# Study on the accuracy of practical functions for R/C wall by a developed database of experimental test results

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## Abstract

Pushover analysis is widely used in modern practical structural design to evaluate collapse modes and member forces at the safety limit state. All members, including shear walls, need to be modeled properly to assure the accuracy of pushover analysis. As such, functions to predict initial stiffness, crack strength, yield or ultimate strength, stiffness degrading ratio to the yielding point, ultimate deformation, and ultimate shear strength must be acceptably accurate. While several theoretical, semi-theoretical, and empirical functions have been proposed and are commonly used in practice, the accuracies of these functions have not been verified comprehensively. The Ministry of Land, Infrastructure, Transportation and Tourism and the Building Research Institute launched a project in Japan from 2012 to 2015 to develop a comprehensive database of experimental data from 507 reinforced concrete wall experiments conducted between 1975 to 2013 to confirm the accuracy of the functions. In this paper, the comparisons between values predicted using the functions against experimental results from the database are presented. It was found that (i) the predicted initial stiffness tends to be higher than that from experimental tests, (ii) the predicted ultimate flexural and shear strengths were generally smaller than those from experimental tests, and its accuracy appears to be dependent on the presence of boundary columns and the existence of openings, and (iii) the predicted deformation corresponding to yield and at ultimate shear strength tends to be smaller than those from experimental tests.

## Keywords

Experimental test data R/C wall Practical design Database## 1 Introduction

Before computer software became a common tool in structural design, structural analyses were generally conducted by hand using virtual work or nodal moment distribution methods. It is impossible to capture the deterioration progress or deformation of structures with these methods, as only the stress or restoring force distribution within the structure can be calculated. Furthermore, as these methods depend on the initial stiffness and ultimate strengths of structural members, these approaches were not very accurate. This process is also time-consuming to perform, particularly for middle to high-rise building structures.

In 1981, a new structural calculation method called the “horizontal load-carrying capacity method” was added to the Japanese building standard law enforcement order for evaluating and verifying the failure mechanism and horizontal load-carrying capacity of a structure. Due to the need for complicated non-linear analysis to meet this requirement, the usage of computer software in Japan propelled in the 1980s.

Pushover analysis is widely used in modern engineering practice to evaluate the mode of collapse and restoring forces acting in each member at the ultimate stage. The minimum required ductility capacity and the horizontal load-carrying capacity of the entire structure is defined in the code, and is dependent on the ductility and failure mode of each member. Thus, there is a great need to perform numerical analysis, and all members including shear walls need to be modeled properly to assure the accuracy of pushover analysis. As such, the predicted initial stiffness, crack strength, yield and ultimate strength, stiffness degrading ratio to the yielding point, ultimate deformation, and ultimate shear strength must be acceptably accurate.

Since the 1970s, many experimental tests with columns, beams, beam-column joints, and shear walls were conducted. A few of these experimental results had been used to develop empirical functions to model the non-linear behavior of structural members, such as member strength and deformation properties. However, the accuracy of these functions had not been verified comprehensively considering a larger group of specimens, and thus there is a possibility that the functions are not applicable for some combinations of parameters. Moreover, high strength materials, which are now commonly used in practice, were not considered when the majority of these functions were developed. There is, therefore, a need to verify the applicability of these functions for a larger range of cases.

The Ministry of Land, Infrastructure, Transportation, and Tourism and the Building Research Institute launched a project in Japan from 2012 to 2015 to develop a comprehensive database of experimental test data between 1975 to 2013 to confirm the accuracy of the empirical functions commonly used in practice (BRI 2016). The databases for columns, beams, columns with wing walls, beams with standing and/or hanging walls, beam-column joints, and shear walls were developed separately.

