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PREFACE

THE following Notes on Plate-Girder Design have

been used by the author to give his c ivil engineeri ng

students the theoretical and practical information nec

essary to enable them to make a design and general

detailed drawing for a through plate-girder railway bridge.

The theory of the plate girder has been developed so

that it may be applied to such structures for any duty .

The notes may easily be given to the average class in

one recitation per week for one semester of the college

year . They have greatly facilitated the computation and

drawing—room work with my own classes .

I t is with the hope that they may be useful to other

teachers, and engim ers,that they are published .

C. W . HUDSON .

POLYTECHN IC I NSTI TUTE, BROOKLYN , N . Y.,

January, 12, 1911 .

CONTENTS

ART I CLE 1 . STRESS D I STRI BUTI ON AND GEN ERAL

Theory of stress distribution . Limits of the use Of the pla tegirder . Design and detail drawing for a single-track deck pla teg irder bridge . Design and detail d rawing for a single

-track throughplate

-

girder bridge. The weight of a plate g irder . Problems .

ARTI CLE 2 . REQUIRED AREA OF CROSS-SECT ION FOR THE FLANGES .

Figures showing make up of cross-section for plate g irders.

Formula for Obtaining the required area of the tension flange.

P roblems .

ARTI CLE 3 . THE DESI GN OF THE CRoss-SECTI ON OF THE FLAN GES

Composition of the flanges . Minimum pitch of rivets in tensionflanges . Formula for a safe unit st ress for the compression flange.

Thickness of material in compression flanges . Problems .

ARTI CLE 4 . LEN GTHS OF COVER PLATES

Formulas for the length of cover plates for g irders car rying uniformand concentrated loads by analyticmethods . Graphicdeterminationof lengths of cover plates . Examples of the application of the

formulas for finding lengths of cover pla tes. P roblems.

ARTI CLE 5. RIVET SPA-CI NG I N GIRDER FLANGES

Rivet spacing a function of the shear . Formulas for spacing

rivets connecting flanges to the web . Formulas for spacing the

rivets connecting the“ componen t parts of the flanges. Formula forrivet spacing at the ends of cover plates. Problems.

vi CONTENTS

PAGE

ARTI CLE 6. WEB PLATES

Stresses on the faces of an element of the web . St resses of Ina-xim inn intensity and the planes on which they act. Compu tat ion of

the stress intensity at various points in the web for a definite case.

Problems .

ARTI CLE 7 . AREA or W EB PLATES

Discussion of stress dist ribution throughout the web pla te . Rela

lation of shearing and compression unit s t resses . Unit stressesfor shear and compression in web plates . P roblems .

ARTICLE 8. S'I ‘I FFEN ERS FOR WEB PLATES

Functions of stifl'

eners . I n termediate stifieners . Concentra tedload stifieners . Formulas forsize of stifleners . Spacing of stifieners .

Problems .

ARTI CLE 9 . SPLI CES FOR WEB PLATES

Arrangement of splices. Duty of a web splice . Thickness of

web splice, plates . Number and arrangement of rive ts in the splice .

Maximum stresses (in the rivets of aweb splice . Problems .

ARTI CLE 10 . SPLI CES FOR THE COM PONENT PARTS OF THE FLANGES .

Loca tion of splices in flange material. Disposition of flangesplicing material . Rivets for splicing the componen t parts of theflanges . Problem .

ARTI CLE 11 . CONNECTI NG ONE GIRDER ToANOTHER

Size and arrangemen t of the material for the connection . Rivetsfor the various parts of the connection .

ARTI CLE 12 . END BEAR I NGS

The du ty of end bearings . Area of bearing on masonry. Properleng th and W idth of bearings . Thickness of bearing plates . Different forms of end bearings . Problem .

CONTENTS vii

PAG E

ARTI CLE 13 . POS I TI ONS or LOAD I NG FOR MAXIMUM SHEAR AN D

Maximum en'

d shea r . Maximum shear at any poin t . Maximum

moment at anydefinite point. Absolute maximum momen t and thepoint at which it occurs . Problems.

ARTI CLE 14 . PREPARATI ON OF A TABLE OF BEN DI N G MOM ENTS , SHEARS ,

AND CON CENTRATED LOADS FOR COOPER’

S E50 LOADIN G

Advantages of tables of momen ts and shears . Equa tions for

maximum moments . Preparation of a table of moments . Tableof, and sketches showing positions of loads for maximum moments .

Problems .

ARTI CLE 15. TABLE OF MOM ENTS,SHEARS

,AN D CONCENTRATED LOADS

FOR COOPER’

S E50

NOTES ON PLATE-G IRDER DESI GN

ART. 1. STRESS DISTRI BUTION AND GENERAL

The theory of stress distribution for plate girders is the

same as that for solid beams of uniform material , it being

generally assumed that the common theory of flexure applies

with sufficient accuracy to the built-up girder . This assumption is certainly not strictly true . The rivets connecting

the various parts together cause a local and small irregular

ity of stress distribution ; splices o f the various parts , and

non-

prismat ic form, produce quite marked irregularity of this

distribution . The Theory Of Flexure properly applied

to plate-girder design , however, leads to a thoroughly good

engineering structure , as an immense tonnage of such work

constructed during the past forty years and doing excel

lent duty witnesses .

This wide use o f the plate girder for service under

greatly varying conditions has increased the knowledge o f

its capacity and given valuable practical in formation as

to the design of certain o f its features .

The requirements for its fabrication,

Shipment and

erection have also affected its design . These requirements

are not fixed in their nature , although in certain respects ,

2 NOTES ON PLATE-GIRDER DESI GN

suCh as minimum thickness of material in web plates,

and minimum edge distances for punching, they fix a limit

to the size of certain parts .

The correct design of plate girders therefore requires

theoretical knowledge and either practical experience or

formulated rules,based on practical experience ,

and so

carefully and c losely drawn as to prevent a poor design .

Plate girders are used for bridges anywhere from 18

deep and 15' long to 126 deep and 130

' long,and even

beyond these limits ; thev are also used in buildings and

other important engineering work to a great extent . I n

general , the lower limit of their use is the I beam ,which

will furnish the proper strength at a less cost per pound

of material ; their upper limit is the truss,whose total

cost is less than the heavier but cheaper per pound girder

I n order to design properly any structure it is necessary

to understand the composition and relat ion o f each to

each o f the various parts . The following drawings are

meant to illustrate some of the most essential features of

plate-girder construction . They are therefore not meant

to be casually inspected , but thoroughly studied , and the

function of the various parts in carry ing a load to the

supports , the makeup of the parts and their connect ion,

each to each clearly understood .

Fig . 1a shows a s ingle-track dec k plate-girder bridge .

Fig . 1b shows a single-trac k through plate-girder bridge .

The computation of the stresses in any structure is

the preliminary step in the des ign . The stresses themselves

are a function o f the weight o f the structure,and hence

an early est imate of the weight o f each portion of the

u I

86x Xx LxH

Lateral Pl

2 Ls s'

i ai yj

Stringer 20 l 65

Cente r Line

N

2 Ls cx s'

§ 9{

nW eb 72 xa;

xCov. PL 18} x

"

: 40,

1 Cov. Pl. 13'x

'

x 35

45—8-Glear

ated We igh tLoad

Live Load Diag ram per T rack

_lI I "

0

3 % Rock bolt'every third tie

3if : 8Guard Rail

gem Pl. 9"x7 d

’

St ringersBend ing Moment inch lbs.

End Shear lbs.

51 I bet.Back W alls

G irderChord StressLoad on end shoe

STANDARD NO. 12

50’—0 THROUGH PLATE GIRDER SPAN . TH E NATI O

LI NES or M EX lCO

Scale A”

1'- 0

" Ap ril, 1907BOLLER HODGE

,

Consulting Engineers , New

STRESS D ISTRI BUTION AND GENERAL 3

structure is necessary . Many valuable compilat ions of

weights o f structures are to be had, but they are not usually

ac companied by full data . Every engineer must be sure

o f the quantit ies which go into his structure , and they

therefore should be estimated as the computations are

carried along and all computations based on wrong assumed

dead loads corrected .

The des ign of a structure Should always begin with

the part receiving the load and fo llow with each successive

part in the path of the load as it travels to the support .

For example, in an ordinary through p late-girder rail

road bridge, the des ign of the various parts should be

made in the following order : Rails , t ies , stringers , floor

beams , lateral brac ing , and main girders .

I n making a first est imate Of the weight Of any girder ,

compute the live- load stress at the center of the bottom

flange and increase this for impact if any is to be added ,

add to this sum 15% for the stress due to the weight o f

the girder , then determine the net area Of this member in

square inches . This net area in square inches multiplied

by 14 will give a weight per foot o f girder which may be

used as a first est imate .

This est imate is based on equal gross areas 15% g reater

than the net area Of the bottom flange , for top flange,

bottom flange , and web , and their sum increased 15% for

details . An error Of 1% in the total stress due to a wrong

first assumed dead load Should require a corrected dead

load to be used .

I t is Very d ifficult to give a good general rule for est i

mat ing the weight of g irders , as the weight obviouslv is a

4 NOTES ON PLATE—GIRDER DESIGN

function of the depth, loading , unit stresses , and many

other things .

PROBLEMS

1a . Sketch the cross-section of a plate g irder with and without

cover plates and write the name of each part on the Sketches .

1h. Sketch the plan of single-track railroad deck plate-girder bridge

and write the names Of the various parts on the sketch .

1c. Sketch the plan of single-track railroad through plate-girderbridge, and write the names Of the various parts in the sketch .

ART. 2. REQUI RED AREA OF CROSS-SECTION FOR THE

FLANGES

The plate girder is a built—up structure , and may be made

with a variety of cross-sect ions depending on the require

ments of the case . The forms of crOSS—section most fre

quently used are those of a,b and c of Fig . 2 .

Top fl ange

7 r

W eb Plate

Bottom Flange

The object of these forms being to take advantage of

the known law of stress distribution in flexure .

The relation between the moment of the outer forces

and inner stresses for homogeneous bodies subject to flexure

is given from the formula

AREA OF CROSS-SECTION FOR FLANGES 5

M is the moment of the outer forces ;S is the unit stress in the extreme fiber of the

cross-section under considerat ion ;I is the moment of inertia of the cross-section ;

c is the distance from the neutral axis to the

extreme fiber o f the cross-section .

Any existing straight prismat ic girder may be examined

for intensity o f flexural s tres s in a plane normal to the

axis means of this formula. plate girder prac

tice however, far

Theory o f Flexure .

mula to the design

it could be greatly facilitated by the use of tables of

for a great variety of eas es , the labor Of its application

to problems of design is not warranted .

The almost universal method of plate-girder flange

des ign wi ll now be developed , with the. aid o f the following

nomenclature and Fig . 2d .

from the ideal prismatic

The application of the preceding

Of girders is very tedious,and while

I

G

6 NOTES ON PLATE-GIRDER DESIGN

Let A be the net area of each flange ;h be the distance 0 . to c . of gravity of flanges ;

S be the allowable unit stress for tension or com

pression produced by flexure ;‘hl be the depth of the web plate ;t be the thickness o f the web plate ;hz be the depth out to out of flanges ;I f be moment of inertia of each flange about own

horizontal axis .

Then the flexure formula becomes

S2 XAX

0 ha

i thl h,M — S -hl -

h—

2

+

right half of this equat ion represents the moment

of the internal stresses ; of the three terms which make

up this half it may be said :

41;The term

hzis small in most cases , but for Shallow

girders it is relatively quite large, its omission from the

Cxpression requires the other terms to be larger .

hThe value of

11—

2

IS always less than unity, and I n shallow

girders considerably less , frequently as much as

hThe value of

h; 18 usually less than un ity,for very

shallow girders without cover plates it is often equal to

unity,and for girders with two or more cover plates it

may be as much as 10% less than unity .

