Shear wall layout optimization of tall buildings using Quantum Charged System Search

Abstract

This paper presents a developed meta-heuristic algorithm to optimize the shear walls of tall reinforced concrete buildings. These types of walls are considered as lateral resistant elements. In this paper, Quantum Charged System Search (QCSS) algorithm is presented as a new optimization method and used to improve the convergence capability of the original Charged System Search. The cost of tall building is taken as the objective function. Since the design of the lateral system plays a major role in the performance of the tall buildings, this paper proposes a unique computational technique that, unlike available works, focuses on structural efficiency or architectural design. This technique considers both structural and architectural requirements such as minimum structural costs, torsional effects, flexural and shear resistance, lateral deflection, openings and accessibility. The robustness of the new algorithm is demonstrated by comparing the outcomes of the QCSS with those of its standard algorithm.

Keywords

Quantum Charged System Search / shear wall / layout optimization / tall buildings

Introduction

Overview

Shear walls are the most commonly used among lateral load-resistant systems. In earthquakes, shear wall structures were shown to perform well [¹]. A concrete shear wall is a structural resistance system combined with reinforcement concrete to withstand the effects of lateral load acting as vertical cantilevers on a structure that may be located in the center of a large building—often surrounding an elevator shaft or a stairwell and the elements that continue vertically along the building. Because of the rigidity and strength of the plane and the effects on the spatial arrangement in the plan, lateral loads and gravity loads can be resisted simultaneously [²]. Therefore, their location in terms of structure and architecture must be considered and predicted. The common goal of optimization in civil engineering is to gain the cost impressive solution in existing resources to achieve the tranquility of society. Since economic issues are the first word in all areas, it is very important to use optimization methods to reduce costs in the construction of such high-rise buildings.

Literature review

Optimization of structures is important and there are several methods for this purpose. Socha and Dorigo [³] presented an uncomplicated way of expanding Ant Colony Optimization (ACO) to continuous domains. Eberhart and Kennedy [⁴] optimized different functions using Particle Swarm Optimization (PSO), a very straightforward method that appears to be useful in optimizing a wide variety of problems. Kaveh and Talatahari [⁵] have developed another robust metaheuristic optimization technique called Charged System Search (CSS), inspired by Coulomb's electricity laws and Newton's mechanical movement laws. Geem et al. [⁶] proposed Harmony Search (HS). To improve the robustness of the CSS algorithm, Kaveh and Talatahari [⁷] introduced a hybrid PSO and CSS algorithm. They have shown that in constructing open channel systems, the efficiency of the CSS approach is greater than the ACO and the genetic algorithm (GA). Erol and Eksin [⁸] proposed a method of optimization inspired from the Big Bang and Big Crunch Theory.

The other researchers used different methods to optimize their structures. Hayalioglu and Degertekin [⁹] worked with the GA to minimize the structural design costs of steel frame buildings. Chan [¹⁰] used the Optimality Criteria (OC) method to demonstrate an optimal sizing method for the lateral rigidity design of high-rise steel and concrete structures. Degertekin and Hayalioglu [¹¹] used the HS-based algorithm to calculate the lowest cost of the steel frame design. They also compared the results of the GA and HS algorithms, indicating that HS is more efficient. Chan and Wong [¹²] used the OC-GA hybrid method to simultaneously optimize topological and element sizes for two tall steel building frameworks. Liang et al. [¹³] used a performance-based optimization technique to optimize the bracing system's topology design including minimum weight and maximum rigidity in multi-story buildings under lateral loads. Liu [¹⁴] presents a structural optimization method that considers cost as its objective function in order to strengthen steel frames against a progressive collapse. PSO is also used by Shamshirband et al. [¹⁵] in order to calculate the depth of scour in vicinity of bridge pier. Talatahari et al. [¹⁶] propose a new risk-based approach to optimizing arch dams using three metaheuristic algorithms, including CSS, Big Bang-Big Crunch (BBBC) and CSS-BBBC methods. Talatahari et al. [¹⁷] proposed an efficient algorithm for shape optimization of double curvature arch dams using The Magnetic Charged System Search (MCSS), as well.

For concrete structures, Kaveh and Behnam [¹⁸] optimized three dimensional multi-story reinforced concrete structures by applying limitations consist of the slenderness ratio of compression members, the maximum acceptable structural drift and the natural frequency of the structure using the CSS and enhanced CSS. Camp et al. [¹⁹] proposed a design process to reduce the cost of concrete buildings using the GA. There are also a series of studies for optimal seismic design of concrete structures, including Kaveh and Zakian [²⁰] accomplished optimal seismic design of dual reinforced concrete systems (shear wall frame as an optimization issue and use of CSS as an optimizer).

This kind of work is remarkable, but addressing the formation of the lateral resistance mechanism in design is also crucial from the architectural point of view. From the architectural point of view, the topology of floor plan layout was investigated by some researchers such as: Peng et al. [²¹] proposed a technique based on integer programming for the design of computational networks for layout computing, such as street networks and floor plan construction. Terzidis [²²] generates a computer application called autoPLAN that develops architectural layouts out of a structure program and a site by using graph theory. Shekhawat [²³] presented an explicit algorithm for optimal using of spaces inside the rectangle that gives acceptable outcomes. Aminnia and Hosseini [²⁴] were looking for obtaining the most reliable seismic behavior in multi-level buildings using shear walls which are shaped like T, Z, U or L as lateral systems. Ghasemi et al. [²⁵] represented a computational method in order to optimize the topology of multi-material based flexoelectric composites. Also, Ghasemi et al. [²⁶,²⁷] provided some numerical examples which demonstrate the considerable improvement in coefficient of electromechanical coupling by help of topology optimization. Nguyen et al. [²⁸] showed the efficiency of the topology optimization in the metamaterials design. Nanthakumar et al. [²⁹] has validated a significant increase in energy conversion using topology optimization.

