Paper No. TS-3-1

NUMERICAL STUDIES OF SEISMIC PERFORMANCE OF SPECIAL
PLATE SHEAR WALLS UNDER LONG DURATION EARTHQUAKES
Carlos MATOS1 and Karina CARBAJAL 2
SYMMARY
Special plate shear wall (SPSW) is a novel Earthquake-resistant alternative, withexcellent ductility and energy dissipation; easy to be replaced if damaged after a major
seismic event. Long duration earthquakes as the last Japan’s earthquake of 2011 present
new challenges. Some of them, relevant to buildings with SPSW are addressed here. This
study tries to quantify SPSW inelastic behavior under long duration earthquakes. In apreviously designed building, concrete shear walls were replaced by steel plates. The new
system was analyzed and designed using response-spectrum obtained from actual groundmotion records from Tohoku Earthquake. Finally, we perform push-over analyses toinvestigate behavior under extreme loads. Steel is modeled as elastoplastic and isotropic
hardening has been considered. SPSW (Beams, columns and walls) are discretized withshells elements, this allows seeing the distribution of plasticity on beams, which cannot
using traditional concentrated plasticity models. Simo’s Radial Return algorithm fornumerical integration of plasticity constitutive equations was used. We compare
numerically obtained capacities against capacity provided by AISC 341. We compute
triaxiality indexes to evaluate probability of fracture, since this is a key parameter todetermine fracture during earthquakes. Results show that steel plates yielded extensively
without signs of brittle fracture.
INTRODUCTION
Shear walls in buildings have been the main lateral resistant system used in our country and in most partof the world. It has been tested during major earthquakes. However, there are several shortcomings of thesystem. Especially there is a concern to repair damaged shear walls. The Special plate shear wall (SPSW)
is a feasible novel alternative, that can be easily repaired or replaced if damaged in a major event, also
allows expediting the construction of buildings since no form-work is required. There is currently an
extensive research of SPSWs in USA and Canada. The concept of the SPSW is based on plate girders
used for long span bridges developing post-buckling, developed during early sixties by Basler [6]. SPSW
is highly resistant to the type of damage that would reduce structural integrity. SPSW is designed todevelop large inelastic deformation and keep elastically boundary elements. Experimental tests of SPSW
show that the system is very ductile and dissipates energy; developing large inelastic deformation before
failure [8]1 Senior Structural Engineer, JEG, USA.
2 Structural Engineer,

Carbajal y Asociados, Perú.

Long duration earthquakes have been very destructive. Not only for the large input of energy, but for thelarger peak accelerations. A particular characteristic of the response spectrum of long duration strong
motions has been reported [8], this consist of a second peak located in the range of 2 to 3 seconds. Thatcan be appreciated on the spectrum of Japan Earthquake depicted in Figure 1. For normal earthquakes
typically design response spectra is composed of a ramp, a plateau and finally a decreasing curve. Thus
for tall buildings and long span bridges the response falls on the decaying part and the lateral system iscontrolled by wind forces, which would not be the case of structures subjected to long duration
earthquakes. The large duration earthquake also produces a large additional number of load cycles, which
in many cases take the structure many more times into the plastic range. This known as low-cycle-fatigue,
and it can onset and develop fatigue cracks, which is not addressed here.
The effort of this paper is two folded. First, in this study we try to present de design advantages of SPSW
system over the traditional Reinforced Concrete shear wall. Second, this study tries to quantify theinelastic effects of long duration earthquakes on SPSW. Our main objectives are to understand theelastoplastic distribution of stresses, investigate the possibility of fracture, and validate AISC design
equations. For the elastoplastic analysis, we use the J2 flow plastic theory. To investigate the possibility ofbrittle fracture we use triaxiality index [1].
This paper is organized as follows. Section 2 exposes the SPSW and the design requirements of AISC
341. We also present a comparative design of seven story building designed using Reinforced concrete
shear walls against the SPSW. In both cases we design for a long duration earthquake spectrum. We
conclude the section presenting some key comparative results.
In section 3 we describe the procedure of inelastic finite element analysis used, a few details of the radial
return algorithm used to integrate the incremental constitutive differential equations. We also describe thepushover analysis of the SPSW presented in section 2.
Finally in section 4 we present the conclusions and recommendations of the present study.

