University of Cambridge Department of Engineering

Concrete Bridge Assessment

C.R.Middleton , BE(Hons), MSc, PhD, CEng, CPEng, MICE, MIE(Aust)
Assistant Director of Research, University of Cambridge, U.K.

First presented at the Bridge Surveyor Conference, 24th March, 1998


In this paper the conventional approach to bridge assessment is examined and the definition and consequences of ‘failure’ are discussed. Alternative methods of assessment are considered and the potential for using plastic collapse or yield-line analysis for the assessment of short-span reinforced concrete slab bridges evaluated. A new technique for performing yield-line analysis has been developed recently at Cambridge University and implemented in a computer program called COBRAS. This approach provides a simple, rapid and practical means of performing yield-line analysis and overcomes many of the difficulties that have previously limited the application of this method in practice.

A large number of concrete bridges, ‘failed’ using conventional assessment methods, have now been re-assessed by the author using the COBRAS yield-line program. Over two-thirds of these "failed" bridges were re-assessed at full 40 tonne HA capacity when plastic rather than elastic analysis methods were used. Independently, fourteen concrete bridges that had failed their original assessment were re-assessed by three County Council bridge consultancies using the COBRAS yield-line program. Thirteen of these bridges were re-assessed at full 40 tonne HA assessment live load capacity, with the other bridge being rated at 38 tonnes.

The adoption of yield-line methods has directly resulted in huge savings to the bridge owning authorities involved. Clearly there is a very strong argument for yield-line analysis to be at least considered, if not actually applied, for evaluating the ultimate load capacity of short-span concrete slab bridges.


Most bridge owning authorities are well progressed with their bridge assessment programmes. However they are now facing up to the task of dealing with the backlog of structures that have failed their original assessment. Many of these have been placed into the basket for "future strengthening or replacement" or for "monitoring and re-assessment" based on the "engineering judgement" of the bridge engineers who have had to prioritise their scarce resources for strengthening and replacement works. However at some stage all of these "failed" bridges must still be re-assessed and a decision taken on what action is required to ensure their structural integrity and safety.

The causes of failure are varied and depend very much on the type of structure and also, to an extent, on age and location. For example most of the motorway and trunk road bridges were built post-1960 during the motorway expansion schemes. These bridges are predominantly concrete and the Highways Agency is concerned with problems such as deficiencies in shear, flexural capacity, inadequate anchorage details, pre-stress corrosion and deterioration of joints, piers and cross-heads. Local authorities, on the other hand, have large numbers of masonry arch bridges, which pose particular analysis problems, and also many older concrete bridges which have often been subject to significant deterioration or were designed with inadequate detailing, little or no top steel, and low percentages of transverse steel.

The Analysis Credibility Gap

There is an underlying realisation that the analytical techniques developed for design are in many cases unable to accurately model the structural behaviour of existing bridges. As a result assessments often significantly underestimate the actual load capacity of bridges. This discrepancy between theoretical predictions and reality has been highlighted by the number of bridges which have ‘failed’ their assessment even though the assessing engineers' experience and intuitive feelings tell them that the bridges are capable of safely carrying significantly higher loads. There are examples cited of bridges which have regularly carried abnormal vehicles weighing 180 tonnes without distress being assessed to have an ultimate load capacity of 7.5 tonnes [1]. Equally most bridge engineers will know of examples of assessment reports in which concrete slab bridges have been rated at zero live load capacity!

In many cases, the bridges exhibit no outward signs of distress. Although this does not, in itself, imply that failure may not be imminent, it is likely that some evidence of damage or significant deformation will precede collapse in cases of ductile flexural failure of concrete slabs.

This brings into question the appropriateness of using elastic analysis for the determination of ultimate strength for many types of bridge, and in particular for short-span concrete slab bridges which have been found deficient in flexure.

Clearly there is a need to review and refine our existing methods and to develop improved techniques which can more realistically model the ultimate load capacity of bridges. These must also be practical to apply and relatively quick to implement due to the large numbers of bridges involved.

It must also be remembered that the current assessment programme for 40 tonne vehicles should be viewed as a prelude to an on-going task of re-assessing load capacity as it is almost inevitable that lorry loads will follow past trends and rise again in the future. Since 1994 the Government has allowed 6-axle articulated vehicles and drawbar-trailer combinations to operate at 44 tonnes when carrying loads to or from rail terminals (although the drive axle limit remains at 10.5 tonnes). Belgium, Denmark, Finland, Italy and Luxembourg all allow 6-axle vehicles of 44 tonnes or more on their roads. France allows axle loads of up to 13 tonnes and in the Netherlands 50 tonne vehicles are permitted! [3]

The Scope For Improved Assessment

What scope is there for refining the existing approach or using alternative methods to re-assess short span concrete bridges that have failed their initial assessment?

