TRANSPORT A N D ROAD RESEARCH LABORATORY Department of Transport

RESEARCH REPORT 198

TESTS ON A HALF-SCALE PRESTRESSED-BEAM AND

SLAB BRIDGE DECK

by K H Withey

The views expressed in this report are not necessarily those of the Department of Transport

Bridges Division Structures Group Transport and Road Research Laboratory Crowthorne, Berkshire, RG11 6AU 1989

ISSN 0266-5247

Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on I st April 1996.

This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.

CONTENTS

Page

Abstract 1

1. Introduction 1

1.1 Limit State Design 1

1.2 Partial Safety Factors 2

2. Testing Facilities 2

3. Details of the Model 3

3.1 Scaling of Physical Properties 3

3.2 Specification for Model 4

3.2.1 Precast, Prestressed Beams 4

3.2.2 Insitu Deck Slab 4

4. Loads and Measurements 5

4.1 Method of loading 5

4.2 Instrumentation 5

5. Test Procedure 6

5.1 Arrangements for the Collapse Test 8

5.2 Loading to Collapse 9

5.3 Description of Failure 9

5.4 Discussion of Failure 11

5.5 Punching Tests 12

6. Experimental Results Compared with Theory 14

6.1 Elastic Analysis--Calculations Using GRIDS, BS5400 and VABFIS 14

6.2 Non-Linear Analysis--NFES 16

7. Conclusions 18

8. Acknowledgements 19

9. References 19

© CROWN COPYRIGHT 1989 Extracts from the text may be reproduced,

except for commercial purposes, provided the source is acknowledged

TESTS ON A HALF-SCALE PRESTRESSED-BEAM AND SLAB BRIDGE DECK

ABSTRACT

Research is currently being carried out at the Transport and Road Research Laboratory to refine methods used for the assessment of existing structures or develop new methods where none suitable exist. Techniques involve the testing of both large scale and full size structures and the development of computer programs. These enable the lessons learned about load redistribution through interlinking structural components to be utilised in assessment. The object of the work reported in the Research Report was to rationalise the knowledge of the behaviour of complex structural systems to allow some quantification of the gratuitous strength thought to exist in these structures. This would allow the assessed strength of many of the older structures to be increased. Currently all older bridges in the United Kingdom are being assessed to ensure that they are capable of carrying increased vehicular loading, and strengthening or replacement of understrength bridges will proceed over the next decade. More precise estimates of the strength of bridges would enable some of this strengthening to be deferred resulting in considerable savings of maintenance funding.

The work described in this Report has shown that, under loading to failure, a simply supported concrete beam and slab deck did not behave in the way in which the design codes assume, in that the mode of failure was not predicted. The load at failure was close to that which was calculated using the design code once the implicit factors of safety were taken out and so extra strength could not be shown. However, as the mode of failure was not predicted, the agreement of failure load was fortuitous.

1 INTRODUCTION

Tests on large scale model bridge structures and components are being undertaken at the Transport and Road Research Laboratory to study the behaviour of decks designed using both BS 5400 and earlier allowable stress codes. BS 5400 is the current code used for the design of 'Steel, Concrete and Composite Bridges' published by the British Standards Institution (1978,1984). This has replaced allowable stress approaches described in BE4/73 (Annex A) published by the Department of Transport (1973), as the code to which new bridges are to be designed.

Bridges designed both to BS 5400 and to earlier design codes are generally considered to be

conservative, and data from this series of tests may quantify the spare capacity suff iciently so that it can be used in the assessment of existing bridges. Such enhancement of assessed strength may make strengthening or replacement less urgent or even unnecessary.

Both elastic and non-linear methods of analysis are being used during the tests to provide predictions and to allow subsequent assessment of behaviour. Grillage analysis using a computer program GRIDS validated by the Department of Transport (1975) and finite strip models as described by Loo and Cusens (1978) represent the elastic methods, and NFES, a non-linear f inite element computer program developed at the Transport and Road Research Laboratory by Crisfield and Wills (1985) is being used to represent the plastic methods.

The models are half-scale with spans of 9 metres and widths of 6.3 metres. They are being tested to give information about the behaviour of specimens of this size, to compare the behaviour with that predicted by BS 5400, especially at the stage of collapse; to provide data with which to calibrate and validate bridge design and analysis programs and to investigate the effects of lateral redistribution of forces as the structure begins to behave non-linearly.

In preparation for the test described in this Report, a section of the model consisting of two beams with a 1.5 m width of deck slab was tested to collapse under a loading which closely represented the HB lane loading used for the main test. The width of the specimen obviated any possibil ity of transverse distribution during the test.

Loading produced cracks at 1.1 times design HB ultimate loading when the central deflection was 21 mm. Failure occurred at a loading of 2.2 times HB ultimate loading at a deflection of 190 ram. The model failed in the manner predicted, in shear- bending alongside the simulated HB bogie positioned near the centre of the span.

This Report describes the results obtained from a model consisting of eight, 9 metre long, prestressed concrete inverted tee beams with a 6.3 metre wide insitu deck slab cast over them (Figure 1).

1.1 L I M I T STATE D E S I G N A structure may reach any one of several conditions at which it becomes unfit for use. Each of these conditions or l imit states must be separately

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examined to ensure that it is not attained (or more precisely that the probabil ity of it being attained is acceptably low given the inherent variability of loads and material properties).

The Bridge Code BS 5400 requires two limit states to be considered, the serviceability limit state and the ultimate limit state. The serviceability limit state denotes a condit ion beyond which loss of utility or cause for public concern may be expected, such that remedial act ion will be needed, whereas the ultimate limit state corresponds to the maximum load carrying capacity of a structure or part of a structure, beyond which actual structural damage may be expected.

1.2 PARTIAL SAFETY FACTORS To take account of statistical variation in Ioadings and material strengths, and design assumptions, the nominal values of these are multiplied by factors called partial safety factors. The magnitude of these factors depends on the certainty of the value being considered, and the consequences of any error. Thus the factors are lower at serviceability limits states than an ult imate limit states, lower for steel strengths than for concrete strengths, and lower for controlled loads (HB loads) than for uncontrol led loading (HA loads). Values of the factors to be applied to Ioadings in particular circumstances are given in BS 5400: part 2.

