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Bulletin of Earthquake Engineering

, Volume 17, Issue 12, pp 6621–6644 | Cite as

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

  • Koichi KusunokiEmail author
  • Masanobu Sakashita
  • Tomohisa Mukai
  • Akira Tasai
S.I.: Nonlinear Modelling of Reinforced Concrete Structural Walls

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

Experimental test data from the Journal of Structural and Construction Engineering of the Architectural Institute of Japan (referred to as AIJ, hereafter), Journal of Structural Engineering of AIJ, Proceedings of the Japan Concrete Institute (referred to as JCI, hereafter), and Concrete Research and Technology of JCI between 1975 to 2013 were sourced. 510 papers on shear walls were obtained in total. However, only 217 papers were included in the database as others lacked the necessary information required for comparisons and where did not publish information in other publications (i.e. Summaries of Technical Papers for Annual Meeting, Research Reports of AIJ). The number of papers and published year is shown in Fig. 1.
Fig. 1

Number of papers related to shear wall members and their published year

684 wall specimens were collated in total from the 217 papers included in the database, regardless of the presence of boundary columns or wall openings. However, several cases were filtered from the database. For example, walls without boundary columns which have a depth to width ratio of less than 2.5 were excluded as they behave more like columns. Other specimens filtered from the database were:
  • 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).

After filtering of the specimens, a total of 507 remained. These were categorized into three groups based on whether boundary columns were present or if there were wall openings as shown in Table 1. Average values of major parameters of specimens in the database are summarized in Table 2.
Table 1

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

Table 2

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, r0

 

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, r0, which is calculated from Eq. (1) (AIJ 2010). The shear stiffness reduction factor, r1, and the shear strength reduction factor, r2, are calculated from Eqs. (2) and (3), respectively. Based on Japanese regulations, if the equivalent opening area ratio of r0 is greater than 0.4, the wall should be modeled as a frame instead without the need for considering r1 and r2.

$$ r_{0} = 1.1 \times \sqrt {\frac{{\sum {h_{0} l_{0} } }}{hl}} $$
(1)
$$ r_{1} = 1 - 1.25r_{0} $$
(2)
$$ r_{2} = 1 - {\mathrm{max}} \left( {r_{0} ,1.1 \times \frac{{l_{0} }}{l}, {\lambda}\frac{{{\sum}{h_{0}} }}{{\sum}{h}}} \right) $$
(3)
where r0: equivalent opening area ratio; l0: projected horizontal length of opening; h0: projected vertical length of opening; h: height of the wall; and l: whole length of the wall including boundary columns.
Figure 2 shows the number of specimens with openings binned according to their r0. The number of specimens with r0 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.
Fig. 2

Number of specimens for each equivalent opening area ratio

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, Kf, and initial shear stiffness, Ks. Kf is calculated by firstly determining the lateral deformation which involves double integrating the curvature obtained from the moment demand, geometrical inertia moment, Iw, and Young’s Modulus, Ec (Eq. (10)), at 50 sections divided along the wall’s height, and then dividing the corresponding lateral force by the calculated displacement. Iw 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. Ks is calculated from Eq. (11), while \( \kappa_{e} \) is calculated based on the energy concept using Eq. (13).

