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1
BEHAVIOR AND DESIGN OF FIN WALLS AT THE PERIPHERY OF 1
STRONG CORE WALL STRUCTURE IN HIGH-RISE BUILDINGS 2
3
Taesung Eoma, Su-Min Kangb*, Seung-Yoon Yuc, Jae-Yo Kimd, Dong Kwan Kime 4
5
aDepartment of Architectural Engineering, Dankook Univ., 152 Jukjeon-ro, Gyeonggi-do, 6
Korea, 448-701, Email: [email protected], M: +82-10-3296-9678. 7
bDepartment of Architectural Engineering, Chungbuk National University, Chungdae-ro 1, 8
Seowon-Gu, Cheongju, Chungbuk, Korea, 361-763, *Corresponding Author, Email: 9
[email protected], M: +82-10-9208-4116. 10
cDepartment of Architectural Engineering, Chungbuk National University, Chungdae-ro 1, 11
Seowon-Gu, Cheongju, Chungbuk, Korea, 361-763, [email protected], M: +82-10-12
6674-4180 13
dDepartment of Architectural Engineering, Kwangwoon University, Nowon-gu, Seoul, Korea, 14
139-701, [email protected], M: +82-10-3215-9437. 15
eDepartment of Architectural Engineering, Cheongju University, Daeseong-ro 298, 16
Cheongwon-gu, Cheongju, Chungbuk, Korea, [email protected], M: +82-10-2886-8216. 17
18
Biography: 19
Tae-Sung Eom is an Associate Professor of the department of architectural engineering at 20
Dankook University in South Korea. He received his BE, MS, and PhD in architectural 21
engineering from Seoul National University. His research interests include nonlinear analysis 22
and seismic design of RC structures. 23
2
Su-Min Kang is an assistant professor of the department of architectural engineering at 24
Chungbuk National University in South Korea. He received his BE, MS, and PhD in 25
architectural engineering from Seoul National University. His research interests include 26
earthquake design of reinforced concrete structures. Corresponding author. 27
Seung-Yoon Yu is a graduate student of the department of architectural engineering at 28
Chungbuk National University in South Korea. He received his BE in architectural 29
engineering from Chungbuk National University. His research interests include earthquake 30
design of reinforced concrete structural walls. 31
Jae-Yo Kim is a professor of the department of architectural engineering at Kwangwoon 32
University in South Korea. He received his BE, MS, and PhD in architectural engineering 33
from Seoul National University. His research interests include inelastic FE analysis of 34
reinforced concrete structures and serviceability of RC flat-plate system. 35
Dong Kwan Kim is an assistant professor of the department of architectural engineering at 36
Cheongju University in South Korea. He received his BE, MS, and PhD in architectural 37
engineering from Seoul National University. His research interests include earthquake design 38
of reinforced concrete structures and seismic wave generation for earthquake design. 39
40
Abstract 41
In high-rise buildings, lateral loads, such as wind and seismic loads, are frequently resisted by 42
reinforced concrete (RC) structural walls. Behavior and design of fin walls at the periphery of 43
strong core wall structure in high-rise buildings were not analyzed seriously despite their 44
structural importance. Using elastic design/analysis methodologies for the design of high-rise 45
RC fin walls, it is shown that elicited reinforcement ratios are too high, the economic 46
feasibility and constructability thus becomes worse and the ductile failure mode can not be 47
3
assured. In the present study, the current design process of these fin walls is investigated by 48
analyzing their structural behavior. According to the investigation of the current design and 49
elicited results, high-rise RC fin walls are coupled by beams although they are located on 50
another line and apart from each other, which is main cause of high reinforcement in high-rise 51
RC fin walls. In the present study, a literature review has been conducted to recommend the 52
alternative design method for high-rise RC fin walls under lateral loads, and inelastic analysis 53
has been performed to verify the design method. 54
Keywords: High-rise RC structural walls, Fin walls, Alternative design, Inelastic analysis, 55
Coupling beam, High-rise building design 56
57
1. Introduction 58
In high-rise buildings, lateral loads, such as wind and seismic loads, are frequently resisted by 59
reinforced concrete (RC) structural walls [1~3]. Since RC structural walls have large stiffness 60
and strength, they can control deformation and efficiently assure the strength of the building 61
structure. In the structural design of RC building structures, the RC structural walls are 62
designed as independent single walls (isolated walls) or coupled walls connected by 63
connecting beams which are located closely on a straight line. However, in the high-rise RC 64
building structures, in many cases, the structural RC walls were located on another line and 65
apart from each other, and they are designed generally as independent single walls (isolated 66
