Feng Li1,2
Tian Lan- 1Jiangxi Provincial Key Laboratory of Traffic Infrastructure Safety, East China Jiaotong university, Nanchang, Jiangxi, China
- 2Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education, Changsha University of Science & Technology, Changsha, Hunan, China
- 3School of Civil Engineering, Hunan University of Science and Technology, Xiangtan, China
- 4National Key Laboratory of Green and Long-Life Road Engineering in Extreme Environment (Changsha), Changsha University of Science & Technology, Changsha, Hunan, China
The anchor frame beams have been widely applied in slope engineering. Using conventional concrete frame beams, their structure is heavy and during the installation process, a large amount of construction machinery and manpower are usually required. However, foam concrete reduces weight but has low strength. To address these issues, steel pipe foam concrete with hydrogen peroxide as the foaming agent was selected to replace the conventional concrete. Through experiments, the addition method of hydrogen peroxide, the mix ratio, and the thickness of the steel pipe were determined. Through finite element analysis of a typical cutting slope, the internal force distribution of the structure and the influence of parameters were obtained. The results show that the foam concrete sample prepared by the hydrogen peroxide solution addition method is better. The strength of 2 mm steel pipe-based steel foam concrete with a 5% hydrogen peroxide content is equivalent to that of C30 concrete, and its density is approximately 30% lower. Simultaneously, the peak moment of the articulated frame beam becomes smaller due to the presence of the hinge, but the shear force is basically unchanged. The optimal length of hinge interval in the articulated frame beam is 1.6m, and the breath and height of the beam are 0.4m and 0.5 m.
1 Introduction
Every year, various geological hazards occur worldwide (Andreini et al., 2014; Xie et al., 2026). China is a country with frequent geological disasters, causing billions of direct economic losses and hundreds of casualties each year (Jian et al., 2012; Hu et al., 2025). Vranken et al. (2013) analyzed the economic valuation of the landslide damage in Flanders, Belgium. The annual cost of direct damage due to landslide amounted to €688,148/year and the annual indirect damage augmented to €3,020,049/year in hilly regions. Winter et al. (2016) presented four Scottish landslide events that occurred in 2004 and 2007 found that Direct costs range between approximately £400,000 and £1,700,000 while consequential costs range between around £180,000 and £1,400,000. Li et al. (2017) found that the number of landslides for 28 provinces in China increases with increasing road density, and the degree gradually increases from west to east. According to the planning goals of the National Road Network Plan, China will add 47,000 km of expressways and nearly 1 million kilometers of ordinary roads by the end of 2030. Roads are influencing slope stability through slope cutting, concentrating surface water, changing hydrological patterns, which are destabilizing upper slopes (Zhang et al., 2022). The stable performance of all aspects of the road can guarantee its safe operation (Yan et al., 2019). Thus, timely and effective support of the cutting slope can effectively reduce the risk of landslides and collapses. The supported areas decreases the number of landslides in the central and eastern regions (Li et al., 2017). At present, many structures are used for slope support, such as retaining walls (Sudmeier-Rieux et al., 2012), quay walls (Ghalandarzadeh et al., 2020), and sheet pile walls (Lin Y.- et al., 2018). Among them, the frame beam with anchor cables structure (Tang et al., 2018; Danziger et al., 2006) had been widely used in slope support engineering owing to their unique advantages of high strength, simple construction, and economic benefits.
