Shear Deformation Beam Research Articles

Dynamic stability analysis of delaminated composite beams in frequency domain using a unified beam theory with higher order displacement continuity

In this paper the dynamic stability behaviour of single delaminated composite beam is investigated using higher order shear deformable beam theories. The system is discretised by the finite element method and the mass, material stiffness and geometric stiffness matrices are formulated by means of higher order displacement continuity in the intact part. The natural frequencies and critical buckling forces are determined for various beam theory orders and delamination lengths and positions. Furthermore, using the multi frequency method the dynamic stability properties of the system are calculated in frequency domain. Based on the stability analysis the sufficient beam theory order and multi frequency method’s corresponding truncated matrix determinant order are determined in order to achieve the desired accuracy on the observed parameter plane. The results indicate that the dependence of the natural frequencies and buckling forces is negligible on the order of the applied beam theory. In contrast the former quantities are significantly influenced by the delamination length and position. Finally, it is shown that the dynamic stability diagrams and the boundary frequencies of parametric instability do not depend on the delamination length and position from the engineering point of view.

Examination of thermal postbuckling of temperature dependent FG-GRMMC laminated beams with negative Poisson’s ratio on elastic foundations

This paper examines the in-plane negative Poisson’s ratio (NPR) effect on the thermal postbuckling of graphene-reinforced metal matrix composite (GRMMC) laminated beams subjected to a uniform temperature rise and supported by an elastic foundation. GRMMC layers having different graphene volume fractions can facilitate the formation of functionally graded (FG) patterns of the beams. The temperature-dependent material properties of GRMMC layers can be estimated by a nonlinear function of temperature from the results of the molecular dynamics simulations. The governing equations for the beam thermal postbuckling problem can be derived based on a higher order shear deformation beam theory and solved by applying the two-step perturbation approach. The von Kármán geometric nonlinearity, the elastic foundation support and the thermal effect are considered in the modeling. Numerical investigations are carried out for (10/-10/10/-10/10)S and (10/-10/10)S GRMMC beams with in-plane NPR. The results show that the buckling temperature of (10/-10/10)S UD beam is slightly higher than that of (10/-10/10/-10/10)S UD beam. The thermal postbuckling capacity for the (10/-10/10)S FG-X beam becomes much higher than that of the (10/-10/10/-10/10)S FG-X beam, when the beam deflection is sufficiently large.

Bending analysis of functionally graded porous nanocomposite beams based on a non-local strain gradient theory

In the present work we study the static response of functionally graded (FG) porous nanocomposite beams, with a uniform or non-uniform layer-wise distribution of the internal pores and graphene platelets (GPLs) reinforcing phase in the matrix, according to three different patterns. The finite-element approach is developed here together with a non-local strain gradient theory and a novel trigonometric two-variable shear deformation beam theory, to study the combined effects of the non-local stress and strain gradient on the FG structure. The governing equations of the problem are solved introducing a three-node beam element. A comprehensive parametric study is carried out on the bending behavior of nanocomposite beams, with a particular focus on their sensitivity to the weight fraction and distribution pattern of GPLs reinforcement, as well as to the non-local scale parameters, geometrical properties, and boundary conditions. Based on the results, it seems that the porosity distribution and GPLs pattern have a meaningful effect on the structural behavior of nanocomposite beams, where the optimal response is reached for a non-uniform and symmetric porosity distribution and GPLs dispersion pattern within the material.

An Efficient Multi-Scale Modeling Method that Reveals Coupled Effects Between Surface Roughness and Roll-Stack Deformation in Cold Sheet Rolling

Abstract In thin-gauge cold rolling of metal sheet, the surface roughness of work rolls (WRs) is known to affect the rolled sheet surface morphology, the required rolling load, and the roll wear. While modeling of rough surfaces using statistical asperity theory has been widely applied to problems involving semi-infinite solids, the application of asperity distributions and their elastic-plastic behavior has not been considered in roll-stack models for cold sheet rolling. In this work, a simplified-mixed finite element method (SM-FEM) is combined with statistical elastic-plastic asperity theory to study contact interference and coupling effects between a rough work roll (WR) surface and the roll-stack mechanics in cold sheet rolling. By mixing equivalent rough surface contact foundations, Hertz foundations, and Timoshenko beam stiffness, an approach is created to efficiently model interactions between the micro-scale asperities and the macro-scale roll-stack deformation. Nonlinearities from elastic-plastic material behavior of the asperities and the sheet, as well as changing contact conditions along the roll length, are also accommodated. Performance of the multi-scale SM-FEM approach is made by comparison with a continuum finite element virtual material model. 3D studies for a 4-high mill reveal new multi-scale coupling behaviors, including nonuniform roughness transfer, and perturbations to the sheet thickness “crown” and contact force profiles. The described multi-scale SM-FEM approach is general and applies to rough surface contact problems involving plates and shear-deformable beams having multiple contact interfaces and arbitrary surface profiles.

