International Journal of Structural Stability and Dynamics

This paper presents the governing equation of motion, boundary conditions and exact vibration frequencies of a segmented rod where the segments are connected by hinges with elastic rotational springs of constant stiffness. The mass of each segment is assumed to be evenly distributed along the length of the rod. Another discrete model called Hencky bar-chain model (short for HBM; which is equivalent to the finite difference model for discretizing continuous rod) assumes the rod mass to be lumped at the ends instead and a different set of boundary conditions are adopted clamped end. The vibration results of a clamped–clamped segment rod are compared with those of the HBM. It is shown that the HBM underestimates the vibration frequencies when compared to the segmented rod model for a finite number of segments while both models furnish vibration solutions that converge to the solutions of Euler beam for infinitely large number of segments.

Precision guided projectiles (PGPs) experience severe shock loads during launch emanating from the propellant gases inside the barrel and the surrounding air. The complex flow environment that exists within the confined space of the barrel and at muzzle exit is greatly influenced by the supersonic speed of the projectile, the compressibility of the air, and the rapid state transition of the projectile from the confined volume of the barrel to the surrounding free-space. In our earlier efforts (X. W. Yin, P. Verberne and S. A. Meguid, Multiphysics modelling of the coupled behaviour of precision-guided projectiles subjected to intense shock loads, Int. J. Mech. Mater. Des. 10 (2014) 439–450; P. Verberne and S. A. Meguid, The coupled behaviour of precision-guided projectiles subject to propellant induced shock loads using multiphysics analysis, in 8th Int. Conf. Mech. Mater. Des. (2019); P. Verberne and S. A. Meguid, Dynamics of precision guided projectile launch: Fluid-structure interaction, Acta Mech. (2020)) examined the fluid–solid interaction problem. In this paper, we expand our earlier effort by examining the underlying mechanisms associated with the solid–solid interaction between the projectile and the barrel walls that severely govern the survivability of the embedded electronic systems (EES). This was achieved by conducting comprehensive finite element (FE) simulations of the dynamics of the entire launch process of a projectile accounting for the intense combustion pressures of the propellant, the large accelerations experienced during the launch and the induced shock waves. Our FE simulations successfully capture the interaction of the projectile with the barrel. Our work reveals that frictional forces due to contact inside the barrel significantly affect the projectile’s acceleration response at muzzle exit. Immediately following muzzle exit, the rapid reduction of the frictional forces inside the barrel results in a rapid increase of the projectile acceleration followed by a rapid reduction due to the free expansion of the propellant gases and air drag, leading to large acceleration fluctuations.

In this research, a new analytical solution is used for finding the buckling loads of rectangular plates with vertically and rotationally restrained edges. The solution method in this study is based on the development of a static solution for a plate. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution fits all the combinations of possible boundary conditions, of the deflection, slope, shear force and bending moment along the edges of the plate. In the case of springs, the edge force and moment boundary conditions are modified to include these effects. Any number of edges, from one to four, with both types of stiffening springs can be solved. Using this new method, the exact buckling loads and modes are found. The results are verified with published data, and many new cases are presented for uni-axially and bi-axially loaded isotropic, orthotropic, and composite plates.

In this study, a new, effective procedure is proposed for identifying the surface roughness from the responses recorded of two connected test vehicles moving over the bridge. Central to this study is the proposal of a simple static correlation formula for relating the dynamic deflections of the two vehicles’s contact points on the bridge, via the displacement influence lines (DILs). With the aid of this relation, the roughness formula for estimating the bridge surface profile is derived using the responses of the leading and following vehicles. It does not require any prior knowledge of the dynamic properties of the bridge. The efficacy of the proposed procedure is validated for both the simple and three-span continuous beams by the finite element method (FEM). Also, a parametric study is conducted for various physical properties of the test vehicles. It is confirmed that the roughness profiles back-calculated from the proposed formula agree excellently with the assumed ones for both the simple and continuous beams. For use in practice, the two connected test vehicles should not be designed too heavy and not to move at too fast speeds, in order to reduce the impact on the bridge.

