Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1990-12 Analysis of thick composite plates using higher order three dimensional finite elements Alon, Yair Monterey, California: Naval Postgraduate School http://hdl.handle.net/10945/27545 AD-A243 188 NAVAL POSTGRADUATE SCHOOL Monterey, California ,,v STAT,,V THESIS STATIC ANALYSIS OF THICK COMPOSITE PLATES USING HIGHER ORDER THREE DIMENSIONAL FINITE F LEMENTS BY YAIR ALON December 1990 Thesis Advisor: Prof. Ramesh Kolar Approved for public release: distribution is unlimited 91-17184 ' '1 I ' . 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ADDRESS (City, State, and ZIP Code) 93943-5000 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER 10 SOURCE OF FUNDING NUMBERS PROGRAM ELEMENT NO PROJECT NO TASK NO WORK UNIT ACCESSION NO 11 TITLE (Include Security Classfication) STATIC ANALYSIS OF THICK COMPOSITE PLATES USING HIGHER ORDER THREE DIMENSIONAL FINITE ELEMENTS 12. PERSONAL AUTHOR(S) Yair Aln 13a TYPE OF REPORT 13b TIME COVERED I Engineers/thesis FROM 14 DATEOF REPORT (Yejir, Month, Day) 1990, TO 15 PAGE COUNT December 100 e Viws expressed in t;'thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. 16 SUPPLEMENTARY NOTATION 17 COSATI CODES FIELD GROUP SUB-GROUP 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number) Finite Element, nonlinear analysis, plate bending thick plates, laminated composites, buckling, fonstant arc length three dimensional element. 19 ABSTRACT (Continue on Ceverse if necessary &(didentify biQk number) A triquadratic isoparametric solielement is developed to study the behavior of thick isotropic and laminated composite plates. The element is a 27 noded Lagrangian element based on three dimensional elasticity. Material characterisics are accounted by either using laminate plate theory or three dimensional anisotropic theory. Element matrices for nonlinear stability analysis are derived based on total Lagrangian formulation. Results are presented to compare with analytical solutions to validate the elements behavior. The effects of various integration schemes on the element performance are presented. Convergence studies for laminated composits for different fiber orientations are provided to illustrate application An analysis for thin plated is carried out and results for thick plates are compared with available higher order plate theories. One row of elements in the thickness directions gives satisfactory results for thick laminates. 20 DISTRIBUTION /AVAILABILITY MUNCLASSIFIED/UNLIMITED OF ABSTRACT 0 SAME AS RPT 0 DTIC USERS 22a NAME OF RESPONSIBLE INDIVIDUAL Prof. 22b TELEPHONE (Include Area Code) Ramesh Kolar DD Form 1473, JUN 86 211 x2936 Previous editions are obsolete S/N 0102-LF-014-6603 i 2ec OFFICE SYMBOL AA/Kj SECURITY CLASSIFICATION OF THIS PAGE Unclassified SECURITY CLASSIFICATION OF THIS PAGE DD Form 1473, JUN 86 (Reverse) SECURITY CLASSio-iCATION OF T-'S :;A3E Approved for public release; distribution is unlimited Analysis of Thick Composite Plates Using Higher Order Three Dimensional Finite Elements by Alon Yair Captain, Israeli Air Force B.S.C., Israel Technion Institute of Technology, 1983 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING and AERONAUTICS AND ASTRONAUTICS ENGINEERS DEGREE from the NAVAL POSTGRADUATE SCHOOL December 1990 L JUBLI ~c at 1.,ci Author: str b t i */ Av¢illbliA Approved by: Rame G. H. Kolar, Thesis Advisor indse ,econ e E. Rol.erts Wood Departmentof Aeroutics and Astronautics Gordon E. Schacher Dean of Faculty and Graduate Studies ii Dist p1 Cui ABSTRACT A triquadratic isoparametric solid element is developed to study the behavior of thick isotropic and laminated composite plates. The element is a 27 noded Lagrangian element based on three dimensional elasticity. Material characteristics are accounted by either using laminate plate theory or three-dimensional anisotropic theory. Element matrices for nonlinear stability analyses are derived based on total Lagrangian formulation. Results are presented to compare with analytical solutions to validate the elements behavior. The effects of various integration schemes on the element performance are presented. Convergence studies for laminated composites for different fiber orientations are provided to illustrate applications. An analysis of thin plates is carried out and results for thick plates are compared with available higher order plate theories. One row of elements in the thickness directions gives satisfactory results for thick laminates. iii TABLE OF CONTENTS INTRODUCTION II. ............................. 