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PROBLEMS

2a. Compute the net area required in the flanges of a plate girder30

’long and 48 deep out to out Of flange angles, when carrying two

loads , Of lbs . each, spaced 5'from the center Of the girder and

a uniform load of 300 lbs . per linear foot. Assume a web plate of

48X

2b. Make up a section for the bottom flange of the girder of Prob .

2a,using only two angles .

20 . Make up a section for the bottom flange Of the girder Of Prob .

2a,using two cover plates and two angles .

ART. 3. THE DESI GN OF THE CROSS SECTION OF THE

FLANGES

Flanges o f plate girders are generally composed o f angles

in pairs or angles in pairs and plates , as is shown in Fig .

2a,2b

,and 20 . The several parts are connected by rivets

The holes for the rivets are generally punched to a diam

eter 115” greater than that Of the rivet , or to a diameter Of

136” less and subsequentlv reamed to 1

3k” greater , or, as in

the best c lass of railway work,drilled to a diameter 55

”

greater than that of the rivet . I t is customarv in des ign

ing tension members to allow for a hole t" greater in

diameter than that of the rivet , that is , for a rivet a

hole 1 " in diameter should be deducted . The number Of

holes to be deducted from any tension flange depends on

the number of rows of rivets and the spacing of the rivets

in the rows . Much might be written on this po int but

here onlv litt le will be given .

For the flange shown in a in section,the Views o f c

and d are longitudinal developments of one of the angles

showing common methods of grouping the rivets .

DESIGN OF THE CROSS-SECTI ON OF FLANGES 9

I t is c lear that a symmetrical arrangement of rivets

such as shown in c is better than that of d, for the center

of gravity of the net area through x—x and y—y of c lies

in the root of the angle, while for the corresponding see

W eb PI. 96"

Ls o'i 31

"

sCov. Pis n'

ix"

FI G . 3 .

t ions of d it is first to the right o f the root and then to the

left , as indicated . The arrangement at d is sometimes

chosen because it permits a little more freedom in driving

one of the inner lines of rivets . Care must be taken in

lo cat ing the sections a: and y so that 2D +H V; for the

case selected the longitudinal pit ch p cannot be less than

I n fact the stresses


pas sing through the space marked D are considerably

bent and thereby increased , and therefore the distance p

for a good stagger for this case should be I t should

be noted that most girders have an excess o f strength

except at the point of maximum bending moment and

at the ends of cover plates , and hence care in staggering

the rivets need only be exerc ised at these points .

The entire flange stress is developed in small incre

ments by the web and transmitted to the flange by the

rows of rivets connecting the vertical legs o f the angles

to the web . I t is clear that these angles then should

comprise a considerable part of the total flange area

some eng ineers require others permit as little as

The girder which has a flange stress developed

in a short distance, requires heavier angles than one in

which the stress is devcloped in a great distance . The

foregoing provision for maintaining net section applies par

ticularly to the tension flange . The compression flange

is usually made equal in gross area to the tension flange .

The compression flange is in somewhat the condition

of a column as far as liability to failure in a sidewise

direction is concerned . For girders with a constant topflange section the maximum unit stress occurs only at the

point of maximum moment,for a flange with cover plates ;

that is,for a flange which varies c losely as the flange stress

,

the unit stress is nearly constant throughout . The web

stiffeners , i f any are used , give considerable lateral stiff

ness to the top flange, reaching and connecting, as they do

for half their length to material in tension . The sketch of

Fig . 36 will help make this point clear . The tension flange

DESIGN OF THE CROSS-SECTI ON OF FLANGES 11

is held in line by virtue o f its stress ; any tendency to side

wise deflection of the compression flange is resisted bv the

tension flange if st iffeners are used .

I f the unit stress for static l oads in tension is lbs .

per square inch , then —5is a corresponding unit.

stress for a compression mem ber, in which the quantities

need no definition . I f it be assumed that a girder flange

Compression Flange

FI G . 3a.

is similar to a column in its action and that its section

a rectangle whose width = b, then

— 70 — 70 . — 242

4 4061,

gives a compressive unit stress for a girder flange cor

responding to lbs . per square inch in tension . As

the st ress in a girder flange is only a maximum for a


small part o f its length a formula for the safe compressive

and unit stress

P — 200

may be used .

Experience has shown that no material Should be

used which is less in thic kness than 115 of the distance

between the rivets in the direction Of the action Of the

stress , 1gof the distance from the center of a rivet to the

edge of the p iece at right angles to the line o f stress or

that any angle leg when used alone in a girder flange shall

be longer than 12 times its thickness , otherwise the girder

flange may fail in detail rather than as a whole .

For simplicity and ease of construction the compression

flange should not be made of greater section than the

tension flange, therefore the compression flange must be

supported in a sidewise direction at frequent intervals , which ,

f rom the preceding formula, will be about every ten t imes

the flange width . I t is customary to consider that the

rivets completely fill the rivet holes in the compression

flange and that the‘

full g ross section o f the flange may

be assumed to act to res ist compression . This assump

t ion with regard to the rivets always fi lling the holes is

open to serious quest ion , and if they do fi ll the ho les

they do not Offer the same res istance to lateral deforma

t ion which takes place under compression as the unpunched

material . However , it is fair to assume that they partially

make up the punched—out material . The formula fo r the

unit stress for the compress ion flange is believed to be

severe enough to allow it to be applied to the gross area

DESIGN OF THE CROSS-SECTION OF FLANGES 13

o f the flange . The depth h (called the effective depth )

o f formula (1) is obtained by taking the gross area o f

both flanges in computing the locat ion Of the center o f

gravity o f the flanges .

I t should be borne in mind that i f there is no lateral

deflection o f the top flange that its maximum unit stress

Idepends entirely on

cthe section modulus , as it is some

t imes called . Whi le rivet ho les affect the posit ion of the

neutral axis for the sections in which they occur, pro bablytwo-thirds o r one-hal f o f the length o f a girder will be

undiminished by holes , hence in applying the fo rmula

M Z —S

E

Ito chec k the results of formula (1) it will be best

t o determine the position o f the neutral axis from the

gross sect ion , and find the moment o f inert ia of the net

sect ion o f the entire girder section fo r I in the formula.

PROBLEMS

3a. Compute the probable maximum unit stress in the tension flangeSI

Of the girder Of Prob . 2b, bymeans of the formula M=~

Z

3b. Compute the probable maximum unit stress in the tensionSI

flange of the girder of Prob . 26, bymeans of the formula M

30 . Compute the probable maximum unit stress in the compression

SIflanges of Probs. 3a and 3b, bymeans of the formula M =7

.


ART. 4. LENGTHS OF COVER PLATES

The plate girder , being a composite structure , may eas ily

be constructed so that its c ross-section may vary approxi

mately as the moments and shears require . The full flange

section being required only where the moment is a maxi

mum,a method o f determining where the parts may be

omitted when no longer required is necessarv to economi

cally des ign the girder . I n general the cover plates are

the only parts o f the flange which do not extend the full

length . Two methods will be developed fo r finding where

cover plates (or any other part o f the flange) may be

omitted .

The first : For girders which carry a unifo rm load,

or a load which may be closely represented by a uniform

load . Dec k plate girders for railway bridges may be

included in this class ificat ion .

I n Art . 2 the approximate moment o f the internal

thlstresses at any section = t A —

6in which A

is the net flange area des ignat ing this area by a for

s impli c ity ,the moment Of the internal stresses becomes

Sha .

Let w= the unifo rm load per foot o f girder , or for

locomotive loading the uniform load which would pro

duce the same end shear,then the bending moment at

wlx was?any point distant a: from the end =

T 2and this

must equal the moment of the internal stresses .

wlx was?

2 2


Another simple formula by the author for finding the

lengths o f cover plates for this class of loading is give n

in Eng . News , XXXI I , pag e 278, the issue Of Oct . 4,1894 .

The second : For girders carry ing loadings whi ch may

not be represented by a uniform load . Girders o f th is

class receive their'

loads through other girders or columns

at definite points . The main girders Of through p late

girder spans with floorbeams are common examp les of

this class .

As before the resist ing moment = Sha,where

a = area of flange at end o f any cover ;h =depth c . to c . of gravity of flanges at end o f any

cover

S = unit st ress .

The bending moment M at any point along the length

of the beam and distant 1; from the left end,where P 1 ,

P2 , P 3 , etc. of Fig . 4b are the concentrated loads,w the

FI G . 4b.

uniform load per foot leng th , and RI left hand reaotion

,is given from

M =R1x— wx-g— P 1 (:c — O) — P 2 (x — b) etc.

LENGTHS OF COVER PLATES 17

Since the moment o f the external forces =moment of

internal stresses

— P 1 (x — 0) —P 2 (x - b) +etc.=Sha, (4)

in which every quantity is known except as.

To apply this to any point n between the first con

centrated and the end the above becomes

I

I

in which R, and Ska are known , and the solution of which

is a very simple matter . I t should be noted that the

proper position for the live load is that which makes R1

a maximum .

To apply it to any point q between two loading po ints .

the formula becomes ,

R10 (R, — o)

The position of the live load must be taken, first , so

as to make the bending moment at P, a maximum ; second ,

so as to make the bending moment at P2 a maximum .

This gives two equat ions of the form of each of

which must be solved for x.

The value of r,which is least , i .e .

, the one requiring

the longest cover , is to be taken

The lengths o f cover plates are readily found by a

graphic method which needs no explanat ion beyond the

following sket ch : 4c. The full line is drawn to represent .


the resist ing moment Of the various portions , and the

d otted line the moments of the outer forces .

The applicat ion Of formulas (3 ) and (4) will now be

made to finding the leng ths Of cover plates ; for thi s pur

pose assume a girder 43 ’ long c . to c . end bearings ,

F I G . 4c.

deep out to Out of flange ang les and of the following compo

s ition :

web plate 00 X3" = 22 .50 sq in . sq . ins .

top angles 0 x6 X19 .0 sqi ns .

top plate 14 Sq .ins .

top plate 14 XQL

+ 21 33 sq . ins .

bottom angles 6 x6 x

Sq . ins .

1 bottom plate 14 X .

1

sq . ins .

1 bottom plate 14 X% + 13 52

sq . ins .

The girder will first be assumed to act as in a s ingletrack dec k railway bridg e , using formula

Maximum bending moment ft .lbs .

Maximum end shear lbs . w — 9000

a = 12 .33 , and sq .in .

h = 4 .77,

and ft .

LENGTHS OF COVER PLATES 19

For first cover 0 1 = y+ 2

For second cover 0 2 y+ 2 X x 16,000

yxzx/wzz

The girder will now be assumed to have floorbeams

attached to it as indicated in Fig . 4d. The react ion R1 ,

when the moment at P I is a maximum,is lbs .

and the own weight o f the girder 300 lbs . per linear foot .

FI G . 4d.

Then for the first cover

2

136,500x— 300 x4 .77 x 941

,

g l 0x —.r

‘

6273 or x2 — 9 lox+ 207025 — 628+ 207025

x — 38

The length of the first cover 0 ; =2 — 15)

For the second cover

— l 5ox2 X x

— 910x= 9483 ;

it . 455 — 445 10 .

The length of the second cover Cg = 2 + (43 — 20 )

2 x x4 .77 x 16,000

4 9000

y— 209 . 1 = y+ 2

2 + 2 X16 = 34’ long .

1833

9000


PROBLEMS

4a. Let each of the flanges of the g irde r of this Art. consist of

angles 6 lbs . sq .ins .

1 cover plate 14 >( 45t

1 cove r plate1 cove r plate

and of the web plate

Find the leng ths of the cover plates when it acts as a part of a

deck railway b ridge .

4b. Find the lengths of the cover plates when the g irder of Prob .

