In-plane Biaxial Loading Research Articles

Survival probability for brittle honeycombs under in-plane biaxial loading

The failure envelopes of brittle honeycombs are affected by the cell-wall modulus of rupture. The variability in the cell-wall modulus of rupture is accounted for by assuming that it follows a Weibull distribution, giving the corresponding modulus of rupture for a prescribed survival probability. Furthermore, the existing model for the failure envelopes of honeycombs under in-plane biaxial loading is modified to take into account the effect of variability in the cell-wall modulus of rupture. Consequently, the failure envelopes of brittle honeycombs with a prescribed survival probability are developed. The effects of cell size, Weibull modulus and prescribed survival probability on the failure envelopes of brittle honeycombs are also evaluated.

A STRAIN-ENERGY BASED FAILURE CRITERION FOR NON-LINEAR ANALYSIS OF COMPOSITE LAMINATES SUBJECTED TO BIAXIAL LOADING

The non-linear responses of a lamina under uniaxial longitudinal tension and compression, transverse tension and compression and shear loadings are used to predict the stress/strain response, failure onset and progression in composite laminates subjected to in-plane biaxial loading. A piecewise cubic spline interpolation scheme has been used to represent the basic material properties along the lamina material axes. A laminated plate theory that includes an iterative incremental constitutive law to account for the non-linear behavior of the laminae is combined with a strain-energy based failure criterion for orthotropic materials. Unloading of the failed laminae and subsequent failure progression of the laminate are included in the model. Results for failure onset and progression are presented for a wide variety of unidirectional and symmetric laminates under biaxial loading.

Natural frequencies of plates with square holes when subjected to in-plane uniaxial, biaxial or shear loading

The finite element method is used to determine the natural frequencies of flat square plates containing eccentrically located square holes. The plates are subjected to in-plane uniaxial, biaxial or uniformly distributed shear along the four outer edges. These edges are either simply supported or clamped. In order to evaluate the stiffness and mass matrices, the non-conforming rectangular displacement element is used to model the out-of-plane behaviour of the plate. The in-plane stress distribution within the plate, required in the analysis, is determined by using a rectangular finite element having the only essential degrees of freedom at each of the four corner nodes. The element displacement functions are based on assumed strains and satisfy the exact requirements of strain free rigid-body modes of displacements. The variation of the natural frequency with the size and location of the hole is first investigated in the absence of any in-plane stresses. This analysis is then repeated for a series of values of the applied in-plane stresses. When uniform shear is applied, tension and compression zones are produced in the plate and hence the effect of locating the hole in each of the regions is also investigated. The values of the applied in-plane stresses ranged up to the point that would cause the plates to buckle. In this way a comprehensive set of results can be obtained.

Dynamic stability of a rectangular plate on non-homogeneous winkler foundation

The dynamic stability of a rectangular plate on non-homogeneous foundation, subjected to uniform compressive in-plane bi-axial dynamic loads and supported on completely elastically restrained boundaries is studied. The non-homogeneous foundation consists of two regions having different stiffnesses but symmetric about the centre lines of the plate. The equation governing the small amplitude motion of the system is derived by a variational method. The use of Galerkin method with reduced beam eigenfunctions transforms the system equations in matrix form. The system of coupled Mathieu-Hill equations thus obtained, are analysed by the method of multiple scales which yields the stability boundaries for different combinations of the excitation amplitude and frequency. The effects of stiffness and geometry of the foundation, boundary conditions, static load factor, in-plane load ratio and aspect ratio on the stability boundaries of the plate for first- and second-order simple and combination resonances are studied.

Strain distributions around fasteners in laminated plates under biaxial in-plane loading

A finite element model is employed to investigate the effect of variations in biaxial in-plane applied loading on the strain distributions around a bolt-filled quasi-isotropic laminated plate. The theoretical model includes the effect of friction between a washer and the plate surface which is generated by a torque loaded bolt. It is this aspect of the investigation that is a novel feature of the work. Contour plots around the hole/washer region for all three in-plane strain components are presented and discussed. The results show that the strain distribution is very much affected by the biaxial loads and that the highest strains occur when biaxial tension and compression stresses of equal magnitude are applied. Washer friction is found to have a beneficial effect, and in some load cases strains are fairly evenly distributed over the hole/washer region.

Failure in Carbon/Polyimide Laminates under Biaxial Compression

The failure mechanisms of a woven 2-D carbon/polyimide (C/P) laminate have been determined under in-plane uniaxial and biaxial compressive loads, and the associated failure envelopes that account for the effect of loading directions were obtained. The overall failure was primarily in the form of axial interply delaminations aligned with the principal axis of loading, with only a few isolated events of kinking in the bundles. While the biaxial strength varied significantly with the ratio of in-plane stresses, R, there was no variation in the local failure mechanisms. Accordingly, it was found that the composite fails upon achieving a maximum strain along the primary axis of loading.