1.4
1.2
1.0

Acceleration Spectra
El Centro
E-030
TohokuSa (g)

0.8
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
T (s)Figure 1. Strong ground motion Tohoku and corresponding Spectrum

SPECIAL PLATE SHEAR WALLS
Description of SPSWThe special steel plate shear wall (SPSW) is a lateral-resisting system consisting of vertical steel plate
infill connected to surrounding beams and columns and installed in one or more bays along the full heightof the structure to form a cantilevered wall presented in Figure 2(a). When subjected to cyclic inelastic
deformations exhibits high initial stiffness, behave in a very ductile manner, and dissipate significant
amounts of energy. These characteristics make them suitable to resist seismic loading. SPSW can be used
not only for the design of new buildings but also for the retrofit of existing construction. Lateral loads aretransferred through the plate by the principal tension stresses. This post buckling behavior is typicallyreferred to as “tension-field action” This is illustrated in Figure 2(c). Lateral loads are resisted throughdiagonal tension in the web plate or steel plate, rather than in pure shear. Boundary elements are designed
to permit the web plates to develop significant diagonal tension; for high seismic design, this is to permit
the web plates to reach their expected yield stress across the entire panel. Beam-to-column connections in
SPSW may be either of the simple type or moment-resisting type. Steel plate shear walls have been used
in a large number of buildings, including in the United States, Canada, Mexico, and Japan. Building types
have ranged from single-family residential to high-rise construction.
Design requirements for SPSW
During the last two decades considerable research on SPSW was carried out in the U.S. Design provisions
were included in the 2005 AISC 341 [4], [9]. Main requirements by AISC 341 are the limits on aspectratio of 0.8<L/h≤2.5 (this allows for fairly wide walls, but not tall and slender walls) and the nominal
strength of the web plate = 0.42

 sin 2. (1)The AISC 341 also requires in order to achieve the high level of ductility expected of SPSW, theboundary members have to be designed to remain elastic as the web plate yields. As a result, size ofbeams and columns are designed based on the plate thickness. HBE’s and VBE’s have to satisfy the weak
beam/strong column requirements for moment frames without consideration of the web plate. The
moments in VBE’s caused by the web plate tension field are fairly severe. Similarly, the axial loads
caused by the overturning moments from web plates yielding are quite high. The required strength of
VBE’s shall be based upon the forces corresponding to the expected yield strength, in tension, of the web
calculated at an angle, α. To reach this level of force the beams are required to develop plastic hinges [9].
Plate design, however, is based on angle of inclination of the tension field, which is dependent on theboundary members. The design process, therefore, becomes iterative. Another factor to consider whenusing thin web plate is the shear that causes the plate to buckle. Plate buckling is expected to occur as thetension field develops, but it is not desirable for service loads
AISC 341 does not limit plate slenderness; instead, the commentary states that story drift limitations will
indirectly limit the plate slenderness. Acceptable plate material includes A36 or A1011 Grade 55.

Design Example of a 7-story building
To expose the design methodology, we re-design a Reinforced Concrete building. Shear walls have been
replaced by SPSW. We are going to compare both lateral resisting systems. Design of structure withSPSW was based on the AISC 341, AISC 360 and ASCE 7 and RNC. A seven story education Building
was selected that has 8-24’ bays in one direction and 3-27’ bays in the other direction. The seven floor
heights are each 9.85’. SAP software was used to analyze the structure. Tohoku Earthquake (longduration) Spectrum was used. Figure 2(b) shows an isometric view of building. An elevation of the SPSW
design is shown in Figure 2(a). Steel used is A36.
In the example building shown in Figure 2(b), the 9.85’ story heights would require a wall 24’ long.
SPSW web plate was 24’ long with 1/2” thickness. On upper floors could reduce the web plate, but wehave worked with the same web plate on all floors, this could be optimized, plate can have more capacity
than needed.
Four SPSW panels were required in large direction and seven in the building direction. The VBE design
considered the moments and axial forces due to all web plates fully yielding and HBE’s developing
plastic hinges.
The columns are W18x97 and the beams are W14x109. Shear walls were determined to be the best
structural system to accommodate the constraints of high design seismic forces and high stiffness(assumed adequate at the time to protect educational building systems from damage so they could stay inservice after a severe earthquake). The shear capacity for the web plate using equation 1 was 931Kips.
We summarize some key findings of this comparative study:
The structure with SPSW system has only 83% of the traditional Reinforced Concrete shear wall system’s
weight.
Consequently, the building’s base shear with SPSW system decreases in 17% and this reduction makesthat the structure’s demand also decreases in all structural system (walls, beam, and columns).
The shear force in the analyzed wall with SPSW system has only 81% of the Reinforced Concrete shear
wall systems’ shear force which is suitable in buildings located in high seismicity regions.
The fundamental period of the building with SPSW system is greater that the fundamental period of thesame building with the Reinforced Concrete shear wall system. Increasing the fundamental period of thebulging SPSW, it also increases the lateral displacement, but they are still within the limits of the
Peruvian seismic code.
On the analyzed shear wall was found that the ratio of lateral stiffness of the Reinforced concrete shear
wall and SPSW building was 2.32, it clearly explains why the Reinforced Concrete shear wall building is
stiffer.
For the fundamental period (T=0.38s) of the SPSW building, the spectral acceleration of “El Centro”
ground motion has been scaled by 1.57, and for “Tohoku” ground motion has been scaled by 1.60. Bothrecords were scaled in order to obtain similar demand levels than the Peruvian seismic code.

