Bolotin's Method Research Articles

Static and dynamic buckling analysis of two-layer shells include auxetic honeycomb core combined with laminated three-phase polymer/GNP/fiber surface resting on Kerr elastic medium in hygro-thermal environments

This article investigates the static and dynamic buckling behavior of a double-curved sandwich shell composed of two layers via improved first-order shear deformation theory without using the shear correction is utilized to model the sandwich shell. The sandwich two-layer shell are held together by elastic pins, each layer made from an ultra-light auxetic honeycomb core layer (negative Poisson’s ratio) and is reinforced by a layer of tri-phase polymer, graphene nanoplatelets (GNP), and fiber. The double-curved sandwich shell is supported by a Kerr elastic medium in hygro-thermal environment assuming that temperature and humidity only cause forces to be applied tangentially to the shell and do not change the mechanical properties of the shell material. The motion equations of sandwich two-layer shell are derived by Hamilton’s principle, then the finite element method and Bolotin method are used to derive the dynamic instability region of sandwich two-layer shell. The accuracy of these results is confirmed by a comparison with established statements within the specific conditions of the problem model. The influence of inputs on the static and dynamic stability of sandwich two-layer shell is fully explored and discussed. Additionally, this serves as a crucial foundation for the calculation and design of structures that can integrate a variety of materials with different properties with low cost and simplification.

Harmonic in-plane patch loading induced nonlinear dynamic instability of unidirectional and bidirectional stiffened composite plates

A finite element framework is established to study the nonlinear dynamic instability of stiffened plates subjected to in-plane harmonic patch loading. An eight-noded isoparametric degenerated shell element and a three-noded curved beam element are used to model the skin and the stiffener, respectively. The Green–Lagrange strain displacement relationship is adopted to formulate the system matrices for the plate and the stiffener, adopting the total Lagrangian approach. The Bolotin method is adopted to trace the boundaries of linear dynamic instability region. The Incremental Harmonic Balance (IHB) method is used to investigate the nonlinear instability behavior of stiffened plates. The effect of varying loading patches along with the number of stiffeners in unidirectional ( x-directional and y-directional) and bidirectional ( x as well as y-directional) stiffened plates on the nonlinear frequency response is studied. It is observed that the cross-stiffened plates with the higher number of stiffeners are dynamically more stable than the unidirectional stiffened plates. Moreover, the stiffened plates with smaller loading patches are dynamically more stable than those with larger loading patches.

Coupled dynamic instability of graphene platelet-reinforced dielectric porous arches under electromechanical loading

Graphene platelet-reinforced dielectric porous (GPLRDP) arches are a kind of multifunctional structures, which can be used to design various dielectric resonators and soft robotic components. This work studies the coupled dynamic instability of GPLRDP arches under electromechanical loading. It is assumed that the presence of voids is uniformly distributed in the arch structure, while graphene platelets (GPLs) are uniformly dispersed inside the skeleton of such voids. The effective medium theory (EMT) is used to determine the effective dielectric permittivity and Young's modulus of such structures. Based on the Hamilton principle, the governing equations of this problem are formulated. The differential quadrature method (DQM) and the Bolotin method are then used for solving the arch system to identify the coupled dynamic instability region. In this study, we observe the coupling effect of parametric and forced resonance phenomena. A series of numerical experiments are also carried out to examine the influence of porosities, GPL weight fractions, boundary conditions, and electrical voltage levels on the critical excitation frequency and coupled dynamic instability behavior of GPLRDP arches. It is found that the coupled dynamic instability region becomes wider and shifts to low frequency as the effect of porosity increases. When the signal amplitude of AC frequency is within a certain range, there is an abrupt increase of the critical excitation frequency under a high level of DC voltage. The numerical results of this work demonstrate that the location and size of the coupled dynamic instability region can be actively controlled by adjusting the material and structural parameters.

