Economics of Construction in Relation to Framed Structures

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ECONOMICS OF CONSTRUCTION IN

RELATiON TO

FRAMED STRUCTURES. BY

R. H.

BOW,

C.E., F.R.S.E.

FX?,ET_,I3VIIISr^. IRTST

PART

F

PA.KTS.

."" !

sst

IIAGR

WSmm mm

;

;,

CORNELL UNIVERSITY LIBRARY

ENGINEERING

TG

260.B782

Economics of construction

in relation

3 1924 004 138 123

to

Cornell University Library

The

original of this

book

is in

the Cornell University Library.

There are no known copyright

restrictions in

the United States on the use of the

text.

http://www.archive.org/details/cu31924004138123

ECONOMICS OF CONSTBUCTION IN RELATION TO

FRAMED STRUCTURES

ECONOMICS OF CONSTBUCTION IN RELATION TO

FEAMED STEUCTUEES

By EOBERT HSTBOW, [/BC

C.E., F.E.S.E.

PRELIMINARY PARTS

PAET I.— CLASSIFICATION OF STEUCTUEES PAET II.—DIAGEAMS OF FOECES

LONDON E. & F. N. SPON, 48 CHAEING CEOSS NEW YORK 446 BROOME STREET EDINBURGH ADAM & CHARLES BLACK :

:

:

1873

[AH rights

reserved.}

PKEFACE. The two Manual

Parts

now published

And

in themselves.

are calculated to form a useful

as they are almost altogether of

a preliminary character in relation to Economics of Construction, no introduction to that subject need here be attempted.

But

as touching

ing Science, I

upon the neglect of that branch of Engineer-

may quote

the following passage with reference

to British Engineering from Applied Mechanics by the late

eminent Professor W. J. Macquorn Rankine cases

we

see the strength

and the

stability

:

—" In too many

which ought to be

given by the skilful arrangement of the parts of a structure, supplied by means of clumsy massiveness, and of lavish ex-

penditure of material, labour, and money

;

and the

evil is

increased by a perversion of the public taste, which causes

works to be admired, not in proportion to their

fitness for

their purposes, or to the skill evinced in attaining that fitness,

but in proportion to their

Part

I. is

size

and

cost."

devoted to the Classification of Structures, but

the illustrations are, in the meantime, confined to

Bridge Trusses.

Roof and

Many previous attempts have been made by

others and myself in this direction, but not one of those that I

am

acquainted with

is

at all satisfactory.

1

trust that the

now given, although no doubt open to some will commend itself as based for the most part

classification

objections,

upon

A

scientific distinctions.

considerable

number of the designs supplied

trate the classification are original.

to illus-

PREFACE. Part

gives the application of the recently

II.

method of

arriving at the stresses in structures

diagrams of the has attained

is

is,

by drawing

The importance to which this method almost altogether due to Professor J. Clerk forces.

Maxwell, who has shown how surprisingly general tion

expanded

and who has placed

it

in quite a

new

its

applica-

aspect by his

discovery or detection of those diagrams of forces which bear

a reciprocal relationship to the relative framed structures.

When

I first

gave particular attention to the method,

was' with a view to preparing an appendix to the general I projected; but having devised an

it

work

improvement in the mode

was led on to devote a considerable amount of time to the subject ; and the notes and particulars of its application accumulating on my hands, I have thought them deserving of separate publication. of lettering the diagrams, I

Some

novelty

may

also

be claimed for the application of

the method to the imperfect structures constituting Class

7

South Gray Street, Edinburgh, October 1873.

II.

ECONOMICS OF CONSTRUCTION IN RELATION TO

FRAMED STRUCTURES

PART

I.

CLASSIFICATION OF

FRAMED STRUCTURES OR TRUSSES.

Note.

—In

my Treatise

on Bracing (1851), the term "Truss" was applied to outline, or which did not require any of the

any framed girder with a triangular

struts or ties to change their characters

tion of

its

under irregular loadings

meaning has not been adopted hy subsequent

;

hut

writers,

this restric-

and now the

term may be used to designate any form of framed structure employed to support a pressure or load over a void.

CLASSIFICATION OF TRUSSES. BETWEEN ROOF AND BRIDGE TRUSSES.

DISTINCTIONS

In roof chiefly

trusses the line or surface on which the loading

imposed

is,

from

its

office

is

of discharging rain and

snow, necessarily a more or less inclined one.

In ordinary

bridge trusses, on the contrary, the line of the roadway at

which the principal part of the loading mates to a perfect to

leTel.

if

situated approxi-

This important distinction leads

another yery marked difference

loading,

is

—a

roof truss and

its

treated as inverted, has no practical value, whereas

any good bridge truss becomes, when inverted, another useful design, the natures of the parts becoming simply reversed as

regards their strut or roofs

and bridges

is

tie action.

Another distinction between

that the former have to withstand import-

ant oblique forces arising from the action of the wind blowing in the direction of the plane of the truss against the inclined surfaces,

whereas the action of the wind in the same plane

in the case of bridge trusses is of

no account.

TIED AND ABUTTING STRUCTURES.

An

abutting truss

is

to be regarded as nearly in the

same

condition as the same structure supplied with a tie-bar and resting vertically

upon the supports, the

chief difference being

that in the abutting arrangement the distance between the

abutments, sometimes assumed theoretically as invariable, practically liable

to uncertain,

and

it

may be

is

important,

changes; whereas, when the horizontal thrust or spreading

tendency of the feet span may

is

resisted

by a

tie-bar, the

be calculated with some certainty.

change of

— CLASSIFICATION OF TRUSSES.

10

CLASSIFICATION OF FRAMED STRUCTURES OR TRUSSES. Class

I.

In tkis class are placed all framed structures or trusses which are theoretically perfect, or which are characterised by the following properties 1st.

:

be supposed to be so con-

If at each joint the parts

nected together as to leave them free to rotate as

though hinged, the frame

will, nevertheless,

from

—that

liability to distortion

is,

be free

no such rotation

at the joints can take place. 2d.

Any one

part of the structure

may be made

longer or

shorter, without thereby inducing stresses.

3d.

Slight irregularities in the lengths of the parts* arising

from imperfect workmanship,

elasticity, or

other cause, have no practical effect upon the distribution of the stresses.

Class includes those trusses which,

if

II.

the joints were

all

regarded

as hinged, could only maintain their shapes under equilibrat-

owe what powers they practically great a distortion when subjected to

ing loadings, and which possess to resist too

non-equilibrating loadings, to the stiffness of the joints, and to the transverse stiffness of the parts

when

distortion takes place.

deficiency of parts as

which become bent

In such structures there

compared with Class

is

a

I.

Class III.

comprises trusses in which there

is

a redundance of parts.

This redundancy renders the particular distribution of the *

This includes the span in abutting structures, liable to change from

various causes.

:

CLASSIFICATION OF TRUSSES. stresses fitting,

more or

less

dependent upon the peculiarities of the

and upon the

parts, arising

from

11

relative changes in the lengths of the

elasticity,

temperature,

teristic test of this class is that

etc.

The charac-

any one part cannot be made

longer or shorter, without thereby inducing longitudinal stresses in other parts of the structure.

Class IV. Into this Class

may be put

all

the defects of both the Classes

other trusses. II.

and

III.

These have

—that

is,

while

they present a redundance of parts in one region, a deficiency is

apparent at another, so that uncertain longitudinal stresses

and a non-equilibrating load makes a demand upon the transverse strength of some of the parts.

are introduced,

GROUPS, SUB-GROUPS, ARRANGEMENTS,

AND MODIFICATIONS. The designs these classes

for roofs

and bridges embraced by any of

may be grouped

according to the number of

sub-spans or stretches into which the whole span

and the

is

divided

trusses belonging to any group placed in sub-groups

according to the number of polygons (usually triangles) contained in them.

The sub-groups sometimes admit of sub-

division into arrangements of the parts ;*

and these again into

modifications, arising from variations in the inclinations of

the members. It is obvious that the

truss will

importance and complexity of the

be roughly indicated by the number of

its

sub-

spans ; and similarly the complexity of the diagram of forces will

be somewhat in proportion to the number of polygons

in the truss. * In the table of designs, the letters A, B, C, etc., indicating a succession of "Arrangements,'' are carried through each growp, instead of sub-group, since

many

sub-groups contain each but one arrangement of parts.

