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UNIVERSITY OF CALIFORNIA

ARCHITECTURAL DEPARTMENT LIBRARY

I

GIFT OF

Llrs. Lydia Barth

COMPOUND RIVETED GIRDERS,

AS APPLIED IN THE

CONSTRUCTION OF BUILDINGS.

WITH NUMEROUS

PRACTICAL ILLUSTRATIONS AND TABLES.

BY

WILLIAM H. BIRKMIRE,

i

ATTTHOR OF "ARCHITECTURAL IRON AND STEEL*' AND

" SKELETON CONSTRUCTION IN BUILDINGS."

THIRD EDITION

NEW YORK:

JOHN WILEY & SONS.

LONDON : CHAPMAN & HALL, LIMITED.

1905

COPYRIGHT, 1893,

BY

WILLIAM H. BIRKMIRE.

ROBFRT DRUMMOND, ELECTROTYPER AND I RINTER, NEW YORK.

PREFACE.

IN order to facilitate the calculation attending the construe-

tion of Wrought Iron and Steel Riveted Girders, the author

has endeavored in this work to supply the link which separates

Theory from Practice. Its object may be briefly stated. A

riveted girder is to be designed ; the span, depth, and loads are

known, the strains are calculated by the well-known bending-

moment formulae, and largely by the graphic method ; lastly,

the details of construction are fully illustrated.

Touching the question of accuracy, it is scarcely necessary

to notice the slight difference that may arise between the two

methods, i.e., working out the usual formulae, or by measuring

from the graphic diagrams. The time consumed in wading

through a complicated series of equations to reach a few meas-

urements is objectionable when at least such measurements

can at once be had by the graphic method.

This work does not investigate exceptional or extremely

scientific riveted girders, but more especially those of a type

now extensively adopted and constructed by well-known archi-

tectural iron workers.

The diagrams and the various examples explaining the

Author's method are submitted to architects and architectural

students with the hope that they will become a medium of use-

fulness to them in the routine of office work.

WILLIAM H. BIRKMIRE.

NEW YORK, 1893.

57270 i

TABLE OF CONTENTS.

PART I.

THE STRAINS IN COMPOUND RIVETED GIRDERS.

PAGE

Compound riveted girders described i

Bending moments 2

Flanges 4

Shearing forces on the webs 5

Buckling of webs 7

Stiff eners 8

Riveting 8

Frict on of plates 10

Proportioning rivets 1 1

Rivets connecting webs with flanges 12

Spacing rivets according to strain produced in the flanges by the bending

moments ...* 15

Proportioning girders 16

Shearing and bearing resistance of rivets (Table) 16

Details of construction 17

Extract from the New York Building Law in relation to riveted girders... . 18

To calculate the approximate weight of girder before its dimensions are

fixed .... 19

Splicing 20

PART II.

QUALITY OF MA TERIAL.

Wrought-iron ^, 21

Limit of elasticity of wrought-iron 21

Ultimate strength of wrought-iron 21

Rivet iron 21

Mild steel... . 21

V

Vi TABLE OF CONTENTS.

PAGB

Ultimate strength and elongation 22

Rivet steel 22

Painting 22

PART III.

EXAMPLE 1.

Girder supporting a concentrated load at centre of span. . . 23

Construction of flanges in a girder supporting a concentrated load at

centre 25

Flanges reduced in area towards the supports in a girder supporting a con-

centrated load at centre 26

Webs proportioned in a girder supporting a concentrated load t centre of

span 28

Stiff eners in a girder supporting a concentrated load at centre of span 28

Rivet spacing in a girder supporting a concentrated load at centre of span. 29

Graphical representation of bending moments and shearing forces in a

girder with a concentrated load at centre of span 30

List of material and details of a girder supporting a concentrated load at

centre 32

Areas of angles with even legs (Table) 33

" " " " uneven legs (Table) 33

Sectional area in inches of rivet-holes in plates of various thicknesses

(Table) , 34

Gross area of plates of various thicknesses (Table) 35

Safe buckling value of web plates in wrought-iron (Table) 35

Shearing value of wrought-iron web plates (Table) 36

" " steel web plates (Table) 37

EXAMPLE If.

Girder supporting one concentrated load not at centre of span 38

Construction of flanges in a girder supporting one concentrated load not at

centre 40

Flanges reduced in area in a girder supporting one concentrated load not

at centre 41

Webs proportioned in a girder supporting a concentrated load not at centre 42

Stiff eners in a girder with one concentrated load not at centre 43

Spacing of rivets in a girder with one concentrated load not at centre 43

Graphical representation of bending moments and shearing forces in a girder

with one concentrated load not at centre of span 44

List of material and details of a girder supporting one concentrated load

not at centre of span 46

TABLE OF CONTENTS. VII

EXAMPLE III.

PAGE

Girder supporting a uniformly distributed load 47

Construction of flanges in a girder supporting a uniformly distributed load. 49

Flanges reduced in area towards the supports of a girder supporting a uni-

formly distributed load 49

Webs proportioned in a girder supporting a uniformly distributed load. ... 50

Stiffeners in a girder supporting a uniformly distributed load 51

Spacing of rivets in a girder supporting a uniformly distributed load 51

Method of drawing parabolas 52.

Parabola by the construction of a diagram 53

Graphical representation of bending moments and shearing forces in a

girder supporting a uniformly distributed load 54

List of material and details of a girder supporting a uniformly distributed

load 55

EXAMPLE IV.

