Study on the Shear Performance of MMOM Stay-in-Place Formwork Beams Reinforced with Perforated Steel Pipe Skeleton

Abstract

The simulation analysis of a novel stay-in-place formwork (SIPF) beam reinforced with perforated steel pipe skeleton was conducted. The SIPF beam consists of a modified magnesium oxysulfide mortar (MMOM) formwork, a square steel pipe skeleton with holes dug on the sides and top, and cast-in-place concrete. The finite element (FE) analysis model of the SIPF beam was established by using the ABAQUS CAE 2021 software, and simulation analysis was conducted with the shear span ratio (SSR), the distance between the remaining steel strips, and the strength of concrete as the variation parameters. The results show that the stiffness and shear capacity of the SIPF beam decrease with the increase in SSR and increase with the decrease in strip spacing. Under the same conditions, when the concrete strength grade is increased from C30 to C50, the shear bearing capacity of the SIPF beam increases by 11.8% to 16.2%. When the spacing of the steel strips is reduced from 200 mm to 150 mm, the shear bearing capacity can be increased by 12.7% to 31.5%. When the SSR increases from 1.5 to 3.0, the shear bearing capacity decreases by 26.9% to 37.3%. Moreover, with the increase in the SSR, the influence of the steel strip spacing on the shear bearing capacity of the SIPF beam improves, while the influence of the concrete strength on the shear bearing capacity decreases. Taking parameters such as SSR, steel strip spacing, and concrete strength as variables, the influence of steel pipe constraining the core concrete on the shear bearing capacity was considered. The calculation formula for the shear bearing capacity of the SIPF beam with perforated steel pipe skeleton was established. The calculation results fit well with the laboratory test and simulation test results and can be used for the design and calculation of engineering structures.

1. Introduction

Formwork has been widely used in civil engineering, but there are still some problems to be improved [1]. Cast-in-place concrete construction consumes a lot of formwork materials, the process is relatively complex, the formwork project cost accounts for about 30% of the concrete project, and the construction period accounts for about 50% [2]. Construction companies must allocate significant funds for the upfront purchase of formworks. Additionally, the turnover rate in construction is low, resulting in high consumption levels and extended timeframes. In the use of formworks, the surface should be coated with oil, which affects the appearance quality of the structure and it is easy to make the surface of the structure a plastered empty drum, and the quality of the structural components is greatly affected when the formwork is removed [3]. The installation and removal of large steel formworks must use large lifting equipment, which increases the difficulty and danger of construction. On the other hand, the reinforcement engineering also involves complex construction technology, large inputs of manpower and material resources, and hidden dangers in construction quality [4]. The stay-in-place formwork (SIPF) concrete belongs to the integrated assembly structure, which eliminates the process of on-site support and removal of molds. Promoting the application of demolding formworks in civil engineering is of great significance for improving project quality, speeding up construction progress, and reducing construction costs [5,6,7]. At the same time, while ensuring the performance of the formwork structure, high-performance materials can be fully developed and utilized to improve the bearing capacity and durability of the structure [8]. Instead of the traditional steel reinforcement as the bearing skeleton, the steel pipe can further simplify the construction process, improve the structural performance, and save financial and material resources [9]. At present, a lot of research work has been carried out on the mechanical properties of SIPF structures. With the development of science and technology, high-performance materials have greatly improved the strength and durability of SIPFs [10,11,12,13]. In the early stage, steel mesh cement mortar was used to make the SIPF, but the test results showed that it had little impact on the structural performance, so only the contribution of cast-in-place reinforced concrete was considered in the calculation of bearing capacity [14,15]. Although an SIPF made of fiber-reinforced polymer has high strength, light weight, and good durability, it has not been widely used in engineering due to its high price [16,17,18]. Some researchers use ultra-high-performance concrete (UHPC) to make an SIPF, which greatly improves its mechanical properties and durability. A significant amount of testing (Wu [19,20], Azizinamini [21], Zhang [22]) indicates that the cracking load of SIPF beams can be enhanced by approximately 50%, while the ultimate load-bearing capacity may rise between 10% and 70% by using UHPC to make an SIPF. However, the problems of high brittleness, high cost, and high preparation conditions of UHPC still need to be solved. The use of engineering cement-based composite (ECC) and textile-reinforced concrete (TRC) can greatly improve the ductility of SIPFs. The test results show that the SIPF made with ECC and TRC can significantly improve the ductility of SIPF beams, and the cracks are thin and dense, but the improvement effect on the bearing capacity is relatively lower than that of UHPC [23,24]. Yin [25] showed that each additional layer of TRC can increase the carrying capacity of SIPF beams by about 3%, and there is no significant improvement beyond four layers.

Modified magnesium oxysulfate cement (MMOC) is a kind of low-carbon high-performance material made by adding an appropriate amount of citric acid admixture (CA) on the basis of the existing preparation method of magnesium oxysulfate cement, which is set and hardened in the air [26]. Compared with Portland cement-based materials, the calcination temperature of MMOC-based materials is only 600–900 °C [27], and MMOC-based materials also have the advantages of simple production process, low manufacturing cost, low energy consumption, good mechanical properties, good fire resistance, and so on [28]. Numerous studies [29] have shown that MMOC is a kind of whisker cement with self-forming, self-strengthening, and self-toughening whiskers. Therefore, the promotion of MMOC-based materials to make the SIPF can not only reduce carbon emissions, but also effectively improve the strength and durability of SIPF components [30].

Our research group previously proposed a novel type of SIPF beam [31], in which the SIPF is made of modified magnesium oxysulfide mortar (MMOM), square steel pipes with holes dug on both sides and the top surface are used as the load-bearing skeleton, and core concrete is poured inside. The four-point moment test of two SIPF beams and one RC control beam was conducted to verify the feasibility of the SIPF beam. Based on this, the ABAQUS finite element (FE) model of the SIPF beam was established to extend the laboratory test data. Laboratory test results were used to verify the accuracy of the model analysis, and then the effects of shear span ratio (SSR), residual steel strip spacing, and cast-in-place concrete strength on the shear performance of SIPF beams were investigated. Based on a truss-arch model, the calculation formula of the shear capacity of SIPF beams was established.

