Analysis of Composite Bridge Superstructures Using Modified Grillage Method

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Eng. & Tech. Journal, Vol.27, No.5, 2009

942

Analysis of Composite Bridge Superstructures

Using Modified Grillage Method

Dr. Sabeeh Z. Al-Sarraf, Dr.Ammar A. Ali & Rana A. Al-Dujaili

Received in 13/7/2008

Received in 4/12/2008

Abstract

The analysis principle was used to analyze anisotropic plates (having different

elastic properties and geometries in different directions), the model consist of four side

beams with flexural rigidity and torsional rigidity and two diagonal beams with only

flexural rigidity.

The substitute grid framework is analyzed to give the same deformations anddeflections of the orthotropic plate element of the modeled bridge. Applicability of the

suggested procedure in the analysis of actual bridge decks is investigated using STAAD

Pro.2006program. The results show that the suggested procedure is an acceptable

procedure which can be adopted to analyze this type of bridge deck. It is found that the

modified grillage method gives simpler method and adequate results when compared with

the Finite Element Method or orthotropic plate theory solved using Finite Difference

Method for this type of bridges.

Keywords: composite bridges, superstructures, grillage, orthotropic.

( )

.

.

STAAD Pro.2006

.

.

Introduction

After the end of World War II, anew method based on the anologybetween a grid system and an orthotropicplate was developed. The fundamental ofthis approach were establishedbyHuber[1] in the twenties of the lastcentury. The most difficult problem wasestablishing a solution to the biharmonicequation governing the plate problem.Guyon [2], in 1964, gave a solution oforthotropic plates of negligible torsional

rigidity. He showedthat any variation inthe loading can be handled if thecoefficients of lateral distribution areemployed. Later on massonnet [3] usedthe principles given by Guyon togeneralize a solution that includes theeffect of torsion.

An orthotropic plate is defined asone which has different speciefied elasticproperties in two orthogonal directions inpractice two forms of orthotropy may beidentified [4]; material orthotropyand shape orthotropy. Most bridge


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Eng. & Tech. Journal, Vol.27, No.5, 2009 Analysis of Composite Bridge Superstructures


943

decks are generally [5] orthotropic due togeomatic (shape) rather than material

differences in two orthogonal directions.More rarely, however, there exists acombination of material and geometricorthotropy.

The concept of considering thebridge as an orthotropic plate for thepurpose of determining the distribution ofthe stress is well established. It wasfirstly used by Huber in (1914) to analyzereinforced concrete slabs [4]. This wasfollowed by Guyon in (1964), who usedthe method to analyze a torsionless deck.

Massonnet in (1950) [3] extends

Guyon analysis to include the effect oftorsional rigidity. He introduces atorsional parameter (), in order with theoriginal flexural parameter () defined byGuyon as follows:

5.0).(2 DyDx

DyxDxy += (Torsional

parameter) (2.1)in which

Dxy and Dyx= Torsionalrigidities in X and Y directions.

25.0

][Dy

Dx

L

b=

(Flexural

parameter ) ... (2.2)in which

b= Half width of the deck.L= Span of the deck.

Massonnet analysis is limited to

decks with torsional parameter ()

ranging from "0" to "1" which representthe limits of no torsional decks and

isotropic decks respectively. Rowe in(1955) extends Massonnet method to

include the effect of Poisson's ratio. He

made a review on the previous methodsand presented applications and

extensions of them in his book [6], at astage before the widespread availabilityof the electronic digital computer. AfterMassonnet, Cusens and Pama 1975 [4]rederived the basic equation oforthotropic plate theory and presentednew design curves for the longitudinaland transverse moments using nineterms of the series expression. Theyalso derived a solution to the case ofbatch loading, statically indeterminate,curved, and skew bridge decks. Thegeneral treatment of orthotrpic plate

element is based on the classicalPoisson Kirchhoff assumptions whichare specified as follows[7]:1. The material is perfectly elasticand homogeneous.2. The thickness is uniform andsmall as compared to the otherdimensions of the plate.3. The normal strain in thedirection transverse to the plane of theplate is negligible and the platethickness does not undergo anydeformation during bending.4. Points of the plate lying on a

normal to middle plane remain on thenormal to the middle surface afterbending.5. Deflection of the plate is smallcompared to the thickness.6. Body forces are eitherdisregarded or assumed as a part ofexternal loads.7. External forces are assumedto act perpendicular to the plane ofthe plate.Grillage Analysis

A grillage is a plane structure

consisting of orthogonally or obliquelyrigidly connected beams with threedegrees of freedom at each connection


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node. The most common method ofanalysis is the stiffness (or

displacement)method based on memberstiffness. Alternatively, the grillage maybe solved by the flexibility (or force)method, which is usually not applicableto large structures. The total stiffnessmatrix may be constructed, expressingthe nodal forces Q, MRxRand MRyRin termsof the corresponding nodaldisplacements w, RxRand RyRrespectively,using member stiffnesses.

