Study on the Flexural Strength of Interior Thick Wall-Thick Slab Joints Subjected to Lateral Force Using Finite-Element Analysis

Abstract

A brand-new structural type, termed thick wall-thick slab structure, has been developed in recent years based on the reinforced concrete wall structure in Japan. This type of structural system can not only maintain high seismic performance, but also provide large interior space and improve the flexibility of the architectural design that is more favored by architects. In the present study, an equivalent cross-sectional area concept in which a coefficient termed the equivalent cross-sectional area ratio is proposed to estimate the flexural strength of the horizontal member of the interior wall-slab joints, which are critical assemblies in the thick wall-thick slab structures. To capture the stress condition of slab flexural reinforcement for calculating the equivalent cross-sectional area ratio, the method of finite-element analysis was employed in this work. The finite-element modeling was calibrated firstly using two isolated interior wall-slab joints from literature and consequently applied to a parametric numerical study. Finally, results from these finite-element analyses are adopted to propose the equivalent cross-sectional area ratio for modifying the current code formula to predict the flexural strength of interior wall-slab joints in TWTS structures subjected to lateral force.

1. Introduction

Reinforced concrete wall structural systems (abbreviated as WRC structures in this work) as shown in Figure 1, which mainly consist of walls, wall beams, and slabs without any columns were developed and commonly used for a long time in Japan as a popular and suitable type for residential apartment housing due to several benefits. It can provide good seismic performance and high fire resistance, which has been observed from past earthquake events, such as the 2011 Tohoku earthquake [1], and the 2016 Kumamoto earthquake [2], due to the wall components being highly solid and can provide higher lateral stiffness than other frame structures. The wall-beam-slab assemblies are the critical load-carry components in this type of structural system, as shown in Figure 2. However, because of the presence of wall beams of which the depth is much larger than the thickness of the slab components, it cannot provide large openings and service areas for interior layout design. Thus, the WRC structural system is less favorable from an architectural viewpoint and is expected to be updated for a more friendly layout design.

Based on these considerations, an attempt to deal with the wall beam components to satisfy the architectural expectation is made in recent years in Japan. A new structural system termed the “thick wall-thick slab” (TWTS) structure by reducing the beam depth to make the general wall-beam to flat wall-beam as thick as the surrounding slabs and completely hidden into the slab components has been developed as shown in Figure 3. There are no columns and conventional wall beams present, and the bearing wall, flat wall beam, and slab elements generally have the same width, by which, the TWTS structures can provide larger openings and interior service space while retaining high seismic resistance provided by walls similar as the WRC structures.

To evaluate the structural performance under earthquake loadings of this new type of structure, one of the critical indexes, the flexural behavior, especially the flexural strength under earthquake loadings is investigated in this study. For the wall-beam-slab joints in the conventional WRC structures or “T-shaped” beam composing of a rectangular beam and a slab integrated on the compression side in reinforced concrete (RC) structures, the current “AIJ Standard for Structural Design of Reinforced Concrete Boxed-Shaped Wall Structures” (abbreviated as the “WRC Standard” in this work) [3] and the “AIJ Standard for Structural Calculation of Reinforced Concrete Structures” (abbreviated as the “RC Standard” in this work) [4], impose the calculation formula as given by Equation (1) to predict the flexural strength of the beam component mainly considering the resistance provided by the wall beam, as shown in Figure 4.

$$M_{flex}=a_t·f_t·j,$$

(1)

Here,

$M_{flex}$: Wall beam allowable ultimate flexural moment (flexural strength).

$a_t$: Total cross-sectional area of flexural reinforcement at the tensile side of the beam component.

$f_t$: Allowable tensile strength of reinforcing steel bars.

$j$: Distance between stress centers and usually employed 0.9 or 7/8 times the effective depth (i.e., depth from the top surface to centroid of tensile reinforcing) of the beam component.

$D$: Depth of the beam component.

$d$: Effective depth of the beam component.