Girder supporting two concentrated loads 56

Construction of flanges in a girder supporting two concentrated loads 58

Flanges reduced in area towards the supports in a girder supporting two

concentrated loads 59

Webs proportioned in a girder supporting two concentrated loads. , 60

Stiffeners in a girder supporting two concentrated loads 61

Spacing of rivets in a girder supporting two concentrated loads 61

Graphical representation of bending moments and shearing forces in a

girder supporting two concentrated loads 62

List of material and details of a girder supporting two concentrated loads. 64

EXAMPLE V.

Girder supporting two concentrated loads and a uniformly distributed load 65

Construction of flanges in a girder supporting two concentrated loads and

a uniformly distributed load 67

Webs proportioned in a girder supporting two concentrated loads and a

uniformly distributed load 68

Stiffeners in a girder supporting two concentrated loads and a uniformly

distributed load 69

Spacing of rivets in a girder supporting two concentrated loads and a uni-

formly distributed load 69

Graphical representation of bending moments and shearing forces in a

girder supporting two concentrated loads and a uniformly distributed

load 70

Flanges reduced in area towards the supports in a girder supporting two

concentrated loads and a uniformly distributed load by the funicular

polygon 71

List of material and details of a girder supporting two concentrated loads

and a uniformly distributed load 73

Vlll TABLE OF CONTENTS.

EXAMPLE VI.

PAGE

Girder supporting three concentrated loads 74

Construction of flanges in a girder supporting three concentrated loads. . . 76

Webs proportioned " " " " " " 77

Stiffened " " " " " "... 78

Spacing of rivets " " " " " "... 78

Flanges reduced in area towards the supports in a girder supporting three

concentrated loads 79

Graphical description of bending moments and shearing forces in a girder

supporting three concentrated loads 80

List of material and details of a girder supporting three concentrated

loads 82

EXAMPLE VII.

Girder supporting four concentrated loads. ... 83

Construction of flanges in a girder supporting fotir concentrated loads 85

Webs proportioned " " 85

Stiffeners " " " " " " 86

Spacing of rivets " " 86

Flanges reduced in area towards the supports in a girder supporting four

concentrated loads 87

Graphical representation of the bending moments and shearing forces in a

girder supporting four concentrated loads 88

List of material and details of a girder supporting four concentrated loads. 90

EXAMPLE VIII.

Steel girder supporting five concentrated loads 91

Determination of bending moments in a girder supporting five concentrated

loads 93

Construction of flanges in a girder supporting five concentrated loads 94

Stiffeners " " " " " " 95

Spacing of rivets " 9 5

Flanges reduced in area towards the supports in a girder supporting five

concentrated loads 96

Girder fixed at one end and loaded with a concentrated load at the other, as

a cantilever 97

Girder fixed at one end supporting a uniformly distributed load, as a canti-

lever 98

Girder fixed at one end supporting more than one load, as a cantilever 99

The relative strength of simple and cantilever girders; maximum vertical

shear, bending moments, and deflection (Table) 99

Modulus of elasticity of wrought-iron and steel in riveted girder as com-

pared with solid sections, as I-beams 99

Moment of Inertia for Rectangular sections 100

TABLE OF CONTENTS. IX

PART IV.

TABLES.

PAGE

Average weight in pounds of a cubic foot of various substances 101

Weight of 100 rivets in pounds 104

Decimal equivalents for fractions of a foot 105

Number of U. S. gallons contained in circular tanks 106

Decimal equivalents for fractions of an inch 106

Weight per lineal foot of cast-iron columns 107

Weight of square cast-iron columns per lineal foot 108

Weight per foot of flat iron 109

Table of squares and cubes in

Table of circles 115

Shearing and bearing resistance of rivets 16

Areas of angles with even legs 33

" " " " uneven legs 33

Sectional area in inches of rivet-holes in plates of various thicknesses 34

Gross area of plates of various thicknesses 35

Safe buckling value of web plates (wrought-iron) 35

Shearing value of wrought-iron web plates 36

" " steel web plates 37

LIST OF ILLUSTRATIONS.

PART I.

WG. . PAGE

1. Plate-girder section I

2. Plate-girder section with single flange plate I

3. Plate-girder section with three flange plates i

4. Box-girder section with three flange plates I

5. Box girder-section with three webs r

6. Girder with two loads supported upon a fulcrum 3

7. A simple girder supported at each end and load in middle 3

8. A lever held up with a weight at either end 5

9. A simple girder with load out of centre 6

10. A simple girder with a specified load out of centre 6

11. Two plates riveted with rivets in single shear 12

12. Plate-girder section with rivets in double shear 12

I2a. Box-girder section with rivets in single shear 12

13. Girder illustrating the strains on rivets connecting flange with web. . . 12

PART III.

14. Diagram of a girder with one concentrated load at centre 23

15. Diagram determining position of flange plates in a girder of one con-

centrated load at centre 26

16. Section of plate girder with stiffeners bent around chord angles 29

17. Section of plate girder with straight stiffeners and fillers 29

18. Diagram of the graphical representation of bending moments and

shearing forces of a plate girder with one concentrated load at

centre 31

19. Detail of girder of one concentrated load at centre 32

20. Diagram of a girder with one concentrated load not at centre 38

21. Diagram determining position of flange plates in a girder of one con-

centrated load not at centre 41

22. Diagram of the graphical representation of bending moments and

shearing forces in a girder with one concentrated load at centre. . 45

23. Detail of girder of one concentrated load not at centre 46

24. Diagram of a girder with a uniformly distributed load 47

25. Diagram determining position of flange plates in a girder supporting a

uniformly distributed load 50

xi

Xii LIST OF ILLUSTRATIONS.