1 A. OVERVIEW .............................. 1 B. LITERATURE REVIEW ....................... 1 C. THESIS OUTLINE 3 .......................... THEORETICAL FORMULATION .................... 4 A. INTRODUCTION ........................... 4 B. GENERAL DERIVATION OF FINITE ELEMENT EQUILIB- C. D. E. RIUM EQUATIONS ......................... 4 INTERPOLATION SCHEME 9 .................... 1. Shape Functions (Displacement Interpolation Functions) . .. 2. Jacobian Transformation Matrix ................. 9 11 STRAIN DISPLACEMENT RELATIONS - [B].............. 12 1. Basic Formulation ......................... 12 2. General Nonlinear Discretization ................ 17 STRESS-STRAIN RELATIONS ................... 25 1. Classical and Higher Order Laminate Theories ........... 25 2. Three-dimensional Anisotropic Theory ............. 34 F. CONSISTENT LOADS ........................ 35 G. INTEGRATION ............................ 36 1. Gauss Quadrature ........................ 39 2. Integration Scheme ........................ 39 H. BUCKLING ANALYSIS 1. ....................... 41 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 iv III. IV. 2. Implementation .......................... 42 3. Constant Arc Length Method [Kolar and Kamel (1985)] 4. Convergence Criterion .... ...................... 45 PROGRAM IMPLEMENTATION ........................ 47 A. INTRODUCTION ................................ 47 B. LINEAR ANALYSIS .... 47 C. NONLINEAR ANALYSIS ....................... 48 D. SOLUTION PROCEDURE ...................... 48 1. Composite Material ........................ 48 2. Linear Case ............................ 48 3. Nonlinear Case .......................... 50 ......................... 54 NUMERICAL EXAMPLES ........................ ............... A. INTRODUCTION AND NOTATIONS 1. B. V. . . . 44 54 54 Material Properties ........................ COLUMNS AND BARS ....................... 54 1. B ars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2. Beam s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 C. CLAMPED PLATES ......................... 59 D. SIMPLY SUPPORTED PLATES ................... 66 CONCLUSIONS AND SCOPE FOR FUTURE RESEARCH A. CONCLUSIONS ............................ B. SCOPE FOR FUTURE RESEARCH ..... 76 76 ................ 77 APPENDIX A - SHAPE FUNCTIONS ..................... 78 APPENDIX B - JACOBIAN MATRIX ..................... 82 APPENDIX C - THEORIES 84 ........................... A. THEORY OF ELASTICITY SOLUTIONS V ............. 84 B. 1. Cantilevered bar under traction ..................... 84 2. Cantilevered Beam under end load .................. 84 CLASSICAL PLATE THEORY (CPT) .................. 84 1. All edges clamped rectangular isotropic plate under central load 84 2. All edges simply-supported, rectangular plate under uniformly 3. distributed load ................................ 85 Composite ................................... 85 LIST OF REFERENCES ....... ............................. INITIAL DISTRIBUTION LIST ...... ......................... vi 87 89 LIST OF TABLES 2.1 SHAPE FUNCTIONS FOR 9 NODED BIQUADRATIC ELEMENT 2.2 SAMPLING POINTS AND WEIGHTS FOR GAUSS QUADRATURE OVER THE INTERVAL-1 to 1 ................. 4.1 36 41 EFFECTS OF REDUCED INTEGRATION AND REFINED MESH, ON THE MAXIMUM DEFLECTION OF CLAMPED ISOTROPIC CANTILEVER BAR UNDER UNIAXIAL LOAD ............. 4.2 58 EFFECTS OF REDUCED INTEGRATION AND MESH CONFIGURATION ON THE MAXIMUM DEFLECTION OF CLAMPED ISOTROPIC CANTILEVER BEAM LOADED AT THE END .... 4.3 CENTER DEFLECTION VS. ASPECT RATIO (1) OF AN ISOTROPIC CANTILEVER CLAMPED BEAM LOADED AT ONE END 4.4 061 .... 62 MESH COMPARISON OF AN ALL EDGES CLAMPED RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD (P = 1000 lb.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 66 INTEGRATION RULES COMPARISON OF AN ALL EDGES CLAMPED RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD (iP = 1000) 4.6 ................................. ALL EDGES SIMPLY SUPPORTED RECTANGULAR PLATE, UNDER UNIFORMLY DISTRIBUTED LOAD ................ 