# Quantifying the cost of suboptimal decisions in daily structural design

## Context

To date the structural built environment is developed following design standards, which contain safety rules that support daily structural engineering decision making based on simple calculus. The major objective that has been followed in their development, was the provision of sufficient safety, and the observed relative low failure rates do proof success in this regard. However, optimization principles, crucial for sustainable development, have only marginally been considered in the establishment of these rules. For assuring an optimal structural material use with minimised environmental impact, well defined adjustments to the decision strategies and rules for structural design are imperative. Conveniently, such developments should be based on risk-informed approaches.

## Objective function

Given its ability to explicitly account not only for uncertainties and consequences, but also for associated safety costs, risk analysis has been identified as a powerful approach for consistent decision making related to the design of structures. Risk-informed approaches facilitate optimized structural design such that a balance is achieved between the costs and benefits associated with a specific structural engineering decision. If the benefits may be considered independent of the decision parameter p (e.g. a cross-section dimension), the optimization problem simplifies to the determination of p = popt that minimizes the expected total costs Ctot associated with a specific structural engineering decision, as expressed by Eq.(1) and illustrated in Fig.1a. The total cost Ctot is constituted by the sum of the expected safety- (Cs) and failure- (Cf ) cost. The former are defined as the sum of the construction cost, Cc and the expected obsolescence cost, Co. The latter are function of the structural failure probability Pf , which depends on p (Fig.1b). Note that the optimum decision popt should be consistent with societal preferences in regard to life safety, what can be assured by imposing a corresponding acceptance criterion pf,acc,LS (Fig. 1b), e.g. based on the marginal life saving principle.

Fig. 1: a) Illustration of the cost optimization principle and of deviations δCtot and δp of a specific design solution from the minimum total cost (Ctot,min) and optimal decision parameter (popt), respectively; b) Illustration of relationship between failure probability (Pf) and p and of the acceptable range for popt limited by life-safety (pacc,LS) requirement.

popt = arg min [Ctot] = arg min [Cc + Cf + Co] (1)

## A case study

An economically optimum design of steel beams in office buildings based on Eq. (1) is performed and the results compared to a Eurocode (EC) Partial Factor design [1]. Simply supported members are assumed, which provide support to one-way RC slabs (Fig. 2). Bending failure at mid-span of the beams is investigated. Decision parameter p corresponds to the beam’s material volume V.

Fig. 2: : Sketch of the analysed floor system consisting of steel beams supporting RC slabs

The saving potential of the optimisation is illustrated in Fig. 3 depending on the ratio of variable to total loads aq (high aq indicate dominant variable loads / light structures). Fig. 3a reveals differences δCtot (Fig.1a) between total costs for, respectively, the optimised and the EC solution of up to 10%. Similarly, Fig.3b shows potential reductions δV (=δp) in material volume/emissions at the design stage of up to 12%. However, due to the large sensitivity of the Eurocode reliability level to aq (Fig. 3c), the savings diminish with increasing realisation

Fig. 3: Saving potential of the optimisation δ: a) expected total costs Ctot and b) material volume V at design stage; c) reliability index (Tref=1y)

This limitation can be overcome if the objective of the optimisation consists in minimizing expected carbon emissions (CE) instead of costs, see Eq. (2). In this case, total emission savings up to 23% (Fig. 4a) and reductions of material amount/emissions at the design stage up to 27% (Fig.4b) are obtained. However, since the corresponding high failure probabilities (Fig. 4c) are likely to exceed the human safety acceptance criterion, pf,acc,LS (Fig. 1b), effective reductions are somewhat lower.

popt = arg min [CEtot] = arg min [CEc + CEf + CEo] (2)

Fig. 4: Saving potential of the optimisation δ: a) expected total emissions CEtot and b) material volume V at design stage; c) reliability index (Tref=1y)

## Need for risk-informed calibration

The shown results underline the suboptimality of everyday decision making in structural design with respect to the resource-saving risk-based approach. However, the inherent complexity of risk-informed design hampers its wide application in practice. One way to deal with this problem is to use such approaches as a basis for the calibration of the standardised decision rules, such as the Partial Factor method. Under consideration of both economic and environmental sustainability objectives, and taking due account of human safety boundaries, the project CoDe-S will conclude with such a calibration.