design study of a three span continuous tied-arch bridge

8/18/2019 Design Study of a Three Span Continuous Tied-Arch Bridge

1/72

Scholars' Mine

M!2%2 %2%2 S$% R%2%!# & C%!% W+2

1939

Design study of a three span continuous tied-arch bridge

George Perry Steen

F6 2 !$ !$$! 6+2 !: 0://2#!2%.2.%$/!2%2%2%2

P! % C E%% C2Department:

R%#%$%$ C!S%%, G%% P%, "D%2 2$ ! %% 20! #2 %$-!# $%" (1939). Masters Teses. P!0% 4754.

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2/72

Design

Study of a Three Span Continuous

Tied Arch Bridge

George

Perry Steen

A

Thesis

Submitted

to the Faculty of the

School of

Mines

and Metallurgy

of

The University

of Missouri

In p r t l ful f i l lment

of the

work

required

for

the

Degree Of

Master of Science in Civil

Engineering

Rolla Missouri

9 9


3/72

KNOWLEDGEMENT

o

Professor E W Carlton for

h is valuable cri t ic ism and to

Mr Howard

Mullins

for

his help and suggestions the

writer owes

an expression

of appreci at ion


4/72

T LE OF CONTENTS

Acknowledgment

Page

Synopsis

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

5

Lis t of l lus t ra t ions

• • • • • • • • • • • • • • • • • • • • • •

6

A General

Discussion

of tat ical ly

Indeterminate Structures

7

A Brief H istory o f

Arch i s t o r y

Description

of

Type

• • • • • • • • • • • • • • • • • • • • • • •

The

Principle

of

Least

Work •••••••••••••••

6

Direct

Stresses

Notation

Influence Lines

• •

8

19

tress

Analysis

Diagram For Main Span Equations

Equations For Design Main

Span

Diagram

For

Side

Span

Equations

Equations For

Design

Side

Span

• • • • • • • • • •

•

• • • • • •

•

• • •

25

6

35

36

Evaluation

of

Evaluation

of

±

x

A

f s e ~ ¢ d x

o At

•

•

• • • • • •

•

•

•

•


5/72

T LE OF

CONTENTS

Page

omputations esign

N o

6

on lusions

• • • • • • • • • • • • • • • • • • • • • • • • •

B i b l i o g r a p h y

57

Tables

and l lustr t ions • • • • • • • • • • • • • • •


6/72

Synopsis

In th is study

of

the continuous

t ied arch

r i g e ~

having one main span and two equal side spans the

follow-

ing items

are endeavored:

1

se t

up

the

to ta l

in ternal

work

equations

and

to find

the values

of

the unknown

forces

or

redundants.

2. compute to maximum moments for design and to

complete

the

design of

the

super s truc tu re for

one

bridge.

Detai ls such

as

the gusset plates

r ive t s

connection

angles e tc are

not

design-

ed;

the effect of such b ein g allowed in the

design

for

dead

load as a certain

per

cent for

deta i l s

3.

se t up hypothetical conditions for other

bridges of different span lengths

and

to con-

s t ruct

the

influence

l ines of

the

unknown forces

of

each.

4 compare the

different

bridges upon

the

basis

of

the i r

influence

diagrams.


7/72

List of I l lus t ra t ions

Diagram for

Main Span

Equations

Diagram For

Side

Span Equation

Evaluation

of Equations Design

N o

Evaluation

of Equations Design No 2

Evaluation of Equations Design N o 3

Evaluation

of Equations Design No 4

Influence Diagrams

Design

N o

Influence Diagrams Design No 2

Influence

Diagrams

Des ign No. 3

Influence Diagrams Design No. 4

Influence

Ordinates For Live

Load

Moment Design No.

aximum

Positive

and

Negative

Live Load Moment Design No.

1

Influence

Ordinates for Dead

Load Moment and aximumMoment

for

Design

No.

1

Section

Summary

Design

No. 1

Section

and Weight

Summary Design No.

1

Plate A

Plate

Plate

Plate 2

Plate 3

Plate 4

Plate

5

Plate 6

Plate

7

Plate

8

Plate 9

Plate 10

Plate

11

Plate

12

Plate

12A


8/72

A General

Discussion

of Stat ical ly

Indeterminate

Structures

The

s ta t ica l ly

indeterminate type of struc-

ture

i s

not g enera lly favored by the American Engineer.

Until

recently

this

type

has

been

actually

shunned

be

cause

of

the

lack of information

and the

lack of t ra in

ing in the design of such. The

majority of

st ructural

engineers prefer

to

use

types

with which they are famil

ia r making many

estimates based

on

judgment and experi

ence.

They

contend too

tha t

for

every indeterminate

st ructure tha t is possible

to

design a determinate

type

tha t w il l

carry the

same load with as l i t t l e or

less

material .

A

few

ci ta t ions fol low giv ing the com-

parative

amount

of

material

needed

for

a

type

of bridge

stepping

from

a s ta t ica l ly determinate one to

one

of in

determinateness the f i r s t

degree

or from one toanoth

er

type that i s

s ta t ical ly

indeterminate a higher degree.

trFor

the

Sciotovi l le Bridge

over

the Ohio r iver

a t Sciotovi l le Ohio con sis tin g o f

two

spans a t

77 feet

the saving favor of continuous

spans over

two simple

spans

was

found to

be about

15 per

cent.

ftAccording to comparative studies reported by


9/72

four

spans

is

16

19 and 21

per

cent

respectively

when the

span

length is about 325 feet ;

and

20 24

and

28 per

cent

respectively

when the

length is

about

500

feet .