4.7 69 72 CENTER DEFLECTION VS. ASPECT RATIO OF SIMPLY SUPPORTED RECTANGULAR PLATE UNDER UNIFORMLY DISTRIBUTED LOAD ............................ vii 73 LIST OF FIGURES 2.1 General 3-D Body .................................... 2.2 Lagrangian Solid Element - 27 nodes ....................... 10 2.3 Motion of a body in a fixed Cartesian coordinate system ......... 14 2.4 Lamina coordinate system (2-D) ......................... 26 2.5 Stress resultants and couples in a lamina .................... 30 2.6 Consistent loads .................................... 37 2.7 Consistent loads on a mesh ............................. 38 2.8 Gauss points ....................................... 40 2.9 Instability and bifurcation points ..... 6 .................... 43 2.10 Constant Arc Length Method ........................... 46 3.1 Thick composite plate-element arrangement .................. 49 3.2 Flow chart - linear analysis ..... 51 3.3 Flow chart - Nonlinear analysis .......................... 53 4.1 Bar Sample Problems ................................. 56 4.2 Clamped Bar Under Uniaxial Load ....................... 57 4.3 End Loaded Beam Bending ..... 60 4.4 End Loaded Beazm Deflection ...... 4.5 Plate Sample Problems ................................ 65 4.6 Clamped Plate Under Central Load ....................... 67 4.7 Clamped Plate Integration Rules ..... 68 4.8 Simply Supported Isotropic Plate ......................... 74 4.9 Simply Supported Laminated Plate ....................... 74 4.10 Isotropic Plate Deflections vs. Aspect Ratio .................. 75 viii ....................... ....................... ...................... .................... 63 4.11 Laminated Plate Deflection vs. Aspect Ratio .................. ix 75 I. INTRODUCTION A. OVERVIEW The finite element method pro,, ides a general tool to solve problems of contin ia such as heat conduction and fluid flow, but it is most widely used in structural mechanics. In structural mechanics, the methodology is applicable for static and dynamic response of structures and in predicting the elastic stability limits. The focus of the present study is to develop tools to analyze thick laminater composite plates and validate the model by comparing with known solutions. More specifically, the objective of the present study is to develop a finite element for both linear and nonlinear analysis using three dimensional elasticity relations. By adopting such theory for thick plates, both isotropic and composite, the solutions account for transverse shear stresses, This approach eliminates the limitations imposed by classical plate theory based on Kirchoff-Love hypothesis [Batoz, 19501 or higher order shear deformation theoies [Reddy, 1984, Lo et al., 19771. B. LITERATURE REVIEW In this section, some literature pertaining to the analysis of thick composite plates is reviewed. The finite element method has been increasingly used as a research tool, as well a& a design analysis tool, and the methodology is rapidly evolving along with the development of faster and more efficient computers. Basic concepts of the theory of finite element analysis are well documented [Cook, et ;a., 1989]. Yang (1987) describes various two dimensional higher order elements as well as three dimensional solid elements. Bathe (1982) discusses the general formulation of finite elements in nonlinear analysis for one, two and three dimensional elements. 1 based on the total Lagrangian formulation and the principle of virtual displacements. A good source for continuum formulation may be found in Malvern (1969). Tsai and Pagano (1968) establish a notation in which composite lamina properties are invariant with respect to lamina direction. The laminate theory is well documented by Vinson (1987), where the elasticity solution for "structures composed of composite materials" is given for various cases, such as bending of thin plates. Based on laminate theory, Hoskin, et al. (1986) outline procedures involved in manufacturing composite components and presents some of its applications. A higher order