Optimization of Wall Dimensions in Earthquake-Resistant Masonry Structure Design Using the Taguchi Method

Abstract

This study investigates seismic loads in single-story masonry buildings with walls of varying heights and thicknesses, and determines optimum wall dimensions for seismic resistance using the Taguchi method. For this purpose, 25 (5 × 5 = 25) different masonry building models were created with thicknesses of 16, 20, 24, 28, and 32 cm and heights of 260, 280, 300, 320, and 340 cm. The building models were analysed using a software package in accordance with the 2018 Turkish Building Earthquake Code (2018 TBEC). C-30 concrete and S-420 steel were used in the designed building models. A 12 cm thick reinforced concrete slab was placed on top of the masonry walls. A live load of 0.2 t/m² was designed on the slab, and the mortar strength of the brick wall was taken as 30 MPa. When a building model with a height of 260 cm and a thickness of 16 cm was used as a reference, it was observed that the seismic resistance of other building models increased by approximately 72%, while shear forces increased by approximately 89% in the “x” direction and approximately 95% in the “y” direction. Furthermore, it was observed that as the ratio of wall height to wall thickness increased, the seismic resistance of the building models decreased. The seismic resistance of 25 different building models was analysed using the Taguchi method, depending on wall thickness and wall height. The analysis revealed that the building model with walls 24 cm thick and 340 cm high was the most resistant to shear forces, while the building model with walls 32 cm thick and 340 cm high provided the best resistance to seismic loads.

1. Introduction

In Türkiye, particularly in rural areas, the masonry construction technique is still used and constitutes a significant portion of the existing building stock. According to 2023 Turkish Statistical Institute (TÜİK) data, approximately 15–20% of the total housing stock in Türkiye consists of masonry structures [1]. This rate can exceed 30% in rural areas. This construction method is a system created by placing materials such as stone, brick, or adobe on top of each other. Although the use of masonry structures has decreased in city centres, it continues in rural areas for both economic and traditional reasons. However, historical masonry structures show that, when the necessary engineering and construction rules are followed, masonry structures can be long-lasting and durable. The Turkish Building Earthquake Code (2018 TBEC) and the Turkish Earthquake Hazard Map, published in 2018, are an important step in assessing the earthquake hazard in Türkiye and improving the earthquake resistance of buildings [2]. According to this map, 26% of the existing building stock in Türkiye is located in the most dangerous areas. This situation increases concerns about the safety of buildings, especially in areas with high earthquake risk. The same concern exists for masonry buildings, which constitute 7% of the existing building stock according to TÜİK’s 2023 data [3]. Masonry buildings are generally constructed with brick, stone, and similar materials and are considered less durable than reinforced concrete building technology. This situation shows that the entire building stock, including masonry structures, should be strengthened against earthquakes, especially in areas with high earthquake risk. Strengthening buildings in earthquake-prone areas and constructing new buildings in accordance with 2018 TBEC standards is extremely important to reduce loss of life and property in possible earthquakes.

