Browsing by Author "Johnson, Eric R."

Now showing 1 - 20 of 142
  • Thumbnail Image
    Accuracy analysis of the semi-analytical method for shape sensitivity analysis
    Barthelemy, Bruno (Virginia Polytechnic Institute and State University, 1987)
    The semi-analytical method, widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements, is studied in this research. The research shows that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Local and global indices are developed to test the accuracy of the semi-analytical method. The local indices provide insight into the problem of large errors for the semi-analytical method. Local error magnification indices are developed for beam and plane truss structures, and several examples showing the severity of the problem are presented. The global index provides us with a general method for checking the accuracy of the semi-analytical method for any type of model. It characterizes the difference in errors between a general finite-difference method and the semi-analytical method. Moreover, a method improving the accuracy of the semi-analytical method (when possible) is provided. Examples are presented showing the use of the global index.
  • Thumbnail Image
    Aeroelasticity of Morphing Wings Using Neural Networks
    Natarajan, Anand (Virginia Tech, 2002-07-03)
    In this dissertation, neural networks are designed to effectively model static non-linear aeroelastic problems in adaptive structures and linear dynamic aeroelastic systems with time varying stiffness. The use of adaptive materials in aircraft wings allows for the change of the contour or the configuration of a wing (morphing) in flight. The use of smart materials, to accomplish these deformations, can imply that the stiffness of the wing with a morphing contour changes as the contour changes. For a rapidly oscillating body in a fluid field, continuously adapting structural parameters may render the wing to behave as a time variant system. Even the internal spars/ribs of the aircraft wing which define the wing stiffness can be made adaptive, that is, their stiffness can be made to vary with time. The immediate effect on the structural dynamics of the wing, is that, the wing motion is governed by a differential equation with time varying coefficients. The study of this concept of a time varying torsional stiffness, made possible by the use of active materials and adaptive spars, in the dynamic aeroelastic behavior of an adaptable airfoil is performed here. A time marching technique is developed for solving linear structural dynamic problems with time-varying parameters. This time-marching technique borrows from the concept of Time-Finite Elements in the sense that for each time interval considered in the time-marching, an analytical solution is obtained. The analytical solution for each time interval is in the form of a matrix exponential and hence this technique is termed as Matrix Exponential time marching. Using this time marching technique, Artificial Neural Networks can be trained to represent the dynamic behavior of any linearly time varying system. In order to extend this methodology to dynamic aeroelasticity, it is also necessary to model the unsteady aerodynamic loads over an airfoil. Accordingly, an unsteady aerodynamic panel method is developed using a distributed set of doublet panels over the surface of the airfoil and along its wake. When the aerodynamic loads predicted by this panel method are made available to the Matrix Exponential time marching scheme for every time interval, a dynamic aeroelastic solver for a time varying aeroelastic system is obtained. This solver is now used to train an array of neural networks to represent the response of this two dimensional aeroelastic system with a time varying torsional stiffness. These neural networks are developed into a control system for flutter suppression. Another type of aeroelastic problem of an adaptive structure that is investigated here is the shape control of an adaptive bump situated on the leading edge of an airfoil. Such a bump is useful in achieving flow separation control for lateral directional maneuverability of the aircraft. Since actuators are being used to create this bump on the wing surface, the energy required to do so needs to be minimized. The adverse pressure drag as a result of this bump needs to be controlled so that the loss in lift over the wing is made minimal. The design of such a "spoiler bump" on the surface of the airfoil is an optimization problem of maximizing pressure drag due to flow separation while minimizing the loss in lift and energy required to deform the bump. One neural network is trained using the CFD code FLUENT to represent the aerodynamic loading over the bump. A second neural network is trained for calculating the actuator loads, bump displacement and lift, drag forces over the airfoil using the finite element solver, ANSYS and the previously trained neural network. This non-linear aeroelastic model of the deforming bump on an airfoil surface using neural networks can serve as a fore-runner for other non-linear aeroelastic problems. This work enhances the traditional aeroelastic modeling by introducing time varying parameters in the differential equations of motion. It investigates the calculation of non-conservative aerodynamic loads on morphing contours and the resulting structural deformation for non-linear aeroelastic problems through the use of neural networks. Geometric modeling of morphing contours is also addressed.