CLASSIFICATION OP TRUSSES.

12

Parts required merely to support the tie-bar, or give lateral stiffness to the long struts,

may be

occasionally indi-

cated, but these parts as usually treated do not appear in the diagram of forces, and should not in strictness be recognised

as affecting the

On

number of the polygons

in the truss.

pages 17 to 39 will be found a collection of designs

arranged according to this

Those

classification.

for

which

diagrams of forces are afterwards given are distinguished by

an

asterisk,

and the same reference numbers

will

be retained

in the lithographed plates of Part II.

CONVERSION TO OTHER CLASSES OF SOME STRUCTURES

BELONGING TO CLASS

When

one or more quadrilateral openings occur in a truss

belonging to Class it

may

II.

II.,

as in Nos. 139, 149, 150, 157,

frequently be reduced to Class

I.

etc.,

by subdividing one

of the quadrilaterals into two triangles by means of a diagonal piece capable of acting in turn as a strut or

tie,

as the changes

in the distribution of the loading or other external forces

require

;

but this addition

shown by

Figs. 11, 17,

may sometimes

and 19;

as

may

prove unsightly, as

an alternative we may

frequently divide the quadrilateral into three triangles, as in Figs. 13, 18, 30, 33, without introducing a redundance of parts.

The most usual mode of

treating

it

is,

however, to

supply two diagonal pieces, as in Figs. 167, 168, 169, 185,

but this at once places the truss in Class

III., for

though

might be assumed that each piece could only act as a as a strut,

and the one only when the other was

etc.,

tie,

it

or

inactive, yet

practically a slight misfitting of the lengths of the pieces

would render such assumptions

incorrect,

and even though

accurately fitted, under one degree of severity of the stresses,

a change in the loading might undo the adjustment.

— CLASSIFICATION OF TRUSSES.

I

13

SOME TYPES OP DESIGNS ARE COMMON

MANY

TO

Some

of the

GROUPS.

modes of arranging the

parts of a truss are

suitable for a variety of divisions of the span

but it will not be necessary to repeat each such type of design in the illustrated lists for every

group to which

it

;

may be applicable.

The following may be noted

as

such general designs

Those in which a brace radiates

:

1st,

some of the best examples of

from a central point in the truss to each otherwise unsupported point, as in Figs. 21 to 28, 63 to 66, 71 and 72,

among

roofs

;

and

Figs. 205, 213,

2d, Designs in which the braces,

and 223, among

etc.,

bridges.

radiate from two points

in the truss, as in Nbs. 40, 74, 106, 230, etc.

And 3c?, Those

having the braces arranged in zig-zag fashion, and dividing the trass into the simplest chain of triangles, as in Figs. 68,

among roofs; and Figs. 258, 259, 260, 327, among bridges. The merits of this last type of designs were little understood

85, 98, 105, etc., etc.,

before the publication of

my

Treatise on Bracing, in January

1851, which was almost wholly devoted to their elucidation.

In

its

simplest form this arrangement of parts had been

patented by Nash, Warren, state of the lattice-bridge

general application

etc.,

and

in the

compounded

by Town in America;

its

more

and the proper proportioning of the

scantlings to the stresses were, however, strangely neglected.

Indeed,

it

had been claimed

as

an advantage in Captain

Warren's bridge that the structure could be built up of triangles all cast

from the same pattern.

MODIFICATION OF ROOF-TRUSS ARRANGEMENTS.

The contour of the the tie-bar

may be

roof-surface,

greatly varied,

and the shape given to

and the appearance of the

CLASSIFICATION OF TRUSSES.

14

design greatly changed, without causing any change in the It is of course impracticable

relative positions of the parts.

to give all the modifications of the arrangements which so arise.

to 53, of

The

series of figures,

may be

21 to 28, 33 to 37, 39 to 45, 48

pointed out as exhibiting the extreme plasticity

some designs

:

in each of these series the designs are

mere

modifications of one arrangement of the parts.

The leading

outlines of roofs are as follow

isosceles triangle (Fig. 338,

when

i.)

with slopes varying from a

to one of 1 in 1|

(ii.)

—the common

rise of 1 in

2£

and sometimes much steeper

architectural effect is sought after

A

(iii.)

more

shape becoming

common than

i

:

formerly

is

that

having the top more or less flattened

(iv.)

may be

part zinc,

or

The

central

covered in with

surmounted

by the

arrangements for lighting and iv.

ventilation..

Occasionally the upper part of the roof has the usual slopes,

but with the lower parts made

^

extremely steep

(v.)

For some of the vii.

been used viii-

largest

roofs an arc of a circle has

rafter,

for the

form of the

and the arc of a larger

circle for the tie (vi.)

A

I' Fig- 338

offer

form that has not yet

been used, but which might

-

some advantages over the

circular, is the parabolic, or

an

approach thereto, by combining an arched crown with the ordinary sloped flanks

The Gothic arch

(Vii.)

(viii.)

might also be found good.

;

CLASSIFICATION OF TRUSSES.

15

The part these different contours would play in an economic point of view must depend greatly upon the shapes given to

More will be said regarding some of the special designs.

the tie-bars. to discuss

Sometimes the end of the trass the left-hand side of Fig. 338

i.,

this

when we come

is finished,

shown

as

to afford a rigid connection with the supporting column.

secondary

office

at

above, and in Fig. 190, so as

This

of giving rigidity to the whole erection will

require additional material to

meet the induced

stresses.

MODIFICATIONS OF BRIDGE-FRAME ARRANGEMENTS.

Much

that has been said regarding the modification of

roof trusses might be repeated as applicable to framed bridges.

In the tabulated designs for bridges few modifications are given, as the student

may

readily supply such.

The leading

modifications result from giving triangular, quadrilateral, and

segmented outlines to the also the fish shape also arise

and 234

truss, as in Figs. 232, 233,

shown by

Fig. 330.

Modifications

from variations in the inclinations of the

may

interior

parts of the trass, as in Figs. 275, 276.

RELATION BETWEEN THE NUMBER OF POLYGONS AND

THE NUMBER OF PARTS IN THE TRUSS. The only

part of the foregoing classification

artificial

the ranging the trusses in the order of the

There

exists,

number of parts

for

be the number of polygons, then in

those belonging to Class

I.,

being regarded as joints)

is

the

number

least

number of polygons.

however, a curious relation between the number

of polygons and the

p

is

of parts

number of

is

it

all

;

thus, let

perfect trusses, or

the number of parts (intersections

=2p +

= 2p +

parts

each Class

1

+

1.

d,

Whereas

in Class II.

where d represents the

would be necessary to add

in order

;;

CLASSIFICATION OF TRUSSES.

16

to convert the trass into one of Class

to abstract to reduce the truss to is = 2 p + 1 = 2p + 1 + d-

parts

e.

is

e.

Thus

III.,

it

And in Class IV.

in Class II., for Fig. 140,

In Class

I.

would be necessary Class I., the number of

putting e for the least number of parts,

the

d=l,

number of parts number of

the

.:

= 2 p + 2 = 8. For Fig. 148 the deficiency number of parts = 2^> + 3 = ll. In Class III., for

parts in Fig. 140

d= 2,

.•.

Fig. 172

it

would only be necessary to take out one of the

central parts, therefore the excess

number of

parts

= 2p,

is

or 12.

e=l, and

accordingly the

In Fig. 178

would be

it

= 2, = 2p + 1-2 = 19.

necessary to remove two parts at least .\ e

and the

number of parts

In Class

IV. there

is

in the truss is

same time deficiency and excess of parts

at the

d= 1 and e = 2, ,\ the number of parts = 2 p + + 1-2 = 2^=18; in Fig. 199, d= 2, and e = 1 number of parts = 2^> + 1 + 2-1 = 2 p + 2 = 18; in Fig. 200, d = 1 and e = l and number of parts = 2 p + 1 = 21. A trass belonging to Class must have 2^ + 1 parts, but a for Fig. 198, 1

.-.

.*.

I.

may

belong to

clearly

marked

trass having that relation of parts to polygons

Class IV.