IG. JACK

26. Diagram of a parabola by ordinates from a tangent to a paiahoia m

its vertex ..-,

27. Diagram of a parabola, by lines to two sides of an isosceles triangle.. 53

28. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting a uniformly distributed

load 54

29. Detail of girder supporting a uniformly distributed load 55

30. Diagram of a girder supporting two concentrated loads 56

31. Diagram determining position of flange plate in a girder supporting

two concentrated loads f 59

32. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting two concentrated loads 62

33. Detail of girder supporting two concentrated loads 64

34. Diagram of a girder supporting two concentrated loads and one uni-

formly distributed load 66

35. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting two concentrated loads and

one uniformly distributed load 70

36. Diagram determining position of flanged plates in a girder supporting

two concentrated loads and a uniformly distributed load 72

37. Detail girder supporting two concentrated loads and a uniformly dis-

tribute^ load 73

38. Diagram of a girder supporting three concentrated loads 75

39. Diagram determining position of flange plates in a girder supporting

three concentrated loads 79

40. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting three concentrated loads. . . 80

41. Detail of girde. supporting three concentrated loads 82

42. Diagram of a girder supporting four concentrated loads 84

43. Diagram determining position of flange plates in a girder of four con-

centrated loads 88

44. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting four concentrated loads. ... 89

45. Detail of girder supporting four concentrated loads 90

46. Diagram of a girder supporting five concentrated loads 92

47. Diagram determining position of flange plates in a girder supporting

five concentrated loads 96

48. Diagram of a girder secured at one end (as a cantilever) and supporting

a concentrated load at the other 98

49. Section of a plate girder, determining the notation used in the calcula-

tion of the section for the moment of inertia 100

50. Section of a box girder, determining the notation used in the calcula-

tion of the section for the moment of inertia . 100

PART I.

THE STRAIN IN COMPOUND RIVETED GIRDERS.

PART I.

THE STRAINS IN COMPOUND RIVETED GIRDERS.

For buildings, as well as railway and highway bridges, there

is probably no other form of girders more extensively used

than those made up of plates and angles, called Compound

Riveted Girders.

Some of the principal reasons for this lies mainly in the

simplicity of their construction ; they can be adopted for any

load or number of loads, and accommodated to any span

usually met with in the construction.

The single web or plate girder is more economical, more

accessible for painting and inspection. Formed of a single web

and four angles, as Fig. I, suitable for light loads and short

spans ; for heavier loads a single plate is added to the top

and bottom flanges, as shown in Fig. 2 ; for still heavier loads

additional plates, as in Fig. 3.

' IT T

. _JL jL

PIG. i. FIG. 2. FIG. 3. FIG. 4. FIG. 5.

Where thick walls are to be supported and lateral stiffness

is required, the double web or box girder, Fig. 4, or the triple

2 COMPOUND RIVETED GIRDERS.

web, Fig. 5, is employed. It also becomes necessary in man*

cases to place two plate or two box girders side by side.

The box girders as represented infection, Fig. 4, is consid-

ered superior to the plate girders represented in Figs, i, 2, and

3 ; but the preference should be given the latter on account

of its simplicity of construction, and although inferior in

strength to the box girder it has nevertheless other valuable

properties to recommend it.

On comparing the strengths of these separate girders, weight

for weight, it will be found that the box girder is as I to .93,

or nearly as 100 to 90. The difference in strength does not

arise from want of proportion in the top and bottom section

of either girder, but from the position of the material ; which

in that of the box girder offers greatly superior powers of re-

sistance to lateral flexure. The box girder, it will be observed,

contains larger exterior sectional area, and is consequently

stiffer and better calculated to resist lateral stress, in which

direction the plate girder generally yields before its other re-

sisting powers of tension and compression can be brought fully

into action. Taking this girder, however, in a position similar

to that in which it is used in supporting floor-beams and floor-

arches of buildings, its strength is very nearly equal to that of

the box shape, and, as previously mentioned, is of more simple

construction, less expensive, and more durable, from the cir-

cumstance that the web-plate is thicker than the web-plates

of the box girder, and it admits of easy access to all its parts

for purposes of painting, etc.

Bending Moments. Generally, the strength of a com-

pound riveted girder is founded on the equality that must

always exist between the resultant of the various loads tending

to cause its rupture and the strength of the material of which

the girder is composed. The former may be resolved horizon-

tally into strains, depending for their value upon what are

known as moments of rupture, bending moments, or leverage, of

THE STRAINS IN COMPOUND RIVETED GIRDERS. 3

greater or less complexity, tending to cause the failure of the

girder by tearing asunder its fibres in the bottom flange, crush-

ing them together in the top flange, and vertically upon the

web into what are known as shearing forces, due to the trans-

mission of the vertical pressure of the loads to the points of

support.

The strain produced in the flanges is resisted by a leverage

equal to the depth of the girder, that is, between the centre of

gravity of the flanges, and the amount per square inch of sec-

tion with which the metal may be safely trusted.

O

FIG. 6.

The bending moment is a compound quantity resulting

from the multiplication of a force by a distance, and desig-

nated by the letter M. The forces are expressed in tons or

pounds, and the distances in feet or inches ; then the bending

moments are in ton-feet or pound-inches.

