Buckling of cracked functionally graded plates under tension

Buckling of functionally graded cracked plates under tension has not been investigated so far. In this paper critical buckling load of functionally graded plates containing a crack has been obtained using classical plate theory through the finite element method. Displacement in vicinity of crack tips has been approximated using previous solutions related to bending of cracked plates. Effect on buckling of plate under uni-axial and bi-axial tension of different parameters, such as plate dimensions and material properties, are studied. Results show that the critical load decreases as material gradient index increases, while bi-axial loading leads to higher critical loads compared to uni-axial case.     

Parametric instability of edge cracked plates

The presented paper deals with the parametric instability behavior of a simply supported rectangular plate with a crack emanating from one edge, subjected to in-plane compressive periodic edge loading. The problem is reduced to computing the free vibration frequencies and the corresponding mode shapes and substituting them into an integral equation based formula, which leads to a compact matrix form. Once the components of this matrix are found, the rest of the computation, i.e., establishing regions of instability, buckling loads and modified frequencies, is straightforward and fast. Several plates, each with a different dimension and crack length size are analyzed using this approach. The comparison of results with those of finite element models is found to be in close agreement.     

Buckling of cracked thin-plates under tension or compression

Plates are easily susceptible to buckling under compression, in particular when plate's thickness becomes sufficiently small with respect to others plate's sizes; such a mode of failure is often prevalent with respect to strength failure. The buckling phenomena under tension loading can also occur, especially in plates containing defects such as cracks or holes; when the buckling load is reached, complex wrinkling deflection patterns in compressed regions develops around such imperfections.In the present paper, the buckling analysis of variously cracked rectangular elastic thin-plates under tension and compression is considered. A short explanation of the buckling phenomena in plates is recalled and several numerical analyses, carried out by using the Finite Element Method (FEM), are performed in order to determine the critical load multiplier, both in compression and in tension, by varying some plates' parameters. In particular, the critical load multiplier is determined for different relative crack length, crack orientation and Poisson's coefficient of the plate's material which is made to range between 0.1 and 0.49.Moreover a simple approximate theoretical model to explain and predict the buckling phenomena in cracked plates under tension is proposed and some comparisons are made with FE numerical results in order to assess its reliability in predicting buckling load multipliers.Finally, the obtained results are graphically summarised (in dimensionless form) in several graphs and some interesting conclusions are drawn.     

Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression

Linear eigenvalue analysis of cracked cylindrical shells under combined internal pressure and axial compression is carried out to study the effect of crack type, size and orientation on the buckling behavior of cylindrical thin shells. Two types of crack are considered; through crack and thumbnail crack. Our calculations indicate that depending on the crack type, length, orientation and the internal pressure, local buckling may precede the global buckling of the cylindrical shell. The internal pressure, in general, increases the buckling load associated with the global buckling mode of the cylindrical shells. In contrast, the effect of internal pressure on buckling loads associated with the local buckling modes of the cylindrical shell depends mainly on the crack orientation. For cylindrical shells with relatively long axial crack, buckling loads associated with local buckling modes of the cylindrical shell reduce drastically on increasing the shell internal pressure. In contrast, the internal pressure has the stabilizing effect against the local buckling for circumferentially cracked cylindrical shells. A critical crack length for each crack orientation and loading condition is defined as the shortest crack causing the local buckling to precede the global buckling of the cylindrical shell. Some insight into the effect of internal pressure on this critical crack length is provided.     

Buckling of plates with strengthenings

This paper studies the buckling behaviour of simply supported square plates, which have weakening or strengthening bands. The weakening/strengthening bands are equally spaced and their thickness is either decreased or increased. The analysis assumes that the stress state in the plate before and during the buckling process remains in the elastic range. Two cases of plate loading are studied, one with compressive forces and one with tangential forces. The buckling coefficients are calculated for different numbers and thicknesses of strengthening/weakening bands. The thickness of strengthening/weakening bands and the thickness of the remaining plate is varied so that the volume/weight of the plate remains constant. In one case it is found that the buckling load at constant weight of the plate can be increased by 118% if an optimal ratio of the thickness of strengthening bands and the thickness of the remaining plate is used.     

