Shear Deformable Beam Model Research Articles

Framework of Physically Consistent Homogenization and Multiscale Modeling of Beam Structures

In this work, a framework of physically consistent homogenization and multiscale modeling (PCHMM) is developed to analyze complex slender structures with homogenized one-dimensional beam models. Firstly, it is highlighted that periodic boundary conditions are not valid for the representative volume element (RVE)-based homogenizations if a shear-deformable beam model is to be obtained. Addressing this issue, physically consistent boundary conditions (PCBCs) are discussed in detail, which accounts for the influence of shear forces on the behavior of the slender structure and the corresponding RVE. A procedure of enforcing such boundary conditions is also given, applying sectional stresses to the RVE with fictitious DoFs. Additionally, we introduce a newly developed correct formula to calculate effective sectional properties which guarantee the cross-scale energy conservation though a modified Hill’s condition. The PCHMM framework, with homogenization/dehomogenization procedures using PCBC and the modified Hill’s condition, is also conveniently integrated into a commercial finite element software. The correctness and effectiveness of the developed framework of homogenization and multiscale analysis are demonstrated through several numerical examples, including lattice beams, twist-morphing metamaterial spars, and a realistic wing. The extension of this theory to plate/shell structures will be presented in our future work.

Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

AbstractThe modal analysis and seismic response of a vibrating cantilever, with or without tip mass and rotary inertia, are investigated in this study using a shear deformable beam model and including the effect of vertical load. Based on existing approaches, an original method is proposed that does not use fourth‐order uncoupled equations to determine modal deflection and rotation. In fact, the approach presented herein transforms the second‐order coupled system into a first‐order system which can be solved more easily using matrix algebra and Laplace transform. Furthermore, the proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self‐adjoint, and the solution in the case of forced response using modal superposition. In addition, even if the solution presented herein is applicable only to the cantilever with a tip mass and rotary inertia, the scope is general, and the approach can be applied to shear deformable beams with other boundary conditions. Finally, the seismic response by modal superposition is shown, and some examples are proposed and discussed for the case of uniform or continuously varying cross‐sectional properties.

Vibration characteristics of arbitrary thick sandwich beam with metal foam core resting on elastic medium

An analysis is performed in this research to obtain the natural frequencies and mode shapes of a thick sandwich beam with metal foam core. Different types of foams are considered for the sandwich beam where distribution of pores is a functionally graded patterns. The governing equations of the sandwich beam are established by means of a shear and normal deformable thick beam model. This theory considers the thickness stretching and admits the nonuniform through-the-thickness shear strain. The governing equations of the sandwich beam are established which are four in number by means of the Hamilton principle. The established equations are solved for the case of thick sandwich beams resting on elastic foundation via the Navier solution method. Results of this study are first compared with the available data in the open literature and after that novel results are given to explore the effects of different parameters on the vibration characteristics of the sandwich beam. It is shown that porosity of the core is an important factor on vibration characteristics of sandwich beam with metal foam core.

Numerical Model for a Geometrically Nonlinear Analysis of Beams with Composite Cross-Sections

This paper presents a beam model for a geometrically nonlinear stability analysis of the composite beam-type structures. Each wall of the cross-section can be modeled with a different material. The nonlinear incremental procedure is based on an updated Lagrangian formulation where in each increment, the equilibrium equations are derived from the virtual work principle. The beam model accounts for the restrained warping and large rotation effects by including the nonlinear displacement field of the composite cross-section. First-order shear deformation theories for torsion and bending are included in the model through Timoshenko’s bending theory and a modified Vlasov’s torsion theory. The shear deformation coupling effects are included in the model using the six shear correction factors. The accuracy and reliability of the proposed numerical model are verified through a comparison of the shear-rigid and shear-deformable beam models in buckling problems. The obtained results indicated the importance of including the shear deformation effects at shorter beams and columns in which the difference that occurs is more than 10 percent.