In this calculation concept, only reinforcement located in the wall beam is considered to determine the flexural strength of the beam component, the contribution of side slabs is ignored. This is reasonable for wall-beam-slab joints, but is not appropriate for wall-slab joints in TWTS cases. To explain this, consider the reinforcing layouts shown in Figure 5. For typical wall-beam-slab joints in WRC structures (Figure 5a), the wall beam is deeper than the slabs, which results in the wall beam’s reinforcing providing the majority of the tensile resistance to flexural demands, and thus the slab’s contribution to providing flexural resistance is relatively small and is ignored. In contrast, for the TWTS cases shown in Figure 5b, the flat wall beams have the same depth as the slabs, resulting in the surrounding slabs having a greater contribution to flexural resistance. Therefore, the contribution from the surrounding slab flexural reinforcement cannot be ignored and needs to be investigated rather than simply applying the existing WRC concept.

Based on the discussion around Figure 5, Equation (1) is not applicable for wall-slab joints in TWTS structures and needs adjustment appropriately to consider the contribution from the side slabs. To model the wall-slab joints analytically, the concept of the effective beam width model (EBWM) is employed in this work. The EBWM was developed first for analyzing the column-slab joints in reinforced flat plate structures. In the EBWM, only the effective beam portion of the slab, which is defined as an effective beam, is considered when determining the strength and stiffness of the joint. Based on this consideration, the EBWM can as well applicable for the wall-slab joints in the TWTS cases. Thus, the contribution to the flexural strength of the beam components from surrounding slabs can thus be considered using the effective beam concept. A coefficient termed the effective beam width factor is accordingly defined as the proportion between the effective beam width and the original slab full width.

A lot of research including analytical, experimental, and numerical studies for estimating the effective beam width has been conducted. Bijan Aalami [5] pointed out that there should be a lower and upper limit to the actual value for the effective beam width of the slab in column-slab joints. The lower value representing the connection portion of column-slab joints was not completely rigid, while the upper value could be calculated by assuming the connection part was completely rigid so that the rotational angle was solely decided by slab rotation. Vanderbilt and Corley [6] compared different techniques to obtain the effective beam width factor and summarized the results of several investigations for square column-slab joints under lateral loads. They indicated that for rigid column joints, it tended to produce similar values for the effective beam width factor, even though the answers were observed to exceed unity when the ratio of width dividing by height c/l greater than about 0.15. Additionally, for the flexible column joints, by which the lower limit for the effective beam width factor could be approximately estimated. Luo et al. [7,8,9] proposed a modified calculation procedure of the effective beam factor based on the results from an experimental study on 40 interior column-slab joints and 41 exterior column-slab joints. They suggested the effective beam width factor was given as 0.963 for rigid columns and 0.434 for flexible columns. Dovich and Wight [10] presented a simplified effective slab width model for column-slab joints to predict flexural strength under lateral loads. They suggested the effective beam width to be estimated by 1/3 times or 1/2 times the full width of the slab for different cases.

In the current provisions recommended by the Japanese RC Standard [4], the EBWM concept is also applied for predicting the flexural strength of the beam components in column-slab joints. By considering the contribution from side slabs, the reinforcement located outside the actual beam but inside the effective beam width (the effective part of the slab, $b_a$) should thus be included. For T-shaped structural components, as shown in Figure 6, $b_a$ can be estimated using Equation (2).

$$b_a=\{\begin{array}{c}\hfill (0.5-0.6\frac{a}{l})a, \frac{a}{l}<0.5\\ \hfill 0.1l, \frac{a}{l}\ge 0.5\end{array},$$

(2)

Here,

$b_a$: Effective part of the slab.

$a$: Distance between two side surfaces of two adjacent T-shaped structural components (such as beams connected by slabs or columns connected by walls).

$b$: Thickness of the beams or walls in T-shaped structural components.

$B$: Effective beam width.

$l$: Span length of frames or continuous beams.

For the wall-slab joints in TWTS structures, Inai et al. [11,12] carried out a series of experimental tests on 4 wall-slab specimens as a first trial to investigate the structural performance of TWTS structures under lateral load. In their study, Specimen 1 and 2 were both comprised of a bearing wall and flat slabs on both sides with identical thickness. The plan view and configuration of the specimens are shown in Figure 7. The fixity condition is shown in Figure 8a, where the bottom of the wall is pin-supported, the top of the wall is free, and both ends of the slab are roller-supported and restrained only in the vertical direction. Figure 8b represents the photograph of the specimen at the laboratory. The entire cyclic loading process was controlled by the displacement applied on the top of the wall to simulate inter-story drift ratios of 0.05%, 0.1%, 0.25%, 0.5%, 0.75%, 1%, 1.5%, 2%, 3% and 4%.