FIG. 7.

If b or a is the arm of leverage, Fig. 6, and a load R or R

acts for a distance from W, M at W is equal to the load R

or R' multiplied by the distance b or a.

Then M '= Rb or R a.

The direction of the stresses upon the girder are vertical,

those at the ends being downwards, while that at the middle is

i 4 COMPOUND RIVETED GIRDERS.

upwards. In Fig. 7 we have a girder supported at both ends,

and a load W resting upon the middle of its length. Compar-

ing this with Fig. 6, we see that the stresses here are also verti-

cal,* but in reversed order, the one at the middle being

downwards, while those at the end are upwards. In other re-

spects we have the same conditions as in Fig. 6.

M does not represent strain, being independent of depth,

but is converted into flange strain by dividing by the

depth ; the strain then found, divided by the maximum unit

strain, determines the number of square inches to be given to

the flanges.

The maximum unit strain herein adopted is 6 tons (12,000

pounds) per square inch for wrought-iron and 7 tons (14,000

pounds) for steel.

Here A = Area of flange d = depth in feet, s = unit strain

in tons, and M = bending moment Then

ds

Flanges. Compound girders are unlike rolled beams, in

which every fibre is connected ; but have strains transmitted

only through rivets which are distributed only at certain dis-

tances apart ; consequently the flange angles are at every point

more or less subjected to strains in addition to their own.

This additional strain will evidently increase with the amount

of plates. It is good practice, therefore, to make the girder so

deep that the flanges do not require a number of plates to be

packed one upon another, and then to choose angles as heavy

as possible consistent with the total flange area required.

* " We have to distinguish between the outer forces which may act at various

portions of the girder tending to cause motion of its parts, and the inner forces

which prevent this motion. The first we may call stresses, and the second

strains. We therefore speak of the ' stresses ' upon a girder and the 'strains'

in a girder."

THE STRAINS IN COMPOUND RIVETED GIRDERS. 5

In order_to give the single-web girders the greatest amount

of resistance, it is usual to use angles with unequal legs with

the longer leg horizontal.

It sometimes becomes important to have the plates of the

top flange extend from end to end, even when angles may be

found which alone are sufficient to make up the required sec-

tion, as it gives great lateral stiffness to the flange, and also

helps to distribute the stress more uniformly than with the

angles alone.

In box girders the flange plate adjoining the angles are

required to extend from end to end.

In making up the bottom flange, rivet-holes must be de-

ducted to obtain the net section, and in so doing the diameter

of the rivet-hole should be taken at least inch larger ; this

latter provides to a certain extent for the damage done to the

strength of the metal in the process of punching or drilling.

For the top flange the gross sectional area may be taken

as making up the same, providing the riveting is well done,

i.e., the rivet completely filling up its own hole.

Shearing Forces on the Webs. It is by the law of the

lever that we are enabled to determine precisely what portion

of a given load resting upon a girder is sustained by either

point of support ; the loads balancing each other at either end

FIG. 8.

of a girder, or lever, on any point are to each other inversely

as their distances (called lever arms) from the point or ful-

crum.

For example: suppose we have a girder held up as in

Fig. 8, with a load at either end, the point of support being to

6 COMPOUND RIVETED GIRDERS.

one side of the centre, say one-fourth of the lever arm from one

end. In order that the lever be balanced, the load at W must

be one-fourth the sum of J'Fand W, and that at W three-

fourths that sum, for W multiplied by JZ, must always equal

^multiplied by \L, and the sustaining force P must of course

equal the sum of J^and W.

Again : supposing that there is one load as in Fig. 9. This

condition is the same as before, only reversed ; and, according

to the law of the lever, we find that for equilibrium a force must

be applied to R equal to ^W. This example is precisely the

same as that of a girder, only R and R' are now called reactions

of the supports, the sum of which must always be equal to the

load or number of loads causing them.

In order, then, to know just how much of the load or num-

ber of loads at any point of the girder is supported by either

support, all that is necessary to be done is to multiply the

shorter or longer distance by the load and divide the product

by the span L.

LE

FIG. 10.

Example : Suppose we have a girder R and R' , Fig. 10, of 25

feet span, and there is a load of 20 tons 5 feet from R'. Then

each sustains a certain amount of this load proportionately to

THE STRAINS IN COMPOUND RIVETD GIRDERS. J

its distance from the load, the sum of the reactions being equal

to the load,

R supports - - - = 4 tons ;

K ?-2 = I6 tons.

The most practical way of proportioning the web is then

to make its section sufficient to resist this entire shearing force

at either end of the girder. Under the supposition that the

flanges alone resist the entire bending moment, and the web

only the shearing action, the following formula can be

adopted :

Let S = shearing stress ;

A = area at point of stress ;

K = effective resistance to shearing ;

t = thickness of web ;

d = depth of web in inches.

c

Atd and S= ktd, or t = -^.

aK.

The safe shear on the webs per square inch herein adopted

is 6000 Ibs. for wrought-iron and 7000 Ibs. for steel.

Example. Suppose we take the above wrought-iron plate

girder, Fig. 10, and have 16 tons (32,000 Ibs.) shear on the

web at R r support, the web being 12 inches in depth.

t = = - > = .44 more than T V of an inch in

dk 12 X 6000

thickness.

