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NASA NTRS2010

Frequencies and Flutter Speed Estimation for Damaged Aircraft Wing Using Scaled Equivalent Plate Analysis

Equivalent plate analysis is often used to replace the computationally expensive finite element analysis in initial design stages or in conceptual design of aircraft wing structures. The equivalent plate model can also be used to design a wind tunnel model to match the stiffness characteristics of the wing box of a full-scale aircraft wing model while satisfying strength-based requirements An equivalent plate analysis technique is presented to predict the static and dynamic response of an aircraft wing with or without damage. First, a geometric scale factor and a dynamic pressure scale factor are defined to relate the stiffness, load and deformation of the equivalent plate to the aircraft wing. A procedure using an optimization technique is presented to create scaled equivalent plate models from the full scale aircraft wing using geometric and dynamic pressure scale factors. The scaled models are constructed by matching the stiffness of the scaled equivalent plate with the scaled aircraft wing stiffness. It is demonstrated that the scaled equivalent plate model can be used to predict the deformation of the aircraft wing accurately. Once the full equivalent plate geometry is obtained, any other scaled equivalent plate geometry can be obtained using the geometric scale factor. Next, an average frequency scale factor is defined as the average ratio of the frequencies of the aircraft wing to the frequencies of the full-scaled equivalent plate. The average frequency scale factor combined with the geometric scale factor is used to predict the frequency response of the aircraft wing from the scaled equivalent plate analysis. A procedure is outlined to estimate the frequency response and the flutter speed of an aircraft wing from the equivalent plate analysis using the frequency scale factor and geometric scale factor. The equivalent plate analysis is demonstrated using an aircraft wing without damage and another with damage. Both of the problems show that the scaled equivalent plate analysis can be successfully used to predict the frequencies and flutter speed of a typical aircraft wing.

Krishnamurthy, Thiagarajan↗
NASA NTRS2007

Frequency Response of an Aircraft Wing with Discrete Source Damage Using Equivalent Plate Analysis

An equivalent plate procedure is developed to provide a computationally efficient means of matching the stiffness and frequencies of flight vehicle wing structures for prescribed loading conditions. Several new approaches are proposed and studied to match the stiffness and first five natural frequencies of the two reference models with and without damage. One approach divides the candidate reference plate into multiple zones in which stiffness and mass can be varied using a variety of materials including aluminum, graphite-epoxy, and foam-core graphite-epoxy sandwiches. Another approach places point masses along the edge of the stiffness-matched plate to tune the natural frequencies. Both approaches are successful at matching the stiffness and natural frequencies of the reference plates and provide useful insight into determination of crucial features in equivalent plate models of aircraft wing structures.

Krishnamurthy, T.↗
NASA NTRS2008

Static and Dynamic Structural Response of an Aircraft Wing with Damage Using Equivalent Plate Analysis

A process to generate an equivalent plate based on an optimization approach to predict the static and dynamic response of flight vehicle wing structures is proposed. Geometric-scale and frequency-scale factors are defined to construct an equivalent plate with any desired scale to use in simulation and wind tunnel experiments. It is shown that the stiffness and the displacements are scaled linearly with the geometric-scale factor, whereas the load is scaled as the square of the geometric-scale factor. The scaled stiffness of the reference flight vehicle is matched first to construct the equivalent plate. Then the frequency-scale factor is defined to scale the flight vehicle frequencies. The scaled flight vehicle frequencies are matched by placing arbitrary point masses along the equivalent plate geometry. Two simple stiffened-plate examples, one with damage and another without damage, were used to demonstrate the accuracy of the optimization procedure proposed. Geometric-scale factors ranging from 0.2 to 1.0 were used in the analyses. In both examples, the static and dynamic response of the reference stiffened-panel solution is matched accurately. The scaled equivalent plate predicted the first five frequencies of the stiffened panel very accurately. Finally, the proposed equivalent plate procedure was demonstrated in a more realistic typical aircraft wing structure. Two scale equivalent plate models were generated using the geometric-scale factors 1.0 and 0.2. Both equivalent plate models predicted the static response of the wing structure accurately. The equivalent plate models reproduced the first five frequencies of the wing structure accurately.

