Nonstationary Stochastic Excitation Research Articles

Structural reliability analysis under stochastic seismic excitations and multidimensional limit state based on gamma mixture model and copula function

A novel method for analyzing the reliability of structures under non-stationary stochastic seismic excitations, considering the combined effect of multiple structural demand extreme values, is proposed. The spectral representation method is employed to establish a non-stationary stochastic seismic excitation model, and based on the theory of first-passage probability, multiple integral formulas for seismic reliability under multidimensional limit states are derived. The extreme value distribution is established using the Gamma mixture model (GMM). The equations for estimating the model parameters are derived based on both fractional moments and moment-generating functions, while the determination of the number of gamma distribution components is guided by the probability distribution and statistical characteristics of the samples. The joint probability density function (JPDF) for multiple demand extreme values is established by incorporating copula functions to account for correlation, and the fitting accuracy of different copula functions is assessed. The proposed method is illustrated using reinforced concrete (RC) frame structures. The results demonstrate that the fitting accuracy of extreme value distribution can be enhanced by adjusting the number of gamma distribution components in the GMM, which exhibits high accuracy in fitting both the main and tail regions. The presence of correlation can induce variations in the JPDF, thereby exerting an influence on the failure probability. Consequently, disregarding correlation is not conducive to reliability analysis.

A KDE-based non-parametric cloud approach for efficient seismic fragility estimation of structures under non-stationary excitation

With the development of performance-based earthquake engineering, the risk-informed assessment framework has received broad recognition over the world, of which the probability seismic fragility analysis is an important step. The classic seismic fragility adopts the lognormal assumption and forms a parametric derivation. With the development of fragility theory, researchers are hoping to seek out non-parametric approaches to express the intrinsic fragility in a pure analytical form without any distribution assumptions. Besides, how to keep the calculation efficiency (e.g., combining with cloud approach) and how to consider the non-stationary stochastic responses (e.g., combining with non-stationary stochastic excitation model) are critical aspects in fragility that deserve further attention of researchers. In this paper, a kernel density estimation (KDE) based non-parametric cloud approach is proposed for efficient seismic fragility estimation of structures under non-stationary excitation. First, the methodology framework of the efficient approach is illustrated. Then, the procedures of non-stationary stochastic seismic response of structures and KDE-based non-parametric cloud approach for efficient seismic fragility are demonstrated. After that, an application example via a three-span-six-story reinforced concrete frame is given for implementation, followed with a parametric analysis of critical factors. During the process, the classic parametric linear-regression based cloud approach (cloud-LR) and benchmark Monte-Carlo-simulation based cloud approach (cloud-MCS) are also incorporated for validation. In general, the analysis verifies the effectiveness of the non-parametric cloud-KDE approach without requiring more computation work (i.e., same as the parametric cloud-LR approach and much less than the benchmark cloud-MCS approach). Meanwhile, the non-parametric cloud-KDE approach indicates a comparable accuracy with the classic fragility approaches (i.e., less deviation than the parametric cloud-LR approach and much closer to the benchmark cloud-MCS approach), and with the increase of stochastic cloud-point number, the corresponding fitting degree of cloud-KDE approach is growing better. The research provides a new sight for the development of non-parametric seismic fragility approach, and the corresponding findings can be further combined with the probabilistic hazard and risk analysis for a non-parametric assessment procedure in performance-based earthquake engineering.

Simulation of non-stationary ground motions and its applications in high concrete faced rockfill dams via direct probability integral method

Many high concrete face rockfill dams (CFRDs) have been constructed in earthquake-prone areas in recent years, and it is essential to evaluate the seismic safety performance of these dams considering the randomness of seismic excitation. In this paper, the relationship between the stochastic ground motion generation method and the direct probability integral method (DPIM) is established by using the number theory point selection method as the bridge. The DPIM is introduced into the field of stochastic dynamic analysis and multi-index reliability evaluation of high CFRD for the first time. First, a non-stationary stochastic excitation model at five sites is established based on the latest standard for the seismic design of hydraulic structures. Then, taking a practical project as an example, stochastic dynamic response analysis and reliability assessment based on dam crest settlement deformation and dam slope cumulative slippage are conducted by using DIPM. Finally, the influence of these two indexes on reliability is explored by drawing the equal failure probability curve. The results show that the stochastic dynamic analysis and reliability assessment framework established in this paper can provide probabilistic seismic safety assessments for high CFRDs. Sliding damage of the dam slope is more likely than the deformation damage of the rockfill from the perspective of probability.

