1028
STRUCTURAL CAPACITIES OF H-SHAPED RC CORE WALL SUBJECTED TO
LATERAL LOAD AND TORSION
Makoto MARUTA1, Norio SUZUKI2, Takashi MIYASHITA3 And Takamasa NISHIOKA4
SUMMARY
This paper describes an experimental and analytical study on H-shaped RC core walls subjected to
simultaneous lateral load and torsion. The torsional stiffness of an open section core wall (OSCW)
is smaller than that of a closed section one (CSCW). It is predicted that the elasto-plastic
capacities of OSCW are inferior to those of CSCW. However, few researches have been carried
out on the elasto-plastic torsional behavior of OSCW. When a high-rise RC structure with OSCW
is designed in an aseismic country, it is important to evaluate the structural capacities of OSCW,
especially H-shaped core wall subjected to simultaneous lateral load and torsion. The correlation
between the maximum lateral strength and the maximum torsional strength is especially important.
Nine H-shaped core walls were tested under simultaneous lateral load and torsion. Two
parameters were studied: the ratio of torsion to lateral load and the lateral load direction of the Hshaped wall. All specimens were designed to reach flexural yielding before shear failure.
Nonlinear finite element analyses were carried out to verify the test results, and to obtain the data
of untested cases.
The following results are obtained: (1) the maximum lateral strength decreases with increase in
torsion. (2) the bending-torsional resistant mechanism of an H-shaped wall varies depending on
the loading direction. (3) the correlation curves at maximum strengths were close to elliptical for
each direction, making the overall correlation spherical.
INTRODUCTION
Recently, several high-rise RC structures with core walls have been designed and constructed in Japan. Due to
architectural planning requirements, these core walls have often been of the open section type, typically Hshaped. They would be subjected to simultaneous lateral load and torsion in an earthquake. The lowest portion
of the core walls is subjected to especially large bending moment and torsion.
The design procedure for a column or a closed section wall subjected to lateral load and torsion is prescribed in
the ACI code [ACI, 1995], etc. The ACI code describes the limit of torsional moment, and the reinforcing
method required when the torsional moment exceed this limit.
However, there are unknown factors concerning the structural capacities of an open section core wall (OSCW)
subjected to simultaneous lateral load and torsion. Each plate of a closed section core wall (CSCW) like a boxsection subjected to pure torsion is likely in the pure shearing state. Because of the warping of OSCW, its flange
walls are subjected to bending moment and shear, so the ACI design method for torsion can not be applied to
OSCW. Therefore, experimental and analytical studies were carried out to determine the elaso-plastic behavior
of OSCW, especially H-shaped core walls.
1
2
3
4
Kajima Technical Research Institute, Kajima Corp., Japan Email: maruta@katri.kajima.co.jp
Kajima Technical Research Institute, Kajima Corp., Japan Email: nsz@katri.kajima.co.jp
Information Processing Center, Kajima Corp., Japan E-mail: miyashita@ipc.kajima.co.jp
Nuclear Power Department, Construction Group, Kajima Corporation E-mail: nisioka@psa.kajima.co.jp
This paper describes the test results and analytical results, and the correlation between maximum bending and
the maximum torsional moment derived from these results. The correlation between bending and torsion is
important for design of a high-rise structure’s core wall, because the lowest portion of the wall is in the most
severe condition under earthquake load. In particular, the bending moment is more severe than the shear force.
STRUCTURAL TEST
Specimens and Test Method
The material properties are shown in Table 2.
The maximum size of concrete coarse
aggregate was 10mm diameter, and the target
compressive strength was 60N/mm2 at the
time of test.
The torsion is born mainly by the flange
walls. Therefore, the elastic moment
distributions of the flange walls are predicted
as shown in Fig.2. The torsional capacities
of Series 1 were assumed to be influenced by
the magnitude of the axial stress in each
flange. If the lateral force became bigger, the
axial stresses between flange 1 and flange 2
varied more. Its stress has a large influence
on torsion resisted by each flange in the inplane direction.
In Series 2, only the flange walls resist the
bending moment and torsion. The moment
distributions of the flanges are assumed to
vary with the torsional ratio, as shown in
Fig.2. The behavior of Series 3 may be
between those of Series 1 and Series 2.
