Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls

Engineering Structures 27 (2005) 963–976
www.elsevier.com/locate/engstruct

Efficient three-dimensional seismic analysis of a high-rise building
structure with shear walls
Hyun-Su Kima, Dong-Guen Leea,∗, Chee Kyeong Kimb
a Department of Architectural Engineering, Sungkyunkwan University, Chun-chun-dong, Jang-an-gu, Suwon, 440-746, Republic of Korea
b Department of Architecture, Sun moon University, Kalsan-ri, Tangjeong-myeon, Asan-si, Chungnam, 336-708, Republic of Korea

Received 1 November 2003; received in revised form 1 December 2004; accepted 17 February 2005
Available online 8 April 2005

Abstract
In many cases, high-rise building structures are designed as a framed structure with shear walls that can effectively resist horizontal
forces. Many of the high-rise apartment buildings recently constructed in the Asian region employ the box system that consists only of
reinforced concrete walls and slabs as the structural system. In most of these structures, a shear wall may have one or more openings
for functional reasons. It is necessary to use a refined finite element model for an accurate analysis of a shear wall with openings. But it
would take a significant amount of computational time and memory if the entire building structure were subdivided into a finer mesh. Thus
an efficient method that can be used for the analysis of a high-rise building structure with shear walls regardless of the number, size and
location of openings in the wall is proposed in this study. The proposed method uses super elements, substructures and fictitious beams.
Static and dynamic analyses of example structures with various types of opening were performed to verify the efficiency and accuracy of the
proposed method. It was confirmed that the proposed method can provide results with outstanding accuracy requiring significantly reduced
computational time and memory.
© 2005 Elsevier Ltd. All rights reserved.
Keywords: Shear wall with openings; Super elements; Substructuring technique; Matrix condensation; Stiff fictitious beam

1. Introduction
It is common to design high-rise building structures in
a framed structure with shear walls to resist horizontal
loads such as wind or seismic loads. This structural
system may have many openings in the shear walls to

accommodate the entrances to elevators or staircases etc.,
as shown in Fig. 1. In the analysis of this kind of building
structure, commercial software such as ETABS [1] and
MIDAS/ADS [2] is generally used. In general, plane stress
elements and beam elements are used to model the shear
walls and frames respectively in the analysis of this kind of
building structures. Drilling degrees of freedom are required
in the plane stress elements for the connection of shear

∗ Corresponding author. Tel.: +82 31 290 7554; fax: +82 31 290 7570.

E-mail address: (D.-G. Lee).
0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2005.02.006

walls and frames. Otherwise, beams cannot be rigidly
connected to shear walls, resulting in the underestimation of
the lateral stiffness of a building structure. For this reason,
the use of plane stress elements with drilling degrees of
freedom was proposed by Allman [3], and Bergan and
Fellipa [4]. The concept has been further elaborated by many
other researchers to obtain improved elements [5–8]. Choi
et al. added non-conforming modes to the translational and
rotational degrees of freedom (DOFs) to obtain an improved
element [9]. Kwan et al. developed a finite element with
rotational DOFs defined as vertical fiber rotations, which
is compatible with beam elements [10,11]. Lee provided
a 12 DOFs plane stress element having two translational
DOFs and one rotational DOF per node based on the
16 DOFs plane stress element proposed by Barber [12,13].

The displacement shape functions along the boundary of the
Lee element are identical to those of a typical beam element
and the Lee element can accurately represent the shear stress


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H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

(a) Floor plan.

(b) A-A section.

Fig. 1. Typical frame structure with shear walls.

(a) Typical plan of apartment.

(b) Window type
opening.

(c) Door type opening.
Fig. 2. Shear wall with openings.

distribution in an element. Therefore, a Lee element can be
used appropriately for the modeling of the shear wall in the
building structures.
Recently, many high-rise apartment buildings have been
constructed in the Asian region using the box system, which
consists only of reinforced concrete walls and slabs. Shear
walls in a box system structure may have openings to

accommodate windows, doors and duct spaces, as shown in
Fig. 2(a), and window and door type openings in shear walls
are shown in Fig. 2(b) and (c). The number, location and size
of these openings would affect the behavior of a structure as
well as stresses in the shear wall. Therefore, it is necessary
to use a refined finite element model for an accurate analysis
of a shear wall with openings. But it would be inefficient
to subdivide the entire apartment building structure into a
finer mesh with a large number of elements because of the
tremendous amount of analysis time and computer memory
costs. Therefore, many researches on the efficient analysis
of a shear wall with openings have been performed [14–17].
Ali and Atwall have presented a simplified method for
the dynamic analysis of plates with openings based on
Rayleigh’s principle of equilibrium of potential and kinetic
energies in a vibrating system [14]. Tham and Cheung
have also presented an approximate analytical method for a