This paper will focus on evaluating the functions commonly used in Japan for both linear and nonlinear analytical modeling of reinforced concrete walls in both new and existing buildings using the database for shear walls. Firstly, the development of the database is outlined. Then, the accuracy of each function, including Japanese and ACI functions, is evaluated. This paper will consider both specimens which failed in flexure (ratio of lateral forces corresponding to peak shear and peak flexural strength greater than 1.25) or in shear (ratio of lateral forces corresponding to peak shear and peak flexural strength less than 1.25).

## 2 Development of shear wall database

Specimens of which concrete strengths were greater than 60 MPa, which is the highest concrete strength allowed by the building code (39 specimens),

Specimen with different concrete strengths for wall and boundary columns (1 specimen),

Specimens of which sizes and/or bar arrangements of both boundary columns were different (5 specimens),

Specimens with circular openings (37 specimens),

Specimen of which the scale was extremely small (3% of full scale, 1 specimen),

Specimens with X-shape bar arrangements (11 specimens),

Specimens that had uncommon bar arrangements (3 specimens),

Specimens loaded bilaterally (24 specimens),

Specimens under eccentric vertical load (29 specimens),

Specimens under varied vertical load (13 specimens), and

Specimens with varied shear span ratio during the loadings (14 specimens).

Number of specimens in the database

Group | Boundary columns | Openings | Number of specimens |
---|---|---|---|

1 | Present | None | 254 |

2 | None | None | 102 |

3 | Present | Present | 144 |

None | Present | 7 | |

Total | 507 |

Major parameters of specimens in the database

Items | Unit | Range | Average | Number of specimens | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Total | 1. W/o Opening but w/boundary col. | 2. W/o opening nor boundary col. | 3. W/opening | Total | 1. W/o Opening but w/boundary col. | 2. W/o opening nor boundary col. | 3. W/opening | ||||||

Material | Concrete strength | MPa | 12.7–59.5 | 28.0 | 28.7 | 27.2 | 27.3 | 506 | 253 | 102 | 151 | ||

Column (confined area) | Main bar strength | MPa | 244–1044 | 421 | 405 | 553 | 363 | 502 | 254 | 97 | 151 | ||

Hoop strength | MPa | 160–1423 | 379 | 346 | 500 | 383 | 441 | 236 | 60 | 145 | |||

Wall | Vertical reinforcement strength | MPa | 160–882 | 352 | 319 | 414 | 365 | 505 | 253 | 101 | 151 | ||

Horizontal reinforcement strength | MPa | 160–1423 | 361 | 319 | 463 | 365 | 505 | 253 | 101 | 151 | |||

Dimensions | Scale | 0.07–1.0 | 0.31 | 0.25 | 0.41 | 0.36 | 279 | 141 | 45 | 93 | |||

Total wall length, L | mm | 300–4740 | 1430 | 1489 | 884 | 1699 | 507 | 254 | 102 | 151 | |||

Clear height of wall, H | mm | 280–4900 | 1223 | 1207 | 1190 | 1284 | 507 | 254 | 102 | 151 | |||

H/L | 0.36–4.00 | 0.97 | 0.83 | 1.60 | 0.77 | 507 | 254 | 102 | 151 | ||||

Wall thickness, tw | mm | 19–250 | 74 | 58 | 126 | 65 | 507 | 254 | 102 | 151 | |||

Wall thickness/boundary column width | 0.10–0.67 | 0.31 | 254 | ||||||||||

Wall length w/o boundary column/wall length | 0.67–0.95 | 0.78 | 254 | ||||||||||

Wall length/wall thickness | 2.5–26.7 | 7.3 | 102 | ||||||||||

Opening area ratio, r | 0.16–0.82 | 0.39 | 151 | 151 | |||||||||

Bars | Boundary col. (confined area) | Equivalent tensile bar ratio | W/boundary columns | % | 0.09–3.54 | 0.85 | 0.77 | 254 | 150 | ||||

W/o boundary col. (confined area w/depth of tw) | % | 0.02–1.13 | 0.39 | 99 | |||||||||