Response and EPSD of rotor-blade nonlinear system with non-stationary non-Gaussian stochastic excitation via PGHW method

Response and EPSD of rotor-blade nonlinear system with non-stationary non-Gaussian stochastic excitation via PGHW method

Survival probability determination of nonlinear oscillators subject to combined deterministic periodic and non-stationary stochastic loads

This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

Design sensitivity analysis of structural systems with damping devices subjected to fully non-stationary stochastic seismic excitations

In the framework of dynamic excitations to be considered during the design phase of structures, the most crucial one is the ground motion acceleration. To increase the structural performances against seismic actions, one of the most effective design criteria is to introduce damping devices.An efficient approach to define the optimal parameters of the damping system is based on the design sensitivity analysis, which provides a quantitative estimate of desirable design change, by relating the available design variables.In this paper a method to evaluate the sensitivities of stochastic response characteristics of structural systems with damping devices subjected to seismic excitations, modelled as fully non-stationary Gaussian stochastic processes, is proposed. The main steps are: i) to define the time–frequency varying response (TFR) function for non-classically damped systems; ii) to evaluate closed form solutions for the first-order derivatives of the TFR function as well as of the one-sided evolutionary power spectral density function of the structural response, with respect to damping parameters of devices; iii) to perform a design sensitivity analysis selecting as performance measure function the non-geometric spectral moments of nodal displacements.A numerical application demonstrates how the proposed approach is suitable to cope with practical problems of engineering interest.

Dynamic reliability analysis of structures under nonstationary stochastic excitations using tail-modified extreme value distribution

In practical engineering, it is usually necessary to compute the small first-passage probability of dynamic systems exposed to nonstationary stochastic excitations. However, it is still a challenging problem to efficiently estimate the super-small failure probability using an approximate extreme value distribution (EVD) based on a small number of simulated samples. In this study, a novel EVD model is proposed to describe the probability distribution of the structural extreme response induced by nonstationary stochastic excitations. This EVD model consists of two parts: the main body is modeled by a truncated-shifted generalized lognormal distribution, and a monotonic exponential model is proposed to fit the tail region. Besides that, a criterion for determining the breakpoint between the main body and tail region is proposed, which ensures the non-negativity and normativity of the proposed EVD model and allows the EVD to be reconstructed accurately and efficiently, particularly in its tail region. In this regard, the corresponding first passage probabilities under different thresholds can be straightly evaluated. The precision and completeness of the proposed EVD model are investigated using two numerical examples that consider linear and nonlinear frame structures subjected to nonstationary ground motion accelerations. The results show that the proposed model is in excellent agreement with the results of Monte Carlo simulation, even when the failure probability is very small.

A harmonic-wavelet-based representation for non-stationary stochastic excitations

Representation of nonstationary stochastic excitations is crucial for stochastic response analyses of (time-varying) linear and nonlinear structural systems. This paper proposes a new representation method of non-stationary stochastic excitations based on the generalized harmonic wavelet (GHW) that takes the phase angles and frequencies as basic random variables. The orthogonal properties of the discrete-form spectral process increments describing non-stationary stochastic processes are formulated. Then the GHW-based representation is derived by using the orthogonal properties. This method can be used to accurately reproduce non-stationary stochastic excitations with the target asymptotic Gaussianity and evolutionary power spectrum density. The effectiveness and accuracy of the proposed method have been validated via numerical examples. This study provides a novel way for the representation of non-stationary processes and deserves to be applied in the stochastic response analyses of structures.

