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soil structure interaction

# soil structure interaction - GG 454 1 RESPONSE OF...

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GG 454 February 20, 2002 1 Stephen Martel 1 6 - 1 University of Hawaii RESPONSE OF STRUCTURES (16) I Main Topics A Acceleration, velocity and displacement spectra B Resonance and natural frequencies C Response of structures I I Displacement, velocity, and acceleration spectra A Spectra represent parameters as a function of wave frequency (or period), not time, and reveal the most energetic/forceful waves B Examples of shaking vs. time and shaking vs. frequency plots C Displacement, velocity, and acceleration spectra y x A λ A = amplitude Displacement as a function of position along a wave (1) y(x,t) = A sin {(2 π / λ )(x + vt)}where v = λ /T = f λ , so λ = v/f (2) y(x,t) = A sin {(2 π f/v)(x + vt)} Consider the displacement at a fixed point on the ground (e.g. x = 0) (3) y(t) = A sin {(2 π f/v)(vt)} or y = A sin (2 π f t ) Displacement given as a function of the frequency and wave amplitude Let ω = 2 π f = angular frequency (4) y = [A] sin ( ω t ) The term in square brackets gives y max The shaking velocity of the ground (not the velocity of the wave) = dy/dt (5) y' = d(A sin ( ω t))/dt = [ ω A] cos ( ω t) The acceleration of the ground = dy'/dt (6) y'' = d( ω A cos ( ω t))/dt = -[ ω 2 A] sin ( ω t) = - ω 2 y

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