Hoop ratio | % | 0.00–1.60 | 0.58 | 0.64 | 0.42 | 0.59 | 492 | 243 | 98 | 151 | |||

Wall | Vertical reinforcement ratio, pvw | % | 0.00–3.75 | 0.65 | 0.59 | 0.81 | 0.64 | 504 | 254 | 99 | 151 | ||

Horizontal reinforcement ratio, phw | % | 0.00–1.76 | 0.62 | 0.60 | 0.63 | 0.66 | 507 | 254 | 102 | 151 | |||

pvw/phw | 0.1–3.6 | 0.97 | 0.96 | 0.97 | 0.97 | 498 | 252 | 81 | 151 | ||||

Loading | Shear span ratio | 0.21–2.96 | 1.01 | 0.98 | 1.19 | 0.95 | 506 | 253 | 102 | 151 | |||

Axial force ratio | − 0.40–0.61 | 0.07 | 0.05 | 0.08 | 0.08 | 505 | 253 | 102 | 150 | ||||

Shear stress ratio | 0.02–0.33 | 0.14 | 0.17 | 0.11 | 0.13 | 487 | 243 | 99 | 145 |

According to the Japanese regulations, walls with opening must have its stiffness and shear strength decreased by a function of the equivalent opening area ratio, *r*_{0}, which is calculated from Eq. (1) (AIJ 2010). The shear stiffness reduction factor, *r*_{1}, and the shear strength reduction factor, *r*_{2}, are calculated from Eqs. (2) and (3), respectively. Based on Japanese regulations, if the equivalent opening area ratio of *r*_{0} is greater than 0.4, the wall should be modeled as a frame instead without the need for considering *r*_{1} and *r*_{2}.

*r*

_{0}: equivalent opening area ratio;

*l*

_{0}: projected horizontal length of opening;

*h*

_{0}: projected vertical length of opening;

*h*: height of the wall; and

*l*: whole length of the wall including boundary columns.

*r*

_{0}. The number of specimens with

*r*

_{0}greater than 0.4 is 71 out of 151, which will be included in this study, too. The applicability of this limit will also be evaluated later in this paper.

## 3 Accuracy of empirical functions

### 3.1 Initial stiffness

The initial wall stiffness, *K*, is defined as the initial slope of the relationship between applied lateral force and lateral deformation. *K* is calculated with Eq. (4), which is a function of initial flexural stiffness, *K*_{f}, and initial shear stiffness, *K*_{s}. *K*_{f} is calculated by firstly determining the lateral deformation which involves double integrating the curvature obtained from the moment demand, geometrical inertia moment, *I*_{w}, and Young’s Modulus, *E*_{c} (Eq. (10)), at 50 sections divided along the wall’s height, and then dividing the corresponding lateral force by the calculated displacement. *I*_{w} is calculated with Eq. (5) which considers the contribution of rebar in boundary columns and vertical reinforcement along with the wall web. The assumption that the plane section remains plane after the deformation is applied regardless of whether wall openings were present. *K*_{s} is calculated from Eq. (11), while \( \kappa_{e} \) is calculated based on the energy concept using Eq. (13).

*b*,

*D*: width and depth of boundary column, respectively; \( t_{w} \), \( l_{w} \): wall thickness and clear length of wall plate, respectively; \( l_{o} \),

*L*, \( L_{e} \): length of opening, whole length of wall, eccentric length of wall, respectively; \( L_{c1} \), \( L_{c2} \): distances from compression edge to center of columns in compression and tension sides, respectively; \( L_{w} \), \( L_{o} \): distance from compression edge to the center of the wall plate and opening, respectively; and \( A_{c} \), \( A_{w} \), \( A_{o} \): areas of boundary column, wall plate, and opening, respectively.