Numerous researches have been conducted on the stability of the supporting slope by the frame beam with anchor cable using various research methods such as laboratory tests, numerical simulations, and on-site monitoring (Lin Y.-l. et al., 2018; Mansouri and Ajalloeian, 2018; Paraskevopoulou et al., 2017). However, most of the existing studies in literature were mainly concentrated at the research of the anchoring force of anchor cable and internal force of frame beam. The frame beams still use cast-in-situ concrete, which has a density of 2,400 kg/m3 and is heavy in weight (Jingliang et al., 2025; John et al., 2025). Due to steep slopes concrete construction requires the installation and removal of formwork (Tang et al., 2018), lashing and binding steel bars, pouring and curing concrete, there are many problems, such as the large number of constructors, difficult concrete pouring (Abdalla et al., 2019), long curing time (Alberti et al., 2016; Zawam et al., 2019), and difficult to ensure concrete strength (Li et al., 2016; Söylev, 2011). Lin et al. (2017) studied the dynamic behavior of anchoring frame beam by means of shaking table test and dynamic numerical simulation and the frame beam model with a size of 12 m × 9.6 m (width × height) was shaped by aerated concrete. Shi et al. (2019) carried out on-site monitoring of the anchoring force of the cutting slope with a maximum excavation depth of 76 m supported by an frame beam with anchor cable, and the interface dimensions of the frame beam using C25 concrete were 0.4 m × 0.4 m. Ping and Yongbiao (2012) built a dynamic simplified model for frame ground beam with anchor cable under horizontal seismic load using large-scale finite element software ANSYS. The transverse frame beams and the vertical frame beams in the grid are considered as continuous beams independent of each other. Ye et al. (2019) found that the dynamic response of a slope reinforced by a frame with prestressed anchors under horizontal earthquake action.
Foam concrete has been the subject of various studies (Liu et al., 2025; Motloq et al., 2024) in many fields (Eswari et al., 2025; Pandey et al., 2025). Foam concrete offers benefits such as reduced weight, lower thermal conductivity, and increased durability compared to conventional concrete. However, there are relatively few reports on the use of lightweight concrete in slope support projects, as its strength is relatively low. Eid et al. (2025) found that lightweight fiber-reinforced concrete exhibits superior tensile strength-to-weight ratio and ductility due to fibers inclusion. Sathvik et al. (2024) found that density reduced 12% using a 100% plastic aggregate mix, achieving 1,358 kg/m3 with compressive stress of 3.92 MPa. Abrahimi et al. found that LWRPC grades achieved 86%–90% of its 28-day compressive stress within 7 days, with an average density of 1893 kg/m3, 22% lower than corresponding normal high-strength concrete grades. Hong et al. (2024) found that pre-wetted lightweight aggregates can enhance the effect of internal curing, thereby reducing the early-age shrinkage of concrete, and weakening the discreteness of strain distribution in concrete, thus theoretically reducing the risk of cracking. Liu et al. (2024) found that the fiber incorporation had no significant or even negative effect on the compressive stress but significantly improved the splitting tensile strength.
Thus, the purpose of this article is to explore the addition method and mix ratio of foam concrete using hydrogen peroxide as an additive, and to propose a steel tube foam concrete structure with equivalent strength to the articulated frame beam, in order to ensure strength and reduce self-weight. This study was organized as follows. First, determine the addition method and mix ratio of the foam concrete, and establish the steel tube foam concrete structure. Then, taking the site slope as an example, a set of finite element numerical models was established using the ABAQUS software. Finally, the internal forces and the structural design parameters of the two structures are analyzed, and the influence of the structural parameters of the articulated frame beam is compared.
2 Laboratory tests of foam concrete
The physical properties of the above two influencing factors are tested by single element experiment method, and the mechanical influence characteristics of foam concrete are studied. Combined with the experimental results, the mathematical statistics analysis is carried out, and the density of each factor on the foam concrete is compared to determine the optimal mix ratio design scheme of foam concrete, which provides the basis and reference for the subsequent mix ratio design of steel tube foam concrete, and obtains the best mix ratio of foam concrete and the best thickness of steel tube.
2.1 Sample material
1. Foam concreteThe foam concrete in this paper is composed of cement, sand, water and hydrogen peroxide. The cement is made of conch brand P·O42.5 ordinary portland cement from a cement plant in Jiangxi province. The fine aggregate is river sand with 2 mm particle size after screening; the experimental water is local tap water; the foaming method of foamed concrete adopts chemical foaming. The foaming agent used is 10% hydrogen peroxide with a self-made concentration. The hydrogen peroxide crystal is provided by Changle Fuqiang Chemical Technology. The content of hydrogen peroxide in the crystal is 60%. The ratio of hydrogen peroxide solution with a concentration of 10% is crystal: hot water = 1:3.