Nonlinear thermo-elastic stability of variable stiffness curvilinear fibres based layered composite beams by shear deformable trigonometric beam model coupled with modified constitutive equations

Nonlinear thermo-elastic buckling characteristics of composite variable stiffness beam with layers making use of curvilinear fibres under thermal environment is attempted here. The model is based on a shear deformable theory introducing trigonometric function, and considering von Kármán’s assumptions based geometrical nonlinear effect. The beam constitutive equation is modified according to the stress-free situation in the width direction of beam-Poisson’s effect in the formulation for predicting the behaviour of general lay-up composite beams. By the principle of minimum total potential energy, the governing equations in terms of incremental stiffness matrices are formed introducing general beam finite element. The global equilibrium equations formulated are solved for envisaging the post-buckling path through eigenvalue analysis iteratively, thus establishing the relationship of thermal temperature against moderate amplitude level of beam deflection. A systematic parametric analysis considering different lamina properties such as curvilinear fibre path angles and including lay-up sequences, thermal expansion coefficient, mixed laminate combining straight and curvilinear fibres-based layers is carried out on thermo-structural stability of curvilinear fibre-based beams. Also, the influence of geometric factors, flexible beam end support, and variation in thermal profile, etc. over the stability behaviour of beam is examined.

Stability analysis of shear deformable cross-ply laminated composite beam-type structures

This paper presents a shear deformable beam model for the nonlinear stability analysis of composite beam-type structures. Each wall of the cross-section is stacked in the cross-ply scheme. The incremental equilibrium equations are derived in the framework of an updated Lagrangian formulation. Hooke’s law is assumed to be valid. The nonlinear displacement field of the composite cross-section accounts for the restrained warping and the large rotation effects. The shear deformation effects are included through Timoshenko’s bending theory and modified Vlasov’s torsion theory. Additional shear correction factors due to the bending-torsion coupling occurring at asymmetric cross-sections are introduced in the analysis. The accuracy and reliability of proposed numerical mode are verified on benchmark examples, and the obtained results confirm that it can be classified as a shear locking-free model.

A four-variable shear and normal deformable quasi-3D beam model to analyze the free and forced vibrations of FG-GPLRC beams under moving load

A four-variable shear and normal deformable quasi-3D beam model to analyze the free and forced vibrations of FG-GPLRC beams under moving load

A 2D warping theory for shear deformable elastic beams of axisymmetric cross section in flexure

Shear deformable elastic beam models in flexure are considered in the hypothesis of cross section symmetric with respect to the flexure axis, whereby the external transverse forces act as distributed body forces. A 2D warping theory is presented where the shear-warping process is driven by a shear-warping parameter, say τ, 0≤τ≤1, in such a way that equilibrium between zero external (t(n)i=0) and internal (σ(n)i=0) tractions upon the lateral surface of the beam is complied with no matters the value of τ∈(0,1). On letting τ vary within (0,1), a continuous beam family, say B≔{Bτ:0≤τ≤1}, is generated, which spans from the Euler–Bernoulli (EB) beam model B0 at τ=0 (no shear strain, nor warping effects), to a series of shear deformable beam models Bτ up to τ=1. For intermediate values of τ, 0<τ<1, every beam model is featured by transversally nonuniform shear stresses accompanied by suitable 2D warping effects, while the boundary equilibrium condition with zero applied tractions is satisfied. The warping theory herein presented is an original generalization of a previous 1D type warping theory devised for rectangular cross sections. Applications are reported in which a circular cross section is considered and a beam problem in static conditions is worked out.

Structural behavior of MDF beams reinforced by unidirectional glass fiber reinforced thermoset polyurethane outer layers

In this study, the structural behavior of medium density fiber board (MDF) beams reinforced by unidirectional glass fiber reinforced thermoset polyurethane outer layers is investigated by conducting beam bending and vibration tests followed by a parametric study. First, the MDF material properties are determined by 3 point bending and cantilever beam vibration tests. Then the same tests are conducted on the composite sandwich beam specimens produced by hand layup technique and the level of increase in overall stiffness and the ultimate failure loads are determined. The experimental results obtained are found out to be in good agreement with the analytical results and the numerical results obtained using a 1D shear deformable beam model. Finally, a parametric study is conducted to determine the influence of FRP reinforcement for different span lengths and fiber volume ratios. The results obtained from this study show that reinforcing MDF beams by FRP outer layers may considerably increase the flexural stiffness for all span lengths but the ultimate failure load may not be increased for short span lengths due to core shear failure, especially when there is no constraint on maximum displacement.