laterally loaded shear wall system with openings [15]. Each
opening is taken into account by incorporating a negative
stiffness matrix into the overall stiffness matrix through the
super element concept. Choi and Bang have developed a
rectangular plate element with rectangular openings [17].
The stiffness matrix of the element was formed by numerical
integration in which the region for the opening in the
element was excluded. But the efficiency and accuracy of
these analysis methods largely depended significantly on the
location, size and number of openings.
Approximate modeling methods for a shear wall with
openings are frequently adopted to avoid the troublesome

preparation of refined models and significant amount of
computational time in practical engineering. When the size
of an opening is significantly smaller than that of the shear
wall, the opening is usually ignored, as shown in Fig. 3(a).
In the case of a door type opening, the lintel may be modeled
by an equivalent stiffness beam, as shown in Fig. 3(b). If
the opening is quite large, the surrounding part of the shear
wall would be modeled using beam elements, as shown
in Fig. 3(c) and (d). However, this type of models may
lead to inaccurate analysis results, especially in dynamic
analyses [18].
An efficient method for an analysis of a shear wall with
openings was proposed by Lee et al. using stiff fictitious
beams to enforce the compatibility at the boundary of super
elements [18,19]. Fig. 4(a) shows the deformed shape of
a shear wall with window type openings due to lateral
loads obtained using a refined finite element model. The
model using super elements derived without stiff fictitious
beams could not satisfy the compatibility condition at the
interfaces, as shown in Fig. 4(b). As could be observed
in Fig. 4(c), stiff fictitious beams used in a super element
could result in the deformed shape of the structure very
close to that of the refined mesh model. A similar result
could be obtained, as shown in Fig. 5, for a shear wall
with door type openings. This method is very efficient for
a two-dimensional analysis of a shear wall with openings.
Therefore, similar results can be expected in a threedimensional analysis of high-rise building structures if a
three-dimensional super element developed in a similar
manner were used.
An efficient method for a three-dimensional analysis of

a high-rise building structure with shear walls is proposed
in this study. Three-dimensional super elements for shear
walls and floor slabs were developed and a substructure
was formed by assembling the super elements to reduce the
time required for the modeling and analysis. The proposed
method turned out to be very useful for an efficient and
accurate analysis of high-rise building structures based on
the analysis of example structures.
2. Use of a fictitious stiff beam
The use of a fictitious stiff beam is one of the
most important techniques used in the proposed analytical


H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

965

Fig. 3. Approximate modeling methods for shear wall with openings.

(a) Refined mesh.

(a) Fine mesh model.

(b) Super element w/o
fictitious beams.

(c) Super element
w/ fictitious beams.

Fig. 4. Deformed shape of a shear wall structure with window type

openings.

(b) Super element w/o fictitious beam.

(c) Super element w/ fictitious beam.
Fig. 6. Deformed shape of box system structure.

(a) Refined mesh.

(b) Super element w/o
fictitious beam.

(c) Super element
w/ fictitious beam.

Fig. 5. Deformed shape of a shear wall structure with door type openings.

method. Therefore, the procedure of the use of a fictitious
beam is theoretically explained in this section. Three types
of modeling methods are used to verify the efficiency of the
proposed method, as shown in Fig. 6. Fig. 6(a) represents the
refined mesh model that is assumed to be the most accurate.
Each shear wall in a story can be modeled with a single
element, as shown in Fig. 6(b), for more efficient analysis.
The proposed model in this study is illustrated in Fig. 6(c).

The equilibrium equation for the refined mesh model can
be rearranged as shown in Eq. (1) by separating the active
DOFs for the corners of shear walls from the inactive DOFs
for the boundary and inner area of shear walls and floor slab

as follows:
SD =
=

Sii Sia
Sai Saa

Di
Da

)
)
S(W
S(S)
S(S)
S(W
ii
ia
ii
ia
+
)
(S)
)
S(S)
S(W
S(W
aa
ai Saa
ai

Ai
Di
=
Da
Aa

(1)

where subscripts a and i are assigned to the DOFs for the
active and inactive nodes respectively, the matrix S(S) is the


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H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

stiffness matrix for a floor slab, and S(W ) is the stiffness
matrix for a shear wall.
A Gaussian elimination process can be employed to
condense Eq. (1) into the equation consisting of only active
DOFs at corner nodes of the slab and wall.
GSD = GA

(2)

where the matrix G makes the stiffness matrix S into an
upper triangular matrix. If the equation is represented by
separating the active and inactive DOFs, then
Gii O

Gai I

Sii Sia
Sai Saa

Di
Gii O
=
Da
Gai I

Ai
.
Aa

=

Gii Sia
Gii Sii
O Gai Sia + Saa

=

Gii Ai
Gai Ai + Aa

Di
Da
Di
Da

(4)

T
.
Sia = −Sii Gai

(5)

The second row of Eq. (4) can be expanded as follows:
(Gai Sia + Saa )Da = Gai Ai + Aa .

