Experimental and numerical investigation of the assembled monolithic spherical-shaped reinforced concrete ribbed folded plate structure

Abstract

To address the insufficient stiffness of the V-shaped reinforced concrete folded plate structure and its construction process causing environmental pollution, a novel assembled monolithic spherical-shaped reinforced concrete ribbed folded plate structure (AMRRFS) was proposed. The advantages of AMRRFS are that its construction process is environmentally friendly while it also exhibits great stability and rigidity. Therefore, an experimental and numerical investigation were conducted on the AMRRFS to investigate its mechanical properties. In addition, the parametric analysis of the AMRRFS was conducted, and some design recommendations were proposed. Under the design load, the experimental findings revealed that AMRRFS possessed excellent mechanical properties. During the overloading phase, the interface between the in situ casting area and the prefabrication area was severely damaged, leading to the loss of the structure’s ability to bear loads. The outcomes from the finite element simulations of AMRRFS closely mirrored the results of the experimental investigation. Based on the parametric analysis, it was recommended that the height of the AMRRFS, the height of the ribs, and the height of the secondary ridge beams shall be 1/7–1/5, 1/65–1/50, and 1/34–1/30 of the span, and that the minimum reinforcing ratio for all types of plates shall exceed 1.0%.

Graphical abstract

Keywords

experimental analysis / numerical analysis / parametric analysis / prefabricated structure / concrete folded plate structure

1 Introduction

Inspired by sea shells, engineers developed curved roofs that blend strength with natural beauty [1–5]. However, the complexity of their design and maintenance procedures leads to high costs [6–9]. As a result, scholars proposed the folded plate structure (FPS), featuring a beautiful structural appearance, simpler structural design, and reducing the adverse effects on the structure caused by initial defects [10,11].

As the most common FPS, the V-shaped FPS has attracted much academic attention owing to its widespread applicability in the resent five years. Basu et al. [12] employed the finite element (FE) methodology to conducte a study on the natural frequency characteristics of functionally graded FPS. Their research revealed that both the thickness and the boundary constraints influenced its natural frequency characteristics. Javani et al. [13] investigated the influence of various factors, including crank angle, plate dimensions, graded patterns, and boundary constraints, on the natural frequency responses of V-shaped FPS using the generalized differential quadrature element method. Lu et al. [14] improved the joints of V-shaped timber FPS using self-tapping screws. Their study revealed that the new joints significantly contributed to an enhancement in the overall performance of the V-shaped timber FPS. Katlav et al. [15,16] studied the performance characteristics of V-shaped FPS featuring varying thicknesses, crafted from self-compacting concrete with or without the inclusion of steel fibers. They found that the thickness of the specimens mixed with hybrid steel fibers could be effectively reduced and their bending stiffness were higher.

However, all of the above studies focused on the V-shaped non-ribbed FPS. Compared with the V-shaped non-ribbed FPS, the inclusion of ribs in the V-shaped FPS resulted in a notable enhancement of both its bearing capacity and stiffness, and ribs could be made into various forms to satisfy the different needs of human beings for construction. Therefore, Wang et al. [17] investigated the dynamic behaviors of the V-shaped ribbed FPS, and found that it exhibited good dynamic characteristics. Nguyen-Minh et al. [18] investigated both the static and dynamic behaviors of the V-shaped stiffened FPS through the application of the cell-based smoothed discrete shear gap method, and suggested that this method provided a high accuracy in studying this structure. As early as 2007, the V-shaped ribbed FPS was adopted in practical projects in China, including the training building and the multi-function auditorium in the vocational school in Wuchuan County [17]. These projects yielded significant economic and social benefits. Compared with the V-shaped ribbed FPS, the spherical-shaped ribbed FPS not only exhibited a more aesthetically pleasing structural form but also demonstrated greater stiffness and stability under both dynamic and static loads. However, the complexity of manufacturing limited the application of spherical-shaped ribbed FPS in practical engineering. To address this issue, the assembled monolithic spherical-shaped reinforced concrete ribbed folded plate structure (AMRRFS) was proposed. This structure has higher strength and stability compared with the V-shaped FPS. Moreover, this structure possesses the advantages associated with prefabricated structures, such as simple construction [19–21], high construction quality [22,23], cost savings [24,25], and little environmental pollution during construction [26], etc. Despite the numerous advantages of AMRRFS, there are few experimental studies on AMRRFS, resulting in a lack of experimental data validation and theoretical support. These data are essential for validating complex numerical models and offer guidance for designing and constructing.

