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This paper proposes a heterogeneous time integration algorithm to analyze the dynamic response of structures with localized hysteretic nonlinearities. The critical point is to treat the whole structure as a combination of a linear substructure governed by the second-order dynamic equation and a nonlinear substructure controlled by the first-order differential formulation. With this partitioning, tailored numerical integration algorithms with heterogeneous time steps are applied directly to calculate the displacements and hysteretic forces from the linear and the nonlinear substructures, respectively. Subsequently, the dynamic responses of the whole structure are solved by a predictor–corrector procedure, where the hysteretic forces and displacement responses are coupled and exchanged to update the partitioning solutions. Furthermore, the energy balance method is derived to verify the stability of the proposed heterogeneous time integration algorithm. Dynamic responses of structures with friction damper, hysteretic models, and bolted joint models are studied to demonstrate the accuracy and efficiency of the proposed algorithm.

In the field of structural dynamic differential equations, although traditional numerical solution methods are mature, they generally have problems with high computational costs and numerous constraints. These constraints include whether the system is linear, sensitivity to time step size, limited ability to handle high-frequency responses, and limited applicability to nonlinear systems and complex dynamic environments. Therefore, this paper attempts to use a recurrent neural network (RNNs) to solve dynamic differential equations. Unlike intelligent models that require a large amount of sample data, this solver does not need samples. Instead, it constrains the loss function using the physics-informed neural network method and provides high-precision predictive solutions based on input parameters and memory parameters. The solver is applied to single-degree-of-freedom and multi-degree-of-freedom systems, including undamped and damped free vibrations as well as undamped and damped harmonic load problems. The feasibility of the model is validated based on the average relative error, with results showing that the average relative error between the model predictions and theoretical values is $10^{-4}$, with a few cases reaching 1%. The relationship between network structure, learning rate, and model performance is also explored. Additionally, common activation functions perform poorly when applied to this problem, so this paper constructs a new activation function, verifying its effectiveness and efficiency through the solver and analyzing the factors influencing the model’s performance.

The Lanczos method has rapidly become the preferred method of solution for the generalized eigenvalue problems. The recent emergence of parallel computers has aroused much interest in the practical implementation of the Lanczos algorithm on these high performance computers. This paper describes an implementation of a generalized Lanczos algorithm on a distributed memory parallel computer, with specific application to structural dynamic analysis.

One major cost in the parallel implementation of the generalized Lanczos procedure is the factorization of the (shifted) stiffness matrix and the forward and backward solution of triangular systems. In this paper, we review a parallel sparse matrix factorization scheme and propose a strategy for inverting the principal block submatrix factors to facilitate the forward and backward solution of triangular systems on distributed memory parallel computers. We also discuss the different strategies in the implementation of mass-matrix-vector multiplication and how they are used in the implementation of the Lanczos procedure. The Lanczos procedure implemented includes partial and external selective reorthogonalizations. Spectral shifts are introduced when memory space is not sufficient for storing the Lanczos vectors. The tradeoffs between spectral shifts and Lanc-zos iterations are discussed. Numerical results on Intel’s parallel computers, the iPSC/860 hypercube and the Paragon machines will be presented to illustrate the effectiveness and scalability of the parallel generalized Lanczos procedure.

This paper presents a simplified approximate analysis of the overall collapse of the towers of World Trade Center in New York on September 11, 2001. The analysis shows that if prolonged heating caused the majority of columns of a single floor to lose their load carrying capacity, the whole tower was doomed. Despite optimistic simplifying assumptions, the structural resistance is found to be an order of magnitude less than necessary for survival.

Magneto-rheological liquids are controllable liquids that under the action of a magnetic field can reversibly pass from the linear viscous liquid state with free-flow to the semi-solid one with a controlled stress-state. They are composed of typically non-colloidal magnetic micronized particles and possess a load carrying capacity higher than other, more controllable, fluids, such as electro-rheological liquids; moreover they are less sensitive to impurities and contaminations that may possibly occur in manufacturing. in the paper, the most suitable models for simulation of such devices are investigated with emphasis on evaluation of their efficiency as structural control systems.

