PSI - Issue 84

Sebastian Thöns et al. / Procedia Structural Integrity 84 (2026) 1310–1317

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The infrastructure system is represented by two basic system states: the intact state I and the failure state F . For each state, corresponding utility components - namely benefit, cost, consequence, and environmental impact models - are assigned. In the intact state, the system remains fully operational, generating operational benefits ( ) op B t , while incurring operational expenditures ( ) op C t and carbon dioxide emissions ( ) em,.op C t . Based on Table 1 and Equ. (1) and (2), the expected utility for the intact system state at time t , ( ) 2 X : I t , can be written as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 I U op op em,.op t E E U I t P I t E B t E C t E C t r           =  + +            + (6) The expected operational benefits, costs and emissions are determined with the intact system state probability ( ) ( ) P I t and discounted with the discount rate r . The failure state gives rise to a range of direct and indirect consequences (e.g., Baker, Schubert and Faber (2008)). Direct consequences ( ) D C t include the loss of physical components, reflecting both their financial value and the embodied carbon emissions associated with the failed elements. Indirect consequences ( ) ind C t are more extensive and may involve human casualties, macroeconomic impacts such as GDP losses due to system downtime, and substantial costs related to failure investigation, demolition, debris removal, and system reconstruction. In addition, failure events result in further carbon emissions associated with these recovery and restoration activities. The expected utility for the failure system state 1 X : F can be written analogously: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 F U D ind t E E U F t P F t E C t E C t r     =  +              + (7) Direct consequences include the loss of the failed bridge components, calculated with the present monetary values ( ) Val C t . Additionally, the failed components result in the loss of the embedded production and construction carbon emissions ( ( ) em,prod E C t     and ( ) em,const E C t     , respectively):   ( ) ( ) ( ) D Val em,prod em,const E C E C t E C t E C t   = + +           (8) Indirect consequences, i.e., follow-up consequences in case of failure, include the investment costs for a new infrastructure ( ( ) Inv,new C t ), economic losses ( ( ) loss C t ) and carbon dioxide emissions ( ) em C t representing the environmental impact (Equ.(9)). ( ) ( ) ( ) ( ) Ind Inv,new loss em E C t E C t E C t E C t = + +                 (9) The losses (Equ. (10)) may involve human casualties among those using or working near the bridge at the time of collapse t ( loss,cas C ). Given a specific size and importance of a traffic infrastructure, there may also be a potential drop in GDP due to disruptions in operations, traffic detours, and delays that affect economic activities ( loss,GDP C , see also Stewart, Thöns and Beck (2025)). Furthermore, there are substantial costs associated with investigating the collapse, demolishing the remaining structure, and cleaning up the site ( ( ) decomm C t ). These costs are in the order of approx. 10% of the lost value according to Mattsson, Sundquist and Stenbeck (2008). ( ) ( ) ( ) ( ) loss loss,cas loss,GDP decomm E C t E C t E C t E C t = + +                 (10) Finally, emissions are generated during the decommissioning ( em,dec C ) and rebuilding processes ( em,prod ,new C and em,const ,new C : ( ) ( ) ( ) ( ) em em,dec em,prod ,new em,const ,new E C t E C t E C t E C t   = + +               (11)