1

Ierriman and Jacoby investigated the resul t

of placing

a

hinge

a t

the

crown

of

a

550-foot

two-hinged

spandrel-braced arch bridge over the Niagara

River

and

found tha t i f jus t

one hinge

were placed in

the

upper

chord the weight of the arches alone would be decreased

11.8

per

cent. 2

Professor

Dietz

of

Munich s tates

that

a

very careful comparison o f mate ria l required for

a

two

hinged and three -h inged arch for

the

110-foot span of

the Hacker

Bridge in

Munich indicated that 11.3 per

cent

excess

was

required

for

the

l a t te r

type.

2

A chief

objection

to the

use

of a continuous

bridge or of a

h inge le ss a rch has

been that i t

is

sensi

t ive to s ettlemen ts of foundations. Needless to say

1 .

Hool and

Kline

Movable

and Long-Span Steel

Bridges p. 200.

2 .

Parcel and Maney

s truct ion

p. 405

Stat ical ly Indeterminate

Con


10/72

th is type

of bridge would not

be considered unless

rock

foundation

was

available

and any

s l ight

se t t l e -

ment of

the

foundation would cause

very

l i t t l e change

in

the

s t ress

Another objection has

been

that the

continu-

ous bridge

would be

seriously

affected by temperature

changes.

The effect

of

the expansion and contraction

of intermQdiate piers wil l

cause

l i t t l e

changes

in

s t resses

according to

several

studies

reported.

I t

has also been determined tha t

the

effec t of unequal

temperature

changes

different

parts

of the

span can

safely

be

neglected providing the members

of the

span

have the

same

coeff ic ient

of expansion.

A continuous

bridge of the

indeterminate type

has advan tages r ig id i ty

and

economy and the

fewer

number of expansion jo in ts I t s use permits a decrease

the width of intermediate piers .

In

addition a

continuous

bridge

and especial ly a continuous arch span

has a l l the advantages

of

a cantilever bridge

for loca

t ions

over

navigable

waters

where

clearance

must be

mainta ined during the

construction

period.

The

cont in

uous s t ructure e lim in ate s th e detai l ing

and expense of

hinges. There i s

no question but

that the design of an


11/72

indeterminate

structure involves great deal more work

and

calculat ions

t

is

usually

necessary

to

m ke more

than one t r i l design However i f small per

cent

of

material

and

weight is saved

the

extra

time spent

in

design is

warranted

the f in l analysis

the resul ts

given by

engineers both pro

and

con

on the

st t ic l ly

indeter-

minate

structure

are l ar ge ly t he o re ti ca l and are not

based on

extensive experiments There are not enough

actual

monuments of

i n e t e r m ~ t e

structures that have

approximately

the

s me

s i te

conditions of the

m ny

sim-

ple

structures

Until more

such

bridges are bui l t

affording

us

more re l i ble comparisons the advantages

of one

or of

the other wil l

be

debated largely on the

basis

of

theory t

is probable that the

st t ic l ly

determinate s t ructure wil l

be

favored for

re la t ively

short spans and

the inde te rminate

bridge

will replace

them

for

longer spans


12/72

A Brief

H istory of Arch

Bridges

The

modern period in arch-bridge building be

gan in the year 1716

when

the French ~ e p r t e m e n t des

Ponts

e t

Chaussies

was

formed

although

the

f i r s t

real

emploYment

of the

t rue-arch

type

was

as

early

as

6

B.C.

The f i r s t

arch span

of

the monolithic

type was a

wrought

iron bridge bui l t in 1808

over

the River

Crow a t

St.

Den-

nis However,

was not unt i l several years l a te r in

1879, tha t Weyrauch

developed

the

fundamental equations

upon which

the

elast ic-arch theory is based. Arches of

iron

have

given way to structural s teel spans, which type

the

form of r ibs hinged and

f ixed),

t russ design, or

braced

spandrel

construction i s

being

ut i l ized to a very

large

extent

in

the

f ield

today.

The

concrete

arch

con

s t i tutes the

l a t es t

development in the arch

and

is

a

very

important

type,

especial ly

for

short-span structures, in

which has

nearly

an exclusive f ie ld

Weare more

part icular ly concerned with the

long-span

structure, in

which the s tee l arch supersedes

the

use

of

concrete

for

material .

I t i s per t inent and

interest ing to

follow th e

several

examples

of arch spans tha t

have

been erected.


13/72

tensively is

the

Eads Bridge

erected

in 1874 across the

Mississippi

River

a t

s t

Louis.

t

consists

of

two

side

spans of 502 feet and a center span

of

520 feet with four

l ines of fixed circular arch trusses

with paral lel

chords.

The

structure

carries an upper deck of a 52 foot highway

and a

lower deck

32 feet wide

for

railway

t raf f ic

In

1889 the

Washington

Arch Bridge was bui l t

over

the

Harlem River a t

w

York. wo spans were erect-

ed

at

509 feet and

had 6

l ines of

circular plate girder

arches 13 feet deep and a 90 foot

r ise

Each girder has

two

hinges.

The

bridge

carries

a highway 80

feet

wide.

The

bridge

Alexander

was constructed

1899 over the River

Seine

a t Paris France. Spans of 352

feet with 15 l ines of circular

plate girder arches

3 feet

deep and a 21

foot r ise

were used. A highway deck was

supported on

top

with a width of 131 fee t .

During 1897 1898 the

Rhine Bridge

a t

Bonn

Germany was erected. t was a through

arch

having a

highway

f loor 46

feet wide. t contained two l ines of

arch

trusses

614

feet

center

to

center

of

hinges with

a

r ise of 97

fee t to

the

bottom chord.

