Formulating building response to Earthquake · PDF fileThe deflected shape of the building due to earthquake loading could ... Table 3: Soil Profile and ... the one obtained using

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 2, No 1, 2011

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

Received on September, 2011 Published on November 2011 305

Formulating building response to Earthquake loading Mohammed S. Al-Ansari

Civil and Architectural Engineering Department, Qatar University,

P.O.Box 2713, Doha, Qatar,

[email protected]

ABSTRACT

This paper presents a numerical method for the computation of building responses under

earthquake loads using a simple closed-form equation, which takes into account the

earthquake zone and the soil profile. The equation was developed using a power best fit

regression using the response data of numerous buildings with different stiffness and heights;

different earthquake zones; and different soil profiles. The closed-form equation can be used

to compute the response of buildings at any desired level based on the building height,

stiffness, earthquake zone, and soil profile. The response results, which were obtained using

the closed-form response equation, were in close agreement with those obtained using the

finite element program STAAD-PRO. The closed-form response equation was also able to

accurately predict the actual top drift of the Bank of California building during San Fernando

earthquake. The presented method represents a simple and practical tool for computing

building responses in earthquake zones with different soil profile.

Keywords: Drift, Response, Numerical Formulation, Earthquake Zones, Soil.

1. Introduction

Structural response, such as building drift, is a key parameter that should be considered in the

performance-based seismic design rather than strength, which is used in conventional design

approaches. This is due to the fact that the building performance is characterized by its level

of damage, which is directly related to its displacement (Mori, Yamanaka, Luco, and Cornell,

2006).

The relative lateral displacement of buildings is measured by an overall drift index that is the

ratio of the maximum lateral displacement to the height of the building. The inter-story drift,

a commonly used drift index, is defined as the ratio of the relative displacement of the floor

to the story height at that level. When subjected to earthquake loading, the lateral

displacement of a structural system must be limited to: 1) preserve its structural stability and

architectural integrity and 2) reduce the structure members’ damage and human discomfort.

Drift limitations are commonly imposed by seismic design codes such as the Uniform

building Code (UBC), the International Building Code (IBC) and many other codes around

the world (Steven,L.K, 1967). The value of the drift index ranges between the values of

0.002 and 0.005 for conventional structures.

The determination of the expected response of a structure when subjected to seismic loading,

is one of the most important task in seismic design. Once the maximum expected seismic

response of the structure has been determined, the adequacy of all structural elements must be

verified. The excessive lateral displacement or inter-story drift causes failure of both

structural and non-structural element. Thus, the top and inter-story drifts at the final structural

design stages must be checked not to exceed the specified index limits (James, A, 2006).

Formulating building response to Earthquake loading

Mohammed S. Al-Ansari

International Journal of Civil and Structural Engineering

Volume 2 Issue 1 2011

306

This paper presents a numerical method for the computation of the response of building

structures in earthquake zones (Carpenter, L 2004). The method consists of computing the

building response under earthquake loads using a simple closed-form equation, which takes

into account of the earthquake zone and the soil profile. The equation was developed using a

power best fit regression using the response data of different buildings with different stiffness

and heights; different earthquake zones; and different soil profiles. The closed-form equation

can be used to compute the response of buildings at any desired level based on building

height, stiffness, earthquake zone, and soil profile. Hence, the earthquake formula (E.F.) can

be used for the preliminary design of buildings in order to save time and money.

2. Problem Formulation

Figure 1 shows the structural model a 12-story building. From the basic structural frame

model and side-sway moments, the following equations can be written:

[1]

[2]

Where:

IB = moment of inertia of the building under consideration

∆∆∆∆B = horizontal displacement of the building under consideration

IS = moment of inertia of the standard building

∆∆∆∆S = horizontal displacement of the standard building

Figure 1: Finite Element Model of a 30-Story Concrete Building





307

The deflected shape of the building due to earthquake loading could be represented by

different regression models such as polynomial, exponential, power …etc. After several

trials, the power function was found to be the best fit equation to model the response of a

structural system under earthquake loads. The power regression function of structural system

response is given by the following equation:

ββββαααα∆∆∆∆ h B = [3]

Where α and β are constants of the power regression model. By combining Eqs 2 and 3, the

following equation can be derived:

[4]

The displacements of all considered buildings are related to the displacements of the standard

building shown in Figure 2. The moment of inertia Is of the standard building, which is equal

to the sum of the moment of inertia of all columns and shear walls in all floors, is equal to

9.7875 m2. On the other hand, its modulus of elasticity Es is 3.5355 10

7 kN/m

2. The

horizontal displacement was determined using the finite element program STAAD PRO

[10].

Figure 2: Standard Building Floor Plan and Elevation

The regression model constants α and β are determined by solving the following equation:

[5]

Where:

i = Building height index

j = Building shape index





308

xi = Height of floor i in meters.

rI = Moment of inertia ratio (B

SI

I

Ir = )

rE = Modulus of elasticity ratio B

SE

E

Er( = )

As shown in Figure 1, the twelve-story square concrete building has a total of 5808 members

and elements and the total degrees of freedom is 35656. All buildings are reinforced concrete

framed structures with shear walls and slabs. A total of 21 equations had to be solved in

order to determine the regression constants α and β. Table 1 lists the buildings used to

generate the response data used to determine the regression constants α and β. Table 2 and 3

list the earthquake zones and soil profiles used for the generation of the response data. Figure

3 summarizes the response of the buildings listed in Table 1 when subjected to different

earthquake loading combinations in earthquake zone 1. On the other hand, Figure 4 shows

the response of the buildings listed in Table 1 when subjected to different earthquake loading

combinations in the earthquake zones and soil profiles listed in Tables 2 and 3, respectively.

Because of the divergence of the iterative solution for certain heights, two regression models

of earthquake building response were developed to accommodate two ranges of structure

height as shown in Eqs 6 and 7. The two equations take into consideration the building drifts

and deflected shapes, the building stiffness and heights, the earthquake zones, and the soil

profiles:

Table 1: Building Structures

Building Number Building

Shape of Stories Height (m)

Rectangular 3 12

5 20

7 28

18 72

20 80

26 104

30 120

Circular 3 12

5 20

7 28

18 72

20 80

26 104

30 120

Square 3 12

5 20

7 28

18 72

20 80

26 104

30 120





309

Table 2: Earthquake Zones and Soil Profiles

Building Number Earthquake Soil

Shape of Stories Zone Profile

Rectangular 5 1, 2, 3, 4 S1, S2, S3

Circular 12 1, 2, 3, 4 S1, S2, S3

Square 25 1, 2, 3, 4 S1, S2, S3

Table 3: Soil Profile and Soil Types

Profile Name Soil Type

S1 Hard Rock

S2 Rock

S3 Very dense soil and soft rock

Building Height ( m)

0 20 40 60 80 100 120 140

Displacement ( mm )

0

10

20

30

40

50

60

5 Stories

12 Stories

20 Stories

25 stories

30 stories

Figure 3: Building Displacements in Earthquake Zone 1





310

All Zones and Soil Profiles

Z1S1 Z1S2 Z1S3 Z2S1 Z2S2 Z2S3 Z3S1 Z3S2 Z3S3 Z4S1 Z4S2 Z4S3

Top Drift (mm)

0

50

100

150

200

250

F.E. 5 Stories

E.F. 5 Stories

F.E.12 Stories

E.F. 12 Stories

F.E. 25 Stories

E.F. 25 Stories

Figure 4: Building Drifts for all Earthquake Zones and Soil Profiles

[6]

And

[7]

Where = Horizontal displacement at the desired level (mm)

x = Height (m)

FS = Factor of safety

= Soil Factor

The soil factor relates building displacements to soil profiles in a specified zone, as shown

in Table 4.

