Effect Of Shear Deformation Research Articles

Accurate solution for size-dependent free vibration analysis of functionally graded micro-plates with Levy boundary conditions

This paper has developed the Levy solution of a size-dependent model for free vibration analysis functionally graded thin rectangular plate according to the theories of first-order shear deformation as well as modified couple stress. The couple stress theory is used to calculate the effects of small-scale parameter, while the first-order shear deformation theory is employed to calculate the effects of shear deformation. By taking the displacement field according to the Mindlin plate theory, application of the Hamilton principle aims at obtaining five governing equations of motion and desired six boundary conditions. The governing equations of motion have been solved based on Levy boundary conditions by using four auxiliary functions. Numerical findings have been used to determine the impacts of various parameters including small-scale parameter changes, power-law index changes, as well as aspect ratios changes with different boundary conditions. As the results show, an increase in the size effects makes that plate tougher and consequently the natural frequency is increased. Also, it can be seen that in little thicknesses of plate, the effects of small-scale parameter are noticeable, but as the plate becomes thicker, the effects will be insignificant.

Free vibration analysis of multi-layer rectangular plate containing magnetorheological fluid and flexible core layers rested on Winkler-Pasternak foundation

Magneto-rheological (MR) fluids viscosity can be varied by changing the magnetic field intensity. Therefore, they can improve structural rigidity and damping property. The current study presents a free vibration analysis of a multilayer rectangular plate with two layers of MR fluid and a flexible core layer, rested on a Winkler-Pasternak foundation based on exponential shear deformation theory (ESDT). In this theory, the exponential functions are applied in terms of thickness coordinates to include the effect of transverse and inertial rotational shear deformation. The flexible core displacement is modeled via the second-order Frostig model. The Hamilton’s principle is used to derive the equations of motion. These equations are solved using the Navier method to obtain the natural frequencies of the plate. The accuracy of the derived equations is validated, and the obtained results are compared with a specific case. Finally, the results show that by applying and increasing the magnetic field intensity, the natural frequencies and loss factor increase. With the increase in the mode number for each specific magnetic field intensity, the natural frequency increases and the loss factor decreases. The natural frequencies and loss factor decrease with the increase of the MR layer thickness to overall thickness ratio and the flexible core layer thickness to overall thickness ratio. The natural frequencies increase when the parameters of the foundation increase.

Effect of elastic deformations on direct polymorphic transformations in BN under pressure

This paper gives an investigation of the effect of shear deformation on direct transitions in BN under pressure. The transition of a hexagonal (graphite-like) phase h-BN into a cubic (“sphalerite”) c-BN through the intermediate rhombohedral phase r-BN, at a pressure P = 3.0 GPa, is shown. With shear deformation applied, the mechanism of h-BN transformation into the hexagonal (wurtzite) phase w-BN changes, and the following planes are parallel in the phases: for P = 5.0 GPa and deformation ε = 0° (001)h|| (001)w; and for P = 5.0 GPa and deformation ε ≅ 7.2 (001)h|| (100)w. The shear deformation at 6–17 GPa leads to the reverse transition of w-BN to h-BN and the observed orientation relationship is different from the one occurring at the thermal transformation of w-BN to h-BN (T ≅ 1600 K) at a pressure P = 0.1 MPa.

On the static deformation and frequency analysis of functionally graded porous circular beams

Due to the availability of limited literature on functionally graded porous circular beams, the static deformation and free vibration analysis of simply supported functionally graded porous circular beams considering the effects of even and uneven porosities are presented in this paper. The Navier type closed-form solutions are obtained for both static and vibration problems within the framework of the higher-order hyperbolic circular beam theory. The theory considers the effects of transverse shear deformation and rotary inertia. The theory satisfies the transverse shear-stress free boundary conditions at the upper and lower surfaces of the beam. The circular beam is made up of the functionally graded material in which material characteristics vary across the thickness direction using the power-law technique which has been adjusted to approximate the porosity effects with even and uneven distributions. A dynamic version of the virtual work principle is employed to derive the governing differential equations of motion, which are then solved analytically using the Navier solution technique. The non-dimensional values of displacements, stresses, and fundamental frequencies are presented for various values of the power-law index (pi), the radius of curvature (R), porosity volume fraction (α), and type of porosity distribution (even and uneven). The present results are compared in few problems with existing literature and found in good agreement with those. This study also highlights some new results for the reference of future researchers, especially on circular beams.

