Research on Mechanical Properties of Core-Filled Desulfurized Gypsum Masonry

Abstract

Desulfurized gypsum, as a by-product of the wet flue gas desulfurization process in power plants, is one of the main pathways for its resource utilization by preparing new-type desulfurized gypsum hollow block with gypsum core. This paper, based on the research of high-precision desulfurized gypsum materials, conducted axial compressive strength tests on five groups of desulfurized gypsum hollow blocks with concrete core-filling and reinforced masonry. Through regression analysis of the test data, compressive strength and elastic modulus calculation formulas were proposed for such desulfurized gypsum hollow blocks with concrete core-filling and reinforced masonry. The calculated values from the formulas are in good agreement with the experimental values, providing a theoretical basis for the application and research of these masonry structures.

1. Introduction

Industrial desulfurized gypsum [1], as the main solid waste generated from the desulfurization treatment of exhaust gases in thermal power generation and smelting industries, is stockpiled in large quantities, occupying land and polluting the environment.

High-precision desulfurized gypsum hollow blocks are one of the important methods for the secondary utilization of gypsum solid waste in recent years. The “high-precision” characteristic is reflected in the production process through stringent control over the dimensional tolerances, geometric consistency, and weight uniformity of the blocks, achieving exceptionally high accuracy standards. These blocks address the shortcomings of traditional blocks, offering advantages such as energy efficiency, environmental friendliness, lightweight properties, and thermal insulation. They particularly outperform traditional blocks in terms of fire resistance, comfort, stability, ease of processing, and construction convenience. The new high-precision desulfurization gypsum core-filled masonry and reinforced masonry structures, as new types of masonry structures, have been widely applied in the construction of low-story prefabricated green residential buildings in rural areas. Therefore, investigating the mechanical properties of gypsum hollow block core-filled masonry and reinforced masonry is of great significance.

The existing methods for calculating the compressive strength of core-filled masonry can be categorized as follows: deformation coordination analysis method [2], elastic theory analysis method [3], strength failure theory-based analysis method [4], finite element analysis method [5], and test statistical analysis method [6,7]. There is limited research both domestically and internationally on the load-bearing performance of gypsum core-filled masonry and reinforced masonry, with most studies drawing on the performance research of concrete core-filled masonry. Zhu [8] compiled test data from standard specimens of concrete core-filled block masonry and fitted the stress–strain relationship using the least squares method, deriving a reasonable constitutive relationship; Liu et al. [9] combined test data and China’s “Code for Design of Masonry Structures” to propose a calculation formula for the load-bearing capacity of masonry walls; Zahra et al. [10] derived and discussed the failure modes, compressive strength, and stress–strain curves based on experimental data from 40 concrete block masonry specimens; Lv et al. [11] derived a formula for calculating the compressive strength of grouted masonry; Cheng [12] conducted axial compression tests on multiple core-filled masonry specimens and obtained a relational expression between the elastic modulus and compressive strength of core-filled masonry.

Sun [13] conducted axial compressive strength tests on 22 concrete hollow block core-filled masonry specimens and 3 core-filled reinforced masonry wall panels, proposing calculation Formula (1) for the axial compressive bearing capacity of core-filled masonry and reinforced masonry.

$$f_{\mathrm{m}}=0.7f_{\mathrm{b}}+0.76\mu f_{cu}$$

(1)

where $f_m$ represents the compressive strength of the core-filled masonry; $f_b$ represents the gross cross-sectional compressive strength of the block; $f_{cu}$ represents the compressive strength of the concrete core column; $\mu$ represents the void ratio of the block. Mahrous et al. [14] concluded that the addition of vertical reinforcement increased the shear capacity of reinforced masonry by 21% and the peak shear strain by 52%; Xu [15] conducted axial compression tests on reinforced masonry and proposed a calculation formula for the axial compressive bearing capacity of reinforced masonry; Hu [16] conducted a comprehensive analysis of the entire compression testing process of grouted block masonry, deriving a calculation formula for the compressive bearing capacity of reinforced masonry wall panels; Wang et al. [17] investigated the influence of steel strength, steel ratio, and steel section form on the axial compressive bearing capacity of short high-strength steel-reinforced concrete (HSSRC) columns and concluded that existing methods can accurately calculate the axial bearing capacity of HSSRC columns. Dhanasekar et al. [18] used a low-cost reinforcement method to increase the compressive strength of grouted masonry by 38%; Qian [19] conducted axial compression tests on concrete columns with stirrup reinforcement and derived stress–strain curves for reinforced masonry; Doty Natalie [20] proposed a new type of concrete-filled masonry with cross-spiral stirrups; Zheng [21] proposed that reinforced masonry exhibits higher deformation capacity, more complete failure, and stronger compressive performance compared to unreinforced masonry; Eltaly et al. [22] analyzed the structural performance of lightweight core steel mesh-reinforced concrete columns and derived a new equation for predicting the ultimate load of steel mesh-reinforced hollow reinforced concrete columns.

