Acta Polytechnica https://doi.org/10.14311/AP.2021.61.0242 Acta Polytechnica 61(1):242–252, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague APPLICATION OF GLOBAL OPTIMIZATION TO PREDICT STRAINS IN RC COMPRESSED COLUMNS Marek Lechmana, ∗, Andrzej Stachurskib a Building Research Institute, Filtrowa 1, 00-611, Warsaw, Poland b Warsaw University of Technology, Institute of Control and Computation Engineering, Nowowiejska 15/19, 00-665 Warsaw, Poland ∗ corresponding author: m.lechman@itb.pl Abstract. In this paper, the results of an application of global and local optimization methods to solve a problem of determination of strains in RC compressed structure members are presented. Solutions of appropriate sets of nonlinear equations in the presence of box constraints have to be found. The use of the least squares method leads to finding global solutions of optimization problems with box constraints. Numerical examples illustrate the effects of the loading value and the loading eccentricity on the strains in concrete and reinforcing steel in the cross-section. Three different minimization methods were applied to compute them: trust region reflective, genetic algorithm tailored to problems with real double variables and particle swarm method. Numerical results on practical data are presented. In some cases, several solutions were found. Their existence has been detected by the local search with multistart, while the genetic and particle swarm methods failed to recognize their presence. Keywords: Global optimization, nonlinear equations, least squares method, RC compressed structure members. 1. Introduction Our problem is to determine the normal strains in the cross-sections of reinforced concrete structure mem- bers subjected to compression. Mathematically, it may be formulated as a task of solving sets of equa- tions with box constraints. The unknown variables are: � ′ – maximum strain in the cross section and ξ – coordinate describing the location of the neutral axis. The presence of the box constraints makes a direct use of numerical methods for solving sets of nonlinear equations impractical. Therefore, our task is reformu- lated by means of the frequently used least squares method. It leads to a nonlinear, nonconvex optimiza- tion problem of finding a minimum of a nonlinear function with the restricted scope of variables. 1.1. Motivation to study the strains in RC compressed structure members Reinforced concrete structure members subjected to the compression are frequently encountered in the en- gineering practice (columns, pillars, tower-like struc- tures etc.). The determination of strains is very impor- tant in the safety assessment of existing RC structures. In order to solve this problem analytically, several physical models of materials and methods were pro- posed. Lechman and Lewiński [1] considered a general- ized linear section model. A simplified approach based on the rectangular stress distribution for concrete was used by Knauff [2] and Knauff et al. [3]. Nieser and En- gel [4] and CICIND [5] applied the parabola-rectangle diagram for the design of cross-sections. For reinforcing steel itself, both linear and nonlin- ear models are used, see for instance Lechman and Stachurski [6], Lechman [7–10], where the ring sec- tions were investigated. The results of FE (finite element) modelling of failure behaviour of RC com- pressed columns were presented by Majewski et al. [11] and Rodriguez et al. [12]. In Kim and Lee [13], a nu- merical method for predicting the behaviour of RC columns subjected to axial force and biaxial bend- ing is proposed and verified in tests. Campione et al. [14] experimentally