Table 4: Soil Factors





311

Seismic Zone Factor

Building Earthquake for Soil Profiles S1, S2, & S3

Height Zone S1 S2 S3

< 48 meters 1 1.10 1.40 2.20

2 2.10 2.60 4.15

3 2.75 3.45 5.40

4 4.30 5.30 7.50

≥ 48 meters 1 1.00 1.35 1.90

2 1.85 2.30 3.50

3 2.45 3.10 4.00

4 3.70 4.60 6.00

3. Result Validation

In order to validate the results of the closed-form response equations, the responses of the

three building models obtained using the proposed closed-form response equations were

compared with those obtained using the finite element program STAAD-PRO. The results

were found to be close.

Rectangular

The closed-form response equation (Eq. 6) was used to compute the top drift of the 5-story

rectangular concrete building with rI = 0.581 and rE = 1 under static equivalent earthquake

loads (Figure 5). The building responses were computed considering four earthquake zones

and three different soil profiles. The top drift results obtained were very close to those

obtained using the closed-form response equation. As shown in Table 5, the top drift of the

five- story building obtained using the closed-form response equation was 1.100 mm while

the one obtained using the finite element program STAAD-PRO was 1.188 mm for

earthquake zone 1 and soil profile S1.

Figure 5: Rectangular Building Plan and Elevation





312

Table 5: Building Top Drifts

Building Number Building variables Earthquake Soil Building Responses

Shape of Floors r1 rE Zone Profile F.E. Method Proposed Method

Rectangular 5 0.581 1.000 1 1 1.10 1.21

2 1.47 1.54

3 1.65 2.42

2 1 2.20 2.31

2 2.75 2.86

3 3.30 4.57

3 1 2.93 3.03

2 3.67 3.80

3 4.40 5.83

4 1 4.40 4.73

2 5.50 5.83

3 6.05 8.25

Circular 12 0.244 1.629 1 1 5.80 7.59

2 7.73 10.24

3 10.33 14.42

2 1 11.59 14.04

2 14.49 17.45

3 19.87 26.56

3 1 15.45 18.59

2 19.32 23.52

3 25.43 30.35

4 1 23.18 28.08

2 28.97 34.91

3 35.77 45.26

Square 25 0.116 1.000 1 1 5.80 7.59

2 7.73 10.24

3 10.33 14.42

2 1 11.59 14.04

2 14.49 17.45

3 19.87 26.56

3 1 15.45 18.59

2 19.32 23.52

3 25.43 30.35

4 1 23.18 28.08

2 28.97 34.91

3 35.77 45.26

Square. The closed-form response equation (Eq. 7) was used to calculate the top drift of a

25-story square concrete building model with and under static equivalent

earthquake loads (Figure 6). The top drifts that were obtained using STAAD PRO were very

close to the one computed using the closed-form response equation. As shown in Table 5, the

top drift of the 25-story building obtained using the closed-form response equation was





313

93.686 mm while the one obtained using STAAD-PRO was 121.062 mm for earthquake zone

2 and soil profile S3.

Figure 6: Square Building Floor Plan and Elevation

Circular

The top drifts that were obtained using STAAD PRO were very close to those computed

using the closed form response equation. As shown in Table 5 and Figure 7, the top drift of

the 12-story building obtained using the closed-form response equation was 19.316 mm while

the one obtained using STAAD PRO was 23.523 mm in earthquake zone 3 and soil profile S2.





314

Figure 7: Circular Building Floor Plan and Elevation

4. Actual Case Example

The example is used to further validate the use of the closed-form response equation for

estimating the response of buildings under earthquake loads. The top drift of the Bank of

California building during San Fernando earthquake was computed using the closed-form

response equation and compared to the one actually measured during the earthquake [11].

The Bank of California building is a 12-story reinforced concrete frame building located at

15250 Ventura Boulevard approximately 23 kilometers from the center of San Fernando

earthquake. Figure 8 and 9 show the transverse section and a typical floor plan of the

building, respectively.

