Transverse Shear Effects Research Articles

Flexoelectric and transverse shear effects on band gaps in periodic microbeams

This paper applies wave equations and boundary conditions to a periodic electro-elastic microbeam, which incorporates transverse shear, microstructure and flexoelectric effects to anticipate the occurrence of band gaps in elastic waves. The new model has been simplified as the flexoelectric model and classical elastic model, as special cases. Meanwhile, compared to Bernoulli-Euler beam model, it is found that transverse shear effect has a significant effect on the band gap predictions of thick beams, but this effect is negligible for slender beams, indicating that this new model can be used to predict band gaps of both thick and slender beams in the high-frequency range. Furthermore, the impacts of microstructure and flexoelectric effects as well as material and microstructural parameters on band gaps are studied. The numerical results show that the influence of flexoelectricity on band gaps is the primary in the submicron beams, while the microstructure effect is the main in the micron beams. Additionally, the band gap frequencies and sizes (bandwidth) are significantly affected by the beam thickness, unit cell length and volume fraction. Thus, these discoveries demonstrate the feasibility of tailoring the band gap frequencies and sizes through the adjustment of material and microstructural parameters.

Quasi-3D flexural analysis of laminated composite plates resting on elastic foundations using higher-order displacement model

This work examines the impact of transverse strains on the static, vibration, and buckling analysis of laminated composite plates that are symmetric and anti-symmetric and rest on elastic foundations. This laminated plate theory, which is called a quasi-3D plate theory since it considers the effects of both transverse shear and normal strains, is provided here. The significance of nonclassical influences on plate deformation, such as shear deformation and thickness stretching, is sufficiently highlighted by the current theory. The theory demonstrates realistic transverse shear stress distributions across the laminated plate's thickness and complies with the traction-free boundary conditions at the upper and bottom surfaces of the plate. Hamilton's principle provides the current theory's equations of motion. For simply supported edges, the laminated plates supported by elastic foundations are investigated. For the aim of verification, the numerical results of the displacements, stresses, frequencies, and buckling load derived using the present theory are, if possible, compared with established literature. The current results and those derived from the other hypotheses found in the literature show a good degree of consistency. Additionally, the distributions of the interlaminar stresses in thickness direction of laminated composite plates sitting on elastic foundations are demonstrated.

A refined shell theory considering the effects of transverse normal strain for the analysis of functionally graded porous shallow shells with double curvature

In this paper, a static and free vibration analysis of functionally graded porous shallow shells with double curvature containing an even distribution of porosity is investigated using a trigonometric shear and normal deformation theory. Both the effects of transverse shear and normal strains are included in the present theory to investigate their effects on the responses of FGM porous shells. The theory satisfies zero transverse shear stress conditions at the top and bottom surfaces of the shell. Hamilton’s principle is used for obtaining governing equations and boundary conditions of the present theory for functionally graded porous shallow shells. The FGM porous shell is assumed isotropic at any point within the shell domain, with its elastic properties varying across its thickness by a power law in terms of the volume fractions of the shell constituents. The simply-supported FG porous shell is analyzed in the present study using the Navier method. The numerical results of FGM porous plates are obtained and compared with other higher-order theories available in the literature to verify the present theory. Further, the numerical results are presented for the FGM porous shells with double curvature. The effects of a/h ratios, porosity distribution factor, and radii of curvature on deflections, stresses, and fundamental frequencies are presented. The solution of hyperbolic and elliptical FGM porous shells is presented for the first time in this paper.