Due to the significant differences between the new high-precision desulfurized gypsum blocks and traditional blocks, calculating the compressive strength of such masonry using existing codes results in substantial errors. Furthermore, there is limited research by domestic and international scholars on the fundamental load-bearing performance of new high-precision desulfurized gypsum core-filled masonry and reinforced masonry, and there is a lack of formulas for calculating the compressive strength and elastic modulus of such masonry. Through experimental research on the mechanical properties of this new high-precision desulfurized gypsum core-filled masonry and reinforced masonry, the failure modes of such masonry under compression were investigated. Based on the test results, calculation formulas for the compressive strength and elastic modulus of such masonry were proposed, providing guidance for the research and engineering application of this new high-precision desulfurized gypsum block.

2. Test Overview

2.1. Specimen Design

The high-precision desulfurized gypsum hollow non-plastering blocks used in this test are made from a mixture of industrial desulfurization gypsum, water, fly ash, calcium hydroxide, and water-reducing agents [23], and are industrially manufactured building infill wall blocks, provided by Spring Building Materials Technology Co., Ltd. (Jinan, China). The main component of industrial desulfurization gypsum is dihydrate calcium sulfate (CaSO₄·2H₂O), which contains impurities introduced by the desulfurization process, such as unburned carbon, soluble salts, and organic matter. Based on the “Standard for Test Methods of Basic Mechanical Properties of Masonry” (GB/T 50129-2011) [24], standard blocks and half blocks were selected. The dimensions of the standard block are 600 mm × 200 mm × 250 mm, and the dimensions of the half block are 298 mm × 200 mm × 250 mm. The gypsum mortar was mixed with water at a water-to-binder ratio of 0.57. The mix design of the core-filling concrete was carried out in accordance with the “Ordinary Concrete Mix Design Code” [25], using the mass method for mix proportion calculation. The amount of materials used and the strength for each test of grouted concrete are shown in

Table 1. The amount of materials used and the strength for each test of grouted concrete.

Specimen Number	Age (In Days)	Cement (kg/m3)	Water (kg/m3)	Gravel (kg/m3)	Sand (kg/m3)	Average Compressive Strength (MPa)
C20	28	15.34	18.57	21.04	21.04	14.95
C25	28	12.63	12.63	12.63	12.63	18.24
C30	28	46.94	45.30	44.03	44.03	22.52
C35	28	45.10	43.52	42.31	42.31	25.66
C40	28	15.34	252.5	832.4	799.8	28.89

.

All test specimens were constructed using a three-course, fully grouted masonry method. The top and bottom courses were made of standard blocks, while the middle course consisted of two half blocks. The core-filled masonry was designed in 5 groups, with each group containing 6 specimens of C20, C25, C30, C35, and C40 strength core-filled masonry, totaling 30 core-filled masonry specimens, as shown in Figure 1. After curing the masonry for 7 days, axial compression tests were conducted. The reinforced masonry was designed in five groups, with each of the four grout holes containing 1 or 2 vertical ribbed reinforcing bars, i.e., 4Φ14 or 8Φ14. In the C25 and C30 groups, the number of reinforcing bars in each of the four grout holes was increased to 3 or 4 vertical ribbed reinforcing bars, i.e., 12Φ14 or 16Φ14. The length of the vertical ribbed reinforcing bars was 700 mm, the stirrup spacing was 200 mm, and the minimum concrete cover thickness for the reinforcing bars was 25 mm. In total, there were 42 reinforced masonry specimens, as shown in Figure 2.

2.2. Test Procedure

The compression test is conducted using a 500 T electro-hydraulic servo hydraulic press, and the test setup is shown in Figure 3. The loading of desulfurized gypsum core-filled masonry and reinforced masonry specimens follows the standard method for masonry testing. All specimens are pre-loaded three times at 5% of the estimated failure load, with simultaneous physical alignment adjustments. The loading is applied in stages according to the standard test method, with simultaneous data recording by both the testing machine and the data acquisition device to ensure a one-to-one correspondence between load and displacement. The first five levels are loaded at 10% of the estimated failure load, with each level completed within 72 s, and after each level, the load is maintained at a constant pressure for 90 s. Subsequently, the load applied at each level is reduced to 5% of the estimated failure load. When the first crack appears in the specimen, the load at each level is immediately adjusted to 10% of the estimated load. This continues until the load reaches 0.8 times the failure load of the core-filled masonry, at which point loading proceeds until the specimen fails.