Buckling analysis of isotropic and laminated plates by radial basis functions according to a higher-order shear deformation theory

The third-order shear deformation theory of Reddy and collocation with radial basis functions is used to predict the buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse strains through the thickness of the plate. It is shown that the collocation method with radial basis functions produces highly accurate critical buckling loads and modes.     

Buckling analysis of stepped plates using modified buckling mode shapes

A new approximate procedure for buckling analysis of simply supported rectangular stepped or perforated plates subjected to uniform edge stresses is formulated. The procedure uses energy method based on modified buckling mode shapes. The change of thickness within a plate is characterized by introducing a stepping index. It is shown that the buckling (vibrational) mode shapes of stepped plates can be predicted by linear combination of various mode shapes of the equivalent flat plates. These buckling mode shapes, in turn, are incorporated to evaluate buckling loads of stepped plates. Some case studies are carried out to demonstrate the accuracy and the versatility of the proposed method by comparing them to the results presented by other researchers.     

Buckling analysis of thin circular FG plates made of saturated porous-soft ferromagnetic materials in transverse magnetic field

This study presents the buckling analysis of soft ferromagnetic FG circular plates made of poro material. Equilibrium and stability equations of a poro circular plate in transverse magnetic field are derived. This study analyzes the poroelastic instability of clamped edge ferromagnetic plates subjected to magnetic loadings. The geometrical nonlinearities are considered in the Love–Kirchhoff hypothesis sense. In this paper the effect of pore pressure on critical magnetic field of plate and the effect of important parameters of poroelastic material on buckling capacity are investigated. Also the compressibility of fluid and porosity on the buckling strength are being investigated.     

Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method

A new version differential quadrature method (DQM) has been proposed to obtain buckling loads of thin anisotropic rectangular and isotropic skew plates. The essential difference from the old version DQM is the introduction of two degrees of freedom for boundary points and from the existing differential quadrature element method (DQEM) is the determination of the weighting coefficients. The methodology is worked out in detail and a variety of buckling problems shown slow convergence earlier by Rayleigh-Ritz method with beam functions, including isotropic skew plates with various skew angles and anisotropic rectangular plates with simply supported or clamped boundary conditions, are solved by the proposed DQM. Numerical results indicate that fast convergence is achieved and excellent results are obtained by the proposed DQM.     

Buckling of simply supported rectangular plates under combined bending and compression using eigensensitivity analysis

In this paper, an approximate quadratic closed-form expression is presented for the critical buckling analysis of a plate subjected to combined bending and compression. The formula is developed by expanding the eigenvalue, the critical buckling load, for a plate under combined bending and compression in a Mauclaurin's series about a plate subjected only to compression. The general expression can be used for all combinations of simply supported and clamped rectangular plates boundary conditions. An explicit formula in terms of the plate aspect ratio R and plate load parameter α is evaluated for simply supported plates. Compared with the Rayleigh–Ritz method, this approximate expression provides an excellent comparison when the load parameter α1.52 for plate aspect ratio between 0.2R2.8.     

Numerical modelling for buckling analysis of cracked shear panels

Utilizing shear panels is a common practice in civil, naval and aerospace engineering structures and the performance of the one with thin steel plate shear walls is an interesting issue of ongoing related researches. It is a well-recognized phenomenon that the behaviour, and consequently the design procedure, of shear panels are mostly dominated by pre-yield buckling occurrence. However the existence of crack defects due to corrosion, fatigue, welding or mishandling may shed more complexities to such panels in terms of bearing capacity degradation.