The tests were conducted using the sixdegree-of freedom apparatus shown in Photo
1. This apparatus can impose a torsional
moment that is proportional to the constant
ratio of bending moment at the lowest portion
of the walls.
The weight of a loading plate and the loading
slab, 118 kN, was applied as the axial force to
the specimen. A cyclic loading was applied at
400
Loading Slab
1200
80
800
Wall
360
Main Rebar 4-D10
Horizontal Rebar D6@150
Wall Rebar D6@100
(Longitudinal&Transverse)
Lappig Joint(240mm)
800
80
Base Mat
(Unit: mm)
80
720
880
360
Concrete Strength : 60N/mm2
500
Nine 1/12 scale specimens as listed in Table
1 were tested. All specimens had the same
H-shaped section, were the same size and
had the same reinforcement. Fig.1 shows
the details of the specimens. They had
column type reinforcement at both flange
edges. Experimental parameters were the
lateral loading direction and the ratio of
torsional moment to bending moment at the
lowest portion of the wall. The loading
directions were the strong axis for Series 1,
the weak axis for Series 2 and the diagonal
axis for Series 3, as shown in Fig.2.
The ratios of torsional moment (T) to
bending moment (M) at the lowest portion of
the wall were varied: 0%, 25%, 50%, 75%
and 100%. These ratios are called the
torsional ratio. The torsional ratio was the
main parameter of this test.
1400
80
1400
Fig. 1: Test Specimens
Table 1: List of Test Specimen
Series
HS2 5
HS5 0
HS7 5
H W 00
1
L oadin g R atio (% )
Lo adin g
Direc tion
S pecimen
*1
(M : T .)
75 : 25
50 : 50
25 : 75
100 : 0
75 : 25
50 : 50
25 : 75
75 : 25
0 : 100
Strong
H W 25
W eak
H W 50
H W 75
3
H D25
Dia gonal
4
H 100
P ure To rsio n
M : B ending M omen t at Lowe st Portion T: To rsion al Mo ment
*1 : To rsio nal Ratio
2
Table 2: Material Properties
(Concrete)
S pecim en
C om pre ssive
S treng th
2
H S25
H S50
H S75
H W 00
H W 25
H W 50
H W 75
H D 25
H 1 00
Sea led in Fie ld
Y oun g's
M o dulus
4
2
T ens il
S treng th
2
(N / m m )
(*10 N /m m )
(N / m m )
57.6
59.3
60.4
56.1
57.9
57.8
64.1
62.5
66.4
3.02
2.98
2.83
3.58
3.46
2.89
3.25
3.13
3.10
3.05
3.29
3.48
2.79
3.74
3.12
2.94
3.47
4.00
(Rebar)
Y eil d Poin t
Te nsile S trength Y ie ildi ng Strai n
2
Ð y (N /m m )
ƒ
Ð t ( N/m m )
ƒ
à y@i *10 j
ƒ
E longa tion
(% )
D6
365
419
1972
17.9
D10
403
595
2145
25.3
D iam e ter
2
2
-6
1028
each target drift angle (R1) that was defined in Fig.3, of 1/800, 1/400, 1/200, 1/100, 1/50, 1/25. δ R is the
average lateral displacement in the loading direction, and δ • is the rotational displacement, as shown in Fig.3.
Lateral Loading
HW00
Torsion
Fla
Fla HW25
nge
2
nge
HW50
HW75
H100 @
Pw T
1
T
Ps
eb
W
M:50% T:50%
M:75% T:25%
M:100%
@ T:0%
M:25% @
T:75%
@
M:Bending Moment at lowest Portion
M:0% T:100%
δ2
δ3
δ1
Flange 1
Pw
T
M/QD=0.75
M:75% T:25%
HS25,50,75
HD25
T:Torsional Moment
Series 1
Series 3
δ3
L
δ1+δ2
2
δR1
δ4
R1=
H
δ2
Weak Axis
P (Ps,Pw,P D)
δ
Flange 2
δR=
1
M/QD=1.13
Strong Axis
δR1=
δ4
δ1+δ2+δ3+δ4
4
H
Web 2
nge
L
T
e
b2
Fig.2: Assumed Moment Distribution of Specimen
Web 1
Ps
Fla
Fla
ng
e2
M/QD=0.75
Series 2
δ1
1
W
M/QD=1.22
M/QD=0.98
M/QD=1.78
M/QD=0.84
M/QD=0.75
(Shear Span Ratio)
M/QD=0.75
M/QD=1.15
M/QD=6.73
M/QD=1.78
M/QD=0.86
T
PD
δR
δR
R=
H
(Lateral Disp.)
