverse products can be performed advantageously through the use of a geometrically based progression introduced by C. Renard. He originally adopted as a basis a rule that would yield a 10th multiple of the value a after every 5th step of the series: a  q5 ¼ 10a or

q¼

p ffiffiffiffiffi 5 10

pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi where the numerical series a, a½ 5 10Š; a½ 5 10Š2 ; a½ 5 10Š3 ; ½ 5 10Š4 , 10a, the values of which, to five significant figures, are a, 1.5849a, 2.5119a, 3.9811a, 6.309a, 10a. Renard’s idea was to substitute, for these values, more rounded but more practical values. He adopted as a a power of 10, positive, nil, or negative, obtaining the series 10, 16, 25, 40, 63, 100, which may be continued in both directions. From this series, designated by the symbol R5, the R10, R20, R40 formed, pffiffiffiffiffi pffiffiffiffiffiwere pffiffiffiffiffiseries each adopted ratio being the square root of the preceding one: 10 10, 20 10, 40 10. Thus each series provided Renard with twice as many steps in a decade as the preceding one. Preferred numbers are immediately applicable to commercial sizes and ratings of products. It is advantageous to minimize the number of initial sizes and also to have adequate provision for logical expansion if and when additional sizes are required. By making the initial sizes correspond to a coarse series such as R5, unnecessary expense can be avoided if subsequent demand for the product is disappointing. If, on the other hand, the product is accepted, intermediate sizes may be selected in a rational manner by using the next finer series R10, and so on. Such a procedure assures a justifiable relationship between successive sizes and is a decided contrast to haphazard selection. The application of preferred numbers to raw material sizes and to the dimensions of parts also has enormously important potentialities. Under present conditions, commercial sizes of material are the result of a great many dissimilar gauge systems. The current trend in internationally acceptable metric sizing is to use preferred numbers. Even here, though, in the midst of the greatest opportunity for worldwide standardization through the acceptance of Renard series, we have fallen prey to our individualistic nature. The preferred number 1.6 is used by most nations as a standard 1.6 mm material thickness. German manufacturers, however, like 1.5 mm of the International Organization for Standardization (ISO) 497 for a more rounded preferred number. Similarly in metric screw sizes, 6.3 mm is consistent with the preferred number series; yet, 6.0 mm (more rounded) has been adopted as a standard fastener diameter. The International Electrochemical Commission (IEC) used preferred numbers to establish standard current ratings in amperes as follows: 1, 1.25, 1.6, 2.5, 3.15, 4.5, 6.3. Notice that R10 series is used except for 4.5, which is a third step R20 series. The American Wire Gauge size for copper p wire ffiffiffiffiffi is based on a geometric series. However, instead of using 1.1220, the rounded value of 20 10, in a  q20 ¼ 10a, the q chosen is 1.123. A special series of preferred numbers is used for designating the characteristic values of capacitors, resistors, inductors, and other electronic products. Instead of using the Renard series R5, R10, R20, R40, R80 as derived from the geometric series of numbers 10N/5, 10N/10, 10N/20, 10N/40, 10N/80, the geometric series used is 10N/6, 10N/12, 10N/24, 10N/48, 10N/96, 10N/192, which are designated respectively E6, E12, E24, E48, E96, E192.

1830

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

It should be evident that any series of preferred numbers can be generated to serve any specific case. Examples taken from the American National Standards Institute (ANSI) and ISO standards are reproduced in Tables 51.65–51.68. TABLE 51.65 Basic Series of Preferred Numbers: R5, R10, R20, and R40 Series Theoretical Values R5

R10

R20

R40

Mantissas of Logarithms

1.00

1.00

1.00

1.00 1.06 1.12 1.18 1.25 1.32 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.12 2.24 2.36 2.50 2.65 2.80 3.00 3.15 3.35 3.55 3.75 4.00 4.25 4.50 4.75 5.00 5.30 5.60 6.00 6.30 6.70 7.10 7.50 8.00 8.50 9.00 9.50 10.00

000 025 050 075 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 000

1.12 1.25

1.25 1.40

1.60

1.60

1.60 1.80

2.00

2.00 2.24

2.50

2.50

2.50 2.80

3.15

3.15 3.55

4.00

4.00

4.00 4.50

5.00

5.00 5.60

6.30

6.30

6.30 7.10

8.00

8.00 9.00

10.00

10.00

10.00

Calculated Values

Differences between Basic Series and Calculated Values (%)

1.0000 1.0593 1.1220 1.1885 1.2589 1.3335 1.4125 1.4962 1.5849 1.6788 1.7783 1.8836 1.9953 2.1135 2.2387 2.3714 2.5119 2.6607 2.8184 2.9854 3.1623 3.3497 3.5481 3.7584 3.9811 4.2170 4.4668 4.7315 5.0119 5.3088 5.6234 5.9566 6.3096 6.6834 7.0795 7.4989 7.9433 8.4140 8.9125 9.4406 10.0000

0 þ0.07 0.18 0.71 0.71 1.01 0.88 þ0.25 þ0.95 þ1.26 þ1.22 þ0.87 þ0.24 þ0.31 þ0.06 0.48 0.47 0.40 0.65 þ0.49 0.39 þ0.01 þ0.05 0.22 þ0.47 þ0.78 þ0.74 þ0.39 0.24 0.17 0.42 þ0.73 0.15 þ0.25 þ0.29 þ0.01 þ0.71 þ1.02 þ0.98 þ0.63 0

STANDARD SIZES

1831

TABLE 51.66 Basic Series of Preferred Numbers: R80 Series 1.00 1.03 1.06 1.09 1.12 1.15 1.18 1.22 1.25 1.28 1.32 1.36 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

1.80 1.85 1.90 1.95 2.00 2.06 2.12 2.18 2.24 2.30 2.36 2.43 2.50 2.58 2.65 2.72 2.80 2.90 3.00 3.07

3.15 3.25 3.35 3.45 3.55 3.65 3.75 3.87 4.00 4.12 4.25 4.37 4.50 4.62 4.75 4.87 5.00 5.15 5.20 5.45

5.60 5.80 6.00 6.15 6.30 6.50 6.70 6.90 7.10 7.30 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

TABLE 51.67 Expansion of R5 Series Preferred Number 1.0 1.6 2.5 4.0 6.3

Divided by 10

Multiplied by 10

Multiplied by 100

Multiplied by 1000

0.10 0.16 0.25 0.40 0.63

10 16 25 40 63

100 160 250 400 630

1000 1600 2500 4000 6300

TABLE 51.68 Rounding of Preferred Numbersa Preferred Number 1.12 1.25 1.60 2.24 3.15 3.55 5.60 6.30 7.10 a

First Rounding

Second Rounding

1.1 1.25 1.6 2.2 3.2 3.6 5.6 6.3 7.1

1.1 1.2 1.5a 2.2 3.0 3.5 5.5 6.0 7.0

Rounded only when using the R5 or R10 series.

1832

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

Applicable Documents Adoption of Renard’s preferred number system by international standardization bodies resulted in a host of national standards being generated for particular applications. The current organization in the United States that is charged with generating American national standards is the ANSI. Accordingly, the following national and international standards are in use in the United States. ANSI Z17.1-1973 ANSI C83.2-1971

American National Standard for Preferred Numbers American National Standard Preferred Values for Components for Electronic Equipment Preferred Values for Components for Electronic Equipment (issued by EIA Standard the Electronics Industries Association; Same as ANSI C83.2-1971) RS-385 ISO 3-1973 Preferred numbers—series of preferred numbers ISO 17-1973 Guide to the use of preferred numbers and of series of preferred numbers ISO 497-1973 Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers Table 51.67 shows the expansibility of preferred numbers in the positive direction. The same expansibility can be made in the negative direction. Table 51.68 shows a deviation by roundings for cases where adhering to a basic preferred number would be absurd as in 31.5 teeth in a gear when clearly 32 makes sense. 51.6.2 Gages 51.6.2.1 Wire Gages The sizes of wires having a diameter less than 12 in. are usually stated in terms of certain arbitrary scales called “gages.” The size or gage number of a solid wire refers to the cross section of the wire perpendicular to its length; the size or gage number of a stranded wire refers to the total cross section of the constituent wires, irrespective of the pitch of the spiraling. Larger wires are usually described in terms of their area expressed in circular mils. A circular mil is the area of a circle 1 mil in diameter, and the area of any circle in circular mils is equal to the square of its diameter in mils. TABLE 51.69 U.S. Standard Gagea for Sheet and Plate Iron and Steel and Its Extensionb

Weight per Square Foot Gage Number

oz.

lb

0000000 000000 00000 0000 000 00 0 1 2 3 4

320 300 280 260 240 220 200 180 170 160 150

20.00 18.75 17.50 16.25 15.00 13.75 12.50 11.25 10.62 10.00 9.375

Weight per Square Meter kg 97.65 91.55 85.44 79.34 73.24 67.13 61.03 54.93 51.88 48.82 45.77

Approximate Thickness Wrought Iron, 480 lb/ft3

Steel and openhearth Iron, 489.6 lb/ft3

in.

mm

in.

mm

0.500 0.469 0.438 0.406 0.375 0.344 0.312 0.2812 0.2656 0.2500 0.2344

12.70 11.91 11.11 10.32 9.52 8.73 7.94 7.14 6.75 6.35 5.95

0.490 0.460 0.429 0.398 0.368 0.337 0.306 0.2757 0.2604 0.2451 0.2298

12.45 11.67 10.90 10.12 9.34 8.56 7.78 7.00 6.62 6.23 5.84

STANDARD SIZES

1833

TABLE 51.69 (Continued )

Weight per Square Foot

Weight per Square Meter

Approximate Thickness Wrought Iron, 480 lb/ft3

Steel and openhearth Iron, 489.6 lb/ft3

Gage Number

oz.

lb

kg

in.

mm

in.

mm

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

140 130 120 110 100 90 80 70 60 50 45 40 36 32 28 24 22 20 18 16 14 12 11 10 9 8 7 6 12 6 5 12 5 4 12 4 14 4 3 34 3 12 3 38 3 14 3 18 3

8.750 8.125 7.500 6.875 6.250 5.625 5.000 4.375 3.750 3.125 2.812 2.500 2.250 2.000 1.750 1.500 1.375 1.250 1.125 1.000 0.8750 0.7500 0.6875 0.6250 0.5625 0.5000 0.4375 0.4062 0.3750 0.3438 0.3125 0.2812 0.2656 0.2500 0.2344 0.2188 0.2109 0.2031 0.1953 0.1875

42.72 39.67 36.62 33.57 30.52 27.46 24.41 21.36 18.31 15.26 13.73 12.21 10.99 9.765 8.544 7.324 6.713 6.103 5.493 4.882 4.272 3.662 3.357 3.052 2.746 2.441 2.136 1.983 1.831 1.678 1.526 1.373 1.297 1.221 1.144 1.068 1.030 0.9917 0.9536 0.9155

0.2188 0.2031 0.1875 0.1719 0.1562 0.1406 0.1250 0.1094 0.0938 0.0781 0.0703 0.0625 0.0562 0.0500 0.0438 0.0375 0.0344 0.0312 0.0281 0.0250 0.0219 0.0188 0.0172 0.0156 0.0141 0.0125 0.0109 0.0102 0.0094 0.0086 0.0078 0.0070 0.0066 0.0062 0.0059 0.0055 0.0053 0.0051 0.0049 0.0047

5.56 5.16 4.76 4.37 3.97 3.57 3.18 2.778 2.381 1.984 1.786 1.588 1.429 1.270 1.111 0.952 0.873 0.794 0.714 0.635 0.556 0.476 0.437 0.397 0.357 0.318 0.278 0.258 0.238 0.218 0.198 0.179 0.169 0.159 0.149 0.139 0.134 0.129 0.124 0.119

0.2145 0.1991 0.1838 0.1685 0.1532 0.1379 0.1225 0.1072 0.0919 0.0766 0.0689 0.0613 0.0551 0.0490 0.0429 0.0368 0.0337 0.0306 0.0276 0.0245 0.0214 0.0184 0.0169 0.0153 0.0138 0.0123 0.0107 0.0100 0.0092 0.0084 0.0077 0.0069 0.0065 0.0061 0.0057 0.0054 0.0052 0.0050 0.0048 0.0046

5.45 5.06 4.67 4.28 3.89 3.50 3.11 2.724 2.335 1.946 1.751 1.557 1.400 1.245 1.090 0.934 0.856 0.778 0.700 0.623 0.545 0.467 0.428 0.389 0.350 0.311 0.272 0.253 0.233 0.214 0.195 0.175 0.165 0.156 0.146 0.136 0.131 0.126 0.122 0.117

a For the Galvanized Sheet Gage, add 2.5 oz to the weight per square foot as given in the table. Gage numbers below 8 and above 34 are not used in the Galvanized Sheet Gage. b Gage numbers greater than 38 were not in the standard as set up by law but are in general use.

1834

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.70 American Wire Gage: Weights of Copper, Aluminum, and Brass Sheets and Plates Approximate Weight,a lb/ft2

Thickness Gage Number 0000 000 00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 a

in.

mm

Copper

Aluminum

Commercial (High) Brass

0.4600 0.4096 0.3648 0.3249 0.2893 0.2576 0.2294 0.2043 0.1819 0.1620 0.1443 0.1285 0.1144 0.1019 0.0907 0.0808 0.0720 0.0641 0.0571 0.0508 0.0453 0.0403 0.0359 0.0320 0.0285 0.0253 0.0226 0.0201 0.0179 0.0159 0.0142 0.0126 0.0113 0.0100 0.00893 0.00795 0.00708 0.00630 0.00561 0.00500 0.00445 0.00397 0.00353 0.00314

11.68 10.40 9.266 8.252 7.348 6.544 5.827 5.189 4.621 4.115 3.665 3.264 2.906 2.588 2.305 2.053 1.828 1.628 1.450 1.291 1.150 1.024 0.9116 0.8118 0.7230 0.6438 0.5733 0.5106 0.4547 0.4049 0.3606 0.3211 0.2859 0.2546 0.2268 0.2019 0.1798 0.1601 0.1426 0.1270 0.1131 0.1007 0.0897 0.0799

21.27 18.94 16.87 15.03 13.38 11.91 10.61 9.45 8.41 7.49 6.67 5.94 5.29 4.713 4.195 3.737 3.330 2.965 2.641 2.349 2.095 1.864 1.660 1.480 1.318 1.170 1.045 0.930 0.828 0.735 0.657 0.583 0.523 0.4625 0.4130 0.3677 0.3274 0.2914 0.2595 0.2312 0.2058 0.1836 0.1633 0.1452

6.49 5.78 5.14 4.58 4.08 3.632 3.234 2.880 2.565 2.284 2.034 1.812 1.613 1.437 1.279 1.139 1.015 0.904 0.805 0.716 0.639 0.568 0.506 0.451 0.402 0.3567 0.3186 0.2834 0.2524 0.2242 0.2002 0.1776 0.1593 0.1410 0.1259 0.1121 0.0998 0.0888 0.0791 0.0705 0.0627 0.0560 0.0498 0.0443

20.27 18.05 16.07 14.32 12.75 11.35 10.11 9.00 8.01 7.14 6.36 5.66 5.04 4.490 3.996 3.560 3.172 2.824 2.516 2.238 1.996 1.776 1.582 1.410 1.256 1.115 0.996 0.886 0.789 0.701 0.626 0.555 0.498 0.4406 0.3935 0.3503 0.3119 0.2776 0.2472 0.2203 0.1961 0.1749 0.1555 0.1383

Assumed specific gravities or densities in grams per cubic centimeter; copper, 8.89; aluminum, 2.71; brass, 8.47.

STANDARD SIZES

1835

TABLE 51.71 Comparison of Wire Gage Diameters in Milsa

Gage No. 7–0 6–0 5–0 4–0 3–0 2–0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

American Wire Gage (Brown & Sharpe)

Steel Wire Gage

Birmingham Wire Gage (Stubs’)

Old English Wire Gage (London)

Stubs’ Steel Wire Gage

(British) Standard Wire Gage

— — — 460 410 365 325 289 258 229 204 182 162 144 128 114 102 91 81 72 64 57 51 45 40 36 32 28.5 25.3 22.6 20.1 17.9 15.9 14.2 12.6 11.3 10.0 8.9 8.0 7.1 6.3 5.6 5.0 4.5

490.0 461.5 430.5 393.8 362.5 331.0 306.5 283.0 262.5 243.7 225.3 207.0 192.0 177.0 162.0 148.3 135.0 120.5 105.5 91.5 80.0 72.0 62.5 54.0 47.5 41.0 34.8 31.7 28.6 25.8 23.0 20.4 18.1 17.3 16.2 15.0 14.0 13.2 12.8 11.8 10.4 9.5 9.0 8.5

— — — 454 425 380 340 300 284 259 238 220 203 180 165 148 134 120 109 95 83 72 65 58 49 42 35 32 28 25 22 20 18 16 14 13 12 10 9 8 7 5 4 —

— — — 454 425 380 340 300 284 259 238 220 203 180 165 148 134 120 109 95 83 72 65 58 49 42 35 31.5 29.5 27.0 25.0 23.0 20.5 18.75 16.50 15.50 13.75 12.25 11.25 10.25 9.50 9.00 7.50 6.50

— — — — — — — 227 219 212 207 204 201 199 197 194 191 188 185 182 180 178 175 172 168 164 161 157 155 153 151 148 146 143 139 134 127 120 115 112 110 108 106 103

500 464 432 400 372 348 324 300 276 252 232 212 192 176 160 144 128 116 104 92 80 72 64 56 48 40 36 32 28 24 22 20 18 16.4 14.8 13.6 12.4 11.6 10.8 10.0 9.2 8.4 7.6 6.8

Metric Gageb

— — — — — — — 3.94 7.87 11.8 15.7 19.7 23.6 27.6 31.5 35.4 39.4 — 47.2 — 55.1 — 63.0 — 70.9 — 78.7 — — — — 98.4 — — — — 118 — — — — 138 — — (continued)

1836

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.71 (Continued )

Gage No. 38 39 40 41 42 43 44 45 46 47 48 49 50

American Wire Gage (Brown & Sharpe) 4.0 3.5 3.1 — — — — — — — — — —

Steel Wire Gage

Birmingham Wire Gage (Stubs’)

8.0 7.5 7.0 6.6 6.2 6.0 5.8 5.5 5.2 5.0 4.8 4.6 4.4

Old English Wire Gage (London)

— — — — — — — — — — — — —

5.75 5.00 4.50 — — — — — — — — — —

Stubs’ Steel Wire Gage

(British) Standard Wire Gage

101 99 97 95 92 88 85 81 79 77 75 72 69

6.0 5.2 4.8 4.4 4.0 3.6 3.2 2.8 2.4 2.0 1.6 1.2 1.0

Metric Gageb — — 157 — — — — 177 — — — — 197

a

Bureau of Standards, Circulars No. 31 and No. 67. For diameters corresponding to metric gage numbers, 1.2, 1.4, 1.6, 1.8, 2.5, 3.5, and 4.5, divide those of 12, 14, etc., by 10. b

TABLE 51.72 Standard Engineering Drawing Sizesa Flat Sizesb Margin Size Designation A (horizontal) A (vertical) B C D E F

Widthc (Vertical)

Length (Horizontal)

Horizontal

Vertical

8.5 11.0 11.0 17.0 22.0 34.0 28.0

11.0 8.5 17.0 22.0 34.0 44.0 40.0

0.38 0.25 0.38 0.75 0.50 1.00 0.50

0.25 0.38 0.62 0.50 1.00 0.50 0.50

Roll Sizes Lengthc (Horizontal) Size Designation G H J K a

Widthb (Vertical)

Min

Max

Horizontal

Vertical

11.0 28.0 34.0 40.0

22.5 44.0 55.0 55.0

90.0 143.0 176.0 143.0

0.38 0.50 0.50 0.50

0.50 0.50 0.50 0.50

See ANSI Y14.1-1980. All dimensions are in inches. c Not including added protective margins. b

Marginc

STANDARD SIZES

1837

TABLE 51.73 Eleven International Paper Sizes International Paper Size

Millimeters

Inches, Approximate

A-0

841  1189

33 18  46 34

A-1

594  841

23 38  33 18

297  420

11 34  16 12

420  594

A-2 A-3

210  297

A-4

148  210

A-5

105  148

A-6

74  105

A-7

52  74

A-8

37  52

A-9

26  37

A-10

16 12  23 38 8 14  11 34 5 78  8 14

4 18  5 78 2 78  4 18 2  2 78

1 12  2

1  1 12

51.6.3 Paper Sizes 51.6.3.1 International Paper Sizes Countries that are committed to the International System of Units (SI) have a standard series of paper sizes for printing, writing, and drafting. These paper sizes are called the “international paper sizes.” The advantages of the international paper sizes are as follows: 1. The ratio of width to length remains constant for every size, namely: Width 1 ¼ pffiffiffi Length 2

or

1 approximately 1:414

Since this is the same ratio as the D aperture in the unitized 35-mm microfilm frame, the advantages are apparent. pffiffiffi 2. If a sheet is cut in half, that is, if the p 2 ffiffilength is cut in half, the two halves retain ffi the constant width-to-length ratio of 1= 2. No other ratio could do this. 3. All international sizes are created from the A-0 size by single cuts without waste. In storing or stacking they fit together like parts of a jigsaw puzzle—without waste.

1838

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

51.6.4 Sieve Sizes TABLE 51.74 Tyler Standard Screen Scale Sieves This screen scale has as its base an opening of 0.0029 in., which is the opening in 200-mesh 0.0021-in. wire, the standard sieve, as adopted by the Bureau of Standards of the U.S. government, the openings increasing in the ratio of the square root of 2 or 1.414. Where a closer sizing is required, column 5 shows the Tyler Standard Screen Scale with intermediate sieves. In this series the sieve openings increase in the ratio of the fourth root of 2, or 1.189. Every Every Every Fourth Other Sieve Other Sieve Tyler StanSieve from from from dard Screen 0.0029 to 0.0029 to 0.0041 to pffiffiffi Scale 2 or 0.742 in., 0.742 in., 1.050 in., 1.414 Open- Ratio of 2 Ratio of 2 Ratio of 4 ings (in.) to 1 to 1 to 1 (1) (4) (3) (2) 1.050 — 0.742 — 0.525 — 0.371 — 0.263 — 0.185 — 0.131 — 0.093 — 0.065 — 0.046 — 0.0328 — 0.0232 — 0.0164 — 0.0116 — 0.0082 — 0.0058 — 0.0041 — 0.0029

— — 0.742 — — — 0.371 — — — 0.185 — — — 0.093 — — — 0.046 — — — 0.0232 — — — 0.0116 — — — 0.0058 — — — 0.0029

1.050 — — — 0.525 — — — 0.263 — — — 0.131 — — — 0.065 — — — 0.0328 — — — 0.0164 — — — 0.0082 — — — 0.0041 — —

— — 0.742 — — — — — — — 0.185 — — — — — — — 0.046 — — — — — — — 0.0116 — — — — — — — 0.0029

For Closer Sizing Sieves Openings from 0.0029 in Fracto 1.050 in., tions of ffiffi ffi p Ratio 4 2 or openings inch Diameter 1.189 (mm) (approx.) Mesh of Wire (5) (6) (7) (8) (9) 1.050 0.883 0.742 0.624 0.525 0.441 0.371 0.312 0.263 0.221 0.185 0.156 0.131 0.110 0.093 0.078 0.065 0.055 0.046 0.0390 0.0328 0.0276 0.0232 0.0195 0.0164 0.0138 0.0116 0.0097 0.0082 0.0069 0.0058 0.0049 0.0041 0.0035 0.0029

26.67 22.43 18.85 15.85 13.33 11.20 9.423 7.925 6.680 5.613 4.699 3.962 3.327 2.794 2.362 1.981 1.651 1.397 1.168 0.991 0.833 0.701 0.589 0.495 0.417 0.351 0.295 0.246 0.208 0.175 0.147 0.124 0.104 0.088 0.074

1 7 8 3 4 5 8 1 2 7 16 3 8 5 16 1 4 7 32 3 16 5 32 1 8 7 64 3 32 5 84 1 16

— 3 64

— 1 32

— — — 1 64

— — — — — — — — — —

— — — — — — — 2 12 3 3 12 4 5 6 7 8 9 10 12 14 16 20 24 28 32 35 42 48 60 65 80 100 115 150 170 200

0.148 0.135 0.135 0.120 0.105 0.105 0.092 0.088 0.070 0.065 0.065 0.044 0.036 0.0328 0.032 0.033 0.035 0.028 0.025 0.0235 0.0172 0.0141 0.0125 0.0118 0.0122 0.0100 0.0092 0.0070 0.0072 0.0056 0.0042 0.0038 0.0026 0.0024 0.0021

STANDARD SIZES

1839

TABLE 51.75 Nominal Dimensions, Permissible Variations, and Limits for Woven Wire Cloth of Standard Sieves, U.S. Series, ASTM Standarda Size or Sieve Designation

mm

No.

mm

in. (approx. equivalents)

Permissible Variations in Average Opening (%)

5660 4760 4000 3360 2830 2380 2000 1680 1410 1190 1000 840 710 590 500 420 350 297 250 210 177 149 125 105 88 74 62 53 44 37

3 12 4 5 6 7 8 10 12 14 16 18 20 25 30 35 40 45 50 60 70 80 100 120 140 170 200 230 270 325 400

5.66 4.76 4.00 3.36 2.83 2.38 2.00 1.68 1.41 1.19 1.00 0.84 0.71 0.59 0.50 0.42 0.35 0.297 0.250 0.210 0.177 0.149 0.125 0.105 0.088 0.074 0.062 0.053 0.044 0.037

0.233 0.187 0.157 0.132 0.111 0.0937 0.0787 0.0661 0.0555 0.0469 0.0394 0.0331 0.0280 0.0232 0.0197 0.0165 0.0138 0.0117 0.0098 0.0083 0.0070 0.0059 0.0049 0.0041 0.0035 0.0029 0.0024 0.0021 0.0017 0.0015

3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7

Sieve Opening

Permissible Variations in Maximum Opening (%)

mm

in. (approx. equivalents)

10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 25 25 25 25 25 40 40 40 40 40 60 90 90 90 90

1.28–1.90 1.14–1.68 1.00–1.47 0.87–1.32 0.80–1.20 0.74–1.10 0.68–1.00 0.62–0.90 0.56–0.80 0.50–0.70 0.43–0.62 0.38–0.55 0.33–0.48 0.29–0.42 0.26–0.37 0.23–0.33 0.20–0.29 0.170–0.253 0.149–0.220 0.130–0.187 0.114–0.154 0.096–0.125 0.079–0.103 0.063–0.087 0.054–0.073 0.045–0.061 0.039–0.052 0.035–0.046 0.031–0.040 0.023–0.035

0.050–0.075 0.045–0.066 0.039–0.058 0.034–0.052 0.031–0.047 0.0291–0.0433 0.0268–0.0394 0.0244–0.0354 0.0220–0.0315 0.0197–0.0276 0.0169–0.0244 0.0150–0.0217 0.0130–0.0189 0.0114–0.0165 0.0102–0.0146 0.0091–0.0130 0.0079–0.0114 0.0067–0.0100 0.0059–0.0087 0.0051–0.0074 0.0045–0.0061 0.0038–0.0049 0.0031–0.0041 0.0025–0.0034 0.0021–0.0029 0.0018–0.0024 0.0015–0.0020 0.0014–0.0018 0.0012–0.0016 0.0009–0.0014

Wire Diameter

a For sieves from the 1000-mm (No. 18) to the 37-mm (No. 400) size, inclusive, not more than 5% of the openings shall exceed the nominal opening by more than one-half of the permissible variation in the maximum opening.

51.6.5 Standard Structural Sizes — Steel Steel Sections. Tables 76–83 give the dimensions, weights, and properties of rolled steel structural sections, including wide-flange sections, American standard beams, channels, angles, tees, and zees. The values for the various structural forms, taken from the eighth edition, 1980, of Steel Construction, by the kind permission of the publisher, the American Institute of Steel Construction, give the section specifications required in designing steel structures. The theory of design is covered in Section 4—Mechanics of Deformable Bodies.

1840

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.76 Properties of Wide-Flange Sections

Flange

Axis X–X

Axis Y–Y

Nominal Size (in.)

Weight per Foot (lb)

Area (in.2)

Depth (in.)

Width (in.)

Thickness (in.)

36  16 12

300

88.17

36.72

16.655

1.680

0.945

20290.2 1105.1 15.17 1225.2 147.1 3.73

280

82.32

36.50

16.595

1.570

0.885

18819.3 1031.2 15.12 1127.5 135.9 3.70

260

76.56

36.24

16.555

1.440

0.845

17233.8

951.1 15.00 1020.6 123.3 3.65

245

72.03

36.06

16.512

1.350

0.802

16092.2

892.5 14.95

944.7 114.4 3.62

230

67.73

35.88

16.475

1.260

0.765

14988.4

835.5 14.88

870.9 105.7 3.59

194

57.11

36.48

12.117

1.260

0.770

12103.4

663.6 14.56

355.4

58.7 2.49

182

53.54

36.32

12.072

1.180

0.725

11281.5

621.2 14.52

327.7

54.3 2.47

170

49.98

36.16

12.027

1.100

0.680

10470.0

579.1 14.47

300.6

50.0 2.45

160

47.09

36.00

12.000

1.020

0.653

9738.8

541.0 14.38

275.4

45.9 2.42

150

44.16

35.84

11.972

0.940

0.625

9012.1

502.9 14.29

250.4

41.8 2.38

240

70.52

33.50

15.865

1.400

0.830

13585.1

811.1 13.88

874.3 110.2 3.52

220

64.73

33.25

15.810

1.275

0.775

12312.1

740.6 13.79

782.4

200

58.79

33.00

15.750

1.150

0.715

11048.2

669.6 13.71

691.7

87.8 3.43

152

44.71

33.50

11.565

1.055

0.635

8147.6

486.4 13.50

256.1

44.3 2.39

141

41.51

33.31

11.535

0.960

0.605

7442.2

446.8 13.39

229.7

39.8 2.35

130

38.26

33.10

11.510

0.855

0.580

6699.0

404.8 13.23

201.4

35.0 2.29

210

61.78

30.38

15.105

1.315

0.775

9872.4

649.9 12.64

707.9

93.7 3.38

190

55.90

30.12

15.040

1.185

0.710

8825.9

586.1 12.57

624.6

83.1 3.34

172

50.65

29.88

14.985

1.065

0.655

7891.5

528.2 12.48

550.1

73.4 3.30

132

38.83

30.30

10.551

1.000

0.615

5753.1

379.7 12.17

185.0

35.1 2.18

124

36.45

30.16

10.521

0.930

0.585

5347.1

354.6 12.11

169.7

32.3 2.16

116

34.13

30.00

10.500

0.850

0.564

4919.1

327.9 12.00

153.2

29.2 2.12

108

31.77

29.82

10.484

0.760

0.548

4461.0

299.2 11.85

135.1

25.8 2.06

177

52.10

27.31

14.090

1.190

0.725

6728.6

492.8 11.36

518.9

73.7 3.16

160

47.04

27.08

14.023

1.075

0.658

6018.6

444.5 11.31

458.0

65.3 3.12

145

42.68

26.88

13.965

0.975

0.600

5414.3

402.9 11.26

406.9

58.3 3.09

114

33.53

27.28

10.070

0.932

0.570

4080.5

299.2 11.03

149.6

29.7 2.11

102

30.01

27.07

10.018

0.827

0.518

3604.1

266.3 10.96

129.5

25.9 2.08

94

27.65

26.91

9.990

0.747

0.490

3266.7

242.8 10.87

115.1

23.0 2.04

160

47.04

24.72

14.091

1.135

0.656

5110.3

413.5 10.42

492.6

69.9 3.23

145

42.62

24.49

14.043

1.020

0.608

4561.0

372.5 10.34

434.3

61.8 3.19

130

38.21

24.25

14.000

0.900

0.565

4009.5

330.7 10.24

375.2

53.6 3.13

120

35.29

24.31

12.088

0.930

0.556

3635.3

299.1 10.15

254.0

42.0 2.68

110

32.36

24.16

12.042

0.855

0.510

3315.0

274.4 10.12

229.1

38.0 2.66

100

29.43

24.00

12.000

0.775

0.468

2987.3

248.9 10.08

203.5

33.9 2.63

36  12

33  15 34

33  11 12

30  15

30  10 12

27  14

27  10

24  14

24  12

Web Thickness (in.)

I (in.4)

S (in.3)

r (in.)

I (in.4)

S (in.3)

r (in.)

99.0 3.48

1841

STANDARD SIZES

TABLE 51.76 (Continued ) Flange Nominal Size (in.)

Weight per Foot (lb)

Axis X–X

Axis Y–Y

Web Thickness (in.)

I (in.4)

S (in.3)

r (in.)

I (in.4)

Area (in.2)

Depth (in.)

Width (in.)

Thickness (in.)

94

27.63

24.29

9.061

0.872

0.516

2683.0

220.9

9.85

102.2

22.6 1.92

84

24.71

24.09

9.015

0.772

0.470

2364.3

196.3

9.78

88.3

19.6 1.89

76

22.37

23.91

8.985

0.682

0.440

2096.4

175.4

9.68

76.5

17.0 1.85

142

41.76

21.46

13.132

1.095

0.659

3403.1

317.2

9.03

385.9

58.8 3.04

127

37.34

21.24

13.061

0.985

0.588

3017.2

284.1

8.99

338.6

51.8 3.01

112

32.93

21.00

13.000

0.865

0.527

2620.6

249.6

8.92

289.7

44.6 2.96

21  9

96

28.21

21.14

9.038

0.935

0.575

2088.9

197.6

8.60

109.3

24.2 1.97

82

24.10

20.86

8.962

0.795

0.499

1752.4

168.0

8.53

89.6

20.0 1.93

21  8 14

73

21.46

21.24

8.295

0.740

0.455

100.3

150.7

8.64

66.2

16.0 1.76

68

20.02

21.13

8.270

0.685

0.430

1478.3

139.9

8.59

60.4

14.6 1.74

62

18.23

20.99

8.240

0.615

0.400

1326.8

126.4

8.53

53.1

12.9 1.71

114

33.51

18.48

11.833

0.991

0.595

2033.8

220.1

7.79

255.6

43.2 2.76

24  9

21  13

18  11 34

S (in.3)

r (in.)