4a acts as apart of a through b ridge . The concentrated load P I

lbs. The loads being spaced as shown in Fig . 4d.

ART. 5. RIVET SPACI NG IN GIRDER FLANGES

The connection between the web and flanges o-f girders ,as in other composite structures, is made by means of rivets .

These rivets are spaced with reference to the hori zontal

component of the stress in the flange, for at the extreme

fiber the direction of the stress is horizontal and the

maximum shearing unit stresses in the flanges are very small,

and hence the horizontal component is the only stress o f

importance .

The one important exception to the foregoing is where

the girder load is applied to one of the flanges,here the

rivets have to transmit the loading which the girder carries

together with the horizontal stress increments between the

flanges and web . The exceptional case will receive spec ial

consideration .

The moment of the external forces at any point in a

girder equals the moment of the internal stresses,there

RIVET SPACI NG I N G IRDER FLANGES 2 1

fore between any two points it is essential that there be

enough rivets to properly develop the stresses produced

by the maximum increase in moment between the points .

The increment in the moment is constantly varving

throughout the girder length . The general equat ion for

moment for a s imple girder as given in Art . 4 is

M = Rl a:-P 1 ( :c

-0 ) + 6) — etc.

,2

The derivat ive of M with respect to x is

dz:=RI

— wx —.P 1

— P 2—etc and is the shear .

I t is therefore seen that the greatest increase in the

bending moment occurs when the shear has the greatest

poss ible value .

The increase in bending moment between two points

so close together that the load between them may be

neglected ,is the shear multiplied by the distance between

the points ; the increase in flange stress is this increase

in moment d ivided by the depth .

V= the maximum shear at any point on the girder ;

h = the effect ive depth, i .e the depth 0 . to c . o f

gravity of flange , in inches ;

R= the least value of the rivet to resist either

crashing or shear , and

p= the space between two adjacent rivets in inches .

= the maximum increase in flange stress in a

space of 1 inch (a)

stress per inch o f girder length carried by

the rivets (b)

22 NOTES ON PLATE—GIRDER DESI GN

For the proper degree of strength (a) should equal

V Ror p

—

V

I n order to show the application of (5) to determining

rivet pitch : Let Fig . 5a represent a portion of the girder

I}

2 Cov'}Pl

lel. 14xK

2~L56x 6 x

eh Pl. so”x

2 L863?2 Cov._Pls. 14 i 34

FI G . 5a .

used to illustrate the method of finding lengths of cover

plates in Art . 4 .

The shear at a at b and ate

The value of a rivet in bearing = 24 ,000 , and in shear

lbs . per sq .in .

The pitch for rivets,at (a)

x 12 x7s76

provided the flange ang les are designed to carry the

entire flange stress at this“

point . I f,however , as was

assumed in finding the len gth of the cover plates , the

portion of the web used as flange area is sq . in .

,then

2 .34 x12 .33The p itch

of would require that two lines o f rivets be used,

as good construction requires that the rivets be not closer


horizontal increment of stress in a length o f

one inch carried by cl ;

vertical load per inch of length of flange affected

by the concentrated load ,

resultant of h, and v, , 112, v2,

Then if R= the value o f the rivet , as before,

Let this be applied to finding the pitch for connecting

the top flange angles to the web for the end of the girder

o f Fig . 5a .

Va ] x9 .52 193 500 x9 .52

ha X

W_5o,ooo

s 42

For the heavier concentrated loads,such as columns

bring when carried on the flanges of the girders,the flange

angles will often not contain enough rivets for security,

and stifleners must be used , as will be explained later .

The foregoing gives a simple method for finding the max

imum permissible pitch o f the rivets connecting the vertical

legs o f the flange angles to the web .

The determination of the p it ch of the rivets for con

RIVET SPACING I N GIRDER FLANGES 25

nectip g the cover plates to the horizontal legs o f the flange

angles is not a simple matter if theoret ical accuracy is

des ired .

At b,Fig . 5a

,the point where the first cover plate

begins the flange angles have in them all the stress they can

carry . I t is c lear,therefore

,that from b to cwith the rivets

connecting the first cover to the flange angles spaced to

take onlv the flange stress increment , the flange angles

s imply transmit the increments of stress to the first cover

plate . At 0 the first cover p late and angles have all the

stress they can carry,and with rivets spaced as before

the increments o f flange stress from c to the point of

maximum flange stress are simply transferred through the

angles and first cover to the second cover . Rivets con

meeting a cover plate to a flange are generally spaced to

take the increments o f flange stress which occur from the

end to the point o f maximum stress in the cover . I f n

be the number of lines of rivets connecting the cover to

the flange then

Assuming two lines o f rivets in the cover plates the p itch

at b for connecting the first cover to the flange angles

2 x7216 X57 .24

be used ; the actual pit ch would be made considerably

less and usually a multiple'

of t, it , or 1 and at the same

time such a p it ch as would stagger well with the rivets

in the vertical legs of the flange angles . I f the maximum


pitch permiss ible were used the stress per square inch in the

first cove r plate would be zero at b and increase to

at c. This is highly undes irable, as there would exist in

the girder in j uxtaposition the angles with a unit stress

of and a cover with an average of 8000 lbs . The

girder flange material cannot act in any such way with

out undue bending stresses on the rivets .

For this important reason cover plates should be made

longer than a me re consideration of their re lation to the

moment polygon would require . The additional length,

designated by y in formula (3) required , is a function of

the number of rivets necessary to equali ze the flange

un it stress in all the flange material,and may be deter

mined as follows :

n = uumber of rows of rivets in the cover plates ;

c = area of flange without the cover plate under

considerat ion , with unit stress 3 ;

al=area of flange

,inc luding the cover plate under

consideration , with unit stress 3 1 ;

p=pitch in inches. of the cover plate rivets

each line ;

R= value of one rivet connecting the cover to the

flange ;

yl= additional length of cover plate required at

(a l 6 03 1each end In ft .

2.

12Rn

S -a -p

p 6a l -Rm (a1

fi

a

RIVET SPACI NG I .N GIRDER FLANGES 27

The value o f y for the first cover o f the first example

used“

to illustrate the method of finding the lengths is :

p 3 p

6 x 18.33 >< 7216‘

n 2 n'

I f the rivets be placed 3”apart in two rows y= 2 .25 .

I f enough rivets are used at the end of a cover plate

as at b o f Fig . 5a to transmit to it its full proportion of

the flange stress , the rivet pit ch required to connect the

cover plates to the flange at any point will be the following'

[11 7 t

I n which AI is the total flange area,and A the area of

all the cover plates , at the point under consideration .

The number of rivets as determined by and

should be increased by 20% to allow for the bendingstresses . To st ill further limit these bending stresses no

rivet pass ing through angles and covers should have a

grip more than five t imes the diameter o f the rivet .

PROBLEMS

5a . Using unit stresses of and lbs . for rivets in shear

and crushing , compute the required rivet pitch for connecting the

bottom flange ang les to the web at c. of Fig . 5a .

5b. For the data of Prob . 5a compute the rivet pitch for connectingthe top flange angle to the web at a point vertically over 0

,including

the effect of a locomotive wheel load.

5c. Using the unit stresses of 5a compute the value of y for the second

cover plate.

5d. Compute the pitch of the rivets in the cover plate of 56 justadjacent to the, portion y.


ART. 6. WEB PLATES

The maximum stresses that the web of a beam or

girder must resist are of varying intensity and direction

throughout its area.

For any elementary parallelopiped cut from the body

of a beam in equilibrium subject to fl-exure, the forces on

the faces of the element perpendicular to the plane of

the drawing Inav be represented for the most general case

FI G . 6a.

as shown . The weight of the elementary particle be

neglected , as it is an infinitesimal of the third order as

compared with the amount o f stress on the faces,which

are infinitesimals of the second order .

As the length of the sides o f the particle approach

zero as a limit , the intensities o f the oppositely directed

forces p,, are equal , and also the forces pr . I f the oppositely

directed normal forces are of equal intensitv the. tan

gential or shearing forces q are also o f equal intensity .

The intensity of the shearing stresses at any point in a

homogeneous prismat ic body subject to flexure is given

WEB PLATES 29

Vq bI

Q’ a well-known formula of Mechanics o f Mate

in which

q=the unit shearing stress ;

V=the total shear ;

b=ths width of the body at the point under con

siderat ion ;

Q= statical moment of the portion o f the body above

the point under consideration with reference

to the neutral axis of the body ,and

I =moment o f inert ia of the body with reference to

the neutral axis .

The intensity of stress px at any point is given from

flexure formula pm IIn which the quant it ies are

too well known to need definition .

Uniform load per inch.

FI G . 6b. FI G . 60 .

The intensity o f stress p, may be found as fo llows :

Let Fig . 6b represent any beam subject to flexure and

a any point in the beam . Assume two planes distant

1 apart on either side of a as shown by the dotted lines .

The vert ical forces on the portion of the beam above

a and between the paralle l lines are as indicated on the

larger scale sketch, Fig . 6c, in which VR is the sum o f


the shearing resistances o f the right surface above a,

VI , is the sum of the shearing res istances o f the left surface

above a .

W = (P I total load between the two

assumed planes above the point a,and A= the area in

square inches o f the horizontal surface through d . Then

VL — VR — W + puA=O,

VR W V,,

A

Care must be taken in giving the proper signs to VR and

Having a method for finding pm, and q for any

point in a beam,there remains to be found the plane

in which their combined action has the greatest intensity .

The stresses per square inch pm, py and q may be pro

duced in any manner , even in an arbitrary manner or p ,

and q, and p,, and qmay be the rectangular components into

which the actual stresses on any elementary parallelopiped

have been resolved ,as far as the following argument is

concerned .

Let Fig . 6d be a redrawing o f 6a . Let the plane OP

be passed through the elementary particle perpendicular

to the face and making the angle 0 with AA’

, consider

the elementary figure to.

have a dimension of unity per

pendicular to the plane of the drawing .

Let it be required to find 0 so that the stresses on

OP will be normal and therefore either the maximum orm inimum stresses in the body .


The two values of p given from (9) correspond to the

maximum and minimum values and occur on either OP

or OP

For an extended treatment of the subject o f combined

stresses see Wood ’s Analytical Mechanics , Chapter V

I f either or all of the quantities p l , p, or q have a

direction oppos ite to that shown in Fig . 6d they should

be entered in the formulas with a minus sign .

The planes of maximum shear make angles o f 45° with

the lines o f maximum direct stress .

The amount of pl, is small, except for large coneen

trated loads , with reference to p , and g . I f py be neglected ,

at the end of the beam where p$= 0 the angle

and p=q, that

‘

is the direct stresses are equal to the shears

in intensitv and make an angle of 45° with them .

I f py be neglected , at all points on the neutral axis

where p , is zero , p = q and makes an angle o f 45° with the

axis .

I n determining the location of the plane on which p ,

the principal stress intensity , is a maximum note,the values

o f the component stresses and inspection will show about

the direction the resultant must have . I f difficulty occurs

in applying formula (9 ) it will be better to derive a special

formula for the case at hand .

PROBLEM

6a. A plate g irder of 14’effective length carries a total load of

lbs . per linear foot of length, the load being applied to the topflange . Consider the structure to be solid and of a cross section com

posed of 4 angles32”

and 1 plate 24x95”,as shown in the

sketch, dete rmine the maximum stress in tensity and the direction of

WEB PLATES 33

its action at each numbered point as is indicated for points9 , 11, and 15 of Fig . 66 .

Eff .

This problem important for a proper understanding

the stress distribution on web plates that it is com

pletelv solved for po in ts Nos . 3 , 7, 9 11 and 15,and much.

of the computat ion made for the remaining points , in what

follows in this art icle . I n Fig . 6c arrows acting toward

a point indicate compression and away from point

tension .

The.computations for p i , p,, and q should made

about as fo llows

X

X4

4. X X

M B 000 in .