2. Steel pipe
The strength index of steel is determined by compression test of 100 mm * 100 mm * 100 mm standard specimens made of the same batch of steel. Three empty steel pipe samples were taken from each group for testing. The relationship between the measured stress and strain, bearing capacity and thickness is shown in Figure 1.
Figure 1. Performance index curve of hollow steel pipe with different thickness. (a) Stress-strain curves. (b) Bearing capacity-steel tube thickness curve.
In this state, it can be found that the failure stress of the hollow steel tube is relatively close, but the ultimate yield strength measured by the hollow steel tube with different thickness is different.
2.2 Mix proportion test of foam concrete
2.2.1 Mortar mix design
The foam concrete to be used in this test needs to be tested for density and compressive stress first, so as to find out the mix ratio of foam concrete suitable for combination with steel tube. Therefore, in the scheme design, the hydrogen peroxide addition method and the hydrogen peroxide content are selected as a single independent variable to explore the characterization of the two physical performance indexes of density and compressive stress of foamed concrete under the influence of the level difference of the two factors, and a more suitable hydrogen peroxide addition method and hydrogen peroxide content are compared to determine the mix ratio of the subsequent combination with the steel pipe.
In this experiment, the mix ratio of M40 cement mortar was used as the initial mix ratio, and the slurry was foamed by different hydrogen peroxide addition methods and different amounts of hydrogen peroxide. In the way of adding foaming agent, a variety of adding methods such as crystal water or solution are adopted. Among them, the first method is cement mortar + (hydrogen peroxide crystal + water) stirring, the second method is cement mortar + (hydrogen peroxide solution) stirring, and the third method is more water cement mortar + (hydrogen peroxide crystal) stirring. Considering that the role of hydrogen peroxide in the mix ratio of this test is not clear and it is necessary to consider the presence or absence of hydrogen peroxide, the level of hydrogen peroxide content is considered to be 0%, 2.5%, 5%, 7.5%, 10%. Among them, more than 3 samples were poured in each level test, and the density and compressive stress of the subsequent samples were averaged to reduce the error.
2.2.2 Specimen preparation and test methods
The specific process is as follows:
1. The river sand was screened with a 2 mm sieve to control the fine aggregate without too large particle size, and to prevent the large particle size aggregate from causing pore rupture during the foaming process. Oil is brushed on the inner side of the mold to facilitate the subsequent sample demoulding.
2. Weigh the corresponding cement, sand, water according to different production methods, and prepare the corresponding hydrogen peroxide crystal or solution. The corresponding amount of hydrogen peroxide in each test group was prepared. According to the preparation of different dosage of hydrogen peroxide solution weighing the corresponding weight of hydrogen peroxide required for hot water and hydrogen peroxide crystal, both poured into the beaker after rapid stirring, until the crystal dissolved in the cup.
3. Pour cement and sand into the mixer for full mixing, and then add water for 2 min of mixing to form M40 cement mortar.
4. The hydrogen peroxide crystal or solution was quickly poured into the prepared M40 cement mortar, and continued to stir for 20 s to form a foam concrete mortar with different dosages or different addition methods of hydrogen peroxide.
5. The foam concrete mortar is poured into a 100 mm × 100 mm × 100mm mold, during which the foam concrete mortar expands naturally, without vibration treatment, and the surface is smoothed after the initial setting.
6. After removing the mold, the samples were put into the standard curing room for 3, 7, 28 days (humidity 95% ± 3%, temperature 25 ± 3 C).
After the sample is cured, the surface moisture is dried, and the samples in each group are marked and weighed, so as to calculate the density of the samples under the action of two factors in each test group.
2.2.3 Uniaxial compression test results
The uniaxial compression test is measured under the universal material testing machine 10T. The displacement-controlled loading method (loading rate is 1 mm/min) is used to stop the loading when the sample is obviously damaged.
