Structural Reliability Sensitivities under Nonstationary Random Vibrations

Abstract

Response sensitivity evaluation is an important element in reliability evaluation and design optimization of structural systems. It has been widely studied under static and dynamic forcing conditions with deterministic input data. In this paper, structural response and reliability sensitivities are determined by means of the time domain covariance analysis in both classically and nonclassically damped linear structural systems. A time integration scheme is proposed for covariance sensitivity. A modulated, filtered, white noise input process is adopted to model the stochastic nonstationary loads. The method allows for the evaluation of sensitivity statistics of different quantities of dynamic response with respect to structural parameters. Finally, numerical examples are presented regarding a multistorey shear frame building.

1. Introduction

The problem of structural response to dynamic nondeterministic actions has to be posed when structures, such as rotating machinery components or constructions subject to sea/wind loads, are designed to operate in random vibration environments. In these cases, due to the intrinsic random nature of inputs, a random dynamic analysis is needed, as documented in random vibration theory (e.g., [1–5]). Random dynamic analysis offers valid information about structural response otherwise impossible to obtain through classic deterministic analysis. A suitable evaluation of structural integrity, fundamental to design, can be obtained through this approach, in which structural safety is evaluated by the probability that the analyzed structure will not collapse during its working or life time. More specifically, this probability regards one or more components of a given structure and the critical state or conditions of failure these components reach in a given time interval. In this work, failure is related to a first crossing threshold, so that the structural crisis is associated with a first overcoming of one or more structural response measures (from a safe domain). Typically, structural response measures are displacements, stresses, buckling loads or natural frequencies. Moreover, statistics of response and structural reliability including their sensitivities (i.e., evaluation of partial derivatives of a performance measure with respect to system parameters), allows us to evaluate structural performances when some design variables have parametric changes. In particular, sensitivity analysis is a suitable way to evaluate the response variation of structures and, due to its importance in structural design, it is actually an important research topic in the field of computational mechanics. The sensitivities of structural responses are a fundamental information for gradient-based optimization methods needed in structural reliability analysis, optimization, and identification. Even random structural analysis with uncertain system parameters, obtained by removing commonly used deterministic assumptions (often extremely unrealistic), needs structural sensitivities. The analysis and design sensitivity computations do not offer quite hard difficulties for static problems, and analytical formulations for design sensitivity analysis are being reported in the literature from early seventies [6–8]. In dynamic problems, sensitivity analysis of structural response under transient dynamic loads has been studied especially in case of deterministic dynamic loads. Conversely not much research [9–11] seems to have been carried out for computation of design sensitivity under stochastic dynamic environment. Recently, an interesting analytical approach has been proposed by Chaudhuri and Chakraborty [12]. It is based on response sensitivity evaluation of structures in seismic reliability evaluation. The formulation has been developed in double frequency domain to take nonstationary earthquake motion for obtaining the analytical sensitivity statistics of various dynamic response quantities with respect to structural parameters. A semianalytical approach to evaluate the sensitivity of the stochastic response of both classically and nonclassically damped structural systems under stationary or nonstationary stochastic Gaussian excitation has been proposed [13]. The equations governing the evolution of the sensitivity of the first two statistical moments of the response for multi-degree-of-freedom (MDoF) structural systems are first derived in analytical form in the time domain, once the Kronecker algebra, the series, expansions, and a suitable modal expansion of the deterministic response are adopted.

Finally in [14] a new closed-form analytical approximation to the first-passage problem is presented in structural reliability by using the exact closed-form solutions for the spectral characteristics of nonstationary random processes.

The main objective of the present work is to propose an analytical method to evaluate in time domain response covariance and reliability sensitivity for general linear structures subjected to nonstationary stochastic loads. This analytical approach presents a serious reduction in computational cost, especially if compared with numerical procedures. The evolutive random input is modelled by a nonstationary modulated filtered white noise process, able to represent many real physic loads, as the earthquake loads here considered. In order to obtain a general nonstationary approach to be used in different contexts, the structural response is evaluated by a covariance approach. By knowledge of evolutionary covariance matrix in state-space, the reliability is derived with reference to first crossing failure events. Sensitivity of covariance matrix and reliability is then obtained. Therefore, the proposed formulation computes the sensitivity gradient of covariance matrix and structural reliability with respect to the structural design parameters. A time integration algorithm for solving this problem is proposed, and finally a multistory building idealized by a shear frame structure (Figure 2) is considered to study the sensitivity of dynamic responses. The rigid floors masses and interstorey lateral stiffness and damping are considered as design variables.

2. Linear Elastic MDoF Subject to Nonstationary Random Vibration

Many real cases of structural problem deal with structural configurations that could be represented by linear viscoelastic lumped masses system, subject to stationary or nonstationary actions. In this work, both of these input conditions are considered to evaluate structural response statistics by means of the covariance approach. This offers serious advantages both in stationary and more in nonstationary conditions if inputs are modelled as white noises, eventually filtered to increase their accordance with real dynamic loads.

The structural response of a deterministic second order 𝑚_𝑠-degree of freedom linear mechanical system subject to a probabilistic dynamic input is determined by solving the dynamic equilibrium system equations: 𝐌𝑋̈_𝑠(𝑡)+𝐂𝑋̇_𝑠(𝑡)+𝐊𝑋_𝑠(𝑡)=𝐆_s𝑓(𝑡), (1) where 𝑋_𝑠, 𝑋̇_𝑠, and 𝑋̈_𝑠 are the structural displacement, velocity, and acceleration process vectors, and 𝐌, 𝐂, and 𝐊 are the mass, viscous and stiffness matrices.

The forcing vector 𝑓(𝑡)^𝑇=𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑛(𝑡) (2) collects 𝑛 stochastic excitations applied to the structure, and 𝐆_𝐬 is a 𝑚_𝑠𝑥𝑛 matrix coupling the excitation components of forcing vector to the structural degrees of freedom. If system excitations vector elements are white noises, the first and second order statistical moments are time invariant, so that 𝑓_𝑖^𝑆𝑇(𝑡)=𝜇_𝑓𝑖,𝑓_𝑖^𝑆𝑇𝑡₁𝑓_𝑗^𝑆𝑇𝑡₂=[𝐉]_𝑖𝑗𝛿𝑡₂−𝑡₁=𝐑^𝑆𝑇_𝑓𝑓𝑡₂,𝑡_2𝑖𝑗, (3)(4) where ⟨⋅⟩ is for the mathematical expectation and 𝛿(⋅) is Dirac’s delta function.

If 𝑓_𝑖^𝑆𝑇(𝑡) are Gaussian excitations, the responses 𝑋_𝑖(𝑡) and their time derivatives 𝑋̇_𝑖(𝑡) constitute a Markov vector in the 2𝑚_𝑠 dimension phase state.

The matrix 𝐉, related to a vector satisfying the shot noise properties (usually denoted as a shot noise vector), has diagonal elements [𝐉]_𝑖𝑖=2𝜋𝑆^𝑖₀ equal to the auto covariance intensity of each force and extra diagonal elements equal to [𝐉]_𝑖𝑗=2𝜋𝑆₀^𝑖𝑗=2𝜋𝑆₀^𝑗𝑖, representing the level of correlation between two generic different forces, so that [𝐉]_𝑖𝑗 can vary from 2𝜋𝑆₀, if 𝑓_𝑖 and 𝑓_𝑗 are completely correlated, to zero, if 𝑓_𝑖 and 𝑓_𝑗 are completely uncorrelated.

Then, in case of complete uncorrelated forcing loads, the matrix 𝐉 is replaced by the more simple diagonal matrix of components [𝐃]_𝑖𝑖=2𝜋𝑆^𝑖₀, where the elements 𝑆^𝑖₀ are the input power spectral density of each entry.

