Experimental study on the axial compression performance of rat-trap bond brick masonry strengthened with steel plates under preload

Abstract

Rat-trap bond brick masonry is commonly found in historic and heritage structures. During reinforcement, achieving complete unloading is often impractical, making it necessary to consider the impact of existing load. Reinforcement of brick masonry with steel plates offers advantages such as ease of construction, safety, reliability, and good reversibility. This study presents axial compression tests on six rat-trap bond brick masonry specimens (including one unreinforced specimen, one steel plate-reinforced specimen without preload, and four steel plate-reinforced specimens under preload). The results demonstrate that steel plate reinforcement effectively enhances the load-bearing capacity and ductility of rat-trap bond brick masonry while improving its failure mode. The preload ratio, defined as the ratio of the axial compressive load applied at strengthening point to the peak load of the unreinforced specimen, is found to have a certain impact on the reinforcement effectiveness. Additionally, increasing the steel plate thickness further enhances load-bearing capacity; however, compatibility must be taken into consideration. A finite element model was developed based on the experimental results and extended for further analysis. Furthermore, an axial load-bearing capacity formula for rat-trap bond brick masonry reinforced with steel plates under preload is proposed, incorporating existing calculation theories for steel plates and masonry.

Keywords

rat-trap bond brick masonry steel plates preload ratio axial compression finite element

Introduction

Masonry structures, among the oldest construction techniques in human history, have been widely utilized due to their cost-effectiveness and ease of construction (Samiullah et al., 2022). Rat-trap bond brick masonry, as a form of brick construction, is widely used in historic and heritage structures across Asia. This construction technique incorporates header bricks, rowlock bricks, and shiner bricks, as illustrated in Figure 1. Compared to conventional brick masonry, the internal cavities formed by the arrangement of rowlock and shiner bricks result in an approximately 20% reduction in brick consumption. This not only decreases construction costs and reduces foundation loads but also improves the thermal insulation performance of the wall (Abbas et al., 2022; Hamna et al., 2020; Nauman Azhar and Ali Qureshi, 2022; Sinha et al., 2020).

Figure 1.

Schematic diagram of rat-trap bond brick masonry.

Currently, common reinforcement techniques for brick masonry components primarily enhance structural performance by incorporating additional materials either within or on the exterior of the masonry. Among these methods, fiber-reinforced polymers (FRP) are extensively employed for strengthening existing masonry structures. When externally bonded to masonry components, FRP significantly improves structural behavior, with its effectiveness largely governed by the bond strength between the composite material and the masonry substrate. However, this bond is highly sensitive to moisture, as environmental humidity can degrade adhesion over time, thereby compromising the long-term reliability of the reinforced masonry. Moreover, FRP exhibits limited effectiveness in enhancing the compressive bearing capacity of brick masonry walls (Aiello et al., 2019; Ghiassi et al., 2013; Ma et al., 2023; Maljaee et al., 2016; Sciolti et al., 2012).

Dadras Eslamlou et al. conducted a comparative study on unreinforced masonry structures strengthened using various techniques and concluded that the application of steel plate bands can significantly enhance structural ductility, lateral load-bearing capacity, and displacement performance. Furthermore, strengthening with steel plate bands introduces little additional weight to the original structure, offers a cost-effective solution, and involves relatively simple construction procedures (Dadras Eslamlou et al., 2019).

Steel exhibits high ductility and strength. Reinforcing masonry structures with steel plates mechanically anchored using through-bolts can effectively compensate for masonry’s low ductility and high brittleness. This reinforcement also significantly enhances both the compressive and shear strength of masonry. Previous studies on the seismic performance of brick walls strengthened with steel plate bands have demonstrated a substantial improvement in lateral stiffness and energy dissipation capacity (Farooq and ElGawady Mohamed et al., 2014a; Farooq et al., 2012; Farooq and Shahid et al., 2014b; Taghdi et al., 2000a; Taghdi et al., 2000b).

In engineering practice, it is often impractical to completely unload existing brick masonry during strengthening. Consequently, masonry structures remain under a certain level of load, resulting in an initial stress state that cannot be neglected. Under preload conditions, reinforcement materials do not immediately participate in load-bearing upon installation; instead, they begin to share the load only after additional deformation develops with increasing external load. As a result, the stress-strain response of the reinforcement lags behind that of the masonry (Cheng and Jing, 2024). During subsequent loading, continuous internal force redistribution occurs between the masonry and the reinforcement, which directly influences the ultimate load-bearing capacity (Ouyang and Liu, 2006). Zhao et al. (Zhao et al., 2022) conducted axial compression tests on welded reinforced circular steel tubes under preload conditions and reported a significant increase in peak load after reinforcement; however, when the preload ratio increased from 0.5 to 0.8, the peak load decreased by 13%.