105

30.86

18.32

11.792

0.911

0.554

1852.5

202.2

7.75

231.0

39.2 2.73

96

28.22

18.16

11.750

0.831

0.512

1674.7

184.4

7.70

206.8

35.2 2.71

85

24.97

18.32

8.838

0.911

0.526

1429.9

156.1

7.57

99.4

22.5 2.00

77

22.63

18.16

8.787

0.831

0.475

1286.8

141.7

7.54

88.6

20.2 1.98

70

20.56

18.00

8.750

0.751

0.438

1153.9

128.2

7.49

78.5

17.9 1.95

64

18.80

17.87

8.715

0.686

0.403

1045.8

117.0

7.46

70.3

16.1 1.93

60

17.64

18.25

7.558

0.695

0.416

984.0

107.8

7.47

47.1

12.5 1.63

55

16.19

18.12

7.532

0.630

0.390

889.9

98.2

7.41

42.0

11.1 1.61

50

14.71

18.00

7.500

0.570

0.358

800.6

89.0

7.38

37.2

9.9 1.59

16  11 12

96

28.22

16.32

11.533

0.875

0.535

1355.1

166.1

6.93

207.2

35.9 2.71

88

25.87

16.16

11.502

0.795

0.504

1222.6

151.3

6.87

185.2

32.2 2.67

16  8 12

78

22.92

16.32

8.586

0.875

0.529

1042.6

127.8

6.74

87.5

20.4 1.95

71

20.86

16.16

8.543

0.795

0.486

936.9

115.9

6.70

77.9

18.2 1.93

64

18.80

16.00

8.500

0.715

0.443

833.8

104.2

6.66

68.4

16.1 1.91

58

17.04

15.86

8.464

0.645

0.407

746.4

94.1

6.62

60.5

14.3 1.88

50

14.70

16.25

7.073

0.628

0.380

655.4

80.7

6.68

34.8

9.8 1.54

45

13.24

16.12

7.039

0.563

0.346

583.3

72.4

6.64

30.5

8.7 1.52

40

11.77

16.00

7.000

0.503

0.307

515.5

64.4

6.62

26.5

7.6 1.50

36

10.59

15.85

6.992

0.428

0.299

446.3

56.3

6.49

22.1

6.3 1.45

426

125.25

18.69

16.695

3.033

1.875

6610.3

707.4

7.26 2359.5 282.7 4.34

398

116.98

18.31

16.590

2.843

1.770

6013.7

656.9

7.17 2169.7 261.6 4.31

370

108.78

17.94

16.475

2.658

1.655

5454.2

608.1

7.08 1986.0 241.1 4.27

342

100.59

17.56

16.365

2.468

1.545

4911.5

559.4

6.99 1806.9 220.8 4.24

314

92.30

17.19

16.235

2.283

1.415

4399.4

511.9

6.90 1631.4 201.0 4.20

287

84.37

16.81

16.130

2.093

1.310

3912.1

465.5

6.81 1466.5 181.8 4.17

264

77.63

16.50

16.025

1.938

1.205

3526.0

427.4

6.74 1331.2 166.1 4.14

246

72.33

16.25

15.945

1.813

1.125

3228.9

397.4

6.68 1226.6 153.9 4.12

237

69.69

16.12

15.910

1.748

1.090

3080.9

382.2

6.65 1174.8 147.7 4.11

228

67.06

16.00

15.865

1.688

1.045

2942.4

367.8

6.62 1124.8 141.8 4.10

219

64.36

15.87

15.825

1.623

1.005

2798.2

352.6

6.59 1073.2 135.6 4.08

211

62.07

15.75

15.800

1.563

0.980

2671.4

339.2

6.56 1028.6 130.2 4.07

18  8 34

18  7 12

16  7

14  16

(continued)

1842

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.76 (Continued ) Flange Nominal Size (in.)

14  14 12

Axis X–X

Weight per Foot (lb)

Area (in.2)

Depth (in.)

Width (in.)

Thickness (in.)

202

59.39

15.63

15.750

193

56.73

15.50

15.710

184

54.07

15.38

176

51.73

167

Axis Y–Y

Web Thickness (in.)

I (in.4)

S (in.3)

r (in.)

I (in.4)

1.503

0.930

2538.8

324.9

6.54

979.7 124.4 4.06

1.438

0.890

2402.4

310.0

6.51

930.1 118.4 4.05

15.660

1.378

0.840

2274.8

295.8

6.49

882.7 112.7 4.04

15.25

15.640

1.313

0.820

2149.6

281.9

6.45

837.9 107.1 4.02

49.09

15.12

15.600

1.248

0.780

2020.8

267.3

6.42

790.2 101.3 4.01

158

46.47

15.00

15.550

1.188

0.730

1900.6

253.4

6.40

745.0

95.8 4.00

150

44.08

14.88

15.515

1.128

0.695

1786.9

240.2

6.37

702.5

90.6 3.99

142

41.85

14.75

15.500

1.063

0.680

1672.2

226.7

6.32

660.1

85.2 3.97

320a

94.12

16.81

16.710

2.093

1.890

4141.7

492.8

6.63 1635.1 195.7 4.17

136

39.98

14.75

14.740

1.063

0.660

1593.0

216.0

6.31

567.7

77.0 3.77

127

37.33

14.62

14.690

0.998

0.610

1476.7

202.0

6.29

527.6

71.8 3.76

119

34.99

14.50

14.650

0.938

0.570

1373.1

189.4

6.26

491.8

67.1 3.75

111

32.65

14.37

14.620

0.873

0.540

1266.5

176.3

6.23

454.9

62.2 3.73

103

30.26

14.25

14.575

0.813

0.495

1165.8

163.6

6.21

419.7

57.6 3.72

95

27.94

14.12

14.545

0.748

0.465

1063.5

150.6

6.17

383.7

52.8 3.71

S (in.3)

r (in.)

87

25.56

14.00

14.500

0.688

0.420

966.9

138.1

6.15

349.7

48.2 3.70

14  12

84

24.71

14.18

12.023

0.778

0.451

928.4

130.9

6.13

225.5

37.5 3.02

78

22.94

14.06

12.000

0.718

0.428

851.2

121.1

6.09

206.9

34.5 3.00

14  10

74

21.76

14.19

10.072

0.783

0.450

796.8

112.3

6.05

133.5

26.5 2.48

68

20.00

14.06

10.040

0.718

0.418

724.1

103.0

6.02

121.2

24.1 2.46

61

17.94

13.91

10.000

0.643

0.378

321.5

92.2

5.98

107.3

21.5 2.45

53

15.59

13.94

8.062

0.658

0.370

542.1

77.8

5.90

57.5

14.3 1.92

48

14.11

13.81

8.031

0.593

0.339

484.9

70.2

5.86

51.3

12.8 1.91

43

12.65

13.68

8.000

0.528

0.308

429.0

62.7

5.82

45.1

11.3 1.89

38

11.17

14.12

6.776

0.513

0.313

385.3

54.6

5.87

24.6

7.3 1.49

34

10.00

14.00

6.750

0.453

0.287

339.2

48.5

5.83

21.3

6.3 1.46

30

8.81

13.86

6.733

0.383

0.270

289.6

41.8

5.73

17.5

5.2 1.41

190

55.86

14.38

12.670

1.736

1.060

1892.5

263.2

5.82

589.7

93.1 3.25

161

47.38

13.88

12.515

1.486

0.905

1541.8

222.2

5.70

486.2

77.7 3.20

133

39.11

13.38

12.365

1.236

0.755

1221.2

182.5

5.59

389.9

63.1 3.16

120

35.31

13.12

12.320

1.106

0.710

1071.7

163.4

5.51

345.1

56.0 3.13

106

31.19

12.88

12.230

0.986

0.620

930.7

144.5

5.46

300.9

49.2 3.11

99

29.09

12.75

12.190

0.921

0.580

858.5

134.7

5.43

278.2

45.7 3.09

92

27.06

12.62

12.155

0.856

0.545

788.9

125.0

5.40

256.4

42.2 3.08

85

24.98

12.50

12.105

0.796

0.495

723.3

115.7

5.38

235.5

38.9 3.07

79

23.22

12.38

12.080

0.736

0.470

663.0

107.1

5.34

216.4

35.8 3.05

72

21.16

12.25

12.040

0.671

0.430

597.4

97.5

5.31

195.3

32.4 3.04

14  8

14  6 34

12  12

65

19.11

12.12

12.000

0.606

0.390

533.4

88.0

5.28

174.6

29.1 3.02

12  10

58

17.06

12.19

10.014

0.641

0.359

476.1

78.1

5.28

107.4

21.4 2.51

53

15.59

12.06

10.000

0.576

0.345

426.2

70.7

5.23

96.1

19.2 2.48

12  8

50

14.71

12.19

8.077

0.641

0.371

394.5

64.7

5.18

56.4

14.0 1.96

45

13.24

12.06

8.042

0.576

0.336

350.8

58.2

5.15

50.0

12.4 1.94

40

11.77

11.94

8.000

0.516

0.294

310.1

51.9

5.13

44.1

11.0 1.94

1843

STANDARD SIZES

TABLE 51.76 (Continued ) Flange Nominal Size (in.)

Weight per Foot (lb)

Area (in.2)

Depth (in.)

Width (in.)

Axis X–X

Thickness (in.)

Web Thickness (in.)

I (in.4)

S (in.3)

Axis Y–Y r (in.)

I (in.4)

S (in.3)

r (in.)

12  6 12

36

10.59

12.24

6.565

0.305

280.8

45.9

5.15

23.7

7.2 1.50

31

9.12

12.09

6.525

0.465

0.265

238.4

39.4

5.11

19.8

6.1 1.47

27

7.97

11.95

6.500

0.400

0.240

204.1

34.1

5.06

16.6

5.1 1.44

10  10

112

32.92

11.38

10.415

1.248

0.755

718.7

126.3

4.67

235.4

45.2 2.67

0540

100

29.43

11.12

10.345

1.118

0.685

625.0

112.4

4.61

206.6

39.9 2.65

89

26.19

10.88

10.275

0.998

0.615

542.4

99.7

4.55

180.6

35.2 2.63

77

22.67

10.62

10.195

0.868

0.535

457.2

86.1

4.49

153.4

30.1 2.60

72

21.18

10.50

10.170

0.808

0.510

420.7

80.1

4.46

141.8

27.9 2.59

66

19.41

10.38

10.117

0.748

0.457

382.5

73.7

4.44

129.2

25.5 2.58

60

17.66

10.25

10.075

0.683

0.415

343.7

67.1

4.41

116.5

23.1 2.57

54

15.88

10.12

10.028

0.618

0.368

305.7

60.4

4.39

103.9

20.7 2.56

49

14.40

10.00

10.000

0.558

0.340

272.9

54.6

4.35

93.0

18.6 2.54

45

13.24

10.12

8.022

0.618

0.350

248.6

49.1

4.33

53.2

13.3 2.00

39

11.48

9.94

7.990

0.528

0.318

209.7

42.2

4.27

44.9

11.2 1.98

33

9.71

9.75

7.964

0.433

0.292

170.9

35.0

4.20

36.5

9.2 1.94

29

8.53

10.22

5.799

0.500

0.289

157.3

30.8

4.29

15.2

5.2 1.34

25

7.35

10.08

5.762

0.430

0.252

133.2

26.4

4.26

12.7

4.4 1.31

21

6.19

9.90

5.750

0.340

0.240

106.3

21.5

4.14

9.7

3.4 1.25

67

19.70

9.00

8.287

0.933

0.575

271.8

60.4

3.71

88.6

21.4 2.12

58

17.06

8.75

8.222

0.808

0.510

227.3

52.0

3.65

74.9

18.2 2.10

48

14.11

8.50

8.117

0.683

0.405

183.7

43.2

3.61

60.9

15.0 2.08

40

11.76

8.25

8.077

0.558

0.365

146.3

35.5

3.53

49.0

12.1 2.04

35

10.30

8.12

8.027

0.493

0.315

126.5

31.1

3.50

42.5

10.6 2.03

31

9.12

8.00

8.000

0.433

0.288

109.7

27.4

3.47

37.0

9.2 2.01

8  6 12

28

8.23

8.06

6.540

0.463

0.285

97.8

24.3

3.45

21.6

6.6 1.62

24

7.06

7.93

6.500

0.398

0.245

82.5

20.8

3.42

18.2

5.6 1.61

8  5 14

20

5.88

8.14

5.268

0.378

0.248

69.2

17.0

3.43

8.5

3.2 1.20

17

5.00

8.00

5.250

0.308

0.230

56.4

14.1

3.36

6.7

2.6 1.16

10  8

10  5 34

88

a

Column core section.

Most of the sections can be supplied promptly steel mills. Owing to variations in the rolling practice of the different mills, their products are not identical, although their divergence from the values given in the tables is practically negligible. For standardization, only the lesser values are given, and therefore they are on the side of safety. Further information on sections listed in the tables, together with information on other products and on the requirements for placing orders, may be gathered from mill catalogs.

1844

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.77 Properties of American Standard Beams

Flange Nominal Size (in.)

Weight per Foot (lb)

Area (in.2)

Depth (in.)

Width (in.)

24  7 78

120.0

35.13

24.00

105.9

30.98

24.00

24  7

100.0

29.25

90.0

20  7

Axis X–X

Axis Y–Y

Thickness (in.)

Web Thickness (in.)

I (in.4)

S (in.3)

r (in.)

8.048

1.102

0.798

3010.8

250.9

7.875

1.102

0.625

2811.5

234.3

24.00

7.247

0.871

0.747

2371.8

26.30

24.00

7.124

0.871

0.624

79.9

23.33

24.00

7.000

0.871

95.0

27.74

20.00

7.200

0.916

85.0

24.80

20.00

7.053

20  6 14

75.0

21.90

20.00

65.4

19.08

18  6

70.0

20.46

54.7

15  5 12

I (in.4)

S (in.3)

r (in.)

9.26 84.9

21.1

1.56

9.53 78.9

20.0

1.60

197.6

9.05 48.4

13.4

1.29

2230.1

185.8

9.21 45.5

12.8

1.32

0.500

2087.2

173.9

9.46 42.9

12.2

1.36

0.800

1599.7

160.0

7.59 50.5

14.0

1.35

0.916

0.653

1501.7

150.2

7.78 47.0

13.3

1.38

6.391

0.789

0.641

1263.5

126.3

7.60 30.1

9.4

1.17

20.00

6.250

0.789

0.500

1169.5

116.9

7.83 27.9

8.9

1.21

18.00

6.251

0.691

0.711

917.5

101.9

6.70 24.5

7.8

1.09

15.94

18.00

6.000

0.691

0.460

795.5

88.4

7.07 21.2

7.1

1.15

50.0

14.59

15.00

5.640

0.622

0.550

481.1

64.2

5.74 16.0

5.7

1.05

42.9

12.49

15.00

5.500

0.622

0.410

441.8

58.9

5.95 14.6

5.3

1.08

12  5 14

50.0

14.57

12.00

5.477

0.659

0.687

301.6

50.3

4.55 16.0

5.8

1.05

40.8

11.84

12.00

5.250

0.659

0.460

268.9

44.8

4.77 13.8

5.3

1.08

12  5

35.0

10.20

12.00

5.078

0.544

0.428

227.0

37.8

4.72 10.0

3.9

0.99

31.8

9.26

12.00

5.000

0.544

0.350

215.8

36.0

4.83

9.5

3.8

1.01

10  4 58

35.0

10.22

10.0

4.944

0.491

0.594

145.8

29.2

3.78

8.5

3.4

0.91

25.4

7.38

10.00

4.660

0.491

0.310

122.1

24.4

4.07

6.9

3.0

0.97

84

23.0

6.71

8.00

4.171

0.425

0.441

64.2

16.0

3.09

4.4

2.1

0.81

18.4

5.34

8.00

4.000

0.425

0.270

56.9

14.2

3.26

3.8

1.9

0.84

7  3 58

20.0

5.83

7.00

3.860

0.392

0.450

41.9

12.0

2.68

3.1

1.6

0.74

15.3

4.43

7.00

3.660

0.392

0.250

36.2

10.4

2.86

2.7

1.5

0.78

6  3 38

17.25

5.02

6.00

3.565

0.359

0.465

26.0

8.7

2.28

2.3

1.3

0.68

12.5

3.61

6.00

3.330

0.359

0.230

21.8

7.3

2.46

1.8

1.1

0.72

53

14.75

4.29

5.00

3.284

0.326

0.494

15.0

6.0

1.87

1.7

1.0

0.63

10.0

2.87

5.00

3.000

0.326

0.210

12.1

4.8

2.05

1.2

0.82 0.65

4  2 58

9.5

2.76

4.00

2.796

0.293

0.326

6.7

3.3

1.56

0.91

0.65 0.58

7.7

2.21

4.00

2.660

0.293

0.190

6.0

3.0

1.64

0.77

0.58 0.59

3  2 38

7.5

2.17

3.00

2.509

0.260

0.349

2.9

1.9

1.15

0.59

0.47 0.52

5.7

1.64

3.00

2.330

0.260

0.170

2.5

1.7

1.23

0.46

0.40 0.53

1845

STANDARD SIZES

TABLE 51.78 Properties of American Standard Channels

Flange Nominal Size (in.)

Weight per Foot (lb)

Area (in.2)

Depth (in.)

Width (in.)

18  4a

58.0

16.98

18.00

51.9

15.18

18.00

45.8

13.38

42.7 15  3 38

12  3

10  2 58

9  2 12

8  2 14

7  2 18

62

Axis X–X

Axis Y–Y

Thickness (in.)

Web Thickness (in.)

I (in.4)

S r I S x r (in.3) (in.) (in.4) (in.3) (in.) (in.)

4.200

0.625

0.700

670.7

74.5 6.29 18.5

5.6

1.04 0.88

4.100

0.625

0.600

622.1

69.1 6.40 17.1

5.3

1.06 0.87

18.00

4.000

0.625

0.500

573.5

63.7 6.55 15.8

5.1

1.09 0.89

12.48

18.00

3.950

0.625

0.450

549.2

61.0 6.64 15.0

4.9

1.10 0.90

50.0

14.64

15.00

3.716

0.650

0.716

401.4

53.6 5.24 11.2

3.8

0.87 0.80

40.0

11.70

15.00

3.520

0.650

0.520

346.3

46.2 5.44

9.3

3.4

0.89 0.78

33.9

9.90

15.00

3.400

0.650

0.400

312.6

41.7 5.62

8.2

3.2

0.91 0.79

30.0

8.79

12.00

3.170

0.501

0.510

161.2

26.9 4.28

5.2

2.1

0.77 0.68

25.0

7.32

12.00

3.047

0.501

0.387

143.5

23.9 4.43

4.5

1.9

0.79 0.68

20.7

6.03

12.00

2.940

0.501

0.280

128.1

21.4 4.61

3.9

1.7

0.81 0.70

30.0

8.80

10.00

3.033

0.436

0.673

103.0

20.6 3.42

4.0

1.7

0.67 0.65

25.0

7.33

10.00

2.886

0.436

0.526

90.7

18.1 3.52

3.4

1.5

0.68 0.62

20.0

5.86

10.00

2.739

0.436

0.379

78.5

15.7 3.66

2.8

1.3

0.70 0.61

15.3

4.47

10.00

2.600

0.436

0.240

66.9

13.4 3.87

2.3

1.2

0.72 0.64

20.0

5.86

9.00

2.648

0.413

0.448

60.6

13.5 3.22

2.4

1.2

0.65 0.59

15.0

4.39

9.00

2.485

0.413

0.285

50.7

11.3 3.40

1.9

1.0

0.67 0.59

13.4

3.89

9.00

2.430

0.413

0.230

47.3

10.5 3.49

1.8

0.97 0.67 0.61

18.75

5.49

8.00

2.527

0.390

0.487

43.7

10.9 2.82

2.0

1.0

13.75

4.02

8.00

2.343

0.390

0.303

35.8

9.0 2.99

1.5

0.86 0.62 0.56

11.5

3.36

8.00

2.260

0.390

0.220

32.3

8.1 3.10

1.3

0.79 0.63 0.58

14.75

4.32

7.00

2.299

0.366

0.419

27.1

7.7 2.51

1.4

0.79 0.57 0.53

12.25

3.58

7.00

2.194

0.366

0.314

24.1

6.9 2.59

1.2

0.71 0.58 0.53

9.8

2.85

7.00

2.090

0.366

0.210

21.1

6.0 2.72

0.98 0.63 0.59 0.55

13.0

3.81

6.00

2.157

0.343

0.437

17.3

5.8 2.13

1.1

10.5

3.07

6.00

2.034

0.343

0.314

15.1

5.0 2.22

0.87 0.57 0.53 0.50

0.60 0.57

0.65 0.53 0.52

8.2

2.39

6.00

1.920

0.343

0.200

13.0

4.3 2.34

0.70 0.50 0.54 0.52

5  1 34

9.0

2.63

5.00

1.885

0.320

0.325

8.8

3.5 1.83

0.64 0.45 0.49 0.48

6.7

1.95

5.00

1.750

0.320

0.190

7.4

3.0 1.95

0.48 0.38 0.50 0.49

4  1 58

7.25

2.12

4.00

1.720

0.296

0.320

4.5

2.3 1.47

0.44 0.35 0.46 0.46

5.4

1.56

4.00

1.580

0.296

0.180

3.8

1.9 1.56

0.32 0.29 0.45 0.46

3  1 12

6.0

1.75

3.00

1.596

0.273

0.356

2.1

1.4 1.08

0.31 0.27 0.42 0.46

5.0

1.46

3.00

1.498

0.273

0.258

1.8

1.2 1.12

0.25 0.24 0.41 0.44

4.1

1.19

3.00

1.410

0.273

0.170

1.6

1.1 1.17

0.20 0.21 0.41 0.44

a

Car and Shipbuilding Channel; not an American standard.

1846

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.79 Properties of Angles with Equal Legs

Axis X–X and Axis Y–Y Size (in.)

Thickness (in.)

Weight per Foot (lb)

88

1 18

56.9

66

55

44

Area (in.2)

I (in.4)

S (in.3)

r (in.)

x or y (in.)

Axis Z–Z r (in.)

16.73

98.0

17.5

2.42

2.41

1.56

1

51.0

15.00

89.0

15.8

2.44

2.37

1.56

7 8 3 4 5 8 9 16 1 2

45.0

13.23

79.6

14.0

2.45

2.32

1.57

38.9

11.44

69.7

12.2

2.47

2.28

1.57

32.7

9.61

59.4

10.3

2.49

2.23

1.58

29.6

8.68

54.1

9.3

2.50

2.21

1.58

26.4

7.75

48.6

8.4

2.50

2.19

1.59

1

37.4

11.00

35.5

8.6

1.80

1.86

1.17

7 8 3 4 5 8 9 16 1 2 7 16 3 8 5 16 7 8 3 4 5 8 1 2 7 16 3 8 5 16 3 4 5 8 1 2 7 16 3 8

33.1

9.73

31.9

7.6

1.81

1.82

1.17

28.7

8.44

28.2

6.7

1.83

1.78

1.17

24.2

7.11

24.2

5.7

1.84

1.73

1.18

21.9

6.43

22.1

5.1

1.85

1.71

1.18

19.6

5.75

19.9

4.6

1.86

1.68

1.18

17.2

5.06

17.7

4.1

1.87

1.66

1.19

14.9

4.36

15.4

3.5

1.88

1.64

1.19

12.5

3.66

13.0

3.0

1.89

1.61

1.19

27.2

7.98

17.8

5.2

1.49

1.57

0.97

23.6

6.94

15.7

4.5

1.51

1.52

0.97

20.0

5.86

13.6

3.9

1.52

1.48

0.98

16.2

4.75

11.3

3.2

1.54

1.43

0.98

14.3

4.18

10.0

2.8

1.55

1.41

0.98

12.3

3.61

8.7

2.4

1.56

1.39

0.99

10.3

3.03

7.4

2.0

1.57

1.37

0.99

18.5

5.44

7.7

2.8

1.19

1.27

0.78

15.7

4.61

6.7

2.4

1.20

1.23

0.78

12.8

3.75

5.6

2.0

1.22

1.18

0.78

11.3

3.31

5.0

1.8

1.23

1.16

0.78

9.8

2.86

4.4

1.5

1.23

1.14

0.79

STANDARD SIZES

1847

TABLE 51.79 (Continued ) Axis X–X and Axis Y–Y Size (in.)

3 12



Thickness (in.)

3 12

33

2 12  2 12

22

1 34



1 34

1 12  1 12

1 14



11

1 14

5 16 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 3 16 1 2 3 8 5 16 1 4 3 16 3 8 5 16 1 4 3 16 1 8 1 4 3 16 1 8 1 4 3 16 1 8 1 4 3 16 1 8 1 4 3 16 1 8

Weight per Foot (lb)

Area (in.2)

I (in.4)

S (in.3)

r (in.)

x or y (in.)

Axis Z–Z r (in.)

8.2

2.40

3.7

1.3

1.24

1.12

0.79

6.6

1.94

3.0

1.1

1.25

1.09

0.80

11.1

3.25

3.6

1.5

1.06

1.06

0.68

9.8

2.87

3.3

1.3

1.07

1.04

0.68

8.5

2.48

2.9

1.2

1.07

1.01

0.69

7.2

2.09

2.5

0.98

1.08

0.99

0.69

5.8

1.69

2.0

0.79

1.09

0.97

0.69

9.4

2.75

2.2

1.1

0.90

0.93

0.58

8.3

2.43

2.0

0.95

0.91

0.91

0.58

7.2

2.11

1.8

0.83

0.91

0.89

0.58

6.1

1.78

1.5

0.71

0.92

0.87

0.59

4.9

1.44

1.2

0.58

0.93

0.84

0.59

3.71

1.09

0.96

0.44

0.94

0.82

0.59

7.7

2.25

1.2

0.72

0.74

0.81

0.49

5.9

1.73

0.98

0.57

0.75

0.76

0.49

5.0

1.47

0.85

0.48

0.76

0.74

0.49

4.1

1.19

0.70

0.39

0.77

0.72

0.49

3.07

0.90

0.55

0.30

0.78

0.69

0.49

4.7

1.36

0.48

0.35

0.59

0.64

0.39

3.92

1.15

0.42

0.30

0.60

0.61

0.39

3.19

0.94

0.35

0.25

0.61

0.59

0.39

2.44

0.71

0.27

0.19

0.62

0.57

0.39

1.65

0.48

0.19

0.13

0.63

0.55

0.40

2.77

0.81

0.23

0.19

0.53

0.53

0.34

2.12

0.62

0.18

0.14

0.54

0.51

0.34

1.44

0.42

0.13

0.10

0.55

0.48

0.35

2.34

0.69

0.14

0.13

0.45

0.47

0.29

1.80

0.53

0.11

0.10

0.46

0.44

0.29

1.23

0.36

0.08

0.07

0.47

0.42

0.30

1.92

0.56

0.08

0.09

0.37

0.40

0.24

1.48

0.43

0.06

0.07

0.38

0.38

0.24

1.01

0.30

0.04

0.05

0.38

0.36

0.25

1.49

0.44

0.04

0.06

0.29

0.34

0.20

1.16

0.34

0.03

0.04

0.30

0.32

0.19

0.80

0.23

0.02

0.03

0.30

0.30

0.20

1848

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.80 Properties of Angles with Unequal Legs

Axis X–X

Axis Y–Y

Axis Z–Z

Size (in.)

Thickness (in.)

Weight per Foot (lb)

94

1

40.8

12.00

97.0

17.6

2.84 3.50 12.0

4.0

7 8 3 4 5 8 9 16 1 2

36.1

10.61

86.8

15.7

2.86 3.45 10.8

3.6

1.01 0.95 0.84 0.208

31.3

9.19

76.1

13.6

2.88 3.41

3.1

1.02 0.91 0.84 0.212

86

84

74

64

Area (in.2)

I S r y I S r x r (in.4) (in.3) (in.) (in.) (in.4) (in.3) (in.) (in.) (in.)

9.6

tana

1.00 1.00 0.83 0.203

26.3

7.73

64.9

11.5

2.90 3.36

8.3

2.6

1.04 0.86 0.85 0.216

23.8

7.00

59.1

10.4

2.91 3.33

7.6

2.4

1.04 0.83 0.85 0.218

2.92 3.31

21.3

6.25

53.2

9.3

6.9

2.2

1.05 0.81 0.85 0.220

1

44.2

13.00

80.8

15.1

2.49 2.65 38.8

8.9

1.73 1.65 1.28 0.543

7 8 3 4 5 8 9 16 1 2 7 16

39.1

11.48

72.3

13.4

2.51 2.61 34.9

7.9

1.74 1.61 1.28 0.547

33.8

9.94

63.4

11.7

2.53 2.56 30.7

6.9

1.76 1.56 1.29 0.551

28.5

8.36

54.1

9.9

2.54 2.52 26.3

5.9

1.77 1.52 1.29 0.554

25.7

7.56

49.3

9.0

2.55 2.50 24.0

5.3

1.78 1.50 1.30 0.556

23.0

6.75

44.3

8.0

2.56 2.47 21.7

4.8

1.79 1.47 1.30 0.558

20.2

5.93

39.2

7.1

2.57 2.45 19.3

4.2

1.80 1.45 1.31 0.560

1

37.4

11.00

69.6

14.1

2.52 3.05 11.6

3.9

1.03 1.05 0.85 0.247

7 8 3 4 5 8 9 16 1 2 7 16 7 8 3 4 5 8 9 16 1 2 7 16 3 8 7 8 3 4

33.1

9.73

62.5

12.5

2.53 3.00 10.5

3.5

1.04 1.00 0.85 0.253

28.7

8.44

54.9

10.9

2.55 2.95

9.4

3.1

1.05 0.95 0.85 0.258

24.2

7.11

46.9

9.2

2.57 2.91

8.1

2.6

1.07 0.91 0.86 0.262

21.9

6.43

42.8

8.4

2.58 2.88

7.4

2.4

1.07 0.88 0.86 0.265

19.6

5.75

38.5

7.5

2.59 2.86

6.7

2.2

1.08 0.86 0.86 0.267

17.2

5.06

34.1

6.6

2.60 2.83

6.0

1.9

1.09 0.83 0.87 0.269

30.2

8.86

42.9

9.7

2.20 2.55 10.2

3.5

1.07 1.05 0.86 0.318

26.2

7.69

37.8

8.4

2.22 2.51

3.0

1.09 1.01 0.86 0.324

9.1

22.1

6.48

32.4

7.1

2.24 2.46

7.8

2.6

1.10 0.96 0.86 0.329

20.0

5.87

29.6

6.5

2.24 2.44

7.2

2.4

1.11 0.94 0.87 0.332

17.9

5.25

26.7

5.8

2.25 2.42

6.5

2.1

1.11 0.92 0.87 0.335

15.8

4.62

23.7

5.1

2.26 2.39

5.8

1.9

1.12 0.89 0.88 0.337

13.6

3.98

20.6

4.4

2.27 2.37

5.1

1.6

1.13 0.87 0.88 0.339

27.2

7.98

27.7

7.2

1.86 2.12

9.8

3.4

1.11 1.12 0.86 0.421

23.6

6.94

24.5

6.3

1.88 2.08

8.7

3.0

1.12 1.08 0.86 0.428

1849

STANDARD SIZES

TABLE 51.80 (Continued ) Axis X–X Size (in.)

6  3 12

5

3 12

53

4  3 12

43

3 12  3

Thickness (in.) 5 8 9 16 1 2 7 16 3 8 5 16 1 2 3 8 5 16 1 4 3 4 5 8 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 5 8 1 2 7 16 3 8 5 16 1 4 5 8 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16

Weight per Foot (lb)

Area (in.2)

Axis Y–Y

Axis Z–Z

I S r y I S r x r (in.4) (in.3) (in.) (in.) (in.4) (in.3) (in.) (in.) (in.)

tana

20.0

5.86

21.1

5.3

1.90 2.03

7.5

2.5

1.13 1.03 0.86 0.435

18.1

5.31

19.3

4.8

1.90 2.01

6.9

2.3

1.14 1.01 0.87 0.438

16.2

4.75

17.4

4.3

1.91 1.99

6.3

2.1

1.15 0.99 0.87 0.440

14.3

4.18

15.5

3.8

1.92 1.96

5.6

1.9

1.16 0.96 0.87 0.443

12.3

3.61

13.5

3.3

1.93 1.94

4.9

1.6

1.17 0.94 0.88 0.446

10.3

3.03

11.4

2.8

1.94 1.92

4.2

1.4

1.17 0.92 0.88 0.449

15.3

4.50

16.6

4.2

1.92 2.08

4.3

1.6

0.97 0.83 0.76 0.344

11.7

3.42

12.9

3.2

1.94 2.04

3.3

1.2

0.99 0.79 0.77 0.350

9.8

2.87

10.9

2.7

1.95 2.01

2.9

1.0

1.00 0.76 0.77 0.352

7.9

2.31

8.9

2.2

1.96 1.99

2.3

0.85 1.01 0.74 0.78 0.355

19.8

5.81

13.9

4.3

1.55 1.75

5.6

2.2

0.98 1.00 0.75 0.464

16.8

4.92

12.0

3.7

1.56 1.70

4.8

1.9

0.99 0.95 0.75 0.472

13.6

4.00

10.0

3.0

1.58 1.66

4.1

1.6

1.01 0.91 0.75 0.479

12.0

3.53

8.9

2.6

1.59 1.63

3.6

1.4

1.01 0.88 0.76 0.482

10.4

3.05

7.8

2.3

1.60 1.61

3.2

1.2

1.02 0.86 0.76 0.486

8.7

2.56

6.6

1.9

1.61 1.59

2.7

1.0

1.03 0.84 0.76 0.489

7.0

2.06

5.4

1.6

1.61 1.56

2.2

0.83 1.04 0.81 0.76 0.492

12.8

3.75

9.5

2.9

1.59 1.75

2.6

1.1

0.83 0.75 0.65 0.357

11.3

3.31

8.4

2.6

1.60 1.73

2.3

1.0

0.84 0.73 0.65 0.361

9.8

2.86

7.4

2.2

1.61 1.70

2.0

0.89 0.84 0.70 0.65 0.364

8.2

2.40

6.3

1.9

1.61 1.68

1.8

0.75 0.85 0.68 0.66 0.368

6.6

1.94

5.1

1.5

1.62 1.66

1.4

0.61 0.86 0.66 0.66 0.371

14.7

4.30

6.4

2.4

1.22 1.29

4.5

1.8

11.9

3.50

5.3

1.9

1.23 1.25

3.8

1.5

1.04 1.00 0.72 0.750

10.6

3.09

4.8

1.7

1.24 1.23

3.4

1.4

1.05 0.98 0.72 0.753

1.03 1.04 0.72 0.745

9.1

2.67

4.2

1.5

1.25 1.21

3.0

1.2

1.06 0.96 0.73 0.755

7.7

2.25

3.6

1.3

1.26 1.18

2.6

1.0

1.07 0.93 0.73 0.757

6.2

1.81

2.9

1.0

1.27 1.16

2.1

0.81 1.07 0.91 0.73 0.759

13.6

3.98

6.0

2.3

1.23 1.37

2.9

1.4

0.85 0.87 0.64 0.534

11.1

3.25

5.1

1.9

1.25 1.33

2.4

1.1

0.86 0.83 0.64 0.543

9.8

2.87

4.5

1.7

1.25 1.30

2.2

1.0

0.87 0.80 0.64 0.547

8.5

2.48

4.0

1.5

1.26 1.28

1.9

0.87 0.88 0.78 0.64 0.551

7.2

2.09

3.4

1.2

1.27 1.26

1.7

0.73 0.89 0.76 0.65 0.554

5.8

1.69

2.8

1.0

1.28 1.24

1.4

0.60 0.90 0.74 0.65 0.558

10.2

3.00

3.5

1.5

1.07 1.13

2.3

1.1

9.1

2.65

3.1

1.3

1.08 1.10

2.1

0.98 0.89 0.85 0.62 0.718

7.9

2.30

2.7

1.1

1.09 1.08

1.9

0.85 0.90 0.83 0.62 0.721

6.6

1.93

2.3

0.95 1.10 1.06

1.6

0.72 0.90 0.81 0.63 0.724

0.88 0.88 0.62 0.714

(continued)

1850

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.80 (Continued ) Axis X–X Size (in.)