The Wupper Railway Viaduct was completed

1897

a t

Mungsten

Germany. The spans were hingeless-

arch

trusses

588 fe·et long

with

a r ise of 219 feet and


14/72

carried

a deck of 28 fee t in width. The

t russes

were

16

feet

apart

a t

the

~ o v v

and

84

feet

apart

a t

the

springing l ine

n

of

the f i r s t bridges of the type which is

under

study was a

three span continuous t russ

with a t ied

arch in the

m iddle span

bui l t over the Angora

River

a t

Irkutsk

Russia. The

s t ructure c on sis te d o f

two

side

spans

278 fee t and a

center span of

5 feet with

the

arch

r i s ing 78

fee t

In 1933 the Lachine Bridge was designed and

la te r b uil t

across the

S t

Lawrence

River

near

Montreal.

t consisted of

a

continuous

structure

of several spans

but the main span was a

continuous

ti ed ar ch tr us s 400

feet

long. This

structure was

designed

by the Lake

S t.

Louis Bridge Corporation and constructed by the Dominion

Bridge

Company Ltd.

There is no bridge known that consists o fa

three span

continuous plate

girder with

a t ied arch

r ib

in the

middle

span.

This

scheme was

suggested by

the

l a s t

competition

a t

Mannheim Germany

but

is

not

known whether any has

been

designed and

erected.

The

t ied

arch

i t se l f has

been

used

frequently

in Europe but

no examples can be ci ted


15/72

D es cr ip tio n o f Type

The type

under study is

a

three span

continu-

ous plate girder bridge with

a t ied arch r ib in

the mid-

dle

span. The s t ructure

i s st t ic l ly indeterminate

in

the th ird degree.

In

an ordinary arch

incl ined react ions

are

exerted

on the

bearings

but

in a

t ied arch the hori-

zontal component of

the reaction is resisted by

the

t ie

connecting

the

two bearings

the t ie

acting as a beam as

far

as

the

react ions

are concerned.

The

stresses

how-

ever are

d ete rm in ed as

in

an

arch.

The

bridge

is

a

through

structure

the roadway being suspended

by

hangers

from the arch

r ib

n

end of

the arch

wil l be

fixed

with

the

remaining bearings

consisting

of

ro l le rs

The arch i t s e l f wil l be parabolic shape

the

r ise

of

the arch being about 1/6 of the span length.

Ample r ig id i ty

must

be secured by making

the

width be-

tween the arch

spans t

le s t 1/20 of the span length.

The side spans are plate girders with

the

roadway fasten-

ed

to

the bottom

of

the

girders . The one

exception is

Design

No The length of the

side spans

do

not allow

the

use of a plate

girder

the weight and s ec ti on r equi r-

ed

being

excessive for

economy


16/72

side-sway

and

windloads

in

the l a t e r a l direct ion by

ample

bra cing bo th

the

upper

chord

and

in

the sec

t ion of the t i e

I t

i s generally agreed tha t the Warren

t russ

system

i s

the

most

economical for such l a t e r a l

bracing.

Types similar to t h i s a re gener ally designed

on the basis

of

e l a s t i c centers or on

the

basis

of

l e a s t work. The method of l e a s t work

wil l be

follow

ed

in th is

study. This method, commonly known

as

Cas

t i g l ia n o s

second theorem,

was discovered independently

by

A

Castigliano

and

was

developed

i n

his

t r e a t i s e

on

the t1Theorie de

l equil ibrede s

systemes elastiques

A

detai led

explanation of

the

method

wil l follow.


17/72

The Principle

of

Least Work

The fundamental conception upon which the

method of l eas t work is based

m y

be

presented in an

ideally simple

statement

namely that

a

structure wil l

deform

n

a

manner

consis tent

with

physical

l imitat ions

and

so

tha t the internal

work

of deformation

will

be a

minimum.

nl

n order to apply the

above me.thod the

structure

must

be severed so as

to

obtain a s ta t ica l ly

determinate condition the unknown forces being called

redundants.

Then an

expression for

the

to ta l internal

work

is

se t up in terms of these redundants. The to ta l

in ternal work

i s

equal to one

half

the pro duct

of

the

square of

the

moment and

the

sub division x

divided by

the

product

of

the

moment

of

iner t ia

and

the

modulus

of

e las t ic i ty 2 Part ia l derivatives

of the

internal

work are taken n respect to each redundant and

the

de

rivatives are se t equal

to zero. The unknowns

m y

be

determined y solving the result ing simultaneous

equa-

t ions.

A

m m mum

condition

is obtained

n

set t ing the

1.

2.

Grunter

Theory

of

Modern Steel Structures p.

64.

See Plate

A

for the

mathematical

development of the

to ta l in ternal work.


18/72

derivative

to

zero.

In

every

case of

s t t ic l inde-

termination

where

an

indefini te nrunber

of

different

values

of the redundant

forces

X

will

s tisfy l l s t t -

ic l

requirements the

t rue

values are

those which

rend-

er the to t l in tern l work of deformation a minimum.

D

The

indeterminate

forces or redundants are

chosen

t

points

where

.any displacements are prevented such as

supports.

Hence

the p r t i l derivative is equal to

zero.

Parcel

and

Maney Elementary

Treatise of t ti-

cal ly

Indeterminate

Stresses p. 123.


19/72

Direct S tr es se s

The

tension

s tress in the

horizontal

t ie

produces a direct stress

in

the arch

ring

equal to

Sec

e

There

i s

also a direct stress produced

in

the

t ie

i t se l f

I t

i s

affected

by

the

camber

the

t i e but

the

angle of

inclination i s so

small

tha t

the

secant of

the

angle i s nearly

unity

There-

fore the d ire ct s tre ss i s equal to The camber

of

the

t ie also causes a direct stress in the

hangers

but

the

value of the stress i s negligible


20/72

Notation

H =

horizontal

thrust t ie

Ml= moment

a t l e f t

end

of arch span over pier

M

moment

a t r ight end

of

arch span over pier

I = moment

of

iner t ia

a t any cross section of arch

1 =

moment of iner t ia of side span cross section

A = arch section in square feet

Ah= hanger sect ion in square feet

At=

horizontal

t ie section in square feet

X =

distance

from

end

of arch

span

to

any

section

Xl= distance

from end of

side

span to any

section

Y

=:

ver t ica l

distance between

arch

and

t ie

axes a t

any

point

= incl inat ion of

arch

rib

¢ = incl inat ion

of

horizontal t ie section

=

Linear

increment

arch

span fee t

£

Sl

=

l inear

increment

side

span

in feet

L

=

length

of

arch span feet

Ll

=

length of

side

span

feet

=:

modull s

of

e las t ic i ty

of

structural s tee l

Eh=

modulus

of

e las t ic i ty

of

hanger i f other

than

s tee l

f

=

camber or

horizontal

t i e feet


21/72

A length of any ha nger

in

feet

A

spacing

of

h n g r s ~

or

panel

length

in feet

do

depth of section over pier

in feet

d

depth

of arch in

fee t

dl

depth

of side span sect ion

feet

k

posit ion r t io of

any

load

x... .