Due to a large number of elements that are available in large structures, this type of building optimization needs a lot of time to run. It is known that optimization algorithms, which require significant iterations, cannot be used to optimize large structurers to achieve good results. For this reason, Kaveh and Talatahari [³⁰,³¹] compared the features of the CSS algorithm with other standard and advanced meta-heuristics in order to demonstrate the efficiency of the CSS. They used some standard frame examples that they had optimized before. It can be observed that comparing the CSS method with other heuristics demonstrates the effectiveness of the present algorithm.

As seen in most of the current papers in this field of study, optimization is considered either in terms of architecture or structure. Although considering both together is less common, Zhang and Mueller [³²] have worked on a new method of developing diverse shear wall layouts which take both structural and architectural aims into consideration. They tried to simplify the structural formulas, while we brought together the formulas with their details in this article. They also handled a computational view of a wide range of possible layouts by reducing the diversity of layout topologies and grouping the walls. In this regard, although it takes time, we allow the program to group the walls without any limitation, especially for every possible layout our program can plot interaction diagrams. This allows the program to evaluate possible shear wall configurations freely by considering minimum structural cost, effects of torsion, flexural and shear strength, lateral deflection, openings and accessibility. In addition, CSS algorithm and its new variant are used to optimize our objective function, while Zhang and Mueller [³²] used a revised algorithm of evolution. The objective function, we used in this article, has a series of differences with previous works. Similarly, reinforcement is presumed to have uniform distribution and cross-section in the aforementioned article and this article.

The research shown in this paper attempts to generate a computational technique to determine the best location of the shear walls in the plan. This will be achieved by considering architectural and structural requirements with optimal structural weight under structural analysis using a developed CSS-based method, so-called Quantum Charged System Search (QCSS). Since finding the best places for shear walls has a great influence on various design aspects, it is important to use an efficient way of optimization. QCSS was introduced as a new optimization algorithm by adding several aspects to the CSS. Such features carry some advantages compared to CSS, which contribute to the rapid performance results convergence.

Overall overview

Definition of example

This article provides an order to discover the optimal layout formation of shear walls with regard to structural costs, considering structural and architectural constraints. In this research, the input plan is assumed to be a 2D rectangular building plan. First, the plan is divided into a four-square mesh. Meanwhile, each of the edges shows a possible location of the shear wall. This mesh is shaped as an unusual ground structure, similar to optimizing the topology of truss structures [³³]. Shear walls on each of the edges can be activated or not. The aim is to optimize structural cost regarding torsion, flexural, shear and drift as structural penalties. In addition, accessibility and openings are considered as basic architectural requirements.

Definition of the optimization design problem

The cost function for this article is considered as below:

(1)\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}:{Cost}_{\mathrm{t}}={\displaystyle \sum _{i=1}^n{V}_{i,\mathrm{t},\mathrm{A}\mathrm{c}}}·{Cost}_{\mathrm{A}\mathrm{c}}+V_{i,\mathrm{t},\mathrm{A}\mathrm{c}}·{Cost}_{\mathrm{A}\mathrm{s}},

where Cos{t}_{\mathrm{t}} is the total cost of the shear walls of the structure; n is the number of floors, V_{i,t,Ac}is the total volume of concrete for each floor, Cos{t}_{\mathrm{Ac}}represents the price of producing 1m³ concrete,V_{i,\mathrm{t},As} is the total volume of steel for each floor, Cos{t}_{\mathrm{As}}represents the price of producing 1m³ steel. Inequality constraints that restraint design variable dimensions and structural responses must be assured in optimal design of the building [³⁴].

Ground structure

Dorn et al. [³³] proposed a ground technique structure that would select the best outcome among all possible structures. This technique is used extensively to optimize topology, especially for trusses with a limited number of bars and nodes. The aforementioned technique was enhanced by Hagishita and Ohsaki [³⁵] to new version named the growing ground structure.

Approximately all the layout of the rectangular structures could be divided into four-square mesh. In the meantime, each of the edges can represent a shear wall. As we can see in (Fig. 1(a)) each of the edges has its own index and each one shows the presence or absence of shear walls by using a binary string (“0 “depicts deactivated and”, 1 “depicts activated, as shown in Fig. 1(b)). This binary string conveys the information about the structure topology. Consequently, a unique binary string could display all the possible position for shear walls in the layout. To make the calculation simple, all potential shear walls were considered either vertical or horizontal.

Assumptions and rules for the analysis of reinforced concrete structures

In this paper, we try to consider both the rigidity and strength of the plan for reinforced concrete shear walls. This included a technique of uncomplicated design with a cracking hypothesis. In the detailed design step, some additional goals such as shrinkage and creep should be considered later.

