Elements of Engineering Statics 9780231881470

Citation preview

E L E M E N T S OF E N G I N E E R I N G STATICS

ELEMENTS OF E N G I N E E R I N G STATICS

by H. Deresìewicz A s s o c i a t e P r o f e s s o r of M e c h a n i c a l E n g i n e e r i n g Columbia University

NEW COLUMBIA

YORK

UNIVERSITY

PRESS

1V

Copyright ©

1957, 1958, 1959,

Columbia University P r e s s , New York First Second

printing

complete

1957

printing

1959

Manufactured in the United States of America

V

PREFACE

The present volume contains e s s e n t i a l l y the material d i s c u s s e d in the course on Engineering Statics given to pre-engineering students at Columbia University. It i s assumed that the student has a working knowledge of elementary differential c a l c u l u s and i s concurrently studying integral c a l c u l u s . The first chapter d e a l s with the algebra of vectors, considered at first without reference to subsequent applications. The purpose i s to instill in the student a facility in manipulating vectors and thinking in terms of vectors. The reason i s twofold: this approach will enhance understanding of some of the basic physical quantities with which the idea of direction is intrinsically associated, and it will simplify greatly the mathematical expression of such quantities and of the relations between them. This i s particularly needed in the subsequent course which d e a l s with kinematics and Newtonian dynamics of particles and rigid bodies. Chapter II i s devoted to consideration of the idea of a force and its moment, followed by a discussion of the concept of, and the conditions for, the equilibrium of a s i n g l e particle and of s y s t e m s of p a r t i c l e s . The conditions arrived at are then applied, in Chapter ID, to problems of equilibrium under concurrent, p a r a l l e l , and general planar force systems, particular emphasis being placed on the idea of the " f r e e - b o d y " diagram. Chapter IV offers a discussion of the concept of couples and of the equivalence of force systems and their reduction to simpler form. The techniques worked out in Chapter II and illustrated in Chapter III are applied, in Chapter V, to s p e c i f i c problems of s t a t i c a l l y determinate plane trusses and frames. Problems involving sliding friction are analyzed in Chapter VI. Chapter VII contains a definition of the concept of work done by a force, which l e a d s , in turn, to a d i s c u s s i o n of the principle of virtual work and its application to problems of s t a t i c equilibrium. Moraover, in this chapter the idea of a conservative system and i t s potential energy i s introduced and then employed in the a n a l y s i s of tiie stability of the equilibrium of snch a system. Chapter VIII h a s a brief exposition of simple spatial s y s t e m s . The volume concludes with an Appendix which

PREFACE

vi

deals

with the

concept

of the c e n t e r of gravity and i t s a p p l i c a t i o n to

problems of d i s t r i b u t e d l o a d i n g . Throughout

the

text theoretical

concepts are

i l l u s t r a t e d by s p e c i f i c

e x a m p l e s in a c c o r d a n c e with the primary p u r p o s e of the c o u r s e , which i s to f o s t e r u n d e r s t a n d i n g of the fundamental p r i n c i p l e s o f s t a t i c s and to d e v e l o p a f a c i l i t y in s o l v i n g p r o b l e m s a r i s i n g in e n g i n e e r i n g p r a c t i c e . I wish to e x p r e s s my gratitude to Mr. C . W. T h u r s t o n , of the D e p a r t m e n t of C i v i l E n g i n e e r i n g , for h i s v a l u a b l e s u g g e s t i o n s .

T o Professor J .

E.

E n g l u n d , of the D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g , my warmest t h a n k s are due for h i s e n c o u r a g e m e n t and support. Columbia U n i v e r s i t y New York, New York July, 1957

H. D e r e s i e w i c z

CONTENTS

I. E L E M E N T S O F V E C T O R A L G E B R A 1. Introduction 2. E q u a l i t y of V e c t o r s 3. Addition of V e c t o r s 4 . Subtraction of V e c t o r s 5. 6. 7. 8. 9. 10. 11.

Multiplication by S c a l a r s P a r a l l e l P r o j e c t i o n of V e c t o r s Unit Coordinate V e c t o r s S c a l a r Multiplication of V e c t o r s V e c t o r Multiplication of T w o V e c t o r s Moment of a V e c t o r Differentiation of V e c t o r s

II. T H E P R O B L E M O F E Q U I L I B R I U M 1. Introduction 2. Composition of F o r c e s and Moments of F o r c e s 3. Newton's L a w s for a P a r t i c l e 4 . Equilibrium

1 1 2 2 3 4 4 5 8 10 12 17 21 21 21 27 28

III. E Q U I L I B R I U M O F S I M P L E P L A N A R S Y S T E M S 1. Introduction 2 . F r e e - B o d y Diagram 3. Concurrent F o r c e s 4. Parallel Forces 5 . General Coplanar F o r c e s 6. S t a t i c D e t e r m i n a t e n e s s and I n d e t e r m i n a t e n e s s

32 32 32 34 37 38 39

IV. E Q U I V A L E N C E O F F O R C E S Y S T E M S 1. Introduction 2. C o u p l e s 3 . Reduction of a System of F o r c e s

41 41 42 43

vili

CONTENTS

V. S I M P L E S T R U C T U R E S 1. T r u s s e s

46

2. T h e Method of Joints

48

3. T h e Method of Sections

51

4. Complex T r u s s e s

52

5. Frames

54

VI. S L I D I N G F R I C T I O N

58

1. Introduction

58

2. T h e L a w s of Friction

58

3. A p p l i c a t i o n s of Coulomb's L a w s of Friction

60

VII. WORK A N D E N E R G Y METHODS

VIII.