Structural behaviours of zigzag and armchair nanobeams using finite element doublet mechanics

This paper presents a comprehensive study on bending, vibration and buckling behaviours of the zigzag and armchair nanobeams. Based on a third order shear deformation beam theory and the doublet mechanics, a finite element model is proposed and employed to solve the problems of zigzag and armchair nanobeams with various boundary conditions for the first time. The verification is performed by comparing the numerical results with those from the previous studies with respect to various boundary conditions. A number of numerical examples on zigzag and armchair nanobeams with four boundary conditions have been carried out. The effects of material length scale parameters, aspect ratio, nanobeam model and boundary conditions on the displacements, natural frequencies and buckling loads of nanobeams are investigated in details. Some new results, which are not available in open literature, are provided as references for the future studies.

Effect of porosity and profile axial loading on elastic buckling and free vibration of functionally graded porous beam.

Elastic buckling and free vibration behaviour of porous functionally graded beam under profile axial mechanical loading is presented analytically. Symmetric and asymmetric porosity distribution with non linear material functional grading is presented. Buckling study is carried out first, then free vibration study under buckling loads is presented with profile loading. The effect of porosity, porosity pattern, slenderness ratio, boundary condition on critical buckling load and fundamental frequency is also studied. Ritz method with shear and normal deformable beam theory (SNDBT) is employed for formulation. It is observed that, with increase in porosity, buckling load decreases for both porosity distribution whereas, fundamental frequency increases for symmetric porosity distribution and is vice versa for asymmetric porosity distribution under all loading conditions.

Buckling and Vibration Behavior of Composite Beam due to Axially Varying In-plane Loads

In this work buckling and vibration behavior of composite beam is analyzed based on the shear and normal deformable beam theory (SNDBT) by using Ritz method. The effects of aspect ratios, boundary conditions, fiber angles on the buckling and vibration subjected to non-uniform axial loads studied in detail. The obtained numerical results in terms of dimensionless fundamental frequencies and dimensionless first critical buckling loads are compared with the results from previous research for convergence studies.

A wave propagation study for porous metal foam beams resting on an elastic foundation

The purpose of present investigation is to study propagation of flexural wave in porous metal foam beams resting on Winkler–Pasternak foundation based upon refined higher-order shear deformable beam theory. Moreover, different types of porosity distribution through the thickness direction are probed namely; uniform, symmetric and asymmetric. The Hamiltonian approach is implemented to catch motion equations of porous metal foam beams. Then, the obtained governing equations of porous metal foam are solved analytically. The influences of different parameters such as various types of porosity distribution, porosity coefficient, slenderness ratio, wave number and elastic foundation coefficients on the variation of wave frequency, escape frequency and phase velocity are covered and presented within the framework of a group of figures which can be observed in detail.

On the piezoelectric effect on stability of symmetric FGM porous nanobeams

In present paper, the piezoelectric effect on bifurcation buckling of symmetric FGM porous nanobeam is presented based on a higher-order nonlocal elasticity and strain gradient theory in conjunction with Reddy third-order shear deformation beam theory. The strain energy density and electric enthalpy density functions are used to derive constitutive relations for porous functionally graded, as well as piezoelectric materials. Equations of motion for elastically supported layered FGM nanobeam with diverse distribution of porosity and electro-elastic coupling are derived based on the modified Hamilton’s variational principle. The formulated boundary value problem is analytically solved. For the first time, the effects of porosity distribution, volume of voids, distance between porosity and FGM core surfaces as well as material gradation, layers size, aspect ratio, stiffness of Kerr foundation, and external electric voltage on critical in-plane force, critical porosity and critical voltage were comprehensively presented and discussed. The investigation enables analysis and precise multi-dimensional control of static response of smart beam-like nanostructures widely used in MEMS and NEMS devices.

Forced Vibration Analysis of Composite Beams Reinforced by Carbon Nanotubes.

This paper presents forced vibration analysis of a simply supported beam made of carbon nanotube-reinforced composite material subjected to a harmonic point load at the midpoint of beam. The composite beam is made of a polymeric matrix and reinforced the single-walled carbon nanotubes with their various distributions. In the beam kinematics, the first-order shear deformation beam theory was used. The governing equations of problem were derived by using the Lagrange procedure. In the solution of the problem, the Ritz method was used, and algebraic polynomials were employed with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method was applied in the time history. In the numerical examples, the effects of carbon nanotube volume fraction, aspect ratio, and dynamic parameters on the forced vibration response of carbon nanotube-reinforced composite beams are investigated. In addition, some comparison studies were performed, with special results of published papers to validate the using formulations.

