Multivariate regression analysis to strength and stiffness evaluation of masonry infills

doi:10.21203/rs.3.rs-3120498/v1

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Multivariate regression analysis to strength and stiffness evaluation of masonry infills

https://doi.org/10.21203/rs.3.rs-3120498/v1

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Reinforced concrete frames with masonry infill are widely utilized in construction worldwide. This present study aimed to investigate the behaviour of masonry infill. Firstly, the compressive strength of the masonry infill and the evaluation methods were examined, specifically prism tests and wallette tests. The study collected a substantial database, revealing that wallette tests indicate higher stiffness compared to prism tests. Based on the findings, it is recommended to employ prism tests for assessing the stiffness of masonry and wallette tests for evaluating the compressive strength of masonry. Furthermore, the study assessed the performance of 18 empirical models to identify the most suitable model for predicting the compressive strength of masonry using the two testing methods (wallette and prism). To evaluate the strength of masonry infills under horizontal loading, a database of reinforced concrete frames with masonry infills was collected. The infill responses were determined by calculating the differences between the infilled frame and the bare frame. By employing multivariate regression analysis, equations were proposed for the various parameters of the quadri-linear relationship governing the behaviour of masonry infill. A comparison of the equations proposed in this present study with those present in the literature demonstrated their reliability.

Civil Engineering

Database

Masonry compressive strength

Elasticity Modulus

Infilled frame structure

Multivariate Regression analysis

Empirical models

Masonry is often treated as a non-structural element, primarily valued for its thermal and sound insulation properties. However, post-earthquake reports by Bertero & Shah [1] and Davidovici [2] have shed light on its significant influence on the behaviour of frame structures. Reinforced concrete frame structures with masonry infill are prevalent worldwide, and they exhibit a complex behaviour. On one hand, the frame structure demonstrates flexible yet ductile characteristics, while the masonry wall exhibits rigid but brittle behaviour. The addition of infill walls to the frame substantially increases the overall stiffness of the structure. This increase primarily stems from the interaction between the frame and the infill wall. Notably, the enhanced stiffness has direct implications for the structure, such as the reduction in its period (El-Ouali et al. [3]). Numerous studies have highlighted that the incorporation of masonry infill enhances the stiffness, lateral strength, and energy dissipation capacity of the structure (C.V.R Murty and K Jain [4], Fiore et al. [5], Wang et al. [6]).

However, it is also important to acknowledge the potential detrimental effects of infill walls. These effects include alterations in the failure mechanism of frame structures (Ye et al. [7]), the occurrence of random damage in the surrounding frame of the infill (Asteris [8], Yuen and Kuang [9]), and the possibility of brittle failure in the infill itself.

Many studies have been conducted on the compressive strength of masonry ; authors such as Monteagudo et al. [10] and Uday & Reddy [11] have shown, using experimental study and finite element analysis, the detrimental influence of increasing the thickness of bedding joints for masonry, Thamboo et al. [12] reported an increase in the compressive strength of masonry with the reduction of bed joint thickness, especially for joint thicknesses less than 4mm. Caldeira et al. [13] found that joint thickness has a significant influence on the compressive strength of masonry only with units with sufficiently high strength. In the second case, increasing the joint thickness was found to significantly decrease the strength of the masonry. According to Lumantra et al. [14], their compression tests on masonry revealed that the presence of vertical joints had a negligible effect on the compressive properties of the masonry. Similarly, in their study, Ramamurthy & Ganesan [15] found that the properties of the vertical joints had no significant impact on the compressive behavior of the tested masonry.

Masonry infill modeling can be categorized into two main groups: micro-modeling and macro-modeling. Within micro-modeling, two sub-categories exist: detailed micro-modeling and simplified micro-modeling. In detailed micro-modeling, each masonry unit and mortar are individually modeled, and discrete elements are introduced to simulate the cohesive behaviour between them (D'altri et al. [16]). On the other hand, simplified micro-modeling involves extending the masonry units by half the thickness of the mortar and introducing discrete elements to simulate the cohesive behaviour between the units (Lourençou & Rots [17], Aref & Dolatshahi [18]).

Micro-modeling offers the advantage of accurately representing the local behaviour and damage of the masonry structure. However, it requires a significantly longer computational time and is typically employed for studying the local behaviour of simple frames. For a more in-depth understanding of micro-modeling of infills, interested readers can refer to the classification proposed by D'altri et al. [19] in recent literature.

In the field of global modeling methods for infilled frames, the diagonal strut method is widely recognized and extensively discussed in the literature. Polyakov [20] introduced this method, while Holmes [21] further expanded upon it by suggesting the replacement of infill with an equivalent diagonal strut. The thickness of this strut matches that of the infill, and its width is set to one-third of the masonry infill diagonal length. Smith [22], along with Smith and Carter [23], later refined this concept by proposing the first approach to calculating the width of the equivalent diagonal strut. According to these authors, this width is determined by the relative stiffness (λ_h) of the masonry infill and the frame structure.

$${\lambda _h}=\sqrt[3]{{\frac{{{E_w}\,{t_w}\,\sin 2\theta }}{{4\,{E_c}{I_c}{h_{\inf }}}}}}$$

Mainstone [24] carried out tests on steel frames with masonry infill and then proposed his model:

$$w=0.175\,{r_{\,\,\inf }}{\left( {{\lambda _h}{h_{col}}} \right)^{ - 0.4}}$$

Mainstone's equation, which was widely used and adopted by FEMA 356, has been the subject of further research by various authors. Kwan and Liauw [25], Decanini and Fantin [26], Dawe and Seah [27], El-Ouali et al. [3], and Durani & Luo [28] are among the authors who have proposed equations to evaluate the width of the equivalent strut. In terms of modeling the equivalent diagonal strut, two main approaches have been employed: the concentric strut model and the eccentric model. Studies have shown the benefits of using a concentric model for the overall response of the infilled frame, while an eccentric model is more suitable for analyzing the local response of the infilled structure (Samoilă [29] and Louzai and Abed [30])

Furthermore, several other strut models have been suggested by various authors ; Chrysotomou et al. [31], Crisafull [32], and more recently Brodsky and Huang [33] have proposed alternative models to further enhance the understanding and analysis of strut behaviour.

Some authors have proposed models for the behaviour of the masonry infill (Fig. 1). Panagiotakos and Fardis [36] and Bertoldi et al. [35] have each proposed quadri-linear models for the behaviour of masonry Fig. 1 (b) Rodrigues et al. [37] propose a penta-linear model Fig. 1 (c) inspired by models for reinforced concrete elements. The model has been calibrated using two experimental tests. Fardis [38] also proposed a simplified quadri-linear model Fig. 1 (d). These models are often developed from limited experimental data. Later authors have proposed models by analyzing larger datasets. De Risi et al. [34] collected more than 75 experimental tests of reinforced concrete frames with masonry infill. By deriving the responses of the different masonry infills, the authors then compared them to the models proposed by Bertoldi et al. [35] and Panagiotakos and Fardis [36]. The authors thereafter proposed a tri-linear model for masonry infill behaviour Fig. 1 (a). Yet again, De Risi et al. [34] did not investigate the different statistical parameters affecting the strength and stiffness of the masonry infill. Huang et al. [39] used a database of 113 tests of RC frames with masonry infill. The authors proposed equations using multivariate regression analysis for the quadri-linear relationship of the strut behaviour after modeling and numerical calibration.