  • Thumbnail Image
    Aerospace Structures
    Johnson, Eric R. (Kevin T. Crofton Department of Aerospace and Ocean Engineering in affiliation with Virginia Tech Publishing, 2022)

    Aerospace Structures by Eric Raymond Johnson is a 600+ page text and reference book for junior, senior, and graduate-level aerospace engineering students. The text begins with a discussion of the aerodynamic and inertia loads acting on aircraft in symmetric flight and presents a linear theory for the statics and dynamic response of thin-walled straight bars with closed and open cross-sections. Isotropic and fiber-reinforced polymer (FRP) composite materials including temperature effects are modeled with Hooke’s law. Methods of analyses are by differential equations, Castigliano’s theorems, the direct stiffness method, the finite element method, and Lagrange’s equations. There are numerous examples for the response of axial bars, beams, coplanar trusses, coplanar frames, and coplanar curved bars. Failure initiation by the von Mises yield criterion, buckling, wing divergence, fracture, and by Puck’s criterion for FRP composites are presented in the examples. Professors, if you are reviewing this book for adoption in your course, please let us know by filling out this form. Instructors reviewing, adopting, or adapting parts or the whole of the text are especially encouraged to fill it out. The EPUB was released early May 2023. The EPUB contains MathML and alternative text. LaTeX sourcefiles were publicly released in October 2023. LaTeX sourcefiles are also available in Overleaf under a CC BY NC SA 4.0 license. Stable link for this resource: https://doi.org/10.21061/AerospaceStructures Report errors
    View errata Table of Contents 1. Function of Flight Vehicle Structural Members 2. Aircraft Loads 3. Elements of Thin-Walled Bar Theory 4. Some Aspects of the Structural Analysis 5. Work and Energy Methods 6. Applications of Castigliano's Theorems 7. Arches, Rings, and Fuselage Frames 8. Laminated Bars of Fiber-Reinforced Polymer Composites 9. Failure Initiation in FRP Compositives 10. Structural Stability of Discrete Conservative Systems 11. Buckling of Columns and Plates 12. Introduction to Aeroelasticity 13. Fracture of Cracked Members 14. Design of a Landing Strut and Wing Spar 15. Direct Stiffness Method 16. Applications of the Direct Stiffness Method 17. Finite Element Method 18. Introduction to Flexible Body Dynamics Appendix A: Linear Elasticity of Solid Bodies Resources
    Problem sets:     http://hdl.handle.net/10919/104169
    LaTeX sourcefiles:     https://www.overleaf.com/read/skqbjscvwhjh#5decd4
    PDFs (book and chapter-level) and EPUB:     Available from the left side of the screen
    Print (paperback - does not include the appendix):     Order here About this book
    This text is evolved from lecture notes by the author for junior and senior students in the aerospace engineering curriculum at Virginia Tech. The subjects covered in the book presume some knowledge of statics, dynamics of rigid bodies, mechanics of deformable bodies, and mechanical vibrations. Several practice exercises in the text require programming, and typically the students use Mathematica1 or MATLAB 2 software to complete them. Examples in the text were programmed in Mathematica. A first semester sequence for junior students includes chapters 1 through 6. Note that chapter 3 on thinwall bar theory maybe too mathematical for some students, but can be used as a reference for the applications of the theory provided in chapter 4. The important topic of work and energy is covered in chapter 5, and chapter 6 is devoted to the application of Castigliano’s theorems to trusses, beams, and frames. A second semester sequence for junior students includes topics selected by the instructor from chapter 7 on curved bars, and chapters 10 through 16. The influence of imperfection sensitivity on the buckling load of discrete systems is presented in chapter 10, followed by buckling of columns and plates in chapter 11. Article 11.2 is optional. Analysis for wing divergence is presented in the introduction to aeroelasticity in chapter 12. The methods of linear elastic fracture mechanics to predict critical loads for crack propagation is discussed in chapter 13. Design of a landing strut, and the optimal design of a spar subject to constraints on yielding, buckling and fracture are presented in chapter 14. Chapters 15 and 16 detail the direct stiffness method for trusses, beams and frames. Topics appropriate for senior students are in chapters 8, 9, 17, and 18, and initial post-buckling in article 11.2. The response of closed and open section bars fabricated from a fiber-reinforced polymer composite (FRP) is presented in chapter 8, and failure initiation of FRP bars is presented in chapter 9. The finite element method applied to the extension and bending of bars is presented in chapter 17, which includes transverse shear deformations. The topic of adaptive mesh refinement in article 17.2.4 is optional. Articles 18.1 to 18.4 cover the dynamic response of lumped mass models, eigenvalue problems, and Lagrange’s equations. The remainder of chapter 18 utilizes the finite element method for the dynamic response of beams, trusses, and frames. In this textbook analytical methods are developed for the response and failure of the primary structural components of aircraft. Newton's laws of motion, Hooke s law, and the first law of thermodynamics are the basis to model the thermoelastic response of thin-walled, straight bars and coplanar curved bars. Analytical methods include energy principles to develop Castigliano s theorems and to develop the cross-sectional material law for transverse shear and torsion. Stiffened shells typical of aircraft structures are analyzed with the thin-walled bar theory. Externally prescribed loads are due to accelerated flight and the thermal environment. Velocity-load factor (V-n) diagrams for maneuvers and gusts are described to evaluate flight loads. Initiation of failure is predicted by one of the following criteria: von Mises yield criterion for ductile metals; the critical load to cause buckling (failure by excessive displacements); fracture criteria for the critical stress to cause crack propagation; Puck’s criterion for the brittle failure modes in fiber-reinforced polymer composites (FRP). The subject of structural stability of discrete conservative systems introduces the methods of stability analysis, classification of bifurcation buckling problems, the concept of imperfection sensitivity, and snapthrough at a limit point. Static instability of an elastic column from pre-buckling equilibrium, buckling, and through initial post-buckling is presented in detail. Buckling of flat rectangular plates subject to compression and shear is presented in a qualitative way using the classic charts from the National Advisory Committee for Aeronautics (NACA). The analysis for the static instability of a wing in steady incompressible flow, or divergence, is part of the discussion of aeroelastic phenomena. - Results from linear elastic fracture mechanics (LEFM) are introduced to illustrate the relation between crack size and the stress to cause crack propagation. Airplane damage-tolerant design is based on LEFM such that subcritical length cracks do not grow to critical length between inspection intervals. - The incentive to study optimal design is