The calculation diagram for single frame-filled wall components is shown in Fig. (2), which assumes that only bracing walls with a certain width participate in the force under the seismic load, where the contact length with the frame column is denoted as α_ch, the bracing wall thickness t is the same as the infill thickness t, and the sloping angle between the medial axis of the equivalent diagonal bracing wall and the horizontal direction is denoted as θ'. The following basic hypothesis is proposed based on the model. First, it is considered that the infill walls are made of ideal homogeneous isotropic materials. Second, for convenience, it is assumed that the border of the infill coincides with the central axis line of the frame in the following force analysis, i.e., h' = h, l' = l, where h' and l' are the height and span of the frame, respectively, h and l are the height and span of the infill. Third, the simplified brace infill wall has no contact with the beam. There is a certain distance Δx between the top of the bracing walls and the frame beam, and the contact length with the frame column may not exceed 0.3 times the height of the infill, as shown in Eq. (1). Fourth, according to the nonlinear finite element analysis results provided by Saneinejad [8] in 1990, the ratio of the maximum elastic moment and the plastic yield moment in the columns and beams may not exceed 0.2, as shown in Eq. (2):

(1)

(2)

Fig. (2). Simplified calculation model and forces in the wall-filled frame.

where α_c is the normalized length of contact of the frame column, M_c and M_b are the maximum elastic moments in the columns and beams, respectively, M_pc and M_pb are the plastic yield moments in the columns and beams, β_c and β_b are the ratios of the maximum elastic moments and plastic yield moments in the columns and beams, and β₀ is the upper-bound value of the ratio of the maximum elastic moment and plastic yield moment in the columns and beams.

Taking the frame column on the left hand side and the frame beam on the top as an isolated body, as shown in Fig. (3):

(3)

(4)

where H is the horizontal bearing capacity of shear, C_c is the normal force at the column–infill interface, S is the column shear force and beam axial force at the unloaded corners, σ_c is the proposed uniform normal stress at the column–infill interface, and t is the infill thickness.

Fig. (3). Forced distortion diagram of a wall-filled frame.

In Fig. (3), taking the bending moment of point A through the internal force of frame column AD according to ΣM_A = 0, and S can be expressed as:

(5)

Where M_j represents the bending moments of points B and D in the infill corner. To simplify the analysis, it is considered that points B and D have equal bending moments, i.e., M_B = M_D = M_j, where the value of M_j is very small in the load that influences the collapse failure of the infill wall and thus it can be neglected if necessary.

Substituting Eq. (4) and (5) into Eq. (3) yields the formula for calculating the shear bearing capacity of the wall-filled frame component H:

(6)

Where the first part of Eq. (6) is the shear strength of infill wall and the second part represents the shear strength of the bare frame. If R is considered to be the shear strength of the central axis of the equivalent diagonal bracing wall, then Eq. (6) can be rewritten as follows.

(7)

Therefore, the formula for calculating the specified crushing load of the diagonal strut in CC mode is as follows.

(8)

At the peak load, the diagonal bracing wall is subjected to failure, which results from the combined normal and shear stresses that act on the surfaces in contact with the frame column. According to Tresca [9], the hexagonal yield criterion is:

(9)

(10)

where σ and τ are the proposed uniform normal and shear stresses of the column–infill interface, respectively, r is the height-span ratio of the frame (r = h'/l'), and µ is the coefficient of friction for the column–infill interface with a general value of 0.45 [10]. Eq. (9) and Eq. (10) lead to Eq. (11)

(11)

where σ_c0 is the proposed nominal value of the contact normal stress of the column–infill interface, f_c is the effective compressive strength of infill, which can by calculate as f_c = 0.6ϕ f'_m, where f'_m is the compressive strength of the infill material, and ϕ is the adjustment coefficient with a general value of 0.65 [10].

As shown in Fig. (3), it is assumed that the equivalent diagonal bracing wall and frame column separate at point E under the horizontal load, and the shear stress distributed uniformly on the contact length range is neglected. Taking the static moment of the forces that act on the column along EA gives the following:

(12)

Where M_c is the maximum elastic moment of the frame column. After substituting Eq. (2) into Eq. (12) and combining with Eq. (1), the equivalent contact length α_ch is:

(13)

Under the horizontal load, the deformation of the frame is shown in Fig. (4) as a dotted line. Assuming that the quality and stiffness of the equivalent bracing wall are concentrated in the center line of the wall, according to the definition of the elastic modulus, the secant elastic modulus of the equivalent diagonal strut is:

(14)

Fig. (4). Deflection of the frame.

