# Comparing the Fully Stressed Design and the Minimum

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Comparing the Fully Stressed Design and the Minimum

Chapter 2 Comparing the Fully Stressed Design and the Minimum Constrained Weight Solutions in Truss Structures David Greiner, José M. Emperador, Blas Galván and Gabriel Winter Abstract The optimization structural design problems of Fully Stressed Design (FSD) and Minimum Constrained Weight (MCW) are compared in this work in a simple truss test case with discrete cross-section type bar sizing, where both optimum designs are coincident. An analysis of the whole search space is included, and the optimization behaviour of evolutionary algorithms are compared with multiple population sizing and mutation rates in both problems. Results of average, best and standard deviation metrics indicate the success and the robustness of the methodology, as well as the fastest and easiest behaviour when considering the FSD case. Keywords Structural design · Truss optimization · Evolutionary algorithms · Fully stressed design · Minimum constrained weight 2.1 Introduction The use of evolutionary algorithms/metaheuristics has allowed the resolution of the global optimum design of many engineering problems (see e.g. [2, 10]), and particularly, in the case of discrete cross-section bar structures since the first nineties of the twentieth century [1, 7, 8]. In this book chapter, it is handled a comparative and relational study of the search algorithm performance in two structural problems: first, the minimization of the constrained weight and, second, the obtainment of the fully stressed design. Results using the above mentioned global search methods in a simple truss structure considering some statistical metrics are D. Greiner (*) · J.M. Emperador · B. Galván · G. Winter Institute of Intelligent Systems and Numerical Applications in Engineering SIANI, Universidad de Las Palmas de Gran Canaria ULPGC, 35017 Las Palmas, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2015 J. Magalhães-Mendes and D. Greiner (eds.), Evolutionary Algorithms and Metaheuristics in Civil Engineering and Construction Management, Computational Methods in Applied Sciences 39, DOI 10.1007/978-3-319-20406-2_2 17 D. Greiner et al. 18 obtained. First, the structural handled problems are described in Sect. 2.2, then the test case is shown in Sect. 2.3. Section 2.4 presents results and discussion, and finally, the conclusions Sect. 2.5 ends this book chapter. 2.2 Structural Problems Two optimization problems of bar structures with discrete section-types are fronted in this book chapter. In first place, the problem of minimization of the constrained structural weight (MCW), which is related with the minimization of raw cost of the structure, is considered (Eq. 2.1) (e.g. see [4, 12]). It is the most common structural optimum design problem. MCW = Nbars Ai · li · ρi (2.1) i=1 where Ai is cross-sectional area, li is length and ρi is specific weight, all corresponding to bar i; subjected under constraints of stresses, displacements and/ or buckling. In this chapter only stresses constraints are taken into account, and treated as in [6]. In second place, the problem of achieving the fully stressed design (FSD) structure is considered (e.g., see [14]), which has been handled since the beginning of the 20th century. The FSD of a structure is defined as the design in which some location of every bar member in the structure is at its maximum allowable stress for at least one loading condition. Nbars FSD = (σ MAX−i − σMAX−Ri )2 (2.2) i=1 where σMAX-i and σMAX-Ri are the maximum stress and the maximum allowable stress, respectively, both corresponding to bar i. Some relation between both previous problems, MCW and FSD, has been established, mainly in trusses structures where the material is allowed to work at its full potential due to the only existence of normal efforts, associated with the cross-sectional area [11, 13]. In this work, we show through the use of metaheuristic global optimization methods in discrete cross section-type trusses, that even in the possible case that both problems (MCW and FSD) share the same optimum solution, the search still has different characteristics and topology, which makes easier or harder to solve for the global search evolutionary algorithm. 