Based on their study, Matsui et al. [13,14] conducted finite-element analyses to investigate the moment transfer around the wall end of the wall-slab joints and the effective scope of slab reinforcement that contributes to the flexural strength. They suggested an effective slab width to be 1 m to predict the flexural strength of the beam member of interior wall-slab joints. However, their proposed value of effective beam width was obtained from limited specimens without considering various parameters, such as the slab length, slab width, etc. Additionally, the value of the effective slab portion they proposed is a certain constant value, which leads to a concern that whether it is reasonable to estimate the effective proportion of slab reinforcement using the same value for different cases in a large scope. Therefore, further investigation of the flexural behavior of the wall-slab joints is still required.

In this present study, an equivalent cross-sectional area concept is proposed for replacing the effective beam width concept in the EBWM approach to evaluate the effective proportion for the flexural strength provided by flexural reinforcement of the slab. Additionally, this concept is discussed using results obtained from an experimental test conducted by Inai et al. [11,12]. To simulate the stress condition of flexural reinforcing bars in specimens of the experiment tests, the finite-element method is employed in this work. The finite-element modeling is calibrated firstly using the results from Inai et al.’s experimental tests. Consequently, the stress conditions of slab longitudinal reinforcement of these specimens have been examined. Next, the impact of several parameters, such as the slab span length and width, the slab flexural reinforcement distribution are investigated carefully by conducting a parametric numerical simulation based on the calibrated finite-element modeling, and influencing factors are evaluated. Furthermore, a proposal to predict the flexural strength of beam components of isolated interior wall-slab joints in TWTS structures under lateral load has been suggested.

2. Methodology

First of all, a concept termed as equivalent cross-section area is established to evaluate the contribution of the slab flexural reinforcement considering that of the flat wall beam integrally to the flexural strength of interior wall-slab joints in TWTS structures subjected to lateral load. Next, the finite-element modeling was calibrated using specimens in the experimental test conducted by Inai et al. [11,12] and stress conditions of the slab longitudinal reinforcement of these specimens are discussed using the equivalent cross-section area concept. Additionally, parametric analyses based on the calibrated finite-element modeling considering several parameters were consequently performed. Finally, a coefficient termed equivalent cross-section area ratio is proposed using the parametric analysis results based on the equivalent cross-section area concept for interior wall-slab joints in TWTS structures.

2.1. Equivalent Cross-Sectional Area Concept

To accurately evaluate the flexural strength of the beam component in wall-slab joints, the total cross-sectional area of flexural reinforcement $a_t$ is re-evaluated. Regarding the different rotation distribution over the slab full width, the yielding condition of slab longitudinal reinforcement can be considered where rebars located close to the core region of the wall-slab joint may have yielded or shown higher stress than other rebars located far from the core region to the slab edge. Thus, a coefficient termed as “equivalent cross-sectional area ratio”, $\beta$, is introduced to evaluate the slab stress condition averagely and represent the effective proportion of reinforcement that needs to be accounted for in the flexural strength. As shown in Figure 9, an equivalent gross cross-sectional area is defined as $a_e$ to represent the effective proportion of total slab reinforcement as given by Equation (3), where $a_s$ represents the total gross cross-sectional area including tensile rebars on the top and bottom mats. Additionally, the equivalent cross-sectional area ratio $\beta$ can be as well obtained by rearranging Equation (3).

$$a_e=\beta ·a_s,$$

(3)

Thus, by applying the yield stress of reinforcing steel bars ${\sigma }_y$ as the allowable tensile strength and substituting $a_t$ by the equivalent gross cross-sectional area $a_e$ shown in Equation (3), the flexural strength calculation formula Equation (1) can be modified to Equation (4).

$$M_{flex}=0.9·a_e·{\sigma }_y·d=0.9·\beta ·a_s·{\sigma }_y·d,$$

(4)

On the other hand, as shown in Figure 10, the flexural strength can be also given by Equation (5) using the tensile stress of each longitudinal rebar engaged in tension, where ${\sigma }_i$ is the actual tensile stress of ith rebar, and $a_{0b}$ and $a_{0t}$ represent the cross-sectional area of each rebar located at the top mat on one face and bottom mat on the opposite surface of the slab, respectively.

$$M_{flex}=0.9·d·a_{0b}·{{\displaystyle \sum }}_{i=1}^n{\sigma }_i+0.9·d·a_{0t}·{{\displaystyle \sum }}_{i=1}^n{\sigma }_i,$$