Fractional Mathieu Differential Equations in Dynamic Stability of Piles

Piles are commonly used in Civil and Mechanical Engineering to provide support to superstructures and to transmit loads to deeper ground layers. During earthquakes, pile instability can result in the collapse of the entire structure. In this paper, the pile is modeled as a column surrounded by Winkler soil foundations with fractional damping. Investigation of the axially loaded pile leads to a fractional Mathieu differential equation of motion. The Bolotin method, employing harmonic balance, is proposed to obtain the approximate instability boundaries of the pile in the stability diagrams. A practical example is presented to conduct a parametric study on pile instability concerning the fractional order. The study observes parametric resonance in the first order, representing the principal instability region, which requires special attention to maintain stability. Furthermore, increasing the fractional damping order leads to a higher critical dynamic load and a slight reduction in the critical frequency ratio in each instability region. Higher-order instability regions exhibit greater sensitivity to changes in the fractional order. These results provided insights into the stability of piles under earthquake conditions.

Dynamic instability characteristics of graphene‐reinforced cellular sandwich plates using a three‐variable isogeometric approach

AbstractThe primary objective of the current study is to introduce a potent computational model for analyzing the dynamic instability response of advanced sandwich plate models subjected to various time‐dependent in‐plane edge loadings. To achieve this, we employ the finite element formulations derived from high‐order shear deformation plate theory (HSDT) involving only three unknowns in conjunction with NURBS‐based isogeometric analysis. Additionally, advanced sandwich plate models are constituted by a middle layer with either a uniform or symmetric porosity distribution and two outer layers which are reinforced with graphene nanoplatelets (GNPs) to boost dynamic structural performance. The fundamental dynamic instability region of the sandwich plates can be then obtained by solving the Mathieu‐Hill equation using the Bolotin's method. Numerous numerical studies are performed to evaluate the impacts of numerous input parameters on the dynamic stability behavior of the cellular three‐layer plate structures. The crucial role of internal porosity distribution as well as GNPs reinforcement phase in enhancing the instability performance of the cellular structures is thoroughly evaluated. With the current results, this research provides important insights into the development and application of cellular materials as well as the GNPs for designing advanced engineering structures in the future.

Nonlinear dynamic instability and dynamic response of stiffened laminated composite plates subjected to in-plane pulsating patch loading

In this article, the nonlinear dynamic instability of stiffened laminated composite plates is studied in the finite element (FE) framework subjected to uniform in-plane harmonic patch loading. The harmonic load is applied to the two opposite sides of the stiffened plate. The linear and nonlinear time-history response analysis is also studied. The skin and the stiffener are modeled using an eight-node isoparametric degenerated shell element and a three-node curved beam element, respectively. A system of matrices is developed by considering the Green–Langrange strain–displacement relationship. In the linear case, the Bolotin method is used to analyze the dynamic instability region (DIR). The nonlinear instability behavior of the laminated composite stiffened plate is studied by applying the Incremental Harmonic Balance Method (IHB). The Newmark-β method is used to solve the linear and nonlinear time-history response equations to understand the instability behavior of the stiffened plates. The effect of the parameters such as the length of the in-plane loading patch, varying number of stiffeners in x -direction and the position of the patch on the nonlinear vibrations and nonlinear dynamic response is examined.

Nonlinear dynamic stability analysis of CNTs reinforced piezoelectric viscoelastic composite nano/micro plate under multiple physical fields resting on smart foundation

