on the design of perforated steel shear walls, an
TRANSCRIPT
On the design of perforated steel shear walls, an experimentaland numerical study
M MANSOURI1, M G VETR2,* and M R JAVAHERI TAFTI3
1Department of Civil Engineering, Taft Branch, Islamic Azad University, Taft, Iran2Department of Civil Engineering, International Institute of Earthquake Engineering and Seismology, Tehran,
Iran3Structural and Earthquake Research Center, Taft Branch, Islamic Azad University, Taft, Iran
e-mail: [email protected]
MS received 17 September 2020; revised 15 January 2021; accepted 18 February 2021
Abstract. Despite the considerable advantages of SPSW, there were some shortcomings concerning the
system. Needing huge columns surrounding the infill plate has been the main dilemma of the SPSW system. This
problem will be due to high imposed stresses to columns by the infill plate. Perforated SPSW has shown a
capable method among the proposed methods to reduce applied forces to SPSW’s columns. In this paper, the
optimum configuration of holes was investigated experimentally and numerically. Results indicated that the
imposed forces to columns are reduced by increasing of hole area. The ultimate strength and energy absorption
of perforated SPSW also are reduced by increasing of hole area. Therefore, the optimum configuration was
proposed to achieve a minimum reduction in structural parameters and maximum reduction in column forces.
Besides, new relations were proposed to calculate the structural parameters of the perforated SPSW. The
proposed relations showed a good agreement with finite element results.
Keywords. Perforated; SPSW; stiffness; drift; ultimate strength; ductility.
1. Introduction
In addition to successful performance in experimental [1–5]
and numerical studies [6–10] conducted by researchers, the
steel shear wall (SPSW) has also performed well in past
earthquakes. Past studies [11, 12] carried out on the system
confirmed that they enjoy excellent energy absorption and
ductility capacity, high initial stiffness, and ultimate strength.
Excellence SPSW in comparison with other lateral bearing
loads systems has persuaded designers to utilize the SPSW in
their projects [13]. Therefore, the SPSW is an economical
and applicable replacement for concrete walls and steel
braces. The main dilemma concerning the system is needing
the huge columns surrounding of infill plate. it is due to high
stress by infill plate applied to columns. Although a thinner
filler plate reduces stress in boundary columns, it does sig-
nificantly reduce lateral stiffness and lateral strength. Also,
even by using a thinner infill plate, the possibility of hinge
formation is not reduced. Moreover, reducing of infill plate is
made slenderness ratio limitation. Therefore, the reduction of
infill plate thickness due to its limitation is not an applicable
option for reducing imposed stress to boundary columns. To
solve the problem, researchers proposed Low Yield Point
(LYP) steel to reduce the imposed stress to columns without
reduction of lateral strength, stiffness, and ductility [14, 15].
Although LYP steel improves the behavior of SPSW, it
increases construction costs. Also, the LYP infill plate makes
complicity during infill plate installing. In line with studies
on LYP-SPSW, the researcher proposed semi-supported
SPSW [16, 17] to reduce imposed stress in themain columns.
Ghamari and co-worker [18] proposed an SPSW with sec-
ondary oblique columns that improves the behavior of con-
ventional semi-supported SPSW. In the semi-supported
SPSW, two used secondary columns cause to reduce the
imposed stress in columns. Although the semi-supported
SPSW is reduced imposed stress considerably, lateral stiff-
ness and strength are considerably reduced as well. Also, in
the semi-supported SPSW concentrated stress is made in the
beam. Among all methods proposed to overcome weakening
the infill plate, the perforated SPSW (shown in figure 1)
presented by Vian [19], showed a more successful behavior
than other presented systems.
It can be said that the first research on the perforated
SPSW was done by Sabouri-Ghomi [20] which led to the
proposition of Equation (1) based on the PFI method for
estimating lateral strength of SPSW with the opening.
Vop ¼ 1� D
dp
� �:Vp ð1Þ
*For correspondence
Sådhanå (2021) 46:132 � Indian Academy of Sciences
https://doi.org/10.1007/s12046-021-01587-3Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
where Vop and Vp are the strength of a perforated and a
solid shear panel, respectively, D is the perforation diam-
eter, and dp is the panel height.
The equation was studied by Purba [21] for perorated
SPSW with several openings. He concluded that Eq. (1)
results in a conservative assessment of the lateral strength
comparing to the perforated wall. Based on the Purba [21]
results, Vian [19] proposed Eq. (2) to determine the lateral
strength of perforated SPSW.
Vop ¼ 1� 0:7D
Sdiag
� �Vp ð2Þ
Where Sdiag is shown in figure 1.
Moreover, Purba [21] reported that an individual perfo-
rated strip analysis can truthfully estimate the behavior of
SPSW without holes provided the DSdiag
� 0:6 be applied.
