Sap SOLID 2

European Journal of Scientific Research

ISSN 1450-216X Vol.51 No.3 (2011), pp.359-371

© EuroJournals Publishing, Inc. 2011

http://www.eurojournals.com/ejsr.htm

Analysis of Shear Wall with Openings using Brick Element

Muhammed Abbas Husain

Department of Civil Engineering, College of Engineering, Tikrit University, Iraq

E-mail: [email protected]

Abstract

The box system structure that consists only of reinforced concrete walls and slabs is

recently used in high – rise apartment buildings. Due to functional requirements such as

doors, windows, and other openings, a shear wall in a building contains many openings.

Some researches on the analysis of shear wall with openings were performed, but the

analysis using a three dimensional element (or brick element), which are used for modeling

the three dimensional structures, did not used in previous researches. This paper presents

the elastic analysis of shear wall with openings using brick element. Three models of shear

walls with various opening size were carried out using brick elements to investigate the

effect of the openings on the behavior of the shear wall. Also three models of shear walls

with openings of various locations were carried out to study the effect of opening locations.

In order to compare the analyses results, a shell element, which is commonly used in

modeling the shear walls, has been used to simulate each of these six models. The method

used in this paper gave an accurate analysis for the shear walls with openings, and

presented a study of the effect of opening locations on the behavior of shear wall.

Keywords: Shear wall with openings; Brick element

1. Introduction

The box system structure which consists only of reinforced concrete walls and slabs is recently used in

high – rise apartment buildings. Due to functional requirements such as doors ,windows, and other

openings, a shear wall in a building contains many openings.

The finite element method is considered to be one of the most methods that used to solve this

problem. Therefore, much research in analyses of shear wall with openings has been undertaken [1-5].

Also, framed structures with shear wall cores which contain openings has been undertaken by previous

research [6-9].

In this study, ideal finite element models were developed using brick element by using

SAP2000 Version 12 [10].

SAP2000 has several element types to simulate the shear wall behavior. But among those, four

- node shell element and brick element were selected for the ideal model as they have the capability to

simulate the shear wall.

The shell element was selected for comparison with the using of brick element since the shell

element is usually used for modeling the shear walls by the researchers. The eight - node brick element

was selected in this study to simulate the shear wall behavior since the brick element is used for

modeling the three dimensional structures. Also, the effect of opening locations in shear wall has been

studied.

Analysis of Shear Wall with Openings using Brick Element 360

The shell element in SAP2000 called area element, while the brick element in SAP2000 is

called solid element.

2. Analysis of Shear Wall with Openings In this section, the plane stress elements for the analysis of shear walls are presented, and also the

analysis method proposed by Choi and Bang [3] were discussed.

2.1. Finite Elements for Analysis of Shear Wall

The four - node element which has two translational degrees of freedom per node as shown in Fig. 1(a)

is commonly used for the analysis of shear wall systems [11].

Figure 1: Plane stress element for shear wall

Because displacement shape functions of this element are expressed in linear functions,

deformation of element edges can be expressed by straight lines and the shear stresses in an element

are constant and cannot represent the actual stress distribution accurately if the finite element mesh is

not fine. Also it is not easy to connect this element with beam elements because the drilling degrees of

freedom do not exist. Therefore, the 12 degrees of freedom plane stress element (Lee element) [12]

with drilling degrees of freedom as shown in Fig. 1(b) was used by Kim and Lee [1] to solve this

problem.

2.2. Previous studies on the analysis of shear wall with opening

The analysis method proposed by Choi and Bang [3] is one of the most efficient methods among

previous research about the analysis of shear wall with openings. A simple rectangular element was

proposed in their research. The stiffness matrix of this plane stress element with openings was derived

as illustrated in Fig. 2.

