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

8/7/2019 Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls

1/14


2/14

964 H.-S. Kim et al. / Engineering Structures 27 (2005) 963976

(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 necessaryto 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 [ 1417].

Ali and Atwall have presented a simplified method for

the dynamic analysis of plates with openings based on

Rayleighs 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 theelement 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 shownin 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 observedin 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 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


3/14

H.-S. Kim et al. / Engineering Structures 27 (2005) 963976 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/ofictitious beam.

(c) Super elementw/ 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).

(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 canbe 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 SiaSai Saa

DiDa

=

S

(S)ii S

(S)ia

S(S)ai

S(S)aa

+

S

(W)ii S

(W)ia

S(W)ai

S(W)aa

DiDa

=

AiAa

(1)

where subscripts a and i are assigned to the DOFs for the

active and inactive nodes respectively, the matrix S(S) is the


4/14


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, thenGii O

Gai I

Sii SiaSai Saa

DiDa

=

Gii O

Gai I

AiAa

. (3)

If Eq. (3) is developed, the stiffness matrix is transformed

to an upper triangular matrix and Sia can be represented as

follows:Gii Sii Gii Sia

Gai Sii + Sai Gai Sia + Saa

DiDa

=

Gii Sii Gii SiaO Gai Sia + Saa

DiDa

=

Gii Ai

Gai Ai + Aa

(4)

Sia = Sii GTai . (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:

(Gai Sii GTai + Saa )Da = Gai Ai +Aa. (7)

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

follows:

(Gai S(S)ii

GTai Gai S(W)ii

GTai + S(S)aa + S

(W)aa )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)ii

S(S)ia

S(S)ai S

(S)aa

+

O O

O S(W A)aa

DiDa

=

AiAa

(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 fromS(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

DiDa

= Hii O

Hai I AiAa (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 Sia can be given as Eq. (12):

Hii S(S)ii

Hii S(S)ia

O Hai S(S)ia + S(S)aa + S(W A)aa Di

Da

=

Hii Ai

Hai Ai +Aa

(11)

S(S)ia = S

(S)ii

HTai . (12)

After expansion of second row of Eq. (11), substitution of

Eq. (10) into that expanded equation leads to Eq. (13).

(Hai S(S)ii H

Tai + S

(S)aa + S

(W A)aa )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 thisdifference, 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

DiDa

=

AiAa

(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

DiDa

=

Jii O

Jai I

AiAa

(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).Jii S

(S B)ii

Jii S(S B)ia

O Jai S(S B)ia + S

(S B)aa + S

(W A)aa S

(G)aa

DiDa

=

Jii Ai

Jai Ai + Aa

(16)

S(S B)ia = S

(S B)ii

JTai . (17)

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

Eq. (16) gives:

(Jai S(S B)ii J

Tai + S

(S B)aa + S

(W A)aa S

(G)aa )Da

= Jai Ai + Aa. (18)


5/14


From the equation S(S B) = S(S) + S(B), Eq. (18) can be

further expanded as follows:

(Jai S(S)ii

JTai Jai S(B)ii

JTai + S(S)aa + S

(B)aa + S

(W A)aa S

(G)aa )

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, thefollowing 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 (20)

Gai S(S)ii

GTai Jai S(S)ii

JTai (21)

Gai S(W)ii G

Tai Jai S

(B)ii J

Tai (22)

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 floorslab 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 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.

(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

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


6/14


(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 computationaltime 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 MiaMai Maa

DiDa

+

Sii SiaSai Saa

DiDa

=

AiAa

. (24)

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:

Maa Da + Saa Da = A

a (25)

where Maa = Maa + TTia Mia + Mai Tia + T

Tia Mii Tia ,

Aa = Aa Sai S1ii

Ai , Saa = Saa Sai S

1ii

Sia and Tia =

S1ii Sia . The matrix Maa is the mass matrix, S

aa 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


7/14


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. [1821]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 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 in Fig. 10(b) using the matrix

condensation technique. The surplus stiffness introduced bythe 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. 1115 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 bemaintained 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 compatibilitycondition 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


8/14


(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).



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.


(c) Subtract fictitious

beams.

(d) Super element.

Fig. 14. Use of fictitious beams for shear wall D 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

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


9/14


(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 inFig. 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 toa 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

(a) Model A. (b) Model B.

(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


10/14


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 in Fig. 19(c), are assembledinto a substructure for shear walls in a typical floor, as shown

in Fig. 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 andstaircase 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 floorsin 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. 21 illustrates the modeling procedure for atypical 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


11/14


(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.

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.