Masonry structures are generally structures made by joining blocks such as natural or artificial stones with binding mortar and are resistant to horizontal and vertical loads [4]. However, due to their rigid properties, these structures have a low energy absorption capacity, which makes them susceptible to large horizontal forces [5]. The main reason why masonry structures exhibit brittle behaviour during earthquakes is the non-ductility of the materials used. This leads to sudden fractures and widespread damage. Therefore, in situations where horizontal loads are intense, such as earthquakes, masonry structures should be designed and strengthened with special care. In our country, especially in rural areas, the use of masonry structures continues to be widespread. Therefore, in the 2018 TBEC regulation, which came into force in 2018, masonry structures were examined as a separate subject. The regulation defines the design and calculation rules and connection details of masonry structures. Masonry structures consist of load-bearing walls made of stone, brick, aerated concrete, adobe, and mortar units that connect them. The building materials used in wall construction are classified according to the TS 699 standard; Brick, concrete blocks, aerated concrete, and mortar are used in the construction of wall structures [6,7,8,9,10,11]. Mortar is a binding structural material obtained by mixing sand, cement/lime, and water in appropriate proportions. Masonry structural elements can be defined as foundations, walls, arches, lintels, domes, vaults, floors, etc. Masonry structures can be designed with or without reinforcement. Recently, many studies have been conducted on the behaviour of masonry structures under various loads, their rigidity, earthquake damage reduction, strengthening, and new design approaches [12,13,14,15,16,17,18,19,20,21]. It was assessed that there was no detailed analysis, modelling and optimization studies on the effect of wall void ratios on the seismic performance of masonry structures. In this context, the effect of wall void ratios on the seismic performance of masonry structures was statistically investigated. Mathematical models based on wall void ratios were created, and prediction models were created for the seismic performance parameters of structural models [21]. According to the analysis results obtained, it was revealed how much the seismic effect changes when void ratios are left in the load-bearing walls of a masonry structure in the architectural design process. Thus, optimal wall void ratios were determined and designers were given the opportunity to prevent damage or collapse of masonry structures designed according to these conditions in a possible earthquake [22]. Traditional Design of Experiments (DOE) approaches aim to determine the effects of multiple factors and their interactions on system performance through full or partial factorial experimental designs. While these methods provide a comprehensive statistical framework and allow for explicit modelling of interaction effects, they typically require numerous experimental studies. This makes them impractical or costly in engineering applications involving complex systems or where data availability is limited [23,24,25]. In structural engineering, where both on-site field tests and laboratory tests are time-consuming and costly, the applicability of classical DOE may therefore be limited. In such cases, the Taguchi Method offers a more economical and practical alternative by using orthogonal array designs to reduce the number of experiments required to obtain the fundamental parameters while extracting their dominant effects [26,27]. Instead of explicitly modelling all interactions, the Taguchi approach focuses on identifying the factor levels that lead to robust performance under varying noise conditions. This simplification may limit the detailed interpretation of higher-order interactions. The method has proven effective in preliminary optimization and sensitivity analyses, particularly in situations characterized by natural variability and uncertainty, such as the seismic behaviour of buildings [28,29]. The Taguchi Method, developed by Dr. Genichi Taguchi, is a statistical and engineering approach widely used to enhance product and process quality through robust design. The primary objective of the method is to reduce performance variability by minimizing the sensitivity of a system to uncontrollable factors [23,28]. The Taguchi experimental procedure consists of the following steps:

*

Problem definition,

*

Determination of the response variable and performance target,

*

Selection of control and noise factors and their respective levels,

*

Selection of the appropriate orthogonal array based on degrees of freedom,

*

Conducting experiments following the OA structure,

*

Calculating S/N ratios for each experimental condition,

*

Analysis of main effects to determine optimal factor levels, and

*

Performing validation tests to confirm the predicted improvements [26,27,28].

This formulation indicates that any deviation, even within specification limits, results in quality loss. Previous research has shown that Taguchi-based experimental designs can successfully identify the relative influence of key soil parameters such as cohesion, internal friction angle, unit weight, relative density, and groundwater conditions on foundation performance. For example, Taguchi optimization has been used to evaluate the sensitivity of bearing capacity to variations in soil strength parameters and footing geometry, as well as to minimize settlement by determining optimal combinations of soil stiffness and loading conditions. In ground improvement studies, the method has been employed to optimize treatment variables (e.g., cement content, curing time, and mixing ratios) with the aim of enhancing strength and reducing compressibility [29,30,31,32]. Although Taguchi-based approaches cannot model the different parameters obtained under different conditions, the method is used in screening the dominant parameters and guiding more detailed analyses. The Taguchi method has been successfully used in many engineering fields, and its results have been evaluated by comparing them with different design approaches [33,34,35]. In the field of civil engineering, it is used especially in the optimization of aggregates, concrete mixes, lightweight concrete mixes, and in the analysis of load-bearing structural elements such as columns and beams. The results obtained have shown that the Taguchi method can be easily used in the optimization of many applications and experimental studies in civil engineering [36,37,38,39,40,41].

This study examines the geometric requirements for the design of masonry structures as specified in Table 11.4 of the 2018 Turkish Building Earthquake Code, titled Special Rules Regarding the Design of Masonry Structure Systems Under Seismic Effects [42]. For brick walls, the minimum wall thickness is given as 24 cm, and the ratio of effective wall height to wall thickness (hef/tef) is given as 12. Beyond this information, the 2018 Turkish Earthquake Code does not define any other geometric requirements regarding the seismic resistance of masonry structures, the appropriate wall thickness for each wall height, or the effects of wall thickness and height on the seismic behaviour of the structure. Therefore, this study aims to analyse the seismic resistance of wall thicknesses and heights not defined in the code and to determine the optimum wall thickness and height that geometrically provides seismic resistance using the Taguchi method. In this study, wall thicknesses and heights in masonry buildings were evaluated by creating 25 different models covering all possible situations, extreme cases, lower and upper values, and situations not mentioned in the regulation, which fall outside the limits of the 2018 Turkish Building Earthquake Code.