Stochastic Optimization and Sensitivity Analysis of the Combined Negative Stiffness Damped Outrigger and Conventional Damped Outrigger Systems Subjected to Nonstationary Seismic Excitation

In recent years, various kinds of damped outrigger systems, particularly with the negative stiffness devices, have been shown to effectively suppress excessive vibration induced by earthquake and wind. To gain insight into such systems, this paper proposes a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. The simplified analysis model of various combinations of damped outriggers is developed, and the state-space representation of outrigger systems is then formulated. The nonstationary seismic excitation is modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following Clough–Penzien spectrum, and combining equations of motions of outrigger systems and seismic excitation gives rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. The optimal designs of damped outriggers are defined via the solution of the differential Lyapunov equation. The multiobjective optimization with Pareto optimal fronts is adopted to deal with conflicting objectives between harmful interstory drift and floor absolute acceleration. The sensitivity of negative stiffness devices on optimal objectives is also investigated. The optimal designs are further examined under real typical and near-fault earthquake records. The results demonstrate the efficacy of the proposed method and provide insight into the systems subjected to nonstationary seismic excitation. While the purely NSDO systems could have better performance with adequate negative stiffness devices, the proposed combined NSDO and CDO systems prove to be more effective with limited negative stiffness devices and thus provide more design flexibilities tailored to different application situations.

Non-stationary response of nonlinear systems with singular parameter matrices subject to combined deterministic and stochastic excitation

A new technique is proposed for determining the response of multi-degree-of-freedom nonlinear systems with singular parameter matrices subject to combined deterministic and non-stationary stochastic excitation. Singular matrices in the governing equations of motion potentially account for the presence of constraints equations in the system. Further, they also appear when a redundant coordinates modeling is adopted to derive the equations of motion of complex multi-body systems. In this regard, the system response is decomposed into a deterministic and a stochastic component corresponding to the two components of the excitation. Then, two sets of differential equations are formulated and solved simultaneously to compute the system response. The first set pertains to the deterministic response component, whereas the second one pertains to the stochastic component of the response. The latter is derived by utilizing the generalized statistical linearization method for systems with singular matrices, while a formula for determining the time-dependent equivalent elements of the generalized statistical linearization methodology is also derived. The efficiency of the proposed technique is demonstrated by pertinent numerical examples. Specifically, a vibration energy harvesting device subject to combined deterministic and modulated white noise excitation and a structural nonlinear system with singular parameter matrices subject to combined deterministic and modulated white and colored noise excitations are considered.

Non-stationary response determination of nonlinear systems subjected to combined deterministic and evolutionary stochastic excitations

A semi-analytical method is proposed for determining the response of a lightly damped single-degree-of-freedom nonlinear system subjected to combined deterministic and non-stationary stochastic excitations. This is attained by combining the stochastic averaging and statistical linearization methodologies. Specifically, first, the system response is decomposed into two components, namely the deterministic and the stochastic parts. This leads to a set of coupled differential sub-equations governing, respectively, the deterministic and the stochastic component of the response. Next, aiming at solving the set of differential sub-equations, an additional expression is derived by applying the statistical linearization methodology, followed by the application of a stochastic averaging step to the stochastic sub-equations. Therefore, an equivalent time-varying linear system is defined for the original nonlinear system. The stochastic averaging method is then applied to the linearized system for reducing its order, and thus, its complexity from a solution perspective. In this regard, an additional equation is derived, which connects the deterministic and stochastic components of the response. The latter and the deterministic differential sub-equations are solved simultaneously for determining the system response. A single-degree-of-freedom nonlinear system exhibiting quadratic and cubic nonlinear stiffness is considered for assessing the reliability of the proposed technique. The obtained results are compared with pertinent Monte-Carlo simulation estimates.

Stochastic vibration suppression of composite laminated plates based on negative capacitance piezoelectric shunt damping

This paper studies the stochastic vibration suppression of the composite laminated plates by using the negative capacitance piezoelectric shunt damping. The energy method and Hamilton’s principle are employed to formulate the dynamic model of the composite laminated plate with negative capacitance piezoelectric shunt damping circuit. The random dynamic response of the coupled system subjected to nonstationary stochastic excitations is further obtained based on the modified Ritz method and pseudo excitation method (PEM). The accuracy of the proposed method is validated by comparing the obtained random results with those from the published literature and satisfactory agreements can be observed between them. Furthermore, the vibration suppression performance of the proposed piezoelectric shunt damping with series negative capacitor or parallel negative capacitor is discussed, respectively. Finally, the effects of key parameters of the piezoelectric shunt damping on the vibration suppression performance are discussed in detail.