This paper is concerned with the dynamic stability and response of an inclined Euler–Bernoulli beam under a moving mass and a moving follower force. The extended Hamilton’s principle is used to derive the governing equation of motion and the boundary conditions for this general moving load/force problem. Considering a simply supported beam, one can solve the problem analytically by approximating the spatial part of the deflection with a Fourier sine series. Based on the formulation and method of solution, sample dynamic responses are determined for a beam that is inclined at 30${}^{\circ }$ with respect to the horizontal. It is shown that the dynamic response of the beam under a moving mass is rather different from an equivalent moving follower force. Also investigated herein are the dynamic stability of inclined beams under moving load/follower force which are described by four key variables, viz. the speed of the moving mass/follower force, concentrated mass to the beam distributed mass, vibration frequency and the magnitude of the moving mass/follower force. The critical axial load and the critical follower force are different when they are located at different positions in the beam; except for the special case when they are at the end of the beam.

This paper presents a numerical investigation on the nonlinear dynamic response of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) beam with open edge cracks in thermal environment. It is assumed that graphene platelets (GPLs) in each GPLRC layer are uniformly distributed and randomly oriented with its concentration varying layer-wise along the thickness direction. The effective material properties of each GPLRC layer are predicted by Halpin-Tsai micromechanics-based model. Finite element method is employed to calculate the dynamic response of the cracked FG-GPLRC beam. It is found that the maximum dynamic deformation of the cracked FG-GPLRC beam under dynamic loading is quite sensitive to the crack location and grows with an increase in the crack depth ratio (CDR) and temperature rise. The influences of GPL distribution, concentration, geometry as well as the boundary conditions on the dynamic response characteristics of cracked FG-X-GPLRC beams are also investigated comprehensively.

In this paper, free vibrations of functionally graded (FG) graphene-reinforced composite blades with varying cross-sections are investigated. Considering the cantilever boundary conditions, the dynamic model of a rotating blade is simplified as a varying cross-sections plate with pre-installed angle and pre-twisted angle. As a reinforcement, the graphene platelets (GPLs) are distributed either uniformly or gradiently on the plate along its thickness direction. The effective Young’s modulus is formulated by the modified Halpin–Tsai model. The rule of mixture is applied to calculate the effective Poisson’s ratio and mass density. The equations of motion are established by using the first-order shear deformation theory and von Karman geometric nonlinear theory. Based on the Rayleigh–Ritz method, the natural frequencies of the rotating FG blade reinforced with the GPLs are obtained. The accuracy of the present method is verified by comparing the obtained results with those of the finite element method and published literature. A comprehensive parametric study is conducted, with a particular focus on the effects of distribution pattern, weight fraction, and geometries size of the GPLs together with dimensional parameters of varying cross-sections blade on the dynamics of the FG blades reinforced with the GPLs.

Carbon nanotube reinforced composite (CNTRC) is one of the novel classes of advanced composite materials. This paper investigates the nonlinear bending and nonlinear vibration responses of hybrid laminated beams made of CNTRC layers bonded with metal layers on the top and bottom surfaces. We proposed a hybrid metal/CNTRC laminated beam for which out-of-plane Poisson’s ratio is negative. The effective material properties of CNTRC layers are graded in a piece-wise pattern across the thickness of the beam. The material properties of both CNTRC layer and metal layer are temperature dependent. The beams are rested on an elastic foundation and are located in thermal environments. Reddy’s higher-order beam model is used to establish the motion equations of the hybrid metal/CNTRC laminated beam. The motion equations include the von Kármán geometric nonlinearity, the thermal effects and the beam-foundation interaction. By employing a two-step perturbation approach, the bending and nonlinear vibration solutions are obtained. A comparison of nonlinear responses of hybrid metal/CNTRCl laminated beams with positive and negative Poisson’s ratios (NPRs) under different thermal environmental conditions is carried out and discussed in detail.