The main objective of this study is to analyse the seismic performance of masonry building models with different wall thicknesses and heights in rural areas inhabited by economically disadvantaged people exposed to earthquake effects, and to determine the optimum wall thickness and height in terms of seismic performance, thereby ensuring that these structures are designed to be earthquake-resistant and preventing loss of life and property. For this purpose, a total of 25 different building models with 5 different wall thicknesses and 5 different wall heights were analysed, their seismic performances were determined, and the optimum wall thickness and height in terms of earthquake-resistant building design were determined using the Taguchi method.

2. Materials and Methods

2.1. Materials

The single-story masonry structure was designed with five different story heights and five different wall thicknesses. The structure was modelled in accordance with the 2018 TBEC. In the structural analyses performed with STA4 V14.1 program, which is widely used in our country and whose analysis results are accepted as accurate by official institutions, concrete class C-30 (compressive strength varies between 30 MPa) and steel class S-420C (tensile strength of steel is 420 MPa) were used in all structure models. A masonry structure model with different story heights and wall thicknesses was created. Fully mixed bricks with load-bearing properties were used in the structure walls. The unit volume weight of the brick is 1.8 t/m³, E = 2100 MPa and f_d = 14 kg/cm², and the cement mortar used in the brick courses is f_uko = 3.5 MPa. The axis spacing of the structures in the plan was selected as 4.5 m for the largest span and 1 m for the smallest span. The number of axes in the “x” and “y” directions is equal, and there are 4 axes. The slab spacing in the building models is 12 cm. The story heights of the buildings were designed as 260, 280, 300, 320 and 340 cm. The behaviour of the building models under earthquake effects was analysed using Sta4cad V14.1 programme [43]. In the analysis, according to Earthquake Level-1 was selected as the reference value for the earthquake ground motion level (probability of exceedance in 50 years is 2%). The soil class is ZC, the structural behaviour coefficient is R = 2.5 and the resistance coefficient is D = 1.5. The spectrum characteristic period is Ta = 0.069 and Tb = 0.347, respectively, and the live load coefficient is 0.3. The bearing capacity and design stress of the soil are taken as 20 t/m², and the bearing coefficient is 3000 t/m³. The floor plan and 3D view of designed building models off the masonry building models were shown in Figure 1 and Figure 2.

2.2. Methods

The single-story masonry structure was designed with five different story heights and five different wall thicknesses A total of 25 different masonry building models with five different wall thicknesses and five different wall heights were created. In the building models, reinforced concrete calculations were made according to the bearing capacity method according to TS-500-2000, and the section reinforcement was calculated according to the gross section [44]. The earthquake calculations were made using Mode Superposition and Dynamic Analysis methods. Foundations were taken into consideration in the earthquake calculations. The soil stress live load reduction coefficient was taken as 1.00. The differences between the earthquake-induced floor drifts and seismic moments in masonry building models with different story heights were analysed and compared using multiple comparison tests at a 95% confidence interval. Similar and different wall structures are shown in the tables. Five different building models constructed with five different Masonry Wall Structures (MAS) were analysed under earthquake effects. The results of these analyses are presented in separate tables, graphs, and figures for each model. Seismic analyses were conducted in accordance with the 2018 TBEC using Sta4Cad. The analysis results were calculated separately for the “x” and “y” directions for seismic loads and displacements in the masonry structures acting on different wall thicknesses for the building models and are presented in tables. The optimum wall thickness and optimum wall height that maximize the earthquake resistance of masonry structures according to their wall thickness and wall height were determined by analysing with the Taguchi method. General information about the Taguchi method is given below under a separate heading his section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

3. Taguchi Method

The Taguchi method is named after its developer, Dr. Genichi Taguchi. Taguchi methods are an approach that uses experimental design techniques to obtain the best quality characteristics to achieve a specific goal. Taguchi’s methods are not only a statistical application of experimental design, but have also transformed statistical experimental design into a powerful engineering tool [45,46,47]. The Taguchi approach is not only about experiments to achieve target values. In fact, the Taguchi method is concerned with quality in general and with statistical techniques and tools. The Taguchi philosophy, also known as Quality Engineering, divides quality control into two main parts: in-production and out-of-cycle quality control [48,49,50]. In quality management systems, achieving the target without any losses is essential. However, in practice, this target is often not met or is exceeded. In this case, loss values arise [26]. There are three types of quality variables in tolerance design. These are:

• Larger is better: In this tolerance study, there is no upper limit to the quality variable and therefore no target value. As the measurement increases, the efficiency will also increase.
• Smaller is better: This type of tolerance is the tolerances where the target value is zero, such as the percentage of scrap in the production process. As the tolerance decreases, the efficiency of the system will increase.
• Target value is best: This is the type of tolerance where deviations can occur in both directions.

