The dimensioning of overhang slabs in bridge decks is usually based on simplified, thus conservative methods. The resulting over-dimensioned overhang bridge slabs can also affect the design of the girders. In this paper, an optimization procedure for the design of this structural element is presented. The aim is to minimize investment cost and global warming potential in the material production stage simultaneously while fulfilling all safety requirements. The design variables used in this study are the thicknesses of the overhang slab and the height of the edge beam. However, a complete detailed design of reinforcement is performed as well. Both a single-objective and a multi-objective formulation of the nonlinear problem are presented and handled with two well-known optimization algorithms: pattern search and genetic algorithm. The procedure is applied to a case study, which is a bridge in Sweden designed in 2013. One single solution minimizing both objective functions is found and leads to savings in investment cost and CO2-equivalent emissions of 4.2% and 9.3%, respectively. The optimization procedure is then applied to slab free lengths between 1 and 3 m. The outcome is a graph showing the optimal slab thicknesses for each slab length to be used by designers in the early design stage.
Notes
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1 Introduction
The typical cross section of a beam bridge deck can be divided into three main parts: one or more girders, one slab and two edge beams. The girders are typically made of steel or reinforced concrete (RC) and can have varying dimensions along the bridge to guarantee enough resistance in the longitudinal direction. A RC slab is placed on top of the girders or is cast together with them. In the case of concrete slabs which are wider than the girders, the two most external portions are uniquely constrained on one side each, therefore they are called overhang bridge slabs or cantilever bridge slabs. In this paper, overhang bridge slab is used to refer to this component of the deck. Finally, the edge beams are connected to the external edges of this slab (Figure 1).
Fig. 1
Overhang bridge slab and edge beam
×
The dimensions of the overhang slab and the edge beams are usually kept constant along the bridge and chosen based on the engineer experience and rules of thumb. During the design, these elements are not considered to contribute much to the longitudinal resistance of the deck. However, they affect the deck self-weight. The reinforcement in these members is designed in order to resist the actions from groups of concentrated axle loads acting eccentrically on the slab. Such analysis can be performed on different levels of modelling sophistication (Plos et al. 2016). Simplified methods for calculating the design forces are normally used by engineers (Veganzones Muñoz 2016).
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These methods permit a fast and simple analysis of the overhang slab, but they are too conservative (Plos et al. 2016 and Shu et al. 2019). Several studies have been performed to estimate the actual load bearing capacity of overhang slabs. In the work by Vaz Rodrigues et al. (2008), six specimens with thicknesses resembling those of actual overhang deck slabs of bridges have been subjected to concentrated loads to determine whether failure is developed in shear or bending. Hoonhee et al. (2010) performed an experimental study on ten specimens to assess the punching and fatigue behaviour of prestressed concrete slabs. Moreover, the load capacity of overhang deck slabs in composite box girders is proved to be enhanced by the compressive membrane action (Amir 2014, Keyvani et al. 2014, Belletti et al. 2015 and Zhu et al. 2020) and the redistribution of shear forces (Shu et al. 2019). However, modelling these effects in the design stage would be too complex and it is supported only by a few Codes (UK Highways Agency BD 81/02, 2002, Transit New Zealand Ararau Aotearoa 2003 and CAN/CSA-S6-06, 2006). The use of simplified conservative methods for the design of overhang slabs leads to over-dimensioned elements. The common use of preassigned dimensions instead of a proper design of this element due to its limited contribution to the resistance of the superstructure in the longitudinal direction cause a further over-dimensioning. The result is unnecessary self-weight that must be borne by the rest of the superstructure and the entire bridge. In turn, both the cost and the environmental impact of the entire bridge are affected.
In the last decades, researchers have focused a lot on the size optimization of bridge decks to reduce CO2 emissions and the cost of the structure. Several optimization algorithms and techniques have been applied to optimize prestressed concrete (Sirca and Adeli 2005, Rana et al. 2013, Yepes et al. 2015, García-Segura and Yepez, 2016 and Kaveh et al. 2016), reinforced concrete (Akin and Saka 2010, Jahjouh et al. 2013 and Khouri Chalouhi et al. 2019) and composite decks (Pedro et al. 2017 and Orcesi et al. 2018). However, most studies focus the optimization process on the longitudinal design of the deck and perform a posterior assessment of the overhang slab. While some research about optimization of RC or prestressed concrete slabs has been performed (Aldwaik and Adeli 2016; El Semelawy et al. 2012; Ahmadkhanlou and Adeli 2005), no article about the optimization of the overhang bridge slab itself has been found in the literature, to the best of the authors’ knowledge.
This work presents an optimization procedure for RC overhang bridge slabs with edge beams. By optimizing the thicknesses of the slab and the height of the edge beams, both investment cost (IC) and global warming potential (GWP measured as CO2-eq emissions) are minimized. Concerning the life cycle stages included in the calculation of GWP, a cradle-to-gate approach, which considers only the material production phase has been used. Two optimization algorithms are applied and compared in order to identify the one most suitable for the problem and guarantee the uniqueness of the solution. Conclusions about the relationship between cost-effective and environment-friendly solutions are drawn. Finally, to cover the gap between research and application of structural optimization in design offices, recommendations about the optimal slab dimensions are presented. To produce the results presented in this paper, a software for the automated design and optimization of the studied structure has been developed. The entire application runs on MATLAB® except for the structural analysis that is performed using Abaqus FEA, a commercial FEM software. The communication between the main MATLAB® code and Abaqus FEA is implemented using Python as coding language.