These factors are represented by the symbol ), and are commonly called 'gamma factors'. In this Report, terms such as 'HA serviceability loading' are used for the nominal HA loading multiplied by the relevant serviceability gamma factors, and 'HB ultimate loading' is the nominal HB loading multiplied by the gamma factors specified for the ultimate limit state. This HB loading is applied on one lane of a bridge at the same time as HA ultimate loading (with modified gamma factors) is applied to other lanes. Multiples of HB ultimate loading will be represented herein by, for example, the notation 1.7 HBU.

2 TESTING FACILITIES

The Structures Laboratory at the Transport and Road Research Laboratory is 32 m long, 17 m wide and 11 m high. It is fully air conditioned with both temperature and humidity control to enable stable conditions to be maintained so that long-term tests on concrete specimens can be undertaken without excess drying of the concrete. Excess drying of the concrete would result in large creep and drying strains that would be untypical of concrete in the open air in the UK. The floor is constructed with anchor points spaced on a 0.75 m grid to which loading framework can be bolted. Each point has a load capacity of about 40 tonnes.

The test rig for loading the half scale bridge models has been assembled in this building and is internally 7.3 m wide, 18 m long and 4 m high. It consists of four longitudinal beams, one above each road lane and one along each footpath of the model bridge. These beams are supported on five rigid portal frames which span transversely. Three more pin- jointed portal frames can be positioned where needed along the length of the test rig to support the longitudinal beams against local Ioadings. The rig was designed as a general purpose test frame capable of applying 200 tonnes at any one point and having a total capacity of over 2000 tonnes.

Loading is applied by low-friction hydraulic compression jacks reacting off the test frame with hydraulic oil being supplied under controlled pressure by a six channel servo control-valve system to a manifold of 14 feed pipes which can be coupled to service any combination of hydraulic jacks from any of the servo valves.

3 DETAILS OF THE M O D E L

The model tested was designed to simulate half of a two-span simply supported bridge deck carrying a two-lane, single-carriageway, rural all-purpose road over a dual three-lane motorway. The design was carried out by a Consulting Engineer with extensive practical experience using conventional methods involving GRIDS for the assessment of load distribution. The details of the design are described by Hambly (1986). The deck consisted of an in-situ reinforced concrete slab on precast prestressed concrete beams (Figure 1). The eight 9 m long beams were of C and CA inverted-Tee section with the prestressing steel positioned to simulate the steel distribution in standard M-beams. The full size T2 section modelled the M3D section well at half scale. Only the web thickness of 105 mm for the T2 section was significantly different from the scale dimension of 80 mm required to scale the M3D section accurately. Thus the shear resistance of the beams was somewhat greater than required for a true scale model. Properties such as depth, cross-sectional area, moment of inertia and height of centroid were from 1 per cent to 10 per cent different from the true scale values. Upstands at the edges of the deck slab were modelled but the parapet rails, deck surfacing and footpath fill were omitted to simplify the model construction and to enable crack formation to be observed during testing. The self weight of these components was included in the applied loading.

A scaling factor of one half was selected as a compromise between having models large enough to use typical commercial materials with structural sections big enough to exhibit elastic and plastic behaviour similar to full size sections, but small enough to be erected in the Structures Laboratory and to be loaded to failure by available equipment.

The yield mechanisms and the redistribution of load between structural members in different types of concrete bridge are of particular interest. During the tests, measurements were made of features such as local loads, strains and deflections. These were used for checking current and future methods of bridge analysis, both elastic and non-linear, and for testing the efficacy of the design rules given in the Bridge Code.

3.1 SCALING OF PHYSICAL PROPERTIES

For simple models subject to static or pseudo-static loading the responses of the full scale prototype and the model are related by the laws of structural mechanics. Dimensional analysis shows that the number of scale factors is related to the number of independent fundamental dimensions involved. Each element E measured in the model is scaled to that in the prototype by a relationship Ep= EroS where S is the relevant scale factor. For statically loaded models there are two fundamental dimensions, force and length. Therefore only two scale factors are required for full definition. The selection of these is somewhat arbitrary but units of length S~ and stress Sf were selected as the basic scaling elements. Table 1 gives the relationship between various quantit ies and the relevant scale factors.

The linear scale of this model was 2.0 and commercial materials were used for the construction, requiring model stresses to be equal to full-scale stresses for simil itude. Thus the scale factor for stress Sf was arranged to be unity so that elements which had units of stress did not have to be scaled.

T A B L E 1

Scale factors for modelling

Element E Dimensions Scale Factor S

Length Area Volume Displacement Concentrated load Line load Uniformly distributed

load Mass density Moment Stress Vlodulus of elasticity

L L 2 L 3

L F FL-1

FL-2 FL-3 FL FL -2 FL-2

S1 s, ~ s? Sl Sf S12 Sf Sl

Sf Sf S£ 1 Sf S~ Sf Sf

2.0 4.0 8.0 2.0 4.0 2.0

1.0 0.5 8.0 1.0 1.0

The greatest problem caused by the scaling was in the representation of the doubled density required for the unit mass and hence the self-weight of the model. As it was not feasible to cast concrete of double density with normal mechanical properties the

extra mass required to give comparable stresses was added by imposed dead-weights. These weights were hung on the beams before casting the deck slab to provide verisimil i tude of dead load stresses during construction, and extra weights were added to the edge beams after casting. When extra weights in the form of loading spiders and jacking equipment were imposed on the deck, sufficient of the dead-load weights were removed to keep the total deadweight constant.

The model was mounted on laminated rubber bearings as its prototype would be, but the rotations and longitudinal movements of the ends of the deck as failure was approached were expected to be greater than could be absorbed by the bearings. The load cells under the bearings had to be protected from excessive torsion and side thrust and so extra rocker bearings were included under the rubber bearings and PTFE sliding plates were included at one end.

3.2 S P E C I F I C A T I O N FOR M O D E L The modell ing of a large concrete specimen is dif f icult , and exceptionally close controls have to be maintained on placement of reinforcement and concrete. As the results obtained from testing are to be compared with mathematical models, the maximum as well as the minimum strength of t h e concrete requires to be controlled.

3.2.1 Precast, prestressed beams The beams were supplied by a recognised precast manufacturer. The requirements for manufacture were stringent, particularly as a previous batch of

similar beams had resulted in a 50 per cent rejection rate caused by poor compaction or low concrete cube strength. The concrete was a limestone mix designed at TRRL to have a characteristic strength of 50 N/mm 2. It turned out to have an average strength of 58 N/mm 2 ___8 N/mm 2 (equivalent to a standard deviation of 4.1 N/mm2). Compaction was good and all the beams were acceptable. Each beam was tested elastically and those at the mean of the ranges of stiffness and static deflection were selected to obtain a set of beams that was matched as nearly as possible.