Taguchi loss functions according to the objective function were shown for three types of quality variables in tolerance design (Figure 3).

Noise Factors and Signal-to-Noise Ratio (S/N)

Taguchi divided the factors affecting the product and process into two groups: controllable and uncontrollable factors. He called the uncontrollable factors that create differences (variations) in the functional characteristics of the product or process “noise factors”. Controllable factors are our design parameters [46]. The aim in experimental designs is to optimize the mean value of the quality variable. Taguchi developed a statistic known as the Signal-to-Noise “S/N” ratio for use in analyses as a performance indicator in experimental design [45]. According to the “S/N” ratio, the best condition can be determined from the experimental design. Taguchi divided the problems in practice into three categories according to the type of objective and defined a different “S/N” ratio for each. For “Best Target Value” type problems, the Signal-to-Noise Ratio (S/N) is expressed as follows;

S/N = 10log(µ²/σ²)

(1)

This is example 2 of an equation:

$$\sigma ={\displaystyle \frac{1}{n}}{\sum }_{i=1}^{n}yi and {\sigma }^2={\displaystyle \frac{1}{n-1}}{\sum }_{i=1}^n{\left(yi-\mu \right)}^2 \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e},$$

(2)

Here, “n” is the number of external noise observation combinations used for each combination of design parameter matrices. For “Smaller is Better” type problems, the signal-to-noise ratio S/N is defined as follows.

$$\mathrm{S}/\mathrm{N}=-10.\mathrm{\lambda }\mathrm{o}\mathrm{g}({\displaystyle \frac{1}{n}}.{\sum }_{i=1}^n{yi}^2)=-10.\left(MSD\right)$$

(3)

The design parameter tuning strategy primarily uses control factors to minimize output variability, followed by signal factors to bring the average value closer to the desired target value. The Taguchi method is an approach that aims to maximize output performance economically while minimizing the impact of output variability.

Figure 3. Taguchi loss functions according to the objective function [49].

Figure 3. Taguchi loss functions according to the objective function [49].

4. Determination of Earthquake Resistance of Buildings

Some studies on the seismic performance of masonry structures have yielded the following results. To improve the seismic performance of existing masonry structures and optimize the thickness of the reinforcement layer, ultra-high-performance concrete (UHPC) can be used as a reinforcement material. To protect existing masonry structures from small to moderate magnitude earthquakes, their structural integrity and seismic performance must be effectively strengthened. Engineered cementitious composites (ECC) have been widely used in engineering enhancement due to their high toughness. Seismic assessment of historical masonry structures studied the linear and non-linear response of a masonry wall with an opening [51,52,53,54,55]. Recently, studies have been conducted on increasing the seismic resistance of masonry structures with different design features, different spans and different strengthening approaches, and the results have been presented in a multifaceted way [56,57,58].

In the single-storey masonry building model, the relationships between the weight of the building, the earthquake loads and shear forces occurring in the building in cases where the building has five different wall thicknesses “16, 20, 24, 28 and 32 cm” and five different wall heights “260, 280, 300, 320 and 340 cm” were examined and comparisons were made depending on these parameters. The earthquake loads and shear force acting on the structures depending on the floor heights and wall thicknesses of the examined building models are shown below in (

Table 1. Earthquake load and shear force according to wall thickness and height (kN).

Earthquake Load (kN)
Wall Height (cm)	Wall 16 cm	Wall 20 cm	Wall 24 cm	Wall 28 cm	Wall 32 cm
260 cm	1166.42	1285.67	1396.69	1515.94	1635.19
280 cm	1215.36	1342.85	1470.24	1597.63	1725.12
300 cm	1256.25	1400.02	1543.79	1679.32	1823.09
320 cm	1305.29	1457.19	1609.10	1761.01	1913.01
340 cm	1354.32	1514.37	1682.65	1842.80	2011.08
The shear forces (kN)
260 cm	203.48	239.59	284.78	330.02	375.23
280 cm	199.36	239.83	285.17	330.46	375.71
300 cm	199.74	246.70	293.60	340.34	387.22
320 cm	200.87	242.95	289.04	335.05	381.03
340 cm	203.44	245.41	291.83	338.05	384.36

) [58].

The analyses revealed that the seismic loads and shear forces acting on the designed masonry models occur most strongly in the x-direction; therefore, the analyses were performed only for the x-direction.