2 Optimization problem
The optimization problem studied in this paper is highly nonlinear and complex. The following formulation (Griva et al. 2009) is used to describe the problem:
The design variables of the current problem represent the dimensions of structural members and are treated as continuous variables. Constraints can be linear and nonlinear. Since in general, nonlinear constrained optimization problems are more difficult to handle than unconstrained ones, the penalty method (Griva et al. 2009) is used. This consists in solving an equivalent unconstrained problem where each objective function fi(x) is replaced by a penalized objective function as in Eq. (1).
The penalty coefficients μj can be assumed constant for every constraint as long as they fulfil the following conditions: μj > 0. The terms Hj[gj(x)] used in this work are defined in Eq. (3).
With this formulation, the penalty term is proportional to the magnitude of the constraint violations.
2.1 Design variables
It is possible to isolate the overhang slab from the rest of the deck without significant loss of accuracy (Bakht 1981). In this study, it is modelled as a slab, which is clamped at one edge and free at the other three edges. The concrete slab has a tapered thickness, linearly varying from the root to the free edge where the edge beam is attached (Figure 2).
Fig. 2
Overhang slab and edge beam model
×
The design variables of the studied problem are: the slab thickness at the clamped edge (tCE), the slab thickness at the free edge (tFE) and the edge beam height (hEB). The remaining dimensions in Figure 2 are treated as preassigned parameters for the reasons listed in the following. The free slab length (lc) is very much dependent on the total deck width and the design of the superstructure in the longitudinal direction. The longitudinal length of the analysed strip (lx) has to be selected in function of lc as explained in section 3.1. The edge beam width (bEB) is supposed not to play a significant role in the structural behaviour of the system compared to its height.
2.2 Objective functions
The two objective functions of this work are investment cost and CO2-eq emissions (i.e. global warming potential) of the system overhang slab-edge beam.
Investment cost includes both the material cost (MC) and the labour cost (LC) as in Eq. (4):
where e stands for element and refers to the slab and the edge beam, while m stands for material and refers to concrete, reinforcement and formwork. MC and LC are computed as in Eqs. (5) and (6).
where \({C}_{m}\) is the unit price for material m (Table 1), \({q}_{m}^{e}\) is the amount of material m in the considered element e, \({C}_{\mathrm{h}}\) is the cost of labour per hour (50 €/h), \({t}_{m}^{e}\) is the time needed to construct/install one unit of material m in element e (Table 2) and \({C}_{\mathrm{extra},m}\) is the cost for extra-work for certain materials (e.g. concrete classes with strength above C50/60 require more work that is quantified as extra 20 €/m3). Assigning specific values to constructability coefficients can be difficult since the actual time for manufacturing each element depends on non-quantifiable factors such as the expertise of workers. Values used in this work (Table 2) have been obtained interviewing experienced designers and construction estimators from companies in Sweden.1
Table 1
Unit prices for materials
Material (m)
\({\mathrm{C}}_{\mathrm{m}}\)
Unit
Concrete up to C30/37
170
€/m3
Concrete from C32/40 to C45/55
180
€/m3
Concrete from C50/60 to C80/95
200
€/m3
Reinforcement
900
€/ton
Formwork
20
€/m2
Concrete prices in the table include 20–30 km of transportation. (Solat Yavari et al. 2016)
Table 2
Time needed to construct/install one unit of material m in element e (\({{\varvec{t}}}_{{\varvec{m}}}^{{\varvec{e}}}\))
Material (m)
Element (e)
\({\mathrm{t}}_{\mathrm{m}}^{\mathrm{e}}\)
Unit
Concrete
Slab
1
h/m3
Edge beam
1
Reinforcement
Slab with constant thickness
20
h/ton
Slab with variable thickness
23
Edge beam
20
Formwork
Slab
2
h/m2
Edge beam
2
Concerning the global warming potential, the life cycle assessment (LCA) technique is used to quantify it. The applied method for life cycle impact assessment (LCIA) is ReCiPe with midpoint approach (Goedkoop et al. 2013). GWP is calculated as in Eq. (7):
where \({f}_{m}\) is the emission of kg of CO2-eq per unit of material m. Unit emissions are taken from the Ecoinvent Database v3 (Swiss Center For Life Cycle Inventories 2013) and summarized in Table 3. Values representing European average are used and the analysis is limited to one life cycle stage: material production stage.