To more accurately reflect the practical conditions of strengthened masonry structures, this study investigates the axial compression performance of preloaded rat-trap bond brick masonry reinforced with steel plates. The research parameters include the preload ratio, steel plate thickness, and masonry strength. An integrated approach combining experimental testing and finite element analysis is adopted, and a calculation formula for predicting the axial compression load-bearing capacity of preloaded rat-trap bond brick masonry reinforced with steel plates is proposed. Different from prior studies, this work investigates standard-sized axial compression masonry specimens, providing data directly applicable to the structural design of individual elements. This study employs a steel plate configuration that differs both in form and in relative steel usage, optimizing the reinforcement for axial compression.

Experimental program

Specimens

In this experiment, a total of six specimens were designed: one unreinforced specimen, one steel plate-reinforced specimen without preload, and four steel plate-reinforced specimens under preload. According to the Chinese Standard Fired Common Bricks (Fired Common Bricks, 2017), the standard design dimensions of the specimens are 240 mm × 370 mm × 720 mm. The construction method adopted a one-shiner-one-rowlock construction style, which provides higher load-bearing capacity and better ductility (Feng et al., 2009). To ensure uniform load distribution and facilitate handling, C30 reinforced concrete pads with dimensions of 120 mm × 440 mm × 570 mm were placed at both the top and bottom of the specimens. The design details of the specimens are shown in Figure 2, while the main design parameters are shown in Table 1.

Figure 2.

Specimen construction details.

Table 1.

Parameters of specimens.

Specimens	Mortar	Preload ratio	Steel plate Thickness (mm)
US	M1.0	0	0
S-0A	M1.0	0	3
S-0.25A	M1.0	0.25	3
S-0.5A	M1.0	0.5	3
S-0.25B	M1.0	0.25	5
S-0.25C	M1.0	0.25	8

Note: U stands for unreinforced; 0/0.25/0.5 stand for preload ratio; A/B/C stand for 3 mm/5 mm/8 mm steel plates.

The selection of the initial load ratio was based on two main considerations: the mechanical performance of rat-trap bond masonry and the appraisal requirements specified in the Chinese standard for the reliability of civil buildings (Chinese Standard for Appraisal of Reliability of Civil Buildings, 2015) Masonry structures with an initial load ratio exceeding 0.4 are required to undergo strengthening interventions to ensure structural safety, according to the standard. To ensure test reliability and comparability, the preload ratios is controlled to the range of 0.25–0.5.

The masonry specimens were constructed using MU20 fired common bricks and M1.0 mortar. The steel plates used for reinforcement were made of Q235b low-carbon steel and fixed to the masonry using M10 through-bolts. The angle steels serving as the top and bottom fixing elements were made of Q235b low-carbon steel with dimensions of 50 mm × 50 mm × 5 mm, and anchored to concrete pads with M12 chemical anchor bolts and subsequently welded to the steel plates.

The pads were pre-fabricated and cured prior to masonry construction. MU20 fired common bricks from the same production batch, ensuring uniform material properties, were laid in an one-shiner-one-rowlock bond. The measured dimensions of the bricks were 235 mm × 115 mm × 50 mm, with mortar joints controlled within a range of 8∼12 mm. Upon completion of the masonry construction, the surface was leveled with mortar before installing the top pads. The specimens were then cured with water for 28 days under natural conditions.

The construction process of specimens are shown in Figure 3. For the reinforced specimens, bolt holes were drilled in the masonry according to the design drawings. The steel plates were then installed, followed by the insertion of through-bolts. Anchor bolts were employed to secure the angle steel at the top and bottom, after which the steel plates were welded to the angle steel. However, for the preloaded specimens, welding was not initially performed. It should be noted that, due to the construction characteristics of rat-trap bond brick masonry, only one face of each specimen could be fully leveled, while minor surface irregularities remained on the opposite face. This condition was identical for all specimens and considered in the analysis. The detailed construction procedure is provided in Section setup and instrumentation.

Figure 3.

Specimen construction process.

Material properties

The bricks, mortar, steel plates, and through-bolts used in specimen fabrication were tested for material properties according to Chinese standards (Metallic Materials-Tensile Testing-Part 1: Methods of Test at Room Temperature, 2021, Standard for Test Method of Basic Mechanics Properties of Masonry, 2011, Standard for Test Method of Performance on Building Mortar, 2009) The results of the material property tests are shown in Table 2.

Table 2.

Material properties.

Material	Properties		Average value (MPa)	Coefficient of variation
Brick	Compressive strength		21.04	0.92
Mortar	Compressive strength		1.66	0.13
Steel plate	3 mm	Yield strength	346.06	4.87
	3 mm	Ultimate strength	462.11	2.44
	5 mm	Yield strength	284.58	5.78
	5 mm	Ultimate strength	421.93	0.76
	8 mm	Yield strength	288.29	10.22
	8 mm	Ultimate strength	428.31	1.97
Bolt	Ultimate strength		555.77	8.60

Note: The material properties of the angle steel are the same as the 5 mm steel plates.

Setup and instrumentation

The axial compression tests were conducted using a 2000 kN hydraulic-pressure testing machine (Figure 4(a)) with a combined force-displacement loading mode applied.