•


22/72

Influence Lines

influence l ine may be

described

as a

curve any ordinate of which

gives

the value of

the

func-

t ion shear moment horizontal thrust s t ress e tc .

for

which the curve is

drawn

when a load of unity is a t

the

ordinate.

influence l ine

for

shear

or

moment

records

graphically

the value

of

the func tio n a t

a

sin-

gle section fo r

loads a t

a l l sect ions

which

contrasts

greatly

with

a

simple shear or

moment curve

that records

graphically the

value

of the function

a t

a l l

sections of

a

structure under

a f ix ed loading . From

i t s manner of

construction

rol1ows tha t

any

influence l ine gives

the effect o f a concentrated. load or of

a

series of

such

loads by the mult ipl icat ion

of each

load in tensi ty

by

the

value

of

the in flu en ce o rd in ate a t

the

load.


23/72

Stress Analysis

Ordinari ly

the

f i r s t step the analysis

of

a continuous

s t ructure is the construction of an

influence

diagram

for one of the unknown

reactions

known as

the el s t ic curve.

The

influence l ines

for

l l members

of the

s t ructure

are constructed

by

draw-

ing s t r ight

l ines

across the curve. In the

case of

the structure under study

not

only are the ver t ic l

reactions unknown

but the

moments t

the piers and the

horizontal

thrust

in

the

t ie

as

well. In

such

a

case

influence

l ines

must be obtained for the

three

redun

After

these values are found

the

stresses may

be

determined by s t t ics

The sections

are

proportioned

for

a uniform

l ive

load

in combination with

one

concent ra ted load

placed where

i t s

effec t

s

maximum

This

is in ac-

cordance with the

bridge

specificat ions

of

the

American

Assoc ia tion of State

Highway

Offic ia ls .

l

This load is

chosen

of

such magni tude

as

to

give

an

effect

equal

to

tha t produced by the actual

vehicle

on the

s t ructure .

1 .

The

American Association of

State Highway

Officials

Standard Specificat ions for

Highway Bridges

p.

174.


24/72

Live

loads

can

be used as a uniform load across

the

span

and

the value

of

th e fu nc tio n

can

be

obtained

by m ultiplying th e a rea of

th e in flu en ce diagram

by

the

uniform load per foot . This has been varied

sl ight ly

by

breaking up the load into

a series

of con-

centrated loads assumed

to

be

acting a t

the panel

points . The l ive load moment is then computed

by

mul-

t iplying the panel load by the

ordinate

a t tha t

point.

A fter the

s imultaneous equations

are

solved

by determinants and

the values

of

H

Ml

and M are

found

we

tabulate

the values

for

each

a t

th e v ariou s

load

points .

By taking

these

values and subst i tut-

ing in our i n i t i a l moment

equations we calculate

the

influence ordinates

for

the

moment

a t

each

load point.

3

The dead

load

moments are computed direct ly from our

original i nf luence ord inates

for

H Ml and M as the

dead

loads are

stationary and wil l always

be acting.

4

For our maximum negative and positive l ive load moments

we

must load

th e in flu en ce diagrams

whose

o rd in at es a re

given

on

Plate

10.

5

For

the

maximum

positive

moment

2·

See

Plate 5 p. 63

3·S

ee

Plate 9

p.

67

4·S

ee

Plate 11

p.

69


25/72


26/72

DIAGRAM

FOR MAIN

PAN EQUATIONS

PLATE A.

o

Q

NOTATION

W TOTAL. INTERNAL woRK

M:: B E ND I NG

MOMEN f

0(

=

ANGULAR. D I S T ORT I ON

S ::

TOTAL.

LINEAR

O :5TORTIO l

C DISTANCE TO

r::-XTR£ME

FIBER

- U N I T STR5 i :SS IN EXTREMe: FIBER

E MOOUl-US OF ELASTIC IT Y


27/72


28/72

The to ta l work

will

be since W=.1..JM2.

d

x

E I

Where

the

l a s t three terms represent the work

done by

the

arch t i e

and hangers i n r es is tin g

the

di

r ec t s tr es se s

Different ia t ing

equation

5

with

respect

to

Ml

and which are the unknown forces:


29/72

f

kl

k l ~ · kl

J . li-k {t-J


30/72

®

1

kt

M2.J.. 1

d

- [U-k f Y- fdx k r

l-1.) ldx.

] =0

t

0

I

0 I

I

1

1 l

f X Idx. - M. \ -xhdx

M2.

\2. r ~ x -

-\

xfdx.

1 I

lz.

I

J

I

1

0 I ,

0

These th ree equations

may be expressed the f o l

lowing form by s U bs ti tu ti ng s in g ul ar co efficien ts fo r one


31/72

Instead o f a ctu ally in te gra tin g in order

to

ob

t in

the

various

values

for a structure we

may divide

the stru ctu re in to

increments

or 65 distances obtain

the values for the

various subdivisions

and

make a

summation

of

l l

individual

values

in

order to obtain

an

evaluation for the entire st ructure.