The

distinctions

between the classes are

but although the examples of Classes

II.

and

III.

always have

the defects appertaining to the class, the degree in which the defect

is likely

greatly.

to be present in a frame of Class III. varies

Thus the

lattice-girder

expected to be very

little

No. 324 may be confidently

affected

by any ordinary

misfitting

of one of the end pieces ; whereas structures of the abutting

kind are not only

likely to

be misfitted to the span

originally,

but the length of the span between the abutments, which plays the part of one of the lines of the framing, after

changes of serious amount.

may undergo

CLASSIFICATION OF ROOF TRUSSES.

CLASS Number

of

I.

— CLASSIFICATION OF ROOF TRUSSES.

18

CLASS Number

of

Sub-spans.

I.

Continued.

— CLASSIFICATION OF ROOF TRUSSES.

CLASS Number

of

Sub-spans.

I.

Continued.

19

— CLASSIFICATION OF ROOF TRUSSES.

20

CLASS Number

of

Sub-spans.

I.

Continued.

CLASSIFICATION OF ROOF TRUSSES.

21

22

CLASSIFICATION OF ROOF TRUSSES.

— CLASSIFICATION OF ROOF TRUSSES.

CLASS Number

of

Sub- spans.

I.

Continued.

23

— 24

CLASSIFICATION OF ROOF TRUSSES.

CLASS Number

of Sub-spans.

I.

Continued.

CLASSIFICATION OF ROOF TRUSSES.

CLASS I.— Concluded. Number

of

Sub-spans.

25

26

CLASSIFICATION OF ROOF TRUSSES.

CLASS Number

of Sub-spans.

II.

CLASSIFICATION OF ROOF TRUSSES.

CLASS II.— Continued. Number

of

Sub-spans.

27

CLASSIFICATION OF ROOF TRUSSES.

28

CLASS Number

of

Sub-spans.

III.

CLASSIFICATION OF ROOF TRUSSES.

CLASS Number

of

Sub-spans.

III.— Continued.

29

30

CLASSIFICATION OF ROOF TRUSSES.

CLASS

Sub-spans.

III.— Concluded.

CLASSIFICATION OF FRAMED BRIDGES.

CLASS

cq"

I.

— 32

CLASSIFICATION OF FRAMED BRIDGES.

CLASS Number

of

Sub-spans.

I.

Continued.

— CLASSIFICATION OF FRAMED BRIDGES.

CLASS Number

of

Sub-spans.

I.

Continued.

33

— 34

CLASSIFICATION OF FRAMED BRIDGES.

CLASS Number

of Sub-spans.

I.

Continued.

— CLASSIFICATION OF FRAMED BRIDGES.

CLASS Note.

I.

Concluded.

—A great variety of designs may be produced by assuming the existence

bracket-action, -as in Figs.

Number

of Sub-spans.

35

202 a, 204, 246, and 254 ;

of efficient

further examples need not be given.

I CLASSIFICATION OF FRAMED BRIDGES.

36

CLASS Number of Sub-spans.

Number

" Arrange-

of

Polygons in each Truss.

ments

" of

II.

Reference

numbers of the

the Parts.

Trusses.

277 1

II.

III.

A

278

B

279

IV.

280 5

A

281

10

B

282

VI.

11

VII.

8

A

284

X.

17

A

285

283

— %

»x of the truss figure infinite area,

partial

but

contiguous lines are given, forming a

five

boundary to

it,

and

in the diagram

we

accordingly

find the five corresponding lines radiating from point their lengths represent the

The space

E

is

magnitudes of the respective

also infinite,

D, and forces.

and has only three defining con-

tiguous lines, and the forces acting in these are given by the

corresponding lines radiating from the point

Example

II.

—

This

Fig. 358.

foregoing, but from

of overlying lines ; thus space vertical lines

in

ii.

nearly similar to the

making the reactions of the supports

vertical, it is apparently simpler.

two

is

E

ED

and

E

E

It will afford us is

examples

here only defined by the

F, and in the diagram these are

found proceeding from the point E, but overlying ; similarly the reaction or supporting force

weight point

FG

in

in the diagram, these

common.

By

DE two

partially overlies the lines

have no terminal

these examples of overlying lines in

DIAGRAMS OF FORCES. the diagram, this

mode

of lettering.

ing lines in Fig. ing lines

be seen that no confusion can arise under

will

it

i.

F E, F

55

;

A, and

sides of the polygon

F

Space

and

in Fig.

F

has three contiguous boundii.

we

find the correspond-

G, radiating from point F.

ABC6FA

in Fig.

forces acting in the lines of the frame Fig.

at the joint, surrounded by the polygons

G and F,

and so

The

represent the

ii.

i,,

which meet

ABC

and spaces

on.

DIRECTIONS FOR THE CONSTRUCTION OF

DIAGRAMS OF FORCES. Assign a letter to each enclosed area of the

1st.

truss,

also to each division of the surrounding space as separated

by the

lines of action of the external forces.

2d.

The

lines in the

diagram are to be drawn parallel to

the corresponding lines or parts of the truss figure. 3d.

The

forces acting in lines radiating from a point in

the truss must in the reciprocal diagram form a closed polygon. Note.

—The

particular order in which the lines representing the

form the closed polygon must be such that, when

forces are taken to

combined with the other polygons of the diagram, the reciprocity will If the polygon

be maintained.

is

complete in

sequence in what order the lines are taken sent-

;

itself, it is

a point acted upon by four forces in equilibrium;

quadrilateral figure constituting the diagram of forces in a variety of ways, each of

these diagrams are

shown by

in each the directions in

are

which will be a Figs.

which the

another round the polygon.

When

one only

of the others

is

may

iii.

iv.

v.

repre-

may be drawn

reciprocal.

and

i.

then the

Some

of

of 359, and

forces act against the point in

shown by the arrows arranged

two of the four

ii.

of no con-

thus let Pig. 359

i.

so as to appear to follow one

We

forces before

require to know the magnitudes of we can draw the diagram definitely.

given, say for instance the vertical one, the values

vary greatly, as shown by the dotted line in

ii.

DIAGRAMS OF FORCES.

56

The

4th.

any uncrossed space,

sides of

triangle, or other

polygon, around or in the truss, must always be represented

by

from a point in the reciprocal diagram, and

lines radiating

that point

to be

is

named by

the letter assigned to the space

or polygon.

In diagrams which are not reciprocal, a triangle in the truss will also generally have its sides represented lines radiating

by three

from a point, but sometimes the lines are

found connected, as in Fig. 360, or 361, or even detached.

An

5th.

important guide and check in constructing any

diagram of forces for a fabric supported by vertical reactions, is

to balance the horizontal

forces cut

by any

line

and

vertical

elements* of the

The sum

such as a-a', Fig. 362.

of

the horizontal elements of the intersected parts, which act as ties,

must be equal

to the

and

its

sum

of the horizontal elements of

In the Figure 362

the intersected struts.

D A acts as a tie,

horizontal element will of course be just equal to the

horizontal stretch of

D A in the diagram

ii.,

and

F B and B

the other parts, eut by a-af, act as struts, and the horizontal elements in the diagram,

is

sum

A,

of their

equal to their united horizontal stretches

and the amount of

this

combined

stretch is

seen by simple inspection to be equal to the horizontal stretch of

AD. Then, as to the vertical elements, we can readily calculate

the excess that must be in course of transmission past the

In the figure

section a-a' towards one or other support.

there

is

evidently a

a-a' equal to half

parts intersected,

E, while *

downward

FF'=FD

FB

A B and A D

is

pressure or sheering force in in the

carrying load

distinction

was

force are taken vertically

first

down

are counteracting the

The term "element "is substituted

components of a

diagram

for

and

;

then of the

to the support

work

;

therefore

"component" when the two horizontally.

employed in Treatise on Bracmg, 1851.

This convenient

DIAGRAMS OF FORCES. the vertical element of

element of

tical

F B = vertical

AD + FD,

and

57

element of

be

this can

A B + ververified

by

attending to the vertical rises of the lines in the diagram.