3.2.2 Insitu deck slab Numerous problems arose with the control of the materials and placement of the concrete of the deck slab. Tolerances of __.2 mm were specified for all dimensions including cover to reinforcement. A characteristic strength of 30 N/mm 2 specified for the design dictated a target strength of 38 N/mm 2 with a allowed variation of __.8 N/mm 2.

Such close control of tolerances was only possible by taking a number of special precautions. The reinforcing steel was fixed from a platform spanning the model to prevent distortion due to walking on the steel. Reinforcement supports and ties were closely spaced. The aggregates and cement used for the concrete were carefully controlled. The concrete was placed using a sequence of machines in the styles of a road paving train for the positioning of a regulated concrete surcharge, uniform compaction and final trimming and finishing. A representation of this casting train is shown in Figure 2. Final tolerances obtained were generally within 1.5 mm.

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Fig. 2 Concreting train for half scale model bridge.

4

Both a readymix supply of concrete and a mobile batching unit were tried but the quality control was not sufficient to keep the strength within the • +8 N/mm 2 required. As a result the TRRL small batching plant was used for the supply of all the concrete and the required strength was achieved during trials with a slump of 100 mm by using an admixture.

Many trials for placement of the insitu concrete were made to ensure that adequate compaction could be achieved and that levels of the finished concrete were sufficiently accurate. It was found that the tolerances of _+2 mm could only be achieved by using the 'paving train', with a separate placement of the edge beams after 24 hours. During the trials the concrete strength was maintained to 38 N/mm 2 _+4 N/mm 2, but a fresh batch of aggregate and different plasticiser used for the construction of the model had the effect of raising the achieved strength to 45 N/mm2.

4 L O A D S A N D M E A S U R E M E N T S

4.1 M E T H O D OF L O A D I N G The loads were applied by means of single acting hydraulic jacks which reacted against longitudinal beams supported by portal frames. The jacks were precalibrated low friction jacks so that the loads applied could be determined from the pressure of the hydraulic oil. The jacks representing the HA and HB

vehicle loads were supplied from a central hydraulic pump and controlled by servo-control valves using servo-amplifiers. The jacks representing the footpath loads and knife-edge loads were supplied from servo- mechanical load maintainers. All jack loads were applied to the bridge deck through load spreading "spiders' which represented the distributed loads for the HA jacks, or modelled the heavy vehicle bogies for the H B loads.

4.2 I N S T R U M E N T A T I O N Displacements along the beams'were measured by linear variable differential transformer (LVDT) transducers carried on supports hung on separate frames mounted on the bearing plinths at each corner of the model. These transducers were positioned at each end, at the centre and at the quarter points of each beam. Strains due to bending were measured by 50 mm electrical resistance strain (ERS) gauges placed on the surface of the concrete at the top and bottom of the section at the beam centre and quarter points. Six extra gauges were positioned at each of four places where the bending or combination of shear and bending were considered to be most critical. These points are marked in Figure 3. Strains due to shear were measured by 50 mm ERS gauges arranged in an equilateral triangle on the neutral axis at the location of maximum shear, also marked in Figure 3. Two patches of the deck reinforcement were instrumented with strain gauges, one under one of the bogies of the HB vehicle and one in an area of the deck which

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would remain relatively undamaged, to be used for subsequent punch tests. These patches are shown in Figure 3.

Each of the six lines feeding the hydraulic jacks included a pressure transducer enabling the applied load to be found from prior jack calibrations. The sixteen bearings supporting the beams were mounted on load cells. Load cells were placed between the rams of the jacks representing the HB vehicle and the load spreading 'spiders' to check the data being obtained from the hydraulic line pressures.

The data from the instrumentation was measured and converted by a pair of data loggers which buffered the information until it was sent back to a mini-computer used for storage and on-line processing to provide information about the progress of the test. This information was presented in both tabular and graphical form on the computer video terminals.

An attempt was made to use the computer to control the loading sequences by command to the servo control amplifiers. However problems with the servo-control amplifiers precluded this possibility in the time available for the test and loads were applied using two manual servo control amplifiers and three mechanical load-maintainer panels.

5 TEST PROCEDURE

BS 5400 requires that the prototype bridge should be capable of supporting HA (Highways A) and HB (Highways B) Ioadings, but as with the majority of relatively short-span bridges the strength requirements needed to satisfy HB loading are greater than those to satisfy HA loading. Thus although HA loading was applied to the model, and used to determine various information about the structural action, the failure was caused by overloading the model HB bogie.

(a) HA Loading. The initial layout was set up to represent the HA loading, and is shown in Figure 4. Four jacks were arranged along each road lane and along each footpath, applying loads through spreading spiders to represent the uniformly distributed loads required by the code. Two jacks applied the knife-edge loads at the centre of the span in each lane. Elastic tests were performed under this loading. The jacks were loaded individually to obtain influence surfaces to investigate how the loads were being distributed through the structure. All the jacks were then loaded to represent firstly HA serviceability and then HA ultimate loading conditions. In no case could any suggestion of structural damage be found and all the instrumentation indicated that the behaviour remained completely elastic.

Neg. no. R725 84 10

Figure 4 Test rig with HA loading ar rangement

The jack loads were finally increased incrementally to 130 per cent of the factored HA ultimate load value. This load was considered to represent the greatest overload of HA that was likely to occur in the field, being almost twice the unfactored HA loading. Load cycles were applied to the bridge. No sign of any damage could be seen, and the maximum deflection, which was at the centre of beams 4 and 5, was 13.6 mm. This was just over half the deflection at which cracks had been observed in the two-beam and slab model which had been tested in longitudinal bending. It was therefore decided that a prototype bridge would be unlikely to suffer cracking to any extent under HA loading, even by grossly overloaded vehicles. No further attempt was made to crack the deck with HA loading before applying the loading representing the HB vehicle.

All data in the HA tests were linear, and the bearing reactions, deflections and strains measured were close to those predicted by both elastic and non- elastic methods at this stage.

