The Seventh International Conference on Vibration Problems, IŞIK University, Istanbul, Turkey, 05-09, September, 2005 TITLE

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1 The Seventh International Conference on Vibration Problems, IŞIK University, Istanbul, Turkey, 05-09, September, 005 TITLE DYNAMIC RESPONSE ANALYSIS OF ROCKING RIGID BLOCKS SUBJECTED TO HALF-SINE PULSE TYPE BASE EXCITATIONS AUTHORS Mutlu OZER 1 G. Füsun ALIŞVERİŞÇİ ABSTRACT The motion of a rocking block subjected to half-sine pulse base excitation is analyzed by the mathematical model of lumped-mass approximation. The governing equation of the motion of the rocking block is derived as a second order ODE. The solutions of the derived equation for different size of rocking blocks are illustrated by mat lab figures in terms of displacement, velocity and acceleration responses. The coefficients of the equation of the minimum required overturning acceleration of rocking blocks are obtained, and the overturning instant of the rocking block is investigated through force vibration and free vibration. Parameter studies are also performed to verify the results accordingly to the Housner s[1] postulation and to the report of N. Makris[] about dynamic response of rocking blocks Holloway Ave. San Francisco State University, School of Engineering, San Francisco, CA 9413, Barbaros Bulvarı Yıldız-İstanbul 1

2 INRODUCTION The motion illustrated in Fig. 1, is called a rocking motion of rigid bodies. In the case of strong ground motion heavy industrial equipment such as heavy electrical transformers and large oil or chemical storage tanks undergo rocking motion that may overturn or slide from their foundation that can cause severe damage to industrial facilities and human lives. In such case, pumps and other devices required for rescue efforts would also not be in service because of delays to restoring power. It is experienced from recent earthquakes that overturning of industrial equipment are very costly to human lives and physical environment. Therefore proper assessment for dynamic response properties of rocking rigid body became a great interest for the engineers. Fig. 1 Schematic of a rocking rigid block The minimum acceleration amplitude of a half-sine pulse that will overturn the rigid block is controversial topic among researchers and further study shall be required. The minimum acceleration amplitude of a half-sine pulse that will overturn the rigid block was postulated by Housner,1963 and later by N. Makris,1998. Although both researchers had used the same model and governing equations, they have reached different conclusion in terms of the amplitude of the minimum acceleration that will overturn rigid body and the overturning instant of the rocking rigid block. The controversy between both researchers would not be easily clarified since the governing equations of the rocking block they used are non-linear and extremely difficult to solve it. In this report the governing equation of the rocking block is derived as a second order ODE by a mathematical called the lumped-mass approximation model.

3 Dynamic response analysis of a rocking rigid blocks by the model of lumped-mass approximation Fig. Schematic of free body diagram of the proposed model of he Lumped-Mass Approximation for analyzing dynamic responses of a rocking rigid block Within the limit of the linear approximation, the conditions for a block to overturn are also studied in this report by the proposed lumped-mass approximation model as illustrated in Fig.. By this model the dynamic equation for the minimum acceleration that block will overturn is derived as follows. Assumptions: Rectangular shape rigid body is modeled to a dimensionless mass that moves forward- and-back through a path of c.g. of rectangular rigid body. When t=0, θ = 0, U = b, U = 0, and the ground acceleration is to left. There is no energy losses through the path. Definitions: m x,y Mass of rigid body Moving axis attached to rigid block 3

4 X,Y Imaginary fix axis h The height of rigid block b The width of rigid block R θ α Position vector of lump mass of rigid block, Rotation angle of lump mass Slenderness angle R = b + U The amplitude of displacement of lump mass (when θ = 0, U = b or θ = α, U = 0 ) U & g (t) Harmonic ground acceleration, U & (t ) = ) g a sin( ϖ pt + ψ where, a po The amplitude of peak ground acceleration, po h ϖ p p Excitation frequency oscillation frequency of a rigid block, (for rectangular rigid block p = 3g ) 4R ψ phase when rocking initiates, ψ = sin 1 ( αg / a ), (Housner-1963) po Dynamic equilibrium of motion: m( U& + U&& ) mg sin( α θ ) = 0 sin( α θ ) g U = R g U& + U = a po sin( ϖ pt + ψ ) R 4 U & + p U = a sin( ϖ t + ψ po p ) (1) 3 Mat Lab figures of the solution of the proposed equation (1) derived by lumpedmass model of rocking rigid blocks The following Mat Lab figures are the solution of equation (1). The figures illustrate dynamic responses of rocking rectangular bodies subjected to harmonic ground excitation with various input parameters. 4

