Differential Quadrature Method for Bending Problem of Plates with Transverse Shear Effects

A differential quadrature (DQ) method for orthotropic plates was proposed based on Reddy' s theory of plates with the effects of the higher-order transverse shear deformations. Wang-Bert's DQ approach was also further extended to handle the boundary conditions of plates. The computational convergence was studied, and the numerical results were obtained for different grid spacings and compared with the existing results. The results show that the DQ method is fairly reliable and effective.

DIFFERENTIAL QUADRATURE METHOD FOR BENDING OF ORTHOTROPIC PLATES WITH FINITE DEFORMATION AND TRANSVERSE SHEAR EFFECTS

Based on the Reddy's theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature (DQ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert (DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitudefrequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.

THE EFFECT OF TRANSVERSE SHEAR DEFORMATION ON STRESS CONCENTRATION FACTORS FOR SHALLOW SHELLS WITH A SMALL CIRCULAR HOLE

Simplified equations are derived for the analysis of stress concentration for shear-deformable shallow shells with a small hole. General solutions of the equations are obtained, in terms of series, for shallow spherical shells and shallow circular cylindrical shells with a small circular hole. Approximate explicit solutions and numerical results are obtianed for the stress concentration factors of shallow circular cylindrical shells with a small hole on which uniform pressure is acting.

EFFECT OF TRANSVERSE SHEAR ON FINITE DEFORMATION OF GENERALLY LAMINATED SHALLOW SHELL WITH DOUBLE CURVATURE

Nonlinear equations of equilibrium for the titled shell of rectangular planform under transverse and inplane edge loads are derived by using the virtual work principle and expressed in terms of a stress function,the transverse displacement and two rotation functions.The sheU is elastically re- strained against rotation.A generalized double Fourier series solution is formulated for nonlinear bending of the shell.The Galerkin technique furnishes an infinite set of simultaneous nonlinear alge- braic equations for the above four variables,which can be truncated to obtain any desired degree of ac- curacy.Numerical results for antisymmetrically laminated angle-ply and cross-ply graphite-epoxy doubly curved panels are presented graphically for the transverse shear effect and various shell parame- ters and boundary conditions.The present results are also compared with available data.

Structural Characteristics of the Spring Transverse Shear Line over the Qinghai-Tibet Plateau and the Influence Mechanism of the Upper-level Jet on Its Evolution

Based on the four-times-daily ERA-Interim data with the resolution of 0.75°×0.75°,the structure and evolution characteristics of a transverse shear line(TSL)over the Qinghai-Tibet Plateau in April 2017 were analyzed,and the influence mechanism of the frontogenesis and frontolysis caused by the upper-level jet on its evolution was also investigated.The results show that the TSL was mainly located near the axis of the positive vorticity zone,which was a low-value area of the wind speed.It was a shallow baroclinic system with weak ascending motion.In the vertical direction,the TSL extended to the lowest height at 00:00 and the highest at 18:00.In the horizontal direction,the length of the TSL in the east-west direction was relatively shorter during 00:00-06:00 and relatively longer during 12:00-18:00.Besides,the position of the TSL was slightly northward at 06:00 and slightly southward at 18:00.The moving direction of the TSL was generally consistent with that of the upper-level jet.In addition,the vertical stretching height of the TSL and the near-surface wind speed were positively correlated with the intensity of the upper-level jet.The calculation by frontogenesis function indicates that the frontogenesis(frontolysis)was conducive to the formation(weakening)and strengthening(dissipation)of the TSL.The horizontal deformation-induced and diabatic heating-induced frontogenesis were favorable for the formation of the TSL,while the middle-level horizontal convergence-induced and diabatic heating-induced frontogenesis were beneficial to its maintenance.Besides,the moving direction and baroclinicity of the TSL over the Qinghai-Tibet Plateau were determined by the horizontal deformation-induced frontogenesis.In the frontogenesis function,the terms of horizontal deformation and horizontal convergence together determined the position of the TSL,and the diabatic heating term was conducive to the upward extension of the TSL.

