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NASA NTRS1988

Application of the Thurston bifurcation solution strategy to problems with modal interaction

The solution of bifurcation problems with closely-spaced critical points is achieved by first separating the singular part of the equation system encountered during a Newton iteration and carrying the Taylor expansion of the reduced system out to higher order. This separation is accomplished by transforming the equation system into an equivalent system in which some of the original unknowns are replaced with an equal number of modal amplitude coefficients. This method was used to continue the analysis of two significant example problems well past multiple fiburcation points, allowing a detailed examination of postbuckling behavior in the presence of modal interaction.

Rankin, C. C.↗

NASA NTRS1988

Compression strength of composite primary structural components

The geometrically nonlinear response and failure of thin-walled structural components made from advanced composite materials is discussed. The materials involved are structural components in flight vehicle structures that are laminated from graphite-epoxy unidirectional tape. The investigation involves the buckling, postbuckling, and failure of structures subject to axial compression. The structural analysis was performed using the Structural Analysis of General Shells (STAGS) computer code. Experimental and theoretical methods are used to study the fundamental mechanisms limiting the load-carrying capacity of these components.

Johnson, Eric R.↗

NASA NTRS1989

Generic element processor (application to nonlinear analysis)

The focus here is on one aspect of the Computational Structural Mechanics (CSM) Testbed: finite element technology. The approach involves a Generic Element Processor: a command-driven, database-oriented software shell that facilitates introduction of new elements into the testbed. This shell features an element-independent corotational capability that upgrades linear elements to geometrically nonlinear analysis, and corrects the rigid-body errors that plague many contemporary plate and shell elements. Specific elements that have been implemented in the Testbed via this mechanism include the Assumed Natural-Coordinate Strain (ANS) shell elements, developed with Professor K. C. Park (University of Colorado, Boulder), a new class of curved hybrid shell elements, developed by Dr. David Kang of LPARL (formerly a student of Professor T. Pian), other shell and solid hybrid elements developed by NASA personnel, and recently a repackaged version of the workhorse shell element used in the traditional STAGS nonlinear shell analysis code. The presentation covers: (1) user and developer interfaces to the generic element processor, (2) an explanation of the built-in corotational option, (3) a description of some of the shell-elements currently implemented, and (4) application to sample nonlinear shell postbuckling problems.

Stanley, Gary↗

NASA NTRS1989

A finite element formulation for the large deflection random response of thermally buckled beams

The effects of temperature and acoustic loading are included in a theoretical finite element large deflection formulation for thin, isotropic beams. Thermal loads are applied as steady-state temperature distributions, and acoustic loads are taken to be ergodic and Gaussian with zero mean and uniform magnitude and phase along the length of the beam. Material properties are considered presently to be independent of temperature. Also, inplane and rotary inertia terms are assumed to be negligible, and all inplane edge conditions are taken to be immovable. For the random response analysis, both auto- and cross-correlation terms are included. The nature of the loads leads to the solution of two separate problems. First, the problem of thermal postbuckling is solved to determine the deflections and stresses due to the thermal load only. These deflections and stresses are then used as initial deflections and stresses for the random vibration analysis. Root-mean-square (RMS) maximum deflections and strains are obtained and compared with previous classical equivalent linearization results.

Locke, James↗

NASA NTRS1989

Sensitivity of buckling loads of anisotropic shells of revolution to geometric imperfections and design changes

Buckling load sensitivity calculations in the shell-of-revolution program FASOR are discussed. This development is based on Koiter's initial postbuckling theory, which has been generalized to include the effect of stiffness changes, as well as geometric imperfections. The implementation in FASOR is valid for anisotropic, as well as orthotropic, shells. Examples are presented for cylindrical panels under axial compression, complete cylindrical shells in torsion, and antisymmetric angle-ply cylindrical panels under edge shear.