(6)

Substitution of Eq. (5) into Eq. (6) leads to the following
result:
T
+ Saa )Da = Gai Ai + Aa .
(−Gai Sii Gai

(7)

This equation can be represented by using the slab stiffness
matrix (S(S)) and the shear wall stiffness matrix (S(W )) as
follows:
(S)

(W )

T
T

(W )
− Gai Sii Gai
+ S(S)
(−Gai Sii Gai
aa + Saa )Da
= Gai Ai + Aa .

(8)

On the other hand, modeling a shear wall using a single
element and joining a shear wall to a slab only at corner
nodes leads to the following equilibrium equation:
(S)
(S)
O O
Sii Sia
(S)
(S) + O S(W A)
Sai Saa
aa

Di
Ai
=
Da
Aa

(9)

A)

where the matrix S(W
is the stiffness matrix for shear walls
aa
that is modeled by a single element. It is different from
)
S(W
aa , which is the stiffness matrix for active DOFs of shear
walls modeled with a refined mesh. In order to make the
equilibrium equation consist of only active DOFs at common
nodes of the slab and wall, a Gaussian elimination process
can be employed as follows:

Hii O
Hai I

O O
S(S)
S(S)
ii
ia
+
A)
(S)
O S(W
Sai S(S)
aa
aa

Hii O
=

Hai I

Ai
Aa

Hii S(S)
Hii S(S)
ii
ia
(S)
(W A)
O
Hai Sia + S(S)
aa + Saa

Di
Da

Hii Ai
Hai Ai + Aa

=
(S)

(3)

If Eq. (3) is developed, the stiffness matrix is transformed
to an upper triangular matrix and Sia can be represented as
follows:
Gii Sia

Gii Sii
Gai Sii + Sai Gai Sia + Saa

where the matrix HHaiii OI makes the stiffness matrix into an
upper triangular matrix. Eq. (10) can be represented as an
upper triangular stiffness matrix by the Gaussian elimination
process and Sia can be given as Eq. (12):

(11)

(S)

T
.
Sia = −Sii Hai

(12)

After expansion of second row of Eq. (11), substitution of
Eq. (10) into that expanded equation leads to Eq. (13).
(S)

T
(W A)
(−Hai Sii Hai
+ S(S)
aa + Saa )Da = Hai Ai + Aa .

(13)

It can be easily noticed that the stiffness in Eq. (13) is
different from that of the equilibrium equation constituted
by the refined mesh model (Eq. (8)). To remove this
difference, a fictitious beam is employed in this study.
From the proposed method using a fictitious stiff beam, the
equilibrium equation can be represented as follows:
O O
S(S)
S(S)
S(B)
S(B)
ii
ia
ii
ia
+
A)
(S)
(B)
(S) +
O S(W
Sai Saa
Sai S(B)
aa
aa

Di
Da

Ai

Aa

=

(14)

where S(B) denotes the stiffness matrix of the fictitious
beam. A Gaussian elimination process was used to make the
equilibrium equation consist of only active DOFs at common
nodes of the slab and wall as follows:
Jii O
Jai I

B)
B)
O O
S(S
S(S
ii
ia
(S B)
(S B) + O S(W A)
Sai
Saa
aa

O O
O S(G)
aa

−

Di
J O
= ii
Jai I
Da

Ai
Aa

(15)

where S(S B) = S(S) + S(B) and S(G)
aa represents the stiffness
matrix of the beam that is to be subtracted.
From the Gaussian elimination process, Eq. (15) can be
(S B)
transformed into an upper triangular matrix and Sia can be
represented as Eq. (17).
B)
B)
Jii S(S
Jii S(S
ii
ia
(S B)
B)
(W A)
O

Jai Sia + S(S
− S(G)
aa + Saa
aa

Di
Da

Jii Ai
Jai Ai + Aa

(16)

B)
B) T
S(S
= −S(S
Jai .
ia
ii

(17)

=

Substitution of Eq. (17) into the second row of expanded
Eq. (16) gives:

Di
Da

(10)

(S B) T
Jai

(−Jai Sii

B)
(W A)
+ S(S
− S(G)
aa + Saa
aa )Da

= Jai Ai + Aa .

(18)


H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

967

From the equation S(S B) = S(S) + S(B), Eq. (18) can be
further expanded as follows:
(B) T
T
(S)
(B)
(W A)

(−Jai S(S)
− S(G)
aa )
ii Jai − Jai Sii Jai + Saa + Saa + Saa

× Da = Jai Ai + Aa .