The characterization results of the two factors in the two mechanical properties of the density and compressive stress of the sample are shown in Figures 2, 3. Among them, due to the test results of different hydrogen peroxide addition methods, under the comprehensive consideration of the two performance indicators, it is considered that the foam concrete sample prepared by the hydrogen peroxide solution addition method is better, and the hydrogen peroxide solution is uniformly used when preparing foam concrete samples with different hydrogen peroxide content.
Figure 2. Mechanical properties index diagram of test blocks with different hydrogen peroxide addition methods. (a) Density. (b) Compressive stress.
Figure 3. The mechanical properties index of the test block under different hydrogen peroxide content. (a) Density. (b) Compressive stress.
In engineering practice, the compressive stress of foam concrete is an important index to measure its mechanical properties, and density is one of the key factors affecting its compressive stress. Therefore, according to the compressive stress of foam concrete under different hydrogen peroxide content reflected in the above experiments, the relationship curve between the density and compressive stress of foam concrete test block is studied, which is of great significance for the rational design and use of foam concrete. Through the collation and analysis of the test data, the relationship between the density and compressive stress of the foam concrete test block is drawn as shown in Figure 4.
Figure 4. The relationship curve between density and compressive stress of foam concrete test block.
2.3 Composite test of steel tube foam concrete
2.3.1 Specimen preparation
The preparation process of the steel tube foam concrete sample is shown in Figure 5. The preparation process of the foam concrete mortar is the same as above, and the other processes are as follows:
1. Put the cut 100 mm * 100 mm * 100 mm steel pipe (thickness of 2 mm, 2.5 mm, 3 mm, 3.75 mm) into the mold in advance, and brush the oil at the bottom of the mold (Note: the inside of the steel pipe does not need to brush oil), and place the empty steel pipe at the top for enclosure;
2. According to the obtained foam concrete mixture ratio of 5% hydrogen peroxide content, the foam concrete mortar is prepared, and the foam concrete mortar is poured into the steel pipe. During the period, the foam concrete mortar is naturally expanded without vibration treatment, and the surface is flattened after the initial setting.
3. After removing the mold, the samples were put into the standard curing room for 28 days.
Figure 5. Steel tube foam concrete test block pouring.
After the curing of the sample is completed, the surface moisture is dried, and the samples of each group are marked and weighed, so as to compare the compressive stress of the steel tube foam concrete test blocks with different thicknesses.
2.3.2 Uniaxial compression test results of foam concrete filled steel tube
The uniaxial compression test is measured under the universal material testing machine 200T. Using a step-by-step loading method (loading rate of 0.5 MPa/s), the loading is stopped when the sample loses greater bearing capacity and is destroyed.
During the test, the damage development process of each specimen is similar. Therefore, the steel tube foam concrete with a thickness of 2 mm is taken as an example to illustrate. Before the peak load, the test block has no obvious damage; when the load rises to 80 kN (25% of the peak load), the upper part of the side of the specimen (5 mm from the top) appears slight bulging visible to the naked eye; when the load increased to 300 kN (90% of the peak load), the other three sides of the specimen showed slight buckling in turn. As the loading continued, the buckling degree of each side of the steel tube increased. When the load drops to 326 kN (peak load), the bearing capacity of the test block drops sharply. The final failure mode of each specimen is shown in Figure 6. It can be found that the bulging of all specimens of steel tube foam concrete test blocks is concentrated in some local areas, and no obvious damage bulging occurs in other locations.
Figure 6. Ultimate failure mode of steel tube foam concrete test block. (a) Final failure mode 1. (b) Final failure mode 2.
The relevant test results of compressive stress and density of steel tube foam concrete under different thicknesses are plotted as curves, as shown in Figure 7.
Figure 7. The mechanical properties of steel tube foam concrete with different thickness. (a) The relationship between compressive stress and strain. (b) Relationship between compressive stress and density.
It can be found that the steel tube foam concrete test block with a steel tube thickness of 2 mm and a hydrogen peroxide content of 5%, while ensuring that its bearing capacity is similar to that of C30 concrete, the density is about 30% lower than that of C30 concrete, and its elastic modulus (E) can be referred to as 30 Gpa.