2.3. Failure Mode

The failure modes of core-filled masonry and reinforced masonry undergo three stages. The first stage, from the beginning of the test until the first visible crack appears in the specimen, is the elastic stage. During this phase, fine hairline cracks become visible to the naked eye. At this stage, both the gypsum outer wall and the core column concrete jointly bear the load, deforming in a coordinated manner. The second stage, from the emergence of the first crack in the specimen to the point of reaching ultimate bearing capacity, is the elastoplastic stage. As loading continues, the initial crack extends and develops, while new vertical cracks appear, some of which develop into through cracks. As the outer wall of the masonry gradually peels off, part of the gypsum outer wall ceases to function, and the load-bearing capacity of the gypsum-filled masonry gradually decreases, with the deformation rate accelerating. A hoop effect appears at the top of the reinforced masonry, causing the vertical strain of the uppermost block to increase, leading to crushing failure of the mortar surface at the top layer of the masonry. However, overall, it is still the combined effect of the core column and the blocks that bears the load. The third stage, from the point when the specimen reaches its ultimate load to the point of failure where it loses its bearing capacity, is the softening stage. As loading continues, cracks develop rapidly, their widths increase, and multiple major cracks appear. Large-scale spalling occurs on the outer walls of both the core-filled masonry and the reinforced masonry. During this stage, the majority of the gypsum block outer walls cease to function, and the concrete core column bears the primary load. Eventually, the concrete core column is crushed, leading to the failure of the core-filled masonry, as illustrated in Figure 4. In the reinforced masonry, the internal steel reinforcement gradually becomes exposed and exhibits bending, as shown in Figure 5.

3. Compressive Strength Calculation Formula

According to the test data, the compressive strength of the new high-precision desulfurized gypsum core-filled masonry is calculated using current national standard calculation Formula (2).

$$f_{c,i}={\displaystyle \frac{1000P_{c,i}}{A}}$$

(2)

where $f_{c,i}$ represents the compressive strength of the gypsum-filled masonry specimens and $P_{c,i}$ represents the ultimate load of the gypsum-filled masonry.

From

Table 2. Comparison between the average test values of compressive strength of grouted masonry and the standard values.

Masonry Group	Average Compressive Strength Test Value (MPa)	Code-Specified Value (MPa)	Ratio of the Average Compressive Strength Test Value to the Code-Specified Value
C20 grouted masonry	5.45	5.78	0.94
C25 grouted masonry	6.05	6.75	0.90
C30 grouted masonry	6.75	8.03	0.84
C35 grouted masonry	7.43	8.96	0.83
C40 grouted masonry	8.00	9.92	0.80

, it can be seen that the ratio of the average compressive strength of each group of core-filled masonry to the code-specified value is less than 1. This indicates that Formula (2) is not suitable for calculating the compressive strength of reinforced masonry, as its calculated values tend to be unsafe.

According to the test data, the compressive strength of the new high-precision desulfurized gypsum reinforced masonry is calculated with reference to Formulas (3) and (4) in the Code for Design of Masonry Structures.

$$N_c=\phi \left(f_{g}A+0.8f_y^{\prime }{A}_s^{\prime }\right)$$

(3)

$$\phi ={\displaystyle \frac{1}{1+0.001{\beta }^2}}$$

(4)

where $f_g$ represents the compressive strength of reinforced masonry (MPa). $A$ represents the cross-sectional area of the masonry specimen, $A_s^{\prime }$ represents the cross-sectional area of the masonry specimen, $f_y^{\prime }$ represents the compressive strength of the vertical steel reinforcement; $\phi$ represents the stability coefficient and $\beta$ represents the height-to-thickness ratio of the masonry.

From

Table 3. Comparison between the average compressive strength test value of reinforced masonry and the code-specified value.

Masonry Group	Average Compressive Strength Test Value (MPa)	Code-Specified Value (MPa)	Ratio of the Average Compressive Strength Test Value to the Code-Specified Value
C20 Reinforced Masonry	6.90	8.40	0.82
C25 Reinforced Masonry	7.96	9.00	0.89
C30 Reinforced Masonry	8.57	9.68	0.88
C35 Reinforced Masonry	9.08	10.35	0.88
C40 Reinforced Masonry	9.72	10.92	0.89

, it can be seen that the ratio of the average compressive strength of each group of reinforced masonry to the standard value is less than 1. This indicates that Formula (3) is not suitable for calculating the compressive strength of reinforced masonry, as its calculated values tend to be unsafe.

The compressive strength of grouted masonry and reinforced masonry exhibits a generally linear relationship with the strength of the grouted concrete, as shown in Figure 6.