δ1−δ4
δθ=
2
M=P*H
(Drift Angle)
θ=
(Torsional Disp.)
δθ
L
(Rotational Angle)
Fig. 3: Measurement and Evaluation
of Displacement
Photo 1: Test Setup
Table 3: Test Results
Elastic Bending Crack
(Flange)
stiffness
Specimen
HS25
HS50
HS75
HW00
HW25
HW50
HW75
HD25
Positive
Negative
KƒÂ *1
M*3
*2
*4
Shear Crack
(Flange)
R*5
M
Yeilding of
Column Rebar
Yeilding of
Transverse
Rebar of Wall
M
R
Maximum
Strength
R
M
R
M
R
KĮ
4973
103200
4916
88400
4690
7900
1990
T
71
6
79
69
38
12
100
ƒ *6
Æ
0.04
0.04
0.28
1.61
0.16
2.24
0.90
T
Æ
ƒ
106
100
50
143
243
0.53
2.43
0.46
4.78
4.61
T
271
89
139
131
60
167
145
ƒ
Æ
2.25
3.51
1.36
4.99
0.67
6.81
2
T
425
124
190
177
79
227
450
ƒ
Æ
13.6
20
2.31
8.16
1.59
13
19.1
T
468
153
235
228
93
283
459
ƒ
Æ
32.7
46.5
7.46
22.2
6.34
32.8
23.8
2120
64100
2030
91500
1630
74800
3591
65200
95
32
57
57
20
54
133
43
-79
-22
0.39
0.59
0.29
0.39
0.12
0.96
0.42
0.73
-0.09
-0.14
315
101
122
117
51
148
240
76
-192
-61
3.35
4.65
0.91
2.89
0.82
4.85
1.3
1.79
-0.98
-2.07
250
82
122
117
61
168
341
109
-223
-69
1.97
2.57
0.91
2.89
0.84
6.93
2.9
4.3
-1.33
-2.8
423
135
172
162
97
197
487
161
15.1
21.3
2.1
6.63
1.17
9.26
9.9
15.1
426
137
214
205
87
251
529
171
-390
-120
12.7
18.6
5.06
16
3.29
18.9
12.5
18.5
-10.9
-23.1
H100
11600
88
1.65
205
*1: Elastic Stiffness of Lateral Load (kN/cm)
*3: Bending Moment at Lowest Portion(kNm)
-3
*6: Rotational Angle(*10 rad)
9.24
216
10.3
216
10.3
268
22.1
*2: Elastic Stiffness of Torsion (kNm/rad)
*4: Torsional Moment(kNm) *5: Drift Angle(*10-3rad)
3
1028
TEST RESULTS
The test results are listed in Table 3. All specimens reached flexural yielding before shear failure.
The load and deflection curves (M - δ R , T- δ •) of HS25, HW25, HW75 and HD25, are shown in Fig.4 for
examples.
For Series 1 (HS25), the both load-deflection curves of M - δ R and T- δ • showed pinching behavior after shear
cracking. No decline in maximum strength was observed after the flange yielded, because of the small torsion
ratio. It showed a good energy absorbing hystereris loop. One of flanges that was under compression due to the
lateral load didn’t yield until the large displacement region.
At Series 2, HW25 showed a good hystereris loop before R=1/100. The damage to the flange was more severe
than that of the HS25’s. Both flanges of HW25 had yielded by R=1/50.
As the torsional effect of HW75 was bigger than that of HW25, HW75 had many shear cracks in both flanges.
The pinching behavior of HW75 after shear cracking was more remarkable than that of HW25.