^{3}); \( \sigma_{B} \): compression strength of concrete (MPa); \( G_{c} \): shear modulus of elasticity of concrete (MPa); \( A_{all} \): total area of wall; \( \kappa_{e} \): shape factor; and \( \nu \): Poisson’s ratio of concrete (= 1/6).

*b*,

*D*: width and depth of boundary column, respectively; \( t_{w} \): wall thickness; and \( l_{w}^{\prime } \): clear length of the wall.

Comparison of elastic stiffness

(1) Wall w/boundary columns w/o opening | (2) Wall w/o boundary column nor opening | (1) + (2) | (3) Wall w/opening | (1) + (2) + (3) | ||
---|---|---|---|---|---|---|

Num. of specimen | 51 | 38 | 89 | 34 | 123 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 0.67 | 0.63 | 0.65 | 0.88 | 0.72 |

SD | 0.27 | 0.19 | 0.24 | 0.27 | 0.26 | |

Coef. of variation | 0.40 | 0.30 | 0.36 | 0.30 | 0.37 |

The predicted initial stiffnesses disperse widely and tend to be stiffer than the test results. It leads to a more lateral force to the wall and less deformation. It is well known that the secant stiffness to the yield point is more effective to the nonlinear lateral deformation than the initial stiffness.

### 3.2 Ultimate flexural moment capacity

The ultimate flexural moment can be calculated from either Eq. (16) or (17) (NILIM and BRI 2011), which were derived with the assumption that the plane section always remains plane.

*D*: whole wall length (mm);

*B*: width of the compression edge; \( l_{w} \): distance between the centers of boundary columns or 90% of whole wall length for rectangular-shaped walls (mm); \( F_{c} \): concrete strength (MPa); \( \sigma_{y} \): tensile strength of rebar in boundary column (MPa); \( \sigma_{wy} \): yield strength of vertical reinforcement in the wall (MPa); \( a_{t} \): total area of tensile bars in the boundary column (mm

^{2}); \( a_{w} \): total area of vertical reinforcement in the wall web (mm

^{2}); and

*N*: vertical load (N).

The whole area of tensile bars, \( a_{t} \), can be calculated using Eq. (18).

*D*

_{c}: boundary column depth,

*p*

_{c}: vertical flexural reinforcement ratio within each boundary column,

*p*

_{w}: vertical reinforcing ratio outside of boundary column area, and

*l*

_{pt}: length of the region where reinforcing bars contributes in tension.

*D*

_{c}was taken as the length of the region where the flexural reinforcement is concentrated, and the regions for calculating

*p*

_{c}and

*p*

_{w}are as shown in Fig. 4. Furthermore, there are several assumptions of \( l_{pt} \) commonly used in practice, such as half of the wall thickness, full wall thickness, or 10% of the whole wall length, the latter of which was adopted in this study.

Comparison of ultimate moment

(1) Wall w/boundary columns w/o opening | (2) Wall w/o boundary column nor opening | (1) + (2) | (3) Wall w/opening | (1) + (2) + (3) | ||
---|---|---|---|---|---|---|

| ||||||

Num. of specimen | 129 | 47 | 176 | 33 | 209 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 1.08 | 1.08 | 1.08 | 0.79 | 1.03 |

Standard deviation (SD) | 0.16 | 0.12 | 0.15 | 0.34 | 0.22 | |

Coef. of variation | 0.15 | 0.11 | 0.14 | 0.43 | 0.21 | |

| ||||||

Num. of specimen | 129 | 47 | 176 | 33 | 209 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 1.09 | 0.99 | 1.06 | 0.80 | 1.02 |

SD | 0.16 | 0.13 | 0.16 | 0.35 | 0.22 | |

Coef. of variation | 0.15 | 0.13 | 0.15 | 0.44 | 0.22 |

The ratios of experimental to calculated lateral force using Eqs. (16) and (17) were almost identical for the walls without opening, of which the average is 1.08 and 1.06, respectively. As shown in Fig. 5, the calculated values were generally within ± 20% of the experimental value.