(5)

As Equations (4) and (5) both represent the flexural strength, that indicates they should be equal, and by setting Equation (5) to be equal with Equation (4), the equivalent sectional area $a_e$ as shown in Equation (6) can be obtained.

$$a_e=\frac{a_{0b}·{{\displaystyle \sum }}_{i=1}^n{\sigma }_i+a_{0t}·{{\displaystyle \sum }}_{i=1}^n{\sigma }_i}{{\sigma }_y},$$

(6)

Thus, the equivalent cross-sectional area coefficient $\beta$ can be obtained according to Equation (3). Finally, Equation (4) employing the coefficient $\beta$ can be used to predict the flexural strength by evaluating the tensile condition of slab reinforcement more accurately.

2.2. Finite-Element Method

To investigate the stress condition of slab flexural reinforcement and evaluate the value of coefficient $\beta$, a finite-element model was developed on the computer program FINAL [15], in which the nonlinear material behavior was taken into account while the nonlinear geometric behavior was not considered. In this modeling, the plasticization of materials (cracking in concrete, reduction in compressive rigidity, strain softening, crushing, etc.) was considered and expressed by different material composition models for the material nonlinear analysis. The rigidity and stress changes due to the plasticization of the material would be considered and recalculated in the analysis. A three-dimensional mesh composed of 8 nodes in which concrete was simulated using hexahedron elements while steel rebars were simulated using truss elements was adopted in this modeling. Model 1, Model 2 corresponding to two specimens (Specimen 1 and Specimen 2) from Inai et al.’s experimental test [11,12] (more details in the following sections) were used for calibrating the FEM modeling.

To simulate the compressive failure condition of the concrete, Ottosen’s 4-parameter model with coefficient proposed by Hatanaka et al. [16] was used for reinforced concrete under low lateral pressure up to about 0.2 times the uniaxial compressive strength, while the same model with Ottosen’s coefficient [16] was employed for the un-reinforced concrete. To model the stress transfer carried out by steel rebars across cracked concrete, the “Naganuma-Yamakuchi” model [17] considering the reinforcing ratio, concrete strength, and stiffness reduction ratio was used for the tensile behavior of reinforced concrete as shown in Figure 11a. In contrast, the Izumo model [18] with a coefficient of C = 1.0, which describes that the tensile stress is unrelated to the reinforcing ratio shown in Figure 11b, was employed for unreinforced concrete. The modified Ahmad model [16] shown in Figure 11c was adopted for the strain-softening for the ascending branch of the compressive stress-strain relationship before reaching the compressive strength point. The method of determining to yield of the reinforcing bar was based on the stress obtained from the local strain of the crack region and the Naganuma method [19] was employed for the descending branch of the compressive stress-strain relationship after cracking, in which the uniaxial compressive strength of concrete and the compressive axial force in the axial direction of the reinforcing bar was considered for both reinforced and unreinforced concrete. In addition, the “Naganuma-Okubo” model [20] as shown in Figure 11d was employed for the cyclic stress-strain relationship of concrete under tensile and compression.

For the reinforcing steel, a bi-linear steel model was employed in this analysis to simulate: (i) the pre-yield elastic and post-yield strain-hardening behavior with a strain-hardening ratio of 0.001 as shown in Figure 12a; and (ii) the kinematic strain hardening behavior as shown in Figure 12b, which was based on considered the Baushinger effects. The pull-out behavior between steel rebar and concrete was not considered in this work, so a truss element representing rebar was assumed to be completely connected to the hexahedron element representing concrete, which could make the results of simulated strength prediction a bit larger than that of real experiments, but not have significant effect on the general performance.

Next, the results of the hysteresis behaviors, the cracking patterns obtained from the FEM analysis were compared with that measured from the experimental study to verify the effectiveness of the FEM modeling. Experimental details, and FEM analysis results, as well as the comparison between the experimental test and FEM analysis, were described in Section 3.

2.3. Parametric Study

Since limited specimens were tested in Inai et al.’s experimental tests [11,12], a parametric study using the calibrated FEM models developed in this work was performed to consider a wider range of potential scenarios.