Therefore, the focal points of this paper’s investigation encompassed. 1) The scaled-down model test for AMRRFS was conducted. 2) The crack pattern, mode of failure, displacement variation, and strain distribution of AMRRFS were observed and studied. 3) A FE model for AMRRFS was established, and a comparison was drawn between the outcomes of the FE analysis and the empirical results obtained from experiments. 4) The height-span ratio, height of the ribs, minimum reinforcing ratio, and plate thickness were considered for conducting parametric analysis on AMRRFS. 5) Several design recommendations of AMRRFS were put forward.

2 Experimental investigation

2.1 Specimen design

To accurately perform the scaled-down model test, the model needs to be able to simulate the response and performance of the prototype structure under loads, and the model should be designed for ease of manufacture and assembly. Accounting for these considerations, Tab.1 lists the similarity ratio existing between the full-scale structure and the scaled-down model.

Tab.1 Similarity analysis

Sample	Parameter	Symbol	Calculation method	Similarity ratio
Material characteristics	elastic modulus	SE	SE	1
	strain	Se	Se	1
	stress	Ss	Ss = SESe	1
	density	Sp	Sp = SE/Sl	6
Geometric characteristics	length	Sl	Sl	1/6
	area	SA	SA = Sl2	1/36
	moment of inertia	SX	SX = Sl4	1/1296
	flexural rigidity	SG	SG = Sl3	1/216
Load characteristics	concentrated load	SF	SF = SESl2	1/36
	surface load	Sq	Sq = SE	1

The AMRRFS was designed with a span and a total height of 3000 and 500 mm, respectively. The AMRRFS was composed of ribbed plates, main ridge beams, secondary ridge beams, ring beams, and edge beams. Considering the limitations of the total load in the laboratory, the specimen employed galvanized iron wires as both its longitudinal reinforcing bars and stirrups. A single layer of galvanized iron wire mesh was arranged in the plates. Several reinforcing bars were inserted at the design position to prevent iron blocks from sliding during the experiment. The specimen, with its detailed geometric dimensions illustrated in Fig.1, was symmetrically arranged atop 12 columns, each having a cross-sectional dimension of 200 mm × 200 mm, and was considered to be simply supported.

Fig.1 Detailed geometric dimensions (unit: mm): (a) plan view; (b) partial details.

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2.2 Manufacturing of the assembled monolithic spherical-shaped reinforced concrete ribbed folded plate structure

The manufacturing process of the AMRRFS is represented in Fig.2. During the prefabrication stage, ribbed plates of various sizes were produced using different stereotyped formworks, and the longitudinal reinforcing bars of the ribbed plates were prepared with sufficient length. During the transportation stage, the ribbed plates were transported to the construction site, and they were accurately placed in their designated locations. During the on-site construction stage, the reinforcing bars in the in situ casting areas were bound, and the reserved longitudinal reinforcing bars were anchored into the in situ casting areas. Subsequently, the in situ casting areas were filled with higher-strength concrete. During the curing stage, AMRRFS was cured for more than 28 d.

Fig.2 Manufacturing process: (a) manufacturing of prefabrication areas; (b) installation of plates; (c) pouring concrete in the in situ casting area; (d) manufacturing completed.

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2.3 Distribution of main measuring points

For observing the displacement of the AMRRFS, a series of linear variable displacement transducers (LVDTs) were affixed to the specimen, as illustrated in Fig.3(a). Furthermore, to measure various strain parameters, strain gauges were attached not only to the concrete but also to the reinforcing bars. Specifically, these gauges were intended to gauge the concrete strain, the reinforcing bar strain at the mid-span of each component, and the reinforcing bar strain at the interface between the in situ casting area and the prefabrication area, as illustrated in Fig.3(b)–Fig.3(d). All main measuring points were only arranged within one sixth of the entire structure.

Fig.3 Distribution of main measuring points: (a) LVDTs; (b) concrete strain gauge; (c) bottom reinforcing bars strain gauge; (d) top reinforcing bars strain gauge.

