2.1. Modeling Method of Beam, Slab, and Column

The actual seismic damage [28], shaking table tests [29, 30], and numerical simulations show that the story failure mode of RC frame structure is one of the main reasons for the overall collapse of buildings due to the column hinge mechanism. The failure phenomenon is that the plastic hinge appears at the column end, and the bottom column is inclined, with the overall structure collapse. Figure 5 shows the final collapse state of the Xuankou middle school building; the seismic damage phenomenon includes column hinge, spalling concrete, and exposed reinforcement. For the abovementioned reasons, the element deletion (erosion) technique (EDT or EET) in finite element method (FEM) can be utilized to simulate RC frame beam, slab, and column. The principle of EDT is that the finite element will be removed from the mesh as soon as the strength vanishes at the end of strain softening process. A single time step of the algorithm is often divided into a series of phases (Figure 6). Figure 7 is the schematic diagram of column (beam) element deletion. However, the disadvantages of the EDT include high dependence on the mesh and nonconservation of mass and momentum in the deleted region, particularly in large-scale element models (Alsos) [31]; hence the finite element size should not be too large. EDT is utilized by many researchers to simulate the collapse of RC frame structures because it is applied in many finite element programs, such as LS-Dyna, Abaqus, and MSC.Marc, etc. Lu et al. [32] proposed a fiber-beam element model and multilayer shell model and utilized EDT in MSC.Marc software to simulate the collapse process of a ten-story RC frame structure. Elsanadedy et al. [33] and Qian et al. [34] analyzed the collapse of single-story and multistory RC frame structures by using EDT in LS-Dyna software, respectively.

2.2. Modeling Method of Infilled Wall

The investigation results of the Wenchuan M8.0 earthquake revealed that the strength of the mortar interface between infilled wall blocks is low, and cracks and failures occur on the contact surface of blocks [35, 36], as shown in Figure 8. For abovementioned reason, finite-discrete element method (F-DEM) can be utilized to simulate infilled wall of RC frame structure. Finite-discrete element method (F-DEM) is a novel computational approach developed by Munjiza et al. [37–40] and is a special branch of the DEM family. The structures are discretized using discrete elements in this approach, with a finite element mesh integrated into each discrete element. The process of fracture and breakage is observed in the finite element mesh, and each discrete element is deformable. F-DEM is widely utilized to predict the entire failure process of masonry structure from continuity to discontinuity. Smoljanovic et al. [41], Munjiza et al. [42], and Chen et al. [43] simulated the process behavior of masonry structure from the initial state to the final collapse by F-DEM. Actually, infilled wall cracks can also appear on blocks themselves; thus, the infilled wall can be meshed diagonally and the contact is considered, as shown in Figure 9. In addition, infilled wall is simplified as the homogeneous material when the strength of mortar is consistent with block, as shown in Figure 10.

2.3. Modeling Method of the Connection between Infilled Wall and Frame

Figure 14(a) is structural drawing between infilled wall and RC frame; the tie bars were set between infilled wall and RC frame column to ensure the in-plane and out-plane seismic performance of infilled wall according to the code for seismic design of building in China [52], and the infilled wall and RC frame beam is filled with mortar. Figure 14(b) is equivalent drawing between infilled wall and RC frame; the tie bars and mortar are simplified as springs with normal force and shear force. Tie-break contacts (in NRA) were set on the contact surface between infilled wall and frame column and beam to realize the simplified spring. In order to realize the combination of infilled wall and RC frame, the frame column and beam element mesh adopted a common node, node1 and node2 were merged together, and similarly, node3 and node4 were also merged together. The same coordinate nodes of infilled wall elements were not merged. Instead, springs were set between nodes. The tie bar between the frame column and the infilled wall was equivalent to the springs, which were set between node 3, node 5, and node 6, as shown in Figure 15.

In summary, according to the actual seismic damage characteristics, structural construction measures, and structural design code of RC frame structure in China, the combined FEM and F-DEM are proposed to simulate the seismic collapse of the multispan and multistory RC frame structure with infilled wall. The actual seismic damage building is taken as the numerical simulation and experimental model, and the accuracy of the simulation method is validated by the shaking table test.

3.1. Model Structure

The Xuankou middle school building is one of the most typical collapsed RC frame structures in the 2008 China Wenchuan M8.0 earthquake; its column grid layout is shown in Figure 16. This paper selects two-span × two-span and four-story from the Xuankou middle school building; the numerical model and shaking table test model were established at the scale of 1/5. The strength grade of concrete, reinforcement, block, and mortar are C10, Q235, Mu3.5, and Mb5, respectively. The architectural drawing (2nd, 3rd, and 4th floor plan and elevation are the same) and structural reinforcement drawing are shown in Figures 17 and 18.