Buckling of Web. The web is still in danger of buckling

under this compression stress ; consequently the web with its

thickness as already proportioned for shearing must now be

ARCHITECTURAL DEPARTMENT LIBRARY

I

GIFT OF

Llrs. Lydia Barth

COMPOUND RIVETED GIRDERS,

AS APPLIED IN THE

CONSTRUCTION OF BUILDINGS.

WITH NUMEROUS

PRACTICAL ILLUSTRATIONS AND TABLES.

BY

WILLIAM H. BIRKMIRE,

i

ATTTHOR OF "ARCHITECTURAL IRON AND STEEL*' AND

" SKELETON CONSTRUCTION IN BUILDINGS."

THIRD EDITION

NEW YORK:

JOHN WILEY & SONS.

LONDON : CHAPMAN & HALL, LIMITED.

1905

COPYRIGHT, 1893,

BY

WILLIAM H. BIRKMIRE.

ROBFRT DRUMMOND, ELECTROTYPER AND I RINTER, NEW YORK.

PREFACE.

IN order to facilitate the calculation attending the construe-

tion of Wrought Iron and Steel Riveted Girders, the author

has endeavored in this work to supply the link which separates

Theory from Practice. Its object may be briefly stated. A

riveted girder is to be designed ; the span, depth, and loads are

known, the strains are calculated by the well-known bending-

moment formulae, and largely by the graphic method ; lastly,

the details of construction are fully illustrated.

Touching the question of accuracy, it is scarcely necessary

to notice the slight difference that may arise between the two

methods, i.e., working out the usual formulae, or by measuring

from the graphic diagrams. The time consumed in wading

through a complicated series of equations to reach a few meas-

urements is objectionable when at least such measurements

can at once be had by the graphic method.

This work does not investigate exceptional or extremely

scientific riveted girders, but more especially those of a type

now extensively adopted and constructed by well-known archi-

tectural iron workers.

The diagrams and the various examples explaining the

Author's method are submitted to architects and architectural

students with the hope that they will become a medium of use-

fulness to them in the routine of office work.

WILLIAM H. BIRKMIRE.

NEW YORK, 1893.

57270 i

TABLE OF CONTENTS.

PART I.

THE STRAINS IN COMPOUND RIVETED GIRDERS.

PAGE

Compound riveted girders described i

Bending moments 2

Flanges 4

Shearing forces on the webs 5

Buckling of webs 7

Stiff eners 8

Riveting 8

Frict on of plates 10

Proportioning rivets 1 1

Rivets connecting webs with flanges 12

Spacing rivets according to strain produced in the flanges by the bending

moments ...* 15

Proportioning girders 16

Shearing and bearing resistance of rivets (Table) 16

Details of construction 17

Extract from the New York Building Law in relation to riveted girders... . 18

To calculate the approximate weight of girder before its dimensions are

fixed .... 19

Splicing 20

PART II.

QUALITY OF MA TERIAL.

Wrought-iron ^, 21

Limit of elasticity of wrought-iron 21

Ultimate strength of wrought-iron 21

Rivet iron 21

Mild steel... . 21

V

Vi TABLE OF CONTENTS.

PAGB

Ultimate strength and elongation 22

Rivet steel 22

Painting 22

PART III.

EXAMPLE 1.

Girder supporting a concentrated load at centre of span. . . 23

Construction of flanges in a girder supporting a concentrated load at

centre 25

Flanges reduced in area towards the supports in a girder supporting a con-

centrated load at centre 26

Webs proportioned in a girder supporting a concentrated load t centre of

span 28

Stiff eners in a girder supporting a concentrated load at centre of span 28

Rivet spacing in a girder supporting a concentrated load at centre of span. 29

Graphical representation of bending moments and shearing forces in a

girder with a concentrated load at centre of span 30

List of material and details of a girder supporting a concentrated load at

centre 32

Areas of angles with even legs (Table) 33

" " " " uneven legs (Table) 33

Sectional area in inches of rivet-holes in plates of various thicknesses

(Table) , 34

Gross area of plates of various thicknesses (Table) 35

Safe buckling value of web plates in wrought-iron (Table) 35

Shearing value of wrought-iron web plates (Table) 36

" " steel web plates (Table) 37

EXAMPLE If.

Girder supporting one concentrated load not at centre of span 38

Construction of flanges in a girder supporting one concentrated load not at

centre 40

Flanges reduced in area in a girder supporting one concentrated load not

at centre 41

Webs proportioned in a girder supporting a concentrated load not at centre 42

Stiff eners in a girder with one concentrated load not at centre 43

Spacing of rivets in a girder with one concentrated load not at centre 43

Graphical representation of bending moments and shearing forces in a girder

with one concentrated load not at centre of span 44

List of material and details of a girder supporting one concentrated load

not at centre of span 46

TABLE OF CONTENTS. VII

EXAMPLE III.

PAGE

Girder supporting a uniformly distributed load 47

Construction of flanges in a girder supporting a uniformly distributed load. 49

Flanges reduced in area towards the supports of a girder supporting a uni-

formly distributed load 49

Webs proportioned in a girder supporting a uniformly distributed load. ... 50

Stiffeners in a girder supporting a uniformly distributed load 51

Spacing of rivets in a girder supporting a uniformly distributed load 51

Method of drawing parabolas 52.

Parabola by the construction of a diagram 53

Graphical representation of bending moments and shearing forces in a

girder supporting a uniformly distributed load 54

List of material and details of a girder supporting a uniformly distributed

load 55

EXAMPLE IV.

