Timoshenko Model Research Articles

Mechanics of Unidirectional Fiber-Reinforced Composites: Buckling Modes and Failure Under Compression Along Fibers

One-dimensional linearized problems on the possible buckling modes of an internal or peripheral layer of unidirectional multilayer composites with rectilinear fibers under compression in the fiber direction are considered. The investigations are carried out using the known Kirchhoff–Love and Timoshenko models for the layers. The binder, modeled as an elastic foundation, is described by the equations of elasticity theory, which are simplified in accordance to the model of a transversely soft layer and are integrated along the transverse coordinate considering the kinematic coupling relations for a layer and foundation layers. Exact analytical solutions of the problems formulated are found, which are used to calculate a composite made of an HSE 180 REM prepreg based on a unidirectional carbon fiber tape. The possible buckling modes of its internal and peripheral layers are identified. Calculation results are compared with experimental data obtained earlier. It is concluded that, for the composite studied, the flexural buckling of layers in the uniform axial compression of specimens along fibers is impossible — the failure mechanism is delamination with buckling of a fiber bundle according to the pure shear mode. It is realized (due to the low average transverse shear modulus) at the value of the ultimate compression stress equal to the average shear modulus. It is shown that such a shear buckling mode can be identified only on the basis of equations constructed using the Timoshenko shear model to describe the deformation process of layers.

Size-dependent modelling of elastic sandwich beams with prismatic cores

Sandwich panel strips with prismatic cores are modelled using the modified couple stress theory and their elastic size-dependent bending behaviour investigated. Compatibility between the discrete sandwich and continuum beam kinematics is first discussed. A micromechanics-based framework to estimate effective mechanical properties is provided and unit cell models constructed with elementary beam elements to determine the stiffness parameters of various prismatic cores. Numerical studies show that the modified couple stress Timoshenko beam enhances the static deflection predictions of the classical Timoshenko model. A sensitivity measure based on structural ratios is proposed to estimate the influence of size effects in the global beam-level response. The parameters governing size effects in elastic sandwich beams are identified: Face-to-core thickness ratio, core density and topology, vertical corrugation order and set of load and boundary conditions. Size effects are shown more pronounced in low-density cores that rely on corrugation bending as shear-carrying mechanism. Based on the external load, boundary conditions and sensitivity factor, one can assess whether size effects are non-negligible in a given engineering structure.

Direct and inverse problems for prestressed functionally graded plates in the framework of the Timoshenko model

In this study, modelling and identification of prestress state in functionally graded plate within the framework of the Timoshenko theory are discussed. With the help of variational principles, statements of boundary problems for stationary vibration of inhomogeneous prestressed plates have been derived taking into account various factors of prestress state. The comparative analysis of classical and nonclassical models has been conducted. The effect of the prestress state factors on the solution characteristics has been estimated. New approaches to solving the inverse problems on a reconstruction of inhomogeneous prestress functions in a functionally graded plate have been proposed on the basis of derivation of reciprocity relations and iterative regularization. The results of numerical reconstruction experiments are presented; practical recommendations on a selection of frequency range for the purpose of getting the highest reconstruction accuracy are given.

On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency

In the present work, we prove that there exists a relation between a physical inconsistence known as second spectrum of frequency or non-physical spectrum and the exponential decay of a dissipative Timoshenko system where the damping mechanism acts on angle rotation. The so-called second spectrum is addressed into stabilization scenario and, in particular, we show that the second spectrum of the classical Timoshenko model can be truncated by taking a damping mechanism. Also, we show that dissipative Timoshenko type systems which are free of the second spectrum [based on important physical and historical observations made by Elishakoff (Advances mathematical modeling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010), Elishakoff et al. (ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11 2015) and Elishakoff et al. (Int J Solids Struct 109:143–151, 2017)] are exponential stable for any values of the coefficients of system. In this direction, we provide physical explanations why weakly dissipative Timoshenko systems decay exponentially according to equality between velocity of wave propagation as proved in pioneering works by Soufyane (C R Acad Sci 328(8):731–734, 1999) and also by Munoz Rivera and Racke (Discrete Contin Dyn Syst B 9:1625–1639, 2003). Therefore, the second spectrum of the classical Timoshenko beam model plays an important role in explaining some results on exponential decay and our investigations suggest to pay attention to the eventual consequences of this spectrum on stabilization setting for dissipative Timoshenko type systems.

