upstand finite element analysis of slab bridges

Upstand ®nite element analysis of slab bridges

E.J. O'Brien a, *, D.L. Keoghb

aUniversity College Dublin, IrelandbRoughan and O'Donovan Consulting Engineers, Ireland

Received 26 May 1997; accepted 6 May 1998

Abstract

For slab bridge decks with wide transverse edge cantilevers, the plane grillage analogy is shown to be an

inaccurate method of linear elastic analysis due to variations in the vertical position of the neutral axis. The upstandgrillage analogy is also shown to give inaccurate results, this time due to inappropriate modelling of in-planedistortions. An alternative method, known as upstand ®nite element analysis, is proposed which is su�ciently simple

to be used on an everyday basis in the design o�ce. The method is shown to give much better agreement than theothers when compared with an elaborate three-dimensional solid ®nite element model. Single- and two-span bridgedecks with solid and voided sections are considered for both longitudinal and transverse bending stresses. # 1998

Elsevier Science Ltd. All rights reserved.

Keywords: Bridge; Bridge deck; Grillage; Finite element analysis; Voided slab

1. Introduction

The plane grillage analogy is a popular method

among bridge designers of modelling slab bridge decks

in two dimensions. It involves the idealisation of the

bridge slab as a mesh of longitudinal and transverse

beams all located in the same plane. Implicit in the

method is an assumption that all parts of the bridge

cross-section bend about a single neutral axis which is

generally taken to pass through the centroid of the

deck. The depth of this neutral axis (below some

datum, usually horizontal) is assumed not to vary

throughout the bridge. Finite element analysis (FEA)

is used extensively by bridge designers, but is most

often limited to planar analysis using plate bending el-

ements, which, like the plane grillage method, assumes

a constant neutral axis depth.

Slab bridge decks with wide edge cantilevers are

known to have neutral axes of variable depth. Fig. 1 il-

lustrates a bridge deck in which the neutral axis rises

signi®cantly towards the edges as the cantilevers

attempt to ¯ex about their own centroids. It has been

reported [8] that the neutral axis depth may also vary

in the longitudinal direction in some cases, such as

close to concentrated loads or adjacent to intermediate

supports in multispan decks.

Variability of neutral axis depth, when signi®cant,

results in the incorrect representation by a planar

analysis of the behaviour of the bridge deck. The pro-

blem can be overcome by the use of a three-dimen-

sional model. Such methods do not require all

members or elements representing parts of the deck to

be located in the one plane. Consequently this

approach does not require a pre-assumed neutral axis

position, and allows for a rational handling of cases in

which the neutral axis depth varies. Many such three-

dimensional techniques have been suggested, ranging

from space frame methods to ®nite element analysis

(FEA) using solid, brick-type elements. While these

methods can be successfully used to model a variable-

depth neutral axis, they are generally excessively com-

plex for everyday use in design o�ces. This paper

addresses the problems by proposing a simpli®ed but

accurate three-dimensional (3D) method, referred to

here as the upstand ®nite element analysis (upstand

Computers and Structures 69 (1998) 671±683

0045-7949/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(98 )00148-5

PERGAMON

* Corresponding author.

FEA) method. The method is simple enough to be

used in design o�ces for everyday design and is signi®-cantly more accurate than plane grillage or plane ®niteelement analysis, particularly for bridge slabs withwide edge cantilevers.

2. Three dimensional ®nite element analysis

In order to assess the accuracies of alternative sim-pli®ed methods of analysis, it was necessary to com-pare them to a considerably more accurate method for

a range of bridge decks. A program from the DYNA3Dsuite, NIKE3D [7], was used for this purpose to con-struct three-dimensional solid models from solid `brick'

elements. The program was tested extensively by com-

paring analysis results for simple beam and slabmodels with known values.

3. Grillage analysis methods

The theory and application of grillage analysis iswell established and has been discussed by manyauthors [3, 5, 12]. Results have been compared tomodels and full-size bridges in the past [2, 12] and the

method has been found to be reasonably accurate formany shapes of structure, loading conditions and sup-port arrangements. While it is widely used by bridge

designers, the plane grillage analogy has been found bythe authors [6] to be inaccurate for slab bridges withwide edge cantilevers.