Analysis of nonlinear wave propagation within architected materials consisting of nonlinear Timoshenko beam structural elements

In the present work, a full nonlinear Timoshenko beam employing nonlinear shape functions is developed. The extended Hamilton principle is employed for deriving the differential equations of motion and the associated boundary conditions. The general form of the boundary conditions is then utilized to determine the static solution of the beam motion. Using this solution for the deformation and rotation of the beam, the nonlinear shape functions of the beam are identified, which leads to the linear and nonlinear mass and stiffness matrices of the Timoshenko beam element. The nonlinear dispersion diagram incorporating the non-linear corrections is obtained using the Linstedt–Poincaré perturbation method. An analysis of the effect of internal transverse shear and bending on the nonlinear dispersion characteristics of wave propagation in two-dimensional periodic network materials made of nonlinear Timoshenko beams is done. The formulated theory shows that the percentage of correction factor of the nonlinear kinematics versus the linear dynamical behavior is inversely proportional to the frequency amplitude. The shear and extension modes are shown to have the higher effect in the non-linear correction term in comparison to the flexural mode.

A modified spring-mass model of low-velocity-impact on composite laminate including shearing deformation at delaminated region and reduction on membrane stiffness

A modified spring-mass model for response prediction of composite laminate subjected to destructive large mass impact was developed based on perturbation analysis on membrane stiffness and modified subsection superposition of bending stiffness. The reduction of the membrane effect caused by impact induced intraply matrix failure was evaluated by substituting the perturbated elastic modulus into the membrane stiffness. The subsection superposition method for calculating the bending stiffness of delamination-damaged laminate was modified by considering the transverse shear effect. According to the results of the finite element simulation, the accuracy of the first-order shear deformation theory (FSDT)-based subsection superposition method in evaluating the bending stiffness of delaminated laminates is obviously higher than the classical plate theory (CPT)-based one with the increase of laminate thickness. Destructive drop weight impact tests on the FRP laminate were conducted to validate the proposed model. The results show that the FSDT-based model is more accurate than the CPT-based one in destructive large mass impact response analysis.

Extension of a shell‐like RVE model to represent transverse shear effects in textile reinforced composites

AbstractTextile reinforcements have long been used in shell‐like components like tires, belts, hoses and, in particular, air spring bellows. These bellows consist of a certain number of cords embedded in soft rubber, resulting in high membrane stiffness to absorb tensile forces and low resistance to changes in curvature. The embedded cords are produced by twisting multifilament yarns. Modelling and simulation of bellows is a challenging task. Thereby, the complex internal geometry, large deformations as well as the strongly anisotropic and nonlinear material behavior have to be taken into account in a computationally efficient manner. One approach is the application of multiscale analyses in conjunction with representative volume elements (RVE). Within this study, a through‐thickness RVE based on in‐plane periodic microstructures is applied. The local behavior of the cords is represented by an anisotropic constitutive model that regards directions and mechanical properties of filaments, which result from previous cord twisting simulations. In this contribution, an extension of the classical periodic boundary conditions (PBC) is presented, which considers curvature, pressure load, and, in particular, transverse shear. The RVE is used to simulate the stresses and strains in cord‐rubber composites under realistic load conditions. Moreover, it is employed for homogenizing the mechanical behavior and, thus, for developing a suitable constitutive model for shells.

Analysis of Catapult-Assisted Takeoff of Carrier-Based Aircraft Based on Finite Element Method and Multibody Dynamics Coupling Method

Catapult-assisted takeoff is the initiation of flight missions for carrier-based aircrafts. Ensuring the safety of aircrafts during catapult-assisted takeoff requires a thorough analysis of their motion characteristics. In this paper, a rigid–flexible coupling model using the Finite Element Method and Multibody Dynamics (FEM-MBD) approach is developed to simulate the aircraft catapult process. This model encompasses the aircraft frame, landing gear, carrier deck, and catapult launch system. Firstly, reasonable assumptions were made for the dynamic modeling of catapult-assisted takeoff. An enhanced plasticity algorithm that includes transverse shear effects was employed to simulate the tensioning and release processes of the holdback system. Additionally, the forces applied by the launch bar and holdback bar, nonlinear aerodynamics loads, shock absorbers, and tires were introduced. Finally, a comparative analysis was conducted to assess the influence of different launch bar angles and holdback bar fracture stain on the aircraft’s attitude and landing gear dynamics during the catapult process. The proposed rigid–flexible coupling dynamics model enables an effective analysis of the dynamic behavior throughout the entire catapult process, including both the holdback bar tensioning and release, takeoff taxing, and extension of the nose landing gear phases. The results show that higher launch bar angle increase the load and extension of the nose landing gear and cause pronounced fluctuations in the aircraft’s pitch attitude. Additionally, the holdback bar fracture strain has a significant impact on the pitch angle during the first second of the aircraft catapult process, with greater holdback bar fracture strain resulting in larger pitch angle variations.