Eq. (2) is vastly used by designers despite its error.
Therefore, it should be modified that is done in this paper.
In the AISC [22] a capacity design method for SPSW has
been presented. The method assumes that all the infill plates
over the height of the wall will yield and plastic hinges will
form at the two ends of beams. The presence of holes in the
infill plates affects the stress in the boundary columns, thus
needing modifications to the current design method. Anjan
and co-workers [23] proposed Eq. (3) tcated SPSW, Vop.
Vop ¼ 0:5rtwLp;eff sin 2a ð3Þwhere tw is the infill plate thickness and r is the stress in
the infill plate tension strips, taken as the material yield
strength for design, the Lp,eff is the effective width of the
perforated infill plate. Also, the a (that is the angle of the
tension field) is calculated from CAN/CSA-S16-09 [24] or
AISC [25].
Moghimi [26] reported that SPSW’s column moment
demand may cause up to 20% with a perforation pattern.
This finding confirms the need to consider the specific
layout of holes rather than simply using a uniformly dis-
tributed stress with a value reduced from that of a solid
plate.
Mazzolani and co-workers indicated that, despite the
presence of holes, the a remains about 45� [27, 28].Vetr and co-workers [29] showed that an aluminum
perforated panel has a better yielding pattern than a per-
forated steel panel. They concluded that the aluminum
panel shows higher ductility and stiffness but lower lateral
strength. Based on the results, the designer can use alu-
minum or steel based on their aims in the design of per-
forated SPSW. Afshari [30] reported that the existing hole
on the compression diameter does not have considerable
effect on the angle of stress. Also, the hole in the mid-span
of the wall has a minimum effect on the SPSW behavior.
2. Method of study
Although perforated SPSW has been introduced as a cap-
able suggestion to improve SPSW behavior, the optimum
configuration of the system has not been investigated
comprehensively. In doing so, in this paper optimum con-
figuration of perforated SPSW is investigated. Therefore,
the configuration of perforated SPSW is investigated in
three aspects including; the effect of the number of the hole,
area of the hole in comparison with infill plate, and con-
figuration of perforated SPSW.
Investigation of the literature review shows that formulas
presented till now to calculate of shear strength of SPSW,
do not account for area ratio. It is caused to errors in the
calculation of shear strength. Therefore, the proposed for-
mula is presented for calculating the shear strength of
perforated SPSW in the case of area ratio.
3. Experimental study
3.1 Test set-up
Figure 2 shows the test setup and experimental geometries.
Based on the laboratory facilities, the pate with 600 9 600
mm was used for experimental testing. The models were
applied loading diagonally to make pure shear in the infill
plate. Four corners of the plate were attached to the
boundary frame by pin to make pined supported. The infill
plate thickness of the models is designed to 5mm. the infill
plate was bolted to the boundary frame. The boundary
frame was designed to resisting against stress imposed by
the infill plate. Also, the distances and diameter of bolts
were designed to prevent any nonlinear behavior at bolted
or preventing fracture of bolts.
Figure 1. Test specimen from Vian [19].
132 Page 2 of 17 Sådhanå (2021) 46:132
3.2 Experimental results
Despite SPSW, the perforated SPSW is fully yielded, fig-
ure 3. The infill plate of SPSW does not yield completely.
In the other words, the infill plate does not contribute to the
absorption of energy due to imposed seismic energy.
The ultimate strength of SPSW was 189 kN. Results
showed that the capacity in perforated SPSW-1 (SPSW
with 4 holes) is reduced by around 50%. Also, the perfo-
rated model with 4 corner holes is reached to ultimate
strength in bigger displacement.
Figure 2. Test specimen.
Sådhanå (2021) 46:132 Page 3 of 17 132
4. Numerical study
4.1 Numerical models
To investigate the effect of hole type on the seismic
behavior of SPSW, 25 FE models were analyzed. Table 1
illustrates the properties of the FE models. In this table, the
tw is the infill plate thickness. Also, D and Sdiag are the hole
diameter and distance between two-hole, respectively. The
sixth column of the table reports the ratio of the sum of hole
areas (Ahole) to the area of the infill plate (A plate). The A
plate is the area of the infill plate without the infill plate. In
the other words, the differences between Ahole and Aplate
measure the pure area of the infill plate.
The numerical models contain three parts. The first part
represents the type of SPSW as shown in figure 4. The
second part represents the infill plate thickness in mm. the
third part shows the presence of a hole on the infill plate.
For example, the S3-4-10 means the S3 model with an infill
plate of 4mm and the ratio of holes to the infill plate equals
10%.
4.2 Materials
The ST37 steel was used for FE models. For the steel, the
yield stress is 235 MPa and the ultimate stress is 370 MPa.