A unified thermal vibration and transient analysis for quasi-3D shear deformation composite laminated beams with general boundary conditions

A unified procedure is proposed in this study to analyze the free and transient vibration behaviors of a composite laminated beam subjected to general boundary conditions under thermal scenario. A general theoretical quasi-3D shear deformation beam model incorporating both the shear deformation and thickness stretching effects is derived using the Hamilton's principle. The coupling of axial–shear–flexural–stretching with the thermal stress and the Poisson's ratio effect is taken into account. The natural frequencies, mode shapes and transient responses of composite laminated beams are evaluated analytically by the method of reverberation ray matrix (MRRM). The inhomogeneous term and the artificial spring boundary technique are introduced in MRRM to acclimatize itself to the general boundary conditions and the impulse loads of arbitrary distributions. The validity, reliability and efficiency of the proposed analytical solutions are verified by comparing with the results obtained from the published studies and finite element simulation. A considerable number of parametrical studies for the composite laminated beams with different elastic restraint parameters, lamination schemes, geometry parameters, material properties as well as various temperature rises are also analyzed and discussed. The present procedure is powerful in terms of its applicability for different high-order shape functions including polynomial and non-polynomial, as well as thin and thick beams subjected to arbitrary boundary conditions in practical engineering.

Nonlinear mechanical model of the shaft of a roll forming mill and parameter identification

Roll forming is a continuous process in which a moving metal sheet passes through numerous pairs of opposing forming rolls. The shafts of the roll forming mill are equipped with these rolls and must be set up and aligned to achieve the required final profile of the sheet. The practically relevant task of predicting the profile geometry of this incremental rolling process with varying characteristics of the metal sheet entering the mill requires an accurate description of the stiffness behavior of the shaft with rolls, which is the most compliant part of the roll forming mill. In this paper, the measured force-deflection characteristic of the shaft without rolls is compared with predictions of various theoretical models, followed by the adoption of the shear deformable beam model of the shaft with nonlinear elastic supports in the bearings. The coefficients of the cubic stiffness characteristics of the rotational springs as well as the effective length between the supports are identified based on the experimental data for the deflections, measured along the shaft for various loading levels. The theoretical predictions are obtained via the nonlinear finite element model of the shaft. The model thus provided shows high accuracy compared with the measurements. The paper’s results serve as a foundation for models to predict the stiffness of shafts with rolls.

Isogeometric configuration design optimization of three-dimensional curved beam structures for maximal fundamental frequency

This paper presents a configuration design optimization method for three-dimensional curved beam built-up structures having maximized fundamental eigenfrequency. We develop the method of computation of design velocity field and optimal design of beam structures constrained on a curved surface, where both designs of the embedded beams and the curved surface are simultaneously varied during the optimal design process. A shear-deformable beam model is used in the response analyses of structural vibrations within an isogeometric framework using the NURBS basis functions. An analytical design sensitivity expression of repeated eigenvalues is derived. The developed method is demonstrated through several illustrative examples.

Buckling and postbuckling of extensible, shear-deformable beams: Some exact solutions and new insights

This paper presents exact solutions for the buckling loads and postbuckling states of extensible, shear deformable beams. The governing equation for the large-amplitude lateral deformation of beams in compression is expanded in Taylor series up to the cubic nonlinearity. Closed-form solutions in terms of the axial and shear stiffnesses are developed for statically determinate and statically indeterminate beams. Namely, pinned–pinned, cantilevered, clamped–clamped, and clamped–pinned beams are considered with the loaded end is a roller that is able to slide. The postbuckling response under a given axial load is exactly derived. The dependence of the buckling load on the length-to-radius of gyration is discussed. It is shown that the extensibility and the shear deformation significantly affect the buckling loads and the postbuckling response. For conventional materials with positive Poisson’s ratio, the inclusion of the axial and shear deformation results in a meaningful reduction of the buckling load. It is further shown that the buckling load can be enhanced by designing artificial metamaterials materials with an effective negative Poisson’s ratio.

Bending analysis of functionally graded graphene oxide powder-reinforced composite beams using a meshfree method

In this paper, the linear bending response of straight, thin to moderately thick functionally graded (FG) composite beams reinforced with graphene oxide powder (GOP) and subjected to two types of static mechanical loads, with various boundary conditions is investigated using a meshfree collocation method based on multiquadric radial basis functions. The weight fractions of the GOP are considered to vary continuously through the thickness direction of the composite beams. To determine the effective Young's modulus of the FG-GOP reinforced composite beams, a modified Halpin–Tsai model is used, whereas the rule of mixture is adopted to estimate the equivalent Poisson's ratio. The first-order shear deformation (FSDT) beam theory is adopted to model the functionally graded graphene oxide powder (FG-GOP) reinforced composite beams. The static equilibrium equations and associated boundary conditions of the FG-GOP reinforced composite beams are obtained by using the principle of minimum potential energy. The governing equations are rewritten in a matrix-vector strong form and discretized using the multiquadric radial basis functions. Various comparisons studies are performed to demonstrate the robustness and accuracy of the proposed numerical model. Parametric studies are conducted to examine the effects of length-to-thickness ratio, GOP distribution patterns, weight fraction, and size of GOP, types of loads, and boundary conditions on the bending behavior of the FG-GOP reinforced composite beams.