46

65

1. Work

65

2. The Principle of Virtual Work (for a Free P a r t i c l e )

70

3. Ideal Constraints

70

4. T h e Principle of Virtual Work (for Systems)

73

5. Potential Energy

79

6. Stability

81

E Q U I L I B R I U M O F S I M P L E S P A T I A L SYSTEMS

86

Appendix.

C E N T E R OF GRAVITY

90

1. Static E q u i v a l e n c e of P a r a l l e l F o r c e s

90

2. Center of Gravity

91

3. Application to Problems Involving Distributed L o a d

94

PROBLEMS

97

chapter I. ELEMENTS OF VECTOR ALGEBRA 1. Introduction

The quantities with which we shall be concerned may be classified into vectors and scalars, depending on whether or not the idea of direction is associated with them. A quantity which is completely specified by a single number, positive or negative, without intrinsic reference to direction in space, is called a scalar. Such quantities as length and time belong to this category. A vector quantity, on the other hand, involves inherently the idea of direction as well as magnitude. For example, the position of a point B relative to another point A may be specified by means of a straight line drawn from A to B (Fig. 1). Moreover, it may equally well be specified by any equal and parallel straight line drawn in the same sense from, say, C to D, since the position of D relative to C is the same as that of B relative to A. A straight line regarded in this way as having a definite magnitude and direction, but no specific location in space, is called a vector — o c c a s i o n a l l y , to emphasize the latter property, a free vector. In short, a vector may be represented geometrically by a directed line segment. For example, "translation" of a body, accomplished by a displacement in which lines joining initial and final positions of various points of the body are all equal and parallel, is specified by a free vector which may be any one of these lines ( F i g . 2).

Fig. 2

In distinction, a sliding vector is one whose point of application is confined to any point on a given line; a bound vector is one with a unique point of application. In general, by the term vector, we will understand a free vector.

Notation: in print, vectors are denoted by boldface letters (e.g., P), scalars by italicized letters (e.g.,m); in manuscript, a variety of notations is employed for vectors, as for example letters with superscribed arrows ( e . g . , P ) or letters with a line doubled (e.g., P). It will be well

2

ELEMENTS OF VECTOR ALGEBRA

for the student to achieve a facility in writing vectors; this may be accomplished most readily by intensive practice. For convenience, sample Latin alphabets are given in capital and lower case letters.

/A lb jIk 'C >m mop + P2 + PI ,

Example 1: the angles Ot = 3 0 ° , /9 express the Given: P Hence, Further, or so that Therefore,

and

It i s known of a certain vector that i t s magnitude i s P = 100, it forms with the positive X and Y a x e s are, r e s p e c t i v e l y , = 7 0 ° , and it is located in the first octant. It is required to vector in terms of the unit coordinate vectors i , j , k . = 100, a = 3 0 ° , j8 = 7 0 ° , vector in first octant. I = cos a = 0.866,

m = c o s ¡3 = 0 . 3 4 2

I1 + m2 + n1 = 1, n 1 = 1 - I 1 - m1 = 0 . 1 3 3 n = cos y = + 0.365 Px = PI = 8 6 . 6 Py = Pm = 3 4 . 2 Pz = Pn = 3 6 . 5 P = 86.6 i + 34.2 j + 3 6 . 5 k .

To find the expression for a unit vector along P , P = Px i + Py \ + Pz k and P = P P„ . Hence, P

Pxi+Py\

+

Pzk

we recall that

8

E L E M E N T S OF V E C T O R A L G E B R A

or

P0«/i + mj+nk.

Example 2: Given a vector whose î , j , It components are of magnitude 5, 0, 12, r e s p e c t i v e l y ; find the magnitude and direction of this vector and the unit vector parallel to it. Given: Px « 5, Py « 0, P , - 1 2 . Hence,

P = sJPl + P$ + P\ = 1 3 . 5 Z = — «= — = 0 . 3 8 4 6 ; a-67°23'. P 13 F

r

m = y -

0;

/3 • 9 0

P 12 « = — = — = 0.9231; P 13 The corresponding unit vector is P„ = / i + m j + n k =

.

y = 22° 37'.

(5 i + 12 k ) .

It is easy to s e e that, if p ^ i + zVi + ^ k , Q = & i + Çyj+