investigated the behaviour of compressed concrete columns subjected to the over- coring technique, see also Campione et al. [15]. The list of researchers working in various directions could be continued. Let’s mention some of them: Lloyd and Rangan [16], Bonet et al. [17], Ye et al. [18], Xu et al. [19], Trapko and Musiał [20], Trapko [21], Hadi and Le [22], El Maddawy et al. [23], Csuka and Kollar [24], Elwan and Rashed [25], Sadeghian et al. [26], Eid and Paultre [27], Wu and Jiang [28], Quiertant and Clement [29], Lee et al. [30], Kumar and Patel [31] and many others. Of course, the list is not complete. Despite the variety of calculation methods and ex- perimental investigations concerning this problem, there are not any appropriate analytical solutions based on the nonlinear material laws for determin- ing the strains in RC externally compressed structure members that considers concrete softening. The aim of our paper is twofold. Firstly, to formu- late equilibrium equations allowing to calculate the strains. Secondly, to investigate the usefulness of some 242 https://doi.org/10.14311/AP.2021.61.0242 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 61 no. 1/2021 Global optimization to predict strains in RC compressed columns Figure 1. Distribution of strain �, stresses in con- crete σc and stresses in steel σs across the section global optimization methods to solve the problem nu- merically. 2. Formulation of the equilibrium equations To get the required equations, we started with the integral equilibrium equations and integrated them. The rectangular RC cross-section is subjected to the axial force N and the bending moment M (see Fig. 1). The content of the current section is an extension of that presented in Lechman and Stachurski [32]. The detailed way of deriving the formulas for the section wholly in compression is included. In the derivation of the governing equations, the following assumptions are made: • plane cross-sections remain plane, • elasto-plastic stress/strain relationships for concrete and reinforcing steel are used, • the tensile strength of concrete is ignored, • the ultimate strains for concrete are determined as �cu and for reinforcing steel as �su. In Fig. 1, the following notation is used: t, b - the thickness and the width of the cross-section, re- spectively, t1, t2 - coordinates describing the locations of rebars, x, x ′ - coordinates describing the location of the neutral axis and the location of any point of the section, respectively. In accordance with the Eu- rocode 2 [33], the stress-strain relation for concrete σc – �c in compression for a short term uniaxial loading is assumed as σc = kηc −η2c 1 + (k − 2)ηc fcm, (1) where: ηc = �c/�c1, �c1 – the strain at peak stress on the σc – �c diagram, k = 1.05 Ecm|�c1|/fcm, fcm – the mean compressive strength of concrete, Ecm – secant modulus of elasticity of concrete, �cu (�cu1) – the ultimate strain for concrete. The reinforcing steel is characterized by yield stress fyk, Es – modulus of elasticity and Eh – coefficient of steel hardening (linear elastic model with hardening). 2.1. Equations for strains in the rectangular sections In further considerations, the corresponding dimen- sionless coordinates are used: ξ = x/t, ξ ′ = x ′ /t, ξ1 = t1/t, ξ2 = t2/t. (2) 2.1.1. Equations for sections wholly in compression Let us consider the section wholly in compression. The strain distribution can be expressed in the form � = �1 + (�2 − �1)ξ ′ , (3) where: �1 – maximum compressive strain in the cross section, �2 – minimum compressive strain in the cross section. Thus, ηc occuring in (1) assumes the form ηc = k2ξ ′ + k1, (4) after including in (3) the