2. Materials and Methods

2.1. Material Constitutive Model

2.1.1. Concrete

The constitutive relations of concrete in tension zones are suggested by the Concrete Structure Design Code GB50010-2010 [32]. Since the concrete beam with an appropriate amount of stirrups is subject to shear failure due to the concrete being crushed in the shear compression zone, the strength of the concrete will affect the shear resistance performance of the concrete beam. In this paper, square steel pipes are used instead of traditional steel bars, which provide a greater constraint effect on the concrete, thereby limiting the lateral deformation and cracks of the core concrete and improving the compressive strength of the concrete. In order to effectively simulate the influence of the constraining effect of square steel pipes on the compressive strength of concrete, we need to introduce a concrete constraining model. The concrete is divided into confined and unconfined zones in the FE analysis. The Mander model [33] was adopted to describe the stress–strain relationship of concrete in confined and unconfined zones, as shown in Figure 1.

According to the characteristics of concrete in the confined area, the material parameters in the concrete plastic damage model were modified according to the suggestion of Bao [34]. The effect of confining stress is considered in the hardening/softening criteria to provide a more precise description of the mechanical behavior of confined concrete. Among them, the proportion of restrained compressive strength to unrestrained compressive strength is 1.16, the expansion angle is 40°, the ratio of tensile and compressive meridional stress is 0.69, and the viscosity coefficient is 0.0005. The definition of damage factors D_c and D_t in ABAQUS is different from the damage parameters d_c and d_t in GB50010-2010 [32]. The d_c and d_t are related to the compressive plastic strain ${\epsilon }_{\mathrm{c}}^{\mathrm{pl}}$ and tensile plastic strain ${\epsilon }_{\mathrm{t}}^{\mathrm{pl}}$. According to the principle of energy equivalence, the conversion formula is shown in Equation (1).

$$\left\{\begin{array}{l}{D}_{\mathrm{c}}=1-{\displaystyle \frac{{\sigma }_{\mathrm{c}}}{{\sigma }_{\mathrm{c}}+{\epsilon }_{\mathrm{c}}^{\mathrm{in}}{E}_0(1-b_{\mathrm{c}})}}\\ D_{\mathrm{t}}=1-{\displaystyle \frac{{\sigma }_{\mathrm{t}}}{{\sigma }_{\mathrm{t}}+{\epsilon }_{\mathrm{t}}^{\mathrm{in}}{E}_0(1-b_{\mathrm{c}})}}\end{array}\right.$$

(1)

where b_c refers to the ratio of plastic strain and inelastic strain of the material, and ${\epsilon }_{\mathrm{c}}^{\mathrm{pl}}$ and ${\epsilon }_{\mathrm{t}}^{\mathrm{pl}}$ are tensile and compressive plastic strains, respectively.

2.1.2. MMOM

The uniaxial compression constitutive relation of MMOM adopts the model proposed by Yang [35], and the uniaxial tension constitutive relation adopts the model proposed by Du [36].

2.1.3. Steel

A simplified complete elastoplastic model was adopted for steel. The stress σ_s–strain ε_s relationship includes two parts: linear elastic stage and complete plastic stage, that is, the σ_s-ε_s relationship of steel before yielding is linear, and the steel has complete plastic deformation after reaching yield strain ε_y, and the stress does not increase. The expression of the σ_s-ε_s relation is:

$$\left\{\begin{array}{l}{\sigma }_{\mathrm{s}}=E_{\mathrm{s}}{\epsilon }_{\mathrm{s}} {\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{y}}\\ {\sigma }_{\mathrm{s}}=f_{\mathrm{y}} {\epsilon }_{\mathrm{s}}>{\epsilon }_{\mathrm{y}} \end{array}\right.$$

(2)

2.2. Establishment of FE Model

2.2.1. Boundary Conditions and Contact Methods

The interaction of the contact surface mainly includes three parts: between the square steel pipe and SIPF, between the SIPF and concrete, and between the plate and concrete. According to the laboratory test results, the steel pipe and SIPF and the SIPF and concrete were never separated, maintaining good integrity. The square steel pipe was placed in the SIPF using the Embedded command, and the interface between the SIPF and the concrete used the Tie command to set constraints. Reference points are set respectively at the loading point and the center point of the support, and then the reference points are coupled onto the pad. The boundary conditions schematic diagram of the FE model is shown in Figure 2. As shown in Figure 2, in the Load command, the boundary conditions are set. At the reference point on the left side of the beam bottom, U1, U2, U3, UR1, and UR3 are set to 0 to limit the displacement and rotation outside the plane, which is a fixed hinge support. U1, U3, R1, and UR3 are set to 0 at the reference point on the right side of the beam bottom to limit lateral displacement and lateral rotation, and it is a movable hinge support, thereby achieving the simply supported state of the beam. Binding constraints are also adopted for the spacers at the loading points on the upper part of the beam. A reference point is set on the upper pad block for applying vertical loads. This reference point is connected to the top pad block of the beam in a coupled manner to achieve the application of force.

2.2.2. Element Type and Mesh Size

The concrete, SIPF, and steel plates are made of eight-node three-dimensional solid units (C3D8R). The reinforcement adopts a two-node linear truss element (T3D2). The square steel tube was modeled using the S4 shell element with four-node full integration. To ensure computational accuracy, Simpson integration with nine integration points was employed along the thickness direction of the shell element. Before grid division, the steel pipe holes are cut along both sides of the remaining steel strip to make the model calculation more accurate and easier to converge. The meshing density of each cell is 15 mm.