Perhaps a point that is sometimes overlooked is that these assessment codes also provide considerable scope for engineers to use a variety of methods for assessing ultimate strength. In particular, although the code is based predominantly on the premise that elastic analysis will be used for analysis, engineers may use other forms of assessment such as plastic methods, non-linear finite elements or load testing in appropriate circumstances.

In the U.K. the assessment code explicitly states that ‘non-linear and plastic methods of analysis (e.g. yield-line methods for slabs) may be used with the agreement of the Overseeing Organisation’ (BD44/95 Clause 4.4.3). The bridge design code also permits these methods to be used stating that ‘plastic or yield-line methods may be adopted when appropriate to the form of construction’ (BS5400:Part 1:1988, Clause 7.3.1).

The Assessment Of Short-Span Concrete Slab Bridge Decks

Although these may seem somewhat unexciting structures, and would also appear to be very simple to assess, short-span concrete slab bridges have in fact created significant problems in the current bridge assessment programme. Statistics from Northern Ireland serve to highlight this point. The Department of the Environment in Northern Ireland is responsible for all of the approximately 6,500 public road bridges in Northern Ireland. Statistics are available for five of the six administrative regions which show that by August 1997, 3558 (75%) out of a total of 4712 bridges have been assessed. Of these, 487 (10%) have failed assessment. It is interesting to note that of the 487 failed structures, 431 (89%) have a span of between 2.0 and 15.0 m. Just under half (194) were concrete of which 185 (95%) were deemed to be inadequate due to insufficient flexural capacity [1].

Clearly it is important that engineers carefully evaluate the methods of analysis employed and ensure that the most realistic and relevant ones are used for determining the strength of bridges. To be overly conservative could result in expensive and often unwarranted remedial action such as replacement or strengthening being undertaken. Even placing some form of traffic or weight restriction on a strategic bridge can impose a substantial economic burden on a community.

Methods of analysis

The fundamental philosophy adopted for assessment, as distinct from design, has been to evaluate only the ultimate strength as the fundamental criterion for passing or failing a structure. Serviceability criteria are not usually considered. The argument given is that an existing structure is likely to have already exhibited evidence of any serviceability problems and these should have been dealt with in maintenance programmes. Thus the methods of analysis employed need to be able to predict realistically this ultimate capacity.

Elastic analysis - the conventional approach to assessing load capacity

Current codes of practice are written with the implicit assumption that the design and assessment of bridges will usually be undertaken using linear elastic analysis techniques. Elastic theory is well established and understood, is supported by many computer software packages, and has been found most satisfactory for the design of bridges. As a lower-bound method the engineer can be confident that the analysis method should be conservative and hence safe.

This approach is quite understandable for design where a certain degree of conservativeness costs relatively little. However is this appropriate for assessing the load carrying capacity of existing short-span concrete slab bridges? What does failure actually mean in an elastic analysis and what are the consequences of such a failure in terms of both risk to life and economic terms?

Firstly, the conventional approach to the assessment of short-span concrete bridges must be reviewed. Typically, the engineer might initially perform a simple elastic beam analysis using a representative strip of the bridge deck. If this ‘quick’ check shows the structure to be inadequate, a more detailed linear elastic analysis allowing for transverse distribution of load would probably be performed using either a grillage or finite-element analysis. These results are then examined to identify individual locations at which the maximum calculated moments or shears exceed the estimated ultimate capacity of the section.

The decision to strengthen or replace a structure is commonly made on the basis of these results. However, many older reinforced concrete bridges in the U.K. were built with little or no top steel and very little transverse steel. Such structures almost inevitably are rated at very low flexural capacities using such an elastic failure criterion when, for example, the live load is positioned to one side of the deck resulting in some hogging or transverse sagging moments.