following assignements: k1 = �1 �c1 and k2 = �2 − �1 �c1 . The equilibrium equation of the axial forces in the cross-section takes the following form∫ Ac σcdAc + σs1Fa1 + σs2Fa2 + N = 0, (5) where: dAc - element of the concrete area Ac, Fa1, Fa2 - areas of the steel in compression and in tension, respectively. The sectional equilibrium of the bending moments about the symmetry axis of the rectangle can be ex- pressed in the form∫ Ac σc(0.5t−x ′ )dAc + σs1Fa1(0.5t− t1)+ σs2Fa2(0.5t− t2) −M = 0. (6) In order to obtain the final form of the equilibrium equations, we integrated formulas in (5) and (6). The 243 Marek Lechman, Andrzej Stachurski Acta Polytechnica most difficult part was to find the antiderivatives of the functions in the integral expressions in (5) and (6). After substituting relations (3) and (4) in relation (1), the function to be integrated in (5) is fN (ξ′) = k(k2ξ ′ + k1) − (k2ξ ′ + k1)2 1 + (k − 2)(k2ξ ′ + k1) , (7) and in (6) is fM (ξ′) = k(k2ξ ′ + k1) − (k2ξ ′ + k1)2 1 + (k − 2)(k2ξ ′ + k1) (0.5 − ξ ′ ). (8) Finally, the following equilibrium equations for strains in the rectangular sections are found. The first one concerns the equilibrium equation of the axial forces n + 1 k−2 { W1 + 0.5k2 + W2W3 (ln W5 − ln W6) } + µ1 fyk fcm { δi1 [ −1 + Eh fyk (((�2 − �1) ξ1 + �1) + �ss)] + δi1+1 (�2 − �1)ξ1 + �1 �ss } + µ2 fyk fcm { δi2 [ 1 + Eh fyk [((�2 − �1) (1 − ξ2) + �1) − �ss]] + δi2+1 (�2 − �1)(1 − ξ2) + �1 �ss } = 0 (9) and the second one represents the sectional equilib- rium of the bending moments −m + 1 k−2 { −k212 + 0.5 W2 W6 (ln W5 − ln W6) − W2 W3 [ 1 − W6 W4 (ln W5 − ln W6) ]} + µ1 fyk fcm (0.5 − ξ1) { δi1 [ 1 + Eh fyk (((�2 − �1) ξ1 + �1) + �ss)] + δi1+1 (�2 − �1)ξ1 + �1 �ss } + µ2 fyk fcm (0.5 − ξ2) { δi2 [ 1 + Eh fyk (((�2 − �1) (1 − ξ2) + �1) − �ss)] + δi2+1 (�2 − �1)(1 − ξ2) + �1 �ss } = 0 (10) where: W1 = k1 −k − 1 k − 2 W2 = k(k − 2) + 1 W3 = (k − 2)(k − 2)k2 W4 = (k − 2)k2 W5 = 1 + k − 2 k2 + k1 n = N btfcm W6 = 1 + (k − 2)(k2 + k1) m = M bt2fcm δi = 0.5((−1i) + 1), i = 1, 2 and: µ1 – the reinforcement ratio of steel in compres- sion, µ2 – the reinforcement ratio of steel in tension. The unknown variables are: �1, �2. 2.1.2. Section under combined compression with bending Let us consider the section under combined compres- sion and bending. Due to the Bernoulli assumption, one obtains (see Fig. 1) � = ( 1 − ξ ′ ξ ) � ′ , (11) where: � ′ – the maximum compressive strain in con- crete. The resulting formulas are given below. Equa- tion (12) (for axial forces) n + 1 k−2 { W1ξ + 0.5k2ξ2 − 1k−2 [ W2 W3 ln W − ξ ]} + µ1 fyk fcm { δi1 [ −1 + Eh fyk (( 1 − ξ1 ξ ) � ′ + �ss )] + δi1+1 � ′ �ss ( 1 − ξ1 ξ )} + µ2 fyk fcm { δi2 [ +1 + Eh fyk (( 1 − 1 − ξ2 ξ ) � ′ − �ss )] + δi2+1 � ′ �ss ( 1 − 1 − ξ2 ξ )} = 0 (12) and equation (13), representing the sectional equilib- rium of the bending moments −m + 1 k−2 { 0.5 ( W1 + 1k−2 ) ξ + 0.5 [ −W1 + 0.5k2 − 1k−2 ] ξ2− 1 3k2ξ 3 − W2(k−2)W3 [ 0.5 ln W + ξ − W W3 ln W ]} + µ1 fyk fcm (0.5 − ξ1) { δi1 [ 1 + Eh fyk (( 1 − ξ1 ξ ) � ′ + �ss )] + δi1+1 � ′ �ss ( 1 − ξ1 ξ )} + µ2 fyk fcm (0.5 − ξ2) { δi2 [ +1 + Eh fyk (( 1 − 1−ξ2 ξ ) � ′ − �ss)] + δi2+1 � ′ �ss ( 1 − 1 − ξ2 ξ )} = 0, (13) where: W1 = k −k2ξ W2 = k(k − 2) + 1 W3 = (k − 2)k2 W = 1 + (k − 2)k2ξ δi = 0.5((−1)i + 1), i = 1, 2 k2 = � ′ �c1ξ The unknown variables are: • � ′ – maximum strain in the cross section, • ξ – coordinate describing the location of the neutral axis. 3. Computational solution and numerical results It is not our first work with models of the processes in the RC structure members. We have already got some experience with the circular RC structure mem- bers [6]. This experience suggests