Delamination buckling of laminated plates

The subject of this paper is the buckling of laminated plates, with a pre-existing delamination, subjected to in-plane loading. Each laminate is modelled as an orthotropic Mindlin plate. The analysis is carried out by a combination of the finite element and asymptotic expansion methods. By applying the finite element method, plates with general delamination regions can be studied. The asymptotic expansion method reduces the number of unknown variables of the eigenvalue equation to that of the equation for a single Kirchhoff plate. Numerical results for the critical buckling load are presented for several examples. The effects of the shape, size and position of the delamination on the buckling load are studied through these examples.     

Thermal buckling of laminated cylindrical plates

Thermal buckling of a laminated cylindrical plate subjected to a temperature change is studied. The governing differential equations for Donnell-type laminated cylindrical shells are used and Galerkin's method is employed to determine the critical buckling temperature. Clamped and simply supported boundary conditions are both considered. The effects of various parameters on thermal buckling are examined.     

Thermal buckling behavior of composite laminated plates

Thermal buckling behavior of composite laminated plates was studied by making the use of finite element method. The thermal buckling mode shapes of cross-ply and angle-ply laminates with various E₁/E₂ ratios, aspect ratios, fiber angle, stacking sequence and boundary condition were studied in detail. The results indicate that the high E₁/E₂ and α₂/α₁ ratios of AS4/3501-6 and T 300/5208 laminates produce higher bending rigidity along the fiber direction and higher in-plane compressive force in a direction perpendicular to the fiber direction. Therefore, the higher thermal buckling mode shapes are formed. The thermal buckling mode that composite laminated plate will buckle into is mainly dependent on the E₁/E₂ ratio, α₂/α₁ ratio, fiber orientation and aspect ratio of the plate.     

Thermomechanical buckling of hybrid cross-ply laminated rectangular plates

A thermomechanical buckling analysis is presented for simply supported rectangular symmetric cross-ply laminated composite plates that are integrated with surface-mounted piezoelectric actuators and are subjected to the combined action of in-plane compressive edge loads, two types of thermal loads, and constant applied actuator voltage. The formulation of equations is based on the classical laminated plate theory and the von-Karman non-linear kinematic relations. The analysis uses an exact method to obtain closed-form solutions for the buckling load. The effects of applied actuator voltage, thermal and mechanical loads, plate geometry, and lay-up configuration of the laminated plates are investigated. The novelty of the present work is to obtain closed-form solutions for electro-thermomechanical buckling of hybrid composite plates, and to cover non-uniform temperature distribution loading. The results for various states are verified with known data in the literature.     

Hygrothermal effects on the buckling of laminated composite plates

The effects of moisture and temperature on the static instability of laminated composite plates are investigated. The analysis is carried out by the finite element method using a quadratic isoparametric element which takes transverse shear deformation into account. The conventional finite element formulation is modified to include hygrothermal effects. The stiffness matrix, geometric stiffness matrix due to residual stresses, geometric stiffness matrix due to applied in-plane loads and load vector of the element are derived based upon the principle of minimum potential energy. The analysis accounts for reduced lamina material properties at elevated moisture concentrations and temperatures. Critical loads are obtained for symmetric and anti-symmetric laminates with simply-supported and clamped boundary conditions at different moisture concentrations and temperatures. Their dependence on moisture and temperature is studied. The effects of aspect ratio and side-to-thickness ratio on the critical loads are also considered. Mode shapes are verified in a few cases for anti-symmetric cross-ply laminate.     

Effects of nonlinear strain components on the buckling response of stiffened shear-deformable composite plates

A detailed investigation of the weight of each non linear term of the Green–Lagrange strain displacement equation is presented, with reference to the buckling of orthotropic, both flat and prismatic, Mindlin plates. Usually in the literature, in buckling analysis only the second order terms related to the out-of-plane displacement are considered. Such heuristic simplification, known as von Kármán hypothesis, starts by the consideration that the buckling mode of a flat plate is described by dominant out-of-plane displacement and disregards the non-linear terms of the Green–Lagrange strain tensor depending on the in plane displacement components, whose role is confined to first order, say pre-critical, deformation. The present paper shows that disregarding the non linear terms related to the in-plane strain–displacement is equivalent to neglect shear induced rotation. In the work, the governing equations are derived using the principle of strain energy minimum and the differential equations solution is gained by using the general Levy-type method. The obtained results show that the von Kármán model overestimates the critical load when, in buckling mode, magnitudes of shear rotation, in-plane and out-of-plane displacements are comparable.     