Circular markers in Fig. 5 represent wall specimens with high strength longitudinal reinforcing bars (greater than or equal to 600 MPa, with the highest being 713 MPa). Although the number of specimens in this category is limited (3 specimens), the results show that Eqs. (16) and (17) are still reasonably accurate. As such, these equations can potentially be used for the walls with high strength lateral reinforcement.

The predicted ultimate flexural moment capacities of shear walls, especially those without openings, are slightly less than the test results, and the dispersion is small. The accuracy is acceptable for practical design. The number of experimental tests with high strength materials is not enough, and more experimental tests are required.

### 3.3 Ultimate shear strength

Equations (19) and (20) (NILIM and BRI 2011) are widely used in Japanese practice to predict the ultimate wall shear strength. The equations are empirically derived from experimental tests by Hirosawa and Goto (1971). Equations (19) and (20) are used to estimate the minimum and mean ultimate shear strength value, respectively.

*D*−

*D*

_{c}/2 for walls with boundary columns, 0.95

*D*for other rectangular walls);

*D*

_{c}: boundary column depth (mm); \( F_{c} \):concrete strength (MPa); \( M/(Q \cdot D) \): shear span ratio (\( 1 \le M/Q \cdot D \le 3 \));

*D*: whole wall length (mm); \( \sigma_{wh} \): yield strength of lateral reinforcement (MPa); \( p_{wh} \): lateral reinforcement ratio with equivalent wall thickness of

*t*

_{e}; \( \sigma_{0} \): vertical load ratio (MPa);

*j*: distance between centers of tension and compression (taken as (7/8)

*d*); and \( r_{2} \): strength reduction factor due to opening (Eq. (3)).

Comparison of ultimate shear strength

(1) Wall w/boundary columns w/o opening | (2) Wall w/o boundary column nor opening | (1) + (2) | (3) Wall w/opening | (1) + (2) + (3) | |||||
---|---|---|---|---|---|---|---|---|---|

W/hoop | W/o hoop | Planar column | Total | ||||||

| |||||||||

Num. of specimen | 115 | 7 | 30 | 15 | 52 | 167 | 108 | 275 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 1.75 | 1.96 | 1.36 | 1.51 | 1.49 | 1.67 | 1.85 | 1.74 |

SD | 0.38 | 0.35 | 0.25 | 0.17 | 0.31 | 0.38 | 0.37 | 0.38 | |

Coef. of variation | 0.22 | 0.18 | 0.18 | 0.11 | 0.21 | 0.23 | 0.20 | 0.22 | |

| |||||||||

Num. of specimen | 115 | 7 | 30 | 15 | 52 | 167 | 108 | 275 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 1.40 | 1.55 | 1.11 | 1.28 | 1.22 | 1.34 | 1.50 | 1.41 |

SD | 0.31 | 0.26 | 0.22 | 0.17 | 0.25 | 0.31 | 0.31 | 0.32 | |

Coef. of variation | 0.22 | 0.17 | 0.19 | 0.13 | 0.21 | 0.23 | 0.21 | 0.23 | |

| |||||||||

Num. of specimen | 116 | 7 | 30 | 15 | 52 | 168 | – | – | |

\( \frac{Experimental}{Calculation} \) | Ave. | 2.38 | 2.39 | 1.18 | 1.51 | 1.44 | 2.09 | – | – |

SD | 0.91 | 0.87 | 0.33 | 0.24 | 0.58 | 0.93 | – | – | |

Coef. of variation | 0.38 | 0.36 | 0.28 | 0.16 | 0.40 | 0.44 | – | – |

Standard deviation and coefficient of variation of the ratios of experimental to calculated ultimate shear strength were almost identical regardless if Eq. (19) or (20) were used, and were also similar for both walls with and without openings. However, the average ratio was around 30% higher for Eq. (19) compared with Eq. (20), which is reasonable since the former is used to estimate the minimum strength value as mentioned previously. Another observation which can be made is that the average of the ratio for walls with opening was only marginally higher than that for walls without opening. As both cases had similar margins of error, the strength reduction factor due to the presence of openings, \( r_{2} \), is reasonable. The average of the ratio for rectangular walls without hoops in both ends is lower than the others. Overall, the majority of the examined specimens had exceeded the calculated strength using Eq. (19). Only 3 rectangular wall specimens had lower strength than expected, all of which had lateral reinforcement ratios lower than the minimum allowable ratio of 0.25% specified in code requirements.