shear deformation theory of laminated composite plates was developed by Lo, et al. (1977). A higher order nonlinear theory of thick plates was suggested by Reddy (1984a, 1984b, and 1985) and presented solutions (Reddy, 1987) and compared numerical results to Pagano's (1969) elasticity solution for the case of cylindrical bending. Other elasticity solutions are given by Timoshenko (1951 and 1959) and Eisley (1989) who discusses the elasticity solutions. The Heterosis finite element was suggested by Hughes, et al. (1978) for thick and thin plate bending problems. "Ihe effect of reduced integration in isoparametric finite elements was presented by Zienkiewicz, et al. (1971). In recent years, much work is concentrated on the analysis of buckling and postbuckling response of laminated plates and shells using nonlinear analysis. Ramm (1982) applies degenerate finite elements to solve buckling of thin shells. Arnold, et al. (1983) presents a theoretical analysis procedure for prediction of buckling and post-buckling in laminated composite plates and compares the results to experimental results. A combined numerical and experimental study of the post-buckling behavior of composite panel is performed by Natsiavas, et al. (1987). Gujbir et al. (1989) use an eight noded biquadratic element to study the effects of transverse shear on the stability of laminated plates. Some solution algorithms for nonlinear 2 analysis of structures by adapting modified Newton-Raphson and arc-length methods are given by Kolar et al (1985) and Ford et al (1987). In the literature reviewed, there appears to be no discussion on the higher-order solid element for the analysis of thick laminated plates. This research addresses the problem of using a tri-quadratic displacement field based finite element based on three-dimensional elasticity equations. A total Lagrangian formulation is used to derive relevant element nonlinear matrices, and numerical examples are included to validate the linear portion of the development. Analysis of typical examples include slender bars under traction and bending loads, thin and thick plates under bending loads and effects of various integration schemes. C. THESIS OUTLINE This section provides an overview of various chapters of the thesis. The total Lagrangian formulation for analyzing structures composed of three-dimensional elements is presented in Chapter II. Element matrices are derived for both linear and nonlinear static analysis using the incremental load method. The material characteristics account for both linear isotropic and anisotropic behavior. Formulas are provided to obtain work-equivalent loads for distributed body and surface forces. Chapter III addresses aspects of computational implementation of the problem formulated in Chapter fl. Test cases, example calculations and comparison with classical solutions and other high order theories are given in Chapter IV. Finally, Chapter V summarizes the results and reflects some suggestions for future work. 3 II. THEORETICAL FORMULATION A. INTRODUCTION In this chapter, using the principle of virtual displacements, the stiffness matrix will be developed for static equilibrium of triquadratic isoparametric solid elements. In the total formulation presented, both small and large displacements are permissible for linear and nonlinear structural analysis. For both cases, small strains and linearly elastic material will be assumed. The element is developed for analysis of both isotropic and composite structures. B. GENERAL DERIVATION OF FINITE ELEMENT EQUILIBRIUM EQUATIONS The principle of virtual work is invoked for the general formulation of equilib- rium [Bathe, 1982; Cook, 19891. The principle of virtual work states that a body is in equilibrium, if and only if, the total virtual work done by the internal forces is equal to the total virtual work done by the external forces. That is, = 6WW.t W.t (2.1) This principle is equivalent to the minimum total potential energy principle [6fl, = 01, and holds at any given time. Consider a three-dimensional body under arbitrary loads as shown in Figure 2.1. Using a Cartesian system, let the loads be given by {f 8 f: f = 4 ,IT (2.2) {f 8 } = [fB fVB fI] 1p, = [p