Figure 2: Composing stiffness matrix of plane stress element with openings

361 Muhammed Abbas Husain

The stiffness matrix of a plane stress element with openings (Kc) is obtained by subtracting the

stiffness matrix of the plane stress element without openings (K0) using equations (1) and (2). But there

is a difference in the formulation procedures for K0 and K1. The general procedure for the stiffness

matrix of a plane stress element is used to obtain K1. 1 1

0

1 1

− −

= ∫ ∫TK t B EB J dζdη (1)

2 2

1

1 1

= ∫ ∫b a

T

b a

K t B EB J dζdη (2)

In the case of K0, the possible nonconforming displacement modes are added to the element by

modifying the shape function and then addition a degrees of freedom due to the addition of

nonconforming modes are eliminated by the static condensation.

Lateral displacements of the model of Choi and Bang were similar to those of a fine mesh

model when the opening was small, but the error in lateral displacements tended to increase as the

opening became larger. Also stress concentrations at the corners of the opening could not be

represented.

Kim and Lee [1] proposed an efficient analysis method for the analysis of shear wall with

openings using super elements derived by introducing fictitious beams. For the efficiency in the

analysis, the stiffness matrix of a super element was developed. The number of nodes in a super

element is identical to that of a conventional plane stress element . This procedure was represented

using the matrix form in equations (3) and (4). The static equilibrium equation for a super element has

been rearranged by separating the degrees of freedom for the corners from those for the inner area of a

super element as follows;

=

ii ic i i

ci cc c c

S S D A

S S D A (3)

Where subscripts i and c represent the inner area and corners of a super element respectively.

Eliminating the degrees of freedom for the inner area by matrix condensation, the static equilibrium

equation can be represented for the degrees of freedom only at the corners only as follows;

{ } { } = * *

cc c cS D A (4)

Where,

{ } { } [ ][ ] { }1*

, c c ci ii iA A S S A and−

= −

{ } { } [ ][ ] { }1*

cc cc ci ii icS S S S S−

= −

The matrix *

ccS is the stiffness matrix for the super element having nodes only at corners of a

shear wall and Eq. (4) is the equilibrium equation for a super element. The compatibility condition at

boundaries of super elements has been enforced by using fictitious beams to improve the efficiency in

the analysis without using nodes at the interfaces of super elements.

3. Modeling of Shear Wall Using Brick Element

The brick element used in this study is an eight – node element. The brick element is used for modeling

three dimensional structures. Since the shear wall is a three dimensional structure , the brick element

may give an accurate simulation for the behavior of it.

This type of element based upon isoparametric formulation that includes nine optional

incompatible bending modes [10] .

The solid element activates the three translational degrees of freedom at each node of its

connected joints. Rotational degrees of freedom are not activated. Fig. 3 shows a typical eight – node

brick element.


Figure 3: Typical eight – node brick element

4. Analysis of Shear Wall with Openings Using Brick Element 4.1. Convergence Study

A convergence study was first carried out for a shear wall without openings subjected to point load of 2

lbs as shown in the Fig. 4

Figure 4: The shear wall used for convergence study of first numerical example

thickness of wall = 1 inch

E = 3 × 106 psi

ν = 0

2 lbs

10 in.

30 in.

wall

A

B

The shear wall was modeled by four different mesh sizes of shell elements and again by four

different mesh sizes of brick elements. The convergence study showed that an element size of (0.5 inch

× 0.5 inch ) for shell element , and (0.5 inch × 0.5 inch × 1 inch ) for brick element gave an accurate

representations of displacements and stresses in shear wall. The lateral displacements at point A, and

the maximum stresses at point B for each mesh size are given in Table 1, and Table 2 respectively .


Table 1: Lateral displacements and maximum stresses for the shear wall in Fig. 4 using shell element

Stress (psi) Displacement (inch × 10

-5) Mesh size (inch × inch )

3.55 7.579 2.5 × 2.5

3.62 7.594 2 × 2

3.78 7.616 1 × 1

3.90 7.622 0.5 × 0.5

Table 2: Lateral displacements and maximum stresses for the shear wall in Fig. 4 using brick element

Stress (psi) Displacement (inch × 10-5

) Mesh size (inch × inch × inch ) 3.56 7.582 2.5 × 2.5 × 1

3.62 7.596 2 × 2 × 1

3.78 7.617 1 × 1 × 1

3.90 7.622 0.5 × 0.5 × 1

4.2. Numerical Example

The example shear wall adopted from the work by Choi and Bang is a three – story shear wall has

square window type openings. The description of the shear wall is shown in Fig. 5. The openings

centers were considered at the centers of stories.