$$ K = \frac{{K_{f} \cdot K_{s} }}{{K_{f} + K_{s} }} $$
(4)
$$ I_{w} = (1 + (n_{c} - 1)p_{g} )I_{c} + (1 + (n_{p} - 1)p_{wv} )I_{p} - (1 + (n_{p} - 1)p_{wv} )I_{o} $$
(5)
$$ I_{c} = 2 \times \frac{1}{12}bD^{3} + A_{c} (L_{c1} - L_{e} )^{2} + A_{c} (L_{c2} - L_{e} )^{2} $$
(6)
$$ I_{p} = \frac{1}{12}t_{w} l_{w}^{3} + A_{w} (L_{w} - L_{e} )^{2} $$
(7)
$$ I_{o} = \sum {\left( {\frac{1}{12}t_{w} l_{o}^{3} + A_{o} \left( {L_{o} - L_{e} } \right)^{2} } \right)} $$
(8)
$$ L_{e} = 0.5L - \frac{{A_{c} (L_{c1} + L_{c2} )(1 + (n_{c} - 1)p_{g} ) + A_{w} L_{w} (1 + (n_{p} - 1)p_{wv} ) - \sum {A_{o} L_{o} (1 + (n_{p} - 1)p_{wv} )} }}{{2 \times A_{c} (1 + (n_{c} - 1)p_{g} ) + A_{w} (1 + (n_{p} - 1)p_{wv} ) - \sum {A_{o} (1 + (n_{p} - 1)p_{wv} )} }} $$
(9)
where \( I_{w} \): geometrical moment inertia of wall with considering steel bars; \( I_{c} \), \( I_{p} \), \( I_{o} \): geometrical moment inertia of boundary column, wall plate, and opening, respectively; \( n_{c} \), \( n_{p} \): ratio of Young’s modulus of steel to concrete in boundary column, and in wall plate, respectively; \( p_{g} \), \( p_{wv} \): rebar ratio of boundary column, and vertical reinforcement ratio of wall plate, respectively; 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.
$$ E_{c} = 33500 \times \left( {\frac{\gamma }{24}} \right)^{2} \times \left( {\frac{{\sigma_{B} }}{60}} \right)^{1/3} $$
(10)
$$ K_{s} = r_{1} \times \frac{{G_{c} A_{all} }}{{\kappa_{e} }} $$
(11)
$$ G_{c} = \frac{{E_{c} }}{2(1 + \nu )} $$
(12)
where \( E_{c} \): Young’s modulus of concrete (MPa); \( \gamma \): unit weight of concrete (= 23 kN/m3); \( \sigma_{B} \): 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).
$$ \begin{aligned} \kappa _{e} & = \frac{{72(1 + 2\alpha \beta )}}{{\left\{ {1 + 2\alpha \beta ^{3} + 6\alpha \beta (1 + \beta )^{2} } \right\}^{2} }}\left\{ {\frac{8}{{15}}\alpha \left( {\beta + \frac{1}{2}} \right)^{5} - \frac{1}{2}\alpha (1 - \alpha )\left( {\beta + \frac{1}{2}} \right)^{4} } \right. \\ & \quad \left. { + \frac{1}{4}\alpha (1 - \alpha )\left( {\beta + \frac{1}{2}} \right)^{2} + \frac{1}{4}(1 - \alpha )\left( {\frac{1}{{15}} - \frac{1}{8}\alpha } \right)} \right\} \\ \end{aligned} $$
(13)
$$ \alpha = \frac{b}{{t_{w} }} $$
(14)
$$ \beta = \frac{D}{{l_{w}^{\prime } }} $$
(15)
where \( A_{all} \): whole wall area; \( \kappa_{e} \): shape factor according to energy concept; b, D: width and depth of boundary column, respectively; \( t_{w} \): wall thickness; and \( l_{w}^{\prime } \): clear length of the wall.
A total of 123 wall specimens in the database had initial stiffness values provided and were considered in this evaluation. The comparison between experimental and calculated initial stiffness is shown in Fig. 3 and summarized in Table 3. Here, the experimental result was lower than the calculated stiffness, and the ratio between the two was 0.72 on average. The coefficient of variation was also large. One possible explanation is that the initial stiffness calculated following Eq. (4) was assuming elastic response. On the contrary, the stiffness of the specimen may be degraded due to dry shrinkage, which may have resulted in the lower observed initial stiffness.
Fig. 3

Comparison of initial stiffness from experimental test and Eq. (4)

Table 3

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.

$$ M_{wu} = 0.9a_{t} \sigma_{y} D + 0.4a_{w} \sigma_{wy} D + 0.5ND\left( {1 - \frac{N}{{BDF_{c} }}} \right) $$
(16)
$$ M_{wu} = a_{t} \sigma_{y} l_{w} + 0.5a_{w} \sigma_{wy} l_{w} + 0.5Nl_{w} $$
(17)
where \( M_{wu} \): ultimate moment of wall; 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 (mm2); \( a_{w} \): total area of vertical reinforcement in the wall web (mm2); and N: vertical load (N).

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

$$ a_{t} = \left\{ {\begin{array}{*{20}l} {p_{c} Bl_{pt} } \hfill & {l_{pt} \le D_{c} } \hfill \\ {p_{c} BD_{c} + p_{w} B(l_{pt} - D_{c} )} \hfill & {l_{pt} > D_{c} } \hfill \\ \end{array} } \right. $$
(18)
where Dc: boundary column depth, pc: vertical flexural reinforcement ratio within each boundary column, pw: vertical reinforcing ratio outside of boundary column area, and lpt: length of the region where reinforcing bars contributes in tension.
For rectangular walls without boundary columns, Dc was taken as the length of the region where the flexural reinforcement is concentrated, and the regions for calculating pc and pw 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.
Fig. 4

Confined area at both ends of the rectangular wall

For this assessment, the maximum restoring moment recorded during experimental testing was compared against the ultimate moments calculated from Eqs. (16) and (17). Only specimens which either failed in (i) flexure, or (ii) shear after first yielding in flexure were considered. 209 specimens matched this criterion, where 129 walls had boundary columns without openings, 47 were rectangular without openings, and 33 had openings. The comparison between experimental and calculated ultimate moment is shown in Fig. 5 and summarized in Table 4.
Fig. 5

Comparison of lateral strength at flexural yielding from experimental test and Eqs. (16) and (17)

Table 4

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)

 

(a) Equation (16)

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

(b) Equation (17)

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.