An analytical model for flexural vibration of composite beams with shear slip based on third order deformation kinematics

In this paper, an analytical solution is derived for accurately predicting the free and forced vibration responses due to time varying as well as moving loads of two layered composite beams which consist of two different materials. For this purpose, the third order deformation kinematics is used in the proposed model, and it allows to take a parabolic (third order) variation of the longitudinal displacement along the beam depth for the two material layers. The partial shear interaction between the two material layers produced by the deformability of shear connectors joining these layers in the form of shear slip at their interface is modelled using distributed shear springs along the beam length. Hamilton’s principle is applied to derive the governing equations of the dynamic system which are solved analytically using a Navier type solution technique. Moreover, a two-dimensional (2D) finite element model is built up in ABAQUS for the validation of the proposed analytical model. Some parametric studies are conducted to investigate the effect of shear deformation on the forced vibration response of a two-layered composite beam partial with different shear stiffness, span-to-depth ratio and elastic-to-shear modulus ratio.

Low frequency band gap for box girder attached IDVAs

Elastic metamaterial is a kind of periodic structures in which the locally resonators are periodically arranged. The most prominent dynamic characteristic of the metamaterials is that they possess the frequency pass bands and band gap. In the band gap, the elastic waves will attenuate significantly and this property is potentially used to control low frequency engineering vibration. Although many efforts have been made to lower the band gap frequency and broaden the band width, the host beams are usually imaginary Euler beam with rectangular section, not Timoshenko beam with box-section in engineering, in which the effects of rotary inertia and shear deformation shall be considered. Moreover, how to design the metamaterial beam with light-weight and low frequency band gap is still a big challenge because the conflict between the frequency band gap and mass of resonators. To address this issue, the inerter-based dynamic vibration absorbers (IDVAs) are adopted to realize the low frequency band gap due to its amplification mechanism. Band gap structure of Timoshenko beam with IDVAs has been developed based on the spectral element method and validated by the test and numerical experiments. And then, a simplify method to predict the starting frequency of band gap has been presented and the band width has been broadened by band gap merging mechanism. Finally, a computer based design method of metamaterial box girder with IDVAs has been presented and applied to control the vertical vibration of footbridge box girder.

Flexural vibration of multistep rotating Timoshenko shafts using hybrid modeling and optimization techniques

In this paper, the flexural vibration of uniform and non-uniform rotating shafts based on the Timoshenko beam theory is investigated. Considering shear deformation and gyroscopic effects to build an exact framework of a solution, the fourth-order differential equation of vibration is solved by an analytical method. The overall transfer matrix of the system formed by a series of flexible distributed in addition to lumped elements is achieved. The results obtained by the distributed lumped modeling technique (DLMT) are verified with other techniques. The damped frequencies obtained by the hybrid modeling method for a multistep gas turbine rotor system using the Euler–Bernoulli and Timoshenko beam theories are compared with the results of the transfer matrix method (TMM). It is shown that the presented method provides highly accurate results, while with no compromise, it can be simply and effectively employed for complex systems. It is also shown how the Hooke and Jeeves and the ant colony optimization (ACO) besides the direct enumeration method can be applied to obtain the damped natural frequencies of complicated vibrational systems such as the gas turbine rotors. The comparison between the frequency and damping ratio values obtained by the optimization and direct enumeration methods shows less than 1% error for different cases.

Finite element formulation for the dynamic analysis of shear deformable thin-walled beams

The present study develops a finite element formulation for the natural vibration and dynamic analysis of thin-walled members with asymmetric cross-sections. The formulation captures shear deformation within the middle surface, global and local warping, viscous damping, translational and rotatory inertial effects, and inertial coupling. The formulation starts with the extended variational form of Hamilton’s principle to account for non-conservative damping effects, in conjunction with a special interpolation scheme aimed at circumventing shear locking. The resulting finite element has two nodes with seven degrees of freedom per node. The discretized equations of motion are integrated using the unconditionally stable Newmark integration scheme with constant average acceleration. For members with doubly symmetric sections, comparisons with shell solutions show that the formulation reliably predicts the natural frequencies and vibration modes associated with flexure/shear and torsion/warping responses. Further comparisons with other non-shear deformable thin-walled beam solutions show that shear deformation effects gain significance in shorter spans, higher vibration modes, and in flexural vibration about the major axis. For monosymmetric and asymmetric sections, comparisons with benchmark solutions showcase the ability of the formulation to capture the inertial coupling arising between the flexure/shear and torsion/warping responses, while keeping the computational cost to a minimum. Comparisons to shell-based solutions demonstrate the ability of the solution to accurately capture the damped time history response of members with doubly symmetric, monosymmetric, and asymmetric sections subjected to harmonic, ramped, and blast-like forces. A separate treatment is provided for degenerate cross-sections with a vanishing global warping function.