Some features, including material properties, should be selected in advance due to its important role in controlling drift and strength constraints. In this article, these two are impressive criteria. Table 1 shows the used material properties. Reinforcement in this paper is assumed to have the uniform cross-section and distribution due to the fulfillment of economic and programming purposes. It can be observed that in each generation of optimization method, some layouts in different positions consisting of different shear walls can be obtained. As shown in Fig. 2, some connected shear walls create a group. Lateral loads are divided according to the relative stiffness of the shear walls’ group [³⁶].

According to its relative stiffness, lateral loads are divided among the group of shear walls by in-plane action of the rigid floor plate diaphragm.

Each group of shear walls must satisfy the following equations [³⁷]:

M_{\mathrm{u}}\le \varphi M_{\mathrm{n}},

P_{\mathrm{u}}\le \varphi P_{\mathrm{n}},

V_{\mathrm{u}}\le \varphi V_{\mathrm{n}}\mathrm{.}

In these equations, M_u, P_u, and V_u are ultimate bending moment, ultimate axial load and ultimate shear load, respectively. Also, M_n, P_n and V_n are nominal bending moment, nominal axial load and nominal shear load, respectively. In all above Eqs. \varphi as the reduction factor has a constant value for each one of equations.

According to American Concrete Institute (ACI) [³⁸], the main assumptions in the design and control of reinforced concrete could be mentioned as follows (Fig. 3).

• After bending, the sections perpendicular to the bending axis remain flat, which are plane before bending.

• The reinforcement and concrete have a good adhesion, so the amount of strain in the concrete and the reinforcement are the same.

• The strain in concrete and reinforcement is assumed to be directly dependent on the distance from the neutral axis.

• The concrete reaches its failure when the strain rate becomes 0.003 in concrete.

• Concrete tensile strength is ignored.

• The tension in concrete and reinforcement can be calculated with regard to the strain rate in concrete and also the stress-strain diagrams in steel.

• It is assumed that the region of pressure distribution for concrete is rectangular, known as the Whitney block.

To determine whether Eqs. (2) and (3) are satisfied, the interaction diagram must be developed. Irregularities in the group of shear walls do not make any differences in the process of developing interaction diagrams compared to regular shear walls. Six stages below [³⁹] involve the development of an interaction diagram (Fig. 4):

• Point1: In this point there is only compression;

• Point 2: The amount of stress on bars could be assumed zero (f_s = 0);

• Point 3: The amount of stress on bars near tension face of member could be assumed 0.5 f_y (f_s = –0.5 f_y);

• Point 4: The amount of stress on bars near tension face of member could be assumed f_y (f_s = –f_y);

• Point 5: In this point there is only bending;

• Point 6: In this point there is only tension;

where f_s, f_y are the stress in steel rebar and yield stress in rebar, respectively. Every point outside the curve is considered flexural failure.

Objective function

The purpose of this article is to optimize structural costs. In addition, to find the best answer, certain constraints such as flexural, torsion, shear and drift are added to the program:

Minimize: Cost of the structure

\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{int}\mathrm{s}:D_{\mathrm{c}{\mathrm{m}}_{-}\mathrm{c}\mathrm{s}}\le D_{\mathrm{allowable}}

M_{\mathrm{u}}\le \varphi M_{\mathrm{n}},P_{\mathrm{u}}\le \varphi P_{\mathrm{n}},V_{\mathrm{u}}\le \varphi V_{\mathrm{n}},drift\le {drift}_{\mathrm{allowable}},

S{W}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}=\mathrm{0.}

In the equations above, M_u, P_u, and V_u are the ultimate moment of bending, ultimate axial load and ultimate shear load respectively. Also, M_n, P_n and V_n are the nominal bending moment, nominal axial load and nominal shear load respectively. \varphias the reduction factors. D_allowable is the maximum allowable distance, while D_{cm_cs} is the space between the stiffness and mass center. drift is horizontal movement of structure at the last floor. drift_allowable is 6.4 cm (0.02×height of the one story). If during optimization process, one generation containing a layout with closed area, SW_{closed area} would be 1. This would be out of architectural requirement constraints criteria.

To find the best location for shear walls, an objective function used in the optimization process is shown below:

(5)f\left(v\right)=Cost·(1+(C_{D_{\mathrm{C}\mathrm{x}}}+C_{D_{\mathrm{C}\mathrm{y}}})+(C_{D_{\mathrm{c}\mathrm{m}-\mathrm{c}\mathrm{x}\mathrm{s}}}+C_{D_{\mathrm{c}\mathrm{m}-\mathrm{c}\mathrm{s}\mathrm{y}}})+(C_{{\mathrm{V}}_{\mathrm{x}}}+C_{{\mathrm{V}}_{\mathrm{y}}})+C_{{SW}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}}+(C_{{drift}_{\mathrm{x}}}+C_{{drift}_{\mathrm{y}}}\left)\right),

where Cost is the total cost for the shear walls of the structure; C_{D_{Cx}}and C_{D_{Cy}}are the maximum value of the demand division into capacity by using X and Y interaction diagrams; C_{D_{\mathrm{cm}-\mathrm{c}\mathrm{s}\mathrm{x}}} and C_{D_{\mathrm{cm}-\mathrm{c}\mathrm{s}\mathrm{y}}} are the spaces between the stiffness and mass center; C_{\mathrm{Vx}} and C_{\mathrm{Vy}}are maximum shear value of the demand division into capacity for X and Y directions; S{W}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}is the number of groups of shear walls which generates a closed-off area; C_{drif{t}_{\mathrm{x}}} and C_{drif{t}_{\mathrm{y}}} are horizontal movements in X and Y directions, respectively.