In reality, concrete structures will crack under heavy loads resulting in a change in the stiffness of the slab. Even when the ultimate moment capacity of a section of the deck is exceeded loads will be redistributed elsewhere in the slab provided sufficient ductility is available and it does not fail prematurely in shear. As a result, a linear elastic analysis will not accurately model the distribution of stresses or the actual behaviour in the post-elastic range where non-linear effects dominate. Elastic methods can be very conservative since failure of one element in the structure is typically used to define failure of the structure as a whole. In the cases of flexural failure, the consequences are likely to be small and may only affect the serviceability of the structure. If one accepts that serviceability criteria do not govern and collapse is the criterion on which to base the assessment, such conservativeness is not warranted for concrete slab bridges for which ductile flexural failure is the critical mechanism of failure. Once an individual section has reached ultimate or yielded, the failure must develop into a full collapse mechanism before the structure will actually fall down.

Despite the enormous cost implications of adopting such an approach, elastic methods are still relied upon as the primary analysis tool for assessing concrete deck slabs. This is despite the fact that there is a wealth of evidence from model experiments and full-scale load tests to show that concrete bridges are often able to carry loads well in excess of the ‘theoretical’ capacity calculated using elastic techniques [4].

It is thus important to investigate the options available to an engineer if, after performing an elastic analysis, the structure still fails to comply with the required standards.

The only practical alternatives to elastic analysis would involve undertaking a more sophisticated analysis of the ultimate strength of the bridge or else carrying out load tests on the bridge itself as a means of verifying the load capacity. In the research environment, where the best possible predictive methods are sought to model the actual behaviour of bridges, researchers have, almost without exception, used yield-line theory, and in more recent years non-linear finite element methods, to predict the flexural collapse behaviour of concrete slabs and concrete bridge decks.

The question is do these provide a practical alternative to elastic analysis for assessing ultimate strength and could they be adopted widely in practice?

Non-Linear Finite Element Analysis (NLFE)

The NLFE method is more suited to in-depth, specialised assessments of major structures or for laboratory research, and is not presently considered to be a practical option for use in assessing large numbers of existing bridges. This situation could well change in the future as computing developments continuously result in decreasing costs and greater speed with NLFE programs, although the sensitivity of results, need for calibration, and specialised expertise required are still likely to limit their application.

Plastic Collapse Analysis -- The Yield-Line Method

The other analytical alternative is to use plastic collapse analysis or yield-line methods for assessing the strength of concrete bridges. Yield-line analysis considers the global collapse of a concrete slab rather than the ‘failure’ of a single element thus utilising the full, distributed strength capacity of a structure. As a result it is usually less conservative than elastic methods.

Yield-Line Analysis: The COBRAS Program

The Highways Agency in the U.K. recognised the potential for applying yield-line analysis to concrete bridge assessment and commissioned the Department of Engineering at Cambridge University to develop such a program. This project has resulted in a novel collapse analysis program called COBRAS (for COncrete BRidge ASsessment).

The solution scheme incorporated in this program is based on the realisation that the yield-line problem can be reduced to what is fundamentally a problem of geometry. Using recent developments in computer graphics and solid modelling theory, an analysis technique has been developed which creates a three-dimensional ‘picture’ of the bridge. This is used to derive all the required geometrical relationships for the failure mechanisms, whilst incorporating features describing the component material properties and the applied loads.

Perhaps the most significant feature of this modelling technique is its ability to analyse rigorously realistic configurations of loading, bridge geometry, support fixity and failure mechanisms without the need to derive mathematical expressions describing the inter-relationship between these parameters. Multi-layered, banded and curtailed reinforcement layers can be included. It is also possible to make some provision for the effects of steel corrosion and concrete deterioration. The rotation capacity of each plastic zone in the collapse mechanism is checked to provide some indication of available ductility. The COBRAS program calculates the ‘theoretical’ moment capacity of the yield-lines in each mechanism allowing for different orientations, depths and types of reinforcement that cross the each yield-line. This overcomes the need to adopt Johansen's stepped yield criterion [6] and also avoids any necessity to use the affinity theorems [6] to account for orthotropic reinforcement layouts.

The user selects from a pre-defined library failure mechanisms (currently 18) and the program then iterates through a large number of possible geometries for each mechanism in search of the lowest, and hence critical, failure mode. As a result, structures that were hitherto impractical to assess by hand can now be analysed automatically. With a modern portable computer a typical concrete bridge assessment can be performed in a couple of minutes.

Validation Of The COBRAS Yield-Line Program

To confirm the validity of the program, a number of forms of calibration were undertaken. Firstly, the program was checked against a number of published analytical solutions to confirm the correct theoretical result was obtained in each case.

Secondly, a calibration study was undertaken in collaboration with the Transport Research Laboratory's (TRL) Structural Analysis Unit to compare predictions for collapse load and failure mode geometry obtained using the COBRAS program with those obtained by the TRL's non-linear finite element program, NFES, for a number of different bridge structures under a variety of load configurations.