Theory of Flexural Shear, Bending and Torsion for a Thin-Walled Beam of Open Section

Aspects of the general Vlasov theory are examined separately as applied to a thin-walled channel section cantilever beam under free-end end loading. In particular, the flexural bending and shear that arise under transverse shear and axial torsional loading are each considered theoretically. These analyses involve the location of the shear centre at which transverse shear forces when applied do not produce torsion. This centre, when taken to be coincident with the centre of twist implies an equivalent reciprocal behaviour. That is, an axial torsion applied concentric with the shear centre will twist but not bend the beam. The respective bending and shear stress conversions are derived for each action applied to three aluminium alloy extruded channel sections mounted as cantilevers with a horizontal principal axis of symmetry. Bending and shear are considered more generally for other thin-walled sections when the transverse loading axes at the shear centre are not parallel to the section = s centroidal axes of principal second moments of area. The fixing at one end of the cantilever modifies the St Venant free angular twist and the free warping displacement. It is shown from the Wagner-Kappus torsion theory how the end constrained warping generates an axial stress distribution that varies with the length and across the cross-section for an axial torsion applied to the shear centre. It should be mentioned here for wider applications and validation of the Vlasov theory that attendant papers are to consider in detail bending and torsional loadings applied to other axes through each of the centroid and the web centre. Therein, both bending and twisting arise from transverse shear and axial torsion applied to each position being displaced from the shear centre. Here, the influence of the axis position upon the net axial and shear stress distributions is to be established. That is, the net axial stress from axial torsional loading is identified with the sum of axial stress due to bending and axial stress arising from constrained warping displacements at the fixing. The net shear stress distribution overlays the distributions from axial torsion and that from flexural shear under transverse loading. Both arise when transverse forces are displaced from the shear centre.

Climatology of Transverse Shear Lines Related to Heavy Rainfall over the Tibetan Plateau during Boreal Summer

Based on ERA-Interim data and precipitation data of 2474 stations in China during May-October from 1981 to 2013, transverse shear lines （TSLs） were identified, and their climatic characteristics and association with torrential rainfall events over the Tibetan Plateau and the region to its east during boreal summer were analyzed statistically, based on three criteria： the meridional shear of zonal wind, the relative vorticity, and the zero contour line of zonal wind. It was found that TSLs are generally west east oriented over the Tibetan Plateau, with the highest occurrence frequency in June, and least occurrence in October. The high frequency axis of TSLs, parallel to the terrain of the Tibetan Plateau, shifts southward from May to August, and then slightly northward from September to October. The annual average TSL frequency is 65.3 days, and there are obvious interannual and interdecal variations of TSLs. The annual fluctuation of TSL frequency is most distinct in the 1980s, followed by the 2000s, with average frequency appearing during 1995 2000. It was found that the occurrence frequency of TSLs and that of heavy rainfall events over the Tibetan Plateau are stable during 1981-2013. However, the occurrence frequency of the heavy rainfall events resulting from TSLs is decreasing. More than 50% of the TSLs can lead to heavy rainfall, while 40% of the heavy rainfall events are caused by TSLs. TSLs are closely related to heavy rainfalls in the flooding season of June-August over the Tibetan Plateau.

SNAP－THROUGH OF LAMINATED ORTHOTROPIC SHALLOW SPHERICAL IMPERFECT SHELLS WITH TRANSVERSE SHEAR

The equations of large deformations of laminated orthotropic spherical shellsare derived. The effects of transverse shear deformation and initial imperfection are considered. on this basis. the semi-analytical solution of the axisymrnetric snap-throughbuckling of laminated orthotropic shallow spherical shells under uniform pressure is obtained using orthogonal collocation method. The effects of material parameters, structuralparameters, initial imperfection and transverse shear deformation are discussed.

DYNAMIC RESPONSE OF LAMINATED ORTHOTROPIC SPHERICAL SHELLSINCLUDING TRANSVERSE SHEAR DEFORMATION AND ROTATORY INERTIA

In this paper.the equations of motion of axisymmetrically laminated cylindrical orthotropic spherical shells are derived.Theeffects of transverse shear deformation and rotatory inertia are considered.On this basis,the dynamic response of spherical shells under axisymmetric dynamic load is calculated using the finite difference method The effects of material parameters.structural parameters and transverse shear dgformation are discussed.