Figure 5: Shear wall with window type openings

2 lbs

10 in.

3@10 in.

A

B

thickness of wall = 1 inch

E = 3 × 106 psi

V = 0

Analysis was performed for the cases of opening size : a = 1 inch, a = 3 inches, and a = 5

inches. For each of these cases, two models of finite elements were used, the first was a model using

shell element , Fig. 6 , and the second was a model using brick element , Fig. 7 .

Figure 6: Shell element models for shear wall with window type openings


Figure 7: Brick element models for shear wall with window type openings

Table 3 shows the lateral displacements at the point A and the maximum stresses at the point B

for shell element models and brick element models.

Table 3: Lateral displacements and maximum stresses for the shear wall with window type opening

Opening size Model Displacement (inch×10

-5) Stress (psi)

a = 1 inch Shell element 7.664 3.912

Brick element 7.665 3.918

a = 3 inches Shell element 8.314 4.112


a = 5 inches Shell element 10.000 5.166


As noticed from Table 3, the values of displacements for brick element models were very close

to those of shell element models, while the values of stresses for brick element models were slightly

larger than those of shell element models.

The comparison between the results of lateral displacements for the brick element models with

the plate element model adopted from the work of Choi and Bang , and the super element models

adopted from the work of Kim and Lee can be shown in Fig. 8 , Fig. 9 , and Fig. 10.

Figure 8: Displacements for shear wall with 1 inch opening size




As shown in Fig. 8 , the lateral displacements along the height of the shear wall obtained from

the brick element model were very close to those obtained from the plate element method, while the

super element model gave larger displacements.

If the opening was larger , Fig. 9, the lateral displacements of the three models were

approximately identical.

As the opening becomes larger , Fig. 10, the lateral displacements obtained from brick element

model were larger than those of the two other models at the opening elevations. The largest

displacement was significantly seen at the center level of the upper opening.

The maximum stress distribution for the brick element models are shown in Fig.11.


Figure 11: Stress distribution for shear wall with window type openings

(a) without opening (b) opening size of 1 inch

(c) opening size of 3 inches (d) opening size of 5 inches

The maximum value of stress occurs at the base of the shear wall (point B in Fig.5 ). This stress

was increased as the opening becomes larger.

5. Effect of Location of Openings in Shear Wall The choice of location of the openings in the shear wall for the designers is very important because the

values of internal stresses and displacements for the shear wall are affected by the location of openings.

To study the effect of the location of the openings in the shear wall, a shear wall of seven

stories with door type openings has been modeled by brick elements.

5.1. Convergence Study

A convergence study was first carried out for a shear wall without openings subjected to point load of 2

kips as shown in the Fig.12.


Figure 12: The shear wall used for convergence study of second numerical example

2 kips

18 ft

84 ft

wall

A

B

thickness of wall = 1 ft

E = 3.3 × 106 psi

ν = 0

The shear wall was modeled by four different mesh sizes of shell elements and again by four

different mesh sizes of brick elements. The convergence study showed that an element size of (0.5 ft ×

0.5 ft ) for shell element , and (0.5 ft × 0.5 ft × 1 ft ) for brick element gave an accurate representations

of displacements and stresses in shear wall. The lateral displacements at point A, and the maximum

stresses at point B for each mesh size are given in Table 4 ,and Table 5 respectively .