Theoretically

these

subdivisions should be infini tesimally

small

but

fo r p ra ctic al purposes we make

each

division

equal

to a

panel length or case

of

the arch r ing

the

amount

subtended by a

panel

length. The main reason for so

doing

is to

allow computations

to be

made

in

tabular

form. Therefore we subst i tute the subdivi sion ~

for the term

dx and the

sign

of summation

for the

in tegra l

sign

.

For

a

symmetric.al

structure

we

write the following. In computing

these values

we have

made

the subst i tut ion that

the moment of

in r t i

varies

d

o

3

as

the cube

of the depth.

For I we subst i tute

where do

i s

the depth over the pier and d

is t

any sec-


32/72

02.=

b =

t t \ - ~ , , 6 5

- -

@

I 1 1 1 2

b

::

c

~

1

l1s

+ - )

~

X ~ 5 1 ]

- -

-

- -

@

3 1 0 I I

k t

\

A =

\- k ) E J< ltls ~

n-XbJ6S

@

0 I u I

I k t A \ z

A ~ - [ l - k ) ~ \ - . ~ u s + k Z

\ -x

A S ] - - - - - - - ®

2

1 0

I k \ I

k l

\


33/72

t should

be

noted that the second

term of

is antisymmetrical to the f i r s t term,

i s

antisymmetrical

to the

and A is antisymmetrical to A2.

The word antisymmetrical is applied to the

condi

t ion in

which the

to ta l

sum of the values

for

a certain

term

is

equal to tha t of another term, but as

the

values

for one

are increasing or

decreasing, the

values

for the

other are

increasing

or decreasing

but

the reverse or-

der .

For example,

the values of

x

a re increasing

from

a minimum a t

the

l e f t side of the

main

span to a m ximum

a t

the

r ig ht p ie r

At the

same time,

the

values for

l-x are

increasing

from a minimum a t the r ight end to

a maximum

a t

the l e f t

pier

where x was a minimum. The

summation

for both terms is equal.

Inter-changing some

of the

coe ff ic ient s for

their

equal in order

to

have

as few

different

coefficients as

possible, we have:

Ha.-M Qz. - Mzaz. = t - - - - - - -

- - - - - -

@


34/72

Solving

for the

unknown

values

by determinants

we

have:

A

Q2. Qz.

A2

b

2 - 03

H

=

A 3 b ~ b 2

=

- Q2.

al.

Q1.. PI. b

3

a 2. b ~ - b

A

b ~ -

b:)+A2- a1.

b

Chbl.)

A 3 Q 1 . b ~

a

2

b; .

C l \ b ~ -

b ~ a ~ { 2 b 3 - 2 b : z . .

Simplifying

and

cancelling

the term

we

obtain:

@

Q A -a2.

Qz. A2..-

b

3

Q31-A

3

b l

M

:= -

e l l -

Qz. - ell.

Q1

b

z.

b

3

D

®


35/72

Where

is the

same

as

for H

The

influence ordinates

for

are

antisymmetrical

to those

of MI hat

is the ordinates e ~ e s e or -

crease from the l t side to the r ight in the same man-

ner

and

value as

the

ordinates for

l

decrease

or -

crease from the r ight to

the

l t and the summation

of

the

ordinates

i s the

same


36/72

DIAGRAM

FOR

5 D E - s P A N

E Q U A T I O N S

P L A T E

B.

kt

•

tv

CJl

I

o

I .k t

3

I-

Z

-;

z

f)

1;;}J

D

..Q.,

t

Q

USING UNITY

LOAD

A N D

T A K I NG

MOMENTS fiT

Z.MRa. :

-

I

l-k).Q.

0

:. R : 1-1

W H E N

< . < ~ . t l

Mrn= I-Je.)X

WHf:N

> ~ ~ , M n ~ I - J e ) x - l x - . k . t ) , . k . . Q - ) )

R2

D E V E L O P M E N T OF REACT ION

M O M E N T S

1*. - - I

I -.II:-}J .

,m

n

x

I

x

J-

R


37/72

Equations for Design

Side Span:

Developing

the

p r l le l theory for the side spans

we

have

t he fol lowing :

For a uni t load t a distance k l from

the

end of

the

side

span

we have:

Loaded

Side

Span

w n k l l il l -k x , M, - - - - - - @

I

Unloaded

Side Span

Mx=

M ~ ~ @

\

MJ

M

1-

t

M

l -

Hy

®

The to t l work wil l be:


38/72

W=

Where

th e l a s t

three terms as

before represent

the work

done by th e

arch t i e

an d

hangers in res i s t -

ing

the d ir ec t s tr es se s.

D i f f e r e n t i a t i n g th e eq ua tio n fo r

to ta l

work with

resp ect

to

th e

unknown

values

H

Ml

an d

respective-

ly we have

fo r

part ia l de r i va t i ve s :

®

\

it

fl l

MJl ( l-)( y d x ~ J d ) . I i

El 0

I El.

0 :

E

0

:

+

ll

sec:l e dx

E G A


39/72

aw

={

M

r ~ x

n-x.h.dx

M2 S\ :L

d

X

11 ItX IdX } ®

dM1 ~ J

E1

J I

I E\ I

o

0 0 0

Setting the above ~ s t

derivatives

equal to zero,

multiplying

through by

E, and rearranging, we have:

@

M . r \ t - l < . y d ~ M

1 1\ xydx

0

\ \ J I 2 1

I

o 0


40/72

These three equations

may

be expressed in the fol

lowing

manner:

Ha M b Mz.c =O -

®

Hal M b3 M;tC3=O

®

In the

same

manner

as

before

we

substi tute

~

for

dx

and

the E

sign

for the

sign

of

integration.

For

a

symmetrical structure

we

have:

\ \

\.

l..E \.