T (2.3) jT 24 FV, where {f}, (2.4) {fB} and {F} are surface tractions, body forces, and concentrated applied forces respectively. The displacements of a finite element in the body due to external load is denoted by {d}, where {d} = [u v WIT (2.5) and the corresponding strains are given by, {4 = [rz 6V ex, f. tzz f ]I (2.6) for which the corresponding stresses are, {0 = [Ozz Opy 47zz O,. ffxs O,]T The total internal virtual work for a finite element in the body is {6e}Tf{}dv (2.7) and for the whole body, 6Wt = j{6c}Tfaldt where the virtual strains, (be, {b} (2.8) are = [be.,,,btyv e, , . (?,]T (2.9) The total external virtual work is given by: 6We, = f 6d}T{fB}dv + fj{6dTI{f°}ds + {di}T {F} 5 (2.10) ~0 LL. CO C3 %U c) a .V4E IL a Figure 2.1: General 3-D Body 6 where {d } denotes a surface displacements and {6d' ) represents point (nodal) displacements corresponding to the applied loads, and the virtual displacements {6d} are {6d}T = [6u 6v bw] (2.11) On substituting equations 2.8 and 2.10 into 2.1, we get, j6e}Tfojdv = j {6d}T{fB }dv + j{&P} T~fhl)ds + {6d1}jTf P} (2.12) It may be noted that the principle of complementary virtual work may have been undertaken, assuming small virtual stresses with true displacements, yielding in an analogous expression for equation 2.12. Introducing the generalized Hooke's law for material constitutive relations, {a = [E] {} + {a} (2.13) where {a.} denotes the element initial stresses and [E] denotes the elasticity matrix of the element material. In general, the strain-displacement relations are given by (e) = [B]{d} (2.14) while the virtual strains are given by I6e = [B] {6d} (2.15) Substituting equations 2.13 and 2.15 into equation 2.12 and simplifying, we have, If{6d)T ([BI T [E] (BI) {d}dv j{d}T~fB}dv + fj{6cP}T{P.ds If fj{6d}T[B] T {a.}dv+ {bd'}{fP} (2.16) The integrations in equation 2.16 are performed over the element volume and surface, i.e., we can evaluate every integral using the element local coordinates and 7 assemble tor the global system coordinates. Thus, we define the global displacement vector and the global virtual displacement vector as follows: {D} = [uIvIwI u 2v 2w 2 ... Unvnwn ]T (2.17) and {6D} = [buj1 v6vw, ... u,,6v,,,,T (2.18) where n is the total number of nodal points in the body. Now we define, for m elements, [K] = [k,] [BIT [E] [B] dv f (2.19) [B]T [E] [B] dv = (2.20) where [K) and [kj] are the global and local stiffness matrices respectively. In addition, we define, in {RB} = {Bj= {R.} = {r,}j = {R,) = V , f [NIT {fB}d jfN] T (2.21) (2.22) {fE}dv . j=1 >21[N']T{l'}da (2.23) j[NI T {f}d (2.24) (2.25) . j=1 rl}j = j[BIT {,}dv where (2.26) {R) and {r} denote the global and local load vectors and [NJ and [N'] are the displacement interpolation (shape functions) matrices for the volume and surface where traction is prescribed. Using these definitions, we obtain {6D} T [KJ(DJ {R} = {6D) T ({R = {R 8 } + {R.}8 3 ) + {R.} {R) - {R,) + {fr) + {P} (2.27) (2.28) By invoking the principle of virtual displacements and noting that {bD} is arbitrary, we get the equilibrium equations in the following form: [K] {D} = {R} (2.29) Equation 2.29 is the basic equation for static equilibrium, which also gives the general form for nonlinear analysis with large displacements and strains. C. INTERPOLATION SCHEME 1. Shape Functions (Displacement Interpolation Functions) In this section, the interpolation scheme for a triquadratic isoparametric solid element will be developed. The one-dimensional Lagrange interpolation function based on parameters is given by q NP= P i=1 NP,+ N2P2 + ...+ NPq (2.30) where Ni, also called the shape functions, are given by M l N,(x) = __ (2.31) 0*j A triquadratic solid element is a three dimensional element in which the displacements u, v, and w are interpolated by quadratic langrangian interpolation functions with 27 nodes. Figure 2.2 depicts an element in the local non-dimensional coordinates (r, a, t). For an isoparametric element, the geometry may be interpolated as, 27 Nizi X= i-I 27 y= 9al II i9I .- , N Ln I U., I. . .. I H . (fl 0 'VI0 0 0 t_, 0 - Figure 2.2: Lagrangian Solid Element - 27 nodes 10 0 k C 27 z = j (2.32) Niz, i=1 or, in a matrix form, y z -[NJ [zjyjzj .... r27y27z2] r (2.33) where the shape function matrix is given by N, 0 ... N2 0 0 N2 0 ... 