Hybrid energy transformation to generalized Reissner–Mindlin model for laminated composite shells

By using the variational-asymptotic method, a two-dimensional mechanical model for laminated composite shells is established from a mathematical perspective, having the energy functional asymptotically correct up to the desired order in the small parameters. However, it is not in a practical form from an engineering perspective because of the appearance of partial derivative terms, which bring unnecessary mathematical complexity and obscure physical interpretation of mechanical boundary conditions in the shell modeling. Therefore, one more procedure is inevitably required – a so-called energy transformation procedure, which constructs a mathematical link between the energy functional derived herein and a simpler engineering model, such as a generalized Reissner–Mindlin model. In a different manner from previous works, this article introduces a hybrid energy transformation procedure composed of two successive steps: an equilibrium transformation via a linear algebraic approach, and an energy transformation via a perturbation approach. During this procedure, the first step is to transform the two-dimensional equilibrium equations from a hyper-static system into an isostatic system by augmenting them with the two-dimensional compatibility equations. Then, for obtaining a generalized Reissner–Mindlin model, the second step is to introduce initial curvature/twist as small parameters (in the sense of perturbations) into the constitutive law. The coupling stiffness terms between the transverse shear generalized strain measures and the remaining generalized Reissner–Mindlin strain measures are shown to be identically zero for laminated composite plates/shells (in contrast to the analogous terms of a similarly constructed generalized Timoshenko model for composite beams). Several examples are presented to demonstrate the capability and accuracy of this new approach.

Refined Models for an Analysis of Internal and External Buckling Modes of a Monolayer in a Layered Composite

For an analysis of internal and external buckling modes of a monolayer inside or at the periphery of a layered composite, refined geometrically nonlinear equations are constructed. They are based on modeling the monolayer as a thin plate interacting with binder layers at the points of boundary surfaces. The binder layer is modeled as a transversely soft foundation. It is assumed the foundations, previously compressed in the transverse direction (the first loading stage), have zero displacements of its external boundary surfaces at the second loading stage, but the contact interaction of the plate with foundations occurs without slippage or delamination. The deformation of the plate at a medium flexure is described by geometrically nonlinear relations of the classical plate theory based on the Kirchhoff–Love hypothesis (the first variant) or the refined Timoshenko model with account of the transverse shear and compression (the second variant). The foundation is described by linearized 3D equations of elasticity theory, which are simplified within the framework of the model of a transversely soft layer. Integrating the linearized equations along the transverse coordinate and satisfying the kinematic joining conditions of the plate with foundations, with account of their initial compression in the thickness direction, a system of 2D geometrically nonlinear equations and appropriate boundary conditions are derived. These equations describe the contact interaction between elements of the deformable system. The relations obtained are simplified for the case of a symmetric stacking sequence.

Frequency equation using new set of fundamental solutions with application on the free vibration of Timoshenko beams with intermediate rigid or elastic span

Stepped beams are crucial power transmission components in many mechanical engineering systems. These beams may be dynamically analyzed using the stepped Timoshenko model or the rigid mass model. In this paper, a new set of fundamental solutions is derived in order to normalize the Timoshenko beam equation at the origin of the coordinates. This set of solutions is used to derive the frequency equation of both stepped and rigid mass models. The validity ranges of these models were investigated by comparing the modal frequency results of both models. In addition, selected cases were compared using mode shape analysis. Three different models with classical end conditions are considered through this work. These are free–free, pinned–pinned and clamped–free beam configurations. The numerical results of the current work show that increasing the intermediate diameter ratio and decreasing the length ratio of the rigid mass results in decreasing the percentage deviation between the rigid mass model results and the elastic model results.