3 Numerical simulation of articulated frame beam
In order to verify the support effect of the articulated precast frame beam with anchor cable, a typical road cutting slope was selected and the simulation was performed using ABAQUS software.
3.1 Project overview
The cutting slope of Jiang-Yu expressway in Guizhou Province, southwest China was selected, and it was reinforced with the traditional anchoring frame beam and the sheet pile walls. The cross section and 3D model were shown in Figure 8. The rock and soil were divided into 3 engineering geological layers, from top to bottom were red clay, strongly weathered dolomite, and weakly weathered dolomite. The maximum height of the cutting slope divided into two levels slope was 18 m. The cutting slope of the first level supported by the sheet pile wall was 10 m height and vertical. And the second level supported by the traditional anchoring frame beams was 8 m height and 45 angle (vertical: horizontal = 1:1). The sheet pile wall was composed of the square piles and plates. Among them, the square pile with a length of 25.0 m was embedded in the 15.0 m deep of the rock and soil to form a cantilever length of 10.0m, with the spacing of 5.0 m and the section size of 3.0 × 4.0 m. And the thick of 0.5 m with reinforced concrete sheet was set behind the pile to prevent the supported soil from sliding down.
Figure 8. Sectional sketch (a) and numerical model (b) of slope.
Three rows of anchor cables with a spacing of 3.0 m were arranged along the height of the second level slope (z direction, 2.0m, 5.0m, 8.0 m away from the first level platform), and they were separated by 3.0 m along x direction, thereby forming a 3.0 m × 3.0 m grid from the elevation view. The average length of the anchor cable extending into the cutting slope was 30.0m, including the anchoring section of 10 m. The anchoring force of each anchor cable was 700kN, the anchoring direction was 20 between the slope and the y direction.
To better transmit the anchoring force to the cutting slope, the grid of the frame beam was the same as that of the anchor cable, which was 3.0 m × 3.0 m from the elevation view, thus forming a reinforced slope system. The length of the stringer should be expanded by 1.4 times, since the frame beam was completely covered the 45 slope. The cross-sectional size of the frame beam was 0.6 m × 0.6 m. The rebars were 4Φ22 in the beam and 6Φ22 in the stringer. In order to reduce the effect of thermal expansion and contraction of concrete, the traditional frame beam was a unit of 9 m, with a slit between each frame unit.
3.2 Numerical model and material parameters
Figure 8b shown that the full scale model of the cutting slope reinforcement with anchoring frame beam and sheet pile wall was established. In the simulation, except for the red clay using the Mohr-Coulomb elastic model, the rest of the materials, such as the frame beam, the pile, strongly weathered dolomite, and weakly weathered dolomite, were all adopted with the clastic model. According to the data of engineering geology in Jiang-Yu expressway and the soil test results, the main physical and mechanical parameters of the soil slope, the rock mass, the pile and the frame beam were shown in Table 1. It was well known to the structural calculations, the position of the boundary conditions imposed by the finite element software was critical. According to Saint-Venant’s principle, it should be far away from the analysis site to avoid the influence of boundary condition on the results. The boundary of the three-dimensional model was determined from two aspects of thickness (x) and cross-section size (y × z). The displacement boundary at the bottom of the slope constrains displacement in three coordinate directions, the front and back sides of the slope constrains displacement in the x direction, and the left and right sides constrain displacement in the z direction, and the slope surface was a free surface. The slope grid used C3D10M (modified 10-node tetrahedron element), and the frame beams and anti-slide piles used C3D8R (8-node linear hexahedron element with reduced integral control). The size of the frame beam structural unit is 0.2m, and the size of the slope unit is 1 m. They directly contact and have friction, with no slippage.