Where E_d is the secant elastic modulus of the equivalent diagonal strut, ε_c is the strain on the equivalent diagonal strut due to horizontal load H, d is the length of the equivalent diagonal strut, and Δd is the axial deformation of the equivalent diagonal strut under horizontal load H.

When the frame produces horizontal deflection Δh, the rod end of equivalent diagonal strut has horizontal displacement a. Thus, Eq. (14) can be changed into Eq. (15):

(15)

(16)

Where s is the distance between the top of the column and the rod end of the equivalent diagonal strut. It should be noted that the derivation process above occurs in elastic–plastic state conditions, where the wall-filled frame components achieve the ultimate bearing capacity and the initial elastic modulus of the infill wall in the elastic state is higher than the secant elastic modulus of the equivalent diagonal strut. Here it is considered that the initial modulus of elasticity for the equivalent diagonal strut E_d0 is two times larger than the secant elastic modulus E_d:

(17)

And when Δh = 1, the result calculated for E_d0 is the equivalent diagonal strut stiffness K_d0 , where the stiffness of the horizontal component is the stiffness that the infill contributes to the overall component.

The derivation above gives the method for calculating the shear bearing capacity of the wall-filled frame component and the actual participation of the infill walls in the stiffness, thereby providing a reference for the structure during the design stage of seismic checking. The effective widths of the equivalent bracing walls are given in the form of α_ch, which provides a method for building the simplified finite element model. When the scantling and material grade of the wall-filled frame component is determined, the relative parameters except Δx are identified. The reasonable value of Δx is determined using the fitting formula method based on the ABAQUS finite element simulation results. In the following, we will introduce the process used to determine the calculation formula Δx and the finite element simulation method.

Y. S. Tong and G. F. Qian [11] tested a certain number of reinforced concrete frame models to study the bearing capacity and deformation performance of infill frame. In this paper, we take the specimen C-1 and C-3 which included block masonry as examples to construct the solid finite element model with the finite element software ABAQUS and compare the results with the data obtained via testing. Main parameters of the framework model and material properties were as follows: specimens were both single layer with single span, the dimension of column section was 150mm×100mm, the dimension of beam section was 180mm×100mm, column spacing was 1500mm, and story height was 1000mm. The internal force steel of framework used HRB 335, and stirrup used HPB235. Concrete strength grade of C-1 was C20, block shear strength grade was M2.5. Concrete strength grade of C-3 was C30, block shear strength was M5. Other parameters were shown in attached A of literature [11].

In ABAQUS, the modeling process was as follows: the frame column, beam, and infill wall used a solid unit (C3D20R), and the steel bar used a truss unit (T3D2). Embedded connections were used in the steel bars of the frame column and beam, and a contact connection was used between the infill and frame column to simulate their interaction. The surface-to-surface contact is one of the interaction types of ABAQUS, the contact property is to define a tangential behavior to simulate interaction between the infill and frame which will separate each other during ultimate bearing capacity. The latest research results show that: this contact method in solid element model could simulate very well. The modeling of the components is shown in (Fig. 5).

Fig. (5). Solid finite element model.

Using ABAQUS's own concrete damage plasticity model to define the nonlinearity of material. In this model, the nonlinear response of the whole structure under earthquake action is simulated by defining the stress-strain relationship of the material under the compression and tensile behavior.

The constitutive relation of concrete is described in literature [12], which gives the uniaxial compression and tensile stress-strain relationship of reinforced concrete members. The stress strain relationship of concrete under uniaxial compression is calculated by the following formulas:

(18)

(19)

Where α_a and α_d are the parameters related to the rise and fall segment of the stress-strain curve of uniaxial compression, f_c^* is uniaxial compressive strength of concrete, ε_c is the corresponding peak compression strain of concrete related to f_c^*.

The following formulae are used for the stress-strain relationship of concrete under uniaxial tension:

(20)

(21)

Where α_t is the parameter related to the rise and fall segment of the stress-strain curve of uniaxial tension, f_t^* is uniaxial tensile strength of concrete, ε_t is the corresponding peak tension strain of concrete related to f_t^*.