Figure 4.

Loading procedure.

For specimens without preload, the test procedure included inspection of bolt connections and welds, surface leveling with sand, application of 30 kN preliminary load to verify system performance, force-controlled loading at 30 kN/min up to 100 kN, followed by displacement-controlled loading at 0.1 mm/min until failure.

For preloaded specimens, through-bolts were initially removed and marked, and a 30 kN preload was applied. The specimens were then loaded at 30 kN/min to the target preload level, which was maintained during steel plate anchorage and welding. After a cooling period of 5 min, displacement-controlled loading at 0.1 mm/min was applied until failure.

All welding procedures complied with relevant standards, with controlled cooling intervals. Welds were minimized in size and located away from the primary load-bearing regions to reduce the influence of residual stresses.

The detailed loading process is illustrated in Figure 4.

During this experiment, the observations include crack development, load, deformation of the steel plates, and the failure mode of the specimens. Strain gauges are attached to the steel plates to measure strain, while displacement gauges are arranged on all four sides of the specimen to measure axial displacement. The arrangement of measurement points is illustrated in Figure 5.

Figure 5.

Location of strain and displacement gauges.

Results and discussion

Failure modes

For the unreinforced specimen US, extrusion and crushing of the rowlock occurred during loading, with multiple through-cracks converging at the vertical mortar joints in the rowlock layer, as shown in Figure 6(a). The final failure mode is characterized by a brittle failure due to crushing and detachment of the bottom rowlock, leading to a sudden loss in load capacity.

Figure 6.

Failure modes.

For the reinforced specimens, regardless of whether local buckling occurs in the steel plates, there is a significant increase in the peak load, and the failure mode shifts from brittle failure to ductile failure.

For the reinforced specimen without preload (S-0A), significant local buckling was observed in the midsection of the steel plates at 85% of the peak load, as shown in Figure 6(b). When reached the peak value of 460 kN, continuous cracks appeared on the north face of the masonry, and the local buckling of the steel plates became more visible. When the load dropped to 80% of the peak load, the through-cracks on the north face of the masonry continued to widen, and local sections of the masonry were extruded. Compared to specimen US, the masonry on the north face exhibited crushing without detachment, demonstrating improved ductility.

Specimens S-0.25A and S-0.5A showed similar experimental behaviors, as shown in Figure 6(c) and (d). When the load reached 80% of the peak load, slight local buckling occurred in the steel plates, followed by the first crack. At peak load, through-cracks appeared in the masonry, and the deflection of local buckling in the steel plates increased. When the load dropped to 90% of the peak load, the cracks continued to widen, and masonry crushing occurred. The final failure mode of the specimens was visible local buckling of the steel plates and crushing of the masonry.

For specimens S-0.25B and S-0.25C, long through-cracks had already developed in the masonry before reaching peak load. After reaching peak load, these cracks widened, and crushing occurred in the masonry, while no noticeable local buckling was observed in the steel plates throughout the test, as shown in Figure 6(e) and (f).

Load-displacement curves

The load-displacement curves for all specimens are shown in Figure 7. The unreinforced specimen demonstrated relatively low axial stiffness, with a gradual load increase followed by a sharp decline after reaching the peak load, indicating a distinctly brittle failure mode.

Figure 7.

Load-displacement curves.

For non-preloaded reinforced specimens, axial stiffness was notably higher, and a pronounced gradual descending phase was observed after the peak load, showing an improvement in the failure mode. The loading curve for the preloaded reinforced specimens exhibited a two-stage ascending segment. Before the strengthening point, the specimen behaved as a plain masonry model, with axial stiffness gradually degrading as masonry damage accumulated. After reinforcement, load sharing between the masonry and steel plates resulted in a significant increase in axial stiffness.

Notably, the ascending segment of the load-displacement curve for specimen S-0.25C exhibited a three-stage pattern. This behavior is likely due to the high stiffness and load-bearing capacity provided by the 8 mm steel plate. When the masonry exceeded its peak load-bearing capacity, its stiffness dropped sharply. However, the steel plate retained substantial axial stiffness, leading to continuous redistribution of internal forces between the masonry and the steel plate during subsequent loading, thereby forming a load plateau.

The main results are presented in Table 3. As shown in Table 3, all reinforced specimens exhibited an increased peak load compared to the unreinforced specimen, indicating a general improvement. However, the cracking load of some reinforced specimens occurred earlier, likely due to the influence of initial imperfections and the weakening of the masonry cross-section caused by drilling during reinforcement.

Table 3.

Main results of the axial compression test.