A widely adopted approach consists in representing a nonstationary input by means of an intensity modulation of a stationary process, also denoted as uniform modulation, in which it is assumed that process intensity changes during time as a deterministic function 𝜑(𝑡), without any variation in spectral contents. Thus, in case of time modulation, a stationary forcing process vector is replaced by the following nonstationary vector: 𝑓^𝑁𝑆(𝑡)=𝜑₁(𝑡)𝑓₁^𝑆𝑇(𝑡),𝜑₂(𝑡)𝑓₃^𝑆𝑇(𝑡),…,𝜑_𝑛(𝑡)𝑓_𝑛^𝑆𝑇(𝑡) (5) with the stochastic characterization 𝑓_𝑖^𝑁𝑆(𝑡)=𝜑_𝑖(𝑡)𝜇_𝑓𝑖, (6) and, in case of uncorrelated excitations, the covariance matrix 𝐑^𝑁𝑆_𝑓𝑓(𝑡₂,𝑡₂) is a diagonal 𝐑^𝑁𝑆_𝑓𝑓𝑡₁,𝑡₂=𝑓_𝑖^𝑁𝑆𝑡₁𝑓_𝑗^𝑁𝑆𝑡₂=2𝜋𝑆^𝑖₀𝜑_𝑖𝑡₁𝜑_𝑖𝑡₂𝛿𝑡₂−𝑡₁if𝑖=𝑗,0if𝑖≠𝑗. (7)

2.1. Prefilters Technique

The prefilter approach is adopted when the autocorrelation function of the input is not known. In this approach, input processes are modelled by solving filter differential equations, whose input is a white noise process. So doing, it is possible to give a representation of physical phenomena whose frequency characterization is subject to seriously variations in time, potentially causing structural resonant consequences, while maintaining the advantages offered by a shot noise input.

In particular, in the prefilter approach, the filter response is described by the 2𝑚_𝑓 filter state-space vector 𝑍_𝑓=(𝑋_𝑓,𝑋̇_𝑓)^𝑇, solution of the 2𝑚_𝑓 set of differential equations 𝑍̇_𝑓=𝐇𝟎𝐈_1𝑓(𝑡)𝐇²_𝑓(𝑡)𝑍_𝑓+𝐆_𝑓𝑊_𝑓(𝑡) (8) that generally could have a time-dependent form, when not only the frequency, but also the amplitude of loads, has an intrinsic evolutive nature. In (8), 𝑊_𝑓(𝑡) is a vector of 𝑛_𝑓 white noise processes (stationary or nonstationary), 𝐇¹_𝑓(𝑡) and 𝐇²_𝑓(𝑡) are two 𝑚_𝑓𝑥𝑚_𝑓 matrices, and 𝐆_𝑓 is a 2𝑚_𝑓𝑥𝑛_𝑓 matrix that couples the excitation components of the forcing vector to the filter with a degree of freedom. Then, adopting the prefilter technique, the following system of motion differential equations is written in the state-space: 𝑋̇_𝑆𝑋̈_𝑠=𝟎𝐈−𝐌⁻¹𝐊−𝐌⁻¹𝐂𝑋_𝑆𝑋̇_𝑆+𝐆𝟎𝟎_𝑠𝜶₁(𝑡)𝐆_𝑠𝜶₂(𝑡)𝑋_𝑓𝑋̇_𝑓,𝑋̇_𝑓𝑋̈_𝑓=𝐇𝟎𝐈_1𝑓(𝑡)𝐇²_𝑓(𝑡)𝑋_𝑓𝑋̇_𝑓+𝐆_𝑓𝑊_𝑓(𝑡), (9) which can be written also as 𝑋̇_𝑠𝑋̇_𝑓𝑋̈_𝑠𝑋̈_𝑓=𝟎𝟎𝐈𝟎𝟎𝟎𝟎𝐈−𝐌⁻¹𝐊𝐆_𝑠𝜶₁(𝑡)−𝐌⁻¹𝐂𝐆_𝑠𝜶₂(𝑡)𝟎𝐇¹_𝑓(𝑡)𝟎𝐇¹_𝑓(𝑡)×𝑋_𝑠𝑋_𝑓𝑋̇_𝑆𝑋̇_𝑓+𝟎𝟎𝟎𝐆_𝑓𝑊_𝑓 (10) and summarized as 𝑍̇(𝑡)=𝐀(𝑡)𝑍(𝑡)+𝐆(𝑡)𝑊_𝑓(𝑡), (11) where 𝑍^𝑇=𝑋_𝑠,𝑋_𝑓,𝑋̇_𝑆,𝑋̇_𝑓^𝑇 (12) is a new global 2(𝑚_𝑠+𝑚_𝑓) state-space vector (structure plus filter).

In a design sensitivity problem, both the structural matrix and the response vector depend on a design parameter vector 𝑏, whose elements may be structural stiffnesses, dampings, or masses, or any other mechanical parameters, as cross-sections, Young modulus, boundary conditions, and so forth. Generally, also filter parameters and input intensity can be taken into account as design parameters. Thus, the matrix equation (11) must be explicitly re-written as: 𝑍̇(𝑏,𝑡)=𝐀𝑏,𝑡𝑍𝑏,𝑡+𝐆𝑊_𝑓(𝑡). (13) In case of zero initial condition, the solution of (13) has the general expression: 𝑍𝑏,𝑡=^𝑡₀Φ𝑏,𝑡,𝜏𝐆𝑊_𝑓(𝜏)𝑑𝜏, (14) where the matrix Φ(𝑏,𝑡₁,𝑡₂) (e.g., see [1]) is usually called a transition matrix. The mean space-state vector 𝜇_𝑧(𝑡) can be determined by the differential vectorial equation: 𝜇̇_𝑧𝑏,𝑡=𝐀𝑏,𝑡𝜇_𝑧𝑏,𝑡+𝐆𝜇_𝑤(𝑡), (15) and the covariance matrix 𝐑_𝑍𝑍(𝑏,𝑡₁,𝑡₂) of the state-space 𝑍(𝑏,𝑡) is 𝐑_𝑍𝑍𝑏,𝑡₁,𝑡₂=𝐑_𝑋𝑋𝑏,𝑡₁,𝑡₂𝐑_𝑋̇𝑋𝑏,𝑡₁,𝑡₂𝐑_̇𝑋𝑋𝑏,𝑡₁,𝑡₂𝐑_̇𝑋̇𝑋𝑏,𝑡₁,𝑡₂. (16) This second-order symmetric statistical moments matrix 𝐑_𝑍𝑍(𝑏,𝑡₁,𝑡₂) has (2𝑚²+𝑚) independent elements and can be evaluated by the well-known Lyapunov differential matrix equation 𝐑̇_𝑍𝑍𝑏,𝑡=𝐀𝑏,𝑡𝐑_𝑍𝑍𝑏,𝑡+𝐑_𝑍𝑍𝑏,𝑡𝐀𝑏,𝑡^𝑇+𝐁𝑏,𝑡, (17) where 𝐁𝑏,𝑡=𝑍𝑏,𝑡𝐆^𝑇𝑊(𝑡)^𝑇+𝐆𝑊(𝑡)𝑍^𝑇𝑏,𝑡=𝐏𝑏,𝑡+𝐏^𝑇𝑏,𝑡, (18) and where 𝐏 can be written as 𝐏𝑏,𝑡=^𝑡₀Φ𝑏,𝑡,𝜏𝐆𝑊(𝜏)𝑊^𝑇(𝑡)𝐆^𝑇=𝑑𝜏_𝑡0Φ𝑏,𝑡,𝜏𝐆𝐑_𝑤𝑤(𝑡,𝜏)𝐆^𝑇𝑑𝜏 (19) The matrix 𝐁(𝑏,𝑡) can be simplified when the forcing vector is a white noise process as defined in (4) or in (7). In these cases, due to the properties of the Dirac function and of the transition matrix, the matrix 𝐏(𝑡) is equal to 𝐏(𝑡)=^𝑡₀Φ𝑏,𝑡,𝜏𝐆𝐑_𝑤𝑤(𝑡,𝜏)𝐆^𝑇𝑑𝜏=Φ𝑏,0𝐆𝐑_𝑤𝑤(𝑡,𝑡)𝐆^𝑇=𝐆𝐑_𝑤𝑤(𝑡,𝑡)𝐆^𝑇. (20) Finally, 𝐁(𝑏,𝑡)  can be written as 𝐁(𝑡)=𝐆𝐑_𝑤𝑤(𝑡,𝑡)𝐆^𝑇+𝐆^𝑇𝐑_𝑤𝑤()𝐆=𝟎𝑡,𝑡_𝑚𝑥𝑚𝟎^𝑚𝑥𝑚𝟎^𝑚𝑥𝑚𝐋(𝑡), (21) where the 𝑚_𝑓𝑥𝑚_𝑓 submatrix 𝐋(𝑡) is a diagonal matrix whose generic elements are [𝐋(𝑡)]_𝑘,𝑘=2𝜋𝑆_0𝑘𝑚²_𝑘𝜑²_𝑘(𝑡). (22) For some applications, covariance information about structural acceleration is needed so that the matrix 𝐑_̈𝑋̈𝑋(𝑏,𝑡)=𝑋̈𝑋̈^𝑇 (23) must be determined. This matrix is easily obtainable by the relation 𝐑_̈𝑋̈𝑋𝑏,𝑡=𝐃𝑏,𝑡𝐑_𝑍𝑍𝑏,𝑡𝐃𝑏,𝑡^𝑇, (24) where 𝐃=𝑏,𝑡−𝐌𝑏⁻¹𝐊𝑏𝐆_𝑠𝜶₁𝑏,𝑡−𝐌𝑏⁻¹𝐂𝑏𝐆_𝑠𝜶₂𝑏,𝑡. (25)