The mechanical properties of masonry materials are more complex than concrete, so the research on the constitutive relation of masonry was proposed by domestic and foreign scholars. In this paper, the masonry constitutive relationship used the research results of C. X. Shi and G. Q. Liu [13] which was based on the results of previous experiments. The rise section is a parabola and the falling section is linear:

(22)

(23)

Where f_m is the average compressive strength of masonry, ε_o is the peak pressure strain of masonry.

Damage variable D is used to describe the stiffness degradation of the material in the ABAQUS damage-plasticity model. Damage variable D is also called damage factor, it is a number which is greater than 0 and less than 1. The closer to 1 the value is, the more severe the stiffness degradation is. D is derived based on the assumption that the equivalent of the material energy, and the formulae are as follows:

(24)

(25)

Where E₀ is the initial elastic modulus, D is damage factor, σ and ε are the stress and strain of materials respectively.

Constructing the solid finite element model of the wall-filled frame component with the finite element software ABAQUS and comparing the results with the data obtained via testing by Y. S. Tong and G. F. Qian. Finally, the load-displacement curves of C-1 and C-3 were obtained using the implicit static analysis method and by applying a lateral load at the beam end. The finite element analysis results and test data are shown in Fig. (6) and (Fig. 7).

Fig. (6). Data comparison of C-1.

Fig. (7). Data comparison of C-3.

Fig. (6) and Fig. (7) compared the finite element simulation values with the solid finite element model based on the load-displacement curves, which both showed that the two curves agreed in the platform segment, thereby demonstrating that the maximum horizontal bearing capacity obtained with the solid finite element model used for the static analysis of the simulation results was close to the actual results. Therefore, the analytical results obtained with the solid finite element model are treated as the actual values for the component in the following process.

Design a series of infill frame components with the same span-height ratio but different heights h' and spans l'. The section dimensions and reinforcements of the frame beam and column were the same, and the span-depth ratio was 1.5. The specific design parameters are shown in (Table 1).

Taking KJ-1 as an example to illustrate the design condition for the infill frame component, it was shown in Fig. (8). The concrete strength grade was C30, the cover thickness was 35 mm, the longitudinal bearing force reinforcement used HRB335 and the structural stirrup used HPB235, the infill thickness was 120 mm, and the axial compression ratio was 0.2. The yield moments of the frame column and beam section were calculated according to Eq. (26) and Eq. (27), respectively:

(26)

(27)

where M_u is the yield moment of the frame beam, f_y is the design value of the tensile strength of reinforcement, A_s is the section area of longitudinal tensile bars, h₀ is the effective height of the frame column and beam, a_s is the distance from the edge of the compressive region to the point of the resultant force of compression reinforcement, M_cy is the yield moment of the frame column, f_yk is the standard tensile strength value of the longitudinal bars in the column, N is the axial pressure of the column, which corresponds to the representative value of the gravity load, h and b are the height and width of the column section, respectively, α₁ is the ratio of the equivalent rectangular stress in the concrete compression block relative to the design value of the axial compressive strength of the concrete, and f_ck is the standard value of the compressive strength of the concrete cube. According to this calculation, the yield moment of the column section is M_pb = 5.79 x 10⁷N.mm and the yield moment of the beam section is M_pc = 9.03 x 10⁷N.mm.

Taking KJ-1 as an example to illustrate the calculations of the relevant parameters. According to Eq. (1), α_ch ≤ 0.3h = 525mm, which is calculated by Eq. (11) as:

Where f_c = 0.6ϕf'_m = 0.585MPa, and this is calculated by Eq. (13) as:

Where M_Rj = min {M_pc, M_pb} = 5.79 x 10⁷N.mm.

Fig. (8). Component design of the infill frame.

Based on the solid finite element model, truss element T3D2 was used to replace the infill with the equivalent diagonal strut. The material properties had the same constitutive masonry relationship using in the infill, and it was stipulated that the truss element T3D2 could only bear the pressure but without tension, corresponding to the physical characteristics of infill walls in actual force. The joint connection in the connector was used between the truss rod and frame column, it meant that there was no relative displacement between the joints, but the relative rotation was allowed. The simplified model is shown in (Fig. 9).

Fig. (9). Equivalent diagonal single-strut bracing model.