Analyses on thermal vibration and stability of sandwich skew plates with functionally graded porous core

In this study, vibration and stability of sandwich skew plates possessing functionally graded porous cores were investigated including significant effect of temperature rise. The modified formulations used for describing the temperature distributions across the plate’s thickness were utilized to carry out the critical buckling temperature and thermos-elastic vibration results. The equations of motion in accordance with the first-order shear deformation theory including the transverse shear effect were solved via the Gram-Schmidt-Ritz method. This method is able to produce accurate solutions of such complex-shaped plates with numerically stable displacement functions generated by Gram-Schmidt procedure for various combinations of boundary conditions. Several important effects such as skew angle, porous coefficient, thickness ratio, aspect ratio, temperature, etc., that have a considerable impact on thermal stability and vibration of such sandwich skew plates are taken into account. According to the numerical experiments, it can be revealed that the critical bucking temperature and frequency of the plates increase considerably with the increase of skew angle. New findings of this study can serve as the benchmarking results for future studies.

Stress analysis of laminated and sandwich beams subjected to concentrated load by using quasi-two-dimensional theory

This study presents a quasi-two-dimensional higher-order shear deformation theory for stress and displacement analysis of isotropic, laminated, and sandwich beams subjected to concentrated load. The assumed displacement field includes the effect of transverse shear and normal deformations. The condition of zero transverse shear stresses on the upper and lower surface of beams is satisfied, hence the present formulation does not require the shear correction factor generally associated with first-order models. By applying the principle of virtual work, the governing equations and boundary conditions for laminated and sandwich beams are derived. Transverse shear and normal stresses are determined from the stress-equilibrium equations of elasticity theory. The transverse stresses obtained using this approach satisfy the continuity condition at layer interfaces and the stress boundary conditions at the external surfaces. The outcomes of the present theory are compared with those of third-order, and first-order shear deformation models, and the classical beam model. The results highlight the significant deviation in displacements and stresses of laminated and sandwich beams when normal strain is included in the model, as compared to the prediction made by lower-order models.

Influence of flexural-torsional couplings and rotational effects on vibration behaviour of tapered thin-walled anisotropic biconvex beam

Accurate modelling and prediction of dynamic characteristics for rotating thin-walled blades have drawn attention as it is an essential part of structural health monitoring for various engineering and civil applications such as wind turbine, helicopter and aeroplane blades. With the advent of structural tailoring techniques, various non-classical effects and couplings implicitly affecting the behaviour of such structures are yet not fully explored. In this regard, the present work proposes a numerical model for finding the free vibration characteristics of anisotropic tapered biconvex closed cross-section beam under rotating conditions. The model is derived based on energy method following the Ritz approximation. Circumferentially asymmetric stiffness (CAS) ply configuration is considered in order to study the impact of flexural torsional coupling on the vibration characteristics. Transverse shear effects, primary and secondary warping were also included in the current model. Mode switching phenomenon is dissected elaborately for the first few frequencies under rotating and nonrotating conditions. The impact of rotational effects like centrifugal acceleration, centrifugal warping, inertial warping on the first few frequencies is also demonstrated. An iterative technique is adapted to solve the equations having these rotational effects. Modal Assurance Criterion (MAC) is done in order to judge the consistency of the modes at different rotating speeds.