Table 3
Emission of LCI items (Swiss Center For Life Cycle Inventories 2013)
Material (m)
\({f}_{m}\)
Unit
Concrete C28/35
295.2
kg CO2-eq/m3
Concrete C32/40
367.7
kg CO2-eq/m3
Concrete C35/45
381.0
kg CO2-eq/m3
Concrete C50/60
456.4
kg CO2-eq/m3
Reinforcement A500HW
2092
kg CO2-eq /ton
For the calculation of both IC and GWP, maintenance and end of life have not been included. Concerning maintenance, the reason is that a change in the slab thicknesses and in the edge beam height will not change the maintenance schedule, nor the related cost and CO2-eq emissions. This phase would play an important role in the comparison of solutions with different edge beam types or with and without edge beams, which is not the case in the current study. Concerning end of life, the quantification would be uncertain since technologies for recycling in 100 years will be different from today (Tromp et al. 2019).
2.3 Constraints
Eurocode 2 (CEN, 2005) requirements concerning ultimate limit state (ULS) and serviceability limit state (SLS) are the structural constraint of this optimization problem. For every set of variables, the structural analysis is performed and reinforcement is designed for bending moments (M) and shear forces (V) as explained in section 3. Serviceability limit states of stress limitation for durability (σ), crack control (w) and deflection control (u) are considered as well in the design, together with minimum and maximum amount of reinforcement defined by Eurocodes and Swedish regulations.
These structural constraints are intrinsically nonlinear, thus the penalty method is used. Once the reinforcement has been designed (including placement/location of each reinforcement bar), the possible constraint violation is quantified as in Eq. (8):
where \(\mathrm{Ed}\) and Rd represent, respectively, the action on the cross section and its resistance, fck is the characteristic compressive cylinder strength of concrete at 28 days and the maximum deflection (umax) is computed according to Eq. (9):
When defining the constraint, constructability is considered as well. Eurocode limits maximum amount of reinforcement (As) and minimum bar spacing and concrete cover. As a consequence, in order to fit at least one layer of reinforcement, the overhang slab and the edge beam (EB) have minimum thicknesses (t). Equation. (10) shows the mathematical formulation of these constraints.
While the penalty method is used to handle the constraints above, limitations on single thicknesses according to user preferences and/or national codes (Trafikverket 2011) are used to narrow down the searching space.
Finally, the linear constraints of several variables shown in Eq. (11) have to be fulfilled as well (Trafikverket 2011).
where tpavement is the thickness of the pavement. Considering x as a vector containing the variables \({t}_{\mathrm{FE}}, {t}_{\mathrm{CE}}\) and \({h}_{\mathrm{EB}}\), Eq. (11) can be rewritten in matrix form as in Eq. (12). This format is implemented in the algorithm used in this work.
The optimization algorithm takes care of the constraints in Eq. (12) while selecting the set of variables to test at each iteration.
2.4 Optimization algorithms
When selecting the optimization algorithm, types of variables and objective functions should be considered together with computational time.
Concerning the objective functions, reinforcement plays an important role but it is not one of the variables. Instead, it is the result of a design procedure which takes into account several load scenarios and verifications. As a consequence, the objective functions cannot be expressed in a closed form and derivative-based optimization algorithms cannot be used. Concerning the variables, in this problem they are the dimensions of the overhang slab and the edge beam, so they are all continuous and represent similar features.
This type of variables and objective functions can be handled by a gradient-free optimization algorithm such as Pattern Search (PS), which is a deterministic local search algorithm employed in similar problems in the literature (Surtees and Tordoff 1981, Yavari et al. 2016 and 2017 and Khouri Chalouhi et al. 2019). The nature of this algorithm makes it sensitive to the starting point; moreover, it might get stuck in a local minimum. A way to avoid these problems is to use an algorithm that combines local and global search using randomization, such as Genetic Algorithm (GA). It is another well-known algorithm in the structural optimization field (Camp et al. 2002, Govindaraj and Ramasamy 2005, Srinivas and Ramanjaneyulu 2007 and Khouri Chalouhi et al. 2019). Both algorithms have been used in this study for the single-objective formulation of the problem and their performances have been compared. For the multi-objective formulation, only GA has been used since PS cannot handle such problems.
Finally, in order to account for the dimension accuracy during construction and speed up the search for the solution without handling discrete variables, a memory system as in the work by Khouri Chalouhi et al. (2019) has been integrated together with the customized stopping criteria for GA presented in that paper.
3 Structural analysis and reinforcement design
When designing the overhang slab, one common approach in design offices is to model this part as a slab with varying thickness, free in correspondence of the edge beam and clamped at the opposite edge (Figure 2). When calculating the internal forces, the effect of the edge beam is accounted for by removing the edge beam and extending the slab. The length of the added portion is computed in such a way that it has the same flexural rigidity of the edge beam (Veganzones Muñoz 2016). The flexural moment at the clamped edge is computed using approximated methods such as the influence surfaces proposed by Homberg and Ropers (1965). Shear forces, instead, are computed in the most critical cross sections considering equilibrium. This approach is conservative and does not consider the beneficial effect of the edge beam, which contribute in distributing the load along the slab. A still conservative but more accurate alternative is to create a 3D FE model of the overhang slab and the edge beam and use it for the structural analysis. By using shell elements for the slab and beam elements for the edge beam, it is possible to capture the different behaviours of the two elements and take into account the structural contribution of the edge beam.