The selection of rock and soil layering and physical and mechanical parameters in the calculation of slope stability is based on the geological survey report of the slope and the recommended values of physical and mechanical properties of rock groups in similar slope projects, as shown in Table 1. The hinge unit in the connector was used to simulate the hinge effect of the articulated precast frame structure. First, set the characteristics of the connector, then create a characteristic line between the two assemblies, and finally assign the defined connector characteristics to the corresponding characteristic line. And divide the connection unit. The comparison and analysis of the supporting effects of traditional frame beam and precast frame beam were taken as the research focus. Therefore, a concentrated force pointing in the slope at the end of the anchor cable (the center position where the frame beam and the anchor cable are connected) can be used to simplify the prestressed anchor cable. The slope model uses the self-weight stress as the initial stress, and after calculating to equilibrium under the action of the self-weight, the displacement is removed and the stress is retained. With this stage as the initial state, the strength reduction method was used to analyze the stability of the frame beam after strengthening the slope, and the traditional frame beam structure was compared with precast frame beam to verify the feasibility of precast frame beam.
Table 1. The parameters of numerical model.
3.3 Structure parameters of frame beam
In order to verify the support effect of the articulated precast frame beam with anchor cable, in the designed precast frame beam with anchor cable system, only the following changes were made in the frame beam part: the frame beam is divided into a number of cross beams and a beam, as shown in Figure 9. The flexible structure was formed by assembling the hinges. The hinges are symmetrically arranged on each transverse and longitudinal beam near the center of the anchor cable, the middle section was a straight beam, the two ends were cross beams, and the distance between the transverse anchor cables was 3 m.
Figure 9. Dimensions of (a) traditional structure (T) and (b) articulated frame beam with anchor cable (H).
4 Results and analysis
According to the selected model size, the safety factors and the internal force of the frame beam of the two structural support slopes were analyzed. At the same time, the influence on the structural parameters of beam length and section size was compared.
4.1 Internal force of the frame beam
The internal force of the frame beam of the traditional frame beam and the articulated frame beam are show in Figure 10. The solid line is the moment and the shear force of traditional frame beam, and the dashed line is that of articulated frame beam. The results indicate that the moments of both traditional and new are symmetrically arranged at x = 4.5 along the length direction. However, their distributions are significantly different. The moment of articulated frame beam is below the traditional structure. For the 1# beam, the moment of the traditional structure is 68.9 kN·m (positive value means that the beam near the soil side is under tension) at the position of 1.5 m along the length and intersects with the 1# stringer where the anchoring force is applied, while the moment of the articulated frame beam is 46.8 kN·m, and decreases 32%. Similarly, at the position of 4.5 m which intersects with the 2# stringer, the moment of the traditional frame beam is 65.8 kN·m, and the moment of the articulated frame beam is 27.3 kN·m, a drop of 58%. And at the position of 7.5 m which intersects with the 3# stringer, the moment is consistent with 1.5m, and will not be repeated. And at the position of the mid-span (X = 3 m), the moment of the traditional structure is 0 kN·m, and the new structure is −46.7 kN·m (negative value means that the beam away from the soil side is under tension).
Figure 10. Moments and shear force of beam: (a) 1#beam; (b) 2#beam; (c) 3#beam.
In Figure 10b, at the two points of 1.5m, 3m, and 4.5 m on the#2 beam, the moments of the traditional structure are 70.1 kN·m and 68.0 kN·m, and that of the new structure are 39.1 kN·m and 27.6 kN·m, respectively. The new structure has a reduction of 33% and 59%. The internal force distribution of the 3# beam is shown in Figure 10c. It can be found that the moments of 1.5m and 4.5 m for the traditional structure are 94.1 kN·m and 82.2 kN·m, and the new structure are 66.6 kN·m and 32.0 kN·m, the decreases are 29% and 61%. The moment reaches the peak value of the positive moment at the position where the anchor cable acts, and the moment that the anchor cable end bears at both ends is slightly larger than the position of the middle anchor cable.