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2.4 Material characteristics

Concrete in prefabrication areas used Portland cement of grade 32.5, while the concrete in in situ casting areas used Portland cement of grade 42.5. Both the in situ casting and prefabrication areas utilized concrete made with lake sand and less than 10 mm macadam. Tab.2 and Tab.3 present the material characteristics of concrete and galvanized iron wire, respectively.

Tab.2 Mechanical properties of concrete

Type	Cement:sand:macadam:water	Elastic modulus, Ec (MPa)	Cubic compressive strength, fc (MPa)
Prefabrication areas	1:1.59:3.39:0.47	2.52 × 104	23.2
In situ casting areas	1:1.23:2.49:0.44	3.12 × 104	35.6

Tab.3 Mechanical properties of galvanized iron wire

Type	Diameter, D (mm)	Elastic modulus, Es (MPa)	Yield stress, fy (MPa)	Ultimate stress, fs,u (MPa)
8#	4.1	1.9 × 105	255.4	347.2
10#	3.5	1.9 × 105	250.3	342.1
12#	2.8	1.9 × 105	269.3	350.3

2.5 Loading method

The vertical uniform loading in this experiment was simulated using iron blocks and sandbags, as illustrated in Fig.4(a). The test consisted of three stages. 1) To achieve the similarity ratio between the density of the specimen and the density of the prototype structure, the specimen needs to be applied 5 times its own weight. The application of the loads was carried out in two steps, with each step set at 1.0 kN/m². 2) The loads were applied in five steps to reach the design load, with each step set at 1.0 kN/m². After completion, these loads were maintained for over 12 h. 3) The load subsequently increased by 0.5 kN/m² for each step until the total load was 10.5 kN/m², at which point the specimen sustained significant damage, prompting the termination of the test. The entire loading process was illustrated in Fig.4(b).

Fig.4 Loading process: (a) loading site; (b) loading diagram.

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3 Experimental results

3.1 Crack distribution and failure mode

Fig.5(a) illustrates the initial cracking distribution. At a load level of 7.0 kN/m², the top surface of the specimen did not observed any cracks. However, cracks were noticed on the secondary ridge beam and the ribs of the Type III plate on the bottom surface of the specimen.

Fig.5 Crack distribution: (a) initial cracking; (b) cracking of the interfaces between the prefabrication area and the in situ casting area; (c) yield of reinforcing bar; (d) structural failure.

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Fig.5(b) illustrates the cracking distribution when the interface between the in situ casting area and the prefabrication area cracked. At a load level of 8.0 kN/m², it was observed that cracks appeared at the interface between the in situ casting area and the prefabrication area. These cracks extended along the interface between the in situ casting area and the prefabrication area.

Fig.5(c) illustrates the cracking distribution when the reinforcing bar yielded. At a load level of 9.5 kN/m², the already cracked interfaces between the in situ casting area and the prefabrication area gradually extended toward the upper part of the specimen along the interface between the in situ casting area and the prefabrication area.

Fig.5(d) illustrates the cracking distribution during structural failure. At a load level of 10.5 kN/m², cracks distribution was symmetric, and the interfaces between the in situ casting area and the prefabrication area were severely cracked.

Based on the above phenomena, the primary cause of the loss of load-bearing capacity of the AMRRFS appeared to be severe damage to the interfaces between the in situ casting area and the prefabrication area.

3.2 Structural displacement analysis

The load–displacement curves of the AMRRFS were presented in Fig.6(a). During the initial loading stage, the specimen exhibited elastic behavior, and the relationship between load and displacement was linear. At the design load, the maximum displacement at D2 recorded was 0.95 mm. The value of displacement to span measured at D2 was 1/3157, falling below the limit, as specified in Ref. [27]. This suggested that this structure possessed a high ability to resist deformation. As the load increased incrementally, the rising number of cracks resulted in a reduction of the structure’s stiffness, leading to an acceleration in the development of displacement. As the reinforcing bars yielded, there was a significant acceleration in the development of displacement. When AMRRFS could no longer bear the load, the maximum displacement at D2 recorded was 9.08 mm.

Fig.6 Displacement: (a) load–displacement curves; (b) D1–D6; (c) D4, D7–D10. (Note: Pu is the ultimate load).

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Fig.6(b) and Fig.6(c) illustrate the variation of displacements along the span. It was noted that bending deformation constituted the main form of deformation in the AMRRFS. The peak deformation was recorded in the central region of the Type II plate. The Type II plate exhibited a bowl-shaped deformation pattern, which was similar to the ribbed flat plate subjected to uniform loading.