According to the principle of equal lateral displacement on the top of the column, the value of Δx was adjusted continuously until the load-displacement curves of the equivalent diagonal single-strut bracing model were consistent with the solid finite element model and the value of Δx was recorded at this point. The simulation results with different contact positions between the equivalent bracing walls and the frame columns are shown in Fig. (10).

Fig. (10). Load-displacement curve of Member 1.

As shown in Fig. (10), the load-displacement curve was most consistent with the results calculated by the solid finite element model when Δx = 350mm, thus the reasonable value of Δx for Member 1 was 350 mm. According to the method above and the analysis of the remaining nine members in the same process, the value of Δx was determined when the load-displacement curves of the equivalent diagonal single-strut bracing model were consistent with the solid finite element model.

The steel frame structure was analyzed using a finite element simulation by Liauw [14] in 1984, thereby obtaining the mechanical characteristics of steel frame structures under a horizontal load, where he considered that the effective width of the equivalent bracing wall w is associated with h' cosθ', and the value of w/h' cosθ' can be written as an expression that contains the function λh'. Through the derivation of Eq. (13), w can be expressed via the expression that contains Δx and they have a direct proportional relationship. When combined with the finite element analysis results, it is considered that Δx is associated with h' cosθ' and the value of Δx/h' cosθ' can be written as an expression that contains the function λh', where λ is the relative stiffness parameter first proposed by Stafford Smith [15] in 1969. The formula for calculating λ is:

(28)

where E_m is the elastic modulus of the filled-wall material, E_c is the elastic modulus of concrete, t is the infill thickness, θ is the sloping angle of diagonal of the infill, I_c is the moment of inertia of the column cross-section, and h is the infill height. It shows that λ is the main parameter to calculate the width of equivalent diagonal brace, and it has been widely used. The relative stiffness parameter λ provides a key to the estimation of an infilled frame's behavior and it therefore assumes a prominent role in the development and presentation of the methods for predicting its stiffness and strength. The length of contact, the stiffness of the infill and the strength and sequence of the modes of failure may all be expressed as functions of λ either in algebraic or graphical form.

By calculating the corresponding values of Δx / h' cosθ' and λh' for the 10 design components described above, when Δx took an appropriate value, a series of points were obtained, which were plotted with Δx / h' cosθ' as the ordinate against λh' as the abscissa in the same rectangular coordinate system, as shown in Fig. (11).

Fig. (11). Distribution points and formula fitting curve.

The empirical formula for calculating Δx was obtained from thedistribution points and the formula fitting curve given above as Eq. (29):

(29)

The relative parameters were identified after determining the scantling and material grade of the wall-filled frame component, where the value of Δx was calculated by Eq. (29). It should be noted that during ABAQUS finite element modeling, for the equivalent diagonal strut concentrated quality and the stiffness of equivalent bracing wall located in the center line of it, the contact position between the rod end and frame column was calculated according to Eq. (16): s = Δx + 0.5α_ch.

As described in Section 2, Y.S. Tong and G.F. Qian who studied the bearing capacity and deformation performance of members in a series of reinforced concrete frame model tests. Simulation verification of C-1 and C-3 was done by solid element model and simplified model described in this paper. Comparing the experimental values with the two finite element modeling method results of the component C-1 and C-3, they are shown in Fig. (12) and Fig. (13).

The calculation results could be seen from the Fig. (12) and Fig. (13). The calculation of maximum horizontal bearing capacity of infill frame though the simplified modeling method described in this article coincided both the experimental data and solid element model results. The rationality and accuracy of the equivalent simplified model were verified.

Fig. (12). Load-displacement curve of Member c-1.

Fig. (13). Load-displacement curve of Member c-3.

Take one bottom frame structure in Wenchuan earthquake as an example to verify the accuracy of the formula for calculating the stiffness of the infilled wall which was derived in this paper. And explain the influence of the stiffness of the infilled wall on the lateral stiffness ratio of the bottom frame structure though analyzing of calculation results.