The circular markers in Fig. 7 represent specimens which use high strength lateral reinforcing bars (greater than or equal to 600 MPa, with the greatest being 1423 MPa). Although the number of the specimens is small (14 specimens), the results show that the predicted ultimate shear strengths with Eqs. (19) and (20) have almost the same error margin as the specimen with the normal strength. As such, these equations are just as conservative for the walls with high strength lateral reinforcement as it is for walls with lower strength reinforcing.

ACI 318-14 provides another equation which can be used to predict the ultimate shear strength of walls without openings, as shown in Eq. (21) (ACI 2014). Note that this equation can only be used if (i) the ratio between the wall height, *h*_{w}, and the wall length, *l*_{w}, (ii) the concrete strength, and (iii) the lateral bar’s strength are all within a given range. However, the *h*_{w}/*l*_{w} requirement is not considered in this study, while the requirements for the other two parameters are described in the following paragraph.

*h*: overall thickness of the wall (in.);

*d*: \( 0.8l_{w} \) (in.); \( N_{u} \): factored axial force normal to cross-section (lb); \( l_{w} \): length of entire wall (in.); \( M_{u} /V_{u} \): shear span ratio (in.); \( A_{v} \): area of shear reinforcement within spacing s (in.

^{2}); \( f_{yt} \): yield strength of transverse reinforcement, (psi, \( f_{yt} \) ≤ 60,000 psi); \( s \):center-to-center spacing of transverse reinforcement.

The comparison of experimental to predicted ultimate wall shear strength using Eq. (21) is shown in Fig. 7c. Here, Eq. (21) also conservatively underestimates the ultimate shear strength. Table 5 (c) shows that the difference in average and SD values between the experimental/calculation ratio for (i) walls with confinement at their both ends with tie or with boundary columns and (ii) other walls was greater compared to using Eqs. (19) and (20). This is due to the effect of longitudinal bars and boundary column on the ultimate shear strength being explicitly considered in Eqs. (19) and (20), but not in Eq. (21).

The predicted ultimate shear strengths are less than the test results and disperse widely. For the practical design, it means conservative, but it is acceptable because shear failure is brittle and need to be avoided.

### 3.4 Deformation at the ultimate shear strength

*r*

_{1}while the ultimate shear strength is reduced by

*r*

_{2}, as shown in Fig. 8. Equation (20) is applied for calculating ultimate shear strength in this study since this provides an estimate of the mean strength value. The flexural deflection is calculated as the deformation corresponding to the initial flexural stiffness

*K*

_{f}, and the secant stiffness degrading factor calculated by Eq. (23), which is explained in detailed later. The flexural deflection is added to the calculated deflection angle.

^{2}); and \( F_{c} \): concrete strength (kg/cm

^{2}).