2.3. Model Verification

2.3.1. SIPF Beam Laboratory Shear Test

The shear failure test [31] results of two porous steel pipe skeleton SIPF beams (numbered B1 and B2) and an RC control beam (numbered B0) conducted by our research group were used as the basis for comparison, to verify the precision of the calculated outcomes of the simulation model. The section size of each test beam was 200 mm × 300 mm, the length was 2000 mm, and the SSR was 1.8. The lateral and transverse cross-sections of each test beam are shown in Figure 3 and Figure 4, respectively. The reinforcement of RC beam B0 is shown in Figure 3a and Figure 4a, and the amount of steel used for bending and shear resistance was equal to the remaining steel strip of beam B1. For the measured reinforcement strength, the upper longitudinal reinforcement and stirrup were 331 N/mm², the lower longitudinal reinforcement was 415 N/mm², and the steel pipe of SIPF beams B1 and B2 was 275 N/mm². The dimensions of the remaining steel strip after digging are shown in Figure 3b,c and Figure 4a,b. The symbol “” indicates the upper longitudinal remaining steel strip after the steel pipe is dug and cut, “” indicates the lower longitudinal remaining steel strip, and “—” indicates the lateral remaining steel strip. Taking beam B1 as an example, 213 × 13 × 5 indicates that the transverse width, vertical height, and thickness of the remaining longitudinal steel strip on both sides of the upper part of the steel pipe after digging and cutting are 13 mm, 13 mm, and 5 mm. 35 × 150 × 35 × 5 indicates that the longitudinal remaining steel strip in the lower part of the steel pipe after digging and cutting is 150 mm in transverse width, 35 mm in vertical height, and 5 mm in thickness. —41 × 5@200 indicates that the width of the remaining steel belts is 41 mm, the thickness is 5 mm, and the spacing is 200 mm. The test beam adopts the four-point bending loading form. The schematic diagrams of the loading device and the laboratory loading diagram are shown in Figure 5 and Figure 6, respectively.

2.3.2. Comparison of Load–Deflection Curves

The load–deflection curve drawn according to the test results and simulation analysis results of the two SIPF beams B1 and B2 is shown in Figure 7, and the ultimate load error value is shown in

Table 1. Simulation and test errors of composite beams.

Number	Test		Simulation
	Ultimate Load	Deflection	Ultimate Load	Deviation	Deflection	Deviation
B1	265	14.9	271	2.3%	15.2	2.0%
B2	290	14.8	301	3.8%	16	8.1%

. On the whole, the load–deflection curve of the FE simulation results is basically consistent with the test results, and the error between the ultimate load and the test results is less than 3.8%, indicating that the simulation accuracy is high, and it can be used for the simulation of the shear analysis of SIPF beams. The ultimate load value predicted by the simulation for the SIPF beams is marginally greater than that obtained from the testing, which is because the internal defects generated in the concrete pouring and curing process cannot be realized by the simulation model, and the relative slip effect between SIPF and steel pipe and between SIPF and concrete is ignored. However, as the load continues to increase during the laboratory test, SIPF cracks continue to occur and develop, resulting in local slippage between the steel pipe in the SIPF beam and the SIPF, which adversely affects the overall stiffness and bearing capacity.

2.3.3. Comparison of Oblique Crack Development

The crack comparison of the two SIPF beams B1 and B2 and the one RC beam B0 is shown in Figure 8. In this paper, the crack development of concrete is reflected by reading the tensile damage values in the FE simulation results. In Figure 8, the red part indicates that the tensile damage value of the concrete has reached 0.9, suggesting that this part of the concrete has reached the cracked state. The simulation results show that cracks first appear at the bottom of the shear span zone of the beam in the initial stage of loading and gradually extend diagonally towards the loading point as the load continuously increases. The damage accumulates continuously along the oblique crack and spreads upward to the shear and compression zone. Moreover, vertical cracks caused by bending moments and uniformly distributed along the longitudinal direction also appeared at the lower part of the beam between the two loading points. It can be seen from Figure 8 that the numerical simulation of the SIPF beam is relatively consistent with the test in terms of crack location, extension length, and development form. It should be noted here that Figure 8 determines the location and extension direction of the cracks through the tensile damage of the concrete. However, on the one hand, the generation of cracks in the beam is rather random. On the other hand, when the main cracks form, stress concentration will occur in the crack cross-section. Although large areas of red areas appear in some parts of the figure, it does not mean that all of them have transformed into cracks. Instead, one or two main cracks will be concentrated. Figure 9 shows the FE analysis results of compressive stress of the three test beams at failure. It can be seen from Figure 9 that certain compressive stresses were generated in the inclined sections of the three beams between the supports and the loading points, but none of them reached the compressive strength of the concrete, indicating that the concrete between the inclined cracks was not crushed. The failure of the beam is caused by the crushing of the concrete in the shear and compression zone at the top of the beam, which is in line with the characteristics of shear and compression failure and is consistent with the laboratory test results.

In conclusion, the results of ABAQUS FE analysis can well reproduce the deformation and crack development of SIPF beams in the shearing process, and the ultimate load error is small, so it can be used to simulate the shear performance of SIPF beams.

3. Results and Discussion

To investigate the influence of parameters on the shear performance of SIPF beams, the laboratory test parameters were extended by FE simulation based on the laboratory test data. The SIPF beams had SSR designs of 1.0, 1.5, 1.7, 1.8, 2.0, 2.5, and 3.0; the strip spacing S was designed as 150 mm, 180 mm, and 200 mm; and the concrete strength grades were designed for C30, C40, and C50. To prevent the occurrence of bending failure in the SIPF beams prior to shear failure, the length of the beams with a large SSR of 2.5 and 3.0 was set at 3050 mm, with other parameters unchanged.

3.1. The Influence of Each Parameter on the Shear Performance of SIPF Beams

3.1.1. Influence on the Strain of Vertical Steel Strip of SIPF Beam

(1)

SSR

The SSR of the test beam was changed by adjusting the position of the loading point, and was designed to be 1.0, 1.5, 1.7, 1.8, 2.0, 2.5, and 3.0, respectively. Here, the SSR is the ratio of the shear span a to the effective section height h₀, that is, a/h₀. The load–steel strip strain curves of each SIPF beam are illustrated in Figure 10. It can be observed that the curve can be approximately segmented into two developmental phases. The initial phase is referred to as the uncracked stage, the SIPF beam shows elastic characteristics, and the shear force acting on the inclined section is supported collectively by the concrete, SIPF, and steel strip. When the SIPF near the steel strip enters the plastic stage and produces inclined cracks, the internal force within the SIPF beam undergoes redistribution. The load–steel strip strain curve has an inflection point and the slope suddenly becomes smaller, and the shear force supported by both the SIPF and the concrete is progressively transferred to the steel strip intersecting the cracks with the development of the inclined cracks.