illustrated by the example of an aluminum wing spar. The objective is to achieve minimum weight by a search for two design variables. Constraints on yielding, buckling, and fracture are evaluated with the thin-walled bar theory. - The analyses are developed for closed and open section bars made from fiber-reinforced polymer composites. The cross-sectional compliance matrix for bars with a closed cross-sectional contour and an open cross sectional contour include shear-extension coupling. The first ply failure envelope for a graphite epoxy circular tube subject to an axial force and torque is determined by Puck’s intralaminar criterion. Interlaminar failure, or delamination, is modeled with fracture mechanics, and the method is illustrated by analyses of standard fracture test specimens. - Numerical methods for static analysis begin with the direct stiffness method, which originated to model skeletal structures consisting of bars connected by joints. Applications include coplanar trusses, beams and coplanar frames. The finite element method is developed from the integral formulation of the ordinary differential equations of an axial bar and a beam. - Analyses for the linear elastic, dynamic response of axial bars, coplanar trusses, beams, and coplanar frames are presented using the finite element method and the mode-separation method. Hamilton’s principle and Lagrange’s equations are developed for discrete mechanical systems. - Numerous examples to illustrate the application of the structural analysis are presented in each chapter using either U.S. customary units. or SI units. Suggested citation
    Johnson, Eric R. (2022) Aerospace Structures. Blacksburg, VA: Kevin T. Crofton Department of Aerospace and Ocean Engineering. https://doi.org/10.21061/AerospaceStructures. Licensed with CC BY NC-SA 4.0. https://creativecommons.org/licenses/by-nc-sa/4.0 Acknowledgments
    The peer-reviewed work is made possible in part by financial and in-kind contributions from the Open Education Initiative at Virginia Tech, Virginia Tech Publishing, and VIVA—The Virtual Library of Virginia. Contributors
    Co-investigators: Mayuresh Patil, Rakesh Kapania
    Managing editor and co-investigator: Anita Walz
    Alt text writer: Joseph Brooks
    Alt text assistant: Claire Colvin
    Cover design and selected graphics: Kindred Grey About the author
    Eric Raymond Johnson is emeritus professor of aerospace and ocean engineering at Virginia Tech. He earned his doctoral degree in applied mechanics from the University of Michigan in 1976, and from 1976 to 2003 was a member of the engineering faculty at Virginia Tech. Dr. Johnson's research area is composite structures. Research activities include the mechanics of the response and failure of advanced composite material structures with applications to flight and land vehicles, buckling and post-buckling of plates and shells, progressive failure analysis for the prediction of energy absorption in laminated composites and in bonded joints, and fracture mechanics. He has sixty-four publications in structural mechanics, and has been awarded research funding from government agencies and industries.. He is a senior member of the American Institute of Aeronautics and Astronautics and a member of the American Society of Mechanical Engineers.
  • Thumbnail Image
    Analyses of Ship Collisions: Determination of Longitudinal Extent of Damage and Penetration
    Sajdak, John Anthony Waltham (Virginia Tech, 2004-07-29)
    The overall objective of this thesis is to develop, validate and assess a probabilistic collision damage model to support ongoing work by the Society of Naval Architecture and Marine Engineering (SNAME) Ad Hoc Panel #6 and IMO working groups. It is generally agreed that structural design has a major influence on tanker oil outflow and damaged stability in grounding and collision, but crashworthiness is not considered in present regulations. The proposed methodology provides a practical means of considering structural design in a regulatory framework, and when implemented would improve the safety and environmental performance of ships. This thesis continues the development and applies a Simplified Collision Model (SIMCOL) to calculate damage extent (transverse, vertical and longitudinal) and oil outflow in ship collisions. The primary contribution of this thesis is the development and validation of a theory for the determination of energy absorbed in longitudinal extent of damage, and the implementation of the theory within SIMCOL. SIMCOL is sufficiently fast to be applied to thousands of collision cases as is required for a probabilistic analysis. The following specific tasks were completed using SIMCOL in support of this project: Completed the development of SIMCOL Version 3.0 including: 1) Deformable Bow sub model 2) Implementation and validation of theory for the determination of energy absorbed in longitudinal extent of damage. • Developed the capability to model collision events using LSDYNA. • Validated Virginia Tech LSDYNA ship collision modeling procedure. • Validated SIMCOL using real collision data, and probabilistic collision data for penetrating collisions.
  • Thumbnail Image
    Analysis and Design of Variable Stiffness Composite Cylinders
    Tatting, Brian F. (Virginia Tech, 1998-10-13)
    An investigation of the possible performance improvements of thin circular cylindrical shells through the use of the variable stiffness concept is presented. The variable stiffness concept implies that the stiffness parameters change spatially throughout the structure. This situation is achieved mainly through the use of curvilinear fibers within a fiber-reinforced composite laminate, though the possibility of thickness variations and discrete stiffening elements is also allowed. These three mechanisms are incorporated into the constitutive laws for thin shells through the use of Classical Lamination Theory. The existence of stiffness variation within the structure warrants a formulation of the static equilibrium equations from the most basic principles. The governing equations include sufficient detail to correctly model several types of nonlinearity, including the formation of a nonlinear shell boundary layer as well as the Brazier effect due to nonlinear bending of long cylinders. Stress analysis and initial buckling estimates are formulated for a general variable stiffness cylinder. Results and comparisons for several simplifications of these highly complex governing equations are presented so that the ensuing numerical solutions are considered reliable and efficient enough for in-depth optimization studies. Four distinct cases of loading and stiffness variation are chosen to investigate possible areas of improvement that the variable stiffness concept may offer over traditional constant stiffness and/or stiffened structures. The initial investigation deals with the simplest solution for cylindrical shells in which all quantities are constant around the circumference of the cylinder. This axisymmetric case includes a stiffness variation exclusively in the axial direction, and the only pertinent loading scenarios include constant loads of axial compression, pressure, and torsion. The results for these cases indicate that little improvement over traditional laminates exists through the use of curvilinear fibers, mainly due to the presence of a weak link area within the stiffness