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INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 1, No 3, 2010

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

635

Estimation of storey shear of a building with Mass and Stiffness

variation due to Seismic excitation Bhattacharya S.P

1, Chakraborty S.K

2

1- Assistant Professor, Department of Architecture, Birla Institute of Technology, Mesra,Ranchi- 835215, India

2- Associate Professor, Department of Applied Mathematics, Birla Institute of

Technology, Mesra, Ranchi- 835215, [email protected]

ABSTRACT

Mass and stiffness are two basic parameters to evaluate the dynamic response of a

structural system under vibratory motion. High rise and multi-storeyed buildings are behaved differently depending upon the various parameters like mass-stiffness

distribution, foundation types and soil conditions. 2001 Bhuj earthquake in Gujrat, Indiademonstrated the damage and collapse of the buildings due to the irregularities in

structural stiffness and floor mass. This paper attempted to investigate the proportionaldistribution of lateral forces evolved through seismic action in each storey level due to

changes in mass and stiffness of building. As per the BIS provisions, a multi-storeysymmetrical building is considered as simplified lump mass model for the analysis with

various mass and stiffness ratios. The sway pattern of multi-storeyed building underseismic excitation is taken under consideration with parabolic shape functions. The result

concludes as a building structure with high mass and stiffness ratio provides instabilityand attracts huge storey shear. A proportionate amount of mass and stiffness distribution

is advantageous to control over the storey and base shear.

Keywords: Mass Ratio, Stiffness Ratio, Storey Drift, Storey Shear, Base Shear

1. Introduction

Regular shape plan of building is one of the basic principles of seismic resistant design.

Inertia force is obvious in the seismic excitation for all the symmetric plan buildings. Butirregular shape buildings attract the twisting couples along with linear vibration. A

symmetric building structure shows very discipline performance during any level ofearthquake. The structural behaviour, member deformations and induced stresses can also

be predicted easily. The building symmetry can be achieved with the even distribution offloor masses, storey stiffness. But the need of complex functional character of the

building generally offers architectural solutions, which are deviated from the prefectsymmetric case. It was found out from several previous earthquake experiences that, the

wave propagation in a particular earthquake under some specified soil character impartsresonance effect for typical tall building structures. The 1995 Mexico earthquake,

Gueguen et.al. shows the maximum damage to the buildings of height 15 to 20 storey. Asimilar survey and analysis of the seismic vulnerability and damage characteristics of the

building stock of San Giuliano have been investigated by Dolce et al . An approximatesolution for non-uniform shear buildings, in which the mass centres lie on one vertical

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axis and the centres of resistance lie on different vertical axes; but which are identicallyoriented, given by Kan and Chopra, Balendra et al, obtained the solution for shear

building on a flexible foundation for the case of the centre of mass and centre ofresistance lying on two vertical axes. Using the continuum approach, the problem of

uniform shear wall-frame buildings in which the centre of resistance, assumed to lie ontwo vertical axes, was investigated by Rutenberg et al..

The present work deals with the estimation of natural frequency of the building modelaccording to lump-mass system. In this respect, variety form of shape functions have

been presumed to measure natural frequency, base shear and storey drift of buildingstructure. Moreover, the variation of mass and property of structural stiffness has been

encountered in the investigation to analyse the nature of the dynamic property of the building.

2. Objectives and Assumptions

The objectives for the current study are

· To determine the effect of the natural frequency of the Moment Resistant Frame

structure under parabolic Shape Functions.

· To study the effect of the natural frequency with mass and stiffness variations.

· To estimate the base shear for above-mentioned variations.

Followings are the assumptions made for the current study:

· The Moment Resistant Frame structural system is considered as a lumped- masswith multi-degree of freedom.

· Buildings are considered to be regular planner symmetry.

· Elastic pseudo-acceleration design spectrum is assumed with 5% damping ratio(x),

and 0.25g peak ground acceleration (a) (g, the acceleration due to gravity).

2.1 Building Model and Mathematical Formulation

A multi-storeyed moment resistant frame is modelled as a string with distributed floormasses at different levels, joined by mass less connectors having different storey stiffness

values. Figure 1 shows a lumped-mass building model, with floor mass as mi , stiffnessas k i and floor displacement as xi.