In this study excellent agreement between the collapse load and failure mode geometry predicted by COBRAS and NFES was obtained for all the bridges analysed. The difference in predicted ultimate capacity was less than 4% in all the examples assessed except for four specific cases where a conservative assumption about the strength of edge-beams in the COBRAS program resulted in a maximum of 13% underestimation of the collapse load [7].

Thirdly, the program was used to predict the failure mode and collapse load for a number of experimental tests on concrete slabs. Although there have been numerous tests over the years to verify yield-line theory [5], a series of tests was conducted at Cambridge University specifically aimed at validating the theoretical predictions of collapse load obtained using the COBRAS program. To date, a total of 13 different tests have been carried out as part of an ongoing validation programme. The model slabs were scaled at approximately 1/10th the size of a full-scale bridge in Scotland that had been tested to destruction by the TRL in 1992 [8]. This resulted in the model slabs being nominally 600 mm in length by 1000 mm wide and 40 mm thick. In one set of tests the slabs were skewed at 30 degrees, and in another the slabs were widened to 1500 mm to examine failure mechanisms contained wholly within the central region of the slab. Various reinforcement configurations were considered, with and without transverse and top steel, and with varying percentages of each.

Truck loading was simulated using a two-wheel axle load which was applied at mid-span in all but one of the 13 tests. In the exception a solitary point load was used. The goal was to force the model structures to fail in some form of complex fan mechanism rather than just a full-width transverse yield-line at mid-span (which is often found to be critical under the uniformly distributed HA lane load pattern that is specified in the assessment code). Such a fan mechanism puts a greater demand on the ductility of the slab as well as on the predictive capabilities of the computer program.

Results from the Cambridge model tests

The results from these model tests are shown in Table 1, which compares the failure loads predicted using the COBRAS plastic collapse program (PCOBRAS) and the actual failure loads measured in the laboratory (Ptest).

In Table 1 it can be seen that in all but the final two tests (A1 & A2), the yield-line method was conservative in predicting the capacity of the model slabs. The mean value of the ratio of the measured failure load to the predicted load was 1.13, with standard deviation 0.13. Values ranged from 0.87 in test A2 to 1.33 in test C2, with the range being between 1.04 and 1.33 for the first 11 tests. By way of example, figures 1(a) and 1(d) show the failure mechanism patterns obtained in two of these tests (K4 & C3) and figures 1(b) and 1(c) show the corresponding critical yield-line pattern predicted using COBRAS overlaid on the observed soffit crack pattern. (Slab K4 was tested twice – once with a single axle load at each side of the slab).

Examination of the specimens in tests A1 and A2 after failure suggest that a breakdown in the bond between the concrete and the smooth, shiny 4 mm diameter bars used to reinforce the slab may have caused these two lower than expected results. This hypothesis was in fact subsequently confirmed when further tests using various degrees of bond between bar and concrete showed that ductile, under-reinforced behaviour could be correctly predicted when sufficient bond was provided to avoid the premature slippage observed in tests A1 and A2.

An important observation in all these experimental tests in the laboratory was that substantial deformation and cracking developed well before the maximum load capacity was reached. Thus if a structure has been in service for many years and there is no visual evidence of distress the assessing engineer can be reasonably confident that the structure is capable of sustaining significantly higher loads than those already experienced by the structure. Clearly this does not mean the structure is necessarily capable of sustaining the full 40 tonne load, as it may never have been subjected to loads near the maximum legal limit. However it does give some reassurance to the assessor that collapse is not imminent at the loads to which the structure has already been subjected.

Fundamental to this statement is the assumption that the critical failure mode will be flexural and the structure is sufficiently ductile to allow such a mechanism to form. Shear failures may occur in a brittle manner and may not give warning of impending failure (although a recent test programme at Cambridge on shear in beam and slab bridges has indicated that significant cracking usually precedes shear failure at loads well below the ultimate collapse load [5].)

Yield-Line Analayis For Concrete Bridge Assessment

Since 1994, a number of bridge authorities and consultants have commissioned Cambridge University Technical Services Limited (C.U.T.S.) to re-assess concrete bridges that had failed their initial assessment. Over twenty such bridges have now been re-analysed using the COBRAS program. In each case, the bridge was first assessed using conventional elastic analysis procedures (or in a few cases simple hand yield-line analysis) and found to have inadequate ultimate flexural load carrying capacity.