0 0 N2 ... 01 N27 0 (2.34) 0 N 27 0 0 [Ni 0 0 N 0 0 N2I The shape functions and their derivatives in local coordinates are pre- 0 NI 0 0 sented in Appendix A. 2. Jacobian Transformation Matrix To obtain equilibrium equations in the global coordinate system and con- struct stiffness matrices, we need the derivatives of the shape functions in cartesian system. Using the chain rule for differentiation, we obtain, Nj,, = Nt z,, y,, z., t z, N,,, t N (2.35) where [J], the Jacobian matrix is given by, [J]= Z.r Y.r z, y.o X't X~t Z,r z, (2.36) Z.1 A comma denotes differentiation, where for example, Ni,, = etc. Using the shape function derivatives, the elements of the Jacobian matrix may be calculated and is given in Appendix B. The global cartesian derivatives may now be obtained as, rN' N,,, N,.. = [J]11 rj~ NI,, 1 N,., N,,] (2.37) where the inverse of the Jacobian, (2.38) is given explicitly in Appendix B. D. STRAIN DISPLACEMENT RELATIONS 1. - (B] Basic Formulation In this section, the basic formulation for nonlinear analysis of a general solid body is presented [Refs. Bathe (1982), and Malvern (1969)]. First, some definitions and notations will be introduced concerning the coordinate system, displacement, stress and strain measures and later on the linearized equilibrium equation will be developed based on section II/B. Consider the motion of a body, or an element within, in a fixed cartesian coordinate system as shown in Figure 2.3. We have the body at time 0, t and t + At for which the upper left superscript corresponds. The displacements at time t and t + At are given as tui=t Z _0 Zi t+Aiu .F At _0 Zi (2.39) (2.40) so that the incremental displacements are % =t+t _ u, (2.41) where, U 1 U X 1 =r U2 -V U3 -W z 2 =s z 3 =t 12 (2.42) we use the following notation for derivatives at time, say t + At, with respect to coordinate at time 0 as, +atuj = & (2.43) In the present approach, we use the total Lagrangian formulation, referencing all variables to the undeformed configuration at time 0, [Ref. Bathe, 1982; Malvern, 1969]. It is assumed that at time 0 and t the equilibrium configuration is known. Basically, equation 2.12 needs to be solved corresponding to time t + At. Since we assume large displacements, and nonlinear constitutive relations [equation 2.14], equation 2.12 may be solved by incremental load methods [Ref. Ford & Stiemer, 1987]. On introducing the 2nd Piola-Kirchhoff stress, it may be shown that the 2nd Piola-Kirchhoff stress tensor is energentically conjugate to the Green-Lagrange strain tensor [Ref. Bathe, 1982; Malvern, 1969]. j { _+At}T {t+&S}Odv = j+A, {6e} T {t+at'}t+Atdv (2.44) where the 2nd Piola-Kirchhoff stress at time t is defined as, I o,= [,X]T {tS} [,Xi det [X] (2.45) such that [tX], the deformation-gradient tensor is a tranformation operator from the coordinates at time 0 to time t. Note that in the equation 2.44, the right hand side represents internal virtual work at time t + At over the volume at that time while the left hand side has the virtual work integrated over known configuration at the reference volume. Assuming linear material behaviour, we may use the linear stress-strain relations (Generalized Hooke's Law) for the 2nd Piola-Kirchhoff stress tensor. [ s] = [E] 13 [] (2.46) $44 *X V -H I, (13 ' , YPO) 0p '4 _- ... ..>.. . E 4-)N CC 1 U414 4 where the Green-Lagrange strain tensor at time t is defined as, Si 1 (i +' Uj, +' Uk,, 'Uk,) (2.47) with i, j, k = 1, 2, and 3. On subs Quting 1+u = ui + ui, we obtain at time t + At, t+At + 1 + +t Uj,i + u (ui, + uji, k,itUk) u k., ukj + Uk,, k,,) (2.48) Uks ut,, as, which may be written t+atfii = f + i (2.49) where, ti is defined earlier and the incremental strain e,, is given by 4, = 4 J + r/ij (2.50) {e} = {e} + {r/} (2.51) In matrix notation, The linear incremental strain is identified as 1 ei= +t (, + U,,i + ,, Uk + Uki Ukj) (2.52) in which tukbj and 'ukj are the known displacement gradients at time t. The non- linear incremental strains, then, are givtn by = 1 uk.i u, (2.53) Rewriting the equilibrium equation as stated in equation 2.12, using the total Lagrangian approach, we have, 5(2.54) where t+AR, the external virtual work, is assumed to be deformation independent. Using the identity form equation 2.44, we may write the equilibrium equation in the undeformed configuration as j{At+dte}T{t+&ts}odv _.