Due to the regular increase in compressive strength of both grouted masonry and reinforced masonry as the strength of the grouted concrete increases, regression analyses were conducted separately for the compressive strengths of grouted masonry and reinforced masonry. In this context, let

$$y={\displaystyle \frac{f_{ci,m}}{f_{ch,m}}}$$

(5)

$$x={\displaystyle \frac{\alpha f_{cu,m}}{f_{ch,m}}}$$

(6)

where $f_{ci,m}$ is the average compressive strength of grouted masonry, $f_{ch,m}$ is the average compressive strength of hollow masonry, and $f_{cu,m}$ is the average compressive strength of grouted concrete. Under the premise that the reinforcement ratio remains constant, let

$$x={\displaystyle \frac{f_{g}A}{f_y^{ \prime }{A}_{\mathrm{s}}^{\prime }}}$$

(7)

$$y={\displaystyle \frac{N_{\mathrm{c}}}{\phi {ƒ}_{\mathrm{y}}^{ \prime }{A}_{\mathrm{s}}^{\prime }}}$$

(8)

where $N_c$ is the compressive strength of reinforced masonry, ${ƒ}_{\mathrm{y}}^{ \prime }$ is the compressive strength of longitudinal reinforcement, and $f_g$ is the average axial compressive strength of masonry.

The fitted data for the compressive strength calculation formulas of grouted masonry and reinforced masonry are presented in

Table 4. Fitting data for the calculation formula of compressive strength of grouted masonry.

Specimen Group	$\mathit{y}=\frac{{\mathit{f}}_{\mathit{c}\mathit{i},\mathit{m}}}{{\mathit{f}}_{\mathit{c}\mathit{h},\mathit{m}}}$	$\mathit{x}=\frac{\mathit{\alpha }{\mathit{f}}_{\mathit{c}\mathit{u},\mathit{m}}}{{\mathit{f}}_{\mathit{c}\mathit{h},\mathit{m}}}$
C20 Grouted Masonry	4.10	5.31
C25 Grouted Masonry	4.55	6.47
C30 Grouted Masonry	5.08	7.99
C35 Grouted Masonry	5.59	9.11
C40 Grouted Masonry	6.02	10.25

and

Table 5. Fitting data for the calculation formula of compressive strength of reinforced masonry.

Specimen Group	$\mathit{x}=\frac{{\mathit{f}}_{\mathit{g}}\mathit{A}}{{\mathit{f}}_{\mathit{y}}^{ \prime }{\mathit{A}}_{\mathbf{s}}^{\prime }}$	$\mathit{y}=\frac{{\mathit{N}}_{\mathit{c}}}{\mathit{\phi }{\mathit{f}}_{\mathbf{y}}^{ \prime }{\mathit{A}}_{\mathbf{s}}^{\prime }}$
C20 Reinforced Masonry	1.42	1.82
C25 Reinforced Masonry	1.57	2.10
C30 Reinforced Masonry	1.75	2.26
C35 Reinforced Masonry	1.93	2.39
C40 Reinforced Masonry	2.08	2.56

, respectively.

The fitted relationship curve between the strength of grouted masonry and the strength of grouted concrete, as well as the relationship curve between the compressive strength of reinforced masonry and the compressive strength of grouted masonry, are shown in Figure 7.

Based on Figure 7, the coefficient of determination R-squared (COD) is 0.9984, indicating a high degree of match between the fitted formula and the actual data. The linear fitting yields the formula

$$y=2.03+0.39x$$

(9)

By substituting $y={\displaystyle \frac{f_{ci,m}}{f_{ch,m}}}$ (5), $x={\displaystyle \frac{\alpha f_{cu,m}}{f_{ch,m}}}$ (6) into the fitted formula, we obtain the compressive strength calculation Formula (10) for grouted masonry.

$$f_{ci,m}=2.03f_{ch,m}+0.39\alpha f_{cu,m}$$

(10)

Based on Figure 3, the coefficient of determination R-squared (COD) is 0.97503. The linear fitting yields the formula $y=1.06x+0.38$ (9). By substituting $x={\displaystyle \frac{{ƒ}_{\mathrm{g}}A}{{ƒ}_{\mathrm{y}}^{ \prime }{A}_{\mathrm{s}}^{\prime }}}$ (7), $y={\displaystyle \frac{N_{\mathrm{c}}}{\phi {ƒ}_{\mathrm{y}}^{ \prime }{A}_{\mathrm{s}}^{\prime }}}$ (8) into the fitted formula, we obtain the compressive strength calculation Formula (11) for reinforced masonry.

$$N_{c,m}=\phi 1.06f_{g,m}A+0.38f_y^{ \prime }{A}_{\mathrm{s}}^{\prime }$$

(11)