Critical velocities of a two-layer composite tube incorporating the effects of transverse shear, rotary inertia and material anisotropy

Critical velocities of a two-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are analytically derived by using a first-order shear deformation shell theory incorporating the transverse shear, rotary inertia and material anisotropy. The composite tube consists of two perfectly bonded axisymmetric circular cylindrical layers of dissimilar materials, which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form expressions for four critical velocities are first derived for the general case by including the effects of transverse shear, rotary inertia, material orthotropy and radial stress. The formulas for composite tubes without the transverse shear, rotary inertia or radial stress effect and with simpler anisotropy are then obtained as special cases. In addition, it is shown that the model for a single-layer, homogeneous tube is included in the current model as a special case. To illustrate the newly derived closed-form formulas, a composite tube with an isotropic inner layer and an orthotropic outer layer is analyzed as an example. The numerical values of the lowest critical velocity of the two-layer composite tube predicted by the new formulas compare well with existing data.

Modelling of laminated glass PVB walls of buildings exposed to vehicle impact with different speeds

This paper presents an analytical model, developed for laminated glass subjected to a low-velocity impact. It has the ability to capture glass cracks as well as large non-linear deformations. It is based mathematically on the first-order deformation concept, which considers the effect of membrane and transverse shear as well as bending. This theory uses damage mechanics to capture the glass cracking. For this purpose, several experiments have been carried out based on PVB laminated glass. The history of acceleration, transverse central displacement and velocity estimated over time is in a favourable relationship with the experimental information. In terms of laminated glass, non-dimensional coefficients have been suggested that regulate both the first peak contact force and the maximum transverse displacement. Laminated glass consists of several layers of soda-lime glass sheets bound together by intermediate layers of polyvinyl butyral (or PVB). Cracking of the glass layer is the main cause of laminated glass damage under both low and high-speed impacts. The main objective of the present article is to conduct experimental studies and numerical analyses of the glass ply cracking mechanism as part of the development of new strength parameters for PVB laminated glass. The non-linear characteristics of PVB are described using the Mooney-Rivlin constitutive model. The present article proves that it is possible to precisely model a wall made of VSG (Verbund Sicherheits Glas) laminated glass reinforced with a vinyl interlayer of appropriate thickness, and further, that such walls can constitute an element absorbing the impact energy of vehicles with specific parameters such as a passenger car, buses, and HGVs (Heavy Goods Vehicle). Based on the results of our study, new parameters were elaborated to determine the properties of PVB laminated glass exposed to vehicle impact. These new parameters were verified qualitatively by comparing the simulation results with experimental observations. We also assessed the strength of a wall of adequate thickness made of laminated glass at the ground floor level of a building exposed to a high-risk terrorist attack. The developed analytical model allows for a quick and reliable assessment during the initial design of safety glass, where a full-scale FE analysis is often too time-consuming.

A failure strength analysis method for fiber reinforcement composite cylindrical shells under hydrostatic pressure based on generalized differential quadrature

In this paper, a numerical model is presented to study the failure strength of moderate thickness composite cylindrical shells under hydrostatic pressure. Considering the transverse shear effect on the mechanics of the moderate thickness cylindrical shell, the numerical model is proposed based on the First-order Shear Deformation Theory (FSDT). The Generalized Differential Quadrature (GDQ) method is applied to solve the 3D displacement and strain field of the static governing equations. Tsai-Wu tensor failure criteria is used to acquire the structural failure strength of the cylindrical shell. The calculated displacement and strain results are compared with the ones of FE analysis and show very good agreement of the distribution. The obtained failure strength based on proposed model is validated with an experimental test and the maximum error of the failure strength is 7.78%. The results and errors indicate that the proposed numerical model is valid and accurate enough for the failure strength analysis of the moderate thickness composite cylindrical shells under hydrostatic pressure.

Buckling analysis of plates using an efficient sinusoidal shear deformation theory

Mechanical buckling response of isotropic and orthotropic plates using the two variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate; hence it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The nonlinear strain-displacement of Von Karman relations are also taken into consideration. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results are presented for the present efficient sinusoidal shear deformation theory, demonstrating its importance and accuracy in comparison to other theories.