Also, the Yang modulus and Poison ratio for ST 37 is 200
GPa and 0.3 respectively. It should be noted, the stress-
Figure 3. The experimental model after testing.
Table 1. FE models properties.
Model tw (mm) D (mm) Sdiag D/Sdiag Ahole/Aplate
SPSW 4 – – – –
S1-4-10 4 35 63 0,56 0,10
S2-4-10 4 44 265 0,17 0,10
S3-4-10 4 52 127 0,41 0,10
S4-4-10 4 48 127 0,38 0,10
S5-4-10 4 62 201 0,31 0,10
S6-4-10 4 36 84 0,43 0,10
S1-4-10 4 35 63 0,56 0,10
S2-4-10 4 44 265 0,17 0,10
S3-4-10 4 52 127 0,41 0,10
S4-4-10 4 48 127 0,38 0,10
S5-4-10 4 62 201 0,31 0,10
S6-4-10 4 36 84 0,43 0,10
S1-4-20 4 48 63 0,76 0,20
S2-4-20 4 60 265 0,23 0,20
S3-4-20 4 70 127 0,55 0,20
S4-4-20 4 66 127 0,52 0,20
S5-4-20 4 84 201 0,42 0,20
S6-4-20 4 50 84 0,60 0,20
S1-4-30 4 61 63 0,97 0,30
S2-4-30 4 74 265 0,28 0,30
S3-4-30 4 89 127 0,70 0,30
S4-4-30 4 83 127 0,65 0,30
S5-4-30 4 105 201 0,52 0,30
S6-4-30 4 63 84 0,75 0,30
132 Page 4 of 17 Sådhanå (2021) 46:132
strain curve of the materials was obtained from experi-
mental tests and then was introduced to ABAQUS. The
Von-Mises yield criterion, known to be the most suit-
able yield function for metals, is used in this research. The
type of hardening to be used in this simulation is isotropic
hardening. And also, the curve diagram is depicted in
figure 5.
4.3 Boundary condition
All FE models are analyzed under applied lateral loads as
pushover analysis. In so doing, pushover analysis with
displacement control was carried out. The ultimate
displacement was limited to a drift ratio of 0.025 per ASCE
7-16 [31].
In FE simulations, to consider the fixed support condition
at the end of columns, the degree of freedoms including
displacements and rotations in all directions were restricted.
Moreover, since the top beam of SPSW in real projects is
restrained from out of plane displacement, its degree of
freedom including out of plane displacement was restricted.
4.4 FE Modeling and validation of results
To modeling and analysis of FE models, the ANSYS pro-
gram was used. This software has a very good capability in
modeling and analysis of structures, especially steel struc-
tures. In the modeling, the shell element was used to model
the steel components. Both geometric nonlinear and non-
linear materials were considered for nonlinear analysis.
Also, to consider the nonlinear geometric analysis due to
initial defects and geometric defects, imperfection was
applied to it as a coefficient of buckling mode. Comparing
FE results with experimental results as shown in figure 6
reveals a good agreement between them.
4.5 Numerical results
4.5a Stress distribution in numerical models: figure 7
illustrates the von Misses stress for numerical models. In
S4, S5, and S6 models no nonlinear behavior in columns
occurs. In models S1 and S2, yield lines are formed hori-
zontally and vertically between the holes. In the S1 and S2
models, the horizontal and vertical stress distribution
between the holes is created, which practically creates an
Figure 4. The schematic view of holes in the SPSW.
Figure 5. Typical tensile curve for ST37 steel.
Sådhanå (2021) 46:132 Page 5 of 17 132
incomplete diameter tension field. The whole infill steel
plate has not yielded and parts of it have remained elastic,
therefore the energy absorption capacity of the whole steel
plate is not used. Also, large stress has been created at the
bottom of the column, so in this case, the column is sus-
ceptible to plastic hinge formation. Besides, these models
are susceptible to plastic hinge formation at the panel zones
and the two ends of the beam. Although the formation of a
plastic hinge at the two ends of the beam is a desirable
yield, the yield of the panel zone is not desirable.
In the S3 model, yield lines are created with the steel
plate in line with the diameter of the tension field action.
The distribution of stress and expansion of steel plate yield
is better in this model than in S1 and S2 models, but the
panel zones and bottom of the columns of this model is
prone to plastic hinge formation that is not appropriate. In
the S4, S5, S6 models, the yield is limited to the steel plate,
and the column, beams, and panel zones are not affected by
the high stresses that lead to the formation of the plastic
hinge. The stress distribution and yield strength of the steel
sheet in the S6 model are wider than in the S4 and S5
models. In the S5 model, the stress distribution and sheet
yield are more limited to near the holes along the tensile
diameter, but in the S4 model, a better trend is observed.
Figure 8 shows the status of stresses multi-story models.