2 Comparing the Fully Stressed Design and the Minimum … 19 2.3 Test Case The test case used is a simple test case with truss bar structures based on one in [15] and [16], solving it with discrete cross-section types variables (as in [9]). The computational domain, loading and boundary conditions are shown in Fig. 2.1, with Load P = 4450 N. This test case has been solved also for simultaneous (that is, multiobjective optimization) minimization of weight and maximization of the reliability index in Greiner and Hajela [3]. Each bar corresponds to an independent variable. Table 2.1 shows the set of cross section types and their geometric properties (area, radius of gyration). Table 2.2 represents the search space of variables, including the lower and upper limit of each variable. An own implemented truss bar structure stiffness matrix calculation has been used to evaluate the structural variables, where articulated nodes (that is Fig. 2.1 Test case. Computational domain, loading and boundary conditions Table 2.1 Cross-section types Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Cross-section C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 Area (cm2) 0.85 0.93 1.01 1.09 1.17 1.25 1.33 1.41 1.49 1.57 1.65 1.73 1.81 1.89 1.97 2.05 Radius of gyration (cm) 0.653 0.652 0.651 0.650 0.649 0.648 0.647 0.646 0.645 0.644 0.643 0.642 0.641 0.640 0.639 0.638 D. Greiner et al. 20 Table 2.2 Search space of variables Table 2.3 Geometric parameters Table 2.4 Material properties (Steel) Bar number 1 2 3 4 5 6 Height (H) Width (W) Parameter Density Young modulus Maximum stress Bar variable v1 v2 v3 v4 v5 v6 Cross-section type set From C1 to C16 From C1 to C16 From C1 to C16 From C1 to C16 From C1 to C16 From C1 to C16 Value (m.) 0.9144 1.2190 Value 7850 kg/m3 2.06 × 105 MPa 276 MPa non-resisting moment capabilities) are considered, elastic behaviour of steel is assumed, and no buckling effect is taken into account in these results. Table 2.3 shows the geometric parameters (height and width) of the structure. Table 2.4 exposes its material properties, -those of standard construction steel-. In order to define the cross-section type sizing of each bar (that is the structural design), the quantities of interest are the values of the fitness function/s (minimum constrained weight and /or fully stress design) and the maximum stress of each bar. 2.4 Results and Discussion 2.4.1 Test Case Analysis This section studies the relationship between the MCW problem and the FSD problem in our test case (as described in Sect. 2.3). Therefore, the whole search space of the previous test case has been explored, evaluating both objectives: (a) Minimum Constrained Weight (MCW, in kg.) as shown in Eq. 2.1, and (b) Square root of the sum of squared stress differences of each bar with Fully Stressed Design (FSD) as shown in Eq. 2.2. Values of the 224 = 16,777,216 designs (corresponding to a 6 bar × 4 bits/bar chromosome = 24 bits) are obtained and shown in Fig. 2.2 (whole search space). 2 Comparing the Fully Stressed Design and the Minimum … 21 Fig. 2.2 Whole search space designs of test case Fig. 2.3 Zommed vision over Fig. 2.2 (search space designs of test case) In addition, a zoomed picture of the best solution designs are shown in Fig. 2.3. In this test case, the minimum of both fitness functions is a coincident design, the most left and bottom point in this Fig. 2.3. Moreover, calculating the population Pearson’s correlation coefficient r between both objectives (MCW and FSD) gives a value r = 0.71089 (where 1.0 means a perfect linear relationship). D. Greiner et al. 22 The detailed best fifty designs of each objective are shown in Table 2.5 (MCW optimum designs) and Table 2.6 (FSD optimum designs). In addition to the optimum design, which is shared by the problem of MCW and the problem of FSD, fifteen designs out of this list of fifty are included in both sets (all shared designs are highlighted in bold type in the tables). Table 2.5 Minimum MCW