All were reinforced concrete slabs ranging in span from 2.5m to 13.1m. They carry roads ranging from small country lanes to the M1 motorway. Most of the structures were simply supported, although three were continuous structures (bridge no. 7,14 & 21 in Table 2).

In each case, the capacity of the structure under the various assessment load configurations specified in the relevant code at the time, BD21/93, was evaluated for the full range of mechanism shapes in the yield-line program library.

Results of assessments undertaken on behalf of C.U.T.S.

The collapse analysis results obtained using the COBRAS program are shown in table 2 and compared with the results obtained in the initial assessment by the bridge authorities and consultants using predominantly elastic analysis methods.

It can be seen in table 2 that all but three of the 21 bridges listed had been assessed initially by the authorities at less than 17 tonne capacity, with many rated at 7.5 tonne or less. By comparison, using plastic analysis, 15 (70%) of the bridges were passed at the required 40 tonne level and two others at the current limit of 38 tonnes.

In many cases the low elastic ratings were due to elastic moments exceeding the transverse section capacities of bridges with low reinforcement percentages in the transverse direction and/or no top steel.

In a few cases the bridge owning authorities or consultants attempted hand yield-line calculations. It was found that, in general, the simplifying assumptions made to enable hand calculations to be undertaken resulted in a significantly lower estimate of the collapse load than was determined using the more rigorous computerised technique in the COBRAS program.

The ability to examine a wide range of potential failure modes other than those typically considered in simple hand yield-line assessments is a very important feature of the methodology that is encompassed in the COBRAS program. In one case (bridge no.4 in Table 2), the initial elastic analysis considered flexural capacity at midspan to be critical resulting in a rating of 16 tonnes. A yield-line analysis of a full-width transverse mode also located at midspan indicated that the bridge would in fact pass at the full 40 tonne load level. However, by checking a large number of possible failure mechanism geometries, the program identified a zone of potential weakness at the location of steel curtailment near the abutment reducing the assessed capacity back to 16 tonnes.

Many of these bridges were narrow, two lane structures for which the critical failure mechanism under the HA assessment lane load was usually found to be a full-width transverse yield-line at midspan. However a mechanism extending only part way across the deck and then spreading out into a fan mechanism, which is modelled in the COBRAS program by a fan comprising 6 wedges, was also found to be critical in a number of cases. Many of these bridges were also assessed under the abnormal heavy vehicle loadcase (HB loading) for which this partial-width fan mode was often found to be critical. It would be impractical to attempt to analyse these complex mechanism shapes and loading configurations using hand yield-line methods.

Results from County Council Trials of COBRAS Yield-line Program

As part of the development and verification procedure for the COBRAS analysis program, three county councils were invited to participate in a three-month evaluation trial of the analysis program. The aim of this trial was to obtain feedback and recommendations for modifications and/or improvements to the current program user-interface and also to obtain independent verification of the yield-line assessment results.

Each council analysed a number of bridges which had previously "failed" using elastic analysis methods at below the required 40 tonne Assessment Live Load level. The results of the re-assessments carried out using plastic yield-line analysis are presented in table 3 and show that thirteen out of the fourteen bridges examined were re-assessed at full 40tonne capacity with the other bridge being upgraded to 38 tonnes.


Overall, the use of elastic analysis methods for assessing the ultimate load capacity of concrete bridges may in many situations result in a significant under-estimate of strength. The development of the COBRAS yield-line program provides a very powerful alternative tool with which plastic collapse analyses of these bridges can be undertaken for a wide selection of possible failure modes and assessment loadcases. As an upper-bound approach, care must be used in applying this technique however there is substantial theoretical and experimental evidence to support its validity for concrete bridge decks in which sufficient ductility exists to justify the assumptions inherent in yield-line theory.

It is likely that many, if not most, of the concrete slab bridges which have failed their initial assessment due to inadequate flexural capacity, would in fact be found to pass the 40 tonne assessment requirements if yield-line analysis was used to re-assess their ultimate capacity. Twenty-eight out of thirty-five bridges (80%) condemned as unsafe and re-assessed using this approach have now been passed with a further 9% passing at 38 tonne. It is evident that this approach can result in very substantial savings to bridge owning authorities if applied in the appropriate circumstances to short-span concrete bridge slabs.