At Series 3, the axial force of flanges fluctuated between positive and negative loading, as shown in Fig.5. In
this figure, the F2B portion was in a full compressive state under positive loading, and in a full tensile state
under negative loading. Therefore, the load-deflection curves of HD25 showed different behavior under positive
HS25
200
0
-200
Bending Crack
Shear Crack (Flange)
-400
Yield of Column Rebar
Maximum Strength
-600
-20
100
0
-100
-200
0
10
20
30
40
-40
50
R=1/25
Torsional Moment T (kNm)
0
-200
-400
Bending Moment M (kNm)
0
10
20
30
40
R=1/25
R=1/200 R=1/100 R=1/50
200
0
-200
-400
-10
Bending Moment M (kNm)
10
20
30
40
R=1/200 R=1/100 R=1/50
600
400
0
200
0
-200
-400
-600
-20
-10
0
10
20
30
40
Lateral Displacement δR (mm)
0
80
100
80
100
80
100
-100
-200
-20
200
0
20
40
θ=1/100 θ=1/50 θ=1/25
60
θ=1/15
HW75
100
0
-100
-200
-20
200
0
20
40
θ=1/100 θ=1/50
60
θ=1/15
HD25
100
0
-100
-200
-300
-40
50
60
θ=1/15
œ
300
HD25
40
100
-300
-40
50
R=1/25
Torsional Moment T (kNm)
-600
-20
20
HW25
200
300
HW75
0
θ=1/100 θ=1/50 θ=1/25
-300
-40
50
Torsional Moment T (kNm)
Bending Moment M (kNm)
HW25
-10
-20
300
200
400
HS25
200
600
-600
-20
600
θ=1/15
-300
-10
R=1/200 R=1/100 R=1/50
400
θ=1/100 θ=1/50
300
Torsional Moment T (kNm)
Bending Moment M (kNm)
400
R=1/25
R=1/200 R=1/100 R=1/50
600
-20
0
20
40
60
80
Rotational Displacement δθ (mm)
100
Fig. 4: Load-Deflection Curves
4
1028
Negative Loading
Positive Loading
PP
P
F2A
F2B
Flange 2
Comp. Comp.
Ten.
Tension
Ten.
Comp.
Comp.
Full Tension
Zone
Tension
Comp.
Comp.
F1B
Flange 1
F1A
F1B
Ten.
T
Ten.
Ten.
Flange 1
F1A
Ten.
Full Compression
Zone
Compression
Compression
T
F2B
Flange 2
F2A
: Te n s i o n
:Compression
*:Diagonal Force is distributed to strong and weak forces
Fig. 5: Axial Force Distribution of Series 3
and negative loading.
The comparison of specimens with the torsional moment ratio of 25•, shows that the specimen subjected to
strong axis lateral loading (HS25) had better ductility than the specimens subjected to weak axis and the diagonal
axis loading.
FINITE ELEMENT ANALYSIS (FEM)
A nonlinear finite element analysis was conducted to verify the test results and, to obtain the data of untested
cases.
Analytical Model
Concrete
The concrete and rebars were modeled as a layered
shell element [Miyashita et all, 1991] as shown in Fig.
6. In this analysis, the stress and strain of the concrete
was calculated for each layer. The rebars were
idealized as a layered plate element that had the
equivalent stiffness in the rebar direction. The stressstrain relationship of the rebars was modeled by as bilinear. The stress-strain relationship of the concrete
was modeled as shown in Fig. 7. Model details are as
follows;
1) The stress-strain relationship is evaluated on each
principal axis assuming orthogonal anisotropy.
2) The deteriorate ratio β of compressive strength is
defined as in Eq. 1.