(b) HB Loading. After the tests were completed using the HA layout of jacks, one lane of the HA loading was replaced by the HB loading. The load was applied by a pair of hydraulic jacks each with a

load spreading spider to represent the bogie of the HB vehicle as required by the code. Two smaller jacks with their spiders were also positioned along the lane to apply the increased dead and superimposed dead Ioadings required because of the partial factors specified at serviceability and ult imate limit states (y factors). The layout of loading for the model was selected to represent that H B load configuration which was predicted to be most critical to the design (Figure 5). This was with one HB bogie positioned in one vehicle lane with its inner wheels on the centre of the span of the bridge. The other bogie was positioned at the minimum vehicle length of 6 metres between bogies (full scale). Thus the HB vehicle was positioned mainly over beams 3 and 4, see Figure 5.

Tests on the elastic behaviour of the structure were performed by loading the jacks several times to 1.0 HBU, checking that there were no progressive deflections or departure from linear behaviour. Some slight longitudinal cracking was found on the soff i t of the insitu deck slab either side of beam 3, under the HB vehicle. This appeared at loads between the 95 per cent and 100 per cent HBU increments, but closed up and disappeared when the load was removed.

• HB vehicle loading [ ] Dead loading [ ] Associated HA loading 1 ~ N

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Figure 6 shows the loads measured on each of the load cells and the beam deflections at 1.0 HBU load. The figure shows that at this stage beam 4 gives the largest reaction and also has a marginally greater deflection.

5.1 A R R A N G E M E N T S F O R T H E C O L L A P S E T E S T

To prepare for final collapse the weights which had been applied to correct the self-weight stresses of the model were removed to enable closer examination of damage during the collapse process, and to minimise danger to personnel due to components dropping or flying off during collapse. A force equivalent to these weights was applied by the jack loading system and held for some time at the beginning of the test. As this represented the dead- loaded state the instrumentation was set so that all the data collected represented the changes caused by the live loads. The same configuration of loading spiders was used as for the tests to 1.0 HBU and the

test itself started similarly to the previous one, all the loads being brought up to those representative of 1.0 HBU using load control. The load in the HB jacks alone was then increased above 1.0 HBU loading, still using load control, taking readings from all the transducers at each increment. Loading was continued until signs of non-linearity were seen on the plots being produced by the on-line data processing computer. At this stage the HB jacks were switched to displacement control using a displacement transducer fixed under the centre of beam 4 to provide the displacement reference. Increments of displacement were applied to the HB jacks, logging data at each increment until failure occurred. Displacement control was used for the final failure to allow the unloading section of the load- displacement relationship to be followed, giving some indication of the behaviour past maximum load. The procedure is also inherently safe because an increase of displacement caused by the failure of a structural component will automatically result in jack pressure being reduced or removed without reliance on secondary safety circuits.

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5.2 L O A D I N G T O C O L L A P S E Loading commenced under load control, increasing all the jack loads in units of one-fifth of the load required by BS 5400 for HB ultimate loading. Once 1.0 HBU was achieved, the loads on the HA jacks were maintained constant, and the loads on the HB jacks were increased in units of one-tenth HBU load.

The first visible sign of distress occurred at an applied load of 1.3 HBU. Cracking was seen on the soffit of beams 3 and 4 below the central HB bogie. The deflection at the centre of both beams was about 15 mm at this stage.

Load increments were added and at a load of 1.6 HBU, longitudinal cracks were seen along the webs of beams 2 and 5. These cracks stretched along most of the length of the webs, but were only on the sides of the beams furthest from the HB bogies. It appeared that the top flange of the web was being rotated by the deck dishing towards the HB bogies, and wi th the laterally and torsionally stiff bottom flange of the beams being restrained by the diaphragm, the web did not have sufficient flexibil ity to absorb the rotation wi thout cracking.

5.3 D E S C R I P T I O N O F F A I L U R E The response was seen to be becoming non-linear at a load of 2.0 HBU. From this stage displacement control was used to control the jacks, applying increments of displacement which corresponded to load increments of 0.1 HBU in the elastic region. A t a displacement of 72.2 mm on beam 3 a loud report was heard and the deflection suddenly increased to

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1000

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Deflection (ram)

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76.6 mm. This was accompanied by a fall in load from 2.91 HBU to 2.79 HBU, shown as point 'a' in Figure 7. The report appears to have been caused by stirrups on beam 3 fail ing. The side of the web of beam 2 facing the HB bogie was spall ing quite badly, especially in the area about 2 metres from the southern bearings, a l though it occurred over half the length of the beam as indicated in Figure 8. The equivalent face of beam 5 was showing similar distress, but to a lesser degree.

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The beams taking the largest share of the load, beams 3 and 4, showed a marked degree of negative curvature next to the south diaphragm at this stage. It appeared that the edge beams and the diaphragms were acting as a frame to restrain the central beams and the deck was behaving like a cont inuous structure. It is likely that this is the reason for the relatively low level of damage caused by bending of the beams, as the effective span was very much reduced by the 'built- in' effect at the ends. The shear stresses were not alleviated by this effect and thus became the principal cause of final failure.

Further increments of displacement were applied to the bridge model to a maximum load of 3.17 HBU at a deflection of 98.9 mm on beam 3 and 93.6 mm on beam 4. This stage is indicated as point 'b' in Figure 7. A period of failure followed during which time the deflection was increased to 112 mm on beam 3 and the load fell to 2.8 HBU. Several reports were heard as further stirrups in beams 3 and 4 failed. Finally a period of increasing deflection fol lowed wi thout any great change in load until final collapse occurred at a deflection of 150 mm and 143 mm on beams 3 and 4 respectively. The load applied immediately prior to collapse was 2.89 HBU.

At this point the flanges of beams 3 and 4 were effectively disconnected as the concrete and all the stirrups of a 2 metre section of the web had broken (see Figure 8). As a result the resistance of this section of beam was insignificant and the deck slab alone carried the HB bogie. The slab was not strong enough to do this and sheared along the side of beam 5, displacing vertically downward. To provide

flexibil i ty for this movement the slab failed in bending about 1.5 metres from the south bearings over the broken section of beams 3 and 4. The final collapse of the slab could be considered as large scale partial punching of the deck around the central HB bogie. The locations of the various types of damage are shown in Figure 8.

Figures 9 and 10 show the sides of beams 3 and 4 after the failure. The extent to wh ich the horizontal cracks have completely disconnected the top and bottom flanges should be noted. Al l the stirrups in a length of about 2 metres have fractured, mostly w i th indications of transverse movement during the break.

Figures 11 and 12 show both sides of beam 2. Figure 11 illustrates how the tension crack on the side away from the HB load has opened up cleanly wi th only a little spalling around some stirrups due to shear movement. Figure 12 illustrates how the reverse face shows marked surface spall ing over a 3 metre length due to the concentrat ion of vertical load at this face.