In the full steel shear wall model, the thickness on the first
and second story is 6 mm, and on the upper two floors, the
thickness of the steel sheet is 4 mm. In multi-story models,
it is not possible to use walls of the same thickness. But in
walls with successive holes, the infill plate thickness is
selected on all 6 mm stories. Because in the height of the
stories, the lateral shear force is reduced, the thickness of
the infill plate should be less, which is perforated walls,
instead of reducing the thickness of the infill plate, holes are
used. The status of the stress of structural components in
figure 7 shows that in a conventional wall (without hole), in
addition to the infill plate, a plastic hinge is formed in the
first and last story columns. Besides, the least surrender
occurs on the top story. Because if lower stories are yielded
before top stores, the possibility of the soft story is
increased. So, in the SPSW without the hole, the form of
the soft story is high possible but it is recused in perforated
SPSW.
4.5b Load displacement curve: The load drift and stiffness
curve are plotted in figure 9. Based on the results, the type
of perforated SPSW does not significantly affect the ulti-
mate strength of models. At drift around 0.4%, the stiffness
of perforated SPSW and SPSW coincide together. It means
that at a drift greater than 0.4%, the hole does have any
effect on the stiffness of models. For perforated SPSW, at a
drift greater than 0.1%, the hole does have any effect on the
stiffness of models.
The ultimate strength and stiffness of analyzed models
have been reported in table 2. Results indicate that perfo-
rated SPSW with a reduction of 20% in the area of infill
plate caused to 15% to 21% reduction in ultimate strength.
The S1 and S2 result in a minimum reduction in ultimate
strength and S5 and S6 result in a maximum reduction in
ultimate strength. The S1 and S2 result in maximum
reduction (35% to 38%) in elastic stiffness and S4 and S6
result in minimum (28% to 29%) reduction in elastic
stiffness. Unlike of ultimate strength, reduction in elastic
stiffness is more.
4.5c The effect of hole type on the imposed stresses tocolumns: The tension stress in the column during lateral
load is plotted in figure 10. Besides model S1-4-20, the hole
configuration does not considerable effect on the imposed
tension stresses to columns. But, utilizing the configuration
of the hole as model S1 reduces the tension stress up to drift
1%.
Unlike tensile stresses, model S1 does not suitable per-
formance in compressive stresses. The compression force in
-200
-150
-100
-50
0
50
100
150
200
-30 -20 -10 0 10 20 30
)N k(
daoL
Displacement (mm)
Experimental
FE
-100
-80
-60
-40
-20
0
20
40
60
80
100
-10 -5 0 5 10
)Nk(
daoL
Displacement (mm)
Experimental
FE
-150
-100
-50
0
50
100
150
-25 -20 -15 -10 -5 0 5 10 15 20 25
)N k(
daoL
Displacement (mm)
Experimental
FE
Figure 6. Comparison of FE results with experimental results.
132 Page 6 of 17 Sådhanå (2021) 46:132
S1 and S2 models are almost like SPSW models, figure 11.
But, the compression column force is considerably reduced
in S3, S5, S6, and S6 models in comparison with SPSW.
The lower forces are created in models S5, S4, S6, and S3,
respectively. The reduction compression forces of the col-
umn for models S6 and S4 are almost close. Before drift
of0.3% to 0.6%, the compression forces of columns for S5,
S4, S6, and S3 are bigger than SPSW. Since in the range of
drift the wall experience of nonlinear behavior, the
increasing not important and can be ignored. It is also
confirmed for the bending moment as shown in figure 10.
4.5d The effect of area ratio on the compression forceimposed on the column: In figure 12, the created compres-
sion force in columns versus the drift ratio is plotted. This
figure shows that by increasing the hole are from 10% to
30% the compression force of the column reduced by 12%
for models S1 and S2. For other models, the compression
force of the column is reduced by only 2% to 3%. The
results indicated that besides models S1 and S2, the con-
figuration of the hole is more important than the area hole.
4.5e The effect of area ratio on the bending momentimposed to column: In figure 13, the created bending
moment in columns versus drift ratio is plotted. The fig-
ure shows that in the elastic zone, the configuration of the
hole does a significant effect on the bending moment of the
column. After adrift around 0.5%, the bending moment is
reduced in columns for all models. More effective is seen in
the S5 model. In these models, by increasing of hole area
from 10% to 30% the bending moment is reduced by 14%.
In the other models, the reduction is between 1% to 8%.
There, it is concluded that by perforated SPSW the bending
moment can be reduced maximum to 14%.
4.5f The effect of area ratio on the pushover curve: In the
previous sections, it showed the forces created (compres-
sion force and moment) in columns were dwindled up to
14% and 16%, respectively. In some models, the configu-
ration did not considerable impact on the reduction in
compression force and bending moment of the column.
Unlike the compression force and bending moment of the
column, the pushover curve of the model is affected by the
Figure 7. Distribution of stress in structural components.