3. Measure of Structural Reliability

As stated previously, a wide class of structural dynamic problems deal with reliability evaluation. Due to the essential random nature of actions, a random dynamic analysis is necessary and the natural way of testing the structural integrity is to evaluate the probability whether a structure may not have a failure during its lifetime. Mechanical safety or reliability 𝑟 at a fixed time 𝑇 is then defined as a failure survival probability, where the failure is a partial or total static damage in a given time [0,𝑇]. It is then clear that the definition of failure plays a central role in the evaluation of reliability, given the many possible interpretations. Normally, only two different kinds of mechanical failure are considered: fatigue failure, due to cumulative damage, and threshold crossing failure, determined by the first time crossing of a structural response parameter 𝑠 (able to describe failure conditions) through a given threshold value 𝛽. The present work deals with threshold crossing failure. Regarding a generic mechanical system subject to a stochastic process, the reliability 𝑟(𝑇) is defined as the survival probability for the exceeding of the given threshold value 𝛽 and, under the assumption that the initial survival probability is equal to one (𝑟(0)=1), is given by 𝑟𝑏,𝑇=exp−^𝑇₀ℎ𝛽,𝑏,𝑡𝑑𝑡. (26) The hazard function ℎ(𝛽,𝑡) is defined as the probability of having a threshold crossing, in a unit time, in absence of previous threshold crossings. Its exact formulation is still an open question. In the Rice original formulation [15] for a single degree of freedom system, the hazard function has the following general form: ℎ𝛽,𝑏,𝑡=₀^+∞𝑠𝑝̇_𝑆̇𝑆𝛽,𝑏,𝑠̇∣𝑄(𝑡)𝑑𝑠,̇ (27) in which 𝑝_𝑆̇𝑆(𝑠,𝑠̇∣𝑄(𝑡)) is the joint probability density of 𝑆(𝑡) and 𝑆(̇𝑡) processes, and 𝑄(𝑡) is the condition of absence of excursions of the fixed barrier 𝛽 before the time 𝑡. Difficulties related to the determination of this joint probability often impose the use of approximate solutions and one of which commonly used is the replacing of the conditional probability of failure with the unconditional probability 𝑝_𝑆̇𝑆(𝑠,𝑠)̇. By means of this assumption, the hazard function ℎ(𝛽,𝑡) is replaced by the threshold crossing rate 𝑣⁺_𝑆(𝛽,𝑡) which, under the further assumption that the above stochastic processes are Gaussian with null mean values, can be written as 𝑣⁺_𝑆𝛽,=1𝑏,𝑡𝑎2𝜋⁽¹⁾𝑏,𝑡𝑎⁽²⁾𝑏,𝑡×𝑎⁽³⁾𝛽,𝑏,𝑡𝜒𝑑_𝑆𝛽,𝑏,𝑡, (28) where 𝑎⁽¹⁾=𝜎𝑏,𝑡^̇𝑆𝑏,𝑡𝜎_𝑆,𝑎𝑏,𝑡₍₂₎𝑏,𝑡=1−𝜌²_𝑆̇𝑆,𝑎𝑏,𝑡₍₃₎𝛽,−1𝑏,𝑡=exp2𝛽𝜎_𝑆𝑏,𝑡²,𝜒𝑑_𝑆𝛽,−𝑑𝑏,𝑡=exp^𝑆𝛽,𝑏,𝑡²2+𝑑_𝑆𝑏,𝑡√2𝜋1−Φ𝑑_𝑥𝛽,,𝑑𝑏,𝑡_𝑆=𝛽𝑏,𝑡𝜎_𝑆𝜌𝑏,𝑡^𝑆̇𝑆𝑏,𝑡1−𝜌²_𝑆̇𝑆𝑏,𝑡. (29)(30) In (30), 𝜎_𝑆(𝑏,𝑡) and 𝜎_̇𝑆(𝑏,𝑡) are the standard deviations of 𝑆(𝑏,𝑡) and 𝑆(̇𝑏,𝑡), and 𝜌_𝑆̇𝑆(𝑏,𝑡) is their correlation factor. For a structural system, the threshold crossing rate 𝑣⁺_𝐬 of a global safety domain 𝐷⊂𝑅^𝑚 can be determined by means of the following extension of the Rice formula [16], regarding a generic multidimensional stochastic process, and here referred to as 𝑛_𝑠𝑋1 internal force vector 𝐬, whose elements are suitable structural response parameters (internal forces or stresses evaluated in a finite number of critical sections or points): 𝑣⁺_𝐬=_{Γ𝐷̇𝑠⋅𝑛≥0}𝑠̇⋅𝑛𝑝_𝐒𝐒𝑠,𝑠,̇𝑏𝑑𝑠̇𝑑𝑠, (31) where Γ_𝐷 is a safety domain boundary and 𝑛 is the outer normal to the boundary.

Though an exact evaluation of reliability based on (31) is conceptually possible, it presents some difficulties. Many different simplified approaches have been proposed in last twenty years, when outcrossing rates of a vector random process have been studied in the context of problems in load combinations and in structural reliability [17–23].

In this work, an approximate and conservative evaluation of 𝑣⁺_𝐬 is preferred as it can be easily adopted when the safe domain 𝐷 is a hyper rectangle in 𝑅^𝑚 (Figure 1) 𝐷=−𝛽₁,𝛽₁×−𝛽₂,𝛽₂×⋯×−𝛽_𝑚,𝛽_𝑚, (32) and 𝑣⁺_𝐬 is evaluated as the sum of the mean crossing rate on each face of the hyper rectangle 𝐷. In this simplified approach, it is assumed that 𝑣⁺_𝐬 is the sum of the crossing rates through each hyper plane on which the faces of 𝐷 lie. For instance, in the two-dimensional space 𝑅², the safe domain 𝐷 is the rectangle 𝐼₁∩𝐼₂, as shown in Figure 1, and (32) is an integral evaluated on the rectangular boundary Γ_𝐷. The safe domain boundary Γ_𝐷 is replaced by Γ_𝐼1∪Γ_𝐼2, and 𝑣⁺_𝐬 is evaluated as the sum of the crossing rates 𝑣⁺₁ and 𝑣⁺₂ out of the regions 𝐼₁ and 𝐼₂, respectively.

Figure 1 

Plane safe domain 𝐷 (a-b-c-d) in a two-dimensional structural response parameters space 𝑠1×𝑠2.

Figure 2 

Mechanical scheme of analysed plane shear type.

Generally, the safe domain can be obtained as 𝐷=𝐼₁∩⋯∩𝐼_𝑖⋯∩𝐼_𝑚, (33) where 𝐼_𝑖=[−∞,∞]×⋯×[−𝛽_𝑖,𝛽_𝑖]×⋯×[−∞,∞]. Since the integrand function in (31) is always nonnegative and given Γ_𝐷⊂^𝑛_𝑖=1Γ_𝐼1, (34) we have that 𝑣⁺_𝐬≤𝑣⁺_𝑖, (35) where 𝑣⁺_𝑖=⁰_−∞𝑠𝑝̇_{𝑆𝑖̇𝑆𝑖}−𝛽_𝑖,𝑏,𝑠_𝑖̇𝑑𝑠_𝑖̇+₀^+∞𝑠𝑝̇_{𝑆𝑖̇𝑆𝑖}𝛽_𝑖,𝑏,𝑠_𝑖̇𝑑𝑠_𝑖̇, (36) and in this particular case with symmetric barriers, we have 𝑣⁺_𝑖=2₀^+∞𝑠𝑝̇_{𝑆𝑖̇𝑆𝑖}𝛽_𝑖,𝑏,𝑠_𝑖̇𝑑𝑠_𝑖̇. (37) In the following, 𝑣⁺_𝐬 will always be evaluated as ∑𝑣⁺_𝑖. It must be noted that (35) is strictly satisfied in case of rare events.