Cohen, Gerald A.↗

NASA NTRS1988

Current research in composite structures at NASA's Langley Research Center

Research on the mechanics of composite structures at NASA's Langley Research Center is discussed. The advantages and limitations of special purpose and general purpose analysis tools used in research are reviewed. Future directions in computational structural mechanics are described to address analysis short-comings. Research results on the buckling and postbuckling of unstiffened and stiffened composite structures are presented. Recent investigations of the mechanics of failure in compression and shear are reviewed. Preliminary studies of the dynamic response of composite structures due to impacts encountered during crash-landings are presented. Needs for future research are discussed.

Card, Michael F.↗

NASA NTRS1989

Design guide for predicting nonlinear random response (including snap-through) of buckled plates

This design guide describes a method for predicting the random response of flat and curved plates which is based on theoretical analyses and experimental results. The plate curvature can be due to postbuckling, in-plane mechanical or thermal stresses. Based on a single mode formula, root mean square values of the strain response to broadband excitation are evaluated for different static buckled configurations using the equivalent linearization technique. The effects on the overall strain response due to instability motion of snap-through are included. Panel parameters include clamped and simply-supported boundaries, aspect ratio, thickness and length. Analytical results are compared with experimental results from tests with 12 in. x 15 in. aluminum plates under thermal loading in a progressive wave facility. Comparisons are also made with results from tests with a 2 in. x 15 in. x 0.032 in. aluminum beam under base mechanical excitation. The comparisons help to assess the accuracy of the theory and the conditions under which deviations from the theory due to effects of imperfection and higher modes are significant.

Ng, Chung Fai↗

NASA NTRS1990

Derivatives of buckling loads and vibration frequencies with respect to stiffness and initial strain parameters

A uniform variational approach to sensitivity analysis of vibration frequencies and bifurcation loads of nonlinear structures is developed. Two methods of calculating the sensitivities of bifurcation buckling loads and vibration frequencies of nonlinear structures, with respect to stiffness and initial strain parameters, are presented. A direct method requires calculation of derivatives of the prebuckling state with respect to these parameters. An adjoint method bypasses the need for these derivatives by using instead the strain field associated with the second-order postbuckling state. An operator notation is used and the derivation is based on the principle of virtual work. The derivative computations are easily implemented in structural analysis programs. This is demonstrated by examples using a general purpose, finite element program and a shell-of-revolution program.

Haftka, Raphael T.↗

NASA NTRS1990

Finite element two-dimensional panel flutter at high supersonic speeds and elevated temperature

A consistent finite element formulation for flutter of two-dimensional panels including effects of temperature variation along panel length is presented. The von Karman nonlinear strain-displacement relation is used to account for the large-deflections, and the quasi-steady first-order piston theory is employed for aerodynamic loading in the finite-element formulation. The panel flutter under combined thermal/aerodynamic loading can be separated into solving two problems in sequence. They are: (1) postbuckling deflection, stresses and buckling stability boundary, and (2) critical dynamic pressure and linear flutter boundary. The solution procedures in solving these problems are presented in detail. Finite element results of two-dimensional panels with uniform temperature change are compared with classic analytical solutions. Influence of nonuniform temperature distributions on critical buckling temperature, flutter deflection and stresses, and flutter boundary is also given.

Xue, David Y.↗

NASA NTRS1990

Repeated buckling of composite shear panels

Failures in service of aerospace structures and research at the Technion Aircraft Structures Laboratory have revealed that repeatedly buckled stiffened shear panels might be susceptible to premature fatigue failures. Extensive experimental and analytical studies have been performed at Technion on repeated buckling, far in excess of initial buckling, for both metal and composite shear panels with focus on the influence of the surrounding structure. The core of the experimental investigation consisted of repeated buckling and postbuckling tests on Wagner beams in a three-point loading system under realistic test conditions. The effects of varying sizes of stiffeners, of the magnitude of initial buckling loads, of the panel aspect ratio and of the cyclic shearing force, V sub cyc, were studied. The cyclic to critical shear buckling ratios, (V sub cyc/V sub cr) were on the high side, as needed for efficient panel design, yet all within possible flight envelopes. The experiments were supplemented by analytical and numerical analyses. For the metal shear panels the test and numerical results were synthesized into prediction formulas, which relate the life of the metal shear panels to two cyclic load parameters. The composite shear panels studied were hybrid beams with graphite/epoxy webs bonded to aluminum alloy frames. The test results demonstrated that composite panels were less fatigue sensitive than comparable metal ones, and that repeated buckling, even when causing extensive damage, did not reduce the residual strength by more than 20 percent. All the composite panels sustained the specified fatigue life of 250,000 cycles. The effect of local unstiffened holes on the durability of repeatedly buckled shear panels was studied for one series of the metal panels. Tests on 2024 T3 aluminum panels with relatively small unstiffened holes in the center of the panels demonstrated premature fatigue failure, compared to panels without holes. Preliminary tests on two graphite epoxy shear panels with small holes in the center showed no similar fatigue life degradation and no shift in failure mode. Further tests on the effect of holes are in progress.