3.2. Numerical Model

3.2.1. Numerical Model Scheme

Figure 19 shows the detailed modeling scheme of RC frame structure with infilled wall. Figure 19(a) shows the modeling scheme of the beam, slab, and column of the RC frame, which adopts the separate common node method. The bond-slip effect between reinforcement and concrete is not considered to improve the overall model’s calculation efficiency. Concrete is simulated using 8-node solid elements with reduced integration. To eliminate the hourglass problem, hourglass control type 5 and hourglass coefficient of 0.05 were used. Reinforcing bars were modeled using 2-node Hughes-Liu beam elements with 2 × 2 Gauss quadrature integration at the cross section. This beam formulation can simulate the behavior of axial force, biaxial bending, and finite transverse shear strains (Hallquist) [53]. Figure 19(b) depicts the modeling scheme of infilled wall, which adopts NRA. According to the code for the design of masonry structures in China [54], the mortar shear failure stress SFLS = k5 $$ = 0.234 × 106 pa, where k5 is 0.125, f2 is the average compressive strength of mortar (3.49 × 106 pa from test), and the tensile strength can be approximately regarded as equal to the shear strength, and the mortar normal failure stress NFLS = 0.234 × 106 pa, as shown in Figure 20. Brameshuber and Schmidt [55] investigated the mechanical properties of mortar via test, using a sliding friction coefficient of 0.7. Figure 19(c) shows the modeling scheme of the connection between infilled wall and RC frame. The failure limit of equivalent normal stress NFLS = N × σy/n = 2.41 × 106 Pa, where N is the number of tie bars (160), σy is the tensile strength of the tie bar (1.15 × 108 Pa from test), and n is the number of normal springs (7630). Similarly, according to the code for design of concrete structures in China [56], the shear strength of reinforced bar can be approximately regarded as equal to the tensile strength; thus the failure limit of equivalent shear stress SFLS is 2.41 × 106 Pa, as shown in Figure 21. The sliding friction coefficient between infilled wall and beam is 0.7.

3.2.2. Numerical Model Material

The Karagozian & Case (K&C) Model (Mat_72REL3 in LS-DYNA R10.0) (Malvar et al.) [57] is utilized to simulate concrete material, and the K&C model has three strength failure surfaces, which include the initial yield strength surface, the maximum strength surface, and the residual strength surface. The strengthening effect, tensile and compression damage effect, volume deformation damage effect, and strain rate effect are observed and can more accurately describe the mechanical properties of concrete materials under seismic load. The K&C model provides a simplified method that uses the algorithm proposed by Malvar et al. [58]. For the uniaxial compressive strength, only density and unit need to be identified; other material parameters can be automatically generated. However, the K&C model predicted that the softening stage of the uniaxial compression stress-strain curve would decreased rapidly, necessitating a modification of K&C model’s parameters. Hong et al. [59] modified the relationship between damage function, λ, and scale factor, μ, as shown in the following equation:

$$ (5)

where λm = 5.6 × 10-5 is the value of λ corresponding to the peak stress; a, ac, and ad are constants estimated as 3, 0.294, and 1.86, respectively. The concrete’s strength is 1.25 × 107 Pa in the RC frame structure with infilled wall, and the stress and strain of the uniaxial unconfined compression strength of the concrete are shown in Figure 22.

The results showed that the strain rate influences the collapse resisting capacity, collapse mode, and collapse path of RC frame structure (Wang) [60]. The dynamic compressive and tensile strength of concrete were given by European Code CEB-FIP (1993) [61] in (6) and (7).

$$ (6)

$$ (7)

where the compressive strength of concrete, fc, is 1.25 × 107 pa, and the tensile strength of concrete, ft, is 1.25 × 107 pa. Figure 23 shows the relationship curve between strain rate and DIF.

It is vital to accurately define the failure criteria of concrete materials to achieve seismic collapse of the RC frame structure and avoid unreasonable element deletion. If the ultimate tensile strain and compressive strain in the code for design of concrete structures [56] are utilized as the concrete failure criteria, the capacity of concrete cracking and closing to continue compression will be ignored, and the structure’s story drift capacity will be underestimated. Thus, from the calculation results without failure criterion, the strain at the expected failure moment of the element is 0.021 and is regarded as the failure criterion of concrete material. The failure of concrete material is realized by adding Mat_add_Erosion in LS-DYNA software.