Girder supporting two concentrated loads 56

Construction of flanges in a girder supporting two concentrated loads 58

Flanges reduced in area towards the supports in a girder supporting two

concentrated loads 59

Webs proportioned in a girder supporting two concentrated loads. , 60

Stiffeners in a girder supporting two concentrated loads 61

Spacing of rivets in a girder supporting two concentrated loads 61

Graphical representation of bending moments and shearing forces in a

girder supporting two concentrated loads 62

List of material and details of a girder supporting two concentrated loads. 64

EXAMPLE V.

Girder supporting two concentrated loads and a uniformly distributed load 65

Construction of flanges in a girder supporting two concentrated loads and

a uniformly distributed load 67

Webs proportioned in a girder supporting two concentrated loads and a

uniformly distributed load 68

Stiffeners in a girder supporting two concentrated loads and a uniformly

distributed load 69

Spacing of rivets in a girder supporting two concentrated loads and a uni-

formly distributed load 69

Graphical representation of bending moments and shearing forces in a

girder supporting two concentrated loads and a uniformly distributed

load 70

Flanges reduced in area towards the supports in a girder supporting two

concentrated loads and a uniformly distributed load by the funicular

polygon 71

List of material and details of a girder supporting two concentrated loads

and a uniformly distributed load 73

Vlll TABLE OF CONTENTS.

EXAMPLE VI.

PAGE

Girder supporting three concentrated loads 74

Construction of flanges in a girder supporting three concentrated loads. . . 76

Webs proportioned " " " " " " 77

Stiffened " " " " " "... 78

Spacing of rivets " " " " " "... 78

Flanges reduced in area towards the supports in a girder supporting three

concentrated loads 79

Graphical description of bending moments and shearing forces in a girder

supporting three concentrated loads 80

List of material and details of a girder supporting three concentrated

loads 82

EXAMPLE VII.

Girder supporting four concentrated loads. ... 83

Construction of flanges in a girder supporting fotir concentrated loads 85

Webs proportioned " " 85

Stiffeners " " " " " " 86

Spacing of rivets " " 86

Flanges reduced in area towards the supports in a girder supporting four

concentrated loads 87

Graphical representation of the bending moments and shearing forces in a

girder supporting four concentrated loads 88

List of material and details of a girder supporting four concentrated loads. 90

EXAMPLE VIII.

Steel girder supporting five concentrated loads 91

Determination of bending moments in a girder supporting five concentrated

loads 93

Construction of flanges in a girder supporting five concentrated loads 94

Stiffeners " " " " " " 95

Spacing of rivets " 9 5

Flanges reduced in area towards the supports in a girder supporting five

concentrated loads 96

Girder fixed at one end and loaded with a concentrated load at the other, as

a cantilever 97

Girder fixed at one end supporting a uniformly distributed load, as a canti-

lever 98

Girder fixed at one end supporting more than one load, as a cantilever 99

The relative strength of simple and cantilever girders; maximum vertical

shear, bending moments, and deflection (Table) 99

Modulus of elasticity of wrought-iron and steel in riveted girder as com-

pared with solid sections, as I-beams 99

Moment of Inertia for Rectangular sections 100

TABLE OF CONTENTS. IX

PART IV.

TABLES.

PAGE

Average weight in pounds of a cubic foot of various substances 101

Weight of 100 rivets in pounds 104

Decimal equivalents for fractions of a foot 105

Number of U. S. gallons contained in circular tanks 106

Decimal equivalents for fractions of an inch 106

Weight per lineal foot of cast-iron columns 107

Weight of square cast-iron columns per lineal foot 108

Weight per foot of flat iron 109

Table of squares and cubes in

Table of circles 115

Shearing and bearing resistance of rivets 16

Areas of angles with even legs 33

" " " " uneven legs 33

Sectional area in inches of rivet-holes in plates of various thicknesses 34

Gross area of plates of various thicknesses 35

Safe buckling value of web plates (wrought-iron) 35

Shearing value of wrought-iron web plates 36

" " steel web plates 37

LIST OF ILLUSTRATIONS.

PART I.

WG. . PAGE

1. Plate-girder section I

2. Plate-girder section with single flange plate I

3. Plate-girder section with three flange plates i

4. Box-girder section with three flange plates I

5. Box girder-section with three webs r

6. Girder with two loads supported upon a fulcrum 3

7. A simple girder supported at each end and load in middle 3

8. A lever held up with a weight at either end 5

9. A simple girder with load out of centre 6

10. A simple girder with a specified load out of centre 6

11. Two plates riveted with rivets in single shear 12

12. Plate-girder section with rivets in double shear 12

I2a. Box-girder section with rivets in single shear 12

13. Girder illustrating the strains on rivets connecting flange with web. . . 12

PART III.

14. Diagram of a girder with one concentrated load at centre 23

15. Diagram determining position of flange plates in a girder of one con-

centrated load at centre 26

16. Section of plate girder with stiffeners bent around chord angles 29

17. Section of plate girder with straight stiffeners and fillers 29

18. Diagram of the graphical representation of bending moments and

shearing forces of a plate girder with one concentrated load at

centre 31

19. Detail of girder of one concentrated load at centre 32

20. Diagram of a girder with one concentrated load not at centre 38

21. Diagram determining position of flange plates in a girder of one con-

centrated load not at centre 41

22. Diagram of the graphical representation of bending moments and

shearing forces in a girder with one concentrated load at centre. . 45

23. Detail of girder of one concentrated load not at centre 46

24. Diagram of a girder with a uniformly distributed load 47

25. Diagram determining position of flange plates in a girder supporting a

uniformly distributed load 50

xi

Xii LIST OF ILLUSTRATIONS.