variation that limits the ultimate load that the structure can withstand. Rigorous optimization studies reveal that even though slight increases in the critical loads can be produced for designs with an arbitrary variation of the fiber orientation angle, the improvements are not significant when compared to traditional design techniques that utilize ring stiffeners and frames. The second problem that is studied involves arbitrary loading of a cylinder with a stiffness variation that changes only in the circumferential direction. The end effects of the cylinder are ignored, so that the problem takes the form of an analysis of a cross-section for a short cylinder segment. Various load cases including axial compression, pressure, torsion, bending, and transverse shear forces are investigated. It is found that the most significant improvements in load-carrying capability exist for cases which involve loads that also vary around the circumference of the shell, namely bending and shear forces. The stiffness variation of the optimal designs contribute to the increased performance in two ways: lowering the stresses in the critical areas through redistribution of the stresses; and providing a relatively stiff region that alters the buckling behavior of the structure. These results led to an in-depth optimization study involving weight optimization of a fuselage structure subjected to typical design constraints. Comparisons of the curvilinear fiber format to traditional stiffened structures constructed of isotropic and composite materials are included. It is found that standard variable stiffness designs are quite comparable in terms of weight and load-carrying capability yet offer the added advantage of tailorability of distinct regions of the structure that experience drastically different loading conditions. The last two problems presented in this work involve the nonlinear phenomenon of long tubes under bending. Though this scenario is not as applicable to fuselage structures as the previous problems, the mechanisms that produce the nonlinear effect are ideally suited to be controlled by the variable stiffness concept. This is due to the fact that the dominating influence for long cylinders under bending is the ovalization of the cross-section, which is governed mainly by the stiffness parameters of the cylindrical shell. Possible improvement of the critical buckling moments for these structures is investigated using either a circumferential or axial stiffness variation. For the circumferential case involving infinite length cylinders, it is found that slight improvements can be observed by designing structures that resist the cross-sectional deformation yet do not detract from the buckling resistance at the critical location. The results also indicate that bucking behavior is extremely dependent on cylinder length. This effect is most easily seen in the solution of finite length cylinders under bending that contain an axial stiffness variation. For these structures, the only mechanism that exhibits improved response are those that effectively shorten the length of the cylinder, thus reducing the cross-sectional deformation due to the forced restraint at the ends. It was found that the use of curvilinear fibers was not able to achieve this effect in sufficient degree to resist the deformation, but that ring stiffeners produced the desired response abmirably. Thus it is shown that the variable stiffness concept is most effective at improving the bending response of long cylinders through the use of a circumferential stiffness variation.
  • Thumbnail Image
    Analysis and optimal design of pressurized, imperfect, anisotropic ring-stiffened cylinders
    Ley, Robert Paul (Virginia Tech, 1992-06-15)
    Development of an algorithm to perform the structural analysis and optimal sizing of buckling resistant, imperfect, anisotropic ring-stiffened cylinders subjected to axial compression, torsion, and internal pressure is presented. The structure is modeled as a branched shell. A nonlinear axisymmetric prebuckling equilibrium state is assumed which is amenable to exact solution within each branch. Buckling displacements are represented by a Fourier series in the circumferential coordinate and finite elements in the axial or radial coordinate. A separate, more detailed analytical model is employed to predict prebuckling stresses in the flange/skin interface region./p> Results of case studies indicate that a nonlinear prebuckling analysis is needed to accurately predict buckling loads and mode shapes of these cylinders, that the rings have a greater influence on the buckling resistance as the relative magnitude of the torsional loading to axial compression loading is increased, but that this ring effectiveness decreases somewhat when internal pressure is added./p> The enforcement of stability constraints is treated in a way that does not require any eigenvalue analysis. Case studies performed using a combination of penalty function and feasible direction optimization methods indicate that the presence of the axisymmetric initial imperfection in the cylinder wall can significantly affect the optimal designs. Weight savings associated with the addition of two rings to the unstiffened cylinder and/or the addition of internal pressure is substantial when torsion makes up a significant fraction of the combined load state./p> Assumption of criticality of the stability constaints and neglect of the stress constraints during the optimal sizing of the cylinders produced designs that nevertheless satisfied all of the stress constraints, in general, as well as the stability constraints. Subsequent re-sizing of one cylinder to satisfy a violated in-plane matrix cracking constraint resulted in an optimal design that was 49% heavier than the optimal design produced when this constraint was ignored. The additional internal pressure necessary to produce a violation of a stress constraint for each optimal design was calculated. Using an unsymmetrically laminated ring flange, a substantial increase in the strength of the flange/skin joint was observed.
  • Thumbnail Image
    Analysis of fiber-reinforced composite plates utilizing curvilinear fiber trajectories
    Fierling, Yannick P. H. (Virginia Tech, 1995-05-05)
    Four plates with centrally located circular holes were manufactured using a fiber placement technique. With two plates the fibers were steered around the holes in curvilinear trajectories. With the two other plates the fibers were placed in the conventional straight line format. For the case of the curvilinear trajectories, the fibers were continuous from one end of the plate to the other, whereas for the straight trajectories the fibers were cut by the presence of the hole. Two plates, a curvilinear fiber plate and a straight fiber plate, were tested in tension. The two other plates were tested in compression. The straight fiber plates were considered as baseline cases. Since the plates were thin, compression testing resulted in buckling and post buckling. The current work focuses on the analysis of these four plates and a comparison between the analysis and experimental results. Because of a spatial dependence of the A and D stiffness matrices for the curvilinear fiber cases, the analyses were conducted using finite element methods, and included a failure criterion. A scheme to improve the plate design is also considered.