(2)

Steel strip spacing

The remaining vertical steel strip after opening the side of the square steel pipe can provide shear strength instead of stirrups. The change in the strain of the steel strip of the SIPF beam with external load under the same SSR and different strip spacing is analyzed below. In general, the maximum stress of the single steel strip is located at its intersection with the oblique crack, and the stress at other locations decreases along its length. Here, the maximum strain of the steel strip intersecting the oblique crack when the SIPF beam is damaged is taken for analysis. The load–strip strain curve is illustrated in Figure 11. It can be observed that the first oblique crack appears when the load–steel strip strain curve shows an inflection point. Under the same SSR, the strain of the steel strip under the ultimate load increases gradually as the distance between the steel strips of the SIPF beam decreases. Moreover, the reduction of steel strip spacing increases the overall stiffness and shear capacity of the SIPF beam. The slope of the load–strain curve of a cracked SIPF beam with 200 mm steel strip spacing is lower than that of beams with 180 mm and 150 mm steel strip spacing. As shown in Figure 11a, the steel strip that crosses the diagonal crack of the SIPF beam with an SSR of 1.0 does not exhibit yielding when the beam experiences shear failure. The shear force is mainly borne by the concrete diagonal short column formed by the section of the diagonal crack in the abdomen, and the failure mode is diagonal failure. From Figure 11b–g, it can be seen that the steel strips of SIPF beams with an SSR of 1.5 to 3.0 have yielded before reaching the ultimate load, and the steel strip strain experienced under the maximum load also rises as the spacing of the steel strips decreases under the same SSR. But the steel strip spacing has little impact on the gradient of the load–steel strip strain graph following the fracture of the SIPF.

(3)

Concrete strength

SIPF beams with an SSR of 1.5 and concrete strength classes C30, C40, and C50 were chosen. The relation curve of SIPF beam load–steel strip strain is illustrated in Figure 12. It can be observed that under different concrete strength levels, the strain of the steel strip intersecting the inclined crack in the SIPF beam has roughly the same variation trend as the load. Nonetheless, the slope of the load–strain curve for the strip rises as the concrete strength improves. When the concrete strength is lower, the inflection point on the strain curve of the steel strip manifests sooner, indicating that the inclined cracks of the low-strength concrete SIPF beam appear earlier and the load at which the inclined section cracks is reduced. When the concrete’s strength increases while maintaining the same load, the strain in the steel strip of the SIPF beam decreases, indicating that the concrete shares more shear force.

3.1.2. Influence on the Shear Capacity of SIPF Beams

(1)

SSR

The SSR is an important parameter affecting the shear performance of a beam, which reflects the proportional relationship between normal stress and shear stress of a beam section. As mentioned above, the steel strip has not yet yielded when the SIPF beam with an SSR of 1.0 is broken, that is, baroclinic failure occurs. Such brittle failure should be avoided. Therefore, SIPF beams with an SSR of 1.5 to 3.0 were selected for analysis. The curve of the relationship between the ultimate load of the SIPF beam and the SSR is illustrated in Figure 13. It can be observed that the SSR is an important parameter affecting the shear capacity of SIPF beams. When the SSR increases from 1.5 to 3.0, the shear strength of beams with steel strip spacing of 200 mm, 180 mm, and 150 mm decreases by 37.3%, 32.7%, and 26.9%, respectively. In the case of the same parameters, such as steel strip spacing and concrete strength, the shear capacity of the SIPF beam decreases significantly with the increase in SSR. When the SSR is less than 2, the shear capacity of the beam decreases more with the increase in the SSR.

(2)

Steel strip spacing

The strip spacing is another important parameter. To analyze the effect of the spacing between steel strips on the shear capacity of SIPF beams, taking an SIPF beam with concrete strength grade C30 as an example, the relationship between the ultimate load and steel strip spacing under each SSR is illustrated in Figure 14. It can be observed that with the identical SSR maintained, a reduction in the spacing of the steel strips results in an increase in the shear capacity of the SIPF beam. When the SSR is 1.5, the shear capacity of the beam with 200 mm steel strip spacing is reduced by 6.1% and 12.7% compared with the beam with 180 mm and 150 mm steel strip spacing, respectively. When the SSR is 2.0, the shear capacity of the beam with 200 mm steel strip spacing is 11.6% and 24.7% lower than that of the beam with 180 mm and 150 mm steel strip spacing, respectively. When the SSR is 2.5, the shear capacity of the beam with 200 mm steel strip spacing is reduced by 12.7% and 27.7% compared with the beam with 180 mm and 150 mm steel strip spacing, respectively. When the SSR is 3.0, the shear capacity of the beam with 200 mm steel strip spacing is 13.7% and 31.5% lower than that of the beam with 180 mm and 150 mm steel strip spacing, respectively. Therefore, the larger the SSR, the greater the influence of steel strip spacing on the shear strength of the SIPF beam.

(3)

Concrete strength

The concrete strength classes of simulated beams were C30, C40, and C50 to investigate the influence of cast-in-place concrete strength on the shear capacity of SIPF beams under the condition of the same SSR and steel strip spacing. Figure 15 shows the relationship between the ultimate load of the SIPF beam and the strength grade of concrete and the distance between steel strips at each SSR. It can be observed from Figure 15 that with the increase in concrete strength, the shear capacity of the SIPF beam increases slightly. With the increase in SSR, the influence of concrete strength on the shear capacity of the SIPF beam decreases slightly. This is because when the SSR increases, the proportion of bending moment also increases, so the impact of concrete strength on the section’s bearing capacity diminishes.