Singer, Josef↗

NASA NTRS1991

A method for the geometrically nonlinear analysis of compressively loaded prismatic composite structures

A method was developed for the geometrically nonlinear analysis of the static response of thin-walled stiffened composite structures loaded in uniaxial or biaxial compression. The method is applicable to arbitrary prismatic configurations composed of linked plate strips, such as stiffened panels and thin-walled columns. The longitudinal ends of the structure are assumed to be simply supported, and geometric shape imperfections can be modeled. The method can predict the nonlinear phenomena of postbuckling strength and imperfection sensitivity which are exhibited by some buckling-dominated structures. The method is computer-based and is semi-analytic in nature, making it computationally economical in comparison to finite element methods. The method uses a perturbation approach based on the use of a series of buckling mode shapes to represent displacement contributions associated with nonlinear response. Displacement contributions which are of second order in the model amplitudes are incorported in addition to the buckling mode shapes. The principle of virtual work is applied using a finite basis of buckling modes, and terms through the third order in the model amplitudes are retained. A set of cubic nonlinear algebraic equations are obtained, from which approximate equilibrium solutions are determined. Buckling mode shapes for the general class of structure are obtained using the VIPASA analysis code within the PASCO stiffened-panel design code. Thus, subject to some additional restrictions in loading and plate anisotropy, structures which can be modeled with respect to buckling behavior by VIPASA can be analyzed with respect to nonlinear response using the new method. Results obtained using the method are compared with both experimental and analytical results in the literature. The configurations investigated include several different unstiffened and blade-stiffening panel configurations, featuring both homogeneous, isotropic materials, and laminated composite material.

Stoll, Frederick↗

NASA NTRS1990

Buckling analysis and test correlation of high temperature structural panels

This paper focuses on the design, analysis, and test of a high temperature structural panel, constructed of Ti-6-4, subjected to a variety of thermal and mechanical load conditions. A follow-on panel, constructed of TMC, is also discussed in less detail. The design constraints and test set-up are discussed, as well as the test methods that were used: the grid shadow moire method and a single gage force stiffness method. The agreement between the test data and analysis for this test program provides confidence in the methods that are currently being used to design structures for hypersonic vehicles. The agreement also suggests that postbuckled strength may potentially be used to reduce the vehicle weight.

Teare, Wendy P.↗

NASA NTRS1990

Thermal buckling and post-buckling of symmetric composite plates

The thermally induced buckling and postbuckling behavior of square and rectangular simply supported composite plates is examined using the Rayleigh-Ritz method in conjunction with the second variations of the total potential energy. Spatially uniform and nonuniform temperatures and laminates with material axes that are not aligned with the plate edges (skewed material axes) are considered. Results of the analysis are presented in graphic form.

Meyers, C. A.↗

NASA NTRS1991

Finite element analysis of thermal post-buckling and vibrations of thermally buckled composite plates

A finite-element formulation is presented for the analysis of thermal postbuckling and free vibrations of thermally buckled, thin, rectangular, arbitrarily laminated, composite plates. The thermal load is considered to be a large steady-state temperature change over the plate. The equations of motion are derived from the principle of virtual work. The solution procedures are briefly discussed. Finite-element results are compared with existing isotropic and composite solutions. Examples studied include symmetric and antisymmetric angle-ply laminates and certain symmetric laminates.