IG. JACK

26. Diagram of a parabola by ordinates from a tangent to a paiahoia m

its vertex ..-,

27. Diagram of a parabola, by lines to two sides of an isosceles triangle.. 53

28. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting a uniformly distributed

load 54

29. Detail of girder supporting a uniformly distributed load 55

30. Diagram of a girder supporting two concentrated loads 56

31. Diagram determining position of flange plate in a girder supporting

two concentrated loads f 59

32. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting two concentrated loads 62

33. Detail of girder supporting two concentrated loads 64

34. Diagram of a girder supporting two concentrated loads and one uni-

formly distributed load 66

35. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting two concentrated loads and

one uniformly distributed load 70

36. Diagram determining position of flanged plates in a girder supporting

two concentrated loads and a uniformly distributed load 72

37. Detail girder supporting two concentrated loads and a uniformly dis-

tribute^ load 73

38. Diagram of a girder supporting three concentrated loads 75

39. Diagram determining position of flange plates in a girder supporting

three concentrated loads 79

40. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting three concentrated loads. . . 80

41. Detail of girde. supporting three concentrated loads 82

42. Diagram of a girder supporting four concentrated loads 84

43. Diagram determining position of flange plates in a girder of four con-

centrated loads 88

44. Diagram of the graphical representation of bending moments and

shearing forces in a girder supporting four concentrated loads. ... 89

45. Detail of girder supporting four concentrated loads 90

46. Diagram of a girder supporting five concentrated loads 92

47. Diagram determining position of flange plates in a girder supporting

five concentrated loads 96

48. Diagram of a girder secured at one end (as a cantilever) and supporting

a concentrated load at the other 98

49. Section of a plate girder, determining the notation used in the calcula-

tion of the section for the moment of inertia 100

50. Section of a box girder, determining the notation used in the calcula-

tion of the section for the moment of inertia . 100

PART I.

THE STRAIN IN COMPOUND RIVETED GIRDERS.

PART I.

THE STRAINS IN COMPOUND RIVETED GIRDERS.

For buildings, as well as railway and highway bridges, there

is probably no other form of girders more extensively used

than those made up of plates and angles, called Compound

Riveted Girders.

Some of the principal reasons for this lies mainly in the

simplicity of their construction ; they can be adopted for any

load or number of loads, and accommodated to any span

usually met with in the construction.

The single web or plate girder is more economical, more

accessible for painting and inspection. Formed of a single web

and four angles, as Fig. I, suitable for light loads and short

spans ; for heavier loads a single plate is added to the top

and bottom flanges, as shown in Fig. 2 ; for still heavier loads

additional plates, as in Fig. 3.

' IT T

. _JL jL

PIG. i. FIG. 2. FIG. 3. FIG. 4. FIG. 5.

Where thick walls are to be supported and lateral stiffness

is required, the double web or box girder, Fig. 4, or the triple

2 COMPOUND RIVETED GIRDERS.

web, Fig. 5, is employed. It also becomes necessary in man*

cases to place two plate or two box girders side by side.

The box girders as represented infection, Fig. 4, is consid-

ered superior to the plate girders represented in Figs, i, 2, and

3 ; but the preference should be given the latter on account

of its simplicity of construction, and although inferior in

strength to the box girder it has nevertheless other valuable

properties to recommend it.

On comparing the strengths of these separate girders, weight

for weight, it will be found that the box girder is as I to .93,

or nearly as 100 to 90. The difference in strength does not

arise from want of proportion in the top and bottom section

of either girder, but from the position of the material ; which

in that of the box girder offers greatly superior powers of re-

sistance to lateral flexure. The box girder, it will be observed,

contains larger exterior sectional area, and is consequently

stiffer and better calculated to resist lateral stress, in which

direction the plate girder generally yields before its other re-

sisting powers of tension and compression can be brought fully

into action. Taking this girder, however, in a position similar

to that in which it is used in supporting floor-beams and floor-

arches of buildings, its strength is very nearly equal to that of

the box shape, and, as previously mentioned, is of more simple

construction, less expensive, and more durable, from the cir-

cumstance that the web-plate is thicker than the web-plates

of the box girder, and it admits of easy access to all its parts

for purposes of painting, etc.

Bending Moments. Generally, the strength of a com-

pound riveted girder is founded on the equality that must

always exist between the resultant of the various loads tending

to cause its rupture and the strength of the material of which

the girder is composed. The former may be resolved horizon-

tally into strains, depending for their value upon what are

known as moments of rupture, bending moments, or leverage, of

THE STRAINS IN COMPOUND RIVETED GIRDERS. 3

greater or less complexity, tending to cause the failure of the

girder by tearing asunder its fibres in the bottom flange, crush-

ing them together in the top flange, and vertically upon the

web into what are known as shearing forces, due to the trans-

mission of the vertical pressure of the loads to the points of

support.

The strain produced in the flanges is resisted by a leverage

equal to the depth of the girder, that is, between the centre of

gravity of the flanges, and the amount per square inch of sec-

tion with which the metal may be safely trusted.

O

FIG. 6.

The bending moment is a compound quantity resulting

from the multiplication of a force by a distance, and desig-

nated by the letter M. The forces are expressed in tons or

pounds, and the distances in feet or inches ; then the bending

moments are in ton-feet or pound-inches.