A model corresponding to Specimen 2 was adopted as the standard model to establish the parametric models, considering only one target parameter varied to different values while other properties of the wall-slab joint were kept identical for each series of models. Three parameters of (i) the slap span length, (ii) the slab width, and (iii) the slab flexural reinforcement distribution, were varied and examined in the parametric simulation. Based on the results of the parametric simulation, the equivalent cross-sectional area ratio β was accordingly adjusted. Details of the parametric numerical simulation and related discussions were presented in Section 4.

3. FEM Modeling and Results

In this section, the FEM analysis modeling was validated against the experimental results obtained from Inai et al.’s experimental test [11,12]. Consequently, the stress condition of each slab flexural reinforcing steel rebar was checked carefully and thus applied to the equivalent cross-sectional area concept for calculating the value of $\beta$.

3.1. Validation of Finite-Element Modeling

The FEM analysis results were validated against the measured results obtained from the experimental test. Figure 13a displays the concrete elements while Figure 13b shows the reinforcement for Models 1 and 2 based on Specimens 1 and 2, respectively. The main difference between the two models is the amount and distribution of shear reinforcement in the slab. All other dimensions and reinforcement details are the same.

Figure 14 and Figure 15 show the comparison of hysteretic response up to a drift of 4% for Model 1 and 2, respectively. Here, both experimental and FEM analysis results showed that the subassembly undergoes a ductile response. The maximum applied force reached was considered as the flexural strength of the subassembly. Consequently, the lateral force began to descend slightly after the flexural strength reached was caused by the crushing of concrete at the compression side. The maximum lateral force measured at the top of the wall from experimental results was 121.9 kN and 117.9 kN at an inter-story drift ratio of 1.5% for Specimens 1 and 2, respectively. In contrast, it was 140.6 kN, and 134.4 kN at the same inter-story drift ratio from the corresponding FEM results. As such, the FEM analysis overestimated the flexural strength by 15.3% and 14.0% for Specimens 1 and 2, respectively. Reasons for the FEM model predicting larger strengths are considered to be caused by the early stage cracking, microcracking, shrinkage, or other material deficiencies and bond-slip adhesion which may have occurred in the real experiment, but could not be captured successfully in the FEM modeling.

In addition, a pinching behavior representing cyclic stiffness and dissipated hysteretic energy degradation could be observed in the experimental hysteretic curves, but was not captured in the FEM results. The pinching effect might be associated with effects that were difficult to model such as the increase in shear capacity, crack opening and closing, shear lock, slippage of reinforcing bars from concrete, etc. However, it can be seen that the pinching behavior did not bring a big influence on the flexural strength. Thus, the result that the FEM analysis not simulating the pinching effect for the wall-slab joints could be ignored when discussing the flexural strength in this work.

Figure 16 shows the crack patterns at the maximum drift ratio (deformation angle) captured and the comparison of the experimental test and FEM analysis for Model 1. The cracking patterns at the top and bottom surfaces of the slab obtained from the experimental test are illustrated in Figure 16a, those obtained from FEM analysis results are illustrated in Figure 16b, while the comparison between those of the experimental test and FEM analysis are shown in Figure 16c. In these figures, the yellow color represents softened (post-peak stress) concrete elements while the red color represents post-softened concrete elements.

It can be seen that for Specimen 1 (Figure 16), even though more cracks are observed on the surface in the FEM analysis than those captured in the experimental test, cracks are mainly observed distributed over the entire surface, particularly close to the corner of the core connection in both, which is considered as, in the FEM analysis, all cracks can be captured while in the actual experimental test only the cracks wide enough that can be seen and recorded by observers are captured, which might make cracks recorded in experimental test results fewer than those obtained from the FEM analysis results, and the same distribution trend can be found where these cracks are starting at the corner of connection of bearing wall and slab, then extending outward to slab edge in both FEM analysis and experimental test results as shown in Figure 16c. Similar findings were observed for Specimen 2 as shown in Figure 17. Thus, even though the FEM results recorded more microcracks than those measured in the experimental test, similar crack locations and patterns were observed in both.

Based on these observations and comparisons, the behaviors simulated in the FEM modeling can be considered relatively representative of their actual performance of specimens in the laboratory. As such, the FEM modeling can be utilized consequently to give an applicable approximation of the simulated seismic performance of the thick wall-thick slab joint for a parametric study.