An

6th.

lettering

important principle which the new method of

makes very evident

If

is this.

we draw a

line join-

ing any two points or letters in the diagram, that line will represent in direction and

amount the

resultant of the forces

acting in the parts of the fabric, and in the lines of action of external forces lying between the spaces corresponding with

the same points or letters ; and frequently

we may take the

passage from the one space to the other in a variety of direc-

Thus

tions.

in Fig. 132

ii.,

Plate

XL,

Y in

the dotted line

the diagram represents the resultant of the forces acting in

CD

and

parts

D Y in the

and

truss

i.

or

;

we may take

lines of force separating the spaces

other sets of

G and

Y

;

thus

C Y = resultant of the forces in C X and X Y, or in C X, X Z, and Z Y, or in C B, B Z, and Z Y. Now, in the figure under review, this principle, taken in a reverse manner, gives us a satisfactory

way of determining the

diagram.

the.

cuts

XY

illustrated

Produce the

C

in

until

it

position of the point

line

X

in Fig. 132

i.

now, by the principle

produced in the point a

;

by Fig. 343, Plate

evident that a line join-

I., it is

ing b and a will give the direction of the resultant of the forces acting in

CD

Y C in the diagram X C gives point C. The

and

D Y. We need only therefore draw

parallel to b a,

and

its

intersection with

positions of the other points, such as E, can be de-

termined in a similar way, or

if

found otherwise, the

correct-

ness of the determination checked.

7th. All the external forces acting

erection, must, taken together,

upon a

truss or other

be represented in the recipro-

cal diagram by the sides of a closed polygon.

—

—

DIAGRAMS OF FORCES.

58

The external

forces are conveniently divided into

two

sets,

the one comprising the forces to be borne by the structure,

namely, the loadings, weight of structure, pressure of wind, etc.

the other, the supporting forces, or the reactions of the

;

supports

:

the former are supposed to be

known from

nature of the structure and the circumstances in which

must be determined before we can complete

placed, the latter

the diagram of forces, and the varieties of the problem

be

classified as follows

When

A.

the it is

may

:

the loading

all

and

acts vertically,

uni-

is

formly distributed, and when the reactions of the supports are vertical.

—In

this,

the simplest case, the two reactions will

be equal to one another, and the two diagram

in the

equal

line,

will

made up

When

the loading

equally distributed, this case the

them

line overlying

an

of the lines representing the various parts

of the loading, as in Fig. 358

B.

lines representing

form one continuous

ii.

all

acts

but

is

not

vertical.

—In

vertically,

and when the reactions are

two reactions or supporting forces

will not

equal to one another, but the lines representing them

be

will, as

above, form one continuous line, overlying the line represent-

ing the loadings.

The

effects of

(Fig. 56, Plate VI.)

each separate portion of the loading upon

the two supports will be inversely as the distances of the supports from

its line

of action, and the total

reactions are obtained by the

amount of the

summation of these

partial

effects.

C. all

of

When it

the loading

acts vertically,

is

distributed in any manner, and

and when the reactions are

inclined.

In this case the reactions must have opposite inclinations

and equal horizontal elements:

their vertical elements

be determined as in case

B

forces will be a triangle, of

which the

;

must

and the polygon of external vertical side will repre-

sent the loadings (see Fig. Ill, Plate IX.)

The

inclinations

DIAGRAMS OF FORCES. of the reactions ports, or

may

result either

59

from the nature of the sup-

from the structure tending to increase or reduce the

distance between

its

bearings

:

this

may be

resisted simply

by

the friction of the bearings, or intentionally as in abutting or

suspension structures.

D.

When

the

set of external forces are irregularly

and some of

distributed,

may proceed by force,

first

where

their lines of action oblique.

—We

ascertaining the point for each supported

its line

of action intersects a line joining the two

supports,

and there resolving

(when the

line joining the supports is horizontal)

it vertically

and

horizontally ;

the vertical

elements so obtained are to be treated like the vertical loadings in case reactions.

B

;

this gives us the vertical elements of the

The

horizontal elements of the forces resolved at

the line joining the supports are to be algebraically added together ; the residual horizontal force,

if

any,

is

the force k

tending to move the whole structure to one side, and must be resisted

by the supports, but the proportions in which they

contribute to this

arrangements

tive

all resisted

work ;

will

be determined by the construc-

thus the horizontal force would be nearly

by one support alone,

if

that were a stout piece of

masonry to which one end of the structure was fixed down, while the other end rested on rollers, or was simply propped

up by a tall column. But if each support were equally stable, and the structure rested firmly thereon at both extremities, we might assume that the distribution of the horizontal force would be more equable, but liable to some variations from the elasticity of the parts and other causes.

Though

it

may

frequently be best,

it is

not always neces-

sary to calculate the proportions in which an irregular loading is

distributed to the

supports.

The diagrams sometimes

determine this at once, as for instance in Figs. 19, 20, 30, 32, etc.

;

or the position of the point corresponding with the space

;

DIAGRAMS OF FORCES.

60

between the two supporting forces may be approximated to

by successive

trials.

APPLICABILITY OF THE METHOD OF DIAGRAMS TO THE

DIFFERENT CLASSES. The Class

I.

stresses in the various parts of trusses belonging to

are obtainable at once from a diagram of forces, and

the examples given will be nearly altogether confined to that class.

For structures belonging to Class will give the stresses

and may further

a diagram of forces

II.,

produced by an equilibrating loading,

assist the designer in estimating the severity

of the transverse and other stresses produced by other loadings

For

or forces.

structures belonging to Class III., the assist-

ance to be derived from a diagram of forces will vary greatly; in

some of

may

these,

at once

by making certain assumptions, the

be arrived at

;

be required, in the carrying out of

may be

assistance stresses.

But the

members of

stresses

may which, however, much

in others tedious calculations

derived from the use of diagrams of the objectionable characteristic of

many of the

this class is the uncertainty of the calculated

results truly representing the actual stresses in the structure

when

erected,

from the conditions being or becoming

different

from those assumed.

The graphic method, tation, will

bridges.

and

as contrasted with numerical

The

first class

that of bridges, and are

common

for

of roofs will be very fully treated

as the consideration of roof-trusses

ment

compu-

be found generally more suitable for roofs than

many

is

made

of the remarks and

to both, this will

still

to precede

modes of treat-

further increase the

apparent disparity in the space allotted to

roofs.

But

if

the

reader familiarize himself with the examples given under roofs,

he

will experience little difficulty in projecting the

diagrams

DIAGRAMS OF FORCES. for other structures.

although examples

For some forms of

may be

•

61

parallel girders,

given, the graphic

method

is

not

so satisfactory as arithmetical computation, which will be

treated of at length in another Part of the " Economics of Construction."

DIAGRAMS OF FORCES.

•

62

EXAMPLES OF DIAGRAMS OF FORCES FOR STRUCTURES BELONGING TO CLASS I. Eoof Wtams. Figures 3 rafters

i.

and 3

ii.,

Plate III., show a pair of untied

under two arrangements of the external

The

forces.

corresponding diagrams of forces are given immediately be-

neath each drawing of the structure. Figs. 3 hi. to 3 ix.

—This couple or triangular frame, the

simplest possible kind of tied truss,

here

is

shown under a

great variety of conditions of the external forces. is

two

there

hi.

C F, to be supported, and this is met by C B and F B of the supports: all these three

only one force,

the

In

reactions

them

external forces being vertical, the polygon representing

in the diagram of forces degenerates into overlying lines. iv.

In

the reactions are assumed to be not quite vertical, and this

assumption amples, as forces

;

will

it

be frequently

made

in the succeeding ex-

shows clearly the polygon formed by the external

but in practice care should be taken to give the

most prejudicial directions

actions the

likely to occur.

re-

From

the friction of the bearings at the supports, the reactions in-

tended to be vertical may become somewhat inclined inwards at one time

and outwards at another, leading to augmentations

in the stresses of

when the

some of the parts above those produced

reactions are vertical.

In

v.

and the succeeding

figures account is taken of the parts of the loading that are

brought directly to the supports.

In

vi.

supposed to be inclined slightly inwards. tion of the reactions

is

made

the reactions are

In

vii.

this inclina-

so great, that the tie-bar

AB

— CLASS is

I.

—ROOF TRUSSES. In

required to act as a strut.

ward

inclinations, or so as to

bar.

In

ix.

some of the

63

the reactions take out-

viii.

augment the tension

in the tie-

made

forces to be supported are

oblique, to represent the combined effects of the loading

of a side wind.

embraces

all

It will

the others except

viii.,

and

AB

and

be made nothing, the diagram

EF

are

ternal forces

CD

and

Figs.

made vertical, it becomes vi. be made vertical, it becomes

E F be

reduced to zero,

it

If the stress

that of

is

ix.

reduced to one or

is

other of these by assuming simpler conditions. in

and

be observed that the diagram to 3

If

ii.