The final structural reliability is evaluated as the product of all reliability evaluated separately for each structtural response parameter. Using this approach, it is possible to define a reliability vector 𝑟(𝑡) as the collection of independent reliability 𝑟_ℎ(𝑡) related to the first time crossing of a structural response parameter 𝑠_ℎ through given symmetric threshold values ±𝛽_ℎ that induce a structural failure.

It is then necessarily required to evaluate the probability 𝑃_𝑓,ℎ that the absolute value |𝑠_ℎ| exceeds 𝛽_ℎ, at least once during all input duration. The generic reliability vector element 𝑟_ℎ(𝑇) is defined as 𝑟_ℎ𝑏,𝑇=𝑃𝑆_ℎ≤𝛽_ℎ;𝑡∈[0,𝑇], (38) where, under the Poisson hypothesis, we have: 𝑟_ℎ−1𝑏,𝑇=exp𝜋^𝑇₀𝜎_̇𝑆ℎ𝑏,𝑡1−𝜌²_{𝑆ℎ̇𝑆ℎ}𝑏,𝑡𝜎_𝑆ℎ−1𝑏,𝑡×exp2𝛽_ℎ𝜎_𝑆ℎ𝑏,𝑡²×𝜒𝑑_𝑆ℎ𝑏,𝑡𝑑𝜏, (39) where 𝜎²_𝑆ℎ, 𝜎²_̇𝑆ℎ, and 𝜌_{𝑆̇𝑆ℎ} are, respectively, the ℎ and the 𝑛+ℎ diagonal elements of 𝐑_{𝑍𝑆𝑍𝑆}(𝑡), where 𝑍_𝑆=(_𝑆^̇_𝑆)^𝑇 and 𝜂_𝑆ℎ=𝛽_ℎ/𝜎_𝑆ℎ.

With reference to a vector of response functions of interest 𝑆(𝑡) in reliability evaluation, related to the state space vector 𝑍(𝑡) by the linear transformation 𝑆(𝑡)=𝐓₁(𝑏)𝑋(𝑡), 𝑆(̇𝑡)=𝐓₂(𝑏)𝑍(𝑡), the general form to express it become 𝑌(𝑡)=𝑆𝑆̇^𝑇=𝐓^⌢𝑏𝑍(𝑡), (40) where 𝐓^⌢𝑏=𝐓₁𝑏𝟎𝟎𝐓₂𝑏, (41) and the covariance matrix in structural response parameters space-state 𝑍_𝑆 is 𝐑_{𝑍𝑆𝑍𝑆}𝑏,𝑡=𝐓^⌢𝑏𝐑_𝑍𝑍𝑏,𝑡𝐓^⌢𝑇𝑏. (42) The final global structural reliability is then 𝑟_global𝑏,𝑡=^𝑛_ℎ=1𝑟_ℎ𝑏,𝑡. (43) The equation represents the probability that the ℎth interfloor displacement will cross the maximum acceptable value 𝛽_ℎ during time interval [0,𝑇]. All the barriers can be collected in the barrier vector 𝛽.

In case of failure due to a double symmetric threshold crossing, the threshold crossing rate 𝑣⁺_𝑖 in the generic 𝑖th critical node is given by 𝑣_𝑖𝑏,𝑡=2𝑣⁺_𝑖𝑏,𝛽_𝑖,𝑡, (44) where 𝑣⁺_𝑖(𝛽_𝑖,𝑡) is determined by (37).

3.1. Response Reliability Sensitivity

After covariance matrix 𝐑_𝑧𝑧 has been determinated, its first order sensitivity, whose determination is necessary in determining the structural reliability sensitivity, is evaluated (in the following, the notations ⋅_,𝑏𝑗={⋅}_,𝑏𝑗=𝜕⋅/𝜕𝑏_𝑗 will be adopted). Consider 𝐑_𝑍𝑍,_𝑏𝑗=𝜕𝐑_𝑍𝑍𝑏,𝑡𝜕𝑏_𝑗 (45) In case of nonstationary conditions, the design sensitivity 𝐑_𝑍𝑍,_𝑏𝑗 of the covariance matrix 𝐑_𝑍𝑍 is determined by solving the following matrix differential equation: 𝐑̇𝑏,𝑡𝜕𝑏_𝑗=𝐑̇,_𝑏𝑗(𝑡)=𝐀𝑏,𝑡𝐑,_𝑏𝑗𝑏,𝑡+𝐑,_𝑏𝑗𝑏,𝑡𝐀𝑏,𝑡^𝑇+𝐀_,𝑏𝑗𝐑+𝐑𝐀^𝑇_,𝑏𝑗. (46) The 2(𝑚_𝑠+𝑚_𝑓)𝑥2(𝑚_𝑠+𝑚_𝑓) matrix 𝐀_,𝑏𝑖 has the following expressions which, without loss of generalities, can been simplified as: 𝐀_,𝑏𝑗𝑏,𝑡=𝟎𝟎𝐈𝟎𝟎𝟎𝟎𝐈−𝐌(𝑏)⁻¹𝐊(𝑏)_,𝑏𝑗𝐆_𝑠𝜶₁(𝑡)−𝐌(𝑏)⁻¹𝐂(𝑏)_,𝑏𝑗𝐆_𝑠𝜶₂(𝑡)𝟎𝐇¹_𝑓(𝑡)𝟎𝐇¹_𝑓(𝑡), (47)in which it has been assumed that input parameters do not depend on the design vector 𝑏. In (47), we have −𝐌⁻¹𝐊_,𝑏𝑗=𝐌⁻¹𝐌_,𝑏𝑗𝐌⁻¹𝐊−𝐌⁻¹𝐊_,𝑏𝑗,−𝐌⁻¹𝐂_,𝑏𝑗=𝐌⁻¹𝐌_,𝑏𝑗𝐌⁻¹𝐂−𝐌⁻¹𝐂,_𝑏𝑗. (48) To determine reliability sensitivity in structural response parameter space 𝑆, it is necessary to obtain 𝐑_{𝑍𝑆𝑍𝑆}𝑏,𝑡_,𝑏𝑗=𝐓^⌢𝑏_,𝑏𝑗𝐑_𝑍𝑍𝑏,𝑡𝐓^⌢𝑇𝑏+𝐓^⌢(𝑏)𝐑_𝑍𝑍𝑏,𝑡𝐓_,𝑏𝑗^⌢𝑇𝑏+𝐓^⌢𝑏𝐑_𝑍𝑍𝑏,𝑡𝐓^⌢𝑇𝑏_,𝑏𝑗, (49) and thus the generic 𝑖th reliability sensitivity is 𝑟_𝑆𝑖𝛽,𝑏,𝑡_,𝑏𝑗=^𝑇₀𝑣_𝑆𝑖𝛽,𝑏,𝜏_,𝑏𝑗𝑑𝜏×exp^𝑇₀𝑣_𝑆𝑖𝛽,𝑏,𝜏𝑑𝜏, (50) where the mean rate crossing failure sensitivity is (see details in Appendix B) 𝑣⁺_𝑆𝑖𝛽,𝑏,𝑡_,𝑏𝑗=1𝜋𝑎¹_,𝑏𝑗𝑎²𝑎³𝜒+𝑎¹𝑎²_,𝑏𝑗𝑎³𝜒+𝑎^1𝑎2𝑎³_,𝑏𝑗𝜒+𝑎¹𝑎²𝑎³𝜒_,𝑏𝑗. (51) In view of (42), the first order sensitivity of the global reliability is then 𝑟_global(𝑏,𝑡)_,𝑏𝑗=^𝑛𝑠_𝑖=1𝑟_𝑖𝑏,𝑡_,𝑏𝑗^𝑛𝑠_{𝑘=1,𝑘≠𝑖}𝑟_𝑘𝑏,𝑡. (52)

4. Numerical Example

In the present application of the proposed method, a multistorey building subject to an earthquake excitation is taken into consideration. For simplicity, and without loss of the generalities in the proposed technique, the building model adopted is a generic shear type plane frame structure, as shown in Figure 2. When floor slabs in a building generally have a very large degree of stiffness in-plane, then they can be regarded as rigid diaphragms. This increases analysis efficiency greatly without any significant loss of accuracy in the assessment of the response resulting from ground excitations. In addition to the rigid floor diaphragm assumption, the efficiency of computation can be further increased by applying the matrix condensation technique.