-lbs.

M Cin .

-l o

At points Nos . 1,

‘7 3 , 4 and 5, p 1

34 NOTES ON PLATE-G IRDER DESI GN

pomts Nos . 7 and 9 ,

At points Nos . 11 and 15, p ,

Qu = 2X3 .75X10 .82

X10 .00

Q7 X3 .75 X

X10

X6 12

Q3 + 2 X2

0993 6960

From formula (7) there is found

at No. 3,

833 4at No. 7,

8335 is) 30

8 1 416at No. 11 ,

33s “4165 121 733

.5

Values of Q,and should be tabulated as follows

4

-s

WEB PLATES

o o o o o o o o o o o

c c c c c c

o o o o o o0 0 0 0 0

o o o o o o

F I G . 69 .

Let A1= the area above division NO . .75

A2=the area above divis ion No.

sq . ins .

,

Values o f qXareas'

A2

A3

A4

A5

A5

A7As

A9

A1 0= VX .401

A1 1=°VX.450

A1 2 l"

X .500

QXAi a VX.550

AI 4 VX.599

Am= Vx .696

A18 VX .790

A19 VX .835

A20 VX .879

An VX .919

A22 VX .954

A23 VX .983

A2 4 = VX . 1000

AREA OF WEB PLATES 37

No . 11 :

1470 000an infinitesimal

+8530]

+ 1470 .

No. 15 :

1 4162 41 .28337?

5

3 65 )879 - 200

200]

+ 200

0 = 90 or 0

ART . 7 . AREA OF WEB PLATES

The preceding art ic le has shown that the stresses in

the web of a plate g irde r are of greatly varying directions

and intensit ies . The tensile stresses need an adequate

amount o f materia l to res ist them , the amount of which

is a direct function of the. stress .

The action of the compressive stresses is augmented

to a much greater degree than for tension , by secondary“stresses if the material be even very slightly lacking in

uniformity . That is,a certain kind of column action is

set up .

The maximum tensile and compressive stresses in the


web cross each other at right angles ; at the neutral axis

they are approximately equal,and make angles of approx

imately 45° with the axis and equal the shearing stresses in

intensity ; above the neutral axis the compressive stress

intensity is increasing while the tensile is decreasing ; below

the axis the opposite relation ex ists .

The shearing stresses are of the greatest intensity at

the neutral axis,on horizontal and vert ical planes , for

any cross-section . At the cross—section where the greatest

resultant shear occurs , there , on the neutral axis will be

found the maximum shearing unit stresses . For simple

spans the shear is a maximum at the ends o f the span .

I t is the practice to design web plates so that they

will resist the shear at the point of maximum shear , and

have enough thickness to provide ample bearing.

for the

rivets connecting flange angles to the web ; this latter very

frequently determines the thickness of the web plate . The

design of the web plate is also governed by practical con

siderat ions , such as requirements for handling in shop and

field and requirements to resist corrosion .

The best practice requires as a minimum for the thick

ness of girder webs : 175” for railway

, 2” for h ighway

,

and 153” for building work . I t is the practically universal

custom to make the web plate , for a girder,of uni form

section throughout its length .

The shearing unit stress should be about les s

than that used for direct tension , that is , where

is used for tension , is a consistent unit for the max

imum shearing stress intensity . Where the shearing stress

is assumed to be uniformly distributed over the girder


stress I ntensity is therefore assumed to act a long the

element of the web having a length ofZin Fig . 7b, and

this must be less than the permissible unit stress for a

column of this length . Assuming that a safe unit stress for

a column is given from the formula

P

the notation for which needs no definition , the

should be reduced to if the average shearing unit

stress is used , as the maximum unit compression . The

formula then becomes

P = — 52 5

Substituting for l and r their values ,

and r= .29t,

Where t= the thic kness o f the web .

The formula then is

x/2P

As the tension in the web below the neutral axis will

dprevent buckling , the value o f d should be 5, and as the

tension in the upper half of theweb helps to prevent buckling

the constant may be divided again by 2, the formula then

being

P — 5

AREA OF WEB PLATES 41

I t should be noted that d equals either the clear distance

between flange angles or stiffeners .

To show the method of applying this formula to see i f

web st iffeners are necessary let the girder of Fig . 5a be

examined .

The average shearing unit stress — 8580 lbs .

per square inch .

The allowable unit stress —65 X

— 8420 = 3580 lbs . per square inch .

As the actual stress exceeds the allowable , the web

must be made thi cker or the stiffeners placed closer together

o r else both combined .

I f the web be made 3” thick

The average shearing unit stress — 6450 lbs .

per square inch .

The allowable shearing unit stress = 12,000

—65

lbs . per square inch,which shows

that the web must be further thickened or the st i ffeners

placed closer together .

I f the web be made 175 thick and stiffeners placed

30 apart in the c lear the average shearing unit stress

= 7380 lbs . per square I nch .

The allowable shearing unit stress65X30

— 4420 = 7580 lbs . per sq .in .


PROBLEMS

7a . The accompanying sketch shows the cross-section of the girder

of Prob . 6a curve of unit shear distribution . Construct

FI G . 70 .

figure to three times the size of the sketch write value

of the ordinates to the cu rve at points 1”

apart from the top to the

bottom of the section .

7b. Are stiffeners required on the web of the girder of Prob . 6a?

If not, show it by computation .

70 . What is the minimum thickness pe rmissible for the web of the

girder in Prob . 6a?

ART. 8. STI FFENERS FOR WEB“

PLATES

St iffeners have several important functions the chief o f

which are :

(a) They keep the web from buckling , due to the com

pressive stresses in it .

(b) They help hold the compression flange from lateral

failure as a whole and from failure in detail“

in

any direction .

(c) They are used to relieve the rivets connecting the

STI FFENERS FOR WEB PLATES 43

loaded flange to the web, by transferring the

load directly to the web .

They are used to reduce to proper amount the

vertical stresses , on horizontal planes , in the web

brought by local concentrated loads .

They help hold the web true to shape during man

ufacture and erection .

According to their principal functions they should be

divided into two general c lasses , intermediate stif feners,which are chiefly used for the purposes o f (a) , (b) and (e) ;

concentrated load stifieners , for (c) and (d) , although each

class performs all the functions of course. I ntermediate

st iffeners act as a beam standing vert ically to resist lateral

d isplacement . They are generally made of angles in pairs

and riveted to the girder web as shown in Fig . 8e . The

FI G . 8a .

legs o f the stiffening angles which are against the fi llers

are near the neutral axis o f the pair and therefore should

be only large enough to receive the connecting rivets ;

a 31W leg for i rivet and 3” leg for {

Ware enough .

The width b o f the outstanding legs is the principal


element of efficiency for resistance to transverse displace

ment . A common rule is to make the width o f the outstand

ing leg (0 ) equal to the depth of the girder (d) in inches

divided by 30 plus or expressed as a formula

The following is a common method of locating inter

mediate stiffeners .

I ntermediate stiffeners should be used at points as

required by (10) or wherever the unsupported depth between

flange angles is more than 160 t imes the web thickness .

Where intermediate st iffeners are required they should

never be further apart in the clear than the clear distance

between the flange angles , with a maximum limit o f

Concentrated load stiffeners act to transfer a concen

trated load into the web or to transfer the girder load to

a support . The sketch of Fig . 8b will illustrate both cases .

FI G . 8b.

The girder here shown being loaded with three con

centrated loads of and its own weight of 250

lbs . per linear foot . The lines of maximum web stress

STIFFENERS FOR WEB PLATES 45

for a girder o f this character are very different from those

of the girder of problem 6a . I f such a girder were con

structed without stiffeners , and formula (7) were applied

to finding the intensity of p”, this intensity would be

found to be very high . The heavy vert ical st iffeners are

added to transfer the point of application of these loads

from the outer edges o f the flanges to an average pos ition

of the center of the web , and thereby decrease the inten

sityo f vert ical stress on horizontal planes j ust at the inner

edge o f the flange angles . These stiffeners cannot fully

accomplish their object , as they are of elastic material and

they are shortened by compression , and,

as they are

fastened to the web , the web and stiffeners move together

except for a variation due to the deformation of the rivets .

The st iffeners also serve to relieve the flange rivets from

the component due to the vert ical load .

Stiffeners supporting concentrated loads may be designed

las columns with the formula P = 16

,000—70f in which l

should be taken as one—half the girder depth and r as the

radius o f gyration about an axis in the longitudinal center

line of the girder .

Wherever the combined stress intensity on a web plate

exceeds the allowable unit stress the stresses may be

reduced by means of side plates , as shown bv dotted lines .

Where the concentrated load sti ffeners are at the end

o f a girder they should have their outstanding leg about

equal to the horizontal leg of the flange for the sake of

good appearance, and where it is desired to face the ends

of such girders the st iffening angles must be made thic ker

46 NOTES ON PLATE—GIRDER DESI GN

by ls" than theoret ical considerat ions determine, to allow

for material removed by such facing .

PROBLEMS

8a . Determine the size of the end stiffeners for the girder of Fig .

5e,using two pairs of angles, one on the oute r and one on the inner

edge of the sole plate .

8b. Assuming the wall plate to be 18"long for Fig . 50 , dete rmine

the size and spacing of the intermediate stiffeners for the girde r .

8e. Determine the sizes of the stiffeners at the poin ts of coneen

trated loading for the girde r of Fig . 8b.

8d . Determine the sizes and spacing of the intermediate s t iffene rsfor the girder of Fig . 8b.

ART . 9 . SPLI CES FOR WEB PLATES

Webs of p late g irders should not be spliced unless neces

sary,but as wide p lates cannot be procured in one p iece ,

in general , for girders over 60 ft . long,splic ing for long

g irders is unavoidable . The manufacturers furnish lists

of the maximum dimensions of their p lates . The number

and location of the web splices should be determined by a

combined consideration of the cost and resulting effic iency

of the possible arrangements . The splice should be de

signed with the idea of transferring the stresses in the

web across the cut'

in the most - direct manner possible ,

j ust as in plate girder design as a whole the effort should

be made to have the stresses developed as indicated by

the theory of flexure for solid beams . Let it be required

to splice the web at the center of the girder which has

an e ffective length of and a composition as shown in

Fig . 9a .

The girder has been designed on the assumpt ion that


as is well known from mechanics,but the proper form

of Splice should be designed with reference to the distri

hutien of these forces . The portion o f the web between

the flange angles is so near the extreme

fiber that the stresses may be assumed to

be horizontal throughout . The flange unit

stress has been taken at lbs . ; the

bearing and shearing values for rivets will

be taken to correspond to this at

and lbs . per square inch respect

ively, or at 50% more and 25% less than for

F1G , 9b_ tension . The thickness o f the Sp lice plates

should be sufficient to transmit the maximum

possible stresses which can occur at the point o f maximum

stress in the splice.

The total stress on the portion of the web between

the flange angles = (8” lbs . ; the

least that can be used for this part o f the splice will be

2 — 7 x% flats for each flange . The net area of these two

plates = 4 .5 sq .in .

, while that o f the strip of the web = 3 .06

sq .in . The bearing value of the rivets is less than that

for double shear , hence one rivet is worth 9190 lbs . at

the edge of the web plate and 9190 XT S

8350 , an

average for this part of the splice . The number of rivets

on each side of the splice for splicing the strip = 8350or 6.

For the condition of maximum moment there is no

shear at the splice and the web stresses are horizontal .

The number o f vertical rows o f rivets is evidently a

SPLICE FOR WEB PLATES 49

function of the vertical pit ch , the more vertical rows the

greater the vertical pit ch .

Let w= the pit ch of the rivets in a vert ical d irection ;n=uumber o f vertical rows o f rivets ;d=distance from the neutral axis to horizontal

row of rivets .