DF

If all the five exv.

;

becomes

and

further,

if,

iii.

4 and 5 show the simple triangular frame when the

supports are at different levels.

This very full treatment of the simple triangular frame,

under give

a' variety

of conditions, will render

more than one or two

it

unnecessary to

There

is

another condition

namely, that of irregularly-distributed loading simple triangle

not suited to

is

be

illustrations of the trusses to

afterwards chosen as examples.

illustrate.

—which

the

Diagrams showing

the effects of this in augmenting the stresses in some of the parts over what they suffer

be found under Figs.

19,

32

when the whole loading iv., etc.

also arise in such structures

is

on

will

Similar results of course

from the

lateral action of the

wind.

No.

7, Fig.

gives the truss under vertical loadings

i.

slightly inclined supporting forces.

Fig.

ii.

shows

it

and

under

the combined effects of dead- weight and side-wind pressure. Fig. 9.

—This

truss

is

here assumed to be subjected to

vertical loadings, placed both on the

The diagram could be drawn

rafters

otherwise,

lower loading as brought to bear on the pressures instead of as trated by Figs. 61

No.

10.

—Fig.

downward pulls.

and the tie-beam. by regarding the tie as

downward

Both modes

are illus-

i.

and

i.

shows the truss under the dead-weight

ii.

DIAGRAMS OF FORCES.

64 In Figs.

alone.

and

ii.

the effects of a side wind are super-

iii.

In ii. the leeward foot of the truss is supposed to be and the windward one movable, as when placed on

added. fixed,

In

rollers.

iii.

arrangement

this

is

Attention

reversed.

should be directed to the greatly augmented stresses under the conditions shown by Fig.

No.

11.

—This

iii.

truss does not require the brace

A B when

placed under a symmetrical loading, but under an irregular

when a side wind has to be resisted, it comes into The diagram shows A B acting as a strut when F G rendered oblique by a side wind. Of course when the wind

loading, or action. is

blows from the opposite quarter,

The two

upon

effects

AB

AB

by Fig.

sides of the truss are illustrated

No. than

19.

H

Diagram

I.

than

HI.

strut

and

Many ment

—Diagram The

ii.

iii.

signs

must do duty

is for

the case of

shows the

tie.

+ and— placed

19.

FG

stresses

being greater

F G is

when

against

AC

less

indicate

tie action respectively.

of the diagrams will be passed over without com-

but as they are generally given to

;

as a

of a difference in the loadings on the

of interest or importance, they

illustrate

may be thought

some point

deserving of

the studious attention of the reader.

No.

32.

—This

when deprived of a tie-bar, is only when the length of span own aid. Diagram ii. is for the case of the design,

suited for an abutting principle, or is

fixed without its

right-hand side of the truss being fully loaded, the left-hand side with half the load removed,

right to load,

left.

iii.

and a

side

Fig.

wind

acting from

gives the diagram for the full uniform

and shows how nearly the arched

under such. is

Fig.

iv. is

rafter is equilibrated

the diagram of forces

when

the load

supposed to be altogether removed from the left-hand half

of the structure. alone,

Fig. v. gives the effects of the wind taken

and of about twice the previous

intensity,

and accord-

ing to the ordinary theory that the pressure at right angles

;

CLASS

I.

—ROOF TRUSSES.

65

to an inclined surface varies as the square of the sine of the

angle

it

makes with the plane of the

horizon.

Fig.

vi. is

the

same, on the assumption that the pressure varies simply as the sine of that angle, an assumption that

more

cally

correct

;

but the subject

is

perhaps practi-

is

one that has been

much

neglected, or unsatisfactorily investigated by experimentalists.

No. 40. work.

G

to

— This

When C and C

stress,

design

is particularly

to

L

in the diagram, the tie

and the case becomes that of No.

No.

61.

—This down

rods going

support a

well suited for iron

the supporting forces are in the directions of

floor.

CF

is

free

from

46.

46

tied,

and supplied with suspension

to the

main

tie,

is

In Fig.

shown the changes

are

ii.

which may be supposed to in the

and form of the diagram caused by treating the loadings supported at points a and a', as though imposed by lettering

downward No. on

thrusts instead of

70.

rollers,

—The leeward

downward

pulls.

foot is here supposed to be placed

which arrangement,

as

we found

in considering

No.

10, gives the greatest stresses.

Nos. 88, 89, and 90.

—The

first

of these

is

somewhat out

of place here, but the three are brought together in order that

a comparison

may be

readily

drawn as to the

by changes in the outline of the main tion 88, is

part,

ED

effects

produced

In the modifica-

will be observed that the lowest part of the rafter

it

AM

tie.

subjected to a ;

much

greater stress than the highest

and as the number of

proportion tends to become as 2

:

triangles is increased, the 1.

Now,

in practice,

found convenient to make the sectional area of the

form throughout, and of course

it

the stress in the lowest part here. rafter is usually calculated with

must be

The

it is

rafter uni-

in proportion to

sectional area of the

an allowance of metal per ton

of stress double that given to the main

tie.

There

is

conse-

quently a considerable amount of material uselessly applied

and

this loss is still

more important when,

as

is

very usual,

DIAGRAMS OF FORCES.

66

an upward curvature 89.

Now,

I

as in No. 90,

is

given to the tie-bar, as in No.

would suggest that the tie-bar should be curved, when doing so would lead to no inconvenience.

Such a form renders the stresses in the rafters more uniform throughout, and the sectional area required is very greatly-

The material required

reduced. also be

for the other parts

would

less.

This suggestion has also been embodied in the modifications Nos. 50, 59, 69, 84, 98, 104,

—The

Nos. 106 and 116,

shows one half made after the other

after the

on Plate X.

for these

one pattern, and the other half

done to save space ; the two

this is merely

:

and 120.

figure given

diagrams of forces are given beneath the figure of the

combined

The

also,

figures

similarly

trass,

but so as need not lead to any confusion.

and diagrams

for several other designs are

shown in combination,

Nos. 127 and 128.

—In the drawings of these

trusses the

lines or arrows to represent the portions of the loading, etc.,

have been omitted ; they are not necessary, since the magnitudes and directions of these external forces appear in the diagrams, and they somewhat disfigure the designs. It

appears unnecessary to

add examples with more

numerous regular subdivisions of the span

:

difficulty

little

should be experienced in applying the method to such,

if

the examples given be properly studied.

No. 131. terised

—This

by the

is

one of the trasses of Class

I.

charac-

irregularity of size of the subdivisions of the

span, and by the

number of these being

for the support of the roof platform.

truss has been omitted, but the

greater than required

The diagram

modes of

for this

delineating

it will

be gathered from the account of the next example.

Under No. 88

it

has been pointed out that the lower end

of the rafter in the roof with outlines such as this severely stressed than the upper end.

is

more

In No. 131 some com-

CLASS pensation rafter

is

I.

—ROOF TRUSSES.

67

provided in the more perfect stiffening of the

towards

caused by the shorter intervals between

its foot,

the supported points.

No. 132.

—From

the minuteness of the triangles at the

we do not get a good starting basis manner exemplified by Nbs. 105 and 118. There methods of overcoming the difficulty; we may

extremities of this design, in the usual

are several

make a temporary assumption

that the truss finishes with a

larger triangle substituted for the smaller terminal ones ; after properly transferring the loading

by

it,

proceed to draw the diagram.

The

lines representing

the stresses in the part of the small triangles

be inserted, the

rubbed

ii.,

page 57; by

this

we begin

is

at

ascertaining the resultant b a of 132 is

obtained

and

tie-

indicated in "Direction"

any part; the mode of i.,

for a weight at b there

equally applicable to any loading confined to one

and

side of b, or any other point.

In 132

CZ

B X parallel

To

is easily

truss.

Another method of proceeding

described,

thereafter

fines parallel to the tangents of the arch

bar at the foot of the

6th,

may

temporary triangle being

The terminal point W, 132

out.

by drawing

letter assigned to the

and

on the part occupied

is parallel

to b

c;.

resultant

i.

ii,

to

the resultant

d e, and

so on.

obtain by this method the stresses for the case of the

whole load being on, two diagrams may be drawn

—the one

representing the stresses produced by the load lying to one side of any point b,

and the other diagram the

effects of

the

load on the other side of the same point, the load or force at b being either divided between the two parts of the loadings,

or included wholly in one of them

;

the two stresses so

obtained for each part are to be algebraically added.