For the fundamental period (T=0.46s) of the Reinforced Concrete shear wall building, the spectral
acceleration
cceleration of “El Centro” ground motion has been scaled by 1.55, and for “Tohoku” ground motion has
been scaled by 1.42. Both records were scaled in order to obtain similar demand levels than the Peruvian
seismic code.
short-period buildings,
The spectral acceleration of “Tohoku” ground motion has not high demands on short
but it has higher demands on long-period
long period buildings, it means that this record should be useful in theanalysis and design of tall buildings in our country.
Advantages of the system
SPSW
W offers significant advantages in terms of cost, performance, and ease of design compared toReinforce concrete shear walls, the reduced thickness (and thus plan area devoted to them) represents a
substantial benefit. The reduced mass can also be significant
significant in the design of the foundation. Thickness ofwalls is important to the architects too [9] and [11].
[11] Most importantly, however, steel plate shear walls
can be erected in significantly less time than concrete shear wall structures, no curing time.
time
The ductility
uctility of steel web plates in SPSW results in good performance under severe seismic loading.
Because SPSW can provide significant strength and stiffness, shorter bays can be used. This results ingreater flexibility for use of the space.

Figure 2. (a) An elevation view of typical Steel plate shear (b) Completed building with special
moment frames the traditional Reinforced Concrete shear wall, and (c) SPSW Tension Field

INELASTIC FEM ANALYSIS OF SPSW
To investigate the inelastic behavior of SPSW system we perform nonlinear finite elements analyses.
Shell elements are the most appropriated element to model the web plate. Alternatively solid elements
could be used if resulting stresses area large and influence of the third dimension is important
important, althoughcomputationally prohibited
ibited since several elements through the thickness are required iin this bending
dominated problem. Shell element used is from the family of nonlinear geometrically exact shells with six
degrees of freedom and full integration [2]. Elastoplastic constitutive model is introduced by a J2 flow
plastic rate-independent
independent analysis. Isotropic hardening of steel has been considered.

In a typical load increment, for a global Newton iteration, we solve: ∆ =  − Where:• KT is the tangent stiffness; mathematically the Gateaux derivative of the internal forces or the
Hessian of the potential energy [3,6].  =    • Dep is the algorithmic-consistent constitutive model, which is lightly different from thecontinuum-consistent constitutive model• P is the external applied load, and  =   !Ω is the internal forces vector. Components of theinternal forces vector #$% =  &$% !'( , )$% =  &$% '( !'( , , + = 1,2 are computed throughnumerical integration using Gauss-Lobatto with five points of integration. Gauss-Lobatto allows
computing the stress on the surface without the necessity of extrapolating results. A minimum of
5 points is required to model elastoplastic loading/unloading.
x3
X2