3.1 Model
The work here presented follows the second approach, i.e. 3D FE model with shell and beam elements. The FEM software used in this study is Abaqus (Simulia 2016). The model consists of a rectangular slab clamped at one edge and free at the others (Mufti et
al. 1993 and Pacoste et al. 2012). The edge beam is attached to the edge opposite to the clamped one. The concrete slab has a
tapered thickness, linearly varying from the root to the free edge. Hence, the geometry of the model entirely depends on the
longitudinal length of the strip, the free slab length, the thicknesses both at the clamped edge and at the free edge, and the edge
beam width and height (Figure 2).
Instead of studying the entire overhang slab of the bridge, only a portion of length l is considered. For this model to reproduce the behaviour of the entire overhang slab, l has to be properly defined. Figure 3 shows the relation between the minimum longitudinal length of the model and the slab length obtained with a convergence study on l (Zelmanovitz Ciulla and García-Brioles Bueno 2018). Note that the range of lengths in the figure is larger than that of realistic bridge deck overhang slabs. Equation. (13) approximates the curve in Figure 3.
Minimum longitudinal length (lx) of the model as a function of the free slab length (lc)
×
Concerning the element type, to deal with the diverse behaviours of the overhang slab and the edge beam, two different element types are used. A 4-node doubly curved thin or thick shell with reduced integration (S4R) is chosen to model the slab, whereas 2-node linear beam elements (B31) describe more accurately the response of the edge beam. The connection between the free edge of the overhang slab and edge beam is performed by tie constraints. Linear elastic analysis is adopted and, to account for the conservative results obtained by this approach, distribution widths recommended by Pacoste et al. (2012) are considered.
The advantage of this model lies in its simplicity since it is not necessary to consider the whole length of the bridge. Such analysis would be necessary for short span bridges where the distance between the abutments is less than lx in Eq. (14). However, for very short spans, other type of bridges, such as slab-frame bridges, are preferred over beam bridges with overhang slabs.
3.2 Loads
The loads considered in the analysis are: self-weight of the structure and the pavement, Load Model 1 (LM1) and Load Model 2 (LM2) (Figure 4) defined by Eurocode 1, sections 4.3.2. and 4.3.3 (CEN, 2003). For the ultimate limit state, the less favourable of the load combinations in Eq. 6.10a and 6.10b in section 6.4.3.2 of Eurocode 0 (CEN, 2005) is considered. For the serviceability limit state, the load combination in Eq. 6.16b in section 6.5.3 of Eurocode 0 is considered instead. Concerning load scenarios, three cases are considered both for LM1 and LM2:
Case 1: Vehicle travelling along the outmost part of the overhang slab, at a minimum distance of 24 cm from the inner side of the edge beam. This design case might give rise to the highest bending moment at the root of the overhang slab.
Case 2: Vehicle travelling at distance d0 (effective depth of the cross section at the root) from the clamped edge. This scenario might give rise to the highest values of shear forces on the deck. Loads even closer to the root of the overhang slab are transferred to the longitudinal system and thus do not contribute to the shear in the overhang slab. However, they contribute to the bending moment at the root of the overhang slab.
Case 3: Vehicle travelling at one metre from the free edge and no uniformly distributed load due to traffic is acting within one metre from the edge beam. This case represents the traffic during replacement of the edge beam.
Fig. 4
a and b Possible location of the tandem system for load model 1 (LM1), c load model 2 (LM2)
×
Since LM1 considers a vehicle which is a tandem system (TS) consisting of double-axle concentrated loads, two possible placement in the longitudinal directions of the axles are considered for the cases listed above. In the first instance, the axles are placed at equal distance from the symmetrical axis of the model; in the second, one of the axle is on top of the symmetrical axis (Figure 4). For every considered load scenario, the flexural moment is computed at the clamped edge. Shear forces, instead, are computed in the most critical cross sections according to Swedish regulations (Trafikverket 2011).
3.3 Reinforcement design
The results of the structural analysis in ULS are used to design the reinforcement, which is also verified according to SLS for stress limitation, crack control and deflection control. In the current practice, shear reinforcement in the overhang slab is avoided. Therefore, bending reinforcement is calculated both for the moment at the clamped edge and to provide enough shear resistance in the absence of stirrups. The output of the design procedure is not the theoretical total area of reinforcement, but the specific bar layout as shown in Figure 5. Reinforcement for the slab consists in a grid of bars at the top and another at the bottom. Whenever several layers of transversal bars would be necessary, bundle bars are used instead. In the following, longitudinal reinforcement will refer to that in the direction of the bridge, transversal reinforcement to that in the direction of overhang slab. The following rules have been used for the design of the transversal reinforcement at the top of the slab:
It is designed according to ULS requirements,
The required steel area varies along the overhang slab,
To reduce complexity during construction, the final design can contain a maximum of two groups of reinforcement bars. One group that runs all along the overhang slab and a shorter group that stops at the midpoint (i.e. lc/2).
Fig. 5
Example of reinforcement layout in the overhang slab and in the edge beam (Zelmanovitz Ciulla and García-Brioles Bueno 2018)
×
Concerning the transversal reinforcement at the bottom of the slab instead, the following rules have been followed:
It is designed according to SLS requirements and minimum reinforcement,
The bar spacing is the same as the transversal top reinforcement,
The lowest suitable commercial diameter is adopted.