3 12  2 12

3

2 12

32

2 12  2

2 12  1 12

2  1 12

1 34  1 14

Thickness (in.) 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 3 16 3 8 5 16 1 4 3 16 3 8 5 16 1 4 3 16 1 4 3 16 1 8 1 4 3 16 1 8

Axis Y–Y

Axis Z–Z

Weight per Foot (lb)

Area (in.2)

5.4

1.56

1.9

0.78 1.11 1.04

1.3

0.59 0.91 0.79 0.63 0.727

9.4

2.75

3.2

1.4

1.09 1.20

1.4

0.76 0.70 0.70 0.53 0.486

8.3

2.43

2.9

1.3

1.09 1.18

1.2

0.68 0.71 0.68 0.54 0.491

7.2

2.11

2.6

1.1

1.10 1.16

1.1

0.59 0.72 0.66 0.54 0.496

6.1

1.78

2.2

0.93 1.11 1.14

0.94 0.50 0.73 0.64 0.54 0.501

4.9

1.44

1.8

0.75 1.12 1.11

0.78 0.41 0.74 0.61 0.54 0.506

I S r y I S r x r (in.4) (in.3) (in.) (in.) (in.4) (in.3) (in.) (in.) (in.)

tana

8.5

2.50

2.1

1.0

0.91 1.00

1.3

0.74 0.72 0.75 0.52 0.667

7.6

2.21

1.9

0.93 0.92 0.98

1.2

0.66 0.73 0.73 0.52 0.672 0.58 0.74 0.71 0.52 0.676

6.6

1.92

1.7

0.81 0.93 0.96

1.0

5.6

1.62

1.4

0.69 0.94 0.93

0.90 0.49 0.74 0.68 0.53 0.680

4.5

1.31

1.2

0.56 0.95 0.91

0.74 0.40 0.75 0.66 0.53 0.684

7.7

2.25

1.9

1.0

0.92 1.08

0.67 0.47 0.55 0.58 0.43 0.414

6.8

2.00

1.7

0.89 0.93 1.06

0.61 0.42 0.55 0.56 0.43 0.421

5.9

1.73

1.5

0.78 0.94 1.04

0.54 0.37 0.56 0.54 0.43 0.428

5.0

1.47

1.3

0.66 0.95 1.02

0.47 0.32 0.57 0.52 0.43 0.435

4.1

1.19

1.1

0.54 0.95 0.99

0.39 0.26 0.57 0.49 0.43 0.440

3.07

0.90

0.84

0.41 0.97 0.97

0.31 0.20 0.58 0.47 0.44 0.446

5.3

1.55

0.91

0.55 0.77 0.83

0.51 0.36 0.58 0.58 0.42 0.614

4.5

1.31

0.79

0.47 0.78 0.81

0.45 0.31 0.58 0.56 0.42 0.620

3.62

1.06

0.65

0.38 0.78 0.79

0.37 0.25 0.59 0.54 0.42 0.626

2.75

0.81

0.51

0.29 0.79 0.76

0.29 0.20 0.60 0.51 0.43 0.631

4.7

1.36

0.82

0.52 0.78 0.92

0.22 0.20 0.40 0.42 0.32 0.340

3.92

1.15

0.71

0.44 0.79 0.90

0.19 0.17 0.41 0.40 0.32 0.349

3.19

0.94

0.59

0.36 0.79 0.88

0.16 0.14 0.41 0.38 0.32 0.357

2.44

0.72

0.46

0.28 0.80 0.85

0.13 0.11 0.42 0.35 0.33 0.364

2.77

0.81

0.32

0.24 0.62 0.66

0.15 0.14 0.43 0.41 0.32 0.543

2.12

0.62

0.25

0.18 0.63 0.64

0.12 0.11 0.44 0.39 0.32 0.551

1.44

0.42

0.17

0.13 0.64 0.62

0.09 0.08 0.45 0.37 0.33 0.558

2.34

0.69

0.20

0.18 0.54 0.60

0.09 0.10 0.35 0.35 0.27 0.486

1.80

0.53

0.16

0.14 0.55 0.58

0.07 0.08 0.36 0.33 0.27 0.496

1.23

0.36

0.11

0.09 0.56 0.56

0.05 0.05 0.37 0.31 0.27 0.506

1851

STANDARD SIZES

TABLE 51.81 Properties and Dimensions of Tees

Tees are seldom used as structural framing members. When so used they are generally employed on short spans in flexure. This table lists a few selected sizes, the range of whose section moduli will cover all ordinary conditions. For sizes not listed, the catalogs of the respective rolling mills should be consulted. Flange

Axis X–X

Section Number

Weight per Foot (lb)

Area (in.2)

Depth of Tee (in.)

Width (in.)

Average Thickness (in.)

Stem Thickness (in.)

I (in.4)

ST 18 WFa

150

44.09

18.36

16.655

1.680

0.945

140

41.16

18.25

16.595

1.570

0.885

130

38.28

18.12

16.555

1.440

122.5

36.01

18.03

16.512

1.350

ST 18 WF

ST 16 WF

ST 16 WF

ST 15 WF

ST 15 WF

ST 13 WF

ST 13 WF

ST 12 WF

ST 12 WF

Axis Y–Y

S r y I (in.3) (in.) (in.) (in.4)

S r (in.3) (in.)

1222.7

85.9

5.27 4.13 612.6

73.6

3.73

1133.3

79.9

5.25 4.07 563.7

67.9

3.70

0.845

1059.2

75.4

5.26 4.07 510.3

61.6

3.65

0.802

994.3

71.1

5.25 4.04 472.3

57.2

3.62

115

33.86

17.94

16.475

1.260

0.765

935.8

67.2

5.26 4.02 435.5

52.9

3.59

97

28.56

18.24

12.117

1.260

0.770

904.0

67.3

5.63 4.81 177.7

29.3

2.49

91

26.77

18.16

12.072

1.180

0.725

844.0

63.0

5.61 4.77 163.9

27.1

2.47

85

24.99

18.08

12.027

1.100

0.680

784.7

58.8

5.60 4.74 150.3

25.0

2.45

80

23.54

18.00

12.000

1.020

0.653

741.0

56.0

5.61 4.76 137.7

22.9

2.42

75

22.08

17.92

11.972

0.940

0.625

696.7

53.0

5.62 4.79 125.2

20.9

2.38

120

35.26

16.75

15.865

1.400

0.830

822.5

63.2

4.83 3.73 437.2

55.1

3.52

110

32.36

16.63

15.810

1.275

0.775

754.1

58.4

4.83 3.71 391.2

49.5

3.48

100

29.40

16.50

15.750

1.150

0.715

683.6

53.3

4.82 3.67 345.8

43.9

3.43

76

22.35

16.75

11.565

1.055

0.635

591.9

47.4

5.15 4.26 128.1

22.1

2.39

70.5

20.76

16.66

11.535

0.960

0.603

551.8

44.7

5.16 4.30 114.9

19.9

2.35

65

19.13

16.55

11.510

0.855

0.580

513.0

42.1

5.18 4.37 100.7

17.5

2.29

105

30.89

15.19

15.105

1.315

0.775

578.0

48.7

4.33 3.31 354.0

46.9

3.38

95

27.95

15.06

15.040

1.185

0.710

520.4

44.1

4.31 3.26 312.3

41.5

3.34

86

25.32

14.94

14.985

1.065

0.655

471.0

40.2

4.31 3.23 275.1

36.7

3.30

66

19.41

15.15

10.551

1.000

0.615

420.7

37.4

4.66 3.90

92.5

17.5

2.18

62

18.22

15.08

10.521

0.930

0.585

394.8

35.3

4.65 3.90

84.8

16.1

2.16

58.0

17.07

15.00

10.500

0.850

0.564

371.8

33.6

4.67 3.94

76.6

14.6

2.12

54.0

15.88

14.91

10.484

0.760

0.548

349.5

32.1

4.69 4.03

67.6

12.9

2.06

88.5

26.05

13.66

14.090

1.190

0.725

391.8

36.7

3.88 2.97 259.4

36.8

3.16

80

23.72

13.54

14.023

1.075

0.658

351.4

33.1

3.87 2.91 229.0

32.7

3.12

72.5

21.34

13.44

13.965

0.975

0.600

316.3

29.9

3.85 2.85 203.5

29.1

3.09

57

16.77

13.64

10.070

0.932

0.570

288.9

28.3

4.15 3.42

14.9

2.11

51

15.01

13.53

10.018

0.827

0.518

257.7

25.4

4.14 3.39

64.8

12.9

2.08

47

13.83

13.45

9.990

0.747

0.490

238.5

23.7

4.15 3.41

57.5

11.5

2.04

80

23.54

12.36

14.091

1.135

0.656

271.6

27.6

3.40 2.51 246.3

35.0

3.23

72.5

21.31

12.24

14.043

1.020

0.608

246.2

25.2

3.40 2.48 217.1

30.9

3.19

65

19.11

12.13

14.000

0.900

0.565

222.6

23.1

3.41 2.47 187.6

26.8

3.13

60

17.64

12.16

12.088

0.930

0.556

213.6

22.4

3.48 2.62 127.0

21.0

2.68

55

16.18

12.08

12.042

0.855

0.510

195.2

20.5

3.47 2.57 114.5

19.0

2.66

50

14.71

12.00

12.000

0.775

0.468

176.7

18.7

3.46 2.54 101.8

17.0

2.63

74.8

(continued)

1852

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.81 (Continued ) Flange

Section Number ST 12 WF

ST 10 WF

ST 10 WFa ST 10 WF

ST 9 WF

ST 9 WF

ST 9 WF

ST 8 WF ST 8 WF

ST 8 WF

ST 7 WF

ST 7 WF

Weight per Foot (lb)

Axis X–X

Axis Y–Y

Width (in.)

Average Thickness (in.)

Stem Thickness (in.)

I (in.4)

S r y I (in.3) (in.) (in.) (in.4)

S r (in.3) (in.)

12.15

9.061

0.872

0.516

185.9

20.3

3.67 2.99

51.1

11.3

1.92

12.04

9.015

0.772

0.470

165.9

18.3

3.66 2.97

44.2

9.8

1.89

11.18

11.95

8.985

0.682

0.440

151.1

16.9

3.68 3.00

38.3

8.5

1.85

71

20.88

10.73

13.132

1.095

0.659

177.3

20.8

2.91 2.18 193.0

29.4

3.04

63.5

18.67

10.62

13.061

0.985

0.588

155.8

18.3

2.89 2.11 169.3

25.9

3.01

56

16.47

10.50

13.000

0.865

0.527

136.4

16.2

2.88 2.06 144.8

22.3

2.96

48

14.11

10.57

9.038

0.935

0.575

137.1

17.1

3.11 2.55

54.7

12.1

1.97

41

12.05

10.43

8.962

0.795

0.499

115.4

14.5

3.09 2.48

44.8

10.0

1.93

36.5

10.73

10.62

8.295

0.740

0.455

110.2

13.7

3.21 2.60

33.1

34

10.01

10.57

8.270

0.685

0.430

102.8

12.9

3.20 2.59

30.2

7.30 1.74

31

9.12

10.49

8.240

0.615

0.400

93.7

11.9

3.21 2.59

26.6

6.45 1.71

57

16.77

9.24

11.833

0.991

0.595

102.6

13.9

2.47 1.85 127.8

21.6

2.76

52.5

15.43

9.16

11.792

0.911

0.554

93.9

12.8

2.47 1.82 115.5

19.6

2.73

48

14.11

9.08

11.750

0.831

0.512

85.3

11.7

2.46 1.78 103.4

17.6

2.71

42.5

12.49

9.16

8.838

0.911

0.526

84.4

11.9

2.60 2.05

49.7

11.3

2.00

38.5

11.32

9.08

8.787

0.831

0.475

75.3

10.6

2.58 1.99

44.3

10.1

1.98

35

10.28

9.00

8.750

0.751

0.438

68.1

9.67 2.57 1.96

39.2

8.97 1.95

32

9.40

8.94

8.715

0.686

0.403

61.8

8.82 2.56 1.93

35.2

8.07 1.93

30

8.82

9.12

7.558

0.695

0.416

64.8

9.32 2.71 2.17

23.5

6.23 1.63

27.5

8.09

9.06

7.532

0.630

0.390

59.6

8.63 2.71 2.16

21.0

5.57 1.61

25

7.35

9.00

7.500

0.570

0.358

53.9

7.85 2.71 2.14

18.6

48

14.11

8.16

11.533

0.875

0.535

64.7

9.82 2.14 1.57 103.6

18.0

2.71

44

12.94

8.08

11.502

0.795

0.504

59.5

9.11 2.14 1.55

92.6

16.1

2.67

39

11.46

8.16

8.586

0.875

0.529

60.0

9.45 2.28 1.81

43.8

10.2

1.95

35.5

10.43

8.08

8.543

0.795

0.486

54.0

8.57 2.28 1.77

38.9

9.11 1.93

32

9.40

8.00

8.500

0.715

0.443

48.3

7.71 2.27 1.73

34.2

8.05 1.91

29

8.52

7.93

8.464

0.645

0.407

43.6

7.00 2.26 1.70

30.2

7.14 1.88

25

7.35

8.13

7.073

0.628

0.380

42.2

6.77 2.40 1.89

17.4

4.92 1.54

22.5

6.62

8.06

7.039

0.563

0.346

37.8

6.10 2.39 1.87

15.2

4.33 1.52

20

5.88

8.00

7.000

0.503

0.307

33.2

5.37 2.37 1.82

13.3

3.79 1.50

18

5.30

7.93

6.992

0.428

0.299

30.7

5.10 2.41 1.90

11.1

105.5

31.04

7.88

15.800

1.563

0.980

102.2

16.2

1.81 1.57 514.3

65.1

4.07

101

29.70

7.82

15.750

1.503

0.930

95.7

15.2

1.80 1.53 489.8

62.2

4.06

96.5

28.36

7.75

15.710

1.438

0.890

90.1

14.4

1.78 1.49 465.1

59.2

4.05

92

27.04

7.69

15.660

1.378

0.840

83.9

13.4

1.76 1.45 441.4

56.4

4.04

88

25.87

7.63

15.640

1.313

0.820

80.2

12.9

1.76 1.42 418.9

53.6

4.02

83.5

24.55

7.56

15.600

1.248

0.780

75.0

12.1

1.75 1.39 395.1

50.7

4.01

79

23.24

7.50

15.550

1.188

0.730

69.3

11.3

1.73 1.34 372.5

47.9

4.00

75

22.04

7.44

15.515

1.128

0.695

64.9

10.6

1.72 1.31 351.3

45.3

3.99

71

20.92

7.38

15.500

1.063

0.680

62.1

10.2

1.72 1.29 330.1

42.6

3.97

68

19.99

7.38

14.740

1.063

0.660

60.0

9.89 1.73 1.31 283.9

38.5

3.77

63.5

18.67

7.31

14.690

0.998

0.610

54.7

9.04 1.71 1.26 263.8

35.9

3.76

59.5

17.49

7.25

14.650

0.938

0.570

50.4

8.36 1.70 1.22 245.9

33.6

3.75

55.5

16.33

7.19

14.620

0.873

0.540

46.7

7.80 1.69 1.19 227.4

31.1

3.73

51.5

15.13

7.13

14.575

0.813

0.495

42.4

7.10 1.67 1.15 209.9

28.8

3.72

47.5

13.97

7.06

14.545

0.748

0.465

39.1

6.58 1.67 1.12 191.9

26.4

3.71

43.5

12.78

7.00

14.5

0.688

0.420

34.9

5.88 1.65 1.08 174.8

24.1

3.70

Area (in.2)

Depth of Tee (in.)

47

13.81

42

12.35

38

7.98 1.76

4.96 1.59

3.17 1.45

1853

STANDARD SIZES

TABLE 51.81 (Continued ) Flange

Section Number ST 7 WF ST 7 WF

ST 7 WF

ST 7 WFa

ST 6 WF

ST 6 WF ST 6 WF

ST 6 WF

ST 6 WF ST 6 Ib ST 6 I ST 5 I ST 4 I ST 3.5 I ST 3 I ST 5 WF

Weight per Foot (lb)

Axis X–X

Axis Y–Y

Width (in.)

Average Thickness (in.)

Stem Thickness (in.)

I (in.4)

S r y I (in.3) (in.) (in.) (in.4)

7.09

12.023

0.778

0.451

37.4

6.36 1.74 1.21 112.7

18.8

7.03

12.000

0.718

0.428

34.8

5.96 1.74 1.19 103.5

17.2

3.00

7.10

10.072

0.783

0.450

36.1

6.26 1.82 1.32

13.3

2.48

Area (in.2)

Depth of Tee (in.)

42

12.36

39

11.47

37

10.88

34

66.7

S r (in.3) (in.) 3.02

10.00

7.03

10.040

0.718

0.418

33.0

5.74 1.81 1.29

60.6

12.1

2.46

30.5

8.97

6.96

10.000

0.643

0.378

29.2

5.13 1.80 1.25

53.6

10.7

2.45

26.5

7.79

6.97

8.062

0.658

0.370

27.7

4.95 1.88 1.38

28.8

7.14 1.92

24

7.06

6.91

8.031

0.593

0.339

24.9

4.49 1.88 1.35

25.6

6.38 1.91

21.5

6.32

6.84

8.000

0.528

0.308

22.2

4.02 1.87 1.33

22.6

5.64 1.89

19

5.59

7.06

6.776

0.513

0.313

23.5

4.27 2.05 1.56

12.3

3.64 1.49

17

5.00

7.00

6.750

0.453

0.287

21.1

3.86 2.05 1.55

10.6

3.15 1.46

15

4.41

6.93

6.733

0.383

0.270

19.0

3.55 2.08 1.59

80.5

23.69

6.94

12.515

1.486

0.905

62.6

66.5

19.56

6.69

12.365

1.236

0.755

60

17.65

6.56

12.320

1.106

53

15.59

6.44

12.230

49.5

14.54

6.38

46

13.53

42.5

11.5

8.77

2.61 1.41

1.63 1.47 243.1

38.9

3.20

48.4

9.03 1.57 1.33 195.0

31.5

3.16

0.710

43.4

8.22 1.57 1.28 172.5

28.0

3.13

0.986

0.620

36.7

7.01 1.53 1.20 150.4

24.6

3.11

12.190

0.921

0.580

33.7

6.46 1.52 1.16 139.1

22.8

3.09

6.31

12.155

0.856

0.545

31.0

5.98 1.51 1.13 128.2

21.1

3.08

12.49

6.25

12.105

0.796

0.495

27.8

5.38 1.49 1.08 117.7

19.5

3.07

39.5

11.61

6.19

12.080

0.736

0.470

25.8

5.02 1.48 1.06 108.2

17.9

3.05

36

10.58

6.13

12.040

0.671

0.430

23.1

4.53 1.48 1.02

97.6

16.2

3.04

32.5

9.55

6.06

12.000

0.606

0.390

20.6

4.06 1.47 0.98

87.3

14.6

3.02

29

8.53

6.10

10.014

0.641

0.359

19.0

3.75 1.49 1.03

53.7

10.7

2.51

26.5

7.80

6.03

10.000

0.576

0.345

17.7

3.54 1.51 1.02

48.0

9.60 2.48

25

7.36

6.10

8.077

0.641

0.371

18.7

3.80 1.60 1.17

28.2

6.98 1.96

22.5

6.62

6.03

8.042

0.576

0.336

16.6

3.40 1.59 1.13

25.0

6.20 1.94

20

5.89

5.97

8.000

0.516

0.294

14.4

2.94 1.56 1.08

22.0

5.50 1.94

18

5.29

6.12

6.565

0.540

0.305

15.3

3.14 1.70 1.26

11.9

3.62 1.50

15.5

4.56

6.04

6.525

0.465

0.265

13.0

2.69 1.69 1.22

9.9

3.04 1.47

13.5

3.98

5.98

6.500

0.400

0.240

11.4

7

2.07

5.96

3.970

0.224

0.200

25

7.29

6.00

5.477

0.660

0.687

20.4

5.92

6.00

5.250

0.660

0.460

17.5

5.10

6.00

5.078

0.544

15.9

4.63

6.00

5.000

17.5

5.11

5.00

12.7

3.69

11.5 9.2

2.39 1.69 1.21

8.3

2.55 1.44

1.83 1.92 1.76

1.13

0.57 0.74

25.2

6.05 1.85 1.84

7.85

2.87 1.03

18.8

4.26 1.77 1.57

6.77

2.58 1.06

0.428

17.2

3.95 1.83 1.65

4.93

1.94 0.98

0.544

0.350

14.9

3.31 1.78 1.51

4.68

1.87 1.00

4.944

0.491

0.594

12.5

3.63 1.56 1.56

4.18

1.69 0.90

5.00

4.660

0.491

0.310

7.81

2.05 1.45 1.20

3.39

1.46 0.95

3.36

4.00

4.171

0.425

0.441

5.03

1.77 1.22 1.15

2.15

1.03 0.80

2.67

4.00

4.000

0.425

0.270

3.50

1.14 1.14 0.94

1.86

0.93 0.83

2.92

3.50

3.860

0.392

0.450

3.36

1.36 1.07 1.04

1.58

0.82 0.73

7.65

2.22

3.50

3.660

0.392

0.250

2.18

0.81 0.99 0.81

1.32

0.72 0.77

8.625

2.51

3.00

3.565

0.359

0.465

2.13

1.02 0.92 0.91

1.15

0.65 0.67

6.25

1.81

3.00

3.330

0.359

0.230

1.27

0.55 0.83 0.69

0.93

56

16.46

5.69

10.415

1.248

0.755

28.8

6.42 1.32 1.21 117.7

22.6

50

14.72

5.56

10.345

1.118

0.685

24.8

5.62 1.30 1.14 103.3

20.0

2.65

44.5

13.09

5.44

10.275

0.998

0.615

21.3

4.88 1.28 1.07

17.6

2.63

10

7.66

90.3

0.56 0.71 2.67

(continued)

1854

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.81 (Continued ) Flange

Section Number

ST 5 WF

ST 5 WFa

ST 4 WF

ST 4 WF ST 4 WF

Weight per Foot (lb)

Area (in.2)

Depth of Tee (in.)

Width (in.)

Average Thickness (in.)

Axis X–X Stem Thickness (in.)

I (in.4)

11.33

5.31

10.195

0.868

0.535

17.7

4.10 1.25 1.00

76.7

15.1

2.60

10.59

5.25

10.170

0.808

0.510

16.4

3.83 1.24 0.97

70.9

13.9

2.59

33

9.70

5.19

10.117

0.748

0.457

14.5

3.39 1.22 0.92

64.6

12.8

2.58

30

8.83

5.13

10.075

0.683

0.415

12.8

3.02 1.21 0.88

58.2

11.6

2.57

27

7.94

5.06

10.028

0.618

0.368

11.2

2.64 1.18 0.84

51.95 10.4

2.56

24.5

7.20

5.00

10.000

0.558

0.340

10.1

2.40 1.18 0.81

46.5

22.5

6.62

5.06

8.022

0.618

0.350

10.3

2.48 1.25 0.91

26.6

6.63 2.00

19.5

5.74

4.97

7.990

0.528

0.318

8.96

2.19 1.25 0.88

22.5

5.62 1.98

16.5

4.85

4.88

7.964

0.433

0.292

7.80

1.95 1.27 0.88

18.2

4.58 1.94

14.5

4.27

5.11

5.799

0.500

0.289

8.38

2.07 1.40 1.05

7.61

2.62 1.34

12.5

3.67

5.04

5.762

0.430

0.252

7.12

1.77 1.39 1.02

6.34

2.20 1.31

10.5

3.10

4.95

5.750

0.340

0.240

6.31

1.62 1.43 1.06

4.87

33.5

9.85

4.50

8.287

0.933

0.575

10.94

3.07 1.05 0.94

44.3

29

8.53

4.38

8.222

0.808

0.510

9.11

2.60 1.03 0.87

37.5

9.10 2.10

24

7.06

4.25

8.117

0.683

0.405

6.92

2.00 0.99 0.78

30.45

7.50 2.08

20

5.88

4.13

8.077

0.558

0.365

5.80

1.71 0.99 0.74

24.5

6.05 2.04

17.5

5.15

4.06

8.027

0.493

0.315

4.88

1.45 0.97 0.69

21.25

5.30 2.03

15.5

4.56

4.00

8.000

0.433

0.288

4.31

1.30 0.97 0.67

18.5

4.60 2.01

14

4.11

4.03

6.540

0.463

0.285

4.22

1.28 1.01 0.73

10.8

3.30 1.62

12

3.53

3.97

6.500

0.398

0.245

3.53

1.08 1.00 0.70

9.10

2.80 1.61

10

2.94

4.07

5.268

0.378

0.248

3.66

1.13 1.12 0.83

4.25

1.61 1.20

2.50

4.00

5.250

0.308

0.230

3.21

1.01 1.13 0.84

3.36

1.28 1.16

Area (in.2)

Depth (in.)

5  3 18

13.6

4.00

3 18

5

11.5

3.37

3

5

11.2

3.29

4 12

4

13.5

3.97

4

4

43

9.2

2.68

3

4

8.5

2.48

2 12

4

33

7.8

2.29

3

3

33

6.7

1.97

3

3

3  2 12

6.1

1.77

2 12

3



6.4

1.87

2 12

2 12

4.6

1.33

2 12

2 12

4.1

1.19

2 14

2 14

22

4.3

1.26

2

2

22

3.56

1.05

2

2

4  2 12

2 12 2 12 2 14

a b

 

2 12 2 12 2 14

Axis X–X

Width Minimum Thickness Flange Flange Stem (in.) (in.) (in.)

Nominal Weight per Size (in.) Foot (lb)

44

S r (in.3) (in.)

36

Dimensions

4  4 12

S r y I (in.3) (in.) (in.) (in.4)

38.5

8.5

53

Axis Y–Y

1 2 3 8 3 8 1 2 3 8 3 8 3 8 5 16 5 16 3 8 1 4 1 4 5 16 1 4

WF indicates structural tee cut from wide-flange section. I indicates structural tee cut from standard beam section.

13 32 13 32 3 8 1 2 3 8 3 8 3 8 5 16 5 16 3 8 1 4 1 4 5 16 1 4

9.30 2.54

1.69 1.25 10.7

2.12

Axis Y–Y

I (in.4)

S (in.3)

r (in.)

y (in.)

I (in.4)

S (in.3)

r (in.)

2.7

1.1

0.82

0.76

5.2

2.1

1.14

2.4

1.1

0.84

0.76

3.9

1.6

1.10

6.3

2.0

1.39

1.31

2.1

1.1

0.80

5.7

2.0

1.20

1.18

2.8

1.4

0.84

2.0

0.90

0.86

0.78

2.1

1.1

0.89

1.2

0.62

0.69

0.62

2.1

1.0

0.92

1.84

0.86

0.89

0.88

0.89

0.60

0.63

1.61

0.74

0.90

0.85

0.75

0.50

0.62

0.94

0.51

0.73

0.68

0.75

0.50

0.65

1.0

0.59

0.74

0.76

0.52

0.42

0.53

0.74

0.42

0.75

0.71

0.34

0.27

0.51

0.52

0.32

0.66

0.65

0.25

0.22

0.46

0.44

0.31

0.59

0.61

0.23

0.23

0.43

0.37

0.26

0.59

0.59

0.18

0.18

0.42

STANDARD SIZES

1855

TABLE 51.82 Properties and Dimensions of Zees

Zees are seldom used as structural framing members. When so used they are generally employed on short spans in flexure. This table lists a few selected sizes, the range of whose section moduli will cover all ordinary conditions. For sizes not listed, the catalogs of the respective rolling mills should be consulted. Dimensions

Nominal Size (in.) 6  3 12 5  3 14

1 4  3 16

3  2 11 16

Weight per Foot (lb)

Area Depth (in.2) (in.)

Width of Flange (in.)

21.1

6.19

6 18

3 58

15.7

4.59

6

3 12

17.9

5.25

5

3 14

16.4

4.81

5 18

3 38

14.0

4.10

1 5 16

5 3 16

11.6

3.40

5

3 14

15.9

4.66

1 4 16

3 18

12.5

3.66

4 18

3 3 16

10.3

3.03

1 4 16

8.2

2.41

4

3 18 1 3 16 11 2 16 2 11 16 2 11 16

12.6

3.69

3

9.8

2.86

3

6.7

1.97

3

Axis X–X

Axis Y–Y

Axis Z–Z

S r I S r Thickness I (in.) (in.4) (in.3) (in.) (in.4) (in.3) (in.) 1 2 3 8 1 2 7 16 3 8 5 16 1 2 3 8 5 16 1 4 1 2 3 8 1 4

34.4

11.2 2.36 12.9

r (in.)

3.8

1.44

0.84

25.3

8.4 2.35

9.1

2.8

1.41

0.83

19.2

7.7 1.91

9.1

3.0

1.31

0.74

19.1

7.4 1.99

9.2

2.9

1.38

0.77

16.2

6.4 1.99

7.7

2.5

1.37

0.76

13.4

5.3 1.98

6.2

2.0

1.35

0.75

11.2

5.5 1.55

8.0

2.8

1.31

0.67

9.6

4.7 1.62

6.8

2.3

1.36

0.69

7.9

3.9 1.62

5.5

1.8

1.34

0.68

6.3

3.1 1.62

4.2

1.4

1.33

0.67

4.6

3.1 1.12

4.9

2.0

1.15

0.53

3.9

2.6 1.16

3.9

1.6

1.17

0.54

2.9

1.9 1.21

2.8

1.1

1.19

0.55

TABLE 51.83 Properties and Dimensions of H Bearing Piles

Flange

Axis X–X

Section Number and Nominal Size

Weight per Foot (lb)

Area A (in.2)

Depth d (in.)

Width b (in.)

Thickness t (in.)

Web Thickness W (in.)

BP 14,

117 102

34.44 30.01

14.234 14.032

14.885 14.784

0.805 0.704

89

26.19

13.856

14.696

73

21.46

13.636

14.586

74 53 57 42 36

21.76 15.58 16.76 12.35 10.60

12.122 11.780 10.012 9.720 8.026

12.217 12.046 10.224 10.078 8.158

14  14 12

BP 12, 12  12 BP 10, 10  10 BP 8, 8  8

Axis Y–Y

I (in.4)

S (in.3)

r (in.)

I0 (in.4)

S0 (in.3)

r0 (in.)

0.805 0.704

1228.5 1055.1

172.6 150.4

5.97 5.93

443.1 379.6

59.5 51.3

3.59 3.56

0.616

0.616

909.1

131.2

5.89

326.2

44.4

3.53

0.506

0.506

733.1

107.5

5.85

261.9

35.9

3.49

0.607 0.436 0.564 0.418 0.446

0.607 0.436 0.564 0.418 0.446

566.5 394.8 294.7 210.8 119.8

93.5 67.0 58.9 43.4 29.9

5.10 5.03 4.19 4.13 3.36

184.7 127.3 100.6 71.4 40.4

30.2 21.2 19.7 14.2 9.9

2.91 2.86 2.45 2.40 1.95

1856

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.84 Square and Round Barsa Square Size Weight/ft (in.) (lb)

Area (in.2)

Weight/ft (lb)

Area (in.2)

0 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

0.013

0.0039

0.010

0.0031

0.053

0.0156

0.042

0.0123

0.120

0.0352

0.094

0.0276

0.213

0.0625

0.167

0.0491

0.332

0.0977

0.261

0.0767

0.478

0.1406

0.376

0.1105

0.651

0.1914

0.511

0.1503

0.850

0.2500

0.668

0.1963

1.076

0.3164

0.845

0.2485

1.328

0.3906

1.043

0.3068

1.607

0.4727

1.262

0.3712

1.913

0.5625

1.502

0.4418

2.245

0.6602

1.763

0.5185

2.603

0.7656

2.044

0.6013

2.988

0.8789

2.347

0.6903

3.400

1.0000

2.670

0.7854

3.838

1.1289

3.015

0.8866

4.303

1.2656

3.380

0.9940

4.795

1.4102

3.766

1.1075

5.313

1.5625

4.172

1.2272

5.857

1.7227

4.600

1.3530

6.428

1.8906

5.049

1.4849

7.026

2.0664

5.518

1.6230

7.650

2.2500

6.008

1.7671

8.301

2.4414

6.519

1.9175

8.978

2.6406

7.051

2.0739

9.682

2.8477

7.604

2.2365

10.413

3.0625

8.178

2.4053

11.170

3.2852

8.773

2.5802

11.953

3.5156

9.388

2.7612

12.763

3.7539

10.024

2.9483

2

13.600

4.0000

10.681

3.1416

1 16 1 8 3 16

14.463

4.2539

11.359

3.3410

15.353

4.5156

12.058

3.5466

16.270

4.7852

12.778

3.7583

1 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

Square

Round

Size Weight/ft (in.) (lb) 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

3 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

4 1 16 1 8 3 16 1 4 5 16 3 8

Round

Area (in.2)

Weight/ft (lb)

Area (in.2)

17.213

5.0625

13.519

3.9761

18.182

5.3477

14.280

4.2000

19.178

5.6406

15.062

4.4301

20.201

5.9414

15.866

4.6664

21.250

6.2500

16.690

4.9087

22.326

6.5664

17.534

5.1572

23.428

6.8906

18.400

5.4119

24.557

7.2227

19.287

5.6727

25.713

7.5625

20.195

5.9396

26.895

7.9102

21.123

6.2126

28.103

8.2656

22.072

6.4918

29.338

8.6289

23.042

6.7771

30.60

9.000

24.03

7.069

31.89

9.379

25.05

7.366

33.20

9.766

26.08

7.670

34.54

10.160

27.13

7.980

35.91

10.563

28.21

8.296

37.31

10.973

29.30

8.618

38.73

11.391

30.42

8.946

40.18

11.816

31.55

9.281

41.65

12.250

32.71

9.621

43.15

12.691

33.89

9.968

44.68

13.141

35.09

10.321

46.23

13.598

36.31

10.680

47.81

14.063

37.55

11.045

49.42

14.535

38.81

11.416

51.05

15.016

40.10

11.793

52.71

15.504

41.40

12.177

54.40

16.000

42.73

12.566

56.11

16.504

44.07

12.962

57.85

17.016

45.44

13.364

59.62

17.535

46.83

13.772

61.41

18.063

48.23

14.186

63.23

18.598

49.66

14.607

65.08

19.141

51.11

15.033

1857

STANDARD SIZES

TABLE 51.84 (Continued) Square Size Weight/ft (in.) (lb) 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

5 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

6 1 16 1 8 3 16 a

Square

Round

Area (in.2)

Weight/ft (lb)

Area (in.2)

66.95

19.691

52.58

15.466

68.85

20.250

54.07

15.904

70.78

20.816

55.59

16.349

72.73

21.391

57.12

16.800

74.71

21.973

58.67

17.257

76.71

22.563

60.25

17.721

78.74

23.160

61.85

18.190

80.80

23.766

63.46

18.665

82.89

24.379

65.10

19.147

85.00

25.000

66.76

19.635

87.14

25.629

68.44

20.129

89.30

26.266

70.14

20.629

91.49

26.910

71.86

21.135

93.71

27.563

73.60

21.648

95.96

28.223

75.36

22.166

98.23

28.891

77.15

22.691

100.53

29.566

78.95

23.221

102.85

30.250

80.78

23.758

105.20

30.941

82.62

24.301

107.58

31.641

84.49

24.850

109.98

32.348

86.38

25.406

112.41

33.063

88.29

25.967

114.87

33.785

90.22

26.535

117.35

34.516

92.17

27.109

119.86

35.254

94.14

27.688

122.40

36.000

96.13

28.274

124.96

36.754

98.15

28.866

127.55

37.516

100.18

29.465

130.17

38.285

102.23

30.069

Size Weight/ft (in.) (lb) 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

7 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

8

One cubic inch of rolled steel is assumed to weigh 0.2833 lb.