3.1.3. Influence on the Deformation of SIPF Beams

(1)

SSR

The load–deflection curve of SIPF beams under various steel strip spacing is shown in Figure 16. When the SSR is 1.0, the SIPF beam is suffering baroclinic failure, so it has the worst ductility. When the SSR is 1.5~3.0, the load–deflection curves for SIPF beams exhibit a similar shape and the development stage can be divided into three stages. The initial stage occurs prior to the cracking of the SIPF, during which the beam remains in the elastic phase, and the load–deflection curve is almost a straight line. The second stage spans from the moment the SIPF begins to crack until the beam reaches its peak load, characterized by a gradual reduction in the slope of the load–deflection curve, and the stiffness of the concrete cracking beam decreases. The third stage is when the beam reaches peak load to failure. At this stage, the steel strip has yielded, the load–deflection curve of the SIPF beam begins to decline, and the stiffness of the beam accelerates. When the SSR is between 1.5 and 3.0, the bearing capacity of the SIPF beam decreases but the ductility increases with the increase in the SSR.

(2)

Steel strip spacing

The load–deflection curve of SIPF beams under various SSRs is shown in Figure 17. Other parameters in each group of comparison diagrams are the same except that the steel strip spacing is different, which is 200 mm, 180 mm, and 150 mm, respectively. It can be observed from Figure 17 that with the same concrete strength grade and SSR and other parameters, the ultimate load of an SIPF beam increases as the steel strip spacing decreases, and the maximum deflection at the time of failure also increases. As mentioned earlier, an SIPF beam with an SSR of 1.0 is suffering baroclinic failure and has the lowest deformation capacity. However, it can be observed from Figure 17a that under this SSR, reducing the steel strip spacing can still slightly improve its deformation ability, but the bearing capacity is not significantly improved. From Figure 17b–g, it can be seen that when the SSR is 1.5 to 3.0, reducing the steel strip spacing can bolster the load-bearing capacity and deformation resilience of the SIPF beam in case of failure. The larger the SSR is, the gentler the decline section of the load–deflection curve is.

3.2. Ductility Analysis

In this paper, the ductility of an SIPF beam is assessed using the ductility coefficient μ (μ = ∆_u/∆_y) under shear failure at an oblique section. ∆_u represents the displacement associated with 85% of the ultimate load on the declining section of the load–deflection curve, while ∆_y denotes the displacement that corresponds to the yield point. According to statistics, the load values associated with the initial yield point for the majority of test beams range from 0.8 to 0.9 V_u. Therefore, the point corresponding to the load of 0.85 V_u in the rising section of the load–displacement curve is uniformly taken as the initial yield point. Figure 18 shows the relationship between the ductility coefficient of the SIPF beam and the SSR. Figure 16 shows that the ductility of SIPF beams is generally poor when SSR < 2.0. The ductility coefficient of SIPF beams reaches a peak value when the SSR is 2.5 and then decreases with the increase in the SSR. This shows that when the SSR is moderate, that is, the ratio of normal stress to shear stress on the section is appropriate, the ductility of the beam is the highest. The above analysis shows that appropriate adjustment of the SSR and reduction of the strip spacing of SIPF beams can effectively improve the ductility and optimize the structural performance.

4. Establishment of the Formula of the Shear Bearing Capacity of SIPF Beams in Oblique Section

Utilizing the truss-arch shear model, this paper presents a method for calculating the diagonal shear capacity of SIPF beams.

4.1. Shear Capacity Analysis

The shear capacity of the SIPF beam is composed of the U-shaped SIPF, the cast-in-place concrete, and the holed steel pipe skeleton, as shown in Figure 19. The SIPF width is calculated by taking the sum of the thickness b_f of the two sides, whose area is 2 b_f h₀ (h₀ is the effective height of the section). The calculated area of the cast-in-place concrete is (b − 2b_f) (h − h_f), where h_f is the bottom thickness of the SIPF.

4.2. Role of Trusses

(1)

Establishment of truss model

Before the oblique crack appears, the stress in the steel strip is generally small, and the shear force borne by the SIPF beam is mainly borne by the SIPF and concrete. After the oblique crack appears, part of the SIPF and concrete withdraw from work, internal force redistribution occurs in the member, and with the continuous development of the oblique crack, the steel strip gradually bears most of the shear force. The beam section is diagonally divided by oblique cracks, but each part is connected into a whole through the belly bar, and the shear force is transferred to the support. It is presumed that the upper residual plate strip of the perforated steel pipe and the compression zone of the SIPF and the concrete located at the beam’s upper section serve as the truss’s upper chord, the residual plate strip at the bottom of the steel pipe is the lower horizontal connecting rod of the truss, the lateral residual plate strip of the steel pipe is the tension belly rod of the truss, and the oblique compression SIPF and concrete between the cracks are the compression oblique belly rod of the truss, thus forming the truss model. The test results show that the concrete blocks between cracks are not parallel to each other, but fan out towards the loading point, as shown in Figure 20. However, to simplify the calculation in the theoretical analysis, it is assumed that the compression belly rods are distributed in parallel with the same cracking angle, and the idealized truss model is shown in Figure 21.

(2)

Shear borne by the truss model

Assuming that the lateral steel strip yields, the oblique section at the crack is shown in Figure 22, and the cross-section of the SIPF beam is shown in Figure 23. Where F_td is the tensile force of the steel strip at the bottom, F_cd and V_cf are the resultant compressive stress and shear capacity of the concrete in the shear zone, respectively. The shear V_t borne by the steel strip in the truss model can be calculated by Equation (3).

$$V_{\mathrm{t}}={\displaystyle \sum A_{\mathrm{sv}}}{f}_{\mathrm{y}}={\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{b}_{\mathrm{w}}z\cot \phi$$

(3)

where b_w is the width of the steel pipe, z is the distance between the upper and lower strings of the steel pipe skeleton (approximately 0.9 h), φ is the angle between the concrete compression inclined rod and the beam axis, ρ_sv is the steel ratio of the lateral steel strip, and A_sv is the total section area of the steel strip in the same section.