Finally, the longitudinal reinforcement both at the top and bottom of the slab is the minimum required by standards. Concerning the edge beam, reinforcement is designed according to the recommendations of the Swedish standard Bro 11 (Trafikverket 2011).
4 Case study
The proposed procedure has been applied to redesign the overhang slab and the edge beam of an existing beam bridge crossing Norrtälje River, Sweden. The bridge had been designed in 2013 by the Swedish company ELU Konsult AB according to Eurocodes requirements.
4.1 The built bridge
The shape of the cross section of the deck and the elevation of the bridge are shown in Figure 6a and b. It has five spans (21.5 + 27 + 27 + 27 + 21.5 m) and three footpaths with a minimum vertical clearance of 3 m. The concrete in the superstructure is of quality C35/45 and the reinforcement is of type B500B. The deck has constant width of 10 m except for the first span from the left and part of the second one where it varies from 15.4 m to 10 m. The total height of the deck cross section is equal to 1150 mm at all mid-spans and in correspondence to the first and last supports, while it is 1650 mm in correspondence to the internal piers. The web inclination (b2/h1) is constant as well and equal to 0.38.
Fig. 6
a Deck cross section of the Bridge over Norrtälje River, b Elevation of the existing Bridge over Norrtälje River. MB movable bearing, FB fixed bearing, FP footpath (Khouri Chalouhi et al. 2019)
×
The overhang slab is constant along the bridge and has: thickness at the free edge of 200 mm, thickness at the clamped edge of 350 mm and free slab length of 2690 mm. The edge beam has a constant cross section of dimensions 400 mm × 410 mm.
4.2 Previous studies
The same bridge has been completely redesigned by Khouri Chalouhi et al. (2019). The aim was to minimize separately the investment cost and the environmental impact of the entire structure with a cradle-to-gate approach. In that study, the solutions of the two single-objective optimization problems did not differ significantly, thus suggesting that cost-effective solutions were also environmentally friendly and vice versa. Concerning size optimization of the bridge deck, the edge beam dimensions were treated as preassigned parameters, while all other dimensions including the slab thicknesses were design variables. Also in that study, to design the reinforcement, the overhang slab has been isolated from the rest of the deck and modelled as a slab clamped at one edge. However, no finite element model of this member or of the edge beam has been employed. The flexural moment at the clamped edge has been computed using the influence surfaces proposed by Homberg and Ropers (1965) taking into account the effect of the edge beam by fictitiously extending the slab as mentioned in section 3. Shear forces have been computed by equilibrium in the most critical cross sections according to Swedish regulations (Trafikverket 2011). The overhang slab of the optimal solution identified by the authors had the following dimensions: hEB = 410 mm (preassigned), tCE = 280 mm and tFE = 180 mm, which corresponds to saving of 1%, 5% and 12% in IC, GWP and, as a consequence, weight.
4.3 Preassigned parameters during the optimization
For all the results presented in the following, the slab length in the transversal direction is 2.69 m as in the built solution, while the length in the longitudinal direction is 6.05 m according to Eq. (13). The edge beam as a width of 400 mm as in the built solution.
In the FE model, S4R elements of size 6.25 cm are used for the slab, while B31 elements of size 50 cm are used for the edge beam. Both element sizes have been chosen after a convergence analysis on all results of the structural analysis for the built solution (Zelmanovitz Ciulla and García-Brioles Bueno 2018).
Concerning the objective function, the penalty factor (μ in Eq. (2)) is set to the value 1.5∙106 after performing several tests. That value is suitable for the formulation of the constraints and the magnitude of the objective functions of this study.
When Pattern Search is used as optimization algorithm, the optimal dimensions obtained in the previous study (hEB = 410 mm, tCE = 280 mm and tFE = 180 mm) are used in the current work as starting point for the optimization. The parameter called initial mesh size is set to 150 cm and the optimization process stops when the improvement in the objective function is less than 0.05 Swedish crowns (~ 0.0050 €) and the mesh size is lower than 1 cm, which is a reasonable value for the dimension accuracy during construction.
When using Genetic Algorithm, one of the individuals in the first generation has the dimensions in the paragraph above, while all the others are randomly obtained within the searching domain. When GA is used in the single-objective formulation, each population is made of 40 individuals among which 5 are elites and 70% of the remaining are so-called crossover children. The optimization process stops when one of the customized stopping criteria for GA presented in the work by Khouri Chalouhi et al. (2019) is reached. In particular, it happens to stop because at least 50% of the individuals in one generation are identical and no improvement in the objective function has been observed from the previous generation. When GA is used in the multi-objective formulation instead, each population is made of 40 individuals among which the Pareto fraction is 35% and 70% of the remaining are crossover children. In this case, the customized stopping criteria are not used and the process continues until the average relative change in the best fitness function value over 20 generations is less or equal to 0.001 (The MathWorks, 2018). Such a low value is chosen to let the process run long enough to properly search in the problem domain.
Finally, the searching domain for the design variables is the following: hEB = 400–900 mm, tCE = 170–900 mm and tFE = 170–900 mm. The upper limits have been arbitrarily chosen to be higher than commonly used dimensions, while the lower limits comply with Swedish regulations (Trafikverket 2011).