The individual parts of the objective function f(v)are illustrated as below.

1)C_{D_{\mathrm{c}\mathrm{m}-\mathrm{c}\mathrm{s}}}: constraint function for torsional effect.

Torsional damage is one of the most significant parameters. In several previous earthquakes, this type of damage gave rise to serious collapse of various types of structures. Inconsistency in mass distribution, stiffness and strength distribution are the main reasons that give a rise to torsional damage [⁴⁰]. Unmatched sidelong movements in the components of the structures are due to the results of floor twisting in ductile structures. Therefore, the capacity of elements’ sidelong ductility would be greater than the capacity of systems’ sidelong ductility. Design codes include related needs to be regarded for the effects of torsion. The effects of torsion could be diminished by coinciding the center of stiffness and mass [⁴¹].

The following Eqs. are used to calculate both stiffness and mass centers:

(6)(x_{\mathrm{c}\mathrm{s}},y_{\mathrm{c}\mathrm{s}})=(\frac{\sum k_{\mathrm{y}}x}{\sum k_{\mathrm{y}}},\frac{\sum k_{\mathrm{x}}y}{\sum k_{\mathrm{x}}}),

(7)(x_{\mathrm{m}},y_{\mathrm{m}})=(\frac{\sum Ax}{\sum A},\frac{\sum Ay}{\sum A}),

where k_x and k_y is considered to be each shear wall group’s lateral stiffness in the direction of X and Y, respectively; each shear walls group’s area is shown by A; (x, y) are the center coordinates of each shear wall member measured from the origin [⁴²].

D_cm-csx is calculated by the Eq. below when the lateral load is in the Y direction:

(8)D_{\mathrm{c}\mathrm{m}-\mathrm{c}\mathrm{s}\mathrm{x}}=x_{\mathrm{c}\mathrm{s}}-x_{\mathrm{m}}\mathrm{.}

D_cm-csy is calculated by the Eq. below when the lateral load is in the X direction:

(9)D_{\mathrm{c}\mathrm{m}-\mathrm{c}\mathrm{s}\mathrm{y}}=x_{\mathrm{c}\mathrm{s}}-x_{\mathrm{m}}\mathrm{.}

If either D_cm-csx or D_cm-csy were less than D_allowable, the number 0 would be substituted in objective function. D_allowable is regarded to be optimal distance which is 30.5cm in this paper.

2) C_DC: Constraint function for interaction diagram

The groups of shear walls are supposed to withstand the lateral loads which stem from wind or earthquake considering their stiffness. Also, it must resist the vertical loads regarding their loading area. M_uiand P_uimust be calculated to determine whether the group of shear walls checks the demand for capacity ratio. To calculate M_ui, the torsional moment (M_i) and shear force (V_i) of each story are calculated and distributed among the group of shear walls based on its relative stiffness [⁴³] (k_j, I_j). Afterward M_ui, overturning moment can be calculated. h_i is height of i^th story; I_j is considered to be the moment of inertia of each shear wall group.

(10)V_{ui}=V_i{\displaystyle \frac{k_j}{{\displaystyle \sum k_j}}}\pm M_i{\displaystyle \frac{I_j}{{\displaystyle \sum I_j}}}

(11)M_{ui}=V_{Ti}\cdot h_i

(12)P_{ui}=Q_{\mathrm{d}}·A_{{loading}_i}

where, P_ui is the axial force divided to each shear wall group, Q_d is the combination load on a layout,A_{loadin{g}_i}is the loading area of i^th shear wall group.

Generally, if an element is exposed to the combination of flexure and axial load, it must be considered and designed as compression components [⁴¹]. For the flexural check of each shear walls’ group, interaction diagram must be provided (as described in section 2.3). One can check if the point (P_ui, M_ui) is within the permitted graph area after providing the interaction diagram or not. Interaction diagram can be provided even in irregular shape for any type of group of shear walls that requires complex calculation. Flexural strength must be checked for each possible orientation (there are four possible directions for rectangular layout buildings) since the formation of each group of shear walls is not symmetric. If and only if the flexural strength check is met by all the shear wall groups, it will be considered as one of the potential candidates for possible shear wall locations. If the demand-to-capacity ratio is less than one, the zero will be placed in objective function, otherwise no change will be required.

3) C_v: Constraint function for shear force

For buildings with a large height-to-length ratio, the incidence of structural failure as a result of the shear strength of walls is less prevalent. Simple technique is used in this paper to check a shear strength. To resist lateral load in one direction, it is only necessary to consider shear walls in that direction. Because the shear strength of each shear wall group is calculated based on its relevant area in its direction, it will be sufficient to check the shear strength in one rectangular shear wall member [³²]. Shear force (V_ui) can be obtained by Eq. (10).