The concept of substructure representation is often employed to reduce the size of large and complex models for efficient structural analysis. In the context of structural system identification, this concept has also been used recently to improve numerical accuracy and convergence characteristics. Several research works have thus far demonstrated the effectiveness and advantages of substructural identification. But interface response measurements are usually required as input to the substructure to be identified in these methods. In this paper, a novel substructural method for stiffness identification is proposed with several advantages including eliminating the need of interface response measurement and even force measurement. The identification problem is converted into one of minimizing the difference between measured and predicted displacements at a check point in the substructure. Analytical formulation and numerical study are presented for both lumped mass system and continuous system, taking into consideration the effects of incomplete and noisy data.

In this paper, the dynamic performance of a controlled high building is numerically investigated considering the effects of different numbers of mass dampers and their interconnection. The numerical analysis is conducted on a 20-story building considered as a shear frame and reduced to an SDOF system by means of the mode-superposition method. The system is subjected to harmonic load. Numerical searches are conducted based on the Min. Max. procedure in order to obtain efficient interconnected (I) multiple tuned mass dampers (MTMD). Comparisons are made among the uncontrolled system, the system controlled by non-interconnected (NI) MTMD, and the system controlled by (I) MTMD in frequency and time domain. Both (NI) and (I) MTMD reduce significantly frequency response peak amplitude. It is observed that the (I) MTMD produces great reductions on maximum displacement, rms displacement, steady-state peak response, and story displacements close to the reductions obtained by (NI) MTMD using review parameters. Mass maximum displacements analysis shows that the space required for (I) MTMD installation is smaller than (NI) MTMD.

In this article, we develop a novel stable time integration scheme for the transient analysis of structural dynamics problems. A second-order (in time) differential operator equation (e.g. obtained after finite element discretization in space) is written as a pair of first-order equations in terms of displacements and velocities. Then the solution is sought by minimizing the inner product of the residuals in the two equations (an unconventional approach) over typical time interval to obtain a symmetric set of algebraic equations involving displacements and velocities at two subsequent intervals. The new time integration scheme is termed the cross weighted-residual (CWR) time integration scheme because each of the two residuals takes the other one as a weight function in the minimization. The CWR time integration scheme is developed by using a uniform linear time approximation of the displacement and velocity fields to yield only a single step time integration scheme, which is comparable to the Newmark family of time integration scheme. A reduced integration technique is used to prevent velocity locking, which is caused by linear approximation of both the displacement and velocity fields. For the verification of the consistency and the stability, the CWR time integration scheme is tested with single-degree as well as multi-degree of freedom problems. The scheme performs extremely well compared with those of the well-known Newmark family of time integration schemes.

Complex structural dynamic problems are normally analyzed by finite element and numerical integration techniques. An explicit time integration algorithm with second-order accuracy and unconditional stability is presented for dynamic analysis. Utilizing weighted factors, the current displacement and velocity relations are defined in terms of the accelerations of two previous time steps. The concept of discrete transfer function and the pole mapping rule from the control theory are exploited to develop the new algorithm. Several linear and nonlinear dynamic analyses are performed to verify the efficiency of the method compared with the well-known Newmark method.

This paper introduces a mathematical model for optimizing the dynamic performance of thin-walled functionally graded box beams with closed cross-sections. The objective function is to maximize the natural frequencies and place them at their target values to avoid the occurrence of large amplitudes of vibration. The variables considered include fiber volume fraction, fiber orientation angle and ply thickness distributions. Various power-law expressions describing the distribution of the fiber volume fraction have been implemented, where the power exponent was taken as the main optimization variable. The mass of the beam is kept equal to that of a known reference beam. Side constraints are also imposed on the design variables in order to avoid having unacceptable optimal solutions. The mathematical formulation is carried out in dimensionless quantities, enabling the generalization to include models with different cross-sectional types and beam configurations. The optimization problem is solved by invoking the MatLab optimization ToolBox routines, along with structural dynamic analysis and eigenvalue calculation routines. A case study on the optimization of a cantilevered, single-cell spar beam made of carbon/epoxy composite is considered. The results for the basic case of uncoupled bending motion are given. Conspicuous design charts are developed, showing the optimum design trends for the mathematical models implemented in the study. It is concluded that the natural frequencies, even though expressed in implicit functions, are well-behaved, monotonic and can be treated as explicit functions in the design variables. Finally, the developed models can be suitably used in the global optimization of typical composite, functionally graded, thin-walled beam structures.