The reactions measured by the load cells showed increases on the bearings of beams 2 to 5 approximately proport ional to the applied loads, unti l the beams lost their stiffness when the webs sheared. A t this stage the reactions at the south end of these beams decreased wi th increasing load. Figure 13 shows the change of loading on each bearing. The ordinate of the x-axis is increment number and the change of slope of the plot at increment 14 is due to an increase in the rate of collecting data rather than any signif icant change of behaviour of the model. Beams 1, 7 and 8 unload as

++++++}+,:+

Neg. no. B103B/86

Figure 9 Side of beam 3 a f t e r f a i l u re

+ ~+++.+~. . , y - ~ * ~.++ +.t:..+~-. _ + + , L _ _ _ . . _ + + , ! ~ ~ + - . + - ~ S - + . : + - ~ ~ + : + . - ~ - I -: . . . . . , _. -.;.+~+~ ~ _ - , + + + + + + + + • • + . + , , + ~ + ,

Neg. no. B102A/86

F i g u r e 1 0 S i d e o f b e a m 4 a f t e r f a i l u r e

10

. _ . - 1 . , = . - .... ~ . Neg. no. B103A/86

Figure 11 Hor izon ta l tension crack in web of beam 2

!

Neg. no. B102B/86 Figure 12 SpaUing due to vertical compression

in beam 2

the HB bogie loading is surcharged, to the extent that beam 8 would actually have lifted off its bearings if the HA component of the HB loading had not been applied to the model. The figure shows that when the stirrups began to fail at a load of 2.9 HBU (see point 'a' in Figure 7) beams 3 to 5 unloaded significantly while the other beams, especially beam 1, picked up a greater share of the load. During the collapse of the beam, shown at point 'b' in Figure 7, all beams 3 to 6 shed their load while beams 1, 7 and 8 increased their reactions. At the north end of beam 3 and 4, where little damage was done, the reactions remained closely proportional to the applied load.

The deflections of the beams are drawn in Figure 14, plotted against applied HBU loading. The figure shows that the deflection, and hence absorbed energy, of all the beams increases while the loading is being increased. This is the case even for beams where the reactions at the bearings do not increase. Hence the transverse stiffness of the slab is enabling it to carry the loading, which should result in extra strength of the deck. The deflections of beams 6, 7 and 8 decrease after failure of the webs of beams 3 and 4 and the deck became further damaged.

5.4 D I S C U S S I O N OF FA ILURE The overall behaviour of the bridge was reassuring. A shear dominated failure of this type would normally be expected to be 'brittle', that is it would fail with little increase in deflections as elements yielded and

hence give little prior warning. The redistr ibution prevented sudden collapse, and failure was reasonably ducti le. Little signif icant damage to beams 3 and 4 was seen up to the failure of the stirrups at a deflection of 70 ram. Redistr ibution absorbed the load shed from these beams and the deck was carrying the same load by the time the deflection had increased to 80 mm. The peak load carried was nearly 10 per cent greater at a deflection of 97 mm, and thus failure of a bridge in the field under excessive loading past first damage wou ld still have this factor of safety. In the test, the model cont inued to carry loads up to def lect ion of 150 mm whi le being pushed down under displacement control of the jacks. However the load did not exceed the 3.17 HBU attained at a displacement of 97 mm, and this load and deflection wou ld represent the collapse state for the prototype bridge. A l though structural action of the model enabled some redistr ibution of load to occur after the structural failure of one component, it would be better if the capacity to carry extra load, and the extra deflection to peak load could have been higher. The tendency for the beams to act as buil t- in to the stiff end diaphragm caused the deflect ion at the time of initial failure to be markedly reduced compared wi th a similar beam tested previously. In that case the beam behaved linearly up to a deflect ion of about 18 ram, but then cont inued to carry increasing loads up to failure at a deflection of 150 mm. The st i f fening effect of the diaphragm reduced the deflect ion capacity of the beams in the test deck, and hence reduced their capacity to shed their load transversely before failure occurred.

11

React ion

kN

120

100

80

~ J

I** s

i

P i J

!

. . tel l ~ m ~ j

60

40

20

- -20

- -40

6

[ 1.0

D •

. f . . _ _ _ . . - - - --

• r

p-

/

J

. . I ® " " " \ .

Q J . . . .

\

* * * * * * * * * * * % .

I I I I In i t ia l fa i lu re Final fa i lure

o f s t i r rups o f beams

I I I ! I t t I 10 14 18 22 26 30 34 38

I I I I I I t I 2.0 2.3 2.5 2.7 2.8 3.0 3.2 2.7 2.8

HB load fac to r

I I 42 45

I I 2.8

Fig . 1 3 C h a n g e o f l o a d o n s o u t h s u p p o r t s (base is H B u l t i m a t e load c o n f i g u r a t i o n )

Loadi n q increment

A less reassuring aspect of the failure was that neither the longitudinal splitt ing nor the shear failure were identified beforehand as being critical in the design calculations. The areas considered to be critical actually showed no significant signs of distress. These were: the region of shear between the north HB bogie and the support; bending at centre span; shear-bending just south of the centre span HB bogie; and interfacial shear between the precast beam and the insitu deck south of the HB bogie. The initial longitudinal splitt ing of the beams appeared to be caused by bending about a longitudinal axis in the beams caused by transverse rotation of the top slab. This mode of failure is not considered in design calculations or in the assessment of bridges to BD21/84 (Department of Transport, 1984). The final shear failure was at about quarter-span rather than the near mid-span area that

had been considered critical in design where bending was expected to have formed cracks in the lower flange of the beam to give a shear-bending mode of failure. The bending appeared to be alleviated by the unexpected 'built-in' effect of the diaphragms causing the failure to be at a point of contraflexure where there was little or no bending.