Specimens	Crack load/kN	Displacement/mm	Peak load/kN	Crack load/Peak load	Peak load enhancement
US	270	2.8	422	63.98%	/
S-0A	446	1.9	460	96.96%	9.00%
S-0.25A	490	1.7	508	96.46%	20.38%
S-0.5A	543	2.2	545	99.63%	29.15%
S-0.25B	305	0.9	780	39.10%	84.83%
S-0.25C	903	1.3	1030	87.67%	144.08%

Influence of preload ratio

When the steel plate thickness remains constant and the preload ratio does not exceed 0.5, an increase in preload ratio delays local buckling in the steel plate, allowing the masonry to sustain higher stress levels before buckling occurs, thereby increasing the peak load of the specimen. The peak load of the non-preloaded reinforced specimen with 3 mm steel plates was approximately 9% higher than that of the unreinforced specimen. Increasing the preload ratio to 0.25 led to a 20% increase, while further increasing it to 0.5 resulted in a 29% increase in peak load compared to the unreinforced specimen.

The load-steel strain curves for specimens reinforced with 3 mm steel plates are shown in Figure 8(a)-(c). At the initial stage, the steel plates remained in a uniform compressive state. For specimen S-0A, the midsection strain in both side plates abruptly reversed after the load reached 380 kN. Similarly, for specimens S-0.25A and S-0.5A, the strain in one side plate suddenly reversed after the load reached 440 kN. Near the peak load, local buckling was observed in one or both steel plates for all specimens reinforced with 3 mm steel plates.

Figure 8.

Load-steel strain curves.

As noted in the specimen construction, the inherent surface irregularity of rat-trap bond masonry introduces unavoidable initial imperfections after steel plate installation. These imperfections may result in slight initial tensile strains in the steel plates and contribute to asymmetric local buckling behavior. In addition, masonry construction deviations and possible loading eccentricity may further exacerbate this phenomenon.

The preload ratio has a significant impact on the occurrence of local buckling in the steel plates, as presented in Table 4. Quantitatively, the increase in preload ratio from 0 to 0.5 led to a 100 kN (≈26%) increase in buckling load, while the peak load rose by 85 kN (≈18%). In addition, the ratio of buckling load to peak load improved from 83% to 88%, indicating that with higher preload ratio, the onset of local buckling is shifted closer to the ultimate capacity by 5%. With preload ratios increased, steel plates exhibited delayed local buckling, indicating better composite action between the steel plates and masonry and resulting in higher reinforcement efficiency.

Table 4.

Results for 3 mm steel plates reinforced specimens.

Specimens	Buckling Load (kN)	Peak Load (kN)	Buckling Load/Peak Load (%)
S-0A	380	460	83%
S-0.25A	440	507	87%
S-0.5A	480	545	88%

Influence of steel plate thickness

When the preload ratio remains constant, increasing the steel plate thickness from 3 mm to 8 mm enhances the axial stiffness of the specimen, gradually increasing the peak load for preloaded reinforced specimens. Compared to specimens reinforced with 3 mm steel plates, those with 5 mm plates exhibited a 43% increase in peak load, while 8 mm plates resulted in an 89% increase, as presented in Table 3.

The load-steel strain curves for specimens reinforced with 5 mm and 8 mm steel plates are shown in Figure 8(d)-(e). For specimens S-0.25B and S-0.25C, the compressive strain in the steel plates increased steadily before reaching the peak load, with no occurrence of local buckling. This indicates that the axial stiffness of the steel plates was not fully mobilized at peak load. As the steel plate thickness increases, the stiffness disparity between the steel plates and the masonry becomes more pronounced, reducing deformation compatibility. Consequently, the steel plates tend to carry a larger proportion of the load in the early stages, while the masonry contribution is limited. This premature load concentration diminishes the composite action of the structure, resulting in a lower reinforcement efficiency compared to that of the 3 mm steel plates.

It should be noted that only six specimens were tested in this study, with limited variations in preload ratios (0, 0.25, and 0.5) and steel plate thicknesses (3, 5, and 8 mm). Although consistent influences of the key parameters were identified, this limited experimental database may affect the statistical reliability of the observed trends.

Finite element simulation and extended analysis

Description of models

The model was developed using the ABAQUS software. The macro-modeling approach considers brick and mortar as a homogeneous material, which reduces computational complexity while still providing accurate results under axial compression. Since this study focuses on the overall compressive performance, especially peak load, of rat-trap bond masonry reinforced with steel plates, an integrated modeling approach was adopted by separately constructing the shiner, rowlock and header brick layers and then merging them into a whole structure. By defining appropriate constitutive relationships for the masonry under compression and tension, the model accurately reflects the actual load-bearing behavior.

A bilinear model was used for the constitutive relationship of the steel. When the steel reaches its yield strength, the slope of the stress-strain curve is reduced to 0.01 times the elastic modulus. The constitutive relationship is expressed in equation (1).

σ_{s} = {\begin{cases} E_{s} ε_{s} ε_{s} \leq ε_{y} \\ 0.01 E_{s} (ε_{s} - ε_{y}) + f_{y} ε_{y} < ε_{s} \leq ε_{u} \end{cases}

(1)

where:

σ_{s}

: Stress in the steel;

$ε_{s}$ : Strain in the steel;

$E_{s}$ : Elastic modulus of the steel before yielding

$ε_{y}$ : Strain corresponding to the yield strength of the steel

$f_{y}$ : Yield strength of the steel

$ε_{u}$ : Strain corresponding to the tensile strength of the steel

The constitutive relationship for masonry under compression uses the expression shown in equation (2) (Liu, 2005). Based on prior simulation analysis results from this research group, this relationship closely approximates actual conditions and facilitates computational convergence.