The main objective of this present study is to establish a comprehensive database of tests conducted on reinforced concrete frames with masonry infill, focusing on their behaviour under lateral loading. The main aim is to develop equations that accurately represent the quadri-linear relationship of the masonry infill. Unlike the approach taken by Huang et al. [39], this present study will not rely on numerical calibration. Instead, the masonry infill response will be evaluated by comparing the frame's response with masonry infill to that of the bare frame. Additionally, this research will investigate different methodologies for assessing the compressive strength of masonry, while also addressing the evaluation of the masonry's modulus of elasticity.

In this present study, a comprehensive dataset comprising 160 experimental tests was gathered. The tests involved single-storey, single-span reinforced concrete frames with masonry infill, which were subjected to lateral loading. The primary objective is to conduct a statistical analysis focusing on the strength and stiffness characteristics of the masonry infill. For a visual representation of the collected experimental tests, please refer to Fig. 2, which provides a concise summary of the collected data.

The references for the collected tests, as well as the type of loading applied and the number of tests performed, are provided in Table 1

Table 1

References of the different tests collected for the infilled frame structures
nº	Test references	Loading type	Number of tests
1	Akhoundi et al. [40]	Quasi-static cyclic	6
2	Alwashali et al. [41]	Quasi-static cyclic	5
4	Basha et Kaushik [42]	Quasi-static cyclic	9
5	Bergami et Nuti [43]	Quasi-static cyclic	2
6	Butenweg et al. [44]	Quasi-static cyclic	3
7	Cavaleri & Di Trapani [45]	Quasi-static cyclic	12
8	Colangelo F. [46]	Pseudo-dynamic	11
9	Da porto et al. [47]	Quasi-static cyclic	6
10	Dautaj et al. [48]	Quasi-static cyclic/ Monotonic	7
11	Gazic & Sigmund [49]	Quasi-static cyclic	11
12	Huang et al. [50]	Quasi-static cyclic	4
13	Kakaletsis et al. [51][52]	Quasi-static cyclic	4
14	Layadi et al. [53]	Monotonic	2
15	Maidiawati et al. [54]	Quasi-static cyclic	3
16	Mehrabi & Shing [55]	Quasi-static cyclic	13
17	Misir et al. [56]	Quasi-static cyclic	2
18	Misir et al. [57]	Quasi-static cyclic	5
19	Morandi et al. [58]	Quasi-static cyclic	5
20	Schwarz et al. [59]	Quasi-static cyclic	7
21	Sigmund & Penava [60]	Quasi-static cyclic	9
22	Suzuki et al. [61]	Quasi-static cyclic	4
23	Tawfik Essa et al. [62]	Quasi-static cyclic	3
24	Tekeli & Aydin [63]	Quasi-static cyclic	9
25	Van & Lau [64]	Quasi-static cyclic / Monotonic	4
26	Verdame et al. [65]	Quasi-static cyclic	2
27	Zhai et al. [66]	Quasi-static cyclic	3
28	Zovkic [67]	Quasi-static cyclic	9
	Total number of collected tests		160

From the collected tests (160 tests), certain tests were removed due to their involvement with out-of-plane loading, confined masonry, or reinforced masonry infills and masonry infills with openings. As a result, only 119 tests were retained from the initial database. A summary of the masonry infill types present in the database can be found in Table 2.

Table 2

Types of masonry used in the database
Masonry unit type	Number of samples
Hollow Clay brick	55
Solid Clay brick	20
Hollow concrete bloc	16
Concrete bloc	7
AAC	7
Calcareneite brick	4
Fly ashes brick	9
Ceramic hollow	2
Pumice bloc	1

As can be observed, the collected tests include the loading type of monotonic loading. Van and Lau [64] demonstrated that, in terms of strength, there was no significant difference in the Infilled frame structures between cyclic and monotonic loading types. However, a noticeable difference was observed in terms of stiffness, with frames subjected to cyclic loading exhibiting higher stiffness compared to those under monotonic loading. Based on these observations, it was decided to incorporate both types of loading for strength determination, while excluding tests conducted under monotonic loading for stiffness determination. In regards to pseudo-dynamic loading, it is important to note that this present study assumes negligible differences in strength and stiffness between cyclic loading and pseudo-dynamic loading for masonry infilled frames. However, it is important to recognize that this assumption represents a limitation of the study.

During the collection of experimental data, several pieces of information, particularly regarding the mechanical characterization of the masonry infill, were not provided by the authors. These include the modulus of elasticity of the masonry (E_w), masonry shear strength (f_wu), and in some cases, masonry compressive strength (f_wv).

2.1. Evaluation of the masonry compressive strength

To characterize masonry in compression, two types of tests can be utilized: the prism compression test and the wallette compression test (refer to Fig. 3). Furthermore, to enhance our understanding of masonry behaviour under compression, a separate database was collected, consisting of masonry compressive tests. This database comprises 260 mean values derived from 1133 individual tests. Notably, this updated database surpasses the ones previously gathered by Sarhat & Sherwood [68], which included 248 mean values from 1092 individual tests, as well as the database compiled by Thaickavil & Thomas [69], which comprised 232 mean values.

The database used in our study is notable for its diverse range of test types. Within the collected database, there are 63 average values derived from tests conducted on wallettes, while 197 values originate from tests performed on prisms. This stands in contrast to the databases gathered by Sarhat and Sherwood [68], as well as Thaickavil and Thomas [69], which predominantly focused on prism tests with only a few cases of wallette tests. Indeed, recent studies [70] [71] have indicated that wallette tests consistently yield significantly lower strengths compared to prism tests.

Table 3 summarizes the references of the collected tests, along with their mechanical characteristics and the type of test conducted.