Vibration analysis of atomic force microscope cantilevers in contact resonance force microscopy using Timoshenko beam model

Timoshenko beam model is employed to investigate the vibration of atomic force microscope (AFM) cantilevers in contact resonance force microscopy (CRFM). Characteristic equation with both vertical and lateral tip-sample contact is derived. The contact resonance frequencies (CRFs) obtained by the Timoshenko model are compared with those by the Euler−Bernoulli model. A method is proposed to correct the wave number obtained by the Euler−Bernoulli model. The forced vibration is compared between the two models. Results reveal that the Timoshenko model is superior to the Euler−Bernoulli model in predicting the vibration characteristics for cantilevers’ higher eigenmodes.

An Analysis of Common Drill Stem Vibration Models

Excessive drill stem (DS) vibration while rotary drilling of oil and gas wells causes damages to drill bits and bottom hole assemblies (BHAs). In an attempt to mitigate DS vibrations, theoretical modeling of DS dynamics is used to predict severe vibration conditions. To construct the model, decisions have to be made on which beam theory to be used, how to implement forces acting on the DS, and the geometry of the DS. The objective of this paper is to emphasize the effect of these assumptions on DS vibration behavior under different, yet realistic, drilling conditions. The nonlinear equations of motion were obtained using Hamilton's principle and discretized using the finite element method. The finite element formulations were verified with uncoupled analytical models. A parametric study showed that increasing the weight on bit (WOB) and the drill pipe (DP) length clearly decreases the DS frequencies. However, extending the drill collar length does not reveal a clear trend in the resulting lateral vibration frequency behavior. At normal operating conditions with a low operating rotational speed, less than 80 RPM, the nonlinear Euler–Bernoulli and Timoshenko models give comparable results. At higher rotational speeds, the models deliver different outcomes. Considering only the BHA overestimates the DS critical operating speed; thus, the entire DS has to be considered to determine the critical RPM values to be avoided.

Refined models of contact interaction of a thin plate with positioned on both sides deformable foundations

We consider a rectangular plate connected along the outer contour with an absolutely rigid and fixed support through a low tough elastic support elements included in the class of transversely soft foundations. It is assumed that the contact interaction of plate at its points of faces of the connection to the support elements there is no relative slip and separation, and the opposite boundary surfaces of the support elements are fixed. Deformation of plate’s mid-surface is described by geometrically nonlinear relationships of the classical plate theory based on the hypothesis of the Kirchhoff–Love (the first option), and refined Timoshenko model taking into account the transverse shear compression (second version). The mechanics of support elements is described by the linearized equations of three-dimensional theory of elasticity, which have been simplified in the framework of transversely soft layer model. By integrating the latter equations along the transverse coordinate and satisfying to the conditions of the kinematic coupling of plate to the support elements at their initial compression in the thickness direction, two-dimensional geometrical nonlinear equations and their corresponding boundary conditions have been introduced, which describe the contact interaction of elements of the concerned deformable system. Simplification of derived relationships for the case when foundations have a symmetrical layer structure is carried out.

Non-uniformly scaled buckling modes of reinforcing elements in fiber reinforced plastic

We consider a problem about non-uniformly scaled buckling modes of isolated fiber (without accounting of interaction with the surrounding epoxy) or bundle of fibers, which are structural elements of fiber reinforced plastics under the transverse tension (compression) and shear stresses in prebuckling state. Such initial state is formed in fibers and bundles of fibers at tension-compression tests of flat specimens from cross ply composites with unidirectional fibers. For problem statement we use equations recently constructed by reduction of consistent version of geometrically nonlinear equations of theory of elasticity to one dimensional equations of rectilinear beams. Equations are based on refined shear S. P. Timoshenko model with accounting of tension-compression stresses in transverse directions. We give theoretical explanation of developed phenomenon as reducing shear modulus of elasticity of fiber reinforced plastic during the increasing of shear strains. We show that under the loading process of specimens under review uninterruptedly structure reconstruction of composite trough implementation and uninterruptedly changing of internal buckling modes at changing wave parameter is feasible.