This paper presents a new family of explicit time integration algorithms with controllable numerical dissipation for structural dynamic problems by utilizing the discrete control theory. Firstly, the equilibrium equation of the implicit Yu-$\alpha$ algorithm is adopted, and the recursive formulas of velocity and displacement for the explicit CR algorithm are used in the algorithms. Then, the transfer function and characteristic equation of the algorithms with integration coefficients are obtained by the $Z$ transformation. Furthermore, their integration coefficients are derived according to the poles condition. It was indicated that the proposed algorithms possess the advantages of second-order accuracy, self-starting, and unconditional stability for linear systems and nonlinear systems with softening stiffness. The numerical dissipation of the algorithms is controlled by the spectral radius at infinity ${\rho }_{\infty }$. It was also shown that the proposed algorithms have the same poles as the Yu-$\alpha$ algorithm, and thus the same numerical properties. Compared with the implicit Yu-$\alpha$ algorithm, the proposed algorithms are explicit in terms of both the displacement and velocity formulas. Finally, the effectiveness of the proposed algorithms in reducing the undesired participation of higher modes for solving the dynamic responses of linear and nonlinear systems has been demonstrated.

In the present paper, a practical superposition method is proposed for complex load-dependent Ritz (CLDR) vectors for use in the dynamic analysis of nonclassically damped systems. In particular, an algorithm for CLDR vector generation is developed and the CLDR vectors are calculated in the physical space, instead of the state space, to reduce the computational effort and storage space, while improving the stability of the algorithm. Moreover, single CLDR vector (i.e. using only one starting vector) and block CLDR vector (i.e. using multi-starting vectors) generation procedures are introduced for the uni and multidirectional loading patterns respectively, and the latter is applied to the system with repeated natural frequencies. In addition, a criterion, which is based on the spatial load distribution, is proposed to determine a proper number of the CLDR vectors prior to their use in the dynamic analysis. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed method. Also, the performance of the cut-off criterion is presented and 10% error or less in the participation loading distribution is recommended for practical applications.

In this paper, a number of recently proposed implicit and explicit composite time integration schemes are reviewed and critically evaluated. To give suitable guidelines of using them in practical transient analyses of structural problems, numerical performances of these schemes are compared through illustrative examples. Meaningful insights into computational aspects of the composite schemes are also provided. In the discussion, the role of the splitting ratio of the recent composite schemes is also investigated through a different point of view, and similarities and differences of various composite schemes are also studied. It is shown that the explicit composite scheme proposed recently by the authors can noticeably increase the efficiency and the accuracy of linear and nonlinear transient analyses when compared with other well-known composite schemes.

Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the $s$-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

Shape memory alloy (SMA) dampers are widely investigated passive control systems for structural vibration mitigation. However, the damping robustness of conventional austenite SMA dampers may be affected by environmental temperature. In this study, an innovative double SMA damper (DSD) system is presented to improve the temperature robustness of the SMA dampers. In the proposed system, double SMA hysteretic elements with different phase transition temperatures are arranged in parallel, where the SMA element with lower transition temperature behaves as austenite under room temperature, and the other with higher transition temperature behaves as martensite. To study the vibration control effect, both single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) structures with DSD systems are employed. The thermal and mechanical behaviors of the SMA elements and the working principle of DSD are also introduced. Thereon, the equivalent linearization method for SMA’s output force and the motion-governing equations for SDOF structure with DSD are derived. Moreover, parametric studies are conducted to investigate the performance of the proposed DSD system in both frequency and time domains. Also, numerical analysis for the MDOF structure with DSD systems is carried out to illustrate the trend in response reduction with an increasing number of degrees of freedom. The analytical results show that the DSD can mitigate the structural seismic response more effectively than the conventional one with acceptable residual deformation, and is capable of delaying the degradation of SMA’s energy dissipation capacity. Less SMA material is required for the proposed DSD to fulfill the same mitigation requirement, and it is suitable for general applications for temperature robustness.

