In-plane Stress Distribution Research Articles

Forced Vibration Analysis of Composite Laminates Using a Discrete Layer Stress Based Model

Although modern displacement based plate theories are capable of predicting the deflected shape and lower frequencies of a loaded plate with a high degree of precision, their ability to predict stresses, particularly interlaminar stresses, has been poor. A discrete layer stress based theory for composite laminates that assumes a linear distribution of in-plane stresses across each lamina was presented and showed good prediction of interlaminar stresses. The present study extends that work to forced vibration response of a composite plate subjected to a time varying transverse load.

Some experimental findings in compression-after-impact (CAI) tests of CF/PEEK (APC-2) and conventional CF/epoxy flat plates

Compression-after-impact (CAI) tests have been conducted for quasi-isotropic thick plates with 48 plies by using the NASA method and on plates with 32 plies by using the SACMA method. Specimens are made of CF/PEEK (APC-2) and conventional CF/epoxy for the NASA plates and CF/epoxy for the SACMA plates. In the NASA CAI tests, the sequence of delamination buckling and its propagation is clearly revealed through various experimental techniques. One major technique is moirétopography, and the other is thermo-mechanical stress analysis with a high-accuracy infrared sensor. The arrest of delamination propagation just before catastrophic failure due to high fracture toughness is clearly captured by the moirécamera. This behavior provides good CAI values of CF/PEEK. The initial buckling properties of the delaminated region by the impact are then extensively discussed. Numerical predictions of initial buckling stress have been obtained by modelled geometry of the delaminated region simplified from its precise structure clarified by ultrasonic C-scanning. They agree fairly well with the experimental results. The in-plane stress distribution in the delaminated region before initial buckling is measured by an infrared stress graphic system. This compared favorably with finite element predictions. Two types of symmetric buckling modes with respect to the central plate surface, twin and single peak ones, are experimentally captured.

Stress-based elastodynamic discrete laminated plate theory

A static laminated plate theory based on an assumed piecewise linear through-the-thickness in-plane stress distribution has been extended to include inertia effects. Based on this in-plane stress distribution assumption, outof-plane shear and normal stress component distributions were derived from the three-dimensional equations of motion, resulting in six nonzero stress components. Hamilton's variational principle was used to derive the plate equations of motion, the plate constitutive relationships, and the interface continuity equations. The governing equations were written in a form that is self-adjoint with respect to the convolution bilinear mapping. The resulting system of equations for a single lamina consists of 25 field equations in terms of 9 weighted displacement field variables, 10 stress and moment resultant field variables, and 6 out-of-plane shear and normal stress boundary field variables. For the laminated system, the mixed formulation enforces both traction and displacement continuity at lamina interfaces as it satisfies layer equilibrium. A finite element formulation based on a specialized form of the governing functional was developed. The method is illustrated with results of a free vibration analysis of sandwich and homogeneous plates for which exact solutions are available.

Stress Analysis of a Silicon-Wafer Slicer Cutting the Crystal Ingot

In-plane stress distribution in a spinning free-clamped slicing blade is studied analytically. The blade is subjected to a stationary, distributed, in-plane slicing load along the inner periphery, while it is clamped and tensioned initially in the radial direction along the outer boundary. The radial, hoop, and shear stresses in the blade are given by sum of the stresses from the tensioning, spinning, and edge loading. The solution is obtained by introducing the Laˆme potential functions and the two-dimensional elasticity theory. Numerical results are presented for an actual SUS301 blade cutting a 6-in.-diameter silicon ingot at 1550 rpm. Results obtained show that the contribution of the slicing load on the stress distribution is not significant; further the overall stress distribution is approximated by that of the initial tensioning except for the case of the shear stress distribution.

Delamination Characteristics around Hole in Laminates with Softening Strip

Graphite/epoxy laminates with and without a softening strip around a hole were investigated. The softening strip is a portion of the graphite critical ply, which is replaced by a glass/epoxy composite. Analysis was conducted with finite element methods: The in‐plane stress distribution was found using a planform two‐dimensional model and interlaminar stress distributions around a hole were obtained from a through‐thickness quasi‐three‐dimensional model. The variation of strain‐energy release rate for a delamination occurring at the critical interface was investigated. The finite element results indicate that the inplane and interlaminar stress distributions, as well as the strain‐energy release rates, are significantly reduced for laminates with a softening strip.

Free Vibration and Tension Buckling of Circular Plates With Diametral Point Forces

This paper presents the first known results for the free vibrations of a circular plate subjected to a pair of static, concentrated forces acting on the boundary at opposite ends of a diameter. The closed-form exact solution of the plane elasticity problem is used to provide the in-plane stress distribution for the vibration problem. A proper procedure using the Ritz method is developed for solving the latter problem for clamped, simply supported, or free boundary conditions. Numerical results are given for the vibration frequencies of a simply supported circular plate, which separate into four symmetry classes of mode shapes. Compressive buckling loads for each symmetry class are determined as a special case as the frequencies decrease to zero with increasing compressive force. Tracking the frequency versus loading data with increasing tensile forces shows that buckling due to tensile force can also occur, and the critical value of the force is found.