The main building mechanical model is assumed to be linear. This is an acceptable hypothesis within the horizontal displacement limitations required to guarantee an operative service level. In other words, this simplified hypothesis is perfectly functional when the assigned maximum interstorey drift is quite close to the elastic limit of the structural displacement (e.g., in case of full operatives demand). The design vector is assumed to be in order composed by each floor mass, stiffness, and damping: 𝑏=𝑏₁,𝑏₂,𝑏₃=𝑚^𝑇,𝑘^𝑇,𝑐^𝑇, (53) where 𝑏₁=𝑚^𝑇=𝑚₁,𝑚₂,𝑚₃,…,𝑚_𝑚𝑠,𝑏₂=𝑘^𝑇=𝑘₁,𝑘₂,𝑘₃,…,𝑘_𝑚𝑠,𝑏₃=𝑐^𝑇=𝑐₁,𝑐₂,𝑐₃,…,𝑐_𝑚𝑠. (54)

4.1. Motion Equation

To preserve the main seismic peculiarity of spectral and time modulation, a nonstationary modulated Kanai-Tajimi process is used for stochastic ground motion. It describes the base acceleration 𝑋̈_𝑔(𝑡) applied at the base of the structure as 𝑋̈_𝑔(𝑡)=𝑋̈_𝑓(𝑡)(𝑡)𝑤(𝑡),𝑋̈+𝜑_𝑓(𝑡)+2𝜉_𝑓𝜔_𝑓𝑋̇_𝑓(𝑡)+𝜔²_𝑓𝑋_𝑓(𝑡)=−𝜑(𝑡)𝑤(𝑡), (55) where 𝑋_𝑓(𝑡) is the response of the Kanai-Tajimi filter, having frequency 𝜔_𝑓 and damping coefficient 𝜉_𝑓, and 𝑤(𝑡) is the white noise whose constant bilateral Power Density Spectral (PDS) function is 𝑆₀. This last parameter is related to the Peak Ground Acceleration (PGA) 𝑋̈_𝑔^max by means of relation [24] 𝑆₀𝜉=0.2222^𝑔𝑋̈_𝑔^max2𝜋𝜔_𝑔1+4𝜉²_𝑔. (56) The nonstationary character of the problem is introduced by the deterministic temporal modulation function 𝜑(𝑡), which controls the intensity variation, without changing earthquake frequency contents. In this case, the modulation function proposed by Jennings [25] is adopted as follows: 𝜑(𝑡)=𝑡𝑡₁²𝑡<𝑡₁,1𝑡₁≤𝑡≤𝑡₂,𝑒^{−𝛽(𝑡−𝑡2)}𝑡>𝑡₂. (57) The system of motion equations for the complete structural system then are 𝑋̈_𝑠(𝑡)+𝐌𝑏⁻¹𝐂𝑏𝑋̇_𝑠(𝑡)+𝐌𝑏⁻¹𝐊𝑏𝑋_𝑠(𝑡)=−𝑟2𝜉_𝑓𝜔_𝑓𝑋̇_𝑓(𝑡)+𝜔²_𝑓𝑋_𝑓(𝑡),𝑋̈_𝑓(𝑡)+2𝜉_𝑓𝜔_𝑓𝑋̇_𝑓(𝑡)+𝜔²_𝑓𝑋_𝑓(𝑡)=−𝑤(𝑡)𝜑(𝑡). (58) It can be written in matrix form as 𝑋̇_𝑠𝑋̇_𝑓𝑋̈_𝑠𝑋̈_𝑓=𝟎𝟎𝐈𝟎0001−𝐌⁻¹𝐊−𝜔²_𝑓𝑟−𝐌⁻¹𝐂−2𝜉_𝑓𝜔_𝑓𝑟0−𝜔²_𝑓0−2𝜉_𝑓𝜔_𝑓×𝑋_𝑠𝑋_𝑓𝑋̇_𝑆𝑋̇_𝑓+000(𝑡)𝑤(𝑡),−𝜑 (59) where 𝑟 = [1,1,1,…,1,1]^𝑇 is the 𝑚_𝑠𝑥1 drag vector, and 𝐌(𝑏), 𝐂(𝑏), and 𝐊(𝑏) are, respectively, the mass, viscosity, and stiffness 𝑚_𝑠𝑥𝑚_𝑠 structural matrices, whose general expression is 𝑘𝐊=¹+𝑘₂−𝑘₂0−𝑘₂𝑘₂+𝑘₃−𝑘₃0−𝑘₃𝑘₃𝑐𝐂=¹+𝑐₂−𝑐₂0−𝑐₂𝑐₂+𝑐₃−𝑐₃0−𝑐₃𝑐₃𝐌=diag𝑚₁,𝑚₂,𝑚₃. (60)

The three vectors 𝑋̈_𝑠(𝑏,𝑡), 𝑋̇_𝑠(𝑏,𝑡), and 𝑋_𝑠(𝑏,𝑡) are acceleration, velocity, and displacement 𝑛_𝑥1 vectors, relative to the ground. In the analyzed structure, the mass matrix is diagonal and the two viscous and stiffens matrices are tridiagonal. The matrix differential equation (59) is also written as 𝑍̇𝑏,𝑡=𝐀𝑏𝑍𝑏,𝑡+𝐹(𝑡). (61)

4.2. Reliability Evaluation

With reference to the proposed reliability problem, the probability 𝑃_𝑓,ℎ that each ℎth storey drift 𝑈_ℎ exceeds the threshold 𝛽_ℎ at least once in a given interval must be evaluated. Then, for each ℎth floor, this failure event is associated with the condition that |𝑥_ℎ+1−𝑥_ℎ|=|𝑢_ℎ|=𝛽_ℎ. The corresponding reliability vector element is 𝑟_ℎ𝑏,𝛽_ℎ,𝑇=𝑃𝑈_ℎ𝑏,𝑡≤𝛽_ℎ;𝑡∈[0,𝑇]. (62) Thus, assuming the Poissonian crossing hypothesis 𝑟_ℎ𝑏,𝛽_ℎ−1,𝑇=exp𝜋^𝑇₀𝜎_̇𝑈ℎ𝑏,𝜏𝜎_𝑈ℎ𝑏,𝜏1−𝜌²_{𝑈ℎ̇𝑈ℎ}−1𝑏,𝑡×exp2𝜂²_ℎ𝑏,𝛽_ℎ𝑑,𝜏×𝜒_𝑈ℎ𝑏,𝛽_ℎ,𝑡𝑑𝜏, (63) and the final global structural reliability is then determined as: 𝑟_global𝑏,𝛽,𝑡=^𝑛_ℎ=1𝑟_ℎ𝑏,𝛽_ℎ,𝑡. (64) Although exact analytical solutions for 𝑟_global(𝑏,𝛽,𝑡) are generally definable from the theoretical point of view, it is known that (64) is an approximate upper-bound of global reliability and provides a conservative estimation of 𝑟_global(𝑡), as previous stated. Moreover, it can be efficiently used for design and predesign purposes in practical engineering problems.

In order to evaluate the reliability vector related to the interfloor relative displacement threshold crossing, the interstorey drift vector is introduced as 𝑢(𝑡)=𝑥₁,𝑥₂−𝑥₁,𝑥₃−𝑥₂,…,𝑥_𝑛−𝑥_𝑛−1,𝑥_𝑓^𝑇 (65) whose covariance matrix 𝑅_{𝑍𝑈𝑍𝑈}(𝑡) must be evaluated.