The grillage is assumed to be atwo-dimensional plane structure and thedisplacements in the plane of the

grillage are ignored. As a result, therotation( RzR) about the axis normal to thegrillage plane is also ignored.Formulation of Rectangular Element

The rectangular grid-framework

model consists of side and diagonalbeams as shown in Figure (1a). Thecross-sectional properties of the beams

are obtained by equating the rotation ofthe nodes of the gird model with those

of a plate element of equal size asshown in Figure (1b), when both are

subjected to statically equivalentmoments and torsion. A rectangular gridmodel with five cross-sectional

properties will define uniquely arectangular element of a plate. Theseproperties are chosen to be the flexural

and torsional rigidities of the sidebeams and the flexural rigidity of the

diagonals.

Evaluation of the Cross-Sectional

Properties

For the grid model to simulate the

behaviour of the plate element,corresponding rotation must be equal forthe two systems. ThusR1R=R6R.. . (3a)R2R=R7R (3b)

R3R=R8R.... (3c)R4R=R9R (3d)

R5R=R10R (3e)In this case hR1Rand hR2Rare not identical inthe Eqs. (3.34.b) and (3.34.d).

A- Case 1 hR1R>hR2RP

Take Eqs. (3a), (3b), and (3c) willprovide the flexural rigidity of the gridbeams as [8]

( )( ) 24

)(3

1

3

2

23

1

3

2

23

1 hE

hEhE

hEkhEEI x

yx

yx

x

= ...(4)

( )( ) 24)(

3

2

23

2

3

1

23

1 hEhEhEk

khEEI y

yx

xy

= ..(5)

( ) 24)(

3

1

3

2

23

1

3

2

3hE

hEhEk

hErEI x

yx

y

d

= .. (6)

The last Eqs. of (3) gives thetorsional rigidity as

( ) dx EI

r

khkEGJ )(

2

124)(

3

3

+

=

.. (7)

the torsional rigidity of the sidebeams of length may be found as [8]

( ) dy EI

r

khEGJ )(

2

124)(

3

3

+

=

. (8)

B- Case 2 h1


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( ) 24)(

3

1

3

1

23

2

3

2

3hE

hEhEk

hErEI x

xy

y

d

= .. (11)

The last Eqs. of (3.34) gives thetorsional rigidity as

( ) dx EI

r

khkEGJ )(

2

124)(

3

3

+

=

.. (12)

The torsional rigidity of the sidebeams of length may be found as:[8]

( ) dy EI

r

khEGJ )(

2

124)(

3

3

+

=

... (13)

A square grid pattern is, in mostcases, preferable to a rectangular gridpattern as the former will provide betterresults. Where rectangular grid patterns

are needed (for instance, to fit thegeometry of the boundaries), the valuesof k should be in the range 1/2 k 2.

For a plate with hR1R=hR2R, and

ERxR=ERyRthe expressions (4) to (13) in twocases reduce to equationsP

P[9]

( )( )1212

3

2

2h

k

kIy

= .... (14)

( )( ) 1212

1 3

2

2hk

Ix

= .. .... (15)

( )1212

3

2

3h

k

rId

= . (16)

( )( )121231 3

2

h

E

GJx

= ... (17)

( )( ) 121231

3

2

hk

E

GJy

= .... (18)

ApplicationThe analytical result curves for

the bridges have been obtained using

the procedure presented in section 3and checked by STAAD Pro.2006

Program. These curves have beencompared with the experimental resultsand predicted analytical result curves

which are obtained (using theorthotropic plate theory) by finite

difference model P

P[10]

Equivalent Plate Rigidities of

Composite Slab-on-Beam DeckThere are different methods to

calculate the values of equivalent plate

regidities Dx, Dy, Dxy and Dyx, fordifferent types of bridge decks.Typical cross section of

composite slab-on-beam deck is shown

in Figure (2a). Figure (2b) shows atypical section of concrete T-beam. Theequivalent plate rigidities of such deckscan be obtained as follows [11,12]

Where "IRgR" is the combined secondmoment of beam area and associatedportion of deck slab in units of beammaterial. The subscript "g" applied to "E"and "G" refers to the material of thebeam. G =E/2(1+), n =ERsR/ERcR where

subscript "s" and "c" refer to steel andconcrete respectively, and "J" is thetorsional constant.