Assuming that the angle α is formed by the steel strip and the beam axis, based on the baroclinic laboratory theory, the “variable angle truss analogy method” is used for analysis. Some studies have shown that the shear capacity is influenced by the angle of steel plate and the strength utilization rate of steel plate when damaged, etc. Through theoretical derivation and analysis, the strain coordination equation of the section can be obtained:

$$\cot \phi ={\displaystyle \frac{\sqrt{1+a/h_0}}{\sin \alpha }}-\cot \alpha$$

(4)

Section balance equation:

$$V_{\mathrm{t}}=f_{\mathrm{y}}{A}_{\mathrm{sv}}\sin \alpha {\displaystyle \frac{z(\cot \phi +\cot \alpha )}{s}}$$

(5)

where s is the distance between steel strips. By substituting Equation (4) into Equation (5), the simplified calculation formula of the V_t contribution of the lateral steel strip to section shear capacity can be obtained as follows:

$$V_{\mathrm{t}}=f_{\mathrm{y}}{A}_{\mathrm{sv}}{\displaystyle \frac{z\sqrt{1+k}}{s}}={\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{b}_{\mathrm{w}}z\sqrt{1+k}=\gamma {\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{b}_{\mathrm{w}}z$$

(6)

where α represents the dip angle of the belly bar, and k is the ratio of the principal tensile strain to the principal compressive strain of the beam in the flexural shear zone under load. The influence coefficient of reaction inclination angle $\gamma =\sqrt{1+k}$ is introduced. When α = 90°, γ = 2.

When the steel strip is set to 90°, according to the stress balance condition of the pressurized isolator in Figure 20, it can be obtained:

$$\left({\displaystyle \sum A_{\mathrm{sv}}{f}_{\mathrm{y}}}{)}^2\right(1+{\cot }^2\phi )=(\eta {\sigma }_{\mathrm{c}}{b}_{\mathrm{w}}z\cos \phi {)}^2$$

(7)

In the equation, η is the effective coefficient, which takes into account the effect of steel strips and the distance between steel strips, reducing the section area of the compression zone. According to the research suggestion $\eta =(1-{\displaystyle \frac{s}{2z}})(1-{\displaystyle \frac{b_{\mathrm{w}}}{4z}})$ of Wang [37], the section size of the beam is shown in Figure 24.

As shown in Figure 24, by balancing conditions and substituting Formula (3), we can obtain:

$$N_{\mathrm{c}}={\sigma }_{\mathrm{c}}{A}_{\mathrm{s}}\sin \phi =\eta {\sigma }_{\mathrm{c}}{b}_{\mathrm{w}}s{\sin }^2\phi$$

(8)

$$V_{\mathrm{t}}=N_{\mathrm{c}}\sin \phi$$

(9)

$$A_{\mathrm{sv}}{f}_{\mathrm{y}}=\eta {\sigma }_{\mathrm{c}}{b}_{\mathrm{w}}s{\sin }^2\phi$$

(10)

where N_c is the axial pressure of the concrete baroclinic bar in the truss model; A_s is the section area of the concrete baroclinic bar; and σ_c is the sectional compressive stress of the soil mixture baroclinic bar. In Figure 24, f_td is the tensile force on the lower chord steel plate and f_cd is the pressure on the winding steel plate. Then, the calculation formula of concrete compressive stress can be obtained, namely:

$${\sigma }_{\mathrm{c}}={\displaystyle \frac{1}{\eta }}{\rho }_{\mathrm{sv}}{f}_y{\displaystyle \frac{1}{{\sin }^2\phi }}$$

(11)

Assuming that the orientation of the concrete main baroclinic beam aligns with the angle of the critical oblique fracture, the stress distribution for the truss model is depicted in Figure 25a. The balance of forces in the horizontal direction is obtained:

$${\sigma }_{\mathrm{c}}bx={\sigma }_{\mathrm{c}}bz{\cos }^2\phi$$

(12)

Solution (12) gives $x_{\mathrm{c}}=h{\cos }^2\phi$. In the arch model, the compression zone height x_c of concrete is the sum of x and the thickness of the protective layer, and the simplified calculation is as follows:

$$x_{\mathrm{c}}=h{\cos }^2\phi$$

(13)

As for the determination of the value of the principal stress dip angle $\phi$, Guo et al. [38] measured the critical oblique crack angle of a beam in the test at 22.2°~38.3°. In this paper, the actual measured oblique crack inclination is 30°~45°. However, the measured fracture inclination angle is formed by the coupling of various actions and is the angle after integration. This is not equal to the oblique crack inclination considered by the truss-arch model alone, and the principal stress inclination is different from the actual measured oblique crack inclination, so the measured value cannot be directly used. This paper is biased towards the safe assumption that $\phi$ = 45°.

4.3. Arch Action

(1)

Establishment of arch model

The arch model consists of the SIPF together with the concrete, which is treated as a whole for shear calculation. The shear transfer mechanism represented in the arch model is illustrated in Figure 26a. In the central part of the arch, there will be a fish-shaped bulge, which is mainly caused by the diffusion of arch compressive stress. The arch model in the beam is simplified into a rectangular compression area. The simplified arch model and the force balance relationship are shown in Figure 26b,c.

(2)

Shear force borne by the arch model

The shear resistance of the SIPF and concrete is not considered in the truss model. The oblique pressure σ_c of concrete is obtained according to the equilibrium condition, as shown in Equation (11). To this end, the arch model was introduced, and from the safety direction, it is assumed that the effective strength of concrete ${f_{\mathrm{c}}}^{\prime }$ is composed of two parts: the compressive stress σ_c of concrete in the truss and the compressive stress σ_a of concrete in the arch model. In this paper, the weighted average strength of SIPF and concrete in the shear compression zone is denoted as ${f_{\mathrm{c}}}^{\prime }$. The stress in the arch model can be obtained from Equation (11) as follows:

$${\sigma }_{\mathrm{a}}=f_{\mathrm{c}}\prime -{\sigma }_{\mathrm{c}}=f_{\mathrm{c}}\prime -{\displaystyle \frac{1}{\eta }}{\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{\displaystyle \frac{1}{{\sin }^2\phi }}$$

(14)

It can be seen from the equilibrium relationship in Figure 26b,c that the shear capacity provided by the arch is:

$$V_{\mathrm{a}}={\sigma }_{\mathrm{a}}b{\displaystyle \frac{h}{2}}\tan \theta$$

(15)

where V_a represents the shear action of the arch model and θ represents the inclination angle of the concrete baroclinic struts in the arch model.