Table 3

References of the collected tests to characterize the masonry in compression
References	Material	f_b (MPa)	f_m (MPa)	Type of test
Basha & Kaushik [72]	- Fly ashes	5.7	[6.9–21.6]	Prism
Bennett et al. [73]	- Clay hollow	35.6	16.7	Wallette
Bergami [74]	- Clay hollow	[4.97–23.39]	[11.72–23.49]	Wallette
Bustos-García et al. [75]	- Clay solid	45	5.2	Wallette
Caldeira et al. [13]	- Concrete hollow	[16.3–45.6]	[5.6–18.3]	Prism
Calderon et al. [71]	- Clay hollow	[19.5–22.3]	[7.57–27.98]	Wallette & Prism
Cavaleri et al. [76]	- Clay hollow - Calcareinite bloc - Concrete hollow	[4.07–37.68]	[3.06–9.89]	Wallette
Da Porto et al. [77]	- Clay hollow	[23.49–25.15]	[11.51–17.68]	Wallette
Da Porto et al. [78]	- Clay hollow	13.5	19.9	Wallette
Esposito et al. [79]	- Clay hollow - Calcium silicate brick	[16–25]	[6.11–7.24]	Wallette
Ferreti et al. [80]	- AAC	3.1	7	Wallette & prism
Furtado et al. [81]	- Clay hollow - Concrete hollow	[1.5–3.25]	[5.54–13.5]	Wallette
Gumaste et al. [82]	- Table mounted bricks - Wire cut bricks	[5.7–23]	[0.86–12.21]	Wallette & prism
Kashik et al. [83]	- Clay solid	[16.1–28.9]	[3.1–20.6]	Prism
Lumantarna et al. [14]	- Clay solid	[8.5–43.4]	[0.69–23.2]	Prism
McNary, & Abrams [84]	- Clay solid	[69.8–101.7]	[3.4–52.6]	Prism
Monteagudo et al. [10]	- Clay solid	59.8	22.9	Wallette & prism
Morandi et al. [85]	- Clay hollow	8.64	7.68	Wallette
Padalu & Singh [86]	- Clay solid	26.16	14.75	Prism
Penava et al. [87]	- Clay hollow	14.5	5.02	Wallette
Radovanović et al. [88]	- Clay hollow - Concrete hollow	[3.26–6.51]	[8.2–23.9]	Wallette
Sarangapani et al. [89]	- Clay solid	[3.17–10.67]	[4.2–10.57]	Prism
Silva et al. [90]	- Clay hollow	14.05	10.38	Wallette
Singhal & Rai [91]	- Clay solid	[21.9–40]	8.5	Prism
Singh & Munjal [92]	- Clay solid - Concrete solid	[8.24–16.71]	[12.66–20.85]	Prism
Thaickavil & Thomas [69]	- Cement stabilised pressed brick - Clay solid	[4.56–6.68]	[13.6–35.5]	Prism
Thamboo & Dhanasekar [70]	- Clay solid - Compressed earth brick	[3.8–15.8]	[3.98–6.46]	Wallette & Prism
Veríssimo-Anacleto et al. [93]	- Stone brick	36.49	6.73	Wallette
Wang et al. [94]	- Clay solid	8.4	32.6	Wallette & prism
Wu et al. [95]	- Hollow shale and coal gangue	5.55	5.34	Wallette
Zhou et al. [96]	- Concrete hollow	[23.15–36.75]	[5.6–13.73]	Prism
Zovkić [67]	- Clay hollow - Concrete hollow - AAC	[2.12–13.21]	[4.04–13.89]	Wallette

2.2. Relationship between elastic modulus and the masonry compressive strength

In literature, various recommendations have been made regarding the E_w/f_wv ratio. However, different values have been reported in different sources. For instance, Eurocode 6 [97] suggests a value of 1000, the Canadian code CSA: S304.1 [98] suggests 850, the MSJC 2013 [99] suggests 700, and the FEMA 306 [100] suggests 550. Several authors have also proposed their own values for this ratio. Calderon et al. [71] propose a value of 580, while Wang et al. [94] consider the average of the recommendations from [101] and use a value of 900. Caldeira et al. [13], based on the collected data, state that the range of values found in the literature is too dispersed to draw conclusive general trends. Table 4 provides a summary of the recommended values found in the literature.

Table 4

Relationship between modulus of elasticity and compressive strength of masonry
Reference	Equation	Reference	Equation
[83][100]	E_w = 550 f_wv	[97]	E_w = 1000 f_wv
[102]	E_w = 500 f_wv	[98]	E_w = 850 f_wv
[71]	E_w = 580 f_wv	[99]	E_w = 700 f_wv
[13]	E_w = 474.5 f_wv	[104][105]	E_w = 750 f_wv
[103]	E_w = 1180 f_wv ^0.83	[72]	E_w = 600 f_wv
[94]	E_w = 900 f_wv	[106]	E_w = 800 f_wv

Based on the collected tests, only 157 tests evaluated masonry stiffness, specifically 47 wallette tests and 110 prism tests. Through statistical analysis, it was found that wallette tests yielded an average stiffness value of E_w = 1080 f_wv (Cov = 75.66%). This value is relatively close to the recommendations provided by EN 1996-1-1 [97], which are based on tests conducted on wallettes. On the other hand, the prism tests resulted in an average stiffness value of E_w = 540 f_wv (Cov = 49.73%), which aligns closely with the recommendations of FEMA 306 [100]. It's important to note that the coefficients of variation obtained are relatively high, particularly for the wallettes. Therefore, relying only on a simple average would lead to significant inaccuracies (see Fig. 4).

As shown in Fig. 4(a), the range of values for the E_w/f_wv ratio is much wider in the wallette tests compared to the prism tests depicted in Fig. 4(b). The wallettes exhibit values ranging from 400 to 2500, whereas the prisms have a narrower range of 350 to 800, despite having a higher number of tests conducted. Conducting a linear regression analysis, we obtained values of 695 (R² = 68%) for wallettes and 426 (R² = 87%) for prisms. In a study by Thamboo and Dhanasekar [70], they observed that wallettes displayed lower values of elastic modulus compared to prisms, while Calderon et al. [71] reported opposite results. However, our study, based on a larger database, allows us to assert that the prism tests indicate lower values compared to the wallette tests.

Gumaste et al. [82] have shown that prism tests exhibited a greater dispersion of strength values compared to wallette tests, indicating that wallette tests were more reliable. However, in this present study, we observe the opposite phenomenon concerning the modulus of elasticity values. Wallette tests demonstrate a substantial dispersion, whereas prism tests exhibit a certain concentration around specific values (refer to Fig. 4). Based on these observations, it appears appropriate to utilize prism tests for assessing masonry stiffness, while wallette tests are more suitable for evaluating its compressive strength.

For this present study, an average value of E_w = 560 f_wv is retained, which is obtained by averaging the values from the linear regressions for the two types of tests. This value is consistent with numerous recommendations found in the literature [59] [83] [100] [71] [72], and it is also close to the average stiffness displayed by the prism tests

2.3. Empirical Relationship for Assessing Masonry Compressive Strength

The prediction of compressive strength has been extensively studied, the topic has been the subject of numerous papers. Some authors have employed artificial neural networks (ANN) [107] [96] [108] to predict the compressive strength of masonry walls. Meanwhile, other authors have proposed empirical equations based on linear or non-linear regression. Table 5 presents the most reliable equations found in the existing literature