Masonry buildings with frame and seismic-wall in the lower stories refers to hybrid bearing buildings which the bottom or the bottom two layers are shear walls, and the upper part is Multistory masonry structure. Because of the more flexible space layout in bottom layer, and it is more cost-effective and shorter construction period than the frame structure, so this kind of building is widely used in street-facing residential or office buildings in China. The structure type is a hybrid structure composed of two different bearing and lateral resistant system, which belongs to a seismic-unfavorable structural system. The relative stiffness between the bottom frame-shear wall story and the transitional masonry layer is controlled by the lateral stiffness ratio. Structural appearance is shown in Fig. (14). The first layer of the structure is infilled frame, there are 33 brick masonry walls in X and Y directions. The first plane graph and infill wall number are shown in Fig. (15).

Fig. (14). One bottom frame structure in Wenchuan earthquake.

Fig. (15). First plane graph and infill wall number.

The calculation formula (17) which was derived in this paper was used for calculating the stiffness of the 33 infill wall. Among them, the infilled walls in X direction were number 1-10 and number 31-33 brick masonry walls. The infilled walls in Y direction were number 11-30 brick masonry walls. In order to verify the accuracy of the Eq. (17), the calculation results were compared with the calculation formula which was from literature [16] for calculating the lateral displacement rigidity of infill wall. The calculation formula which is presented by literature [16] as:

(30)

(31)

Where K_w is interlayer lateral stiffness of infill walls, φ_k is stiffness reduction factor, the upper part of the house uses 1.0, the middle of the house uses 0.6, the lower part of the house uses 0.3, E_w is the elastic modulus of infill wall, H_wis the height of infill wall, A_w and I_w is the cross section area and inertia moment of infill wall, φ_m and φ_v is opening influence coefficient. A comparison of the two methods for calculating the stiffness of the infill walls were carried out, and the results are shown in Table 2.

It can be seen from Table 2, both two algorithms show that only some part of the infill wall as a non-structural component is involved in the lateral stiffness. The contribution of the stiffness is less than 20% of their total stiffness, and it is same with the conventional understanding. The calculation results of the two algorithms are close. Considering the whole stiffness of infill wall, the effect of 10% error can be neglected. It also verifies the rationality and accuracy of the derived calculation formula of the infilled stiffness in this paper.

According to the seismic code, the ratio of lateral stiffness of the structure was checked. When the stiffness of infill wall was not included, the lateral stiffness ratio of the structure was shown in (Table 3). And when the stiffness of infill wall was included, the lateral stiffness ratio of the structure was shown in (Table 4).

In our country, the lateral stiffness ratio of the bottom frame structure should be controlled between 1.0 and 2.5. It can be seen from Table 4 that the lateral stiffness ratio of the structure in Y direction is less than 1.0 when the stiffness contribution of the infill wall in first layer is included. The lateral stiffness ratio is less than 1.0, which means that the seismic capacity of the bottom layer of the structure is too strong, so that the weak floor is transferred from the bottom floor to the masonry floor in the upper part of the structure. It is also in line with the actual damage that the first layer is slightly damaged, and the second layer is damaged seriously. Therefore, it will produce erroneous results when the stiffness contribution of the infill wall is ignored completely during the design of the building structure. So it is suggested that the effect of the infill wall on the stiffness should be considered.

The calculation formula (8) which was derived in this paper was used for calculating the shear resistance of the 33 infill wall. The first plane graph and infill wall number were same with (Fig. 15). The calculation results were compared with the calculation formula which was from literature [16] for calculating the shear resistance of infill wall. The calculation formula which was presented by literature [16] as:

(32)

(33)

Where V_my is shear resistance of infill wall, f_vE is design value of seismic shear strength of masonry along the stepped cross section, f_v is non-seismic design value of shear strength of masonry, ζ_N is normal stress influence coefficient of shear strength of brick masonry, A_m is the cross section area of infill wall. A comparison of the two methods for calculating the shear resistance of the infill wall was carried out, and the results were shown in Table 5.

From the results of calculation, the calculation of the code formula of infill wall shear resistance is high, and the reduction factor 0.7 in code formula is not clear. The value is based on engineering experience in a large extent, and it is lacked in strict theoretical derivation. In contrast, the formula which is derived in this paper is more rigorous, the form is more simple and intuitive, the concept is clearer and the results are more consistent with the actual situation. Compared with the bare frame, 1024kN is increased in X direction and 1458kN is increased in Y direction. This part of increasing shear resistance is provided by the infill wall. It can be seen that the shear bearing capacity of the infill wall should not be ignored completely. And the shear resistance of the infill wall should be included in the calculation of the ultimate bearing capacity of frame-shear wall stories.