3.3 Strain analysis

The load–strain curves of key points on AMRRFS were presented in Fig.7. Positive values indicated tensile strains, whereas negative values signified compressive strains. As the load increased during the initial loading phase, all load–strain curves displayed a linear pattern. Meanwhile, the slow development of the strain at each measuring point indicated that the structure was still within its elastic stage. However, as cracks began to form and propagate, the structural stiffness diminished, leading to an accelerated rate of strain development. This indicated that some of the measured points entered the elastic–plastic stage. When the load reached 9.5 kN/m², the yield strain was reached at the measuring point SZP6, which was located at the interface between the in situ casting area and the prefabrication area, indicating that this structure entered the plastic phase. At a load level of 10.5 kN/m², the measuring point SSR6, SZP6, and SSM2 reached yield strains, and the structure is no longer capable of bearing the load.

Fig.7 Load–strain curve: (a) concrete in the prefabrication area; (b) concrete in the in situ casting area; (c) bottom reinforcing bars of ribs in the prefabrication area; (d) bottom reinforcing bars in the in situ casting area; (e) top reinforcing bars of ribs in the prefabrication area.

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4 Finite element model

4.1 Element type, surface contact and boundary condition

The test was simulated using the general FE software ABAQUS [28,29], and the FE model is shown in Fig.8. Within this model, all concrete components were constructed with the C3D8R, and all reinforcing bars were constructed with the T3D2. The analysis did not account for the bond-slip. Therefore, the reinforcment bars were embedded in the concrete.

Fig.8 FE model.

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The surface-to-surface contact was used to simulate the interface interaction between in situ casting and prefabrication areas. Primary surfaces were the in situ casting area, while subordinate surfaces were the prefabrication area. A hybrid model that combines cohesive behavior and Coulomb friction was adopted. During the initial stage of this model, the Coulomb friction was inactive, and the interaction between the in situ casting area and the prefabrication area depended only on the cohesive behavior. As the cohesive behavior decreased, the Coulomb friction became active and increased proportionally. During this stage, the interaction between the in situ casting area and the prefabrication area was provided by the cohesive behavior and Coulomb friction until the cohesive behavior was fully degraded. After this, only the Coulomb friction model operated. The normal and tangential behaviors between surfaces used ‘hard contact’ and ‘penalty friction’ with a friction coefficient of 0.4, respectively.

The stiffness ‘$K_{\text{ss}}$’ and ‘$K_{\text{tt}}$’ were calculated as follows [30]:

(1)$\begin{array}{rl}{K}_{\text{ss}}=& K_{\text{tt}}=0.0726+2.06\frac{{\sigma }_0}{f_{\text{ca}}}-2.47{\left(\frac{{\sigma }_0}{f_{\text{ca}}}\right)}^2\\ & +0.0055H+0.092{\left(\frac{{\sigma }_0}{f_{\text{ca}}}\right)}^{2}H,\end{array}$

(2)$f_{\text{ca}}=\frac{\left(f_{\text{c,o}}+f_{\text{c,n}}\right)}{2}\times 0.67,$

where ${\sigma }_0$, H, $f_{\text{c,o}}$, and $f_{\text{c,n}}$ are the compressive stress, surface roughness (the value was taken according to Ref. [31]), compressive strength of concrete cubes in prefabrication areas, and compressive strength of concrete cubes in in situ casting areas, respectively. The stiffness ‘$K_{\text{nn}}$’ was taken as 10⁵. The calculation of ‘Normal Only’ could be as follows [30]:

(3)$f_{\text{n}}=(0.00381H+0.004361)f_{\text{c,m}},$

where $f_{\text{c,m}}$ is the average compressive strength of the cubic concrete of the in situ casting area and the prefabrication area.

The values of ‘Shear-1 Only’ and ‘Shear-2 Only’ both could be calculated by the following equation [32]:

(4)$f_{\text{s1}}=f_{\text{s2}}=0.06956\xi f_{\text{c,m}}+\gamma \mu {\rho }_{\text{e}}{f}_{\text{y}},$

where $\xi$, $\gamma$, $\mu$, and ${\rho }_{\text{e}}$ are the influence of interface agent type on the bonding surface, the coefficient of influence of pin bolt action on shear bearing capacity, the friction coefficient at the connection interface, and the minimum reinforcing ratio of embedded reinforcing bars at the connection interface. The value of ‘Total/Plastic Displacement’ was taken from the test results, and the value of viscosity coefficient was taken as 0.05. A total of 12 columns were designated as rigid bodies, with constraints imposed on their bottom surfaces to prevent any displacement or rotation. While, their top surfaces were designated as simple supports.