The performance of selected equivalent single-layer (ESL) models is evaluated within several classical benchmark tests for small amplitude free vibration analysis of multi-layered plates. The authors elaborated their own Finite Element software based on the first-order shear deformation (FOSD) theory with some modifications incorporated including a correction of the transverse shear stiffness and an application of zigzag type functions. Seven different ESL models were considered in the study; beside the classical FOSD model, there were three FOSD models with various transverse shear corrections and three ESL models enhanced by the application of zigzag functions and based on Reissner’s Mixed Variational Theorem.

This paper addresses the design problem of piezoelectric actuators for multimodal active vibration control. The design process is carried out by a topology optimization procedure which aims at maximizing a control performance index written in terms of the controllability Gramian, which is a measure that describes the ability of the actuator to move the structure from an initial condition to a desired final state in a finite time interval. The main work contribution is that independent sets of design variables are associated with each modal controllability index, then the multi-objective problem can be split into independent single-objective problems. Thus, no weighting factors are required to be tuned to give each vibration mode a suitable relevance in the optimization problem. A material interpolation scheme based on the Solid Isotropic Material with Penalization (SIMP) and the Piezoelectric Material with Penalization (PEMAP) models is employed to consider the different sets of design variables and the sensitivity analysis is carried out analytically. Numerical examples are presented by considering the design and vibration control for a cantilever beam and a beam fixed at both ends to show the efficacy of the proposed formulation. The control performance of the optimized actuators is analyzed using a Linear-Quadratic Regulator (LQR) simulation.

This paper deals with the evaluation of time response analyses of typical aerospace metallic structures. Attention is focussed on detailed stress state distributions over time by using the Carrera Unified Formulation (CUF) for modeling thin-walled reinforced shell structures. In detail, the already established component-wise (CW) approach is extended to dynamic time response by mode superposition and Newmark direct integration scheme. CW is a CUF-based modeling technique which allows to model multi-component structures by using the same refined finite element for each structural component, e.g. stringers, panels, ribs. Component coupling is realized by imposing displacement continuity without the need of mathematical artifices in the CW approach, so the stress state is consistent in the entire structural domain. The numerical results discussed include thin-walled open and closed section beams, wing boxes and a benchmark wing subjected to gust loading. They show that the proposed modeling technique is effective. In particular, as CW provides reach modal bases, mode superposition can be significantly efficient, even in the case of complex stress states.

In this paper, we discuss the following inverse problem: how to reconstruct the mass distribution function of a two-span beam with an overhang via its polynomial fundamental mode and polynomial stiffness function. This leads to a basic equations group constituted by the coefficients of the mass distribution and the stiffness distribution function. To make the basic equations group match, one way is to divide it into two sub- equations, and solve them. We specify the method for solving this inverse problem, and research the existence and rationality of the positive solutions.

This paper presents a thermodynamically consistent and kinematic assumption free formulation for dynamics of thermoviscoelastic plates/shells based on the conservation and balance laws of classical continuum mechanics (CCM) in which dissipation mechanism has been incorporated through ordered rate constitutive theory for deviatoric stress tensor. In this paper, we consider small deformation, small strain. The conservation and balance laws of CCM in ${ℝ}^3$ in Lagrangian description using Cauchy stress tensor ($\sigma$) and linearized Green strain tensor ($𝜖$) constitute the mathematical model. The constitutive theory for the deviatoric Cauchy stress ($d^{\sigma }$) is derived using conjugate pairs in entropy inequality in conjunction with representation theorem. The argument tensors of $d^{\sigma }$ are $𝜖$ and rates of $𝜖$ up to order $n$. This yields a constitutive theory with dissipation mechanism based on rates of strain up to order $n$. Constitutive theory for heat vector is also derived using the conjugate pairs in the entropy inequality and representation theorem. Finite element method is used to obtain solutions of the initial value problems descried by the balance of linear momenta (BLM), energy equation and the constitutive theories.