5. Determination of the Optimum Masonry Building Model Using the Taguchi Method

To determine the optimum wall thickness and height for earthquake-resistant building design, the Taguchi method was used, with wall thicknesses of 16, 20, 24, 28, and 32 cm and wall heights of 260, 280, 300, 320, and 340 cm. In the analyses, wall thicknesses were designed as “factor A” and wall heights as “factor B”. Wall thicknesses and wall heights were defined at 5 different levels. In this case, the Taguchi design scheme became a 2-level system with 5 × 5 = 25 factors. The objective of this study is to determine the optimum wall thickness “A” and optimum wall height “B” factors that will maximize the seismic load and shear forces acting on the structure. Seismic load and shear force are results that vary depending on wall height and wall thickness factors, and factors A and B also influence each other. The optimum values of factors A and B were determined according to the maximum signal-to-noise ratios determined as a result of the analyses. They are also shown graphically in Figure 4a,b and Figure 5a,b. Design summary for Earthquake load and Shear forces were given below as a table (

Table 2. Taguchi design summary.

Taguchi Array	L25 (52)
Factors:	2
Runs:	25
Columns of L25 (56) array: 1 2

).

Taguchi divided the factors affecting the product and process into two groups: controllable and uncontrollable factors. He called the uncontrollable factors that create differences (variations) in the functional characteristics of products or processes “noise factors”. Controllable factors are our design parameters. Taguchi divided the problems in the application into three categories according to the type of objective and defined a different S/N ratio for each. These are the S/N ratios defined for “Objective Value Best” type problems, “Smaller Better” type problems, and “Bigger Better” type problems. The model coefficients of the S/N ratios predicted for wall thickness A and wall height B as a result of the analyses are shown below (

Table 3. Linear Model Analysis: S/N ratios versus A; B.

Term	Coefficient	Standard Error of Coefficient	T Value	p Value
Estimated Model Coefficients for S/N ratios
Constant	90.378	0.009179	9846.257	0.000
A 16	−0.1081	0.018358	−5.891	0.000
A 20	−0.1130	0.018358	−6.154	0.000
A 24	0.1299	0.018358	7.074	0.000
A 28	−0.0034	0.018358	−0.186	0.855
B 260	−2.9667	0.018358	−161.603	0.000
B 280	−1.3005	0.018358	−70.843	0.000
B 300	0.2009	0.018358	10.941	0.000
B 320	1.4771	0.018358	80.460	0.000
Model Summary
S	R2	Adjusted R2
0.0459	99.97%	99.95%
Estimated Model Coefficients for S/N ratios of Earthquake Loads for Means
Constant	71,025.5	218.6	324.852	0.000
A 16	−4985.3	437.3	−11.401	0.000
A 20	−2617.0	437.3	−5.985	0.000
A 24	306.2	437.3	0.700	0.494
A 28	2328.5	437.3	5.325	0.000
B 260	−15294.1	437.3	−34.976	0.000
B 280	−7783.1	437.3	−17.799	0.000
B 300	−34.1	437.3	−0.078	0.939
B 320	7654.7	437.3	17.505	0.000
Model Summary
S	R2	Adjusted R2
1093.1985	99.42%	99.13%
Estimated Model Coefficients for S/N ratios of Shear Forces
Constant	89.1909	0.01360	6559.759	0.000
A 16	−0.0801	0.02719	−2.946	0.009
A 20	−0.1070	0.02719	−3.935	0.001
A 24	0.0995	0.02719	3.659	0.002
A 28	0.0004	0.02719	0.015	0.988
B 260	−2.9415	0.02719	−108.171	0.000
B 280	−1.3137	0.02719	−48.310	0.000
B 300	0.1923	0.02719	7.073	0.000
B 320	1.4732	0.02719	54.173	0.000
Model Summary
S	R2	Adjusted R2
0.0680	99.92%	99.89%

).

Table 3 presents the constant coefficients of the prediction model obtained by Linear Model Analysis for S/N ratios depending on wall thickness (A) and wall height (B), the coefficients of variables A and B, the standard errors of these coefficients, the “T” comparison test values, and the “P” significance levels. The same table also shows the prediction model coefficients for S/N ratios of average earthquake loads and the predicted S/N ratio coefficients for shear forces. In three separate prediction models obtained from non-experimental but numerical simulations, the correlation coefficients were found to have an “R²” value above 99%, almost perfect.