:L

Q

~

L: sec l.fA5

sec-:l.0A5

f r l£ _ 1:

A f).s

\ 0 I 0 0 A

\ 4

Eh 0 A ; ;

- @

\.

b

= a = l l

\ x . v ~ s

- - - - -

, 1

\.L.r I

o

l

c

:

a -

1

I6os

, l I

®

t

l..£i

l

J. t. 2 ]

b

=C := l 1 [L

Cl

-

x

)

+/_)

~ ~ @

2

3 l

0

I \

t, 0

II


41/72

Again

we have the term for

b and

aZ antisymmetri-

cal to

that

for

c

and

a3.

Changing

some of

the coefficients for their equals

we

have:

Hal M

b

3

- M2 bl : 0 ®

Solving for

the unknown value

by

the method

of

d e t e r m i n n ~ s

we

have:

Q2.

Q;t

A ~ b 1 b 3

o -

b

3 -

b

H

=

~

a G \ 1 . Q ~

Ct1

b

1

- b

3

- b

a

-

bz.

=

Multiplying through

by

1 and

cancelling

terms of

~ - 0-3 we

have:

A4 t

®


42/72

M=

a O 02

C 2

A

4

b

3

Q2. 0 b:z.

Q I Q2. 02 -

0

1

b

b

3

Q 2 b ~ b:z.

Where D

has

the

same

value

as

for

H.

Q

2

Q:2 .

Q b2 . b

3

Q : 2 . b

3

b 2 .

For the values of the

various

unknowns we wil l

use a tabular form.


43/72

o

t

Evaluation of sec} edx

A

The arch axis

is assumed

to

be in the form

of a

parabola.

f so, the vert ic l

ordinates,

or

y distances,

varies as the square of the horizontal

measurement,

or x

distances, from the vertex of the

curve,

or t the crown

of the arch. In accordance, assume

that

the equation

of

the

parabola

i s

x 1 = l y

•

At

the

springing

l ine,

when x = 110 and

y

=

36.67, which i s the tot l

use

of the

arch, have by

substitution

the equation ~ ~ ~ l a y

l\OZ

Q

. 36 67

a

::

\2. \00

= \65

78 31

Then,

the equation

becomes

The

slope

of

the

curve

t any point

i s

the tangent

or

value

of

the

curve

equation

From the trigOm Qlb et:eie

r e l a t i o n l l i ~ s w e 1lavethe


44/72

which becomes

h n the

value

of x

a t

any point is substi tuted

in

th is equation

the

value of

sec 2 g is

obtained.

For

example

a t

the

mid point

of

the

f i f th

panel

x

= 2 0

sec:J.e = \ { ~ o o 5 Y = \ O.\2.12.Y·

=

\ .0 \47 .

In

tabulating

the val ies w

substi tute s

the

len gth a rch axis subtended by

the

panel.

feet and A

=

108 sq. in . or

0.75 sq.

f t

w

have

sec

1

A s = 1 0141

2 0 5 :.

2.7.75

0 75

Securing

the values for ~ h panel

the

summation

across the entire

span

gives us the value of s e c ~ e A s

o

Since w multiplied a l l our terms by

w

must

\

multiply

our summation of

~

sec.:J.e /15 by

to

ab

o

ta in

the

f ina l resul t


45/72

Evalua

t ion

of \sec

A

dx

o

T

The

horizonta l

t ie

between

the bearings of the

arch i s cambered but

the angle of incl inat ion

i s

so

small

tha t

the

secant of the

angle

approaches unity.

For a l l

pract ica l purposes it i s

safe

to cal l

the

value

of sec

¢

as one.

The cross-sect ion

of

the

t ie is

constant so the

term

may be integrated direct-

ly and evaluated.

I f

At

= 40 sq. n or 0.278

f t

and the l imi t s

are

from to 220 f t have

This value

must

also

be multiplied

by d

to

obtain the

f ina l

correct

resul t

6 ~ f

E

} ~ h

Evaluation of E

h

The ra t ioE /E

h

i s used so tha t

the

value will

apply

to

a l l condit ions. I f the

hangers

were

compos-

ed of a di f ferent material the moduli

of

elas t ic i ty

would

not be the

same and

the ra t io

would

be

used.


46/72

In t hi s p ar tic ula r design

we

have

assumed that

the

hanger

and

arch sect ion are

of ident ical

material

s t ruc tura l

s tee l so the

ra t io

i s equal

to unity.

I f we evaluate the term a t panel point

L o

where

the t ie

is

cambered 0.25 f ee t A

0.21 sq. f t the

panel

spacing i s .20

fee t and the length

of the

hang-

er

i s

36.14

f ee t

we

have

the

following:

0 0000\ \5

The

term

i s negl igible as

fa r as

affecting

the

resul t s

of the

design.

Even af te r multiplying by

the

term

d

,

we

obtain 7.5)3 0.0000115= 0.00485

the value

i s

s t i l l

very

small and can safely

be

neg

l ec ted


47/72

Design No.1 .

Data:

Main span

=

220

-0

center

to

center of

bearings

(11

panels

a t :20

on

Two

side

spans =

100

-0

(5

panels

a t 20

1

on

Rise

ra t io

o ~

parabolic

arch

=

1

6

Theore ti ca l he igh t

o ~

arch

=

36

-8

Thiclmess

of

arch a t piers

=

7

-6

Thiclmess

of

arch a t

crown

=

3

1

-6

Clear w id th of roadway = 24 -0

Impact

= 50

1+125

Floor slab

=

8 slab crowned

to

U

a t

center

With 25 lb .

per

sq. f t . wearing

surface

total ing 150 b p ~ r

sq . f t .

Stringers

assumed to weigh 40 per f t . and spaced

4 -0 c. to c.