  • Thumbnail Image
    An analysis of interlaminar stresses in unsymmetrically laminated plates
    Norwood, Donald Scott (Virginia Tech, 1990)
    The results of a numerical study of interlaminar stresses within unsymmetrically laminated plates is presented. The focus of the study is upon the linear thermoelastic response of thin square laminated composite plates subjected to extensional, compressive, or thermal loading. Symmetric and unsymmetric 0/90, +45/-45, and 0/+45 laminate stacking sequences are examined to determine the effects of mismatch between adjacent layers in Poisson’s ratio, coefficient of mutual influence, and coefficients of thermal expansion. Since the out-of-plane (transverse) deflections of unsymmetric laminates are typically large, a geometrically nonlinear kinematic description is used to account for the large displacements and rotations. The geometrically nonlinear three-dimensional boundary value problems are formulated from nonlinear elasticity theory and approximate solutions are determined using the finite element method. A total Lagrangian, displacement-based, incremental finite element formulation is implemented using Newton’s method. Geometrically nonlinear global/local finite element analysis is used to obtain improved free edge stress predictions. For laminates subjected to external loading, the mismatch in material properties between adjacent layers causes interlaminar stresses to arise near the free edges. For unsymmetric laminates under external loading, the mismatch in material properties about the geometric midplane causes out-of-plane deflections. For the laminates and loading conditions considered, the results of this study show that the out-of- plane deflections of unsymmetric laminates reduce interlaminar shear stresses. In addition, the out-of-plane deflections reduce interlaminar normal stresses for some laminates and increase these stresses for others. For the two-layer unsymmetric laminates considered, the effect of out-of-plane deflections upon interlaminar normal stress was shown to be dependent upon the type of in-plane strain mismatch (i.e., normal and/or shear) caused by the dissimilar material properties. The results also show that as the out-of-plane deflections become large, the effects of geometric nonlinearity upon this stress-deformational response become important. These conclusions apply to extensional, compressive (prior to a change in mode shape), and thermal loading. The numerical results include interlaminar stresses for laminated plates which have buckled as a wide column under compressive loading.
  • Thumbnail Image
    Analysis of metal matrix composite structures using a micromechanical constitutive theory
    Arenburg, Robert Thomas (Virginia Polytechnic Institute and State University, 1988)
    The nonlinear behavior of continuous-fiber-reinforced metal-matrix composite structures is examined using a micromechanical constitutive theory. Effective lamina and laminate constitutive relations based on the Aboudi micromechanics theory are presented. The inelastic matrix behavior is modeled by the unified viscoplasticity theory of Bodner and Partom. The laminate constitutive relations are incorporated into a first-order shear deformation plate theory. The resulting boundary value problem is solved by utilizing the finite element method. · Computational aspects of the numerical solution, such as the temporal integration of the inelastic strains and the spatial integration of bending moments are addressed. Numerical results are presented which illustrate the nonlinear response of metal matrix composites subjected to extensional and bending loads. Experimental data from available literature are in good agreement with the numerical results.
  • Thumbnail Image
    An analytical and experimental investigation of the response of elliptical composite cylinders
    Meyers, Carol Ann (Virginia Tech, 1996)
    An analytical and experimental investigation of the response of composite cylinders of elliptical cross-section to axial compression and internal pressure loadings is discussed. Nine eight-ply graphite-epoxy elliptical cylinders, three layups for each of three cross sectional aspect ratios, are specifically examined. The lay-ups studied are a quasi-isotropic (±45/0/90)g, an axially-stiff (±45/0₂)g, and a circumferentially-stiff (±45/90₂)g. The elliptical cross sections studied are characterized by semi-minor axis (b) to semi-major axis (a) ratios of b/a = 0.70, 0.85, and 1.00 (circular). The cross sections are obtained by holding the semi-major axis constant for all cross sections, and only varying the semi-minor axis. The nominal semi-major axis for all specimens was 5.00 in. (127 mm), and all specimens were cut to the same length, which provided a length-to-radius ratio of 2.9 for the circular cylinders. For the elliptical cross section cylinders, the length to- radius ratios, L/R(s), ranged from two to slightly greater than six, where R(s) is the function describing the circumferential variation of the radius. A geometrically nonlinear special-purpose analysis, based on Donnell’s nonlinear shell equations, is developed to study the prebuckling responses of geometrically perfect cylinders. In this analysis the circumferentially-varying radius of curvature of the cylinder is expanded in a cosine series. While elliptical sections are studied here, it should be noted that such an expansion will accommodate any cross section with at least two axes of symmetry. The displacements are likewise expanded in a harmonic series using the Kantorovich method. The total potential energy, written in terms of the displacements, is then integrated over the circumferential coordinate. The variational process then yields the governing Euler-Lagrange equations and boundary conditions. This process has been automated using the symbolic manipulation package Mathematica ©. The resulting nonlinear ordinary differential equations are then integrated via the finite difference method. A geometrically nonlinear finite element analysis is also utilized to compare with the prebuckling solutions of the special-purpose analysis and to study the prebuckling and buckling responses of geometrically imperfect cylinders. The imperfect cylinder geometries are represented by an analytical approximation of the measured shape imperfections. An accompanying experimental program is carried out to provide a means for comparison between the real and theoretical systems using a test fixture specifically designed for the present investigation to allow for both axial compression and internal pressurization. A description of the test fixture is included. Three types of tests were run on each specimen: (1) low internal pressure with no axial end displacement, (2) low internal pressure with a low level compressive axial displacement and, (3) compressive axial displacement to failure, with no internal pressure. The experimental data from these tests are compared to predictions for both perfect and imperfect cylinder geometries. Prebuckling results are presented in the form of displacement and strain profiles for each of the three sets of load conditions. Buckling loads are also compared to predicted values based upon classical estimates as well as linear and nonlinear finite element results which include initial shape imperfections. Lastly, the postbuckling and failure characteristics observed during the tests are described.