The work undertaken on the 13 Cambridge model tests described in this paper was performed by Jolyon Antill, Isaac Hudson and Steve Kite, now with Ove Arup & Partners, and Elisabeth Collins, now with Whitby & Bird, London. The support of the Highways Agency and the Transport Research Laboratory in the development of the COBRAS analysis program described here is also gratefully acknowledged. The assistance of Dr C.T.Morley and Mr P.Fidler of Cambridge University, Mr J.Wills of TRL (now retired) and Mr B.Sadka and Mr S.Chakrabarti of the Highways Agency is also gratefully acknowledged. Any views expressed are not necessarily those of the supporting organisations.

Tables and Figures

(These tables are also available preformatted)

Table 1: Predicted versus actual ultimate loads for Cambridge model bridge slabs

  Hudson tests Kite tests Collins tests Antill tests
Test No. H1 H2 H3 H4 K1 K2 K3 K4 C1 C2 C3 A1 A2
Ptest (kN) 29.2 28.3 27.9 19.0 23.4 23.5 24.1 22.7 36.8 37.2 31.3 18.0 25.6
PCOBRAS (kN) 24.5 25.3 23.0 17.2 22.5 22.3 20.5 17.7 N/A 28.5 25.7 19.0 29.3
Ptest/PCOBRAS 1.19 1.12 1.21 1.10 1.04 1.05 1.18 1.28 - 1.33 1.22 0.95 0.87


Table 2: Revised load capacities of bridges re-assessed using COBRAS by the author

Bridge No. Span (m) Elastic Rating (tonne) Plastic Rating COBRAS (tonne)
1 4.85 7.5 38
2 6.50 7.5 16
3 3.89 38 40
4 5.70 16 16
5 6.55 7.5 16
6 5.70 17 40
7 3.66 7.5 40
8 3.72 38 40
9 5.51 7.5 40
10 3.71 7.5 38
11 6.60 7.5 40
12 6.60 7.5 40
13 6.50 3 40
14 13.10 7.5 40
15 6.10 7.5 40
16 5.40 7.5 40
17 2.48 17 40
18 3.29 17 40
19 7.62 38 40
20 3.00 7.5 16
21 3.06 7.5 40


Table 3: Revised load capacities of bridges re-assessed by County Council's using COBRAS

Bridge No. Span (m) Skew angle (degrees) Elastic Assessment Live Load Rating (tonne) Plastic Assessment Live Load Rating (tonne)
Bedfordshire County Council
1 3.6 5 25 40
2 5.5 3 7.5 40
3 6.8 7 7.5* 40*
4 4.9 25 17* 40*
Northumberland County Council
1 5.0 0 Group 1 FE 40*
2 5.0 0 Group 1 FE 38*
3 4.65 10.5 Group 1 FE 40*
4 3.51 0 38 40*
5 2.95 43.6 25 40*
6 10.7 0 Dead load only 40*
Cumbria County Council
1 7.33 31.5 7.5 40
2 3.81 0 7.5 40
3 7.71 14.5 25 40
4 18.5 54.2 Dead plus superimposed dead load only 40

* provisional results since not checked as yet.

Figure 1: Examples of actual & predicted failure mechanisms for concrete bridge models

Image of K4 (329 KB)
a) Test slab K4 – actual crack pattern

Image of predicted K4 (301 KB)
b) Test slab K4 – predicted yield-line pattern

Image of C3 (317 KB)
c) Test slab C3 – actual crack pattern

Image of predicted C3 (529 KB)
d) Test slab C3 – predicted yield-line pattern



  1. Kirkpatrick,J.: Personal communication, Roads Service, Northern Ireland, October 1997
  2. Brodie,A.: 'Progress and issues arising from the assessment and strengthening programme in Scotland', 5fth Annual Surveyor Bridge Conference, 1997
  3. Department of Transport: 'Lorry weights - a consultation document', 1996
  4. Clark,L.A.: 'Collapse Analysis of Short-to-Medium Span Concrete Bridges', Contractor's Report CRR 528/577/124, TRRL, Crowthorne, 1984
  5. Ibell,T.J., Morley,C.T., Middleton,C.R.: 'A plasticity approach to the assessment of shear in concrete beam-and-slab bridges', Institution of Structural Engineers, 75, No.19, October 1997, pp331-338
  6. Johansen,K.W.: Yield line theory, C&CA, London, 1962
  7. Middleton,C.R.: 'Strength and safety assessment of concrete bridges', PhD Thesis, Cambridge University, Cambridge, December 1992
  8. Cullington,D.W., Beales,C.: 'Is your strengthening really necessary?', Proceedings of Conference on Bridge Modifications - Stronger and safer, London, Institution of Civil Engineers, 1994

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