Comparison of deflection angle at the ultimate shear strength

(1) Wall w/boundary column w/o opening | (2) Wall w/o boundary column nor opening | (1) + (2) | (3) Wall w/opening | (1) + (2) + (3) | ||
---|---|---|---|---|---|---|

Num. of specimen | 62 | 16 | 78 | 58 | 136 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 2.09 | 3.26 | 2.33 | 2.29 | 2.32 |

SD | 0.84 | 1.63 | 1.14 | 0.88 | 1.04 | |

Coef. of variation | 0.40 | 0.50 | 0.49 | 0.38 | 0.45 |

The deformations recorded from the experiments were much larger than the calculated values. However, the coefficients of variation were almost identical as shown in Table 6 regardless of the presence of boundary columns or openings, and thus *r*_{1} and *r*_{2} are reasonable since the resulting level of accuracy is consistent. As shown in Fig. 9a, the ultimate shear deformation for most specimens were calculated in the narrow band of 1/500 to 1/200 (0.2% to 0.5%). Furthermore, the comparisons for specimens with high strength steel (circular markers in Fig. 9a) had the same tendency as those with normal strength steel.

One important note is that the accuracy of the calculated ultimate shear deformation is highly dependent on the accuracy of calculated ultimate shear strength, as shown in Fig. 8. Since the calculated ultimate shear strength using Eq. (20) is about 40% higher than the experimental results as shown in Fig. 9b, the maximum restoring force and its deformation are normalized by the calculated ultimate shear strength and the calculated ultimate shear deformation, respectively, and their relationship is shown in Fig. 9b. However, no significant tendency was observed regardless if high strength steel was used. Nonetheless, the recorded displacements of the majority of specimens exceeded the calculated value, with only 2 specimens for walls with boundary columns without openings and 4 specimens for walls with openings having lower ultimate shear deformations than expected.

Comparison of ultimate shear strength and its deflection angle against pre-peak behavior of wall specimens w/o opening

70% of the ultimate strength | 80% of the ultimate strength | 90% of the ultimate strength | Ultimate strength | ||||||
---|---|---|---|---|---|---|---|---|---|

Deflection angle | Restoring force | Deflection angle | Restoring force | Deflection angle | Restoring force | Deflection angle | Restoring force | ||

Num. of specimen | 102 | 102 | 103 | 103 | 104 | 104 | 110 | 110 | |

\( \frac{Experimental}{Calculation} \) | Ave. | 0.92 | 0.93 | 1.21 | 1.06 | 1.57 | 1.19 | 2.37 | 1.32 |

SD | 0.39 | 0.17 | 0.51 | 0.19 | 0.69 | 0.22 | 1.24 | 0.25 | |

Coef. of variation | 0.42 | 0.18 | 0.42 | 0.18 | 0.44 | 0.18 | 0.53 | 0.19 |

The predicted deformations at the ultimate shear strengths are less than the test results and disperse widely. For the practical design, it leads that the lateral capacity of the building at the shear failure becomes smaller, which is on the safety side.

### 3.5 Deformation at the yield moment

The flexural deformation at flexural yielding is calculated with the initial stiffness and the secant stiffness degrading factor shown in Eq. (23) (NILIM and BRI 2011).

*E*: Young’s modulus of concrete calculated with Eq. (10); \( I_{w} \): geometrical moment inertia of wall with considering steel bars calculated with Eq. (5) (mm

^{4}); and \( \varepsilon_{y} \): yield strain of rebar in the boundary column.

*ε*

_{y}in Eq. (23). As shear deformation is generally predominant for walls and not negligible, the deflection angle at yielding is calculated as the total deflection angle, which is added to the shear deflection angle. The shear deformation is taken as the deformation corresponding to the lateral force at the ultimate moment on the calculated shear behavior model shown in Fig. 8. Specimens where the outermost rebar yields (38 specimens) or where all tensile rebar yields (12 specimens) are considered in this evaluation. The comparison between the experimental and predicted deformations at the yielding moment are shown in Fig. 10 and summarized in Table 8. Here, Eq. (23) underestimates the deformation at the yield moment. This is because the secant shear stiffness is also underestimated as mentioned earlier. In order directly to confirm the accuracy of Eq. (23), experimental test data on flexural deformation is necessary.