Table 5

Empirical models for assessing the compressive strength of masonry
Empirical model reference	Empirical model
Engesser [109]	${f_{wv}}=\frac{1}{3}\,\,{f_b}+\,\,\frac{2}{3}\,{f_m}$
Mann [110]	${f_{wv}}=0.83\,{f_b}^{{0.66}}\,\,{f_m}^{{0.18}}$
Kaushik et al. [83]	${f_{wv}}=0.63\,f_{b}^{{0.49}}\,\,{f_{mc}}^{{0.32}}$
Lumatarna et al. [14]	${f_{wv}}=0.75\,f_{b}^{{0.75}}\,\,f_{m}^{{0.31}}$
Kumavat [111]	${f_{wv}}=0.69\,f_{b}^{{0.6}}\,\,f_{m}^{{0.35}}$
Garzón-Roca et al. [108]	${f_{wv}}=\,\,0.53\,{f_b}\,+0.93\,\,{f_m} - 10.32$
Gumaste et al. [82]	For prism ${f_{wv}}=0.317\,f_{b}^{{0.866}}\,\,f_{m}^{{0.134}}$ For wallette ${f_{wv}}=\,\,1.242\,\,f_{b}^{{0.531}}\,\,f_{m}^{{0.208}}$
Christy et al. [112]	${f_{wv}}=\,\,0.35\,\,f_{b}^{{0.65}}\,\,f_{m}^{{0.25}}$
Bennett et al. [73]	${f_{wv}}=0.3\,\,{f_b}$
Basha & Kaushik [72]	${f_{wv}}=\,\,1.34\,\,f_{b}^{{0.1}}\,\,f_{m}^{{0.33}}$
Thamboo & Dhanasekar [70]	For prism ${f_{wv}}=\,\,0.2\,\,f_{b}^{{1.26}}\,\,f_{m}^{{0.15}}$ For wallette ${f_{wv}}=0.25\,\,f_{b}^{{1.09}}\,\,f_{m}^{{0.12}}$
Bröcker [113]	${f_{wv}}=0.68\,\,{f_b}^{{0.5}}\,\,{f_m}^{{0.333}}$
Dymiotis & Gutlederer [114]	${f_{wv}}=0.3266\,\,{f_b}\left( {1 - 0.0027\,\,{f_b}+0.0147\,\,{f_m}} \right)$
Calderon et al. [71]	${f_{wv}}=\,\,\frac{{2.042\,\,{f_b}^{{0.193}}\,\,{f_m}^{{0.549}}}}{{{{\left[ {\frac{{\left( {{\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h t}}\right.\kern-0pt}\!\lower0.7ex\hbox{$t$}}} \right)}}{3}} \right]}^{0.166}}}}$
Hendry & Malek [115]	${f_{wv}}=0.317\,\,{f_b}^{{0.531}}\,\,{f_m}^{{0.208}}$
Dayaratnam [116]	${f_{wv}}=0.275\,\,{f_b}^{{0.5}}\,\,{f_m}^{{0.5}}$
Veríssimo-Anacleton et al. [93]	${f_{wv}}=\,\,\,\left( {1 - {e^{ - \left( {0.059\,\,x\,\,{n^{0.205}}\,x\,7.943\,\,\,{h^{0.3}}} \right)}}} \right)\,\,\,\left( {0.193\,\,{f_b}^{{1.3}}\,\,{f_m}^{{0.28}}} \right)$

In the existing literature, there is a lack of comparative studies evaluating the performance of these equations in predicting masonry resistance using prism and wallette tests. Previous authors have primarily focused on prism tests [86] [69]. However, considering the findings presented in [70] and [71], conducting a comparative study appears to be highly relevant.

Figure 5 provides a comparison of the 17 empirical models presented in Table 5. Additionally, a regression analysis was conducted, resulting in the derivation of Eqs. (3) and (4) for the prism and wallette tests, respectively.

$${f_{wv}}=\,\,0.155\,\,f_{b}^{{1.05}}\,\,f_{m}^{{0.236}}$$

$${f_{wv}}=0.35\,\,f_{b}^{{0.891}}\,\,f_{m}^{{0.133}}$$

Figure 5 and 6 illustrate the comparison of the empirical model's predictions with experimental test results. The diagonal line on each graph represents perfect prediction, while the scattered points represent the values obtained for each model. A point below the diagonal line indicates an underestimation of resistance by the model, whereas a point above indicates an overestimation. To facilitate comparison between the models, statistical parameters were calculated and are presented in Table 6. The table is divided into two sections, one for the models evaluated using prism tests and the other for those evaluated using wallette tests.

Table 6

Evaluation of empirical models on (a) prism tests (b) wallette tests
(a)	Average*	Min*	Max*	SD	Cov (%)	MAD	RMSE	MAPE	MPE
Equation (3)	1.00	0.287	2.542	0.449	44.9	2.273	3.674	36.2	0.305
Engesser [109]	5.15	0.655	27.08	5.310	103.0	9.037	10.93	418.3	415.47
Mann [110]	1.92	0.470	5.314	1.186	61.6	3.024	3.848	105.1	92.556
Kaushik et al. [83]	1.38	0.273	4.814	1.107	74.7	3.741	6.049	85.2	48.122
Lumatarna et al. [14]	2.97	0.731	9.061	1.798	60.4	6.234	7.969	199.1	197.560
Kumavat [111]	2.21	0.452	7.232	1.542	69.8	3.517	4.186	133.2	120.921
Garzón-Roca et al. [108]	3.39	-5.19	27.00	4.984	146.8	8.048	11.07	308.8	239.526
Gumaste et al. [82]	1.03	0.308	2.619	0.512	49.8	2.830	4.733	42.1	2.791
Christy et al. [112]	0.95	0.228	2.823	0.611	64.0	3.890	6.606	52.4	-4.520
Bennett et al. [73]	0.94	0.281	2.299	0.405	42.8	2.555	4.379	34.3	-5.478
Basha & Kaushik [72]	1.49	0.106	5.446	1.407	93.9	5.174	8.335	108.2	49.845
Thamboo & Dhanasekar [70]	1.06	0.310	2.761	0.441	41.7	2.187	3.517	35.1	5.629
Bröcker [113]	1.69	0.317	5.553	1.262	74.6	3.572	5.481	97.8	69.119
Dymiotis & Gutlederer [114]	1.21	0.376	3.156	0.552	45.5	2.126	3.365	44.7	21.363
Calderon et al. [71]	4.96	0.317	18.747	4.698	94.6	8.561	9.919	404.9	396.50
Hendry & Malek [115]	0.60	0.120	1.755	0.418	69.3	5.086	8.581	51.8	-39.731
Dayaratnam [116]	1.07	0.128	4.072	0.863	80.2	4.319	7.041	69.7	7.563
Veríssimo-Anacleton et al. [93]	1.88	0.339	6.327	0.986	52.4	5.051	12.44	98.3	88.342
(b)	Average*	Min*	Max*	SD	Cov (%)	MAD	RMSE	MAPE	MPE
Equation (4)	1.01	0.485	2.350	0.384	38.1	1.267	1.894	28.8	0.889
Engesser [109]	3.13	0.842	10.823	2.090	66.6	6.244	7.652	214.1	213.52
Mann [110]	1.57	0.653	3.241	0.537	34.2	1.916	2.570	59.48	57.054
Kaushik et al. [83]	1.11	0.369	2.372	0.445	40.0	1.355	2.135	33.58	11.324
Lumatarna et al. [14]	2.25	0.998	4.689	0.811	35.9	4.827	6.411	125.6	125.63
Kumavat [111]	1.63	0.609	3.407	0.605	37.1	2.113	2.864	65.78	63.04
Garzón-Roca et al. [108]	0.67	-5.28	8.850	2.455	36.4	5.436	6.849	182.9	-32.61
Gumaste et al. [82]	1.90	0.679	3.927	0.706	37.2	2.632	3.111	91.11	89.63
Christy et al. [112]	0.75	0.303	1.489	0.258	34.6	1.663	2.564	31.54	-25.26
Bennett et al. [73]	0.86	0.337	2.175	0.375	43.7	1.390	1.942	33.16	-14.169
Basha & Kaushik [72]	0.64	0.287	0.991	0.155	24.4	2.124	3.169	54.25	15.681
Thamboo & Dhanasekar [70]	1.13	0.423	3.083	0.551	48.8	1.871	3.093	39.12	12.89
Bröcker [113]	1.26	0.422	2.713	0.502	39.8	1.434	2.037	40.95	26.08
Dymiotis & Gutlederer [114]	1.02	0.433	2.453	0.425	41.7	1.374	2.057	31.89	2.026
Calderon et al. [71]	3.10	0.647	9.942	1.943	62.7	5.487	6.395	212.1	210.08
Hendry & Malek [115]	0.48	0.173	1.002	0.180	37.2	2.667	3.705	51.61	-51.60
Dayaratnam [116]	0.73	0.231	1.857	0.325	44.7	1.920	2.992	36.82	-27.26
Veríssimo-Anacleton et al. [93]	2.05	0.564	7.87	1.389	67.7	6.609	11.44	113.42	105.16