The Cowper-Symonds model (Mat_003 in LS-DYNA R10.0) [62] (Jones, 1983) is utilized to simulate reinforcement bars material. This model is suited to model isotropic and kinematic hardening plasticity as well as rate effects. The main parameters of the model include young’s modulus, yield stress, tangent modulus, ultimate strain, and strain rate. Lin et al. [63] referring to the mechanical property test of reinforcement bars chose 0.015 as the ultimate tensile strain of HPB235.

The Winfrith-Concrete model (Mat_084 in LS-DYNA R10.0) (Broadhouse) [64] is utilized to simulate the infilled wall material. This model is a smeared crack (sometimes known as pseudocrack), smeared rebar model, implemented in the 8-node single integration point continuum element. The advantage of this material model over other models is that it allows for a graphical representation of the cracks within the finite element in postprocessors of LS-DYNA. The model’s main parameters include density, young’s modulus, uniaxial compressive strength, uniaxial tensile strength, fracture energy, or crack width.

Modified K&C model parameters, Cowper-Symonds model parameters, and Winfrith-Concrete model parameters are mentioned in Tables 1–3.

1. Modified K&C model parameters (unit: N, m, Pa, s).

Density ρ	Compressive strength fc	Tensile strength ft	Damage parameter b1	Damage parameter b2
6.9 E + 02 kg/m3	1.25 E + 07 Pa	1.2 + 06 Pa	1.6	2
Damage parameter b3	Peak damage λm	a	ac	ad
1.15	5.6 E − 05	3	0.294	1.86

2. Cowper-Symonds model parameters (unit: N, m, Pa, s).

Density ρ	Young’s modulus E	Yield strength σy	Tangent modulus Et	Ultimate strain
2.27 E + 05 kg/m3	1.2 E + 11 Pa	1.15 E + 08 Pa	1.2 E + 10 Pa	0.015

3. Winfrith-Concrete model parameters (unit: N, m, Pa, s).

Density ρ	Young’s Modulus E	Uniaxial compressive strength fc	Uniaxial tensile strength ft	Crack width
5.39 E + 04 kg/m3	6.12 E + 09 Pa	3.58 E + 06 Pa	3.58 E + 05 Pa	0.0024 m

The element mesh size should not be too large because the RC frame adopts the element deletion technique in FEM. However, the mesh size should not be too small because the calculation time is inversely proportional to the time step in LS-DYNA; the time step is proportional to the element characteristic length according to the Courant Friedrichs Lewy criterion [65]. Therefore, the element size of beams, columns, and infilled walls is 20 mm, the element size of slabs is 40 mm, and the element of the foundation beam is 100 mm. The number of elements is 140489. The numerical model of RC frame structure is shown in Figure 24.

3.3. Shaking Table Test Model

According to 3.1, the shaking table test model is the same as the numerical model, and the scale is 1/5. The shaking table test model takes the length, concrete elastic modulus, and acceleration similarity ratio as the basic quantities, and the test similarity ratio is deduced according to the similarity criterion (Table 4). The beam, slab, and column of test model are cast by C10 microconcrete and Q235 iron wire, and the infilled wall is poured by MU3.5 concrete hollow block and Mb5 composite mortars. The test model needs 5.73 tons’ additional weight, which is distributed to each floor. The additional weight of the top floor is 1.23 tons, and the additional weight of other floors is 1.5 tons. The shaking table test model is established at the scale of 1/5; the test model materials are obtained according to the test similarity ratio in Table 4. Table 5 shows the materials of full-scale structure and 1/5 scale model. The front view and right view of shaking table test model of RC frame structure are shown as Figure 25.

3.4. Sensor Arrangement and Loading Scheme

As shown in Figure 26, three (2 horizontal and 1 vertical) displacement sensors and three (2 horizontal and 1 vertical) acceleration sensors are installed on each floor of the test model; in addition, three acceleration sensors and three displacement sensors are installed on the shaking table. A classical earthquake acceleration record of site II was selected, which is the EI-Centro wave of the 1940 Valley earthquake (normalization). The earthquake wave needs to be processed because the test model is scaled. The acceleration similarity ratio is 1 and time similarity ratio is 0.447. Therefore, time was compressed to obtain the acceleration time history, as shown in Figure 27; the load case inputs are provided in Table 6. Firstly, the scan frequency test of the shaking table test model was carried out with white noise in order to obtain the natural period of the test model; then 3-direction (3D) ground motion was input, according to the code for seismic design of building in China [51]; PGA ratio of X, Y, Z is 1 : 0.85 : 0.65.