A harmonic input method for computing time-varying response PSD of linear time-invariant structures subjected to nonstationary stochastic excitations

A number of stochastic response analysis methods have been successfully developed for computing the time-varying response power spectral density (PSD) of structures subjected to nonstationary stochastic excitations. However, how to compute the time-varying response PSD when transient response components have significant impacts on the stochastic responses remains largely unsolved. To this end, this paper proposes a harmonic input method (HIM) for accurately and efficiently computing time-varying response PSD of linear time-invariant structures subjected to nonstationary stochastic excitations, which can consider the transient response impact. The accuracy of the proposed HIM method is verified via comparison against the Monte Carlo simulations of an SDOF system and an MDOF system, which outperforms the classical methods if the transient response components are of significance. This study offers a high-efficient and high-accurate approach for response PSD estimation, particularly for a low-damping structural system subjected to short-duration nonstationary stochastic excitations.

Efficient approach for dynamic reliability analysis based on uniform design method and Box-Cox transformation

This paper proposes a novel approach for dynamic reliability analysis of uncertain structures subjected to non-stationary stochastic excitations. This approach calculates the first-passage probabilities by evaluating the transformed extreme value distribution (TEVD) instead of the original extreme value distribution (EVD). The Box-Cox transformation is first used to symmetrize the extreme value, and then the generalized Gaussian distribution (GGD) is suggested to approximate the main body and tails of the TEVD. An absolute skewness minimization criterion is proposed to find the optimal Box-Cox transformation parameter. Furthermore, the random-function spectral representation method (RFSRM) and uniform design method (UDM) are successfully combined to estimate the first four central moments of the transformed extreme value with a small sample size. Two numerical examples involving linear and nonlinear shear-frame structures under nonstationary ground motions are investigated. The results indicate that the proposed method can be used for dynamic reliability analysis with high accuracy and efficiency.

Non-stationary approximate response of non-linear multi-degree-of-freedom systems subjected to combined periodic and stochastic excitation

An approximate method is presented for determining the non-stationary response of multi-degree-of-freedom non-linear systems subjected to combined periodic and stochastic excitation. Specifically, first, decomposing the system response as a combination of a periodic and of a zero-mean stochastic component, transfers the equation of motion into two sets of equivalent coupled differential sub-equations, governing the deterministic and the stochastic component, respectively. Next, the derived stochastic sub-equations under non-stationary stochastic excitation are cast into equivalent linear equations by resorting to the non-stationary statistical linearization method. Further, the related Lyapunov differential equations governing the second moment of the linear stochastic response, and the deterministic sub-equations governing the periodic response, are solved simultaneously using standard numerical algorithms. Pertinent Monte Carlo simulation demonstrates the applicability and accuracy of the proposed semi-analytical method.

Efficient reliability analysis based on deep learning-enhanced surrogate modelling and probability density evolution method

An improved method, termed as the AL-DLGPR-PDEM, is presented to address high-dimensional reliability problems. The novelty of this work lies in developing a complete framework for combining the deep learning (DL) architectures, serving as the utility of dimension reduction, and the Gaussian process regression (GPR), resulting in the so-called DLGPR model. First, the parameters of both the DL and the GPR are inferred using a joint-optimization scheme, rather than the traditional two-step, separate-training scheme. Second, the network configuration of the DLGPR is optimally determined by using a grid-search procedure involving cross-validation, instead of an empirical setting manner. On this basis, the DLGPR is adaptively refined via an active learning (AL)-based sampling strategy, so as to gain the desired DLGPR using as fewer training samples as possible. Eventually, the finalized DLGPR is evaluated at the whole representative point set, thereby the probability density evolution method (PDEM) is conducted accordingly. Two numerical examples are investigated. The first one tackles with the static reliability analysis of a planner steel frame, where the case of small failure probabilities is also considered; the second one addresses the dynamic reliability analysis of the steel frame under fully non-stationary stochastic seismic excitation. Comparisons against other existing reliability methods are conducted as well. Results demonstrate that the proposed AL-DLGPR-PDEM achieves a fair tradeoff between accuracy and efficiency for dealing with high-dimensional reliability problems in both static and dynamic analysis examples.

