Stakeholders of civil infrastructures have to usually choose among several design alternatives in order to select a final design representing the best tradeoff between safety and economy, in a lifecycle perspective. In this framework, the paper proposes an automated procedure for the estimation of lifecycle repair costs of different bridge design solutions. The procedure provides the levels of safety locally guaranteed by the selected design solution and the related total lifecycle cost. The method is based on the finite element modeling of the bridge and uses design traffic models as suggested by international technical standards. Both the global behavior and the transversal cross section of the bridge are analyzed in order to provide local reliability indexes. Several parameters involved in the design, such as geometry and loads and materials’ characteristics, are considered as uncertain. Degradation models are adopted for steel carpentry and rebars. The application of the procedure to a road bridge case study shows its potential in providing local safety levels for different limit states over the entire lifetime of the bridge and the lifecycle cost of the infrastructure, highlighting the importance of the local character of the lifecycle cost analysis.
Road infrastructures represent an important public asset and require large public expenditure for their construction and maintenance. Several failures of bridges all over the world demonstrate that many pieces of the road networks in western countries need strengthening or renovation [
In the traditional design method, the first most economic design alternative, providing the prescribed safety level, is selected. In the last decade, a new design approach has become popular, designated as the lifecycle cost analysis (LCCA) method, that evaluates the expected cost for the whole life cycle of the bridge, accounting for initial costs, maintenance costs and direct and indirect costs related to repair and disposal in a probabilistic framework [
The evaluation of repair costs requires the computation of failure probability that, for large structures such as road bridges, can be a very costexpensive operation. Indeed, the failure probability has to be computed for several structural elements and for numerous limit states, accounting for a large number of uncertainties and load conditions. Lifecycle cost has been used to optimize structural solutions and bridge design [
In the present paper, an automated procedure is proposed for the LCCA of road girder bridges. The lifecycle cost is related to failure probability which is locally and automatically computed over the entire bridge deck for several limit states. The procedure accounts for the main peculiarities involved in the girder bridge design. Material’s degradation and several sources of uncertainty in load and strength quantification are considered. Design traffic actions on the bridge suggested by uptodate standards are also adopted considering the interaction between deck and girder loads. Different construction phases of composite steelreinforced concrete beams are analyzed. The procedure leads to the evaluation of the timedependent reliability index and failure probability for all the considered limit states and in correspondence of each location along the deck and the girder. It also provides the expected lifecycle cost, thus allowing a quantitative comparison between different design alternatives. The main progress of the methodology with respect to the existing literature is the local character of the reliability assessment and the lifecycle cost analysis that allows to decide the priority of intervention that is strongly dependent on the structural details of the deck and on the material’s degradation model. The rest of the paper is organized as follows:
The proposed procedure is a general automated tool for the LCCA of girder bridges which accounts for the main peculiarities involved in the girder bridge design. It can take into consideration the real actions on a structure, also accounting for interaction between deck and girder loads, different construction phases of composite steelreinforced concrete beams, uncertainties in load and strength quantification, and materials’ degradation models. The procedure leads to the evaluation of the reliability index and failure probability for all the considered limit states and in correspondence of each location along the deck and the girder. It also allows for the computation of the expected lifecycle cost, thus allowing quantitative comparison between different design alternatives.
The expected lifecycle cost for a bridge can be expressed as
The initial cost
The maintenance cost
Rehabilitation or repair has to be carried out when the structure reaches a critical limit state [
The outline of the procedure is shown in Figure
Outline of the proposed automated procedure.
In the proposed procedure, as it is common in a girder bridge design at the stage of preliminary structural analysis for comparing different design alternatives, a transversal analysis of the deck is performed at first (local analysis), followed by a longitudinal analysis of the girders (global analysis). With this aim, a simplified modeling of the structure is carried out in which the structure of the bridge is ideally divided into two interconnected parts: the deck and the girder, representing its local and global behavior, respectively. The deck part is a modelization of the transverse section of the bridge, while the girder part is a modelization of the longitudinal, composite, reinforced concretesteel girder. In the preliminary design phase, when the goal is to evaluate which is the best design alternative, the use of a simplified model instead of a complex finite element model is computationally advantageous and does not lead to relevant errors in the lifecycle cost evaluation.
In the deck analysis block, the FE model of the deck is assigned as input for the structural analysis which is repeated for each possible traffic load condition and position, to provide automatically the envelope of the internal forces.
In the girder analysis block, the FE model of the longitudinal beam is built and analyzed. It is loaded with constraint reactions of the deck structure corresponding to forces acting on girders provided by the deck analysis block, in addition to the other loads acting on the bridge, such as thermal actions and transverse loads. Then, a set of uncertain parameters, such as geometric data, materials’ characteristics, and load intensities, are modeled as random variables with assigned probability distributions. Material’s degradation can be considered by varying with time the geometric data and the properties of materials’ strength. The reliability analysis is performed providing, locally in each section, the probability of failure and the corresponding reliability index for each output section of the deck and the girder. Finally, the lifecycle cost analysis is carried out by relating the probability of failure to the expected lifecycle cost of the analyzed bridge.