5 Fig. 3(a) Dynamic responses of a rocking rigid block: b=0. m, h=0.6m ( o a po = g ), (* a po = g,Housner) 5

6 Fig. 3(b) Dynamic responses of a rocking rigid block: b=0.5 m, h=1.5m ( o a po = g ), (* a po = g, Housner) 6

7 Table 1. Comparison of the minimum required peak ground acceleration that rigid block will overturn Studies on min. req. peak acc. that rigid block will overturn Housner N. Makris Lumped-Mass Approximation Model Parameters of the rigid rectangular block: b = 0.m, ϖ = * pi, o α = 18.4 Minimum Required Peak Ground Acceleration ( a po ) that rocking block will overturn a po α g 1+ ( ϖ / p) a po α g( 1 + β ( ϖ / p)) β 0.5 a po α g( 1 + β ( ϖ / p)) β = See Fig *g *g 0.60*g Overturning condition of the rocking rigid block: Rigid block will overturn at time t = ( π ψ ) / ϖ, which is the time that half-sine pulse expires(housner- 1963) Yes No, later. During free vibration regime. Yes Coefficient (Beta) 1*pi *pi Beta (1/)width of rigid blocks Fig. 4 β obtained by lumped mass model at α = and varies width, b(m) 7

8 DISCUSSION AND CONCLUSION 1. This analytical and numerical study by lumped-mass model confirms Housners postulation about the overturning condition of a rocking rigid blocks. A rigid block will overturn( θ = α ) at the time ( t = ( π ψ ) / ϖ p ) when half-sine pulse expires as postulated by Housner. However, the minimum acceleration amplitudes that rocking rigid blocks will overturn are less what Housner approximated by the expression of a po α g 1+ ( ϖ / p) ). The outcomes of lumped-mass approximation model confirms that the equation, a po α g( 1+ β ( ϖ / p)) in the report of PEER-1998/05 by N. Makris and Y. Rouses is the best formulation to define the required minimum acceleration amplitude that rocking rigid blocks will overturn. The coefficient β of the required minimum acceleration that rocking rigid blocks will overturn is approximated by N. Makris as β It is indeed a very accurate o approximation for small size rocking blocks with b 0. 3 when α = However, since β varies by the size of rocking blocks and excitation frequency, the amplitude of the required minimum acceleration that rocking rigid block will overturn are not accurate o when slender angels are different than α = 18.4 and also for the larger size rocking blocks withb > Therefore the coefficients of β s for different size of blocks are obtained at two different excitation frequency of w = pi and w = pi. As an example, o the corresponding result plotted in the Fig 4. at α = It is also confirmed that larger block will overturn with longer period of pulse while small block is more sensitive to higher peak ground acceleration as indicated in the report of N. Makris. If we consider two different half-sine acceleration pulses with the same product as ( a pot n ) s = ( a pot n ) l, then larger block will overturn at the condition of a ) ( a ) /. ( po l po s 3. By the model of lumped-mass approximation, the dynamic analysis of the rocking blocks subjected to half-sine pulse base excitation is pretty simple, and proposed governing equation (1) would help engineers to obtain quick and reliable dynamic properties of rocking rigid bodies. References [1] G. W. Housner, The behavior of inverted pendulum structures during Earthquakes, Bull. Seism. Soc. Am., Vol. 53, , (1963) [] Makris,N. Roussos Rocking Response and Overturning of Equipment Under Horizontal Pulse-Type Motions, Report PEER-1998/05 Pacific Earthquake Research Center, University of California,Berkeley October 1998,USA 8

Pacific Earthquake Engineering Research Center

Pacific Earthquake Engineering Research Center Rocking Response and Overturning of Equipment under Horizontal Pulse-Type Motions Nicos Makris Yiannis Roussos University of California, Berkeley PEER 1998/05