\frac{σ}{f_{m}} = {\begin{cases} 1.96 \frac{ε}{ε_{0}} - 0.96 {(\frac{ε}{ε_{0}})}^{2} ε \leq ε_{0} \\ 1.2 - 0.2 \frac{ε}{ε_{0}} ε_{0} < ε < 1.6 ε_{0} \end{cases}

(2)

where:

f_{m}

: Peak compressive stress of the masonry, taking test value

$ε_{0}$ : Peak compressive strain of the masonry, taking test value

$σ$ : Masonry stress

$ε$ : Masonry strain

Masonry has low tensile strength and failure characteristics similar to concrete. Therefore, the tensile constitutive relationship for masonry adopts the modified concrete tensile constitutive model as shown in equation (3). (Zheng, 2010),

\frac{σ}{f_{tm}} = {\begin{cases} \frac{ε}{ε_{t}} ε \leq ε_{t} \\ \frac{ε / ε_{t}}{2 {(ε / ε_{t} - 1)}^{1.7} + ε / ε_{t}} ε > ε_{t} \end{cases}

(3)

where:

f_{tm}

: Peak tensile stress of the masonry,

f_{tm} = 0.141 \sqrt{f_{2}}

$f_{2}$ : Compressive strength of the mortar

$ε_{t}$ : Peak tensile strain of the masonry, $ε_{t} = f_{tm} / E$

A hard contact was applied between the steel plates and the masonry, while a tie constraint was used to simulate the bolted connections. The bottom was modeled with a fully fixed constraint.

The Model Change module can deactivate and reactivate the steel plates; however, using only Model Change can lead to deformation inconsistencies between the reactivated steel plates and the already-deformed masonry. Therefore, to simulate reinforcement under load, both Model Change and Elcopy were used to manage activation and deactivation of the steel plates.

A new ultra-light elastic material (steel0) was defined, with an elastic modulus equal to 1% of standard steel, serving as a numerical treatment to simulate geometric tolerance effects and minimize its influence on the simulated bearing capacity. The steel plates were defined as the geometric set “steelset” in the model. To implement tracking in the corresponding component of the inp file, add the following code:

*elcopy, oldset=steelset1, newset=trace1, element shift=20000, shift nodes=0

*elset, elset=trace1

*Solid Section, elset=trace1, material=steel0

The steel plates exhibit a combination of two material properties in the imported model: steel and steel0, as shown in Figure 10(a).

The simulation steps are as follows:

STEP-1: Use Model Change to deactivate the original steel plate sets, steelset1 and steelset2.

STEP-2: Apply the corresponding displacement load to bring the masonry to the target preload ratio.

STEP-3: Use Model Change to reactivate the steel plates while deactivating the tracking elements, and then continue loading until failure.

As the finite element model cannot simulate crushing of the masonry, detachment of bricks, and similar phenomena, this section only focus on the behavior of the specimen prior to reaching the peak load.

Results and discussions

The finite element simulation of the unreinforced specimen is shown in Figure 9. The midsection of the masonry side is subjected to tensile stress, where accumulated greater damage, aligning with the observed detachment on the side of the masonry during testing. The load-displacement curve obtained from the finite element simulation is shown in Figure 11(a), demonstrating good agreement with the experimental curve.

Figure 9.

Simulation for unreinforced specimen.

The finite element simulation of the reinforced specimens under preload is shown in Figure 10. Before the strengthening point, the steel plate remains deactivation and does not bear load, only the masonry bears load, as illustrated in Figure 10(b). After the strengthening point, the steel plate is reactivated and begins to bear load, exhibiting local buckling in the midsection, as shown in Figure 10(c). At this stage, the load on the masonry is relatively reduced, aligning well with the observed experimental behaviors.

Figure 10.

Simulation for reinforced specimen.

The load-displacement curve obtained from the finite element simulation is compared with experiment in Figure 11. Rat-trap bond brick masonry exhibit relatively stable peak loads under compressive experiments, whereas the peak displacements show substantial variability due to the unique masonry construction techniques. Replicating the observed displacement behavior poses significant challenges, as highlighted in previous studies (Cheng and Jing, 2024; Hou et al., 2021, 2022; Samiullah et al., 2022). Therefore, the finite element models should prioritize accurately predicting the load-bearing capacity rather than displacement. The relative error in peak load is mostly controlled within 10%, as shown in Table 5.

Figure 11.

Finite element simulation validation.

Table 5.

Finite element simulation validation.