that we have to 244 vol. 61 no. 1/2021 Global optimization to predict strains in RC compressed columns expect many global and local solutions of the least squares problem. Therefore, we decided to compare three different algorithms: local search method (trust region reflective) started many times from all points from a net of points equally distributed on the fea- sible box and genetic and particle swarm algorithms designed for searching a global optimum. For the verification of the obtained formulae, two rectangular cross-sections 0.3 m × 0.3 m under the compression have been considered: the unreinforced one and that of reinforced (µfyk/fcm = 0.1). Both sections had the following characteristics: the concrete grade C20/25, the yield stress of steel fyk = 500MPa (reinforced), reinforcement ratios of the steel in com- pression and in tension µ1 = µ2 = µ, t1/t = t2/t = 0.1, Eh = 0. It is assumed that the resistance of the cross-section is reached when the compressive strain in concrete �cu = −3.5‰ or the ultimate strain in the reinforcing steel equals �su = 10‰. After some rearrangements and substituting � ′ = x and ξ = y, the set of equations (12–13) takes the forms (14–15) for the unreinforced and (16–17) for the reinforced cross-sections, respectively. Due to the appearance of the term y − x in the denominator in some sets of equations, a danger of the division by 0 occurs. For this reason, fmin- con has been finally applied with the algorithm op- tion set to ”interior point method”, that allowed to include special constraints eliminating this dan- ger. Moreover, the existence of multiple minima may not be avoided. However, the least squares formu- lation of the problem itself may, in general, involve extra local solutions (such a counterexample may be found in Stachurski [34]). This has been confirmed through computational results. Many local minima and sometimes several global minima that resulted from the numerical properties of the optimization problem were encountered. Therefore, the clusteriza- tion idea imported from clusterization the methods of the global optimization was incorporated (see, for instance, Thorn and Żilinstas [35]). The size of the problem and computation time were of secondary im- portance. We have also tested the genetic and particle swarm algorithms from the global optimization Mat- lab’s toolbox, comparing them with a local search method started from all points of the net covering the whole set Ω of feasible points. For testing purposes, the sets of equations were used that described the reinforced or unreinforced concrete sections subjected to the compression. The equations for the concrete without the reinforce- ment – the subject to the compression with bending are r1(x,y) = −a1 + (2.25 + 0.5x)y− 0.25xy− 4 [ −12.5 y x ln(1 − 0.125x) −y ] = 0, (14) r2(x,y) = −a2 + (3.125 + 0.25x)y+[ −3.125 + 0.25x− 0.125 x y ] y2+ 0.16667xy2+ 50 y x [0.5 ln(1 − 0.125x) + y + 8 (1 − 0.125x)y x ln(1 − 0.125x) ] = 0, (15) where: x – maximum compressive x ∈ [−5,−10−10] strain in concrete y – coordinate specifying y ∈ [10−10, 1] location of the neutral axis of the cross-section Different values of constants a1 and a2 correspond to different axial forces N and bending moments M. Parameters a are collected in table 1 The corresponding equations for the reinforced con- crete section subjected to the compression with bend- ing are given below r1(x,y) = (2.25 + 0.5x)y − 0.25xy− 4 [ −12.5 y x ln(1 − 0.125x) −y ] + 0.01x ( 1 − 0.9 y ) −a1 = 0, (16) r2(x,y) = (3.125 + 0.25x)y−[ 3.125 + 0.25x + 0.125 x y ] y2+ 0.16667xy2+ 50 y x [0.5 ln(1 − 0.125x) + y + 8(1 − 0.125x) y x ln(1 − 0.125x) ] + 0.004x(1 − 0.9 y ) −a2 = 