Failure mechanisms in laminated carbon/carbon composites under biaxial compression

The failure mechanisms of 2D carbon/carbon (C/C) woven laminates have been determined under inplane biaxial compression loads, and the associated failure envelopes that account for the effect of matrix-type and loading directions were also obtained. The failure was in the form of micro-kinking of fiber bundles, interspersed with localized interply delaminations to form an overall shear fault. The shear fault was aligned with the major axis of loading except at above 75% of balanced biaxial compressive stress, where failure occurred along both axes. Although the biaxial strength varied significantly with the ratio of in-plane principal stresses, R, there was no variation in the local failure mechanisms. Accordingly, it was found that the samples fail upon achieving a critical strain along the primary axis of loading.

Biaxial Bearing/Bypass Testing of Graphite/Epoxy Plates

Bolted graphite/epoxy plates were subjected to in-plane biaxial loading. Cruciform shape samples were used. Testing was performed on a biaxial machine developed and built at Concordia University. The bolt hole was constrained using a rigid structure. Four arms of the cruciform samples were independently loaded. Equal and opposite lateral loads were applied along one direction (y). Along the other direction (x), load is applied on one arm (right arm of the cruciform), while the opposite arm (left arm of the cruciform) was held fixed in the grip without applied load. This was possible because the bolt hole at the center of the cruciform sample was held fixed. This opposite arm measured whichever load that was bypassed around the constrained hole (called bypass load). Acoustic emission was used to measure the onset of failure. The applied load in the x direction at the onset of failure is called the joint strength. Subsequent to the onset of failure, the bypass load increased significantly. Also after onset of failure, the bypass load on reloading was larger than the bypass load on previous loading. Results obtained showed that the existence of lateral loads (load along the y direction) had significant effect on the joint strength. Increasing the lateral load decreased the joint strength. These results show that data obtained using uniaxial test are nonconservative.

Genetic algorithms with local improvement for composite laminate design

This paper describes the application of a genetic algorithm to the stacking sequence optimization of a composite laminate plate for buckling load maximization. Two approaches for reducing the number of analyses are required by the genetic algorithm are described. First, a binary tree is used to store designs, affording an efficient way to retrieve them and thereby avoid repeated analyses of designs that appeared in previous generations. Second, a local improvements scheme based on approximations in terms of lamination parameters is introduced. Two lamination parameters are sufficient to define the flexural stiffness and hence the buckling load of a balanced, symmetrically laminated plate. Results were obtained for rectangular graphite-epoxy plates under biaxial in-plane loading. The proposed improvements are shown to reduce significantly the number of analyses required for the genetic optimization.

Stress concentrations at an oblique hole in a thick flat plate under an arbitrary in-plane biaxial loading

The results of an extensive ‘frozen-stress’ photoelastic investigation of the stresses at isolated oblique holes in thick wide plates subjected to uniform uniaxial tension are used to provide stress concentration factors at holes resulting from any form of biaxial in-plane loading. The work covers plate thickness/hole diameter ratios from 1.3 to 3.0 and hole obliquity angles up to 60 degrees. Over these ranges the effects of changes in the plate thickness/hole diameter ratio are not of major importance but the effects of changes in the angle of obliquity are considerable.

Failure Strength and Damage Mechanisms of E-Glass/Epoxy Laminates under In-Plane Biaxial Compressive Deformation

A strength and failure mechanism study of E-glass/epoxy laminates sub jected to uniaxial and in-plane biaxial compression loadings is presented. Specimen con figuration was found to have a large influence on the damage mechanism and resulting strength of the laminates. Standard compression test specimens with a gage section of 12.5 mm (0.5 in.) fail in a shear damage or a kink band mode. In large 125 mm (5 in.) square biaxial specimens, significant delamination not observed in the small specimens reduces the failure strength by 20-35%. The biaxial compressive failure envelope for E-glass/ epoxy is adequately described by a maximum stress or a maximum strain criteria that con siders the different failure modes. Estimates of the failure envelope based on small com pression test specimens where the failure mode is a shear or a kink band overestimates the strength of the large square biaxial test specimens that delaminate at a lower stress.

Buckling of unsymmetric laminates under linearly varying, biaxial in-plane loads, combined with shear

Classical lamination theory in conjunction with the Rayleigh-Ritz method is used for the determination of the critical buckling load of simply supported, unsymmetric cross-ply and antisymmetric angle-ply laminates, under linearly varying in-plane biaxial loads, combined with shear. The algorithms that produce each element of both the stiffness and load matrices in the final generalized eigenvalue problem are obtained. No limitation in the number of terms of the displacement field expansion considered exists. The validity of the presented method is evaluated by comparing its results with those of other investigators. Finally, a parametric study for both types of laminates is given, changing parameters such as the form of the applied load, the E 1 E 2 ratio and the angle of fiber orientation.

