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NASA NTRS1991

Variational finite element-tensor formulation for the large deflection random vibration of composite plates

A numerical integration routine is derived from a set of unified single step integration algorithms using a weighted satisfaction of the equilibrium equation governing the large deflection random response of laminated composite plates. The equilibrium equation is derived using a constant matrix large deflection finite-element formulation. In-plane inertia terms are considered in the formulation, however, rotary inertia terms are assumed negligible. Probability density, spectral density and autocorrelation functions of the maximum displacement and strain responses are presented for three acoustic excitation levels. Classical thin plate boundary conditions and pseudo white noise excitation are used in this investigation.

Robinson, Jay H.↗

NASA NTRS2005

Pressures in Tumuli: A Study of Tumuli Formation

Tumuli form via localized inflation in surface lava flows. These domed features have widths of 10-20 m, lengths of 10-150 m, and heights of 1-9 m. The axial fracture exposes a brittle crust overlying a ductilely deformed layer. The total crustal thickness is typically less than lm. Tumuli are observed on both terrestrial and martian lava flow surfaces, and provide insight on the flow formation processes and rates. Past studies have estimated the inflation pressure using a bending model for a circular, thin elastic plate, assuming small deflection (Rossi and Gudmundson, 1996). This formulation results in unrealistic pressures for some tumuli. We thus examine alternative models, including those with different shapes, bending of the ductile crust, large deflection, plastic deformation, and thick plate bending. Using the thickness of the ductile crust in the equations for thin, circular plates reduces most pressures to reasonable values. Alternative plate shapes do not cause a significant reduction in inflation pressure. Although the large deflection equations should be applicable based on the plate thickness to tumuli height ratios, they give even less realistic pressures. Tumuli with unrealistic pressures appear to have exceeded the critical bending moment, and have relatively thick crusts, requiring thick plate bending models.

Hansen, James E.↗

NASA NTRS1974

A theory for predicting composite laminate warpage resulting from fabrication

Linear laminate theory is used with the moment-curvature relationship to derive equations for predicting end deflections due to warpage without solving the coupled fourth-order partial differential equations of the plate. Composite micro- and macrohyphenmechanics are used with laminate theory to assess the contribution of factors such as ply misorientation, fiber migration, and fiber and/or void volume ratio nonuniformity on the laminate warpage. Using these equations, it was found that a 1 deg error in the orientation angle of one ply was sufficient to produce warpage end deflection equal to two laminate thicknesses in a 10 inch by 10 inch laminate made from 8 ply Mod-I/epoxy. Using a sensitivity analysis on the governing parameters, it was found that a 3 deg fiber migration or a void volume ratio of three percent in some plies is sufficient to produce laminate warpage corner deflection equal to several laminate thicknesses. Tabular and graphical data are presented which can be used to identify possible errors contributing to laminate warpage and/or to obtain an a priori assessment when unavoidable errors during fabrication are anticipated.

Chamis, C. C.↗

NASA NTRS1975

A theory for predicting composite laminate warpage resulting from fabrication

Linear laminate theory is used in conjunction with the moment-curvature relationship to derive equations for predicting end deflections due to warpage without solving the coupled fourth-order partial differential equations of the plate. Using these equations, it is found that a 1 deg error in the orientation angle of one ply is sufficient to produce warpage end deflection equal to two laminate thicknesses in a 10 inch by 10 inch laminate made from 8-ply Mod-I/epoxy. From a sensitivity analysis on the governing parameters, it is found that a 3 deg fiber migration or a void volume ratio of three percent in some plies is sufficient to produce laminate warpage corner deflection equal to several laminate thicknesses. Tabular and graphical data are presented which can be used to identify possible errors contributing to laminate warpage and/or to obtain an a priori assessment when unavoidable errors during fabrication are anticipated.

Chamis, C. C.↗

NASA NTRS1967

Nonlinear oscillations of a fluttering plate. II.

Quasi-steady aerodynamic and von Karman large deflection plate theory equations of nonlinear oscillations of fluttering plate for single mode subsonic and sonic or coupled mode supersonic oscillations

VON KARMAN EQUATION↗

NASA NTRS1967

Nonlinear oscillations of a fluttering plate. II.

VON KARMAN EQUATION↗

NASA NTRS1968

Prediction of the interfacial stress and gap in a bolted joint.

Interfacial stress and gap in bolted joint predicted by programmed equations of plate deflection

Fontenot, J. E., Jr.↗

NASA NTRS1971

ELAS8 - Computer program for linear structure equilibrium problems

Program generates and solves governing equations for unknown deflection of mesh points as if problem were to locate stationary point of total potential function associated with given loading and unknown deflections. Solution is obtained by means of displacement method and finite element technique.

Utku, S.↗

NASA NTRS1977

Composite laminate warpage

Set of simplified equations predicts warpage that often occurs upon removal from fabrication mold. Equations predict corner deflection and are based upon micromechanics and macromechanics of composites, as well as laminate theory. Factors considered include ply misorientation, fiber migration, and void/volume ratio nonuniformity.

Chamis, C. C.↗

NASA NTRS1983

Postbuckling of long orthotropic plates in combined shear and compression

The nonlinear large-deflection partial differential equations of von karman for orthotropic plates loaded in combined shear and compression are converted into a set of first-order nonlinear ordinary differential equations by assuming trigonometric functions in one direction. These equations are solved numerically using a two point boundary problem solver which makes use of Newton's method. Results are obtained which determine the postbuckling behavior of rectangular plates with loading up to about three times the buckling load. Both isotropic and orthotropic composite plates are considered. Results show that orthotropic plates may behave quite differently than isotropic plates and that in-plane boundary conditions are important for plates loaded in shear.