The values (*) represent the average, minimum and maximum values for (F_pred/F_exp), SD represents the standard deviation of the (F_pred/F_exp) ratio and Cov represents its coefficient of variation, MAD represents the mean absolute deviation, RMSE represents the root mean square error, MAPE represents the mean absolute percentage error and MPE represents the mean percentage error.

Table 6 presents the performance of the 18 empirical equations outlined in Table 5. It can be observed that, in terms of average (F_pred/F_exp), Eq. (3) and the equation proposed by Gumaste et al. [82] exhibit the best values for prism tests. They are closely followed by the equations developed by Thamboo & Dhanasekar [70] and Dayaratnam [116], with coefficients of variation of 0.498, 0.417, and 0.863, respectively. Notably, for the Dayaratnam [116] equation, it should be noted that Fig. 5 (n) reveals an underestimation of strength for f_wv greater than 15 MPa. The equations proposed by Christy et al. [112] and Bennett et al. [73] also demonstrate average values that are relatively close to one. These equations exhibit particularly low MAPE and MPE indicators compared to the other equations examined. However, it should be noted that the Christy et al. [112] equation, as depicted in Fig. 5 (i), underestimates masonry strength when f_wv exceeds 20 MPa

On the other hand, the equation developed by Hendry & Malek [115] significantly underestimates the compressive strength, with an average value (F_pred/F_exp) of 0.60. Consequently, it displays the lowest error index (MPE), as clearly illustrated in Fig. 5 (l), where a substantial number of points fall below the diagonal line.

In contrast, the equations proposed by Engesser [109], Calderon et al. [71], and Garzón-Roca et al. [108] exhibit a significant overestimation of masonry strength. These models have the highest average values (F_pred/F_exp) among the examined equations, as well as RMSE, MPE, and MAPE error indicators that are considerably higher than those of other models. This observation is evident in Fig. 5 (g), (h), and (r), respectively. Thaickavil & Thomas [69] also noted an overestimation of strength for the Engesser [109] and Garzón-Roca et al. [108] equations.

Regarding the wallette tests, Eq. (4) as well as the equations proposed by Bennett et al. [73], Dymiotis & Gutlederer [114], Kaushik et al. [83], and Thamboo & Dhanasekar [70] exhibit the best values in terms of average (F_pred/F_exp). Notably, the equation presented by Kaushik et al. [83], which originally had an average (F_pred/F_exp) of 1.38 for the prism tests, decreases and approaches to one in the wallette tests. In contrast, the equation proposed by Christy et al. [112] clearly underestimates the compressive strength for the wallette tests, despite performing well in the prism tests.

Moreover, the equations proposed by Lumatarna et al. [14] and Calderon et al. [71] exhibit a significant overestimation of compressive strength, which was observed in the prism tests but not in the wallette tests. Conversely, the equation by Garzón-Roca et al. [108] clearly underestimates the compressive strength in the wallette tests, while it overestimates the compressive strength in the prism tests

Among the collected equations, only Gumaste et al. [82] and Thamboo & Dhanasekar [70] proposed different equations to evaluate the compressive strength using prism and wallette tests. The equation proposed by Gumaste et al. [82] for prism tests demonstrates excellent results. However, the equation for wallette tests tends to overestimate the compressive strength, with a mean value (F_pred/F_exp) that is almost twice the expected value, along with relatively high statistical parameters (MPE, MAPE, and RMSE). On the other hand, the equations presented by Thamboo & Dhanasekar [70] yield satisfactory results for both types of tests, making them suitable for evaluating the compressive strength of masonry.

Padalu and Singh [86] evaluated the performance of empirical equations on Indian masonry and concluded that the equations proposed by Kaushik et al. [83], Dymiotis and Gutlederer [114], and Christy et al. [112] exhibited the best performance for masonry in India. The results of our present study closely align with these findings, as Padalu and Singh [86] also examined masonry samples with strengths below 15 MPa. It is evident that these equations yield satisfactory results for masonry with strengths below 15 MPa. Therefore, for this present study, Eq. (4) has been chosen to calculate the compressive strength of masonry.

2.4. Masonry Shear Strength

Regarding shear strength, numerous experimental tests did not provide this value. Therefore, equations have been proposed in the literature to establish a relationship between compressive strength and shear strength (f_wu). In this present study, the equation proposed by Liberatore et al. [117] was selected to calculate the missing values.

$${f_{wu}}=0.285\sqrt {{f_{wv}}}$$

One of the most commonly used models to describe masonry infill behaviour is the quadri-linear relationship, as shown in Fig. 7.

For this present study, the various parameters that constitute the quadri-linear relationship of masonry infill will be evaluated, namely: V_max, V_crack, V_resi, K_e, K_peak, and K_soft.

The experimental response of the masonry infill is determined by calculating the difference between the response of the infilled frame and the response of the bare frame. This methodology has been extensively adopted in the literature [34] [43] [41] [118] [78].

Among the 119 experimental tests, several tests on the bare frame were not performed. Authors such as Alwashali et al. [118] used a bi-linear relationship to simulate the behaviour of the bare frame. However, using such a method requires numerical calibration. In the case of this present study, only the tests with experimental curves of the bare frame were retained. In the end, 87 tests were used to develop the equations. Table 7 provides a comprehensive overview of the key structural parameters for the database of infilled frame structures employed in this present study.