Upstand grillage modelling is a simpli®ed three-dimensional technique, which is in e�ect an extensionof the plane grillage method. It has been suggested inthe past for the analysis of bridges with variable-depth

neutral axes [5]. Fig. 2(a) shows a single-span slabbridge deck with edge cantilevers. The length to depthratio of the cantilevers is 6:1 which is the maximum

considered practical. Fig. 2(b) shows an equivalent

Fig. 1. Slab bridge deck with wide edge cantilevers showing

non-uniform neutral axis.

Fig. 2. (a) Single span bridge deck and (b) upstand grillage model.

E. O'Brien, D. Keogh / Computers and Structures 69 (1998) 671±683672

upstand grillage model of this bridge deck. The model

is similar to a plane grillage except that each member

is located in the plane containing the centroid of the

portion of bridge it represents. Therefore it is not

necessary to assume an overall neutral axis position.

For the example illustrated, all members are located

on two planes, one for the edge cantilevers and one forthe main part of the deck. The grillage members in

each plane are connected by rigid vertical beams. Withsome programs, it may be necessary to use beams withvery large ¯exural rigidities, being e�ectively in®nite

while not being so large as to cause round-o� errors.A three-dimensional model such as the upstand gril-

lage, develops signi®cant forces in the plane of the

deck. Unfortunately a grillage is ine�ective at provid-ing a realistic representation of a slab subjected to in-plane forces due to the tendency for the beams to bend

individually in a similar manner to a Vierendeel girder.This behaviour is clearly not consistent with that of aslab. To overcome this problem, as has been suggestedby other authors [5], all of the nodes in the upstand

grillage were restrained against in-plane rotation.The upstand grillage of Fig. 2(b) was analysed using

the STRAP computer program [1], under the action of

opposing uniform moments applied at each end of thebridge deck. Typical concrete material properties werechosen for the model. A plane grillage analysis using

the STRAP program, and a 3D FEA of the bridge deckwere also carried out. Fig. 3(a) and (b) show the longi-tudinal bending stress at the top ®bre of the bridge

deck at 1/2 span and 1/8 span, respectively, for each ofthese three analyses. A fourth method is also shownwhich is discussed later. Only half of the bridge deck isshown as it is symmetrical; the cross section is included

in these ®gures for clarity. Further details and resultsfrom these analyses have been presented elsewhere [6].Assuming that the results from the 3D FEA model

are the most accurate, it can be seen from Fig. 3 thatthe upstand grillage is generally more accurate thanthe plane grillage. This is most evident in the edge can-

tilever at 1/2 span and in the main part of the deck at1/8 span. An exception to this general trend occurs inthe edge cantilever at 1/8 span. In this region, theupstand grillage makes a poor prediction of stress,

which is, in fact, worse than that of the plane grillage.Investigation of the 3D FEA model revealed that sig-ni®cant in-plane distortion occurs in this region. The

restraining of nodes against in-plane rotation in theupstand grillage is counter-productive in such a caseand is the reason for the poor prediction.

4. Finite element analysis

The ®nite element method has been used extensivelyin the analysis of bridge decks for some time. It is

regarded by many bridge designers to be the most ver-satile method available and indeed the only onecapable of dealing with certain bridge forms [4, 5].

Although the method is used by bridge designers pri-marily for plate bending problems, it was pioneeredby [11] amongst others, for in-plane analysis of two-

Fig. 3. (a) Longitudinal bending stress at top of bridge deck

from various analyses at 1/2 span and (b) at 1/8 span.

E. O'Brien, D. Keogh / Computers and Structures 69 (1998) 671±683 673

dimensional elastic structures. The method wasextended to the bending of slabs by Zienkiewicz andCheung [13] who also demonstrated the ability of the

method to deal with various boundary conditions,variable slab thicknesses and slab orthotropy. In recentyears, software has become available for personal com-puters which allows the use of plate ®nite element

models for everyday analysis of bridges in the designo�ce.