On the contrary, the ratios of some specimens with openings were much lower than 1.0 as shown in Fig. 5, which indicate that the equations overestimate the flexural strength. Figure 6 shows the relationship between the ratios of experimental to calculated lateral force with Eqs. (16) and (17), and the equivalent opening ratio calculated with Eq. (1). Here, the accuracy of the calculation decreases as the equivalent opening ratio becomes greater. One reason is that a wall with a big opening would behave more like a frame than a single monolithic wall, resulting in the total lateral restoring force decreasing. As mentioned previously, the Japanese building regulations require that walls with an opening ratio greater than 0.4 be modelled as frames (columns and beams), which appears reasonable in light of results shown in Fig. 6.
Fig. 6

Effect of the openings on the accuracy of the ultimate moment

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.

$$ Q_{wsu} = r_{2} \times \left\{ {\frac{{0.053p_{te}^{0.23} (F_{c} + 18)}}{M/(Q \cdot D) + 0.12} + 0.85\sqrt {\sigma_{wh} \cdot p_{wh} } + 0.1\sigma_{0} } \right\}t_{e} \cdot j $$
(19)
$$ Q_{wsu} = r_{2} \times \left\{ {\frac{{0.068p_{te}^{0.23} (F_{c} + 18)}}{{\sqrt {M/(Q \cdot D) + 0.12} }} + 0.85\sqrt {\sigma_{wh} \cdot p_{wh} } + 0.1\sigma_{0} } \right\}t_{e} \cdot j $$
(20)
where \( Q_{wsu} \): ultimate shear strength of the wall (N); \( p_{te} \): equivalent tensile rebar ratio; \( p_{te} \): \( 100a_{t} /\left( {t_{e} \cdot d} \right) \), which is calculated with Eq. (18) for rectangular walls; \( t_{e} \): equivalent wall thickness so that the equivalent wall has the same length and area (mm, no more than 1.5 times of the thickness); \( d \): effective depth (mm, D − Dc/2 for walls with boundary columns, 0.95D for other rectangular walls); Dc: 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 te; \( \sigma_{0} \): 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)).
The maximum restoring force recorded from experimental tests was compared against Eqs. (19) and (20) considering only specimens which failed in shear without prior flexural yielding. The number of the specimens which fit this criterion was 275, where 115 walls had boundary columns with no openings, 52 were rectangular without any openings, and 108 walls had openings. The comparison between experimental and calculated ultimate shear strength is shown in Fig. 7 and summarized in Table 5.
Fig. 7

Comparison of ultimate shear strength from experimental test and Eqs. (19), (20) and (21)

Table 5

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

(a) Equation (19) for a minimum value

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

(b) Equation (20) for mean value

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

(c) Equation (21)

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, hw, and the wall length, lw, (ii) the concrete strength, and (iii) the lateral bar’s strength are all within a given range. However, the hw/lw requirement is not considered in this study, while the requirements for the other two parameters are described in the following paragraph.

$$ V_{n} = V_{c} + V_{s} = Min\left\{ {3.3\lambda \sqrt {f^{\prime}_{c} } hd + \frac{{N_{u} d}}{{4l_{w} }},\left(0.6\lambda \sqrt {f^{\prime}_{c} } + \frac{{l_{w} \left( {1.25\lambda \sqrt {f^{\prime}_{c} } + 0.2\frac{{N_{u} }}{{l_{w} h}}} \right)}}{{\frac{{M_{u} }}{{V_{u} }} - \frac{{l_{w} }}{2}}}\right)hd} \right\} + \frac{{A_{v} f_{yt} d}}{s} $$
(21)
where \( V_{n} \): nominal shear strength of the wall (lb); \( V_{c} \): nominal shear strength of the wall provided by concrete (lb); \( V_{s} \): nominal shear strength of the wall provided by shear reinforcement (lb); \( \lambda \): modification factor to reflect the reduced mechanical properties of lightweight concrete; \( f^{\prime}_{c} \): compressive strength of concrete, (psi, \( \sqrt {f^{\prime}_{c} } \) ≤ 100); 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