Specimens	Peak load
Specimens	Experimental value/kN	Simulated value/kN	Relative error
US	423	413	2.36%
S-0A	460	452	1.74%
S-0.25A	507	495	2.37%
S-0.5A	545	528	3.12%
S-0.25B	780	812	4.10%
S-0.25C	1030	1144	11.07%

This modeling approach introduces a relatively large error for the peak load of the 8 mm specimen. The thicker steel plates provide effective confinement, allowing the masonry to continue carrying load after minor damage occurs. However, the significant difference in axial stiffness between the steel plates and the masonry, which are connected solely by bolts, prevents fully coordinated composite action. Furthermore, the accumulation of internal damage within the masonry can shift its load-bearing centroid, resulting in eccentric compression, a behavior inherent to masonry with a certain degree of stochasticity. As a result, continuous internal force redistribution occurs during loading, which is only partially captured in the finite element analysis. Consequently, the experimental peak loads tend to be lower than the simulated results, leading to the observed discrepancies.

Considering that masonry structures inherently possess significant discreteness, this modeling method is considered reliable and applicable for further extended analysis. The relative error is calculated using equation (4).

δ = | \frac{F_{S} - F_{E}}{F_{E}} | \times 100 %

(4)

where:

F_{S}

:Simulated peak load

$F_{E}$ :Tested peak load

As shown in Figure 12, varying the preload ratio in the reinforced specimen model reveals that, within a certain range, the peak load of the 3 mm steel plate model (where local buckling has occurred) increases with the preload ratio. In contrast, for the 5 mm steel plate model (where no local buckling occurs), the peak load decreases as the preload ratio increases.

Figure 12.

Peak load according to preload ratio.

Similarly, Figure 13 reveals that as the compressive strength of the rat-trap bond brick masonry increases prior to reinforcement, the peak load of the specimen reinforced with a 3 mm steel plate also increases. Moreover, increasing the preload ratio continues to contribute to an increase in peak load. However, compared to masonry with lower compressive strength, the reinforcement efficiency of the steel plate exhibits a relative decline.

Figure 13.

Peak load according to compressive strength.

Figure 14 presents the simulation results for specimens reinforced with steel plates of different thicknesses under a preload ratio of 0.25, along with a cost–benefit analysis comparing the increase in steel usage and the load-bearing capacity relative to the 3 mm steel plate specimen. Considering both economic efficiency and reinforcement effectiveness, steel plates with the thickness not exceeding 5 mm provide relatively high reinforcement efficiency for this type of specimen. Figure 15 shows the peak load of specimens reinforced with 4 mm steel plates under a preload ratio of 0.25 with varying bolt spacings. It can be observed that, under single steel plate thickness, reducing the bolt spacing significantly enhances the reinforcement effectiveness, until the failure mode of the steel plate shifts from plate buckling to plate yielding.

Figure 14.

Peak load according to steel plate thickness.

Figure 15.

Peak load according to bolt spacing.

Calculation method of bearing load

Load borne by the rat-trap bond brick masonry

According to Code for design of masonry structures in China (2011b), the average compressive strength $f_{m}$ of solid masonry built with fired common bricks is calculated as follows. For rat-trap bond brick masonry, a correction factor for brick size $k_{b}$ (Shi, 2017) is applied to adjust the peak load calculated ${F_{m}}^{'}$ by the code. The calculations are performed using equations (5)–(7).

f_{m} = 0.78 f_{1}^{0.5} (1 + 0.07 f_{2})

(5)

{F_{m}}^{'} = f_{m} A_{m} = 0.78 f_{1}^{0.5} (1 + 0.07 f_{2}) k_{b} A_{m}

(6)

k_{b} = \sqrt{\frac{b + 7}{h + 7}}

(7)

Where:

f_{1}

: The average compressive strength of the bricks, taking test value 21.04 MPa

$f_{2}$ : The average compressive strength of the mortar, taking test value 1.66 MPa

$k_{b}$ : Brick size correction factor, where b and h are the width and height of the bricks, taking as 1.5 based on the dimensions of 235 mm (length) × 115 mm (width) × 50 mm (height)

$A_{m}$ : Equivalent compressive area of the rat-trap bond masonry, $A_{r} \leq A_{m} \leq A_{s}$ , where $A_{r}$ is the cross-sectional area of the rowlock course, $A_{s}$ is the cross-sectional area of the shiner course. The value of $A_{m}$ varies based on different size and bonding method. For this study, $A_{m} = (A_{r} + A_{s}) / 2 \approx 0.8 A_{s}$ .

Load borne by the steel plates

For the steel plates in the specimen, the side edges are free and unconstrained. At the buckling locations, the upper and lower ends experience minimal out-of-plane displacement due to bolt constraints. The deflection curve of the steel plate between the two rows of bolts resembles the deformation curve of a compression member with fixed ends. The average buckling stress of the steel plate cross-section $σ_{cr}$ can be calculated using the compression member buckling theory model, as expressed in equation (8).