Krishnamurthy, T.↗
NASA NTRS1994

Analytic Sensitivities for Shape Optimization in Equivalent Plate Structural Wing Models

Equivalent plate modeling techniques based on Ritz analysis with simple polynomials prove to be efficient tools for structural modeling of wings in the preliminary design stage. Accuracy problems are encountered, however, when these models are used to obtain finite difference behavior sensitivities with respect to planform shape. The accuracy problems are associated with the poor numerical conditioning of static and eigenvalue equations. As higher-order polynomials are being used to Improve the analysis itself, the more sensitive is the finite difference derivative to the step size used. This article describes a formulation of wing equivalent plate modeling in which it is simple to obtain analytic, explicit expressions for stiffness and mass matrix elements without the need to perform numerical integration. This formulation leads naturally to analytic expressions for the derivatives of displacements, stresses, and natural frequencies with respect to shape design variables. This article examines the accuracy of finite difference derivatives compared with the analytic derivatives, and shows that In some cases it is impossible to obtain any information of value by finite differences. Analytic sensitivities, in this case, are still sufficiently accurate for design optimization.

Livne, Eli↗
NASA NTRS2006

Equivalent Plate Analysis of Aircraft Wing with Discrete Source Damage

An equivalent plate procedure is developed to provide a computationally efficient means of matching the stiffness and frequencies of flight vehicle wing structures for prescribed loading conditions. First, the equivalent plate is used to match the stiffness of a stiffened panel without damage and the stiffness of a stiffened panel with damage. For both stiffened panels, the equivalent plate models reproduce the deformation of a corresponding detailed model exactly for the given loading conditions. Once the stiffness was matched, the equivalent plate models were then used to predict the frequencies of the panels. Two analytical procedures using the lumped-mass matrix were used to match the first five frequencies of the corresponding detailed model. In both the procedures, the lumped-mass matrix for the equivalent plate is constructed by multiplying the diagonal terms of the consistent-mass matrix by a proportionality constant. In the first procedure, the proportionality constant is selected such that the total mass of the equivalent plate is equal to that of the detailed model. In the second method, the proportionality constant is selected to minimize the sum of the squares of the errors in a set of pre-selected frequencies between the equivalent plate model and the detailed model. The equivalent plate models reproduced the fundamental first frequency accurately in both the methods. It is observed that changing only the mass distribution in the equivalent plate model did not provide enough flexibility to match all of the frequencies.

Krishnamurthy, T.↗
NASA NTRS1999

Static and Vibration Analyses of General Wing Structures Using Equivalent Plate Models

An efficient method, using equivalent plate model, is developed for studying the static and vibration analyses of general built-up wing structures composed of skins, spars, and ribs. The model includes the transverse shear effects by treating the built-up wing as a plate following the Reissner-Mindlin theory, the so-called First-order Shear Deformation Theory (FSDT). The Ritz method is used with the Legendre polynomials being employed as the trial functions. This is in contrast to previous equivalent plate model methods which have used simple polynomials, known to be prone to numerical ill-conditioning, as the trial functions. The present developments are evaluated by comparing the results with those obtained using MSC/NASTRAN, for a set of examples. These examples are: (i) free-vibration analysis of a clamped trapezoidal plate with (a) uniform thickness, and (b) non-uniform thickness varying as an airfoil, (ii) free-vibration and static analyses (including skin stress distribution) of a general built-up wing, and (iii) free-vibration and static analyses of a swept-back box wing. The results obtained by the present equivalent plate model are in good agreement with those obtained by the finite element method.

Kapania, Rakesh K.↗
NASA NTRS2011

A Treatise on Equivalent-Plate Stiffnesses for Stiffened Laminated-Composite Plates and Plate-Like Lattices

A survey of studies conducted since 1914 on the use of equivalent-plate stiffnesses in modeling the overall, stiffness-critical response of stiffened plates and shells is presented. Two detailed, comprehensive derivations of first-approximation equivalent-plate stiffnesses are also presented that are based on the Reissner-Mindlin-type, first-order transverse-shear deformation theory for anisotropic plates. Equivalent-plate stiffness expressions, and a corresponding symbolic manipulation computer program, are also presented for several different stiffener configurations. These expressions are very general and exhibit the full range of anisotropies permitted by the Reissner-Mindlin-type, first-order transverse-shear deformation theory for anisotropic plates. The expressions presented in the present study were also compared with available, previously published results. For the most part, the previously published results are for special cases of the general expressions presented herein and are almost in complete agreement. Analysis is also presented that extends the use of the equivalent-plate stiffness expressions to sandwich plates.