In this paper, a procedure for modelling and analysis of shear panels containing central or edge cracks, using the finite element method, is presented. Results can be incorporated to recognize the required mesh refinements around the crack tips and edges. In addition, the effect of relative crack length on buckling capacity of shear panels is investigated.     

Buckling analysis of skew laminate plates subjected to uniaxial inplane loads

Elastic stability of skew composite laminate plates subjected to uniaxial inplane compressive forces has been studied. The critical buckling loads of the skew laminate plates are carried out by the bifurication buckling analysis implemented in finite element program ABAQUS. The effects of skew angles, laminate layups, plate aspect ratios, plate thicknesses, central circular cutouts, and edge conditions on the buckling resistance of skew composite laminate plates are presented.     

Residual ultimate strength of cracked steel unstiffened and stiffened plates under longitudinal compression

This paper numerically deals with the influence of cracks (in terms of length and location) on the ultimate compressive strength characteristics of unstiffened and stiffened plate elements used in thin-walled structures. The cracks were presumed to be through-thickness, having no contact between their faces and no propagation was allowed. A series of nonlinear finite element analyses was conducted using ANSYS commercial finite element code in which the Newton–Raphson method has been employed to solve the nonlinear governing equations.This study indicates that the length of cracks and especially its location can significantly affect the ultimate strength characteristics of unstiffened and stiffened plate elements subjected to axial compressive action.     

A new stiffened plate element for the analysis of arbitrary plates

In spite of the large number of finite elements developed so far, most of these lack in generality, and are found to be inadequate and inefficient in some way or other, when it comes to analyzing plates of arbitrary geometrical configurations. So far the isoparametric element has been the most successful among available elements because of its ability to model a curved boundary successfully. However, the shear-locking problem inherent in the isoparametric element makes it unsuitable for analyzing thin plates of arbitrary shapes. Though research has been conducted using reduced integration and stabilization to overcome the problem, the formulations either do not converge to the correct solution in the thin-plate limit or they make the stiffness matrix a singular one. In this paper, a four-noded stiffened plate element is developed. This has the advantages and elegance of an isoparametric element in modelling arbitrary shaped plates, but without the disadvantage of shear-locking phenomena. Though this element is a high-order element, only the usual degrees of freedom have been considered, and performance is superior to that of the low-order ones. The stiffened plate element has the feature of accommodating the arbitrary shape of the plate geometry, and the stiffener modelling has been done in a general manner, with the stiffener lying anywhere with arbitrary orientation, and not necessarily following the nodal lines. The new element has been successfully used for the static, free vibration and stability analyses of arbitrary bare and stiffened plates. The results are found to agree quite satisfactorily with those of previous investigators.     

Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite-element method

The trigonometric p-version of the finite-element method is formulated and applied to free-vibration analysis of arbitrarily shaped membranes. This is accomplished using a trigonometrically enriched curved quadrilateral element. New shape functions are expressed in terms of trigonometric sine functions. These functions are numerically more stable than orthogonal polynomials as the order is increased. In this method, large elements are used and curved segments are represented accurately by using the blending function method. The accuracy of the solution is sought by increasing the trigonometric order while fixing the mesh. Frequency values are first found for a closed sectorial membrane and comparisons are made with exact values. Frequency values are also found for an open sectorial membrane and comparisons are made with exact values and with values from linear and isoparametric quadratic quadrilateral finite-element solutions. Furthermore, highly accurate frequency values are found for a square membrane with a central circular opening. It is first shown that the method converges rapidly downwards as the trigonometric order is increased. It is also shown that, for equivalent numbers of d.o.f.s, the method gives a much higher accuracy than linear and isoparametric quadrilateral finite-element solutions.     