4.1. Upstand ®nite element analysis

The upstand FEA method, which is proposed here,

is similar to the upstand grillage method discussed inthe previous section, but with the grillage beamsreplaced by ®nite elements. The elements are locatedon di�erent planes, according to the geometry of the

bridge deck, and these planes of elements are con-nected, once again, by vertical beams with e�ectivelyin®nite ¯exural rigidity. As the model is three-dimen-

sional, the elements have to resist in-plane forces anddeformations as well as out-of-plane bending. Themethod is introduced here by means of a simple

example. It will then be further developed to coverother forms of bridge deck.An upstand FEA model of the single-span bridge

deck of Fig. 2(a) was analysed using the STRAP compu-

ter program for the same loading conditions. Themodel is shown in Fig. 4 with dimensions in metres.As with the upstand grillage model, the vertical mem-

bers were given e�ectively in®nite ¯exural rigidities.The ®nite elements, which were capable of modellingboth in-plane distortion and out-of-plane bending,

were isotropic and hence only their depth was requiredto de®ne their sti�ness. The depth of the elements wastaken to be equal to the depth of the portion of slab

which they represented and the same material proper-ties were used as in the previous model.The upstand FEA gave both bending moments and

axial forces in each element and the total stress wasarrived at by adding the stress components of each ofthese e�ects. Fig. 3(a) and (b), which shows the stressesfrom the 3D FEA, plane grillage and upstand grillage,

also shows the same quantity predicted by the upstandFEA model.The upstand FEA is seen to compare almost exactly

with the 3D FEA and the lines in the ®gures arealmost indistinguishable. This provides a greatimprovement over the upstand grillage and the plane

grillage with little increase in complexity. The tech-nique is extended to bridge decks with more complexgeometries in the following sections.

4.2. Upstand FEA of two-span bridge deck

To further develop the upstand FEA method, a two-span bridge deck with wide edge cantilevers was ana-

lysed under the action of its self-weight. The formu-lation of the upstand FEA model for the two-spancase follows directly from the single-span model, and

uses the same element dimensions and properties, asthe cross-sections of these two models are the same.The upstand FEA model is illustrated in Fig. 5.The longitudinal stress at the top of the bridge deck

predicted by this model was compared to that pre-dicted by a 3D FEA using the NIKE3D program. Thetop stress was taken this time at 0.1 m below the top

surface of the bridge as this location was convenientfor data processing purposes. As the bridge deck issymmetrical, results are only presented here for one

half of the deck (transversely) and for one span. Inorder to simplify the presentation of results, the (half)bridge deck was notionally divided transversely into 6

Fig. 4. Upstand FEA model of single span bridge deck.


longitudinal strips of equal width and labelled as

shown in Fig. 6. The average top stress predicted bythe 3D FEA for each longitudinal strip is presented inFig. 7(a) while Fig. 7(b) shows the same quantity as

predicted by the upstand FEA model. A plane FEA ofthis bridge deck was also carried out and the same

results from this are shown in Fig. 7(c). It was con-sidered more appropriate to compare the results of theupstand FEA with a plane FEA rather than a plane

grillage. This shows that the improved accuracy is notdue to the use of ®nite elements, but rather due to the

three-dimensional nature of the model.The predictions of stress from the 3D FEA and the

upstand FEA follow the expected pattern, with zero

stress at the end support, maximum compressive stress(here shown negative) at 3/8 span, zero stress near 3/4

span and maximum tensile stress at the central sup-port. In addition, the ®gures show how the longitudi-nal stress varies transversely across the bridge deck.

The maximum stress occurs towards the centre of thebridge and diminishes towards the edge cantilever. Thepredictions of longitudinal stress from the plane FEA

also follow the expected pattern along the span, butdoes not show the longitudinal stress varying transver-

sely across the deck to the extent it does in the predic-tions of the other analyses. In fact, the plane FEApredicts an almost uniform longitudinal stress across

the width of the bridge deck, except for the edge of the

cantilever (strip C1) where it drops slightly over the

central support. This is due to the inability of theplane FEA to model the variation in neutral axis lo-cation.

Fig. 8 presents the same results in a form which fa-cilitates a direct comparison of the methods for each

longitudinal strip (C1±M4) in turn. By comparing thestress at C1 and C2 predicted by each of the three ana-lyses (Fig. 8(a) and (b)), the 3D FEA and upstand

FEA predictions are seen to be in very close agreementalong the entire span. The stress predicted by the plane

FEA, on the other hand, does not agree as well withthese. This disagreement is more pronounced near theedge of the cantilever at C1 (Fig. 8(a)) than at the

inside of the cantilever at C2 (Fig. 8(b)). The planeFEA predicts signi®cantly higher stresses in the cantile-ver than the other two analyses. This is due to the fact

that the plane FEA assumes a neutral axis which doesnot rise in the edge cantilever. This causes the second

moment of area of the cantilever to be too large, andthus to attract too much moment. Secondly, themoment which it attracts induces a higher stress

because the distance from the assumed neutral axis tothe top ®bre is too large.