walls) without the consideration for their structural behavior. 67
Fig. 1 shows a typical example of a structural system for a 35 story residential building. As 68
shown in Fig. 1, the gravity load is resisted by the RC flat plate slab and the RC column. 69
Currently, RC flat plates are used frequently as a slab system since they can decrease the floor 70
height and construction time, and increase space utilization and constructability. As shown in 71
4
Fig. 1(b), the lateral load is frequently resisted by RC structural walls since they have large 72
stiffness and strength owing to their long length. Fig. 2 shows the plan of the building drawn 73
in Fig. 1. As mentioned above, RC structural walls are located in the center of the plan to 74
resist the lateral load. In the plan, some RC structural walls extend from the center to the 75
boundary of the plan, which were called as “Fin walls” (Fin walls indicate planar walls such 76
as W1 ~ W5 which were installed along the perimeter of the rectangular core wall structure, 77
as shown in Fig. 2). These RC structural walls, fin walls, are used frequently to divide the 78
space for architectural design and to control the lateral deflection of the high-rise building 79
because they can enlarge effectively the moment inertia of the building system. Generally 80
they are located on another line and apart from each other, and they are designed as 81
independent single walls without the sufficient consideration for their structural behavior. 82
However, as shown in Table 1, the reinforcement ratio of these RC fin walls at floor B3 83
ranges from 0.59% to 3.38%. In accordance to the current structural design code, such as ACI 84
318-14 [4], the reinforcement ratio of the RC structural walls is limited to 1% when they are 85
not enclosed by transverse ties. Therefore, some of these structural walls violate the 86
requirement prescribed by the design code if they are not enclosed by transverse ties. 87
Furthermore, in the cases where the reinforcement ratio is more than 1%, its value is 88
considered to be too high to assure economic feasibility and constructability. Moreover, the 89
application of the transverse ties to the RC structural walls having more than 1% 90
reinforcement ratio decreases significantly the constructability. And, if the flexural 91
reinforcement in high-rise RC structural walls was placed excessively without the 92
consideration for their structural behavior, it may lead to unexpected brittle shear failure of 93
high-rise RC structural walls. In this study, structural behavior of fin walls installed along the 94
periphery of strong core wall structure in high-rise buildings were studied. Elastic analysis of 95
5
a 35-story high-rise building with core walls and fin walls was performed and the design 96
loads acting on each fin wall such as axial load and bending moment were investigated. 97
Further, flexure-compression design of the fin walls were carried out and the reason why the 98
design results (i.e. reinforcement amounts) significantly differed among the fin walls was then 99
discussed in detail. Based on these investigations, their alternative design method was 100
investigated based on literature review. And this alternative design method was verified using 101
the latest structural analysis method for RC structural wall system. 102
103
2. Investigation on the results from the current design 104
As mentioned above, the values of the flexural reinforcement ratios of RC fin walls that 105
extend from the center to the boundary of the plan are too high (Table 1). In this study, 106
applied loads and strengths of the walls W2 and W3 at floor B3 that yielded the largest 107
flexural reinforcement ratio, were investigated through elastic analyses using the Midas Gen 108
program [5]. Recently, new theories on the structural analysis of beams and plates have been 109
developed to improve accuracy and convergence. Bourada, M., Kaci, A., Houari, M.S.A. and 110
Tounsi, A. (2015) developed a simple and refined trigonometric higher-order beam theory for 111
functionally graded beams [6]. Tounsi et al. (2016) proposed a new 3-unknowns non-112
polynomial plate theory for functionally graded sandwich plate [7]. Hebali et al (2014) 113
developed a new quasi-3D hyperbolic shear deformation theory for functionally graded plates 114
[8]. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016) devised a novel five variable refined 115
plate theory for functionally graded sandwich plates [9]. Ait Yahia, S., Ait Atmane, H., 116
Houari, M.S.A. and Tounsi, A. (2015) investigated wave propagation in functionally graded 117
plates [10]. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Bég, O. 118
(2014) invented an efficient and simple higher order shear and normal deformation theory 119
6
[11]. Houari, M.S.A., Tounsi, A., Bessaim, A., and Mahmoud, S.R. (2016) developed a new 120