5 Results
In the following sections, results of optimization are shown. Initially, only one objective function at a time is considered and Pattern Search is used. Then, Genetic Algorithm is used instead to solve the two single-objective problems. Finally, the multi-objective problem is considered and GA is applied again.
In the comparison between optimization algorithms, no exact computational time (average of around 2 hours per optimization using PS) is shown since it depends on computer characteristics, multitasking, etc. Instead, the number of evaluated solutions before reaching the optimal is used as performance metric when two processes converge at the same result. The complete evaluation of one individual, including structural analysis, design of reinforcement, computation of material quantities and evaluation of investment cost and global warming potential, does not depend on the chosen optimization algorithm. Therefore, what makes a difference in terms of computational time is the number of individuals each algorithm has to evaluate before reaching the optimal solution. Moreover, for the comparison of the two optimization algorithms, the number of iterations is not relevant. Indeed, each iteration of GA produces 40 individuals (35 new and 5 elites from the previous generation), while PS produces only 6 individuals per iteration. For the sake of completeness, the number of iterations will be shown. Finally, due to the introduction of the memory system based on the dimension accuracy during construction, not all the solution proposed by the optimization algorithms (so-called analysed solutions) result in a new complete evaluation (so-called evaluated solutions). Therefore, both numbers are shown to measure the reduction in the computational time thanks to the memory system.
5.1 Optimal solutions of the single-objective optimization problem
Table 4 summarizes the results of the two single-objective optimization problems solved with PS. Figure 7 shows the variation in material amounts, IC and GWP during the process.
Table 4
Single-objective optimization using Pattern Search
Built solution
Starting point
Optimal solution minimizing GWP
Optimal solution minimizing IC
tCE (mm)
350
280
290
290
tFE (mm)
200
180
170
170
hEB (mm)
410
410
400
400
GWP/GWPbuilt_solution
1
0.953
0.907
0.907
IC/ICbuilt_solution
1
0.989
0.958
0.958
no. iterations
–
–
33
33
no. eval. solutions
–
–
56
59
no. analysed solutions
–
–
198
198
Considered objective functions: Global warming potential (GWP) and Investment cost (IC)
Fig. 7
Normalized quantities during the single-objective optimization using Pattern Search
×
In both cases, as a consequence of the reduction in material quantities, IC and GWP are reduced by 4.2% and 9.3%, respectively, with respect to the built solution. Moreover, the slenderer overhang slab leads to a decrease of the concrete amount and the self-weight of 13.8%. The weight reduction in the overhang slab-edge beam system is the starting point for a reduction in the material in the rest of the superstructure and substructure in the bridge.
One can notice that the optimal solution for the system overhang slab-edge beam is not far from that for the entire bridge (Khouri Chalouhi et al. 2019). The agreement between the two results has two consequences: (1) it is possible to optimize the overhang slab-edge beam system and the rest of the bridge separately without significantly affecting the results. In this way, the amount of design variables in the entire bridge optimization would be greatly reduced with a consequent significant reduction in the total computational time. (2) In the superstructure optimization of Khouri Chalouhi et al. 2019, no 3D FE model of the deck is used and the flexural moment is computed using approximated methods. The optimization of the overhang slab-edge beam system could be performed using the same simplification and thus further reducing the total computational time.
The results in Table 4 show that the solutions obtained with the two objective functions are identical and the number of iterations is the same. Instead, Figure 7 shows that the convergence paths that the two processes follow differ a little. In particular, when the algorithm tries to minimize IC, in the first iterations there is a phase in which IC and GWP are conflicting. In other words, there is a temporary increase in GWP due to the decrease in IC. However, for the rest of the process, the two quantities decrease together and converge to the same result.
Pattern Search is a deterministic local search algorithm; therefore, one could argue that the solution is a local minimum influenced by the choice of the starting point. Unless the objective function is known in close form and the absolute minimum can be calculated, there is no guarantee that the solution found with any optimization algorithm is not a local minimum. However, there are some strategies to mitigate this doubt. If one fears that the starting point is affecting the final solution, simulations with several starting points can be run. In this case, one additional simulation where the built solution has been used as starting point for the case with investment cost as objective function has been performed: the same optimal solution as Table 4 has been found. To confirm the hypothesis that it is not a local minimum, more simulations would be necessary. A less time-consuming strategy would be to use a different optimization algorithm that combines local and global search using randomization, such as GA. When applying GA, the same optimal solutions as in the case with PS has been obtained; thus confirming that this solution is probably not a local minimum.
Concerning the computational effort, Table 5 compares the evaluated and analysed solutions with PS and GA. Both when minimizing IC and GWP, GA requires around four times as many analysed solutions as PS to get to the same results. It is possible to conclude that PS performs way faster than GA in this optimization problem.
Table 5
Comparison between Genetic Algorithm (GA) and Pattern Search (PS) in the single-objective optimization with respect to Global warming potential (GWP) and Investment cost (IC)
GA-GWP
PS-GWP
GA-IC
PS-IC
No. iterations
21
33
21
33
No. evaluated solutions
254
56
459
59
No. analysed solutions
740
198
740
198
Finally, comparing the evaluated solutions with the analysed ones with both objective functions and optimization algorithms, the memory system allows to save between 28% and 55% of the computational time. One can notice that the difference between analysed and evaluated solutions in the case of GA changes significantly for the two objective function. This is a consequence of the stochastic nature of GA.