(19)

If the stiffness (S(B)
aa ) of the fictitious beam is the same
)
as the stiffness (S(W
aa ) of the refined shear wall, the
following relationships can be noticed from comparison of
the equation of the refined mesh model (Eq. (8)) and that
of the proposed model (Eq. (19)). In conclusion, it can
be expected that the proposed method can approximately
represent the behavior of the refined mesh model.
G≈J
(S) T
(S) T
Gai Sii Gai
≈ Jai Sii Jai
(W ) T
(B) T
≈ Jai Sii Jai
Gai Sii Gai
A)
(W )

(B)
≈ S(G)
S(W
aa
aa → Saa ≈ Saa

(a) Refined mesh.

(20)
(21)
(22)
A)
+ S(W
aa

−

S(G)
aa .

(23)

Generally, the in-plane stiffness of a shear wall or floor
slab is significantly large compared with the out-ofplane stiffness. Therefore, a fictitious beam can employ
sufficiently large stiffness for the compatibility condition
as long as it may not cause numerical errors in the matrix
condensation procedure.
As stated previously, it would be more efficient to model
each shear wall in a story with one element to minimize the
number of nodal points used, which is shown in Fig. 6(b).

In this case, however, the compatibility condition will not
be satisfied at the interface of the slabs and the shear walls,
because most of the nodes at the boundary of the slabs
are not shared with those in the shear walls. The lateral
stiffness of this model becomes smaller than that of the
refined model. The stress distributions in the floor slab for
these two models are significantly different from each other,
as shown in Fig. 7(a) and (b). The number of elements used
in the proposed model shown in Fig. 6(c) is identical to the
model in Fig. 6(b), but much less than that of the refined
mesh model in Fig. 6(a). The deformed shape and stress
distribution of the model with fictitious beams are, however,
similar to those of the refined mesh model in Figs. 6(a)
and 7(a), which are considered to be the most accurate.
3. Modeling of a shear wall structure with openings
3.1. Finite element for modeling of shear walls and floor
slabs
The plane stress element used by Lee et al. for the
development of 2D super elements for the analysis of a
shear wall structure with openings was the Lee element [12]
with 12 DOFs, as shown in Fig. 8(a). Because the edge
of the Lee element deforms in a cubic curve just like the
beam element, the in-plane deformation of the edge of a
slab or shear wall including fictitious beams will be nearly
consistent with that of the neighboring shear wall or slab.

(b) Super element w/o fictitious beam.

(c) Super element w/ fictitious beam.
Fig. 7. Von-Mises stress distribution in slab.

The finite element to be used in this study should be able
to represent the out-of-plane deformation as well as the inplane deformation of walls and slabs for a three-dimensional
analysis of building structures with shear walls. For a threedimensional analysis of a high-rise building structure with
shear walls, a shell element with 6 DOFs per node shown
in Fig. 8(c) was introduced by combining the Lee element
and a plate bending element. For this purpose, the MZC
element [22] with a rectangular shape as shown in Fig. 8(b)
was selected because of the convenience in the combination
of stiffness matrices.
3.2. Modeling of a shear wall structure using super
elements
The efficiency in the modeling and analysis of a building
structure can be significantly improved by using super
elements. A super element derived from the assemblage of


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H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

(a) 12 DOFs plane stress element (Lee element).

(b) 12 DOFs plate bending element (MZC element).
(a) Refined model.

(b) Separate blocks.

(c) 24 DOFs shell element.
Fig. 8. Finite element for shear walls and floor slabs.

several finite elements for a shear wall or floor slab in the
structure has much fewer DOFs compared to the original
assemblage of finite elements. Therefore, the computational
time and memory can be significantly reduced. And the
modeling of the building structure would be more efficient
since a super element can be used repeatedly in many
places. Fig. 9(a) illustrates a refined mesh model of a shear
wall structure. This refined mesh model can be separated
into several blocks of finite elements having the same
configuration in each story, as shown in Fig. 9(b). Super
elements for shear walls and floor slabs can be generated, as
shown in Fig. 9(c), if all of the DOFs for the inactive nodes
are eliminated by using the matrix condensation technique
to have only active nodes in the super element. The active
nodes indicated by solid circles in Fig. 9(c) and (d) are used
to connect the shear walls and floor slabs. Then, the entire
structure is assembled by joining the active nodes of super
elements, as shown in Fig. 9(d).
The equation of motion for a block of finite elements can
be rearranged as shown in Eq. (24). The subscripts a and i
are assigned to the DOFs for the active and inactive nodes
respectively.
Mii Mia
Mai Maa

¨i
D
Sii Sia
¨ a + Sai Saa

D

Di
Ai
=
.
Da
Aa

(24)

Eliminating the DOFs by the matrix condensation procedure [23], the equation of motion for the super element can

(c) Generate super elements.