Buckling behavior of a central cracked thin plate under tension

The buckling characteristics of cracked plates subject to uniaxial tensile loads are analysed by the aid of the finite element method. Owing to the fact that crack buckling behavior is affected by the in-plane stress distribution around a crack, to get more accurate results, pre-buckling in-plane stress fields are analysed by the finite element method. For the critical loads calculation, the finite element approach adopted is based on Von Karman's linearize theory for buckling of plates subjected to pre-buckling state of plane stress. Several singular elements based on the Willian series are used in this plate bending approach. In this study, the effect of crack length, the effect of boundary condition, the effect of Poison's ratio and the effect of biaxial force on critical loads are analysed and discussed. Furthermore, the effect of initial imperfection is also discussed. There is a good agreement between other researcher's work and present results.

Large deflection analysis of flat plate using nonlinear programming theory. 3rd Report. Axisymmetric annular plates with constant thickness and with simply supported edges.

The large deflection of the annular plates with simply supported boundary conditions is analyzed by the finite element method using the nonlinear programming theory. Both fixed-hinged and movable-hinged supports are taken into account, while the line loads or the uniform distributed loads are applied. The distributions of the out-of-plane and in-plane displacements and stress distributions in the radial cross section are presented for some load ratios in six cases. It is concluded that the out-of-plane displacements with movable-hinged supports were smaller than those with the fixed-hinged support, and that the shrinkages between the inner and outer edges in the radial direction were almost the same under the two simply supported boundary conditions.

Three-dimensional effects in beams: Isodyne assessment of a plane solution

The stress distribution in a homogeneous beam subjected to three-point bending is investigated using the method of optical isodynes. The three stress componentsσxx,σyy andτxy acting in the planes formed by the longitudinal and vertical axes of the beam are determined in three planes situated at different through the thickness locations with respect to the beam's midplane. The experimental results are subsequently correlated with the two-dimensional elasticity solution. It is illustrated that at locations sufficiently removed from the centrally applied concentrated load, good correlation between theory and experiment is obtained. In the regions where high stress gradients exist however, differences are observed in the in-plane stress distributions in the different planes. These differences are explained by the presence of the out of plane normal stressσzz using the relations of optical isodynes. Greatest differences between theory and experiment are obtained for the in-plane shear stress componentτxy.

Flexural vibration of rotating rectangular plates of variable thickness

Free vibration of rotating rectangular plates with an exponentially varying thickness is studied. The plate is clamped at the rim of a central hub and is free along the other three edges. In the general formulation, the plate is considered to be made of orthotropic materials. The differential equations governing the in-plane displacements before vibration takes place are solved by a finite difference method. The curve-fitted previbratory in-plane stress distribution based on the numerical results is used in the equation for the flexural vibration of the plate, and the natural frequencies are subsequently determined by the extended Galerkin procedure. Some numerical results are presented for illustrative purposes.

The elastic wrinkling of rectangular sheets

Two models for predicting the elastic behaviour of rectangular sheets pulled in tension are investigated. In one, a parabolic loading distribution is assumed; in the other, the normal displacement is taken to be uniform on the loaded edges. Collocation techniques are used to find the wrinkling modes and the associated eigenvalues for the tensile loads. In the second problem the in-plane stress distribution is found using finite difference procedures. The stress singularities in the corners are allowed for. Buckling under compression is also studied for both models.

Stress Analysis of a Spinning Annular Disk to a Stationary Distributed, In-Plane Edge Load

In-plane stress distribution in a spinning annular disk is studied analytically. The disk is clamped at the inner boundary and subjected to a stationary distributed load along the outer boundary. The solution is determined through a Galilean transformation by using the Lame´ potential functions and the two-dimensional elasticity theory. Numerical results are presented for WA vitrified grinding wheels with outer radii of 50 mm and inner radii of 25 mm rotating at 3000 rpm and 15000 rpm.

The natural frequencies of in-plate stressed rectangular plates

The stress distributions in some practical in-plane loaded plates have been obtained either directly via strain gauges or indirectly from the measurement of transverse deflections or initial imperfection profiles. These stress distributions have been incorporated into a purpose-written finite difference computer program set up to evaluate the natural frequencies of the plates. A comparison has been made between these frequencies and those measured directly in the laboratory. A further comparison has been made with the natural frequencies calculated on the assumption of a uniform in-plane stress distribution. The results explain, in part, the discrepancies between theory and experiment that have been observed by previous workers.