3.2. Application of Equivalent Cross-Sectional Area Concept on Experimental Results

The stress and yield condition of slab reinforcement in Model 1 and 2 corresponding to Specimen 1 and 2 [11,12] obtained from FEM analysis results were applied to calculate the equivalent cross-sectional area ratio $\beta$.

As shown in Figure 18 and Figure 19, the yield condition and stress of the elements located at the surface of the wall-slab connection for each flexural reinforcing steel bar (marked by black dotted line border) when the flexural strength reached for Models 1 and 2 at 1.5% drift ratio in the positive loading direction are illustrated, respectively. It can be observed that for each model, the slab flexural reinforcing rebars located closer to the bearing wall had already yielded while several reinforcing rebars located at the edge region of the slab have not yielded, either for the top mat on one face or the bottom mat on the opposite face. The calculated β based on the observed stress condition was obtained as 0.98 and 0.96 for Models 1 and 2, respectively. Even though there is not a big difference to estimate the flexural strength using the original formula as shown in Equation (1) by considering all slab reinforcement accounted because of the values of $\beta$ showing close to and almost equal to 1.00, there is a trend suggesting the less yield and stress for reinforcing steel bars as located further from the joint core. In other words, the flexural strength could be predicted more accurately by considering the effective cross-sectional area ratio. Thus, a further investigation considering the effects of various parameters using the calibrated FEM modeling was described in the following section.

4. Parametric FEM Analysis

This section discussed the influence of (i) the slab width, (ii) the slab span length, (iii) and reinforcement distribution in the slab by adopting a parametric FEM analysis.

4.1. Slab Span Length

Six models namely Models S1~S6 with different span lengths of 1500 mm, 1900 mm, 2300 mm, 2700 mm, 3100 mm, and 3500 mm were modeled to discuss the factor of slab span length. Figure 20 shows the relationship between the lateral force and lateral drift ratio for each model. It can be observed that all these models showed a typical flexural failure pattern. Additionally, in these specimens, as the slab span length increased, the measured lateral force from FEM analysis slightly decreased.

The stress and yield conditions of slab flexural reinforcing rebars at the deformation angle when the ultimate flexural moment reached for each model are illustrated in Figure 21a–f. It can be seen that for models with different slab span lengths, slab flexural reinforcing rebars located close to the joint core were observed almost all yield while other rebars located away from the joint core and near the slab edge were not yielded yet, which indicated a reducing trend of the stress as the reinforcing bar gets farther from the joint core. However, even though stress values were not identical, the yield condition did not show a significant difference for these models with different slab span lengths.

Table 1. FEM results for slab span length models.

Model	Slab Span Length (mm)	Simulation Lateral Force (kN)	Corrected Simulation Lateral Force (kN)	$\mathit{\beta }$	Calculated Lateral Force (kN)
S1	1500	134.40	114.70	0.94	103.17
S2	1900	126.90	108.30	0.98	107.71
S3	2300	118.30	100.96	0.98	108.03
S4	2700	116.20	99.17	0.99	108.70
S5	3100	111.00	94.73	0.97	107.11
S6	3500	110.30	94.13	0.97	107.06

summarizes values of the simulation flexural strength, the corrected flexural strength considering the overestimation of the FEM simulation, the corresponding calculated values of $\beta$, and the calculated flexural strength using Equation (6) considering $\beta$ for each model. It can be seen that values of $\beta$ were observed close and ranging from 0.94~0.99, which was in coincidence with that observed in the stress condition of reinforcing bars. Thus, the calculated value is not identical, but still close to that measured in the FEM analysis for all models. Additionally, a comparison between relationships of slab span length-corrected simulation flexural strength and slab span length-calculated flexural strength for slab span length models is shown in Figure 22. It can be seen that even though there was still a slight deficiency between the calculated and simulated values, the calculated values by applying the $\beta$ coefficient can give a reasonable and conservative prediction for the flexural strength for these models.

4.2. Slab Width

Nine models with different values of slab width ranging from 350 mm to 1850 mm were established to discuss the effect on the slab flexural behavior. Figure 23 shows the relationship between lateral force and drift ratio while Figure 24a–i presents the stress and yield conditions of slab longitudinal rebars at the deformation angle when the ultimate flexural moment reached for each model. It could be observed that lateral strength increased while the amount of lateral strength increment reduces as the slab width increased, which indicates that increasing the slab width indeed brings a positive influence on the lateral strength. It can be seen that when the slab width is smaller, such as Model W1~Model W3, almost all slab longitudinal rebars yielded. When the slab width increases moderately, such as Model W4~Model W6, the stresses of slab longitudinal rebars located further from the joint core were observed reducing. When the slab was much wider, such as Model W7~Model W9, a similar trend of the slab longitudinal rebars were observed.