The secant shear stiffness at the ultimate shear strength can be obtained with the shear stiffness degrading factor (Sugano 1973), \( \beta_{u} \), which is calculated following the empirical function shown in Eq. (22). If the wall has openings, the secant stiffness at the ultimate shear strength is reduced by r1 while the ultimate shear strength is reduced by r2, 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 Kf, 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.
Fig. 8

Shear deformation at the shear failure of shear wall with opening

$$ \beta_{u} = 0.46p_{w} \sigma_{y} /F_{c} + 0.14 $$
(22)
where \( \beta_{u} \): shear stiffness degrading factor at shear failure; \( p_{w} \): wall reinforcement ratio; \( \sigma_{y} \): yield strength of wall reinforcement (kg/cm2); and \( F_{c} \): concrete strength (kg/cm2).
Similarly to the evaluation of ultimate shear strength, this evaluation will only consider walls that failed in shear prior to yielding in flexure. The number of the specimens is 136 in total, where 62 walls had boundary columns without any openings, 16 were rectangular without openings, and 58 walls had openings. The comparison between deformation at the maximum restoring force recorded from the experiment and the calculated deflection angle is shown in Fig. 9 and summarized in Table 6.
Fig. 9

Comparison of deflection angle at ultimate shear strength from experimental test and Eq. (20)

Table 6

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 r1 and r2 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.

A further evaluation was performed considering the restoring force and deformation corresponding to \( {{\upalpha}} \)% of the maximum strength along with the lateral force vs deformation envelope prior to the peak strength being reached. Since no digital data is available, the deformation was read directly from the figures in the original research papers. These are then normalized by the predicted ultimate shear strength and deformation values, and the results for the walls without opening are summarized in Table 7. Here, the calculated ultimate shear strength and deformation are consistent with the average test results at \( {{\upalpha}} \) of 70% and 80%. The calculated shear deformation becomes smaller than the experimental results at \( {{\upalpha}} \) of 90% and 100%.
Table 7

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).

$$ \alpha_{y} = \frac{{{}_{w}M_{y} \cdot C_{n} }}{{EI_{w} \cdot \varepsilon_{y} }} $$
(23)
where \( {}_{w}M_{y} \): ultimate moment of the wall calculated with Eq. (16); \( C_{n} \): distance between neutral axis to the center of tensile boundary column (mm, derived from section analysis); 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) (mm4); and \( \varepsilon_{y} \): yield strain of rebar in the boundary column.
The deformation at the yield moment is calculated based on yielding at the rebar in the boundary column as shown by the use of εy 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.
Fig. 10

Comparison of deflection angle at yielding from experimental test and Eq. (23)

Table 8

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

The ultimate deflection angle of the specimen recorded from the experiment, Ru,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.
Fig. 11

Ultimate deflection angle from experimental test result (expRu)

The relationships between Ru,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.
Fig. 12

Relationship between shear strength margin and ultimate deflection angle

Fig. 13

Relationship between shear stress ratio and ultimate deflection angle

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

In order to evaluate the accuracies of the functions commonly used in Japanese engineering practice to estimate initial stiffness, ultimate moment, ultimate shear strength, deflection angles at ultimate moment and ultimate shear strength of shear wall, a database of experimental tests on shear walls was developed, and the properties of these walls were compared against predicted values using the functions. In conclusion, it was found that:
  1. 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. 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. 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. 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. 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. 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. 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. 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

  1. American Concrete Institute (2014) Building code requirements for structural concrete (ACI318-14) and commentary (ACI 318R-14)Google Scholar
  2. Architectural Institute of Japan (2010) AIJ standard for structural calculation of reinforced concrete structures—based on allowable stress method—revised 2010Google Scholar
  3. Building Research Institute (2016) Investigation on structural performance evaluation of RC members using comprehensive experimental database. Building Research Data, No. 175Google Scholar
  4. Hirosawa M, Goto T (1971) Strength and ductility of reinforced concrete column under gravity load, part 2 study with database. IN: Proceedings of AIJ annual meeting, vol 46, pp 819–820 (in Japanese) Google Scholar
  5. National Institute for Land and Infrastructure Management (NILIM), Building Research Institute (BRI) (2011) Commentary on structural regulations in the building standard law of JapanGoogle Scholar
  6. Sugano S-S (1973) Hysteresis model of reinforced concrete members—experimental study on strength and stiffness of girders, columns and walls without openings that fail in flexural manner. Concr J 11(2):1–9 (in Japanese) Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Earthquake Research InstituteThe University of TokyoTokyoJapan
  2. 2.National Institute for Land and Infrastructure ManagementTsukuba CityJapan
  3. 3.Building Research InstituteTsukuba CityJapan
  4. 4.Yokohama National UniversityYokohama CityJapan

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