Nonlocal excitation and potential instability of embedded slender and stocky single-walled carbon nanotubes under harmonically vibrated matrix

Until now various aspects of vibrations of single-walled carbon nanotubes (SWCNTs) have been explored; however, their dynamics and possible instabilities because of the excitation of matrix have not been addressed methodically. By considering a harmonic transverse excitation, the explicit expressions of elastic fields are obtained based on the nonlocal Rayleigh, Timoshenko, and higher-order beam models. The roles of the nonlocality, slenderness ratio, amplitude and frequency of matrix excitation and interactional behavior of the embedded nanotube on the dynamic transverse displacements of SWCNTs are comprehensively displayed. The capabilities of the Rayleigh model as well as the Timoshenko model in capturing the deflection of the nanostructure based on the higher-order beam theory are also explained in some detail. The nonlocal elastodynamic fields of the nanostructure in the resonance state as well as the critical values of lateral and rotational stiffness of the matrix are also introduced and the influences of crucial factors on such parameters are explained and discussed carefully.

Differential transform method and Adomian decomposition method for free vibration analysis of fluid conveying Timoshenko pipeline

The free vibration analysis of fluid conveying Timoshenko pipeline with different boundary conditions using Differential Transform Method (DTM) and Adomian Decomposition Method (ADM) has not been investigated by any of the studies in open literature so far. Natural frequencies, modes and critical fluid velocity of the pipelines on different supports are analyzed based on Timoshenko model by using DTM and ADM in this study. At first, the governing differential equations of motion of fluid conveying Timoshenko pipeline in free vibration are derived. Parameter for the nondimensionalized multiplication factor for the fluid velocity is incorporated into the equations of motion in order to investigate its effects on the natural frequencies. For solution, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Timoshenko beam theory. After the analytical solution, the efficient and easy mathematical techniques called DTM and ADM are used to solve the governing differential equations of the motion, respectively. The calculated natural frequencies of fluid conveying Timoshenko pipelines with various combinations of boundary conditions using DTM and ADM are tabulated in several tables and figures and are compared with the results of Analytical Method (ANM) where a very good agreement is observed. Finally, the critical fluid velocities are calculated for different boundary conditions and the first five mode shapes are presented in graphs.

Differentiation of the energy functional in the equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.

Nonlinear behaviour of different flexible size-dependent beams models based on the modified couple stress theory. Part 1: Governing equations and static analysis of flexible beams

In the first part of the paper we employ the Sheremetev-Pelekh-Reddy-Levinson hypotheses, which yield a non-linear mathematical model of a beam taking into account geometric and physical non-linearity as well as transverse shear based on the modified couple stress theory. The general model includes both Bernoulli-Euler and Timoshenko models with/without geometric/physical non-linearity, and the size-dependent beam behaviour.In addition, we present results of the development of the relaxation method for solution to numerous static problems. The influence of the size-dependent coefficient on the load-deflection and stress-strain states of the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson mathematical models has been also studied.

Flexural Vibration of Prestressed Concrete Bridges with Corrugated Steel Webs

Prestressed concrete bridges with corrugated steel webs have emerged as a new form of steel-concrete composite bridges with remarkable advantages compared with the traditional ones. However, the assumption that plane sections remain plane may no longer be valid for such bridges due to the different behavior of the constituents. The sandwich beam theory is extended to predict the flexural vibration behavior of this type of bridges considering the presence of diaphragms, external prestressing tendons and interaction between the web shear deformation and flange local bending. To this end, a [Formula: see text] beam finite element is formulated. The proposed theory and finite element model are verified both numerically and experimentally. A comparison between the analyses based on the sandwich beam model and on the classical Euler–Bernoulli and Timoshenko models reveals the following findings. First of all, the extended sandwich beam model is applicable to the flexural vibration analysis of the bridges considered. By letting [Formula: see text] denote the square root of the ratio of equivalent shear rigidity to the flange local flexural rigidity, and L the span length, the combined parameter [Formula: see text] appears to be more suitable for considering the diaphragm effect and the interaction between the shear deformation and flange local bending. The diaphragms have significant effect on the flexural natural frequencies and mode shapes only when the [Formula: see text] value of the bridge falls below a certain limit. For a bridge with an [Formula: see text] value over a certain limit, the flexural natural frequencies and mode shapes obtained from the sandwich beam model and the classical Euler–Bernoulli and Timoshenko models tend to be the same. In such cases, either of the classical beam theories may be used.