Selection of size dependency theory effects on the wave's dispersions of magneto-electro-thermo-elastic nano-beam resting on visco-elastic foundation

Current research develops a comprehensive wave propagation analysis of a magneto-electro-thermo-elastic (METE) nano-beam (NB) resting on the visco-elastic medium. To model the size dependency effects, modified couple stress (MCS) and Eringen's nonlocal (ENL) theories are employed to analyze and describe the wave propagation behaviors for those nano-beams. These theories were the most used in the literature due to the inclusion of one additional size-dependent length scale parameter. Those theories are compared side by side in this investigation and their impacts/differences on the wave propagation of specific materials are explored. Sinusoidal shear deformation beam model with Hamilton's principle is adopted to develop the governing equations of motion. Then, an analytical solution is executed to extract numerical results for transverse wave propagation in both elastic and METE configurations of the nano-beam. The effects of size-dependent length scale of both theories, thickness of NB, Winkler-Pasternak coefficients, thermal gradient, and magnetic potential and external electric voltage are illustrated and discussed in details. It is concluded that wave frequency decreases with increment of nonlocal parameter for ENL model. On the other hand, a stiffening effect takes place for the wave frequency when the MCS model is considered. Hence, the results indicate that there is a significant difference between ENL and MCS theories in the estimation of the behavior of wave propagation in small-scale structures. One of the main results of this investigation indicates that the MCS theory has similar nonlocal effects as the Eringen's theory for specific conditions.

Analytical and meshless DQM approaches to free vibration analysis of symmetric FGM porous nanobeams with piezoelectric effect

The current study is an first attempt to develop a vibrational-based piezoelectrically actuated smart nanoactuator model based on assumptions of Reddy third-order shear deformation beam model together with nonlocal strain gradient theory. The considered model is a sandwich structure consisting of piezoelectric layers under an external electric field, and symmetric to the midplane functionally graded (FG) porous core. Expressed by displacements equations of motion are derived based on a dynamic version of Hamilton's principle. Eigenvalue problem of free vibration of simply supported nanobeam is solved analytically by using properties of Fourier series, and numerically via meshless Differential Quadrature Method (DQM). The study includes a convergence study for the numerical approach, as well detailed discussion on obtained results involving effects of nonlocal parameters, external mechanical and electrical loads, material gradation coupled with diverse porosity distribution, and influence of external environment on dynamic behavior of the nanostructure. The present paper, in view of included optimization study, may positively affect nanodevices reliability and extend its operation by avoiding resonance phenomenon.

A geometrically nonlinear shear deformable beam model for piezoelectric energy harvesters

An electromechanical model for beam-like piezoelectric energy harvesters based on Reissner’s beam theory is developed in this paper. The proposed model captures first-order shear deformation and large displacement/rotation, which distinguishes this model from other models reported in the literature. All governing equations are presented in detail, making the associated framework extensible to investigate various piezoelectric energy harvesters. The weak formulation is then derived to obtain the approximate solution to the governing equations by the finite element method. This solution scheme is completely coupled, and thus allows for two-way interaction between mechanical and electrical fields. To validate this model, extensive numerical examples are implemented in the linear and nonlinear regime. In the linear limit, this model produces results in excellent agreement with reference data. In the nonlinear regime, the large amplitude response of the piezoelectric beam induced by strong base excitation or fluid flow is considered, and the comparison of results with literature data is encouraging. The ability of this nonlinear model to predict limit cycle oscillations in axial flow is demonstrated.

Vibration analysis of functionally graded beams using a higher-order shear deformable beam model with rational shear stress distribution

This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams. In the element model proposed, the stress equilibrium equation is employed to derive the rational distribution of transverse shear stress. The beam element formulation oriented with vibration analysis is derived on the basis of the variational principle with mixed energy. To visualise the effectiveness and applicability of the proposed mixed beam element model in vibration analysis, numerical examples have been studied. According to the numerical results, in comparison with the classical beam element model and the beam element models based on the conventional first-order or higher-order shear deformation theory, the vibration frequencies obtained using the mixed beam element model are significantly more satisfactory. Moreover, the comparative study demonstrates that, for vibration analysis of FG sandwich beams, the rational distribution of transverse shear stress is of decisive significance to achieve high-precision solutions. With the mixed beam element used, the load–frequency relationship of FG sandwich beams is discussed. As indicated by the results, in addition to the axial force, the bending moment exerts a significant difference upon the vibration frequencies of the FG beams.

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.