=+t R (2.55) Noting that I{l} is displacement invariant, {b+A'e = {&e} (2.56) The 2nd Piola-Kirchhoff stress at time t + At may be expressed as { 3 } + {1) {t+AS} = (2.57) where {s} is the incremental 2nd Piola-Kirchhoff stress. On substituting Equations 2.56, 2.51, and 2.57 into 2.55 yields, j{6 eTfs}Odv + I {60{'.1v+f6 Its)Odv 17 )T (2.58) The incremental stresses are expressed using equations 2.47 and 2.57 as Is} = [E] I +4'e} - [E] {'e} (2.59) which in view of Equation 2.50 yields {s} = [El {e} (2.60) Referring to Equation 2.51 and neglecting the nonlinear strain contribution, we get the linearized approximation as I1 - [E] {e} (2.61) {6e}T (2.62) and {6C}T - 16 On substituting these into Equation 2.58 and rearranging yields the linearized incremental equilibrium equation, [E e 0 j{Se}T~ v+j{,}{ts}lodv = 4t R j6eTftSOdv -v (2.63) 2. General Nonlinear Discretization The general nonlinear finite element discretization for the 27-noded element is presented based on the total Lagrangian formulation discussed in the previous section. Equation 2.52 for the linear incremental strain in cartesian form yields er= evy UI 9UI ' tVXVX+ -= U'Y +}t U'Y uY -+-tV'W V' +O t eVI - U, 1 eX = U, 2e = V, v= + WV +t UU 2e, = u, + w. +t 2erv = u.1, + v" +9 ' u'Y + u U U, 1 UI U. +' wW wWz W IV . U~i pl Uw, 1 WV V13 V,+ W," W, + U1 'U's +t V,1 V's + V,1 tv,5 +W u,.UV+ u= t .,+t vS v, + VIC t . Y+tV'. v W+ v 'y w,5 + W.V tW.X ' ?+ 1 W w) tWp t WI. oy' + w" t'Y~ (2.64) which in matrix form is given by {c} = {eLa) + {eL,} (2.65) The first term on the right hand side is displacement independent while the second term is displacement dependent with the engineering strains {e} represented by {e} = le evy e.. 2ey, 2e. 2 e,,]T (2.66) We define the incremental displacement gradient inthe global coordinates by w'T w,W Wo. {UG} = [uu.,U u, v, V,1 v,., 1 17 T (2.67) Equations 2.64, 2.65, 2.66, and 2.67 result in eLo) = [ALo] {UG} (2.68) {eL1} = [ALI] {UG} (2.69) so that [ALo] and [ALI] are given by, 1 0 0 0 0 0 0 0 0 0 0 010 0 00 000 0 0 0 001 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 100 0 0 0 [Au] = tu, 0 0 [ALI] 0 t (2.70) U,. 0 0 0 t u., tv, 0 0 tvV 0 0 tw. 0 0 0 tv,, 0 tv,y 0 tw, 'w, tv, 0 tw, 1wv 0 tw , tw,. 0 0 w 0 0 0 Sw, (2.71) - 0 tu, tU,y 'U, 5 tu, 0 t v, 0 tu., tu. 0 v., 9v,y 0 tv.: 5 It may be noted that the values of t uij are known at the new configuration at time t + At. With the displacements interpolated by 27 Nkuk U = k-l 27 v = E Nkv, k= 27 w = ENwk (2.72) kal the local displacement gradients are obtained from 27 U =r = N,,. uk kni 27 U's = E Ni,,. uk k-l 27 U's N,,t uk = k=i 18 (2.73) and similar expressions may be attributed to v and w. These gradients may be represented as U! N,,. Ni., 0 0 0 0 0 0 N2 ,. N 2 .. N 2.# 0 0 0 0 0 0 N2 7., N 27'. N27 , 0 0 0 0 0 0 0 N1 . 0 0 N,, 0 0 N27., 0 N,., 0 0 N2,. 0 0 N27.. 0 N,., U.T U.8 1,2 V2 U's V., V W = 0 V,, w.,. W, 0 Ns., 0 0 N2, t 0 0 N27,, 0 0 N1 ., 0 0 0 0 0 0 0 0 0 0 0 N2 ,, N2 .. 0 ,,,, . 0 0 0 0 N., NI,, ,, . . w2 0 N27 ,, N2 7,. N27 ,, J • 527 P27 (2.74) or alternatively, {UL} = [DH] {d} (2.75) where the nodal displacement vector is given by, {d} =[u, VI W1 u2 v2 ... (2.76) w 2 7]T and the local incremental displacements gradient are given by {UL) = [u., us u,t ,Vv,,. v,, w,, w. .,lT (2.77) As previously mentioned, in isoparametric finite elements, the same interpolation functions are used for approximating the geometry and the displacements. Using these definitions, we may transform the displacement in global and local coordinates by similar transformations used for the geometry. Furthermore, we can define arbitrarily the global and local coordinates to coincide at time 0 configuration, thereby, the transformation from local coordinates at time 0 to local coordinates at any other time, say t or t + At is identical to the transformation that relates the global coordinates to the local coordinates at any configuration. In other words, we 19 can use the same jacobian matrix