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Figure 1: Lumped-mass Building Model

The lump mass in the n-storey illustrates n-degrees of freedom (x1,x2,x3 ....., xn), in which

the equation of motion is established neglecting the damping. However, damping of thestructure is more or less introduced in the mathematical formulation on the basis of the

experimental and the observed data prescribed by the Indian seismic code of practices,IS: 1893 (Part-I), 2002 and the standard literatures.

Thus, the total displacement in the, ith

floor, Xi (t) is given by

)()()( t g

xt i

xt i

X += (1)

where, the floor displacement relative to the ground motion is )(t i

x

)(t g

x is the excitation due to the earthquake ground motion. Further, the storey shear in

the ith floor can be determined as

)1

( --= i xi xikiV (2)

The equation of motion of the lumped-mass at i th floor is given by (D’Alembert’s principle),

÷÷ ø

öççè

æ

¶

¶+

¶

¶-=

¶

¶-=

2

2

2

2

2

2

t

x

t

xm

t

Xm f

gi

i

i

ii (3)

The external work done, W E , by the inertia force f i is given by

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638

åååúúû

ù

êêë

é

¶

¶+

¶

¶-===

n

i

i

gii

n

i

ii

n

i

E E xt

x

t

xm x fWW ddd

2

2

2

2

(4)

Internal work done, W I , by the building, due to the storey shear is given by

( ) ( )( )ååå --- ---== n

i

iiiii

n

i

iii

n

i

I I x x x xk x xVWW 111 ddddd (5)

The internal and external virtual work can further be modified and expressed in terms of

the generalised coordinate z(t) as

)()( t zt x ii y = (6)

Moreover, a shape function ψ is introduced, so that the virtual displacements of the

structure become consistent with the assumed deformed shape of the building. Here inthis paper the shape function is taken as a parabolic one and its mathematical expression

is shown in the Figure 1. The relation between the storey drifts and the shape functioncan be written as

2

2

2

2

t

z

t

x ii

i

¶

¶=

¶

¶ y and z x ii dyd = (7)

Using Equation (4) and the shape function component ψ i , from Equation (7), the external

work done yields

zmt

xm

t

zW

n

i

n

i

ii

g

ii E dyy úúû

ù

êêë

é

¶

¶+

¶¶

-= å å 22

2

2

2

(8)

Similarly, the internal work done, from Equation (5) and using Equations (6) and (7), can be rewritten as

( ) zk zWn

i

iii I dyy úúû

ù

êêë

é-= å -

2

1 (9)

Finally, equating the external and the internal work done from Equations (8) & (9), thegeneralised equation of motion becomes

0)(....

=++ t g x L zk zm (10)

Considering

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639

generalised mass as å= n

i

iimm 2 y (10a)

generalised stiffness as ( )

2

1 å --= n

i

iiikk yy (10b)

generalised excitation as å= n

i

iim L y , (10c)

Also, the natural frequency (w n) and the natural time period (T n) of the excitation can be

deduced from Equation (10) as

m

k

n = w and nnT wp 2

= (11)

The generalised single degree of freedom system is further analyzed to evaluate the

seismic response parameters. The peak responses of the frame structure due to earthquakeexcitation are determined by design spectrum. The generalised equation of motion,

Equation (10), can be rearranged as,

0)(..2.. =++ t g xC z n z w (12)

Once the deformation history x (t) is being known through the dynamic analysis of the

given structure, the internal forces in the structural system can then be evaluated by staticanalysis. The equivalent static force or storey shear force V(t) can be expressed as

)()()()(2

tmt xmtkxtV n gw === (13)

where )()( 2 t xt n wg = is the pseudo-acceleration.

This pseudo-acceleration is computed from the natural time period (T n ) and the specific

damping ratio of the structural system. The elastic pseudo-acceleration design spectrumfor ground motion with 5% damping [xx] is presented in figure 2 is used to determine γ(t).