Evaluation of static and dynamic responses considering thickness stretching effect for layered composite and sandwich arches using exponential shear and normal deformation theory

Present theory investigates the dimensionless vibration frequencies, deformations, and stresses for multilayered and sandwich arches using exponential shear and normal deformation theory (ESNDT). Present studies consider the effect of [1+(z/R)]radius of curvature while selecting the displacement field of arches under the action of uniform load. It includes the effects of transverse shear (γxz≠0) and transverse normal deformation(εz≠0)i.e., effect of thickness stretching. The governing relations are obtained using the Navier's method, and Hamilton's virtual work principle. The proposed layered arches are completely free from shear correction, and its top and bottom surfaces is satisfies zero traction free end conditions. Present study obtained very accurate natural frequencies for layered sandwich arches than other theories because, available literature not considered the thickness stretching effect i.e.,(εz=0). Hence, the present scientific investigation accounts the effect of thickness stretching i.e.,(εz≠0). However, the non-availability of exact elasticity results of bending and dynamic responses for layered sandwich arches. Present theory is obtained vibration frequencies, displacements, and stresses of layered composite and sandwich arches; the results are compared and validates through prior published literature.

Experimental and numerical investigation of the damage characteristics of a ship side plate laterally punched by a scaled raked bow indenter

Experimental and numerical simulation studies are conducted on a scaled ship side-shell quasi-statically punched at the mid-span by a raked bow indenter to investigate its damage characteristics from deformation to a large opening. A scaled stiffened panel and a raked bow indenter are designed in model tests. Numerical simulation is performed to simulate the indentation responses. The experimental and numerical results are compared well in terms of resistance-penetration curves, final damage shapes and especially the three failure models, i.e., initial fracture, model “I” crack and model “III” crack. In addition, the whole damage process, the energy dissipation characteristics of the specimen and failure related parameters in three specific failed elements corresponding to the three failure models are analyzed through numerical simulation. The results demonstrate that the energy dissipated by the stiffener and frictional effect is substantial during the crack propagation process. In addition, shell elements involved with three failure models prior to fracture are mainly subjected to tension and bending effects, as shell elements cannot satisfactorily simulate the transverse shear effect. Finally, the influences of collision parameters, the failure sequence of integration points, modeling of weld seams, mesh resolution and failure criteria on the numerical simulation results are discussed.

Size-dependent thermal bending of bilayer microbeam based on modified couple stress theory and Timoshenko beam theory

The bilayer microbeam can be assigned with two different thermal expansion coefficients, so that the microbeam can be used as a thermal actuator, which can provide bending with thermal loading. As a microdevice, the size effect plays an important role. In addition, the length height ratio L/h of the microbeam has to be smaller than 20, to provide sufficient support. With this ratio, the shear effect of the beam can not be ignored. In this work, considering both the size and shear effects, we modified the couple stress and Timoshenko beam theories to study the thermal bending of bilayer microbeams. By using the principle of minimum potential energy, we derived the higher-order governing equations under thermal load. Then with specified boundary conditions, we adopted the differential quadrature method to solve the governing equations. Finally, using a bilayer microbeam made of aluminum and polysilicon as an example, we verified effects of transverse shear and beam size. Results show that the size effect on beam stiffness is significant when the beam height is on the order of equivalent length scale (leq). Moreover, considering the transverse shear effect indicates that the lateral deflection of the Timoshenko beam depends on the ratio L/h: as L/h increases, the tip deflection increases and converges to the classical solution. Specifically, for a Timoshenko beam with h/leq = 15, the change in deflection can reach 44.6% when L/h is increased from 5 to 25.

Higher order computational model considering the effects of transverse normal strain and 2-parameter elastic foundation for the bending of laminated panels

In this study, the authors analyze laminated composite panels supported on an elastic foundation considering the effects of transverse normal strain. A 2-parameter, i.e., Winkler and Pasternak foundation model is assumed to represent the interaction between the panels and the foundation. The theory presented here takes into account the effects of transverse shear and normal strains. The theory plots realistic distributions of the transverse shear stress through the plate thickness and satisfies the shear-free conditions at the extreme surfaces of the panel. The differential equations of the present model are obtained from the principle of virtual work. The laminated composite panel resting on the elastic foundation is analyzed for simply supported boundary conditions. For the verification purpose, the presented problems are also solved using the Reddy's model, Mindlin's model, and the classical model. Good agreement is observed between the numerical results obtained using the present model and the other models.