The procedure, built in a Matlab environment [
Courbon’s method of analysis is adopted to properly allocate traffic loads between the girders. The girders are loaded with the maximum reactions obtained from the deck analysis.
The structural response is computed through static analyses for the three typical life phases of the steelconcrete mixed structure: construction, short term, and long term. Several types of loads are considered in addition to dead and traffic loads, like thermal action, transverse wind load, and centrifugal force. Results are directly provided for a preselected number of output stations along the deck and the girder.
Without loss of generality, to compute the failure probability
The general procedure described in
Two design alternatives of the viaduct are available: the first one corresponds to the original design, while the second one is a variant proposed by the contractor, leading to a reduction of the initial investment of about one million euros.
In alternative 1, the deck is a 30 cm thick concrete slab, supported by two steel box girders. The beams are stiffened and connected to each other at the supports with a steel plate. In alternative 2, the deck is a 26 cm thick concrete slab, supported by two doubleT steel girders connected by transverse beams. The slab is designed as an equivalent beam with a plate behavior, thus guaranteeing that a thickness of 26 cm is sufficient to support the design loads. Figure
Cross sections of the two bridge design alternatives (D1 and D2).
Longitudinal sections with span length of the two bridge design alternatives (G1 and G2).
The strength class of concrete is C28/35 according to European guidelines [
In order to carry out structural and reliability analyses, finite element models of the deck and the girders are built, for both design alternatives. Separate models have been adopted for the deck and for the main girder, as described in the following sections.
The deck has been modeled as a continuous reinforced concrete beam. For the deck analysis, the girders are considered as fixed supports, located in correspondence with the girders’ axes. Figure
Schematic representation of the deck FE models in the two design alternatives. (a) Alternative 1: D1. (b) Alternative 2: D2.
Three different cross sections are assigned to the frames in D1, differing from each other in terms of slab thickness and reinforcements. Five different cross sections are assigned to the frames in D2, according the available preliminary design.
The girders are modeled as continuous beams, as shown in Figure
Schematic representation of the girder FE models in the two design alternatives. (a) Alternative 1: G1. (b) Alternative 2: G2.
Stiffness properties have been calculated considering the different life phases of the steelconcrete composite structure:
Phase1 (P1): the construction phase in which steel girders only contribute to resistance; loads due to girders and deck selfweight are applied
Phase2 (P2): the shortterm phase in which the concrete deck contributes to section resistance (modular ratio
Phase3 (P3): the longterm phase in which the concrete deck contributes to section resistance (modular ratio
Therefore, for each model (G1 and G2), three different model variants are considered, corresponding to the first, the second, and the third phases (G11, G12, G13, G21, G22, and G23). The effect of shrinkage is neglected, as shrinkagereducing admixtures are used for the deck concrete.
To properly describe the probabilistic nature of traffic loads, the load cases (selfweight, dead, and variable loads) have been treated separately.
Selfweights correspond to the weight of the reinforced concrete slabs and the steel girders. The dead loads include the weight of the pavement, the guard rail, and the parapet. The first one is a uniformly distributed load, while the other ones are concentrated loads. Variable loads are traffic loads and thermal fluctuation. The seismic action has not been considered at this stage of the work.
According to European guidelines [
Road traffic loads:
Load model 1 (LM1): concentrated and uniformly distributed loads, which cover most of the effects of the traffic of lorries and cars
Load model 2 (LM2): a singleaxle load applied on specific tyrecontact areas which represents an exceptionally heavy vehicle and also the dynamic effects of the normal traffic on short structural members
Actions on footways:
Uniformly distributed load (UDL): a uniformly distributed load, equal to
Concentrated load (CL): a concentrated load on a square imprint having a 0.1 m side, equal to 10 kN, for local verifications.
In LM1, the roadway is divided into 3 equivalent lanes, 3 m wide. Any equivalent lane contains a distributed load acting over the whole lane width and two distributed loads, representing wheels of heavy vehicles, acting in a 0.4 m side square imprint to be diffused till half the deck thickness. LM2 consists of two 200 kN loads, on footprints of 0.6 m × 0.35 m, 2 m spaced, to be diffused till half the deck thickness. The worst load position along the roadway has to be considered. Values of lane loads, heavy vehicles wheel loads, number of equivalent lanes, and any other details on load models are given in [
For the deck analysis, the selfweight, the dead loads, and the load models LM1, LM2, UDL, and CL are considered. Figure
Schematic representation of the applied loads for the deck analysis. (a) Alternative 1: D1. (b) Alternative 2: D2.
Figure
Schematic representation of the applied loads for the girder analysis. (a) Alternative 1: G1. (b) Alternative 2: G2.
With the aim of computing failure probabilities, it is necessary to consider the probabilistic nature of strength properties and loads and the uncertainties in geometrical properties of the members. Without loss of generality of the procedure, in order to comply with the HasoferLind hypothesis (
For some quantities, a degradation model is adopted to reproduce the progressive decrease of mechanical characteristics due to damage. In this application, in order to account for corrosion mechanisms, the damage is considered to affect the steel rebars’ yield strength (
It is assumed that
Degradation model as a function of time (
Table
Random variables used in the case study.
No.  Variable  Symbol  Mean value  CoV 