simple three-unknown sinusoidal shear deformation theory for functionally graded plates 121
[12]. Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015) proposed a new hyperbolic shear 122
deformation theory for functionally graded, sandwich and laminated composite plates [13]. 123
However, in this study, to investigate the overall behavior of high-rise RC structural wall 124
system in an overall framework and to use commercial program, the Midas Gen program [5] 125
which is FEM program using conventional beam theory was used. 126
In Fig. 3, the P-M relationship curves of walls W2 and W3 at floor B3 are shown as the 127
strength capacity of these wall sections. Moreover, round dots in Fig. 3(b) and 3(c) represent 128
the applied axial load and moment according to load combinations. As shown in Fig. 3(b) and 129
3(c), the flexural reinforcement ratio of walls W2 and W3 was determined based on the load 130
case, including the wind load (W1, W2), which is a common phenomenon in high-rise 131
buildings. Therefore, walls W2 and W3 are designed to resist the wind load as the critical 132
lateral load. In Fig. 3(b) and 3(c), the axial load is proportional to the moment load. This is 133
not a typical feature for isolated RC structural walls in general. In Fig. 4, the axial load 134
applied on walls W2 and W3 was investigated according to the load case. As shown in this 135
figure, the axial load of wall W2 is inversely proportional to that of wall W3. This means that 136
wall W2 and W3 are coupled with each other by the coupling beams, although the two walls 137
are physically located far apart from each other and on another line. 138
As shown in Fig. 4(b), walls W2 and W3 are connected by center walls and coupling beams. 139
Since center walls have relatively large stiffness, the coupling level of walls W2 and W3 may 140
be determined by the stiffness of the coupling beams. To verify this coupling behavior of 141
walls W2 and W3, their strain distribution under the wind load was investigated according to 142
the stiffness of the coupling beams (Fig. 5). In the present study, strain distribution of coupled 143
7
walls under the wind load was calculated by cross-sectional analysis. The cross-sectional 144
analysis was based on the plane hypothesis and the equilibrium condition. The stress-strain 145
relation for the concrete and steel rebar was assumed to be linear because, in the majority of 146
cases, the maximum concrete stress was less than half the concrete strength and the maximum 147
tensile stress of steel rebar was much less than the yield stress. As shown in Fig. 5, the strain 148
distribution of walls W2 and W3 with stronger coupling beams under the wind load shows an 149
increased linearity in the distribution of strain. This means that as the stiffness of the coupling 150
beams becomes large, the walls W2 and W3 will behave as one unified wall. Therefore, the 151
walls W2 and W3 are coupled by the coupling beams although they are located far apart with 152
respect to each other and on another line, and the stiffness of the coupling beams is a key 153
factor that determines the coupling level. 154
As shown in Fig. 6(a), when the walls are coupled by coupling beams, these beams 155
experience large deformation under lateral loads owing to the geometric shape of beams and 156
walls. Because of this large deformation elicited under a lateral load, large shear forces occur 157
in coupling beams. Correspondingly, these forces are then transferred to the coupled walls as 158
an axial load [1]. In the case of high-rise buildings, the number of coupling beams is very 159
large and the axial load that is transferred to the coupled walls is also large. As shown in Fig. 160
6(b), when the wind load is applied from the left direction to the right, the additional axial 161
tensile load owing to the coupling beams is applied to wall W2, and the additional axial 162
compressive load exerted by the coupling beams is applied to wall W3. As shown in Fig. 6(c), 163
when an additional axial tensile load is applied to RC structural walls, the moment capacity of 164
these walls decreases and additional flexural reinforcement is needed to resist the applied 165
moment load. As shown in Fig. 6(d), sometimes, when the additional axial compressive load 166
that is applied is too high, the strength reduction factor for the structural RC walls would be 167
8
decreased. Therefore, in this case, additional flexural reinforcement or high-strength concrete 168
should be used. Accordingly, the high flexural reinforcement ratio of the walls (which extend 169
from the center to the boundary of the plan) is caused by the additional axial load from the 170
coupling of the beams. Moreover, these additional axial loads are relatively very high in value 171
because of the large deformation of the coupling beams, shown in Fig. 6(a). 172
However, the additional axial load from the coupling beam, which is calculated as part of the 173