From the geometric relationship in Figure 26b, we can obtain:

$$\tan \theta ={\displaystyle \frac{h-x_{\mathrm{c}}}{a+x_{\mathrm{c}}\tan \theta }}$$

(16)

Solved:

$$\tan \theta ={\displaystyle \frac{\sqrt{a^2-4x_{\mathrm{c}}^2+4h{x}_{\mathrm{c}}}-a}{2x_{\mathrm{c}}}}$$

(17)

z = 0.9 h₀, h₀ = 0.9 h, then z = 0.81 h. Bring into Equation (17) together with Equation (13):

$$\tan \theta ={\displaystyle \frac{\sqrt{a^2-4{(h{\cos }^2\phi )}^2+4h^2{\cos }^2\phi }-a}{2h{\cos }^2\phi }}$$

(18)

The same simplified method is used in the Japanese seismic code for concrete, “Calculation Model of Shear Strength of Reinforced Concrete Members in the Japanese Seismic Guide” [37], and the calculated theoretical results are relatively safe. Therefore, by bringing Equations (14) and (18), $\phi$ = 45° into Equation (15), the shear capacity of the arch model can be calculated as follows:

$$V_{\mathrm{a}}={\displaystyle \frac{1}{2}}{\sigma }_{\mathrm{a}}b(\sqrt{h^2+a^2}-a)$$

(19)

Given ${\sigma }_a={f_c}^{\prime }-{\sigma }_c$, the above formula can be written as:

$$V_{\mathrm{a}}={\displaystyle \frac{({f_{\mathrm{c}}}^{\prime }-{\sigma }_{\mathrm{c}})bh}{2}}.{\displaystyle \frac{1}{\sqrt{{(a/h)}^2+1}+a/h}}=(1-\beta ){\displaystyle \frac{{f_{\mathrm{c}}}^{\prime }bh}{2}}.{\displaystyle \frac{1}{\sqrt{{(a/h)}^2+1}+a/h}}$$

(20)

where $\beta ={\displaystyle \frac{1}{\eta {f_{\mathrm{c}}}^{\prime }}}{\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{\displaystyle \frac{1}{{\sin }^2\phi }}$.

4.4. Restraint Effect of Holed Steel Pipe on Core Concrete

According to the truss-arch model, concrete plays the role of the arch, while steel pipes restrain the deformation of the arch outside the plane, especially the longitudinal steel plate at the lower chord and the belly bar. The analysis reveals that the computed value is significantly less than the tested value, suggesting that the influence of the steel pipe on the concrete should not be overlooked. This study attempts to apply the concrete constraint theory to the truss-arch model. According to the calculation theory of restrained concrete, the principle of superposition is utilized to account for the enhanced strength of restrained concrete, which arises from the constraining influence of the steel pipe, in the calculation of the concrete’s shear capacity as “extra strength”. The concrete diagram of the confined area is shown in Figure 27. The shaded part represents the effective confinement zone of concrete, and it is considered that the binding force is evenly distributed in the concrete, and its value is equal to the yield force of the steel pipe.

Suppose that the angle between the tangent line of the curve and the horizontal position of the longitudinal steel plate is θ, and take θ = 45°, according to the geometric relationship:

$$y_1={\displaystyle \frac{r_1}{2\sin \theta }}-{\displaystyle \frac{r_1}{2\tan \theta }}=0.21r_1$$

(21)

$$A_1=0.14r_1^2$$

(22)

where r₁ is the clear distance in the horizontal direction between the upper chord and the lower chord. Similarly, in the beam height direction:

$$A_2=0.14r_2^2$$

(23)

where r₂ is the clear distance in the vertical direction between the upper chord and the lower chord. Therefore, the area of concrete in the confined area of SIPF beam cross-section is:

$$A_{\mathrm{c}\mathrm{c}}=A_{\mathrm{cor}}-{\displaystyle \sum _{i=1}^n{A}_1-{\displaystyle \sum _{j=1}^n{A}_2}}$$

(24)

where A_cc is the concrete area of the cross-section constraint area of the composite beam at the location of the steel strip, A_cor is the concrete area of the core area, A_cor = b_ic_i, and b_i and c_i are the width and length of the inner surface of the steel pipe, respectively.

The stress diagram of steel pipe and core concrete is shown in Figure 28. The lateral pressure of concrete is balanced with the belly bar and the upper and lower chord connecting rod. The concrete (unrestrained concrete) between the two belly bars is taken as the isolation body, that is

$$f_{\mathrm{kx}}s{H}_{\mathrm{c}}=A_{\mathrm{svx}}{f}_{\mathrm{y}},f_{\mathrm{ky}}s{B}_c=A_{\mathrm{svx}}{f}_{\mathrm{y}}$$

(25)

$$f_{\mathrm{kx}}={\displaystyle \frac{A_{\mathrm{svx}}{f}_{\mathrm{y}}}{s{H}_{\mathrm{c}}}},f_{\mathrm{ky}}={\displaystyle \frac{A_{\mathrm{svy}}{f}_{\mathrm{y}}}{s{B}_{\mathrm{c}}}}$$

(26)

$$f_{\mathrm{k}}={\displaystyle \frac{f_{\mathrm{kx}}{B}_{\mathrm{c}}+f_{\mathrm{ky}}{H}_{\mathrm{c}}}{B_{\mathrm{c}}+H_{\mathrm{c}}}}$$

(27)

In the formula, f_kx, f_ky, and f_k are the x, y direction and average concrete confinement stress, respectively. A_svx and A_svy are the steel plate area and the steel strip area of the upper and lower chord, respectively.

Therefore, the shear capacity provided by the additional strength of the core concrete can be calculated by the following:

$${V_a}^{\prime }={\displaystyle \frac{f_{cc}^{\prime }{A}_{cc}}{2}}\tan {\theta }^{\prime }$$

(28)

$$f_{cc}^{\prime }=f_{cc}-f_c^{\prime }$$

(29)

$$f_{cc}=f_c^{\prime }[-1.254-2{\displaystyle \frac{f_k}{f_c^{\prime }}}+2.254\sqrt{1+7.94{\displaystyle \frac{f_k}{f_c^{\prime }}}}]$$

(30)

where ${\theta }^{\prime }$ represents the inclination angle of the restrained concrete baroclinic belly bar in the arch model. To simplify the calculation, $\tan {\theta }^{\prime }=\tan \theta$.

