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Bridge Engineering 1

Chapter 6Simplified Methods of Analysis


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Introduction

Loads (Dead + Live loads) aretransmitted from deckto thesuperstructure and then to the

supporting sub-structure Example: slab-on-girder bridge

Slab transfers the load to the girders

The transferred load is primarily proportional torelative stiffness ofgirders and the slab and to alesser extent to that ofdiaphragms, cross frames andbearings


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Force response in a bridge (a) longitudinal moment

(b) longitudinal shear

(c) Transverse moment

Not required in slab-on-girder bridges if deckslab design by empirical method

(d) Transverse shear

Required only in multi-beam bridges

Always required to be calculated


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Beam analogy and load distribution


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Load distribution


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Introduction

Deflection patterns in two bridges


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Introduction

Effect of diaphragms on load distribution


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Introduction

Transverse Distribution of Longitudinal Moments


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Determination of load distribution factorsusing orthotropic Plate Theory

),()( 4

4

22

4

214

4

yxqdy

wd

Ddydx

wd

DDDDdx

wd

D yyxxyx=+++++

Where, Dx, Dy, Dxyand Dyxare the longitudinaland transverse flexural rigidities, as well as the

longitudinal and the transverse torsional rigiditiesper unit width or per unit lengthD1andD2arethe longitudinal and the transversecoupling rigidities per unit width or per unit length

b= the half-width of the plateL= the length of a plateq(x,y)= applied loadw= vertical displacement (deflection)


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Dimensionless characterizing parameters

For a simply supported rectangular plate underconcentrated load, the 8 structural parameterscan be reduced to the following 2dimensionless characterizing parameters(Massonnet)

Plates with same and have the same loaddistribution pattern

ratio of torsional to flexural rigidity andratio of longitudinal to transverse flexuralrigidity

( ) 5.021

2 yx

yxxy

DD

DDDD +++=

25.0

=y

x

D

D

L

b


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Orthotropic Plate Theory


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Finite Element Results


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Finite Element Results


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Effect of rigidities on load distribution


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Effect of rigidities on load distribution


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Plate rigidities for different types of bridges

Non-composite slab-on-girder bridges No shear stud, so the slab can rotate around its

own neutral axis freely

n and are ratio of the moduli ofelasticity and shear moduli

sn


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Composite slab-on-girder bridges With shear connectors


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Concrete slab bridges


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Simplified method of analysis

Fatigue limit state One truck only

Serviceability limit state

More than one truckto produce max. forceeffects

Ultimate limit state

More than one truckto produce max. Forceeffects


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Lane load correction factor

Lane widthcorrection factor

60.0

3.3

)

100

1(

=

+=

e

f

d

W

CDD

But not greater than 1.0

a correction factor used to adjust the D values fordifferent lane widths

fC


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and charts are developed for 1,2,3 and4 lane bridges for the various loaded laneconditions.

Steps,

a) Obtain the initial from the table b) Calculate the initial load distribution fraction

where s is the actual girder spacing in slab-on-girder bridge and the spacing of the webs in voidedslabs and cellular structures or 1m for solid slabs,

transversely prestressed laminated wood bridgesand concrete-wood composite bridges composed ofwood laminate and concrete overlays

dDdD

s


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c) Treat the bridge as one dimensional beam,obtain bending moment (M) diagrams dueto one halfof the truckor lane loading

d) Multiply M by , to obtain the

initial live load moment, where DLAisDynamic Load Allowance

e) Assume the calculated moment issupported by width s of bridge, whetherinside or outside

( )DLA

D

s

d

+1


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f) Calculate and

g) Calculate

but not greater than1.0

where is the design lane width in meters

h) Corresponding to and and number of

lanes, obtain the values ofD separately forexternal and internal portion and the value offrom relevant charts

6.0

3.3= eW

eW

fC


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i) obtain the design value of separatelyfor external and internal portion from

j) for each of the external and internal portion,obtain the final live load design moments bymultiplying the live load moment due to oneline of wheels or half lane load obtained in (c)

above by where has thevalue appropriate to external or internalportion.

dD

+=

1001

f

d

CDD

( )DLADs

d+1

dD


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ULS, SLS II, SLS I, 1 lane bridge


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ULS, SLS II, SLS I, 1 lane bridge


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ULS and SLS II, 2 lane bridge


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CHBDC simplified method of analysis

Condition 5.7.1.1 a) the width is constant

b) the support conditions at ends and in

between are close to line support c) for slab bridges and slab on girder ones with

skew, the skewness provisions are to be met

d) for bridges with curved span the radius ofcurvature, span and width provisions to be met


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e) for solid slab or voided one, with substantiallyuniform depth across a section or tapered at thefree edge, where the length of taper in transversedirection does not exceed 2.5 m

f) for slab-on-girder, at least 3 girders of the samerigidity and distance or with max. 10% differencefrom mean

g) for bridges with longitudinal girders and with

overhang, the overhang max. 60% of mean spacingof the girders, or the spacing of 2 outermost websin box girders or 1.80 m.


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h) for continuous span bridges, provisions of

App. A5.1 (a) is to be met i) for multispine bridges, each spine has only

two webs and conditions of 10.12.5.1. shallapply for steel and steel-composite multispine

bridges.

Fatigue limit state: Stress range, lower stress level, (only truck

may govern Ultimate and serviceability limit state

Max force effects (more than one truck)


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Method of analysis


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Method of analysis


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Method of analysis


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Example


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