Description of elastic forces for the ANCF shear deformable beam element by using the interpolation of rotation

Description of elastic forces for the ANCF shear deformable beam element by using the interpolation of rotation

Postbuckling analysis of nonlocal functionally graded beams

Abstract The main goal of this research is to study the postbuckling behavior of nonlocal functionally graded beams. Eringen’s nonlocal differential model is used to evaluate the influence of the material length scale in the bending response. An improved shear deformation beam theory with five independent parameters is utilized, which is suitable for the use of 3D constitutive equations. A finite element model is derived with spectral high-order interpolation functions to avoid shear locking. The formulation is verified by comparing the present results with the ones found in the literature. Functionally graded beams with different boundary conditions, nonlocal parameters, and power law indices are analyzed. It is shown that the present model can accurately predict the behavior of nonlocal beams due to the use of high-order terms in the displacement field in comparison with classical beam formulations. Finally, new benchmark problems are analyzed to show the capabilities of the present model to evaluate the effect of the nonlocal parameter and the power law index on postbuckling beam behavior.

On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations

This paper introduces a comprehensive buckling response of cross-ply orientation of carbon nanotube reinforced composite (CNTRC) multilayered nanobeams with different boundary conditions. The nonlocal strain gradient (NLSG) stress-strain governing relations are utilized to include the size-dependence and microstructure effects. Novel hyperbolic higher shear deformation beam theory including thickness stretching effect is used to fulfill both parabolic shear distribution through the thickness and the zero-shear at free boundaries. Parametric studies are performed to inspect the influences of arrangement of reinforcement material distributions functions, different functionally graded (FG) functions, and uniform distribution (UD). The balance equilibrium equations are derived, and Fourier functions are utilized to obtain the critical buckling loads of nanobeam under mechanical loadings. Mechanical properties are assumed to be temperature-dependent by using Touloukian principal. An exact solution is performed satisfying the edge boundary conditions. A detailed numerical analysis is illustrated to examine the impact of CNTs patterns, lamination, side-to-thickness, aspect ratios, microstructure and size scale parameters on critical buckling loads of CNTRC laminated nanobeams.

Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam

This work focused on the novel numerical tool for the bending responses of carbon nanotube reinforced composites (CNTRC) beams. The higher order shear deformation beam theory (HSDT) is used to determine strain-displacement relationships. A new exponential function was introduced into the carbon nanotube (CNT) volume fraction equation to show the effect of the CNT distribution on the CNTRC beams through displacements and stresses. To determine the mechanical properties of CNTRCs, the rule of the mixture was employed by assuming that the single-walled carbon nanotubes (SWCNTs)are aligned and distributed in the matrix. The governing equations were derived by Hamilton's principle, and the mathematical models presented in this work are numerically provided to verify the accuracy of the present theory. The effects of aspect ratio (l/d), CNT volume fraction (Vcnt), and the order of exponent (n) on the displacement and stresses are presented and discussed in detail. Based on the analytical results. It turns out that the increase of the exponent degree (n) makes the X-beam stiffer and the exponential CNTs distribution plays an indispensable role to improve the mechanical properties of the CNTRC beams.

The nano scale buckling properties of isolated protein microtubules based on modified strain gradient theory and a new single variable trigonometric beam theory

Microtubules (MTs) are the main part of the cytoskeleton in living eukaryotic cells. In this article, a mechanical model of MT buckling, considering the modified strain gradient theory, is analytically examined. The MT is assumed as a cylindrical beam and a new single variable trigonometric beam theory is developed in conjunction with a modified strain gradient model. The main benefit of the present formulation is shown in its new kinematic where we found only one unknown as the Euler-Bernoulli beam model, which is even less than the Timoshenko beam model. The governing equations are deduced by considering virtual work principle. The effectiveness of the present method is checked by comparing the obtained results with those reported by other higher shear deformation beam theory involving a higher number of unknowns. It is shown that microstructure-dependent response is more important when material length scale parameters are closer to the outer diameter of MTs. Also, it can be confirmed that influences of shear deformation become more considerable for smaller shear modulus and aspect ratios.