The shell element geometry is described by the middle surface and the nodal vectors at the middle surface defining bottom and top surfaces of the element. The local approximation for the displacement field is $p$ - version hierarchical in the plane of the element as well as in the transverse direction. A space-time decoupled finite element formulation using Galerkin Method with Weak Form (GM/WF) in space is constructed for BLM as well as energy equation, both resulting in ordinary differential equations (ODEs) in time. The ordinary differential equations (ODEs) in time resulting from the finite element formulation of BLM are used to study: (i) natural undamped modes of vibration (ii) the transient dynamic response using the ODEs in time recast in modal basis: (a) using Rayleigh damping (b) using the ordered rate damping proposed in this paper. Time response is calculated using modal damping based on Rayleigh damping as well as using the proposed ordered rate damping mechanism. Model problem studies are presented to demonstrate: (1) accuracy of the natural frequencies obtained from the present formulation for thin and thick plates/shells (in which shear deformation is significant) and the results are compared with the currently used plate formulations (2) accuracy of damped transient response using proposed damping mechanism is compared with time response using Rayleigh damping (3) it is shown that Rayleigh damping has no physical basis and leads to spurious stationary states. The proposed damping yields accurate stationary states that are in exact agreement with the solution of corresponding BVP. A single formulation presented in this paper remains valid and accurate for very thin as well as very thick plates/shells and correctly simulates 3D state of deformation regardless of plate/shell thickness and is free of shear locking problems as well as need for shear corrections. When obtaining the time response, solution for an increment of time alternates between the solution of BLM followed by the solution of the energy equation. Details are presented in the paper.

We present a stabilized finite element method for thermomechanical problems in the class of materials with discrete microstructural interfaces that undergo interfacial fatigue and dominant interfacial damage. This formulation is applicable to polycrystalline solids, fibrous composites, filled elastomers, and additively manufactured layered materials. A finite strain formulation for monolithically coupled thermomechanical fields is presented where interfacial kinematic models for low-cycle fatigue and for strong interfacial discontinuities are variationally embedded at the interfaces. Formulation is written in the spatial configuration to account for large local strains and finite rotations of the interfaces. The method is implemented employing the family of low-order 3D Lagrange elements comprised of linear hexahedra and linear tetrahedra. A set of benchmark problems is presented to show the mathematical and algorithmic attributes of the method.

This paper presents a comprehensive study and it concludes that the resonance of forest trees with properly aligned conditions precisely working as naturally available locally resonant metamaterials that are equipped with wonderful capability of generating low frequency extremely wide bandgaps in the earthquake frequency range of interest. At the geophysical scale, the propagation of Rayleigh wave in the soft sedimentary soil basin experiences strong wave attenuation when the longitudinal resonant modes of trees are coupled with vertical component of the Rayleigh wave that mimic wave hybridization phenomena. A finite element-based numerical technique is adopted and we considered a total of 10 cases where spacing, height, thickness and mechanical properties of resonant trees are varied to study the Rayleigh wave propagation and attenuation mechanism. The trapping and/or mode conversion of Rayleigh wave by resonant trees is observed as dominant phenomena for wave attenuation. A time history analysis is conducted based on an actual earthquake record to validate the performance and efficiency of the bandgaps. The effects of ground stiffness, resonant tree mechanical and geometric properties on the bandgaps are also discussed. The study explores another peculiar characteristic of the forest trees that controls the propagation of seismic wave to protect a region from earthquake hazards. Our study may motivate the relevant organizations, authorities and global communities on the needs of forestation to reduce the earthquake catastrophe.

A plasticity-based approach has been proposed for reinforced concrete (RC) subjected to impact and blast loading. The proposed approach includes a new failure surface together with modified descriptions related to equation of state, strain rate effect and damage taken from literature. These selective descriptions in numerical analysis have been found to give better results than the existing models. A user-defined UMAT subroutine for the commercial software LS-DYNA has been developed for the proposed model. Dynamic analysis for both blast and impact type of loading on RC slabs are performed. The results are compared with an in-built material model in commercial software LS-DYNA, namely the Winfrith concrete model (WCM) and with experimental results taken from literature. A parametric study is carried out by analyzing the mechanical and damage response by varying parameters such as concrete strength, slab thickness and reinforcement ratio. The proposed approach is found to be effective in predicting the behavior under both impact and blast analysis.