elastic analysis, is larger than the value elicited from the actual behavior since the strength of 174
the coupling beam is limited by the actual details of the beams [1, 14]. Therefore, 175
consideration of the actual behavior of the coupled wall system can lead to a proposition of a 176
reasonable design of these fin walls. 177
178
3. Actual behavior and reasonable design of coupled fin walls 179
To investigate the actual behavior of the coupled wall system, a literature review and an 180
inelastic analysis was performed. According to Paulay and Priestley [1], the strength of a 181
coupled wall system under lateral load can be determined by controlling the strength of 182
individual walls as indicated below: 183
𝑀 = 𝑀1 +𝑀2 + 𝑙𝑆 × 𝑇 (1)
where 𝑀 is the strength of the coupled wall system, 𝑀1(𝑀2) is the strength of the individual 184
wall with additional axial tensile(compressive) load, 𝑇 is the axial load caused by the 185
coupling beams and 𝑙𝑠 is the length between the centroids of the coupled walls (See Fig.7). In 186
the coupled wall system, the shear force generated by the coupling beams was transferred to 187
the axial load of the wall. Generally, as shown in Fig. 7, the shear force (V) generated by the 188
coupling beams is limited by the moment strength of coupling beams (Mb,i) because the beams 189
are designed to yield prior to failure by shear. Therefore, the axial load in the wall base (T) 190
9
can be calculated by summating the shear force of the coupling beams at each story as 191
follows[1]: 192
𝑇 =∑𝑀𝑏,𝑖/2𝑙𝑏
𝑛
𝑖=1
(2)
where 𝑀𝑏,𝑖 is the moment strength of the coupling beams, 𝑙𝑏 is the beam length and 𝑛 is the 193
number of beam. 194
Therefore, the axial load transferred to the walls can be determined from the strength of 195
the coupling beams. According to Paulay and Priestley [1], Paulay [14, 15], and Santhakumar 196
[16], the moment loading of an individual wall in a coupled wall system can be redistributed 197
by maintaining the strength of the system [Eq. (1)]. Therefore, using this design concept, the 198
reinforcement ratio of the wall with an additional axial tensile load can be decreased by 199
sending its moment load to the wall with an additional axial compressive load [1]. 200
Correspondingly, the moment strength capacity of the wall is enlarged owing to the additional 201
axial compressive load. In the present study, in order to verify this moment redistribution 202
method for the coupled wall system and investigate its actual behavior, inelastic analysis was 203
performed. 204
According to Kim et al. [17] and Park and Eom [18], the ultimate behavior of a coupled 205
wall system can be accurately simulated by the longitudinal-and-diagonal-line-element-model 206
(LDLEM) method. The LDLEM method has excellent stability for numerical analysis of 207
structural wall systems because it consists of simple trusses as a macro model as shown in Fig. 208
8. Therefore, in this study, the LDLEM method was adopted for the inelastic analysis of a 209
coupled wall system consisting of a number of structural members. 210
In the LDLEM method, the unit wall between the adjacent floors can be simulated using a 211
simple longitudinal and diagonal truss model (see Fig. 8). The diagonal truss was used to 212
10
represent the shear resisting element and the diagonal truss angle (θc) was defined to be the 213
diagonal crack angle in the web of the RC wall. The diagonal crack angle could be calculated 214
using the modified compression field theory proposed by Vecchio and Collins [19]. Moreover, 215
according to Kim et al. [17], the diagonal truss angle could be simplified to 45° for practical 216
analysis. In this study, the material model of the concrete and steel for a truss element was 217
defined as the stress–strain relationship to reflect the inelastic behavior of the coupled wall 218
system, as shown in Fig. 9. 219
As shown in Fig. 10, the prototype model of the coupled wall system (model A, model B) 220
is selected for the inelastic analysis. Model A is representative model for the typical design 221
result of the coupled wall system under elastic analysis. In this case, a high reinforcement 222
ratio value is assumed for the wall with an additional axial tensile load (𝜌𝑓 = 0.015 ) and a 223
low reinforcement ratio value is assumed for the wall with an additional axial compressive 224
load (𝜌𝑓 = 0.005 ). On the other hand, in the case of model B, the reinforcement ratio of the 225
coupled wall is unified as 𝜌𝑓 = 0.01 and this reinforcement ratio is determined by equalizing 226
the sum of the individual moment strengths of model B to that of model A [𝑀′1 +227
𝑀′2(model B) = 𝑀1 +𝑀2(model A), see Fig.10.]. Except for the reinforcement ratio of the 228
coupled walls, the other design parameters of model A and model B are the same. 229
The LDLEM analysis for model A and model B in Fig. 10 was conducted by using 230