4.2 Material models

The concrete damaged plasticity (CDP) model was used to simulate the material behavior of concrete. This model is suitable for simulating the material behavior of concrete in all types of structures under monotonic, cyclic, and/or dynamic loading [33]. The model treats concrete as an isotropic continuous medium, uses elastic damage and tensile-compressive plasticity to characterize the nonlinear behavior of concrete, and considers the damaging effect of concrete to account for the stiffness degradation due to plastic strain. For the uniaxial compressive constitutive relationship of concrete, the Hognestad model [34] was chosen due to its wide applicability, and its calculation method was detailed in Eq. (5), as shown in Fig.9(a). The plasticity parameters of flow potential eccentricity, dilation angels, ratio of the biaxial compressive strength to the uniaxial compressive strength, K, and viscosity coefficient in the CDP model were 0.1, 30°, 1.16, 0.6667, and 0.0005, respectively.

Fig.9 Constitutive model: (a) concrete; (b) reinforcing bars.

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(5)$\sigma =\{\begin{array}{ll}\left(2\left(\epsilon /{\epsilon }_{\text{c}}\right)-{\left(\epsilon /{\epsilon }_{\text{c}}\right)}^2\right)f_{\text{c}}^{\mathrm{\prime }},& 0⩽\epsilon <{\epsilon }_{\text{c}},\\ \left(1-0.15\left(\epsilon -{\epsilon }_{\text{c}}\right)/\left({\epsilon }_{\text{u}}-{\epsilon }_{\text{c}}\right)\right)f_{\text{c}}^{\mathrm{\prime }},& {\epsilon }_{\text{c}}⩽\epsilon ⩽{\epsilon }_{\text{u}},\end{array}$

where ε, $f_{\text{c}}^{\mathrm{\prime }}$, ${\epsilon }_{\text{c}}$, ${\epsilon }_{\text{u}}$, and $f_{\text{u}}$ are the strain of the concrete, the compressive strength of concrete cylinders, the strain corresponding to the peak compressive stress, the strain corresponding to the ultimate compressive stress, and the ultimate compressive stress. ${\epsilon }_{\text{c}}$, ${\epsilon }_{\text{u}}$, and $f_{\text{u}}$ took values of 0.002, 0.0038 and 0.85$f_{\text{c}}^{\mathrm{\prime }}$. The concrete tensile behavior was modeled using the constitutive relationship proposed in Ref. [27]. The constitutive relationship could be calculated by Eq. (6). The ascending linear stage represents the linear behavior of concrete before cracking, and the descending curved stage represents the nonlinear behavior after concrete cracking, as shown in Fig.9(a).

(6)$\sigma =\{\begin{array}{ll}\left(1.2\epsilon /{\epsilon }_{\text{t}}-0.26{\left(\epsilon /{\epsilon }_{\text{t}}\right)}^6\right)f_{\text{t}},& 0⩽\epsilon <{\epsilon }_{\text{t}},\\ {\displaystyle \frac{\epsilon /{\epsilon }_{\text{t}}}{{\alpha }_{\text{t}}{\left(\epsilon /{\epsilon }_{\text{t}}-1\right)}^{1.7}+\epsilon /{\epsilon }_{\text{t}}}}{f}_{\text{t}},& {\epsilon }_{\text{t}}⩽\epsilon ⩽{\epsilon }_{\text{u}},\end{array}$

where ${\epsilon }_{\text{t}}$ and $f_{\text{t}}$ are the strain corresponding to the tensile strength of concrete and the tensile strength of concrete. ${\alpha }_{\text{t}}$ represents the parameters that define the descending trend of the curve under tensile conditions, and takes 0.312${f_{\text{t}}}^2$.