Elastic buckling strength of corroded steel plates

Corrosion makes structures more vulnerable to buckling and yielding failures. It is common practice to assume a uniform thickness reduction for general corrosion. To estimate the remaining strength of corroded structures, typically a much higher level of accuracy is required, since the actual corroded structures have irregular surfaces. Elastic buckling of simply supported rectangular corroded plates are studied with one- and both-sided irregular surfaces. Eigenvalue analysis by using finite element method (FEM) is employed for computing Euler stress. The influence of various geometric and corrosion characteristics are investigated and it is found that the aspect ratio of the plate, the average thickness diminution, the standard deviation of thickness diminution and the amount of corrosion loss have influence on the reduction of buckling strength of the corroded plates. Buckling strength of one- and both-sided corroded plates are the same. In plates with low value of aspect ratio, reduction of buckling strength is negligible. Reduction of buckling strength is more prominent in plates with higher aspect ratio. Reduction of buckling strength is very sensitive to the amount of corrosion loss; the higher the amount of corrosion loss, the more reduction of buckling strength. Reduction of buckling strength is less sensitive to the standard deviation of thickness diminution.     

Thermal buckling behavior of laminated composite plates with temperature-dependent properties

The thermal buckling behavior of composite laminated plates subjected to a uniform temperature field is investigated by the finite element method. Temperature-dependent elastic and thermal properties are considered. The stiffness and geometry matrices are derived based on the principle of minimum potential energy. The assumed displacement state over the middle surface of the plate element is expressed as the product of one-dimensional, first-order Hermitian polynomials. An iterative method is employed to determine the thermal buckling load. It is shown by numerical results that the influence of temperature-dependent mechanical properties on the thermal buckling behavior is significant.     

Designing laminated plates specified strain and strength characteristics

The article explains the method of designing laminated plates with a specified set of rigidity and strength characteristics on the basis of the use of a finite number of materials. An algorithm is presented for solving problems of designing which yields partial solutions of the corresponding inverse problem. Examples are discussed.Translated from Problemy Prochnosti, No. 2, pp. 85–88, February, 1990.     

Residual compressive strength of laminated plates with delamination

A study of residual compressive strength in delaminated laminates is presented. A methodology is proposed for simulating the whole compressive failure responses, such as initial buckling, postbuckling, contact of delamination front region, delamination propagation, fiber breakage, and matrix cracking etc. An finite element analysis (FEA) of the residual compressive strength is conducted on the basis of the Von Karman's nonlinearity assumption and the first-order shear deformation plate theory, combined with a stiffness degradation scheme. The numerical analysis models and methods are briefly introduced in this paper and some numerical examples are presented to illustrate it. From numerical results and discussion, it is clear that the compressive failure response involves complex multi-failure modes during compressive process. The method and numerical conclusions provide in this paper should of great value to engineers dealing with composite structures.     

Study of critical buckling loads and modes of cross-ply laminated annular plates

Buckling of composite annular plates under uniform internal and external radial edge loads have been investigated using energy method. Trefftez rule is used in the stability equations. The symmetric buckling of symmetric cross-ply laminates is considered. In this paper, buckling behavior for the three laminates (90/0)_2s, (90/0₂/90)_s and (90₂/0₂)_s are studied. Influence of some parameters such as thickness, stacking sequence, type of supports and the ratio of hole to sheet radius on buckling loads and modes are investigated. The results of the energy method are compared with the results of numerical method. Based on the results, in the plates with clamped boundary conditions the symmetric buckling assumption is not accurate, contrary to other boundary conditions.     