OpenSEES program (PEER 2001[22]). Static pushover analysis was performed by 231
displacement control. The coupling beams in model A and model B were modeled with 232
equivalent beam elements (nonlinear Beam-Column Element of OpenSEES (PEER 2001) 233
with fiber section). And concrete elements resist only compressive force, which structural 234
behavior was determined by material model in Fig 9(a). 235
Using the LDLEM method that simulated the ultimate behavior of the coupled wall system 236
11
(model A and model B), the relationship of the total moment capacity at the base and the drift 237
at the roof was investigated, as shown in Fig. 11. According to Fig. 11, model A and model B 238
show almost the same behavior under a lateral load. The structural performances (stiffness, 239
strength and deformability) of the coupled wall systems before/after the moment 240
redistribution are almost identical. Therefore, the moment redistribution concept [1, 14, 16] 241
for the effective design of the coupled fin walls in high-rise buildings was verified through 242
LDLEM analysis method. Particularly, the high reinforcement ratio of the walls with an 243
additional axial tensile load can be reduced when the moment redistribution concept is used as 244
Paulay and Priestley proposed [1]. If the structural design was performed according to elastic 245
analysis, the flexural reinforcement might be placed such as the pattern of Model A, where 246
the amount of flexural reinforcement in tension-side wall is larger than that of Model B. If 247
lateral load in the opposite direction is applied, the existing compression-side wall will 248
become tension-side wall and the amount of flexural reinforcement will increase. As a result, 249
in the structural design based on the elastic analysis, the amount of the flexural reinforcement 250
is determined by the amount of the flexural reinforcement of the tension-side wall. And 251
according to the result of the analysis of the model B, the design by the elastic analysis has a 252
greater flexural performance than the flexural performance required for the design load. These 253
design results decrease the economic feasibility and constructability. Moreover, excessive 254
flexural performance of high-rise RC structural walls may lead to their unexpected brittle 255
shear failure. Therefore, to assure structural safety and economic feasibility, the use of 256
moment redistribution method should be considered seriously. However, further studies are 257
still required to verify the moment redistribution concept under various design parameters and 258
to identify the limitations of this method. 259
260
12
4. Conclusions 261
In high-rise buildings, lateral loads, such as wind and seismic loads, are frequently resisted by 262
RC structural walls since they have a large stiffness and strength. In the high-rise RC building 263
structures, in many cases, the structural RC walls were located on another line and apart from 264
each other, and they are designed generally as independent single walls (isolated walls) 265
without the consideration for their structural behavior. Especially, behavior and design of fin 266
walls at the periphery of strong core wall structure in high-rise buildings were not analyzed 267
seriously despite their structural importance. Sometimes, these RC fin walls, which extend 268
from the center to the boundary of the plan, have a large reinforcement ratio that is more than 269
1%. Such a value cannot assure economic feasibility and constructability. In the present study, 270
the current design process of these fin walls is investigated, and inelastic analysis and 271
literature review is performed to study the actual behavior of these RC fin walls, and to 272
recommend the alternative design method for them. According to the investigation of the 273
current design and elicited results, the walls, which are located far apart from each other and 274
which are connected by center walls and beams on another line, are coupled by the beams. 275
Therefore, owing to the large deformation of the coupling beams under the lateral load, large 276
shear forces occur in coupling beams that are subsequently transferred to the coupled walls as 277
an axial load. Moreover, in the case of high-rise buildings, the number of coupling beams is 278
very large and the axial load that is transferred to the coupled walls is also large. These 279
additional axial forces result in a significant reinforcement of the coupled walls. However, the 280
additional axial load from the coupling beam, which is calculated using the current elastic 281
analysis, is larger than that elicited actually since the strength of the coupling beam is limited 282
by the actual detail of the beams. Accordingly, the high-rise RC fin walls, which are located 283
far apart from each other and which are connected by center walls and beams on another line 284
13