Table 2. FEM results for slab width models.

Model	Slab Span Length (mm)	Simulation Lateral Force (kN)	Corrected Simulation Lateral Force (kN)	$\mathsf{\beta }$	Calculated Lateral Force (kN)
W1	350	73.64	60.41	1.03	55.15
W2	450	94.29	80.47	1.03	79.41
W3	650	116.10	99.08	1.02	101.69
W4	850	135.80	114.70	0.97	117.85
W5	1050	156.40	134.47	0.99	141.95
W6	1250	162.00	134.33	0.89	147.95
W7	1450	172.50	147.21	0.79	148.48
W8	1650	173.09	148.41	0.67	140.86
W9	1850	175.50	149.77	0.57	130.93

summarizes values of the simulated flexural strength, the corrected flexural strength considering the overestimation of the FEM simulation, the corresponding calculated values of $\beta$, and the calculated flexural strength using Equation (6) considering $\beta$ for each model. The value of $\beta$ showed an evident reduction from 1.03 to 0.57 as the slab width increased (the amount of reinforcement as well increased but the reinforcing spacing was kept the same), which was in coincidence with observations obtained from the stress condition of reinforcing bars. Additionally, a comparison between relationships of slab span length-corrected simulation flexural strength and slab span length-calculated flexural strength for slab span length models is shown in Figure 25. It can be known that even though there still is a slight difference between the calculated and simulated values, the calculated values can give a reasonable prediction for flexural strength, which is in coincidence with the above-mentioned observations.

Based on these results, the slab width does bring a significant influence on the stress and yield conditions of slab bending reinforcement. As the slab is being wider, slab flexural reinforcing rebars located far from the joint core are not being yielded and their stress decreases as the distance is further. Thus, the slab width should be treated as a significant influencing factor on slab flexural behavior and consequently taken into account in $\beta$ estimation for predicting the flexural strength of the interior wall-slab joints.

4.3. Slab Flexural Reinforcement Distribution

The slab flexural reinforcement distribution was investigated here by arranging the distribution of reinforcing bars, but the amounts of slab longitudinal reinforcing bars including bars within the flat wall-beam scope were kept the same. Five models Model R1~R5 with different distributions of reinforcement in the flat wall beam in which numbers of steel bar arranging from 6 to 14 and corresponding numbers of steel bar in the slab arranging from 14 to 6 were established. Figure 26 shows the relationship between lateral force and lateral drift ratio. It can be seen that the flexural strength shows no significant deficiency for all these models, which indicates that the different distributions of reinforcement in the flat wall beam and slab components do not bring effect on the flexural behavior.

The stress and yield conditions of slab longitudinal rebars at the deformation angle when the ultimate flexural moment reached for each model are illustrated in Figure 27a–e. It can be observed that for models with different slab reinforcement distribution, even though the steel bars located close to the joint core showed larger stress than those located away from the joint core, the yield condition and stress distribution did not show a big difference.

Table 3. FEM results for reinforcement distribution models.

Model	Steel Bar in Flat Wall Beam (No.)	Steel Bar in Slab (No.)	Corrected Simulation Lateral Force (kN)	$\mathsf{\beta }$	Calculated Lateral Force (kN)
R1	6	14	132.50	0.94	103.17
R2	8	12	134.50	0.94	97.18
R3	10	10	134.40	0.88	103.17
R4	12	8	135.60	0.94	103.17
R5	14	6	140.20	0.99	109.28

summarizes values of the flexural strength obtained from the FEM simulation, the corrected flexural strength considering the overestimation of the FEM simulation, the corresponding calculated values of $\beta$, and the calculated flexural strength using Equation (6) considering $\beta$ for each model. It can be seen that values of $\beta$ were observed close and ranging from 0.88~0.99, which was in coincidence with that observed in the stress condition of reinforcing bars. Additionally, a comparison between relationships of slab span length-corrected simulation flexural strength and slab span length-calculated flexural strength for slab span length models is shown in Figure 28. It can be known from these values that even though with the slight difference, the calculated values can give a reasonable conservative prediction for flexural strength, which is in coincidence with the above-mentioned observations.