Third-order shear deformation beam model for flexural waves and free vibration of pipes

A third-order shear deformation beam model is proposed to analyze dynamic behavior of straight hollow cylinders of annular cross-section, in which shear stress vanishes on the inner and outer surfaces of the pipe. Shear deformation, warping, and rotational inertia of cross-section are all considered, and the shear correction factor is not needed. A single governing differential equation is derived for analyzing flexural wave propagation and free vibration of straight pipe-beams. The phase and group speeds of flexural waves propagating in pipes are determined for acoustic and optical modes. The dispersion of flexural waves is analyzed. The frequency equations are obtained explicitly for pipe-beams with ten typical boundary conditions including clamped, pinned, guided, and free ends. The natural frequencies of clamped-free, clamped-clamped, and pinned-pinned pipe-beams are evaluated for the first four vibration modes. A comparison of this paper's numerical results of the natural frequencies with the previous ones is made and turns out the effectiveness of the suggested method. The influences of the pipe's thickness and length on the natural frequencies and mode shapes for a cantilever pipe are presented.

Large Deflection of Elastic Beams under Impact by Rigid Particles

The structural response of a slender beam subjected to impact by several rigid particles is investigated. A higher-order shear deformation beam model and the von-Karman geometric measure are employed. The equations of motion, continuity conditions at the points of impact by particles, and geometric and natural boundary conditions for the beam are derived by means of Hamilton’s principle. The generalized differential quadrature and Newmark time integration methods are utilized to carry out the numerical solution of the system of nonlinear partial differential equations comprised of the equations of motion and stress resultants. The durations of contact, displacements, and stress resultants are obtained for beams with various types of boundary conditions.

Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniformly distributed moving mass

The aim of the present work is to develop a high-order shear deformable beam model and investigate the transient response of a sandwich porous beam when acted upon by a non-uniformly distributed moving mass. It is assumed that the total mass is distributed within a specified distance and has various distribution patterns. The sandwich beam is composed of two functionally graded (FG) facesheets and an FG porous core. The modulus of elasticity of the porous core is graded owing to the continuously variation in porosity in the thickness direction. The open-cell metal foam model is used to evaluate the mass density of the core. A new high-order shear deformation theory is proposed and implemented to develop a reliable and accurate model that can address the vibration responses of the beam. The equations of motion are derived using the Lagrange method, whereas a standard Newmark-β method is employed to solve the time-dependent response. A few numerical examples are discussed to study the effects of various sandwich configurations and different types of mass distributions on the vibration behaviors of the sandwich beam. The results indicate that dispersing the mass toward the two ends of the load to the extent feasible is suitable for preventing large dynamic deflections.

A brief history of first-order shear-deformable beam and plate models

This paper presents a brief history of beam and plate models in elasticity, that take into account the rotary inertia and shear contribution. We show that Bresse (1859) already in the mid XIXth century rigorously derived the set of equations for curved shear and axially extensible beams in dynamics, although without shear correction factor. When restricted to straight beams, Bresse (1859) obtained a two-field beam kinematics composed of independent deflection and rotation variables. These equations have been generalized by Timoshenko (1913; 1916; 1920; 1921; 1922), without direct reference to the works of Bresse, by a slight modification of the shear stiffness, that incorporates a shear correction factor generally different from unity. The calibration of the shear correction factor with respect to three-dimensional elasticity solutions has been elaborated jointly by Ehrenfest and Timoshenko in the second decade of the XXth century (the calibration of the shear correction factor for rectangular cross section has been finalized by Timoshenko in 1922, even if the basic results were available in his letter dated from 1913, in collaboration with Ehrenfest). Whereas Bresse-Timoshenko–Ehrenfest equations have been elaborated mainly during more than half a century between 1859 and 1922 (even if researches on shear beam theories were still active), the Uflyand–Mindlin plate theory has been built in a more compact period, mainly between 1944 and 1951. Whereas Reissner (1944, 1945) developed a static (or stress-based) formulation, the kinematic theory of Uflyand-Mindlin plate model has been first elaborated by Bolle (1947) and Hencky (1947) for static setting, before the full generalization to dynamics by Uflyand a year later, in 1948, and its complete variational derivation by Mindlin (1951). A lot of efforts were spent since almost half of the century to calibrate the shear correction factor and to enrich the kinematics of Uflyand-Mindlin plate theory. It is quite surprising that the shear correction factors of both the Bresse–Timoshenko–Ehrenfest beam model and its Uflyand-Mindlin plate analogy were already implicitly available in the paper of Timoshenko (1922), performed, in his own testimony with Ehrenfest.