For three Vert ical rows

(to x xd/48= 3 xd/48

w= 34 .570/7000 say 4;

For two vert ical rows

(w x Xd/48= 2 x9190 xd/48

w say 3

For th is splice 3 vert ical rows on each side spaced

45will be used , as it gives a web plate with less reduct io n

o f strength at the splice.

The spli ce should now be examined for the case o f

shear and simultaneous moment o f ft .—lbs .

9190 x96

for no part of the web acting as flange . The

flange as designed uses sq .ins . of the Web for flange

area; the pitch of the rivets connecting the flange angles

96to the web is

9190 XX3

75 000

The 7x§ flats on the vert ical legs of the flange angles

will therefore be made to take about 6 or 8 rivets on each

side o f the splice .


For a. web splice at some point other than the center of

the girder, the number o f rivets in the flange plates on

the side toward the center will be the number required

for splic ing the web ; on the s ide toward the end the

number will be the sum of those,for connecting flange to

web plus those for web splice .

The extreme fiber stress on the web for a moment

of ft .-lbs . was for a moment o f

ft .- lbs . it is 8630 . For the strip of web between the flange

angles the previous determinat ion is evidently ample . A

study of the stress intensity at different points in the

web for this condition of loading would show that the

stress intensity was nowhere as great as for the case for

which the Splice was designed .

A safe resultant rivet bearing of 9190 should not be

exceeded .

The vert ical component on a rivet at e ither edge of

the main splice plates will not exceed

lbs .,the horizontal

3480 per rivet . The resultant 13902

+ 34802= 3800 .

PROBLEMS

90 . What is the relative efficiency of the unriveted plate of Fig .

9a to that of the section of the plate through the first vertical line of

rivets in the Splice .

9b. Compute and compare the resisting moment of the rivets in thesplice of Fig . 9a with that of the net section of the web plate throughthe first vertical line of rivets .

9c. Make a sketch of a Splice for the girder , using only twovertical

rows of rivets on each side of the splice .

SPLICES FOR THE FLANGES 51

ART . 10 . SPLICES FOR THE COMPONENT PARTS OF THE

FLANGES

Splic ing any of the component parts o f a g irder flange

should be avoided,as it is never necessary except to meet

some emergency ,or reduce the cost slight ly . The flange

angles and cover plates , for girders of the maximum length

now used ,may be obtained in one piece . The cost o f a

few very long pieces , however , may be so high ,due to the

expense of Shipment in less than proper amounts,or the

need of the completed structure may be so urgent , as to

make it advisable to permit splic ing of component parts

of the flange .

Flange angle splices should be located when poss ible

as follows :

Where there is an excess of flange sect ion over the

required amount ;

Bet-ween adjacent pairs o f st iffeners , where these

pairs o f stiffeners are far enough apart to permit

a splice of proper length to be made ;

So that not more than two flange angles shall be

Spliced in any cross—section of the girder :

d. So that the flange angles on opposite sides o f the

girder flange will be spliced on opposite Sides o f

the center o f the girder .

The splic ing material Should be disposed with reference

to the shape of the section spliced . The splice for an

angle with equal legs Should be made by a cover angle

with two equal legs . The net area of each leg of the Splice


should equal the net area of the leg spliced . I n general

in splic ing or connecting material under stress the splice

or connection Should be arranged to avoid any redistri

bution of stress . This requires that ang les with unequal

legs be spliced by cover angles with unequal legs , and

that the net area of each leg of the one be equal to the

net area of each leg of the other .

The number of rivets in the splice on each s ide of the

point where the angles are out should be determined with

care . The duty of the rivets in developing the flange

stress as well as in connecting the sp liced parts should

be clearly understood . For the purpose of illustrat ion let

it be supposed that for the g irder o f Fig . 5a the only

material available for the flange ang les is 4 ang les 6 X6 X%—16

’44

” long and 4 ang les 6 X0 X%— 28’ long . This

makes it necessary to splice the bottom flange at a point

6’ from the center of the girder . Fig . 10 shows the Splice

in cons iderable detail . The net area of the two angles

sq . ins .

,and this at lbs . as the strength

which must be developed by both the rivets and ang les

of the Splice . Two 6X6Xf’g angles with the legs cut to

each and the corner of the ang le planed so to fit the fillet

of the flange ang les will be used for the splic ing material .

The Shear , at the point of splic ing , Simultaneous with

maximum flange stress,is lbs . The pit ch of the

rivets cannot be less than 6 in each line without reducing

the flange section (see Art .

The horizontal flange increment I per rivet from Formula

59x3 = a54o when none of the web is assumed

SPLI CES FOR THE FLANGES 53

to act as flange area ; for this girder, however , sq . in .

of the web was assumed to act as flange area . The value

of I therefore is 2540 — 2540 X24 33

= 2540 290 = 2250 lbs .

FI G . 10 (a and b) .

Fig . (c) shows the forces on a rivet through the web

and vert ical legs of the splice and flange angles due to

increment I .

Flg .Ls

FI G . 10 (c, d and c) .

Fig. 10 (d) shows the forces on a rivet through the

web and vertical legs o f the splice and flange angles due

to Splic ing , for the left portion of the splice.

Fig. 10 (e) shows the forces on a rivet through the


web and vert ical legs o f the splice and flange angles due

to spli c ing , for the right portion of the splice .

The unit stresses on the rivet will be taken at

lbs . per square inch for shear , and lbs . per square

inch for bearing .

Consideration of the forces on the rivets and the thic k

ness of flange and splice material shows that the Shearing

value of the rivets is the determining one along the plane

number requ i red ln the vert i cal legs

or 6 .

FI G . 10 (f , g and h) .

Fig . 10 (f ) , (g ) and (h) Show the forces on the rivets

the horizontal legs of the flange angles of the topflange splice . I t is readily seen that the number of rivets

4 X7216

2naCov.

FI G . 107i.

Cover plate splices should be located so that a s imple

lengthening of an outer cover plate will form the splice .

For example if a splice were required in the first cover


The stringer is composed of 1 web plate 27 x%;

4 angles 5 X 3%X 13 .6

The floorbeam is composed of 1 web plate 38X 7};

4 angles 5 x35x19 .8

The maximum stringer reaction is lbs . (ll= 48,800

I mp . and dl = 2400) .

The maximum reaction for two stringers is lbs .

I mp . and dl = 4800 ) .

The requirements of construction will be assumed to

demand that the bottom flange of the stringer be

above the bottom flange of the floorbeam . The connection

between the stringer and floorbeam is made bv two ver

t ical angles which are riveted onto the stringer in the

bridge shop . The rivets through the outstanding legs of

the connection angles and the floorbeam web must be

field driven . The unit stresses are and for

shear and bearing for shop rivets and 20% less than these

values for field-driven rivets ; rivets will be used and

the connection angles will be made i" thic k so that after

they are faced on the back of their outstanding legs they

will not be less than thic k on these legs .

Fig . 11 (d) shows how the load W is applied to each

rivet and carried to the st i ffeners . The rivet should be

examined for crushing by the web for W and for shear

W .

2

97 800For shear the number =

2 x7216or 7 . The upper

and lower rivet in the lines xx and zz of (b) cannot be

CONNECTI NG ONE GIRDER TO ANOTHER 57

counted for this connect ion , as they have other duty to

perform . Therefore , in order that the rivets be not closer

than a leg for the connection angles which will contain2 rows of rivets must be Used . As 10 rivets are required

,

a symmetrical arrangement demands 11 . These rivets pas s

through fillers and a good requirement is to extend the

fillers such cases and put 50% addit ional rivets through

Conn LE

them . Many good designers consider it allowable to assume

that fillers connected to web plates by a line of rivets

such as yy increase the bearing value of rivets in lines

and se by the entire bearing value of the rivets in yy.

Consideration of (c) of Fig . 11 Shows that it will be

poss ible to get

16 rivets in the two lines 0 0 and hav e them a litt le

over 3” apart , and


18 rivets in the two lines 0 0 and have part of them

as and part 3”apart .

Fig . 11 (c) shoWs how the forces act on a rivet con

necting the outstanding legs of the connection angles to

the floorbeam web .

The bearing value of a field rivet in the g (net thickness )

st iffener leg is 6300 .

The shearing value of a fie ld rivet is 5770 .

For the maximum stringer load the number o f rivets

5770

two lines of rivets to be used , i f part of the number are

required for shear or 18,which would enable

23”apart .

The rivets must also be examined for bearing on the

floorbeam web when both stringers are bringing their cem

bined maximum loads at this connection . The bearing

value of a field rivet in the floorbeam web = 8400 lbs '.

8400

which is one less than“

the number which are required for

The number of rivets required = or 16,

maximum shear on the rivets for one stringer loaded .

The number of rivets by the second computation is

usually,though not in this case , the larger and hence the

governing one . The rivets should be located on the lines

0 0 in (c) of Fig . 11 . The two fillers under the connection

angles and on the stringer web must be t” thic k and 3

”

wider than the angle leg . They Should therefore be 9” x é”

flats .

The requirement of two lines of rivets for the leg o f

the connection angle against the stringer web demands a

END BEARINGS 59

6" leg . One line between each leg of the connection

angle and the fioorbeam web fixes the width of the out

standing leg at 3 71; or

The connection angles cannot be less than 3 in thic k

ness. Examination of this net section through lines 332:

and 0 0 shows them to be ample to resist shear . Some

t imes they must be increased in thicknes s for this .

The connection angles are therefore made 6X4 x16 .2

PROBLEM

11 . Design the connection between adjacent stringers

floorbeam . The stringer consisting of

1 web 24XIQB ;

4 ang les 5X3}X (195)

floorbeam consisting of

1 web 34Xt;

4 ang les 5X3§X

2 covers 11x1}.

shears being the same as for the example solved this article .

ART. 12. END BEARI NGS

The end bearings of a girder must receive the load

brought to the end o f the girder and distribute the same

over the masonry or other support . For the single-trac k

girder o f Art . 4,whi ch was taken to illustrate the method of

finding lengths o f cover plates an end reaction of

lbs. was assumed . For girders less in length than 60’ one

end of the girder is bolted to the masonry and the other


allowed to slide on a plate . This arrangement makes the

determination of the reactions poss ible .

The area of the bearing on the wall , assuming the bridge

seats to be o f granite and 600 lbs . per s quare inch as a

600

I t is not advisable to make an end bearing of the s imple

nature indicated in Fig . 12 too long . The tendency of

unit stress , should be = 323 sq .in .

such a bearing is to overstress the bearing along its inner

edge .

The upper plate o f the two shown at each end is

called the sole p late, and is connected by rivets , counter

sunk on the bottom , to the girder . The bottom o f thesole plate at (b) , the expansion end

,should be planed .

END BEARI NGS 61

The lower plate of the two shown at each end is called

the wall plate .

The upper surface of the wall plate at (b) , the expan

sion end,should be planed .

The wall plates are held in position by the two anchor

bolts at each end

The bearings at the ends of the girder , when of two

simple plates , should not be too wide , as the tendency

of the bearing is to overload the masonry along the portion

covered by the bottom flange of the girder .

The bearing for this case will be made 18x18

The two plates which pro ject beyond the bottom flange

angles must be strong enough in flexure to distribute the

load on the bridge seat . Each inch of length of the plates

may be considered acted on in a transverse direction by

the forces shown in (d) .

The bending moment then

= 5400 X1 .41 = 7614 in .-lbs .

I f the sole and wall plates be made of equal thic kness,then the resist ing moment of each plate must be 3807

in .

- lbs .

The depth (or thickness ) of plate required may be

SI S ~bd2 3807obtamed from M =—

c

= 3807 or6

and d= l .25"about .

These plates should neither of them be less than

thick, even if a less thickness would furnish proper strength .

For small spans generally only two pairs of st iffeners ,

one over the outer and the other on the inner edge of the

bearing plates , are used .