Table 7

Structural parameter summary for the database of infilled frame structures
	H_w (mm)	L_w (mm)	D_w (mm)	t_w (mm)	tan θ	f_wv (MPa)	f_wu (MPa)	E_w (MPa)	λ_h
Min	705	900	1357	44	0.454	0.73	0.04	508	1.517
Max	2950	4220	5148	365	1.555	18.5	1.225	9520	6.275
Mean	1482	2021	2516	128	0.752	4.80	0.453	3365	3.538
SD	487.66	711.43	834.81	74.27	0.158	3.70	0.271	2413.78	1.166
Cov (%)	32.9	35.19	33.18	58.01	20.99	77.13	59.80	71.72	32.95

All the values obtained after extracting the curves of the masonry infills are shown in Figs. 8 and 9.

By comparing the values obtained in this present study with those obtained by De Risi et al. [34], it is evident that the resistances obtained in this present study are slightly higher. The resistances range from 16.4 KN to 400.6 KN in this present study, whereas De Risi et al. [34] reported resistances ranging from 16.0 KN to 351.8 KN. It can be observed that the lower limits of the resistances obtained are nearly identical. However, for the upper limits, this study presents higher values. This difference can be attributed to the fact that the database collected for this present study is larger than the one used by De Risi et al. [34].

Regarding the initial stiffness of the masonry infill, this present study reports values ranging from 10.75 KN/mm to 511.6 KN/mm, while De Risi et al. [34] reported values ranging from 10.9 KN/mm to 510.2 KN/mm. We note that the two ranges exhibit almost equal values.

In terms of peak stiffness, this present study revealed a wide range of values, spanning from 0.9 KN/mm to 254 KN/mm. In contrast, De Risi et al. [34] reported a narrower range of 4.0 KN/mm to 89 KN/mm.

The general form of the equations for the parameters that constitute the quadri-linear relationship of masonry infill is as follows:

$$Ln\left( Y \right)={\alpha _1}\,Ln\left( {{X_1}} \right)+{\alpha _2}\,Ln\left( {{X_2}} \right)+{\alpha _3}\,Ln\left( {{X_3}} \right)+{\alpha _4}\,Ln\left( {{X_4}} \right)+.....+{\alpha _n}\,Ln\left( {{X_n}} \right)+{\alpha _{n+1}}$$

Let Y be the dependent variable representing the stiffness and strength of masonry infill, and let x₁, x₂, ..., x_n be the set of variables that explain the dependent variable. Huang et al. [39] used this form to determine the parameters of the quadri-linear diagonal strut relationship. Once the equation is derived, it can be transformed to obtain the desired parameter value.

$$Y=\,\,{e^{{\alpha _{n+1}}}}\,\,X_{1}^{{{\alpha _1}}}\,\,\,X_{2}^{{{\alpha _2}}}\,\,\,X_{3}^{{{\alpha _3}}}.......\,.......X_{n}^{{{\alpha _n}}}$$

For the selection of explanatory variables, we will refer to different equations available in the literature. Additionally, we will utilize the F-test statistic to identify the variables that can adequately explain the variability of the dependent variable we are investigating.

4.1. Strength of the masonry infill

Various parameters that potentially impact the strength of the masonry infill were taken into account. To analyze their influence, an ANOVA analysis was performed, and the outcomes of the static F-test are displayed in Table 8.

Table 8

Results of F-Test statistics for parameters affecting masonry infill strength
Characteristic	F observed	p value
Ln (D_w)	2.863	9.45 x 10^− 2
Ln (t_w)	13.606	4.07 x 10^− 4
Ln (tan θ)	9.169	3.30 x 10^− 3
Ln (f_wv)	0.569	0.453
Ln (f_wu)	7.241	8.65 x 10^− 3

After performing a regression analysis, the masonry infill strength can be expressed as follows:

$${V_{\hbox{max} }}=\,\,0.0732\,\,\left( {\tan \theta } \right){\,^{ - 0.9301}}\,D_{w}^{{0.5274}}\,\,f_{{wu}}^{{0.2636}}\,\,t_{w}^{{0.6177}}$$

The strength of the masonry infill depends on various factors, including its geometric characteristics (D_w, t_w, tan θ) and its mechanical properties (f_wu). This differs from the findings of Huang et al. [39], where the strength of the masonry infill depends on its compressive strength (f_wv). However, the results of this present study align with the proposals made by Panagiotakos and Fardis [36] and De Risi et al. [34]. To compare the findings, Eq. (8) is referenced alongside the equation proposed by Huang et al. [39], based on the experimental strengths obtained.

Table 9

Comparison of masonry infill strength equations with experimental values obtained
Statistical parameter	Equation (8)	Huang et al. [39]
Average (F_pred/F_exp)	1.001	1.729
Min (F_pred/F_exp)	0.319	0.389
Max (F_pred/F_exp)	4.817	7.704
Standard deviation (SD)	0.648	1.226
Coef. variation (%)	64.7	70.9
MAD	42.52	73.14
RMSE	55.36	113.62
MAPE (%)	41.76	85.39
MPE (%)	0.132	72.93

We can observe a significant improvement in the results using Eq. (8) compared to the values obtained from the equation proposed by Huang et al. [39]. Eq. (8) demonstrates a mean (F_pred/F_exp) value equal to one, indicating a more accurate prediction. Conversely, the equation by Huang et al. [39] tends to overestimate the masonry infill strength, with a mean value (F_pred/F_exp) nearly twice as high as Eq. (8). Notably, the equation proposed in this present study exhibits superior statistical parameters such as standard deviation, MAPE, MPE, MAD, and others.

4.2. Elastic stiffness of masonry infill

Utilizing a similar approach used for assessing the strength of the masonry infill, we examined the parameters that could potentially influence the elastic stiffness of the masonry infill. Following an ANOVA analysis, the results of the static F-test are outlined in Table 10

Table 10

Result of the F-test statistics for parameters affecting masonry stiffness
Characteristic	F observed	p value
Ln (D_w)	3.422	6.9 x 10^− 2
Ln (t_w)	0.121	0.729
Ln (L_w)	3.921	5.2 x 10^− 2
Ln (λ_h)	7.531	7.69 x 10^− 3
Ln (H_w)	2.779	0.1
Ln (E_w)	38.995	2.88 x 10^− 8
Ln (f_wv)	11.792	1 x 10^− 3

The resulting equation is given by the following form:

$${K_e}=0.00697\,\,L_{w}^{{1.6663}}\,D_{w}^{{ - 1.3952}}\,\,f_{{wv}}^{{ - 0.7374}}\,\,E_{w}^{{1.148}}\,\lambda _{h}^{{ - 0.8830}}$$

The elastic stiffness of masonry is influenced by both its geometric characteristics (D_w, L_w) and mechanical properties (f_wv, E_w), which are determined through compression tests, as illustrated in Fig. 2. Among these characteristics, the modulus of elasticity holds the highest significance. Moreover, it is worth noting that the masonry stiffness depends on the relative stiffness between the frame and the masonry infill, as recommended by Smith [22]. To provide further insights, a comparison is made with the equation proposed by Huang et al. [39].