In the single-storey masonry building model, the relationships between the weight of the building, the earthquake loads and shear forces occurring in the building in cases where the building has five different wall thicknesses “16, 20, 24, 28 and 32 cm” and five different wall heights “260, 280, 300, 320 and 340 cm” were examined and comparisons were made depending on these parameters. The earthquake loads acting on the structures depending on the floor heights and wall thicknesses of the examined building models are shown below in kN (Table 1) [58]. The analyses revealed that the seismic loads and shear forces acting on the designed masonry models occur most strongly in the x-direction; therefore, the analyses were performed only for the x-direction.

The aim of experimental designs is to optimize the mean value of the quality variable. In many applications, the goal is also to minimize the variance of the quality variable. In Taguchi’s experimental design, the Signal-to-Noise (S/N) ratio statistic is used as a performance indicator in the analyses. The Signal-to-Noise ratio measures the effects of noise factors on the controlled factors, showing the sensitivity of the change in the quality characteristic. Based on the S/N ratio, the best condition can be determined from the experimental design. For this purpose, the highest S/N ratio is selected from the results of any experimental design. According to Taguchi, in product design, the results with the highest S/N ratio will always determine the product with the smallest variance and the highest quality. The results of the prediction model, analysis of variance for the average values of wall thickness and wall height, variance analysis of the S/N ratios for wall thickness (A) and wall height (B) are shown below (

Table 4. Analysis of Variances.

Analysis of Variance for SN Ratios
Source	DF	Seq SS	Adj SS	Adj MS	F Value	p Value
A	4	0.2515	0.2515	0.0629	29.85	0.000
B	4	97.0955	97.0955	24.2739	11,524.33	0.000
Residual Error	16	0.0337	0.0337	0.0021
Total	24	97.3807
Analysis of Variance for Means
A	4	309477325	309477325	77369331	64.74	0.000
B	4	2959961654	2959961654	739990414	619.20	0.000
Residual Error	16	19121326	19121326	1195083
Total	24	3288560306
Analysis of Variance for SN ratios for Shear Force
A	4	0.279	0.2794	0.0699	74.48	0.000
B	4	99.894	99.8938	24.9734	26,626.26	0.000
Residual Error	16	0.015	0.0150	0.0009
Total	24	100.188

).

The “Response Table for Signal-to-Noise Ratios” values, showing the wall thickness and wall height that can withstand the seismic load for 5 different wall thicknesses “A” and 5 different wall heights “B” are shown below (

Table 5. SN ratios for wall thickness (A) and wall height (B).

Response Table for Signal to Noise Ratios for Prediction Model of the Shear Force Target: Larger Is Better			Response Table for Signal to Noise Ratios for Earthquake Loads Target: Larger Is Better
Level	A	B	A	B
1	90.27	87.41	103.0	102.2
2	90.27	89.08	103.5	103.1
3	90.51 * (24 cm)	90.58	103.8	103.9
4	90.37	91.86	104.2	104.7
5	90.47	92.97 * (340 cm)	104.6 * (32 cm)	105.4 * (340 cm)
Delta	0.24	5.56	1.6	3.2
Rank	2	1	2	1

* The numbers in parentheses are the optimum wall thicknesses and optimum heights determined as a result of Taguchi analysis.

).

The following figures “Figure 4a,b and Figure 5a,b” show the distribution of average wall heights according to wall thicknesses and the effect plots and S/N ratios showing the wall heights and thicknesses that can withstand the highest seismic load.

6. Results and Recommendations

This study, which examines the effects of wall heights and thicknesses on seismic resistance in masonry structures which are economical, inexpensive, and easy to construct in rural areas of earthquake zones, mostly inhabited by disadvantaged people—and includes 25 different wall models, determined the optimum wall thicknesses and heights in terms of resistance to seismic loads and shear forces using the Taguchi method. Since the largest seismic loads occur in the “x” direction, the analysis results were evaluated in this direction.

• It was observed that the weakest wall model in terms of seismic resistance was the one with a thickness of 16 cm and a height of 260 cm, and a minimum seismic load of 118,940 kN. The highest seismic load occurred at wall heights of 32 cm and 340 cm with a thickness of 205,070 kN. When the wall thickness was doubled and the wall height was increased by approximately 1.3 times, the seismic resistance of the wall models increased by approximately 72.4%.
• When the wall thickness was fixed at 16 cm, it was observed that the 340 cm high wall model had a resistance of 138.101 kN, showing an increase of approximately 16.1%.
• When the wall thickness was 32 cm, the seismic resistance of the 260 cm high wall model was 166.747 kN, while the resistance of the 340 cm high wall increased by approximately 23% to 205.070 kN.
• In terms of sheer force, the shear force was 38.262 kN in the 16 cm thick and 260 cm high wall model, while this value increased by 2.43% to 39.193 kN in the 16 cm thick and 340 cm high wall model.