A semi-analytical analysis of the elastic buckling of cracked thin plates under axial compression using actual non-uniform stress distribution

An improved hybrid semi-analytical method for calculating elastic buckling load of a thin plate with a central straight through-thickness crack subject to axial compression is proposed. In the study, the actual non-uniform in-plane stress distribution is firstly conducted by using Muskhelishvili's complex variable formulation in conjunction with boundary collocation method. A deflection shape function, satisfying not only the outer boundary conditions but also the inner boundary conditions of the crack edges, is obtained by using domain decomposition method. Finally the buckling load of a cracked plate using Raleigh–Ritz energy method is calculated based on the actual in-plane stress distribution and the reasonable deflection shape function obtained. The effects of crack length, plate's aspect ratio are studied for thin plates with different boundary conditions. Results obtained from the proposed method are in good agreement with the existing numerical results and experimental ones. It is finally shown that the proposed method, based on a correct non-uniform in-plane stress distribution, is more accurate than the few existing analytical methods based on a uniform in-plane stress distribution.     

Free vibration analysis of bridges

Most bridges are designed using a static analysis, adjusted by a dynamic amplification factor which is a function of the first flexural frequency. This method is imprecise but continues to be used, partly because of the apparently complex processes required to estimate accurately and to provide for the levels of vibration in a bridge. This paper investigates the natural frequencies and associated mode shapes of bridge superstructures. It compares field observations with theoretical idealizations and finds that, while a single beam idealization is accurate for straight, nonskewed bridges and for some continuous superstructures, many other bridges require an eigenvalue analysis of a finite beam element grillage. A simplified method for estimating the natural frequency of vibration is developed. An application of the Rayleigh method to a grillage model of the bridge, it is quick to apply and accurate to within 10%. The paper also compares the effects of using the static and dynamic moduli of elasticity of concrete in estimating the natural frequency of vibration, and concludes that the dynamic modulus is more appropriate. Finally, it debates the significance of certain types of support stiffness in estimating the fundamental frequency, and finds that their effect is negligible.     

Strengthening of perforated plates under uniaxial compression: Buckling analysis

This paper focuses on the cutout-strengthening of perforated steel plates subjected to uniaxial compressive loads. The square plates considered each has a centrally placed circular hole and four simply supported edges in the out-of-plane direction. Four types of stiffeners named ringed stiffener (RS), flat stiffener (FS), longitudinal stiffener (LS) and transverse stiffener (TS) are mainly discussed. The finite element method (FEM) has been employed to analyse the elastic and elasto-plastic buckling behaviors of strengthened and unstrengthened perforated plates. The results show that the strengthened perforated plates have higher buckling strengths than those of the unstrengthened ones, while the elevations in elastic buckling stress and elasto-plastic ultimate strength are closely related to stiffener types (i.e., RS, FS, LS and TS) as well as plate geometric parameters (i.e., a plate slenderness ratio and a hole diameter ratio). Furthermore, comparisons of strengthening efficiency considering the variations of buckling stress with stiffener weight are carried out, and recommendations on the most efficient cutout-strengthening methods for the uniaxially compressed perforated square plates with centric circular holes are proposed.     

Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method

The free vibration analysis of cracked thin plates via a quasi-convex coupled isogeometric-meshfree method is presented. This formulation employs the consistently coupled isogeometric-meshfree strategy where a mixed basis vector of the convex B-splines is used to impose the consistency conditions throughout the whole problem domain. Meanwhile, the rigid body modes related to the mixed basis vector and reproducing conditions are also discussed. The mixed basis vector simultaneously offers the consistent isogeometric-meshfree coupling in the coupled region and the quasi-convex property for the meshfree shape functions in the meshfree region, which is particularly attractive for the vibration analysis. The quasi-convex meshfree shape functions mimic the isogeometric basis function as well as offer the meshfree nodal arrangement flexibility. Subsequently, this approach is exploited to study the free vibration analysis of cracked plates, in which the plate geometry is exactly represented by the isogeometric basis functions, while the cracks are discretized by meshfree nodes and highly smoothing approximation is invoked in the rest of the problem domain. The efficacy of the present method is illustrated through several numerical examples.