Based on these results, it can be considered that the reinforcement distribution mostly does not affect the flexural strength and is not taken into account for predicting the flexural strength under the lateral force of wall-slab joints in TWTS structures.

5. Flexural Strength Prediction

Based on the parametric analysis results, only the slab width showed a big difference at the moment transferred from slab to the joint, which determined the beam component flexural strength. Therefore, a relationship between the slab width and $\beta$ can thus be derived as shown in Figure 29, in which $b_s$ represents the slab width. It can be observed that when $b_s$ was less than about 1.0 m, $\beta$ was approximately 1.0, when $b_s$ was greater than 1.0 m, $\beta$ was roughly inversely proportional to $b_s$. Thus, the relationship between $b_s$ and $\beta$ can be simply approximated using two lines: one horizontal line where $\beta$ almost equals to 1.0, and another decreasing line capturing the reduction in $\beta$ as $b_s$ increases. This can be considered that when the slab width is less than 1.0 m (approximately calculated using curve fitting when $\beta$ equals 1 and corresponding to the intersection point in Figure 29), the behaviors of slab longitudinal reinforcement as similar to beam flexural reinforcement in general column-beam joints, in which total reinforcement yield inside both the wall scope, slab beam, even side slabs are taken into account when calculating the flexural strength. In other words, when the slab width is less than 1.0 m, the moment provided by slab flexural behavior to the joint can be considered as general beams and predicted by considering all slab longitudinal reinforcement. However, when the slab width is getting larger than 1.0 m, the behaviors of the slab longitudinal reinforcement are distributed to a large region of the slab, which makes the steel bars located far away from the joint core not yield so that the value of $\beta$ begins to decrease.

As a result, the equivalent cross-sectional area ratio β can be given by Equation (7) using a two-line model based on the results obtained from the parametric study on slab width.

$$\beta =\{\begin{array}{c}1.0, b_s<1.0 m\hfill \\ -0.53·\left(b_s\right)+1.55, b_s\ge 1.0 m\hfill \end{array},$$

(7)

It is noticed that because the models applied in the parametric study are 1/2 scaled, the results obtained need to be adjusted considering the scale. Thus, Equation (7) in practical design should be adjusted to Equation (8).

$$\beta =\{\begin{array}{c}1.0, b_s<2.0 m\hfill \\ -0.53·\left(b_s\right)+1.55, b_s\ge 2.0 m\hfill \end{array},$$

(8)

Thus, the value of $\beta$ obtained by this two-line model can be applied to Equation (3) to correct the total cross-sectional area of all slab flexural reinforcement $a_t$ substituted by the equivalent gross cross-sectional area $a_e$ for providing a more reasonable and conservative estimation of the flexural strength of the interior wall-slab joints in TWTS structures.

6. Conclusions

In this study, a concept termed the equivalent cross-section area has been proposed to evaluate the slab flexural strength of the interior wall-slab joints in TWTS structures. Additionally, the results of the FEM analysis of the interior wall-slab joints were calibrated according to their reference data from an experimental study conducted by Inai et al. [11,12]. The FEM analysis shows that it is reasonable to effectively estimate the actual structural behaviors of wall-slab joints with a certain correction coefficient for flexural strength. Thus, a parametric study based on the calibrated FEM modeling has been performed in order to investigate the influence of several slab properties of the slab span length, the slab width, and the slab flexural reinforcement distribution on the flexural strength provided by slab bending. The main findings of this work are:

• The slab span length and the slab flexural reinforcement distribution showed a slight influence, but did not appear to have a significant impact on the flexural strength of interior wall-slab joints.
• The slab width is found to significantly influence the flexural behavior and should be taken into account in predicting the moment transferred from slab to the joint and increasing the slab width leads to an increment of the value of the effective cross-sectional area ratio $\beta$.
• The coefficient $\beta$ considering the influence of slab width can be approximately modeled by a two-line model and for use in predicting the flexural strength of the interior wall-slab joints in TWTS structures.

This study provides the basic concept of a building type in Japan that was newly developed in recent years, as well as proposed a reasonable method for predicting the flexural strength using the FEM analysis approach calibrated according to actual experimental tests from literature. Besides, it presents a new conceptual approach to discuss the flexural behavior considering the reinforcement contribution of slab components in building structures.