By comparing the stresses at other transverse lo-cations (Fig. 8(c)±(f)) it can be seen that all three ana-lyses predict stresses which are generally in close

agreement, but that those predicted by the plane FEA

Fig. 5. Upstand FEA model of two-span bridge deck.

Fig. 6. Subdivision of two-span bridge deck into longitudinal strips (`C' indicates cantilever strip, `M' indicates main deck strip).


Fig. 7. (a) Longitudinal stress at top of two-span bridge deck from 3D FEA; (b) from upstand FEA; and (c) from plane FEA.


deviate a little from the other two analyses, especially

above the support. This is due to the fact that in the

plane FEA, the incorrect neutral axis location in the

cantilever caused an increased moment to be predicted

at that location. As the total moment at a section must

remain constant, the moment, and hence stress, pre-

dicted in the remainder of the deck must be lower than

the true value.

A comparison has also been made between the

transverse stress predicted by both the 3D FEA model

and the upstand FEA model. The upstand FEA gave

bending moments and axial forces in the transverse

direction and the stresses due to these were added to

arrive at the total stress. Fig. 9 shows the transverse

stress, again at 0.1 m below the top surface of the

bridge deck, at a number of span locations, as pre-

dicted by the two analyses.

It can be seen from this ®gure that, for both analysis

methods, similar transverse stresses are predicted for

all span locations at C1, C2, M1 and M2 (refer to

Fig. 6). At M3 and M4, there is a signi®cant variation

in transverse stress at di�erent span locations. This is

predicted by both analyses which are in reasonable

agreement here. Although a signi®cant variation in

stress is predicted by the two methods of analysis from

C1 to M2, it can be seen that both predict a similar

maximum value, and both predict a similar distri-

bution of stress across the width of the deck. The main

variation appears to be in the prediction of the lo-

cation of the maximum transverse stress. The 3D FEA

predicts this to occur further into the deck than the

upstand FEA does. It is felt by the authors that this

discrepancy is due in part to the e�ect of the rapid

transition from the cantilever to the main part of the

Fig. 8a and bÐCaption on page 679


deck. This would be expected to have a more pro-nounced e�ect on the transverse stresses than on the

longitudinal stresses. The comparison of shear stressbetween the two models is not made here as it is thesubject of ongoing research [10].

4.3. Upstand FEA of voided slab bridge deck

Longitudinal voids are incorporated into slab bridgedecks in some countries to reduce their self-weightwhile maintaining a relatively high second moment of

area. In order to determine the accuracy of theupstand FEA method for this form of construction,the two-span bridge deck of Fig. 5 was modi®ed to in-

corporate longitudinal voids in the main part of thedeck. Solid sections were maintained above the endand central supports to simulate the presence of dia-phragm beams. This structure was then analysed using

a 3D FEA model under the action of its own self-

weight. Fig. 10(a) shows the longitudinal elevation of

the bridge indicating the position of the solid sections

above the supports. Fig. 10(b) shows a cross-section

through the voided part of the bridge deck and gives

dimensions in metres. A portion of the 3D FEA model

can be seen in Fig. 11 which indicates the voids. As

the 3D FEA program uses ®nite elements with straight

sides, the circular voids were approximated with octag-

onal voids. The diaphragms were also incorporated

into the 3D FEA model.

This bridge deck was also analysed using an upstand

FEA model. The modelling procedure was similar to

that used in previous sections except for the addition

of the voids. The process generally adopted for model-

ling voided slabs with isotropic elements is to calculate

an `equivalent depth' of element. The depth is that

which gives a second moment of area for the element

Fig. 8c and dÐCaption opposite


equal to the second moment of area of the voided sec-

tion. It is clear that the area of such an equivalent el-

ement is not equal to that of the corresponding voided

section. This is generally not a problem in a two-

dimensional plane analysis, as in-plane forces are not

modelled. However, in an upstand FEA model, in-

plane forces do exist. The sti�ness of each portion of

the deck results from a combination of the element

second moment of area about its centroid, its area and

its distance from the bridge neutral axis at that point.

