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This note presents closed-form solutions for axisymmetrical and axially unsymmetrical buckling of angle-ply laminated circular cylindrical shells under axial compression. The axisymmetrical and axially unsymmetrical buckling stress are found to be different from each other, and the best lamination angles which give the highest buckling stress are obtained.
Equations are developed for the buckling of a general lattice structure that has repetitive geometry. Equilibrium at a typical node is expressed using finite element techniques, and the only assumption is that the response is periodic. By basing the stiffness matrix on the exact solution of the beam column equation, accurate results are obtained for complex buckling behavior that would require a very large system of equations using conventional techniques. The present method requires the eigenvalues of only a 6x6 determinant. The results are used to study the buckling of isogrid cylinders, three-element truss columns and polygonal rings. Details of the analysis including expressions for all terms in the governing stability determinant are given.
The buckling of a truncated elastic conical shell subjected to an axial compression is a classical problem in shell structures. The paper reinvestigates the buckling of an axially compressed truncated conical shell with rigid bulkheads. Two improvements are achieved. First, the condition that the total horizontal displacement must vanish due to rigid bulkhead and axisymmetry is treated as a constraint. This constraint is incorporated into the system through the use of the Lagrange multiplier; then the variational method is used to derive a complete set of boundary conditions for conical shells. Second, the stability is evaluated in the deformed state using the asymptotic solutions of the pair of Donnell-type equations for axisymmetric configuration. The results indicate that the buckling strength of conical shells depends mainly on the condition of the smaller end. In addition to the vertex angle, the distance ratio plays, at least, an equally important role.
The shear buckling analyses used in PASCO are summarized. The PASCO analyses include the basic VIPASA analysis, which is essentially exact for longitudinal and transverse loads, and a smeared orthotropic solution which was added to alleviate a shortcoming in the VIPASA analysis. Buckling results are presented for six stiffened panels loaded by combinations of longitudinal compression and shear. The buckling results were obtained with the PASCO, EAL, and STAGS computer programs. The EAL and STAGS solutions were obtained with a fine finite element mesh and provide calculations for the entire range of combinations of longitudinal compression and shear loadings.
Time-dependent lateral deflection of flat rectangular plates is predicted by the Norton-Bailey (Norton 1929, Bailey 1935) power law for material creep. The plates have a through-thickness steady-state temperature distribution, and the effects are considered by using Maxwell's law to modify the power creep law. Equations are derived for creep exponents of 3 and 5, using the sandwich plate element to predict creep buckling of plates. Predictions of creep buckling with a temperature variation between the inner and outer plate surfaces are found to be somewhat dependent on the creep buckling relationship assumed. When significant scatter justifies a variation in the creep constants up to an order of magnitude, discrepancies in predictions using the two exponents are reasonable, and for one engineering material, the predictions have the same degree of agreement with experimental data as have the respective creep laws.
Advances and current trends in plate bucklin research are summarized. The field is divided into three parts: (1) classical buckling studies, including plates of rectangular, circular and other shapes; (2) classical complicating effects, including elastic foundation, anisotropic material, variable thickness, shear deformation and nonhomogeneous material; and (3) nonclassical considerations, including postbuckling, imperfections, parametric excitation, follower forces, magnetoelastic buckling and inelastic buckling.
An efficient scheme utilizing standard nonlinear programming methodology is presented for the optimal design of elastic, redundant structures with strength and buckling constraints. Numerical examples are given which demonstrate how this method, involving a constraint redefinition, can successfully overcome certain problems typical to such structures. The scheme is applied to the design of a cantilever truss with strength and buckling constraints, in which optimal nonuniform cross section columns are used for all elements critical in buckling. It is found that permitting node movement and adding redundancies can lower the weight of the structure for the prescribed design constraints. In addition, the node excursions in this configuration are shown to be relatively small.
An experimental investigation was conducted to study the effect of specimen size on the buckling strains of composite laminates subjected to low-velocity projectile impact. The specimens were fabricated from a T300/5208 graphite/epoxy material with laminate configurations (+ or - 45,0,90)2s and (+ or - 45,0,90)4s. These 16- and 32-ply quasi-isotropic laminates were side-supported during testing. Specimens of three different length-to-width aspect ratios of 1, 1.5, and 2 were studied. The low-velocity projectile impact was provided by an air gun. The preload and impact energy combination necessary to cause catastrophic buckling failure was determined. The residual strength of specimens that survived the impact damage was also measured. The results were compared with those of a 48-ply laminate tested elsewhere. The present experimental results show that (1) specimen length does not seem to have any significant influence on the buckling strains at failure caused by projectile impact and (2) the influence of specimen thickness on strain at failure decreases as the velocity of the projectile increases.
Recent advances in the study of interactive buckling of thin walled structural components achieved with the aid of finite strip technique used in conjunction with the theory of mode interaction are summarized. The interaction of the primary local mode with Euler buckling (in columns) and flexural torsional buckling (in columns and beams) is of primary interest. The interaction of two companion local modes with the overall mode is also considered briefly for the columns with doubly symmetric cross sections. The effect of dynamic loads in the form of suddenly applied and compression is also investigated.