Compared with the traditional structure, it found that the moment of the beam in the new structure at the intersection of 1# and 2# stringers are reduced by 29%–32% and 58%–61%, respectively, and the shear force is basically the same, which means that the moment value of the new structure is small and the internal force distribution of the structure is more reasonable. Due to its unique shape, the cantilever section without the longitudinal beam above shares some of the internal force, so the moment and shear force are greater than those of the 1# beam and the 2# beam. Comparing the moment and shear force values of the 1#beam, 2#beam, and 3# beam in Figure 10, it is found that the positive moment value of the precast structure is smaller than that of the traditional frame beam, and the negative moment value is larger than that of the traditional frame beam. The shear forces at each point on the beam are slightly smaller than those of traditional frame beams, and the internal force distribution of the structure is more reasonable. At the hinge position (x = 2m, 4m, 5m, and 7 m), the moment of the articulated precast frame beam is zero, no moment is transmitted, and the moment and shear force of the cantilever end at both ends are also zero. The hinges can be rotated within a small range to release a certain amount of soil. The deformation of the body is more reasonable than that of the traditional frame beam.
4.2 Influence of hinge position and structure size on factor of safety and internal force of frame beam
Through the previous section, it was found that the addition of hinges in the beam effectively changes the distribution of moments in the beam. However, this is only the result of a length of 2m, width and height of 0.6 m. And the location of the hinge and the cross-sectional size of the frame beam structure will affect the distribution of the moment of the structure. However, the sum of the positive and negative moments at this time is less than zero, indicating that the current hinge position and cross-sectional size are not the optimal position and optimal structural size. This study will be discuss and analyze influence of length (L), breadth (b) and height (h) of the beam with hinge on the factor of safety and internal force of frame beam, as shown in Figure 11. The safety factor and internal force under 5 beam lengths of 2.0m and 1.8 m were carried out to find the best beam length working condition. And in this case, combined with the minimum cross-sectional size of the beam in the construction of 0.3 m × 0.3 m (b × h), a total of 16 calculation simulations are combined for the breadth (b) and height (h) of the cross-beam structure from 0.6m, 0.5m, 0.4m and 0.3 m.
Figure 11. Schematic diagram of 2# beam with length and cross-sectional size.
4.2.1 Influence of length
Figure 12 shows factor of safety and the moment distribution and the positive and negative values of the 2# beam under the different lengths. The safety factors for different beam lengths are 1.211 to 1.243, shown in Figure 12a, which are all greater than 1.20, and the amplitude of change is very small. It shows that the change of hinge position has little effect on slope stability. From Figure 12b, it can be found that the moment distribution in the beam is still symmetrically distributed with x = 4.5 m due to the symmetry of the frame beam. At the same time, it can be seen that the distribution of moments with different lengths is significantly different. As the length becomes shorter, the positions of the two hinges are closer and farther away from the point of action of the anchor cable, its moment curve is moving up. Both the positive moment at the anchor point and the negative moment in the span become larger. As the beam length decreases from 2.0 m to 1.2m, the moment values at three typical points in the beam (anchor cable (1.5m, 4.5 m) and mid-span (3.0 m)) range from 39 kN·m to 70 kN·m, 28 kN·m to 63 kN·m and −47 kN·m to −12 kN·m. The typical moment values of three points are drawn as shown in Figure 12C. The moment of the side beam (x = 1.5 m) is greater than that of the middle beam (x = 4.5 m), and all are inversely related to the beam length. The absolute value of is positively correlated with the beam length. When the beam length is l = 1.6m, the moments of x = 4.5 m and x = 3.0 are basically 38 kN·m, and the moment of x = 1.5 m is 61 kN·m. It shows that the optimal beam length (L) between the two hinges in this study is 1.6 m. At this time, the moment used for design is reduced by 50% compared with the traditional structure, effectively reducing the amount of tensile steel.
Figure 12. FOS of slope (a) and moments of the different lengths: (a) FOS; (b) moment distribution and (c) three typical points in the 2# beam.