On the circumferential wave responses of connected elliptical-cylindrical shell-like submerged structures strengthened by nano-reinforcer

This paper is devoted to estimate the dynamic performance related to Connected Elliptical-Cylindrical Shells (CECSs), which can be applicable in submerged structures. Conducive to progress the vibrational behavior associated with the homogenous CECS, nanocomposite materials including Graphene Nano Platelets (GNP) nano-reinforcer are used. It is obvious that the corresponding mechanical criteria of nanocomposite counting polymer and GNP nano-reinforcer are required for obtaining the governing formulation. Halpin-Tsai and the rule of mixture schemes are utilized for predicting the equivalent density, Poisson's ratio, and Young's modulus. Furthermore, various patterns are registered for the distribution of GNP nano-reinforcer along with the thickness of the CECSs. According to the obtained equivalent mechanical characteristics, by means of Hamilton's concept, the governing motion equations of CECSs are achieved as well. Moreover, the effect of shear deformation is considered subject to the First Order Shear Deformation Hypothesis (FOSDH). Afterward, a professional semi-analytical approach, licensed the Generalized Differential Quadrature (GDQ) scheme, is employed for discretizing the differential equations at different separator points throughout the longitudinal (for the cylindrical part) and meridional (for the elliptical section) direction of the structure. By solving an eigenvalue problem, the natural frequencies of CECSs having different types of Boundary Conditions (BCs) can be evaluated numerically. First, a validation study is performed to prove the correctness and accuracy of the proposed formulation. Then, several novel and complex problems are analyzed numerically and the effects of different parameters including boundary conditions, geometric properties, and material characteristics on the frequency parameter of CECSs are also investigated.

Vibration analysis of laminated planar frame structures

A wave-based model is developed to analyze the free vibrations of symmetrically and asymmetrically laminated cross-ply planar frame structures. The model includes the effects of shear deformation, rotary inertia and the elastic coupling due to structural anisotropy. The frame is modeled as an assemblage of beam segments, whereby each segment acts as a one-dimensional waveguide. The system matrices governing wave propagation, reflection and transmission along the connected segments are derived. An advantage of the proposed model over numerical methods, such as the finite element method, is that exact solutions for the equations of motion are used to derive the system matrices of the frame. The matrices are subsequently assimilated to give a complete description of the wave train present during free vibration. The approach, which is well suited for implementation in conventional computing programs, can be used to determine the exact natural frequencies and mode shapes of the frame. By using the proposed model, the free vibration characteristics of inclined laminated portal frames are analyzed. The effects of stacking sequence, frame angle, relative frame length, orthotropicity ratios and boundary conditions on the exact natural frequencies, and in certain cases the mode shapes, are evaluated and discussed. The numerical results presented in this work are in close agreement with finite element (FE) simulations and where possible, published results.

The effect of the Najd shear deformation on the Pan-African belt of the central Egyptian Nubian Shield: a synthesis for the post-collision tectonic events

This study concerns the structural setting of the central Egyptian-Nubian Shield (El Shalul area) utilizing field-structural, remote sensing, petrological and geochemical data. The exposed basement comprises ophiolitic-mélange, arc-related metavolcanics, metasediments, metagabbro-diorites and granitoids. The area experienced two stages of deformation, pre-Najd (∼850–630 Ma) and Najd-related (∼630–580 Ma). The pre-Najd stage is represented by the assembly of arc-terranes and their N-ward extrusion while the Najd-related stage encompasses three deformation phases. D1 is post-collision extensional event, depicting lateral spreading of tectonic terranes and NW-ejection of ophiolites. The emplacement of El Shalul granite (∼630 Ma) and deposition of molasse sediments in E-W and NW-SE extensional basins (Zeidun and Meesar) are D1- related. Following extension is a protracted phase of compression, shearing and transpression. It was commenced with NW-SE shortening (D2) and deformed the extensional basins by folding and thrusting with mild effect on the other basement units. Sinistral shearing (D3a) and transpression (D3b) along the NW-trending faults superseded the D2 compression, while dextral shearing (D3c) on the ENE-WSW and NE-SW faults overprinted the NW-SE penetrative structures and controlled the emplacement of post-granitic dykes. The significant conclusions of this study include (1) El Shalul granite complex is a large alkaline granitic sheet emplaced during a post-collision extensional regime and suffered the subsequent ∼630–580 Ma top-to-NW sinistral shearing and SW-directed thrusting (not a gneissic core complex) and (2) The NW-SE and NE-SW structural trends are not conjugated.

Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique

Abstract Theoretical and experimental large deflection and stress analysis of a master leaf spring considering stress concentration effect of clamping is reported. The non-uniformly curved master leaf spring under three point bending subjected to moving boundaries is modeled. Geometrically nonlinear strain-displacement relations, as necessary for the theoretical analysis, are derived through visualization of physics behind the large deformation problem. An embedded curvilinear coordinate system is considered, to study the combined effects of non-uniform curvature, bending, stretching and shear deformation including cross-sectional warping. Governing equation of the non-uniformly curved beam system is derived in variational form using energy method, based on linear material constitutive relations and the derived nonlinear kinematic relations. An iterative solution scheme through successive geometry updation is developed and executed in MATLAB® software to solve the governing equation involving strong geometric nonlinearity together with complicating moving boundary effect. Experimental deflection profiles under static loading are obtained through manual image processing technique using AutoCAD® software. Whereas, strain measurements are performed using strain gauges with data acquisition system (HBM-MX840B). Comparison between the theoretical and experimental results lead towards observation on stress concentration effect due to presence of geometric discontinuity in form of a small hole in the physical system. A modified formulation is proposed using domain decomposition method incorporating effect of geometric discontinuity through an equivalent curved beam geometry of variable cross-section. The modified theoretical model is validated successfully with the experimental results, and observations on stress characteristics and effect of hole diameter to beam width ratio are made.

Wave propagation and directionality in two-dimensional periodic lattices considering shear deformations

The effects of shear deformation on analysis of the wave propagation in periodic lattices are often assumed negligible. However, this assumption is not always true, especially for the lattices made of beams with smaller aspect ratios. Therefore, in the present paper, the effect of shear deformation on wave propagation in periodic lattices with different topologies is studied and their wave attenuation and directionality performances are compared. Current experimental limitations make the researchers focus more on the wave propagation in the direction perpendicular to the plane of periodicity in micro/nanoscale lattice materials while for their macro/mesoscale counterparts, in-plane modes can also be analyzed as well as the out-of-plane ones. Four well-known topologies of hexagonal, triangular, square, and Kagomé are considered in the current paper and their wave propagation is investigated both in the plane of periodicity and in the out-of-plane direction. The finite element method is used to formulate the governing equations and Bloch’s theorem is used to solve the dispersion relations. To investigate the effect of shear deformation, both the Timoshenko and Euler-Bernoulli beam theories are implemented. The results indicate that including shear deformation in wave propagation has a softening effect on the band diagrams of wave propagation and moves the dispersion branches to lower frequencies. It can also reveal some bandgaps that are not predicted without considering the shear deformation.

Dynamic analysis on axially functionally graded plates resting on elastic foundation

In this paper, the dynamics of functionally graded plates resting on elastic foundation is investigated using finite element method. The properties of the plate material like modulus of elasticity and material density are considered to be varying exponentially along the axial direction. The elastic foundation is of Winkler type which is modelled as a combination of evenly spaced linear elastic springs. The formulation of the plate is based on the Mindlin plate theory where the effect of rotary inertia and shear deformation are incorporated. The governing equations for the free vibration problem are derived from Hamilton's principle. Two discrete boundary conditions viz. all sides clamped and all sides simply supported are considered. The results of the present methodology are compared with already established results in the literature and good agreement is observed. The effect of different parameters like foundation stiffness and material variation parameter is reported. It is found that the natural frequency of the plate increases with increasing foundation stiffness which suggests that the elastic foundation adds to the stiffness of the system.

A semi analytical method for in-plane free vibrations of arches with a variable curvature

Rayleigh-Ritz method (RRM) is used in this paper in order to investigate the in-plane vibrations of arches with a variable curvature in particular, the frequency parameters and mode shapes. The trial functions are particular solutions of the sixth order differential equation of arch vibrations corresponding to an opening angle equal to one rad. The investigations are made with the follow’s hypotheses: the constant dimensions of the cross-section are small in comparison with the radius, the arch axis is inextensible and the effect of rotary inertia and shear deformation are neglected. The parabolic, catenary, spiral, circular and cycloid arches with different boundary conditions are investigated. The frequency parameters are presented for different types of curved arches with various geometrical properties and boundary conditions and the corresponding mode shapes are plotted. The results are compared with the published literature. The accuracy and relative simplicity of the RRM, applied herein a systematic way with the new choice of trial functions, is established, making it ready to use in more complex situations, such as those of arches with added masses, with non-uniform cross sections or with one or more point supports.