should be designed as coupled wall system instead of isolated walls. 285
In the present study, the moment redistribution method is introduced as alternative design 286
method for high-rise RC fin wall system through a literature review. And it was verified by 287
the inelastic analysis using the LDLEM method which was newly-developed and specialized 288
structural analysis method for RC wall systems. According to the results of the inelastic 289
analysis and the literature review, the moment redistribution concept can be used for the 290
effective design of the RC fin walls in high-rise buildings. Particularly, by using the moment 291
redistribution concept, the high reinforcement ratio of the walls with additional axial tensile 292
load can be reduced and the unexpected shear failure of high-rise RC structural systems can 293
be avoided by protecting their excessive flexural performance. However, further studies are 294
still required to verify the moment redistribution concept under various design parameters and 295
to identify the limitations of this method. 296
297
Acknowledgement 298
This work was supported by the research grants of the National Research Foundation of 299
Korea (NRF) (No. 2015R1C1A1A01053471 and Code R-2015-00441 [Mid-career Research 300
Program]). The authors are grateful to the authorities for their support. 301
302
Reference 303
[1] Paulay, T. and Priestley, M. J. N. “Seismic Design of Reinforced Concrete and Masonry 304
Buildings”, John Wiley & Sons, New York (1992). 305
[2] Taranath B. S. “Reinforced Concrete Design of Tall Buildings”, CRC Press, london 306
(2009). 307
[3] Kaltakci, M.Y., Arslan, M.H. and Yavuz, G., “Effect of Internal and External Shear Wall 308
14
Location on Strengthening Weak RC Frames”, Scientia Iranica. Transaction A, Civil 309
Engineering; Vol. 17, No.4, pp. 312-323 (2010). 310
[4] ACI Committee 318. “Building code requirement for structural concrete (ACI 318-14)”, 311
American Concrete Institute, Farmington Hills. MI (2014). 312
[5] Midasit, http://www.midasit.com. 313
[6] Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A., “A new simple shear and normal 314
deformations theory for functionally graded beams”, Steel and Composite Structures, 315
18(2), pp. 409 – 423 (2015). 316
[7] Tounsi, A., Houari, M.S.A. and Bessaim, A. , “A new 3-unknowns non-polynomial plate 317
theory for buckling and vibration of functionally graded sandwich plate”, Struct. Eng. 318
Mech., Int. J., 60(4), pp.547 – 565 (2016). 319
[8] Hebali, H. Tounsi, A., Houari, M.S.A. Bessaim, A and Bedia, E.A.A, “New quasi-3D 320
hyperbolic shear deformation theory for the static and free vibration analysis of 321
functionally graded plates”, ASCE J. Engineering Mechanics, 140(2), pp. 374 – 383 322
(2014). 323
[9] Bennoun, M., Houari, M.S.A. and Tounsi, A., “A novel five variable refined plate theory 324
for vibration analysis of functionally graded sandwich plates”, Mechanics of Advanced 325
Materials and Structures, 23(4), pp. 423 – 431 (2016). 326
[10] Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A., “Wave propagation in 327
functionally graded plates with porosities using various higher-order shear deformation 328
plate theories”, Structural Engineering and Mechanics, 53(6), pp. 1143 – 1165 (2015). 329
[11] Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Bég, O., "An 330
efficient and simple higher order shear and normal deformation theory for functionally 331
graded material (FGM) plates”, Composites: Part B, 60, pp. 274–283 (2014). 332
15
[12] Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R., "A new simple three-333
unknown sinusoidal shear deformation theory for functionally graded plates", Steel and 334
Composite Structures, 22(2), pp. 257 – 276 (2016). 335
[13] Mahi, A., Adda Bedia, E.A. and Tounsi, A., "A new hyperbolic shear deformation theory 336
for bending and free vibration analysis of isotropic, functionally graded, sandwich and 337
laminated composite plates”, Applied Mathematical Modelling, 39, pp. 2489–2508 (2015). 338
[14] Paulay, T. “Seismic Design in Reinforced Concrete-The State of The Art in New 339
Zealand”, Proceeding of 9th World Conference on Earthquake Engineering, Tokyo, Japan 340
(1988). 341
[15] Paulay, T. “ The Philosophy and Application of Capacity Design”’ Scientia Iranica, 342
Vol.2, No.2, pp.117−143 (1995). 343
[16] Santhakumar, A.R. “The Ductility of Coupled Shear Wall”, PhD thesis, University of 344
Canterbury, Chrischurch, New Zealand (1974). 345
[17] Kim, D. K., Eom, T. S., Lim, Y. J., Lee, H. S. and Park, H. G. “Macro model for 346
nonlinear analysis of reinforced concrete walls”, Journal of the Korea Concrete Institute, 347
Vol. 23, No. 5, pp. 569-579, (in Korean) (2011). 348
[18] Park, H. G. and Eom, T. S. “Truss model for nonlinear snalysis of RC members subject 349
to cyclic loading”, Journal of Structural Engineering, Vol. 133, No. 10, pp. 1351-1363 350
(2007). 351