Experimental and numerical studies on buckling of laminated composite skew plates with circular holes under uniaxial compression

This article deals with experimental and finite element studies on the buckling of isotropic and laminated composite skew plates with circular holes subjected to uniaxial compression. The influence of skew angle, fiber orientation angle, laminate stacking sequence, and aspect ratio on critical buckling load are evaluated using the experimental method (using Methods I through V) and finite element method using MSC/NASTRAN. Method I yields the highest experimental value and Method IV the lowest experimental value for critical buckling load in the case of isotropic skew plates with circular holes. For all laminate stacking sequences considered, Method V yields the highest experimental value for critical buckling load for skew angle = 0° and Method IV yields the highest experimental value for critical buckling load for skew angles = 15° and 30°. For all laminate stacking sequences and skew angles considered, Method II yields the lowest experimental value for critical buckling load. The maximum discrepancy between the experimental values given by Method IV and the finite element solution is about 10% in the case of isotropic skew plates. The maximum discrepancy between the experimental values given by Method II and the finite element solution is about 21% in the case of laminated composite skew plates considered. The percentage of discrepancy between the numerical or finite element solution and experimental value increases as the skew angle increases. The critical buckling load decreases as the aspect ratio increases.     

Multi-objective design for vibration and buckling of laminated plates by statistical treatment

A design approach is presented for dealing with multiple (six or even more) object functions in the optimal design problem of laminated composite plates. The proposed approach is divided into some steps. First, the design variables are discretized with a certain increment that depends on the required accuracy, and all the values of physical parameters (used as object functions) are calculated for the discrete values of design variables. The second step is based on statistical treatment of the data. The calculated values of each physical parameter for discrete design variables are normalized and modified to be the deviated values on the assumption that the values are subjected to the Gaussian distribution with the prescribed standard deviation and average. At the last step, the best design in global sense is determined by considering the weighted sum of the deviated object functions and also graphical presentations. In the application, a three-layered rectangular plate with angle-ply stacking sequence [θ/−θ/θ] is considered where a representative fiber orientation angle θ is a single design variable. The object functions are taken to be six physical parameters of vibration and various buckling problems.     

Thermal buckling behaviour of composite laminated plates with a circular hole

Thermal buckling of antisymmetric cross-ply laminates having central circular holes subjected to uniform or non-uniform temperature distribution are analysed by the finite element method. To account for the transverse shear deformation, the thermo-elastic Mindlin plate theory is used. Numerical results show that significant reduction of the critical buckling temperature occurs for laminates having a central hole.     

A new finite element for buckling analysis of laminated stiffened plates

A new stiffened plate element for stability analysis of laminated stiffened plates has been presented. The basic plate element is a combination of Allman's plane stress triangular element and a Discrete Kirchhoff–Mindlin plate bending element. The element includes transverse shear effects. The model accommodates any number of arbitrarily oriented stiffeners within the plate element and eliminates constraints on the mesh division of the plate. The element has no problem associated with shear locking – a phenomenon usually encountered in isoparametric elements. The stability analysis of laminated stiffened plates has been carried out under different loading conditions with the present element.     

Optimization of laminated composites subject to uncertain buckling loads

Optimal design of composite laminates under buckling load uncertainty is presented. The laminates are subjected to biaxial compressive loads and the buckling load is maximized under worst case in-plane loading which is computed using an anti-optimization approach. The magnitudes of the in-plane loads are not known a priori resulting in load uncertainty subject to the only constraint that the loads belong to a given uncertainty domain. Results are given for continuous and discrete fibre orientations which constitute the optimization problem coupled to load anti-optimization problem leading to a nested solution method. It is observed that the stacking sequence of a laminate designed for a deterministic load case only differs considerably from that of a robust laminate designed taking load uncertainties into account. Consequently the buckling load carried by a deterministic design is considerably less than the one carried by a robust design when both are subjected to uncertain loads.     

On the analysis of the buckling behaviour of laminated composite plates and shells

In general, plates and shells out of composite laminates are more susceptible to transverse shear than those out of homogeneous isotropic material. By means of different shear flexible theories the effect on buckling and postbuckling behaviour is studied. It turns out that there is a rather limited range of plates where the transverse shear is of considerable influence. Results obtained with Mindlin- or Reissner-type theories prove almost as adequate as those obtained with Reddy's theory. For the buckling analysis of shells the Kirchhoff–Love theory is precise enough. Snap-through buckling and imperfection sensitivity appear in composite shells, too, especially if they are optimized with respect to high bifurcation buckling loads.