1  Steel bar yield strength 

570 MPa  0.052 
2  Steel bar area 

Nominal value  0.020 
3  Steel girder yield strength 

381.26 MPa  0.070 
4  Deviation from concrete cover measure 

1 cm  1.000 
5  Concrete compression strength 

36.55 MPa  0.180 
6  Internal forces due to deck selfweight 

Nominal values  0.010 
7  Internal forces due to deck dead loads 

Nominal values  0.025 
8  Internal forces due to deck traffic loads 

Nominal values  0.200 
9  Internal forces due to girder selfweight 

Nominal value  0.080 
10  Internal forces due to girder dead loads 

Nominal value  0.025 
11  Internal forces and deflection due to global traffic loads 

Nominal values  0.200 
12  Internal forces and deflection due to wind 

Nominal values  0.200 
13  Internal forces and deflection due to thermal action 

Nominal values  0.200 
Since the loads vary around nominal (mean) values, the internal forces obtained through the structural analysis vary accordingly. The envelopes of the deformed shape, of the shear forces, and of the bending moments, obtained by varying the positions of the traffic loads along the deck and the girder, are computed from the FE models for each load case and for each one of the three construction phases. Figure
Envelopes of the bending moment and shear force in the deck for the two design alternatives (D1 and D2).
The achievement of the ultimate bending moment and the ultimate shear force are the limit states taken into account for the reliability analysis of the deck.
The limit state function for bending is computed in any section of the deck as follows:
The limit state function for shear, is computed adopting the shear strength for concrete beams without stirrups, taken from the Eurocode 2 [
For girders analysis, three limit states have been considered: ultimate bending, ultimate shear (sheartorsion for model G1), and excessive deflection in longterm conditions (P3).
Bending strength corresponds to yield deformation of steel sections. The corresponding limit state equation can be written as follows:
Shear limit state for girders can be expressed as follows:
Deformability limit state concerns deflections due to variable load that in long terms leads to road surface damage. The limit state equation for deformability is the following:
Total expected lifecycle cost is computed according to equations (
The initial cost
Initial costs of the two design alternatives for the deck.
Cost item  Cost in alternative 1 ( 
Cost in alternative 2 ( 

Construction  695.509  1.011.634 
Safety burdens  52.017  102.348 
Terrain purchase  12.378  12.378 
Testing  20.865  30.349 
Design  48.685  70.814 
Total initial cost 
829.454  1.227.523 
Initial costs of the two design alternatives for the girders.
Cost item  Cost in alternative 1 ( 
Cost in alternative 2 ( 