Finally, the base shear of the structural system can be expressed as

å= n

i

ib tVtV )()( (14)

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Figure 2: Elastic Pseudo-Acceleration Design Spectrum for Ground Motion with 5%

Damping (Source: Dynamics of Structures, 2nd

Ed., by Chopra. A.K., 2003)

3. Analysis and Discussions

The base model (the building having equal floor mass and storey stiffness), is

numerically computed with the uniform mass and the stiffness. They are assumed as250KN and 4000KN-m respectively. The seismic responses due to the variation of massand stiffness is computed according to the mathematical formulation. Initially four

different types of multi-storied buildings are selected for the study. The four types are

classified as four-storied, six-storied, eight-storied and ten-storied buildings. The massand stiffness ratios are varied from 0.25 to 2.0, with an increment of 0.25.

Figure 3 is drawn to compare the variation of stiffness ratio and base shear of the

structure. The plot shows a positive sloping indicating the increment of base shear withincrement of stiffness ratio. If the lower stories are comparatively weak in stiffness, the

building structure attracts much more lateral force and subsequently the amount of baseshear is also increased. It can be also noted from the figure that the increment is

absolutely proportional with the increase of storey height. The average increment in theamount of base shear is about 3.4 times over an increment of stiffness ratio from 0.25 to

2.0. Plot of Figure 4 is performed with varying mass ratio keeping stiffness ratio as

constant. The nature of the graph is reverse and flatted with respect to the earlier plot. Inthe Figure 5 the variation in base shear is computed by variation of stiffness and mass ofthe building model. The two sloping lines of different nature indicating the relation

between the changing pattern of base shear with the two ratios. Further, the figure issubdivided into three zones. The zone-I should be avoided as it attracts high base shear

due to low mass ratio. Similarly, higher stiffness ratio (zone-III) also makes the buildingmodel vulnerable. The central zone-II provides a controlled base shear with an optimum

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combination of mass and stiffness. The present mathematical analysis indicating thatthese ratios should be in between 0.75 to 1.25 to minimize the level of base shear.

Figure 3: Variations of Base Shear and Stiffness Ratio

Figure 4: Variations of Base Shear and Mass Ratio

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Figure 5: Variations of Base Shear due to Mass and Stiffness Ratio

4. Conclusions

The present investigation reveals that both the mass and stiffness are very basic

parameters to estimate the nodal force and the base shear of the building. Irregulardistribution of Mass and structural stiffness of the building plays a vital role in seismic

environment. Any abrupt change in these two basic parameters increases the amount of base shear of the building structure. The present study suggested that in the earthquake

prone zones, it is always safe to construct a high-rise building with nearly uniform floormass and storey stiffness. Uniformity in mass and stiffness produces a optimum amount

of seismic forces.

5. References

1. Gueguen, P., Bard, P.Y. and Chavez-Garcia, F.J., Site-City-Seismic Interaction inMexico City-Like Environments: An Analytical Study, Bull. Seism. Soc. Am., Vol.

92, No. 2, 2002, pp. 749-811.

2. Dolce, M., Masi, A., Zuccaro, G., Cacace, F., Samela, L., Santarslero, G. and Vona,

M., Survey and Analysis of Damage in San Giuliano di Puglia, GNDT TechnicalReport (in Italian), 2004.

3. Kan, C.L. and Chopra, A. K., Elastic Earthquake Analysis of Torsionally CoupledMultistorey Buildings, Earthquake Engg. and Struct. Dyn., Vol. 5, 1977, pp. 395-

412.

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4. Balendra, T., Swaddiwudhipong, S., Quek, S.T. and Lee, S.L., Free Vibration ofAsymmetric Shear Wall-Frame Buildings, Earthquake Engg. and Struct. Dyn., Vol.

12, 1984, pp.629-650.

5. Rutenberg, A., Tso, W.K. and Heidelbrecht, A.C., Dynamics Properties of

Asymmetric Wall-Frame Structures, Earthquake Engg. and Struct. Dyn., Vol. 5,1977, pp. 41-51.

6. Chopra, A. K., Dynamics of Structures-Theory and Applications to EarthquakeEngineering, 2

ndEd., Prentice Hall, New York, 2000.

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