Sådhanå (2021) 46:132 Page 7 of 17 132
Figure 8. Distribution of stress in four-story structures.
132 Page 8 of 17 Sådhanå (2021) 46:132
hole area, figure 14. The reduction of ultimate strength is
39%, 40%, 10%, 12%, 21%, and 21%, respectively, for S1,
S2, S3, S4, S5, and S6. Therefore, by increasing of area
hole the more reduction is made respectively in S1 and S2,
S5 and S6, S4, and S3. So, the configuration of S1 and S2 is
not suggested due to the high reduction of the ultimate
strength of SPSW.
4.5g Optimum configuration of perforated SPSWs: In the
previous sections, it showed the compression force and a
bending moment of the column are reduced maximum up to
14% and 16%, respectively. In some models, the configu-
ration did not considerable impact on the reduction in
compression force and bending moment of the column.
Unlike the compression force and bending moment of the
column, the pushover curve of the model is affected by the
hole area, figure 14. The reduction of ultimate strength is
39%, 40%, 10%, 12%, 21%, and 21%, respectively, for S1,
S2, S3, S4, S5, and S6. Therefore, by increasing of area
hole the more reduction is made respectively in S1 and S2,
S5 and S6, S4, and S3. So, the configuration of S1 and S2 is
not suggested due to the high reduction of the ultimate
strength of SPSW.
5. The proposed method to achieve the responsecurve of perforated SPSPW
The following equation is proposed to compute the ultimate
strength of perforated SPSW.
Vop ¼ Vpð1� 0:6D
Sdiag� 0:5
Ahole
AplateÞ ð4Þ
In table 3, the results of the proposed relation are com-
pared with Eq. (2). Results indicated that the proposed
relation has better agreement with FE results than Eq. (2).
To achieve the load-displacement curve of the perforated
SPSW, first, the load-displacement curve for the frame and
infill plate is measured separately. Then the measured
curves are summed together. An SPSPW is measured as a
separate frame (figure 15). The mentioned curve is obtained
(a) (b)
0
500
1000
1500
2000
2500
3000
3500
4000
0 0.5 1 1.5 2 2.5
Base
shea
r (KN
)
Dri� angle (%)
S6-4-20S5-4-20S4-4-20S3-4-20S2-4-20S-1-4-20
0
50
100
150
200
250
300
350
0 0.5 1 1.5 2 2.5
K (k
N/m
m)
Dri� angle (%)
S6-4-20
S5-4-20
S4-4-20
S3-4-20
S2-4-20
S-1-4-20
Figure 9. Comparing the response of the FE models with different hole pattern a) pushover curve, b) stiffness versus drift ratio.
Table 2. The ultimate strength.
Model Fu Fui/FuSPSW K (kN/mm) Ki/KSPSW
SPSW 4685,28 479,66
S1-4-20 3999,24 0,85 310,0477 0,65
S2-4-20 3992,56 0,85 302,2565 0,63
S3-4-20 3826,87 0,82 359,238 0,75
S4-4-20 3833,68 0,82 338,8559 0,71
S5-4-20 3682,88 0,79 360,9753 0,75
S6-4-20 3805,57 0,81 343,41 0,72
0
1000
2000
3000
4000
5000
6000
0 0.5 1 1.5 2 2.5
)NK(
nmuloc
n iecrof
no isneT
Dri� angle (%)
SSW
S6-4-20
S5-4-20
S4-4-20
S3-4-20
S2-4-20
S-1-4-20
Figure 10. The tension tress created on the columns due to
lateral loading.
Sådhanå (2021) 46:132 Page 9 of 17 132
0100200300400500600700800900
0 0.5 1 1.5 2 2.5
)NK(
nmuloc
niecrof
noi serpmoC
Dri� angle (%)
SSWS6-4-20S5-4-20S4-4-20S3-4-20S2-4-20S-1-4-20 0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5
Bend
ing
mom
ent (
kN.m
)
Dri� angle (%)
SSWS6-4-20S5-4-20S4-4-20S3-4-20S2-4-20S-1-4-20
(a) (b)
Figure 11. The stress created on the columns due to lateral loading a) compression stress, b) bending moment.