Construction  4.673.660  3.331.431 
Safety burdens  349.548  337.046 
Terrain purchase  83.182  83.182 
Testing  140.209  99.942 
Design  327.156  233.200 
Total initial cost  5.573.755  4.084.801 
For the computation of the repair costs, only direct costs are considered, as it is assumed that indirect repair costs are almost the same in the two design alternatives.
Expected values of direct repair costs
The cost
The reliability index is computed according to the FORM method, as illustrated in Section
With reference to the deck model,
Reliability index
Regarding the girder model,
Reliability index
Table
Results of the reliability analysis for the two design alternatives.
Limit state  D1  D2  









Bending deck  12.39 

4.75  8.08 

3.75 
Shear deck  5.10 

4.75  5.02 

9.78 


Limit state  G1  G2  









Bending girder  6.88 

491.00  5.48 

494.66 
Shear girder  12.74 

345.30  8.06 

539.64 
Deformability girder  11.81 

378.60  10.50 

418.94 
In order to highlight the effect of considering the materials’ degradation, the reliability analysis has been repeated by taking into account the timedependent random variables, as in equations (
Reliability index for the deck for different lifetimes.
Reliability index for the girder for different lifetimes.
Minimum
Limit state 


 
D1  D2  D1  D2  D1  D2  


Bending deck  12.39  8.08  10.27  6.16  6.14  2.41 
Shear deck  5.10  5.02  5.04  4.95  4.83  4.75 


Limit state  G1  G2  G1  G2  G1  G2 


Bending girder  6.87  5.48  5.80  4.46  3.72  2.30 
Shear girder  12.74  8.06  11.34  6.81  8.68  4.43 
Deformability girder  11.81  10.50  10.16  8.94  7.02  5.96 
In this Section, the total expected lifecycle cost is computed trough the proposed procedure, based on the results of the reliability analysis. Maintenance costs are neglected, as they do not vary significantly in the two design alternatives. Therefore, without loss of generality, equation (
The timedependent probabilities of failure
Expected repair costs over a lifetime for the deck (a) and the girder (b), for the two design alternatives.
(a) Expected total costs of the two design alternatives compared with the corresponding initial costs; (b) total cost increase with respect to initial cost in design alternative 1 and (c) and total cost increase with respect to initial cost in design alternative 2.
In order to study the influence of the degradation model described in equations (
Moreover, to account for the fact that corrosion can have a different effect on the steel area reduction
Numerical results reported in Table
Minimum
Limit state  DM1  DM2  DM3  DM4  DM5  
D1  D2  D1  D2  D1  D2  D1  D2  D1  D2  


Bending deck  8.69  4.72  10.27  6.16  11.20  6.84  11.41  7.19  10.99  6.73 
Shear deck  4.97  4.89  5.04  4.95  5.07  4.98  5.08  4.99  5.04  4.95 


Limit state  G1  G2  G1  G2  G1  G2  G1  G2  G1  G2 


Bending girder  5.00  3.62  5.80  4.46  6.27  4.83  6.38  5.02  6.34  5.01 
Shear girder  10.32  5.89  11.34  6.81  11.95  7.25  12.09  7.48  12.09  7.48 
Deformability girder  8.95  7.79  10.16  8.94  10.88  9.48  11.04  9.77  11.04  9.77 
Expected repair costs for different deterioration models and for
The present paper has proposed a procedure for the lifecycle cost analysis of road girder bridges. The lifecycle cost is related to the failure probability which is locally and automatically computed along the entire bridge deck for several limit states. The procedure accounts in a simplified way for material degradation by introducing the dependency on time of several damagedependent parameters. Load and strength parameters are considered uncertain with a normal probability distribution. Design traffic actions on the bridge suggested by international technical standards are also adopted considering the interaction between deck and girder loads. Different construction phases of composite steelreinforced concrete beams are analyzed. Reliability analysis is carried out for all the considered limit states and in correspondence of each location along the deck and the girder, to provide local values of the reliability index and failure probability. The procedure is applied to a case study of a road bridge for which, in the design phase, two different design alternatives were proposed. Results show that the reliability index is strongly dependent on the considered limit state and on the structural details of the deck, thus emphasizing the importance of the local reliability assessment, in order to estimate the priority and timing of interventions. The computation of the expected lifecycle cost allows a quantitative comparison between the two different design alternatives in a lifecycle perspective. The choice of the most economic solution strongly depends of the lifetime expected for the structure and on the degradation model adopted to account for material’s deterioration.
The numerical data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.