Equivalent plate rigiditiescorresponding to reinforced concrete arecalculated by ignoring the steelreinforcement and by assuming that, theconcrete is uncracked. It is used to

)22(6

tGDyx

)21(6

tG

Py

JGDxy

)20(

12

tE

)1(12

tEDy

)19(IPy

EDx

3c

3cg

3c

2

c

3c

gg

=

+=

=

=

=


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ignore the reduction of equivalent plate

rigidities due to cracking. The torsionalconstant of a beam section is oftencalculated by dividing the section into a

number of rectangles. The torsionalconstant of a rectangle of sides bRwRand

d , where d is the smaller side ,is givenas [11,12]

J=kwd.(23)

where "k" can be obtained from Table(1)

Application: Analysis of Bridge with

Steel GirdersFor further study of the reliability of themethod and its applicability in actualanalysis, a small bridge deck sealed

with high molecular weightmethacrylate (slab and overlay) isanalyzed for four concentrated loads. Inan effort to stop or slow down thecorrosion process in existing bridge

decks, the bridge deck may be sealed bya high molecular weight methacrylate or

gravity fed epoxy resin prior to the

application of portland cement overlay.In addition to sealing the existing

cracks, the sealer stops the intrusion ofwater and chlorides if the overlay

cracks P

P[14].This test was done by Cole et alP

P[14]. The presence of a sealer at thedeck overlay interface is expected toreduce the available bond strength. Thistest was carried out to investigate theperformance of the overlays placed oversealed bridge decks (to examine the

level of bond strength). Test resultsindicate that the sealer reduces theavailable bond strength by as much as50 %. Up to 85 % of the bond strengthcan be restored if sand is broadcast overthe sealer while it is curing or if driedsealed surface is lightly sanded.

In calculating bridge deckrigidities, the bond strength is assumed

to be equal to 100%. From beam tests,bond strength between overlay andbridge deck becomes more criticalwhen the overlay is subjected topositive moments, therefore, a simplysupported specimen was selected [14].

Geometry and StructureThe test deck is a right simply

supported deck, of 6.096 m span and2.438 m width. The clear span is 5.486m. Plan and crosssectional dimensions

are shown in Figure (3). The deck wasconstructed from three W12x19 steel

beams and L1x1x1/8 cross braces. Thecompressive strength of deck was 61MPa at the time of testing and thecompressive strength of overlay was 52MPa when the specimen was loaded .The Poissons ratio of concrete wastaken as 0.18[14] .

The flexural and torsionalrigidities of the equivalent orthotropicplate (Dx, Dy, Dxy and Dyx) needed for

analysis are calculated using formulaswhich were suggested by Al-Dawar[12] P Pand Flaih [10]. Details ofdimensions, material properties,flexural and torsional rigidities for thisbridge deck are tabulated in Table (2)P

P[10].The deck is assumed to actcomposite with longitudinal beams andwith horizontal cross braces, while theeffect of diagonal cross braces isneglected. The moduli of elasticity ofthe longitudinal beams and cross braces

are 205 GPa and 215 GPa respectively.The equivalent rigidities of the

test deck (EI)R xR, (EI)RyR, (GJ)RxR, (GJR)yR and(EI)Rd Rneeded for the analysis arecalculated using Eqs.(4) through (8).Details of dimensions (=406mm,k=499mm), material properties, flexuraland torsional rigidities for this bridgedeck are tabulated in Table (3).