Comparison of deflection angle at flexural yielding

Outermost rebar yields | All tensile rebar yield | ||
---|---|---|---|

Num. of specimen | 38 | 12 | |

\( \frac{Experimrental}{Calculation} \) | Ave. | 1.53 | 1.69 |

SD | 0.49 | 0.43 | |

Coef. of variation | 0.32 | 0.25 |

The predicted deformations at the yield moment are less than the test results and disperse widely. It leads to a more lateral force before flexural yielding, which is not on the safe side.

### 3.6 Ultimate deflection angle

*R*

_{u,exp}, is digitized from the force–displacement relationship curves retrieved from relevant papers in the database. The ultimate deflection angle is the angle when the restoring force of the wall dropped down to 80% of the maximum strength. The smaller angle among positive and negative directions was applied. If the deflection angle when the strength dropped to 80% of the maximum had been reached, this value is taken as the ultimate deflection angle as shown in Fig. 11a. Linear interpolation is performed for other cases as shown in Fig. 11b.

*R*

_{u,exp}and key parameters for the ultimate deflection angle, shear strength margin (ratio of lateral forces corresponding to ultimate shear strength and the ultimate flexural strength), and shear stress ratio (ratio of shear stress to concrete strength) are shown in Figs. 12 and 13, respectively, considering only cases where the failure mode was either “shear failure after yielding” or “flexural failure”. Equations (16) and (20) were used for calculating the ultimate flexural strength and ultimate shear strength, respectively, in order to derive the shear strength margin.

From Fig. 12a, b, the ultimate deflection angle becomes larger as the shear margin increases due to flexural failure, which is more ductile than shear failure, becoming more predominant. It was also found that if the shear strength margin is greater than 1.25, which is code requirement to ensure flexural failure mode, the ultimate deflection angle is able to exceed 15/1000 which is the minimum deflection capacity required according to the Japanese code. However, a handful of test results (shown by the dotted circle marker in Fig. 12b did not exceed 15/1000 despite the shear strength margin being greater than 1.5. These are cases where the specimen had no confinement in both ends of the wall, or where vertical bars had yielded in compression prior to tension.

In Fig. 13, the ultimate deflection angle becomes smaller as the shear stress ratio increases. It can also be seen that the shear stress ratio of walls with boundary columns tended to be greater than that of the rectangular wall.

## 4 Concluding remarks

- 1.
The predicted initial stiffness was greater than the experimental test results. The average and the coefficient of variation of the ratio between the experimental test result and the predicted value is 0.72 and 0.37, respectively.

- 2.
The prediction of the ultimate moment showed good accuracy for shear walls without openings as the ratio between experimental and calculated values was in the range of 80% to 120%. On the contrary, the predicted ultimate moment of walls with openings is overestimated as the walls behaved more like frames than a single monolithic wall.

- 3.
The average ratios between the ultimate shear strengths recorded from experimental tests with the predicted value for both minimal and mean values were 174% and 141%, respectively, indicating that the prediction provides a conservative estimation of the actual ultimate shear strength.

- 4.
The predicted shear deflection angle at the ultimate shear strength is smaller than that recorded from the experimental studies. The restoring force and deformation when the restoring force reached 70 to 80% of the maximum restoring force on the envelope curve agree well with the calculated values.

- 5.
The calculated total deflection angle at flexural yielding is smaller than the experimental result, which might be caused by the underestimation of the shear deformation.

- 6.
The flexural strength with high strength steel bars can be predicted accurately with conventional functions. The predicted ultimate shear strength with high strength steel bars have the same error margin as that of the specimen with normal strength steel bars.

- 7.
The ultimate deflection angle becomes larger than 15/1000 if the shear strength margin to the flexural yielding is greater than 1.25.

- 8.
The shear stress ratio of walls with column tends to be larger than that of the rectangular wall to reach the same shear deformation.

## Notes

### Acknowledgements

The database used in this study was developed under a project funded by the Ministry of Land, Infrastructure, Transportation and Tourism in 2012 and 2013 and also by Building Research Institute.

## References

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