The end sti ffeners for this case should be 5 X3%angles

their radius of gyration in a direction transverse to

web is approximately . The required area for

10800

FI G . 12 .

193 500thes e st i ffeners sq . in . [P = 16,000 — 70

— 750

Therefore 4 angles sq . in . will be

The addition of another pair o f stiffeners over the

center o f the bearings would help greatly in distributing

pressure over the wall plates as their outstanding

legs would prevent an upward deflectiOn of the plates

angles between the pairs of st i ffeners . The additional

sti ffeners are shown by dotted lines in Fig . 12 (a) and (b) .


end being 1 web plate 48X1’g and 4 angles the end

reaction being lbs.

APPENDIX

ART . 13. POSITI ONS OF LOADI NG FOR MAXIMUM SHEAR

AND MOMENT

The shears and moments , which are necessary for the

design of plate girder bridges ,are required for such a

great variety of span lengths within the limiting lengths

of span for such structures,that special methods o f

computing and tabulating them are advisable . I n the

principal Railway Engineers’B ridge Works and Consult ing

Engineers ’ o ffices the maj or part of the structures designed

will be made for some standard loading . The loadingknown as Cooper E50 is perhaps more widely used than

any other for railway bridges . The quantit ies given in

Table No. 1 are for Cooper’s E50 and are similar to those

which should be determined and kept on record for any

standard loading to enable design to be made with proper

facility

SHEARS

I n computing shears it should be noted that for deck

bridges the maximum end shear is given for the engine

located so that either the first or last driving wheel stands

at the end of the span .

The maximum shear at an interior point on the left

of the center of the span is given when the load extends

from the right and up to the point and perhaps a little

beyond . Every load that pas ses from just to the right

LOADING FOR MAXIMUM SHEAR AND MOMENT 65

of the point to just the left o f the point under consideration

causes a decrease in shear by the amount of that load .

Any further movement of the system to the left increases

the shear until another load passes from the right to the

left o f the point . The maximum shear must therefore be

determined by trial and generally will occur for the first

or second engine wheel just to the right o f the given point .

For through bridges the maximum shear in any panel

Wis given when the well-known criterion P

7nis sat isfied ,

in which P is the load in the panel , W the total load on

the bridge and m the number of equal panels in the span .

Sometimes time or even three posit ions of the load satisfy

the criterion,in which case the maximum shear is determined

by comupting the shear for all posit ions whi ch satis fy

the criterion .

For maximum concentrated load brought to a floor

beam or trestle bent by two adjacent stringers or girders

nWn +m’

m are the lengths of the adjacent spans and P the load

on the span o f length n,and W the load on both spans .

The criterion for position is P = in which n and

BEND ING MOMENTS

The bending moment , due to a certain number o f

moving loads at any definite point in a girder either deck

nWor through

,is a maximum when P =—

min whi ch P is

the load on the left of the point , W the total load on the

structure and n the distance from the point to the left


end and m the span length . This criterion enables any

group of loads to be placed to produce a maximum moment .

The criterion will somet imes be sat isfied by more than

one position of the l ive load , generally that pos it ion which

has the heaviest load at the point under considerat ion

and in addition the greater load on the structure is the

one giving maximum moment . The moment for all posi

t ions of lead which satisfies the criterion must be com

puted for a certain determ ination of the maximum .

For a deck structure the point o f maximum bending

moment occurs at or near the cente r , generally a little

distance away from the center , the locat ion of this point

of maximum moment being different for different systems

of loading . To aid in determining the position of the

loading for absolute maximum it should be remembered

that :

The maximum moment must occur where the shear

passes through zero ;

For a system of concentrated loads the shear must

pass through zero at one of the loads '

The amount of load on either segment, into whichthe point of maximum moment divides the span ,is to the total load on the span as the lengthof the segment is to the span length .

These requirements for pract ical cases fix the point ofmaximum bending moment under one of the two Whee ls

adjacent to the resultant of the system . The criterion for

the exact position‘bf the loading to produce maximum

bending moment folloWs :

For any given system of loading , the toads should be

nearest to the resultant. This criterion may

as; follows :

Fig . 130 show any carry ing

.movmg

FI G . 13a .

center of the span ;

unde r 0“

0 1: a

the resultant of the system W

0 be the distance of W from the right

the. resultant o f the loads to th e left o f 0 be

.r be the distance from 0 to 0 ;

Then the bending moment at“

o=M = R1— 1:

— Pb=

68 NOTES. ON PLATE-GIRDER DESIGN

Now suppose the load-ing to advance a small distance

to the left o f dx, then b= b, az= x+ dro, and c= c and

the bending moment

— [m+d — Pb;

neglecting term-s containin g din?

W0. Wm: Wade: d d ft

Subtracting (a ) from (0)

W

z

dx Winds Wade

which establishes the criterion. as stated .

PROBLEMS

130 . Compute the maximum bending moment

girder of 18"e ffective Span length for Cooper

’s E

130 . Campu-te the maximum bending moment for a 2 01

’span ,

sameloading as 130 .

loading as 130 .

TABLE OF BEND I NG MOMENTS, ETC. 00

ART . 14. PREPARATION OF A TABLE OF BENDI NG MOMENTS,

SHEARS, AND CONCENTRATED LOADS FOR COOPER’S E50

LOAD ING

The great advantage of a table of moments and shears is

that it maybe prepared in a few days for all spans for which

it is at all likely to be needed . Computations for a system

of quantit ies made at one t ime show by the law of the

in crease or decrease of the quantities any error , and the

similarity o f the computat ions enables them to be very

rapidly made . When a system of manv loads moves over

spans o f varying length it is evident that one load produces

maximum moment for spans up to a certain length , two

loads for a certain other length ,and so on .

FI G . 140 .

For one load on a span see Fig . 140 .

M R(l — .r) —.r ) 50 ,000x

1dM 00 ,000 t

dx l

or M = a max ., when

which, as is well known, is the general

expression for max. moment for any span with one load


The general equation for moment for two loads spaced

5’apart is

M Rd _ 5 71000 001:

(I _ x _ 5)

2

X100 ,000x— 1oo,000 w ,

200 ,000x

l l

F I G . 14b.

Substituting for 3: its vlaue, we have fo r

+ 1

+ 37 5

By plott ing the curves o f moments for different lengths

of spans we see that somewhere between 8’and 9' one

load and two leads produce the same moment .

To find this point exactly we make the equations fo r

moments simultaneous and solve to find the value of l

for which the moments are equal .

50 ,000lFor this case

4

whence l =

The next step is to get the general equat ion of moments

for three loads . Make it simultaneous with that for two


Maximum bending moment

+ 225 0001:

+225,000x

2,075,000x+ 225,000x

l

$2 ,250 ,000x

I l l

450,000x

l

450 ,000x

Ix=§

Substituting this value for x in the follow-ing simplified

e xpression for the moment

l l

there is given

2

_20

,75

l

0 ,000

+ 112,500l

56,250l 56,250l

1

l

TABLE OF BENDING MOMENTS,ETC. 73

Mak ing the moments for four and five loads equal, we havw

+34

’

l

13 l

— 6250l = — fi

6250i2

12 — 26l

22 — 26l + 169

1 — 13 =

+ 56,25oz

as the limiting length of span for which four loads produce maximum moments .

The tabulat ion following shows the. moments and the

positions o f the loads that produce then for the spans

from 9 to 27’ for intervals of 1’

9x :

s 132-333

I 4 332888 r) 0 015 5131500 M = -5

,oocz 0+

16l ltS ' l

1718

”5z

19 to R=75,000

262

330 18-66’

M =37,5DOl

a 53 fr :

5 28826 R= 100

,000

27

M = 50 ,0001


140 . Compute the foregoing table up to

14b. Compute the maximum fl end shear for one girder of a 38'

effective single-track deck plate

-

g irder bridge . Two girders carry one

track . Cooper’

s E

ART. 15 . TABLE OF MOMENTS, SHEARS AND CONCENTRA‘TED

LOADS FOR COOPER’S

“

E

The method of writing equations for b ending moments

of Art . 14 need not be followed fer spans over 30’ long .

For s‘p-ans over 30’ in length the maximum center moment

is only a small fraction of one per cent less than the

absolute maximum moment . The following table gives a

convenient arrangement of the quantities for spans up to

75'

e ffective length .

Fer through spa ns tables o f moments at the panel

points and shears in all the panels for varying numbe r

o f pan els and panel length are readily prepared and should

be made for anv loading as much used as Cooper’s systems .

PROBLEMS. Loading Cooper’

s E

15a . Compute the live load concentration for“ a tres tle bent which

car ries adjacen t spans of 30’and Show by a sketch the position

of loads .

15b. Compu te the maximum cen ter moment for a 60’ span . Show

by sketch the position of loads .

150 . Compu te the maximum shear at the cen ter of a 60’span.

Showby ske tch the posit ion of loads .

15d. Compute the maximum shear at, a point 15’

from one end of

a 60’span . Show by sketch the position of the loads .

TABLE OF MOMENTS, SHEARS, ETC.

Q. Pt. C. O. Come . Span End Q. Pt .

Sh . Sh . Mom . F1. Bm . in ft . Sh . Sh.

l

Mom .

SHORT -T ITLE CATALOGUEOF THE

PUBLICAT IONS

JOHN W I LEY SON S

N EW YORK

LONDON : CHAPMAN HALL,LIMI TED

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Manua l of Qua lita tive Chem ica l Ana lysis . Part I Descrip tive . (Wells . )8voQuan titative Chem ica l Ana lysis . (Cohn . ) 2 vols 8vc ,

When Sold Separately , Vol I , $6 Vol I I 88.

Fuertes '

s Water and Public Hea l th .

Furman and Pardoe '

s Manua l of Practical AssayingGetman

’

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G ill 's Gas an d Fuel Ana lys is for Engineers .

Gooch and Brown ing'

s Ou tlines of Qual ita tive Chem ica l Ana lysis .

Large 12mo,

Grotenfelt'

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Groth'

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Hanausek'

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Herrick '

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Hinds '

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Laboratory Manual for Studen ts . 12mo,

Hol leman'

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(Wa lker .

Text book of I norganic Chem is try . (Cooper ) . 8vo ,

Text book of Organ ic Chem istry (Wa lker and Mott . ) . 8v0 ,

Holley '

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5 D irections for LaboratoryWork 1n Physiologica l Chem istry . .8voJohnson ’

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Landauer '

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. 12mo,

Leach '

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. . 8vo ,

Lob'

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Lodge'

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5 First Princip les of Chem ical Theory” . . 8vo.

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10

1 00

Wait ’s Law of Operations Preliminary to Construction in Engineering andArchitecture

Sheep ,

MATHEMATICS.

Baker ’5 E l liptic Functions . 8yo,

B riggs ’5 Elemen ts of P lane Ana lytic Geometry . ( .Bocher 12mo ,

Buchanan '

s P lane and Spherica l Trigonometry 8vo,

Byerly’

s Harmon ic Functions . 8vo,

Chandler ’s Elements of the Infinitesimal Ca lcu lus 12mo,

Cofiin'

s Vector Ana lys is . 12mo,

Compton ’

s Manua l of Logarithm ic Compu tations 12mo ,

D ickson '

s Col lege Algebra Large 12mo ,

I n troduction to the Theory of Algebra ic Equa tions Large 12mo,

Emch’

s I n troduction to Projective Geometry and its App lication 8vo,

Fiske '

s Functions of a Comp lex Variable 8vo,

Ha lsted '

s E lemen tary Syn thetic Geometry .