(d) Assemble super elements.

Fig. 9. Modeling procedure using super elements.

be obtained as follows:
¨ a + S∗aa Da = A∗a
M∗aa D

(25)

M∗aa

T
T
= Maa + Tia

Mia + Mai Tia + Tia
Mii Tia ,
where
−1
−1
∗
∗
Aa = Aa − Sai Sii Ai , Saa = Saa − Sai Sii Sia and Tia =
∗
∗
−S−1
ii Sia . The matrix Maa is the mass matrix, Saa is the
∗
stiffness matrix, Aa is the reduced action vector and Da is

the vector of nodal degrees of freedom for a super element
with only active nodes. If this super element is used in the
numerical model, the compatibility condition will not be
satisfied at interfaces of super elements because the nodes
only at the corners of the super elements are shared by
adjacent super elements. Therefore, the lateral stiffness of
the entire structure may be underestimated in comparison to
that of the refined model. Thus, it is necessary to enforce


H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

969

the compatibility without using additional nodes along the

interface of super elements for an accurate and efficient
analysis.
3.3. Super elements for shear walls and floor slabs
Stiff fictitious beams introduced by Lee et al. [18–21]
were used to enforce the compatibility at the interface of
super elements in this study. The use of fictitious beams
in the development of a super element for the floor slab
shown in Fig. 9(b) is illustrated in Fig. 10. Fictitious beams
are added to the interface of the floor slab and five shear
walls, as shown in Fig. 10(a). Because the analysis is
expanded from two dimensions [18] into three dimensions,
the fictitious beams used in this procedure are threedimensional elements. Then, all of the DOFs except those
for the active nodes located at the ends of each fictitious
beam are eliminated as shown in Fig. 10(b) using the matrix
condensation technique. The surplus stiffness introduced by
the fictitious beams should be eliminated by subtracting the
stiffness of fictitious beams from the stiffness matrix of the
super element, as shown in Fig. 10(c). It should be noticed
that the fictitious beams in Fig. 10(a) are subdivided into
many elements to share nodes with the refined mesh of the
floor slab, while the fictitious beam in Fig. 10(c) has nodes
only at both ends. Finally, a super element with the effect of
fictitious beams can be generated, as shown in Fig. 10(d).
Figs. 11–15 illustrate the use of fictitious beams in the
development of super elements for shear walls A, B, C, D
and E shown in Fig. 9(b). The location of fictitious beams
added to the refined model for a shear wall depends on the
location of the shear walls, and the selection of nodes to be
maintained in the super element depends on the type and
location of the openings in the shear wall. In a 2D analysis

of a shear wall structure, the compatibility condition is to
be satisfied on the boundary between the shear walls in
the adjacent stories. However, the compatibility condition
on the boundary between the neighboring shear walls in a
floor or between floor slabs and shear walls in addition to
the boundary between the shear walls in the adjacent stories
should be satisfied in a 3D analysis.
A fictitious beam is added to each side of the shear wall
A as shown in Fig. 11 to enforce the compatibility between
this shear wall and the shear wall B and D. The compatibility
condition between this shear wall and the slab in this floor
or the floor above can be approximately satisfied by the
fictitious beam added at the top or bottom of this wall. The
short fictitious beam added in between two openings is to
enforce the compatibility with the shear wall C.
The fictitious beams on both sides of the shear wall B
are to enforce the compatibility between this shear wall and
the shear wall A and E. The compatibility at the boundary
between this wall panel and the floor slab is enforced by two
short fictitious beams at the bottom of the wall, and the same
fictitious beams are added at the top, as shown in Fig. 12.
Since the opening is located at the left edge of the shear

(a) Add fictitious beams.

(b) Condense matrices.

(c) Subtract fictitious beams.

(d) Super element.

Fig. 10. Use of fictitious beams for floor slab.

C, as shown in Fig. 13, a short fictitious beam is added on
the left side of the wall for a similar reason of using a short
fictitious beam at the bottom of the wall. The short fictitious
beam used for the shear wall A in Fig. 11 and this fictitious
beam will enforce the compatibility at the boundary between
the shear walls A and C.
The fictitious beams on the perimeter of the shear walls
D and E, as shown in Figs. 14 and 15, are to enforce
compatibility at the boundary with shear walls or floor slabs


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H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

(a) Add fictitious beams.

(c) Subtract fictitious beams.

(b) Condense matrices.
(a) Add fictitious beams.

(b) Condense matrices.