The

effects of any other distribution of the external forces or

loading

may be

arrived at in a similar

the stresses given

by

its

manner by combining

subdivisions.

A third method is to ascertain the position of a convenient

DIAGRAMS OF FORCES.

68

starting-point, such as

AX

or

B

Z, from the

A

or B, by calculating

known

points

moments measured round any

may

its distance,

X or Z, by equating the

suitable point

;

otherwise

we

use the result thus obtained as a check upon other

methods.

132

iii.

and

iv. illustrate

the case of a side wind acting in

combination with a uniform loading. that

when the whole

brace acts as a this tie-action

tie

;

It is obvious

from

iv.

truss is not loaded very unequally each

and before a

strut action can

must be neutralised

—that

is,

be induced,

each brace can act

virtually as a strut to the extent of its previous tension, with-

out suffering compression in doing the work of distributing

any superadded irregular loading.

This

is

a source of economy

in wrought-iron work, since it is cheaper in such to provide ties

than

struts, especially

amount of

tion to the

No. 137

is

when the

stress to

latter are long in propor-

be conveyed.

the roof-principal of the Drill Hall recently

erected at Forrest Koad, Edinburgh

;

I supplied the architect

with this design in 1871, and assisted him in carrying

The span

for calculation is

Plate XI.

is for

97£

feet.

it

out.

The diagram given

in

a uniform loading, and drawn by assuming that

the smaller triangles towards the extremity are replaced by triangle

s,

according to the method described under No. 132.

CLASS

I.

—BRIDGES.

69

EXAMPLES OF DIAGRAMS OF FORCES FOR STRUCTURES BELONGING TO CLASS I.

It has been already noted that a design for a framed bridge may be used in two positions if the one be considered as ;

the erect position, the other

one diagram of

is

forces-is sufficient to give the stresses in the

parts of the truss whether in If

we

its original

or inverted state.

turn the truss simply over, so that what was the top

becomes the bottom, the diagram,

if

the same lettering be

retained, requires to be turned so that

hand

Now

called the inverted one.

side

becomes the

left,

what was the

as in Figs. 201

i.

to

right-

Plate

iv.,

But if, in addition to turning the truss upside down, we also turn it over right to left, as shown by Figs. 205 i. and ii., the one unchanged diagram 205 iii. will be suitable XII.

in all respects for both arrangements.

Although the design be inverted, the directions

in

which

the external forces act are of course not reversed, except relatively to the design

struts into ties,

and

;

but to suit the resulting changes of

ties into struts in

the truss, the external

forces that appear to be applied as thrusts in the one, may, if

thought desirable, be changed in appearance to pulls in the

other drawing, as shown in Nos. 205 and 219.

THE DIRECTION OF THE REACTION OF THE SUPPORT. In

all

cases of girder action in which the truss rests

horizontal surfaces,

it is

upon

usual to assume that the reaction

is

DIAGRAMS OP POKCES.

70 vertical. it is

Now, although it always approximates to this, jet when there are any influences at work tend-

evident that

ing to cause the girder to slide on

which the

friction offers to this

bearings, the resistance

its

may produce a

considerable

when no

deviation in the direction of the supporting force friction rollers are used.

In the case of a simple parallel

girder with its bearings at the level of the lower

when

acting under

its

own weight

from the circumstances of

its

boom,

may be

alone,

erection, to

this,

supposed,

rest vertically

on the supports at ordinary temperatures; but when the movable loading comes upon the

bottom boom part taken

is

increased, and

up by the curvature

girder therefore

simple girder,

girder, the length of its

this will usually :

the bottom part of the

becomes lengthened

but

be only, in

in span if

it still

act as a

this lengthening is resisted

by the

friction

on the bearings, and may be altogether or in part prevented,

and the reaction

in consequence will be inclined inwards,

thereby reducing the tension in the lower it

would be were the supporting force

When

boom below what

truly vertical.

a girder that strongly evinces this tendency to

lengthen the distance between so by pinning

its

its

bearings

ends, the reactions

inclination inwards

when the

may

is

prevented doing

take a considerable

extra load comes on, especially

the structure be of iron, and the temperature be

than when the pinning was effected.

when the load

is off,

a

fall

if

higher

In other circumstances,

of temperature

actions to incline outwards.

much

may

cause the re-

Of course when the ends

are

restrained by the friction alone, these inclinations cannot

exceed the angle of repose, and would probably indeed be confined to a

much

smaller deviation in the presence of the

tremor caused by a passing load. In the case of the triangular girder, Fig. 205

i.,

the length

between the bearings tends to become reduced from the compression of the upper member.

This will cause the reaction

;

CLASS

I.

—BRIDGES.

71

of the supports to slope more or less as shown

the stress in

;

A F, will be thereby reduced, while the tension in the tie A D remains unaltered. When the same truss is the top, as at

used in the inverted form shown by Fig. 205

ii.

the action of

the load will tend to increase the span, so that the friction of the bearings will cause the reactions to assume the inward slopes, relieving

now

the tie

without affecting the upper

A F of

member

some of

tension, but

its

A D.

It is fortunate that in all these cases the effect of the in-

clination given to the reactions from this cause is in favour of

the strength, of parts at least, of the truss.

It

must be borne

in mind, however, that a change of temperature

an opposite

may produce

effect.

No. 202, A, B, and C.

—Three arrangements of the

forces,

and corresponding diagrams of stresses, are here given to represent the supporting of the central weight

in

:

A, through

simple bracket-action ; in B, through simple abutting action

and

in C,

by the structure acting as a tied

truss.

In actual structures the mode of action

No. 211. of truss

A B,

is

is

com-

often a

compounded of these.

plicated one

—The unsatisfactory character of

shown by the excessive

stresses

this description

brought upon

C

I,

etc.

No. 220

A is the

case of the loading being placed on the

Here, for simplicity, the parts of the loading

longer member.

which would be imposed directlyupon the supports are omitted. 220

A

i.

is

the diagram of forces

when

all

the other points

are fully loaded.

220

A

ii.

is

the diagram

removed from point

of the loading

is

L.

when two-thirds of the loading removed from each of the points K L and J K. 220

is

A

K

when two-thirds

iii.

No. 220

B

is

the diagram

gives the case of the loading being imposed at

the level of the shorter member.

—

DIAGRAMS OP FORCES.

72

B B

220 220

i.

when

all

when

three-quarters of the length

the diagram

is

the diagram

ii.

the length

is fully

loaded.

loaded, two-thirds of the loading being removed from

is fully

the other quarter.

B

220

when

the diagram

iii.

half the length

is

fully

loaded, the other half being loaded to one-third.

B

220 is fully

the diagram

iv.

when one-quarter of the length

loaded, two-thirds of the loading being removed from

the remaining part.

No. 231.

—This, compared

with No. 230, has double the

depth of structure, but the saving of material when designed with similar angles

less

is

than

five

per cent, No. 230

therefore the better design of the two, as

its

is

depth could be

increased without inconvenience.

The diagram

No. 261 (and a modification of No. 262). of 261

may be

261

of the part of the frame shown by 261

iv.

constructed by

No. 263£.

(This

is

first

composing the partial one iii.

just No. 266 with eight instead of

nine subdivisions of the span.)

This design

much

are

is

only interesting

when the supporting forces when the structure be-

inclined, or, in other words,

But when

comes abutting.

tion of the reactions

is

abutting, the degree of inclina-

a subject practically of

the structure belonging then to Class

III.,

defects of that class in a high degree

usual way.

But

for

much

doubt,

and exhibiting the

when

erected in the

any definite assumption as to the direc-

tions of the reactions of the supports, the structure is to be

dealt with as belonging to Class

263£

h. is

I.

the diagram of forces

when the supporting

forces are vertical.

263J that

is,

iii.

is

when

the diagram it is

acting in the line

when the

reactions are horizontal

subjected alone to two horizontal forces

R r,

and tending

structure nearer together.

to bring the feet of the

CLASS 263£ iv. and a joint

I.

—The diagram of in I

R is

—BRIDGES. stresses,

when loaded uniformly,

opened to allow of a more accurate ad-

justment of the structure to the span, as suggested by

some years

73

me

ago.

No. 267.