Regarding ACI [²²], shear strength of concrete is calculated by:

(13)\varphi V_{\mathrm{c}}=\varphi 2\sqrt{{f^{\prime }}_{\mathrm{c}}}h0.8l_{\mathrm{w}}\mathrm{.}

In addition, a part of shear strength is resisted by horizontal reinforcement computed by:

(14)\varphi V_{\mathrm{s}}=\varphi \frac{A_{\mathrm{v}}{f}_{\mathrm{y}}d}{s_2}\mathrm{.}

Moreover, the summation of above Eqs. should be lesser than available shear forces (V_ui) [²⁵]:

(15)\varphi V_{\mathrm{n}}=\varphi V_{\mathrm{c}}+\varphi V_{\mathrm{s}}\ge V_{ui},

where A_V is the whole area of the horizontal shear reinforcement which their separation distance is shown by s₂; \varphi= 0.75, f_y = 400MPa and the effective horizontal length of shear wall is shown by d. Shear strength must be checked for each potential orientation (there are two potential directions for rectangular layout buildings) as the formation of each shear wall group is not symmetric. If and only if all shear wall groups satisfy the shear strength check, it will be considered as one of the potential candidates for possible shear wall locations. If the demand-to-capacity ratio is less than one, the zero will be placed in objective function, otherwise no change will be required.

4) C_drift: Constraint function for lateral forces

Lateral forces such as winds and earthquakes can cause lateral deflections that must remain within acceptable range. This enables designers to reduce the damage of structural and non-structural members [⁴⁴].

(16){\delta }_i={\displaystyle \frac{P{h}^{\mathrm{3}}}{\mathrm{3}EI}}+{\theta }_j(h_i-h_j),

(17){\theta }_j={\displaystyle \frac{P{h}^{\mathrm{2}}}{\mathrm{2}EI}},

where E is concrete's elastic modules; P is the shear force for each story; I is the moment of inertia of the whole building; h is the stories’ height from ground. The principle of superposition is used to calculate deflection of the building influenced by lateral loads (as shown in Fig. 5). drift\le drif{t}_{\mathrm{allowable}}must be satisfied. Like other constraints such as shear strength check and flexural strength check, this limitation should also be checked for all possible directions. If drift\le drif{t}_{\mathrm{allowable}}is satisfied, the zero will substitute in objective function, otherwise no change will be applied.

5) C_{S{W}_{\mathrm{closed}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}}Constraint function for architectural requirements

There must be a constraint that prevents the optimization method from creating a group of shear walls with a closed area. This constraint, which considers the architectural requirement in the design procedure, is provided by controlling the ratio of number of different coordinates (n_{\mathrm{d}\mathrm{i}\mathrm{f}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}) and number of members (n_{\mathrm{m}\mathrm{e}\mathrm{n}}) in a shear wall group.

(18)S{W}_{\mathrm{closed\; area}}={\displaystyle \frac{n_{\mathrm{dif}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}}{n_{\mathrm{mem}}}}.

If the above ratio is greater than one, the layout will not have a closed area. C_{S{W}_{\mathrm{closed}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}} in objective function will be the summation of the S{W}_{\mathrm{closed\; area}} for each shear wall group which will be substituted.

Optimization method

Charged System Search (CSS)

The CSS method is a meta-heuristic algorithm that was first developed by Kaveh and Talatahari [⁵]. The pseudo-code for the aforementioned method could be mentioned as follows:

Step 1: CSS starts the initial population of solutions by randomly altering and selecting them as Charged Particles (CP). Due to each CP has its own enlargement of charge (q_i), electrical field was created surrounding its space. The enlargement of the charge is calculated regarding the quality of its solution, as follows:

(19)q_i={\displaystyle \frac{w(i)-w_{\mathrm{worst}}}{w_{\mathrm{best}}-w_{\mathrm{worst}}}},i=\mathrm{1},\mathrm{2},\dots ,N,

where w_best and w_worst are the best and the worst fitness of all the CPs; w(i) represents the objective function value; and N is the whole number of CPs. The separation distance r_ijbetween any of two CPs is measured as follow:

(20)r_{ij}={\displaystyle \frac{\Vert X_i-X_j\Vert }{\Vert (X_i+X_j)/\mathrm{2}-X_{\mathrm{best}}\Vert +\epsilon }},

where X_i and X_j are the the positions of the ith and jth CPs, X_best is the position of the finest present CP, and e is a small positive number.

In the search space, the primary locations of each CP are calculated randomly. It is assumed that the primary velocities of charged particles are zero.

(21)x_{i,j}^{(\mathrm{0})}=x_{i,\min }+rand\cdot (x_{i,\max }-x_{i,\min }),i=\mathrm{1},\mathrm{2},\dots ,n.

Step 2: Creation of CM. A number of finest CPs and their enlargement of objective function are stored in the charged memory (CM).

Step 3: The probability of attraction for each CP is considered as the following Eq.:

(22)p_{ij}=\{\begin{array}{l}\mathrm{1},{\displaystyle \frac{w(i)-w_{\mathrm{best}}}{w(j)-w(i)}}>\mathrm{rand}\vee w(j)>w(i)\hfill \\ \mathrm{0},\mathrm{else}\hfill \end{array}

Step 4: The enlargement of the consequent electrical force acting on each CP is calculated utilizing Eq. (23):

(23)

where F_jis the consequent force acting on the jth CP, as illustrated in Fig. 6a. In this method, each CP is regarded as a charged sphere with radius a having a uniform volume charge density. In this paper, a is set to a unit.