The elastic modulus degradation of concrete, stemming from damage, can be described and analyzed through the application of compressive and tensile plasticity damage factors. Based on the energy equivalence principle, the tensile plastic damage factor $d_{\mathrm{t}}$ and compressive plastic damage factor $d_{\mathrm{c}}$ for concrete can be caculated as follows:

(7)$d_{\text{t}}=\{\begin{array}{ll}0,& x_{\text{t}}⩽1,\\ 1-\sqrt{{\displaystyle \frac{1}{{\alpha }_{\text{t}}{\left(\epsilon /{\epsilon }_{\text{t}}-1\right)}^{1.7}+\epsilon /{\epsilon }_{\text{t}}}}},& x_{\text{t}}>1,\end{array}$

(8)$d_{\text{c}}=\{\begin{array}{ll}0,& x_{\text{c}}⩽0.65,\\ 1-\sqrt{{\displaystyle \frac{E_0{\epsilon }_{\text{c}}}{f_{\text{c}}}}\left({\alpha }_{\text{a}}+\left(3-2{\alpha }_{\text{a}}\right)\left(\epsilon /{\epsilon }_{\text{c}}\right)+\left({\alpha }_{\text{a}}-2\right){\left(\epsilon /{\epsilon }_{\text{c}}\right)}^2\right)},& 0.65<x_{\text{c}}⩽1,\\ 1-\sqrt{{\displaystyle \frac{E_0{\epsilon }_{\text{c}}}{f_{\text{c}}^{\mathrm{\prime }}}}{\displaystyle \frac{1}{{\alpha }_{\text{d}}{\left(\epsilon /{\epsilon }_{\text{c}}-1\right)}^2+\epsilon /{\epsilon }_{\text{c}}}}},& x_{\text{c}}>1,\end{array}$

where ${\alpha }_{\text{t}}$, ${\alpha }_{\text{a}}$, and ${\alpha }_{\text{d}}$ are the descending parameter of the uniaxial tensile stress–strain curve, the ascending parameters of the uniaxial compressive stress–strain curve, and the descending parameters of the uniaxial compressive stress–strain curve.

Considering the elastic and hardening characteristics of reinforcing bars, a bilinear model was adopted to simulate the material behavior of reinforcing bar, as shown in Fig.9(b). This model not only encompasses the elastic and hardening stages of the reinforcing bar but also accurately simulates its nonlinear behavior during loading, while maintaining relatively low computational costs. The constitutive model of reinforcing bars could be expressed as follows:

(9)${\sigma }_{\text{s}}=\{\begin{array}{ll}{E}_{\text{s}}{\epsilon }_{\text{s}},& \epsilon ⩽{\epsilon }_{\text{y}},\\ f_{\text{y}}+0.01E_{\text{s}}\left({\epsilon }_{\text{s}}-{\epsilon }_{\text{y}}\right),& \epsilon >{\epsilon }_{\text{y}},\end{array}$

where ${\epsilon }_{\text{s}}$, ${\epsilon }_{\text{y}}$, and ${\epsilon }_{\text{s,u}}$ are the strain of the reinforcing bar, the yield strain of the reinforcing bar, and the ultimate strain of the reinforcing bar.

4.3 Effect of different mesh sizes

The mesh size affects the accuracy and convergence of FE results [35]. Therefore, it is necessary to conduct a mesh sensitivity analysis for FE model. The mesh size of the model ranged from 30 to 80 mm. Fig.10 presents the comparative results of damage and curves for structures, showcasing the effects of varying mesh sizes. It could be observed that the mesh size within the range of 30 to 60 mm did not have a significant impact on the FE analysis results. But when the mesh size was greater than 60 mm, there were some deviations in the calculation results. Taking into account both computational efficiency and accuracy, the final mesh size was chosen to be 40 mm.

Fig.10 Effect of different mesh sizes: (a) concrete damage; (b) load–displacement curve.

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4.4 Finite element results

4.4.1 Crack development and failure mode

Fig.11(a) illustrates the initial cracking distribution. The experimental and FE results were basically consistent. However, the initial cracking load of the FE was 6.8 kN/m², which was lower compared to the test results. This difference was that FE models produced cracks once tensile stress exceeded uniaxial strength, while in experiments, initial cracks were imperceptible due to their minuteness.

Fig.11 Crack distribution: (a) initial cracking; (b) cracking of the interface between the in situ casting area and the prefabrication area; (c) yield of reinforcing bar; (d) structural failure.

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Fig.11(b) illustrates the cracking distribution when the interface between the in situ casting area and the prefabrication area cracked. In this case, the distribution of cracks and the load were similar in both the test and FE analysis.

