Seismic analysis of structural systems with uncertain damping N. Impollonia, G. Muscolino, G. Ricciardi Dipartirnento di Costruzioni e Tecnologie Avanzate, Universita di Messina Contrada Sperone, 31 - Villaggio S. Agata, 98166 Messina, Italy Email: muscolin@ingegneria. unime. it Abstract The propagation of uncertainty in the damping ratio on the correlation coefficients is investigated. By means of a perturbation procedure, the correlation coefficients are determined as a function of the mean value and the variance of damping ratio. Moreover, a numerical example shows the influence on the structural response of the uncertain damping ratio. 1 Introduction The response spectrum approach is a fundamental tool to carry out the design of structures in seismic areas. The response spectrum is evaluated for assumed damping ratio versus the natural period of the oscillator. So operating, the seismic response of structures is evaluated by considering the model parameters as deterministic. Experimental studies have shown that unlike the inertial and stiffness properties, the uncertainty associated with damping is significant, due to the complexity of damping mechanisms (material damping, interfacial damping) (Lazan*). This uncertainty introduces variability in representing the seismic response, that have to be evaluated in probabilistic sense by means of stochastic analysis (Kareem & Sun^, D'Aveni, Muscolino & Ricciardi^). Aim of this paper is the study of the propagation of uncertainty of the damping ratio on the seismic response of structures. In order to do this, a method for evaluating the cross-correlation coefficients by means of the response spectrum approach for classically damped systems with uncertain damping ratio is proposed. A perturbation technique is utilised to examine the effect of damping variability on the evaluation of the maximum response quantities (Muscolino, Ricciardi & Impollonia *).
448 Earthquake Resistant Engineering Structures 2 Equation of motion with uncertain damping The equation of motion of an %-DOF base excitation can be given by discretized structural system subjected to a u(0) = 0, u(0) = 0 in which M and K are deterministic mass and stiffness matrices, C is a proportional uncertain damping matrix, u(f) is the response vector which represents relative displacements with respect to the foundation ; the dot over a variable denotes time differentiation; u(0) and u(0) are the initial condition vectors in terms of displacements and velocities; u^(t} is the base acceleration and T is the influence vector. By employing the standard transformation of coordinates involving undamped eigenvectors normalised with respect to the mass matrix, that is: u(f) = 7=1 we can write the following uncoupled differential equations: q(0 + 2Mq(f) + Q'q(/) =pm/r), q(0) = 0, q(0) = 0 In eqs.(2) and (3), 0 is the modal matrix, whose y'-th column is the y'-th eigenvector <^. of undamped structure, Q^ is the spectral matrix and p is the vector of participation factors, given respectively as follows: (4) In eq.(3), E(ct) is the modal damping matrix, that for seismically excited structures is usually assumed a diagonal one and can be written as follows: in which is the mean damping ratio of the structure and a is an uncertain parameter, which is supposed small enough in comparison to the unity. Without loss of generality, the parameter a is a zero mean stochastic variable.
Earthquake Resistant Engineering Structures 449 3 Evaluation of the peak response by the response spectrum If the excitation Ug(t) is considered to be a zero mean Gaussian stationary process, a generic quantity of interest s(t), connected to the structural response, such as the stress at a point or the internal force in a member, can be expanded as a linear combination of the nodal displacements as follows: For zero mean input process the statistics of the stochastic response can be evaluated once the variance of s(t) is known: in which p^ is the correlation coefficient, given by: J_ E[d.d,} and In eq.(8) rf,-(0 can be evaluated by solving the following differential equation, having zero start initial conditions: Moreover, for assigned certain damping ratio, it is well recognised that the mean peak of structural response s(t) can be expanded as a combination of the mean value of the maximum modal responses, where each maximum is obtained in terms of the coordinate of the mean response spectrum associated with the corresponding modal frequency: max j(f) = y=l k=\
450 Earthquake Resistant Engineering Structures where D((,oo,) is the conventional response spectrum representing the mean peak response of the oscillator characterised by the radian frequency co. and damping ratio ^. Notice that in obtaining eq.(l 1), the following assumption has been made: a) the maximum value of the structural response is considered to be proportional to its standard deviation; b) the coefficient relating the standard deviation with the maximum of the response, that is the so-called peak factor, is considered to be approximately the same for the response of interest s(t) and for the modal responses. Moreover the correlation coefficients p^. are evaluated assuming the stationary Gaussian zero mean input process as a white one. 4 Cross-correlation coefficients with uncertain damping The cross-correlation function for white noise input can be evaluated by a closed form solution. However here is presented a general procedure which can be extended to non-white excitations and to structural systems with uncertain parameters. In order to do this we write the equation of motion of a single oscillator in terms of state variables (Falsone & Muscolino^): z. =D.z. (12) in which: *-=u. - (13) For stationary white input process iig(t) = w(t), we can write: (14) in which S^ is the strength of the white noise process, the symbol means Kronecker product, the exponent in square brackets means Kronecker power and (15) where 1% is the identity matrix of order two. By using the previously defined quantity, it is easily to show that the correlation coefficient can be written as follows:
Earthquake Resistant Engineering Structures 451 in which r is the first element of the following vector: 07) For the particular form of vector v^, it follows that rfj is the first element of the last column of the matrix D^ multiplied by (- 2nS^ ). After simple algebra, the correlation coefficient for white noise input process can be written in the following form (Der Kiureghian^): where p^ =(o^ /co*. For uncertain damping, we can write eq.(14) as follows: According to the improved approach (Muscolino et al.*), we can write: D.(a) = D + ad- (20) where 0 1 ] - [0 0 Then we can write: (22) where: D^ = D^. I, + 1 j D,, D^t = D. I, + 1, 0 D, (23)
452 Earthquake Resistant Engineering Structures Moreover, we can expand in similar form the vector r^(cc) as follows: (24) where r\& = [r^(a)]. By substituting eqs.(22) and (24) into eq.(19) and making the stochastic average, we obtain: t* = D^f,, + a^d,,,,f,, + v^27i\ (25) The differential equation of r^(t) can be determined by multiplying eq.(19) by a and making the stochastic average, obtaining: ^=D,.,&+D^f,, (26) The stationary solution leads to the following expression of P.*. : (27) After simple algebra, we obtain the following expression of the correlation coefficient: Eq.(28) requires probabilistic information in terms of mean and variance of the damping ratio, usually the only ones available by experimental investigations. If one assume that the probability density function is given, the exact mean correlation coefficient can be sought. In the case of uniform distribution of the random variable a in the range [-a, a], with 0 < a < 1, a closed form solution for the mean correlation coefficient is readily available by performing a stochastic average of eq.(19) for the stationary case. It reads: fv=7^hlt7 b (29) with a =CT^VI. Obviously, eqs.(28) and (29) reduce to eq.(18) for
Earthquake Resistant Engineering Structures 453 In order to check the influence of uncertainty in the damping ratio, in Fig. I the correlation coefficients p^, P,* and p*& are plotted as a function of p^, for different values of ^ and a. p^ DerKiureghian jk eq.(28) i Exact jt DerKiuregman jk eq.(2 Exact Figure 1: Correlation coefficient versus modal frequency ratio for different values of the mean damping ratio: a) a = 0.5 ; b) a - 0.7.