40,000

= bending moment due

to

concent ra ted load

0.345

- 50

=

impact

rat io

20+125

13,800

= 40,000

x

0.345

=

bending moment

due

to impact

_.2

32,000 =

150x.4+40

x

10

=

bending

moment due

to

dead

2

load


48/72

85,800

tot l bending moment

12

x 85,SOO

=

18,000

S

S

57.2 (a 16' -

40

wf

beam

is ample)

Use l l stringers

as

of this section.

Design

of Floor

Beams: (Spaced 28' -0

c.

to c.)

Maximum

concentration

on

floor

beam

at

stringer

point

nearest

the

center comes from

15- ton t ruck

when one

12,000 wheel load

s

over the floor beam

and a 3000

load 14 feet

away.

This

makes

a concentration

of

12,000 +

3000 x

12,300

20

I 50 0.327

,28+125

R

= 12,300 (5.75 + 11.75 + 14.75 +

,20.75)

+ 28 =

.23,300

234,900

(23,300

x

13

25 -

12,300

x

6)

=

bending

moment

due

to

l ive load.

71,800 = (234,900

x

0.327)

bending

moment

due to

impact.

3352

=

(150 +

1 )

20 ] 5 2

equivalent

dead

load

per

4

foot .

(152

section

assumed)

3.27,000

=

(3352 ,£ . 2) (13 •

25

x 14.75) bending moment due

•

to dead load.


49/72

633,700

=

to t l

bending moment

12

x

633,700 = 18000 S

S = 425 (a 33 -152 wf beam

is

sufficient)

Equivalent l ive load = 480 x

12

x

0.94)

= 600

lb .

9

per

foot .

Total

panel

load =

600 x 20 =

Impact

= 0.30

x

600

=

180

lb

per

foot.

Total

panel

load

due

to impact =

Total

Concentrated load for moment =

l2,000 lb .

3,600 lb .

15,600 lb.

13,500 x 12 x

0.94

=

16,900

lb .

9

Concentrated load

for

shear

=

19,500

x 16 x 0.94 = 24,400

lb.

9

Dead load

from each

f loor beam =

3352

x

8

=

2

46,900

lb .

Lateral

Forces:

Assume 150

per

l ine r

foot on unloaded

chord

Assume 300

per

l ine r

foot on

loaded

chord

Assume a

l te r l

force

of

200

lb per l inear

foot

acting

6

f t

above

the

f loor due

to moving

l ive load and

wind

pressure upon

the

load.

Load per panel = 500 x

.20

= 10,000

lb

on loaded

chord

Load per panel = 150 x

20

=

3000 lb

on unloaded

chord


50/72

s t

coeff icient

1 2 3 4 5 6 7 8 9 10

= 2i

11 11

Bottom Chord:

Wind Load Stresses

p /0 , ooo . tan =

0.834-;

sec f = 3 5 ; : Psec e = 1/80:f

/

/

Pian

=

8 3 4

Stress

in

chords =

coeff icient x P t ane

Stress

in

webs coeff icient x P sec e

double

system

of bracing wil l be used in each panel

but

each

diagonal

wil l

be assumed

capable

of

resist ing

tens ion only.

The maximum

shear in

any panel

occurs

when

every panel

point to the r ight of the section is

loaded.

The to ta l

wind load =

110

kips .

Design:

Diagonals

Maximum unsupported

length

may be

taken

as

Ii

feet

less

than the

diagonal distance

c.

to c.

of panel points

be-

cause of the support which is provided

by

flanges of

main girders and gusset

plates

a t connection.

1 1.305

x

4

x

12 - 18 = 357 in .


51/72

This unsupported length f i r ly

large

so wil l be

considered

as each member developing tension only.

The net

a re a r equir ed

is 65,.200/ 16000

=

4.07

Q

n

using

7/8 r iv ts t 3

spacing.

Panel

ab

Net a rea furni shed by

two

5

x

3t X 3/8

angles,

with

5

fl

legs back to back.

Net a rea furnished is 80%

x

6.10 = 4.88 lJ

n

Number

of

r iv ts =

65,200

= 11 r ivets Field driven

bOlO

r iv t

value

single

shear) 12 wil l be used. 6 each

angle.

Net area for

panel bc

53,400/16000

3.34

Area

furnished

by two

4

X

3

X 3/8 angles is

(.2

x

2.48) .

.80

=

3.97

n

53400/6010 9

r iv ts

10

wil l

be used - 5

in

each angle).

Panel

cd

42,700/16000 = .2.67

0

n

Area furnished by two

3

x

,2

x 3/8 angles is

2 x 2.11) . .80 3.38 0 n

No.

of r iv ts

=

42,700/6010

=

7 8

wil l be

used

-

4

each angle).


52/72

Panel de

33,200 ;:

16000

=: .2.08 a

Area

furnished by two 3 x : 2 ~ X 5/l6

n

angles

is

.2

x

1.6.2). .80 =: 2.59 c

Number of r ive ts

33, .200} 6010

=: 6 r ivets (8 wil l be

used

4 in each

angle) .

This

same

s ize

angle

wil l

be

used

panels

remaining

as

it

i s the

minimum

angle

allowed fo r

bracing.

Chords:

Panel

AB

41,700/ 16000 =: .2.60 an required

A 10 [15 .3

wi l l be

used

although

furnishing more

area

than necessary. (4.47 a

nG

. ) (4.47-2xO.24)

=

3.99 °nN.

Panel

e

75,100/ 16000 4.70

required

10

[.20 wi l l

be used

(5.86 °nG.)(5.86-2xO.38)

=

5.08

On

N

fane1

CD

100,100/16000

6.25

n required

10 [

.25

wil l be

used

(7.33

2 x 0.53)

= 6.27 sq . in .

N

Panel

DE


53/72

A 10

[30 will be

used

(8.80 0

G.)

(8.80

.2 x

0.67)

=

7.46 sq.

n

N.

Panel EF

141.8/16000

8.85

A 10 [ 35 wil l be used (10.27 [

G.)