0, (17) where x and y have the same meaning and scope as in equations (14) and (15). We used two sets of constant parame- ters a1 and a2 for that case specified below. Set No. a1 a2 1 0.143445 0.0292155 2 0.129182 0.0348055 We have to solve sets of two nonlinear equations with two unknowns x and y specified above{ r1(x,y) = 0 r2(x,y) = 0 where [ x y ] ∈ Ω Ω = {[ x y ] ∈ R2 ∣∣∣∣ xL ≤ x ≤ xUyL ≤ y ≤ yU } (18) xL, xU are the lower bound and upper bound on variable x and similarly yL, yU are the lower bound and upper bound on variable y. Due to their presence, a direct use of numerical methods for solving sets of nonlinear equations seems to be impractical. Therefore, our task was reformulated by means of the frequently used the least squares method. It has 245 Marek Lechman, Andrzej Stachurski Acta Polytechnica Set No. a1 a2 1 0.17157 0.01717 2 0.13345 0.02522 3 0.10918 0.026805 4 0.08579 0.02574 5 0.07065 0.02369 6 0.04448 0.01763 7 0.02571 0.01138 Table 1. Sets of parameters for unreinforced concrete subjected to compression with bending lead to a nonlinear, nonconvex optimization problem min (x,y) f(x,y) = 12 ( r21 (x,y) + r22 (x,y) ) s.t. x L ≤ x ≤ xU yL ≤ y ≤ yU (19) Below are given the results of the local search with multiple starting points, genetic algorithm and the particle swarm optimization method. In the first approach, we selected the fmincon func- tion from the MATLAB Optimization Toolbox as a tool to solve the least squares problem (19), because it allows the introduction of of the box constraints. The steps of the procedure may be summarized as follows: Set S – set of solution clusters to be an empty set. while (there are non used points in the net cover- ing Ω) • take a new point x0 ∈ Ω, • solve the least squares problem by means of the fmincon function from the MATLAB’s toolbox OP- TIMIZATION starting from the point x0, • denote the found solution by x, • if f(x) < resTOL if x belongs to some cluster in S compare function value f(x) with the best in the cluster and save the better of the two points as the seed of the cluster else save the current point as the seed of a new cluster endif endif We assumed the threshold value resTOL = 1.0e− 20. The only exception was the set of sample problems for the reinforced concrete section subjected to the compression with bending where resTOL = 1.0e − 10. In the clusterization, we treated a new point as a structure member of the cluster if the following inequality was verified ‖x̂ − xseed‖≤ distTOL where xseed is the seed point of the current cluster. We assumed distTOL = 1.0e−10. The need of the clusterization is fully justified by the table 2 presented below. We can evidently observe four different clusters in the table. First seven examples are connected with the con- crete sections without reinforcement. They are sub- jected to compression and bending. Parameters a are collected in table 1 and the calculated solutions are put to table 3. Consecutive table 4 contains the solutions for two sets associated with the situation, when the sections are reinforced. The results of calculations with the genetic algo- rithm are summarized in table 5 (for the sections without reinforcement) and in table 6 (for sections with reinforcement). Unfortunately, the implementa- tion of the genetic algorithm from the Matlab’s global optimization toolbox has found only one global solu- tion, even for sets where the local minimizer detected more global solutions. Furthermore, the accuracy of the ga solution is definitely poorer compared with that found by the local minimizer. Tables 7 and 8 summarize the results obtained by means of the particle swarm algorithm implementation in the Matlab’s