Nemeth, Michael P.↗
NASA NTRS1987

Further generalization of an equivalent plate representation for aircraft structural analysis

Recent developments from a continuing effort to provide an equivalent plate representation for aircraft structural analysis are described. Previous work provided an equivalent plate analysis formulation that is capable of modeling aircraft wing structures with a general planform such as cranked wing boxes. However, the modeling is restricted to representing wing boxes having symmetric cross sections. Further developments, which are described, allow modeling of wing cross sections having asymmetries that can arise from airfoil camber or from thicknesses being different in the upper and lower cover skins. An implementation of thermal loadings, which are described as temperature distributions over the planform of the cover skins, has been included. Spring supports have been added to provide for a more general set of boundary conditions. Numerical results are presented to assess the effect of wing camber on the static and dynamic response of an example wing structure under pressure and thermal loading. These results are compared with results from a finite element analysis program to indicate how well a cambered wing box can be represented with an equivalent plate formulation.

Giles, Gary L.↗
NASA NTRS1987

Further generalization of an equivalent plate representation for aircraft structural analysis

Recent developments from a continuing effort to provide an equivalent plate representation for aircraft structural analysis are described. Previous work provided an equivalent plate analysis formulation that is capable of modeling aircraft wing structures with a general planform such as cranked wing boxes. However, the modeling is restricted to representing wing boxes having symmetric cross sections. Further developments, which are described in this paper, allow modeling of wing cross section having asymmetries that can arise from airfoil camber or from thicknesses being different in the upper and lower cover skins. An implementation of thermal loadings, which are described as temperature distributions over the planform of the cover skins, has been included. Spring supports have been added to provide for a more general set of boundary conditions. Numerical results are presented to assess the effect of wing camber on the static and dynamic response of an example wing structure under pressure and thermal loading. These results are compared with results from a finite element analysis program to indicate how well a cambered wing box can be represented with an equivalent plate formulation.

Giles, Gary L.↗
NASA NTRS1990

Comparison of equivalent plate and finite element analysis of a realistic aircraft structural configuration

This paper presents a comparison study to assess how well a realistic aircraft structure can be represented with an equivalent plate model. A wing box and full wing structure of a high-speed civil transport aircraft configuration are analyzed. Static and vibration results from the equivalent plate analysis are compared with results from a finite element model. The results from these comparisons show that, for the simpler wing box models, the static displacements and first vibration frequency agree to within 1.5 percent. The full wing models only agree to within 10 percent, for both models, the higher frequencies vary by as much as 26 percent. These differences are due to inadequate refinement of the equivalent plate model.

Kao, Pi-Jen↗
NASA NTRS1986

Equivalent plate analysis of aircraft wing box structures with general planform geometry

A new equivalent plate analysis formulation is described which is capable of modeling aircraft wing structures with a general planform such as cranked wing boxes. Multiple trapezoidal segments are used to represent such planforms. A Ritz solution technique is used in conjunction with global displacement functions which encompass all the segments. This Ritz solution procedure is implemented efficiently into a computer program so that it can be used by rigorous optimization algorithms for application in early preliminary design. A direct method to interface this structural analysis procedure with aerodynamic programs for use in aeroelastic calculations is described. This equivalent plate analysis procedure is used to calculate the static deflections and stresses and vibration frequencies and modes of an example wing configuration. The numerical results are compared with results from a finite element model of the same configuration to illustrate typical levels of accuracy and computation times resulting from use of this procedure.

Giles, G. L.↗
NASA NTRS1995

Equivalent plate modeling for conceptual design of aircraft wing structures

This paper describes an analysis method that generates conceptual-level design data for aircraft wing structures. A key requirement is that this data must be produced in a timely manner so that is can be used effectively by multidisciplinary synthesis codes for performing systems studies. Such a capability is being developed by enhancing an equivalent plate structural analysis computer code to provide a more comprehensive, robust and user-friendly analysis tool. The paper focuses on recent enhancements to the Equivalent Laminated Plate Solution (ELAPS) analysis code that significantly expands the modeling capability and improves the accuracy of results. Modeling additions include use of out-of-plane plate segments for representing winglets and advanced wing concepts such as C-wings along with a new capability for modeling the internal rib and spar structure. The accuracy of calculated results is improved by including transverse shear effects in the formulation and by using multiple sets of assumed displacement functions in the analysis. Typical results are presented to demonstrate these new features. Example configurations include a C-wing transport aircraft, a representative fighter wing and a blended-wing-body transport. These applications are intended to demonstrate and quantify the benefits of using equivalent plate modeling of wing structures during conceptual design.