Fig.11(c) presents the cracking distribution when the reinforcing bar yielded. The crack distributions in FE and test were consistent, but the yield load of the FE was 10.4 kN/m², which was higher than the experimental yield load of 9.5 kN/m². This disparity stemmed from construction errors in the test model, which affected the structural rigidity.

Fig.11(d) displays the cracking distribution during structure failure. The differences between the FE results and the test results lie in the different phenomena exhibited at the interfaces between the in situ casting area and the prefabricated area in the upper half of the specimen. During the test, there were no cracks at the upper half of the specimen, while the FE analysis revealed the presence of cracks at these locations. The reason was that considering the safety of the test instrument and loading personnel, the test could not be fully loaded as in the FE analysis, and the cracks did not crack as completely as in FE analysis.

4.4.2 Stress distribution of reinforcing bars

Fig.12 depicts the stress distribution of longitudinal reinforcing bars. The areas, where the stress of the reinforcing bar was relatively high, mainly occurred in the mid-section of the Type III plate and secondary ridge beam, the end of the main ridge beam and the secondary ridge beam, and the interface between the in situ casting area and the prefabrication area.

Fig.12 Stress distribution of reinforcing bars: (a) initial cracking; (b) cracking of the interface between the in situ casting area and the prefabrication area; (c) yield of reinforcing bar; (d) structural failure.

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4.4.3 Displacement analysis

The load–displacement curves are presented in Fig.13. The differences in load–displacement curves between FE and test were identified after the specimen being cracked. As the reinforcing bars reached the yield load, the differences between FE and test further increased. This was due to the initial defects in the experimental specimen that reduced its structural stiffness. Under the ultimate load, the maximum displacement at D2 recorded in the FE was 8.46 mm, which exceeded the experimental value by 7.3%. The reason was that the constitutive model used in the FE model might be conservative.

Fig.13 Load–displacement curve.

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4.4.4 Strain analysis

The load–strain curves of key points on AMRRFS are presented in Fig.14. The experimental data were in close agreement with the simulations that were obtained through FE analysis specifically during the elastic phase. After the specimen entered the nonlinear stage, there were some deviations between the experimental and FE stress–strain curves. The main reason for this was that the manufacturing defects in the experimental specimen that reduced its structural stiffness and the constitutive relationship of reinforcing bars was relatively conservative, which resulted in a higher stiffness in the FE model.

Fig.14 Stress−strain curve: (a) concrete in the prefabrication area; (b) concrete in the in situ casting area; (c) bottom reinforcing bars of ribs in the prefabrication area; (d) bottom reinforcing bars in the in situ casting area; (e) top reinforcing bars of ribs in the prefabrication area.

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5 Parametric analysis and design recommendations

To provide some design references for the future practical engineering of AMRRFS, this paper conducted the parametric analysis. Based on the above research, the research parameters included height-span ratio, height of ribs, minimum reinforcing ratio, and height of secondary ridge beams. The specific parameters for each model were provided in Tab.4.

Tab.4 The parameters analysis of model

Specimen	Height-span ratio	Height of the rib (mm)	Minimum reinforcing ratio (the Type I plate, the Type II plate and the Type III plate) (%)	Height of the secondary ridge beam (mm)
AMRRFS-K1	1/7	50	1.3, 1.0, and 1.0	95
AMRRFS-K2	1/8
AMRRFS-K3	1/5
AMRRFS-K4	1/4
AMRRFS-R1	1/6	55	1.3, 1.0, and 1.0	95
AMRRFS-R2		60
AMRRFS-R3		45
AMRRFS-R4		40
AMRRFS-S1	1/6	50	1.3, 1.3, and 1.3	95
AMRRFS-S2			1.6, 1.6, and 1.6
AMRRFS-S3			0.8, 0.8, and 0.8
AMRRFS-S4			0.6, 0.6, and 0.6
AMRRFS-F1	1/6	50	1.3, 1.0, and 1.0	100
AMRRFS-F2				105
AMRRFS-F3				90
AMRRFS-F4				85

As illustrated in Fig.15(a) and Fig.15(b), as the height-span ratio decreased, the load-bearing capacity of the structure gradually decreased, and the development of structural deformation and strain accelerated. This was primarily due to the decrease in the height-span ratio, which led to an increase in the span of each plate. As a result, the decrease in structural stiffness led to an increase in bending deformation, subsequently resulting in a diminished load-bearing capacity and accelerated structural deformation. Therefore, it was recommended that the height of the AMRRFS shall be between 1/7 and 1/5 of the span.

