Highly orthotropic slab

The automated lower bound approach routinely makes use of a linear programming algorithm for optimisation purposes. To accommodate the algorithm, a linearization of the yield criterion (Wolfensberger, 1964) is commonly undertaken, so introducing an approximation that tends to become more significant as slabs become more highly orthotropic. To test the degree of approximation involved and also to test an alternative “maximum twist” yield criterion, the highly orthotropic “bridge” type slab shown in Figure 6 has been examined (Johnson, 1999).

Figure 6: Highly orthotropic slab example

The slab shown in Figure 6 was analysed by the hand yield line approach; the automated yield line method using both a fine and a coarse mesh; and the automated lower bound method using a linearized (Wolfensberger) yield criterion and a revised “maximum twist” approximation to the yield criterion. The results of the various analyses are shown in Table 2, from which it may be seen that the automated yield line method (coarse analysis) and the automated lower bound (maximum twist yield criterion) provide acceptably close bounds on the collapse load. For this case, likely collapse mechanisms are well established and so the hand yield line analysis should also provide an acceptable collapse load estimate. This is indeed the case if the calculated value of 1152 kN is reduced by a conventional 15% to produce a revised value of 979 kN. On the other hand, the automated yield line method (fine analysis) and the automated lower bound (Wolfensberger yield criterion) result in significant over – and under-estimates, respectively, of the collapse load.

Table 2: Comparative analyses for highly orthotropic slab


Collapse Load (kN)

Hand yield line analysis


Automated yield line – fine mesh


Automated yield line – coarse mesh


Lower bound analysis (maximum twist yield criteria)


Lower bound analysis (Wolfensberger yield criteria)



• The hand approach to yield line (upper bound) analysis can lead to significant unsafe approximations due to the critical mechanism not being examined, as in the Shoemaker and practical design examples. This approach is therefore only reasonably trustworthy in the case of conventional slab layouts and loadings for which the critical yield line patterns have been reliably established.

• The automated yield line analysis will generally be more reliable in identifying critical mechanisms but it involves a somewhat cumbersome two-stage process and can involve uncertainty on the unsafe side. The practical design example, for instance, resulted in a significant discrepancy between automated upper and lower bound solutions.

• The strip (lower bound) hand method will lead to safe designs but can produce conservatism of up to 20-30% if applied to complex configurations and loadings.

• The automated lower bound method has the overriding virtue of guaranteeing a safe solution and is also a single stage analysis, without the need for manual intervention. It is perhaps slightly less intuitive than the automated yield line method, since, by its nature, it does not provide a yield line system. However, a contour plot of the associated collapse mode can be generated. The technique has been shown to produce conservative solutions for highly orthotropically reinforced slabs if the linearized Wolfensberger yield criterion is used, but this can be improved either by using a “maximum twist” linearization of the yield criterion (Johnson, 1999) or by using a formulation that employs the non-linear form of the yield criterion (Krabbenhoft et al, 2002).


Although, in favourable circumstances, the upper and lower bound approaches can be one and the same (just as Heraclitus’ roads), in the sense that the same unique collapse load can be found, the chances of going astray and becoming unsafe are substantially greater with the upper bound (yield line) approaches. The lower bound methods are definitely securer and generally quicker (just as the Traditional Song’s low road).


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Johnson, D. (1999) Lower Bound Collapse Analysis of Concrete Slabs, Proc. Concrete Communication Conference (held at the University of Cardiff, July 1999), BCA, Crowthorne, pp 299-310.

Johnson, D. (2001) On the safety of the strip method for reinforced concrete slab design, Computers and Structures, Vol. 79, pp 2425-2430.

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Ming Liu and Dan M. Frangopol

Department of Civil, Environmental and Architectural Engineering,
University of Colorado, Boulder, Colorado 80309-0428, USA
E-mail: dan. frangopol@colorado. edu


Bridge maintenance planning as a part of public policy management is not only a scientific-analytic task (Heineman, 2002), but also involves political, subjective, and sometimes other factors. Decision­makers such as administrators, managers and/or politicians are the key players in the planning process, supported by policy analysts and other technical experts. The ultimate goal of bridge maintenance planning is to find the “best” strategies and/or operational plans that are not only technically feasible, but also are considered optimal by decision makers. This can be achieved by better understanding the real-world situations, identifying all possible objectives and conflicts, evaluating as many alternatives as possible, and finally reaching rational plans. Therefore, Decision Support System (DSS) is necessary for optimal bridge maintenance planning.

This paper presents a DSS for bridge network maintenance planning that involves a group of existing highway bridges with various remaining service lifetimes. The proposed DSS considers five bridge maintenance alternatives, including “do nothing”, and the associated cost. Based on the annual bridge maintenance budget and the probabilities that each of the five maintenance alternatives may be conducted at that year on individual bridges in the network, the ultimate goal of this DSS is to find the best combination of the five maintenance alternatives applied to all bridges in the network during certain years. Since the mutual preferential independence requirement can be easily satisfied in this case, the additive form of the multiple attribute utility function can be used to establish the single­objective function for optimization with the weight assignment from the Reliability Importance Factor (RIF) of individual bridges in the network. The RIF reflects the sensitivity of the bridge network reliability in terms of the network connectivity to the change in the individual bridge system reliability due to maintenance actions. The optimization problem in the proposed DSS can be solved by either traditional mathematical programming for combinatorial optimization or the advanced heuristic search methods such as Genetic Algorithms (GAs).