Step 5: Construction new solutions. Each particle will locate in its new position as (Fig. 6(b)):

(24)V_{j,\mathrm{n}\mathrm{e}\mathrm{w}}={rand}_{j1}·k_{\mathrm{a}}·\frac{F_j}{m_j}·\mathrm{\Delta }{t}^2+{rand}_{j2}·k_{\mathrm{v}}·V_{j,\mathrm{o}\mathrm{l}\mathrm{d}}·\mathrm{\Delta }t+X_{j,\mathrm{o}\mathrm{l}\mathrm{d}},

(25)V_{j,\mathrm{n}\mathrm{e}\mathrm{w}}={\displaystyle \frac{X_{j,\mathrm{n}\mathrm{e}\mathrm{w}}-X_{j,\mathrm{o}\mathrm{l}\mathrm{d}}}{\mathrm{\Delta }t}},

where k_a is the coefficient of acceleration; k_v is the coefficient of velocity; and rand_j1 and rand_j2 are two numbers divided randomly in the range of (0,1).

Step 6: The position of invalid CPs should be corrected by the HS-based mechanism approach as defined by Kaveh and Talatahari [⁵].

Step 7: Process of updating CM. CM is updated by using some good CPs.

Step 8: Conditions of terminating criterion. The above steps should be repeated until a stopping criterion occurs.

CSS algorithm for structural optimization

Some special constraints in terms of structural view optimization should be considered in the previous section. The penalty function technique is the most conventional technique in order to fulfill this purpose. In using the penalty functions, if the result of the constraints meets the criteria, the above-mentioned value becomes zero; otherwise, the final penalty amount will be calculated by distributing the permissible criteria to the limit itself.

Figure 7 shows the entire stages involved in optimizing structure design in the CSS method. With regard to the steps illustrated in section 3.1, the algorithm could be described in three levels as follows.

Level 1: Initializing. The parameters of CSS algorithm are defined. According to step 1, random initial positions and velocities for CPs are calculated. The values of the fitness function are then evaluated. The best CPs and their related fitness function values are saved in CM.

Level 2: Searching. The probability of the motion of each CP is calculated with the help of step 2. Hence it was possible to create the array of attracting force related to CPs (Step 3) and CPs move to their new positions (Step 5). If each CP denies its permissible searching space, its new position is refined by utilizing the harmony-based algorithm (Step 6). They then evaluate their fitness function and replace the worst CPs with good ones in CM (Step 7).

Level 3: Controlling of terminating conditions. Above levels will be repeated until a terminating condition is reached.

Quantum Charged System Search (QCSS)

A new attitude is proposed in this article by using quantum infusion in the CSS. Here, a new offspring is generated using the quantum principle in QCSS. We intend to increase the CP’s convergence of the CSS method by adding an expression inspired by quantum infusion to an expression previously used (Eq. (24)) in the CSS method. The fitness of the new generation is assessed and the new CPs take the place of the ex-one merely if it has a superior fitness. This ensures that by using this method, the fitness of the CM has ameliorated. Therefore, the outcomes have improved in their fitness value and lead to best results over repetition. By combining the quantum theory with the standard CSS, a newly developed algorithm is introduced that includes the main characteristics of the other related algorithms and thus achieves adequate fitness. In QCSS, swift convergence feature achieved by CSS in the early iteration and therefore the efficiency is enhanced considerably.

In the CSS method, we use the Eq. (24) to create new CPs in a new position. Now an expression can be added to the above Eq. using the quantum method, which brings the convergence of CPs closer to better designs. Quantum behaved PSO was proposed by Sun et al.[⁴⁵,⁴⁶]. Regarding to the uncertainty principle, a particle’s position and velocity couldn’t be calculated simultaneously. In mechanic of quantum, each particle has a wave function as a substitute of having a velocity and position which is provided by:

(26)\psi (r,t).

Eq. (26) has no actual meaning but the amplitude squared of the equation brings the probability calculation of particle’s position in each dimension r at time t. Schrodinger Equation is chief equation of quantum mechanics which is provided by

(27)jh{\displaystyle \frac{\partial }{\partial t}}\psi (r,t)=\stackrel{\wedge }{H}(r)\psi (r,t),

where H (called Hamiltonian operator) is a variable that is not time-dependent and is calculated by:

(28)\stackrel{\wedge }{H}(r)=-{\displaystyle \frac{h^{\mathrm{2}}}{\mathrm{2}m}}{\nabla }^{\mathrm{2}}+V(r),

where h is constant of Planck, particles’ mass is shown by m and V_P(r)is the distribution of potential energy [⁴⁷]. Regarding the density probability function, probability of particle in order to emerge in a direction could be calculated. To stop explosion and make particles easy to converge, in one-dimensional space, we consider P = (p₁, p₂, …, p_D) as the potential center [⁴⁵]. Presume that all particles have quantum functioning and one-dimensional Delta potential well is available on each dimension at point P. The following density probability function Q and distribution function D can be standardized by making Schrodinger Equation.

(29)Q\left(y\right)=\frac{1}{L}{\mathrm{e}}^{-2\left|y\right|/L},

(30)D\left(y\right)={\int }_{-\infty }^{y}Q\left(y\right)\mathrm{d}y={\mathrm{e}}^{-2\left|y\right|/L},

where L is one the chief variables, which is known as particle’s imagination or the creativity and relies on intensity of energy. It determines the potential scope’s length. The separation space between the current location of particles and point P can be considered as L as below:

(31)L=\mathrm{2}\alpha \cdot |P-x|.