(10.27-2 x

0.82

8.63

•

Panel

FE'

150.1/16000

938 •

A 10

[35

wil l be used (10.27

On

G.) (10.27 .2

x

0.82)

8.63

sq.

in .

N.

See

diagram

No. 12

for

sect ions.

Design of Side Span:

Stringers and f loor beams of

the

same section as for

the main span

wil l

be used.

Lateral Bracing

The

l ter l

bracing in

the plane

of

the

lower flanges

must provide for l l the l te r l forces due to the wind

and to the

effe t

of sway, including the wind force which

i s

normally assigned to the

upper

flanges.

Wind force moving load equal

to

30 per a on one

and

one-half times area

in elevation.

It

x

30

x 5 x I 225

per l inear foot.


54/72

Also a 200 load a s a

l a te ra l

force

is

act ing aga in st

the l ive load.

Total

horizontal

force = 425 per l inear foot .

The l a te ra l

system

moving

panel

load

is

425 x2 8500

lb

A double

system

of

bracing

wil l

be

used, but each

diagonal

wil l

be

assumed

capable

of

resis t ing

tension

only.

Firs t coeffic ient

1

2

3

4 =

10 =

2

5 5

P = 8500 ; tan-&= 20

0.834;

sec tt = 1.305;

5

P

sec

24

~

=

2220 ;

P

tan

f r-;707o#

•

Stresses in chords

coeff icient x P s ~

The to ta l

wind

load

i s

42.5

kips.

A minimum

size angle of 3

x

.2t

x 5/16

n

was

used

for

a l l diagonals which furnished

a

net area of 2.59 sq. n


55/72

Conclusions

In combining

our

influence

diagrams for

dead load moment our

in i t i l

equations are used. The

f i r s t term

l-k

x or

k l-x

in

the

equations for the

main

span is

equivalent

to

the

moment at any section,

t reating

the arch

as

though

were a

simple

beam.

The

sum of the values

for

the

l s t

two terms

Ml l-x

and

1

M2 produces a s tr ai gh t li ne

diagram

for a symmetri-

1

cal

design since Ml

and M2 are

antisymmetrical

and

the

totals are

identical .

In superimposing the diagram for

the simple beam moment upon

that

for

the

end moment

we

obtain

points

of zero

stress t

the

intersections

of the

two

diagrams.

t

is

evident that there can be a considera-

ble

error

the

values

for direct stress

and

the

de-

sign

wil l

not be materia lly af fec ted. The

tension

the t ie causes direc t s tress in the hangers, but the

values are negligible.

can have considerable error

assuming

the areas of the arch ring and horizontal

t ie

and our design wil l not be affected a g rea t deal.

The

arch

ring area may

be

error as much as 5 and

yet the values of

the

redundants

be

affected

but by a

small

percent.


56/72


57/72

Increasing the

side

span

past

a certain

point affects the type

of

section used A plate gird-

er section may

be used economical ly

up to lengths

of

125 f t Above tha t box girders

must

be

used

Above

a certain point box

girders

would prove uneconomical

and would be feasible to use a truss for the

side

span In

Design

No 4· the main span is

3 5

feet

and

the side spans are

200

feet

in length The

side

span

section would

of

necessi ty be composed of a box

gird-

er or of a t russ Above

the

point

where a box girder

section proves economical and

a

truss section

must be

used would prove

more satisfactory

to use

a

truss

for the main span

also

The design as

followed

would

be equally applicable

BIBLIOGRAPHY


58/72

McCullough C. B. and

Thayer

E. S. Elastic Arch

Bridges.

New York City.

John

Wiley

Sons

Inc.

1931. pp. 8 14; 33 42;

125 141.

Spofford Chas.

M

The Theory of Continuous Struc-

tures and Arches. New York and London. Graw

Hill

Book Company Inc. 1937.

pp. 1

57 58;

184-.203.

Krivoshein G G Simplified

Calculation

of Stat ic-

al ly Indeterminate Bridges. Prague Czechoslov-

akia. Published by author 1930.

pp.

77 80;

86 95;

110 113.

Parcel J and Maney G A An Elementary Treatise

on Stat ical ly Indeterminate Stresses.

New

York

and London. John Wiley Sons Inc. ; Chapman

Hall

Ltd.

1936. pp. 121 122;

320 338; 403 420.

Cross

Hardy and Morgan

Reinforced Concrete.

Wiley Sons Inc. ;

pp. 247 270.

N D

Continuous Frames

Of

New

York and London. John

Chapman

Hall

Ltd. 193.2.

Sutherland Hale

and Bowman

H.

L. An Introduction

to

Structural

Theory and

Design.

ew York nd

London.

John

Wiley

and Sons

Inc.;

hapman

nd

Hall Ltd.

1930.

pp. 184;·217 18; 231 37.

Seely

Fred B.

York City.

.283-.284.

Advanced Mechanics

of

Materials.

John Wiley and

Sons

Ind. 1932.

New

pp

•

The

American

Association

of State

Highway Officials.

Standard Specifications for

Highway

Bridges.

Wash

ington

D.C.

pUblished

y

the Association

1935.

pp. 165... 205.

Kunz F. C.

London.

312 366.

Design of

Steel Bridges. New

York and

McGraw Hil l Book Company Inc. 1915. pp.


59/72

BIBLIOGR PHY

Hool G A. and Kinne

W S.

Movable and Long Span

Steel

Bridges. New

York and London. McGraw Hill

Book Company Inc. 1923. pp• .218 258; 393 482.

Hool G A. and Kinne W S. Structural Members and

Connections. New York

and

London. McGraw Hill

Book Company Inc. 1923. pp. 1 9 9 ~ 2 3

£.VALUATION

or

EQUATIONS

DES

GN NO J

PLATE

J


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EVALUATION

EQU TIONS

ESiGN

NO l

PLATE 2.


61/72

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design study of a three span continuous tied-arch bridge

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