Formula (10) shares the same expression form as the compressive strength calculation formula for grouted masonry in the “Code for Design of Masonry Structures”. By determining the compressive strength of hollow masonry, the compressive strength of grouted concrete, and the grouting ratio, the compressive strength of the grouted masonry can be calculated using this formula. This formula is significant for practical engineering applications, particularly in calculating the compressive strength of load-bearing walls constructed with high-precision desulfurized gypsum blocks combined with grouted concrete. Formula (11) shares the same expression form as the formula for calculating the axial compressive capacity of reinforced masonry in the “Code for Design of Masonry Structures”. By determining the axial compressive strength of grouted masonry, the compressive strength of longitudinal reinforcement, the cross-sectional area of the masonry, the cross-sectional area of the longitudinal reinforcement, and the height-to-thickness ratio of the masonry, the compressive strength of the reinforced masonry can be calculated using this formula. This formula can be used to calculate the compressive strength of reinforced masonry in practical engineering applications, particularly when it is used in load-bearing walls.

A comparison between the test values of the compressive strength of grouted masonry and the calculated values from the fitting formula is detailed in

Table 6. Comparison of test values and fitted formula values for compressive strength of grouted masonry.

Specimen Group	Specimen Number	Grouting Ratio α	Compressive Strength of Hollow Masonry ${\mathit{f}}_{\mathit{c}\mathit{h},\mathit{m}}$ (MPa)	Compressive Strength of Core Columns ${\mathit{f}}_{\mathit{c}\mathit{u},\mathit{m}}$ (MPa)	Test Value of Compressive Strength of Grouted Masonry ${\mathit{f}}_{\mathit{c},\mathit{i}}$ (MPa)	Calculated Value of Compressive Strength of Grouted Masonry ${\mathit{f}}_{\mathit{c}\mathit{i},\mathit{m}}$ (MPa)	${\mathit{f}}_{\mathit{c},\mathit{i}}/{\mathit{f}}_{\mathit{c}\mathit{i},\mathit{m}}$
C20	20-1	0.472	1.33	14.95	5.77	5.45	1.06
	20-2				5.23		0.96
	20-3				5.5		1.01
	20-4				4.94		0.91
	20-5				5.26		0.96
	20-6				5.82		1.07
C25	25-1	0.472	1.33	18.24	6.00	6.06	0.99
	25-2				6.42		1.06
	25-3				6.11		1.01
	25-4				6.07		1.00
	25-5				5.69		0.94
25-6					6.03		1.00
C30	30-1	0.472	1.33	22.52	6.41	6.85	0.94
	30-2				6.68		0.98
C30	30-3	0.472	1.33	22.52	6.90	6.85	1.01
	30-4				7.20		1.05
	30-5				6.87		1.00
	30-6				6.41		0.94
C35	35-1	0.472	1.33	25.66	7.40	7.42	1.00
	35-2				7.98		1.07
	35-3				7.45		1.00
C35	35-4	0.472	1.33	25.66	7.51	7.42	1.01
	35-5				7.39		1.00
	35-6				6.83		0.92
C40	40-1	0.472	1.33	28.89	7.98	8.02	1.00
	40-2				8.08		1.01
	40-3				8.12		1.01
	40-4				8.05		1.00
	40-5				8.19		1.02
	40-6				8.27		1.03

.

As can be seen from Table 6, the average ratio of the calculated values using Formula (10) to the test results obtained in this paper is close to 1, with a coefficient of variation of only 4.3%. This indicates that Formula (10) can accurately calculate the compressive strength of grouted masonry.

A comparison between the test values of the compressive strength of reinforced masonry and the calculated values from the fitting formula is detailed in

Table 7. Comparison of test values and fitted formula values for compressive strength of reinforced masonry.

Specimen Group	Specimen Number	Height-to Thickness Ratio $\mathit{\beta }$	Stability Factor of Reinforced Masonry $\mathit{\phi }$	Vertical Reinforcement Cross-Sectional Area $\mathit{A}$ (mm2)	Cross-Sectional Area of Vertical Reinforcement ${\mathit{A}}_{\mathbf{s}}^{\prime }$ (mm2)	Test Value of Compressive Strength of Reinforced Masonry ${\mathit{f}}_{\mathit{c}}$ (MPa)	Calculated Value of Compressive Strength ${\mathit{N}}_{\mathbf{c},\mathbf{m}}$ (MPa)	${\mathit{f}}_{\mathit{c}}/{\mathit{N}}_{\mathit{c},\mathit{m}}$
C20	C20-2-1	3.85	0.985	120,000	1231.5	6.90	7.13	0.97
	C20-2-2					6.94		0.97
	C20-2-3					6.87		0.96
C25	C25-2-1	3.85	0.985	120,000	1231.5	8.02	7.13	1.03
	C25-2-2					7.98		1.03
C25	C25-2-3	3.85	0.985	120,000	1231.5	7.87	7.13	1.01
C30	C30-2-1	3.85	0.985	120,000	1231.5	8.67	7.13	1.02
	C30-2-2					8.55		1.01
	C30-2-3					8.49		1.00
C35	C35-2-1	3.85	0.985	120,000	1231.5	9.12	7.13	0.99
	C35-2-2					9.18		1.00
	C35-2-3					8.95		0.97
C40	C40-2-1	3.85	0.985	120,000	1231.5	9.77	7.13	1.00
	C40-2-2					9.73		0.99
	C40-2-3					9.67		0.99

.