global optimization toolbox. The same comment as for the ga Matlab function is valid for the particle swarm one. 4. Comparison of experimental and numerical results In order to verify the calculated results, 175 mm × 175 mm × 1680 mm (the height) column specimens under eccentric compression were considered, the re- sults of which were presented in detail by Lloyd at al. [16]. The longitudinal steel reinforcement of the columns consisted of three rebars φ12 mm, fyk = 430 MPa, Es = 200 GPa and they were made of concrete fcm = 44.78 MPa, Ecm = 32 GPa. The static diagram and test specimen are shown in Fig. 2. In the above mentioned tests, the following failure loads and corre- sponding eccentricities were measured: P1 = 1476 kN, e1 = 15 mm; P2 = 830 kN, e2 = 50 mm; P3 = 660 kN, e3 = 65 mm. The ultimate strain in concrete at failure was assumed in calculations as −2.4 ‰, which corresponds to the peak stress on the σc −�c diagram (Fig. 1). The values collected in table 9 confirm a good 246 vol. 61 no. 1/2021 Global optimization to predict strains in RC compressed columns x(1) x(2) f(x) -2.7790923390672915e+00 -3.0371456600817055e+00 9.8607613152626476e-32 -2.7790923390376046e+00 -3.0371456601087723e+00 2.2186712959340957e-29 -3.5000157890703703e+00 -4.9999510298477368e-01 1.2325951644078309e-29 -2.7790923390051390e+00 -3.0371456601383064e+00 9.1507865005637369e-29 -2.7790923391531197e+00 -3.0371456600033153e+00 1.9721522630525295e-29 -2.7790923392382747e+00 -3.0371456599104514e+00 8.4692264556708872e-25 -1.2763043306571440e+00 -1.0595130305186657e+00 4.9303806576313238e-31 -1.2763043306472746e+00 -1.0595130305290790e+00 6.9364539396083568e-27 -2.7790923389334106e+00 -3.0371456602237172e+00 5.7569619331116098e-25 -1.2763043307376327e+00 -1.0595130304452089e+00 4.7302072029314920e-28 -1.2763043305691497e+00 -1.0595130306000431e+00 2.0523202525456148e-27 -6.2452876553920056e-01 -3.6491369858491312e+00 3.6484816866471796e-30 Table 2. Sample table of results without clusterization x(1) x(2) f Set 1 a(1) = 1.7157000000000000e-01 a(2) = 1.7170000000000001e-02 -3.5001760272980009e+00 8.9999746818148496e-01 1.6308774769071113e-29 Set 2 a(1) = 1.3345000000000001e-01 a(2) = 2.5219999999999999e-02 -3.4992749633586557e+00 6.9999666861370502e-01 1.9772367181057118e-29 -1.7538910914819157e+00 8.3132002867829691e-01 4.3364238627823002e-29 Set 3 a(1) = 1.0918000000000000e-01 a(2) = 2.6804999999999999e-02 -3.5000264028206489e+00 5.7271595174704082e-01 8.1807341061747740e-29 -1.7532806514095229e+00 6.8025870700809510e-01 2.0954117794933126e-29 Set 4 a(1) = 8.5790000000000005e-02 a(2) = 2.5739999999999999e-02 -3.4999421856987043e+00 4.5001889527019823e-01 2.6437933681383566e-28 -1.7533490610068840e+00 5.3451336316461950e-01 2.9496002284279395e-29 Set 5 a(1) = 7.0650000000000004e-02 a(2) = 2.3689999999999999e-02 -3.5002461261714624e+00 3.7060720525856033e-01 6.1800472650662032e-28 -1.7531021761012342e+00 4.4021717424283136e-01 4.8915539099524771e-29 Set 6 a(1) = 4.4479999999999999e-02 a(2) = 1.7630000000000000e-02 -3.4981410371676795e+00 2.3329954055348942e-01 3.9372571632525573e-27 -1.7548124428383809e+00 2.7700765058540189e-01 8.0396019598500773e-29 Set 7 a(1) = 2.5710000000000000e-02 a(2) = 1.1379999999999999e-02 -3.2265678368900650e+00 1.3352149291135310e-01 4.2148274638856933e-26 -1.9823327248432530e+00 1.5060554010027433e-01 2.5005850693336105e-28 Table 3. Results for non-reinforced concrete subjected to compression with bending 247 Marek Lechman, Andrzej Stachurski Acta Polytechnica x(1) x(2) f Set 1 a(1) = 1.4344499999999999e-01 a(2) = 2.9215499999999998e-02 -2.2782425251491150e+00 7.7232501367591999e-01 2.2709743058168710e-29 -3.4999345396782120e+00 6.9999699585013297e-01 3.2106687592596898e-29 -3.4997710707333067e+00 6.9999415656504049e-01 6.6164575189148840e-13 Set 2 a(1) = 1.2918199999999999e-01 a(2) = 3.4805500000000003e-02 -2.8855483058111950e+00 5.9743506676707159e-01 5.7336006495255961e-30 -3.4999204514197872e+00 5.7272335751221370e-01 3.6503587461192280e-28 -2.8894696731050842e+00 5.9713482908132576e-01 4.5482057578604896e-11 -2.8850298075074248e+00 5.9747474535630740e-01 8.0528227557234545e-13 -2.8819290341073800e+00 5.9771169963840232e-01 3.9597006753181514e-11 Table 4. Results for reinforced concrete subjected to compression with bending x(1) x(2) f Set 1 a(1) = 1.7157000000000000e-01 a(2) = 1.7170000000000001e-02 -2.0055257755736533e+00 9.9999993722360547e-01 4.9367829047643648e-06 Set 2 a(1) = 1.3345000000000001e-01 a(2) = 2.5219999999999999e-02 -1.7533927171625709e+00 8.3145002503412480e-01 1.1683502858156642e-11 Set 3 a(1) = 1.0918000000000000e-01 a(2) = 2.6804999999999999e-02 -3.5000168287943323e+00 5.7271604685964617e-01 4.2733490746564677e-15 Set 4 a(1) = 8.5790000000000005e-02 a(2) = 2.5739999999999999e-02 -3.5000605612775848e+00 4.5002358354890987e-01 1.0830715796115672e-13 Set 5 a(1) = 7.0650000000000004e-02 a(2) = 2.3689999999999999e-02 -1.7531661968763803e+00 4.4020811931145998e-01 1.4944588897123600e-14 Set 6 a(1) = 4.4479999999999999e-02 a(2) = 1.7630000000000000e-02 -3.4974023122097648e+00 2.3328811178613765e-01 1.9405016478273012e-13 Set 7 a(1) = 2.5710000000000000e-02 a(2) = 1.1379999999999999e-02 -3.8846165545827720e+00 1.3993640821435832e-01 1.8945631674275027e-08 Table 5. Results for non-reinforced concrete subjected to compression with bending obtained by Matlab’s genetic algorithm function x(1) x(2) f Set 1 a(1) = 1.4344499999999999e-01 a(2) = 2.9215499999999998e-02 -2.2782188146226554e+00 7.7232945260092078e-01 1.9240836428670515e-14 Set 2 a(1) = 1.2918199999999999e-01 a(2) = 3.4805500000000003e-02 -2.8837525833446813e+00 5.9758208926874357e-01 1.0366597163520163e-11 Table 6. Results for reinforced concrete subjected to compression with bending obtained by Matlab’s genetic algorithm function 248 vol. 61 no. 1/2021 Global optimization to predict strains in RC compressed columns x(1) x(2) f Set 1 a(1) = 1.7157000000000000e-01 a(2) = 1.7170000000000001e-02 -3.5002095622539371e+00 8.9992251419894265e-01 1.1469551926718323e-10 Set 2 a(1) = 1.3345000000000001e-01 a(2) = 2.5219999999999999e-02 -3.4988243017354601e+00 6.9999134440783828e-01 8.0833419633913090e-12 Set 3 a(1) = 1.0918000000000000e-01 a(2) = 2.6804999999999999e-02 -3.5005599263496445e+00 5.7271548379815385e-01 9.7034824624575985e-12 Set 4 a(1) = 8.5790000000000005e-02 a(2) = 2.5739999999999999e-02 -3.4996043294305919e+00 4.5005795509525537e-01 4.3542554256757773e-11 Set 5 a(1) = 7.0650000000000004e-02 a(2) = 2.3689999999999999e-02 -3.5287486517409783e+00 3.7098837709011562e-01 3.6828714611703899e-09 Set 6 a(1) = 4.4479999999999999e-02 a(2) = 1.7630000000000000e-02 -1.8974393623306212e+00 2.6652417435303338e-01 1.4855280597896296e-08 Set 7 a(1) = 2.5710000000000000e-02 a(2) = 1.1379999999999999e-02 -3.2244553544557881e+00 1.3346506124150329e-01 5.6375712128260839e-11 Table 7. Results for non-reinforced concrete subjected to compression with bending obtained by Matlab’s particle swarm function x(1) x(2) f Set 1 a(1) = 1.4344499999999999e-01 a(2) = 2.9215499999999998e-02 -3.5011205692965111e+00 6.9993319470774185e-01 1.0180508409446182e-10 Set 2 a(1) = 1.2918199999999999e-01 a(2) = 3.4805500000000003e-02 -3.5003741045884293e+00 5.7271843442355697e-01 8.6633784199345278e-13 Table 8. Results for reinforced concrete subjected to compression with bending obtained by