  • Thumbnail Image
    Analytical Modeling of the Mechanics of Nucleation and Growth of Cracks
    Goyal, Vinay K. (Virginia Tech, 2002-11-15)
    With the traditional fracture mechanics approaches, an initial crack and self-similar progression of cracks are assumed. In this treatise, theoretical and numerical tools are developed to mathematically describe non-self-similar progression of cracks without specifying an initial crack. A cohesive-decohesive zone model, similar to the cohesive zone model known in fracture mechanics as Dugdale-Barenblatt model, is adopted to represent the degradation of the material ahead of the crack tip. This model unifies strength-based crack initiation and fracture based crack progression. The cohesive-decohesive zone model is implemented with an interfacial surface material that consists of an upper and lower surface connected by a continuous distribution of normal and tangential nonlinear elastic springs that act to resist either Mode I opening, Mode II sliding, Mode III sliding, or mixed mode. The initiation of fracture is determined by the interfacial strength and the progression of fracture is determined by the critical energy release rate. The material between two adjacent laminae of a laminated composite structure or the material between the adherend and the adhesive is idealized with an interfacial surface material to predict interfacial fracture. The interfacial surface material is positioned within the bulk material to predict discrete cohesive cracks. The proper work-conjugacy relations between the stress and deformation measures are identified for the interfacial surface theory. In the principle of virtual work, the interfacial cohesive-decohesive tractions are conjugate to the displacement jumps across the upper and lower surfaces. A finite deformation kinematics theory is developed for the description of the upper and lower surface such that the deformation measures are invariant with respect to superposed rigid body translation and rotation. Various mechanical softening constitutive laws thermodynamically consistent with damage mechanics are postulated that relate the interfacial tractions to the displacement jump. An exponential function is used for the constitutive law such that it satisfies a multi-axial stress criterion for the onset of delamination, and satisfies a mixed mode fracture criterion for the progression of delamination. A damage parameter is included to prevent the restoration of the previous cohesive state between the interfacial surfaces. In addition, interfacial constitutive laws are developed to describe the contact-friction behavior. Interface elements applicable to two dimensional and three dimensional analyses are formulated for the analyses of contact, friction, and delamination problems. The consistent form of the interface element internal force vector and the tangent stiffness matrix are considered in the formulation. We investigate computational issues related to interfacial interpenetration, mesh sensitivity, the number of integrations points and the integration scheme, mathematical form of the softening constitutive law, and the convergence characteristics of the nonlinear solution procedure when cohesive-decohesive constitutive laws are used. To demonstrate the predictive capability of the interface finite element formulation, steadystate crack growth is simulated for quasi-static loading of various fracture test configurations loaded under Mode I, Mode II, Mode III, and mixed-mode loading. The finite element results are in agreement with the analytical results available in the literature and those developed in this work. A progressive failure methodology is developed and demonstrated to simulate the initiation and material degradation of a laminated panel due to intralaminar and interlaminar failures. Initiation of intralaminar failure can be by a matrix-cracking mode, a fiber-matrix shear mode, and a fiber failure mode. Subsequent material degradation is modeled using damage parameters for each mode to selectively reduce lamina material properties. The interlaminar failure mechanism such as delamination is simulated by positioning interface elements between adjacent sublaminates. The methodology is validated with respect to experimental data available in the literature on the response and failure of quasi-isotropic panels with centrally located circular cutouts. Very good agreement between the progressive failure analysis and the experiments is achieved if the failure analyses includes the interaction of intralaminar and interlaminar failures in the postbuckling response of the panels. In addition, ideas concerning the implementation of a fatigue model incorporated with a cohesive zone model are discussed.
  • Thumbnail Image
    An Application of Anti-Optimization in the Process of Validating Aerodynamic Codes
    Cruz, Juan Ramón (Virginia Tech, 2003-04-04)
    An investigation was conducted to assess the usefulness of anti-optimization in the process of validating of aerodynamic codes. Anti-optimization is defined here as the intentional search for regions where the computational and experimental results disagree. Maximizing such disagreements can be a useful tool in uncovering errors and/or weaknesses in both analyses and experiments. The codes chosen for this investigation were an airfoil code and a lifting line code used together as an analysis to predict three-dimensional wing aerodynamic coefficients. The parameter of interest was the maximum lift coefficient of the three-dimensional wing, CL max. The test domain encompassed Mach numbers from 0.3 to 0.8, and Reynolds numbers from 25,000 to 250,000. A simple rectangular wing was designed for the experiment. A wind tunnel model of this wing was built and tested in the NASA Langley Transonic Dynamics Tunnel. Selection of the test conditions (i.e., Mach and Reynolds numbers) were made by applying the techniques of response surface methodology and considerations involving the predicted experimental uncertainty. The test was planned and executed in two phases. In the first phase runs were conducted at the pre-planned test conditions. Based on these results additional runs were conducted in areas where significant differences in CL max were observed between the computational results and the experiment — in essence applying the concept of anti-optimization. These additional runs were used to verify the differences in CL max and assess the extent of the region where these differences occurred. The results of the experiment showed that the analysis was capable of predicting CL max to within 0.05 over most of the test domain. The application of anti-optimization succeeded in identifying a region where the computational and experimental values of CL max differed by more than 0.05, demonstrating the usefulness of anti-optimization in process of validating aerodynamic codes. This region was centered at a Mach number of 0.55 and a Reynolds number of 34,000. Including considerations of the uncertainties in the computational and experimental results confirmed that the disagreement was real and not an artifact of the uncertainties.