By substituting the parameters from Table 7 into Formula (11), the ratio of the calculated values to the test results for reinforced masonry is close to 1, indicating that Formula (11) can accurately calculate the compressive strength of reinforced masonry.

4. Elastic Modulus Calculation Formula

Currently, there is no unified constitutive model for the compressive stress–strain relationship of masonry structures, and research on the constitutive relationship of gypsum masonry by domestic and foreign scholars is relatively limited. In contrast, the study of the constitutive relationship of concrete is more extensive and well established.

Based on the constitutive relationship models in the form of a parabola proposed by Hognestad [26], Jiao Zhenzhen, and Guo Zhenhai [27], all stress–strain data for grouted masonry and reinforced masonry were normalized, as illustrated in Figure 8.

The normalization process for grouted masonry involves setting the horizontal coordinate

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{c,i}}}$$

(12)

and the vertical coordinate

$$y={\displaystyle \frac{\sigma }{f_{c,i}}}$$

(13)

based on test data. Similarly, for reinforced masonry, the normalization process involves setting the horizontal coordinate

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}$$

(14)

and the vertical coordinate

$$y={\displaystyle \frac{\mathsf{\sigma }}{f_{\mathrm{c},\mathrm{m}}}}$$

(15)

based on test data. All stress–strain data for grouted masonry and reinforced masonry were normalized, and referencing the constitutive relationship model for concrete proposed by Guo Zhenhai, a regression analysis of the stress–strain relationship for grouted masonry and reinforced masonry was conducted using the least squares method, as depicted in Figure 9.

In summary, the fitting formula for grouted masonry yields an R-squared value of 0.99 for the first segment and 0.98 for the second segment. For reinforced masonry, the fitting formula achieves an R-squared (COD) value of 0.976 for the first segment and 0.977 for the second segment, indicating that the fitted formulas align well with the statistical test data. That is, the stress–strain curve equation for grouted masonry is

$${\displaystyle \frac{\sigma }{f_{c,i}}}=\left\{\begin{array}{c}1.158{\displaystyle \frac{\epsilon }{{\epsilon }_{c,i}}}+0.684{\left({\displaystyle \frac{\epsilon }{{\epsilon }_{c,i}}}\right)}^2-0.842{\left({\displaystyle \frac{\epsilon }{{\epsilon }_{c,i}}}\right)}^3,{\displaystyle \frac{\epsilon }{{\epsilon }_{c,i}}}\le 1\\ {\displaystyle \frac{\frac{\epsilon }{{\epsilon }_{c,i}}}{3.73{\left(\frac{\epsilon }{{\epsilon }_{c,i}}-1\right)}^2+\frac{\epsilon }{{\epsilon }_{c,i}}}},{\displaystyle \frac{\epsilon }{\epsilon }}\ge 1\end{array}\right.$$

(16)

The stress–strain curve equation for reinforced masonry is

$${\displaystyle \frac{\sigma }{f_{\mathrm{c},\mathrm{m}}}}=\left\{\begin{array}{c}1.514{\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}+0.0042{\left({\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}\right)}^2-0.502{\left({\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}\right)}^3,{\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}\le 1\\ {\displaystyle \frac{\frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}{11.579{\left(\frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}-1\right)}^2+\frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}},{\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{m}}}}\ge 1\end{array}\right.$$

(17)

According to Equations (16) and (17), the elastic modulus of grouted masonry and reinforced masonry should be taken as the secant modulus at the point where the stress is 0.43 on the stress–strain curve.

The elastic modulus of grouted masonry:

$$E={\displaystyle \frac{d\sigma }{d\epsilon }}={\displaystyle \frac{1.158f_{c,i}}{{\epsilon }_{c,i}}}+{\displaystyle \frac{1.368f_{c,i}{\epsilon }_{0.4}}{{\epsilon }_{c,i}^2}}-{\displaystyle \frac{2.526f_{c,i}{\epsilon }_{0.4}^2}{{\epsilon }_{c,i}^3}}$$

(18)

The elastic modulus of reinforced masonry:

$$E={\displaystyle \frac{1.514f_{\mathrm{c},\mathrm{m}}}{{\epsilon }_{\mathrm{c},\mathrm{m}}}}+{\displaystyle \frac{0.0084f_{\mathrm{c},\mathrm{m}}{\epsilon }_{0.4}}{{\epsilon }_{\mathrm{c},\mathrm{m}}^2}}-{\displaystyle \frac{1.506f_{\mathrm{c},\mathrm{m}}{\epsilon }_{0.4}^2}{{\epsilon }_{\mathrm{c},\mathrm{m}}^3}}$$

(19)

A comparison between the calculated values from the elastic modulus Formula (18) for grouted masonry and the test data is presented in

Table 8. Comparison of test results and calculated values for the elastic modulus of grouted masonry.