Giles, Gary L.↗
NASA NTRS1993

Recent developments in equivalent plate modeling for wing shape optimization

A new technique for structural modeling of airplane wings is presented taking transverse shear effects into account. The kinematic assumptions of first order shear deformation plate theory in combination with numerical analysis based on simple polynomials which define geometry, construction and displacement approximations lead to analytical expressions for elements of the stiffness and mass matrices and load vector. Contributions from the cover skins, spar and rib caps and spar and rib webs are included as well as concentrated springs and concentrated masses. Limitations of current equivalent plate wing modeling techniques based on classical plate theory are discussed, and the improved accuracy of the new equivalent plate technique is demonstrated through comparison to finite element analysis and test results. Analytical derivatives of stiffness, mass and load terms with respect to wing shape lead to analytic sensitivities of displacements, stresses and natural modes with respect to planform shape and depth distribution. This makes the new capability an effective structural tool for wing shape optimization.

Livne, Eli↗
NASA NTRS1994

Equivalent plate structural modeling for wing shape optimization including transverse shear

A new technique for structural modeling of airplanes wings is presented taking transverse shear effects into account. The kinematic assumptions of first-order shear deformation plate theory in combination with numerical analysis, where simple polynomials are used to define geometry, construction, and displacement approximations, lead to analytical expressions for elements of the stiffness and mass matrices and load vector. Contributions from the cover skins, spar and rib caps, and spar and rib webs are included as well as concentrated springs and concentrated masses. Limitations of wing modeling techniques based on classical plate theory are discussed, and the improved accuracy of the new equivalent plate technique is demonstrated through comparison with finite element analysis and test results. Expressions for analytical derivatives of stiffness, mass, and load terms with respect to wing shape are given. Based on these, it is possible to obtain analytic sensitivities of displacements, stresses, and natural frequencies with respect to planform shape and depth distribution. This makes the new capability an effective structural tool for wing shape optimization.

Livne, Eli↗
NASA NTRS1994

Equivalent Plate Structural Modeling for Wing Shape Optimization Including Transverse Shear

A new technique for structural modeling of airplane wings is presented taking transverse shear effects into account. The kinematic assumptions of first-order shear deformation plate theory In combination with numerical analysis, where simple polynomials are used to define geometry, construction, and displacement approximations, lead to analytical expressions for elements of the stiffness and mass matrices and load vector. Contributions from the cover skins, spar and rib caps, and spar and rib webs are included as well as concentrated springs and concentrated masses. Limitations of wing modeling techniques based on classical plate theory are discussed, and the Improved accuracy of the new equivalent plate technique is demonstrated through comparison with finite element analysis and test results. Expressions for analytical derivatives of stiffness, mass, and load terms with respect to wing shape are given. Based on these, it is possible to obtain analytic sensitivities of displacements, stresses, and natural frequencies with respect to planform shape and depth distribution. This makes the new capability an effective structural tool for wing shape optimization.

Livne, Eli↗
NASA NTRS1994

Coupling equivalent plate and finite element formulations in multiple-method structural analyses

A coupled multiple-method analysis procedure for use late in conceptual design or early in preliminary design of aircraft structures is described. Using this method, aircraft wing structures are represented with equivalent plate models, and structural details such as engine/pylon structure, landing gear, or a 'stick' model of a fuselage are represented with beam finite element models. These two analysis methods are implemented in an integrated multiple-method formulation that involves the assembly and solution of a combined set of linear equations. The corresponding solution vector contains coefficients of the polynomials that describe the deflection of the wing and also the components of translations and rotations at the joints of the beam members. Two alternative approaches for coupling the methods are investigated; one using transition finite elements and the other using Lagrange multipliers. The coupled formulation is applied to the static analysis and vibration analysis of a conceptual design model of a fighter aircraft. The results from the coupled method are compared with corresponding results from an analysis in which the entire model is composed of finite elements.

VIBRATION ANALYSIS↗
NASA NTRS1986

Equivalent plate analysis of aircraft wing box structures with general planform geometry

A new equilvalent plate analysis formulation is described which is capable of modeling aircraft wing structures with a general planform such as cranked wing boxes. Multiple trapezoidal segments are used to represent such planforms. A Ritz solution technique is used in conjunction with global displacement functions which encompass all the segments. This Ritz solution procedure is implemented efficiently into a computer program so that it can be used by rigorous optimization algorithms for application in early preliminary design. A direct method to interface this structural analysis procedure with aerodynamic programs for use in aeroelastic calculations is described. This equivalent plate analysis procedure is used to calculate the static deflections and stresses and vibration frequencies and modes of an example wing configuration. The numerical results are compared with results from a finite element model of the same configuration to illustrate typical levels of accuracy and computation times resulting from use of this procedure.

Giles, G. L.↗