σ_{cr} = \frac{P_{cr}}{A} = \frac{π^{2} E}{{(μ l / i)}^{2}} = \frac{π^{2} E t^{2}}{12 {(μ l)}^{2}}

(8)

Where:

E

: Elastic modulus of the steel plates, taking 206,000 MPa

$μ$ : Effective length factor, taking 0.5

$t$ : Thickness of the steel plates

$l$ : Vertical spacing between bolts;

The method for calculating the maximum load-bearing capacity contributed by the steel plate $F_{s}$ is provided in equation (9).

F_{s} = {\begin{cases} n σ_{cr} b_{s} t σ_{cr} < f_{y} \\ n f_{y} b_{s} t σ_{cr} \geq f_{y} \end{cases}

(9)

Where:

n

: The number of steel plates for reinforcement, taking 2 for this specimen

$b_{s}$ : Width of the steel plate allowed to undergo buckling, taking the total width of 370 mm for this specimen

Load-bearing capacity of rat-trap bond brick masonry reinforced with steel plates under preload

In the reinforced specimens, the load is shared between the steel plates and the masonry. The displacement at peak load for the reinforced specimens is smaller than that of the unreinforced masonry (Δx), indicating that the load borne by the masonry in the reinforced specimens may be less than the peak load of the unreinforced masonry (ΔF), as shown in Figure 16.

Figure 16.

Load-displacement curves between unreinforced and reinforced specimens.

However, due to the complex constitutive behavior and discrete nature of rat-trap bond brick masonry, and the difficulties of directly obtaining the corresponding load data from experiments. It is noted that the Free Body and NFORC functions in finite element model were used to observe the load borne by the masonry at the specimen’s peak load. The finite element model was validated against the available experimental results and demonstrated reliable performance. Therefore, the data obtained from the finite element simulations can be reasonably used for theoretical fitting, and the resulting fitted formula shows good agreement with the experimental data.

Based on finite element analysis results, a reduction factor $γ_{c} (β)$ sensitive to preload ratio is introduced to account for the decreased peak load-bearing capacity of the masonry. As the preload ratio increases, the contribution of the masonry to the peak load of the reinforced specimen increases in a nonlinear manner, exhibiting a pronounced exponential trend.

γ_{c} (β) = \frac{F_{m} (β)}{F_{m} (0)} = 0.74 + 0.009 e^{β / 0.2}

(10)

where:

F_{m} (β)

: The load borne by masonry when the reinforced specimen reached peak load under the preload ratio of

β

in ABAQUS

$F_{m} (0)$ : The peak load of unreinforced masonry in ABAQUS

For the steel plates, when local buckling occurs prematurely, the load-bearing capacity at the peak load is lower than the maximum capacity the plates could potentially contribute. The decline in stress is non-linear and influenced by various factors. By extracting the load borne by the steel plates in finite element models under different preload ratios, it was observed that the ratio to its ultimate capacity remains approximately constant at 0.8. Validation was conducted by adjusting the strength ratio between the masonry and steel plates, as well as the peak displacement load. The results demonstrated good agreement with the theoretical predictions using this reduction factor.

For specimens in which no local buckling of the steel plates occurs at peak load, the significant disparity in strength between the steel plates and the masonry hinders fully coordinated composite behavior, especially when the connection is achieved solely through four-point bolting. Therefore, a coordination factor $η$ is introduced, with the calculation method defined as follows. The proposed threshold is established based on both experimental calibration and theoretical reasoning.

η = {\begin{cases} 1 & , 1 < \frac{E_{c} b_{c} ψ}{k E_{s} t_{s}} \\ \frac{E_{c} b_{c} ψ}{k E_{s} t_{s}} & , 0.5 \leq \frac{E_{c} b_{c} ψ}{k E_{s} t_{s}} \leq 1 \\ 0.5 & , \frac{E_{c} b_{c} ψ}{k E_{s} t_{s}} < 0.5 \end{cases}

(11)

where:

E_{c}

: Elastic modulus of masonry, taking 6800 MPa

$b_{c}$ : Width of the unreinforced side of the masonry, taking 240 mm

$ψ$ : reduction factor for the rat-trap masonry, taking $A_{m} / A_{s} = 0.8$

$k$ : Single-side reinforcement takes 1; Double-side reinforcement takes 2

$E_{s}$ : Elastic modulus of the steel plate, taking 206000 MPa;

$t_{s}$ : Thickness of steel plate;

Based on the compressive stress-strain relationship for masonry and the preload ratio $β$ , the strain difference between the strengthening point and the peak load point of the masonry is calculated by equations (12) and (13).

β = \frac{σ}{f_{m}} = 2 \frac{ε_{r}}{ε_{0}} - {(\frac{ε_{r}}{ε_{0}})}^{2}

(12)

Δ ε = ε_{0} - ε_{r} = \sqrt{1 - β} ε_{0}

(13)

where:

ε_{r}

: Strain of masonry at strengthening point:

Therefore, the steel plate capacity reduction factor $λ$ is calculated through equation (14).

λ = \sqrt{1 - β}

(14)

For specimens where local buckling has occurred in the steel plates at peak load, the axial compressive bearing capacity should be calculated using equation (15).