designs (cross-section types as in Table 2.1) Design order 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11st 12nd 13rd 14th 15th 16th 17th 18th 19th 20th 21st 22nd 23rd 24th 25th 26th 27th 28th 29th 30th 31st 32nd 33rd FSD value 11480.7 12709.5 12709.5 11645.8 11645.8 13834.7 12449.8 12840.5 12875.2 12840.5 12875.2 12163.8 12038.5 12038.5 12163.8 12312.0 12312.0 12036.9 11861.9 12036.9 14232.6 14232.6 14861.9 14861.9 13148.3 13148.3 13645.3 13645.3 13977.5 13977.5 13930.1 13930.1 13970.5 MCW value 5.26338 5.3581 5.3581 5.35907 5.35907 5.37822 5.41648 5.4165 5.4165 5.4165 5.4165 5.43563 5.43563 5.43563 5.43563 5.43998 5.43998 5.45477 5.45477 5.45477 5.45478 5.45478 5.46076 5.46076 5.46882 5.46882 5.47391 5.47391 5.47392 5.47392 5.47392 5.47392 5.47392 Unconstrained weight 5.26338 5.3208 5.3208 5.35907 5.35907 5.37822 5.41648 5.4165 5.4165 5.4165 5.4165 5.43563 5.43563 5.43563 5.43563 5.43563 5.43563 5.45477 5.45477 5.45477 5.45478 5.45478 5.43565 5.43565 5.39735 5.39735 5.47391 5.47391 5.47392 5.47392 5.47392 5.47392 5.47392 Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6 C1 C1 C1 C1 C1 C1 C2 C1 C1 C1 C1 C2 C2 C1 C1 C1 C2 C1 C1 C1 C2 C1 C1 C1 C2 C1 C3 C1 C1 C1 C1 C1 C1 C1 C1 C2 C1 C1 C2 C1 C2 C1 C1 C2 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C3 C2 C3 C1 C2 C2 C1 C1 C3 C1 C3 C2 C3 C3 C3 C4 C3 C3 C3 C3 C3 C4 C4 C4 C3 C4 C3 C5 C2 C5 C4 C3 C3 C3 C3 C3 C3 C3 C2 C4 C3 C4 C4 C3 C4 C3 C3 C3 C3 C4 C3 C3 C4 C4 C3 C3 C3 C4 C3 C4 C2 C5 C3 C4 C5 C3 C3 C3 C3 C3 C3 C4 C2 C4 C3 C3 C4 C3 C1 C2 C1 C1 C1 C2 C1 C1 C2 C2 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C3 C1 C3 C2 C2 C1 C1 C2 C3 C1 C3 C1 C2 C1 C1 C1 C1 C1 C1 C2 C1 C1 C1 C1 C1 C1 C2 C2 C2 C1 C1 C1 C1 C1 C2 C1 C1 C1 C2 C1 C3 C1 C1 C1 C1 C1 (continued) 2 Comparing the Fully Stressed Design and the Minimum … 23 Table 2.5 (continued) Design order 34th 35th 36th 37th 38th 39th 40th 41st 42nd 43rd 44th 45th 46th 47th 48th 49th 50th 51th FSD value 13970.5 13818.0 13818.0 13278.3 13278.3 13330.9 13249.4 13175.6 13175.6 13249.4 13381.0 13381.0 13330.9 13328.1 13328.1 15823.2 11944.9 11944.9 MCW value 5.47392 5.48927 5.48927 5.49303 5.49303 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49307 5.49621 5.49621 Unconstrained weight 5.47392 5.37822 5.37822 5.49303 5.49303 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49305 5.49307 5.33993 5.33993 Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6 C1 C1 C1 C4 C1 C2 C2 C1 C2 C1 C2 C1 C1 C2 C1 C1 C2 C1 C2 C1 C3 C1 C1 C1 C1 C1 C2 C2 C2 C1 C2 C2 C1 C3 C1 C1 C3 C3 C3 C2 C4 C4 C3 C4 C3 C4 C2 C5 C3 C4 C3 C3 C3 C3 C4 C3 C3 C4 C2 C3 C4 C3 C4 C3 C5 C2 C4 C3 C4 C3 C3 C3 C2 C3 C1 C1 C1 C2 C2 C2 C1 C1 C1 C2 C1 C1 C2 C3 C1 C1 C1 C1 C1 C1 C4 C1 C1 C2 C1 C2 C1 C2 C2 C1 C2 C1 C1 C2 Table 2.6 Minimum FSD designs (cross-section types as in Table 2.1) Design order 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11st 12nd 13rd 14th 15th 16th 17th FSD value 11480.7 11645.8 11645.8 11649.7 11649.7 11735.5 11735.5 11861.9 11922.0 11944.9 11944.9 12026.5 12026.5 12036.9 12036.9 12038.5 12038.5 MCW value 5.26338 5.35907 5.35907 6.24373 6.24373 5.77910 5.77910 5.45477 7.32865 5.49621 5.49621 6.31897 6.31897 5.45477 5.45477 5.43563 5.43563 Unconstrained weight 5.26338 5.35907 5.35907 5.16768 5.16768 5.26338 5.26338 5.45477 5.07198 5.33993 5.33993 5.24423 5.24423 5.45477 5.45477 5.43563 5.43563 Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C2 C2 C1 C1 C1 C2 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C3 C3 C4 C3 C2 C4 C2 C4 C2 C3 C3 C2 C3 C5 C3 C3 C4 C3 C4 C3 C2 C3 C2 C4 C4 C2 C3 C3 C3 C2 C3 C5 C4 C3 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C2 C1 C1 C2 C1 C1 C1 C2 (continued) D. Greiner et al. 24 Table 2.6 (continued) Design order 18th 19th 20th 21st 22nd 23rd 24th 25th 26th 27th 28th 29th 30th 31st 32nd 33rd 34th 35th 36th 37th 38th 39th 40th 41st 42nd 43rd 44th 45th 46th 47th 48th 49th 50th FSD value 12041.2 12041.2 12069.7 12069.7 12163.8 12163.8 12186.1 12186.1 12248.0 12248.0 12271.6 12271.6 12281.2 12281.2 12309.8 12309.8 12312.0 12312.0 12324.0 12324.0 12364.8 12364.8 12371.7 12371.7 12432.3 12432.3 12449.8 12491.7 12491.7 12525.0 12525.0 12562.8 12562.8 MCW