The elements of the linear space of stochastic processes are then 𝑍_𝑈(𝑏)=[𝑈^𝑇(𝑏),𝑈̇^𝑇(𝑏)]^𝑇. The linear equation 𝑈(𝑏,𝑡)=𝐓𝑋(𝑏,𝑡) relates the interstorey drift vector 𝑈(𝑏,𝑡) and the storey displacement vector 𝑋(𝑏,𝑡), where the transform matrix 𝐓 is bidiagonal and is independent from design vector 𝐓=1000⋯0−11000−110⋮−11⋱⋱100⋮0−11000−1100⋯01. (66) The covariance matrix 𝐑_{𝑍𝑈𝑍𝑈}(𝑏,𝑡) is then related to 𝐑_𝑍𝑍(𝑏,𝑡) through the following relations: 𝐑_{𝑍𝑈𝑍𝑈}𝑏,𝑡=𝐓^⌢𝐑_𝑍𝑍𝑏,𝑡𝐓^⌢𝑇,_⌢𝐓=.𝐓𝟎𝟎𝐓 (67) The reliability vector 𝑟_𝑈 previously defined can be evaluated as the collection of 𝑟_𝑈ℎ(𝑇)=𝑟₀𝑒^{−∫𝑇0𝜐+𝑈ℎ(𝜏)𝑑𝜏}, (68) where 𝑣⁺_𝑈ℎ(𝜂_𝑉ℎ) is function of 𝜎²_𝑈ℎ, 𝜎²_̇𝑈ℎ, 𝜌_{𝑈ℎ̇𝑈ℎ}, and 𝜂_𝑈ℎ=𝛽_ℎ/𝜎_𝑈ℎ. The first two are, respectively, the ℎ and the 𝑛+ℎ main diagonal elements of 𝐑_{𝑍𝑈𝑍𝑈}(𝑡), the third is the ℎ,𝑛+ℎ element of the same matrix, and finally 𝛽_ℎ is the ℎth barrier. Equation (68) represents the probability that the ℎth interfloor displacement will cross the maximum acceptable value 𝛽_ℎ during time interval [0,𝑇]. All the barriers are collected in the vector 𝛽. In the example here assumed its elements are all equal to 3.0 cm for each floor, which corresponds to a lateral drift of 1.0% in case of 3 m interstorey.

4.3. System Parameters

In the building configurations here adopted, there are three equal mass storeys 𝑚_𝑖=1.50×10⁵(kg). First floor lateral stiffness is 𝑘₁=6.0⋅10⁷(N/m), and it is assumed that a linear stiffness reduction of 30% takes place at the upper floors, so that 𝑘₂=5.1⋅10⁷(N/m) and 𝑘₃=4.2⋅10⁷(N/m). Finally, damping is evaluated by setting 𝑐_𝑖=2𝑚_𝑖𝑘_𝑖, so that 𝑐₁=3×10⁵Nsec/m,𝑐₂=2,8×10⁵Nsec/m,𝑐₃=2.5×10⁵Nsec/m. The seismic characterization assumes a peak ground acceleration (PGA) of 0.45 (g), and 4 different total earthquake durations 𝑡_𝑑=𝑡₂−𝑡₁ are evaluated. They are, respectively, equal to 10, 20, 30 and 40 seconds.

In Figure 3 the structural interstorey covariances are shown, in terms of displacement (a) and velocity (b). Continuous lines are for 𝑡_𝑑=10 sec., dashed lines are for 𝑡_𝑑=20 sec., dash-dot lines are for 𝑡_𝑑=30 sec., and finally dotted lines are for 𝑡_𝑑=40 sec. Blue lines are for first level covariance, magenta lines are for second level, and red lines are for the 3^dh level. It must be noticed that structural responses reach a stationary level if 𝑡_𝑑 is longer than 10 sec. Moreover, for this specific structural configuration, covariance responses of the first level are the greatest; the second is only slightly smaller than the first and the third is about the half.

(a)

(b)

Figure 3 

Interstorey displacement ((a), 𝜎𝑢) and velocities ((b), 𝜎𝑣) response of a 3-DoF (different colors for different levels) system subject to modulated filtered white noise with dissimilar time durations (different line-types).

In Figure 4, reliabilities are shown and structural safeties are evaluated for each lateral interstorey drift threshold crossing, and global reliability is evaluated as approximate upper-bound global reliability. Also in this figure, line type and line colours are used as in Figure 3. Moreover, here black lines are for global system reliability. It should be noticed that the first interstorey drift failure is the more probable, the second one has just a smaller failure probability, and finally the third one has a quite negligible failure probability (i.e., 𝑟₃=1).

Figure 4 

Structural reliability of a 3-DoF system, evaluated as the probability of maximum interstorey drift exceeds over a given threshold of 3 cm.

In Figure 5, first order sensitivities of lateral interstorey drifts evaluated by the proposed approach are shown, for different time durations and storey levels. Results are plotted by using a nondimensional measure, represented by the parameter 𝜎_𝑢𝑖(𝑡)^AD_,𝑏𝑗=𝜕𝜎_𝑢𝑖(𝑡)𝜕𝑏_𝑗𝑏_𝑗𝜎_𝑢𝑖MAX. (69) Sensitivities are evaluated with regards to three sets of mechanical parameters: the three floor masses (in graphs a, d, and g), the three floor stiffness (in graphs b, e, and h), and finally the three floor dampings (in graphs c, f, and i).

(a)

(b)

(c)

Figure 5 

Covariance displacement sensitivity of a 3-DoF subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural masses (different line colours for each storey mass) evaluated at each interstorey drift ((a) for the first, (b) for the second, and (c) for the third).

With reference to the first set of mechanical parameters, it must be noticed that the increase of floor masses (blue for the first one, magenta for the second, and red for the third one) has the effect of increasing lateral drift covariance responses. This effect takes place in any analysed case. Moreover, it must be also noticed that the first interstorey drift sensitivity has about the same values with respect to each three structural masses, as it is extremely evident in Figure 5(a). On the contrary, the second interstorey drift Figure 5(b) is more sensible to 𝑚₂ (magenta) and 𝑚₃ (red), then to 𝑚₁ (blue). Finally, the third interstorey drift Figure 5(c) is more sensible with respect to 𝑚₃ than to the other two masses. This is due to the fact that in general, a change of masses in a position lower than that of the analyzed interstorey drift produces a limited response variation.

A different sensitivity of interstorey drifts could be noticed in analysing the effect of stiffness variations, as shown in Figures 6(a), 6(b), and 6(c). It is evident that at each stiffness variation, a reduction of the interiorly variance 𝜎_𝑢𝑖 of the 𝑖^𝑠𝑡-level occurs only when the stiffness 𝑘_𝑖 is increased (blue for 𝑘₁, magenta for 𝑘₂, and red for 𝑘₃). On the contrary, for each 𝑖-level, the increase of a rigidity 𝑘_𝑗 (𝑗≠𝑖) in levels different from those investigated has an opposite effect, by increasing the lateral displacement variance 𝜎_𝑢𝑖. For example, in Figure 6(b), it should be noticed that 𝜎_𝑢2 decreases when 𝑘₂ increase (magenta line) and meanwhile grows up (with slow velocity) when 𝑘₁ (blue) and 𝑘₃ (red) are increased.

(a)

(b)

(c)

Figure 6 

Covariance displacement sensitivity of a 3-DoF subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural stiffness (different line colours for each storey 𝑘𝑖) evaluated at each interstorey drift ((a) for the first, (b) for the second, and (c) for the third).

A further effect on 𝜎_𝑢𝑖 has the variation of interstorey dampings. It is evident in Figures 7(a), 7(b), and 7(c) that increasing each of them (blue for 𝑐₁, magenta for 𝑐₂, and red for 𝑐₃ ) has the effect of inducing a reduction in all floors lateral displacement. This is a direct effect of the energy dissipation that is induced by increasing structural damping. Then, the growing up of dissipated energy induces in any case the reduction of elastic and kinetic one, and, therefore, of elastic deformations. With regards to this specific structural system, maximum sensitivities in (𝜎_𝑢𝑖)^AD_,𝑏𝑗 have about the same magnitude (0.4–1.0) in case of masses and stiffness variations, and a smaller one (0.20–0.25) for damping variations.

(a)

(b)

(c)

Figure 7 

Covariance displacement sensitivity of a 3-DoF subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural damping (different line colours for each storey 𝑐𝑖) evaluated at each interstorey drift ((a) for the first, (b) for the second, and (c) for the third).

First order sensitivities of reliability associated with interstorey drift exceeding over a given barrier, evaluated by (64), are shown in Figures 8, 9, and 10. Also in this case, they are evaluated for different time durations and storey levels, with the same legend structure of previous figures. Results are plotted by using the non-dimensional measure represented by 𝑟_𝑢𝑖(𝑡)^AD_,𝑏𝑗=𝜕𝑟_𝑢𝑖(𝑡)𝜕𝑏_𝑗𝑏_𝑗𝑟_𝑢𝑖(𝑡). (70) As in Figures 5, 6, and 7, also in this case, the sensitivities are evaluated with regards to the three floor masses, stiffnesses, and dampings.