0
1000
2000
3000
4000
5000
6000
0 0.5 1 1.5 2 2.5
Com
pres
ion
forc
e (k
N)
Dri� ra�o (%)
S1-4-30
S1-4-20
S2-4-10
0500
100015002000250030003500400045005000
0 0.5 1 1.5 2 2.5
Com
pres
ion
forc
e (k
N)
Dri� ra�o (%)
S2-4-30
S2-4-20
S2-4-10
0500
10001500200025003000350040004500
0 0.5 1 1.5 2 2.5
Com
pres
ion
forc
e (k
N)
Dri� ra�o (%)
S3-4-30
S3-4-20
S3-4-10
0500
10001500200025003000350040004500
0 0.5 1 1.5 2 2.5
Com
pres
ion
forc
e (k
N)
Dri� ra�o (%)
S4-4-30
S4-4-20
S4-4-10
0500
10001500200025003000350040004500
0 0.5 1 1.5 2 2.5
Com
pres
ion
forc
e (k
N)
Dri� ra�o (%)
S5-4-30
S5-4-20
S5-4-10
0500
10001500200025003000350040004500
0 0.5 1 1.5 2 2.5
Axia
l for
ce (k
N)
Dri� angle (%)
S6-4-30
S6-4-20
S6-4-10
Figure 12. Compression load versus drift curve.
132 Page 10 of 17 Sådhanå (2021) 46:132
for each separated steel plate, and the frame, and then by
summing their effects, the overall curve is plotted.
In thin SPSW, at the first stage of loading, buckling
occurs. Therefore, yielding may occur after buckling of the
plate. The elastic shear buckling stress is:
scr ¼ Kv:p2:E
12:ð1� #2Þ ð5Þ
Kv ¼ 5:34þ 4
ðd=bÞ2d
b� 1
Kv ¼ 4þ 5:34
ðd=bÞ2d
b[ 1
8>><>>:
ð6Þ
Uwe ¼ scrG
þ 2rt
E:sin2h
� �d ð7Þ
where rt is the infill plate’s yielding stress, E is the Yang
modulus, G is the shear modulus, and h is the diagonal
tension field angle. Figure 16 illustrate the state of stresses
in the infill plate, before and after buckling that is as:
rxx ¼ rtysin2h ð8Þ
ryy ¼ rtycos2h ð9Þ
rxy ¼ ryx ¼ scr þ 1
2rtysin
2h ð10Þ
Thus, the shear force in the plate due to buckling is
0
10
20
30
40
50
0 0.5 1 1.5 2 2.5
)m .
Nk(tnemo
mgnidneB
Dri� ra�o (%)
S1-4-30
S1-4-20
S1-4-10
0
10
20
30
40
50
0 0.5 1 1.5 2 2.5
Bend
ing
mom
ent (
kN.m
)
Dri� ra�o (%)
S2-4-30
S2-4-20
S2-4-10
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5
)m.
Nk(tnemo
mgnidneB
Dri� ra�o (%)
S3-4-30
S3-4-20
S3-4-10
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5
Bend
ing
mom
ent (
kN.m
)
Dri� ra�o (%)
S4-4-30
S4-4-20
S4-4-10
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5
)m.
Nk(tnemo
mgnidneB
Dri� ra�o (%)
S5-4-30
S5-4-20
S5-4-10
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5
Bend
ing
mom
ent (
kN.m
)
Dri� angle (%)
S6-4-30
S6-4-20
S6-4-10
Figure 13. bending moment versus drift curve.
Sådhanå (2021) 46:132 Page 11 of 17 132
Fwcr ¼ scr:b:t ð11ÞThe displacement corresponding to the plate’s buckling
under pure shear is Uwcr ¼ Fwcr=G. Using these shear and
buckling forces, and displacement, the shear force-dis-
placement curve of the plate is plotted as shown in fig-
ure 17. For simplicity, lines OC and CD are substituted by
line OD. Researchers confirmed that such a change has a
small effect on the predicting behavior of SPSW [18].
Considering the von Misses criterion, if Eq. (12) be
satisfied, the infill plate will yields.
ðrxx � ryyÞ2 þ ðrzz � ryyÞ2 þ ðrxx � rzzÞ2þ 6 sxy
2 þ sxz2 þ syz
2� �
¼ 2Fy2 ð12Þ
Considering the infill plate as plane stresses results:
rz ¼ ryz ¼ rxz ¼ 0 ð13ÞTherefore
rx2 þ ry
2 � 2rxry2 þ 3sxy
2 ¼ Fy2 ð14Þ
By considering all stresses (buckling and yielding) and
substituting the stress values, it results:
3scr2 þ 3scrrtsin2hþ rt
2 � Fy2 ¼ 0 ð15Þ
Thus, the equivalent yielding stress of the plate is given:
rt ¼ 1:5scrsin2h�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1:5scrsin2hÞ2 � ð3scr2 � Fy
2Þq
ð16Þ
0
1000
2000
3000
4000
5000
0 0.5 1 1.5 2 2.5
Base
shea
r (kN
)
Dri� ra�o (%)
S1-4-30
S1-4-20
S1-4-100
1000
2000
3000
4000
5000
0 0.5 1 1.5 2 2.5
Base
shea
r (kN
)
Dri� ra�o (%)
S2-4-30
S2-4-20
S2-4-10
0
1000
2000
3000
4000
5000
0 0.5 1 1.5 2 2.5
Base
shea
r (kN
)
Dri� ra�o (%)
S3-4-30
S3-4-20
S3-4-10
0
1000
2000
3000
4000
5000
0 0.5 1 1.5 2 2.5Ba
se sh
ear (
kN)
Dri� ra�o (%)
S4-4-30
S4-4-20
S4-4-10
0500
10001500200025003000350040004500
0 0.5 1 1.5 2 2.5
Base
shea
r (kN
)
Dri� ra�o (%)
S5-4-30
S5-4-20
S5-4-10
0
1000
2000
3000
4000
5000
0 0.5 1 1.5 2 2.5
Base
shea
r (KN
)
Dri� angle (%)
S6-4-30
S6-4-20
S6-4-10
Figure 14. Load displacement curve.