  • Thumbnail Image
    Calculation of skin-stiffener interface stresses in stiffened composite panels
    Cohen, David (Virginia Polytechnic Institute and State University, 1987)
    A method for computing the skin-stiffener interface stresses in stiffened composite panels is developed. Both geometrically linear and nonlinear analyses are considered. Particular attention is given to the flange termination region where stresses are expected to exhibit unbounded characteristics. The method is based on a finite-element analysis and an elasticity solution. The finite-element analysis is standard, while the elasticity solution is based on an eigenvalue expansion of the stress functions. The eigenvalue expansion is assumed to be valid in the local flange termination region and is coupled with the finite-element analysis using collocation of stresses on the local region boundaries. In the first part of the investigation the accuracy and convergence of the local elasticity solution are assessed using a geometrically linear analysis. It is found that the finite-element/local elasticity solution scheme produce a very accurate interface stress representation in the local flange termination region. The use of 10 to 15 eigenvalues, in the eigenvalue expansion series, and 100 collocation points results in a converged local elasticity solution. In the second part of the investigation, the local elasticity solution is extended to include geometric nonlinearities. Using this analysis procedure, the influence of geometric nonlinearities on skin-stiffener interface stresses is evaluated. It is found that in flexible stiffened skin structures, which exhibit out-of-plane deformation on the order of 2 to 4 times the skin thickness, inclusion of geometrically nonlinear effects in the calculation of interface stresses is very important. Thus, the use of a geometrically linear analysis, rather than a nonlinear analysis, can lead to considerable error in the computation of the interface stresses. Finally, using the analytical tool developed in this investigation, it is possible to study the influence of stiffener parameters on the state of interface stresses.
  • Thumbnail Image
    A CFD/CSD Interaction Methodology for Aircraft Wings
    Bhardwaj, Manoj K. (Virginia Tech, 1997-09-15)
    With advanced subsonic transports and military aircraft operating in the transonic regime, it is becoming important to determine the effects of the coupling between aerodynamic loads and elastic forces. Since aeroelastic effects can contribute significantly to the design of these aircraft, there is a strong need in the aerospace industry to predict these aero-structure interactions computationally. To perform static aeroelastic analysis in the transonic regime, high fidelity computational fluid dynamics (CFD) analysis tools must be used in conjunction with high fidelity computational structural dynamics (CSD)analysis tools due to the nonlinear behavior of the aerodynamics in the transonic regime. There is also a need to be able to use a wide variety of CFD and CSD tools to predict these aeroelastic effects in the transonic regime. Because source codes are not always available, it is necessary to couple the CFD and CSD codes without alteration of the source codes. In this study, an aeroelastic coupling procedure is developed which will perform static aeroelastic analysis using any CFD and CSD code with little code integration. The aeroelastic coupling procedure is demonstrated on an F/A-18 Stabilator using NASTD (an in-house McDonnell Douglas CFD code)and NASTRAN. In addition, the Aeroelastic Research Wing (ARW-2) is used for demonstration of the aeroelastic coupling procedure by using ENSAERO (NASA Ames Research Center CFD code) and a finite element wing-box code (developed as a part of this research). The results obtained from the present study are compared with those available from an experimental study conducted at NASA Langley Research Center and a study conducted at NASA Ames Research Center using ENSAERO and modal superposition. The results compare well with experimental data. In addition, parallel computing power is used to investigate parallel static aeroelastic analysis because obtaining an aeroelastic solution using CFD/CSD methods is computationally intensive. A parallel finite element wing-box code is developed and coupled with an existing parallel Euler code to perform static aeroelastic analysis. A typical wing-body configuration is used to investigate the applicability of parallel computing to this analysis. Performance of the parallel aeroelastic analysis is shown to be poor; however with advances being made in the arena of parallel computing, there is definitely a need to continue research in this area.
  • Thumbnail Image
    Characterization and Development of General Material Models for Use in Modeling Structures Bonded with Ductile Adhesives
    Cassino, Christopher (Virginia Tech, 2005-04-22)
    Structural adhesives are materials that are capable of bearing significant loads in shear, and sometimes tension, over a range of strains and strain rates. Adhesively bonded structures can dissipate large amounts of mechanical energy and can be lighter and more efficient than many bolted or vibration welded parts. The largest barrier to using structural adhesives in more applications is the many challenges engineers are presented with when designing and analyzing adhesively bonded structures. This study develops, characterizes and compares several material models for use in finite element analysis of adhesively bonded structures, in general, and a bonded tongue and groove (TNG) joint in particular. The results indicate that it is possible to develop a general material model for ductile adhesives used in structural applications under quasi-static conditions. Furthermore, the results also show that it is also possible to take bulk material data and apply it to an adhesively bonded specimen provided that the mode of failure of the bulk test specimen closely approximates the mode of failure of the bonded joint.
  • Thumbnail Image
    Characterization of damage mechanisms and behavior in two-dimensionally braided composites subjected to static and fatigue loading
    Burr, Scott T. (Virginia Tech, 1996-05-20)
    In the present research project, four braided composite architectures consisting of graphite fibers in an epoxy matrix were tested under static and fatigue loading conditions to determine damage mechanism types and progressions. The braided architectures consisted of straight axial fiber bundles, which were surrounded by braider fiber bundles oriented at ±a 0 with respect to the axial fiber bundles. Static tension and compression testing was completed first to determine material strengths and basic damage modes for each of the architectures. Under static tension loading, cracking in the braider fiber bundles occurred first, and was followed by splitting in curved regions of the axial fiber bundles. Matrix cracking and kink band formation were found to occur under static compression loading.