[19] Vecchio, F. J. and Collins, M. P. “The modified compression-field theory for reinforced 352
concrete elements subjected to shear”, ACI Structural Journal, Vol. 83, No.22, pp. 219-353
231 (1986). 354
[20] Kent, Dudley Charles and Park, Robert., "Flexural- Members with Confined concrete", 355
Proceeding, ASCE, vol.97, No.ST7, pp.1969-1990 (1971). 356
16
[21] Menegotto, M., and Pinto, P. E. “Method of analysis for cyclically loaded reinforced 357
concrete plane frames including changes in geometry and non-elastic behavior of elements 358
under combined normal force and bending.” Proc., IABSE Symp. of Resistance and 359
Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads, IABSE, 360
Libson, Portugal, Vol. 13: 15-22 (1973). 361
[22] Pacific Earthquake Engineering Research Center (PEER), “Open system for earthquake 362
engineering simulation”, Berkeley, CA: University of California at Berkeley (2001). 363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
17
Figure list 380
Fig. 1 Typical example of structural system for high-rise residential buildings 381
Fig. 2 Structural plan 382
Fig. 3 Applied load and strength of the wall W2 and W3 at floor B3 383
Fig. 4 Axial load applied to the wall W2 and W3 at floor B3 according to each load 384
combination 385
Fig. 5 Strain distribution of wall W2 and W3 under the wind load according to the stiffness of 386
the coupling beam 387
Fig. 6 Increase of flexural reinforcement of coupled walls owing to additional axial loading 388
Fig. 7 Strength of coupled wall system [1] 389
Fig. 8 Longitudinal-and-diagonal-line-element-model (LDLEM) for the simulation of the 390
actual behavior of a coupled wall system [17] 391
Fig. 9 Material model 392
Fig. 10 Coupled wall model for inelastic analysis 393
Fig. 11 Result of inelastic analysis 394
395
Table list 396
Table 1 Reinforcement detail of RC structural walls 397
398
399
400
401
402
403
404
18
405
Fig. 1 Typical example of structural system for high-rise residential buildings 406
407
408
409
Fig. 2 Structural plan 410
411
Thirty-five stories
Three basement level
RC structure
a. RC Flat-plate
b. RC Structural wall
c. RC Column
Location : Seoul, Korea
(a) 35 stories residential building (b) Lateral load resisting system
RC Structural wall RC Flat-plate + Column
(C) Gravity load resisting system
W2W3
W1
W4 W5
19
412
Fig. 3 Applied load and strength of the wall W2 and W3 at floor B3 413
414
415
416
417
Fig. 4 Axial load applied to the wall W2 and W3 at floor B3 according to each load 418
combination 419
W2 W3
W2 H : 8.15m, T=0.3m, D25@100 (3.38%)
f’c=30MPa, fy=600MPa
-60000
-40000
-20000
0
20000
40000
60000
80000
100000
120000
-150000 -100000 -50000 0 50000 100000 150000A
xia
l lo
ad
(k
N)
Bending moment (kN-m)
1.2D+1.0L-1.3W1
1.2D+1.0L-1.3W2
0.9D-1.3W1
0.9D+1.3W1
1.2D+1.0L+1.3W1
-60000
-40000
-20000
0
20000
40000
60000
80000
100000
-150000 -100000 -50000 0 50000 100000 150000
Ax
ial
loa
d (
kN
)
Bending moment (kN-m)
0.9D+1.3W1
1.2D+1.0L+1.3W1
1.2D+1.0L-1.3W1
0.9D-1.3W1
W2 H : 7.65m, T=0.3m, D22@100 (2.58%)
f’c=30MPa, fy=600MPa
(a) Building plan (b) Applied load and capacity of wall (W2) (c) Applied load and capacity of wall (W3)
(a) Axial load to W2
-30,000
0
30,000
60,000
0 10 20 30 40 50 60 70 80
W2 벽체 축력
W3 벽체 축력
Load Combination
Axia
l lo
ad(k
N)
1.2D+1.0L+1.3W1
1.2D+1.0L-1.3W1 0.9D+1.3W1 0.9D-1.3W1
(a) Applied load and load combination (b) Axial load applied to wall W2 and W3 at B3
Lateral load
- Earthquake load(E:Rx,Ry)
- Wind load (W)
Gravity load
- Dead load(D)
- Live load (L)
- Snow load (S)
(b) Axial load to W3
No. Load combination1 1.4D2 1.2D + 1.6L3 1.2D + 1.3W1 + 1.0L4 1.2D + 1.3W2 + 1.0L5 1.2D + 1.3W3 + 1.0L6 1.2D - 1.3W1 + 1.0L7 1.2D - 1.3W2 + 1.0L::
::
78
0.9D -1.0(1.0(1.14)[RY(RS)-
RY(ES)]-0.3(1.45)[RX(RS)+RX(ES)])
(C) Axial load applied to wall W2 and W3 at B3 according to each load combination
(a) Axial load to W2
(b) Axial load to W3
Coupling beam
20
420
Fig. 5 Strain distribution of wall W2 and W3 under the wind load according to the stiffness of 421
the coupling beam 422
423
WindLoad
WindLoad
(a) Strain distribution of W2 and W3 with existing coupling beams under the wind load
(b) Strain distribution of W2 and W3 with stronger coupling beams under the wind load
existing coupling beams: 0.6ⅹ0.5 (depth ⅹ thickness, m)
strong coupling beams: 2.4ⅹ0.5 (depth ⅹ thickness, m)
0.00025
-0.00025
030th floor
0.00025
-0.00025
020th floor
0.0005
-0.0005
0
10th floor
B3 floor
0.001
-0.001
0
W2 W3
W2 W3
0.00025
-0.00025
030th floor
0.00025
-0.00025
020th floor
0.0005
-0.0005
0
10th floor
B3 floor
0.001
-0.001
0
21
424
Fig. 6 Increase of flexural reinforcement of coupled walls owing to additional axial loading 425
426
427
Fig. 7 Strength of coupled wall system [1] 428
(a) Deformation of coupling beam (b) Axial load applied to wall W2 and W3 at B3
(1) Axial load to W2
(2) Axial load to W3
(c) Flexural rebar increase of wall
with tensional load
‘ ‘ ‘
‘
θ
P(a
xia
l lo
ad)
Com
p.