132 Page 12 of 17 Sådhanå (2021) 46:132
By ignoring of buckling capacity of the infill plate and
accounting for the hole effect on lateral strength of the infill
plate and, its shear strength will obtain as:
Fwu ¼ t:Fy sin 2h2
ð1� 0:6D
Sdiag� 0:5
Ahole
AplateÞ ð18Þ
In this formula, it can be assumed h = 45� that yield
sin 2h ¼ 1.
The shear plastic displacement of infill, Dwp, is obtained
by equating the work of the shear force to the strain energy
of the plate (U).
U ¼ZZZ
1
2Eðrx2 þ ry
2 � 2#rxryÞ þ 1
2Gsxy
2
� �dxdydz
ð19Þ
Table 3. Comparison of the proposed formula with FE results.
Model Vop (FE) Vop- Eq. (4) error of Eq. (4) Vop-Eq. (4) error of Eq. (4)
SPSW 4685.28 – – – –
S1-4-10 4470.67 2848.65 36.28 2876.76 35.65
S2-4-10 4470.67 4127.732 7.67 3973.12 11.13
S3-4-10 4145.98 3340.605 19.43 3298.44 20.44
S4-4-10 4152.76 3438.996 17.19 3382.77 18.54
S5-4-10 4116.49 3668.574 10.88 3579.55 13.04
S6-4-10 4134.4 3275.011 20.79 3242.21 21.58
S1-4-20 3999.24 2192.711 45.17 2080.264 47.98
S2-4-20 3992.56 3930.95 1.54 3893.468 2.48
S3-4-20 3826.87 2881.447 24.70 3443.681 10.01
S4-4-20 3833.68 2979.838 22.27 3485.848 9.07
S5-4-20 3682.88 3307.808 10.18 3626.407 1.53
S6-4-20 3805.57 2717.462 28.59 3373.402 11.36
S1-4-30 3498.00 1503.975 57.00 2619.07 25.13
S2-4-30 3498.00 3766.965 -7.69 3588.92 -2.60
S3-4-30 3408.26 2389.493 29.89 2998.58 12.02
S4-4-30 3541.4 2553.478 27.90 3068.86 13.34
S5-4-30 3476.19 2979.838 14.28 3251.58 6.46
S6-4-30 3446.01 2225.508 35.42 2928.30 15.02
Figure 15. Assuming a separate frame.
Figure 16. Conditions of stresses in the steel shear wall.
Sådhanå (2021) 46:132 Page 13 of 17 132
Disregarding the buckling strength of the plate,
rx ¼ rtcos2a ð20Þ
ry ¼ rtsin2a ð21Þ
sxy ¼ 0:5rtsin2a ð22ÞFinally, the strain energy is attained.
U ¼ 1
2Ert
2cos4aþ rt2sin4a� 2#rt
2cos2asin2a� �þ 1þ #
4Ert
2sin2a
� �b:d:t
ð23ÞWith Poisson’s ratio= 0.3 and using the trigonometric
relations, Eq. (23) is shortened as:
U ¼ 1:3rt2
8E3þ sin22a� �
bdt ð24Þ
Equaling the external work (W ¼ FwDw) to the strain
energy is resulted in the shear displacement (Dwp).
Dwp ¼ 0:65rtE
3þ sin22hsin2h
d ð25Þ
If h ¼ 45, the Dwp will be 1:95rtE d.
The elastic stiffness of the frame is expressed as
Kf ¼ 24Ec
h312qd þ 1
12qd þ 4ð26Þ
displacement curve of frame, qd ¼ Ibd4Icd
, and IbdandIcd are,
respectively, the moments of inertia of the beam and col-
umns sections.
The frame’s shear force obtained after the formation of
plastic hinges in the columns, Ff , represents the shear
capacity or the ultimate shear force. Displacement Df cor-
responds to the frame’s elastic limit displacement (onset of
plastic displacement) and is equal to:
Df ¼ bMpd
2
6EIcð27Þ
Ff ¼ 4Mfu
dð28Þ
The load-displacement curve of frame, as shown in fig-
ure 18, is obtained from Eqs. (27) and (28).