This study analyzes the nonlinear dynamic stability of carbon nanotube reinforced composite (CNTRC) piezoelectric viscoelastic nano/micro plate under time dependent harmonic compressive biaxial mechanical loading. The nano/micro plate is single-layer with a uniform and rectangular cross section. Polyvinylidene fluoride (PVDF) is the material used for the nano/micro plate. Also, it is located on a nonlinear viscoelastic piezoelectric foundation made of Zinc Oxide (ZnO). Single-walled carbon nanotubes (SWCNTs) have been used to strengthen nano/micro plate. Using Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Nano/micro plate is subjected to 2D magnetic field and electric potential. Magnetic field effects are incorporated using Maxwell’s relations. An alternating and direct voltage is applied to the nano/micro plate and also smart foundation in thickness direction. Eshelby-Mori-Tanaka approach is used to estimate the effective elastic properties. Additionally, follower force and uniform thermal gradient is applied to nano/micro plate in this study. The nonlinear strain–displacement relations are based on the Von-Kármán theory. According to classical, first order, third order, sinusoidal, parabolic, hyperbolic and exponential shear deformation plate theories, a new formulation is proposed incorporating surface stress effects through the Gurtin-Murdoch elasticity theory. In order to consider small scale parameters, this research is based on modified strain gradient theory (MSGT). Using the energy method and Hamilton’s principle, the non classical governing equations are derived. For nano/micro plate, simply supported boundary conditions are considered. For the purpose of solving differential equations, an analysis based on Galerkin procedure and finally the Bolotin method will be used to obtain the dynamic instability region (DIR). It is the objective of this study to investigate the impact of such parameters as small-scale parameters, alternating and direct applied voltage, intensity of magnetic field, surface effects, thermal environment and static load factor on the DIR. This study revealed that taking into account the smart foundation reduces the area of dynamic instability region by more than 60% for a constant intensity of magnetic field. The stability responses are also more convergent with the smart foundation. Additionally, the range is moved toward higher excitation frequencies and greater system stability. Validation of answers based on reliable scientific articles is one of the final stages of this research. These results can be used to design nano/micro electromechanical systems.

A refined quasi-3D isogeometric model for dynamic instability of graphene nanoplatelets-reinforced porous sandwich plates

This study pays attention to the dynamic instability behavior of advanced sandwich plate structures subjected to periodic in-plane compressive loads. By utilizing a powerful and efficient computational approach based on a four-variable refined quasi-3D plate theory and NURBS-based isogeometric analysis (IGA), the dynamic behavior of structures can be accurately evaluated while accounting adequately for the thickness stretching effect of normal deformation. To investigate the dynamic instability behavior, this study examines sandwich plate models that consist of two graphene nanoplatelets (GNPs)-reinforced skin layers and a porous core layer with two typical distributions of internal porosity: uniform and symmetric dispersion. Bolotin's method is applied in this study to solve the Mathieu–Hill equation and obtain the boundary of dynamic instability zones. Numerical examples are performed to examine the influence of various input parameters on the instability regions, as well as to provide new insights into the dynamic instability behavior of advanced sandwich plates under compressive loading which has not been comprehensively studied in the literature. The findings of this study suggest that the thickness stretching effect should be carefully evaluated for moderate to thick plate structures, such as sandwich plates.

Assessment of dynamic instability of thin nanoplates considering size and surface energy effects

The stated goal of this study is to comprehensively understand the dynamic instability characteristics of thin nanoplate structures, taking into account the combined effects of both size and surface energy. To achieve this, we present a computational approach based on the non-classical Kirchhoff plate theory within the framework of NURBS-based isogeometric analysis (IGA). The modified couple stress theory, which includes one material length scale parameter, is implemented to adequately account for the size-dependent effect. Additionally, the Gurtin–Murdoch surface elasticity is effectively utilized to assess the influence of surface energy. For the first time, we explore the dynamic instability regions of thin nanoplates, accounting for both size and surface energy effects simultaneously. To obtain solutions from the Mathieu-Hill equations, the Bolotin method is applied in this research. A number of numerical investigations on square, circular and annular thin nanoplate models under periodic in-plane compressive loads are conducted to assess the influence of several crucial input factors on the regions of dynamic instability. The present outcomes reveal that both size and surface energy effects have a remarkable influence on the dynamic instability characteristics of the nanoplates. Accordingly, these effects contribute to an enhancement in structural rigidity at the small-scale level. Furthermore, we explore that in the case of ultra-thin nanoplates, the influence of surface energy becomes dominant. The obtained findings provide valuable insights into the behavior of thin nanoplates under dynamic in-plane loading conditions as well as can contribute to the design of more robust engineering structures.