(a)

(b)

(c)

Figure 8 

Level reliability 𝑟𝑖 sensitivity of a 3-DoF subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural masses (different line colours for each storey 𝑚𝑖) evaluated at each interstorey drift ((a) for the first, (b) for the second, and (c) for the third).

(a)

(b)

(c)

Figure 9 

Level reliability 𝑟𝑖 sensitivity of a 3-DoF subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural stiffness (different line colours for each storey 𝑘𝑖) evaluated at each interstorey drift ((a) for the first, (b) for the second, and (c) for the third).

(a)

(b)

(c)

Figure 10 

Level reliability 𝑟𝑖 sensitivity of a 3-DoF subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural damping (different line colours for each storey 𝑐𝑖) evaluated at each interstorey drift ((a) for the first, (b) for the second, and (c) for the third).

Variations are evaluated assuming that 𝑏_𝑗=𝑚_𝑗 in Figures 8(a), 8(b), and 8(c), 𝑏_𝑗=𝑘_𝑗 in Figures 9(a), 9(b), and 9(c), and finally 𝑏_𝑗=𝑐_𝑗 in Figures 10(a), 10(b), and 10(c). Also in these figures Blue lines are for 𝑗=1, magenta lines are for 𝑗=2, and red lines are for 𝑗=3.

Keeping in mind that positive sensitivities indicate an increase of survival probability, it is evident that the above sensitivities have the same trend of lateral drift covariance shown in Figure 3. Besides this, it has to be observed that the reliability sensitivities of the first two floors have comparable values (Figures 8(a), 9(a), and 10(a) and Figures 8(b), 9(b), and 10(b) ), while the sensitivity of 𝑟₃, regarding the top floor, (Figures 8(c), 9(c) and 10(c) ) is smaller. This is related to the circumstance that 𝑟₃ practically has a unitary value, and so its variation due to changing in structural parameters is practically negligible.

Finally in Figure 11, the first order sensitivities of global reliability are reported. These are evaluated with respect to variations in structural masses in Figure 11(a), stiffiness in Figure 11(b), and dampings in Figure 11(c). It should be noticed that the global reliability has almost the same behaviour of 𝑟₁ in Figures 8, 9, and 10, being it the smallest one compared with the other two.

(a)

(b)

(c)

Figure 11 

Global reliability sensitivity of a 3-DoF system subject to different seismic duration (different line-types, continuous for 𝑡𝑑=10, lines for 𝑡𝑑=20, dot-line for 𝑡𝑑=30, and dotted for 𝑡𝑑=40) with respect to structural masses (a), structural stiffness (b), and structural damping (c).

5. Conclusions

A new method for evaluating the gradient with respect to structural parameters of stochastic response and structural reliability of a multi-degree-of-freedom system subject to nonstationary random loads has been proposed. The random input is modelled by a nonstationary modulated filtered white noise process, able to represent many real physic loads. In order to obtain a general nonstationary approach to be used in different contexts, the structural response is evaluated by a covariance approach, and the time domain differential matrix equations governing the sensitivity of the stochastic response are derived. In particular, the Lyapunov equation is differentiated with respect to the assigned design variables, which could be assumed as structural or input parameters. Based on this result, the structural reliability is also determined. This is defined as the first passage failure probability, expressed by the standard nonstationary expression based on Poissonian first crossing events. Sensitivity of covariance and reliability are then obtained. The proposed formulation computes the sensitivity gradient of covariance matrix and structural reliability with respect to the structural design parameters. A time integration algorithm to solve this problem is proposed.

Finally, a multistorey building idealized by a shear frame structure is considered to study the sensitivity of dynamic responses to a systemic action. The rigid floors masses and interstorey lateral stiffness and damping are considered as design variables. The structural behavior after a modification of these significant parameters has been evaluated by means of non-dimensional sensitivity parameters of covariance response and structural reliability.

Appendices

A. Time Integration Algorithm

Even if many numerical standard codes exist for the stationary Lyapunov equation 𝐴𝑋+𝑋𝐴^𝑇+𝐵=0, there are few examples of solution of (21). A simple numeric implicit integration method is then here proposed. This is a modified Euler method in which the time period [0,𝑇] is divided in 𝑚 equal subperiods Δ𝑡=𝑡^(ℎ+1)−𝑡^(ℎ)  (ℎ=0,1,…,𝑚−1) and in each subperiod Δ𝑡 a linear variation of the time derivative covariance matrix 𝐑̇(𝑡) is assumed (standard implicit Euler method). Under this assumption, we have 𝐑^(ℎ+1)=𝐑^(ℎ)+12Δ𝑡𝐑̇^(ℎ+1)+𝐑̇^(ℎ), (A.1) where the symbol 𝑎^(ℎ) denotes the generic quantity 𝑎 evaluated at time 𝑡=ℎΔ𝑡. By using the matrix equations (17) evaluated at times 𝑡^(ℎ+1) and 𝑡^(ℎ), we get the following 𝑚 algebraic matrix Lyapunov type equations: 12(𝐈−Δ𝑡𝐀)𝐑^(ℎ+1)+𝐑^(ℎ+1)12(𝐈−Δ𝑡𝐀)^𝑇=12(𝐈+Δ𝑡𝐀)𝐑^(ℎ)+𝐑^(ℎ)12(𝐈+Δ𝑡𝐀)^𝑇+Δ𝑡2𝐁^(ℎ)+𝐁^(ℎ+1), (A.2) which are solved in sequence for each time value 𝑡^(ℎ), starting from the initial time value 𝑡⁽⁰⁾=0 and from the initial covariance matrix value 𝐑⁽⁰⁾=𝟎.

In this way, the 𝑚 unknown matrices 𝐑^(ℎ), (ℎ=1,…,𝑚) are determined. By assuming the matrices as constant or time variable (depending on filter parameters variation) 𝐏_𝐵=12(),𝐏𝐈−Δ𝑡𝐀_𝐹=12(𝐈+Δ𝑡𝐀), (A.3) we have a more compact form of (A.2):𝐏_𝐵𝐑^(ℎ+1)+𝐑^(ℎ+1)𝐏^𝑇_𝐵=𝐏_𝐹𝐑^(ℎ)+𝐑^(ℎ)𝐏^𝑇_𝐹+Δ𝑡2𝐁^(ℎ)+𝐁^(ℎ+1), (A.4) and each step can be solved by a standard stationary Lyapunov equation solver; for example, the the solver lyap in standard Matlab toolbox, 𝐏_𝐵𝐑^(ℎ+1)+𝐑^(ℎ+1)𝐏^𝑇_𝐵+𝐂^(ℎ+1)=0, (A.5) where 𝐂^(ℎ+1)=−𝐏_𝐹𝐑^(ℎ)+𝐑^(ℎ)𝐏^𝑇_𝐹+Δ𝑡2𝐁^(ℎ)+𝐁^(ℎ+1). (A.6) A similar numerical approach could be used for the design sensitivity analysis of the covariance matrix 𝐑(𝑏,𝑡). The matrix equations (A.4) are differentiated with respect to design variables 𝑏_𝑗 to, obtain 𝐏_𝐵𝐑^(ℎ+1)_,𝑏𝑗+𝐑^(ℎ+1)_,𝑏𝑗𝐏^𝑇_𝐵=𝐏_𝐹𝐑^(ℎ)_,𝑏𝑗+𝐑^(ℎ)_,𝑏𝑗𝐏^𝑇_𝐹+Δ𝑡2𝐀_,𝑏𝑗𝐑^(ℎ+1)+𝐑^(ℎ)+𝐑^(ℎ+1)+𝐑^(ℎ)𝐀^𝑇_,𝑏𝑗+Δ𝑡2𝐁^(ℎ+1)_,𝑏𝑗+𝐁^(ℎ)_,𝑏𝑗. (A.7) In the case of matrix 𝐁 being independent from the design vector, often a realistic assumption, a more compact form of (A.7) is 𝐏_𝐵𝐑^(ℎ+1)_,𝑏𝑗+𝐑^(ℎ+1)_,𝑏𝑗𝐏^𝑇_𝐵=+𝐏_𝐹𝐑^(ℎ)_,𝑏𝑗+𝐑^(ℎ)_,𝑏𝑗𝐏^𝑇_𝐹+𝐌^(ℎ+1), (A.8) where 𝐌^(ℎ+1)=+(Δ𝑡/2)[𝐀_,𝑏𝑗(𝐑^(ℎ+1)+𝐑^(ℎ))+(𝐑^(ℎ+1)+𝐑^(ℎ))𝐀^𝑇_,𝑏𝑗] and can be solved by a standard stationary Lyapunov solver in the form 𝐏_𝐵𝐑^(ℎ+1)_,𝑏𝑗+𝐑^(ℎ+1)_,𝑏𝑗𝐏^𝑇_𝐵+𝐂^(ℎ+1)=𝟎, (A.9) where 𝐂^(ℎ+1)=−𝐏_𝐹𝐑^(ℎ)_,𝑏𝑗+𝐑^(ℎ)_,𝑏𝑗𝐏^𝑇_𝐹+𝐌^(ℎ+1). (A.10) Then, for each design variable 𝑏_𝑗, a sequence of 𝑚 Lyapunov type matrix algebraic equations are solved in sequence, starting from the initial time 𝑡⁽⁰⁾=0 and from the initial values 𝐑⁽⁰⁾_,𝑏𝑗=𝟎. In this way, for each time 𝑡^(ℎ) and for each design variable 𝑏_𝑗=[𝑏]_𝑗, the unknown design sensitivities 𝐑^(ℎ)_{,𝑏𝑗(ℎ=1,…,𝑛𝑡;𝑗=1,…,𝑛𝑏)} are determined (Algorithm 1).