5.5 PUNCHING TESTS Various supporting tests were performed on the components and materials of the structure, including punching shear tests on a section of the deck which had been left substantially undamaged by the main test. A 150 mm square pressure plate was loaded by a single jack to the point that a punching failure occurred. This test was repeated at three places,

12

Z _Y. v

" t3

8

- Q

CO "r

100C

500

° 0

0

®@ .® ® ® ®

• "Ji / I / \ / ~ ' - ' " ~ ' k ~1/ p / W / t ~ ) / : . ~ , " ~ . ' W , " X---; "

I; / /~;f B*mnumberseircled 1 1 . 4 " "

/

I I I I I I I

20 40 60 80 100 120 140 Def lect ion (mm)

- 3

- 2

160

Fig. 14 Applied load vs. deflections at the centre of each beam

0

oa

CO -r x

200

z v

8 -J 100

Beam 718

earn 617

Beam 5/6

0 1 2 3 4

Def lect ion (mm)

Fig. 15 Load/deflec'tion curves from punch tests

13

firstly on a section of slab which was surrounded by at least 1.5 m of undamaged slab, then on the slab next to the edge beam, and finally at a point between the other two where there was little sideways support to the beams either side of the punch because of the holes made by the previous tests. The loads which were applied to cause failure were over five times the design ultimate load level, although the final failure was precipitate. The first punching test, which had the greatest lateral support, actually produced the lowest load at failure. The load-deflection relationships obtained from the tests are shown in Figure 15.

6 EXPERIMENTAL RESULTS COMPARED WITH THEORY

The data obtained from the collapse test of the deck has been compared with values obtained using code calculations with the implicit factors of safety (7 factors) removed, and with results obtained from programs GRIDS, VABFIS and NFES. Some of these comparisons have been reported separately by Wills et al (1989). Therefore only a few selected examples are presented here.

6.1 ELASTIC ANALYSIS-- CALCULATIONS USING GRIDS, BS 5400 AND VABFIS

GRIDS is a computer formulation of a grillage analysis for bridge deck design which is extensively used by bridge engineers in the UK. It was validated and supported by the Department of Transport. VABFIS is a finite strip computer program which was developed at the Transport and Road Research Laboratory, originally for the analysis of steel box- girder bridges at the time of the Merrison Enquiries, but has subsequently been developed and extended to enable it to be used for other aspects of bridge analysis.

Program GRIDS was used to predict the deflections and distribution of reactions which would occur under loading up to 1.0 HBU. Figure 16 shows the predicted load-deflection relationship for beam 3 compared with the experimental values. It can be seen that the initial stiffness is predicted fairly closely, up to the point at which the longitudinal cracks appeared along beams 2 and 5. After this point the prediction gets rapidly more inaccurate as the model begins to respond non-linearly to increasing load.

Figure 17 shows the predicted loads on the ends of the beams for the HBU loading, compared with the measured values. GRIDS predicts a slightly smoother distribution of loads across the bearings than was measured in the test but the correlation is generally good up to about 1.5 HBU load, well within the accuracy needed for simple elastic design.

Load (kN) 1200

1000

800

s s ~,

f J

GRIDS j~ Experimental

value /

x H B ultimate load

600 [ - t/1/

0 30

I I I 60 90 120

Deflection (mm)

- 3

2

- 1

I 0 150 180

Fig. 16 Load deflection curves from GRIDS and from experiment (beam 4)

Load (kN)

180

140

100

60 North load cells

• South load cells o GRIDS-- North

• GRIDS -- South

20 I I I I I 1 2 3 4 5

Beam no.

6 7

Fig. 17 Bearing reactions from GRIDS and from experiment

The data from GRIDS were extrapolated to determine the theoretical elastic shear and bending stresses which would occur in the model when the design loading was increased in the same manner as the model loading was increased. Initially the partial factors of safety (), factors) for the material strength were included as this gives an indication of the reserves of strength from a designer's point of view.

14

The load factors at which these stresses just became critical were calculated for different positions along the length of the critical beams 3 and 4. The load factor at which the HB vehicle just caused critical stresses for bending was calculated to be 1.76, using material properties as assumed in the design. This calculated factor increases to 2.55 when the measured material strengths are used.

There are two possible modes of shear failure, firstly due to vertical shear forces only, where the shear resistance is termed Moo, and secondly due to vertical shear combined with bending forces and associated cracking, where the resistance is termed Vc,. Figure 18 shows these values, and it can be seen that the region which would become critical first is between the HB spider and the north bearings as the applied shear exceeds the shear resistance above a load of 2.2 HBU loading. In the test slight shear cracking appeared at this point at a load of 2.3 HBU. The cracking did not develop from that point as the cracks ran in one direction into a region of the beams which was more heavily reinforced towards

the bearings and in the other direction into a region where the shear stress rapidly diminished under the pads of the HB bogie. Further slight cracking occurred when the first stirrups failed in the other (southern) end of beams 3 and 4 at a load factor just below 3.0. A second critical region was just south of the HB vehicle, where the applied shear stress exceeds the design resistance at a load of 2.8 HBU. From this elastic analysis the region midway between the mid-span HB spider and the south bearing would not be expected to show any distress until a load of 3.2 HBU, and failure actually occurred at this load.

Calculations were also performed using normal methods of design as specified in BS 5400, but with the y factors for both Ioadings and materials taken as unity, as the applied loads and material strengths were closely controlled. This gives an indication of the reserves of strength that may be considered to exist during an assessment of the deck. Using design strengths of the materials the deck was expected to fail in bending at 2.7 HBU loading, but using the measured strengths of materials the expected failure

HB Vehicle

N°rth T T f Changes in VST

i A

/ v

- " V P r

South

r - - I .

I !

I ! !

I I- 500k N

500kN

I L.I

r"

,/

Fig. 18 Comparison of maximum shear force and shear strength for DL + H A + 300% HB ult imate loads -- Beam 3

Load of 300% HB

VCO + VST

VCR + VST

15

load was calculated to be 3.5 HBU. The shear strength of the deck was investigated, and the critical region was predicted to be at the north end of the deck just beyond the HB bogie. The estimated load at failure was between 2.2 and 3.3 HBU, the calculated range being caused by uncertainty of local shear values because of the coarseness of the GRIDS model used for design and the high sensitivity of the factors to precise positioning of the loads. However the length of the beam which was at risk was short, as shear cracks would run either into a region containing more reinforcement toward the bearings, or under the pads of the HB bogie and into a region of lower shear stress. In the test the limited cracking in this region never developed significantly. Near the centre of the span, just south of the mid-span HB bogie the estimate of failure load under shear- bending was 2.8 HBU, and at a distance of 2.5 metres from the south support the load at failure was estimated to be between 3.2 and 3.5 HBU. However this calculation of shear capacity of the beams is sensitive to the assumed value of prestress loss. The critical loading for interfacial shear between the beams and the insitu slab is diff icult to predict. Calculations using code criteria gave a lower bound load of 1.7 HBU, but these criteria have been developed from tests which have given a very wide range of results. Results from the test of a similar two-beam configuration of beam, slab and loading suggested that no distress would be expected at the failure load experienced. Making some allowance for redistribution and unaccounted reserves of strength, the deck was expected to fail at between 3.5 and 5.0 HBU, and the rig was prepared to apply loads up to 7.0 HBU. In the event failure occurred at 3.2 HBU.