THE FINITE DISPLACEMENT BEHAVIOR OF HORIZONTALLY CURVED ELASTIC I-SECTION PLATE GIRDERS UNDER BENDING

A study of behavior of curved plate girders under bending is made, emphasizing mainly the interactions between the flange plates and the web plate. The analyses are carried out by using finite element method based on the finite displacement theory. In this paper, the behavior of curved plate girders under bending is described, such as out-of-plane displacemnent of web plate, in-plane stress distribution and corresponding effective web width, resisting moment of girders and so on. The results shows that so called vertical buckling of flange plate does not arise from loss of the bending capacity of the plate girder. Moreover discussions about the more reasonable boundary conditions of the web panel as partial structural element for analyzing curved plate girders are described.

鋼構造部材構成板要素相互の弾塑性連成局部座屈挙動の解析法

In this paper, a geometric and material nonlinear analytical method by the finite element method is presented in order to understand the maximum strength and elasto-plastic postbuckling behavior of plate elements of structural members. And then a simplified numerical method for coupled plate elements is established. The latter method is considered that plate finite element is used for the first buckling plate element and the restraining plate element is replaced with equivalent spring stiffness. In-plane stress distribution is constant because a constant strain and curvature triangular finite element is used in the present method. Therefore the progress of a plastic range has to be investigated only along the depth of the finite element divided into a number of layers. The validity of the present method is examined by the experiment on H-section stub-columns under uni-axial compression. In comparing the method using equivalent spring stiffness with the method using only plate finite element, elastic load-deflection curves are in good agreement, although the difference between them becomes large in the more advanced postbuckling range. As the maximum strength and plastic behavior after it can be understood by pursuing postbuckling behavior in the range of small deflection, it is shown that the method using equivalent spring stiffness is efficient in elasto-plastic problems.

Finite element analysis of rotating disks

A finite element formulation is presented for the analysis of rotating disks in either a body-fixed or a space-fixed co-ordinate system. The in-plane stress distribution resulting from the in-plane body force due to rotation is determined first by a plane stress finite element analysis. This stress distribution is then used in calculating the out-of-plane geometric stiffness which in turn is added to the linear bending stiffness. In the space-fixed co-ordinate system, inertia and a viscous type damping also contribute to the out-of-plane stiffness, even in the steady state case. The formulation presented here places no restrictions on the disk geometry if the problem is solved in a body-fixed co-ordinate system, although only disks of axisymmetric geometry may be considered in the space-fixed co-ordinate system. A direct method of determining the critical speeds through an eigenvalue analysis in space-fixed co-ordinates is presented. Then the undamped steady state response to a space-fixed transverse point load is examined. The effects of a viscous type damping are also presented.

Vibration and buckling calculations for rectangular plates subject to complicated in-plane stress distributions by using numerical integration in a rayleigh-ritz analysis

A Rayleigh-Ritz approach, in which appropriate beam characteristics functions are used in the deflection series, is presented for the solution of the elastic buckling and flexural vibration problems of thin rectangular plates which may be subject to any practical in-plane stress field. The stress distribution may be described in the form of mathematical expressions or by means of a set of values known only at discrete points within the plate; typically the former description may be derived from a classical two-dimensional elasticity solution and the latter from a finite difference or finite element analysis. The integrations associated with the in-plane stresses in the strain energy expression are performed numerically by using a technique based upon natural bicubic spline interpolation, the computer subroutines for which are available to the general scientific public. Several numerical examples of varying complexity are presented, illustrating the accuracy and applicability of the proposed approach.

Static Stability Behaviour of Plate Elements with Non-Uniform, in-Plane Stress Distribution

The results of analytically and experimentally determined buckling loads of a rectangular plate, subjected to partial edge loading and mixed boundary conditions, are presented.

Buckling of webs with openings

The buckling of the webs of steel beams is investigated analytically and experimentally. The influence of circular and rectangular holes on the buckling strength of beams of the dimensions normally used in building structures is considered. The buckling analysis is based on an energy method, and relies on finite element analysis for the in-plane stress distributions in the perforated webs. For the proportions of the beams considered, buckling takes place after the initiation of plasticity in the web and this is considered in the analysis. Comparison of analytical results with results of tests on five beams shows excellent agreement. It is shown that for these beams, the range of slenderness for which elasto-plastic buckling occurs is quite limited, and that below a web slenderness of 50, or in some cases more, holes of the shape and size considered will not lead to buckling prior to the development of full plastic strength.

Vibration of Plates Subject to Arbitrary In-Plane Loads—A Perturbation Approach

Perturbation theory is used to obtain the natural frequencies and mode shapes of lateral vibration of rectangular plates under arbitrary in-plane loading in terms of a power series in a loading parameter, commencing from known approximate or “exact” solutions of the same plate under no in-plane loading. The range of loading over which the solution is applicable is shown to be extendible to practically any limit by the introduction of a stepwise perturbation technique. Numerical results demonstrating the applicability and accuracy of the analysis are presented for plates subject to uniform, linear, and parabolic in-plane stress distributions. Simple approximate frequency expressions are also given.