Matlab’s particle swarm function Experimental Numerical Failure load Pi [kN] �cu = �c1 � ′ ξ strain in �1/�2 [‰] eccentricity ei [mm] [‰] [‰] steel �s [‰] P1 = 1476; e1 = 15 -2.4 -2.20 -2.20 / -0.35 P2 = 830; e2 = 50 -2.4 -2.20 0.63 0.92 P3 = 660; e3 = 65 -2.4 -2.20 0.49 1.85 Table 9. Comparison of the experimental and numerical results – 1 249 Marek Lechman, Andrzej Stachurski Acta Polytechnica Experimental Numerical Failure load Pi [kN] �cu = �c1 �1 �2 eccentricity ei [mm] [‰] [‰] [‰] P1 = 1548; e1 = 0 -2.1 -2.1 -2.1 P2 = 1386; e2 = 16 -2.1 -1.95 -0.31 P3 = 1098; e3 = 32 -2.1 -1.95 -0.12 Table 10. Comparison of the experimental and numerical results Figure 2. Static diagram and test specimen conformity between the numerical solution and the ex- perimental data given by Lloyd and Rangan [16]. It is worth noting that the theoretical values are lower than those obtained from the experiment due to neglecting the effect of confinement of the column. As the next example, the results of tests conducted by Trapko at al. [20] on unstrenghtened column spec- imens 200 mm × 200 mm × 1500 mm (the height) under eccentric compression were analyzed. The lon- gitudinal reinforcement of the column consisted of two rebars φ12 mm, steel grade A-IIIN, fyk = 608 MPa, Es = 224 GPa and the transverse reinforce- ment consisted of stirrups φ6 mm, steel grade A-I. The columns were made of concrete fcm = 31.9 MPa. Ecm = 31 GPa. The failure loads and the correspond- ing eccentricities were determined in these tests as: P1 = 1548 kN, e1 = 0 mm; P2 = 1386 kN, e2 = 16 mm; P3 = 1098 kN, e3 = 32 mm. The ultimate strain in concrete at failure was assumed in calculations as −2.1 ‰. The characteristic failure mechanisms of the tested specimens occurred in the form of crushing the concrete in the upper part of the structure members and yielding the longitudinal reinforcing steel. A good conformity between the calculated and experimental results are confirmed by the values collected in Ta- ble 10. In author’s opinion, further experimental work is needed concerning the post-critical behaviour of RC columns under eccentric compression. 5. Conclusions and comments Our numerical results have confirmed that the elab- orated analytical deformation model (taking into ac- count the effect of concrete softening) may be used to determine the strains in rectangular cross-sections of RC compressed structure members. It can be applied to predict the behaviour of such structure members. The current Matlab’s implementations of the global optimization algorithms (ga and particle swarm) do not seem to be suitable for our application. In genetic algorithm (ga), elitism is used (part of the previous population survives to the next one). But it does not ensure finding the correct solution. The particle swarm procedure also does not guarantee the compu- tation of the correct solution. Of course, we may tune some of their parameters, but we do not expect to gain much from that. Our experiments with the local search method have frequently shown the existence of several global minima. 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Springer Verlag, Berlin, Heidelberg, Germany, 1989. 252 http://dx.doi.org/10.1063/1.5019133 Acta Polytechnica 61(1):242–252, 2021 1 Introduction 1.1 Motivation to study the strains in RC compressed structure members 2 Formulation of the equilibrium equations 2.1 Equations for strains in the rectangular sections 2.1.1 Equations for sections wholly in compression 2.1.2 Section under combined compression with bending 3 Computational solution and numerical results 4 Comparison of experimental and numerical results 5 Conclusions and comments References