CO2-optimization of reinforced concrete frames by simulated annealing
Introduction
Reducing CO2 emissions is undoubtedly the greatest challenge of the present century [1]. The growing concern about these emissions has spread to all areas of human activity, including the construction sector, where the cement industry contributes 5% of the total global emissions [2]. Additionally, the construction sector is responsible for 17% of greenhouse emissions in India and between 8% and 12% in Western Europe [3], [4]. Another concern is the need for new housing in both developing and developed countries [5], [6], which puts extra pressure towards the efficient use of construction materials regarding their environmental footprint. Although alternative and more environmental systems and materials, such low- CO2 cements, are now key areas of research and development [7], [8], [9], [10], reductions should involve the mitigation of CO2 emissions by the efficient use and optimization of structural designs. This approach is possible since data on the environmental impact of most construction materials have been compiled by distinct organizations [11], [12] and, hence, the CO2 impact of a given structure can now be computed.
The traditional approach to design does not fully optimize the use of materials. Most designs are based on the prior experience of the engineer, who selects cross-section dimensions and material grades by comparing past designs. This gives rise to fixed rules-of-thumb for preliminary design. Once the main design variables are selected according to experience, it follows the analysis of the stress resultants and the computation of the active and passive reinforcement required to fulfill the limit states prescribed by the codes of practice. Should the structural type be a new one or should the engineer lack experience, a trial-and-error procedure is usually implemented. A first tentative design is analyzed and modified when the dimensions and material grades are insufficient or excessive. This approach leads to safe designs, but materials are not optimized and, based on the authors’ experience in structural concrete design, the amount of redundant materials may be estimated about as 10%. This estimation is lower when designers are more experienced, but may be much higher for novel designers.
Structural optimization methods are a clear alternative to designs based on experience. A thorough review of structural optimization methods can be found in the 1994 study by Cohn and Dinovitzer [13], who reported on the gap between theoretical studies and the practical application of optimization methods in civil and aeronautical engineering. This gap is evident when comparing the numerous studies on algorithms and the scarcity of reports on practical applications. Additionally, they affirmed that most studies focused on steel structures while only a small fraction dealt with reinforced concrete (RC) structures. Finally, they highlighted the great potential of heuristic optimization methods, which were then in their origins. A review of non-heuristic structural concrete optimization studies can be found in Sarma and Adeli [14], while studies on RC building frames are discussed in Moharrami and Grierson [15]. Early heuristic applications include those by Jenkins [16] as well as Rajeev and Krisnamoorthy [17], who applied genetic algorithms (GA) to steel structures. Recent steel applications include the studies of Rahami et al. [18] and Lagaros et al. [19]. As regards concrete structures, pioneering applications include the 1997 work by Coello et al. [20], who applied GA to sections of simply supported RC beams, together with the study of prestressed concrete beams by Leite and Topping [21]. The number of studies devoted to structural concrete is relatively low. Additional studies describing examples of beams, columns, walls, building frames and flat slab buildings, include those by Rafiq and Southcombe [22], Rajeev and Krishnamoorthy [23], Ceranic et al. [24], Leps and Sejnoha [25], Lee and Ahn [26], Camp et al. [27], Sahab et al. [28] and Govindaraj and Ramasamy [29]. Recently, the authors’ research group has presented studies of earth retaining walls, frame bridges and building frames [30], [31], [32], where additional bibliographical references can be found.
Optimization methods aim to minimize the objective function, which is usually the cost or the weight of the structure rather than environmental factors. However, Paya et al. [32] conducted a multiobjective comparison of four objectives for RC building frames, including the environmental impact measured by Ecoindicator 99. In this research, the structures analyzed are building frames which feature horizontal beams to sustain the vertical loads of the floors and transfer these loads to vertical columns (see Fig. 1). Moderate horizontal loads are usually included in the design, but high levels of horizontal loading are transferred to adjacent shear walls. Building frames are calculated to sustain the loads prescribed by the codes and must satisfy all the limit states required for a RC structure. The method followed in the present study consisted in the development of an evaluation computer module that checks all the relevant limit states. Dimensions, materials and steel reinforcement were taken as variables. Then the embedded CO2 and cost objective functions were calculated. Simulated annealing was then used to search the solution space so as to identify sets of solutions of optimized values for the designer. Finally, the results from this analysis allowed for the conclusions to be drawn.
Section snippets
Structural evaluation module
The structural evaluation module is the basic module of the optimization analysis. Given all the data necessary to define a given structure, this module calculates the stress envelopes and checks all the limit states. Structures that comply with all limit states are called feasible solutions and those that do not are called unfeasible solutions. The structural evaluation module has much in common with computer-aided design (CAD) software. The latter normally includes (1) a user-friendly input
objective function
Environmental concerns regarding CO2 emissions are of increasing importance nowadays. The CO2 objective function measures the embedded CO2 emissions resulting from the use of materials which involve emissions at the different stages of production and placement. The higher this value is, the lower its sustainability will be. Different structural alternatives can be assessed and compared from an environmental point of view. Alternative life-cycle assessment (LCA) approaches have previously been
Proposed optimization methodology
The problem described in Section 3 is what is usually referred to as a constrained optimization problem. It involves the minimization of a given function subject to a set of constraints. There are two methods for solving this problem: exact and approximate methods. Exact methods are usually based on the calculation of optimal solutions following iterative techniques of mathematical programming [38], [39]. Approximate methods include the heuristic methods, whose recent development is tied to the
Numerical examples
The algorithm SA was programmed in Fortran and used to optimize the RC 2-bay frames in Fig. 1, the 2b2f, 2b4f, 2b6f and 2b8f frames corresponding to 2-bay frames of 2, 4, 6 and 8 floors, respectively. Note that the bay span is 5.00 m while the spacing considered between adjacent parallel frames is 5.00 m as well. Hence, these four frames correspond to 100, 200, 300 and 400 m2 of floor slab. The number of variables and the size of the solution space for the different frames are given in Table 3.
Concluding remarks
Reducing CO2 emissions in structural concrete designs is possible since the basic data for emission assessment are now available. This research aimed to evaluate the potential of the SA algorithm in the optimization of emissions of real building frames. The study involved a thorough structural verification module with vertical and horizontal loads including full verification of the Spanish code of practice. The analysis was applied to six typical frames of two, three and four bays and from two
Acknowledgements
This study was funded by the Spanish Ministry of Education (research project BIA2006-01444). The authors are grateful to Debra Westall for her thorough revision of the manuscript.
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