A stochastic analysis method of transient responses using harmonic wavelets, Part 1: Time-invariant structural systems

The transient response components have significant impacts on the stochastic response analysis of structural systems subjected to nonstationary stochastic excitations. However, available stochastic analysis methods focus on the steady-state response components. This paper proposes a novel stochastic analysis method of the transient responses for multi-degree-of-freedom (MDOF) structural systems subjected to nonstationary stochastic excitations based on periodic generalized harmonic wavelets (GHW). Firstly, the importance of transient response components on structural systems is discussed in detail. The equations of motion of the MDOF structural systems are established by using the periodic GHW with considering the rigid displacement components. The linear algebraic equations with respect to the wavelet coefficients of responses and excitations are established, respectively. Then the connection equations of wavelet coefficients in all scales are established to include the transient components in the responses. A stochastic analysis method of transient responses for structural systems is proposed based on the periodic GHW, and the time-varying power spectral density functions of the structural responses are obtained in a semi-analytical form. The proposed method is validated by two numerical examples, which illustrate the high efficiency and high accuracy of the novel method.

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Online damage detection of earthquake-excited structure based on near real-time envelope extraction

A robust real-time damage detection technique of earthquake-excited structures based on a new demodulation technique for nonlinear and non-stationary vibration signals through the identification of signal envelopes in real time is presented. In the present work, the need for the detection of envelope in a vibration signal in real time is addressed by reformulating the concept of Hermitian interpolation functions to a recursive Hermitian polynomial, which is a key entitlement of the present work. Once, the near real-time demodulation is achieved, the proposed framework proceeds to the newly developed error-adapted framework by addressing the errors accrued between the standard and generalized eigen perturbation formulation in the context of real-time estimation of proper orthogonal modes and linear normal modes. In the adaptive framework, the error is modeled as a feedback, which is constructed to account for the truncation in the order of eigen perturbation. In addition to the improved accuracy due to the envelope extraction, the proposed error-adapted eigen perturbation further improves the detectability over the currently available eigen perturbation–based real-time algorithms. To ensure robustness of the proposed algorithm, a new real-time damage indicator based on the maximum of principal eigenvector of the evolving transformed covariance matrix is proposed. The proposed modules together not only improve the detectability of the damage detection in real-time but also enhance the overall performance in presence of non-stationary excitation, that often mask the damage information in the higher energy zones of the amplitude and frequency-modulated response. Simulations for the proposed framework is performed by considering a 5 degrees-of-freedom linear and base-isolated nonlinear structural system driven by non-stationary stochastic excitations with damage simulated at intermediate floor at a particular time instant. Evidence of the near real-time demodulation and/or envelope removal from the signal and improved damage identification is also provided. An examination of the proposed framework using experimental data further validates the robustness of the proposed scheme.

Optimal design of nonlinear energy sinks for mitigation of seismic response on structural systems

This paper proposes an approach for the optimal design of nonlinear energy sinks (NES) installed in structures subjected to stochastic excitation. The response of these nonlinear systems subjected to stationary or non-stationary stochastic excitations has been characterized by the time-history of its covariance matrix. However, time-invariant equivalent linearization methods of NES equipped structures will not always converge due to its essential nonlinearity. A solution strategy is proposed using equivalent linearization at each time step of the governing matrix differential equation, obtained through a sequence of Lyapunov equations. The adequacy of this method for optimization problems is evaluated by comparison with Monte Carlo simulations of two representative NES examples. Moreover, the influence of the Gaussian assumption of the equivalent system response is studied in these examples. An optimization formulation to design systems equipped with NES devices is proposed, in which the performance function is defined in terms of the maximum variance over the time-history. This result is obtained by using the proposed method with a Gaussian assumption of the equivalent linear system. To illustrate the proposed methodology, parametric optimization of an NES installed at the top of a shear building structure is performed.