Toplayer-dependent crystallographic orientation imaging in the bilayer two-dimensional materials with transverse shear microscopy

Nanocontact properties of two-dimensional(2D)materials are closely dependent on their unique nanomechanical systems,such as the number of atomic layers and the supporting substrate.Here,we report a direct observation of toplayer-dependent crystallographic orientation imaging of 2D materials with the transverse shear microscopy(TSM).Three typical nanomechanical systems,MoS_(2) on the amorphous SiO_(2)/Si,graphene on the amorphous SiO_(2)/Si,and MoS_(2) on the crystallized Al_(2)O_(3),have been investigated in detail.This experimental observation reveals that puckering behaviour mainly occurs on the top layer of 2D materials,which is attributed to its direct contact adhesion with the AFM tip.Furthermore,the result of crystallographic orientation imaging of MoS_(2)/SiO_(2)/Si and MoS_(2)/Al_(2)O_(3) indicated that the underlying crystalline substrates almost do not contribute to the puckering effect of 2D materials.Our work directly revealed the top layer dependent puckering properties of 2D material,and demonstrate the general applications of TSM in the bilayer 2D systems.

TWO COPLANAR CRACKS IN A TRANSVERSELY ISOTROPIC ELASTIC SLAB SUBJECTED TO ANTIPLANE SHEAR STRESSES

The problem of a transversely isotropic elastic slab containing two coplanar cracks subjected to an antiplane deformation is considered. With the aid of an integral transform technique, we formulate the problem in terms of a finite-part singular integral equation which can be solved numerically, Once the integral equation is solved, relevant quantities such as the crack energy can be readily computed.

Dynamic Stability of Viscoelastic Plates with Finite Deformation and Shear Effects

Based on Reddy's theory of plates with higher order shear deformations and the Boltzmann superposition principles, the governing equations were established for dynamic stability of viscoelastic plates with finite deformations taking account of shear effects. The Galerkin method was applied to simplify the set of equations. The numerical methods in nonlinear dynamics were used to solve the simplified system. It could be seen that there are plenty of dynamic properties for this kind of viscoelastic plates under transverse harmonic loads. The influences of the transverse shear deformations and material parameter on the dynamic behavior of nonlinear viscoelastic plates were investigated.

Climatology of Shear Line and Related Rainstorm over the Southern Yangtze River Valley Based on an Improved Intelligent Identification Method

Based on four reanalysis datasets including CMA-RA,ERA5,ERA-Interim,and FNL,this paper proposes an improved intelligent method for shear line identification by introducing a second-order zonal-wind shear.Climatic characteristics of shear lines and related rainstorms over the Southern Yangtze River Valley(SYRV)during the summers(June-August)from 2008 to 2018 are then analyzed by using two types of unsupervised machine learning algorithm,namely the t-distributed stochastic neighbor embedding method(t-SNE)and the k-means clustering method.The results are as follows:(1)The reproducibility of the 850 hPa wind fields over the SYRV using China’s reanalysis product CMARA is superior to that of European and American products including ERA5,ERA-Interim,and FNL.(2)Theory and observations indicate that the introduction of a second-order zonal-wind shear criterion can effectively eliminate the continuous cyclonic curvature of the wind field and identify shear lines with significant discontinuities.(3)The occurrence frequency of shear lines appearing in the daytime and nighttime is almost equal,but the intensity and the accompanying rainstorm have a clear diurnal variation:they are significantly stronger during daytime than those at nighttime.(4)Half(47%)of the shear lines can cause short-duration rainstorms(≥20 mm(3h)^(-1)),and shear line rainstorms account for one-sixth(16%)of the total summer short-duration rainstorms.Rainstorms caused by shear lines are significantly stronger than that caused by other synoptic forcing.(5)Under the influence of stronger water vapor transport and barotropic instability,shear lines and related rainstorms in the north and middle of the SYRV are stronger than those in the south.

APPROXIMATE VIBRATION ANALYSIS OF LAMINATED CURVED PANEL USING HIGHER-ORDER SHEAR DEFORMATION THEORY

An approximate analysis for free vibration of a laminated curved panel(shell)with four edges simply supported(SS2),is presented in this paper.The transverse shear deformation is considered by using a higher-order shear deformation theory.For solving the highly coupled partial differential governing equations and associated boundary conditions,a set of solution functions in the form of double trigonometric Fourier series,which are required to satisfy the geometry part of the considered boundary conditions,is assumed in advance.By applying the Galerkin procedure both to the governing equations and to the natural boundary conditions not satisfied by the assumed solution functions,an approximate solution,capable of providing a reliable prediction for the global response of the panel,is obtained.Numerical results of antisymmetric angle-ply as well as symmetric cross-ply and angle-ply laminated curved panels are presented and discussed.