—As

in 261, so here, a little calculation is re-

quired, or a subordinate diagram of forces must be drawn for

any irregularly loaded sub-frame, as

H

H M L here.

DIAGRAMS OF FORCES.

74

APPLICATION OP THE GEAPHIC METHOD TO STRUCTURES OF THE SECOND CLASS. SOME EXAMPLES OF THIS CLASS OF IMPERFECT STRUCTURES ARE PRACTICALLY USEFUL FOR ROOFS.

Although

the designs of this class

may not, when considered

in the usual theoretical manner, be capable of acting as rigid

or stable

trusses

under a non-equilibrating

loading, yet,

through being connected, by means of the planking and other parts running at right angles to the plane of the truss, with partitions

and gable

yery

against turning,

stiff

walls,*

and by having

their joints

and receiving a very

ance of extra material to compensate for induced cross they

may become practically very And when the increased

stability.

made

liberal allow-

satisfactory

as

stresses,

regards

outlay thus incurred

is

repaid by the value of the uninterrupted openings given by

such forms as Nos. 139, 145, 153, 159, portant,

and deserving of

etc.,

they become im-

scientific attention.

As, however,

the graphic method can only be of subordinate use in dealing

with such structures,

it

would not be

suitable to enter very

fully into their consideration here.

The

stiffness of the joints

distortion

adds to the power of

by constraining the parts to assume much more

complicated flexures.

I

form on the top

may

however, in what follows,

shall,

* In the casea of Nos. 138, 139, 149,

distortion.

resisting

and other

be made to act as an

In iron structures

it

designs, the boarded plat-

efficient horizontal

might in many

bracing against

cases be advisable to supply

proper horizontal or vertical bracings running at right angles to the planes of the trusses.

CLASS

75

II.

neglect the advantage to be so gained, and take note only

of the simpler flexures that would be produced at the ends of the pieces meeting the principal

if

the joints

members of

the frame were hinged, and regard the stability under a nonequilibrating loading as altogether resulting from the trans-

verse stiffness of such principal parts.

DIAGRAMS OF FORCES FOR STRUCTURES OF CLASS I shall

now

describe a

mode

II.

of drawing a diagram to give

the longitudinal stresses in an imperfect truss subjected to a distorting load,

and therefore dependent

for its stability

the transverse resistances of some of the parts. in supplying imaginary forces acting

upon

This consists

on the frame, which

shall

represent the reactions against bending, and then proceeding

with the diagram as though these were real forces, and the

The severity member must be dealt with

parts freed from all transverse stress.

of the

transverse stress in any

subse-

quently by considering the part as subjected to forces equal

and opposite to those imaginary

forces in addition to the

longitudinal stresses given by the diagram.

This applying i.,

Plate

mode it

of treatment will be most clearly explained by

to a suitable example, such as

shown by

Fig.

339

XVI.

The black arrows show the

external forces having a real

existence, the dotted arrows the imaginary forces to represent

GH now half of F

the reactions of the tie-beam against being bent.

taken as the unit of force, and acting along with

G

let

H, would

F G=2

;

Let

is

the half of

F

G, and

the duty to be performed indirectly by the transverse ness of the tie-beam

F G

G,

constitute an equilibrating

loading, so that the distorting force

is

be

stiff-

to convey one-third of a unit from

to the right-hand support.

The necessary values

to

be

assigned to the imaginary forces are therefore such as written against them.

The diagram of the

full

longitudinal stresses

—

DIAGRAMS OF FORCES.

76 next to be drawn

The

results are

stress in

shown by

is

Fig.

ii.

;

and the transverse

on the tie-beam are represented in Fig.

forces acting

somewhat remarkable

A F is somewhat,

siderably, greater than

stress in

B

H

is

very con-

would be the case were the structure from the lifting

efficiently braced, arising

beam acted upon by

and the

iii.

the longitudinal

:

the tension in

effect

of the

tie-

A B.

In more complicated cases, or generally,

it will

be most

convenient to treat such structures in two operations

1st,

by

drawing a diagram of the longitudinal stresses produced by

all

that part of the loading that would produce equilibrium 2d,

by a second diagram to show the separate

disturbing part of the loading finally arrived

results.

As

;

and

effects of the

the stress for each part being

;

by the algebraic addition of the separate

at

there can be no difficulty in drawing the

diagram, I shall in the following examples consider the

first

effects

of the distorting force alone.

Example 2.—Truss No. Fig. 139

139 (Plate XVI.)

shows in an exaggerated manner the nature of

i.

the distortion under an irregular loading. It

may be assumed

that the sides of the opening

B

will

always form a parallelogram, and therefore that the stresses in

AB The

and

BC

will

be equal.

position of the point

gram iii. depends upon the

The only

of the supports.

D

or

GH

and

horizontally in the dia-

effect of increasing or reducing

the relative width of the opening the values of

F

horizontal elements of the reactions

B

is

to increase or reduce

J compared with

I

FE

or

ED

;

thus

GH = FE— whole span. width of B.

The

stress in

=\ HI

AB

vertical

2 length

element of

A B in of A F in

Length of

ii.

ii.

H I + horizontal

element of

;

CLASS

The

vertical

instead of the

II.

77

H A = the difference

element of the stress in

sum

of the above two quantities.

H A is inclined at an angle of 45°, and H I is H A, the stress in H A = 0.

When

at

right angles to

As stout,

in timber structures the tie-beam usually supplied is

and well calculated to

resist

the flexures shown by Fig.

the comparative efficiency of this truss

iv.,

is easily

accounted

for.

some

It is of

in

C

practical importance to note that the stress

I is greater here

than when a diagonal brace

inserted

is

in the quadrilateral opening.

Example 3.—Truss No. Fig. Fig.

ii.

141 (Plate XVI.) shows the nature of the distortion to be

i.

applied to

and

iv.

it.

Fig.

iii.

the diagram of longitudinal stresses

the forces causing flexures in the rafter.

The

central bending force

half the

F

component

direction of,

When and

resisted.

the truss with the distorting and imaginary forces

is

and

on each

a of the

rafter will

F

force

be equal to

Gr resolved in

the

at right angles to, the rafter.

the supporting force

C

K

A will coalesce in the diagram.

is vertical,

If

the points

C and B

C

coalesce in

the diagram, there will be no stress in the tie-beam, and the

supporting forces will act in the directions of the

Example 4.—Truss No. When the resistance to neglected, the case

is

that of

rafters.

145 (Plate XVI.) bending of the

Example

2,

rafters

i

may be

Truss No. 139.

When the resistance to bending of thetie-beam is neglected, is the same as Example 3, Truss No. 141. One portion, say X of K L, will be supported on the first mode of action, and the other, say Y, on the second; the

the case

relative

amounts of

X and Y will depend

stiffnesses of the

tie-beam and the

tures, as usually

made,

X will

rafters.

upon the

relative

In wooden struc-

be several times the value of

;

DIAGRAMS OP FORCES.

78

Y

;

but in iron work, where a round bar of iron

the main

tie,

and as the

X will

be of are

rafters

ill

used for

Y

suited to withstand the simple

transverse flexure, the design is a

bad one

but a passably good one when wood

When

is

value compared with

trifling

the width of opening

C

is

is

for a fabric of iron,

the material employed. one-third of the span,

QEorHI = £EF. Note.

—A

transverse stress applied to a long strut is very preju-

A transverse stress

dicial to its strength as a strut.

beam

is

much

applied to a tie-

less objectionable.

ANOTHER METHOD OF APPLYING DIAGRAMS OF STRESSES TO STRUCTURES OF CLASS In dealing with any truss of Class verted into Class

mode

I.

of proceeding

II.,

II.

which could be con-

by the addition of one brace, another

may sometimes be

be used for cases in which there

is

adopted, and

it

may

a complicated departure

from equilibration in the arrangement of the loading or other external forces.

Let the brace requisite to convert the truss into Class

I.

be supposed to be temporarily added, and draw out a diagram of forces for the structure as though

Next, instead of the real external

it

were of Class

I.

forces, insert the imaginary

ones to represent the reactions against cross-bending, and

draw out a second diagram, and

let the scale of this

be such

that the stress on the temporary brace will be the same in

magnitude as in the opposite character.

first

diagram;

it

will,

Then, to arrive at the

final stresses,

the results in the two diagrams algebraically. piece will

come out of course

free

from

however, be of

stress,

add

The temporary and so may be

supposed to be withdrawn. This method might perhaps be extended to cases in which

more than one brace would be required to render the structure perfect.