N = γ_{c} (β) {F_{m}}^{'} + 0.8 F_{s} = γ_{c} (β) \times 0.78 f_{1}^{0.5} (1 + 0.07 f_{2}) k_{b} A_{m} + 0.8 n σ_{cr} b_{s} t

(15)

For specimens where local buckling has not occurred in the steel plates at peak load, the axial compressive bearing capacity should be calculated using equation (16).

N = η ({F_{m}}^{'} + λ F_{s}) = η (0.78 f_{1}^{0.5} (1 + 0.07 f_{2}) k_{b} A_{m} + \sqrt{1 - β} n f_{y} b_{s} t)

(16)

For steel plates which yield stress is close to buckling stress, the ultimate bearing capacity of the reinforced specimen should be determined as the minimum value obtained from equations (15) and (16). Considering both experimental findings and practical engineering applications, this study recommends a preload ratio range of 0 to 0.6.

The above equations are verified through experiments, finite element simulation, and calculation results from literature (Cheng and Jing, 2024), shown in Tables 6 –8. Given the discrete nature of the masonry materials and the differences in steel plate form between this study and references, the error is totally acceptable.

Table 6.

Comparison between calculated values and experimental values.

Specimens	Calculated value	Experimental value	Relative error
US	385.8	423	8.80%
S-0A	471.1	460	2.41%
S-0.25A	486.5	507	4.04%
S-0.5A	505.8	545	7.19%
S-0.25B	803.1	780	2.96%
S-0.25C	930.9	1030	9.63%

Table 7.

Comparison between calculated values and simulated values.

Specimens	Calculated value	Simulated value	Relative error
F3	320.54	324	1.07%
F5	443	435	1.84%
F6	485.12	456	6.39%
F8	601.34	607	0.93%
F9	659.06	670	1.63%
SS-0.5B	699.6	760	7.94%

Table 8.

Comparison between calculated values and experimental values in (Cheng and Jing, 2024).

Specimens	Calculated value	Experimental value	Relative error
SW3-0.25	1185.1	1565	24.28%
SW3-0.5	1249.1	1453	14.03%
SW5-0.25	1950.4	1770	10.19%
SW5-0.5	1799.3	1542	16.69%

Conclusion

This study comprehensively investigates the axial compressive performance of steel plate-reinforced rat-trap bond brick masonry under different preload ratios. The results demonstrate that steel plates can effectively reinforce rat-trap bond brick masonry, while the preload ratio prior to reinforcement has a certain influence on the effectiveness of the reinforcement. Based on the analysis, the following conclusions are drawn:

• Reinforcement with steel plates significantly enhances the load-bearing capacity, improves the ductility, and optimizes the failure mode of rat-trap bond brick masonry.

• Local buckling of the steel plates at peak load has a pronounced impact on the load-bearing capacity of the reinforced specimens. For specimens experiencing local buckling, the peak load increases with the preload ratio. Conversely, in specimens without local buckling, the peak load decreases as the preload ratio increases.

• While the experimental results indicate a positive correlation between reinforcement effectiveness and steel plate thickness, excessive thickness may adversely affect the coordination between the steel plate and the masonry, resulting in reduced efficiency. In addition, excessively high preload ratios can cause substantial internal damage within the masonry, compromising both practical safety and reinforcement performance. Considering both economic efficiency and reinforcement effectiveness, it is therefore recommended to select steel plate thickness with a coordination factor greater than 0.6 (which is the coordination factor for S-0.25B), and a preload ratio below 0.5.

• A validated finite element model was developed for steel plate-reinforced rat-trap bond brick masonry under preload, demonstrating high reliability in predicting structural performance.

• Based on experimental findings, finite element simulations, and existing theories, a formula is proposed to predict the axial compressive load-bearing capacity of rat-trap bond brick masonry reinforced with steel plates under preload conditions.

Given the limited number of specimens and the validated parameter range, the proposed axial load-bearing capacity formula should be applied with caution, especially for higher preload ratios or thicker steel plates.

This study did not explicitly consider initial masonry imperfections or residual stresses induced by welding. Although welding was limited to small regions near the plate ends, residual stresses may still slightly influence local stress transfer between the steel plates and masonry, potentially affecting the initiation or propagation of local yielding or buckling.

Future research should test more specimens over a wider range of steel plate thicknesses (3–10 mm) and preload ratios (0–0.8), and account for construction and steel plate imperfections, including welding-induced residual stresses, to improve reliability and generality.

Footnotes

Jingwen Peng

Haoran Cheng

Zhongyi Qin

Denghu Jing

Author contributions

Jingwen Peng: Data curation, Formal analysis, Investigation, Visualization, Writing - original draft. Haoran Cheng: Data curation, Investigation, Methodology. Zhongyi Qin: Data curation, Investigation, Visualization. Denghu Jing: Conceptualization, Funding acquisition, Resources, Supervision, Writing - review & editing.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial support of the National Natural Science Foundation of China (NSFC) (Grant No. 52178118) is greatly appreciated.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be made available on request.*

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