Specimen Group	Specimen Number	Strain ${\mathit{\epsilon }}_{\mathit{c},\mathit{i}}$ (%)	Axial Strain of 0.4 ${\mathit{\epsilon }}_{0.4}$ (%)	Test Value of Elastic Modulus ${\mathit{E}}_{\mathit{c},\mathit{i}}$ (MPa)	Average Test Value of Elastic Modulus (MPa)	Compressive Strength ${\mathit{f}}_{\mathit{c},\mathit{i}}$ (MPa)	Calculated Value of Elastic Modulus $\mathit{E}$ (MPa)	${\mathit{E}}_{\mathit{c},\mathit{i}}/\mathit{E}$
C20	C20-1	0.215	0.065	3675.34	3985.44	5.77	3597.46	1.02
C20	C20-2	0.174	0.044	4754.55	3985.44	5.23	4034.93	1.18
	C20-3	0.166	0.044	4966.14		5.50	4450.29	1.12
	C20-4	0.177	0.067	2946.43		4.94	3665.29	0.80
	C20-5	0.182	0.050	4208.00		5.26	3881.93	1.08
	C20-6	0.215	0.069	3362.19		5.82	3617.89	0.93
C25	C25-1	0.136	0.049	4887.98	4764.21	6.00	5835.19	0.84
	C25-2	0.147	0.047	5598.29		6.42	5841.31	0.96
	C25-3	0.126	0.053	4620.04		6.11	6241.38	0.74
	C25-4	0.125	0.057	4274.65		6.07	6109.10	0.70
	C25-5	0.112	0.052	4321.71		5.69	6361.09	0.68
	C25-6	0.126	0.049	4882.59		6.03	6250.44	0.78
C30	C30-1	0.130	0.044	6090.09	6410.23	6.41	6562.08	0.93
	C30-2	0.145	0.048	5778.78		6.68	6144.84	0.94
	C30-3	0.138	0.037	7439.35		6.9	6716.03	1.11
	C30-4	0.142	0.046	6315.79		7.20	6778.21	0.93
	C30-5	0.127	0.040	6870.00		6.87	7239.39	0.95
	C30-6	0.127	0.043	5967.36		6.41	6722.29	0.89
C35	C35-1	0.136	0.040	7344.91	7200.77	7.40	7299.71	1.01
	C35-2	0.140	0.038	8461.54		7.98	7656.30	1.11
	C35-3	0.124	0.040	7450.00		7.45	8029.41	0.93
	C35-4	0.125	0.043	6986.05		7.51	7988.69	0.87
	C35-5	0.122	0.045	6642.70		7.39	8001.25	0.83
	C35-6	0.115	0.043	6319.44		6.83	7812.55	0.81
C40	C40-1	0.121	0.031	10,165.61	9697.42	7.98	8856.45	1.15
	C40-2	0.107	0.034	9478.01		8.08	10,099.39	0.94
C40	C40-3	0.114	0.065	10,684.21	9697.42	8.12	9567.16	1.12
	C40-4	0.097	0.044	8923.08		8.05	11,060.72	0.81
	C40-5	0.120	0.044	9829.55		8.19	9158.68	1.07
	C40-6	0.105	0.067	9104.05		8.27	10,510.78	0.87

.

A comparison between the calculated values from the elastic modulus Formula (19) for reinforced masonry and the test data is presented in

Table 9. Comparison of test results and calculated values for the elastic modulus of reinforced masonry.

