Modified parameter-setting-free harmony search (PSFHS) algorithm for optimizing the design of reinforced concrete beams

Abstract

The design of RC members with nontraditional methods is demanding due to the large number of unknown variables inherent in the design process. The complexity of the RC beam design optimization problem has led to many oversimplified models, so that the current metaheuristic search algorithms can deal with it efficiently. In this paper, the optimization design model of RC beams has been introduced by new design variables, while augmented others; accordingly enhanced the solving algorithm. A new enhanced parameter-setting-free harmony search algorithm has been proposed to solve the model. Furthermore, the tackled optimization objectives were the minimization of cost, weight, and cost–weight simultaneously for designing regular or high-strength concrete beams.

Appendices

Appendix A: Flexural reinforcement

In this section, a sound direct, computationally less expensive, and a non-iterative bending capacity algorithm has been derived. The following approach, for doubly reinforced sections, replaces the conventional and iterative strain compatibility method (Beeby and Narayanan 2005; Bhatt et al. 2018) in this paper. The first attempt of this algorithm is to evaluate the neutral axis position, refer to Fig. 16, assuming that both the strains in compression and tension reinforcement yielded simultaneously, while the concrete reached its crushing strain, see (A.1) to (A.4).

Fig. 16

Doubly reinforced rectangular beam section

$$ T = A_{s} f_{yd} $$

(A.1)

$$ C_{c} = \eta f_{cd} \lambda X b $$

(A.2)

$$ C_{s} = A^{\prime}_{s} f_{yd} $$

(A.3)

$$ X = \frac{f_{yd}(A_{s} - A^{\prime}_{s})}{\lambda \eta f_{cd} b } $$

(A.4)

After that, the tension and compression steel strains are evaluated from (A.5) and (A.6) respectively, that were derived in the first place form the triangulation of the strain distribution assumed in Fig. 16.

$$ \epsilon_{s} = \epsilon_{cu3}\left( \frac{d}{X}- 1\right)~\leq \frac{f_{yd}}{E_{s}} $$

(A.5)

$$ \epsilon^{\prime}_{s} = \epsilon_{cu3}\left( 1- \frac{d^{\prime}}{X}\right) ~\leq \frac{f_{yd}}{E_{s}} $$

(A.6)

If the resulted strain in one of them or both, (A.5) and (A.6), is less than the yield strain it means that the acting forces on the cross section are not in equilibrium; the first assumption is violated. Attempt two is required, where the general solution is described by (A.7), and varies as described in the following cases:

$$ X = \frac{\sqrt{A + B + C + D} + E + F}{2\lambda \eta b f_{cd}} $$

(A.7)

• Only the strain of the compression reinforcement yielded. In this case, the tension force, T, in steel must be modified in (A.1) to be T = A_sE_s𝜖_s, and by solving for the neutral axis depth, the coefficients A → F of the general real positive root are depicted in (A.8) to (A.13);

  $$ A = A^{\prime2}_{s} f_{yd}^{2} $$

  (A.8)
  $$ B = 2000 A^{\prime}_{s} A_{s} E_{s} f_{yd} \epsilon_{cu3} $$

  (A.9)
  $$ C = 10^{6} {A^{2}_{s}} {E^{2}_{s}} \epsilon_{cu3}^{2} $$

  (A.10)
  $$ D = 4000 \lambda \eta b d f_{cd} A_{s} E_{s} \epsilon_{cu3} $$

  (A.11)
  $$ E = -10^{3} A_{s} E_{s} \epsilon_{cu3} $$

  (A.12)
  $$ F = -A^{\prime}_{s} f_{yd} $$

  (A.13)

• Only the strain in tension reinforcement yielded. In this case, the compression force in steel, C_s, must be modified in (A.3) to be $C_{s} = A^{\prime }_{s} E_{s} \epsilon ^{\prime }_{s}$ and (A.14) to (A.19) describe the coefficients to find the real positive root of (A.7);

  $$ \begin{array}{@{}rcl@{}} A &=& A^{\prime2}_{s} {E_{s}^{2}} \epsilon_{cu3}^{2} \end{array} $$

  (A.14)
  $$ \begin{array}{@{}rcl@{}} B &=& -2000 A^{\prime}_{s} A_{s} E_{s} f_{yd} \epsilon_{cu3} \end{array} $$