Exact finite beam element for open thin walled doubly symmetric members under torsional and warping moments

Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with those based on exact solution and ABAQUS finite beam element solution at a small fraction of the computational and modelling cost involved.

On the buckling and bending analysis of FG straight-sided quadrilateral nanoplates using a continuum mechanics-based surface elastic model

In this research, an elastic model based on the continuum mechanics is developed to study the static behaviors of functionally graded (FG) arbitrary straight-sided quadrilateral nanoplates. The model is constructed in the framework of Gurtin-Murdoch’s surface and Mindlin’s plate theories to account for the surface energy and shear deformation effects, simultaneously. The variational differential quadrature (VDQ) method is used along with a mapping technique to do the discretization process in a variational framework by means of differential and integral operators. Consequently, a weak form of governing equations is obtained from the energy quadratic representation of the problem. The solution method is of a distinguished feature as it involves just the first-order derivative of the field components in the mapping and discretization. After assuring the effectiveness of presented model by doing comparative studies, the critical buckling load and static deflection of the FG nanoplates with different shapes in geometry are investigated considering the surface effects. It is found that the surface energies effect on the static behavior of the rectangular nanoplates is more significant as compared to the non-rectangular nanoplates.

Computational analysis of shear deformation effects on open thin-walled beams

When subjected to eccentric lateral or vertical loads, Open Thin-Walled Members (OTWMs) will suffer the risk of torsional failure for their relatively low torsional rigidity. The classic torsional theory, Vlasov’s theory, was found to greatly underestimate the torsional rotation especially for OTWMs with lower span-depth ratio or lower shear modulus. Shear deformation along the middle line of cross section, which is neglected in Vlasov’s theory, was considered and quantified in this paper. Two examples (U-beams and I-beams) were taken to be calculated by the proposed method, Vlasov’s theory and finite element analysis. Results of average errors 2.90% and 1.77% guaranteed the applicability and accuracy of the proposed method. Besides, parametric analysis including different configurations, boundary conditions and span-height ratio were conducted to evaluate the effects of shear deformation on the torsional rotation of OTWMs. It was revealed that stronger boundary constraint and smaller span-depth ratio lead to greater effects of shear deformation, which cannot be neglected. The critical criteria to consider or ignore the shear effects under different boundary conditions were also defined.

FREE VIBRATION OF SKEW LAMINATES – A BRIEF REVIEW AND SOME BENCHMARK RESULTS

This study investigates and reviews prior research works on skew composite laminates. The equivalent single layer theories are explored and discussed. An exhaustive review on static and dynamic analysis of composite skew laminates is also presented. Subsequently, a nine node isoparametric plate bending element is used for free vibration analysis of laminated composite skew plate with central skew cut out. The effect of shear deformation is incorporated in the formulation considering first order shear deformation theory. Two types of mass lumping schemes are analysed to study the effect of rotary inertia. Certain numerical examples of plates having different skew angles, skew cut out sizes, boundary conditions, thickness ratios (h/a), aspect ratios (a/b), fiber orientations and number of layers are solved which will be useful for benchmarking of future studies.

A new practical method for the flexural analysis of thin-walled symmetric cross-section box girders considering shear effect

This study proposes a new analytical flexure theory that considers the shear deformation of the web and flange of a thin-walled box girder. Starting with the relationship between the strain and displacement of each slab, the flexural displacement function of thin-walled box girders considering the shear deformation effect is derived by using the deformation continuity condition. Compared with the traditional flexural displacement model, this function obtained by the theoretical analysis is more rigorous and can be transformed into the flexural displacement function of the traditional shear lag or Timoshenko beam by eliminating some parameters. Then, according to the balance conditions of axial force and bending moment, the shear deflection deformation state and Euler–Bernoulli beam deflection deformation state are decoupled. On this basis, a new convenient method for calculating the warping stress and shear deflection of thin-walled girders is proposed. Furthermore, a simplified analysis method for converting continuous box girders into simply supported beams is proposed on the basis of continuity condition of shear deflection at middle supports. Taking into account the shear deflection deformation state, a beam-type finite element model considering the shear deformations of each box wall is developed by using Hermite polynomials to analyze the complex variable cross-section box girder. Finally, numerical examples are used to verify the effectiveness of the simplified method and the beam-type finite element model. Numerical examples show that the flexural stress and deflection calculated by the proposed method agree well with the three-dimensional finite element model analysis results. The shear warping normal stress has a larger value at the shear force discontinuity point. In the control cross-section, the normal warping stress and shear deflection caused by shear deformation both exceed 75% and 38% of stress and deflection of the Euler–Bernoulli beam, respectively.