  • Thumbnail Image
    Combined mechanical loading of composite tubes
    Derstine, Mark S. (Virginia Tech, 1988-06-05)
    An analytical/experimental investigation was performed to study the effect of material nonlinearities on the response of composite tubes subjected to combined axial and torsional loading. An elasticity based analytical model was developed to predict stresses and deformations in composite tubes subjected to combined thermomechanical loading. Material nonlinearities were modeled using the Endochronic Theory. The effect of residual stresses on subsequent mechanical response was included in the investigation. Subsequently, experiments were performed on P75/934 graphite/epoxy tubes with a stacking sequence of [15/1/ ± 10/0/-15], using pure torsion and combined axial/torsional loading. The in-plane material properties needed for incorporation into the analytical model were determined using tests on flat coupons made from P75/934. In the presence of residual stresses. the analytical model predicted a reduction in the Initial shear modulus of a tube subjected to torsional loading. Experimentally. a difference in the nonlinearity of the stress-strain response was found between pure torsion loading and combined proportional loading. This difference is due to coupling between axial loading and shear strain. These phenomena were predicted by the nonlinear analytical model where a linear model did not. The experimentally observed linear limit of the global shear response was found to correspond to the analytically predicted first ply failure. The observed nonlinear response thus appears to be due to a combination of material response at the ply level and gradual damage accumulation. Further, based on cyclic torsion tests, the failure of the tubes was found to be path dependent above a certain critical load level.
  • Thumbnail Image
    Combined structural and manufacturing optimization of stiffened composite panels
    Henderson, Joseph Lynn (Virginia Tech, 1996-07-05)
    Manufacturing considerations have been incorporated into the design optimization of a blade-stiffened composite panel. For the manufacturing analysis, a one-dimensional resin film infusion model is developed to compute the infiltration time of the resin into a fabric preform of the panel. Results are presented showing the effects of structurally important design variables, such as cross-sectional geometry and material properties, on the manufacturing performance of the panel. In addition, the effects of manufacturing process variables, such as pressure and temperature, on the structural performance are studied. The structural problem is formulated to minimize the panel mass subject to buckling constraints. A simplified buckling analysis model for the panel is used to compute the critical buckling loads. The objective of the manufacturing problem is to minimize the resin infiltration time. Optimum panel designs for the manufacturing and structures problems alone, as well as for the combined problem, are generated using a genetic algorithm. These results indicate a strong connection between the structures and manufacturing design variables and trade-offs between the responses, illustrating that a multidisciplinary approach to the problem is essential to incorporating manufacturing into the preliminary design stage.
  • Thumbnail Image
    Computation of interlaminar stresses from finite element solutions to plate theories
    Foster, John L. (Virginia Tech, 1991-04-01)
    Interlaminar stresses are estimated from plate theories by equilibrium. The elasticity equations of equilibrium are integrated with respect to the thickness coordinate z using the linear distribution in z of the in-plane stresses. This procedure, for example, requires fourth order derivatives of the out-of-plane displacement w with respect to the in-plane coordinates x and y to compute the interlaminar normal stress. Since compatible elements for the plate bending problem at most require the displacement and its first derivatives to be continuous across element boundaries, low degree interpolation polynomials are used. Thus, fourth order derivatives of the finite element polynomials are either meaningless, or at least inaccurate. In order to compute high order derivatives, an approximate polynomial solution of high degree to the governing partial differential equation for w(x,y) is determined using the finite element solution as a first approximation. A rectangular subdomain that may consist of several elements is selected from the finite element model. The displacement w(,y) over the subdomain is expanded in a Chebyshev series. Then collocation is used to determine the unknown Chebyshev coefficients such that the Chebyshev series matches displacement w and its normal derivative from the finite element solution at discrete points on the boundary of the subdomain, and the partial differential equation is enforced at discrete points within the subdomain. Interlaminar shear and normal stresses are computed from the third and fourth derivatives, respectively, of the Chebyshev series at the collocation points.
  • Thumbnail Image
    Consequences of Simultaneous Local and Overall Buckling in Stiffened Panels
    Ghosh, Biswarup (Virginia Tech, 2003-04-18)
    In this thesis improved expressions for elastic local plate buckling and overall panel buckling of uniaxially compressed T-stiffened panels are developed and validated with 55 ABAQUS eigenvalue buckling analyses of a wide range of typical panel geometries. These two expressions are equated to derive a new expression for the rigidity ratio (EIx/Db)CO that uniquely identifies ¡°crossover¡± panels ¨C those for which local and overall buckling stresses are the same. The new expression for (EIx/Db)CO is also validated using the 55 FE models. Earlier work by (Chen, 2003) had produced a new step-by-step beam-column method for predicting stiffener-induced compressive collapse of stiffened panels. An alternative approach is to use orthotropic plate theory. As part of the validation of the new beam-column method, ABAQUS elasto-plastic Riks ultimate strength analyses were made for 107 stiffened panels ¨C the 55 crossover panels and 52 others. The beam-column and orthotropic approaches were also used. A surprising result was that the orthotropic approach has a large error for crossover panels whereas the beam-column method does not. Some possible reasons for this are suggested. Collapse patterns for the crossover panels are studied and classified from von Mises stress distribution at collapse. The collapse mechanism and load-deflection diagrams suggest stable inelastic post collapse behavior for most panels and an abrupt drop in load carrying capacity in only nine of the 55.