Area (in.2)

Round Weight/ft (lb)

Area (in.2)

132.81

39.063

104.31

30.680

135.48

39.848

106.41

31.296

138.18

40.641

108.53

31.919

140.90 143.65

41.441 42.250

110.66 112.82

32.548 33.183

146.43

43.066

115.00

33.824

149.23

43.891

117.20

34.472

152.06

44.723

119.43

35.125

154.91

45.563

121.67

35.785

157.79

46.410

123.93

36.450

160.70

47.266

126.22

37.122

163.64

48.129

128.52

37.800

166.60

49.000

130.85

38.485

169.59

49.879

133.19

39.175

172.60

50.766

135.56

39.871

175.64

51.660

137.95

40.574

178.71

52.563

140.36

41.282

181.81

53.473

142.79

41.997

184.93

54.391

145.24

42.718

188.07

55.316

147.71

43.445

191.25

56.250

150.21

44.179

194.45

57.191

152.72

44.918

197.68

58.141

155.26

45.664

200.93

59.098

157.81

46.415

204.21

60.063

160.39

47.173

207.52

61.035

162.99

47.937

210.85

62.016

165.60

48.707

214.21

63.004

168.24

49.483

217.60

64.000

170.90

50.265

1858

TABLE 51.85 Dimensions of Ferrous Pipe Cross-Sectional Area Nominal Pipe Size (in.) 1 8

1 4

3 8

1 2

3 4

1

Outside Diameter (in.) 0.405

0.540

0.675

0.840

1.050

1.315

Schedule No.

Wall Thickness (in.)

Inside Diameter (in.)

Metal (in.2)

Circumference, ft, or surface, ft2/ ft of Length

Flow (ft2)

Outside

Inside

Capacity at 1 ft/sec Velocity

U.S. gal/min

lb/hr water

Weight of Plain-End Pipe (lb/ft)

10S

0.049

0.307

0.055

0.00051

0.106

0.0804

0.231

115.5

0.19

40ST, 40S 80XS, 80S

0.068 0.095

0.269 0.215

0.072 0.093

0.00040 0.00025

0.106 0.106

0.0705 0.0563

0.179 0.113

89.5 56.5

0.24 0.31

10S

0.065

0.410

0.097

0.00092

0.141

0.107

0.412

206.5

0.33

40ST, 40S 80XS, 80S

0.088 0.119

0.364 0.302

0.125 0.157

0.00072 0.00050

0.141 0.141

0.095 0.079

0.323 0.224

161.5 112.0

0.42 0.54

10S

0.065

0.545

0.125

0.00162

0.177

0.143

0.727

363.5

0.42

40ST, 40S 80XS, 80S

0.091 0.126

0.493 0.423

0.167 0.217

0.00133 0.00098

0.177 0.177

0.129 0.111

0.596 0.440

298.0 220.0

0.57 0.74

5S

0.065

0.710

0.158

0.00275

0.220

0.186

1.234

617.0

0.54

10S 40ST, 40S 80XS, 80S 160 XX

0.083 0.109 0.147 0.188 0.294

0.674 0.622 0.546 0.464 0.252

0.197 0.250 0.320 0.385 0.504

0.00248 0.00211 0.00163 0.00117 0.00035

0.220 0.220 0.220 0.220 0.220

0.176 0.163 0.143 0.122 0.066

1.112 0.945 0.730 0.527 0.155

556.0 472.0 365.0 263.5 77.5

0.67 0.85 1.09 1.31 1.71

5S

0.065

0.920

0.201

0.00461

0.275

0.241

2.072

1036.0

0.69

10S 40ST, 40S 80XS, 80S 160 XX

0.083 0.113 0.154 0.219 0.308

0.884 0.824 0.742 0.612 0.434

0.252 0.333 0.433 0.572 0.718

0.00426 0.00371 0.00300 0.00204 0.00103

0.275 0.275 0.275 0.275 0.275

0.231 0.216 0.194 0.160 0.114

1.903 1.665 1.345 0.917 0.461

951.5 832.5 672.5 458.5 230.5

0.86 1.13 1.47 1.94 2.44

5S 10S 40ST, 40S 80XS, 80S

0.065 0.109 0.133 0.179

1.185 1.097 1.049 0.957

0.255 0.413 0.494 0.639

0.00768 0.00656 0.00600 0.00499

0.344 0.344 0.344 0.344

0.310 0.287 0.275 0.250

3.449 2.946 2.690 2.240

1725 1473 1345 1120

0.87 1.40 1.68 2.17

1 14

1 12

1.660

1.900

2

2.375

2 12

2.875

3

3.500

160 XX

0.250 0.358

0.815 0.599

0.836 1.076

0.00362 0.00196

0.344 0.344

0.213 0.157

1.625 0.878

812.5 439.0

2.84 3.66

5S

0.065

1.530

0.326

0.01277

0.435

0.401

5.73

2865

1.11

10S 40ST, 40S 80XS, 80S 160 XX

0.109 0.140 0.191 0.250 0.382

1.442 1.380 1.278 1.160 0.896

0.531 0.668 0.881 1.107 1.534

0.01134 0.01040 0.00891 0.00734 0.00438

0.435 0.435 0.435 0.435 0.435

0.378 0.361 0.335 0.304 0.235

5.09 4.57 3.99 3.29 1.97

2545 2285 1995 1645 985

1.81 2.27 3.00 3.76 5.21

5S

0.065

1.770

0.375

0.01709

0.497

0.463

7.67

3835

1.28

10S 40ST, 40S 80SX, 80S 160 XX

0.109 0.145 0.200 0.281 0.400

1.682 1.610 1.500 1.338 1.100

0.614 0.800 1.069 1.429 1.885

0.01543 0.01414 0.01225 0.00976 0.00660

0.497 0.497 0.497 0.497 0.497

0.440 0.421 0.393 0.350 0.288

6.94 6.34 5.49 4.38 2.96

3465 3170 2745 2190 1480

2.09 2.72 3.63 4.86 6.41

5S 10S 40ST, 40S 80ST, 80S 160 XX

0.065 0.109 0.154 0.218 0.344 0.436

2.245 2.157 2.067 1.939 1.687 1.503

0.472 0.776 1.075 1.477 2.195 2.656

0.02749 0.02538 0.02330 0.02050 0.01552 0.01232

0.622 0.622 0.622 0.622 0.622 0.622

0.588 0.565 0.541 0.508 0.436 0.393

12.34 11.39 10.45 9.20 6.97 5.53

6170 5695 5225 4600 3485 2765

1.61 2.64 3.65 5.02 7.46 9.03

1859

5S

0.083

2.709

0.728

0.04003

0.753

0.709

17.97

8985

2.48

10S 40ST, 40S 80XS, 80S 160 XX

0.120 0.203 0.276 0.375 0.552

2.635 2.469 2.323 2.125 1.771

1.039 1.704 2.254 2.945 4.028

0.03787 0.03322 0.02942 0.02463 0.01711

0.753 0.753 0.753 0.753 0.753

0.690 0.647 0.608 0.556 0.464

17.00 14.92 13.20 11.07 7.68

8500 7460 6600 5535 3840

3.53 5.79 7.66 10.01 13.70

5S 10S 40ST, 40S 80XS, 80S 160 XX

0.083 0.120 0.216 0.300 0.438 0.600

3.334 3.260 3.068 2.900 2.624 2.300

0.891 1.274 2.228 3.016 4.213 5.466

0.06063 0.05796 0.05130 0.04587 0.03755 0.02885

0.916 0.916 0.916 0.916 0.916 0.916

0.873 0.853 0.803 0.759 0.687 0.602

27.21 26.02 23.00 20.55 16.86 12.95

13,605 13,010 11,500 10,275 8430 6475

3.03 4.33 7.58 10.25 14.31 18.58

(continued)

1860

TABLE 51.85 (Continued ) Cross-Sectional Area Nominal Pipe Size (in.) 3 12

Outside Diameter (in.) 4.0

Schedule No.

Wall Thickness (in.)

Inside Diameter (in.)

Metal (in.2)

Circumference, ft, or surface, ft2/ ft of Length

Capacity at 1 ft/sec Velocity

Flow (ft2)

Outside

Inside

U.S. gal/min

lb/hr water

Weight of Plain-End Pipe (lb/ft)

5S

0.083

3.834

1.021

0.08017

1.047

1.004

35.98

10S 40ST, 40S 80XS, 80S

0.120 0.226 0.318

3.760 3.548 3.364

1.463 2.680 3.678

0.07711 0.06870 0.06170

1.047 1.047 1.047

0.984 0.929 0.881

34.61 30.80 27.70

17,305 15,400 13,850

4.97 9.11 12.51

1.152 1.651 3.17 4.41 5.58 6.62 8.10

0.10245 0.09898 0.08840 0.07986 0.07170 0.06647 0.05419

1.178 1.178 1.178 1.178 1.178 1.178 1.178

1.135 1.115 1.054 1.002 0.949 0.900 0.825

46.0 44.4 39.6 35.8 32.2 28.9 24.3

23,000 22,200 19,800 17,900 16,100 14,450 12,150

3.92 5.61 10.79 14.98 19.01 22.52 27.54

17.990

3.48

4

4.5

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

0.083 0.120 0.237 0.337 0.438 0.531 0.674

4.334 4.260 4.026 3.826 3.624 3.438 3.152

5

5.563

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

0.109 0.134 0.258 0.375 0.500 0.625 0.750

5.345 5.295 5.047 4.813 4.563 4.313 4.063

1.87 2.29 4.30 6.11 7.95 9.70 11.34

0.1558 0.1529 0.1390 0.1263 0.1136 0.1015 0.0900

1.456 1.456 1.456 1.456 1.456 1.456 1.456

1.399 1.386 1.321 1.260 1.195 1.129 1.064

69.9 68.6 62.3 57.7 51.0 45.5 40.4

34,950 34,300 31,150 28,850 25,500 22,750 20,200

6.36 7.77 14.62 20.78 27.04 32.96 38.55

6

6.625

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

0.109 0.134 0.280 0.432 0.562 0.719 0.864

6.407 6.357 6.065 5.761 5.501 5.187 4.897

2.23 2.73 5.58 8.40 10.70 13.34 15.64

0.2239 0.2204 0.2006 0.1810 0.1650 0.1467 0.1308

1.734 1.734 1.734 1.734 1.734 1.734 1.734

1.677 1.664 1.588 1.508 1.440 1.358 1.282

100.5 98.9 90.0 81.1 73.9 65.9 58.7

50,250 49,450 45,000 40,550 36,950 32,950 29,350

7.60 9.29 18.97 28.57 36.42 45.34 53.16

8

8.625

5S 10S 20 30 40ST, 40S 60 80XS, 80S 100 120 140 XX 160

0.109 0.148 0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.875 0.906

8.407 8.329 8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.875 6.813

2.915 3.941 6.578 7.260 8.396 10.48 12.76 14.99 17.86 19.93 21.30 21.97

0.3855 0.3784 0.3601 0.3553 0.3474 0.3329 0.3171 0.3017 0.2817 0.2673 0.2578 0.2532

2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258

2.201 2.180 2.127 2.113 2.089 2.045 1.996 1.947 1.882 1.833 1.800 1.784

173.0 169.8 161.5 159.4 155.7 149.4 142.3 135.4 126.4 120.0 115.7 113.5

86,500 84,900 80,750 79,700 77,850 74,700 71,150 67,700 63,200 60,000 57,850 56,750

9.93 13.40 22.36 24.70 28.55 35.66 43.39 50.93 60.69 67.79 72.42 74.71

1861

10

10.75

5S 10S 20 30 40ST, 40S 80S, 60XS 80 100 120 140, XX 160

0.134 0.165 0.250 0.307 0.365 0.500 0.594 0.719 0.844 1.000 1.125

10.842 10.420 10.250 10.136 10.020 9.750 9.562 9.312 9.062 8.750 8.500

4.47 5.49 8.25 10.07 11.91 16.10 18.95 22.66 26.27 30.63 34.02

0.5993 0.5922 0.5731 0.5603 0.5475 0.5185 0.4987 0.4729 0.4479 0.4176 0.3941

2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814

2.744 2.728 2.685 2.655 2.620 2.550 2.503 2.438 2.372 2.291 2.225

269.0 265.8 257.0 252.0 246.0 233.0 223.4 212.3 201.0 188.0 177.0

134,500 132,900 128,500 126,000 123,000 116,500 111,700 106,150 100,500 94,000 88,500

15.23 18.70 28.04 34.24 40.48 54.74 64.40 77.00 89.27 104.13 115.65

12

12.75

5S 10S 20 30 ST, 40S 40 XS, 80S 60 80

0.156 0.180 0.250 0.330 0.375 0.406 0.500 0.562 0.688

12.438 12.390 12.250 12.090 12.000 11.938 11.750 11.626 11.374

6.17 7.11 9.82 12.88 14.58 15.74 19.24 21.52 26.07

0.8438 0.8373 0.8185 0.7972 0.7854 0.7773 0.7530 0.7372 0.7056

3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338

3.26 3.24 3.21 3.17 3.14 3.13 3.08 3.04 2.98

378.7 375.8 367.0 358.0 352.5 349.0 338.0 331.0 316.7

189,350 187,900 183,500 179,000 176,250 174,500 169,000 165,500 158,350

22.22 24.20 33.38 43.77 49.56 53.56 65.42 73.22 88.57

(continued)

1862

TABLE 51.85 (Continued ) Cross-Sectional Area Nominal Pipe Size (in.)

Outside Diameter (in.)

Circumference, ft, or surface, ft2/ ft of Length

Capacity at 1 ft/sec Velocity

Schedule No.

Wall Thickness (in.)

Inside Diameter (in.)

Metal (in.2)

Flow (ft2)

Outside

Inside

U.S. gal/min

100 120, XX 140 160

0.844 1.000 1.125 1.312

11.062 10.750 10.500 10.126

31.57 36.91 41.09 47.14

0.6674 0.6303 0.6013 0.5592

3.338 3.338 3.338 3.338

2.90 2.81 2.75 2.65

299.6 283.0 270.0 251.0

149,800 141,500 135,000 125,500

107.29 125.49 139.68 160.33

lb/hr water

Weight of Plain-End Pipe (lb/ft)

14

14

5S 10S 10 20 30, ST 40 XS 60 80 100 120 140 160

0.156 0.188 0.250 0.312 0.375 0.438 0.500 0.594 0.750 0.938 1.094 1.250 1.406

13.688 13.624 13.500 13.376 13.250 13.124 13.000 12.812 12.500 12.124 11.812 11.500 11.188

6.78 8.16 10.80 13.42 16.05 18.66 21.21 25.02 31.22 38.49 44.36 50.07 55.63

1.0219 1.0125 0.9940 0.9750 0.9575 0.9397 0.9218 0.8957 0.8522 0.8017 0.7610 0.7213 0.6827

3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665

3.58 3.57 3.53 3.50 3.47 3.44 3.40 3.35 3.27 3.17 3.09 3.01 2.93

459 454 446 438 430 422 414 402 382 360 342 324 306

229,500 227,000 223,000 219,000 215,000 211,000 207,000 201,000 191,000 180,000 171,000 162,000 153,000

22.76 27.70 36.71 45.68 54.57 63.37 72.09 85.01 106.13 130.79 150.76 170.22 189.12

16

16

5S 10S 10 20 30, ST 40, XS 60 80

0.165 0.188 0.250 0.312 0.375 0.500 0.656 0.844

15.670 15.624 15.500 15.376 15.250 15.000 14.688 14.312

8.18 9.34 12.37 15.38 18.41 24.35 31.62 40.19

1.3393 1.3314 1.3104 1.2985 1.2680 1.2272 1.1766 1.1171

4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189

4.10 4.09 4.06 4.03 3.99 3.93 3.85 3.75

601 598 587 578 568 550 528 501

300,500 299,000 293,500 289,000 284,000 275,000 264,000 250,500

27.87 31.62 42.05 52.36 62.58 82.77 107.54 136.58

100 120 140 160

1.031 1.219 1.438 1.594

13.938 13.562 13.124 12.812

48.48 56.61 65.79 72.14

1.0596 1.0032 0.9394 0.8953

4.189 4.189 4.189 4.189

3.65 3.55 3.44 3.35

474 450 422 402

237,000 225,000 211,000 201,000

164.86 192.40 223.57 245.22

18

18

5S 10S 10 20 ST 30 XS 40 60 80 100 120 140 160

0.165 0.188 0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781

17.670 17.624 17.500 17.376 17.250 17.124 17.000 16.876 16.500 16.124 15.688 15.250 14.876 14.438

9.25 10.52 13.94 17.34 20.76 24.16 27.49 30.79 40.64 50.28 61.17 71.82 80.66 90.75

1.7029 1.6941 1.6703 1.6468 1.6230 1.5993 1.5763 1.5533 1.4849 1.4180 1.3423 1.2684 1.2070 1.1370

4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712

4.63 4.61 4.58 4.55 4.52 4.48 4.45 4.42 4.32 4.22 4.11 3.99 3.89 3.78

74 760 750 739 728 718 707 697 666 636 602 569 540 510

382,000 379,400 375,000 369,500 364,000 359,000 353,500 348,500 333,000 318,000 301,000 284,500 270,000 255,000

31.32 35.48 47.39 59.03 70.59 82.06 93.45 104.76 138.17 170.75 208.00 244.14 274.30 308.55

20

20

5S 10S 10 20, ST 30, XS 40 60 80 100 120 140 160

0.188 0.218 0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969

19.624 19.564 19.500 19.250 19.000 18.812 18.376 17.938 17.438 17.000 16.500 16.062

11.70 13.55 15.51 23.12 30.63 36.21 48.95 61.44 75.33 87.18 100.3 111.5

2.1004 2.0878 2.0740 2.0211 1.9689 1.9302 1.8417 1.7550 1.6585 1.5763 1.4849 1.4071

5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236

5.14 5.12 5.11 5.04 4.97 4.92 4.81 4.70 4.57 4.45 4.32 4.21

943 937 930 902 883 866 826 787 744 707 665 632

471,500 467,500 465,000 451,000 441,500 433,000 413,000 393,500 372,000 353,500 332,500 316,000

39.76 45.98 52.73 78.60 104.13 123.06 166.50 208.92 256.15 296.37 341.10 379.14

1863

(continued)

1864

TABLE 51.85 (Continued ) Cross-Sectional Area Nominal Pipe Size (in.)

Outside Diameter (in.)

Circumference, ft, or surface, ft2/ ft of Length

Schedule No.

Wall Thickness (in.)

Inside Diameter (in.)

Metal (in.2)

Flow (ft2)

Outside

Inside

Capacity at 1 ft/sec Velocity

U.S. gal/min

lb/hr water

Weight of Plain-End Pipe (lb/ft)

24

24

5S 10, 10S 20, ST XS 30 40 60 80 100 120 140 160

0.218 0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.531 1.812 2.062 2.344

23.564 23.500 23.250 23.000 22.876 22.624 22.062 21.562 20.938 20.376 19.876 19.312

16.29 18.65 27.83 36.90 41.39 50.39 70.11 87.24 108.1 126.3 142.1 159.5

3.0285 3.012 2.948 2.885 2.854 2.792 2.655 2.536 2.391 2.264 2.155 2.034

6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283

6.17 6.15 6.09 6.02 5.99 5.92 5.78 5.64 5.48 5.33 5.20 5.06

1359 1350 1325 1295 1281 1253 1192 1138 1073 1016 965 913

679,500 675,000 662,500 642,500 640,500 626,500 596,000 569,000 536,500 508,000 482,500 456,500

55.08 63.41 94.62 125.49 140.80 171.17 238.29 296.53 367.45 429.50 483.24 542.09

30

30

5S 10, 10S ST 20, XS 30

0.250 0.312 0.375 0.500 0.625

29.500 29.376 29.250 29.000 28.750

23.37 29.10 34.90 46.34 57.68

4.746 4.707 4.666 4.587 4.508

7.854 7.854 7.854 7.854 7.854

7.72 7.69 7.66 7.59 7.53

2130 2110 2094 2055 2020

1,065,000 1,055,000 1,048,000 1,027,500 1,010,000

79.43 99.08 118.65 157.53 196.08

Schedule Nos. 5S, 10S, and 40S American National Standards Institute (ANSI)/American Society of Mechanical Engineers (ASME) B.36.19-1985, “Stainless Steel Pipe.” ST ¼ standard wall, XS ¼ extra strong wall, XX ¼ double extra strong wall are all taken from ANSI/ASME, B.36.10M-1985, “Welded and Seamless Wrought-steel Pipe.” Wrought-iron pipe has slightly thicker walls, approximately 3%, but the same weight per foot, because of lower density. Decimal thicknesses for respective pipe sizes represent their nominal or average wall dimensions. Mill tolerances as high as 12 12 % are permitted. Plain-end pipe is produced by a square cut. Pipe is also shipped from the mills threaded, with a threaded coupling on one end, or with the ends beveled for welding, or grooved or sized for patented couplings. Weights per foot for threaded and coupled pipe are slightly greater because of the weight of the coupling, but it is not available larger than 12 in., or lighter than Schedule 30 sizes 8 through 12 in., or Schedule 40 6 in. and smaller. Source: From Chemical Engineer’s Handbook, 4th ed., New York, McGraw-Hill, 1963. Used by permission.

TABLE 51.86 Properties and Dimensions of Steel Pipea Dimensions

Couplings

Properties

Weight per Foot (lb) Nominal Outside Inside Thickness Diameter (in.) Diameter (in.) Diameter (in.) (in.)

Plain Ends

Thread and Coupling

Threads Outside per Inch Diameter (in.)

Length (in.)

Weight (lb)

I (in.4)

A (in.2)

k (in.)

1865

1 8 1 4 3 8 1 2 3 4

0.405

0.269

0.068

0.24

Schedule 40ST 0.25 27

0.562

7 8

0.03

0.001

0.072

0.12

0.540

0.364

0.088

0.42

0.43

0.685

1

0.04

0.003

0.125

0.16

0.07

0.007

0.167

0.21

0.12

0.017

0.250

0.26

0.21

0.037

0.333

0.33

1

1 18 1 38 1 58 1 78 2 18 2 38 2 58 2 78 3 18 3 58 3 58 4 18 4 18 4 58 4 58 6 18 6 18

0.35

0.087

0.494

0.42

0.55

0.195

0.669

0.54

0.76

0.310

0.799

0.62

1.23

0.666

1.075

0.79

1.76

1.530

1.704

0.95

2.55

3.017

2.228

1.16

4.33

4.788

2.680

1.34

5.41

7.233

3.174

1.51

18

0.675

0.493

0.091

0.57

0.57

18

0.848

0.840

0.622

0.109

0.85

0.85

14

1.024

1.050

0.824

0.113

1.13

1.13

14

1.281

1.315

1.049

0.133

1.68

1.68

11 12

1.576 1.950

1 14 1 12

1.660

1.380

0.140

2.27

2.28

1.900

1.610

0.145

2.72

2.73

2

2.375

2.067

0.154

3.65

3.68

11 12 11 12 11 12

2 12

2.875

2.469

0.203

5.79

5.82

8

3.276

3

3.500

3.068

0.216

7.58

7.62

8

3.948

3 12

4.000

3.548

0.226

9.11

9.20

8

4.591

4

4.500

4.026

0.237

10.79

10.89

8

5.091

5

5.563

5.047

0.258

14.62

14.81

8

6.296

6

6.625

6.065

0.280

18.97

19.19

8

7.358

8

8.625

8.071

0.277

24.70

25.00

8

9.420

8

8.625

7.981

0.322

28.55

28.81

8

9.420

10

10.750

10.192

0.279

31.20

32.00

8

11.721

10

10.750

10.136

0.307

34.24

35.00

8

11.721

2.218 2.760

9.16

15.16

4.300

1.88

10.82

28.14

5.581

2.25

15.84

63.35

7.265

2.95

15.84

72.49

8.399

2.94

9.178

3.70

33.92

125.4

33.92

137.4

10.07

3.69

(continued)

1866

TABLE 51.86 (Continued ) Dimensions

Couplings

Properties

Weight per Foot (lb) Outside Inside Thickness Nominal Diameter (in.) Diameter (in.) Diameter (in.) (in.)

Plain Ends

Thread and Coupling

Threads Outside per Inch Diameter (in.)

Length (in.)

Weight (lb)

10

10.750

10.020

0.365

40.48

41.13

8

11.721

12

12.750

12.090

0.330

43.77

45.00

8

13.958

12

12.750

12.000

0.375

49.56

50.71

8

13.958

6 18 6 18 6 18

I (in.4)

A (in.2)

k (in.)

33.92

160.7

11.91

3.67

48.27

248.5

12.88

4.39

48.27

279.3

14.38

4.38

Schedule 80XS 1 8 1 4 3 8 1 2 3 4

0.405

0.215

0.095

0.31

0.32

27

0.582

1 18

0.05

0.001

0.093

0.12

0.540

0.302

0.119

0.54

0.54

18

0.724

1 38

0.07

0.004

0.157

0.16

0.675

0.423

0.126

0.74

0.75

18

0.898

1 58

0.13

0.009

0.217

0.20

1 78 2 18 2 38 2 78 2 78 3 58 4 18 4 18 4 58 4 58 5 18 5 18 6 18 6 58 6 58

0.22

0.020

0.320

0.25

0.33

0.045

0.433

0.32

0.47

0.106

0.639

0.41

1.04

0.242

0.881

0.52

1.17

0.391

1.068

0.61

2.17

0.868

1.477

0.77

3.43

1.924

2.254

0.92

4.13

3.894

3.016

1.14

6.29

6.280

3.678

1.31

0.840

0.546

0.147

1.09

1.10

14

1.085

1.050

0.742

0.154

1.47

1.49

14

1.316

1

1.315

0.957

0.179

2.17

2.20

11 12

1.575

1 14

1.660

1.278

0.191

3.00

3.05

11 12

2.054

1 12

1.900

1.500

0.200

3.63

3.69

11 12

2.294 2.870

2

2.375

1.939

0.218

5.02

5.13

11 12

2 12

2.875

2.323

0.276

7.66

7.83

8

3.389

3

3.500

2.900

0.300

10.25

10.46

8

4.014

3 12

4.000

3.364

0.318

12.51

12.82

8

4.628

4

4.500

3.826

0.337

14.98

15.39

8

5.233

5

5.563

4.813

0.375

20.78

21.42

8

6.420

6

6.625

5.761

0.432

28.57

29.33

8

7.482

8

8.625

7.625

0.500

43.39

44.72

8

9.596

10

10.750

9.750

0.500

54.74

56.94

8

11.958

12

12.750

11.750

0.500

65.42

68.02

8

13.958

8.16 12.87 15.18

9.610 20.67 40.49

4.407

1.48

6.112

1.84

8.405

2.20

26.63

105.7

12.76

2.88

44.16

211.9

16.10

3.63

51.99

361.5

19.24

4.34

Schedule XX 1 2 3 4

0.840

0.252

0.294

1.71

1.73

14

1.085

1 78

0.22

0.024

0.504

0.22

1.050

0.434

0.308

2.44

2.46

14

1.316

2 18

0.33

0.058

0.718

0.28

1

1.315

0.599

0.358

3.66

3.68

11 12

1.575

2 38

0.47

0.140

1.076

0.36

2.054

2 78 2 78 3 58 4 18 4 18 4 58 4 58 5 18 5 18 6 18

1.04

0.341

1.534

0.47

1.17

0.568

1.885

0.55

2.17

1.311

2.656

0.70

3.43

2.871

4.028

0.84

4.13

5.992

5.466

1.05

6.29

9.848

6.721

1.21

8.101

1.37

1 14 1 12

1.660

0.896

0.382

5.21

5.27

1.900

1.100

0.400

6.41

6.47

2

2.375

1.503

0.436

9.03

9.14

11 12 11 12 11 12

2 12

2.875

1.771

0.552

13.70

13.87

8

3.389

3

3.500

2.300

0.600

18.58

18.79

8

4.014

3 12

4.000

2.728

0.636

22.85

23.16

8

4.628

4

4.500

3.152

0.674

27.54

27.95

8

5.233

5

5.563

4.063

0.750

38.55

39.20

8

6.420

6

6.625

4.897

0.864

53.16

53.92

8

7.482

8

8.625

6.875

0.875

72.42

73.76

8

9.596

2.294 2.870

8.16

15.28

12.87

33.64

11.34

1.72

15.18

66.33

15.64

2.06

21.30

2.76

26.63

162.0

Large Outside Diameter Pipe Pipe 14 in. and larger is sold by actual outside step diameter and thickness. 1 in. from 14 to 1 in., inclusive. Sizes 14, 15, and 16 in. are available regularly in thicknesses varying by 16

All pipe is furnished random length unless otherwise ordered, viz: 12–22 ft with privilege of furnishing 5 % in 6–12-ft lengths. Pipe railing is most economically detailed with slip joints and random lengths between couplings. a

Steel Construction, 1980, A.I.S.C.

1867

1868

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

51.6.6 Standard Structural Shapes—Aluminum12 TABLE 51.87 Aluminum Association Standard Channels—Dimensions, Areas, Weights, and Section Propertiesa

Section Propertiesd

Size

Axis X–X Fillet Fange Web Thickness Thickness Radius R (in.) t1 (in.) t (in.)

Axis Y–Y

Depth Width B A (in.) (in.)

Areab (in.2)

Weightc (lb/ft)

2.00

1.00

0.491

0.557

0.13

0.13

0.10

0.288

0.288 0.766

2.00

1.25

0.911

1.071

0.26

0.17

0.15

0.546

0.546 0.774

0.139 0.178 0.391 0.471

3.00

1.50

0.965

1.135

0.20

0.13

0.25

1.41

0.94

1.21

0.22

0.22

0.47

0.49

3.00

1.75

1.358

1.597

0.26

0.17

0.25

1.97

1.31

1.20

0.42

0.37

0.55

0.62

4.00

2.00

1.478

1.738

0.23

0.15

0.25

3.91

1.95

1.63

0.60

0.45

0.64

0.65

4.00

2.25

1.982

2.331

0.29

0.19

0.25

5.21

2.60

1.62

1.02

0.69

0.72

0.78

5.00

2.25

1.881

2.212

0.26

0.15

0.30

7.88

3.15

2.05

0.98

0.64

0.72

0.73

5.00

2.75

2.627

3.089

0.32

0.19

0.30

11.14

4.45

2.06

2.05

1.14

0.88

0.95

6.00

2.50

2.410

2.834

0.29

0.17

0.30

14.35

4.78

2.44

1.53

0.90

0.80

0.79

6.00

3.25

3.427

4.030

0.35

0.21

0.30

21.04

7.01

2.48

3.76

1.76

1.05

1.12

7.00

2.75

2.725

3.205

0.29

0.17

0.30

22.09

6.31

2.85

2.10

1.10

0.88

0.84

7.00

3.50

4.009

4.715

0.38

0.21

0.30

33.79

9.65

2.90

5.13

2.23

1.13

1.20

8.00

3.00

3.526

4.147

0.35

0.19

0.30

37.40

9.35

3.26

3.25

1.57

0.96

0.93

8.00

3.75

4.923

5.789

0.41

0.25

0.35

52.69

13.17

3.27

7.13

2.82

1.20

1.22

9.00

3.25

4.237

4.983

0.35

0.23

0.35

54.41

12.09

3.58

4.40

1.89

1.02

0.93

9.00

4.00

5.927

6.970

0.44

0.29

0.35

78.31

17.40

3.63

9.61

3.49

1.27

1.25

10.00

3.50

5.218

6.136

0.41

0.25

0.35

83.22

16.64

3.99

6.33

2.56

1.10

1.02

10.00

4.25

7.109

8.360

0.50

0.31

0.40

116.15

23.23

4.04

13.02

4.47

1.35

1.34

12.00

4.00

7.036

8.274

0.47

0.29

0.40

159.76

26.63

4.77

11.03

3.86

1.25

1.14

12.00

5.00

10.053

11.822

0.62

0.35

0.45

239.69

39.95

4.88

25.74

7.60

1.60

1.61

I (in.4)

S (in.3)

r (in.)

I (in.4)

S r (in.3) (in.)

x (in.)

0.045 0.064 0.303 0.298

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061. d I ¼ moment of inertia; S ¼ section modulus; r ¼ radius of gyration.

12

Tables 51.87–51.101 are from Aluminum Standards and Data. Copyright # 1984 The Aluminum Association.

TABLE 51.88 Aluminum Association Standard I Beams—Dimensions, Areas, Weights, and Section Propertiesa

Section Propertiesd

Size Depth

Width

A (in.)

B (in.)

Areab (in.2)

3.00 3.00 4.00 4.00 5.00 6.00 6.00 7.00 8.00 8.00 9.00 10.00 10.00 12.00 12.00

2.50 2.50 3.00 3.00 3.50 4.00 4.00 4.50 5.00 5.00 5.50 6.00 6.00 7.00 7.00

1.392 1.726 1.965 2.375 3.146 3.427 3.990 4.932 5.256 5.972 7.110 7.352 8.747 9.925 12.153

1869

a

Weightc (lb/ft)

Flange Thickness t1 (in.)

Web Thickness t (in.)

Fillet Radius R (in.)

Axis X–X I (in.4)

S (in.3)

r (in.)

I (in.4)

S (in.3)

r (in.)