Adaptive negative stiffness device is one of the promising seismic protection devices since it can generate seismic isolation effect through negative stiffness when it is mostly needed and achieve similar vibration mitigation as a semi-active control device. However, the adaptive negative stiffness device generally combined with linear viscous damping underpins the drawback of degrading the vibration isolation effect during the high-frequency region. In this paper, a modified adaptive negative stiffness device (MANSD) with the ability to provide both lateral negative stiffness and nonlinear damping by configuring linear springs and linear viscous dampers is proposed to address the above issue. The negative stiffness and nonlinear damping are realised through a linkage mechanism. The fundamentals and dynamic characteristics of a SDOF system with such a device are analyzed and formulated using the Harmonic Balance Method, with a special focus on the amplitude–frequency response and transmissibility of the system. The system with damping nonlinearity as a function of displacement and velocity has been proven to have attractive advantages over linear damping in reducing the transmissibility in the resonance region without increasing that in the high-frequency region. The effect of nonlinear damping on suppressing displacement and acceleration responses is numerically verified under different sinusoidal excitations and earthquakes with different intensities. Compared with linear damping, the MANSD with nonlinear damping could achieve additional reductions on displacement and acceleration under scaled earthquakes, especially intensive earthquakes.

Digital twin aims to create a virtual model for a physical structure by combining measurement data in structural health monitoring. The most important feature is to achieve the physical structure-monitoring data synchronization. For this purpose, a physics-data hybrid framework to develop the bridge digital twin model in structural health monitoring is proposed in the paper. The physical base is firstly formed by the finite element model of the digital representation for the physical bridge that can fully incorporate both structural geometry and structural state. The data base is then built by all measurement data of the monitored bridge. By defining the context that is common to both physical base and data base, the mirror relationship between physical base and data base for the specified context is formulated. To achieve the best matching of the mirror relationship by minimizing process, the digital twin model in terms of the specified context can be developed. In such a way, the proposed framework integrates physical knowledge and data intelligence into one model. A demonstration of a simulated simply supported beam is provided to show how the digital twin model is developed by using proposed physics-data hybrid framework. It is found that the generated digital twin model is consistent with the current structural state of the beam. The presented physics-data hybrid framework helps in clearer understanding of the realization of digital twin model in structural health monitoring, providing a new perspective for smart bridge solutions.

The current practice of real-time hybrid simulation (RTHS) often requires specialized finite element programs for computational modeling of the analytical substructures. Considering the limited nonlinear modeling capacity or the increasing computation cost for complex modeling, surrogate models of the analytical substructure provide novel alternatives for RTHS to avoid finite element analysis with fast computation. This study explores the use of arbitrary polynomial chaos expansion (APC) and nonlinear autoregressive with exogenous input (NARX) model to emulate the dynamic behavior of analytical substructures in RTHS. The NARX model training can be conducted numerically in an off-line mode using existing general purpose finite element analysis software, and its implementation presents minimum computational demands on the RTHS equipment. RTHS of a single-degree-of-freedom structure with a self-centering viscous damper is conducted as proof of concept to experimentally demonstrate the effectiveness and superiority of the proposed APC-NARX-based approach. The APC is further compared with other metamodeling techniques including polynomial chaos expansion (PCE) and Kriging to surrogate NARX model coefficients to account for ground motion uncertainties in RTHS. It is demonstrated that APC-NARX modeling with optimal order enables better accuracy of RTHS results than those of Kriging- and PCE-NARX.

A family of structure-dependent methods is proposed based on discrete control theory. Although the displacement and velocity expression of this family method are similar to those of the previously published method developed by Mohammad Rezaiee-Pajand, the structure-dependent parameters of this family are different from the previously published method. The family of structure-dependent methods is named the MUSE algorithm method. Based on discrete control theory, a new family of integration algorithms is proposed by using the poles of the Newmark-$\beta$ method. Theoretical analysis indicated that the MUSE algorithm method possesses properties of zero amplitude decay and is self-starting. Also, its Period Elongation can be reduced by parameter ‘$s$’. Numerical examples show that parameter ‘$s$’ introduced in this paper can improve control Period Elongation and improve the accuracy of this method.

In this study, an explicit time integration scheme based on cubic B-spline interpolation strategy and momentum corrector is extended for structural seismic response analysis. The acquisition strategy for the required seismic load is provided. The cubic B-spline-based scheme with moment corrector is generalized for the nonlinear hysteresis system. The analysis shows that the explicit scheme has more desirable stability and accuracy properties than other competitive schemes. The effectiveness of the scheme is demonstrated by the linear and nonlinear numerical examples.