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Load and Analysis

The specimen was loadedstatically up to 48.9 kN. This load wasapplied to the slab through a transfer

beam positioned on four 152 mm x152mm x 12.7mm steel plates placed at

the intended load points as seen inFigure (3)P

P[14]. Concentrated loads are

linearly distributed to adjacent nodesbased on the location of the loading aspresented in Figure (4) [13]. The

modified grillage mesh which is

adopted for the analysis consists of "84"nodes represented by "7" transversenodes in "12" longitudinal rows equallyspaced along the span of the bridge

deck model as shown in Figure. (5).By using modified grillage

technique, the values of deflections (w)are obtained. The distributed load andmaximum deflection of the bridge are as

shown in Figures. (6) and (7)respectively. These results arecompared with those found from the

test of bridge deck and finite differencemethod. The deflection profile along the

length of the middle girder is shown inFigure. (8).

Discussion of ResultsFor modified grillage method, it

can be seen that acceptable values are

obtained for the deflections of themiddle girder if compared with those

found during field test and finitedifference analysis. The obtained resultsare compared in Table (4).

The examination of theapplicability, limitation, accuracy and

economy of the modified grillageanalysis has been the main concern ofthe present work. This was achieved

through a comparison with othercommonly used but rather sophisticated

analytical techniques namely, the finite

difference and the finite element

methods.The analysis of bridge decks

can be made by using differentapproaches; the modified grillagetheory is one of these approaches, the

effect of Poisson's ratio is taken intoconsideration.

ConclusionThe following conclusions based

upon the findings of this study:

1. The modified grillage technique

is easy to use and gives anacceptable accuracy for theelastic analysis of simplysupported right bridge decks.

2. The proposed technique for theanalysis of slab-on-beam bridgedeck type including supportedge deflections given in thiswork has shown slightdifference compared with thoseneglecting support edge

deflections.

3.

Right bridges with steel girderswith cross braces may besuccessfully analyzed usingmodified grillage method with

good accuracy:

References[1] Hubber M. T., Teari PolytProstokaatnie Roznokierunkovych,Lwow, 1921,[2] Guyon Y., Calcul De PontsLarges a Prouties MultiplesSolidarisees Par des Entretoises,

Ann. De Ports et Chavsees deFrance, 1949, Vol. 10, No. 9, pp .553-612,[3] Massonnet Ch., "Methods ofCalculation of Bridges with SeveralLongitudinal Beams Taking Intoconsideration Their TorsionalResistance", InternationalAssociation for Bridge and


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Structural Engineering Publications,

Vol. 10, 1950, pp. 147-182.[4] Cusens A. R. and Pama R. P.,"Bridge Deck Analysis", John Wiley

and Sons, Ltd., 1979., pp.29-30.[5] Clark, L. A., "Concrete Bridge

Design to BS 5400", LongmanGroup Limited, England, 1983,pp .13-14

[6] Rowe R. E., "Concrete BridgeDesign", John Wiley & Sons Inc.,

New York, 1962.

[7] Timoshenko S.; and Woinowsky-Kreeiger S., "Theory of Plates and

Shells", 2Pnd

P Ed., McGraw-Hill, NewYork, 1959, pp.364-377[8] Hussein H. H.., "Bridge DeckAnalysis by Modified Grillage",M.Sc. Thesis presented to theUniversity of Technology, 2006,pp.35-39.[9] Husain M. H., "Analysis ofRectangular Plates and Cellular

Structures", Ph.D. Thesis, Presented

to the University of Leeds, Leeds,England December, 1964, pp.37-62

[10] Flaih, R. H., "Bridge DeckAnalysis using Orthotropic PlateTheory", M. Sc. Thesis Presented tothe University of Technology, 2005,pp. 33-47.

[11] Bakht B.; and Jaeger L.G.R"Bridge Analysis Simplified",McGraw, 1987, pp.5-175.[12] Al Dawar .M. "A Study onThe Use of Orthotropic Plate Theory

in Bridge Deck Analysis , Ph. D.Thesis Presented to the University ofBaghdad, 1998, PP 1 75.

[13] Eom J. and Nowak A. S., LiveLoad Distribution for Steel Girder

Bridges, Journal of BridgeEngineering, ASCE,November/December 2001, PP.489-497.[14] Cole.J, Gillum .A .J andShahrooz .B.M, "Performance ofOverlays Placed over Sealed Decksunder Static and Fatigue Loading",Journal of Bridge Engineering,ASCE, /July/August 2002, PP.206-

214.


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Analysis of Composite Bridge Superstructures Using Modified Grillage Method

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