SHEAR EFFECTS ON LARGE AMPLITUDE FORCED VIBRATIONOF SYMMETRICALLY LAMINATED RECTILINEARLYORTHOTROPIC CIRCULAR PLATES

In this paper, nonlinear forced vibration of symmetrically laminated rectilinearly orthotropic circular plates excited by a harmonic force q(0)cos omega t including effects of transverse shear deformation is discussed. The analytical solution for the relationship between forcing frequency and amplitude of vibration is obtained by Galerkin's method. Finally, the paper analyses the effect of the transverse shear on the vibration of the plate and gives the ratio of nonlinear period to linear period for nonlinear free vibration of the plate.

BENDING THEORIES FOR BEAMS AND PLATES WITH SINGLE GENERALIZED DISPLACEMENT

Classical bending theories for beams and plates can not be used for short, stubby beams and thick plates since transverse shearing effect is excluded, and ordinary theories with multiple generalized displacements can not be used for long, slender beams and thin plates since the innate relation between rotation angle and deflection is ignored. These two types of theories are not consistent due to the contradiction of dependence and independence of the rotation angle. Based on several basic assumptions, a new type of theories which not only include the transverse shearing effect is presented, but also the relation between potation angle and deflection is obtained. Analytical solutions of several simple beams are given. It has been testified by numerical examples that the new theories can be used for either long, slender beams and thin plates or short, stubby beams and thick plates.

SOLITARY WAVES IN FINITE DEFORMATION ELASTIC CIRCULAR ROD

A new nonlinear wave equation of a finite deformation elastic circular rod simultaneously introducing transverse inertia and shearing strain was derived by means of Hamilton principle. The nonlinear equation includes two nonlinear terms caused by finite deformation and double geometric dispersion effects caused by transverse inertia and transverse shearing strain. Nonlinear wave equation and corresponding truncated nonlinear wave equation were solved by the hyperbolic secant function finite expansion method. The solitary wave solutions of these nonlinear equations were obtained. The necessary condition of these solutions existence was given also.

An advanced higher-order theory for laminated composite plates with general lamination angles

This paper proposes a higher-order shear deformation theory to predict the bending response of the laminated composite and sandwich plates with general lamination configurations.The proposed theory a priori satisfies the continuity conditions of transverse shear stresses at interfaces.Moreover,the number of unknown variables is independent of the number of layers.The first derivatives of transverse displacements have been taken out from the inplane displacement fields,so that the C 0 shape functions are only required during its finite element implementation.Due to C 0 continuity requirements,the proposed model can be conveniently extended for implementation in commercial finite element codes.To verify the proposed theory,the fournode C 0 quadrilateral element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate.Numerical results show that following the proposed theory,simple C 0 finite elements could accurately predict the interlaminar stresses of laminated composite and sandwich plates directly from a constitutive equation,which has caused difficulty for the other global higher order theories.

Geometrically exact nonlinear analysis of pre-twisted composite rotor blades

Modeling of pre-twisted composite rotor blades is very complicated not only because of the geometric non-linearity, but also because of the cross-sectional warping and the transverse shear deformation caused by the anisotropic material properties. In this paper, the geometrically exact nonlinear modeling of a generalized Timoshenko beam with arbitrary cross-sectional shape,generally anisotropic material behavior and large deflections has been presented based on Hodges＇ method. The concept of decomposition of rotation tensor was used to express the strain in the beam. The variational asymptotic method was used to determine the arbitrary warping of the beam cross section. The generalized Timoshenko strain energy was derived from the equilibrium equations and the second-order asymptotically correct strain energy. The geometrically exact nonlinear equations of motion were established by Hamilton＇s principle. The established modeling was used for the static and dynamic analysis of pre-twisted composite rotor blades, and the analytical results were validated based on experimental data. The influences of the transverse shear deformation on the pre-twisted composite rotor blade were investigated. The results indicate that the influences of the transverse shear deformation on the static deformation and the natural frequencies of the pre-twisted composite rotor blade are related to the length to chord ratio of the blade.