In Eq. (27) the b assumed equals 1.5. This confection
was obtained by fitting in FE results. By Eqs. (18), (25),
(27), and (28), the load-displacement diagram of the SPSW
ones are drawn.
6. Accuracy of proposed formulas
In this section as shown in figure 19, to evaluate the
accuracy of the proposed relationships, the pushover curve
obtained from the proposed relationships is compared with
Figure 17. The shear load-displacement diagram of the plate
[18].
Figure 18. Requirement properties for sketch load-displacement of frame.
132 Page 14 of 17 Sådhanå (2021) 46:132
the FE results. According to this figure, the proposed
relations in the elastic region correspond to the results of
the FE. Therefore, it predicts this zone with very good
accuracy. Also, in the inelastic zone, an error of less than
8% is observed, which indicates that the proposed rela-
tionships are of acceptable accuracy.
7. Conclusions
In the present paper, the behavior of perforated SPSW was
investigated experimentally and numerically. Based on the
numerical and experimental results, optimumperforated SPSW
was proposed.Also, a formulawas presented to calculate lateral
shear strength. The results are summarized as follows:
0
1000
2000
3000
4000
5000
6000
0 20 40 60 80
Base
shea
r (kN
)
Displacement (mm)
S4-6-10 Wall
Plate Frame0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 20 40 60 80
Base
shea
r (kN
)
Displacement (mm)
S4-4-10 Wall
Plate Frame
0500
100015002000250030003500400045005000
0 20 40 60 80
Base
shea
r (kN
)
Displacement (mm)
Wall Plate
Frame S4-6-200
500100015002000250030003500400045005000
0 20 40 60 80
Base
shea
r (kN
)
Displacement (mm)
Wall Plate
Frame S4-6-20
0500
10001500200025003000350040004500
0 20 40 60 80
Base
shea
r (kN
)
Displacement (mm)
S4-6-30 Wall
Plate Frame0
500
1000
1500
2000
2500
3000
3500
4000
0 20 40 60 80
Base
shea
r (kN
)
Displacement (mm)
S4-4-30 WallPlate Frame
Figure 19. Comparison of pushover curve with the proposed model.
Sådhanå (2021) 46:132 Page 15 of 17 132
• The configuration of perforated SPSW does not
significantly affect the ultimate strength of the system.
Therefore, the proposed formula can be used for all
types of perforated SPSW.
• At drift around 0.4%, the stiffness of perforated SPSW
and SPSW coincide together. For perforated SPSW, at
drift greater than 0.1%, the hole type does have any
effect on the lateral stiffness.
• In the elastic zone, the configuration of the hole does a
significant effect on the bending moment of the
column. After adrift around 0.5%, the bending moment
is reduced in columns for all models. More effective-
ness is seen in the S5 model. In these models, by
increasing of hole area from 10% to 30% the bending
moment is reduced by 14%. In the other models, the
reduction is between 1% and 8%. Therefore, it is
concluded that by perforated SPSW the bending
moment can be reduced maximum to 14%.
• Perforated SPSW with a reduction of 20% in infill
plate area cause to 15% to 21% reduction in ultimate
strength. The S1 and S2 result in maximum reduction
(35% to 38%) in elastic stiffness and S4 and S6 result
in minimum (28% to 29%) reduction in elastic
stiffness. Unlike of ultimate strength, reduction in
elastic stiffness is more.
• The hole type does not have considerable effect on the
imposed tension stresses to columns. But, utilizing the
configuration of the hole as model S1 reduces the
tension stress up to drift 1%. Unlike tensile stresses,
model S1 does not suitable performance in compres-
sive stresses.
• The lower compression forces are created in models
S5, S4, S6, and S3, respectively. The reduction
compression forces of the column for models S6 and
S4 are almost close. Before drift of0.3% to 0.6%, the
compression forces of columns for S5, S4, S6, and S3
are bigger than SPSW. Since in the range of drift the
wall experience of nonlinear behavior, the increasing
not important and can be ignored.
• By increasing of hole area from 10% to 30% the
compression force of the column reduced by 12% for
models S1 and S2. For other models, the compression
force of the column is reduced only 2% to 3%. The
results indicated that besides models S1 and S2, the
configuration of the hole is more important than the
area hole.
• Reduction of ultimate strength is 39%, 40%, 10%,
12%, 21%, and 21%, respectively, for S1, S2, S3, S4,
S5, and S6. Therefore, by increasing of area hole the
more reduction is made respectively in S1 and S2, S5
and S6, S4, and S3. So, the configuration of S1 and S2
is not suggested due to the high reduction of the
ultimate strength of SPSW.
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Sådhanå (2021) 46:132 Page 17 of 17 132