Static and dynamic instability of functionally graded graphene origami-enabled auxetic metamaterial beams with variable thickness in fluid

Functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) structures have shown great potential for various engineering applications due to their exceptional mechanical and physical properties such as high strength-to-weight ratio, tuneable stiffness and strength, and negative Poisson's ratio (NPR). This paper aims to investigate the buckling and dynamic instability behaviours of FG-GOEAM beams with variable thickness immersed in a fluid, with a particular focus on the influence of NPR. The material properties such as Young's modulus and Poisson's ratio of the GOEAM are determined by using a genetic programming (GP)-based micromechanics model. Within the framework of the first-order shear deformation theory and by employing Hamilton's principle and modelling the fluid effect as added mass, the governing equations of motion are established and are then discretised by means of differential quadrature (DQ) method to obtain a linear system of Mathieu-Hill equations from which the principal instability regions of the FG-GOEAM beam are determined by Bolotin's method. A comprehensive parametric study is conducted to reveal the effects of GOri's folding degree, distribution, weight fraction, as well as fluid density and beam dimensions on the static and dynamic instability behaviours of the FG-GOEAM beam immersed in fluid. Numerical results show that the FG-GOEAM beam with NPR considerably outperforms its pristine metallic counterpart in terms of resistance against static buckling and dynamic stability.

Nonlinear dynamic instability of laminated composite stiffened plates subjected to in-plane pulsating loading

A nonlinear finite element dynamic instability analysis of laminated composite stiffened plates subjected to in-plane harmonic edge loading is presented in this article along with the linear and nonlinear dynamic response study. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the stiffened plates. Bolotin method is applied to analyze the dynamic instability regions in linear case. Incremental Harmonic Balance (IHB) method is applied to solve the nonlinear frequency response equations and Newmark-β method is used to solve the linear and nonlinear time history response equations.

Static and Dynamic Stability Analyses of Functionally Graded Beam with Inclined Cracks

The focus of this paper is to examine the static and dynamic instabilities of functionally graded beam that contains multiple inclined cracks under the influence of an axial force comprising both static and time-varying harmonic components. The elasticity modulus and mass density of the functionally graded beam are assumed to vary exponentially along its thickness direction. Local stiffness matrix model-based finite element analysis (FEA) is conducted to determine the bending stiffness and tensile stiffness of the section with a crack, and the coupled effect of tensile and bending loadings. Two-node beam elements with three degrees-of-freedom per node are utilized. By combining the Euler–Bernoulli beam theory with Lagrange method, we derive the governing equations that describe the static and dynamic instabilities of a functionally graded beam with multiple inclined cracks. These equations can be solved as eigenvalue problems to obtain the natural frequency and static critical buckling load of the beam. Furthermore, to investigate the dynamic instability of the system, we use the Bolotin method to determine the boundary between the regions of instability and stability based on the same governing equations. By adopting this approach, the study comprehensively investigates the impacts of crack position, inclination angle, and length, as well as elasticity modulus ratio, static and dynamic load factors on both static and dynamic stabilities of a cracked functionally graded beam to gain valuable insights into the stability and performance of cracked functionally graded structures.

The impact of periodic axial loads on nonlinear dynamic instability behavior of Inconel 625 pipes

In various engineering fields like aerospace and aircraft structures or marine and offshore platforms, constitutive material of critical components should be made of specific materials that can work properly in the required workspace. Such materials must have excellent properties such as high mechanical strength as well as great resistance to corrosion, oxidation, and creep. Inconel 625 is a superalloy that is chosen as constitutive material of critical components due to its great abilities. On the other hand, since investigating Inconel 625 pipe has not been done yet, different mechanical characteristics of using structures made of Inconel 625 should be assessed. Additionally, doing so would be necessary to gather information for current industrial affairs and also future investigations. Therefore, the nonlinear dynamic instability response of axially loaded Inconel 625 pipes is investigated in the current article. The pipe structure is modeled via the Donnell shell theory and nonlinear von Kármán theory. The motion equations of pipes are established by applying the Hamiltonian approach. Then, in order to alter the nonlinear derived partial differential equations into the Mathieu-Hill equation, both Navier's solution and Airy stress function are implemented. Additionally, the amplitudes of steady-state oscillation of the Inconel 625 pipe are determined by employing Bolotin's method. Eventually, the impacts of various effective parameters on the nonlinear dynamic behaviors of Inconel 625 pipes are evaluated. The results indicate static and dynamic load factors possess a remarkable effect on instability exciting areas and steady-state vibration amplitudes of pipe. Moreover, the dynamic instability response of the pipe is dependent on the radius-to-thickness and length-to-radius ratios, and also how the ratios are affected depends on the wave number.