VABFIS was used init ial ly to determine the strains and deflections of the model at 1.0 HBU loading, but was particularly useful to investigate the cause of the longitudinal cracking along beams 2 and 5. It was found that the necessary rotations of the top flanges of the beams could only be replicated by reducing the stiffness of beams 3 and 4 and of the top slab. With these changes the computer model results showed close agreement with the experimental results. It may be that the beams and deck were suffering degradation which was not immediately obvious from the damage that was exhibited during the test, or that VABFIS was not representing what was actually happening to the model. Correlation of deflections and support Ioadings across the model were not obtained wi thout considerable ' tuning' of the VABFIS data, and the results were not entirely satisfactory.

Two mechanism methods for determining the collapse loads were also employed. Firstly, fol lowing Pillau and Lash (1969) the deck was assumed to hinge across the slab and torsional hinges in the beams were included separately. This gave a collapse mode wi th a lowest failure load of 3.5 HBU, involving hinges in two beams as shown as case 5 in

Figure 19a. The load at failure was predicted closely but the mode of failure was not. Secondly a method following Hendry and Jaegar (1958) was used where the beams were treated as part of an 'equivalent slab'. This method gave a lowest collapse factor of 4.4 HBU with a very similar pattern of hinges to case 5 of the previous method. Figure 19b shows the various modes and loads at predicted failure. As can be seen comparing these with Figures 7 and 8, the method did not predict either load or mode of failure satisfactorily.

These methods are shown to be inadequate for design in this case, partly because of the great difference in the stiffness of the beams compared with the stiffness of the slab, but also because only bending failure is considered. Thus the method should be regarded as an upper bound solution, and was used mainly to give guidance on the maximum loads which were likely to be required of the test rig. The actual behaviour of the deck, especially the way in which the bending stresses were alleviated by the 'built in' effect of the edge beams and diaphragm, was not foreseen.

6.2 NON-LINEAR ANALYSIS--NFES NFES is a non-linear finite element computer program developed at TRRL. The model deck was idealised as a plate with eccentric stiffeners, with plane sections assumed to remain plane through the depth of the structure. The prestressed beams were represented as layers of concrete and steel, as was the reinforcing steel in the slab. The concrete properties of the slab were integrated through the thickness.

Finite element analyses carried out at the time the deck was tested used a 4-node discrete Kirchhoff bending element supplemented by a quadratic membrane element (Crisfield and Wills, 1986). Two idealisations were used, first, a coarse mesh of 4 x 6 plate elements with the beam properties smeared to the plate Gauss points, and second, a finer mesh of 9 x 10 elements coupled with discrete beam modelling. Figure 20 shows the material properties adopted.

Figure 21 shows the calculated load-deflection curve for the centre of beam 3. The stiff response provided by the coarse idealisation relates partly to the coarser mesh and partly to the lower number of integration points used through the depth of the slab. Thus compression damage in the slab was recorded later. The second idealisation correlated very closely with the behaviour of the test up to 3.0 HBU. At this load the model started to exhibit shear-type failure in the two most heavily loaded beams. The discrete Kirchhoff element does not model such modes and thus NFES went on to predict a failure at a live load of 3.4 HBU involving compression softening of the deck above beam 3 at the centre span.

16

Base HBU factor

6.08 (6.08)

6.62 (6,49)

Proposed mechanism

IL Y

T , A

T

$ A w

Y ' - " - - " o ~ e • e q ~ . •

1 "

,y

6 o

v

Base HBU factor

6.08

6.45

Metres Proposed mechanism

6 ,

3.75

1 4.20 (3.37)

3.76 (2.53)

° ° o o o . _ ~ . e o o o o e

Y

V

T.';.'.-.-.- . . . . . . . . . . . . . .-- \ \~:~x ° e e o o . _ ~ . . o e • •

$ • e e o ~ ° e e e e o e e e * e e • e o e * •

l e e •

5.15

4.65

0 . 7 5 !

0"751~5

0.75"-~

3.53 (1.86) ~ . : . : : : . . . . . . . . . . . . . . _ _ . ~ : ' . . ' : : : : . ' " ; ~

v

1 • Moment O Torque I

hinges hinges I Fig. 19 (a) Results of hinge mechanism method.

(Values in brackets exclude slab resistance)

4.42

0.75..~_ 0.75 j . .

Fig. 19(b) Results of equivalent slab method

17

A

t ~

E E z b

10

0

--10

--20

--30

--40

Tension

~\\\ ~ Compression

\ \ Beam // ", ~ concrete //

- / ' _ , J

-8 --6 --4 -2 0 2 St rain ( 1000#(5 )

Uniaxial stress-strain c u r v e f o r c o n c r e t e

A

"E

2000

z 1ooo

' 0 4 0

J

Prestressing steel

t " " " ~',,-R~ iforcing steel

I I I I 10 20 30 40

Strain/4(5

Stress -s t ra in curves f o r steel

50

Fig. 20 Material properties used in NFES

Subsequent analyses also used a mesh of 9 x 10 elements and discrete beam modelling but the discrete Kirchhoff element was replaced by an 8-node Constrained Mindlin element in order to explicit ly consider the shear applied to the beam sections. In addition, sl ightly higher material strengths were adopted and the torsional stiffness of the beams was included in the formulation (Wills et al, 1989). The best conservative estimate of the experimental load/deflect ion response and of the maximum load (Figure 21) was obtained by assuming somewhat speculative elasto-plastic properties derived from BS 5400 (1984) for shear and torsion. The effects due to these actions were found to be the most likely causes of failure.

7 CONCLUSIONS A model prestressed beam and slab deck was loaded to 1.3 HA ultimate loading with no indication of damage. It was then loaded under HB ultimate loading as defined in BS 5400 part 2. This loading was then surcharged until collapse occurred, at a load of 3.17 HBU.

The first major damage occurred at a load of 1.6 HBU. Longitudinal splitt ing of the webs of the beams either side of the HB bogies was caused by inward rotation of the top flanges as the deck dished. This concentrated the vertical loads on the inner face of the webs giving rise to spalling at higher load levels; This damage would be unlikely to cause failure but would give great serviceabil i ty problems. Signif icant yield of the stirrups in these beams had occurred at collapse of the model. This mode of failure has not been documented before for a concrete beam and slab deck, and is not considered during the design or assessment of such a deck.