The lateral load resisting system is composed of several components. Much of the resistance

is provided by reinforced concrete moment resisting perimeter frames that extend the full

height of the structure and are composed of columns and spandrel beams. Interior frames to

the third floor were also designed as moment resisting. Above the third floor, the interior

columns and joists were designed to resist only vertical loads. Two shear walls two stories

high also provide lateral resistance in the longitudinal direction. The foundation system

considered of drilled cast-in-place piles ranging from 12 to 17 meters long.

The modulus of elasticity EB of the building is equal to 2.285 107 kN/m

2 while its moment of

inertia IB is equal to 9.087 m4. The total height hT of the building is equal to 52.727 m. The

calculated maximum acceleration of San Fernando earthquake was computed to be 0.3 g,

which corresponds to an earthquake zone of 3. The soil profile is S3 because the soil is silt

all the way down to the bottom of the pile foundation.





315

Figure 8: Bank of California Building Transverse Section

Figure 9: Bank of California Building Typical Floor Plan





316

Based on the above information, the top displacement of the building under San Fernando

earthquake was computed using the closed-form response equation and was found to be equal

to ∆=277 mm. On the other hand, the actual top displacement of the Bank of California due

to San Fernando earthquake was measured to be 279 mm. This shows that the closed-form

equation was able to accurately predict the actual response of the building even though it was

developed based on equivalent static earthquake loading and linear structural behavior.

5. Conclusions

This paper introduces a numerical method for the computation of the response of building

structures under earthquake loads. The developed closed-form response equations are stable

for buildings with heights up to 120 meters. They gave good drift results when compared

with those obtained using finite element programs for buildings with different height,

stiffness, modulus of elasticity, earthquake zones and soil profiles. The closed-form response

equation was able to accurately predict the actual top drift of the Bank of California building

during San Fernando earthquake. In spite of its simplicity, the closed-form response

equation yielded accurate to conservative results that are acceptable approximation of the top

drift of the building in all earthquake zones and soil profiles.

Acknowledgement

The author would like to acknowledge the financial support provided by Qatar University

through the Internal Grant # QUUG-CENG-CA-09/10-2.

References

1. Mori, Yamanaka, Luco, and Cornell (2006), “A static predictor of seismic demand on

frames based on a post-elastic deflected shape", Earthquake Engineering & Structural

Dynamics, 35(10), pp 1295-1318.

2. Chan, Chun-Man, and Zou (2004), “Elastic and inelastic drift performance

optimization for reinforced concrete buildings under earthquake loads", Earthquake

Engineering & Structural Dynamics, 33(8), pp 929-950.

3. Scholl,R.E (1975), “Effect Prediction Guidelines for Structures Subjected to Ground

Motion”, Report No. JAB-99-115, URS/ Blume Engineers, San Francisco, USA.

4. Steven,L.K (1967), “Elastic Stability of Practical Multi-Story Frames”, Proc. Inst.

Civil Engineers, 36, pp 99-117.

5. International Conference of Building Officials (1997), “Uniform Building Code”,

Whittier, California, U.S.A.

6. James, A (2006), "Comparison of methods to predict response spectra at instrumented

sites given the magnitude and distance of an earthquake”, Journal of Earthquake

Engineering, 10(6), pp 887- 902.

7. Al-Ansari, M. S (2009), “Drift Optimization of High-Rise Buildings in Earthquake

Zones”, Journal of tall and special building, 2, pp 291-307.





317

8. Carpenter, L (2004), “High Rise Building Cladding Drift Accommodation”, Journal

of Tall and Special Buildings, 13, pp 439-456.

9. Al-Ansari, M. S (1996), “Earthquake Response of Structures by Structural Mixture

Theory”, Journal of Structural Engineering, ASCE, 122, pp 1198-1207.

10. Bentley System Inc. (2009), “STAAD PRO V8i”, Three Dimensional Static and

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Yorba Linda, CA 92887- 4608.

11. Murphy, L. M (1973), “San Fernando, California. Earthquake of February 9, 1971”,

U.S. Department of Commerce, Washington D.C.

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Formulating building response to Earthquake · PDF fileThe deflected shape of the building due to earthquake loading could ... Table 3: Soil Profile and ... the one obtained using