Rebar
Idealized
t1
t2
i
k
j
t
Layered Shell Element
Fig. 6: Finite Element Model
Fig. 8: Mesh Layout
Compressive Strength
Ð
ƒ
Compressive Strength
Ð
ƒ
after Cracking
y
(ƒ
Ð c,ƒ
à c)
y'
0.1ƒ
Ð
Ð c /2
ƒ
E0
ƒ
Ð
Ð
ƒ
Ã
ƒ
0
1
Ã
ƒ
Ã
ƒ
à t /ƒ
à )co*1
Ð =ƒ
ƒ
Ð t (ƒ
Ð
ƒ
Ð
ƒ
À
y/ƒ
Ã
ƒ
Constant
p
Ã
ƒ
y' =
E0
E0
t
Ð
ƒ
Ð
ƒ
c
y' =
t
y'
Ã
ƒ
Ã
ƒ
y
Tensile Strength
Ð
Ð 1 = -ƒ
ƒ
(ƒ
À +1)
2ƒ
À
Ã
ƒ
y
Ã
ƒ
p
max
Maximum Strain
t
Ð
ƒ
0
= 0.2ƒÐ
= 0.13 ƒÃ c+0.145 ƒÃ
y
2 Ã
c /ƒ
y
Fig.7: Analytical Model of Concrete
5
1028
β = 0.8 + 0.6{(ε 1 + 0.0002) × 103 }
0.39
. . . . . . . . .. (Eq. 1)
ε 1 : Orthogonal strain to crack direction
3) The shear stiffness G after cracking is based on Aoyagi’s proposal [Aoyagi et all, 1981] as shown in Eq. 2.
 1
1 

+
G = 1 / 
 Ge 3.6 / ε 1 
(N / mm )
. . . . . . . . . (Eq. 2)
2
Ge : Elastic shear modulus
4) The tensile response of cracked concrete*1 is shown in Fig. 7. The multiplier CO in Fig. 7 is a coefficient
depending on bond characteristics. The tension stiffness of an element is determined by CO. The value of
CO is 2. 0 for a bending-cracked portion and 0. 4 [Okamura et all, 1987] for a shear-cracked portion.
Because CO=0. 4 was proposed for shear, it can’t accurately express the bond behavior and the rebar stress in
the bending cracked portion. CO=2. 0 is used to suppress the effect of tension stiffness.
Fig. 8 shows the mesh layout for the specimen.
Analytical Results
Fig. 9 compares the load-deflection curves obtained from the tests and analyses on specimens HS25, HW25 and
HD25. The analysis was stopped at the first unstable step under a downgrade. The analytical results follow the
experimental results well until the maximum strength, but doesn’t represent the pinching behavior well in the
large displacement region. The remaining analytical results not shown in Fig. 9 follow the test results for each
specimen well. However, the analyzed stiffness is slightly higher than the experimental stiffness in the small
loading region.
CORRELATION OF MAXIMUM LOADS
The untested cases were analyzed by the same FEM method. Four cases were analyzed: HS00 (strong axis),
200
600
HS25
Torsional Moment T (kNm)
Bending Moment M (kNm)
400
200
0
FEM Analysis
-200
Test
-400
Maximum Strength of Analysis
-600
-20
-10
0
10
20
30
40
T orsional M om ent T (kN m )
B ending M om ent M (kN m )
H W 25
0
-20 0
-40 0
-20
-10
0
10
20
30
40
Torsional Moment T (kNm)
Bending Moment M (kNm)
-100
-150
-20
0
20
40
60
80
150
H W 25
100
50
0
-50
-100
-150
-20
0
20
40
60
0
20
40
60
200
HD25
200
0
-200
-400
-600
-20
0
-50
-200
-40
50
600
400
50
200
20 0
-60 0
-3 0
HS25
100
-200
-40
50
60 0
40 0
150
-10
0
10
20
30
40
150
50
0
-50
-100
-150
-200
-40
50
Lateral Displacement δR (mm)
HD25
100
-20
Rotational Displacement δθ (mm)
Fig.9: Load-Deflection Curves (Comparisons of Tests and Analyses)
6
1028
HD00 (diagonal axis), HD50 (diagonal axis) and HD75 (diagonal axis).