5. Analysis of example structures

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 staticanalysis 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 analyseswere 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 historiesof 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

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 is20 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


12/14


Table 1

Comparison of DOFs and computational time required for analysis

Models Number of DOFs Computational time (s)

Assembly Static Eigenvalue Time history

M&K analysis analysis analysis Total

Model A 56 640 78 983 16 623 387 18 071

Model C 60 10 6 97 12 125Model D 60 156 6 97 13 272

(a) Typical floor.

(b) Floor plan.

(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

(a) Lateral displacements. (b) Natural periods.

(c) Displacement time histories.

Fig. 23. Seismic analysis results of the example structure from the models

A, C and D.

only 60 DOFs because they applied the rigid diaphragmassumption 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


13/14


(a) Typical floor.

(b) Floor plan.

(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 asimilar 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 high-

rise building structure with shear walls. A personal computer

with Pentium 3 500 MHz processor and 512 MB RAM was

employed in this study.

(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.

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 buildingstructure 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


14/14


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, theproposed 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).

References

[1] Wilson EL, Habibullah A. ETABS-three dimensional analysis of

building systems users manual. Berkeley (CA): Computers and

Structures Inc.; 1995.

[2] Lee HW, Park IG. MIDAS/ADS-shear wall type Apartment

Design System. MIDAS Information Technology Co., Ltd; 2002

(http://[email protected] ).

[3] Allman DJ. A compatible triangular element including vertex rotation

for plane elasticity problems. Computers and Structures 1984;19:18.

[4] Bergan PG, Felippa CA. A triangular membrane element with

rotational degrees of freedom. Computer Methods in Applied

Mechanics and Engineering 1985;50:2569.

[5] Ibrahimbegovic A, Taylor RL, Wilson EL. A robust quadrilateral

membrane finite element with drilling degrees of freedom.

International Journal for Numerical Methods in Engineering 1990;30:

44557.

[6] MacNeal RH, Harder RL. A refined four-noded membrane element

with rotational degrees of freedom. Computers and Structures 1988;

28:7584.

[7] Hughes TJR, Brezzi F. On drilling degrees of freedom. Computer

Methods in Applied Mechanics and Engineering 1989;72:10521.

[8] Cook RD. Four-node flat shell element: drilling degrees of freedom,

membrane-bending coupling, warped geometry and behavior.

Computers and Structures 1994;50:54955.

[9] Choi CK, Lee PS, Park YM. Defect-free 4-node flat shell element:

NMS-4F element. Structural Engineering and Mechanics 1999;8(2):

20731.

[10] Kwan AKH. Rotational DOF in the frame method analysis of coupled

shear/core wall structures. Computers and Structures 1992;14(5):

9891005.

[11] Kwan AKH, Cheung YK. Analysis of coupled shear/core walls using

a beam-type finite element. Engineering Structures 1994;16(2):1118.

[12] Lee DG. An efficient element for analysis of frames with shear walls.

In: ICES88, 1987.

[13] Weaver Jr W, Lee DG, Derbalian G. Finite element for shear walls inmultistory frames. Journal of the Structural Division ASCE 1981;107:

13659.

[14] Ali R, Atwall SJ. Prediction of natural frequencies of vibration of

rectangular plates with rectangular cutouts. Computers and Structures

1980;12:81923.

[15] Tham LG, Cheung YK. Approximate analysis of shear wall

assemblies with openings. The Structural Engineer 1983;61B(2):

415.

[16] Choi CK, Bang MS. Plate element with cutout for perforated shear

wall. Journal of Structural Engineering 1987;133(2):295306.

[17] Amaruddin M. In-plane stiffness of shear walls with openings.

Building and Environment 1999;34:10927.

[18] Kim HS, Lee DG. Analysis of shear wall with openings using super

elements. Engineering Structures 2003;25(8):98191.

[19] Lee DG, Kim HS. Analysis of shear wall with openings using superelements. In: Proceeding of EASEC-8, 2001. Paper No. 1378.

[20] Lee DG, Kim HS, Chun MH. Efficient seismic analysis of high-

rise building structures with the effects of floor slabs. Engineering

Structures 2002;24(5):61323.

[21] Lee DG, Kim HS. The effect of the floor slabs on the seismic response

of multi-story building structures. In: Proceeding of APSEC2000.

2000. p. 45361.

[22] Zienkiewicz OC, Cheung YK. The finite element method for analysis

of elastic isotropic and orthotropic slabs. Proceedings of the Institution

of Civil Engineers 1964;28:47188.

[23] Weaver Jr W, Johnston PR. Structural dynamics by finite elements.

Prentice Hall; 1987. p. 28290.
http://[email protected]/http://[email protected]/http://[email protected]/http://[email protected]/

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

Documents