5.2 Optimal solutions of the multi-objective optimization problem
The single-objective optimization led to the same optimal solution using the two objective functions. To confirm that they are not conflicting, the same problem has been solved using GA and a multi-objective formulation. The expected result of a multi-objective optimization problem with conflicting objectives is a set of Pareto optimal solution. However, in the studied case, only one solution is found after 24 iterations and 198 evaluated individuals. The optimal solution coincides with that of the single-objective formulation.
During the optimization, some generations present couples of Pareto solutions. A representative example is in Table 6, where GWP and IC are normalized with respect to the minima for the two Pareto solutions. This allows to clearly show which solution minimizes each objective function and to see the relative difference between the solutions. The presence of Pareto solutions in intermediate generations agrees with what observed in Figure 7. Initially, IC and GWP can appear conflicting; the reason is that IC is more sensitive to changes in reinforcement amount than GWP, which, instead, is more sensitive to changes in concrete amount. However, if the increase in the reinforcement amount is significant, despite of a significant reduction in concrete, GWP increases. Therefore, at the end of the optimization process, the reinforcement amount is the leading factor for both objective functions.
Table 6
Pareto optimal solution for generation 10
Pareto optimal solution 1
Pareto optimal solution 2
tCE (mm)
290
290
tFE (mm)
170
180
hEB (mm)
410
400
GWP/ GWPmin. 1–2
1
1.0077
IC/ICmin. 1–2
1.0007
1
Merging the results of the individuals evaluated in all the optimization processes mentioned in this section, it is possible to have an idea of how IC and GWP varies with the variable values. Figure 8a shows IC and GWP against tCE and tFE for all the evaluated individuals with hEB = 41 cm. Figure 8b shows IC and GWP against tCE and hEB for all the evaluated individuals with tFE = 17 cm. One can notice that the optimal solution corresponds to the global minimum of the two functions.
Fig. 8
Normalized Investment cost (IC) and Global warming potential (GWP) as a function of a the overhang slab thicknesses and b the overhang slab thickness at the clamped edge (tCE) and edge beam height (hEB). The values are normalized with respect to the minimum found IC and GWP
×
All the results above are obtained considering the height of the edge beam to be at least equal to 40 cm, which is the value recommended by Swedish standards. Simulations with shorter allowed edge beams have been performed to draw some preliminary conclusions on the role of the edge beam in the optimal solution. The results show that the optimization procedure tends to minimize the edge beam height arriving to the same thickness as the slab. This corresponds to having no edge beam at all. The latter cannot be completely removed from the bridge deck because of its non-structural functions. However, it is suggested that alternative solutions to the classical RC edge beam should be considered.
6 Parametric study
In addition to the specific case of the previous section, this design approach based on optimization has been applied to a set of overhang slab lengths going from 1 m to 3 m. Slabs shorter than 1 m would be treated as flanges of the deck cross section, while the upper limit guarantees that only one traffic lane is on the overhang slab. Whenever the bridge deck needs to be particularly wide, instead of extending the overhang slab, a different solution is chosen (e.g. several girders). Due to the results of the previous section, a single-objective formulation is implemented here and PS is used as optimization algorithm. Two classes of concrete are studied: C35/45 and C50/60.
Figure 9 and Figure 10 show the optimal overhang slab thicknesses obtained while varying the free length. The optimal edge beam height is not shown since no interesting trend has been observed: all optimal solutions present an edge beam with height of 40 cm.
Fig. 9
Optimal overhang slab thicknesses for overhang slab lengths in the range 1–3 m and concrete classes C35/45
Fig. 10
Optimal overhang slab thicknesses for overhang slab lengths in the range 1–3 m and concrete classes C50/60
×
×
For both concrete classes and for free lengths above 170 cm, the optimal solutions obtained with the two objective functions coincide, as expected considering the case study. For shorter overhang slabs, instead, two different solutions are obtained. In particular, solutions for minimum GWP, present more tapered slabs with respect to those for minimum IC.
In the studied optimization problem, for the concrete class C35/45, for a length of 100 cm, a decrease of 5% in concrete volume leads to an increase of only 0.07% in the reinforcement. For a length of 160 cm, a decrease of 1.6% in concrete volume leads to an increase of 0.12% in the reinforcement. For longer slabs, a reduction in concrete would require an increase in reinforcement such that the final GWP would increase. Thus, the ratio between concrete decrease and reinforcement increases changes with the overhang slab length. This has to do with the leading design action: for short overhang slabs, shear is dominant over bending moment, vice versa for long overhang slabs. Even though the couples of optimal solutions are geometrically different, they do not differ that much in terms of investment cost and global warming potential. In particular, the biggest differences are found in correspondence of lc = 105 cm and lc = 140 cm. In the first case, an increase in IC of 0.73% leads to a reduction of 1.18% in GWP. For the longer overhang slab, to save 2.70% of GWP, IC has to increase of 0.33%. The non-significant differences in terms of IC and GWP leave the choice between the two optimal solutions to the designer. However, depending on the deck type, one solution could be preferred over the other. When designing reinforced concrete decks, only the part of the overhang slab that is closest to the girder is included in the design cross section. The more external portion only adds self-weight. Therefore, the slab should be as tapered as possible to increase the resistance at the clamped edge and decrease the load at the free edge. Thus, the optimal solution obtained by minimizing GWP should be preferred over that for minimum IC. In composite decks, the design routine is different and thus the designer can choose between the two optimal solutions more freely.