The relationships between the maximum torsional moments and the maximum bending moments are plotted in
Fig. 10, Fig. 11 and Fig. 12 for each loading direction. These figures include the test results and the analytical
results. It is understood from these figures that the bending strength doesn’t deteriorate significantly until 25%
H100
H100
HW75
150
150
HS25
Experimental Results
100
FEM Result
50
HS00
0
100
200
300
400
500
5%
HW25
Experimental Results
FEM Result
50
HW00
0
600
0
Bending Moment Ms (kNm)
100
200
300
400
500
600
Bending Moment Mw (kNm)
Fig. 11: Correlation between T and Mw
(Weak Axis Loading) Series 2
Fig. 10: Correlation between T and Ms
(Strong Axis Loading)•Series 1
H100
300
Torsional Moment T (kNm)
HD75
250
HD50
200
HD25
150
100
Experimental Results (Positive)
Experimental Results (Negative)
FEM Results (Positive)
FEM Results (Negative)
50
HD00
0
0
100
200
300
400
500 600
Bending Moment M D (kNm)
Fig. 12: Correlation between T and MD•(Diagonal Axis Loading) Series 3
T max / T Emax
Tosional Moment
Negative
0.6
Negative
£ Positive
£
0.6
0.4
M
m
ax
/M
0.0
M max
/ MW E
m ax
M
m
D
ax
ia
/M
go
D
na
Em
lA
ax
x
0.2
ax
is
Strong Axis
0.0
0.4
0.0
0.0Weak Axis
0
1.
0.2
m
SE
0.8
Positive
0.8
t
0.4
0.6
0.2
Bending0.4 0.6
Momen
0.2
£
0.8
1.0
en
t
0.8
1.0
om
1.0
££
££
Positive
Negative
M
0
Τ: 2
in
g
100
HW50
Τ:5
200
0%
HS50
200
Τ:75%
250
nd
250
Torsional Moment T (kNm)
Torsional Moment T (kNm)
300
HS75
Be
300
Fig.13: Correlation between Torsional Moment and
Bending Moment
7
1028
of the torsional ratio.
For series 3, the load-deflection curves for positive and negative loading are different, as shown in Fig.4. This is
caused by the difference in the flange’s axial stress, as shown in Fig. 5. The maximum values for positive and
negative loading are also different. They become close as the torsional ratio becomes large, as shown in Fig.12.
This is because the fluctuation of axial stress, as shown in Fig.5, become smaller when the torsional ratio become
larger.
Fig. 13 shows the relationship between the normalized bending moments in each direction and the normalized
torsional moment. The maximum bending moments (M max) in each direction obtained from the tests and the
analyses, are normalized by the maximum bending moments in each loading direction (M SEmax, M WEmax
and M DEmax) of the non-torsional specimens. The torsional moments are normalized by the non-bending
specimen’s maximum moment (TE max).
The correlation between bending and torsion of the H-shaped core wall is likely to be spherical, i.e., similar to
the M-N interaction curves of a column.
From this relationship, it is possible to design an H-shaped core wall subject to simultaneous lateral load and
torsion. In practical design of an H-shaped core wall, the bending moment and pure torsional moment strengths
are calculated under the assumption of external force distribution. Next, the flexural capacities are reduced
according to the spherical correlation, as shown in Fig. 13.
CONCLUSIONS
Through tests and analyses of H-shaped RC core walls, the following conclusions were reached:
(1) The resistant mechanism of an H-shaped wall subjected to simultaneous bending and torsion varies
depending on the loading direction. The web can’t resist torsion. Torsinal capacities are influenced by the
torsional moment ratio. Torsional capacities under a strong axial loading deteriorate with increasing flange
axial stresses. Those under a weak axis loading depend on the changing moment distribution of the flanges.
(2) The bending strength scarcely deteriorate while the torsional moment is under 25%.
(3) The nonlinear finite element method can adequately simulate the hysteric behavior of the test specimens.
The maximum bending moment and torsional moment obtained from these analyses correspond closely with
the test results.
(4) The correlation of lateral load and torsion was determined for each loading direction. The correlation curves
at maximum strengths are very nearly elliptical for each direction. The overall correlation is close to
spherical.
REFERENCES
American Concrete Institute 318 (1995), “Building Code Requirements for Reinforced Concrete”
Hayami, Y. , Miyashita, T. and Maeda, T. (1991), “Nonlinear analysis of shear walls”, 4th International
Conference on Nonlinear Engineering Computations, September.
Aoyagi, Y. , Ohmori, S. and Yamada, K. (1981), “Strength and deformation characteristics of orthogonally
reinforced concrete containments models subjected to lateral forces”, 6th SMIRT Conference, J4/5,Paris,
France.
Okamura, H. , Maekawa, K. and Izumo, J. (1987), “Reinforced Concrete Plate Element Subjected to Cyclic
Loading”, IABSE Colloquium, Delft, pp575~pp590.
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