The coefficient of creativity is shown by the parameter \alpha and it is in charge of convergence rate of the particle. Since the search and solution space are two separate areas, it is imperative to interpret the position of the particle in solution space using mapping mechanism. Using Monte Carlo method, the particles’ new position is calculated as:

(32)x=P\pm {\displaystyle \frac{L}{\mathrm{2}}}\ln \left({\displaystyle \frac{\mathrm{1}}{u}}\right),u=\mathrm{rand}(\mathrm{0},\mathrm{1}).

New formula is proposed regarding Eqs. (31) and (32), as follows:

(33)x=P\pm \alpha \cdot |P-x|\cdot \ln \left({\displaystyle \frac{\mathrm{1}}{u}}\right).

The performance of Delta-Potential-well is enhanced by introducing a global point (which is named Mainstream Thought) into CSS. The outcome of the average CPs position is shown with the mbest. That is:

(34)mbest={\displaystyle \frac{\mathrm{1}}{M}}{\displaystyle \sum _{i=\mathrm{1}}^N{p}_i}=\left({\displaystyle \frac{\mathrm{1}}{M}}{\displaystyle \sum _{i=\mathrm{1}}^M{p}_{i\mathrm{1}}},{\displaystyle \frac{\mathrm{1}}{M}}{\displaystyle \sum _{i=\mathrm{1}}^M{p}_{i\mathrm{2}}},\dots ,{\displaystyle \frac{\mathrm{1}}{M}}{\displaystyle \sum _{i=\mathrm{1}}^M{p}_{iD}}\right),

where N is the number of CPs, number of problem’s dimension is shown by D and p_i is the CP’ position. Then L and new positions are assessed by using equations below [⁴⁸]:

(35)L=\mathrm{2}\alpha \cdot |mbest-x|,

(36)x=P\pm \alpha \cdot |mbest-x|\cdot \ln \left({\displaystyle \frac{\mathrm{1}}{u}}\right).

New positions are calculated regarding Eqs. (24) and (36), as follows:

(37)X_{j,\mathrm{n}\mathrm{e}\mathrm{w}}=ran{d}_{j\mathrm{1}}\cdot k_{a\cdot }{\displaystyle \frac{F_j}{m_j}}\cdot \Delta t^{\mathrm{2}}+ran{d}_{j\mathrm{2}}\cdot k_{v\cdot }{V}_{j,\mathrm{o}\mathrm{l}\mathrm{d}}\cdot \mathrm{\Delta }t+\alpha \cdot \left|mbest-X_{j,\mathrm{o}\mathrm{l}\mathrm{d}}\right|\cdot \ln \left({\displaystyle \frac{\mathrm{1}}{u}}\right)+X_{j,\mathrm{o}\mathrm{l}\mathrm{d}}.

Numerical result

Evaluation of the Methodology

In this article, a habitable structure in Tabriz, with dimensions of 20 m×14 m, is introduced as an uncomplicated example to depict the technique. The height of each story is 3.2 m and the architectural layout of this structure is divided into a 3×4 grid.

In a certain site, when the sizes of the structure layout and buildings’ material are given, consequently the values of loads can be determined (in this paper, the dead and live loads as vertical loads and wind as lateral load are considered) [⁴⁹]. In this paper, the combination used to calculate P_u (ultimate axial load) is: 1.2D + 1.0L + 1.6W where D, L, and W are dead load, live load, and wind load respectively [³⁸], as presented in Table 2.

The layout optimization can be defined by using 4 or 32 variables. In the 4-variables model, the first 3 digits will be converted from decimal to binary which is equivalent to the same 31 variables (number of all possible shear walls in plan). The 4th variable represents the type of wall which illustrate the thickness and type and spacing of the shear rebar used in the walls. In 32-variables mode, the presence or absence of walls in the plan is indicated by 31 variables (number of all possible shear walls in plan). The 32th variable represents the type of walls which illustrate the thickness and type and spacing of the shear rebar used in the walls.

Figures 8 and 9 demonstrate the final best layouts obtained by the CSS and QCSS algorithms using 4- or 32-varibales models. The continuous lines, shown in the figures, presented the selected location of shear walls. The outcome of the 4-variables using the CSS (Fig. 8(a)) shows 3 groups of walls located at middle and both ends of the plan, while 4 groups of walls suggested by QCSS (Fig. 8(b)) are located farthest from the center of the plan. The outcome of the 32-variables model using the CSS and QCSS is shown in Figs. 9(a) and 9(b), respectively. Tables 3–6 compare the final results and their related constraint values obtained by these algorithms. Clearly, the QCSS showed more robustness in finding the shear walls locations compared to the CSS. The 4-variable mode gives better results than the 32-variable model, as reported in Tables 3–6.