Stein, M.↗

NASA NTRS1994

Sensitivity Analysis of Flutter Response of a Wing Incorporating Finite-Span Corrections

Flutter analysis of a wing is performed in compressible flow using state-space representation of the unsteady aerodynamic behavior. Three different expressions are used to incorporate corrections due to the finite-span effects of the wing in estimating the lift-curve slope. The structural formulation is based on a Rayleigh-Pitz technique with Chebyshev polynomials used for the wing deflections. The aeroelastic equations are solved as an eigen-value problem to determine the flutter speed of the wing. The flutter speeds are found to be higher in these cases, when compared to that obtained without accounting for the finite-span effects. The derivatives of the flutter speed with respect to the shape parameters, namely: aspect ratio, area, taper ratio and sweep angle, are calculated analytically. The shape sensitivity derivatives give a linear approximation to the flutter speed curves over a range of values of the shape parameter which is perturbed. Flutter and sensitivity calculations are performed on a wing using a lifting-surface unsteady aerodynamic theory using modules from a system of programs called FAST.

Issac, Jason Cherian↗

NASA NTRS2002

Determination of Ice Crust Thickness from Flanking Cracks Along Ridges on Europa

We use equations describing the deflection of an elastic plate below a line load to estimate ice crust thickness below ridges on Europa. Using a range of elastic parameters, ice thickness is calculated to fall in the range 0.2 2.6 km. Additional information is contained in the original extended abstract.

Billings, S. E.↗

NASA NTRS2009

Analysis of Sting Balance Calibration Data Using Optimized Regression Models

Calibration data of a wind tunnel sting balance was processed using a search algorithm that identifies an optimized regression model for the data analysis. The selected sting balance had two moment gages that were mounted forward and aft of the balance moment center. The difference and the sum of the two gage outputs were fitted in the least squares sense using the normal force and the pitching moment at the balance moment center as independent variables. The regression model search algorithm predicted that the difference of the gage outputs should be modeled using the intercept and the normal force. The sum of the two gage outputs, on the other hand, should be modeled using the intercept, the pitching moment, and the square of the pitching moment. Equations of the deflection of a cantilever beam are used to show that the search algorithm s two recommended math models can also be obtained after performing a rigorous theoretical analysis of the deflection of the sting balance under load. The analysis of the sting balance calibration data set is a rare example of a situation when regression models of balance calibration data can directly be derived from first principles of physics and engineering. In addition, it is interesting to see that the search algorithm recommended the same regression models for the data analysis using only a set of statistical quality metrics.

Ulbrich, Norbert↗

NASA NTRS2010

Analysis of Sting Balance Calibration Data Using Optimized Regression Models

Calibration data of a wind tunnel sting balance was processed using a candidate math model search algorithm that recommends an optimized regression model for the data analysis. During the calibration the normal force and the moment at the balance moment center were selected as independent calibration variables. The sting balance itself had two moment gages. Therefore, after analyzing the connection between calibration loads and gage outputs, it was decided to choose the difference and the sum of the gage outputs as the two responses that best describe the behavior of the balance. The math model search algorithm was applied to these two responses. An optimized regression model was obtained for each response. Classical strain gage balance load transformations and the equations of the deflection of a cantilever beam under load are used to show that the search algorithm s two optimized regression models are supported by a theoretical analysis of the relationship between the applied calibration loads and the measured gage outputs. The analysis of the sting balance calibration data set is a rare example of a situation when terms of a regression model of a balance can directly be derived from first principles of physics. In addition, it is interesting to note that the search algorithm recommended the correct regression model term combinations using only a set of statistical quality metrics that were applied to the experimental data during the algorithm s term selection process.

Ulbrich, N.↗

NASA NTRS1989

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

By using the linear mode shapes of a thermally buckled structure, a governing system of finite element equations for large-deflection random vibrations is reduced to a system of nonlinear modal equations. This system is then solved using the method of equivalent linearization. The methodology developed in this study is applicable to the design of aerospace vehicles subjected to combined thermal/acoustic loads.

Locke, James↗

NASA NTRS1989

Nonlinear strain-displacement relations and flexible multibody dynamics

Dynamics of chains of flexible bodies undergoing large rigid body motions, but small elastic deflections are considered. The role of nonlinear strain-displacement relations in the development of the motion equations correct to first order in elastic deflections is investigated. The general form of these equations linearized only in the small elastic deflections is presented, and the relative significance of various nonlinear terms is studied both analytically and through the use of the numerical simulations. Numerical simulations are performed for a two link chain constrained to move in the plane, subject to hinge torques. Each link is modeled as a thin beam. Slew maneuver simulation results are compared for models with and without properly modeled kinematics of deformation. The goal of this case study is to quantify the importance of the terms in the equations of motion which arise from the inclusion of nonlinear strain-displacement relations. It is concluded that unless the consistently linearized equations in elastic deflections and speeds are available and necessary, the inconsistently (prematurely) linearized equations should be replaced in all cases by ruthlessly linearized equations: equations in which all nonlinear terms involving the elastic deflections and speeds are ignored.

Padilla, Carlos E.↗

NASA NTRS1984

Analytical results for postbuckling behavior of plates in compression and shear

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Stein, M.↗

NASA NTRS1985

Analytical results for post-buckling behaviour of plates in compression and in shear

Stein, M.↗