  (A.15)
  $$ \begin{array}{@{}rcl@{}} C &=& {A^{2}_{s}} f^{2}_{yd} \end{array} $$

  (A.16)
  $$ \begin{array}{@{}rcl@{}} D &=& 4000 \lambda \eta b d^{\prime} f_{cd} A^{\prime}_{s} E_{s} \epsilon_{cu3} \end{array} $$

  (A.17)
  $$ \begin{array}{@{}rcl@{}} E &=& A_{s} f_{yd} \end{array} $$

  (A.18)
  $$ \begin{array}{@{}rcl@{}} F &=& -10^{3} A^{\prime}_{s} E_{s} \epsilon_{cu3} \end{array} $$

  (A.19)

• Both steel strains do not yield. This case is considered impractical and out of the scope of this paper, besides, a well-controlled design algorithm can skip such a case easily.

Note that (A.7) to (A.19) have been derived where the input units for stresses are in (MPa); hence f_cd and f_yd, and the moduli of elasticity are expressed in (GPa). In addition, the provided areas for both tension and compression reinforcements are in (mm²) and all the geometrical dimensions such as b, d, and d^′ are in (mm). Eventually, the resulting depth of the neutral axis was declared in (mm).

Appendix B: Small-scale benchmarking

In this section, a small-scale benchmarking was used to compare the modified PSFHS with the Original Harmony Search (OHS) algorithm (Geem et al. 2001), the standard PSFHS (Geem and Sim 2010), and Cuckoo Search (CS) algorithm (Yang and Deb 2010). For this purposes, standard Schwefel function (see Surjanovic and Bingham (2018)) has been chosen with 100 design variables (dimensions).

First, both the standard PSFHS and the proposed modified PSFHS have been compared (refer to Table 17 to see the parameter settings). Figure 17 reveals a comparison between the modified PSFHS and standard PSFHS algorithms in solving the Schwefel function.

Fig. 17

Convergence of the modified vs. the standard PSFHS

Table 17 Parameter settings for the modified and standard PSFHS

Elaborately, the modified PSFHS result for 10 independent runs was − 40, 180.400 ± 229.672, and the best achieved answer was − 40, 510.588. Note that the exact answer of the function, using 100 dimensions, is − 41, 898.290. Regarding the standard PSFHS, the overall result was − 17, 202.608 ± 1, 870.637, and the best run scored was 20,091.100.

The OHS has been tested and adjusted accordingly to solve this problem. Table 18 explains different solutions obtained with different adjustments and evaluations, where it functioned potentially better than the standard PSFHS for the same problem size.

Table 18 Original Harmony Search (OHS) adjustments

Finally, the CS algorithm has been used to solve the same problem in this section. Fine-tuning has been implemented to obtain the best answer using different generation numbers in order to keep the number of computational efforts nearly equal and to get a sounds-fair comparison. Table 19 shows the different results obtained with different settings.

Table 19 Cuckoo Search (CS) algorithm adjustments

Appendix C: Miscellaneous results

In this section, the results of the fourth design case have been shown in detail. Figure 18 illustrates the ultimate and serviceability envelopes as per the Eurocodes (EN 1992-1-1 Eurocode 2 2005; Gulvanessian 2001; du béton 2008; Hibbeler 2018; Logan 2012), while Table 20 explains the results of the maximum deflection for each span taking into account the shrinkage and creep effects for long-term time spans.

Fig. 18

Design case (4)—Ultimate Limit State (ULS) and Serviceability Limit States (SLS) envelops

Table 20 Design case (4)—the final deflection results obtained

1.1 C.1 Replication of results

The datasets generated during the current study are not publicly available due to the dependency of such algorithms on the time seeds used by the pseudo-random number generator to generate the results. However, we published C++ 14, Python 3.7, and Matlab codes that allow the reader to reproduce and replicate the results of the Schwefel function (refer to Appendix B). The complete manuscripts, source codes, that were generated during the current study are not publicly available; they could be subjected to commercial copyrights soon, yet they are available from the corresponding author as per reasonable request. The abovementioned codes shall be found on the online repository: https://doi.org/10.5281/zenodo.2573261 (Shaqfa and Orbán 2018).