4.2.2 Influence of breadth and height
Figure 13 shows FOS and the moment of the typical point at x = 3.0 of the 2# beam in the different breadths and heights. The safety factors for different breadths and heights are 1.153 to 1.233, as shown in Figure 13a. As the breadth or height of the beam becomes smaller, the FOS gradually decreases, which is more disadvantageous to the supported slope. In some cases, the FOS is less than 1.20, indicating that these conditions have not met the requirements of slope support. Except for b × h is 0.4 m × 0.4 m, the FOS with a width and height not less than 0.4 m is greater than the value required by the specification. The minimum cross-sectional size (b × h) is 0.4 m × 0.5 m or 0.5 m × 0.4 m. It can be seen from Figure 13b that the moments at x = 3.0 of the 2# beam with different breadth or height is significantly different. As the breadth or height decreases from 0.6 m to 0.3m, the moment values range from 39.4 kN·m to 37.1 kN·m. As the breadth or height becomes shorter, its moment is getting smaller. For example, when the breadth is 0.6m, as the height decreases from 0.6 m to 0.3m, the moment value gradually decreases from 39.4 kN·m to 37.9 kN·m, and the reduction in moment is 4% less than the 50% decrease in height. However, the amount of reinforcement needs to be increased by 40%, because the height is reduced and the bending performance is reduced. Similarly, when the height is 0.6m, as the breadth decreases from 0.6 m to 0.3m, the moment value gradually decreases from 39.4 kN·m to 38.9 kN·m, and the reduction in moment is only 2% less than the 50% decrease in breadth and the amount of reinforcing steel can also be reduced by 2%. Therefore, when the interface area is the same, the structural interface size form with a height greater than the width requires less reinforcement and is more economical. In summary, in this study, the recommended structural size for b × h is 0.4 m × 0.5 m.
Figure 13. FOS of slope (a) and moments of the different breadths and heights: (a) FOS; (b) moment of x = 3.0 in the 2# beam.
In conclusion, the slope stability of the articulated frame beams made of lightweight concrete is comparable to that of traditional beams, and the bending moment values are smaller, allowing for the selection of smaller structural dimensions. The content of this study can serve as a reference for slope support projects in other fields, such as geotechnical engineering, hydraulic and hydropower engineering, ocean engineering, etc.
5 Conclusion
The main findings of this study were summarized as follows.
1. The foam concrete sample prepared by the hydrogen peroxide solution addition method is better.
2. The strength of 2 mm steel pipe-based steel foam concrete with a 5% hydrogen peroxide content is equivalent to that of C30 concrete, and its density is approximately 30% lower.
3. Compared with the internal force of the traditional structure, the shear force distribution of the new structure is basically the same, the moment is smaller due to the effect of the hinge. In this study, the optimal hinge spacing length is 1.6m, and the minimum size (breadth × height) of the beam is 0.4 m × 0.5 m.
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
FL: Methodology, Conceptualization, Resources, Funding acquisition, Writing – review and editing, Writing – original draft. WL: Validation, Data curation, Writing – original draft. YH: Software, Writing – original draft, Visualization. TL: Writing – review and editing, Funding acquisition, Methodology. TX: Writing – review and editing, Formal Analysis, Validation.
Funding
The author(s) declared that financial support was received for this work and/or its publication. The National Natural Science Foundation of China (NO.52208436), and it was supported by the Open Fund of Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education (Changsha University of Science & Technology) (NO. kfj210402), National Key Laboratory of Green and Long-Life Road Engineering in Extreme Environment (Changsha) (Changsha University of Science & Technology) (NO. kfj230106).
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Keywords: anchoring slope, foam concrete, frame beam, laboratory test, steel tube
Citation: Li F, Lian W, Hu Y, Lan T and Xiao T (2026) Laboratory test and simulation for anchoring slope of steel tube foam concrete frame beam. Front. Mater. 13:1747622. doi: 10.3389/fmats.2026.1747622
Received: 16 November 2025; Accepted: 02 January 2026;
Published: 15 January 2026.
Edited by:
Miao Yu, Chongqing Jiaotong University, ChinaReviewed by:
Salvatore Verre, University of eCampus, ItalyXiaoning Zhang, Chongqing University, China
Copyright © 2026 Li, Lian, Hu, Lan and Xiao. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Tian Lan, bGFudGlhbkBobnVzdC5lZHUuY24=