Table 4: Lateral displacements and maximum stresses for the shear wall in Fig. 12 using shell element

Mesh size (ft × ft ) Displacement (inch ) Stress (psi)

2.5 × 2.5 0.0210 20.8

2 × 2 0.0210 22.21

1 × 1 0.0211 22.70

0.5 × 0.5 0.0211 23.07

Table 5: Lateral displacements and maximum stresses for the shear wall in Fig. 12 using brick element

Mesh size (ft × ft × ft ) Displacement (inch ) Stress (psi)

2.5 × 2.5 × 1 0.0210 21.88

2 × 2 × 1 0.0210 22.24

1 × 1 × 1 0.0211 22.72

0.5 × 0.5 × 1 0.0211 23.09

5.2. Numerical Example

The example shear wall is a seven – story shear wall has door type openings. The description of the

shear wall is shown in Fig.(13). Three cases have been considered in this shear wall. The first case is a

shear wall with door openings vertically allocated to the left side. The second case is a shear wall with

door openings vertically allocated at the vertical center line of the shear wall. The third case is a shear

wall with door openings vertically allocated to the right side. The distance e (between the vertical

center line of the shear wall and that of the opening ) has been taken as 6.5 ft to the right and to the left

of the center line for the first and the third case respectively, while the distance e was zero for the

second case. Also the door opening dimensions was (3ft width × 7ft height ).


Figure 13: Shear wall with door type openings

thickness of wall = 1 ft

E = 3.3 × 106 psi

ν = 0

2 kips

18 ft

84 ft

A

B

e

For each of these cases, two models of finite elements were used, the first was a model using

shell element , Fig. 14, and the second was a model using brick element , Fig. 15.

Figure 14: Shell element model for shear wall with door type openings


Figure 15: Brick element model for shear wall with door type openings

Table 6 shows the lateral displacements at the point A and the maximum stresses at the point B

for shell element models and brick element models.

Table 6: Lateral displacements at the point A and the maximum stresses at the point B for shell element

models and brick element models for the shear walls with door type openings .

Case Model Displacement (inches) Stress (psi)

Left opening Shell element 0.0294 40.53


Middle opening Shell element 0.0221 25.46


Right opening Shell element 0.0294 29.01


As noticed from Table 6, the values of displacements for brick element models were very close

to those of shell element models, while the values of stresses for brick element models were slightly

larger than those of shell element models.

The comparison between the results of lateral displacements for the brick element models for

the three cases can be shown in Fig. 16.

Figure 16: Displacements for shear wall with door type openings


As shown in Fig. 16 , the lateral displacements along the height of the shear wall obtained from

the brick element model for the center opening case were smaller than those obtained from the left and

right opening cases. Also it can be noticed that the results of displacements for the left and the right

opening cases were very close.

The maximum stress distribution for the brick element models are shown in Fig.17.

The maximum stresses at the base of the wall obtained from the shear wall with left door

openings were the largest compared with those of middle and right door openings.

Figure 17: Stress distribution for shear wall with door type openings

6. Conclusions

This paper has presented the analysis of shear wall with openings using the brick element. Also the

effect of the openings location has been studied. The efficiency of this proposed analysis were

investigated by performing analysis of various example structures.

The major observations are summarized as follows:

1. The lateral displacements of the model using a brick elements were very close to those of a

shell element model.

2. The maximum stresses obtained from the brick element method were slightly larger than those

of shell element model.

3. The error in lateral displacements obtained from the brick element model with those of plate

element and super element models increased for the large openings.

4. As the openings along the shear wall were close to the center line (or middle ) of the shear wall,

the lateral displacements and the maximum stresses at the base became smaller.


References

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[5] Ali R, Atwall SJ. Prediction of natural frequencies of vibration of rectangular plates with

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[8] Thomas H. On drilling degrees of freedom. Computer Methods in Applied Mechanics and

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[9] Weaver Jr. W, Lee DG, Derbalian G. Finite Elements for Shear walls in multistory frames.

Journal of the Structural Divisions ASCE 1981;107:1365-9.

[10] CSI Analysis Reference Manual for SAP2000. Berkeley ,USA, 2008.

[11] Weaver Jr. W, Jhonson PR. Structural Dynamics by Finite Elements. Prentice Hall, 1987.

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