1.637 2.030 2.311 2.793 3.700 4.030 4.692 5.800 6.181 7.023 8.361 8.646 10.286 11.672 14.292

0.20 0.26 0.23 0.29 0.32 0.29 0.35 0.38 0.35 0.41 0.44 0.41 0.50 0.47 0.62

0.13 0.15 0.15 0.17 0.19 0.19 0.21 0.23 0.23 0.25 0.27 0.25 0.29 0.29 0.31

0.25 0.25 0.25 0.25 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.40 0.40 0.40 0.40

2.24 2.71 5.62 6.71 13.94 21.99 25.50 42.89 59.69 67.78 102.02 132.09 155.79 255.57 317.33

1.49 1.81 2.81 3.36 5.58 7.33 8.50 12.25 14.92 16.94 22.67 26.42 31.16 42.60 52.89

1.27 1.25 1.69 1.68 2.11 2.53 2.53 2.95 3.37 3.37 3.79 4.24 4.22 5.07 5.11

0.52 0.68 1.04 1.31 2.29 3.10 3.74 5.78 7.30 8.55 12.22 14.78 18.03 26.90 35.48

0.42 0.54 0.69 0.87 1.31 1.55 1.87 2.57 2.92 3.42 4.44 4.93 6.01 7.69 10.14

0.61 0.63 0.73 0.74 0.85 0.95 0.97 1.08 1.18 1.20 1.31 1.42 1.44 1.65 1.71

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061. d I ¼ moment of inertia; S ¼ section modulus; r ¼ radius of gyration. b

Axis Y–Y

1870

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.89 Standard Structural Shapes—Equal Anglesa

A

t

R

R1

3 4 3 4

1 8 3 16 3 32 1 8 3 16 1 4 1 8 3 16 1 4 1 8 3 16 1 4 1 8 3 16 1 4 5 16 1 8 3 16 1 4 5 16 3 8 1 8 3 16 1 4 5 16 3 8 3 16 1 4 5 16 3 8 7 16 1 2 1 4 5 16 3 8 1 2

1 8 1 8 1 8 1 8 1 8 1 8 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 5 16 5 16 5 16 5 16 5 16 5 16 3 8 3 8 3 8 3 8

3 32 3 32 3 32 3 32 3 32 3 32 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4

1 1 1 1 1 14 1 14 1 14 1 12 1 12 1 12 1 34 1 34 1 34 1 34 2 2 2 2 2 2 12 2 12 2 12 2 12 2 12 3 3 3 3 3 3 3 12 3 12 3 12 3 12

Areab (in.2)

Weight per Footc (lb)

0.171 0.246 0.179 0.234 0.340 0.437 0.292 0.434 0.558 0.360 0.529 0.688 0.423 0.622 0.813 0.996 0.491 0.723 0.944 1.160 1.366 0.616 0.910 1.194 1.470 1.714 1.084 1.432 1.770 2.104 2.428 2.744 1.691 2.093 2.488 3.253

0.201 0.289 0.211 0.275 0.400 0.514 0.343 0.510 0.656 0.423 0.619 0.809 0.497 0.731 0.956 1.171 0.577 0.850 1.110 1.364 1.606 0.724 1.070 1.404 1.729 2.047 1.275 1.684 2.082 2.474 2.855 3.227 1.989 2.461 2.926 3.826

STANDARD SIZES

1871

TABLE 51.89 (Continued ) A

t

R

R1

4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 8 8 8

1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 3 8 7 16 1 2 5 8 3 8 7 16 1 2 5 8 1 2 3 4

3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 5 8 5 8 5 8

1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8

1

Areab (in.2)

Weight per Footc (lb)

1.941 2.406 2.862 3.310 3.753 4.187 4.613 5.032 5.441 3.603 4.177 4.743 5.853 4.353 5.052 5.743 7.102 7.773 11.461 15.023

2.283 2.829 3.366 3.893 4.414 4.924 5.425 5.918 6.399 4.237 4.912 5.578 6.883 5.119 5.941 6.754 8.352 9.141 13.478 17.667

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

TABLE 51.90 Standard Structural Shapes—Unequal Anglesa

A 1 14 1 14 1 12 1 12 1 12 1 12 1 12

B

t

R

R1

Areab (in.2)

3 4

3 32 1 8 1 8 3 16 5 32 1 4 1 8

3 32 1 8 1 8 1 8 5 32 3 16 3 16

3 64 1 16 1 16 3 32 5 64 1 8 1 8

0.180 0.267 0.267 0.386 0.368 0.563 0.329

1 3 4 3 4

1 1 1 14

Weight per Footc (lb) 0.212 0.314 0.314 0.454 0.433 0.662 0.387 (continued)

1872

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.90 (Continued ) A

B

t

R

R1

Areab (in.2)

Weight per Footc (lb)

1 12 1 12 1 34 1 34 1 34 2 2 2 2 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 3 3 3 3 3 3 3 3 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 4 4 4 4 4 4 4 4 5

1 14 1 14 1 14 1 14 1 14 1 12 1 12 1 12 1 12 1 12 1 12 1 12 2 2 2 2 2 2 2 2 2 2 2 12 2 12 2 12 2 12 2 12 2 12 2 12 3 3 3 3 3 3 3 3 3 3 3 12 3 12 3

3 16 1 4 1 8 3 16 1 4 1 8 3 16 1 4 3 8 3 16 1 4 5 16 1 8 3 16 1 4 5 16 3 8 3 16 1 4 5 16 3 8 7 16 1 4 5 16 3 8 1 4 5 16 3 8 1 2 1 4 5 16 3 8 1 2 1 4 5 16 3 8 7 16 1 2 5 8 3 8 1 2 3 8

3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 1 4 1 4 3 16 1 4 1 4 1 4 1 4 1 4 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8

1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 3 16 3 16 3 16 3 16 3 16 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 5 16 5 16 5 16

0.481 0.624 0.358 0.528 0.688 0.422 0.622 0.813 1.172 0.723 0.944 1.152 0.554 0.817 1.069 1.314 1.554 0.911 1.193 1.471 1.740 2.001 1.307 1.614 1.916 1.432 1.770 2.104 2.744 1.566 1.937 2.300 3.003 1.691 2.091 2.488 2.874 3.253 3.988 2.660 3.488 2.848

0.566 0.734 0.421 0.621 0.809 0.496 0.731 0.956 1.378 0.850 1.110 1.355 0.652 0.961 1.257 1.545 1.828 1.071 1.403 1.730 2.046 2.353 1.537 1.898 2.253 1.684 2.082 2.474 3.227 1.842 2.278 2.705 3.532 1.988 2.459 2.926 3.380 3.826 4.690 3.128 4.102 3.349

STANDARD SIZES

1873

TABLE 51.90 (Continued ) A

B

t

R

R1

Areab (in.2)

Weight per Footc (lb)

5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 8 8 8

3 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 4 4 4 4 4 4 6 6 6

1 2 5 16 3 8 7 16 1 2 5 8 5 16 3 8 1 2 3 8 7 16 1 2 9 16 5 8 3 4 5 8 11 16 3 4

3 8 7 16 7 16 7 16 7 16 7 16 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 3 8 3 8 3 8 3 8 3 8 3 8 5 16 3 8 3 8

3.738 2.558 3.046 3.527 4.000 4.921 2.878 3.433 4.512 3.603 4.179 4.743 5.298 5.853 6.931 8.371 9.152 9.931

4.396 3.008 3.582 4.148 4.704 5.787 3.385 4.037 5.306 4.237 4.915 5.578 6.230 6.883 8.151 9.844 10.763 11.679

a Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098lb/in.3, which is the density of alloy 6061.

TABLE 51.91 Channels, American Standarda

A

B

C

t

t1

R

R1

Areab (in.2)

3 3 3 4 4 4 5 5 5 6 6

1.410 1.498 1.596 1.580 1.647 1.720 1.750 1.885 2.032 1.920 1.945

1 34 1 34 1 34 2 34 2 34 2 34 3 34 3 34 3 34 4 12 4 12

0.170 0.258 0.356 0.180 0.247 0.320 0.190 0.325 0.472 0.200 0.225

0.170 0.170 0.170 0.180 0.180 0.180 0.190 0.190 0.190 0.200 0.200

0.270 0.270 0.270 0.280 0.280 0.280 0.290 0.290 0.290 0.300 0.300

0.100 0.100 0.100 0.110 0.110 0.110 0.110 0.110 0.110 0.120 0.120

1.205 1.470 1.764 1.570 1.838 2.129 1.969 2.643 3.380 2.403 2.553

Weight per Footc (lb) 1.417 1.729 2.074 1.846 2.161 2.504 2.316 3.108 3.975 2.826 3.002 (continued)

1874

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.91 (Continued ) A

B

C

t

t1

R

R1

Areab (in.2)

Weight per Footc (lb)

6 6 7 7 7 8 8 8 8 9 9 10 10 12 12 12 15 15

2.034 2.157 2.110 2.194 2.299 2.290 2.343 2.435 2.527 2.430 2.648 2.600 2.886 2.960 3.047 3.170 3.400 3.716

4 12 4 12 5 12 5 12 5 12 6 14 6 14 6 14 6 14 7 14 7 14 8 14 8 14 10 10 10 12 38 12 38

0.314 0.437 0.230 0.314 0.419 0.250 0.303 0.395 0.487 0.230 0.448 0.240 0.526 0.300 0.387 0.510 0.400 0.716

0.200 0.200 0.210 0.210 0.210 0.220 0.220 0.220 0.220 0.230 0.230 0.240 0.240 0.280 0.280 0.280 0.400 0.400

0.300 0.300 0.310 0.310 0.310 0.320 0.320 0.320 0.320 0.330 0.330 0.340 0.340 0.380 0.380 0.380 0.500 0.500

0.120 0.120 0.130 0.130 0.130 0.130 0.130 0.130 0.130 0.140 0.140 0.140 0.140 0.170 0.170 0.170 0.240 0.240

3.088 3.825 3.011 3.599 4.334 3.616 4.040 4.776 5.514 3.915 5.877 4.488 7.348 6.302 7.346 8.822 9.956 14.696

3.631 4.498 3.541 4.232 5.097 4.252 4.751 5.617 6.484 4.604 6.911 5.278 8.641 7.411 8.639 10.374 11.708 17.282

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

TABLE 51.92 Channels, Shipbuilding, and Carbuildinga

A

B

C

t

t1

R

R1

Slope

Areab (in.2)

Weight per Footc (lb)

3 3 4 5 6 6 8 8 10 10 10

2 2 2 12 2 78 3 3 12 3 3 12 3 12 9 3 16 5 38

1 34 1 78 2 38 3 4 12 4 5 34 5 34 7 12 7 12 7 12

0.250 0.375 0.318 0.438 0.500 0.375 0.380 0.425 0.375 0.438 0.500

0.250 0.375 0.313 0.438 0.375 0.412 0.380 0.471 0.375 0.375 0.375

0.250 0.188 0.375 0.250 0.375 0.480 0.550 0.525 0.625 0.625 0.625

0 0.375 0.125 0.094 0.250 0.420 0.220 0.375 0.188 0.188 0.188

12:12.1 0 1:34.9 1:9.8 0 1:49.6 1:14.43 1:28.5 1:9 1:9 1:9

1.900 2.298 2.825 4.950 4.909 5.044 5.600 6.682 7.298 7.928 8.548

2.234 2.702 3.322 5.821 5.773 5.932 6.586 7.858 8.581 9.323 10.052

STANDARD SIZES

1875

TABLE 51.93 H Beamsa

A

B

C

t

t1

R

R1

Slope

Areab (in.2)

Weight per Footc (lb)

4 5 6 8 8

4 5 5.938 7.938 8.125

2 38 3 38 4 38 6 14 6 14

0.313 0.313 0.250 0.313 0.500

0.290 0.330 0.360 0.358 0.358

0.313 0.313 0.313 0.313 0.313

0.145 0.165 0.180 0.179 0.179

1:11.3 1:13.6 1:15.6 1:18.9 1:18.9

4.046 5.522 6.678 9.554 11.050

4.758 6.494 7.853 11.263 12.995

a Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

TABLE 51.94 I Beamsa

A

B

C

t

t1

R

R1

Areab (in.2)

Weight per Footc (lb)

3 3 4 4 5 5 6 6 7 8 8 10 12

2.330 2.509 2.660 2.796 3 3.284 3.330 3.443 3.755 4 4.262 4.660 5

1 34 1 34 2 34 2 34 3 12 3 12 4 12 4 12 5 14 6 14 6 14 8 9 34

0.170 0.349 0.190 0.326 0.210 0.494 0.230 0.343 0.345 0.270 0.532 0.310 0.350

0.170 0.170 0.190 0.190 0.210 0.210 0.230 0.230 0.250 0.270 0.270 0.310 0.350

0.270 0.270 0.290 0.290 0.310 0.310 0.330 0.330 0.350 0.370 0.370 0.410 0.450

0.100 0.100 0.110 0.110 0.130 0.130 0.140 0.140 0.150 0.160 0.160 0.190 0.210

1.669 2.203 2.249 2.792 2.917 4.337 3.658 4.336 5.147 5.398 7.494 7.452 9.349

1.963 2.591 2.644 3.283 3.430 5.100 4.302 5.099 6.053 6.348 8.813 8.764 10.994

a Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

1876

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.95 Wide-Flange Beamsa

A 6.000 6.000 8.000 8.000 8.000 9.750 9.900 11.940 12.060

B

t

t1

R

R1

Areab (in.2)

Weight per Footc (lb)

4.000 6.000 5.250 6.500 8.000 7.964 5.750 8.000 10.000

0.230 0.240 0.230 0.245 0.288 0.292 0.240 0.294 0.345

0.279 0.269 0.308 0.398 0.433 0.433 0.340 0.516 0.576

0.250 0.250 0.320 0.400 0.400 0.500 0.312 0.600 0.600

— — — — — — 0.031 — —

3.538 4.593 5.020 7.076 9.120 9.706 6.205 11.772 15.593

4.161 5.401 5.904 8.321 10.725 11.414 7.297 13.844 18.337

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

TABLE 51.96 Teesa

A

B

C

D

t

R

Areab (in.2)

Weight per Footc (lb)

2 2 14 2 12 3 4

2 2 14 2 12 3 4

0.312 0.312 0.375 0.438 0.438

0.312 0.312 0.375 0.438 0.438

0.250 0.250 0.312 0.375 0.375

0.250 0.250 0.250 0.312 0.500

1.071 1.208 1.626 2.310 3.183

1.259 1.421 1.912 2.717 3.743

a Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

STANDARD SIZES

1877

TABLE 51.97 Zeesa

A 3 3 4 1 4 16 1 44 5 1 5 16

B

t

R

R1

Areab (in.2)

Weight per Footc (lb)

2 11 16 2 11 16 1 3 16 3 18 3 3 16 1 34 5 3 16

0.250 0.375 0.250 0.312 0.375 0.500 0.375

0.312 0.312 0.312 0.312 0.312 0.312 0.312

0.250 0.250 0.250 0.250 0.250 0.250 0.250

1.984 2.875 2.422 3.040 3.672 5.265 4.093

2.333 3.381 2.848 3.575 4.318 6.192 4.813

a Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3, which is the density of alloy 6061.

TABLE 51.98 Aluminum Pipe—Diameters, Wall Thicknesses, and Weights Nominal Pipe Sizea (in.) 1 8 1 4 3 8

1 2

3 4

Outside Diameter (in.)

Inside Diameter (in.)

Wall Thickness (in.)

Weight per Foot (lb)

Schedule Numbera

Noma

Minb

Maxb

Nom

Noma Minb Maxb Nomd Minb

40 80 40 80 40 80 5 10 40 80 160 5 10 40 80 160 5 10

0.405 0.405 0.540 0.540 0.675 0.675 0.840 0.840 0.840 0.840 0.840 1.050 1.050 1.050 1.050 1.050 1.315 1.315

0.374 0.374 0.509 0.509 0.644 0.644 0.809 0.809 0.809 0.809 0.809 1.019 1.019 1.019 1.019 1.019 1.284 1.284

0.420 0.420 0.555 0.555 0.690 0.690 0.855 0.855 0.855 0.855 0.855 1.065 1.065 1.065 1.065 1.065 1.330 1.330

0.269 0.215 0.364 0.302 0.493 0.493 0.710 0.674 0.622 0.546 0.464 0.920 0.884 0.824 0.742 0.612 1.185 1.097

0.068 0.095 0.088 0.119 0.091 0.091 0.065 0.083 0.109 0.147 0.188 0.065 0.083 0.113 0.154 0.219 0.065 0.109

0.060 0.083 0.077 0.104 0.080 0.080 0.053 0.071 0.095 0.129 0.164 0.053 0.071 0.099 0.135 0.192 0.053 0.095

— — — — — — 0.077 0.095 — — — 0.077 0.095 — — — 0.077 0.123

0.085 0.091 0.109 0.118 0.147 0.159 0.185 0.200 0.196 0.212 0.196 0.212 0.186 — 0.232 — 0.294 0.318 0.376 0.406 0.453 0.489 0.237 — 0.297 — 0.391 0.422 0.510 0.551 0.672 0.726 0.300 — 0.486 — (continued)

1878

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.98 (Continued ) Nominal Pipe Sizea (in.) 1

1 14

1 12

2

2 12

3

3 12

4

5

Outside Diameter (in.)

Inside Diameter (in.)

Wall Thickness (in.)

Weight per Foot (lb)

Schedule Numbera

Noma

Minb

Maxb

Nom

Noma Minb Maxb Nomd Minb

40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 5 10 40 80 120 160 5.563 10 40 80 120 160

1.315 1.315 1.315 1.660 1.660 1.660 1.660 1.660 1.900 1.900 1.900 1.900 1.900 2.375 2.375 2.375 2.375 2.375 2.875 2.875 2.875 2.875 2.875 3.500 3.500 3.500 3.500 3.500 4.000 4.000 4.000 4.000 4.500 4.500 4.500 4.500 4.500 4.500 5.532 5.563 5.563 5.563 5.563 5.563

1.284 1.284 1.284 1.629 1.629 1.629 1.629 1.629 1.869 1.869 1.869 1.869 1.869 2.344 2.344 2.351 2.351 2.351 2.844 2.844 2.846 2.846 2.846 3.469 3.469 3.465 3.465 3.465 3.969 3.969 3.960 3.960 4.469 4.469 4.455 4.455 4.455 4.455 5.625 5.532 5.507 5.507 5.507 5.507

1.330 1.330 1.330 1.675 1.675 1.675 1.675 1.675 1.915 1.915 1.915 1.915 1.915 2.406 2.406 2.399 2.399 2.399 2.906 2.906 2.904 2.904 2.904 3.531 3.531 3.535 3.535 3.535 4.031 4.031 4.040 4.040 4.531 4.531 4.545 4.545 4.545 4.545 5.345 5.625 5.619 5.619 5.619 5.619

1.049 0.957 0.815 1.530 1.442 1.380 1.278 1.160 1.770 1.682 1.610 1.500 1.338 2.245 2.157 2.067 1.939 1.687 2.709 2.635 2.469 2.323 2.125 3.334 3.260 3.068 2.900 2.624 3.834 3.760 3.548 3.364 4.334 4.160 4.026 3.826 3.624 3.438 0.109 5.295 5.047 4.813 4.563 4.313

0.133 0.179 0.250 0.065 0.109 0.140 0.191 0.250 0.065 0.109 0.145 0.200 0.281 0.065 0.109 0.154 0.218 0.344 0.083 0.120 0.203 0.276 0.375 0.083 0.120 0.216 0.300 0.438 0.083 0.120 0.226 0.318 0.083 0.120 0.237 0.337 0.438 0.531 0.095 0.134 0.258 0.375 0.500 0.625

0.116 0.157 0.219 0.053 0.095 0.122 0.167 0.219 0.053 0.095 0.127 0.175 0.246 0.053 0.095 0.135 0.191 0.301 0.071 0.105 0.178 0.242 0.328 0.071 0.105 0.189 0.262 0.383 0.071 0.105 0.198 0.278 0.071 0.105 0.207 0.295 0.383 0.465 0.123 0.117 0.226 0.328 0.438 0.547

— — — 0.077 0.123 — — — 0.077 0.123 — — — 0.077 0.123 — — — 0.095 0.135 — — — 0.095 0.135 — — — 0.095 0.135 — — 0.095 0.135 — — — — 2.196 0.151 — — — —

0.581 0.751 0.984 0.383 0.625 0.786 1.037 1.302 0.441 0.721 0.940 1.256 1.681 0.555 0.913 1.264 1.737 2.581 0.856 1.221 2.004 2.650 3.464 1.048 1.498 2.621 3.547 4.955 1.201 1.720 3.151 4.326 1.354 1.942 3.733 5.183 6.573 7.786 — 2.688 7.057 7.188 9.353 11.40

0.627 0.811 1.062 — — 0.849 1.120 1.407 — — 1.015 1.357 1.815 — — 1.365 1.876 2.788 — — 2.164 2.862 3.741 — — 2.830 3.830 5.351 — — 3.403 4.672 — — 4.031 5.598 7.099 8.409 — — 5.461 7.763 10.10 12.31

STANDARD SIZES

1879

TABLE 51.98 (Continued ) Nominal Pipe Sizea (in.) 6

8

10

12

a

Outside Diameter (in.)

Inside Diameter (in.)

Schedule Numbera

Noma

Minb

Maxb

Nom

5 10 40 80 120 160 5 10 20 30 40 60 80 100 120 140 160 5 10 20 30 40 60 80 100 5 10 20 30 40 60 80

6.625 6.625 6.625 6.625 6.625 6.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 10.750 10.750 10.750 10.750 10.750 10.750 10.750 10.750 12.750 12.750 12.750 12.750 12.750 12.750 12.750

6.594 6.594 6.559 6.559 6.559 6.559 8.594 8.594 8.539 8.539 8.539 8.539 8.539 8.539 8.539 8.539 8.539 10.719 10.719 10.642 10.642 10.642 10.642 10.642 10.642 12.719 12.719 12.622 12.622 12.622 12.622 12.622

6.687 6.687 6.691 6.691 6.691 6.691 8.718 8.718 8.711 8.711 8.711 8.711 8.711 8.711 8.711 8.711 8.711 10.843 10.843 10.858 10.858 10.858 10.858 10.858 10.858 12.843 12.843 12.878 12.878 12.878 12.878 12.878

6.407 6.357 6.065 5.761 5.501 5.187 8.407 8.329 8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.813 10.482 10.420 10.250 10.136 10.020 9.750 9.562 9.312 12.438 12.390 12.250 12.090 11.938 11.626 11.374

Wall Thickness (in.)

Weight per Foot (lb)

Noma Minb Maxb Nomd Minb 0.109 0.134 0.280 0.432 0.562 0.719 0.109 0.148 0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.906 0.134 0.165 0.250 0.307 0.365 0.500 0.594 0.719 0.156 0.1580 0.250 0.330 0.406 0.562 0.688

0.095 0.117 0.245 0.378 0.492 0.629 0.095 0.130 0.219 0.242 0.282 0.355 0.438 0.520 0.629 0.710 0.793 0.117 0.144 0.219 0.269 0.319 0.438 0.520 0.629 0.136 0.158 0.219 0.289 0.355 0.492 0.602

0.123 0.151 — — — — 0.123 0.166 — — — — — — — — — 0.151 0.186 — — — — — — 0.176 0.202 — — — — —

2.624 3.213 6.564 9.884 12.59 15.69 3.429 4.635 7.735 8.543 9.878 12.33 15.01 17.62 21.00 23.44 25.84 5.256 6.453 9.698 11.84 14.00 18.93 22.29 26.65 7.258 8.359 11.55 15.14 18.52 25.31 30.66

— — 7.089 10.67 13.60 16.94 — — 8.354 9.227 10.67 13.31 16.21 19.03 22.68 25.31 27.90 — — 10.47 12.69 15.12 24.07 28.78 28.78 — — 12.47 16.35 20.00 27.33 33.11

In accordance with ANSI Standards B36.10 and B36.19. Based on standard tolerances for pipe. c For schedules 5 and 10 these values apply to mean outside diameters. d Based on nominal dimensions, plain ends, and a density of 0.098 lb/in.3, the density of 6061 alloy. For alloy 6063 multiply by 0.99, and for alloy 3003 multiply by 1.01. b

1880

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.99 Aluminum Electrical Conduit—Designed Dimensions and Weights Nominal or Trade Size of Conduit (in.) 1 4 3 8 1 2 3 4

1 1 14 1 12 2 2 12 3 3 12 4 5 6

Nominal Inside Diameter (in.)

Outside Diameter (in.)

Nominal Wall Thickness (in.)

0.364 0.493 0.622 0.824 1.049 1.380 1.610 2.067 2.469 3.068 3.548 4.026 5.047 6.065

0.540 0.675 0.840 1.050 1.315 1.660 1.900 2.375 2.875 3.500 4.000 4.500 5.563 6.625

0.088 0.091 0.109 0.113 0.133 0.140 0.145 0.154 0.203 0.216 0.226 0.237 0.258 0.280

Length without Coupling (ft and in.)

Minimum Weight of 10 Unit Lengths with Couplings Attached (lb)

9 11 12 9 11 12 9 11 14 9 11 14 9–11 9–11 9–11 9–11 9 10 12 9 10 12 9 10 14 9 10 14 9–10 9–10

13.3 17.8 27.4 36.4 53.0 69.6 86.2 115.7 182.5 238.9 287.7 340.0 465.4 612.5

TABLE 51.100 Equivalent Resistivity Values Equivalent Resistivity at 68 F Volume Conductivity, Percent International Amended Copper Standard at 68 F 52.5 53.5 53.8 53.9 54.0 54.3 55.0 56.0 56.5 57.0 59.0 59.5 61.0 61.2 61.3 61.4 61.5 61.8 62.0 62.1 62.2 62.3 62.4

Volume Ohm – Circular Mil/ft

Microhm – in.

19.754 19.385 19.277 19.241 19.206 19.099 18.856 18.520 18.356 18.195 17.578 17.430 17.002 16.946 16.918 16.891 16.863 16.782 16.727 16.700 16.674 16.647 16.620

1.2929 1.2687 1.2617 1.2593 1.2570 1.2501 1.2341 1.2121 1.2014 1.1908 1.1505 1.1408 1.1128 1.1091 1.1073 1.1055 1.1037 1.0983 1.0948 1.0931 1.0913 1.0896 1.0878

STANDARD SIZES

1881

TABLE 51.101 Property Limits–Wire (Up to 0.374 in. Diameter) Ultimate Strength (ksi) Alloy and Temper

Min

1350-O 1350-H12 and H22 1350-H14 and H24 1350-H16 and H26

8.5 12.0 15.0 17.0

8017-H212b

15.0

Max

Electrical Conductivitya percent IACS at 68 F min

14.0 17.0 20.0 22.0

61.8 61.0 61.0 61.0

21.0

61.0

22.0

61.0

20.0

61.0

22.0

61.0

1350

8017 8030 8030-H221

15.0 8176

8176-H24

15.0 8177

8177-H221

Alloy and Temper

15.0

Specified Diameter (in.)

Ultimate Strength (ksi min)

Elongation Percent min in 10 in.

Individuala Averagec

Individuala Averagec

Electrical Conductivitya min percent IACS at 68 F

1350 1350-H19

0.0105–0.0500 0.0501–0.0600 0.0601–0.0700 0.0701–0.0800 0.0801–0.0900 0.0901–0.1000 0.1001–0.1100 0.1101–0.1200 0.1201–0.1400 0.1401–0.1500 0.1501–0.1800 0.1801–0.2100 0.2101–0.2600

23.0 27.0 27.0 26.5 26.0 25.5 24.5 24.0 23.5 23.5 23.0 23.0 22.5

25.0 29.0 28.5 28.0 27.5 27.0 26.0 25.5 25.0 24.5 24.0 24.0 23.5

— 1.2 1.3 1.4 1.5 1.5 1.5 1.6 1.7 1.8 1.9 2.0 2.2

— 1.4 1.5 1.6 1.6 1.6 1.6 1.7 1.8 1.9 2.0 2.1 2.3

0.0601–0.0700 0.0701–0.0800 0.0801–0.0900 0.0901–0.1000 0.1001–0.1100 0.1101–0.1200 0.1201–0.1400 0.1401–0.1500 0.1501–0.1600 0.1601–0.2100

38.0 37.5 37.0 36.5 36.0 35.5 35.0 35.0 34.5 32.5

40.0 39.5 39.0 38.5 38.0 37.5 37.0 36.5 36.0 34.0

1.3 1.4 1.5 1.5 1.5 1.6 1.7 1.8 1.9 2.0

— — — — — — — — — —

0.2101–0.2600

31.5

33.0

2.2

—

5005 5005-H19

5005-H19

g g

61.0

53.5

1882

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.101 (Continued ) Ultimate Strength (ksi min) Alloy and Temper

Specified Diameter (in.)

Individuala

Averagec

6201-T81

0.0612–0.1327 0.1328–0.1878

46.0 44.0

48.0 46.0

8176-H24

0.0500–0.2040

15.0

17.0

Electrical Conductivitya min percent IACS at 68 F

Elongation Percent min in 10 in. Individuala

Averagec

6201 3.0 3.0

— —

10.0

—

8176

g

52.5

61.0

a

To convert conductivity to maximum resistivity use Table 51.100. Applicable up to 0.250 in. c Average of all tests in a lot. b

51.7 STANDARD SCREWS13

Standard Screw Threads The Unified and American Screw Threads included in Table 51.102 are taken from the publication of the American Standards Association, ASA B 1.1—1949. The coarsethread series is the former United States Standard Series. It is recommended for general use in engineering work where conditions do not require the use of a fine thread. 13

This section is extracted, with permission, from EMPIS Materials Selector. Copyright # 1982 General Electric Co.

STANDARD SCREWS

1883

TABLE 51.102 Standard Screw Threads

Sizes

Minor Basic Threads Basic Diameter Pitch per Major External Diametera Threads Inch Diameter n D (in.) Ks (in.) E (in.)

Minor Diameter Internal Threads Kn (in.)

Section at Minor Diameter at D 2hb) (in.2)

Stress Areab (in.2)

1 (0.073) 2 (0.086) 3 (0.099) 4 (0.112)

Coarse-thread Series – UNC and NC (Basic Dimensions) 0.0730 64 0.0629 0.0538 0.0561 0.0860 56 0.0744 0.0641 0.0667 0.0990 48 0.0855 0.0734 0.0764 0.1120 40 0.0958 0.0813 0.0849

0.0022 0.0031 0.0041 0.0050

0.0026 0.0036 0.0048 0.0060

5 (0.125) 6 (0.138) 8 (0.164) 10 (0.190) 12 (0.216)

0.1250 0.1380 0.1640 0.1900 0.2160

40 32 32 24 24

0.1088 0.1177 0.1437 0.1629 0.1889

0.0943 0.0997 0.1257 0.1389 0.1649

0.0979 0.1042 0.1302 0.1449 0.1709

0.0067 0.0075 0.0120 0.0145 0.0206

0.0079 0.0090 0.0139 0.0174 0.0240

1 4 5 16 3 8 7 16

0.2500

20

0.2175

0.1887

0.1959

0.0269

0.0317

1 2 1 2 9 16 5 8 3 4 7 8

0.3125

18

0.2764

0.2443

0.2524

0.0454

0.0522

0.3750

16

0.3344

0.2983

0.3073

0.0678

0.0773

0.4375

14

0.3911

0.3499

0.3602

0.0933

0.1060

0.5000

13

0.4500

0.4056

0.4167

0.1257

0.1416

0.5000

12

0.4459

0.3978

0.4098

0.1205

0.1374

0.5625

12

0.5084

0.4603

0.4723

0.1620

0.1816

0.6250

11

0.5660

0.5135

0.5266

0.2018

0.2256

0.7500

10

0.6850

0.6273

0.6417

0.3020

0.3340

0.8750

9

0.8028

0.7387

0.7547

0.4193

0.4612

1

1.0000

8

0.9188

0.8466

0.8647

0.5510

0.6051

1 18 1 14 1 38 1 12 1 34

1.1250

7

1.0322

0.9497

0.9704

0.6931

0.7627

1.2500

7

1.1572

1.0747

1.0954

0.8898

0.9684

1.3750

6

1.2667

1.1705

1.1946

1.0541

1.1538

1.5000

6

1.3917

1.2955

1.3196

1.2938

1.4041

1.7500

5

1.6201

1.5046

1.5335

1.7441

1.8983

2

2.0000

1.8557

1.7274

1.7594

2.3001

2.4971

2 14 2 12 2 34

2.2500

4 12 4 12

2.1057

1.9774

2.0094

3.0212

3.2464

2.5000

4

2.3376

2.1933

2.2294

3.7161

3.9976

2.7500

4

2.5876

2.4433

2.4794

4.6194

4.9326

3

3.0000

4

2.8376

2.6933

2.7294

5.6209

5.9659

3 14

3.2500

4

3.0876

2.9433

2.9794

6.7205

7.0992

3 12 3 34

3.5000

4

3.3376

3.1933

3.2294

7.9183

8.3268

3.7500 4.0000

4 4

3.5876 3.8376

3.4433 3.6933

3.4794 3.7294

9.2143 10.6084

9.6546 11.0805

4

(continued)

1884

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.102 (Continued ) Section at Minor Diameter at D 2hb) (in.2)

Stress Areab (in.2)

Fine-Thread Series – UNF and NF (Basic Dimensions) 0.0600 80 0.0519 0.0447 0.0465 0.0730 72 0.0640 0.0560 0.0580 0.0860 64 0.0759 0.0668 0.0691 0.0990 56 0.0874 0.0771 0.0797 0.1120 48 0.0985 0.0864 0.0894 0.1250 44 0.1102 0.0971 0.1004 0.1380 40 0.1218 0.1073 0.1109 0.1640 36 0.1460 0.1299 0.1339 0.1900 32 0.1697 0.1517 0.1562 0.2160 28 0.1928 0.1722 0.1773

0.0015 0.0024 0.0034 0.0045 0.0057 0.0072 0.0087 0.0128 0.0175 0.0226

0.0018 0.0027 0.0039 0.0052 0.0065 0.0082 0.0101 0.0146 0.0199 0.0257

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

28 24 24 20

0.2268 0.2854 0.3479 0.4050

0.2062 0.2614 0.3239 0.3762

0.2113 0.2674 0.3299 0.3834

0.0326 0.0524 0.0809 0.1090

0.0362 0.0579 0.0876 0.1185

1 2 9 16 5 8 3 4 7 8

0.5000 0.5625 0.6250 0.7500 0.8750

20 18 18 16 14

0.4675 0.5264 0.5889 0.7094 0.8286

0.4387 0.4943 0.5568 0.6733 0.7874

0.4459 0.5024 0.5649 0.6823 0.7977

0.1486 0.1888 0.2400 0.3513 0.4805

0.1597 0.2026 0.2555 0.3724 0.5088

1 1 18 1 14 1 38 1 12

1.0000 1.1250 1.2500 1.3750 1.5000

12 12 12 12 12

0.9459 1.0709 1.1959 1.3209 1.4459

0.8978 1.0228 1.1478 1.2728 1.3978

0.9098 1.0348 1.1598 1.2848 1.4098

0.6245 0.8118 1.0237 1.2602 1.5212

0.6624 0.8549 1.0721 1.3137 1.5799

12 (0.216)

Extra-Fine-Thread Series – NEF (Basic Dimensions) 0.2160 32 0.1957 0.1777 0.1822

0.0242

0.0269

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

32 32 32 28

0.2297 0.2922 0.3547 0.4143

0.2117 0.2742 0.3367 0.3937

0.2162 0.2787 0.3412 0.3988

0.0344 0.0581 0.0878 0.1201

0.0377 0.0622 0.0929 0.1270

1 2 9 16 5 8 11 16

0.5000 0.5625 0.6250 0.6875

28 24 24 24

0.4768 0.5354 0.5979 0.6604

0.4562 0.5114 0.5739 0.6364

0.4613 0.5174 0.5799 0.6424

0.1616 0.2030 0.2560 0.3151

0.1695 0.2134 0.2676 0.3280

3 4 13 16

0.7500 0.8125

20 20

0.7175 0.7800

0.6887 0.7512

0.6959 0.7584

0.3685 0.4388

0.3855 0.4573

Sizes 0 (0.060) 1 (0.073) 2 (0.086) 3 (0.099) 4 (0.112) 5 (0.125) 6 (0.138) 8 (0.164) 10 (0.190) 12 (0.216)

Minor Basic Threads Basic Diameter Pitch per Major External Diametera Threads Inch Diameter n D (in.) Ks (in.) E (in.)

Minor Diameter Internal Threads Kn (in.)

1885

STANDARD SCREWS

TABLE 51.102 (Continued ) Minor Diameter Internal Threads Kn (in.)