Finally, considering the load cases described in section 3.2, up to lc = 224 cm, only one wheel per axle acts on the overhang slab. For slightly longer slabs, only the bending moment at the clamped edge is influenced by the additional wheel. Finally, for lc around 250 cm and higher (i.e. lc > 224 cm + d0 (effective depth of the cross section at the root)), also the shear force is affected by the presence of the additional wheel. This leads to the need of a thicker clamped edge as shown in Figure 9 and Figure 10 and a higher amount of reinforcement.
All the considerations above are valid for both concrete classes. When comparing optimal solutions obtained with the two concrete classes for a given overhang slab length, the C50/60 solution is slenderer than that for C35/45. However, in the studied range on overhang slab lengths, the solutions for C35/45 happen to be cheaper (savings in the range 1.71-5.13% with mean of 2.63%) and more environment-friendly (savings in the range 4.53-10.54% with mean of 6.91%). Indeed, the increase in unit price and unit CO2-eq emissions obtained by increasing the concrete strength counteracts the savings in concrete amount, without significantly reducing the reinforcement amount.
7 Conclusions
In this paper, structural optimization is applied to overhang slabs of bridge decks: a component that has not got much attention in the optimization field. A design and optimization procedure for RC overhang bridge slabs is presented. Investment cost and global warming potential (measured as CO2-eq emissions) considering material production as life cycle stage are minimized by optimizing the slab thicknesses and the edge beam height. Reinforcement is designed as well to fulfil Eurocodes requirements for ULS and SLS. IC includes material cost and labour cost, while GWP during material production stage is quantified using the life cycle assessment technique and the ReCiPe method with midpoint approach. Two optimization algorithms are used and compared in terms of performance: Pattern Search and Genetic Algorithm.
A case study regarding a reinforced concrete beam bridge in Norrtälje, Sweden, is presented. Both optimization algorithms are used separately to solve the single-objective optimization problem with the two objective functions. GA is also used for the multi-objective formulation of the problem. The same optimal solution is found by the two algorithms and with the two objective functions used separately or simultaneously, i.e. no Pareto optimal solutions are identified. This suggests that the two objective functions are not conflicting in the studied problem. In the single-objective optimization, PS reaches the optimal solution in shorter computational time compared to GA. The obtained optimal value of the edge beam height is the lowest recommended by Swedish standards. When this limit is disregarded, the optimization procedure minimizes the edge beam height arriving to the same thickness as the slab, i.e. it removes the edge beam. The latter cannot be completely removed from the bridge deck because of its non-structural functions. However, the result of the optimization suggests that alternative solutions to the classical RC edge beam should be considered. Pettersson and Sundquist (2014) suggest several alternatives such as steel solutions or prefabricated concrete ones. Such conclusion is expected to be even stronger if the cost and environmental impact due to the maintenance of edge beams is included in the quantification of the objective functions. Edge beams indeed contribute significantly to the overall maintenance of the bridge.
The proposed optimization procedure is also applied to a set of overhang slab lengths going from 100 cm to 300 cm, considering two classes of concrete: C35/45 and C50/60. Optimal solutions for overhang slab thickness are presented in graphs. These can be used by engineers in early design stage as a replacement or in parallel to current rules of thumb. No differences between cost-effective and environment-friendly solutions are observed for overhang slabs longer than 170 cm. For shorter cases, two solutions are found. Even though they differ geometrically, the values of their objective functions are similar: the maximum difference between the two solutions in terms of GWP and IC are, respectively, 2.70 % and 0.73%. This is a confirmation of the fact that climate-friendly and cost-effective solutions do not differ significantly in the studied problem. Finally, it is shown that solutions for C35/45 are cheaper (savings up to 5.13%) and more environment-friendly (savings up to 10.54%) than those for C50/60. In RC decks, common practice is to use the same concrete class for the girder, the overhang slab and the edge beam. Therefore, it can happen that the choice of concrete class has to follow the design of the entire deck. In such a case, it is even more important to opt for optimal dimensions to avoid waste of economic resources and increase of environmental impact.
This paper shows how very conventional designs with common materials and following current regulations can and should be improved to reduce material waste, cost and global warming potential. Further reductions could be obtained using alternative materials such as GFRP reinforcement (Khouri Chalouhi 2022) or designs such as ribbed slabs.
Declarations
Conflict of interest
The authors have no conflict of interest to declare that are relevant to the content of this article.
Replication of results
Due to the nature of this research, authors of this paper did not agree for their code to be shared publicly. However, the data and information for the reproduction of the presented results are available within the article.
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