Orthotropic plates with dynamic vertical seismic load modeled as multi line

Calculation plate floor concrete, using a static load which is a gravity load consisting of a live load and a dead load, with various of boundary conditions, floor slabs are orthotropic plate, and rarely account for dynamic loads due to vertical seismic loads, with other boundary conditions, such as Clamped, simply supported, ES (Elastic Support), ER (Elastic Restraint), and ESR (Elastic Support and Restraint). Analytical solution based on the Modified Bolotin Method to analyze floor slab under Vertical Peak Ground Acceleration (PGAv), the natural frequency solution based on auxiliary Levy’s type problems. Dynamic vertical seismic loads using multiline, first line at 0 < t < 0.5 is linear equation, second line at 0.05 < t < 0.15 is quadratic equation, third line 0.15 > t < 0.6 is sextic equation, last line, t > 0.6 is linear equation, vertical seismic load with two conditions far fault and near fault, multi-line equation are depending on (PGAv/g). A numerical example is given, for various boundary conditions, and far fault, translational stiffness (kx, ky) and rotational stiffness (cx ,cy), from the results of plate calculations due to dynamic vertical seismic loads with 5 types of edge support, ES (elastic support) is the best result.

Dynamic Instability of Functionally Graded Graphene Platelet-Reinforced Porous Beams on an Elastic Foundation in a Thermal Environment.

Under thermal environment and axial forces, the dynamic instability of functionally graded graphene platelet (GPLs)-reinforced porous beams on an elastic foundation is investigated. Three modes of porosity distributions and GPL patterns are considered. The governing equations are given by the Hamilton principle. On the basis of the differential quadrature method (DQM), the governing equations are changed into Mathieu-Hill equations, and the main unstable regions of the porous composite beams are studied by the Bolotin method. Thermal buckling and thermo-mechanical vibration problems are also studied. The effects of porosity coefficients and GPL weight fraction, dispersion pattern, initial thermal loading, slenderness ratio, geometry and size, boundary conditions, and foundation stiffness are discussed. The conclusions show that an elastic foundation has an obvious enhancement effect on thermal buckling, free vibration, and dynamic instability, which improves the stiffness of the beam.

Dynamic Stability Analysis of Periodic Loaded Rotating Conical Shells using Floquet Exponent Method

The dynamic stability/parametric stability characteristics of rotating truncated conical shells under periodic axial loads is investigated. The partial differential equations of rotating conical shells are transformed into Mathieu-Hill ordinary differential equations based on Haar wavelet discretization method. In general, Bolotin method can be used to analyze the stability of parametrically excited systems. This method is a boundary tracing method, and the stability boundary can be easily obtained. However, Bolotin method is not suitable for analyzing the stability of gyroscopic system under parametric excitation, because this method is not satisfied with the hypothesis of the Floquet multiplier for gyroscopic system. In this article, Floquet exponent method, which is widely used in parametric excitation system, is used to analyze the parametric stability of rotating conical shells. The correctness of the proposed model and stability analysis method is verified by comparing with the results of the other literature. The results show that due to the rotational effect, only the combination instability zone exists in the rotating shell, but no principal instability zone is found. When the circumferential wave number is 1 and other wave numbers are compared, it is found that the rotational speed has different effects on the stability under the two circumferential wave numbers. Communicated by Francesco Tornabene.