(c) Subtract fictitious
beams.

(d) Super element.

(d) Super element.

Fig. 11. Use of fictitious beams for shear wall A in Fig. 9(b).

Fig. 14. Use of fictitious beams for shear wall D in Fig. 9(b).

(a) Add fictitious
beams.

(b) Condense matrices.

(c) Subtract fictitious
beams.
(a) Add fictitious beams.

(b) Condense matrices.

(c) Subtract fictitious beams.

(d) Super element.

(d) Super element.
Fig. 12. Use of fictitious beams for shear wall B in Fig. 9(b).

Fig. 15. Use of fictitious beams for shear wall E in Fig. 9(b).

3.4. Use of coarse mesh super elements

(a) Add fictitious beams.

(c) Subtract fictitious beams.

(b) Condense matrices.

(d) Super element.

Fig. 13. Use of fictitious beams for shear wall C in Fig. 9(b).

connected to this wall panel. The compatibility condition at
the boundary between shear walls C and E is approximately
satisfied by the fictitious beam located inside the shear
wall E.

In general, building structures have various arrangements
of shear walls and columns in plan. And the size, type and
location of openings in shear walls and floor slabs may vary
depending on their use. Therefore, the finite element mesh
for each block of a structure such as a floor slab or wall
panel is modeled to account of the location of openings,
shear walls and columns. The nodes on the boundary of
neighboring blocks should be shared in each block, as shown
in Fig. 16(a), to satisfy the compatibility condition. Thus,
it is necessary to use a finer mesh finite element model
to consider various openings and locations of structural
members for an accurate analysis of building structures.
However, when super elements with a limited number of
nodes are used, coarse mesh models for shear walls and
floor slabs can be used, as shown in Fig. 16(b), because
the compatibility at the boundary of the super elements is

enforced by the fictitious beams. Therefore, the location
of nodes except the nodes shared with neighboring super


H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

(a) Model A.

971

(b) Model B.

(a) Fine mesh model.

(c) Model C.

(d) Model D.

(b) Coarse mesh model.
Fig. 16. Mesh type of proposed analysis method.

elements and the mesh size are not restricted. Thus, it would
be very efficient to use super elements in modeling as well
as in the analysis of a building structure.
Static analysis of the 5-story example structure shown in
Fig. 9 was performed to verify the accuracy of the proposed
method using five types of models, as shown in Fig. 17.
Model A is a fine mesh model which is assumed to provide
the most accurate results. Models B and C replace the link
beam above the opening by an equivalent stiff beam, as

shown in Fig. 17(b) and (c). The rigid diaphragm assumption
is applied to each floor in model C and the flexural stiffness
of the floor is ignored. Model D employs the super element
proposed in this study generated from a fine mesh model
while model E is derived from a coarse mesh model, as
shown in Fig. 17(d) and (e).
The lateral displacements of each model subjected to
a lateral load of 10 000 kg at roof level in the transverse
direction are compared in Fig. 18. In the case of models
B and C, the lateral displacements were significantly
larger than those of model A. This overestimation in
displacements was introduced by the overestimation of the
shear deformation in the upper part of the shear wall at both
sides of the opening because the lintel is modeled by an
equivalent beam element. Since the flexural stiffness of the
floor slab was ignored in model C, the lateral displacements
were even larger than those of model B. Model D could
provide lateral displacements very close to those of model
A, indicating that the compatibility is well enforced at the

(e) Model E.
Fig. 17. Name of analytical models.

Fig. 18. Lateral displacement of example structure.

boundary of super elements by the effect of fictitious beams.
Since the lateral stiffness of a coarse mesh model is usually


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H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

overestimated compared to that of a fine mesh model, model
E resulted in slightly smaller displacements compared to
those of model D.

4. Three-dimensional modeling of a building structure
using substructures
(a) Refined mesh model of shear walls.

Most of the high-rise buildings may have the same plan
repeatedly in many floors. Thus, it may be very efficient to
apply the substructuring technique in the preparation of the
numerical model. In this section, the procedure in modeling
a building structure using substructures is presented for the
case of a high-rise apartment building. Shear walls in a story
are modeled as a substructure by assembling super elements,
and a floor slab is modeled by combining super elements for
the floor slab of each residential unit and staircase.

(b) Blocks for shear walls.

4.1. Modeling of shear walls using substructures
The modeling procedure for shear walls in a story using
a substructure is illustrated in Fig. 19. The refined finite
element model for the shear walls in a typical floor shown in
Fig. 19(a) is to be modeled as a substructure. As illustrated
in Fig. 19(b), the refined mesh model is separated into
many blocks for the generation of super elements. The

separated blocks for shear walls can be classified into
several types according to their configuration. If several
shear walls are of the same type, they can be modeled by the
same super element. Then, the super elements derived from
corresponding blocks, as shown in Fig. 19(c), are assembled
into a substructure for shear walls in a typical floor, as shown
in Fig. 19(d).