Specimen Group	Specimen Number	Strain ${\mathit{\epsilon }}_{\mathit{c},\mathit{m}}$ (%)	Compressive Strength ${\mathit{f}}_{\mathit{c},\mathit{m}}$ (MPa)	Axial Strain of 0.4 ${\mathit{\epsilon }}_{0.4}$ (%)	Test Value of Elastic Modulus ${\mathit{E}}_{\mathit{c},\mathit{m}}$ (MPa)	Calculated Value of Elastic Modulus $\mathit{E}$ (MPa)	${\mathit{E}}_{\mathit{c},\mathit{m}}/\mathit{E}$
C20	C20-2-1	0.16	6.90	0.047	5843.47	5889.17	0.99
	C20-2-2	0.15	6.94	0.045	6130.13	6564.56	0.93
	C20-2-3	0.10	6.87	0.047	5833.10	8032.26	0.73
C25	C25-2-1	0.15	8.02	0.039	8231.79	7519.19	1.09
	C25-2-2	0.17	7.98	0.038	8480.44	6908.01	1.23
	C25-2-3	0.15	7.87	0.038	8205.24	7320.58	1.12
C30	C30-2-1	0.16	8.67	0.038	9147.92	7669.78	1.19
	C30-2-2	0.15	8.55	0.038	9108.05	8044.58	1.13
	C30-2-3	0.19	8.49	0.039	8818.03	6584.28	1.34
C35	C35-2-1	0.18	9.12	0.035	10,298.04	7584.92	1.36
	C35-2-2	0.19	9.18	0.041	8932.55	6951.85	1.28
	C35-2-3	0.15	8.95	0.042	8549.05	8161.21	1.05
C40	C40-2-1	0.16	9.77	0.033	11,804.76	9018.46	1.31
	C40-2-2	0.17	9.73	0.042	9246.68	7986.89	1.16
	C40-2-3	0.14	9.67	0.045	8598.36	9298.32	0.92

.

From Table 8, it can be observed that the ratio of the test values of the elastic modulus of grouted masonry to the calculated values from Formula (18) has an average of 0.94 and a coefficient of variation of 0.14. Overall analysis shows that the elastic modulus of grouted masonry calculated using Formula (18) aligns well with the test values. From Table 9, it can be seen that the ratio of the test values of the elastic modulus of reinforced masonry to the calculated values from Formula (19) has an average of 1.12. Overall analysis indicates that the elastic modulus of reinforced masonry calculated using Formula (19) aligns well with the test values.

5. Conclusions

Through axial compressive strength tests on five groups of new high-precision desulfurized gypsum core-filled masonry and reinforced masonry, the test results were analyzed and studied, leading to the following conclusions:

(1)

The calculation of the compressive strength of new high-precision gypsum core-filled masonry blocks and reinforced masonry blocks using the standard formula shows significant errors and tends to be unsafe.

(2)

Based on the test observations, the grouted masonry and reinforced masonry specimens undergo three main stages during the entire compression process: the elastic stage, the elastoplastic stage, and the softening stage. The failure process of grouted masonry and reinforced masonry exhibits distinct phases, with crack development and outer wall spalling being the key factors leading to the reduction in load-bearing capacity. The performance of the core column and steel reinforcement plays a decisive role in the final failure mode.

(3)

Based on the test results, this paper proposes the compressive strength calculation Formula (10) $f_{ci,m}=2.03f_{ch,m}+0.39\alpha f_{cu,m}$ for core-filled masonry and the compressive strength calculation Formula (11) $N_{c,m}=\phi 1.06f_{g,m}A+0.38f_y^{ \prime }{A}_{\mathrm{s}}^{\prime }$ for reinforced masonry. The calculated values from Formulas (10) and (11) are in good agreement with the experimental values, providing a new basis and reference for calculating the compressive strength of such masonry. This solves the problem of compressive strength design for new high-precision desulfurized gypsum masonry blocks combined with core-filled concrete when used in load-bearing walls.

(4)

Since there is no unified constitutive model for the compressive stress–strain relationship of masonry structures and research on the constitutive relationship of gypsum masonry is relatively limited, a regression analysis of the test data was conducted, resulting in the elastic modulus calculation Formula (18) $E={\displaystyle \frac{d\sigma }{d\epsilon }}={\displaystyle \frac{1.158f_{c,i}}{{\epsilon }_{c,i}}}+{\displaystyle \frac{1.368f_{c,i}{\epsilon }_{0.4}}{{\epsilon }_{c,i}^2}}-{\displaystyle \frac{2.526f_{c,i}{\epsilon }_{0.4}^2}{{\epsilon }_{c,i}^3}}$ for grouted masonry and the compressive strength calculation Formula (19) $E={\displaystyle \frac{1.514f_{\mathrm{c},\mathrm{m}}}{{\epsilon }_{\mathrm{c},\mathrm{m}}}}+{\displaystyle \frac{0.0084f_{\mathrm{c},\mathrm{m}}{\epsilon }_{0.4}}{{\epsilon }_{\mathrm{c},\mathrm{m}}^2}}-{\displaystyle \frac{1.506f_{\mathrm{c},\mathrm{m}}{\epsilon }_{0.4}^2}{{\epsilon }_{\mathrm{c},\mathrm{m}}^3}}$ for reinforced masonry. The calculated values from Formulas (18) and (19) align well with the test values. This provides a new basis and reference for calculating the elastic modulus of such masonry. This provides a theoretical foundation for the better application of masonry blocks in practical engineering.