Fig.15 Effect of different parameters: (a) effect of height-span ratio on displacement; (b) effect of height-span ratio on strain; (c) effect of rib height on displacement; (d) effect of rib height on strain; (e) effect of minimum reinforcing ratio on displacement; (f) effect of minimum reinforcing ratio on strain; (g) effect of secondary ridge beam height on displacement; (h) effect of secondary ridge beam height on strain.

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As illustrated in Fig.15(c) and Fig.15(d), when the height of the ribs exceeded 60 mm, the further increase in the sectional height of the ribs did not result in significant alterations to the load-bearing capacity and development of displacement. This could be primarily attributed to the fact that the increase in the height of the ribs alone not only failed to effectively enhance the stiffness of the interface between the in situ casting area and the prefabrication area, but also increased the self-weight of the structure, resulting in negative effects on the overall structure. Conversely, when the height of the ribs was less than 45 mm, there was a relative reduction in the load-bearing capacity and an acceleration in the development of the structural deformation. This could be mainly due to the decrease in the overall stiffness of the structure as the height of the ribs decreased. Therefore, it was recommended that the height of the ribs shall be between 1/65 and 1/50 of the span.

As revealed in Fig.15(e) and Fig.15(f), it was observed that as the minimum reinforcing ratio increased, the load-bearing capacity of the structure gradually increased, and the development of displacements gradually slowed down. The main reason for this was that the increase in the minimum reinforcing ratio effectively increased the stiffness of the interface between the in situ casting area and the prefabrication area, thus slowing down its damage. Therefore, it was recommended that the optimal minimum reinforcing ratio of all types of plates shall exceed 1.0%.

As demonstrated in Fig.15(g) and Fig.15(h), when the height of the secondary ridge beams exceeded 100 mm, the load-bearing capacity and the development of displacements failed to exhibit notable variations. The primary reason for this was that the increased stiffness of the secondary ridge beams failed to effectively mitigate the destruction of the interface between the in situ casting area and the prefabrication area. However, when the height of the secondary ridge beams was reduced to below 90 mm, a decrease in load-bearing capacity was observed, accompanied by an acceleration of the displacement development. This was primarily attributed to the reduction in the overall stiffness of the structure caused by the decrease in the stiffness of the secondary ridge beams. In addition, the decreased stiffness of the secondary ridge beams led to the reduction in its constraint on the Type II plate, resulting in the acceleration in the deformation of the plate. Therefore, it was recommended that the height of the secondary ridge beams shall be between 1/34–1/30 of the span.

6 Conclusions

A scaled-down model experimentation of the AMRRFS was performed in this paper. The failure mode, displacement, and strain development of the AMRRFS were obtained. In addition, the FE model of the AMRRFS was established, and the FE results were compared with the experimental results. Finally, the effects of the height-span ratio, stiffness of the ribs, minimum reinforcing ratio, and stiffness of the secondary ridge beams on the AMRRFS were studied. The main conclusions are as follows.

1) The AMRRFS had a high load-bearing capacity. Under uniform vertical loads, the primary deformation exhibited by the AMRRFS was bending.

2) The main reason for the loss of load-bearing capacity of AMRRFS might be the severe damage to the interface between the in situ casting area and the prefabrication area.

3) The FE model utilized a hybrid model that combines the cohesive behavior model and the Coulomb friction model to simulate the interaction between the in situ casting area and the prefabrication area. This FE model could accurately predict the failure mode, displacement, and strain of the AMRRFS.

4) It was recommended that the height of the AMRRFS, the height of the ribs, and the height of the secondary ridge beams shall be 1/7–1/5, 1/65–1/50, and 1/34–1/30 of the span, and the minimum reinforcing ratio of all types of plates shall exceed 1.0%.

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Acknowledgements

The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51568012) and the Scientific Research Foundation of Guizhou University (GuiDaRenJiHeZi [2023]14), the Science Foundation for Youths of Education Commission of Guizhou Province (QianJiaoJi [2024]020), the Basic Research Project of Guizhou University (GuiDajiChu [2024]18).

Competing interests

The authors declare that they have no competing interests.

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