4.5. Practical Formula of SIPF Beam Shear Bearing Capacity

As mentioned above, concrete strength, SSR, and steel strip spacing are all important parameters that affect the shear capacity of SIPF beams. In this study, we propose that the shear strength of the SIPF beam is influenced by the vertical steel strip in the truss model, the diagonal bar after the combination of the SIPF and concrete in the arch model, and the extra strength of restrained concrete, namely:

$$V_{\mathrm{u}}=V_{\mathrm{t}}+V_{\mathrm{a}}+V_{\mathrm{a}}^{\prime }$$

(31)

By substituting Equations (6), (20) and (28) into the above Equation, the regression model can be obtained:

$$V_{\mathrm{u}}=R_1\gamma {\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{b}_{\mathrm{w}}z+R_2[(1-\beta ){\displaystyle \frac{f_{\mathrm{c}}^{\prime }bh}{2}}+{\displaystyle \frac{f_{\mathrm{c}\mathrm{c}}^{\prime }{A}_{\mathrm{c}\mathrm{c}}}{2}}]{\displaystyle \frac{1}{\sqrt{(a/h{)}^2+1}+a/h}}$$

(32)

The existing load capacity data of the simulated composite beam were input into the formula for calculating linear regression, resulting in the determination of the coefficient value:

$$R_1=0.85, R_2=1.22$$

By bringing all the coefficients into Formula (32), the shear strength of the SIPF beam in an oblique section can be determined as outlined below:

$$V_{\mathrm{u}}=0.85\gamma {\rho }_{\mathrm{sv}}{f}_{\mathrm{y}}{b}_{\mathrm{w}}z+1.22[(1-\beta ){\displaystyle \frac{f_{\mathrm{c}}^{\prime }bh}{2}}+{\displaystyle \frac{f_{\mathrm{c}\mathrm{c}}^{\prime }{A}_{\mathrm{c}\mathrm{c}}}{2}}]{\displaystyle \frac{1}{\sqrt{(a/h{)}^2+1}+a/h}}$$

(33)

4.6. Formula Precision Comparison

To confirm the reliability of the formula developed in this study, the current Chinese code GB 50010-2010 [32], the United States code ACI318-19, and the European code EN 1992-1-1:2014 were selected and compared with the calculation accuracy of the formula established. The results of the laboratory test and simulation test of the SIPF beam were taken as the basis for comparison, and the mean value, standard deviation (SD), and coefficient of variation (CV) of the calculated value/test value of each formula were counted (see

Table 2. Comparison of formulas of shear capacity of composite beams.

Computing Method	Load Patterns	Calculated Value/Test Value	SD	CV
GB 50010-2010	Concentrated	0.92	0.084	0.091
ACI318-08	Concentrated	1.12	0.044	0.039
EN 1992-1-1:2014	Concentrated	0.77	0.024	0.042
Proposed method	Concentrated	1.01	0.090	0.089

).

It can be observed from Table 2 that the theoretical value of the calculation formula for the shear strength of the SIPF beam proposed in this paper is the most consistent with the test value. The calculated value/test value is 1.01, the SD is 0.09, and the CV is 0.089, indicating that the dispersion is small. The calculation result of the US specification ACI318-08 is too high, while the calculation result of the European specification EN 1992-1-1:2014 is much lower than the actual value, which will cause waste of materials. However, the calculation results of the Chinese standard GB 50010-2010 are relatively high in accuracy and tend to be safe. The contribution of the SIPF and core concrete to the shear capacity calculated by the formula proposed in this paper is about 70% of the total shear capacity, and the lateral steel strip provides about 30% of the shear capacity, which is close to the actual situation. Therefore, the formula established in this paper has a high calculation accuracy and can provide a theoretical basis for the research and practice of SIPF beams with perforated steel pipe frame.

5. Conclusions

The FE analysis model of the SIPF beams reinforced with perforated steel pipe skeleton was established by using ABAQUS CAE 2021 software. The effects of the SSR, steel strip spacing, and concrete strength on the shear performance of SIPF beams were analyzed. Based on the findings from both laboratory tests and simulations, a formula to determine the shear capacity of SIPF beams was developed using a truss-arch model. The key conclusions are outlined below.

• Compared with the laboratory test results, the maximum error of the simulation model for the shear bearing capacity of SIPF beams with steel tube skeleton with holes is 3.8%, which indicates that the FE model has a high calculation accuracy and can be used for the simulation analysis of this type of structure.
• The FE simulation results show that the yield, peak load, and stiffness of SIPF beams decrease with the increase in SSR and steel strip spacing. For an SIPF beam with a steel strip spacing of 200 mm, the ultimate load at SSR = 1.5 is 15.9% higher than that at SSR = 1.8. When SSR = 1.5, the shear strength of the SIPF beam with a steel strip spacing of 180 mm and 150 mm increases by 6.1% and 12.7%, respectively, compared with the SIPF beam with a steel strip spacing of 200 mm. When the concrete strength increases from C30 to C50, the shear capacity of SIPF beam decreases slowly with the increase in SSR.
• The relationship between the SSR and the steel strip spacing has great influence on the ductility of the SIPF beam. When the SSR is too small, that is, SSR = 1.0, the SIPF beam will suffer baroclinic failure. When the SSR increases from 1.5 to 2.5, the ductility coefficient of the SIPF beam with 200 mm strip spacing increases from 1.72 to 2.34. The ductility coefficient of the SIPF beam with a steel strip spacing of 180 mm is increased from 1.75 to 2.45; the ductility coefficient of an SIPF beam with a steel strip spacing of 150 mm increases from 1.92 to 2.60.
• Taking SSR, steel strip spacing, and concrete strength as variables, and considering the improvement of the shear strength of the steel pipe confined core concrete, a formula for calculating the shear strength of the steel pipe skeleton SIPF beam with holes is established. Compared with the standard formulas of other countries, the calculated results of the formulas established in this paper are in the best agreement with the experimental results and have higher calculation accuracy. The formula established in this paper can provide reference for the calculation of the shear capacity of SIPF beams in similar projects.