Mei, Chuh↗

NASA NTRS1992

Thermally-induced, geometrically nonlinear response of symmetrically laminated composite plates

This paper discusses the thermally-induced geometrically nonlinear response of symmetrically laminated composite plates. The plate response is due to a temperature increase that is uniform in the plane of the plate but has a slight gradient through the thickness. The case of a completely uniform temperature increase but with an initial out-of-plane imperfection in the plate is also considered. Because they are closely allied problems, thermal buckling and postbuckling are discussed. Using variational methods in conjunction with a Rayleigh-Ritz formulation, these responses are investigated for two laminates, a (+/- 45/0/90)s and a (+/- 45/02)s, under two different simple support conditions.

Meyers, C. A.↗

NASA NTRS1992

Mode interaction in stiffened composite shells under combined mechanical and thermal loadings

Stiffened shells of various configurations fabricated out of composite materials find extensive applications in aircraft structures. Two distinctive modes of buckling dominate structural response of stiffened panels, viz. the short-wave local mode in which the shell skin buckles essentially between the stiffeners and the long-wave overall mode in which the shell skin buckles carrying the stiffeners with it. In optimized designs, the critical stresses corresponding to these modes of buckling would be close to each other. This leads to a nonlinear mode interaction which is recognized to be the principal cause of the failure of stiffened structures. If the structure is subjected to through-the-thickness thermal gradients, then large-wave bending effects would begin to occur well below the overall critical load and these would play the role of overall imperfections. The load carrying capacity would be significantly diminished as a result of interaction of local buckling with overall thermal distortions. The analysis of this problem using standard finite element techniques can be shown to be prohibitively expensive for design iterations. A concept which would greatly facilitate the analysis of mode interaction is advanced. We note that the local buckling occurs in a more or less periodic pattern in a structure having regular spacings of stiffeners. Thus it is a relatively simple matter to analyze the local buckling and the second order effects (which are essential for modeling postbuckling phenomena) using a unit cell of the structure. Once analyzed, these dormations are embedded in a shell element. Thus, a shell element could span several half-waves of local buckling and still be able to depict local buckling effects with requisite accuracy. A major consequence of the interaction of overall buckling/bending is the slow variation of the local buckling amplitude across the structure - the phenomenon of 'amplitude modulation' - and this is accounted for in the present model by letting the scaling parameter of the local mode vary according to a 'slowly varying' function. The construction of the analytical model involves essentially two stages and these are described briefly.

Sridharan, Srinivasan↗

NASA NTRS1990

The behavior of elastic anisotropic laminated composite flat structures subjected to deterministic and random loadings

Within this research project, the following topics were studied: (1) foundation of the refined theory of flat cross-ply laminated composite flat and curved panels as well as their static and dynamic response analysis; (2) foundation of a geometrically-nonlinear shear-deformable theory of composite laminated flat panels including the effect of initial geometric imperfections and its application in the postbuckling analysis; (3) the study of the dynamic response of shear deformable elastic laminated composite panels to deterministic time-dependent external excitations as the sonic boom and explosive blast type-loadings; (4) the study of the dynamic response of shear deformable elastic laminated composite panels to random excitation as e.g. the one produced by a jet noise or by any time-dependent external excitation whose characteristics are expressed in a statistical sense; and (5) the dynamic stability of fiber-reinforced composite flat panels whose materials (due to e.g. an ambient high temperature field) exhibit a time-dependent physical behavior.

Librescu, Liviu↗

NASA NTRS1991

Prediction of stiffener-skin separation in composite panels

A methodology was developed to predict the failure of stiffened composite panels by stiffener-skin separation. The methodology is applicable to curved or flat panels under combined uniaxial compression and in-plane shear loads. The analytical crux of the methodology are two different analysis packages: PACL1, which predicts the stress and displacement fields in a stiffened postbuckled panel; and WEBSTER, which predicts the local interfacial (shear and normal) stresses at the skin and stiffener flange junction. PACL1 is a Rayleigh-Ritz analysis of curved composite panels loaded into a post-buckling range. WEBSTER is a two-dimensional elasticity analysis for the skin-stiffener interface. Results of the predictive methodology are given.

Kan, Han-Pin↗