(c) Generation of super elements.

(d) Generation of substructure.

4.2. Modeling of floor slabs by using substructures
The procedure to model floor slabs in a floor into
a substructure is illustrated in Fig. 20. The refined finite
element model for floor slabs in a floor is shown in
Fig. 20(a). The floor slab in a floor can be separated
into three blocks for the residential units and staircase, as
illustrated in Fig. 20(b), to develop super elements. Super
elements are derived for corresponding residential units and
staircase respectively, as shown in Fig. 20(c). Since super
element SE-A’ is the mirror image of super element SE-A,
the stiffness and mass matrices for this super element can
be obtained easily by rearranging the DOFs and changing
the algebraic sign of terms correspondingly. The number of
super elements to be used in modeling the floors in a building
structure will be limited, because the type of residential units
in a high-rise apartment building is usually limited to one
or two. A substructure for the floor slab in a floor can be
formed by assembling the super elements, as illustrated in

Fig. 20(d). The nodes in the substructure are selected for the
connection of the slab and shear walls.

Fig. 19. Modeling process of shear walls by using a substructure.

4.3. Three-dimensional modeling of building structures
using substructures
The entire structure can be modeled by assembling the
substructures representing the floor slabs and the shear walls,
respectively. Fig. 21 illustrates the modeling procedure for a
typical story by combining the floor slab substructures with
the shear wall substructures. This substructure can be used
repeatedly for all of the stories with the same floor plan
in a building structure. If the rigid diaphragm assumption
is applied, the number of in-plane DOFs in a floor can be
reduced to three, and out-of-plane DOFs can be eliminated
by the matrix condensation procedure again. Therefore,
building structures, for which the slab and the shear wall
are subdivided into plate elements, can be modeled as a
stick having 3 DOFs per story. Therefore, the computational
time and memory for the analysis can be significantly
reduced in comparison with the refined mesh model when


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973

5. Analysis of example structures

(a) Refined mesh model of floor slab.

(b) Division of floor slab.

(c) Generation of super element.

(d) Generation of substructure.
Fig. 20. Modeling process of floor slab by using a substructure.

Analyses of two example structures were performed to
verify the efficiency and accuracy of the proposed numerical
method. A framed structure with a shear wall core and a
box system structure were used as example structures in the
analyses. Equivalent lateral forces were applied for the static
analysis and the ground acceleration record of the El Centro
(1940, NS) was used as input ground motion for the dynamic
analysis.
5.1. A framed structure with a shear wall core
Recently, many high-rise buildings have been constructed
using a frame with shear wall cores. Static and dynamic
analyses of a 10-story building structure with door type
openings in the shear wall core, as shown in Fig. 22, were
performed.
Equivalent static, eigenvalue and time history analyses
were performed and the results are shown in Fig. 23. Models
A, C and D were prepared in the same manner as explained
in Fig. 17. Model C is frequently used by many practical
engineers, and the method proposed in this study was
used in model D. The lateral displacements of model D
are similar to those of model A, as could be observed in

Fig. 23(a), while model C significantly overestimated the
lateral displacements for the same reason as explained in
Section 3.4 for the similar overestimation in Fig. 18. The
natural periods of model C were longer than those of the
other models as expected based on the lateral displacements,
as shown in Fig. 23(b). The roof displacement time histories
of models A and D are very close, while model C resulted in
somewhat different displacement from the others, as shown
in Fig. 23(c), because of the difference in the natural periods.

5.2. A shear wall structure

Fig. 21. Modeling process of typical story by using substructures.

the proposed method is used in the analysis, because the
model can represent the behavior of a refined mesh model
using only a limited number of DOFs. Furthermore, the
time and effort required for the preparation of a numerical
model can be significantly saved if the building has an
identical plan in many floors. This kind of stick model is
employed by conventional analysis software such as ETABS
or MIDAS/ADS. However, this conventional stick model
does not include the flexural stiffness of a floor slab or the
effects of openings in shear walls.

