Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls 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.
Trang 1Engineering Structures 27 (2005) 963–976
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Efficient three-dimensional seismic analysis of a high-rise building
structure with shear walls Hyun-Su Kima, Dong-Guen Leea,∗, Chee Kyeong Kimb
aDepartment of Architectural Engineering, Sungkyunkwan University, Chun-chun-dong, Jang-an-gu, Suwon, 440-746, Republic of Korea
bDepartment 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 inFig 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: dglee@skku.ac.kr (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
Trang 2964 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 inFig 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 inFig 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
inFig 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 three-dimensional 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
Trang 3H.-S Kim et al / Engineering Structures 27 (2005) 963–976 965
Fig 3 Approximate modeling methods for shear wall with openings.
(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.
(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 inFig 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 inFig 6(b), for more efficient analysis
The proposed model in this study is illustrated inFig 6(c)
(a) Refined mesh.
(b) Super element w/o fictitious beam.
(c) Super element w/ fictitious beam.
Fig 6 Deformed shape of box system structure.
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(S)
ii S(S) ia
S(S)
ai S(S) aa
+
S(W )
ii S(W ) ia
S(W )
ai S(W ) aa
Di
Da
=
Ai
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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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
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
Da
=
Gii O
Gai I
Ai
Aa
If Eq (3) is developed, the stiffness matrix is transformed
to an upper triangular matrix and Sia can be represented as
follows:
GiiSii GiiSia
GaiSii+ Sai GaiSia+ Saa
Di
Da
=
GiiSii GiiSia
O GaiSia+ Saa
Di
Da
=
GiiAi
GaiAi+ Aa
(4)
Sia= −SiiGT ai (5)
The second row of Eq (4) can be expanded as follows:
Substitution of Eq (5) into Eq (6) leads to the following
result:
This equation can be represented by using the slab stiffness
matrix(S (S) ) and the shear wall stiffness matrix (S (W ) ) as
follows:
ii Gai T − GaiS(W )
ii GT ai+ S(S)
aa + S(W )
aa )D a
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)
ii S(S)
ia
S(S)
ai S(S)
aa
+
O O
O S(W A) aa
Di
Da
=
Ai
Aa
(9)
where the matrix S(W A)
aa is the stiffness matrix for shear walls 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
S(S)
ii S(S)
ia
S(S)
ai S(S)
aa
+
O O
O S(W A) aa
Di
Da
=
Hii O
H I
Ai
A
(10)
where the matrix
Hii O
Hai I
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 Siacan be given as Eq (12):
HiiS(S)
ii HiiS(S)
ia
O HaiS(S)
ia + S(S) aa + S(W A) aa
Di
Da
=
HiiAi
HaiAi + Aa
(11)
S(S)
ia = −S(S) ii HT ai (12) After expansion of second row of Eq (11), substitution of
Eq (10) into that expanded equation leads to Eq (13)
ii HT ai+ S(S) aa + S(W A) aa )D a= HaiAi+ 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:
S(S)
ii S(S) ia
S(S)
ai S(S) aa
+
S(B)
ii S(B) ia
S(B)
ai S(B) aa
+
O O
O S(W A) 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
S(S B)
ii S(S B) ia
S(S B)
ai S(S B) aa
+
O O
O S(W A) aa
−
O O
O S(G) aa
Di
Da
=
Jii O
Jai I
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
transformed into an upper triangular matrix and S(S B)
ia can be represented as Eq (17)
JiiS(S B)
ia
O JaiS(S B)
ia + S(S B) aa + S(W A) aa − S(G) aa
Di
Da
=
JiiAi
JaiAi + Aa
(16)
S(S B)
Substitution of Eq (17) into the second row of expanded
Eq (16) gives:
(−J aiS(S B)
ii JT ai + S(S B) aa + S(W A) aa − S(G) aa )D a
Trang 5H.-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:
(−J aiS(S)
ii JT ai − JaiS(B)
ii JT ai+ S(S)
aa + S(B)
aa + S(W A)
aa − S(G)
aa )
If the stiffness (S (B)
as the stiffness (S (W )
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
GaiS(S)
ii GT ai ≈ JaiS(S)
GaiS(W )
ii GT ai ≈ JaiS(B)
S(W A)
aa ≈ S(G) aa → S(W ) aa ≈ S(B) aa + S(W A) aa − S(G) aa (23)
Generally, the in-plane stiffness of a shear wall or floor
slab is significantly large compared with the
out-of-plane 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 inFig 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 inFig 7(a) and (b) The number of elements used
in the proposed model shown inFig 6(c) is identical to the
model inFig 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)
and7(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
(a) Refined mesh.
(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 in-plane deformation of walls and slabs for a three-dimensional analysis of building structures with shear walls For a three-dimensional analysis of a high-rise building structure with shear walls, a shell element with 6 DOFs per node shown
inFig 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 inFig 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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(a) 12 DOFs plane stress element (Lee element).
(b) 12 DOFs plate bending element (MZC element).
(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 inFig 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 inFig 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 inFig 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
¨Di
¨Da
+
Sii Sia
Sai Saa
Di
Da
=
Ai
Aa
Eliminating the DOFs by the matrix condensation
proce-dure [23], the equation of motion for the super element can
(a) Refined model (b) Separate blocks.
(c) Generate super ele-ments.
(d) Assemble super ele-ments.
Fig 9 Modeling procedure using super elements.
be obtained as follows:
M∗
aa¨Da+ S∗aaDa= A∗a (25)
where M∗
aa = Maa + TT
iaMia + MaiTia + TT
iaMiiTia,
A∗
a = Aa− SaiS−1
ii Ai, S∗
aa = Saa− SaiS−1
ii Sia and Tia =
−S−1ii Sia The matrix M∗
aa is the mass matrix, S∗
aa is the
stiffness matrix, A∗
a 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
Trang 7H.-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 inFig 9(b) is illustrated inFig 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
three-dimensional elements Then, all of the DOFs except those
for the active nodes located at the ends of each fictitious
beam are eliminated as shown inFig 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 inFig 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 inFig 10(c) has nodes
only at both ends Finally, a super element with the effect of
fictitious beams can be generated, as shown inFig 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 inFig 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 inFig 11to 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 inFig 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 inFig 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 inFig 11and 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
Trang 8970 H.-S Kim et al / Engineering Structures 27 (2005) 963–976
(a) Add fictitious beams (b) Condense matrices.
(c) Subtract fictitious beams (d) Super element.
Fig 11 Use of fictitious beams for shear wall A in Fig 9 (b).
(a) Add fictitious
beams.
(b) Condense ma-trices.
(c) Subtract fictitious beams.
(d) Super element.
Fig 12 Use of fictitious beams for shear wall B in Fig 9 (b).
(a) Add fictitious beams (b) Condense matrices.
(c) Subtract fictitious beams (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
(a) Add fictitious beams (b) Condense matrices.
(c) Subtract fictitious beams.
(d) Super element.
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 (d) Super element.
Fig 15 Use of fictitious beams for shear wall E in Fig 9 (b).
3.4 Use of coarse mesh super elements
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
Trang 9H.-S Kim et al / Engineering Structures 27 (2005) 963–976 971
(a) Fine mesh model.
(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 9was 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 inFig 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 inFig 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 inFig 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
(c) Model C (d) Model D.
(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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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
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
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 inFig 19(c), are assembled
into a substructure for shear walls in a typical floor, as shown
inFig 19(d)
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
(a) Refined mesh model of shear walls.
(b) Blocks for shear walls.
(c) Generation of super elements.
(d) Generation of substructure.
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 21illustrates 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