Consequently, it is essential that the upstand FEA has

both the correct area and second moment of area. This

may be achieved by the addition of extra longitudinal

beams to the upstand FEA model to make up the

shortfall of area or second moment of area or by the

use of a program which allows independent de®nition

of these two properties. For this study, the depth of

the elements were chosen to give the correct area and

the shortfall in second moment of area was assigned to

additional longitudinal beams. Fig. 12 shows the

upstand FEA model incorporating the additional

longitudinal beams.

According to Hambly [5], slabs with a void diameterless than 60% of the slab depth can be modelled as

isotropic, the transverse sti�ness being equal to the

longitudinal sti�ness. This approach was adopted in

the current work. For voids of larger diameter, the

orthotropic nature of the slab should be incorporated

in the model, such as by using orthotropic ®nite el-

ements. This is discussed further by O'Brien and

Keogh [9].

Fig. 13(a) shows the top longitudinal stress predicted

by the 3D FEA for each of the longitudinal strips C1

to M4. Fig. 13(b) shows the same quantity predicted

by the upstand FEA (with additional beams). Both of

these graphs show the same pattern as was observed

Fig. 8. Longitudinal stress at top of two-span bridge deck for each longitudinal strip. (a) Cantilever strip C1; (b) cantilever strip

C2; (c) main deck strip M1; (d) main deck strip M2; (e) main deck strip M3; and (f) main deck strip M4.


for the solid bridge deck; that is zero stress at the end

support, maximum compressive stress at 3/8 span, zero

stress near 3/4 span and maximum tensile stress at the

central support. The variation of longitudinal stress

transversely across the bridge from M4 to C1 which

was observed in the solid bridge deck is again observed

in both of these ®gures.

Fig. 14 provides comparisons of the stresses of

the preceding ®gure for two typical longitudinal strips,

C1 and M1. The predictions from the two analyses

Fig. 9. Transverse stress at top of two-span bridge deck.

Fig. 10. (a) Longitudinal elevation and (b) cross-section of two-span voided bridge deck.

Fig. 11. Portion of 3D FEA model of two-span voided bridge deck.


Fig. 12. Upstand FEA of voided bridge deck with additional longitudinal beams.

Fig. 13. (a) Longitudinal stress at top of voided bridge deck from 3D FEA and (b) from upstand FEA.


compare very well with each other. Some disagreement

is evident at the central support in M1 but this is quitesmall and localized. One possible cause for this dis-agreement is the complexity incurred by the rapid

change from voided to solid section at this location.

5. Conclusions

An upstand FEA technique has been proposed for

the analysis of slab bridge decks with edge cantilevers.In this technique, the edge cantilever and the mainbridge deck are idealized by separate layers of ®nite el-

ements, each located at the centroid of that part of thedeck which they represent and connected by verticalbeams of e�ectively in®nite ¯exural rigidity. The ®nite

elements must be capable of in-plane distortion and

out-of-plane bending.

For a single-span bridge deck with wide edge cantile-

vers, both the plane grillage and the upstand grillage

methods are shown to be inaccurate in predicting

longitudinal bending stresses when compared to a 3D

FEA. The proposed upstand FEA method, on the

other hand, gives excellent agreement. The di�erences

in accuracy are attributed to the inability to model

both variations in the depth of the neutral axis and as-

sociated in-plane distortions. Similar results are

reported for longitudinal stresses in a two-span bridge

deck.

Predictions of transverse stresses from the upstand

FEA and the 3D FEA were found to compare reason-

ably well. Both analyses predicted a similar maximum

Fig. 14. (a) Longitudinal stress at top of voided bridge deck for cantilever strip C1 and (b) for main deck strip M1.


transverse stress and a similar distribution across thewidth of the deck. However, a discrepancy was noted

in the location of the point of maximum transversestress.A voided slab bridge deck, again with wide edge

cantilevers, was also analysed using the upstand FEAtechnique. The presence of the voids complicated themodelling and, for the software used, required the ad-

dition of extra beams in order to maintain both thecorrect area and second moment of area of the voidedsections. Isotropic ®nite elements were used as the void

depth was less than 60% of the slab depth. The modelwas analysed under the action of self-weight and a 3DFEA was carried out for comparison. The predictionsof top longitudinal stress from the upstand FEA once

again compared very well with those from the 3DFEA.

Acknowledgements

The authors would like to thank the developers of

the STRAP program and the Lawrence LivermoreNational Laboratory for providing the analysis soft-ware used in this paper.

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upstand finite element analysis of slab bridges

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