value 5.63246 5.63246 5.67125 5.67125 5.43563 5.43563 6.35272 6.35272 6.77926 6.77926 7.42623 7.42623 5.55047 5.55047 5.53132 5.53132 5.43998 5.43998 6.07500 6.07500 5.53132 5.53132 7.43926 7.43926 6.46520 6.46520 5.41648 6.84455 6.84455 7.50132 7.50132 5.74984 5.74984 Unconstrained weight 5.33993 5.33993 5.35907 5.35907 5.43563 5.43563 5.24423 5.24423 5.16768 5.16768 5.07198 5.07198 5.55047 5.55047 5.53132 5.53132 5.43563 5.43563 5.33993 5.33993 5.53132 5.53132 5.14853 5.14853 5.24423 5.24423 5.41648 5.26338 5.26338 5.14853 5.14853 5.45477 5.45477 Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6 C2 C1 C1 C1 C2 C1 C2 C1 C1 C1 C1 C1 C1 C1 C2 C1 C2 C1 C2 C1 C1 C2 C2 C1 C2 C1 C2 C1 C1 C2 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C2 C4 C5 C2 C4 C3 C3 C2 C1 C4 C1 C3 C5 C4 C4 C4 C2 C5 C4 C2 C5 C3 C2 C2 C1 C4 C3 C5 C1 C1 C3 C6 C2 C4 C2 C2 C5 C3 C4 C2 C3 C4 C1 C3 C1 C4 C5 C4 C4 C5 C2 C2 C4 C3 C5 C2 C2 C4 C1 C3 C1 C5 C3 C1 C2 C6 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C2 C1 C1 C1 C2 C1 C2 C1 C1 C1 C1 C1 C1 C1 C2 C1 C2 C1 C2 C2 C1 C1 C2 C1 C2 C2 C1 C1 C1 C2 C1 C1 2.4.2 Optimization In this section, both the MCW and the FSD problems are optimized. An evolutionary algorithm with two population sizes (80 and 160 individuals), two mutation rates (1.5 and 3 %), uniform crossover and gray codification 2 Comparing the Fully Stressed Design and the Minimum … 25 (as in [5]) has been executed in 100 independent runs with 30,000 evaluations as stopping criterion. Results are shown in terms of average, best value and standard deviation of the fitness functions. (a) Solving the problem of minimum constrained weight MCW optimum design: Figure 2.4 shows the average fitness value, Fig. 2.5 shows the best fitness value and Fig. 2.6 shows the standard deviation fitness value; each containing the population size of 80 in black lines and the population size of 160 in pink (gray) lines; Fig. 2.4 Constrained minimum weight evolution over 100 independent runs in cR00 test case Fig. 2.5 Constrained minimum weight evolution over 100 independent runs in cR00 test case 26 D. Greiner et al. Fig. 2.6 Constrained minimum weight evolution over 100 independent runs in cR00 test case Fig. 2.7 FSD evolution over 100 independent runs in cR00 test case the thicker line belongs to the 1.5 % mutation rate and the thinner line belongs to the 3.0 % mutation rate. (b) Solving the problem of fully stressed design FSD optimum design: Figure 2.7 shows the average fitness value, Fig. 2.8 shows the best fitness value and Fig. 2.9 shows the standard deviation fitness value; each containing the population size of 80 in black lines and the population size of 160 in pink (gray) lines; the thicker line belongs to the 1.5 % mutation rate and the thinner line belongs to the 3.0 % mutation rate. 2 Comparing the Fully Stressed Design and the Minimum … 27 Fig. 2.8 FSD evolution over 100 independent runs in cR00 test case Fig. 2.9 FSD evolution over 100 independent runs in cR00 test case (c) Comparing FSD and MCW problems with population size 80: In this case, fitness values have been scaled between 0 and 1, in order to easily compare visually the behaviour of both problems (FSD and MCW). Figure 2.10 shows the scaled average fitness value, Fig. 2.11 shows the scaled best fitness value and Fig. 2.12 shows the scaled standard deviation fitness value; each containing the FSD optimization problem in black lines and the MCW optimization problem in pink (gray) lines; the thicker line belongs to the 1.5 % mutation rate and the thinner line belongs to the 3.0 % mutation rate. 28 D. Greiner et al. Fig. 2.10 Constrained minimum weight evolution over 100 independent runs in cR00 test case Fig. 2.11 Constrained minimum weight evolution over 100 independent runs in cR00 test case (d) Comparing FSD and MCW problems with population size 160: In this case, fitness values have been scaled between 0 and 1, in order to easily compare visually the behaviour of both problems (FSD and MCW). Figure 2.13 shows the scaled average fitness value, Fig. 2.14 shows the scaled best 2 Comparing the Fully Stressed Design and the Minimum … 29 Fig. 2.12 Constrained