Deflection of an Isotropic All-Round Simply Supported Rectangular Plates Incorporating Shear Effect Using Characteristic Orthogonal Polynomial Function

The breadth and dexterity with which plates are used in the vast majority of engineering structures necessitates an ever-increasing and deeper study focus on plate strength and stiffness at the ultimate and serviceability limit states of response. From Kirchhoff’s hypothesis, thin plates when subjected to transverse loading, bend and experience transverse deflections that are typically minor in comparison to the plate thickness. However, for thicker plates there is an observed limitation in the application of Kirchhoff’s hypothesis, as this theory ignore the effect of transverse shear on the deformation of plates. This study therefore analytically examines the effect of induced shear on the deflection indices of plates with varying aspect ratios, using the characteristic orthogonal polynomial function. Result obtained shows a close agreement between present study and Kirchhoff’s hypothesis for membrane and thin plates. However, a significant difference was observed for moderately thick plates and thick plates, which clearly shows the effect of transverse shear as the plate thickness increases, which further validates the limitations of Kirchhoff’s hypothesis for moderately thick as well as thick plates. For an aspect ratio of 1.0 – 2.0 at 0.1 interval, results obtained indicated a percentage difference in deflection between the Present study and Kirchhoff’s hypothesis to range between -0.040 – 3.508%, 0.527 – 3.552%, 4.266 – 5.858%, and 13.980 – 17.011% for membrane, stiffened, moderately thick, and thick plates respectively. The validation of the Kirchhoff’s hypothesis for membrane as well as stiffened plates by the present study, indicates the suitability of the application of the characteristic orthogonal polynomial function in the evaluation of the deflection of plates regardless of thickness.

On Effects of Shear Deformation on the Static Pull-in Instability Behaviour of Narrow Rectangular Timoshenko Microbeams

MEMS devices utilize electrostatics as preferred actuation method. The accurate determination of pull-in instability parameters (i.e., pull-in voltage and pull-in displacement) of such devices is critical for their correct design. It should be noted that similar to parallel plate capacitors, the electrostatic force between the surface of the deformable microbeam and stationary ground is non-linear in nature. Hence the analysis associated with MEMS devices is always inherently non-linear. In the literature, these devices have been majorly analyzed as Bernoulli-Euler microbeams with cantilever or clamped-clamped beam end conditions. However, Dileesh et al. (doi: 10.1115/ESDA2012-82536) have studied the static and dynamic pull-in instability behavior of slender cantilever microbeams by developing a six-nodded spectral finite element based on the Timoshenko beam theory (TBT-SFE). They have demonstrated the accuracy of the TBT-SFE by comparing their results with corresponding results of COMSOL-based three-dimensional finite element simulations. In addition, effects of shear deformation also start to play significant role as the beam thickness-to-length ratio increases. In this paper, authors have developed the TBT-SFE based on the work by Dileesh et al. for the case of statics. However, unlike Dileesh et al. where they have developed a six-nodded TBT-SFE, authors have investigated the best combination of number of nodes per element and total number of elements to carry out the study. For this purpose, authors have first calculated results of the maximum beam transverse displacement, for a shear deformable propped-cantilever microbeam under the action of uniformly distributed transverse load, obtained by utilizing the developed TBT-SFE with different combinations of number of nodes per element and total number of elements. These results are then compared with corresponding analytical results available in the literature to arrive at the best combination of number of nodes per element and total number of elements for the electrostatic-elastic analysis. In the second step, the finalized TBT-SFE is utilized to determine static pull-in instability parameters of narrow microbeams with various fixity conditions and beam thickness-to-length ratios. This study highlights the importance of transverse shear effects on pull-in instability parameters of Timoshenko microbeams.