4.1. Comparison of Natural Period between the Test and Numerical Simulation

The natural period of the test and numerical model was compared to verify the accuracy of the numerical model. The natural period of the test model was obtained by the scan frequency test and the natural period of the numerical model was obtained by the structural dynamic property analysis, as shown in Table 7. The difference value of the natural period was 0.013 s and 0.021 s in X direction and Y direction, which shows that the numerical model has a certain reliability.

4.2. Comparison of Collapse Mode between the Test and Numerical Simulation

The four main failure stages of the numerical simulation and shaking table test of RC frame structure were compared (Figures 28–43). In the first stage, the first-story columns of the numerical model and test model were inclined, and plastic hinges appear at the top and bottom of columns, which leads to structural instability {Figures 28, 29, 36, and 37}. In the second stage, the first story of the numerical model and test model collapsed completely; however, the deformation of the upper structure was not obvious and the vertical member did not lose stability {Figures 30, 31, 38, and 39}. In the third stage, the second story of the numerical model and test model collapsed completely; the upper structure was inclined to the front right direction {Figures 32, 33, 40, and 41}. In the fourth stage, the third story of the numerical model and test model collapsed completely; however, the fourth story was intact {Figures 34, 35, 42, and 43}.

4.3. Comparison of Structural Response between the Test and Numerical Simulation

The story drift and acceleration time-history response of the numerical simulation and shaking table of RC frame structure were compared. The story drift was taken as the envelope value of maximum relative displacement during El-Centro earthquake with PGA of 0.06 g, 0.26 g, and 0.39 g (Figure 44). The acceleration time-history response of first and fourth story of RC frame structure is obtained during EI-Centro earthquake with PGA of 0.39 g (Figure 45).

As shown in Figures 27–42, the collapse mode and type of the numerical simulation correlate well with the shaking table test result in the four main failure stages. As shown in Figure 43, the story drift ratio of the shaking table test and numerical simulation were 0.0026 and 0.0036, 0.011 and 0.012, 0.026 and 0.03 under the action of earthquake with PGA of 0.06 g, 0.25 g, and 0.39 g, and the difference of story drift ratio was about 0.002, which is relatively small. The maximum story drift ratio of 0.026 and 0.03 can be used as the near-collapse story drift ratio of RC frame structure, which can provide reference for standard design for anticollapse RC frame structure. Meanwhile, the maximum story drift ratio appeared in the first story, which can be classified as the collapse vulnerable story of RC frame structure. As shown in Figure 44, the variety laws of acceleration response of the shaking table test and numerical simulation are similar, and the main acceleration peak is similar.

In summary, it shows that the combined FEM and F-DEM can accurately simulate the seismic collapse of the multispan and multistory RC frame structure with infilled wall.

5.1. Collapse Mechanism

As shown in Figures 28–31, the greater the load resistance ratio of the column, the easier it is to break first and collapse toward the longer-span direction with torsion. The characteristics of the collapsed RC frame structures are broken-bone-with-connected-tendons and pancake type, as shown in Figures 46 and 47.

The shear failure of the frame column was the main reason for the collapse of RC frame structure and it was due to the influence of low or half-height infilled wall. The RC frame structure shows the failure mechanism of “strong beam and weak column.” The main reason was that “frame beam + upper half-height infilled wall” increases the actual stiffness of the frame beam and shortens the height of frame column, which forms the short column. The plastic hinges appeared at column top or the intersection of column and half-height infilled wall, and the frame column is shear failure, as shown in Figures 48–53.

5.2. Survival Space

People tend to take refuge when strong earthquake event occurs. The survived people often hide in some small spaces in the ruins, which are called survival space. It is necessary for rescuers to determine the distribution of survival spaces quickly to construct the life-saving passage safely, which can improve the rescue efficiently and reduce casualties. The distribution of survival space is determined through the method of the shaking table test and numerical simulation; however, it is difficult to observe the small survival spaces inside the collapsed ruins through the shaking table test. Therefore, the distribution of survival spaces of RC frame structure’s ruins was briefly analyzed by numerical simulation.

As shown in Figure 46, there are infilled walls in the survival spaces of the collapsed ruins, and the infilled walls play a very crucial role in the formation of the survival spaces. It can be seen that the infilled wall cannot be ignored to simulate the RC frame structure’s collapse mode, particularly the survival spaces, and it can also provide a basis to study the life-saving passage accurately.