Curvature effects in the buckling of symmetrically-laminated rectangular plates with transverse shear deformation

It is shown that the effect of stress discontinuities from ply-to-ply must be taken into account when curvature terms are included along with shear deformation in the buckling analysis of rectangular, symmetrically-laminated plates. Such ply-stress discontinuities lead to curvature terms in the governing equations which differ considerably from those derived for homogeneous plates. Critical buckling loads are determined for orthotropic laminates subjected to biaxial inplane loading and for cylindrical bending of anisotropic plates subjected to uniaxial compression loading. Simply-supported boundary conditions are considered in conjunction with the rectangular, orthotropic laminate, while simply-supported and clamped boundaries are considered for the case of cylindrical bending of anisotropic plates. Numerical results indicate that the curvature terms have little effect on critical buckling loads for the laminates investigated. The effect of transverse shear deformation is shown to depend on the degree of boundary constraint, laminate stacking geometry, and inplane load ratio.

Optimal Design of Laminated Composite Plates Under Axial Compression

The study is carried out for the optimum design of laminated fiber reinforced composite plates, subjected to multiple in-plane loadings. Angle-ply laminates with orthotropic laminae are considered. Thickness of plies and corresponding fiber orientations are incorporated as design variables. The constrained optimization problem is transformed into a series of unconstrained optimization problems, using an interior penalty function approach. The results have been obtained for different aspect ratios and uniform biaxial in-plane loading ratios. This study shows that the fiber orientations of the plies near mid-plane have little effect on the optimum design. There exists a particular fiber orientation angle for the over all thickness of laminate, which results in the optimum design for a plate of a given aspect ratio under a given set of loadings.

On the interaction between two inclined cracks under biaxial load

An analysis of the in-plane biaxial loading of an infinite homogeneous and isotropic elastic plate with two inclined collinear cracks of equal length is performed. The boundary value problem is solved by using the complex potentials method. The influence of the lateral load on the local stress distributions as well as on the crack tips behaviour is analyzed and the features of the interaction are described.

Fracture behaviour by two cracks around an elliptic rigid inclusion

An analysis is presented for the in-plane biaxial loading of an elastic system consisting of a partially bonded rigid elliptic inclusion embedded in an infinite matrix where the bond imperfections are two symmetric cracks. The boundary value problem is formulated through the complex variable technique in the case of linearly elastic matrix under general biaxial loading at infinity. The elastic solution is obtained for a simplified model by assuming incompressibility and plane strain conditions for the matrix. In particular, stress and displacement fields are represented and discussed. Moreover, a stress criterion, which permits to take into account, either the crack extension at the interface or its deviation into the matrix, is considered to study the fracture response of the elastic system.

Buckling of orthotropic plates due to bi-axial in-plane loads taking rotational restraints into account: Johnston, G.S. Fibre Science and Technology Vol 12 No 6 (November 1979) pp 435–443

Buckling of orthotropic plates due to bi-axial in-plane loads taking rotational restraints into account: Johnston, G.S. Fibre Science and Technology Vol 12 No 6 (November 1979) pp 435–443

Biaxial load effects on a crack between dissimilar media

An analysis of the in-plane biaxial loading of two dissimilar materials with a crack along their common interface is considered. The elastic solution is obtained by the application of the complex variable technique coupled with the principle of superposition. The stress and displacement fields near the tip of the crack are described and the influence of the load parallel to the crack direction is shown. Then a failure criterion is formulated to discuss the debonding along the interface or the brittle fracture of the elastic system. In particular the effect of the non-singular terms on the critical applied tensile stress and on the deviation of the interfacial crack into one of the adjacent media is analyzed.

Buckling of orthotropic plates due to biaxial in-plane loads taking rotational restraints into account

Stability analysis based on the solution of the differential equation for long, orthotropic plates requires little computer time and capacity. Hence, it is useful for routine calculations and synthesis. Full biaxial loads and finite rotational restraints are taken into account. Results in the form of buckling coefficients are presented for sample cases.

Buckling and Postbuckling of Orthotropic Plates

Theme T stability is considered of flat rectangular plates when in-plane biaxial loads are applied. The in-plane and bending elastic properties are orthotropic and the plate axes of symmetry and the elastic axes of symmetry coincide. Both the initial buckling and the reduction in stiffness occurring at the instant of buckling are considered. Numerous authors have considered the stability of orthotropic plates under biaxial load. Wittrick, * for example, has shown for the simply supported case among others that the parameters involved can be combined into one curve giving the buckling load for any plate. Reference 2 considers the buckling and postbuckling of a class of biaxially loaded laminated plates in the presence of bending-extensional elastic couplings. The present paper treats the stability of specially orthotropic plates with these couplings absent; a further rationalization is then possible, in which for any plate the proportional loss of stiffness at buckling can also be expressed in a single curve. For a particular plate, the combination of these universal curves, for buckling load and the loss of stiffness at buckling, allows unified consideration of the stability of simply supported biaxially loaded orthotropic plates. It follows that the relative stiffness at buckling (i.e., the ratio of plate stiffness immediately after buckling to that immediately before) depends on one parameter involving the elastic constants and the transverse load; this fact may be exploited in design.