Instability analysis of fluid-filled angle-ply laminated circular cylindrical shells subjected to harmonic axial loading

This paper presents the dynamic instability and nonlinear vibration analysis of fluid-filled laminated composite circular cylindrical shells subjected to harmonic axial loading. The shells are assumed to be simply supported and filled with still fluid. The mathematical model is prepared by using higher-order shear deformation theory (HSDT) which considers skew modes for accurate results in the analysis of angle-ply laminated composite cylindrical shells. The contained fluid is assumed to be non-viscous and incompressible. The nonlinear governing partial differential equations (PDEs) are obtained using Hamilton's principle, and these equations are discretised into ordinary differential equations (ODEs) by employing Galerkin's method. The pseudo-arclength continuation method along with the incremental harmonic balance (IHB) method is used in order to compute the nonlinear responses, which are plotted in the frequency domain. The regions of instability are determined by adopting Bolotin's method. The time history responses and phase portraits are also studied for empty and fluid-filled laminated cylindrical shells using the Newmark-beta method. A complete set of results that examine the effect of fluid filling, static load factor, dynamic load factor and lamination scheme are obtained.

Dynamic Stability and Responses of Beams on Elastic Foundations Under a Parametric Load

This paper is concerned with numerical simulation of both dynamic stability and responses of beams on elastic foundations under a pulsating axial parametric load in a single matrix method. First, the equation of motion of a beam on an elastic foundation with damping is derived and decoupled into a Mathieu–Hill equation. Three different elastic foundations are considered and compared: Winkler, Pasternak, and Hetenyi models. Then a novel numerical simulation algorithm is proposed to investigate both the dynamic stability and the responses of the beam simultaneously. Accurate instability diagrams are obtained by the numerical simulation and are substantiated by vibration response curves obtained from the same method. These numerically accurate diagrams are used to calibrate the approximate instability boundaries of various orders of Hill infinite determinants from the classical Bolotin method for the first time. A detailed discussion is presented on effects of various aspects including elastic foundation models, damping, and static and dynamic loads. The results provide insights into the efficient and safe application of beams on elastic foundations in engineering. The proposed numerical method can be extended to analyze dynamic stability and vibrations of systems under arbitrary parametric excitations where Mathieu–Hill equations are involved.

In-plane dynamic instability of functionally graded porous arches reinforced by graphene platelet under a vertical base excitation

The in-plane dynamic instability of functionally graded porous (FGP) circular arches made of graphene platelets reinforced composites (GPLRC) under a vertical base excitation is investigated in this paper. The elastic modulus and mass density of the arch are predicated according to Halpin-Tsai micromechanics model and rule of mixture, respectively. The equilibrium differential equations for dynamic stability of the arch are derived based on Hamilton’s principle. Dynamic unstable regions corresponding to period T and period 2T are determined by using Bolotin method. A comprehensive parametric study is then carried out, with a particular focus on the effects of porosity distribution, porosity coefficient, GPL weight fraction and geometry size, and damping ratios on the parametric resonance and the resonance behavior of the FGP-GPLRC arch. The results indicate that the GPL arch with symmetric porosity distribution has the narrowest unstable region in both periods, followed by the asymmetric and uniform distributions. It is also found that a stronger vertical base excitation is required to trigger the dynamic instability for the FGP-GPLRC arch with a larger damping ratio.

Static and Dynamic Stability of Carbon Fiber Reinforced Polymer Cylindrical Shell Subject to Non-Normal Boundary Condition with One Generatrix Clamped

In this paper, static and dynamic stability analyses taking axial excitation into account are presented for a laminated carbon fiber reinforced polymer (CFRP) cylindrical shell under a non-normal boundary condition. The non-normal boundary condition is put forward to signify that both ends of the cylindrical shell are free and one generatrix of the shell is clamped. The partial differential motion governing the equations of the laminated CFRP cylindrical shell with a non-normal boundary condition is derived using the Hamilton principle, nonlinear von-Karman relationships and first-order deformation shell theory. Then, nonlinear, two-freedom, ordinary differential equations on the radial displacement of the cylindrical shell are obtained utilizing Galerkin method. The Newton-Raphson method is applied to numerically solve the equilibrium point. The stability of the equilibrium point is determined by analyzing the eigenvalue of the Jacobian matrix. The solution of the Mathieu equation describes the dynamic unstable behavior of the CFRP laminated cylindrical shells. The unstable regions are determined using the Bolotin method. The influences of the radial line load, the ratio of radius to thickness, the ratio of length to thickness, the number of layers and the temperature field of the laminated CFRP cylindrical shell on static and dynamic stability are investigated.