CLASS

Example 5.—Truss

79

II.

279 (Plate XVI.)

The following example of

this

second method

is

simple character, having only one distorting force,

279

i.

ing Fig.

Fig. i.

ii.

is

B

it,

Fig.

B'.

ary forces ; and

in

first

iv. is

drawn

magnitude to

iii.

I.

by the temporary addition

shows the frame under the imagin-

the second diagram of forces correspond-

to such a scale that

B

B'

in Fig.

the same parts in diagrams the final stresses in

F, Fig,

diagram of forces obtained by treat-

as converted into Class

of the brace

ing with

the

of a very

E

AD

ii.

and

ii.

On

and

CD

B

B' in

it will

be equal

adding the stresses for

iv., it

will

are equal

;

be found that for (attending

to the vertical elements alone since the parts are equally inclined)

CD = £ + £ =

accounted

=

1.

£,

and

A D = f- J = £,E F being

;

DIAGRAMS OF FORCES.

80

STRUCTURES BELONGING TO CLASS OR

III.

THOSE HAVING A REDUNDANCE OF PARTS, AND THEREBY OFFERING MORE THAN ONE CHANNEL FOR THE CONVEYANCE OF THE FORCES.

Abutments and Tie-bars.

— These play

part in Structures of Class

nearly the same

Their differences in

I.

Class III.

In the structures of the

first class it is

not necessary to take

into account the elongation of the tie-bar in tied trusses, or

any moderate change of span in the case of abutting ones

and therefore the horizontal elements of the reactions of the abutments and the tension of a straight tie-bar at the

level of

the supports may, for such structures, be regarded as exactly interchangeable means of withstanding the horizontal thrusts. If to any such structures as 105, 266, etc., of Class

which are complete substitute stable

supports,

it

is

girders,

and

much

perfectly fitted

In Class

III.,

is

employed,

for simple

tie-bars,

and

either addition re-

into Class III.

however, an important distinction must be

drawn between these arrangements in bar

abutments

the same theoretically as though we

added very stout horizontal

moves the design

I.,

but tend to spread somewhat, we

its elastic

practice.

elongation under a

may be exactly calculated, and the may therefore, when the fitting is

When known

a

tie-

stress

stresses in the structure

carefully attended to, be

estimated with some degree of confidence.

But when

abut-

ments are used, we can place no trust in calculations based

CLASS

81

III.

on the misleading theoretical assumption that the magnitude of the span

may be

regarded as known and permanent.

PERFECT FITTING IN SOME STRUCTURES BELONGING TO CLASS

III.

AN

IS

UNDER DIFFERENT

IMPOSSIBILITY

DEGREES OF LOADING. Let us take No. 167 as a is

No. 15 of Class

iron

work

it

is

I.,

example of Class

first

In

almost necessary to treat the diagonals as

capable of acting both as struts and

The

III.; this

with a redundant tie-bar added.

ties.

distribution of the stresses in this Class

by the changes of length of the parts from

is

determined

elasticity

when

acting under the stresses, and as no one part can be length-

ened or shortened without inducing

may result from

stresses

the combined effect of misfitting and

the action of the loading.

when of

stresses in other parts, the

For the

truss

under consideration,

the proportions shown in Plate XV., the following

statement of results will give some idea of the interaction of the parts. If the structure

be put together so

that,

when unloaded,

there shall be no initial stress caused by misfitting of the diagonals, etc., then, on adding equal weights,

FG

D

H,

to carry each unit of stress,

allowance in

C

F A, B G,

compared with the

First, if the first-named parts

E.

G H,

be produced on the diagonal braces,

different effects will

according to the allowance of material in the parts

and

and

be made

excessively stout, while the allowance of metal in the tie

CE

is

small,

we may

consider that of the parts forming the

boundary of the frame,

CE

is

the only one that undergoes a

change of length when the truss will evidently

is

be that the diagonals must become lengthened,

and consequently both the braces tension.

loaded ; the result of this

Second,

If,

as

is

AC

and

CD

subjected to

very usual in iron roofs, the stress

DIAGRAMS OF FORCES.

82

per sectional inch on the rafters and top piece

upon

that

from

the tie-bar, the diagonal pieces will

and

tie pieces

Third,

be at the

rate per ton of stress, the diagonal braces will be sub-

jected to compression

Again,

F

be almost free

be loaded or unloaded.

stress -whether the truss

If the allowance of metal to the strut

same

only half

is

if

when a uniform loading

is

added.

there be an excess of loading at one side, say at

Q, about seven twenty-fourths of this excess must be con-

veyed to the support

A

and

as a tie

H E by means of A B acting as a strut,

the arithmetical

;

sum

of the vertical ele-

ments of the stresses in these must be = ^j(FG-G H), but in

what proportion

this is distributed

upon the allowance of metal

between them depends

and main

in the rafters

tie.

If

the rafters have twice the allowance of the tie-bar, the stresses in the diagonal braces will be nearly equal; if the

ance, the stress in

cannot assume the existence of perfect larly in large structures

lying flat

same allow-

A B will be greater than in A C. fitting,

But we

more

particu-

not previously put together when

on the ground, and while properly supported

every point.

The

tie-bar

C

at

E, for instance, may be put on

too loosely, or too tightly, and thus cause the braces to be subjected to tensions or compressions under a symmetrical

loading from that cause alone all

;

indeed,

put on too

if

tightly,

the parts are subjected to stresses independent of any

load.

OBJECTIONS TO CLASS

The detect,

fact of misfitting

and

its

influence

must frequently be so

upon the

portant, that the judicious engineer

aversion to employing class,

when

III.

many

stresses

may

difficult to

may be

so im-

well entertain an

of the designs belonging to this

those of the former classes will

the examples of Class III. are, however,

suffice.

much

less

objection than others; and whether they should be

must depend upon the practical

difficulties

Some

of

open to

condemned

and sources of

CLASS error connected with them.

83

III.

There might be no good grounds

for entertaining apprehension of evil in employing such a

form as No. 174, which

is

indeed only a modification of a

girder with two series of triangular bracings

;

and the

evil

influence of misfitting almost vanishes in such structures as

Nos. 323, 324,

when a

etc.

;

but the danger becomes much more serious

structure such as

No. 132

to be used as a large

is

abutting roof principal, or No. 266 as an abutting bridge.

There

is, first,

the abutments

the difficulty of fitting ;

it

to the span between

second, the uncertainty of that distance be-

tween the abutments remaining of the estimated extent under the various influences of the thrusts, the settlements of the

masonry and foundations, the action of a watercourse or drain passing the footings, the pressure of earth behind the

abutments,

etc. etc.

;

and, third, the expansion or contraction

of the superstructure from change of temperature.

DIRECT USE OE DIAGRAMS OF FORCES IN CLASS

We

NO. 195.

III.,

might go through the form of drawing diagrams of

the stresses for such structures as 195,

etc.,

the diagonals can only act as struts or as

by assuming that ties,

and

that only one of each pair can act at one time.

further,

The

first

assumption would generally be materially incorrect in iron work, and the second of

all

is

only compatible with perfect fitting

the parts, and not always even then, as

the case of truss No. 167. tion,

The

error,

from the

would, however, be on the side of

these assumptions

Class

I.

is

we

safety.

have seen in first

assump-

But making

really just reducing the structure to

for each operation,

and there should now be no need

to give further illustrations of that class.

No. 195 at

is

the truss or principal used for the great roof

Birmingham; the span of the

When

largest truss is 211 feet.

occupied in taking out the stresses in this for the

DIAGRAMS OF FORCES.

84 Messrs.

Fox and Henderson

(Sept. 1851), I

drew Mr. C. H.

Wild's attention to the fact that, with the circular curves adopted, and with the structure uniformly loaded, the stress

on the main

tie

extremities.

(This

was greater at the centre than near the

may be seen S X or

me

137

in Fig.

W X.)

evidently greater than

ii.,

where

B X is

In reply, he showed

the following interesting relation that subsists between

structures with arched

and

we draw beneath

If

straight ties.

the diagram 340

showing the truss with arched straight tie-bar

tie,

i.

another, 340

and same span, and

also the

XV.)

(Plate ii.,

having a

same height

of framing at corresponding points in the span, as for instance