The effects of pretwist, precone, setting angle and Coriolis forces on the vibration and buckling behavior of rotating, torsionally rigid, cantilevered beams were studied. The beam is considered to be clamped on the axis of rotation in one case, and off the axis of rotation in the other. Two methods are employed for the solution of the vibration problem: (1) one based upon a finite-difference approach using second order central differences for solution of the equations of motion, and (2) based upon the minimum of the total potential energy functional with a Ritz type of solution procedure making use of complex forms of shape functions for the dependent variables. The individual and collective effects of pretwist, precone, setting angle, thickness ratio and Coriolis forces on the natural frequencies and the buckling boundaries are presented. It is shown that the inclusion of Coriolis effects is necessary for blades of moderate to large thickness ratios while these effects are not so important for small thickness ratio blades. The possibility of buckling due to centrifugal softening terms for large values of precone and rotation is shown.
The dynamic response of buckled composite plates impacted by a hard object is studied. A nine-node isoparametric finite element, developed based on the Mindlin plate theory and von Karman large deflection assumptions, is used for both postbuckling and impact analyses. In order to evaluate the contact force, an experimentally established contact law which accounts for the permanent indentation is employed. The static postbuckling problem of laminated plates is first considered. The impact responses include contact force histories, dynamic deflections, and dynamic strains in the plate for various buckling conditions. In addition, free vibration of buckled laminated plates is also solved to determine the range of natural frequencies which are needed to choose a proper time increment for time integration in the impact analysis.
The stiffened composite structural panel analysis and sizing code designated 'PASCO' encompasses both the generality required for the exploitation of composite materials' design flexibility and an accurate buckling analysis for the detection of complex buckling modes. PASCO can accordingly design for buckling, frequency, material strength, and panel stiffness requirements. Attention is given to an additional thermal loading design capability. Design studies illustrate the importance of the multiple load condition capability when thermal loads are present.
The buckling loads of laminated plates are predicted using a new theory which takes into account transverse shearing effects. This new theory assumes trigonometric terms through-the-thickness in the displacements to take into account transverse shearing effects in thick plates. Buckling loads predicted by the new theory and by traditional theories are compared for isotropic and laminated plates. The effect of ply orientation on the buckling loads predicted by each theory is demonstrated.
The effects of pretwist, precone, setting angle and Coriolis forces on the vibration and buckling behavior of rotating, torsionally rigid, cantilevered beams were studied. The beam is considered to be clamped on the axis of rotation in one case, and off the axis of rotation in the other. Two methods are employed for the solution of the vibration problem: (1) one based upon a finite-difference approach using second order central differences for solution of the equations of motion, and (2) based upon the minimum of the total potential energy functional with a Ritz type of solution procedure making use of complex forms of shape functions for the dependent variables. The individual and collective effects of pretwist, precone, setting angle, thickness ratio and Coriolis forces on the natural frequencies and the buckling boundaries are presented. It is shown that the inclusion of Coriolis effects is necessary for blades of moderate to large thickness ratios while these effects are not so important for small thickness ratio blades. The possibility of buckling due to centrifugal softening terms for large values of precone and rotation is shown.
Buckling of a fiber bundle embedded in epoxy resin was studied to gain insight into compressive failure mechanisms in unidirectional composites. The fibers used were E-glass, T300 graphite, T700 graphite, and P75 graphite. These fibers were combined with two different resins: Epon 815/V140 and Epon 828/Z. In both resins the failure mode of the bundle was found to be microbuckling of fibers for the first three types of fibers; however, the high-modulus P75 fibers failed in shear without any sign of microbuckling. The strains at which microbuckling occurred were higher than the compressive failure strains of the corresponding unidirectional composites. In the soft resin, Epon 815/V140, fibers buckled at lower strains than in the stiff resin, Epon 828/Z. The buckling strains and the segment lengths followed the trends predicted for a single filament embedded in an infinite matrix.
Procedure for mathematical modeling of dynamic delamination buckling and propagation of delamination, with plate bending elements and multipoint constraints, developed and incorporated into finite-element computer program. Predicts time at which delamination buckling occurs, shape of dynamic-buckling mode, and strain-energy-release rate due to extension of delamination crack. Method extended to handle such other defects as transply and edge cracks.
The shear loading of a composite rectangular plate with a centrally located circular cutout was analyzed in order to predict the buckling load of the plate. The first step in this analysis is to calculare the in-plane stress distribution of the plate prior to buckling. This problem can be solved using finite element methods, but a quicker and more cost efficient method is desired. The method chosen to determine the pre-buckling stresses is that of boundary collocation using complex variables. The real valued force functions are written in terms of two complex valued functions, each of which is a function of a different complex variable. The solution of the generalized biharmonic function is a superposition of functions of these two complex variables. For this analysis, the force functions are represented with a Laurent series. The constants in these series are found using boundary collocation. Two force equations for complex variables are satisfied over the plate boundaries to obtain these constants. The stresses in the plate are then known for any particular locations on the plate given the applied shear on the edges of the plate.
The buckling load of laminated plates having midplane symmetry is maximized for a given total thickness. The thicknesses of the layers are taken as the design variables. Buckling analysis is carried out using the finite element method. The optimality equations are solved by a homotopy method which permits tracing optima as a function of total thickness. It is shown that for any design with a given stacking sequence of ply orientations, there exists a design associated with any other stacking sequence which possesses the same bending stiffness matrix and same total thickness. Hence, from the optimum design for a given stacking sequence, one can directly determine the optimum design for any rearrangement of the ply orientations, and the optimum buckling load is independent of the stacking sequence.