454 Earthquake Resistant Engineering Structures Figure 2: Percentage difference on the correlation coefficient versus modal frequency ratio It can be drawn the following considerations: i) p^ decreases as the uncertainty on the damping ratio increases; ii) the approximate eq.(28), based on the first order improved perturbation approach (Muscolino et al/), well represents the exact solution relative to the uniform distribution of the uncertainty, even for large values of a (high uncertainty). In Fig.2, the percentage difference 8% = [(pjk -p#)/py*]xloo is plotted versus of (3^ for different values of and a. It reaches a maximum value about 13% for a = 0.5 and 27% for a = 0.7 5 Numerical example The influence of uncertainty in the damping ratio on the structural response, via modification of the correlation coefficient, is analysed. The studied system is a five-story shear frame whose mass at the first four floor are m. = m = 2.6xl0^g whereas the lateral stiffness of each of the first four stories is k. =k = 2.26x10* N/cm. Let the mass and the lateral stiffness of the fifth story assume alternatively the following values: case I) m ^ = m and k$=k; case II) m^ =0.lm and k^ =0.016; case III) m^ = 0.01m and &, =0.0016. The first natural frequencies are reported in Tab.l, along with the frequency ratios, for the three cases considered. The case I) possesses well separated first two natural frequencies, while the case III), that is representative of a four-story frame with a light appendage, is characterised by the first two natural frequencies
Case I Case II Case III Earthquake Resistant Engineering Structures 455 Table 1. Natural modal frequencies and modal frequency ratios CO, (rad/s) 2.659 2.771 1926 #2 (rad/s) 7.762 3.451 3,275 0)3 (rad/s) 12.236 9360 9.344 Pl2 2.919 1.245 1.119 P,3 4.601 3.377 3.194 P23 1.576 2.712 2.853 0.2 0.4 0.6 0.8 o -5 I %(9-10 > -15 (b) 0.2 0.4 0.6 3. 0.8 Figure 3: Percentage difference on thefifth-storyshear (a) and on the base shear (b), versus the amplitude of uncertainty range.
456 Earthquake Resistant Engineering Structures The damping ratio C, = (1 + a) is assumed a random variable with mean value = 0.05 and a uniformly distributed in the range (-a, a). The seismic analysis has been performed by assuming the response spectrum given by the ECS code, with reference to B type soil, maximum ground acceleration a^ = O.lg and behaviour coefficient q = 3.75. The analysis has been focused on the base shear ^ and fifth-story shear V^, where the upper bar stands for the analysis by considering the uncertainty on the damping ratio using the perturbation approach eq.(28). The plotted results (Fig.3) concern with the percentage difference e% = [(Vj - ^.)/^.]xloo (7 = 1,5), where Vj is the shear evaluated in absence of uncertainty in the damping ratio by means of eq.(18). From these figures it appears that the uncertainty on the damping ratio weakly propagates for the case I, characterised by the first two decoupled natural frequencies, whereas the case III, with coupled first two natural frequencies, evidences high propagation of uncertainty in the damping ratio on the structural response, especially for the fifth-story shear V^, for which the percentage difference for a = 1 (high uncertainty) exceeds 15%. References 1. Lazan, B.J., Damping of materials and members in structural mechanics, Pergamon Press, Oxford, 1968. 2. Kareem, A.., Sun, W.J., Dynamic response of structures with uncertain damping, Engineering Structures, 12, pp. 2-8, 1990. 3. Muscolino, G., Ricciardi, G. & Impollonia, N., Improved dynamic analysis of structures with mechanical uncertainties under deterministic input, to appear on Probabilistic Engineering Mechanics, 1999 4. D'Aveni, A.., Muscolino, G. & Ricciardi, G., Stochastic analysis of a simply supported beam with uncertain parameters subjected to a moving load, Proc. of EuroDyn '96, Augusti, Borri & Spinelli Eds., vol. 1, pp. 439-446, 1996. 5. Falsone, G. & Muscolino, G., Cross-correlation coefficients and modal combination rules for non-classically damped systems, accepted for pubblication on Earthquake Engineering and Structural Dynamics, 1999. 6. Der Kiureghian, A., Structural response to stationary excitations, Journal of Engineering Mechanics Division (ASCE), 106, pp. 1195-1213, 1980.