Regarding the overturning moment in the walls of structural models, it can be assumed that the wall model producing the least moment is the one with a height of 260 cm and a thickness of 32 cm. However, the walls of masonry structures are not expected to resist moment as a whole. This is because the walls are assembled from small wall elements such as bricks, blocks, and Ytong blocks, bonded together with mortar, and thus carry the load. Therefore, the resistance to overturning moment in the walls of masonry structures depends on the adhesion properties of the mortar. The seismic loads acting on the structure also depend on the strength properties of the mortar, which is placed between the bricks and other elements in the walls and resists shear forces acting in the horizontal direction. For the mortar to resist shear forces, the compressive stress resulting from the wall’s own height also has a strong positive effect. The compressive stress resulting from the wall’s own weight also increases the adhesion rate of the mortar and, due to the increased friction effect, naturally increases its resistance to shear forces.

Using the Taguchi method, it was observed that the optimum wall dimensions for achieving maximum seismic resistance in masonry models should be 32 cm thick and 340 cm high. When the wall thickness is 32 cm, the moment of inertia of the wall in plan increases. The vertical compressive stresses caused by the shear strength of the mortar used in the wall and the wall’s own weight also increase depending on the wall height. These two factors increase the shear resistance of the wall. According to Article 11.2.9 of the 2018 Turkish Building Earthquake Regulations, the characteristic shear strengths of masonry walls are expressed below;

$$fvk=f{vk}_o+0.4{\sigma }_d\le 0.1f_b$$

(4)

It is calculated as follows;

• $fvk$: Wall characteristic shear strength obtained using the average vertical stresses on the wall,
• $f{vk}_o:$ Initial shear strength values of wall materials (brick, concrete, aerated concrete, etc.) [41].
• ${\sigma }_d:$ Vertical compressive stress calculated under the combined effect of vertical loads multiplied by load coefficients and seismic loads.
• $f_b$: Standardized average compressive strength of the masonry unit (equivalent to a 100 mm × 100 mm sample free from dimensional effects).

In masonry walls, wall rigidity is also an important strength parameter. According to Article 11.3.4 of the 2018 Turkish Building Earthquake Regulations [40], the rigidity of a masonry wall is calculated taking into account shear and bending deformations. Walls with higher rigidity compared to other walls have a relatively greater capacity to carry seismic loads. For a rectangular section of wall, the elastic rigidity is calculated using the following expression, assuming both ends are fixed;

$$k_{wall}={\displaystyle \frac{1}{(\frac{H^3}{12E_{wall}I}+\frac{H}{1.2G_{wall}A})}}$$

(5)

where;

• $k_{wall}$: Wall Stiffness
• $H$: Free height of the wall, length from the top of the slab to the bottom of the slab (or beam, if any)
• $E_{wall}$: Elasticity modulus of the wall,
• I: Moment of inertia of the solid wall segment, G_wall: Shear modulus of the wall
• A: Horizontal cross-sectional area of the solid wall segment
• $E_{wall}$ and $G_{wall}$ values were taken as $E_{wall}$: 750f_k and $G_{wall}$: 0.4$E_{wall}$ according to [40].
• f_k: Characteristic compressive strength of masonry wall [40]. stiffness expression, when the wall thickness increases, the moment of inertia and cross-sectional area of the wall in plan increase. In this case, the stiffness of the wall also increases.

In a single-story structure, the natural vibration of the structure is calculated using the formula $T={\displaystyle \frac{2\pi }{w}}$, and the angular frequency w is calculated using $w^2={\displaystyle \frac{k}{m}}$ As seen in this expression, as the rigidity of the structure increases, the angular frequency increases, and naturally, the period of the structure decreases. A decrease in the period of the structure reduces the duration of the earthquake’s effect on the structure, and all these factors increase the structure’s resistance to earthquakes.