Table 11

Comparing the masonry elastic stiffness equations with the experimental results obtained
Statistical parameter	Equation (9)	Huang et al. [39]
Average (F_pred/F_exp)	1.02	1.68
Min (F_pred/F_exp)	0.167	0.126
Max (F_pred/F_exp)	4.438	6.530
Standard deviation (SD)	0.834	1.394
Coefficient of variation (%)	81.56	82.94
MAD	63.80	71.41
RMSE	108.52	113.39
MAPE (%)	60.56	99.85
MPE (%)	2.309	68.08

The equation proposed in this present study demonstrates significantly better statistical parameters compared to the one introduced by Huang et al. [39]. Furthermore, it is evident that the equation put forward by Huang et al. [39] consistently overestimates the elastic stiffness of masonry, as indicated by an average (F_pred/F_exp) that significantly exceeds one.

4.3. Stiffness at peak of masonry strength

Regarding the stiffness at the peak of the masonry strength, we followed the same procedure to derive the following Eq. (10):

$${K_{peak}}=1.3233\,\,L_{w}^{{3.1142}}\,\,D_{w}^{{ - 3.6385}}\,\,f_{{wv}}^{{ - 0.9254}}\,\,E_{w}^{{0.9840}}\,\,\lambda _{h}^{{ - 0.7440}}$$

Another alternative approach would involve conducting a linear regression analysis between the elastic stiffness and the peak stiffness, obtaining the following equation:

$${K_{peak}}=\,\,\,0.155\,\,{K_e}$$

The analysis reveals a significant reduction of 84.5% from the initial stiffness at the peak strength of the masonry. To evaluate the accuracy of the results, a comparison is carried out between the two equations (Eq. 10 and Eq. 11) and the one proposed by Huang et al. [39].

Table 12

Comparing the masonry stiffness equations with the obtained experimental values
Statistical parameter	Equation (10)	Equation (11)	Huang et al. [39]
Average (F_pred/F_exp)	1.05	1.01	0.433
Min (F_pred/F_exp)	0.114	0.132	0.054
Max (F_pred/F_exp)	20.86	4.84	1.95
Standard deviation (SD)	2.380	0.956	0.352
Coefficient of variation (%)	226.8	94.9	81.28
MAD	14.53	12.49	14.97
RMSE	31.59	25.03	30.25
MAPE (%)	84.77	67.51	61.33
MPE (%)	4.952	0.697	-56.67

We can observe that Eq. (11) demonstrates superior performance compared to Eq. (10) and the equation proposed by Huang et al. [39]. Notably, the equations proposed by Huang et al. [39] consistently underestimates the post-peak stiffness, in contrast to equations (11) and (12) proposed in this present study. It is important to highlight that Eq. (10) occasionally overestimates the stiffness at the peak of the masonry strength. Therefore, Eq. (11) will be selected as the preferred model for this specific parameter.

4.4. Stiffness of the descending branch

In the set of curves obtained, some curves do not exhibit a descending branch (reduced displacement). For the remaining curves (69 tests), a linear regression was performed to establish a relationship between the elastic stiffness and the stiffness of the descending portion (post-peak). The relationship is expressed as follows :

$${K_{soft}}=\,\, - 0.03849\,{K_e}$$

For the descending branch, researchers have put forward equations as documented by Noh et al. [119], based on the model developed by Panagiotakos and Fardis [36]. They recommend selecting a coefficient ranging from 0.5–10% of the masonry's elastic stiffness. Additionally, Liberatore and Decanini [120] propose a value between 2–4% of the peak stiffness (K_peak) as a suitable range.

4.5. Resistance to cracking and residual strength

To assess the resistance of the masonry to cracking, a linear regression analysis was conducted, enabling us to establish the following relationship:

$${V_{crack}}=0.722\,\,{V_{\hbox{max} }}$$

This equation is identical to what was proposed by De Risi et al. [34] and Huang et al. [39], this demonstrates the relevance of this equation.

In terms of residual strength, a series of tests revealed diverse outcomes. In 9 tests, no residual strength was observed, exhibiting a tri-linear curve behaviour. However, in other cases, the residual strength approached a significant portion of the maximum strength, reaching up to 85% of its value. To establish a relationship between residual strength and maximum strength, a linear regression analysis was performed.

$${V_{resid}}=0.311\,\,{V_{\hbox{max} }}$$

A comparison with the equation proposed by Huang et al. [39] is given in Table 13:

Table 13

Comparison of the equations for the residual strength of masonry with the experimental values obtained
Statistical parameter	Equation (14)	Huang et al. [39]
Average (F_pred/F_exp)	1.00	1.29
Min (F_pred/F_exp)	0.316	0.407
Max (F_pred/F_exp)	6.124	7.877
Standard deviation (SD)	0.955	1.229
Coefficient of variation (%)	95.26	95.26
MAD	25.539	22.548
RMSE	35.37	30.57
MAPE (%)	58.02	66.40
MPE (%)	0.30	29.00

The equation proposed in this present study displays a better mean value (F_pred/F_exp) compared to the equation proposed by Huang et al. [39]. However, the latter equation tends to slightly overestimate the residual strength in relation to the experimental values. It is noteworthy that the equation of Huang et al. [39] exhibits notably better statistical parameters, such as RMSE and MAD. Although there is this disparity, both equations can be employed for evaluating the residual strength of masonry, considering the relatively small difference observed in this comparison.

For this present study, we developed two databases. The first database consists of 260 average values obtained from 1,133 masonry compression tests, which were conducted using two types of tests: prism tests and wallette tests. The second database contains data from 119 experimental tests involving reinforced concrete frames with masonry infill that were subjected to horizontal loading.

From the first database, several important findings have emerged:

Stiffness : The wallette tests exhibited a wider range of values compared to the prism tests. Surprisingly, the wallette tests showed higher stiffness values than the prism tests, contradicting previous studies that relied on a limited number of tests. Based on these findings, this present study recommends the use of prism tests for evaluating the stiffness of masonry.
Strength: The wallette tests demonstrated lower strengths compared to the prism tests. Assessing the compressive strength using wallettes helps prevent an overestimation of the strength. Considering the literature and the comparisons made in this present study, it is recommended to employ wallette tests for evaluating the strength of masonry.
Evaluation of empirical models: More than 17 empirical models were evaluated for assessing the compressive strength of masonry under both types of tests. This present study recommends using the equations proposed by Thamboo & Dhanasekar [70], Gumaste et al. [82], and Dayaratnam [116] for evaluating masonry strength following prism tests. For wallette tests, the equations of Bennett et al. [73], Dymiotis & Gutlederer [114], Kaushik et al. [83], and Thamboo & Dhanasekar [70] are recommended. Additionally, this present study has proposed its own equations and demonstrated their reliability for all types of tests.