X1

Figure 3. Shell integration points: 1-2 axes (2x2 Gauss points), Axel 3 (5 Gauss-Lobatto points)
Displacements are updated directly by  = ∆ +  and the increment of strains by ∆. = ∆ . Stresses cannot be updated in the same fashion; instead we use the radial return algorithm tointegrate plasticity equations i.e. flow rule, hardening law, Kuhn-Tucker loading/unloading conditions(yielding criteria), etc. To compute the current stresses σ = ℜ 0& , ∆. , .111,  2∆. 34. For

details see [2], [3], and [5]Elastoplasticity is a strain driven problem. For 3D or 2D plane-strain, a close-form solution for elasticperfectly-plastic and for bi-linear plasticity can be derivated. However, this is not the case for 2D-planestress, where the constrain σ33=0 must be satisfied. Simo [3] introduced the projection of stresses concept,
which make easier and efficient the algorithm. Local Newton iterations are required to satisfied plane̅stress constrain. Factors 5 67! 5are introduced in equation (2) to assure the quadratic rate ofconvergence of global Newton iterations.
 = 9: ⊗ : + 2<5 => − ( 1 ⊗ 1? − 25̅ @ ⊗ 7 (2)AB2C3

κ and λ are elastic constants, 7 = ‖AB2C3‖ a normal vector, and dev is the operator defined by!EF2∘3 = 2∘3 − ( H2∘3

Web plates are typically made of A36 or A572 steel. These are ferritic steels, which are fracture prone if
certain adverse conditions are present, e.g. high triaxiality, low temperature, or high loading rates. To
quantify the possibility of fracture, the triaxiality index has been used [1]. Once known the stress fields,
the triaxiality index can be easily computed.

 =Where:&L = MN2& + &O + &( 3 is the hydrostatic stress

CI
CJK

(3)


O

&PQ = R 2&O + &OO − & &O 3 is the Von Mises stress.
Stresses σH and σVM are computed at each Gaussian point from an incremental iterative process. A
triaxiality Index IT of less than one is required to preclude fracture.

APPLICATION TO A 7-STORY SPSSW BUILDINGSPSW designed in previous section has been studied. A push-over analysis has been performed. A
monotonically increasing loading has been applied each story level. Distribution of loads comes fromresulting forces from spectral analysis. Loads have been applied up to reach the collapse of the SPSW.
Numerically, this occurs when the tangent stiffness matrix become singular. Analysis program applied theload increment in an adaptive fashion.
1400
Applied Shear (k)

1200

φVn
=931k
AISC

1000
800
600

Elastic-perfectly-plastic

400

Elasto-plastic
w/Hardening

200
0
0

1

2

3
4
5
6
7
Displacement (in)89Figure 4. (a) Finite Element Mesh, (b) Deformed at failure load, and c) Applied Shear vs Lateral
Displacement
Figure 4(c) describes the global force-displacement at the seventh story level. It can be seen theelastoplastic global response. The inclusion of hardening adds a bit of capacity. We compare the capacity
load obtained by AISC 341, this value falls near the elastic limit.

Figure 5. Evolution of Von Mises stresses at 24%, 47%, 71%, 95%, and 100% of failure load

Figure 6. Evolution of Triaxiality Index at at 24%, 47%, 71%, 95%, and 100% of failure load
Figure 5 depicts the evolution of the Von Mises stress of the SPSW. At 95% can be seen that plasticity
has spread out extensively. At limit load the first floor SPSW has completely yielded and it become aplastic-hinge. Figure 6 shows the distribution of the triaxiality index for different levels of loading.
Maximum value of this index is less than 0.6, this occurs at the welding of the SPSW with the foundation.
This maximum value of the index of triaxiality tells us that we are far from possibility of brittle fracture
for the level of stresses applied.

CONCLUSIONS AND RECOMMENDATIONS
The important conclusions and recommendations of these numerical studies of seismic performance ofspecial plate shear walls under long duration earthquakes are summarized as follows:
SPSW is a viable alternative to replace Reinforced concrete shear walls. For new buildings and to
retrofit earthquake damaged buildings.
o The Peruvian seismic code spectrum cannot provide adequate safety for structures of periods
between 2.5 y 3.0 seconds when they are subjected to long duration earthquakes; Future versions
should take into account this.
o Due to second peak, more modes could necessary to satisfy dynamic mass participation.
o Capacity predicted by AISC 341 is adequate; results show that SPSW yield extensively under
extreme lateral loads, without signs of fracture in web plates.
o

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