Ten..
M
P
Axial load by
lateral load
Axial load by
lateral load
Axial load by
gravity load
Axial load by
gravity load
W2
W3
M
W2
OK
Increase of flexural re-bar
for strength compliment
(d) Decrease of strength reduction factor of wall with compressive load
Decrease of strength
reduction factor
0 002. t
.085
0 005.
065. Comp.-
controlled
sections
Tension-
controlled
sectionsNG
=( ′
)
Strength
decrease
1M
2M
TT
sl
MV
BMD
ibM ,
ibM ,
bl
∑⇒∴≤n
i bib
b
iblMT
l
MV
1= ,
,2/=
2
M
22
429
Fig. 8 Longitudinal-and-diagonal-line-element-model (LDLEM) for the simulation of the 430
actual behavior of a coupled wall system [17] 431
432
433
434
435
(a) Stress-strain relation for concrete [20] (b) Stress-strain relation for concrete [21] 436
Fig. 9 Material model 437
438
wh
4v
Dimensionless rigid beam
Longitudinal truss element (concrete & re-bar)
Diagonal truss element (concrete)
Tributary areas of longitudinal concrete and re-bar elements A
4u
dcA
mh
ml
b
ls wA bs lc lsA bs A
s
Longitudinal element A
dcA
Web concrete
ccosdc w w cA h b
dcAwh
Concrete stress for positive loading
cc
ml
3v
3u
1v
1u
2v
2u
Re-bar ratiosb bw
c
wh
Concrete stress for negative loading
Definition of web depthwh
wh
c
( cos )/dc w w cA h b n
3n
(a) Longitudinal-and-Diagonal-Line-Element-Model (LDLEM) element
(b) Tributary areas of concrete and re-bar
of each longitudinal uniaxial element
(c) Diaongal concrete strut of wall web concrete
Single set of X-type diagonal concrete struts
( cot )/( 1)d w m cs h l n
ml
(d) Multiple sets of X-type
diagonal concrete struts
ds ds
(a) Stress-strain relation for concrete (b) Stress-strain relation for steel reinforcement
compressive strength of concrete
strain corresponding to concrete strength
secant elastic modulus of concrete
yield strength of steel
:0Cf
:0Cε
:0CE
:yf
23
439
440
Fig. 10 Coupled wall model for inelastic analysis 441
442
2nd
8th
7th
6th
5th
4th
3rd
1st
(a) Coupled wall model A ( (tension wall), (compression wall)) 015.0=fρ 005.0=fρ
2nd
8th
7th
6th
5th
4th
3rd
1st
(b) Coupled wall mode B ( (tension wall), (compression wall)) 010.0=fρ 010.0=fρ
Wall ①
Wall ②
)156.0=( '
1 cg fAPM
)244.0=( '
2 cg fAPM
Wall ①, Wall ②
)156.0=( ' '
1 cg fAPM
)244.0=( ' '
2 cg fAPM
′ + ′ ( ) = + ( )
■ f’c=30MPa, fy=400MPa, Story height=3.1m, Wall thickness=0.3m
■ Beam width-depth = 0.3m-0.5m, Flexural reinforcement ratio of beam = 1%
■ Constant axial load at the base =0.2Agf’c
015.0=fρ
5m 2m 5m
Wall ① Wall ②
005.0=fρ
Inelastic beam-
column element
010.0=fρ
5m 2m 5m
010.0=fρ
Wall ① Wall ②
Inelastic beam-
column element
24
443
Fig. 11 Result of inelastic analysis 444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
25
Table 1 Reinforcement detail of RC structural walls 465
Wall List (at floor B3)
f'c=30 MPa, fy=500 MPa (under D13), fy=600 MPa (over D16)
Wall Story
Thickness
(mm)
Vertical
Reinforcement
Horizontal
Reinforcement
W1 B3F-B2F 300 D22 at 100 (2.58%) D10 at 150
W2 B3F-B1F 300 D25 at 100 (3.38%) D13 at 200
W3 B3F-B1F 300 D22 at 100 (2.58%) D10 at 130
W4 B3F-B2F 300 D13 at 150 (0.59%) D10 at 190
W5 B3F-B2F 300 D16 at 125 (1.07%) D10 at 190
466