FIG. 7.

If b or a is the arm of leverage, Fig. 6, and a load R or R

acts for a distance from W, M at W is equal to the load R

or R' multiplied by the distance b or a.

Then M '= Rb or R a.

The direction of the stresses upon the girder are vertical,

those at the ends being downwards, while that at the middle is

i 4 COMPOUND RIVETED GIRDERS.

upwards. In Fig. 7 we have a girder supported at both ends,

and a load W resting upon the middle of its length. Compar-

ing this with Fig. 6, we see that the stresses here are also verti-

cal,* but in reversed order, the one at the middle being

downwards, while those at the end are upwards. In other re-

spects we have the same conditions as in Fig. 6.

M does not represent strain, being independent of depth,

but is converted into flange strain by dividing by the

depth ; the strain then found, divided by the maximum unit

strain, determines the number of square inches to be given to

the flanges.

The maximum unit strain herein adopted is 6 tons (12,000

pounds) per square inch for wrought-iron and 7 tons (14,000

pounds) for steel.

Here A = Area of flange d = depth in feet, s = unit strain

in tons, and M = bending moment Then

ds

Flanges. Compound girders are unlike rolled beams, in

which every fibre is connected ; but have strains transmitted

only through rivets which are distributed only at certain dis-

tances apart ; consequently the flange angles are at every point

more or less subjected to strains in addition to their own.

This additional strain will evidently increase with the amount

of plates. It is good practice, therefore, to make the girder so

deep that the flanges do not require a number of plates to be

packed one upon another, and then to choose angles as heavy

as possible consistent with the total flange area required.

* " We have to distinguish between the outer forces which may act at various

portions of the girder tending to cause motion of its parts, and the inner forces

which prevent this motion. The first we may call stresses, and the second

strains. We therefore speak of the ' stresses ' upon a girder and the 'strains'

in a girder."

THE STRAINS IN COMPOUND RIVETED GIRDERS. 5

In order_to give the single-web girders the greatest amount

of resistance, it is usual to use angles with unequal legs with

the longer leg horizontal.

It sometimes becomes important to have the plates of the

top flange extend from end to end, even when angles may be

found which alone are sufficient to make up the required sec-

tion, as it gives great lateral stiffness to the flange, and also

helps to distribute the stress more uniformly than with the

angles alone.

In box girders the flange plate adjoining the angles are

required to extend from end to end.

In making up the bottom flange, rivet-holes must be de-

ducted to obtain the net section, and in so doing the diameter

of the rivet-hole should be taken at least inch larger ; this

latter provides to a certain extent for the damage done to the

strength of the metal in the process of punching or drilling.

For the top flange the gross sectional area may be taken

as making up the same, providing the riveting is well done,

i.e., the rivet completely filling up its own hole.

Shearing Forces on the Webs. It is by the law of the

lever that we are enabled to determine precisely what portion

of a given load resting upon a girder is sustained by either

point of support ; the loads balancing each other at either end

FIG. 8.

of a girder, or lever, on any point are to each other inversely

as their distances (called lever arms) from the point or ful-

crum.

For example: suppose we have a girder held up as in

Fig. 8, with a load at either end, the point of support being to

6 COMPOUND RIVETED GIRDERS.

one side of the centre, say one-fourth of the lever arm from one

end. In order that the lever be balanced, the load at W must

be one-fourth the sum of J'Fand W, and that at W three-

fourths that sum, for W multiplied by JZ, must always equal

^multiplied by \L, and the sustaining force P must of course

equal the sum of J^and W.

Again : supposing that there is one load as in Fig. 9. This

condition is the same as before, only reversed ; and, according

to the law of the lever, we find that for equilibrium a force must

be applied to R equal to ^W. This example is precisely the

same as that of a girder, only R and R' are now called reactions

of the supports, the sum of which must always be equal to the

load or number of loads causing them.

In order, then, to know just how much of the load or num-

ber of loads at any point of the girder is supported by either

support, all that is necessary to be done is to multiply the

shorter or longer distance by the load and divide the product

by the span L.

LE

FIG. 10.

Example : Suppose we have a girder R and R' , Fig. 10, of 25

feet span, and there is a load of 20 tons 5 feet from R'. Then

each sustains a certain amount of this load proportionately to

THE STRAINS IN COMPOUND RIVETD GIRDERS. J

its distance from the load, the sum of the reactions being equal

to the load,

R supports - - - = 4 tons ;

K ?-2 = I6 tons.

The most practical way of proportioning the web is then

to make its section sufficient to resist this entire shearing force

at either end of the girder. Under the supposition that the

flanges alone resist the entire bending moment, and the web

only the shearing action, the following formula can be

adopted :

Let S = shearing stress ;

A = area at point of stress ;

K = effective resistance to shearing ;

t = thickness of web ;

d = depth of web in inches.

c

Atd and S= ktd, or t = -^.

aK.

The safe shear on the webs per square inch herein adopted

is 6000 Ibs. for wrought-iron and 7000 Ibs. for steel.

Example. Suppose we take the above wrought-iron plate

girder, Fig. 10, and have 16 tons (32,000 Ibs.) shear on the

web at R r support, the web being 12 inches in depth.

t = = - > = .44 more than T V of an inch in

dk 12 X 6000

thickness.

Buckling of Web. The web is still in danger of buckling

under this compression stress ; consequently the web with its

thickness as already proportioned for shearing must now be

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