Section at Minor Diameter at D 2hb) (in.2)

Stress Areab (in.2)

0.8209 0.8834

0.5153 0.5979

0.5352 0.6194

Fine-Thread Series – UNF and NF (Basic Dimensions) 1.0000 20 0.9675 0.9387 0.9459 1.0625 18 1.0264 0.9943 1.0024 1.1250 18 1.0889 1.0568 1.0649 1.1875 18 1.1514 1.1193 1.1274

0.6866 0.7702 0.8705 0.9770

0.7095 0.7973 0.8993 1.0074

1 14 5 1 16 3 18 7 1 16

1.2500 1.3125 1.3750 1.4375

18 18 18 18

1.2139 1.2764 1.3389 1.4014

1.1818 1.2443 1.3068 1.3693

1.1899 1.2524 1.3149 1.3774

1.0895 1.2082 1.3330 1.4640

1.1216 1.2420 1.3684 1.5010

1 12 9 1 16 5 18 1 11 16 1 34 2

1.5000 1.5625 1.6250 1.6875 1.7500 2.0000

18 18 18 18 16 16

1.4639 1.5264 1.5889 1.6514 1.7094 1.9594

1.4318 1.4943 1.5568 1.6193 1.6733 1.9233

1.4399 1.5024 1.5649 1.6274 1.6823 1.9323

1.6011 1.7444 1.8937 2.0493 2.1873 2.8917

1.6397 1.7846 1.9357 2.0929 2.2382 2.9501

Sizes 7 8 15 16

1 1 1 16 1 18 3 1 16

Minor Basic Threads Basic Diameter Pitch per Major External Diametera Threads Inch Diameter n D (in.) Ks (in.) E (in.) 0.8750 0.9375

20 20

0.8425 0.9050

0.8137 0.8762

Note: Bold type indicates unified threads – UNC and UNF. a British: effective diameter. b The stress area is the assumed area of an externally threaded part which is used for the purpose of computing the tensile strength.

TABLE 51.103 ASAa Standard Bolts and Nuts Thickness Nominal Size 1 4 5 16 3 8 7 16 1 2 9 16 5 8

Across Flats (in.)

Across Square Corners (in.)

3 8 1 2 9 16 5 8 3 4 7 8 15 16

0.498

Across Hex Corners (in.)

Regular Bolt Heads 0.413

0.665

0.552

0.747

0.620

0.828

0.687

0.995

0.826

1.163

0.966

1.244

1.033

Unfinished (in.)

Semifinished (in.)

11 64 13 64 1 4 19 64 21 64 3 8 27 64

5 32 3 16 15 64 9 32 19 64 11 32 25 64

(continued)

1886

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.103 (Continued ) Thickness Nominal Size

Across Flats (in.)

Across Square Corners (in.)

Across Hex Corners (in.)

Unfinished (in.)

Semifinished (in.)

1 18

1.494

1.240

5 1 16

1.742

1.447

1.991

1.653

1 18 1 14 1 38 1 12 1 58 1 34 1 78

1 12 1 11 16 1 78 1 2 16 2 14 7 2 16 2 58 2 13 16

2.239

1.859

2.489

2.066

2.738

2.273

1 2 19 32 21 32 3 4 27 32 29 32

2.986

2.480

1

3.235

2.686

3 1 32

3.485

2.893

5 1 32

3.733

3.100

1 14

2

3

3.982

3.306

1 11 32

2 14 2 12 2 34

3 38 3 34 4 18 4 12

4.479

3.719

1 12

4.977

4.133

1 21 32

5.476

4.546

1 53 64

5.973

4.959

2

15 32 9 16 19 32 11 16 25 32 27 32 15 16 1 1 32 3 1 32 3 1 16 7 1 32 1 38 1 17 32 1 11 16 1 78

7 8 15 16 1 1 16 1 14 7 1 16 1 58 1 13 16

1.167

3 4 7 8

1

3 1 2 9 16 5 8 3 4 7 8

1 1 18 1 14 1 38 1 12 1 58 1 34 1 78 2 2 14 2 12 2 34 3

Heavy Bolt Heads 0.969

1.249

1.037

1.416

1.175

1.665

1.383

1.914

1.589

2.162

1.796

7 16 15 32 17 32 5 8 23 32 13 16 29 32

13 32 7 16 1 2 19 32 11 16 3 4 27 32 15 16 1 1 32 1 18 7 1 32 5 1 16 1 13 32 7 1 16 1 58 1 13 16

2.411

2.002

2

2.661

2.209

1

3 2 16 2 38 9 2 16 2 34 2 15 16 3 18 3 12 3 78 4 14 4 58

2.909

2.416

3 1 32

3.158

2.622

3 1 16

3.406

2.828

9 1 32

3.655

3.036

1 38

3.905

3.242

1 15 32

4.153

3.449

9 1 16

4.652

3.862

1 34

5.149

4.275

1 15 16

5.646 6.144

4.688 5.102

2 18

2

5 2 16

3 2 16

1887

STANDARD SCREWS

TABLE 51.103 (Continued) Width Across Corners Nominal Size

1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

Width Across Flats (in.)

7 16 9 16 5 8 3 4 13 16 7 8

Square (in.)

Hex (in.)

Thickness Unfinished, Regular Nuts (in.)

Jam Nuts (in.)

Regular Nuts and Regular Jam Nuts 7 5 0.584 0.484 32 32

1

1.330

1.104

1.494

1.240

1.742

1.447

1.991

1.653

2.239

1.859

1

1 14 1 38 1 12 1 58 1 34 1 78

1 18 5 1 16 1 12 1 11 16 1 78 1 2 16 2 14 7 2 16 2 58 2 13 16

17 64 21 64 3 8 7 16 1 2 35 64 21 32 49 64 7 8

2.489

2.066

3 1 32

2.738

2.273

1 13 64

2.986

2.480

5 1 16

3.235

2.686

1 27 64

3.485

2.893

1 17 32

1

3.733

3.100

1 1 16

2

3

3.982

3.306

2 14

3 38

4.479

3.719

2 12 2 34

3 34 4 18 4 12

4.977

4.133

5.476

4.546

5.973

4.959

1 41 64 1 34 1 31 32 3 2 16 2 13 32 2 58

1 1 18

3 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

1

1 2 19 32 11 16 25 32 7 8 15 16 1 1 16 1 14 7 1 16 1 58

0.751

0.624

0.832

0.691

1.000

0.830

1.082

0.898

1.163

0.966

3 16 7 32 1 4 5 16 11 32 3 8 7 16 1 2 9 16 5 8 3 4 13 16 7 8 15 16

1 18 1 14 1 12 1 58 1 34

Heavy Nuts and Heavy Jam Nuts 1 3 0.670 0.556 4 16 0.794

0.659

0.919

0.763

1.042

0.865

1.167

0.969

1.249

1.037

1.416

1.175

1.665

1.382

1.914

1.589

5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

2.162

1.796

1

7 32 1 4 9 32 5 16 11 32 3 8 7 16 1 2 9 16

Thickness Semifinished, Regular Nuts (in.)

Jam Nuts (in.)

13 64 1 4 5 16 23 64 27 64 31 64 17 32 41 64 3 4 55 64 31 32 1 1 16 1 11 64 9 1 32 25 1 64 1 12 1 39 64 1 23 32 1 59 64 9 2 64 2 23 64 2 37 64

9 64 11 64 13 64 15 64 19 64 21 64 23 64 27 64 31 64 35 64 39 64 23 32 25 32 27 32 29 32 31 32 1 1 32 3 1 32 1 13 64 1 29 64 1 37 64 1 45 64

15 64 19 64 23 64 27 64 31 64 35 64 39 64 47 64 55 64 63 64

11 64 13 64 15 64 17 64 19 64 21 64 23 64 27 64 31 64 35 64

(continued)

1888

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.103 (Continued) Width Across Corners

Thickness Unfinished, Regular

Thickness Semifinished, Regular

Width Across Flats (in.)

Square (in.)

Hex (in.)

Nuts (in.)

Jam Nuts (in.)

Nuts (in.)

Jam Nuts (in.)

1 18

1 13 16

2.411

2.002

1 18

7 1 64

1 14

2

2.661

2.209

1 14

1 38 1 12 1 58 1 34 1 78

3 2 16 2 38 9 2 16 2 34 2 15 16 3 18 3 12 3 78 4 14 4 58

2.909

2.416

3.158

2.622

3.406

2.828

3.656

3.035

1

1 23 32

3.905

3.242

1 38 1 12 1 58 1 34 1 78

5 8 3 4 13 16 7 8 15 16

1 1 16

1 27 32 1 31 32

3 14

5

1 18 1 14 1 12 1 58 1 34 1 78

3 12 3 34

5 38 5 34 6 18

39 64 23 32 25 32 27 32 29 32 31 32 1 1 32 3 1 32 1 13 64 1 29 64 1 37 64 1 45 64 1 13 16 1 15 16 1 2 16 3 2 16

Nominal Size

2 2 14 2 12 2 34 3

4

4.652

3.862

2 14

5.149

4.275

2 12

5.646

4.688

2 34

6.144

5.102

3

6.643

5.515

3 14

7.140

5.928

2

7 3 16

7.637

6.341

3 12 3 34

2 18

3 11 16

8.135

6.755

4

2 14

3 15 16

1 1 18

Across Corners (in.)

7 16 9 16 5 8 3 4 13 16 7 8

0.485 0.691 0.830 0.898 0.966

1

1.104

1 18

1.240

5 1 16 1 12 1 11 16

1.447 1.653 1.859

1 19 32

2

Across Flats (in.)

0.624

1 15 32

3.449

2 13 64 2 29 64 2 45 64 2 61 64 3 3 16

Heavy Slotted Nuts Semifinished

Width

1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

1 11 32

4.153

Regular Slotted Nuts Semifinished

Nominal Size

7 1 32

Width

Thickness (in.) 13 64 1 4 5 16 23 64 27 64 31 64 17 32 41 64 3 4 55 64 31 32

Across Flats (in.)

Across Corners (in.)

1 2 19 32 11 16 25 32 7 8 15 16 1 1 16 1 14 7 1 16 1 58 1 13 16

0.556 0.659 0.763 0.865 0.969 1.037 1.175 1.382 1.589 1.796 2.002

Slot

Thickness (in.)

Width (in.)

15 64 19 64 23 64 27 64 31 64 35 64 39 64 47 64 55 64 63 64 7 1 64

5 64 3 32 1 8 1 8 5 32 5 32 3 16 3 16 3 16 1 4 1 4

Depth (in.) 3 32 3 32 1 8 5 32 5 32 3 16 7 32 1 4 1 4 9 32 11 32

1889

STANDARD SCREWS

TABLE 51.103 (Continued) Regular Slotted Nuts Semifinished

Heavy Slotted Nuts Semifinished

Width

Width

Across Flats (in.)

Across Corners (in.)

Thickness (in.)

Across Flats (in.)

Across Corners (in.)

1 14

1 78

2.066

1 1 16

2

1 38

1 2 16

2.273

1 11 64

1 12 1 58 1 34 1 78

2 14 7 2 16 2 58 2 13 16

2.480

3.100

2

3

3.306

2 14

3 38

3.719

2 12 2 34

3 34 4 18 4 12

4.133

9 1 32 1 25 64 1 12 1 39 64 1 23 32 1 59 64 9 2 64 2 23 64 2 37 64

Nominal Size

3 a

0.686 2.893

4.546 4.959

Slot

Thickness (in.)

Width (in.)

2.209

7 1 32

3 2 16

2.416

1 11 32

2 38 9 2 16 2 34 2 15 16 3 18 3 12 3 78 4 14 4 58

2.622

1 15 32

2.828

1 19 32

3.035

1 23 32

3.242

1 27 32

3.449

1 31 32

3.862

2 13 64

4.275

2 29 64

4.688

2 45 64

5.102

2 61 64

5 16 5 16 3 8 3 8 7 16 7 16 7 16 7 16 9 16 9 16 5 8

Depth (in.) 3 8 3 8 7 16 7 16 1 2 9 16 9 16 9 16 11 16 11 16 3 4

ANSI standards B18.2.1 – 1981, B18.2.2 – 1972 (R1983), B18.6.3 – 1972 (R1983).

The fine-thread series is the former “Regular Screw Thread Series” established by the Society of Automotive Engineers (SAE). The fine-thread series is recommended for general use in automotive and aircraft work and where special conditions require a fine thread. The extra-fine-thread series is the same as the former SAE fine series and the present SAE extra-fine series. It is used particularly in aircraft and aeronautical equipment where (a) thin-walled material is to be threaded; (b) thread depth of nuts clearing ferrules, coupling flanges, and so on, must be held to a minimum; and (c) a maximum practicable number of threads is required within a given thread length. The method of designating a screw thread is by the use of the initial letters of the thread series, preceded by the nominal size (diameter in inches or the screw number) and number of threads per inch, all in Arabic numerals, and followed by the classification designation, with or without the pitch diameter tolerances or limits of size. An example of an external thread designation and its meaning is as follows: EXAMPLE 1 1/4′′—20UNC—2A Class of screw thread Thread series Number of threads per inch (n) Nominal size

1890

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

A left-hand thread must be identified by the letters LH following the class designation. If no such designation is used, the thread is assumed to be right hand. Classes of thread are distinguished from each other by the amounts of tolerance and allowance specified in ASA B 1.1—1949. Selection of Screws By definition, a screw is a fastener that is intended to be torqued by the head. Screws are the most widely used method of assembly despite recent technical advances of adhesives, welding, and other joining techniques. Use of screws is essential in those applications that require ease of disassembly for normal maintenance and service. There is no real economy if savings made in factory installation create service problems later. There are many types of screws, and each variety will be treated separately. Material selection is generally common to all types of screws. Material. Not all materials are suitable for the processes used in the manufacture of fasteners. Large-volume users or those with critical requirements can be very selective in their choice of materials. Low-volume users or those with noncritical applications would be wise to permit a variety of materials in a general category in order to improve availability and lower cost. For example, it is usually desirable to specify low-carbon steel or 18-8-type stainless steel14 rather than ask for a specific grade. Low-carbon steel is widely used in the manufacture of fasteners where lowest cost is desirable and tensile strength requirements are 50, 000 psi. If corrosion is a problem, these fasteners can be plated either electrically or mechanically. Zinc or cadmium plating is used in most applications. Other finishes include nickel, chromium, copper, tin, and silver electroplating; electroless nickel and other immersion coatings; hot dip galvanizing; and phosphate coatings. Medium-carbon steel, quenched, and tempered is widely used in applications requiring tensile strengths from 90,000 to 120,000 psi. Alloy steels are used in applications requiring tensile strengths from 115,000 to 180,000 psi, depending on the grade selected. Where better corrosion resistance is required, 300 series stainless steel can be specified. The 400 series stainless steel is used if it is necessary to have a corrosion-resistant material that can be hardened and tempered by heat treatment. For superior corrosion resistance, materials such as brass, bronze, aluminum, or nickel are sometimes used in the manufacture of fasteners. If strength is no problem, plastics such as nylons are used in severe corrosion applications. Drivability. When selecting a screw, thought must be given to the means of driving for assembly and disassembly as well as the head shape. Most screw heads provide a slot, a recess, or a hexagon shape as a means of driving. The slotted screw is the least preferred driving style and serves only when appearance must be combined with ease of disassembly with a common screwdriver. Only a limited amount of torque can be applied with a screwdriver. A slot can become inoperative after repeated disassembly destroys the

14

Manufacturer may use UNS—S30200, S30300, S30400, S30500 (AISI type 302, 303, 304, or 305) depending upon quantity, diameter, and manufacturing process.

STANDARD SCREWS

1891

edge of the wall that the blade of the screwdriver bears against. The hexagon head is preferred for the following reasons: Least likely to accidentally spin out (thereby marring the surface of the product) Lowest initial cost Adaptable to high-speed power drive Minimum worker fatigue Ease of assembly in difficult places Permits higher driving torque, especially in large sizes where strength is important Contains no recess to become clogged with dirt and interfere with driving Contains no recess to weaken the head Unless frequent field disassembly is required, use of the unslotted hex head is preferred. Appearance is the major disadvantage of the hex head, and this one factor is judged sufficient to eliminate it from consideration for the front or top of products. The recessed head fastener is widely used and becomes the first choice for appearance applications. It usually costs more than a slot or a hexagon shape. There are many kinds of recesses. The Phillips and Phillips POZIDRIV are most widely used. To a lesser extent the Frearson, clutch-type, hexagonal, and fluted socket heads are used. For special applications, proprietary types of tamper-resistant heads can be selected (Figure 51.1). The recessed head has some of the same advantages as the hex head (see preceding list). It also has improved appearance. The Phillips POZIDRIV is slowly replacing the Phillips recess. The POZIDRIV recess can be readily identified by four radial lines centered between each recess slot. These slots are a slight modification of the conventional Phillips recess. This change improves the fit between the driver and the recess, thus

FIGURE 51.1 Recessed head fasteners.

1892

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

minimizing the possibility of marring a surface from accidental spinout of the driver as well as increasing the life of the driver. The POZIDRIV design is recommended in highproduction applications requiring high driving torques. The POZIDRIV recess usually sells at a high-production applications requiring high driving torques. The POZIDRIV recess usually sells at a slightly higher price than the conventional Phillips recess, but some suppliers will furnish either at the same price. The savings resulting from longer tool life will usually justify the higher initial cost. A conventional Phillips driver could be used to install or disassemble a POZIDRIV screw. However, a POZIDRIV driver should be used with a POZIDRIV screw in order to take advantage of the many features inherent in the new design. To avoid confusion, it should be clearly understood that a POZIDRIV driver cannot be used to install or remove a conventional Phillips head screw. A Frearson recess is a somewhat different design than a Phillips recess and has the big advantage that one driving tool can be used for all sizes whereas a Phillips may require four driving tools in the range from no. 2 (0.086-in.) to 3/8 (0.375-in.) screw size. This must be balanced against the following disadvantages: Limited availability. Greater penetration of the recess means thinner walls between the bottom of the recess and the outer edge of the screw, which tends to weaken the head. The sharp point of the driver can easily scratch or otherwise mar the surface of the product if it accidently touches. Although one driver can be used for all sizes, for optimum results, different size drivers are recommended for installing various screw sizes, thus minimizing the one real advantage of the Frearson recess. The hexagon and fluted socket head cap screws are only available in expensive highstrength alloy steel. Its unique small outside diameter or cylindrical head is useful on flanges, counterbored holes, or other locations where clearances are restricted. Such special applications may justify the cost of a socket head cap screw. Appreciable savings can be made in other applications by substitution of a hexagon head screw. Despite any claims to the contrary, the dimensional accuracy of hexagon socket head cap screws is no better than that of other cold-headed products, and there is no merit in close-thread tolerances, which are advocated by some manufacturers of these products. The high prices, therefore, should be justified solely on the basis of possible space savings in using the cylindrical head. The fluted socket is not as readily available and should only be considered in the very small sizes where a hexagon key tends to round out the socket. The fluted socket offers spline design so that the key will neither slip nor be subject to excessive wear. Many types of special recesses are tamper resistant. In most of these designs, the recess is an unusual shape requiring a special tool for assembly and disassembly. A readily available driving tool such as a screwdriver or hexagon key would not fit the recess. The purpose of a tamper-resistant fastener is to prevent unauthorized removal of parts and equipment. Their protection is needed on any product located in public places to discourage vandalism and thievery. They may also be necessary on some consumer products as a safety measure to protect the amateur repairman from injury or to prevent him from causing serious damage to equipment. With product liability mania what it is today, the term “tamperproof” has all but disappeared. Now the fasteners are called “tamper resistant.”

STANDARD SCREWS

1893

FIGURE 51.2 Pan head.

They are the same as they were under their previous name, but the new term better reflects their true capabilities. Any skilled thief with ample time and proper tools can saw, drill, blast, or otherwise disassemble any tamper-resistant fastener. Therefore, these fasteners are intended only to discourage the casual thief or amateur tinkerer and make it more difficult for a skilled professional. Whatever the choice of fastener design, it is essential that hardened material be specified. No fastener is ever truly tamperproof, but hardened steel helps. Fasteners made of soft material can be disassembled easily by sawing a slot, hammering with a chisel, or drilling a hole and using an extraction bit. Head Shapes The following information is equally applicable to all types of recesses as well as a slotted head. For simplification only slotted screws are shown. The pan head is the most widely used and is intended to replace the round, binding, and truss heads in order to keep varieties to a minimum. It is preferred because it presents the best combination of appearance with adequate head height to minimize weakness due to depth of penetration of the recess (Figure 51.2). The round head was widely used in the past (Figure 51.3). It has since been delisted as an American National Standard. Give preference to pan heads on all new designs. Figure 51.4 shows the superiority of the pan head: The high edge of the pan head at its periphery, where driving action is most effective, provides superior driver-slot engagement and reduces the tendency to chew away the metal at the edge of the slot. The flat head is used where a flush surface is required. The countersunk section aids in centering the screw (Figure 51.5).

FIGURE 51.3 Round head.

FIGURE 51.4 Drive-slot engagement.

1894

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

FIGURE 51.5 Flat head.

FIGURE 51.6 Oval head.

The oval head is similar to a flat head except that instead of a flush surface it presents a low silhouette that improves the appearance (Figure 51.6). The truss head is similar to the round head except that the head is shallower and has a larger diameter. It is used where extra bearing surface is required for extra holding power or where the clearance hole is oversized or the material is soft. It also presents a low silhouette that improves the appearance (Figure 51.7). The binding head is similar to the pan head and is commonly used for electrical connections where an undercut is usually specified to bind and prevent the fraying of stranded wire (Figure 51.8). The fillister head has the smallest diameter for a given shank size. It also has a deep slot that allows a higher torque to be applied during assembly. It is not as readily available or as widely used as some of the other head styles (Figure 51.9).

FIGURE 51.7 Truss head.

FIGURE 51.8 Binding head.

FIGURE 51.9 Fillister head.

STANDARD SCREWS

1895

FIGURE 51.10 Hex head.

The advantages of a hex head are listed in the discussion on drivability. This type head is available in eight variations (Figure 51.10). The indented design is lowest cost as the hex is completely cold upset in a counterbore die and possesses an identifying depression in the top surface of the head. The trimmed design requires an extra operation to produce clean sharp corners with no indentation. Appearance is improved and there is no pocket on top to collect moisture. The washer design has a larger bearing surface to spread the load over a wider area. The washer is an integral part of the head and also serves to protect the finish of the assembly from wrench disfigurement. The slot is used to facilitate field service. It adds to the cost, weakens the head, and limits the amount of tightening torque that can be applied. A slot is unnecessary in highproduction factory installation. Any given location should standardize on one or possibly two of the eight variations. Types of Screws Machine Screws Machine screws are meant to be assembled in tapped holes, either into a product or into a nut. The screw threads of a machine screw are readily available in American National Standard Unified Inch Coarse and Fine Thread series. They are generally considered for applications where the material to be joined is too hard, too weak, too brittle, or too thick to take a tapping screw. It is also used in applications where the assembly requires a fastener made of a material that cannot be hardened enough to make its own thread, such as brass or nylon machine screws. Applications requiring freedom from dust or particles of any kind cannot use thread-cutting screws and, therefore, must be joined by machine screws or a tapping screw which forms or rolls a thread. There are many combinations of head styles, shapes, and materials. Self-Tapping Screws There are many different types of self-tapping screws commercially available. The following three types are capable of creating an internal thread

1896

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

by being twisted into a smooth hole: 1. Thread-forming screws 2. Thread-cutting screws 3. Thread-rolling screws The following two types create their own opening before generating the thread: 4. Self-drilling and tapping screws 5. Self-ex truding and tapping screws 1. Thread-Forming Screws. Thread-forming screws create an internal thread by forming or squeezing material. They rely on the pressure of the screw thread to force a mating thread into the workpiece. They are applicable in materials where large internal stresses are permissible or desirable to increase resistance to loosening. They are generally used to fasten sheet metal parts. They cannot be used to join brittle materials, such as plastics, because the stresses created in the workpiece can cause cracking. The following types of thread-forming screws are commonly used: Types A and AB. Type AB screws have a spaced thread. This means that each thread is spaced further away from its adjacent thread than the popular machine screw series. They also have a gimlet point for ease in entering a predrilled hole. This type of screw is primarily intended to be used in sheet metal with a thickness from 0.015 in. (0.38 mm) to 0.05 in. (1.3 mm), resin-impregnated plywood, natural woods, and asbestos compositions. Type AB screws were introduced several years ago to replace the type A screws. The type A screw is the same as the type AB except for a slightly wider spacing of the threads. Both are still available and can be used interchangeably. The big advantage of the type AB screw is that its threads are spaced exactly as the type B screws to be discussed later. In the interest of standardization it is recommended that type AB screws be used in place of either the type A or the type B series (Figure 51.11). Type B. Type B screws have the same spacing as type AB screws. Instead of a gimlet point, they have a blunt point with incomplete threads at the point. This point makes the type B more suitable for thicker metals and blind holes. The type B screws can be used in any of the applications listed under type AB. In addition the type B screw can be used in sheet metal up to a thickness of 0.200 in. (5 mm) and in nonferrous castings (Figure 51.12).

FIGURE 51.11 Type AB.

FIGURE 51.12 Type B.

STANDARD SCREWS

1897

Type C. Type C screws look like type B screws except that threads are spaced to be exactly the same as a machine screw thread and may be used to replace a machine screw in the field. They are recommended for general use in metal 0.030–0.100 in. (0.76–2.54 mm) thick. It should be recognized that in specific applications, involving long thread engagement or hard materials, this type of screw requires extreme driving torques. 2. Thread-Cutting Screws. Thread-cutting screws create an internal thread by actual removal of material from the internal hole. The design of the cavity to provide space for the chips and the design of the cutting edge differ with each type. They are used in place of the thread-forming type for applications in materials where disruptive internal stresses are undesirable or where excessive driving torques are encountered. The following types of thread-cutting screws are commonly used: Type BT (Formerly Known as Type 25). Type BT screws have a spaced thread and a blunt point similar to the type B screw. In addition they have one cutting edge and a wide chip cavity. These screws are primarily intended for use in very friable plastics such as urea compositions, asbestos, and other similar compositions. In these materials, a larger space between threads is required to produce a satisfactory joint because it reduces the buildup of internal stresses that fracture brittle plastic when a closer spaced thread is used. The wide cutting slot creates a large cutting edge and permits rapid deflection of the chips to produce clean mating threads. For best results all holes should be counterbored to prevent fracturing the plastic. Use of this type screw eliminates the need to use tapped metallic inserts in plastic materials (Figure 51.13). Type ABT. Type ABT screws are the same as type BT screws except that they have a gimlet point similar to a type AB screw. This design is not recognized as an American National Standard and should only be selected for large-volume applications (over 50,000 pieces of one size and type). It is primarily intended for use in plastic for the same reasons as listed for type BT screws (Figure 51.14). Type D (Formerly Known as Type 1). Type D screws have threads of machine screw diameter-pitch combinations approximating unified form with a blunt point and tapered entering threads. In addition a slot is cut off center with one side on the

FIGURE 51.13 Type BT.

FIGURE 51.14 Type ABT.

1898

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

FIGURE 51.15 Type D.

center line. This radial side of the slot creates the sharp serrated cutting edge such as formed on a tap. The slot leaves a thinner section on one side of the screw that collapses and helps concentrate the pressure on the cutting edge. This screw is suitable for use in all thicknesses of metals (Figure 51.15). Type F. Type F screws are identical to type D except that instead of one slot there are several slots cut at a slight angle to the axis of the thread. This screw is suitable for use in all thicknesses of metals and can be used interchangeably with a type D screw in many applications. However, the type F screw is superior to the type D screw for tapping into cast iron and permits the use of a smaller pilot hole (Figure 51.16). Type D or Type F. Because in many applications these two types can be used interchangeably with the concomitant advantages of simpler inventory and increased availability, a combined specification is often issued permitting the supplier to furnish either type. Type T (Formerly Known as Type 23). Type T screws are similar to type D and type F except that they have an acute rake angle cutting edge. The cut in the end of the screw is designed to eliminate a pocket that confines the chips. The shape of the slot is such that the chips are forced ahead of the screw as it is driven. This screw is suitable for plastics and other soft materials when a standard machine screw series thread is desired. It is used in place of type D and type F when more chip room is required because of deep penetration (Figure 51.17). Type BF. Type BF screws are intended for use in plastics. The wide thread pitch reduces the buildup of internal stresses that fracture brittle plastics when a smaller thread pitch is used. The screw has a blunt point and tapered entering threads with several cutting edges and chip cavity (Figure 51.18).

FIGURE 51.16 Type F.

FIGURE 51.17 Type T.

STANDARD SCREWS

1899

FIGURE 51.18 Type BF.

FIGURE 51.19 Thread-rolling screws.

3. Thread-Rolling Screws. Thread-rolling screws (see Figure 51.19) form an internal thread by flowing metal and thus do not cut through or disrupt the grain flow lines of materials as do thread-cutting screws. The screw compacts and work hardens the material, thereby forming strong, smoothly burnished internal threads. The screws have the threads of machine screw diameter–pitch combinations. This type screw is ideal for applications where chips can cause electrical shorting of equipment or jamming of delicate mechanism. Freedom from formation of chips eliminates the costly problem of cleaning the product of chips and burrs as would otherwise be required. The ratio of driving torque to stripping torque is approximately 1:8 for a thread-rolling screw as contrasted to 1:3 for a conventional tapping screw. This higher ratio permits the driver torque release to be set well over the required driving torque and yet safely below the stripping torque. This increased ratio minimizes poor fastening due to stripped threads or inadequate seating of the screws. Plastite is intended for use in filled or unfilled thermoplastics and some of the thermosetting plastics. The other three types are intended for use in metals. At present, there are no data to prove the superiority of one type over another. 4. Self-Drilling and Tapping Screws. The self-drilling and tapping screw (Figure 51.20) drills its own hole and forms a mating thread, thus making a complete fastening in a single operation. Assembly labor is reduced by eliminating the need to predrill holes at assembly and by solving the problem of hole alignment. These screws must complete their metal-drilling function and fully penetrate the material before the screw thread can engage and begin its advancement. In order to meet this requirement, the unthreaded

FIGURE 51.20 Self-drilling and tapping screws.

1900

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

FIGURE 51.21 Self-extruding screw.

point length must be equal to or greater than the material thickness to be drilled. Therefore, there is a strict limitation on minimum and maximum material thickness that varies with screw size. There are many different styles and types of self-drilling and tapping screws to meet specific needs. 5. Self-Extruding Screws. Self-extruding screws provide their own extrusion as they are driven into an inexpensively produced punched hole. The resulting extrusion height is several times the base material thickness. This type screw is suitable for material in thicknesses up to 0.048 in. (1.2 mm). By increasing the thread engagement, these screws increase the differential between driving and stripping torque and provide greater pull-out strength. Since they do not produce chips, they are excellent for grounding sheet metal for electrical connections (Figure 51.21). There is almost no limit to the variety of head styles, thread forms, and screw materials that are available commercially. The listing only shows representative examples. Users should attempt to keep varieties to a minimum by carefully selecting those variations that best meet the needs of their type of product. Set Screws. Set screws are available in various combinations of head and point style as well as material and are used as locking, locating, and adjustment devices. The common head styles are slotted headless, square head, hexagonal socket, and fluted socket. The slotted headless has the lowest cost and can be used in a counterbored hole to provide a flush surface. The square head is applicable for location or adjustment of static parts where the projecting head is not objectionable. Its use should be avoided on all rotating parts. The hexagonal socket head can be used in a counterbored hole to provide a flush surface. It permits greater torque to be applied than with a slotted headless design. Fluted sockets are useful in very small diameters, that is, no. 6 (0.138 in.) and under, where hexagon keys tend to round out the socket in hexagonal socket set screws. Set screws should not be used to transmit large amounts of torque, particularly under shock torsion loads. Increased torsion loads may be carried by two set screws located 120 apart. The following points are available with the head styles discussed: The cup point (Table 51.104) is the standard stock point for all head shapes and is recommended for all general locking purposes. Flats are recommended on round shafts when close fits are used and it is desirable to avoid interference in disassembling parts because of burrs produced by action of the cup point or when the flats are desired to increase torque transmission. When flats are not used, it is recommended that the minimum shaft diameter be not less than four times the cup diameter since otherwise the whole cup may not be in contact with TABLE 51.104 Holding Power of Flat or Cup Point Set Screws d (in.) P (lb)

1 4

5 16

3 8

7 16

1 2

9 16

5 8

3 4

7 8

100

168

256

366

500

658

840

1280

1830

1 2500

1 18 3388

1 14 4198

STANDARD SCREWS

1901

FIGURE 51.22 Cup point.

FIGURE 51.23 Oval point.

FIGURE 51.24 Flat point.

the shaft. The self-locking cup point has limited availability. It has counterclockwise knurls to prevent the screw from working loose even in poorly tapped holes (Figure 51.22). When oval points are used, the surface it contacts should be grooved or spotted to the same general contour as the point to assure good seating. It is used where frequent adjustment is necessary without excessive deformation of the part against which it bears (Figure 51.23). When flat points are used, it is customary to grind a flat on the shaft for better point contact. This point is preferred where wall thickness is thin and on top of plugs made of any soft material (Figure 51.24). When the cone point is used, it is recommended that the angle of countersink be as nearly as possible the angle of screw point for the best efficiency. Cone point set screws have some application as pivot points. It is used where permanent location of parts is required. Because of penetration, it has the highest axial and torsional holding power of any point (Figure 51.25).

FIGURE 51.25 Cone point.

1902

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.105 Lag Screws Diameter of screw (in.)

1 4

5 16

3 8

7 16

1 2

5 8

No. of threads per inch

10

9

7

7

6

5

Across flats of hexagon and square heads (in.)

3 8 3 16

15 32 1 4

9 16 5 16

21 32 3 8

3 4 7 16

15 16 17 32

Thickness of hexagon and square heads (in.)

3 4 4 12 1 18 5 8

7 8

1

4

3 12

5 1 16

1 12

3 4

7 8

Length of Threads for Screws of All Diameters Length of screw (in.)

1 12

2

2 12

3

3 12

4

4 12

Length of thread (in.)

To head

1 12

2

2 14

2 12

3

3 12

Length of screw (in.)

5

6

7

8

9

10–12

Length of thread (in.)

4

5 12 4

4 12

5

6

6

7

The half-dog point should be considered in lieu of full-dog points when the usable length of thread is less than the nominal diameter. It is also more readily obtained than the full-dog point. It can be used in place of dowel pins and where end of thread must be protected (Figure 51.26). Lag Screws. Lag screws (Table 51.105) are usually used in wood but also can be used in plastics and with expansion shields in masonry. A 60 gimlet point is the most readily available type. A 60 cone point, not covered in these drawings, is also available. Some suppliers refer to this item as a lag bolt (Figure 51.27). A lag screw is normally used in wood when it is inconvenient or objectionable to use a through bolt and nut. To facilitate the insertion of the screw especially in denser types of wood, it is advisable to use a lubricant on the threads. It is important to have a pilot hole of proper size and following are some recommended hole sizes for commonly used types of wood. Hole sizes for other types of wood should be in proportion to the relative specific gravity of that wood to the ones listed in Table 51.106. Shoulder Screws. These screws are also referred to as “stripper bolts.” They are used mainly as locators or retainers for spring strippers in punch and die operations and have

FIGURE 51.26 Half-dog point.

FIGURE 51.27 Lag screws.