4.0 ~ e mesh

3.2 ~ e s h

"~ " Experiment - / 2.4

e n

T 1.6 x

0 , 8

0 1 I I I I

0 40 80 120 160 200 240 D e f l e c t i o n ( m m )

Fig. 21 NFES load/deflection curve vs. experimental curve (midspan of beam 3)

The failure giving final collapse occurred at a load of 3.17 HBU in an area which in theory should be a region of both bending and shear. In the test the beams were seen to be supported by the diaphragm so that the region of the failure was at a point of contraflexure and almost entirely in shear. The collapse was primarily caused by the stirrups of the two most heavily loaded beams failing and allowing the top and bottom flanges of the beams to act independently. This reduced the stiffness so much that the load carrying capacity of these two beams became small. The insitu slab was not able to carry the redistributed loads on its own and failed in bending at one point and shear along a supporting beam to cause collapse of the structure.

18

The elastic methods of analysis model the behaviour well, up to a load of about 1.3 HBU. The biggest difficulty is in prediction of the properties of the materials. Once the effects of the properties are known, 'effective' values can always be justified to explain and model the behaviour. Prediction is much more difficult. Above 1.3 HBU predictions of the elastic techniques are not satisfactory, even using techniques involving the reduction of stiffness of various elements to represent extending damage.

The predictions obtained using mechanism methods are even less satisfactory as the techniques do not adequately model the beam and slab configuration.

The finite element model NFES with a fine mesh predicted the behaviour extremely well up to the point where bending was no longer the major mode of displacement. A different type of model is required to consider the shear failure of the specimen. Thus finite element methods of prediction can only be considered as an upper bound solution to the problem. The prediction of the 'effective' properties of the materials is as difficult as with the elastic methods.

The mode of failure emphasises that if arbitrarily large factors of safety are applied to the principal (or desired) mode of failure, (bending in this case) then other modes of failure (shear in this case) will become more likely to occur. Similarly the effect of extra strengthening (a stiff diaphragm in this case) which will strengthen against primary modes of failure, can cause more unsafe brittle modes of failure to become likely. Arbitrary stiffening can also reduce factors of safety by hindering redistribution which may relieve overstressed elements of a structure. Thus care must be taken when increasing individual factors of safety that modes which are undesirable do not predominate, even though these may lead to failure at a higher load level than would otherwise be the case.

It must be emphasised that this model was designed to be close to the limits allowed by the code. As far as was practicable there were no arbitrary increases in the strength of any item. The design was entirely practical and followed the guidelines of the MOT Standard Bridges Designs. Many bridges of similar design have been constructed and are operating satisfactorily.

BS 5400 contains clauses which allow the use of more efficient methods of analysis provided supporting evidence for their use is available. The results of this research could permit the use of these options. The consequent improvement in the assessment of existing bridge decks may allow their load carrying capacity or width to be increased without the need for expensive strengthening. However, this test has not indicated that there are any fortuitous factors of safety above those given by the y factors implicit in designing to BS 5400. Thus

reliance placed on these margins during assessment may be eroding the explicit safety factors enjoined by the bridge code. The implicit factors of safety given by the design y factors would give an expected failure load on this deck of about 3 HBU loading with failure being caused by a shear-bending (Vc,) mode. The extra stiffening given by the diaphragm ensured that this mode did not occur by reducing mid-span bending, but a more precipitate shear mode predominated. It may be that the stiffness produced by the diaphragm reduced the ability for transverse distribution and actually reduced the ultimate strength of the deck. Although failure did not occur in the expected components and these could therefore be considered to be stronger than designed, the alternative mode of failure does not allow for an increase of the assessed strength of the deck.

8 ACKNOWLEDGEMENTS

The work described in this Report was carried out in the Bridges Division of the Structures Group of the TRRL. The concept of the project was developed by the late Mr C A K Irwin. Preparation of the rig, th'e~ loading hardware and the models was carried out under the direction of Mr J W Grainger. Thanks are due to all the members of the Design and Workshop Units without whose efforts this project could never have been completed.

9 REFERENCES

British Standards Institution, (1978). BS 5400: 1978, Steel, Concrete and Composite Bridges (parts 1 to 10). London: BSI.

British Standards Institution, (1984). BS 5400: 1984, Steel, Concrete and Composite Bridges; Part 4 Code of practice for the design of concrete bridges. London: BSI.

Crisfield, M A and Wills, J (1985). Nonlinear analysis of concrete and masonry structures. In: Proceedings of the Europe-US symposium on finite element methods for non-linear problems. Trondheim, Norway: The Norwegian Institute of Technology, section 122, pp 1-14.

Crisfield, M A and Wills, J (1986). Non-linear analysis of concrete and masonry structures, in: Finite Element Methods for Non-Linear Problems, ed. P G Bergan et al, Springer-Verlag, Berlin, pp 639-652.

Department of Transport, (1973). Technical memorandum BE4/73 Annex A. Schedule of standards and memoranda. London: DTp.

19

Department of Transport, (1975). BE2/75 Suite of bridge design and analysis programs--Program HECB/B/9 (GRIDS). London: DTp.

Department of Transport, (1984). The assessment of highway bridges and structures, Departmental Standard BD21/84. Dept of Transport, Roads and Local Transport Directorate. London: Dtp.

Hambly, E C (1986). Design and calculations for the single span half-scale model bridge to BS 5400: 1978. TRRL Contractor Report 25. Crowthorne: Transport and Road Research Laboratory.

Hendry, A W and Jaeger, C G (1958). The analysis of grid frameworks and related structures. Chatto and Windus, London, 1958.

Loo, Y C and Cusens, A R (1978). The Finite-Strip Method in Bridge Engineering. Wexham Springs: Viewpoint Publications (Cement and Concrete Association), Alden Press.

Pillau, S U and Lash, S D (1969). Ultimate strength of reinforced concrete grid and slab bridges. Proco first Int. Symp. on concrete bridge design. ACI Special Publication 23, 1969.

Wills, J, Crisfield, M A and Cheung, J T (1989). Numerical analysis of a half-scale prestressed-beam and slab bridge deck. TRRL Research Report RR218. Crowthorne: Transport and Road Research Laboratory.

Printed in the United Kingdom for Her Majesty's Stationery Office (3789/89) Dd8222758 3/90 C5 G476 10170

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