E lemen ts of 8yo ,

Rationa l Geometry 12mo,

Syn thetic Projective Geometry 8vo,

Hancock'

s Lectures on the Theory of E l lipt ic Functions 8vo,Hyde ’

s Grassmann'

s Space Ana lysis . 8vo,

Johnson ’

s (J. B . ) Three-p lace Logarithm ic Tables : Vest-pocket s ize , paper ,100 cop ies ,

Moun ted on heavy cardboard , 8X10 inches ,10 cop ies ,

Johnson '

s (W. W . ) Abridged Editions of D ifierential and I n tegra l Ca lcu lus .

Large 12mo, 1 vol.

Curve Tracing in Cartes ian Co-ordina tes 12mo ,

D iff eren tia l Equations .

E lementary Treatis e on D iff erentia l Ca lcu lus . Large 12mo ,

E lemen tary Treatise on the I n tegra l Ca lcu lus Large 12mo,

Theoretica l Mechan ics 12mo ,

Theory of Errors and the Method of Least Squares 12mo,

Treatise on D iff erential Ca lcu lus . Large 12mo,

Trea tise on the I ntegra l Ca lcu lus . Large -12mo ,

Treatise on Ordinary and Partia l D ifferen tia l Equations . Large 12mo,

Karapetofi'

s Engineering App l ications of Higher M athematics . Large 12mo ,

Koch '

s Practical M athematics . ( I n Press .)Lap lace ’

s Phi losophica l Essay on Probabilities . (Truscott and Emory . ) 12mo ,

Le M essurier'

s Key to Professor W . W . Johnson ’

s D ifferential Equations .

Smal l 8vo,

Ludlow'

s Logarithm ic and Trigonometric Tables . 8vo,

Ludlow and Bass '

s Elements of Trigonome try and Logari thm ic and OtherTables 8vo ,

Trigonometry and Tables published separa te ly Each ,

Macfarlane ’

s Vector Ana lysis and Quatern ions 8vo,

McMahon'

s Hyperbol ic Functions 8vo,

Mann ing’s I rrationa l Numbers and their Represen ta tion by Sequences and

Series 12mo ,

Mathematica l Monographs Edited by Mansfield Merriman and Rober tS. Woodward . Octavo, each

No. 1 . H istory of Modern Mathema tics , by Dav id Eugene Sm i th .

No. 2 . Syn thetic Projective Geometry , by George Bruce Halsted .

No. 3 . Determ inan ts , by Laonas Gifi'ord Weld . No. 4 . Hyper~bolic Functions , by James McMahon . No. 5 . Harmon ic Functions , by Wil liam E . Byerly . No . 6 . Grassmann

'

s Space Ana lys is ,

by Edward W . Hyde. No. 7 . Probabil ity and Theory of Errors,by Robert S. Woodward . No. 8. Vector Ana lysis and Quatern ions,by Alexander Macfarlane . No. 9 . D iff eren tia l Equations , byWi l liam Woolsey Johnson . No. 10 . The Solu tion of Equationsby Mansfie ld Merriman . No. 11 . Functions of a Comp lex Variableby Thomas S. Fiske .

M aurer 's Techn ica l Mechan ics .

Merriman'

s Method of Least SquaresSolution of Equations .

35 00

5 50

1 25

1 00

‘l' Moritz 's E lements of P lane TrigonometryRice and Johnson'

s D ifferential and I n tegra l Ca lcu lus . 2 vols . in one.

Large 12mo ,

E lemen tary Treatise on the D iff eren tial Ca lcu lus. .Large 12mo,

Sm i th '

5 His tory of Modern Mathematics . 8vo,

Veblen and Lonnes’

5 I n troduction to the Rea l I nfinites imal Analysis of OneVariable svc , 2

Waterbury’

s Vest Pocket Hand book of Ma thematics for Engineers .

x5i inches , mor.

En larged Edition , I ncluding Tables mor.

Weld '

s Determ inan ts 8vo ,

Wood ’

s E lemen ts of Co-ordinate Geometry. . Sro ,

Woodward ’

s Probabil ity and Theory of Errors 8ve ,

MECHANICAL ENGINEERING.

MATERIALS OF ENGI NEERING . STEAM-ENG I NES AND BOI LERS.

Bacon '

s Forge Practice. 12mo,

Ba ldwin '

s Steam Heating for Bui ldings 12mo,

Barr and1Wood’

s Kinematics of M achinery 8yo ,

Bart le tt '5 Mechanica l D rawing 8vo ,

Abridged Ed 8vo,

Bartle tt and Johnson ’

8 Engineering D escriptive Geometry 8yo ,

Burr's Ancien t and Modern Engineering and the I sthmian Cana l 8ve ,

Carpenter ’s Heating and Ventilat ing Bu il dings 8ve ,

Carpen ter and D iederichs ’s Experim ental Engineering . 8vo,

Clerk'

s The Gas, Petrol and Oil En gine . . 8vo,

Compton ’

s First Lessons in Meta l Working 12mo,

Comp ton and De Groodt’

s Speed Lathe 12mo ,

Cool idge '

s Manua l of Drawing sve , paper,Coo l idge and Freeman ’

s E lemen ts of Genera l D rafting for Mechanical Eugineers Oblong 4to,

Cromwel l 's Treatise on Belts and Pu l leys . 12mo,

Trea tise on Toothed Gearin g. 12mo ,

D ingey's Machinery Pattern Making . 12mo,

Durley'

s Kinematics of Machines .

Flanders '

s Gear-cutt ing Machinery. .Large 12mo ,

Fla ther ’s D ynamometers and the Measuremen t of Power. 12mo ,

12mo ,

G i l l ’s Gas and Fuel Analysis for Engineers . 12mo,

Goss'

s Locomotive Sparks . 8vo ,

G reene ’

s Pumping Machinery 8vc ,

Hering’s Ready Reference Tables (Convers ion Factors) . 16mo, mor.

Hobart and E l lis ’s Hig h Speed D ynamo E lectri c Machinery. 8vo ,

Hutton ’

5 Gas EngineJam ison ’

5 Advanced Mechan ica l D rawin g.

E lemen ts of Mechanica l D rawing .

Jones 's Gas Engine.

Machine DesignPart I . Kinematics of Machinery.

Part I I . Form. Strength , and Proportions of Parts. 8vo ,

Kaup‘

s M achine Shop Practice .Large 12mo ,

Kent ’s Mechanical Engineer's Pocket-Book 16mo, mor.

Kerr ’

s Power and Power Transmiss ion. 8vo,

Kimba l l and Barr 's M achine D esign 8vc .

King'

s Elements of the Mechan ics of M aterials and of Power of Transm ission . c o a o o o o o o o o o o o o o o o o o o c o c o-i v o o o o c c o o e o o o o . 8V0

* Lan za’

s Dynam ics of M achinery . . 8vo,

Leonard ’

s Machine Shop Tools and Me thods . 8vo,

c o c o -o n e . o o o o o oLoren z ’s Modern Refri geratin g Machinery. ( Pope, Haven , and Dean) . .8vo,

MacCord’

s Kinema tics ; or. Practical Mechan ism . 8vo,

Mechanica l D rawing.

o o o o o o o o o t o o o o i e c b e t b e o e e 8vo,

MacFarland'

s S tandard Reduction Factors for Gases . 8vo

Mahan '

s I ndustria l Drawing. (Thompson . )Mehrtens

'

s Gas Engine Theory and Design Large 12mo,

M il ler , Berry , and Riley '

s Problems in Thermodynam ics and Heat Engineer. 8vo, paper ,

Oberg ’

s Han dbook of Sma l l Tools . Large 12mo ,

Parshal l and Hobart ’s Electric Machine Des ign . Sma l l 4to. ha lf lea ther ,Peele

'

s Com pressed Air Plant . Second Edition , Revised and En larged . 8vo,

Perk ins'

s General Thermodynam ics . ( I n Press . )Poole '

s Calorific Power of Fuels .

Porter 's Engineering Rem in iscences , 1855 to 1882Randal l ’s Treatise on Hea t . ( I n Press .)Reid '

s Mechan ica l D rawing. (E lemen tary and Advanced . ) 8vo,

Text-book of Mechan ica l D rawing and E lementary Machine D esign .8vo,

Richards'

s Compres sed Air 12mo,

Robinson '

s Princip les of Mechan ism .

Schwamb and Merril l ’s E lemen ts of Mechan ism 8vo ,

Sm i th (A. W . ) and Marx '

s Mach ine Design . 8yo ,

Sm i th '

s (O. ) Press-working of Meta ls 8vo,

Sorel 's Carbureting and Combustion in Alcoho l Engines . (Woodward and

Preston . ) .Large 12mo ,

Stone ’

s Practica l Testing of Gas and Gas Meters . 8vo,

Thurston ’

s An imal as a Machine and Prime Motor , and the Laws of Energe tics .

12mo,

Treatise on Friction and Lost Work in Machinery and M il l Work . . 8vo,

Ti l lson ’

s Com p lete Au tomobile I nstructor. 16mo,

Titsworth’

s Elemen ts of M echan ica l D raw ing . Oblong 8vo ,

Warren '

5 E lements of Machine Construction and D raw ing” . 8vo,

Waterbury s Vest Pocket Hand book of Ma thematics forEngineers .

2aX5§ inches , mor.

En larged Edition , I ncluding Tables . mor.

Weisbach '

s Kinematics and the Power of Transm 1ssion . (Herrmann

Klein ) 8vo ,

Machinery of Transm iss ion and Governors . (Hermann— K le in ) . . 8vo ,

Wood '

s Turbines 8vo ,

MATERIALS OF ENGI NEERI NG.

Burr 's E lastici ty and Resistance of the Materia ls of Engin eering 8vo,

Church '

s Mechan ics of Engineering. 8vo,

M echanics of Sol ids (Being Parts I , I I . I I I of M echan ics of Engineering) .8vo ,

Greene ’

s Structural Mechan icsHol ley ’

s Analysis of Paint and Varnish Products. ( I n Press . )Lead andZinc Pigments . gLar e 12mo ,Johnson ’

5 (C. M . ) Rap id M ethods for the Chem ica l Ana lys is ofg

Specia lSteels Steel Making Al loys and Graphi te Large 12mo ,Johnson '

s (J B Materia ls of Con s truction 8vo ,Keep '5 Cast I ron .. 8vo ,

King '

5 Elements of the M echan ics of M ateria ls and of Power of T ransm ission 8vo ,

Lanza '

5 App lied M echanics .

Lowe’

s Pa in ts for Steel Structu res .

Maire '

5 Modern Pigments and their VehiclesMaurer '5 Techn ica l Mechan ics .

Merriman’

s Mechan ics of MaterialsStrength of Materia ls .

Metca lf '5 Steel . A Manua l for Stee l users .

M urdock '

5 Strength of M aterials .

Sabin '

5 I ndustria l and Artistic Technology of Pa in t and Varn ishSm ith '

s (A. W. ) Materia ls of MachinesSm i th '

s (H . E Strength of Ma teria lThurston '

5 Ma teria ls of Engineering . 3 vols , 8vo,Part I . Non meta l lic Materia ls o

n

i Engineering , . 8vo,Part I I . I ron and Steel 8vo ,Part I I I . A Treat1se on B rasses , _Bronzes , and Other Al loys and their.

,8vo

7 50

6 00

4 50

2 50

3 00

2 50

Dana'

s Text-book of Elementary Mechanics for Col leges and Schools . 12mo, $1 50Du Bois '

s Elemen tary Prin cip les of Mechan icsVol. I . KinematicsVol. I I . Sta tics . . 8v0 ,

Mechan ics of Engineering. Vol. I Sma l l 4toVol I I Small 4to,

Greene'

s Structural Mechan ics . 8vo.

Hartmann’

s Elemen tary Mechan ics for Engineeri ng Studen ts 12mo,

James ’s Kinematics of a Poin t and the Rationa l Mechan ics of a Particle .