data input 𝐌,𝐂,𝐊,𝑆0,𝜔𝑓,𝜉𝑓,𝜑(𝑡) 𝐏𝐵=(1/2)(𝐈−Δ𝑡𝐀) 𝐏𝐹=(1/2)(𝐈+Δ𝑡𝐀) 𝐑(1)=𝟎; (definition of initial condition for 𝑡=0 both on 𝐑 and on (𝐑,𝑏) for 𝑗=1 to 𝑛𝑏  𝐒(𝑗,1)=0; end; (of initial condition definition) for 𝑖=1 to ( 𝑛𝑡−1) (beginning of time integration)   𝐍(𝑖)=(Δ𝑡/2)(𝐁(𝑖+1)+𝐁(𝑖))   𝐂(𝑖)=𝐏𝐹𝐑(𝑖)+𝐑(𝑖)𝐏𝑇𝐹+𝐍(𝑖)   𝐑(𝑖+1)= lyap (𝐏𝐵,−𝐂(𝑖)) (matlab command for stationary Lyapunov eq.)   𝐑(𝑖+1)𝐏𝐵+𝐏𝐵𝐑𝑇(𝑖+1)−𝐂(𝑖)=0   for 𝑗=1 to 𝑛𝑏 (cycle for each design vector element)    𝐌(𝑖,𝑗)=(Δ𝑡/2)𝐀𝑏(𝑗)(𝐑(𝑖)+𝐑(𝑖+1))+(𝐑(𝑖)+𝐑(𝑖+1))𝐀𝑇𝑏(𝑗)    𝐃(𝑖,𝑗)=𝐏𝐹𝐒(𝑖,𝑗)+𝐒(𝑖,𝑗)𝐏𝑇𝐹+𝐌(𝑖,𝑗)    𝐒(𝑖+1,𝑗)= lyap (𝐏𝐵,𝐃(𝑖,𝑗))   end (of cycle on design vector elements) end (of time integration)

Algorithm 1 

B. Derivatives of Structural Reliability

The first order sensitivities of the quantities reported in (29) and (30) are 𝑎¹(𝑏,𝑡)_,𝑏𝑗=𝜎_̇𝑆(𝑏,𝑡)_,𝑏𝑗𝜎_𝑆()−𝜎𝑏,𝑡^𝑆(𝑏,𝑡)_,𝑏𝑗𝜎_̇𝑆(𝑏,𝑡)𝜎²_𝑆(),𝑎𝑏,𝑡₂(𝑏,𝑡)_,𝑏𝑗𝜌=−^𝑆̇𝑆(𝑏,𝑡)𝜌_𝑆̇𝑆(𝑏,𝑡)_,𝑏𝑗1−𝜌²_𝑆̇𝑆(),𝑎𝑏,𝑡₃(𝑏,𝑡)_,𝑏𝑗=𝛽𝜎_𝑆(𝑏,𝑡)_,𝑏𝑗𝜎³_𝑆()−1𝑏,𝑡exp2𝛽𝜎_𝑆(𝑏,𝑡)²,𝜒𝑑_𝑆(𝑏,𝑡)_,𝑏𝑗=−𝑒^{−(1/2)𝑑𝑥(𝑏,𝑡)2}𝑑_𝑥()+√𝑏,𝑡𝑑2𝜋1−1+erf_𝑥(𝑏,𝑡)/√22√×𝑑2𝜋_𝑥(𝑏,𝑡)_,𝑏𝑗,𝑑_𝑆(𝑏,𝑡)_,𝑏𝑗=𝛽𝜎_𝑆(𝑏,𝑡)𝜌_𝑆̇𝑆(𝑏,𝑡)_,𝑏𝑗+𝜌_𝑆̇𝑆()×𝑏,𝑡1−𝜌²_𝑆̇𝑆(𝑏,𝑡)𝜎_𝑆(𝑏,𝑡)_,𝑏𝑗×𝜎²_𝑆(𝑏,𝑡)1−𝜌²_𝑆̇𝑆(𝑏,𝑡)^3/2−1. (B.1)(B.2)(B.3)(B.4)(B.5) Furthermore, the following relations have been used in determining reliability sensitivities 𝜎_𝑆𝑖(𝑏,𝑡)_,𝑏𝑗=12𝜎²_𝑆𝑖(𝑏,𝑡)_,𝑏𝑗𝜎_𝑆𝑖(),𝜎𝑏,𝑡_̇𝑆𝑖(𝑏,𝑡)_,𝑏𝑗=12𝜎²_̇𝑆𝑖(𝑏,𝑡)_,𝑏𝑗𝜎_̇𝑆𝑖(),𝜌𝑏,𝑡_{𝑆𝑖̇𝑆𝑖}(𝑏,𝑡)_,𝑏𝑗=1𝜎_𝑆𝑖(𝑏,𝑡)𝜎_̇𝑆𝑖()×𝛾𝑏,𝑡_{𝑆𝑖̇𝑆𝑖}(𝑏,𝑡)_,𝑏𝑗−𝛾_{𝑆𝑖̇𝑆𝑖}()×𝜎𝑏,𝑡_𝑆𝑖(𝑏,𝑡)_,𝑏𝑗𝜎_𝑆𝑖()+𝜎𝑏,𝑡^̇𝑆𝑖(𝑏,𝑡)_,𝑏𝑗𝜎_̇𝑆𝑖(),𝑏,𝑡 (B.6)(B.7)(B.8) where the relations with matrices 𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡) and {[𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)]_𝑖,𝑖}_,𝑏𝑗 are 𝜎²_𝑆𝑖(𝑏,𝑡)=𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)_𝑖,𝑖,𝜎²_̇𝑆𝑖(𝑏,𝑡)=𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)_{𝑛𝑠+𝑖,𝑛𝑠+𝑖},𝛾_{𝑆𝑖̇𝑆𝑖}(𝑏,𝑡)=𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)_{𝑖,𝑛𝑠+𝑖},𝜎²_𝑆𝑖(𝑏,𝑡)_,𝑏𝑗=𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)_{𝑖,𝑖,𝑏𝑗},𝜎²_̇𝑆𝑖(𝑏,𝑡)_,𝑏𝑗=𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)_{𝑛𝑠+𝑖,𝑛𝑠+𝑖,𝑏𝑗},𝛾_{𝑆𝑖̇𝑆𝑖}(𝑏,𝑡)_,𝑏𝑗=𝐑_{𝑍𝑆𝑍𝑆}(𝑏,𝑡)_{𝑖,𝑛𝑠+𝑖,𝑏𝑗}. (B.9)

Acknowledgment

The authors are extremely grateful to Professor Giuseppe Carlo Marano for his enthusiastic support in developing and writing this paper.