minimum weight evolution over 100 independent runs in cR00 test case Fig. 2.13 FSD evolution over 100 independent runs in cR00 test case fitness value and Fig. 2.15 shows the scaled standard deviation fitness value; each containing the FSD optimization problem in black lines and the MCW optimization problem in pink (gray) lines; the thicker line belongs to the 1.5 % mutation rate and the thinner line belongs to the 3.0 % mutation rate. 30 D. Greiner et al. Fig. 2.14 FSD evolution over 100 independent runs in cR00 test case Fig. 2.15 FSD evolution over 100 independent runs in cR00 test case 2.4.3 Discussion (a) Solving the problem of minimum constrained weight MCW optimum design (Figs. 2.4, 2.5 and 2.6): The lower the population size (80 vs. 160), the faster the convergence of the algorithm without worsening the quality of the obtained solution in this test case as shown in Figs. 2.4, 2.5 and 2.6 (the chromosome length of this problem is easily 2 Comparing the Fully Stressed Design and the Minimum … 31 handled by the evolutionary algorithm). The behaviour of average and standard deviation values are slightly better in the lower mutation rate (1.5 % vs. 3 %). It is remarkable that the range of evaluations required to obtain the best design in all the 100 independent executions in this MCW problem vary from 1610 evaluations (population size 80, mutation rate 1.5 %) to 2272 evaluations (population size 160, mutation rates 1.5 and 3 %). (b) Solving the problem of fully stressed design FSD (Figs. 2.7, 2.8 and 2.9): Here also, the lower the population size (80 vs. 160), the faster the convergence of the algorithm without worsening the quality of the obtained solution in this test case as shown in Figs. 2.7, 2.8 and 2.9 (the chromosome length of this problem is easily handled by the evolutionary algorithm). As well, the behaviour of average and standard deviation values are slightly better in the lower mutation rate (1.5 % vs. 3 %). It is remarkable that the range of evaluations required to obtain the best design in all the 100 independent executions in this FSD problem vary from 1172 evaluations (population size 80, mutation rate 1.5 %) to 2020 evaluations (population size 160, mutation rate 3 %). (c) Comparing FSD and MCW problems with population size 80 and 160 (Figs. 2.10, 2.11, 2.12, 2.13, 2.14, and 2.15): Figures 2.10, 2.11, 2.12, 2.13, 2.14, and 2.15 show clearly through the scaled fitness, that when comparing the fitness behaviour, in all cases of average, best and standard deviation, the FSD problem has a faster convergence than the MCW problem in this test case (except in the best fitness of MCW problem with population size 80 and 3 % mutation rate), requiring lower number of fitness evaluations to achieve the same optimum design. 2.5 Conclusions The relation of the problems MCW and FSD has been shown through a simple truss test case in discrete cross-section types sizing optimization. This relation has been evidenced by having a coincidental optimum design, a high number of coincidental best designs among the best ones and showing a high correlation coefficient when considering the whole search space of each problem. The search process of the optimum design of both problems has been possible by using evolutionary algorithms, showing a high robust behaviour where in 100 out of 100 independent runs, this metaheuristic optimization was able to find the best solution in the range of 1000–2000 evaluations -versus a search space of tens of millions-. When comparing them, the fully stressed design (FSD) problem has shown an easier topology for the evolutionary algorithm optimization versus the MCW problem, that is, requiring less number of fitness function evaluations to achieve the same shared best design. Application of this analysis to an increased number of test cases and generalization in other types of bar structures, -e.g. in the case of frame bar structures-, should be 32 D. Greiner et al. developed in the future to provide more light about the comparison and relationship of the fully stressed design and the minimum constrained weight problems. 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