Deterministic and stochastic seismic analysis of buildings with uncertain-but-bounded mass distribution
Introduction
Methods of predicting the seismic response of structural systems have a relevant importance in engineering design. Indeed it is possible to affirm that progress in the theory of structural analysis is closely related to progress in the understanding of earthquake ground motion and dynamic properties of structures. It is now recognized that the realistic effects of the ground motion on the structures can be appropriately performed by using recorded seismic accelerations or by modelling the earthquake seismic input as a zero-mean Gaussian stochastic process [1]. Furthermore, in structural analysis it is also recognized another class of uncertainties related to the uncertainties of the structural parameters, e.g. due to manufacturing errors, measurement errors and other factors. Therefore, in seismic engineering it is possible summarize two types of uncertainties: the first type arises from the random nature of the seismic action; while the second type is internal and it is related to uncertainties in both the structural geometric and mechanical properties.
Due the above sources of uncertainty, the elastic response of buildings subjected to seismic excitation generally exhibits discrepancies between the computed and actual response as evidenced in [2]. The discrepancies are generally manifested in terms of adjunctive torsional effects underestimated in the numerical model. This is due mainly to the uncertainty in mass and in stiffness distributions and ground motion spatial variability. To take into account of these uncertainties, building codes introduce the so-called accidental eccentricity by introducing in every storey of the buildings an artificial eccentricity, perpendicularly to the direction of ground motion, of the centre of mass with respect to its nominal value. Clearly, for multi-storeys buildings the number of analyses necessary due to all the possible permutations of the position of the centre of mass could be prohibitive. Therefore, in order to reduce the number of the analyses seismic codes propose to consider only the cases in which the accidental eccentricity at each floor possesses the same sign so neglecting various permutations of the accidental eccentricity at each floor. However, it has been shown [3] that by using various reference centres only some cases included in provisions of seismic codes are in agreement with those of dynamic analysis. As a consequence to include the worst scenario, which is encompassed in the bounds of the accidental eccentricity, the seismic analysis problem should be posed as the finding of the response of a structural system with uncertain-but-bounded parameters forced by deterministic or stochastic loads. In this regard, a suitable vehicle for the analysis belongs to the class of non-probabilistic methods for coping with the uncertainty in the mass distribution or to a combination of probabilistic and non-probabilistic methods for coping with both randomness in the excitation and uncertainty in the mass distribution. It has to be emphasized that non-probabilistic approaches are strictly necessary since the uncertainty in the mass distribution is represented through its bounds and not through a proper probability density function. In the framework of non-probabilistic approach starting from the pioneering study of Ben-Haim and Elishakoff [4], the analyses are usually based on theoretical formulations by adopting convex models or interval analysis methods [5], [6], [7], [8], [9], [10] in which only the bounds of parameters magnitude are required. Interval models seem, today the most efficient analytical tool. These models are derived from the interval analysis, in which the number is treated as interval variable with lower and upper bounds. The main advantage of the interval analysis is that it provides analytically rigorous enclosures of solution. However, without rigorous bound on computational errors, a comparison of numerical results with physical measurement does not tell how realistic the model is [11], [12], [13], [14], [15]. Indeed, often the rigorous analytical approach could lead to not very sharp hull for the solution. It follows that sometimes the analytical approaches are not convenient for dealing with the structural uncertainties in practical engineering problems. In this regard, a new interval method has been recently proposed [16] for determining the dynamic response of structures with uncertain parameters by using the Laplace transformation. Furthermore, in the case in which the uncertainty is slight the so-called interval perturbation method and first-order interval Taylor series expansion have been successfully adopted in both static and dynamic analysis [17], [18], [19], [20]. The main advantages of these methods, which avoid the use of the more challenging rigorous interval analysis, are the flexibility to deal with systems with uncertain mass, damping and stiffness matrices as well as the simplicity of the mathematical formulation by using the linear sensitivity.
In this paper, in the framework of the interval perturbation analysis, a method for evaluating the upper and lower bounds of the structural response of buildings with uncertain-but-bounded masses subject to deterministic and stochastic ground motion acceleration is proposed. The procedure requires the definition of a unique structural model so reducing the number of analyses to be performed. Moreover, by the proposed approach, according to the philosophy of vertex theorem [21], [22], all the possible permutations are implicitly considered so to include the worst condition. Numerical results are very promising and show a very good accuracy in the estimation of the upper/lower bounds of the dynamic response.
Section snippets
Problem formulation
Let us consider a linear quiescent n-degree of freedom (n-DOF) classically damped structural system subjected to seismic input , whose equations of motion can be cast by the finite element method in the formwhere M, C, and K are the n × n mass, damping, and stiffness matrices of the structure, u(t) is the vector of nodal displacement, τ is the n order influence vector, and a dot over a variable denotes differentiation with respect to time.
Let us consider now a
Deterministic input
In this sub-section, classical modal analysis is adopted in order to evaluate the vectors y(α0, t) and sy,i(α0, t) required to define the upper and lower bounds of the response under deterministic excitations (see Eq. (11)). To this aim, the modal coordinate transformation is introducedwhere q(α, t) is the vector collecting the m modal coordinates and is a matrix of order m × n ( being a suitable integer) collecting the m eigenvectors, normalized with respect to the mass
Numerical application
In this section the proposed approach is applied to two illustrative structures subjected to both deterministic and stochastic excitations.
Concluding remarks
Modelling the dynamic behaviour of buildings under seismic excitation is a public safety issue that still deserves in depth studies. Randomness in the seismic action, in mass and stiffness distributions could lead to discrepancies between actual and computed responses. Adjunctive torsional effects are in general manifested and have been managed by the building codes by introducing the so-called accidental eccentricity.
This paper addressed the problem how to manage the uncertainties in mass
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