Size-dependent vibration and bending analyses of the piezomagnetic three-layer nanobeams

Vibration and electro-magneto-elastic bending analysis of a three-layer nanobeam with a nanocore and two piezomagnetic face sheets are studied in this paper. Timoshenko model of beam as well as nonlocal magneto-electro-elastic relations are used for analysis of this problem. The nanoface sheets are subjected to applied electric and magnetic potentials. The nanobeam rests on Winkler–Pasternak foundation. Electric and magnetic potentials are assumed as combination of linear function along the thickness direction that reflects applied electric and magnetic potentials and a cosine function that satisfies boundary conditions. Numerical results of this problem investigate the effect of some important parameters of nanobeam, such as nonlocal parameter, applied electric and magnetic potentials, and parameters of foundation on the vibration and magneto-electro-mechanical bending behaviors of the problem.

Bending, buckling and free vibration analyses of functionally graded curved beams with variable curvatures using isogeometric approach

A study on the bending, buckling and free vibration of functionally graded curved beams with variable curvatures using isogeometric analysis is presented here. Non-uniform rational B-splines, known from computer aided geometric design, are employed to describe the exact geometry and approximate the unknown fields of a curved beam element based on Timoshenko model. Material properties of the beam are assumed to vary continuously through the thickness direction according to the power law form. The numerical examples investigated in this paper deal with circular, elliptic, parabolic and cycloid curved beams. Results have been verified with the previously published works in both cases of straight functionally graded beam and isotropic curved beam. The effects of material distribution, aspect ratio and slenderness ratio on the response of the beam with different boundary conditions are numerically studied. Furthermore, an interesting phenomenon of changing mode shapes for both buckling and free vibration characteristics corresponding to the variation in the parameters mentioned above is also examined.

ИДЕНТИФИКАЦИЯ СВОЙСТВ НЕОДНОРОДНОЙ ПЛАСТИНЫ В РАМКАХ МОДЕЛИ ТИМОШЕНКО

ИДЕНТИФИКАЦИЯ СВОЙСТВ НЕОДНОРОДНОЙ ПЛАСТИНЫ В РАМКАХ МОДЕЛИ ТИМОШЕНКО

ИССЛЕДОВАНИЕ ДИНАМИЧЕСКОГО УПРУГОПЛАСТИЧЕСКОГО ДЕФОРМИРОВАНИЯ ИСКРИВЛЕННЫХ СТЕРЖНЕЙ НЕРЕГУЛЯРНОЙ СЛОИСТО-ВОЛОКНИСТОЙ СТРУКТУРЫ

In the Karman approximation the problem of elastic-plastic deformation of flexible laminated- fibrous rods of the small curvature and irregular structure is formulated. The rods consist of two belt-bearing layers (flanges) connected by a thin wall. In the framework of the Timoshenko model the weakened resistance of the wall to shear in its plane is taken into account. Bearing layers are reinforced in the longitudinal direction, and flanges longitudinally or cross-reinforced in its plane. The elastoplastic behavior of components of the composition is described by the equations of the theory of plasticity with isotropic hardening. For the numerical integration of the considered problem the explicit “cross” scheme was used. The calculations are carried out for the dynamic bending deformation of homogeneous and composite, straight and curved flat bars beam of double-T cross section under the action of loads of explosive type. It is shown that the classical theory of bending of these structures can be unacceptable not only for the analysis of metal-composite but homogeneous layered metal rods with unified rolling-sections of double-T type. It is demonstrated that the rational structure of reinforcement of walls for straight rods can be ineffective in the case of curved rods. It is obtained that the dynamic inelastic behavior of curved rods depends on the direction of action of external lateral loads of explosive type. Keywords: layered curved rods, reinforcement, Timoshenko model, geometric nonlinearity, bending deformation, theory of plastic flow, isotropic hardening, dynamic loads, “cross” scheme.