Subsequently, a comprehensive analysis was conducted using an initial database of 119 experimental tests. The responses of the masonry infills were extracted from these tests. The adopted methodology involved distinguishing between the response of the frame with infill and that of the bare frame. Through a rigorous multivariate regression analysis, the parameters of the quadri-linear relationship of the masonry infill were calculated. The findings of this present study demonstrated that :

In terms of strength ; the lateral strength of the masonry infill was more closely related to its shear strength (f_wu) rather than its compressive strength (f_wv).
When considering stiffness ; the elastic stiffness of the masonry infill is predominantly determined by the modulus of elasticity obtained from compression tests on the masonry. Moreover, it has been observed that the relative stiffness exerts a substantial influence on the elastic stiffness of the masonry infill
At the point of peak strength ; it has been observed that there is a significant loss of stiffness, reaching up to 85% of the elastic stiffness of the masonry infill.
This study presents several equations proposed to describe the different parameters of the quadri-linear relationship of the masonry infill. By conducting a thorough comparison with existing equations in the literature, we have successfully demonstrated the significance and reliability of our proposed equations.

Among the limitations of this present study, it should be noted that further investigation is required to consider pseudo-dynamic and cyclic loading as types of loading that may affect the strength and stiffness of the masonry infill. Future studies in this area will provide more insight into this subject. Additionally, the relative strength between the masonry infill and the frame was not taken into account in this present study. However, previous research has shown that the presence of a strong infill generally leads to shear failure of the columns. Therefore, a potential research perspective would be to incorporate a relative strength parameter in future studies.

Furthermore, different evaluation methods were employed to assess the mechanical characteristics of the masonry (such as modulus of elasticity and shear strength) in the various tests conducted. It has been observed that some methods tend to overestimate the strength or stiffness of the masonry. Standardizing the values across the evaluation methods would enable the development of more reliable empirical models, and this serves as a future research topic.

D_w

Diagonal length of the masonry infill

f_b

Compressive strength of the masonry unit

f_m

Compressive strength of mortar

f_wu

Shear strength of masonry

f_wv

Compressive strength of masonry

F_exp

Experimental value of the masonry strength

F_pred

Predicted value of the masonry strength using the empirical model

E_w

Masonry modulus of elasticity

E_c

Concrete modulus of elasticity.

H_w

Masonry infill height

I_c

Inertia of column

K_e

Masonry elastic stiffness of masonry under lateral loading

K_peak

Masonry stiffness at the peak strength

K_soft

Masonry stiffness of the descending slope of the load-displacement curve

L_w

Masonry infill length

t_w

Thickness of the masonry infill.

Tanθ

Height to length ratio of masonry infill

V_crack

Pre-cracking strength of masonry infill under lateral loading

V_max

Maximum strength of masonry infill under lateral loading

V_rési

Residual strength of masonry infill under lateral loading

λ_h

Relative stiffness of the masonry infill to the frame structure

δ_r

Displacement corresponding to the maximum strength of the masonry infill

δ_cr

Displacement corresponding to the pre-cracking resistance of the masonry infill

δ_u

Residual displacement of masonry infill

Competing interest:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Multivariate regression analysis to strength and stiffness evaluation of masonry infills

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Abstract

Figures

1. Introduction

2. Experimental Database Characterization

2.1. Evaluation of the masonry compressive strength

2.2. Relationship between elastic modulus and the masonry compressive strength

2.3. Empirical Relationship for Assessing Masonry Compressive Strength

2.4. Masonry Shear Strength

3. Extraction of masonry infill curves

4. Multivariate regression analysis

4.1. Strength of the masonry infill

4.2. Elastic stiffness of masonry infill

4.3. Stiffness at peak of masonry strength

4.4. Stiffness of the descending branch

4.5. Resistance to cracking and residual strength

5. Conclusion

Abbreviations

Declarations

Competing interest:

References

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Empirical model reference	Empirical model
Engesser [109]	\({f_{wv}}=\frac{1}{3}\,\,{f_b}+\,\,\frac{2}{3}\,{f_m}\)
Mann [110]	\({f_{wv}}=0.83\,{f_b}^{{0.66}}\,\,{f_m}^{{0.18}}\)
Kaushik et al. [83]	\({f_{wv}}=0.63\,f_{b}^{{0.49}}\,\,{f_{mc}}^{{0.32}}\)
Lumatarna et al. [14]	\({f_{wv}}=0.75\,f_{b}^{{0.75}}\,\,f_{m}^{{0.31}}\)
Kumavat [111]	\({f_{wv}}=0.69\,f_{b}^{{0.6}}\,\,f_{m}^{{0.35}}\)
Garzón-Roca et al. [108]	\({f_{wv}}=\,\,0.53\,{f_b}\,+0.93\,\,{f_m} - 10.32\)
Gumaste et al. [82]	For prism \({f_{wv}}=0.317\,f_{b}^{{0.866}}\,\,f_{m}^{{0.134}}\) For wallette \({f_{wv}}=\,\,1.242\,\,f_{b}^{{0.531}}\,\,f_{m}^{{0.208}}\)
Christy et al. [112]	\({f_{wv}}=\,\,0.35\,\,f_{b}^{{0.65}}\,\,f_{m}^{{0.25}}\)
Bennett et al. [73]	\({f_{wv}}=0.3\,\,{f_b}\)
Basha & Kaushik [72]	\({f_{wv}}=\,\,1.34\,\,f_{b}^{{0.1}}\,\,f_{m}^{{0.33}}\)
Thamboo & Dhanasekar [70]	For prism \({f_{wv}}=\,\,0.2\,\,f_{b}^{{1.26}}\,\,f_{m}^{{0.15}}\) For wallette \({f_{wv}}=0.25\,\,f_{b}^{{1.09}}\,\,f_{m}^{{0.12}}\)
Bröcker [113]	\({f_{wv}}=0.68\,\,{f_b}^{{0.5}}\,\,{f_m}^{{0.333}}\)
Dymiotis & Gutlederer [114]	\({f_{wv}}=0.3266\,\,{f_b}\left( {1 - 0.0027\,\,{f_b}+0.0147\,\,{f_m}} \right)\)
Calderon et al. [71]	\({f_{wv}}=\,\,\frac{{2.042\,\,{f_b}^{{0.193}}\,\,{f_m}^{{0.549}}}}{{{{\left[ {\frac{{\left( {{\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h t}}\right.\kern-0pt}\!\lower0.7ex\hbox{$t$}}} \right)}}{3}} \right]}^{0.166}}}}\)
Hendry & Malek [115]	\({f_{wv}}=0.317\,\,{f_b}^{{0.531}}\,\,{f_m}^{{0.208}}\)
Dayaratnam [116]	\({f_{wv}}=0.275\,\,{f_b}^{{0.5}}\,\,{f_m}^{{0.5}}\)
Veríssimo-Anacleton et al. [93]	\({f_{wv}}=\,\,\,\left( {1 - {e^{ - \left( {0.059\,\,x\,\,{n^{0.205}}\,x\,7.943\,\,\,{h^{0.3}}} \right)}}} \right)\,\,\,\left( {0.193\,\,{f_b}^{{1.3}}\,\,{f_m}^{{0.28}}} \right)\)