STANDARD SCREWS

1903

TABLE 51.106 Recommended Diameters of Pilot Hole for Types of Wooda Screw Diameter (in.) 0.250 0.312 0.375 0.438 0.500 0.625 0.750 a

White Oak

Southern Yellow Pine, Douglas Fir

Redwood, Northern White Pine

0.160 0.210 0.260 0.320 0.375 0.485 0.600

0.150 0.195 0.250 0.290 0.340 0.437 0.540

0.100 0.132 0.180 0.228 0.280 0.375 0.480

Pilot holes should be slightly larger than listed when lag screws of excessive lengths are to be used.

found some application as fulcrums or pivots in machine designs that involve links, levers, or other oscillating parts. Consideration should be given to the alternative use of a sleeve bearing and a bolt on the basis of both cost and good design (Figure 51.28). Thumb Screws Thumb screws have a flattened head designed for manual turning without a driver or a wrench. They are useful in applications requiring frequent disassembly or screw adjustment (Figure 51.29). Weld Screws Weld screws come in many different head configurations, all designed to provide one or more projections for welding the screw to a part. Overhead projections are welded directly to the part. Underhead projections go through a pilot hole. The designs in Figures 51.30 and 51.31 are widely used. In projection welding of carbon steel screws, care should be observed in applications, since optimum weldability is obtained when the sum, for either parent metal or screw, of one-fourth the manganese content plus the carbon content does not exceed 0.38. For good

FIGURE 51.28 Shoulder screw.

FIGURE 51.29 Thumb screws.

1904

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

FIGURE 51.30 Single-projection weld screw.

FIGURE 51.31 Underhead weld screws.

weldability with the annular ring type, the height of the weld projection should not exceed half the parent metal thickness as a rule of thumb. Copper flash plating is provided for applications where cleanliness of the screw head is necessary in obtaining good welds. Wood Screws Wood screws are (Table 51.107) readily available in lengths from 14 to 5 in. for steel and from 14 to 3 12 in. for brass. Consideration should be given to the use of type AB thread-forming screws, which are lower in cost and more efficient than wood screws for use in wood. Wood screws are made with flat, round, or oval heads. The resistance of wood screws to withdrawal from side grain of seasoned wood is given by the formula P ¼ 2850G2D, where P is the allowable load on the screw (lb/in. penetration of the threaded portion), G is specific gravity of oven-dry wood, and D is the diameter of the screw (in.). Wood screws should not be designed to be loaded in withdrawal from the end grain. The allowable safe lateral resistance of wood screws embedded seven diameters in the side grain of seasoned wood is given by the formula P ¼ K D2, where P is the lateral resistance per screw (lb), D is the diameter (in.), and K is 4000 for oak (red and white), 3960 for Douglas fir (coast region) and southern pine, and 3240 for cypress (southern) and Douglas fir (inland region). TABLE 51.107 American Standard Wood Screwsa Number Threads per inch Diameter (in.) Number Threads per inch Diameter (in.) a

0 32 0.060 9 14 0.177

1 28 0.073 10 13 0.190

2 26 0.086 11 12 0.203

Included angle of flathead ¼ 82 ; see Figure 51.18.

3 24 0.099 12 11 0.216

4 22 0.112 14 10 0.242

5 20 0.125 16 9 0.268

6 18 0.138 18 8 0.294

7 16 0.151 20 8 0.320

8 15 0.164 24 7 0.372

STANDARD SCREWS

1905

The following rules should be observed: (a) The size of the lead hole in soft (hard) woods should be about 70% (90%) of the core or root diameter of the screw; (b) lubricants such as soap may be used without great loss in holding power; (c) long, slender screws are preferable generally, but in hardwood too slender screws may reach the limit of their tensile strength; and (d) in the screws themselves, holding power is favored by thin sharp threads, rough unpolished surface, full diameter under the head, and shallow slots. SEMS The machine and tapping screws can be purchased with washers or lock washers as an integral part of the purchased screws. When thus joined together, the part is known as a SEMS unit. The washer is assembled on a headed screw blank before the threads are rolled. The inside diameter of the washer is of a size that will permit free rotation and yet prevent disassembly from the screw after the threads are rolled. If these screws and washers were purchased separately, there would be an initial cost savings over the preassembled units. However, these preassembled units reduce installation time because only one hand is needed to position them, leaving the other hand free to hold the driving tool. The time required to assemble a loose washer is eliminated. In addition, these assemblies act to minimize installation errors and inspection time because the washer is in place, correctly oriented. Also the use of a single unit, rather than two separate parts, simplifies bookkeeping, handling, inventory, and other related operations. 51.7.1 Nominal and Minimum Dressed Sizes of American Standard Lumber Table 51.108 applies to boards, dimensional lumber, and timbers. The thicknesses apply to all widths and all widths to all thicknesses. TABLE 51.108 Nominal and Minimum Dressed Sizes of American Standard Lumber Thicknesses

Face Widths

Minimum Dressed Item

Nominal

Drya (in.)

Green (in.)

Boardsb

1

3 4

1 14 1 12

1 1 14

Minimum Dressed

25 32 1 1 32 9 1 32

f

Nominal

Drya (in.)

Green (in.)

2

1 12

9 1 16

3

2 12

9 2 16

4

3 12

9 3 16

5

4 12 5 12 6 12 7 14 8 14 9 14 10 14 11 14 13 14 15 14

4 58

6 7 8 9 10 11 12 14 16

5 58 6 58 7 12 8 12 9 12 10 12 11 12 13 12 15 12 (continued)

1906

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

TABLE 51.108 (Continued ) Thicknesses

Face Widths

Minimum Dressed Item

Nominal

a

Dry (in.)

Minimum Dressed

Green (in.)

Dimension

1 12

9 1 16

2 12

2

3

2 12

3 12

3

1 2 16 9 2 16 1 3 16

2

Dimension

Timbers a

4

3 12

9 3 16

4 12

4

1 4 16

5 and thicker

1 2

off

5 and

f f

Nominal

Drya (in.)

Green (in.)

2

1 12

9 1 16

3

2 12

9 2 16

4

9 3 16

16

3 12 4 12 5 12 7 14 9 14 11 14 13 14 15 14

2

1 12

9 1 16

3

2 12

9 2 16

4

3 12

9 3 16

5

4 12 5 12 7 14 9 14 11 14

4 58

5 6 8 10 12 14

6 8 10 12

4 58 5 58 7 12 9 12 11 12 13 12 15 12

5 58 7 12 9 12 11 12

14

13 12

16

15 12

wider

1 2

off

Maximum moisture content of 19 % or less. Boards less than the minimum thickness for 1 in. nominal but 58 in. or greater thickness dry (11 16 in. green) may be regarded as American Standard Lumber, but such boards shall be marked to show the size and condition of seasoning at the time of dressing. They shall also be distinguished from 1-in. boards on invoices and certificates. Source: From American Softwood Lumber Standard, NBS 20–70, National Bureau of Standards, Washington, DC, 1970, amended 1986 (available from Superintendent of Documents). b

52 MEASUREMENT UNCERTAINTY DAVID CLIPPINGER 52.1 Introduction 52.1.1 Uncertainty as a property 52.1.2 Misuse and inadequacy of significant digits 52.1.3 Modern expressions of uncertainty 52.2 Literature 52.3 Evaluation of uncertainty 52.3.1 ISO GUM methodology 52.3.2 ASME performance test code methodology 52.4 Discussion References

52.1 INTRODUCTION 52.1.1 Uncertainty as a Property The uncertainty of a measurement is a property of the measurement, and serves as an indication of the degree of reliability in the measurement or its usefulness in a given application (Taylor and Kuyatt,1994). ISO (1995) rigorously defines uncertainty as “[the] parameter, associated with the result of a measurement, that characterizes the dispersion of values that could be reasonably attributed to [the particular quantity subject to measurement].” For all but the crudest of applications, the uncertainty associated with a measurement is as important as the units of the measurement itself (i.e., kilogram, millimeter, etc.). That the units of a measurement should be reported with the measurement is beyond debate: there are numerous examples from history where a measured quantity was reported without its units, only to be mistaken by others for a value of a different dimension, often with disastrous results. Despite it’s frequent omission, the uncertainty of a measurement is actually important in all situations where a measured quantity must be trusted with a certain level of Handbook of Measurement in Science and Engineering. Edited by Myer Kutz. Copyright Ó 2013 John Wiley & Sons, Inc.

1907

1908

MEASUREMENT UNCERTAINTY

confidence. A manufacturing process that requires that a component have a mass of very close to 10 kg is one such example. Would it be acceptable to measure the mass of these objects with an inexpensive spring scale, or should a laboratory-calibrated electronic balance be used for such a measurement? The answer depends upon the definition of “very close” in the manufacturing specification, and the level of what is colloquially called the “accuracy” of the two mass measurement systems. To be more specific, the measured value must have an uncertainty that is less than the tolerance specified by the process. For example, when the mass is measured using the spring scale, it might be reasonable to attribute mass values between 9.73 kg and 10.27 kg to any mass that displays “10 kg” on the scale. In this case, the uncertainty of the measurement would be considered to be 0.27 kg. The acceptability of the spring scale as a mass measurement system can now be assessed quantitatively: if an error of 0.27 kg is “close enough” to 10 kg to satisfy requirements, the spring scale is perfectly satisfactory for this purpose. Of course, the ability to make such a quantitatively based decision implies that the uncertainty of measurements made on the spring scale can be evaluated as 0.27 kg (or some other value) in the first place. Fortunately, the uncertainty of a measurement can be estimated with a high degree of confidence. Methods to do this are presented later in this chapter, after brief sections describing unacceptable and acceptable ways to report uncertainty, respectively. 52.1.2 Misuse and Inadequacy of Significant Digits The concept of uncertainty is often first introduced to students couched in terms of significance, where the number of digits (or figures) in a numerical quantity is used to indicate the level of reliability of the quantity: the more digits past the decimal place, the more reliable (or precise) the measurement. It is understood that the maximum error associated with a quantity reported using significant digits is 1/2 the right-most (smallest) digit. This is sometimes called round-off error. However, significant digits are only valid when expressing the approximate value of exact quantities, and (perhaps surprisingly!) this validity does not extend to computations performed with these numbers (Bragg, 1974). Bragg (1974) uses computations with the value of p as an illustration, and the same will be done here. This quantity is often approximated as 3.14 (using three significant digits), which is taken to mean that the true value of p lies between 3.135 and 3.145 (i.e., 0.005). This is in fact the case, as p ¼ 3.141592654 . . . . Now consider the computation p2. Performing the computation 3.142 results in the quantity 9.8596. Expressed to three significant digits, this quantity is 9.86, which would imply that the true value of p2 should be somewhere between 9.855 and 9.865. Unfortunately it is not, (it is 9.869604401 . . . , or 9.87 to three significant digits), and while the error is arguably small (0.01 difference), the purpose of this illustration is to show that simply “carrying” significant digits through computational problems does not necessarily produce results with the same number of significant digits. It is sometimes proposed that carrying an “extra” significant digit through all computations and rounding off at the end will alleviate the problem described above. However, this too is erroneous. Such an approach is described in Hibbeler (2005), but is merely presented as the format for worked example problems, with no claims made about its validity as a computational practice. Kirkup and Frenkel (2006) list several “rules” for working with significant digits, at the same time observing that the use of significant digits represents no more than

INTRODUCTION

1909

“common sense” on the part of the experimentalist and cautioning that their use is “not a substitute for the detailed calculation of uncertainty.” As an illustration, consider the computation p10 as an example. The “true” value of this quantity is 93648.047476 . . . (or 93.6  103 to three significant digits). However, if the four significant digit quantity 3.142 is used as approximation for p and carried through all computations, only rounding off to three significant digits at the very end, the resulting quantity is 93.8  103. While this result is reasonably close to the true value, it does not meet the definition of a number with three significant digits: its quantity is not within 50 (0.05  103) of the true quantity. On top of all this, Bragg (1974) also points out a serious limitation to the use of significant digits as a means of reporting the uncertainty of a quantity: such a method implies that the uncertainty of every number is simply 1/2 the right-most digit. Such an approach is excessively simplistic and assumes that the uncertainty of experimentally derived values can always be reduced to simple round-off error. While round-off error is valid when reporting the approximate value of exact numbers (such as p, e, or square roots of integers, all of which may be computed to a high degree of precision) it is not valid to consider that a measurement recorded to three decimal places is automatically accurate to 0.0005. (If this were the case, why buy sophisticated test equipment when cheaper instruments could simply be fitted with extra-long digital displays!) Returning to the spring scale example, it can be seen that an uncertainty of 0.27 kg cannot be readily expressed using significant digits alone. Instead, a more modern notation is required. 52.1.3 Modern Expressions of Uncertainty As described above, the significant-digit method is very limited in its application and of little use to the experimentalist. Rather, the preferred methods today are the four that are contained in the ISO’s publication Guide to the Expression of Uncertainty in Measurement (1995). Briefly, they are as follows (paraphrased from the original, Section 7.2.2): (1) state the uncertainty in the text accompanying the measurement. For example, “mass¼10.00 kg, with an uncertainty of 0.27 kg,” (2) provide the uncertainty in parentheses after the reported quantity, specifically stating that the digits used for the uncertainty are the right-most digits of the reported measurement. For example, “mass¼10.00(27) kg, where the number in parentheses is the numerical value of the uncertainty corresponding to the last digits of the reported result,” (3) provide the uncertainty in parentheses after the reported quantity, specifically stating that the quantity in parentheses is the uncertainty of the measurement. For example, “mass¼10.00(0.27) kg, where the number in parentheses is the uncertainty of the measurement,” (4) provide the uncertainty following a plus or minus sign that appears after the reported quantity, specifically stating that the quantity following the plus or minus is the uncertainty (and not a confidence interval). For example, “mass¼(10.00  0.27) kg, where the number following the plus or minus is the uncertainty of the measurement, and not a confidence interval.”

1910

MEASUREMENT UNCERTAINTY

A quick check of the engineering experimentation texts published by the leading textbook publishers in the United States seems to indicate that fourth style is the most popular for textbook use (the “not a confidence interval” clause is omitted in many textbooks). The fourth style is also used extensively in the otherwise “harmonized” ASME Test Uncertainty (1998). Ironically, ISO (1995) actually discourages this style, citing the ease with which this notation may be confused with that of a confidence interval (ISO makes special mention that the confusion may occur if the “not a confidence interval” clause is omitted). ISO (1995) also provided further guidance for reporting uncertainties, such as specifying the coverage factor, the level of confidence used, and distribution used to obtain the coverage factor. These terms will be defined, and their application explained, in later sections of this chapter as the need appears.

52.2 LITERATURE Uncertainties may be computed with a high degree of confidence with access to proper information. Methods for doing this may be found in several places, but the ISO (1995) Guide to the Expression of Uncertainty in Measurement (the “Guide” or GUM) cited earlier should be viewed as the authoritative reference for reporting uncertainty. The development of this reference was actually supported by a host of authoritative scientific and engineering organizations—BIPM, IEC, IFCC, IUPAC, IUPAP, and OIML—and is published in their name. As such it can be considered the “official” guide to those working in the fields of chemistry, physics, electrotechnology, and many others. For engineers, ASME (1998) has harmonized their Performance Test Code Test Uncertainty (PTC 19.1-1998) with the ISO GUM, extending the guidance of the Guide to the Expression of Uncertainty in Measurement (ISO, 1995) to the Mechanical Engineering discipline as well. While “harmonized” with ISO the ASME test code does differ from the ISO guidelines in the actual methodology used to evaluate the overall uncertainty of a measurement, and has slight differences in terminology and notation. Both approaches are explained in this chapter and yield almost identical results. NIST has adopted the ISO approach as standard practice in their Administrative Manual, Subchapter 4.09. This subchapter is included as an appendix to NIST’s Technical Note 1297 (Taylor and Kuyatt,1994). The NIST technical note (Taylor and Kuyatt,1994) is particularly convenient since it is concise and (as a U.S. Government publication) accessible free of charge to anyone with an internet connection. (The ISO standard was actually published in 1993, and was corrected and re-printed in 1995, which is the version cited in this chapter. Taylor and Kuyatt (1994) reference the 1993 printing.) Organizations such as the Instrumentation, Systems, and Automation Society, NATO, and others have also adopted methods consistent with the ISO approach (Dieck, 2007). In short, for engineers and scientists working in the United States, the ISO Guide to the Expression of Uncertainty in Measurement and the ASME Test Uncertainty (1998) are the de-facto “official” documents addressing the expression of uncertainty. Engineers or scientists working in other countries should consult their appropriate government ministry, professional society, or corporate leadership for clarification before using a standard other than the ISO Guide or one of its “harmonized” counterparts when evaluating measurement uncertainty.

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52.3 EVALUATION OF UNCERTAINTY 52.3.1 ISO GUM Methodology The ISO (1995) guidelines for the evaluation of uncertainty center on the evaluation of the various standard uncertainties, their addition into what is called combined uncertainty, and (where desired) the development of what is called extended uncertainty. The procedure described below is for cases where the measurements may be collected independently (i.e., the readings of different instruments are uncorrelated) and exhibit random errors that are approximately normally distributed. ISO ( 1995) should be consulted for special cases of correlated readings or asymmetrical error distributions. 52.3.1.1 Standard Uncertainties The current guidance from ISO (1995) describes that evaluation of the components of uncertainty in two methods, which it designates Types A and B. Type A refers to the “method of evaluation of uncertainty by the statistical analysis of a series of observations” (ISO, 1995). Type B refers to the “method of evaluation of uncertainty by means other than the statistical analysis of a series of observations” (ISO, 1995). Taylor and Kuyatt (1994) emphasize that these two methods are not the same as “random” and “systematic” components of error that are commonly described in other references on the subject (e.g., Wheeler and Ganji, 2010; also ASME, 1998). ISO’s guidance should not be taken as implying that measurement errors are no longer thought to have random and systematic components. Rather, the two types (A and B) refer to the means by which the component of the uncertainty is obtained for any given measurement, and are introduced for “convenience of discussion only” (ISO, 1995). The key difference between types A and B uncertainties is that type A uncertainties are evaluated directly from data collected during the experiment, while type B uncertainties are evaluated from data collected beforehand or made available from some other means. Type A standard uncertainty is most commonly computed using the definition of standard deviation. Examples of sources of information from which type B uncertainty may be estimated include “previous measurement data, . . . manufacturer’s specifications, data provided in calibration and other certificates, uncertainties assigned to reference data taken from handbooks,” and from “general knowledge” of the experimental procedures and instruments used (ISO, 1995). Examples of types of uncertainty typically evaluated in a Type B manner include uncertainties due to hysteresis, readability, linearization of the sensor’s response, and resolution uncertainty, all of which are typically obtained prior to performing a measurement. In both cases, the uncertainty so evaluated is designed u, multiple contributions to uncertainty typically indicated with subscripts (e.g., u1, u2, etc.). If it is a type A uncertainty, it is called a Type A Standard Uncertainty; otherwise it is called a Type B Standard Uncertainty. It is these standard uncertainties that are the building blocks that are used when evaluating what is called combined uncertainty. As mentioned above, Type A standard uncertainty is usually the familiar statistical standard deviation, although other statistical means may be used to estimate its value (ISO, 1995). Typically, the standard uncertainty of the mean measurement is desired, in which case the standard deviation of the mean is the appropriate quantity to compute. In Equation (52.1), u is the standard uncertainty, xi are the individual  is the values of the measured quantity, n is the number of measurements, and x

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average value of the n measurements. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn 1 Þ2 ðxi x u¼ i¼1 nðn 1Þ

ð52:1Þ

In contrast, Type B standard uncertainty may be obtained from a variety of different sources. It may be reported as a standard uncertainty (in which case it may be used as the value of u directly), or (more commonly) it may be obtained by dividing a reported confidence interval size by the appropriate coverage factor, assuming that the confidence interval is based on the standard normal distribution. For example, a manufacturer-supplied hysteresis uncertainty of 0.02 reported at the 95% confidence level is divided by 1.96 (the coverage factor or “z-value” for a 95% confidence, although 2 is also used as an approximation for this value) to obtain a Type B standard uncertainty of about 0.010 (to three significant digits). Taylor and Kuyatt (1994) describe situations where other techniques may be used, such as when a rectangular or triangular distribution more suitably describes the dispersion of values than the normal distribution. A real measurement system will have multiple sources of uncertainty in both the type A and B categories. For example, a mass measurement using an electronic balance may (if taken a total of 10 times) return a mean measurement of 10.01 kg, with a standard deviation of the means (Type A standard uncertainty) of 0.022 kg, and have a manufacturer-specified (Type B) uncertainty due to linearization of the sensor response of 0.0003 kg. If the measurement device displays two digits past the decimal point, it will have a resolution uncertainty of 0.005. It is often convenient to arrange these different standard uncertainties in a table. Two such tables are illustrated below. Table 52.1 shows a sample table for a mass measurement with an electronic balance with digital display. In this case the manufacturer’s data provides the linearization uncertainty, but no details as to its exact definition (i.e., whether the value provided is a standard uncertainty or a confidence interval). If the manufacturer cannot be contacted for clarification, one approach suggested by Wheeler and Ganji (2010) is to assume that the coverage factor is 2 for the linearization error since the manufacturer’s data was likely based on a large sample of normally-distributed data and likely reported at the 95% confidence level. Such an approach is used here. Table 52.2 shows a sample table for a power measurement in which the power consumed by a resistor is computed from its resistance and the voltage drop across it while in use. The resistance and voltage drop are both measured with a digital multimeter, but not simultaneously. This is

TABLE 52.1 Standard Uncertainties of a Mass Measurement Name

Symbol u1

Standard deviation of the means Linearization error

u2

Resolution error

u3

a

Source Experimental data Manufactuer’s data Inspection of display

Type Value Units A 0.010

Coverage Factor

Standard Uncertainty

kg

1

0.01

B 0.0003 kg

2

0.0015

1.732 (assumed)a

0.0025

B 0.005

kg

Assumes resolution uncertainty may be modeled with a rectangular distribution.

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TABLE 52.2 Standard Uncertainties of a Power Measurement Name Standard deviation of the means-voltage Linearization uncertainty-voltage Resolution uncertainty-voltage Standard deviation of the meansresistance Linearization uncertaintyresistance Resolution uncertaintyresistance a b

Symbol u11 u12 u13 u21

u22

u23

Source

Coverage Factor

Standard Uncertainty

Type Value

Units

Experimental data Manufacture’s data Inspection of display A priori experimental data Manufacturer’s data

A

0.010

V

1

B

0.0001

V

2 (assumed)a

B

0.005

V

B

0.040

W

1.732 (assumed)b 1

B

0.0001

W

2 (assumed)a

0.0005

Inspection of display

B

0.050

W

1.732 (assumed)b

0.029

0.01 0.0005 0 0.04

Assumes reported value is based on a large-sample normal distribution at 95% confidence. Assumes resolution uncertainty may be modeled with a rectangular distribution.

appropriate since resistance readings cannot typically be directly obtained while a resistor is passing current; in this example the resistance readings were taken in advance, while the system was un-powered. Accordingly, the associated uncertainty is a Type B standard uncertainty (the actual classification of Type A or Type B ultimately has no bearing on the evaluation of uncertainty). The conversion of resolution uncertainty to standard uncertainty deserves special mention. Wheeler and Ganji (2010) propose that resolution uncertainty be treated the same as a 95% confidence interval as an “arbitrary rule.” If this rule is adopted, and it is assumed the error associated with resolution uncertainty may be modeled as a normal distribution the resolution uncertainty should be divided by 2 (or 1.96) to obtain the standard uncertainty. However, resolution uncertainty (which is essentially due to round-off error of the display) is arguably best modeled as a random process with a rectangular (or uniform) probability distribution. If this model is adopted, the guidance from Taylor and Kuyatt (1994) applies, in that case pffiffiffi the appropriate divisor to convert resolution uncertainty to a standard uncertainty is 3 (or about 1.732). Table 52.1 shows the resolution uncertainty of an electronic digital balance that displays two digits past the decimal point. Table 52.2 shows the resolution uncertainty of a multimeter that displays two digits past the decimal point when measuring voltage and one digit past the decimal when measuring resistance. In both tables resolution uncertainty is treated as if arising from a random process with a rectangular distribution. 52.3.1.2 Combined Uncertainty The individual standard uncertainties are then combined using the “root sum square” method of addition (ISO, 1995). In the simplest cases, where the contributions to the uncertainty all have the same dimension, (such as the example shown in Table 52.1) the individual uncertainties add as shown in Equation (52.2). uc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u21 þ u22 þ . . .

ð52:2Þ

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In Equation (52.2), uc is the combined standard uncertainty. The situation becomes more involved when the standard uncertainties have different dimensions. In these cases the experimental measurement is said to be a result. Numerous engineering measurements are actually results. For example, electrical resistance strain gage measurements are really millivolt measurements, which are then converted to strain with knowledge of the appropriate gage factor and excitation voltage. Hence, three independent factors—two voltages and a gage factor, all of which have uncertainties—are needed to compute strain. These different uncertainties are said to propagate to the uncertainty of the final result. In such situations the individual standard uncertainties are multiplied by their appropriate sensitivity coefficients to compute the combined uncertainty. The sensitivity coefficients are the partial derivatives of the measurement function with respect to each of the variables that possess uncertainty, evaluated at the operating point, or mean value of the measured variables. To use the power analogy from Table 52.2, the appropriate expression for power is P ¼ V2/R. Therefore, the appropriate expression for the combined uncertainty of the power measurement is given by Equation (52.3), where bar overscores indicate average values of voltage (V) and resistance (R). The uncertainties u1, u2, and so on that appear in Equation (52.3) are the uncertainty values listed in Table 52.2. The minus sign that would normally appear in the denominator of the last three sensitivity coefficients has been dropped in Equation (52.3) for clarity (the minus signs ultimately drop out as all terms are squared). vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 !2ffi u 2  2  2 2 2 2 u 2V       V V V 2V 2V uc ¼ t  u11 þ  u12 þ  u13 þ  u21 þ  u22 þ  u23 R R R R R R

ð52:3Þ

Note that the uncertainties with common first subscripts (the u1i and u2i) are dimensionally the same. It is sometimes convenient to add these using Equation (52.2) to obtain combined uncertainties u1, u2, and so on (without the double subscript), as shown in Equations (52.4a) and (52.4b). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u211 þ u212 þ u213 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 ¼ u221 þ u222 þ u223

u1 ¼

ð52:4aÞ ð52:4bÞ

The general form of Equation (52.3) is shown in Equation (52.5), where xi represents the individual measured values that contribute to the result, X now represents an ordered n-tuple of the measured values (x1, x2, x3, . . . ), the ui represent the n individual standard combined uncertainties (combined using Equation (52.2)), and bar overscores indicate average values. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  2   2 @R  @R  uc ¼ u1 þ u2 þ . . . ð52:5Þ @x1 X @x2 X

As a check to ensure the correct pairings of the partial derivatives with the appropriate uncertainties, it is helpful to verify that all of the squared terms inside the square root sign

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of Equation (52.5) are dimensionally identical. This can be seen in Equation (52.3), where uncertainties u11, u12 and u13 have the units “Volts,” and are multiplied by sensitivity coefficients that are dimensionally “Volts per Ohm” to yield “Volts squared per Ohm” or “Watts.” The remaining three uncertainties have units of “Ohms” are multiplied by sensitivity coefficients that are dimensionally “Volts squared per Ohm squared” to yield (as expected) terms that are dimensionally “Watts.” 52.3.1.3 Extended Uncertainties It is perfectly acceptable to simply report combined standard uncertainties in one of the formats shown in the section titled Modern Expressions of Uncertainty that appeared earlier in this chapter. However, in some situations it may be desirable to report an extended uncertainty. An extended uncertainty is a combined standard uncertainty multiplied by an appropriate coverage factor (ISO, 1995). Popular coverage factors are 2 (in which case the extended uncertainty represents approximately 95% of the dispersion of the values that may be reasonably attributed to the measured quantity) or 3 (where over 99% is represented). In other cases, a coverage factor between 2 and 3 may be desired. (A coverage factor of 1.65 is used if about 90% of the dispersion is desired to be represented). NIST uses a coverage factor of 2 in all cases, unless specific needs require otherwise (Taylor and Kuyatt,1994). Examples of such situations include cases with small sample sizes where use of the “Student t” distribution rather than the standard normal distribution is more appropriate. In these situations, the appropriate “t-value” should be used as the coverage factor, taking care to report this (along with the number of degrees of freedom and confidence level used) in the text accompanying the measurement. In all cases, the extended uncertainty is computed from the combined uncertainty as shown in Equation (52.6), where U is the extended uncertainty and k is the coverage factor (ISO, 1995). U ¼ kuc

ð52:6Þ

For the situations where a small sample size is used to compute some (or all) of the Type A uncertainties, a complication arises when selecting the appropriate number of degrees of freedom to use as the entering argument for a table of critical t-values. In these cases, the effective number of degrees of freedom veff may be estimated using the Welch–Satterthwaite formula, Equation (52.7), veff ¼

u4c PN c4i ui i¼1 vi

ð52:7Þ

where ci ¼ @R=@xi jX (i.e., the sensitivity coefficients), vi are the degrees of freedom associated with the uncertainties ui, and the number of effective degrees of freedom is less than the sum of the individual degrees of freedom (Taylor and Kuyatt, 1994), Equation (52.8). (Note that the sensitivity coefficients in Equation (52.7) are all equal to 1 if the measurement is not a result, as was the case in Table 52.1.) veff 

XN

v i¼1 i

ð52:8Þ

A conservative approximation is to truncate (rather than round) the value of veff (i.e., to favor the lower number of degrees of freedom).

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52.3.2 ASME Performance Test Code Methodology 52.3.2.1 Types of Uncertainty The approach favored by ASME ( 1998) and many textbooks (e.g., Figliola and Beasely, 2000 or Wheeler and Ganji, 2010) is to categorize sources of uncertainty into two types: Systematic and Random. As mentioned above, these terms are not synonymous with Types A and B uncertainties, but rather refer to the nature of the contribution to the overall uncertainty. If the source of uncertainty is expected to contribute to a scattering of the measurement values, it is considered to be a random contribution to uncertainty. If the contribution to the overall uncertainty is constant, it is considered to be systematic (Rabinovich, 1995). There may be multiple contributors to each of these types of uncertainty (Kirkup and Frenkel, 2006). One way to distinguish between systematic and random sources of uncertainty is by determining if the overall uncertainty can be reduced by increasing the sample size. If the overall uncertainty may be reduced in this method, the uncertainty source is clearly random. Uncertainty that is always present and cannot be reduced by increasing the sample size (which effectively reduces the “scatter” in the estimate of the mean value) is considered systematic (Wheeler and Ganji, 2010). The presence of the systematic uncertainties makes increasing the sample size to huge values in an attempt to reduce uncertainty to zero not only impractical but futile (Kirkup and Frenkel, 2006). The systematic uncertainties can be thought of as forming a “floor,” below which the total uncertainty cannot be lowered due to the limitations of the measuring system. Examples of systematic uncertainties are those due to the limitations of the calibration techniques, which merely reduce, but do not eliminate systematic uncertainty (Figliola and Beasely, 2000; also Rabinovich, 1995), uncertainties due to hysteresis and linearization of the sensor output (commonly called “linearity” uncertainty), and repeatability uncertainty. These three contributors to systematic uncertainty (and occasionally others) are sometimes combined and reported in manufacturers specifications as a sensor’s accuracy (Wheeler and Ganji, 2010). ASME (1998) denotes random uncertainty with the variable P, and systematic uncertainty with the variable B, reflecting the older terms for these two components (“Precision” and “Bias” uncertainty, respectively). The ASME notation will also be used here, as will the terms random and systematic, consistent with the “harmonization” of ASME’s Test Uncertainty (1998) with ISO’s Guide to the Expression of Uncertainty in Measurement (1995). 52.3.2.2 Combined Systematic Uncertainty Even a simple measurement will typically have multiple contributions to the systematic uncertainty. These individual contributions may be combined using the root sum square method as shown in Equation (52.9), B¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 þ b22 þ b23 þ . . .

ð52:9Þ

where the bi are the individual contributions to the systematic uncertainty and B is the total systematic uncertainty of the measured value. Care must be taken when combining the bi to ensure that they are all evaluated at the same level of confidence (Wheeler and Ganji, 2010). For example, it is inconsistent to combine an uncertainty due to hysteresis that is provided at the 90% confidence level with an uncertainty due to linearity that is provided at the 95% confidence level. In the cases where the different components of systematic uncertainty are provided at different levels of

EVALUATION OF UNCERTAINTY

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confidence, the 95% confidence level should be used (ASME, 1998). Uncertainties provided at other levels of confidence should be converted to standard uncertainties in the manner described in the ISO GUM Methodology section of this chapter, and then multiplied by 2 to obtain an approximate 95% confidence interval (ASME (1998) uses 2—not 1.96—as its 95% coverage factor). In cases where the systematic uncertainty of a result is desired, the individual contributions to systematic uncertainty will be dimensionally different and have different “weights” in the computation of total systematic uncertainty. As was the case when combining standard uncertainties when using ISO GUM methodology, the systematic uncertainty components are multiplied by their corresponding sensitivity coefficients before combining, as shown in Equation (52.10), ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 @R  @R  BR ¼  B1 þ  B2 þ . . . @x1 X @x2 X

ð52:10Þ

where BR is the systematic uncertainty of the result, and the Bi are the overall systematic uncertainties of the different measurements that together contribute to the result, X is an ordered n-tuple of the various measured quantities, and bars indicate average values. For example, if the measurement of power dissipated by a resistor is desired to be computed from values of voltage and resistance using the relationship R ¼ V2/ R, the expression for total systematic uncertainty of the power measurement is computed as shown in Equation (52.11), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2    V 2V B B ð52:11Þ BPower ¼ þ Res  V 2 R R where bars indicate average values, BV is the systematic uncertainty of the voltage reading and BRes is the systematic uncertainty of the resistance reading. The required values of BV and BRes are obtained using Equation (52.9). As was the case when computing combined uncertainty of a result using ISO GUM methodology, each of the squared terms in Equation (52.11) can be seen to be dimensionally consistent: they all have units of power. 52.3.2.3 Combined Random Uncertainties For a direct measurement, the random uncertainty is the standard deviation of the means, shown as Sx in Equation (52.12). This equation is identical to Equation (52.1), but uses the notation from ASME (1998). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn 1  Þ2 Sx ¼ ðx x ð52:12Þ i¼1 nðn 1Þ In cases where the measurement is a single result, the individual standard deviations of the means are weighted by the appropriate sensitivity coefficients and combined using the root-sum-squares method, as shown in Equation (52.13), ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 @R  @R  SR ¼ ð52:13Þ  S1 þ  S2 þ . . . @x1 X @x2 X

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where SR is the standard deviation of the result, the Si are the standard deviations of the means of the individual measured parameters that contribute to the result (ASME, 1998),  represents the average value of an n-tuple of the measured parameters (i.e., and X 2 ; x 3 . . .Š). For example, to return to the example of a power measurement ½ x1 ; x based on resistances and voltages, the standard deviation of the power dissipated by a  and average voltage drop V  is computed as shown in resistor of average resistance R Equation (52.14),

SPower

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u 2 u 2V 2  V t ¼ S  SV þ R 2 R R

ð52:14Þ

where bars indicate average values, SV is the standard deviations of the means of the voltage (computed using Equation (52.12)) and SR is the standard deviations of the means of the resistance, computed in identical manner. Again, the squared terms that appear in Equation (52.14) can all be seen to have the dimension of power. 52.3.2.4 Multiple Tests A special case exists for what ASME (1998) calls “multiple tests.” A multiple test is one where more than one set of measurements is made with the same instrument package. An important feature that must be present for this technique to be applied is a reasonable simultaneity of the measurements. To illustrate the difference between a single result with multiple readings and a multiple test, again consider the measurement of power dissipated through a resistor. One such method to measure this power is the previously described measurement of voltage drop and resistance, and the computation of power dissipation from the relationship R ¼ V2/ R. However, as mentioned earlier, the resistance measurements are typically taken in advance—while the system is in an unpowered state—and the voltage measurements are made afterward (the sequence could also be reversed, or the system could be repeatedly powered up and powered down to alternate voltage and resistance readings). However, regardless of the method chosen, no resistance measurement is ever collected at the same instant as any voltage measurement. The value of random uncertainty is then obtained from Equation (52.13). The net result is that there is one average value of voltage obtained (with an uncertainty) and one average value of resistance obtained (with its own uncertainty), which allows one value of average power to be computed (which has its own uncertainty). Now consider an alternative instrumentation of the same measurement. In this case, the circuit is fitted with an ammeter and voltmeter, so that an amperage reading may be taken at the same instant as each of the voltage measurements (this is easily done with modern computer controlled data acquisition systems but could also be reasonably approximated by a trained observer provided that both measurements changed slow enough to be recorded manually with reasonable simultaneity). The data may then be arranged in a table and power computed for each (nearly) simultaneous set of measurements as shown in Table 52.3. From Table 52.3, it can be seen that eight distinct values of power may be computed, which in turn have a mean and standard deviation. Such a test procedure provides the opportunity to consider the eight computed values of power as a sample of results, for which the mean and standard deviation of the result may be computed. In

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TABLE 52.3 Simultaneous Voltage and Amperage Measurements Reading Number

Voltage (V)

1 2 3 4 5 6 7 8 Average power (W) Standard deviation of power (W)

Amperage (A)

Power (W)

1.99 2.02 1.95 2.05 2.02 2.03 2.02 1.98

19.89 20.18 19.58 20.43 20.24 20.26 20.23 19.75 20.07 0.29

10.01 10.00 10.02 9.98 10.01 10.00 10.01 9.98

short, since multiple individual results can be computed from multiple sets of simultaneously collected data, the results themselves can be treated much like individual measurements. In these special cases, the standard deviation of the result is computed in a manner very similar to that used for the standard deviation of a direct measurement, Equation (52.12). Modifying the notation to reflect that the appropriate uncertainty is now that of a result, the appropriate computation is shown in Equation (52.15), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN 1  2 SR ¼ ðR RÞ i¼1 nð n 1Þ

ð52:15Þ

 is the average value of the result, and SR is the standard deviation of the means of where R the result. From the above discussion it can be seen that ASME ( 1998) provides two methods to evaluate the uncertainty of a result: a method employing Equation (52.14) that is essentially identical to the ISO GUM method employing Equation (52.5) which is applicable to measurement procedures that produce a single result, and a method employing Equation (52.15) that treats multiple results as independent measurements. 52.3.2.5 Total Uncertainty The ASME (1998) method combines the systematic and random components of uncertainty to produce the total uncertainty. Note that since the systematic component of uncertainty was developed with the presumption of a 95% level of confidence, the random component needs to be treated in an identical manner. The value of total uncertainty is obtained from Equation (52.16),

U 95

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 B þ S2x ¼2 2

ð52:16Þ

where U95 is the total uncertainty expressed at the 95% level of confidence and B and Sx are defined in Equations (52.9) and (52.12), respectively.