The second example structure is a 20-story reinforced
concrete shear wall building with window type and door type
openings, as shown in Fig. 24. The example structure has
two residential units arranged symmetrically with a staircase
in between. The thickness of shear walls and floor slabs is

20 cm and 15 cm, respectively. Analyses of the example
structure were performed, and the results shown in Fig. 25
were obtained.
The lateral displacements of the proposed model D turned
out to be almost identical to those of model A while model
C significantly overestimated the lateral displacements,
as shown in Fig. 25(a), because of the underestimation
of the lateral stiffness. The natural periods of vibration
are overestimated by model C, as shown in Fig. 25(b).
Therefore, the displacement time history from model C is
somewhat deviated from the others, as shown in Fig. 25(c).
Model D could provide a roof displacement time history


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Table 1
Comparison of DOFs and computational time required for analysis
Models

Number of DOFs

Model A
Model C
Model D

56 640
60

60

Computational time (s)
Assembly
Static
M&K
analysis

Eigenvalue
analysis

Time history
analysis

Total

78
10
156

16 623
97
97

387
12
13

18 071
125

272

983
6
6

(a) Typical floor.

(a) Lateral displacements.

(b) Natural periods.

(b) Floor plan.

(c) Displacement time histories.
Fig. 23. Seismic analysis results of the example structure from the models
A, C and D.

(c) A-A section.
Fig. 22. Example structure.

almost identical to that of model A, as could be expected
from the accuracy in the periods of vibration.
The computational time and the number of DOFs of
each numerical model used for the analysis of the example
structure are compared in Table 1. Models C and D used

only 60 DOFs because they applied the rigid diaphragm
assumption to reduce the DOFs in a floor to 3, while model
A, which is a refined finite element model, used more than

900 times the number of DOFs compared to the others.
Models A and C required 78 and 10 s to obtain stiffness
and mass matrices while model D required 156 s because of
the additional computation required to derive super elements
and substructures. The total computational time for the
model A was 18 071 s including static and dynamic analyses
while model C required only 125 s, demonstrating the
reason why this model is commonly used by practicing
engineers. However, static and dynamic responses obtained
using model C were significantly different from those of


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975

(a) Typical floor.

(b) Floor plan.

(a) Lateral displacements.

(b) Natural periods.

(c) Roof displacement time history.
Fig. 25. Seismic analysis results of the example structure from the models
A, C and D.

(c) 3D view of example
structure.

Fig. 24. Example structure.

model A. Model D could perform the analysis in 272 s
because the computational time required for the procedure
except for the formulation of mass and stiffness matrices is
almost the same as that of model C, because of the same
number of DOFs used in the analysis. The accuracy in the
static and dynamic analysis results of model D was at a
similar level to that of model A, while the computational
time required by model D is about 1.5% of that for model
A. In the case of larger building structures such as 30- or
40-story buildings with 4 or 6 residential units in a floor, the
efficiency of the proposed model will be more significant
because the same super elements can be used for the
additional residential units and the same substructures can
be used for the additional stories. Therefore, the proposed
method can be an efficient means for the analysis of a highrise building structure with shear walls. A personal computer
with Pentium 3 500 MHz processor and 512 MB RAM was
employed in this study.

6. Conclusions
An efficient three-dimensional model for the analysis
of building structures with shear walls was proposed in
this study using super elements and substructures. The
super elements were derived by introducing fictitious beams
to satisfy the compatibility condition at the interfaces of
super elements. The accuracy and the efficiency of the
proposed method were investigated by performing analyses
of example structures. Based on this study, the main features
of the proposed method considered are summarized below:

1. The refined finite element model of a high-rise building
structure with shear walls is expected to cost a significant
amount of computational time and memory while it
would provide the most accurate results. Thus the refined
mesh model may not be feasible for practical engineering
purpose.
2. The model using equivalent beams for the lintel
above the openings and ignoring the flexural stiffness
of the floor slab may lead to analysis results with
somewhat deteriorated accuracy while computational
time is significantly reduced. Thus, it is undesirable
to use this model for the analysis of an important or
complicated building structure. Therefore, it is desirable


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H.-S. Kim et al. / Engineering Structures 27 (2005) 963–976

for the engineers in practice to be aware of the limitation
in the accuracy of the results obtained by this model.
3. The proposed method could provide static and dynamic
analysis results with an accuracy comparable to that
of a refined mesh model with the cost of slightly
increased computational time compared to the model
using equivalent beams for the lintel. Therefore, the
proposed method can be an efficient means for the
analysis of a high-rise building structure with shear
walls.
4. The super elements are connected only through the

active nodes and fictitious beams are used to enforce the
compatibility at the boundary of super elements because
the inactive nodes at the boundary are eliminated in the
proposed method. Thus, the location of inactive nodes in
the finite element mesh to be used for a super element
is not required to coincide with the counterpart in a
neighboring super element. Therefore, a super element
can be developed easily, accounting for the location of
active nodes independently.

Acknowledgements
The Brain Korea 21 Project supported this work, and
this work was partially supported by the Korea Science
and Engineering Foundation (KOSEF) through the Korea
Earthquake Engineering Research Center (KEERC) at the
Seoul National University (SNU).
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