Table 1 shows the seismic loads and shear forces acting on the structure according to wall height and wall thickness. As can be seen from this table, for walls of the same height, as wall thickness increases, or in other words, as the wall height/wall thickness ratio decreases, the seismic resistance of the structure increases significantly. This increase is observed for a 260 cm high wall with a height/thickness ratio of 260/16 = 16.25, while the seismic load is 1166.42 kN. For a height/thickness ratio of 260/32 = 8.125, the seismic resistance ratio of the structure increases by 40% (1635.19/1166.42 = 1.40). Similarly, it was observed that when the wall thicknesses were increased from 16 cm to 32 cm for building heights of 280 cm, 300 cm, 320 cm, and 340 cm, the seismic resistance of the structures increased by 41.9% for 280 cm, 45.12% for 300 cm, 46.55% for 320 cm, and 36.06% for 340 cm. However, it was observed that when the wall thickness was increased from 16 cm, 20 cm, 24 cm, 28 cm, and 32 cm respectively, and the wall height was increased from 260 cm to 340 cm, the earthquake resistance of the structure increased by 16.1% for a thickness of 16 cm (1354.32/1166.42 = 1.161), by 17.78% for 20 cm, by 20.04% for 24 cm, by 21.56% for 28 cm, and by 22.98% for 32 cm. It was observed that when the wall thickness was increased from 16 cm to 32 cm for a height of 260 cm, the shear resistance of the structure increased by 84.4% (375.23/203.48 = 1.844), by 41.94% for a height of 280 cm, by 45.12% for 300 cm, by 46.55% for 320 cm, and by 48.49% for 340 cm.

The designed structural models utilize the material and strength parameters defined in Section 11.2 (Materials and Strength) of the TBDY-2018 Regulation. C-30 concrete and S-420 steel were used in the designed structural models. A 12 cm thick reinforced concrete slab was placed on top of the walls. A live load of 0.2 t/m² was designed on the slab. Brick was selected for the load-bearing walls in accordance with TS EN 771-1. According to [7], the minimum standardized compressive strengths of brick are fb,min = 5.0 MPa perpendicular to the horizontal joints and fbh,min = 2.0 MPa parallel to the horizontal joints. According to TS EN 1015-11, the minimum cubic compressive strength values of the mortar to be used cannot be lower than fm,min = 5.0 MPa for unreinforced walls and fm,min = 10.0 MPa for reinforced walls. Table 11.2 of the TBDY-2018 [42] Regulation defines that the compressive strength of the mortar to be used in the wall varies between 1 MPa and 20 MPa, and the characteristic compressive strengths of load-bearing walls can vary between 5 MPa and 30 MPa.

In the designed structural models, the compressive strength of the bricks was selected as 30 MPa and the compressive strength of the mortar as 20 MPa to ensure maximum strength. It is known that if lower compressive strength values are chosen for bricks and mortar, the structure’s resistance to shear forces, especially in the horizontal direction, can be significantly reduced, and the existing compressive stresses also have a significant effect on the calculation of shear stresses. In wall models, it is clear that changes will occur in the seismic resistance of structures when the compressive strength of the bricks is between 5 and 30 MPa and the compressive strength of the mortar is between 1 and 20 MPa, and it is thought that this issue should be examined in detail in subsequent studies. In addition, it is thought that comparative analyses on how wall thicknesses and wall heights affect construction costs should be carried out in subsequent studies, given that the created structural models have five different wall thickness values between 16 cm and 32 cm and five different wall height values between 260 cm and 340 cm. This study examines the seismic resilience of design models that can prevent the collapse of masonry structures during earthquakes, particularly in high-risk and low-income communities. Future studies could further investigate the effects of variations in brick and mortar material strength, as well as changes in wall thickness and height, on the costs of these designed structural models.

However, considering the wall cross-sectional area affected by the wall’s own vertical loads and the increasing compressive stress depending on the wall height, it was determined that the wall dimensions should be 24 cm thick and 340 cm high in terms of resistance to shear forces. Furthermore, if the structural elements forming the wall, such as brick, can carry horizontal loads as a whole, like a wall panel, it is thought that the most suitable wall dimensions for moment effects might be 32 cm thick and 260 cm high.

Similar studies on the strengthening of load-bearing walls with the same wall thickness and height, and similar studies on strengthened wall models, are considered beneficial in terms of design, application, and optimization of seismic resistance conditions. In this study, the effect of wall height and wall thickness on the seismic resistance of single-story masonry structures was investigated, and the optimum wall thickness and wall height in terms of seismic resistance were determined. Conducting similar studies for 2 or 3-story masonry buildings could make significant contributions to practical applications. In addition, by creating different dimensions of door and window openings in single-story, two-story, and three-story masonry buildings, the optimum door and window opening dimensions in terms of seismic resistance can also be determined. Thus, not only the effects of wall thickness and wall height, but also the effects of door and window openings in masonry buildings can be determined in a multifaceted way. It is thought that determining the optimum door and window openings will be quite useful, especially in the architectural design process.