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In the case of a single result, the appropriate computation is given by Equation (52.17),

U 95

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 BR ¼2 þ S2R 2

ð52:17Þ

where BR is computed using Equation (52.10) and SR is computed using Equation (52.13). In the case of multiple results, Equation (52.17) is still used, but with SR from Equation (52.15) taking the place of SR.

52.4 DISCUSSION As mentioned in the Section 52.1, uncertainty is a property of a measurement. Its value allows the engineer or scientist to quantify the expected level of dispersion in the quantity being measured. It is an important parameter for quantitatively determining the suitability of a measurement technique. The two methods of evaluating or computing uncertainty presented here are consistent with each other while each possessing its own particular advantages. The ISO approach has the advantage of ease of application and flexibility. All uncertainties are treated identically, allowing for simplicity in approach (the distinction between Types A and B uncertainties is only made for purposes of discussion). Indeed, combined uncertainty for any measurement can be computed with only two equations: Equations (52.1) and (52.4a). Further, the experimentalist may select an appropriate coverage factor k, or simply report the combined uncertainty uc and allow the end user of the measurement to determine the suitability of the measurement. However, the use of the standard uncertainty as the building block of this approach requires that the user have at least a basic knowledge of statistics and probability distributions to properly compute this value, particularly in Type B cases where the uncertainty data may only be available in a nonstandard or ambiguous format. The ASME approach requires that the sources of measurement uncertainty be sorted into systematic and random categories. There are also more equations involved, each applying to a specific measurement method. The coverage factor is set at 2, effectively fixing the level of confidence at 95%. One key advantage of the ASME approach is that it provides a simplified method for the computation of the random component of uncertainty in multipletest cases. Another advantage, as noted in (Coleman and Steele, 1999), is that the ASME approach allows for the separate consideration of systematic and random components of uncertainty separately. As such, it allows the experimentalist to compare the relative contributions of each source of uncertainty, and make design or procedure decisions (such as to obtain a more reliable sensor to lower the systematic uncertainty or to proceed with the same sensor but to increase the sample size to reduce the random uncertainty).

DISCLAIMER The views and opinions expressed in this chapter are entirely those of the author and are not to be considered official statements of policies of the U.S. Government or any of its agencies.

REFERENCES

1921

REFERENCES ASME, Test Uncertainty: Performance Test Code 19.1-1998, New York: The American Society of Mechanical Engineers; 1998. Bragg G. Principles of Experimentation and Measurement, New Jersey: Prentice Hall, Englewood Cliffs; 1974. Coleman H, Steele W. Experimentation and Uncertainty Analysis for Engineers, 2nd Edition, New York: John Wiley & Sons; 1999. Dieck R. Measurement Uncertainty: Methods and Applications, 4th Edition, Research Triangle Park, North Carolina: The Instrumentation, Systems, and Automation Society; 2007. Figliola R, Beasely D. Theory and Design for Mechanical Measurement 3rd Edition, New York: John Wiley and Sons; 2000. Hibbeler R. Principles of Statics, 10th Edition, New York: Prentice Hall; 2005. ISO, Guide to the Expression of Uncertainty in Measurement, Geneva: International Organization for Standardization; 1995. Kirkup L, Frenkel R. An Introduction to Uncertainty in Measurement, Cambridge: Cambridge University Press; 2006. Rabinovich S. Measurement Errors: Theory and Practice, New York: American Institute of Physics; 1995. Taylor B, Kuyatt C. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, National Institute of Standards and Technology. 1994. Wheeler, A, Ganji, A. Introduction to Engineering Experimentation3rd Edition, New York: Prentice Hall; 2010.

53 MEASUREMENTS E. L. HIXSON and E. A. RIPPERGER 53.1 Standards and accuracy 53.1.1 Standards 53.1.2 Accuracy and precision 53.1.3 Sensitivity and resolution 53.1.4 Linearity 53.2 Impedance concepts 53.3 Error analysis 53.3.1 Internal estimates 53.3.2 Use of normal distribution to calculate probable error in X 53.3.3 External estimates References

53.1 STANDARDS AND ACCURACY 53.1.1 Standards Measurement is the process by which a quantitative comparison is made between a standard and a measurand. The measurand is the particular quantity of interest—the thing that is to be quantified. The standard of comparison is of the same character as the measurand, and so far as mechanical engineering is concerned the standards are defined by law and maintained by the National Institute of Standards and Technology (NIST, formerly known as the National Bureau of Standards). The four independent standards which have been defined are length, time, mass, and temperature (Wildhack, 1961). All other standards are derived from these four. Before 1960, the standard for length was the international prototype meter, kept at Sevres, France. In 1960, the meter was redefined as 1,650,763.73 wavelengths of krypton light. Then in 1983, at the Seventeenth General Conference on Weights and Measures, a new standard was adopted: A meter is the distance traveled in a

Handbook of Measurement in Science and Engineering. Edited by Myer Kutz. Copyright Ó 2013 John Wiley & Sons, Inc.

1923

1924

MEASUREMENTS

vacuum by light in 1/299,792,458 s (Giacomo, 1984). However, there is a copy of the international prototype meter, known as the national prototype meter, kept by NIST. Below that level, there are several bars known as national reference standards and below that there are the working standards. Interlaboratory standards in factories and laboratories are sent to NIST for comparison with the working standards. These interlaboratory standards are the ones usually available to engineers. Standards for the other three basic quantities have also been adopted by NIST, and accurate measuring devices for those quantities should be calibrated against those standards. The standard mass is a cylinder of platinum–iridium, the international kilogram, also kept at Sevres, France. It is the only one of the basic standards that is still established by a prototype. In the United States, the basic unit of mass is the basic prototype kilogram No. 20. Working copies of this standard are used to determine the accuracy of interlaboratory standards. Force is not one of the fundamental quantities, but in the United States the standard unit of force is the pound, defined as the gravitational attraction for a certain platinum mass at sea level and 45 latitude. Absolute time, or the time when some event occurred in history, is not of much interest to engineers. Engineers are more likely to need to measure time intervals, that is, the time between two events. The basic unit for time measurements is the second. At one time, the second was defined as 1/86,400 of the average period of rotation of the Earth on its axis, but that is not a practical standard. The period varies and the Earth is slowing up. Consequently a new standard based on the oscillations associated with a certain transition within the cesium atom was defined and adopted. That standard, the cesium clock, has now been superceded by the cesium fountain atomic clock as the primary time and frequency standard of the United States (NIST-F1 Cesium Fountain Atomic Clock). Although this cesium “clock” is the basic frequency standard, it is not generally usable by mechanical engineers. Secondary standards such as tuning forks, crystals, electronic oscillators, and so on are used, but from time to time access to time standards of a higher order of accuracy may be required. To help meet these requirements, NIST broadcasts 24 h per day, 7 days per week time and frequency information from radio stations WWV, WWVB, and WWVL located in Fort Collins, Colorado, and WWVH located in Hawaii. Other nations also broadcast timing signals. For details on the time signal broadcasts, potential users should consult NIST (NIST Time and Frequency Services, 2002). Temperature is one of four fundamental quantities in the international measuring system. Temperature is fundamentally different in nature from length, time, and mass. It is an intensive quantity, whereas the others are extensive. Join together two bodies that have the same temperature and you will have a larger body at that same temperature. If you join two bodies which have a certain mass, you will have one body of twice the mass of the original body. Two bodies are said to be at the same temperature if they are in thermal equilibrium. The international practical temperature scale, adopted in 1990 (ITS-90) by the International Committee on Weights and Measurement is the one now in effect and the one with which engineers are primarily concerned. In this system, the kelvin (K) is the basic unit of temperature. It is 1/273.16 of the temperature at the triple point of water, the temperature at which the solid, liquid, and vapor phases of water exist in equilibrium (Bentley, 1998). Degrees celsius ( C) is related to kelvin by the equation t¼T

273:15

where t is the degrees Celsius and T is the kelvin.

STANDARDS AND ACCURACY

1925

53.1.2 Accuracy and Precision In measurement practice, four terms are frequently used to describe an instrument. They are accuracy, precision, sensitivity, and linearity. Accuracy, as applied to an instrument, is the closeness with which a reading approaches the true value. Since there is some error in every reading, the “true value” is never known. In the discussion of error analysis which follows, methods of estimating the “closeness” with which the determination of a measured value approaches the true value will be presented. Precision is the degree to which readings agree among themselves. If the same value is measured many times and all the measurements agree very closely, the instrument is said to have a high degree of precision. It may not, however, be a very accurate instrument. Accurate calibration is necessary for accurate measurement. Measuring instruments must, for accuracy, be from time to time compared to a standard. These will usually be laboratory or company standards which are in turn compared from time to time with a working standard at NIST. This chain can be thought of as the pedigree of the instrument, and the calibration of the instrument is said to be traceable to NIST. 53.1.3 Sensitivity and Resolution These two terms, as applied to a measuring instrument, refer to the smallest change in the measured quantity to which the instrument responds. Obviously the accuracy of an instrument will depend to some extent on the sensitivity. If, for example, the sensitivity of a pressure transducer is 1 kPa, any particular reading of the transducer has a potential error of at least 1 kPa. If the readings expected are in the range of 100 kPa and a possible error of 1% is acceptable, then the transducer with a sensitivity of 1 kPa may be acceptable, depending on what other sources of error may be present in the measurement. A highly sensitive instrument is difficult to use. Therefore, a sensitivity significantly greater than that necessary to obtain the desired accuracy is no more desirable than one with insufficient sensitivity. Many instruments today have digital readouts. For such instruments, the concepts of sensitivity and resolution are defined somewhat differently than they are for analog-type instruments. For example, the resolution of a digital voltmeter depends on the “bit” specification and the voltage range. The relationship between the two is expressed by the equation R¼

V ; 2n

where R is the resolution in volts, V is the voltage range, and n is the number of bits. Thus, an 8-bit instrument on a 1-V scale would have a resolution of 1/256, or 0.004, volt. On a 10-V scale that would increase to 0.04 V. As in analog instruments, the higher the resolution, the more difficult it is to use the instrument, so if the choice is available, one should use the instrument which just gives the desired resolution and no more. 53.1.4 Linearity The calibration curve for an instrument does not have to be a straight line. However, conversion from a scale reading to the corresponding measured value is most convenient if it can be done by multiplying by a constant rather than by referring to a nonlinear

1926

MEASUREMENTS

calibration curve or by computing from an equation. Consequently instrument manufacturers generally try to produce instruments with a linear readout, and the degree to which an instrument approaches this ideal is indicated by its linearity. Several definitions of linearity are used in instrument specification practice (Doebelin, 2004). The so-called independent linearity is probably the most commonly used in specifications. For this definition, the data for the instrument readout versus the input are plotted and then a “best straight line” fit is made using the method of least squares. Linearity is then a measure of the maximum deviation of any of the calibration points from this straight line. This deviation can be expressed as a percentage of the actual reading or a percentage of the full-scale reading. The latter is probably the most commonly used, but it may make an instrument appear to be much more linear than it actually is. A better specification is a combination of the two. Thus, linearity equals þA percent of reading or þB percent of full scale, whichever is greater. Sometimes the term independent linearity is used to describe linearity limits based on actual readings. Since both are given in terms of a fixed percentage, an instrument with A percent proportional linearity is much more accurate at low reading values than an instrument with A percent independent linearity. It should be noted that although specifications may refer to an instrument as having A percent linearity, what is really meant is A percent nonlinearity. If the linearity is specified as independent linearity, the user of the instrument should try to minimize the error in readings by selecting a scale, if that option is available, such that the actual reading is close to full scale. A reading should never be taken near the low end of a scale if it can possibly be avoided. For instruments that use digital processing, linearity is still an issue since the analogto-digital converter used can be nonlinear. Thus, linearity specifications are still essential.

53.2 IMPEDANCE CONCEPTS Two basic questions which must be considered when any measurement is made are how has the measured quantity been affected by the instrument used to measure it? Is the quantity the same as it would have been had the instrument not been there? If the answers to these questions are no, the effect of the instrument is called loading. To characterize the loading, the concepts of stiffness and input impedance are used (Harris and Piersol, 2002). At the input of each component in a measuring system there exists a variable qi1 which is the one we are primarily concerned with in the transmission of information. At the same point, however, there is associated with qi1 another variable qi2 such that the product qi1qi2 has the dimensions of power and represents the rate at which energy is being withdrawn from the system. When these two quantities are identified, the generalized input impedance Zgi can be defined by Z gi ¼

qi1 qi2

ð53:1Þ

if qi1 is an effort variable. The effort variable is also sometimes called the across variable. The quantity qi2 is called the flow variable or through variable. In the dynamic case, these variables can be represented in the frequency domain by their Fourier transform. Then the quantity Z is a complex number. The application of these concepts is illustrated by the example in Figure 53.1. The output of the linear network in the black box

IMPEDANCE CONCEPTS

1927

FIGURE 53.1 Application of Thevenin’s theorem.

(Figure 53.1a) is the open-circuit voltage Eo until the load ZL is attached across the terminals A–B. If Thevenin’s theorem is applied after the load ZL is attached, the system in Figure 53.1b is obtained. For that system the current is given by im ¼

Eo Z AB þ Z L

ð53:2Þ

and the voltage EL across ZL is EL ¼ im Z L ¼

Eo Z L Z AB þ Z L

or EL ¼

Eo : 1 þ Z AB =Z L

ð53:3Þ

Equations (53.2–53.7) are frequency-domain equations. In a measurement situation, EL would be the voltage indicated by the voltmeter, ZL would be the input impedance of the voltmeter, and ZAB would be the output impedance of the linear network. The true output voltage, Eo, has been reduced by the voltmeter, but it can be computed from the voltmeter reading if ZAB and ZL are known. From Equation (53.3) it is seen that the effect of the voltmeter on the reading is minimized by making ZL as large as possible. If the generalized input and output impedances Zgi and Zgo are defined for nonelectrical systems as well as electrical systems, Equation (53.3) can be generalized to qim ¼

qiu 1 þ Z go =Z gi

ð53:4Þ

where qim is the measured value of the effort variable and qiu is the undisturbed value of the effort variable. The output impedance Zgo is not always defined or easy to determine; consequently Zgi should be large. If it is large enough, knowing Zgo is unimportant.

1928

MEASUREMENTS

If qi1 is a flow variable rather than an effort variable (current is a flow variable, voltage an effort variable), it is better to define an input admittance Y gi ¼

qi1 qi2

ð53:5Þ

rather than the generalized input impedance Z gi ¼

effort variable : flow variable

The power drain of the instrument is P ¼ qi1 qi2 ¼

q2i2 : Y gi

ð53:6Þ

Hence, to minimize power drain, Ygi must be large. For an electrical circuit Im ¼

Iu ; 1 þ Y o =Y i

ð53:7Þ

where Im is the measured current, Iu is the actual current, Yo is the output admittance of circuit, and Yi is the input admittance of meter. When the power drain is zero and the deflection is zero, as in structures in equilibrium, for example, when deflection is to be measured, the concepts of impedance and admittance are replaced with the concepts of static stiffness and static compliance. Consider the idealized structure in Figure 53.2. To measure the force in member K2, an elastic link with a spring constant Km is inserted in series with K2. This link would undergo a deformation proportional to the force in K2. If the link is very soft in comparison with K1, no force can be transmitted to K2. On the other hand, if the link is very stiff, it does not affect the force in K2 but it will not provide a very good measure of the force. The measured variable is an effort variable, and in general, when it is measured, it is altered somewhat. To apply the impedance concept, a flow variable whose product with the effort variable gives power is selected. Thus, Flow variable ¼

power : effort variable

FIGURE 53.2 Idealized elastic structure.

IMPEDANCE CONCEPTS

1929

Mechanical impedance is then defined as force divided by velocity, or Z¼

force ; velocity

where force and velocity are dynamic quantities represented by their Fourier transform and Z is a complex number. This is the equivalent of electrical impedance. However, if the static mechanical impedance is calculated for the application of a constant force, the impossible result Z¼

force ¼1 0

is obtained. This difficulty is overcome if energy rather than power is used in defining the variable associated with the measured variable. In that case, the static mechanical impedance becomes the stiffness:

In structures

Stiffness ¼ Sg ¼ R

Sg ¼

effort : flow dt

effort variable : displacement

When these changes are made, the same formulas used for calculating the error caused by the loading of an instrument in terms of impedances can be used for structures by inserting S for Z. Thus qim ¼

qiu ; 1 þ Sgo =Sgi

ð53:8Þ

where qim is the measured value of effort variable, qiu is the undisturbed value of effort variable, Sgo is the static output stiffness of measured system, and Sgi is the static stiffness of measuring system. For an elastic force-measuring device such as a load cell Sgi is the spring constant Km. As an example, consider the problem of measuring the reactive force at the end of a propped cantilever beam, as in Figure 53.3.

FIGURE 53.3 Measuring the reactive force at the tip.

1930

MEASUREMENTS

According to Equation (53.8), the force indicated by the load cell will be Fm ¼ Sgi ¼ K m

Fu 1 þ Sgo =Sgi and

Sgo ¼

3EI : L3

The latter is obtained by noting that the deflection at the tip of a tip-loaded cantilever is given by d¼

PL3 : 3EI

The stiffness is the quantity by which the deflection must be multiplied to obtain the force producing the deflection. For the cantilever beam Fm ¼

Fu 1 þ 3EI=K m L3

ð53:9Þ

or   3EI Fu ¼ Fm 1 þ : K m L3

ð53:10Þ

Clearly, if K m  3EI=L3 , the effect of the load cell on the measurement will be negligible. To measure displacement rather than force, the concept of compliance is introduced and defined as

Then

Cg ¼ R

flow variable : effort variable dt

qm ¼

qu : 1 þ C go =Cgi

ð53:11Þ

If displacements in an elastic structure are considered, the compliance becomes the reciprocal of stiffness, or the quantity by which the force must be multiplied to obtain the displacement caused by the force. The cantilever beam in Figure 53.4 again provides a simple illustrative example. If the deflection at the tip of this cantilever is to be measured using a dial gage with a spring constant Km, C gi ¼

1 Km

and C go ¼

L3 : 3EI

ERROR ANALYSIS

1931

FIGURE 53.4 Measuring the tip deflection.

Thus,   K m L3 dm ¼ du 1 þ 3EI

ð53:12Þ

Not all interactions between a system and a measuring device lend themselves to this type of analysis. A pitot tube, for example, inserted into a flow field distorts the flow field but does not extract energy from the field. Impedance concepts cannot be used to determine how the flow field will be affected. There are also applications in which it is not desirable for a force-measuring system to have the highest possible stiffness. A subsoil pressure gage is an example. Such a gage, if it is much stiffer than the surrounding soil, will take a disproportionate share of the total load and will consequently indicate a higher pressure than would have existed in the soil if the gage had not been there. 53.3 ERROR ANALYSIS It may be accepted as axiomatic that there will always be errors in measured values. Thus, if a quantity X is measured, the correct value q, and X will differ by some amount e. Hence, ðq

XÞ ¼ e

or q ¼ X  e:

ð53:13Þ

It is essential, therefore, in all measurement work that a realistic estimate of e be made. Without such an estimate the measurement of X is of no value. There are two ways of estimating the error in a measurement. The first is the external estimate, or D E, where D ¼ e/q. This estimate is based on knowledge of the experiment and measuring equipment and to some extent on the internal estimate D I. The internal estimate is based on an analysis of the data using statistical concepts. 53.3.1 Internal Estimates If a measurement is repeated many times, the repeat values will not, in general, be the same. Engineers, it may be noted, do not usually have the luxury of repeating

1932

MEASUREMENTS

measurements many times. Nevertheless the standardized means for treating results of repeated measurements are useful, even in the error analysis for a single measurement (Cook and Rabinowicz, 1963). If some quantity is measured many times and it is assumed that the errors occur in a completely random manner, that small errors are more likely to occur than large errors, and that errors are just as likely to be positive as negative, the distribution of errors can be represented by the curve FðXÞ ¼

Y o e ðX 2s 2

UÞ

ð53:14Þ

;

where F(X) is the number of measurements for a given value of (X U), Yo is the maximum height of curve or number of measurements for which X ¼ U, and U is the value of X at point where maximum height of curve occurs s determines lateral spread of the curve. This curve is the normal, or Gaussian, frequency distribution. The area under the curve between X and dX represents the number of data points which fall between these limits and the total area under the curve denotes the total number of measurements made. If the normal distribution is defined so that the area between X and X þ dX is the probability that a data point will fall between those limits, the total area under the curve will be unity and

FðXÞ ¼

exp

and Px ¼

Z

exp

ðX UÞ2 =2s 2 pffiffiffiffiffiffiffi s 2P ðX UÞ2 =2s 2 pffiffiffiffiffiffiffi dx: s 2P

ð53:15Þ

ð53:16Þ

Now if U is defined as the average of all the measurements and s as the standard deviation,

s¼

"P

ðX UÞ2 N

#1=2

;

ð53:17Þ

where N is the total number of measurements. Actually this definition is used as the best estimate for a universe standard deviation, that is, for a very large number of measurements. For smaller subsets of measurements the best estimate of s is given by

s¼

P

ðX UÞ2 n 1

!1=2

;

ð53:18Þ

where n is the number of measurements in the subset. Obviously, the difference between the two values of s becomes negligible as n becomes very large (or as n ! N). The probability curve based on these definitions is shown in Figure 53.5.

ERROR ANALYSIS

1933

FIGURE 53.5 Probability curve.

The area under this curve between s and þs is 0.68. Hence, 68% of the measurements can be expected to have errors that fall in the range of d. Thus, the chances are 68=32, or better than 2 to 1, that the error in a measurement will fall in this range. For the range 2s the area is 0.95. Hence, 95% of all the measurement errors will fall in this range and the odds are about 20:1 that a reading will be within this range. The odds are about 384:1 that any given error will be in the range of 3s. Some other definitions related to the normal distribution curve are as follows: 1. Probable Error: The error likely to be exceeded in half of all the measurements and not reached in the other half of the measurements. This error in Figure 53.5 is about 0.67d. 2. Mean Error: The arithmetic mean of all the errors regardless of sign. This is about 0.8s. 3. Limit of Error: The error that is so large it is most unlikely ever to occur. It is usually taken as 4d. 53.3.2 Use of Normal Distribution to Calculate Probable Error in X The foregoing statements apply strictly only if the number of measurements is very large. Suppose that n measurements have been made. That is a sample of n data points out of an infinite number. From that sample U and s are calculated as above. How good are these numbers? To determine that, we proceed as follows. Let U ¼ FðX 1 ; X 2 ; X 3 ; . . . ; X n Þ ¼ eu ¼

X @F

@X i

P

Xi n

ð53:19Þ ð53:20Þ

exi ;

where eu is the error in U, exi is the error in Xi ðeu Þ2 ¼

X  @F

@X i

exi

2

þ

X  @F

dX i

exi



 dF exj ; dX j

ð53:21Þ

1934

MEASUREMENTS

where I 6¼ j. If the errors ei to en are independent and symmetrical, the cross-product terms will tend to disappear and 2

ðeu Þ ¼

X  @F

@X i

2

:

ð53:22Þ

#1=2

ð53:23Þ

exi

Since @F=@X i ¼ 1=n, eu ¼

"

X 12 n

e2xi

or "  #1=2 1 2X 2 ðexi Þ eu ¼ n

ð53:24Þ

from the definition of s

and

X

ðexi Þ2 ¼ ns 2

ð53:25Þ

s eu ¼ pffiffiffi : n

This equation must be corrected because the real errors in X are not known. If the number n were to approach infinity, the equation would be correct. Since n is a finite number, the corrected equation is written as eu ¼

s ðn

ð53:26Þ

1Þ1=2

and q¼U

s ðn

1Þ1=2

:

ð53:27Þ

This says that if one reading is likely to differ from the true value by an amount s, then the average of 10 readings will be in error by only s=3 and the average of 100 readings will be in error by s=10. To reduce the error by a factor of 2, the number of readings must be increased by a factor of 4. 53.3.3 External Estimates In almost all experiments several steps are involved in making a measurement. It may be assumed that in each measurement there will be some error, and if the measuring devices

ERROR ANALYSIS

1935

are adequately calibrated, errors are as likely to be positive as negative. The worst condition insofar as accuracy of the experiment is concerned would be for all errors to have the same sign. In that case, assuming the errors are all much less than 1, the resultant error will be the sum of the individual errors, that is DE ¼ D1 þ D2 þ D3 þ




ð53:28Þ

It would be very unusual for all errors to have the same sign. Likewise, it would be very unusual for the errors to be distributed in such a way that D E ¼ 0: A general method follows for treating problems that involve a combination of errors to determine what error is to be expected as a result of the combination. Suppose that V ¼ Fða; b; c; d; e;


; x; y; zÞ;

ð53:29Þ

where a; b; c;


x; y; z represent quantities which must be individually measured to determine V. Then dV ¼

X @F  dn @n

DE ¼

X @F 

and

@n

ð53:30Þ

en :

The sum of the squares of the error contributions is given by e2E ¼

X   2 @F en : @n

ð53:31Þ

Now, as in the discussion of internal errors, assume that errors en are independent and symmetrical. This justifies taking the sum of the cross products as zero: X @F  @F  @m

en em ¼ 0

ðD E Þ2 ¼

X @F 2

@n

n 6¼ m:

Hence,

@n

e2n

ð53:32Þ

1936

MEASUREMENTS

or eE ¼

"

X @F 2 @n

e2n

#1=2

:

ð53:33Þ

This is the most probable value of eE. It is much less than the worst case: D e ¼ ½jD a j þ jD b j þ jD c j


þ jD z jŠ:

ð53:34Þ

As an application, the determination of g, the local acceleration of gravity, by use of a simple pendulum will be considered: g¼

4P2 L ; T2

ð53:35Þ

where L is the length of pendulum and T is the period of pendulum. If an experiment is performed to determine g, the length L and the period T would be measured. To determine how the accuracy of g will be influenced by errors in measuring L and T write @g 4P2 ¼ 2 @L T

and

@g ¼ @T

8P2 L : T3

ð53:36Þ

The error in g is the variation in g written as follows: dg ¼



   @g @g DL þ DT @L @T

ð53:37Þ

or dg ¼

 2 4P DL T2

 2  8P L DT: T3

ð53:38Þ

It is always better to write the errors in terms of percentages. Consequently Equation (53.38) is rewritten as dg ¼

ð4P2 L=T 2 ÞDL L

2ð4P2 L=T 2 ÞDT T

ð53:39Þ

2DT T

ð53:40Þ

or dg DL ¼ g L Then eg ¼ ½e2L þ ð2eT Þ2 Š1=2 ;

ð53:41Þ

ERROR ANALYSIS

1937

where eg is the most probable error in the measured value of g. That is, g¼

4P2 L  eg ; T2

ð53:42Þ

where L and T are the measured values. Note that even though a positive error in T causes a negative error in the calculated value of g, the contribution of the error in T to the most probable error is taken as positive. Note also that an error in T contributes four times as much to the most probable error as an error in L contributes. It is fundamental in measurements of this type that those quantities which appear in the functional relationship raised to some power greater than unity contribute more heavily to the most probable error than other quantities and must, therefore, be measured with greater care. The determination of the most probable error is simple and straightforward. The question is how are the errors, such as DL/L and DT/T, determined. If the measurements could be repeated often enough, the statistical methods discussed in the internal error evaluation could be used to arrive at a value. Even in that case it would be necessary to choose some representative error such as the standard deviation or the mean error. Unfortunately, as was noted previously, in engineering experiments it usually is not possible to repeat measurements enough times to make statistical treatments meaningful. Engineers engaged in making measurements will have to use what knowledge they have of the measuring instruments and the conditions under which the measurements are made to make a reasonable estimate of the accuracy of each measurement. When all of this has been done and a most probable error has been calculated, it should be remembered that the result is not the actual error in the quantity being determined but is, rather, the engineer’s best estimate of the magnitude of the uncertainty in the final result (Kline and McClintock, 1953; Taylor and Kuyatt, 1994). Consider again the problem of determining g. Suppose that the length L of the pendulum has been determined by means of a meter stick with 1-mm calibration marks and the error in the calibration is considered negligible in comparison with other errors. Suppose the value of L is determined to be 91.7 cm. Since the calibration marks are 1 mm apart, it can be assumed that DL is no greater than 0.5 mm. Hence the maximum DL ¼ 5:5  10 4 : L Suppose T is determined with the pendulum swinging in a vacuum with an arc of 5 using a stopwatch that has an inherent accuracy of one part in 10,000. (If the arc is greater than 5 , a nonisochronous swing error enters the picture.) This means that the error in the watch reading will be no more than 10 4 s. However, errors are introduced in the period determination by human error in starting and stopping the watch as the pendulum passes a selected point in the arc. This error can be minimized by selecting the highest point in the arc because the pendulum has zero velocity at that point and timing a large number of swings so as to spread the error out over that number of swings. Human reaction time may vary from as low as 0.2 s to as high as 0.7 s. A value of 0.5 s will be assumed. Thus the estimated maximum error in starting and stopping the watch will be 1 s (0.5 s at the start and 0.5 s at the stop). A total of 100 swings will be timed. Thus, the estimated maximum error in the period will be 1/100 s. If the period is determined to be 1.92 s, the estimated maximum error will be 0.01/1.92 ¼ 0.005. Compared to this, the

1938

MEASUREMENTS

error in the period due to the inherent inaccuracy of the watch is negligible. The nominal value of g calculated from the measured values of L and T is 982.03 cm/s2. The most probable error (Equation 53.29) is h i1=2 4ð0:005Þ2 þ ð5:5  10 4 Þ2 ¼ 0:01: ð53:43Þ

The uncertainty in the value of g is then 9.82 cm/s2, or in other words the value of g will be somewhere between 972.21 and 991.85 cm/s2. Often it is necessary for the engineer to determine in advance how accurately the measurements must be made in order to achieve a given accuracy in the final calculated result. For example, in the pendulum problem it may be noted that the contribution of the error in T to the most probable error is more than 300 times the contribution of the error in the length measurement. This suggests, of course, that the uncertainty in the value of g could be greatly reduced if the error in T could be reduced. Two possibilities for doing this might be (1) find a way to do the timing that does not involve human reaction time or (2) if that is not possible, increase the number of cycles timed. If the latter alternative is selected and other factors remain the same, the error in T timed over 200 swings is 1/200 or 0.005, second. As a percentage the error is 0.005/1.92 ¼ 0.0026. The most probable error in g then becomes eg ¼ ½4  ð2:6  10 3 Þ2 þ ð5:5  10 4 Þ2 Š1=2 ¼ 0:005:

ð53:44Þ

This is approximately half of the most probable error in the result obtained by timing just 100 swings. With this new value of eg the uncertainty in the value of g becomes 4.91 cm/s2 and g then can be said to be somewhere between 977.12 and 986.94 cm/s2. The procedure for reducing this uncertainty still further is now self-evident. Clearly, the value of this type of error analysis depends on the skill and objectivity of the engineer in estimating the errors in the individual measurements. Such skills are acquired only by practice and careful attention to all the details of the measurements.

REFERENCES Wildhack WA. NBS source of American standards. ISA Journal 1961;8(2). Giacomo P. News from the IBPM. Metrologia 1984;20(1):171. NIST Time and Frequency Services, NIST Special Publication 432;2002. Bentley RE. editor. Handbook of Temperature Measurement. CSIRO Springer; 1998. Doebelin EA. Measurement Systems–Application and Design. 5th ed. New York: McGraw Hill; 2004. p 85–91. Harris CM, Piersol AG. Mechanical impedance. In: Shock and Vibration Handbook, Chap. 10. 5th ed. New York: McGraw-Hill; 2002. p 10.1–10.14. Cook NH, Rabinowicz E. Physical Measurement and Analysis. Reading (MA): Addison Wesley; 1963. p 29–68. Kline SJ, McClintock FA. Describing uncertainties in single sample experiments. Mechanical Engineering 1953;75(3):3–8. Taylor BN, Kuyatt CE.Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297;1994.