Large 12mo,

Johnson ’

s (W . W . ) Theoretica l Mechan ics . 12mo ,

King '

s Elemen ts of the M echanics of M aterials and of Power of T ransm ission 8yo,

Lanza ’

s App lied Mechan ics 8vo ,

Martin '

s Text Book on Mechan ics , Vol. I , Statics 12mo,

Vol . I I . Kinematics and Kinetics . 12mo ,

Vol . I I I . M echan ics of M aterials 12mo,

Maurer's Techn ica l Mechan ics . .8vo .

Merriman’

s E lemen ts of Mechan ics .

Mechanics of MaterialsM ichie '

s Elemen ts of Ana lytica l Mechan ics . . 8vo ,

Robinson ’

s Princip les of Mechan ism . . 8vo,

Sanborn '

s Mechan ics Problems . Large 12mo ,

Schwamb and Merrill 's Elem en ts of Mechan ism . 8vo,

Wood '

s Elemen ts of Ana lytica l Mechan ics . 8vo ,

Princip les of E lemen tary Mechan ics 12mo,

MEDICAL.

Abderhalden ’

s Physiological Chem istry in Thirty Lectures. (Ha l l and

von Behring's Suppress ion of Tubercu losis . 12mo,

B‘

olduan'

s Immune Sera 12mo ,

Bordet 's Studies in Immun i ty . (Gay. ) 8vo ,

Chap in '

s The Sources and Modes of I nfection .Large 12mo,

Davenport 's Statistica l Methods wi th Specia l Reference to B iological Variations 16mo, mor.

Ehrlich '

s Col lected Studies on Immun i ty. 8vo,

Fischer ’s Nephritis Large 12mo ,

Oedema 8vo ,

Physiology of Alimen tation Large 12mo ,

de Fursac’

s M anual of Psychiatry . (Rosanofi and Col l ins.) .Large 12mo ,

Hammarsten'

s Text-book on Physiologica l Chem istry . (Mande l ) 8vo ,

Jack son '

s D irections for Laboratory Work in Phys iologica l Chem istry . . 8vo.

Lassar-Cohn ’

s Praxis of Urinary Ana lysis . (Lorena ) 12mo,

Mandel ’s Hand-book for the Bio-Chem ical Laboratory . 12mo.

Nelson '

s Ana lysis of D rugs and Medicines 12mo,

Pau li ’s Phys ica l Chem istry in the Serv ice of Medicine. (Fischer. ) . 12mo ,

Pozzi-Escot ’s Toxins and Venoms and their An tibodies. 12mo ,

Rostosk i’

s Serum D iagnosis . (Bolduan ) 12mo,

Ruddiman'

s I ncompatibi lities in Prescriptions . 8vo,

Whys in Pharmacy 12mo ,

Sa lkowski 's Phys iologica l and Pathological Chem istry. (Orndorff ) . .8vo,

Satterlee'

s Ou tlines of Human Embryology 12mo,

Sm i th ’

s Lecture Notes on Chem is try for D en ta l Studen ts 8vo,

Whipp le'

s Tyhpoid Fever Large 12mo ,

Woodhu l l 's M ili tary Hygiene for Officers of the Line Large 12mo,

Persona l Hygiene 12mo ,

Worcester and Atkinson '

s Small Hosp i tals Establishmen t and Main tenance.

and Suggestions for Hosp i ta l Architectu re , w i th P lans for a Smal lHosp ital 12mo, l 25

METALLURGY.

Betts’

s Lead Refin ing by E lectrolysis .

Bol land '

s Encyclopedia of Founding and D ictionary of Foundry Terms usedin the Practice of Mou lding 3 00

I ron Founder 12mo, 32Supp lemen t 12mo ,Borchers's Metal lurgy . (Ha l l and Hayward 8yo,

Burgess and Le Chatelier'

s M easurement of High Temperatures . Th irdEdition . ( I n Press .)

Douglas ’

s Un techn ical Addresses on Techn ica l Subjects” 12mo,

Goesel’

s M inera ls and Metals A Reference Book . 16mo mor.

I les ’5 Lead smel ting . 12mo ,

Johnson 5 Rap id Methods for the Chem ica l Ana lys is of Specia l Stee ls ,Stee l making Al loys and G raphite. Large 12mo,Keep ’5 Cast I ron .

. 8vo,

Metcalf '5 Steel . A Manua l for Steel-users . 12mo,

M inet ’s Production of Alum inum and its I ndustrial Use . 12mo,

Palmer '5 Foundry Practice . ( I n Press .)Price and M eade ’

s Techn ical Analysis of Brass 12mo ,

Ruer ’5 Elemen ts of Meta l lography. (Ma thewson . 8vo ,

Sm i th '

s Materia ls of Machines 12mo ,

Tate and Stone ’

s Foundry Practice 12mo,

Thu rston ’

s Materia ls of Engineering. I n Three Parts . . 8vo,

Part I . Non -meta l l ic Materials of Engineering , see Civil Engineering ,page 9 .

Part I I . I ron and Steel 8vo, 3 50Part I I I . A Treatise on Brasses , Bronzes , and Other Al loys and their

Con stituen tsUlke

’

s Modern E lectrolytic Copper Refin ingWes t ’s American Foundry Practice

Mou lders ' Text Book

M I NERALOGY.

Baskervil le'

s Chem ica l E lemen ts . ( I n Preparation . )B rown ing'

s I n troduction to the Rarer E lemen ts 8vo, 1 50B rush '

s Manua l of Determ ina tive M inera logy . (Penfield . ) 8vo, 4 00

Bu tler 's Pocket Hand-book of M inera ls l o , mor. 3 00Ches ter 's Cata logue of M inera ls . . 8vo , paper , 1 00

Cloth , 1 25

Crane'

s'

Gold and Silver . 8vo , 5 00

Dana’

s Firs t Appendix to D ana’

s N ew Sys tem of M inera logy .Large 8vo, 1 00

Dana’

s Second Appendix to Dana’

s New System of M inera logy .

Large 8vo , 1 50

Manua l of M ineralogy and Pe trography 12mo , 2 00

M inera ls and How to Study Them r. 12mo, 1 50

Sys tem of M ineralog y Large 8vo ha lf leather , 12 50Text book of 8vo, 4 00

Douglas ’s Untechn ica l Addresses on Techn ica l Subjects 12mo , 1 00

Eakle ’

5 M inera l Tables 1 25

Eckel 's Bu il ding Stones and Clays . ( I n Press .)Goesel

'

s M inera ls and Meta ls : A Reference Book 16mo, mor.

G roth'

s The Op tica l Properties of Crys ta ls . ( Jackson ) . 8vo,

Groth'

s I n troduction to Chem ica l Crysta l lography (Marsha l l) . . 12mo,

Hayes ’s Han dbook for Field Geologists . 16mo , mor.

I dding s’

s I gneous Rocks 8vo ,

Rock M inera ls 8yo ,

Johannsen'

s Determination of Rock-form ing M in era ls in Thin Sections . 8vo,

Wi th Thumb I ndex 5

Martin ’

s Laboratory Gu ide to Quali tative Ana lysis wi th the B lowp ipe 12mo,

Merri ll 's Non metal lic M inera ls . Their Occurrence and Uses 8vo,

Stones for Bu i lding and Decoration . 8vo,

Penfield'

s Notes on De terminative M inera logy and Record of M inera l Tests .

8vo , paper ,Tables of M inera ls , I ncluding the Use of M inerals and Statistics of

Domestic Production 8vo,

Pirsson'

s Rocks and Rock M inera ls 12mo,

Richards’

s Synopsis of M inera l Characters . . 12mo, mor.

Ries'

s Clays : Their Occurrence, Properties and Uses .8vo.

17

Ries and Leighton '

s History of the Clay-working I n d u stry of the Uni tedStates 8vo , 82 50

Rowe'

5 Practica l M inera logy Simp lified 12mo, I 25

Tillman’

5 Text book of Importan t M inera ls and 2 00

Washington '

5 Manual of the Chem ica l Analysis of . 8vo , 2 00

M INING .

Beard '

3 M ine Gases and Exp losions.

Crane '

5 Gold and Silver.Index of M in ing Engineering Literature

Ore M in ing MethodsDana and Saunders

’

s Rock D rill ingDouglas '

s Un techn ical Add resses on Techn ica l SubjectsEissler

'

s Modern High Exp los ivesGoesel

'

s M inera ls and Metals : A Reference Book 16mo, mor.

I h lseng’

s Manua l of M in ing. . 8vo ,

I ies 's Lead Smel ting . 12mo ,

Peele'

s Compressed Air Plant . 8vo,

Riemer’

s Shaft Sinking Under D ifficu l t Condition s . (Com ing and Pee le . )8vo ,

Weaver ’s M il itary Explosives . 8vo,

Wi lson ’

s Hydrau lic and P lacer M in ing . 2d edition , rewri tten . 12mo ,

Treatise on Practica l and Theoretica l M ine Ven tila tion 12mo,

SANI TARY SCI ENCE.

Association of State and Nationa l Food and Dairy D epartmen ts , HartfordMeeting , 1906 . . 8vo ,

Jamestown Meeting , 1907 . 8vo ,

Bashore'

s Outlines of Practica l San itation . 12mo,

San i tation of a Coun try House . 12mo ,

San i ta tion of Recreation Camps and Parks . 12mo,

Chap in '

s The Sources and Modes of I nfection . Large 12mo ,

Folwell'

s Sewerage . (Design ing , Construction , and Ma in tenance . ) 8vo ,

Water-supp ly Eng ineering . . 8vo,

Fowler ’s Sewage Works Ana lyses 12mo ,

Fuertes '

s Water-nltra tion Works 12mo ,

W'

ater and Public Hea lth . 12mo ,

Gerhard ’

s Gu ide to San i tary I nspections 12mo ,

Modern Baths and Bath Houses . 8vo,

San i tation of Public Bu ildings . 12mo ,'

L

* The Water Supp ly , Sewerage, and Plumbing of Modern Ci ty Bu ild ings .

8vo,

Hazen '

s Clean Water and How to Get I t. Large 12mo ,

Filtration of Public Water-supp l ies . 8vo ,

Kinn icu tt , W ins low and Pratt 's Sewage D isposa l . . 8vo,

Leach ’

s I nspection and Ana lysis of Food wi th Specia l Reference to Sta teCon trol 8vo ,

Mason'

s Exam ination of Wa ter. (Chem ica l and 12mo,

Water-supp ly . (Considered principa l ly from a San i tary Standpoin t ) .

8vo,

Mast's Light and the Behav ior of Organ isms . Large 12mo,

Merriman'

s E lemen ts of San i tary Eng ineering 8yo,

Ogden '

s Sewer Construction 8vo ,

Sewer Des ign 12mo,

Parsons 's D isposal of Mun icipa l Refuse 8vo,

Prescott and Wins low ’

5 E lemen ts of Wa ter Bacteriology , w ithSpecia l Reference to San itary Water Ana lys is . 12mo ,

Price '

s Handbook on San i tation 12mo,

Richards'

s Conserva tion by San i tation .. 8vo,

Cost of Cleanness 12mo,

Cost of Food . A Study in D ietaries. 12mo ,

Cost of Liv ing as Modified by San i tary Science . 12mo,

Cost of Shel ter. 12mo ,

Richards and W illiams '

s D ietary Compu ter. syo,

18

Cast Stee l Pedestal

I

Rollers

Expansion shoe other end

int imated Weig h tDead Load 1700# p.

Live Load

R oi R.

lowest Cover plate

LSC’ p“End I nter .

Cross Framel l

Bend ing ,

at 10 000 sq . in.

End shear , LiveDead

STANDARD NO. 17

80'—0 OVER ALL. DECK pu rr: GIRDER,

ssr .

THE NATI ONAL LINES or MEXI CO

Scale Ap ril, 1907BOLLER HODGE

,

Consul t ing Engineers. New York

notes on plate-girder design - forgottenbooks.com the following notes on plate-girder design have...

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