Complex Cases

Considering void space

The rectangular plan is considered in the previous subsection. If the building’s plan has an unusual contour or designers need an area to be with no walls, the idea of “void space” could be applied. In this part, the walls’ information of each layout is converted to binary system considering each number showing a line segment in the plan and related coordinates. The void spaces can be introduced to the program by using 0 (depicts deactivated wall) in its binary string. As a result, the program always disables these walls in the building layout. Eliminated shear walls will not consider in loading calculations such as shear forces or lateral deflection. The final results of such structure with s void space are presented in Figs. 10 and 11 obtained by the CSS and QCSS methods. In these figures, “Potential Shear Wall” indicates all potential walls that the program can select while “Shear Wall” shows the selected walls in the final design. “Void Space” were already predefined by the user showing that all the walls from this area have been removed. The numerical results are provided in Tables 7–10 for the 4-variables and 32-variable models, respectively. Wall free area positioned at the bottom right corner allows the shear walls to be concentrated in the middle or on other sides. For instance, in office buildings, an external void pushes the formation of walls in the central part.

Considering predefined fixed shear walls

Although the procedure explained in section 3 can define different types of layouts for an imaginary design, the architect may intend to select some walls in the architectural plan as the default. For example, peripheral walls and walls around stairs and elevators are appropriate places for shear walls. This is mainly due to the fact that these walls exist continuously at the height of the building. For this reason, users can select some places as predefined walls by placing the default number 1 in the layout binary string. Figures 12 and 13 illustrate the results of optimization process for the both CSS and QCSS methods. In addition, Tables 11–14 demonstrate beyond doubt that the QCSS show better results compared to the CSS.

In this subsection, both scenarios mentioned in previous parts were considered together in order to show the flexibility of the program. While one rectangular void space has been placed at the center of layout, two peripheral walls were selected as a predefined shear wall. Figures 14 and 15 present how the shear walls are placed in plan using the CSS and QCSS. The numerical results have been summarized in Tables 15–18 for this case.

Assessment of the CSS and QCSS methodology

Above example is modeled in this paper to evaluate the robustness of the QCSS algorithm and to compare outcomes with the CSS algorithm in identifying the best potential places for shear walls. Tables 3–18 represent the constraint values of objective function in 4-variable and 32-variable model. The results in all cases and models show that the superiority of the QCSS algorithm compared to the CSS. It will take about 180 min on a standard computer to find the best location for shear walls using the CSS method. Since the QCSS has shown more efficiency, there is a faster convergence to the final result in approximately 110 min. Also, it demonstrates the best layouts in different generations. We can obviously observe how the algorithm improves structural performance and minimizes structural costs.

Conclusions

Contribution summary

A new technique is presented in this paper that considers both structural and architectural views. Through this, we can find high-performance shear wall layout that can withstand lateral forces and movements. With regard to architectural views, there is a constraint that prevents the creation of a closed area by shear walls. It could be widely used by engineers either before or after the design stage of the architectural floor layout that will help the designers reach agreement on the potential position of the shear walls. This paper can provide engineers with a variety of optimized outcomes to select from by combining architectural design and structural performance. Furthermore, by choosing any of these layouts, they could be confident that their structural and architectural efficiency will be the best performing. After the optimization process, the shear wall layout could be thoroughly analyzed by structural engineers for more information and any possibility of changes in the location of the shear walls. Furthermore, this method could be used for a wide variety of structures, from low-rise to high-rise, varied ratio of width to height, from commercial to house, and for any type of house and shape.

As far as structural performance is concerned, this paper attempts to store significant computational time processing throughout the optimization processes by specifying a straightforward auto-calculation method for reinforced concrete architecture. Moreover, with regard to the optimization algorithms, the QCSS was used in this paper to evaluate the orientation and enlargement of the movement of CPs. The QCSS is introduced by adding quantum mechanics to the CSS method to increase the efficiency of the CSS so that the convergence of results is accelerated. This new algorithm uses less computational time than other meta-heuristics to achieve the best responses. Because of robustness and efficiency of the QCSS algorithm, it wisely reduces the computing time. By comparing the CSS and QCSS outcomes, it can be observed that the QCSS was able to provide better results. Ultimately, by filling the gap between architects and structural engineers, this paper attempted to reduce the trial-and-error design procedure for structures.

Suggestions for future works

The work demonstrated in this article concentrates on architectural design and structural performance optimization. However, in terms of architecture, due to the complexity of architectural considerations, the proposed plans may not provide adequate accessibility for spaces. For example, optimized outcomes can show an elevator well in an apartment or a living room right next to the door of the house. Moreover, this research describes only the design of the shear walls in the floor plan, which ignores changes in the shear wall dimensions along the structure height. Therefore, it is suggested that consideration of accessibility should be given to the placement of spaces in the plan and has the ability to determine the dimensions along the structure height in future work. Furthermore, since engineers may not have enough computer science to work with MATLAB software, it is best to provide this research with user-friendly software to make this approach more general.

Although, we expect that the performance of the algorithms did not change with using different parameters or at least their changes are small (and in this condition, the algorithms are more reliable); however, the effect of choosing good parameters on the performance of algorithms is not irrefutable. On the other hand, to have a fair compaction between the CSS and QCSS, we should use the same values for the parameters of these algorithms. Since the QCSS does not add a parameter to the CSS, one can find suitable values for one of the algorithms and use them for both. Considering a large number of previous studies on applying or improving the CSS, we use the reported parameter values of previous works [⁵] in this paper. However, utilizing suitable approaches for finding values of parameters such as Vu-Bac et al. [⁵⁰] and Anitescu et al. [⁵¹], can improve the performance of the proposed method more than presented in this paper.

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