defined previously for all configurations. Writing the relation in accordance to equations 2.37 and 2.38 we have, {uG} = [3] {UL} such that, [(F] [p [0]o [o [r [0] [o0 [o1 [r] and substituting equation 2.75 into 2.78 yields, [r 3]= {uG} = (2.78) [r3l] [DH] {d} (2.79) (2.80) The incremental strains, then, may be expressed in terms of nodal displacements, and substituting equation 2.77 in 2.68 and 2.69 {e,} = [An,] [I31 [DHI] {d) (2.81) {eLll = [ALI] (2.82) [r 3] [DHI] {d} The strain displacement operator may be identified as [BLo] = [ALo] [13] [DH] [BLI] = [ALI] (r 3 [DHI] (2.83) (2.84) such that, [BL] = [BLA] + [BL1I (2.85) {e) = [BL {d) (2.86) and, It should be underscored that using only the displacement independent strains in equations 2.63 and 2.64 results in the linearized problem (same as linear small 20 displacement - small strain formulation) with Ni," 0 0 N2.. 0 0 N2 7 , 0 0 N,y 0 0 N 2,1 0 0 N2 7,. 0 0 N27 . 0 0 Nj,. 0 0 0 N27 ,1 0 N1 ,. 0 N,y N1 ,= 0 N 2,. N 2,1 N 2 ,y N 2,. 0 0 N 27 ,, N27 ,, 0 0 N2, N2 . 0 N27 ,y N 27 . 0 0 [BLO) = N1 ,, N, 1, N. N2 r.y, N 27 ,. (2.87) The displacement dependent contribution to the strain-displacement operator is given by 'U.. Nt 'u., N",. tu. [BL. 'U. N, N1+1 , N,,, III , N , 4- u, iNI a NW4 ta. Nil Ni, .' M. ' Nf.,, , 't ' uI Vi , N ,, + ', .,U Ni, M ,, .N,+' V., N,, N. 'u,, U, MW.. NaT' 'U., N, 'i. 1W..N,,+W, IV ,, V NJ,, N+, Nwo l , N ,, I .N ,, 'W, N,, +t W., N,' , in, N 'u,' N2,, a N 4 NJ, U . N ,, +, U., NU, NW,. ** . iNV. Hr , N ". ~ W. NW,, (2.88) To obtain the incremental nonlinear strain contribution, consider equation 2.53, which has the cartesian components, 2 S1(2 + W ) 2 (uI+*l + 2a = 1 3 +22) 'lpv = 'Iv (uyiU + V,,v + = 2 W ) 1 17X= 77, 77v = -(u,,u, + V.WV., + w.,w.,) 2 -(u.,u, + V.zv, 1 + ww,,) 1 -- i(u.xu., + Uv.v + WxW.Y) 22 21 (2.89) With the variations at time t given by, t v, 6?7, = 6ul,: tu, + 6v, br/ = 67,, = bu,, t u,, + 6v., t v., + 6w., 'w., 6 + 6w,. 'v, uLUI 'u. , + 6vb iv,V + bwV tw., 2b%. = 6u,, 26r7., = U,, + uV 6 u,, + bv., iV, + *'v.6v,, + 6 w'V 'W. + 'w11w ut, t u., + tus6u, + 6v, Iv., + t v.,bv, + 6w,: tw. + tw6w., 2,"v = 6u. tu,, + CuX6u. + 6,v, tv , + 'v,:v,, + 6xw,: t w, + 'w bw.2.90) (2.91) It is worth noting that equations 2.90 are in exactly the same form as the displacement dependent strains {qIt} given in equations 2.64 and 2.65 with the incremensal displacements derivatives replaced by their variations. Thus, we may write, (6bq) = [ALI] {6UG} (2.92) (7) = [ALI] {UG} (2.93) such that the nonlinear incremental strain variation vector is defined as, = [61,, 6q, 671, 2671,, 26%, 26,~v { b}T 6 (2.94) and [ALI] is as given in 2.71. Observing relation 2.80 for JUG), we have, {6uG} = [r3] [DH] {6d) (2.95) and substituting for the global variations into the nonlinear strains in equation 2.90, we get {6,7 }T = {6d)T [DHIT [r3 lT [ArlIT 22 (2.96) Using the strain-displacement relations defined thus far, we formulate the linearized incremental equilibrium equation, given by 2.63 to take the general form as stated in 2.29, to give, tR} d"+)= [VkL] +{t+A) - {t+AtFI(I) (2.97) where [t kLl, in view of equation 2.85 is seen to be [tkL] : [BOn [E] [BL] dv (2.98) which is the linear stiffness matrix, and i is the iteration number. This includes the displacement dependent and independent contributions. When small displacements are assumed, the [tkL] reduces to standard linear stiffness matrix, given by fov[BL oIT [E] [B o]0 dv. In what follows, the derivation of [kNLI is described. The 2nd-Piola stress vector at time t is accumulated such that, {t s} = {+'+s} + [E] {} (2.99) Using equations 2.51, 2.85, 2.86, and 2.91, the incremental strains take the form {E} = (2 [ALI] + [ALo]) [IF3 [DHI {d} (2.100) or alternatively, {c = (2 [BLI] + [B,] Il){d} (2.101) Noting that equations 2.99 and 2.100 are valid at any time t, and by using equation 2.46, the 2nd Piola-Kirchhoff stress may be written as { s} = [E] (2 [AL)] + [ALo]) IV3] [DH] {d} (2.102) and the nonlinear part of equation 2.63 becomes, using the relations 2.95 and 2.101, ]T [ALIT) [E] ({6d}T [DH]T f{b 71 r {ts} ° dv = f (2 [ALI] + [A,]) [173] [DH] {d}°dv 23 (2.103) we define the nonlinear strain contribution, [kNLI, as [kNL] = jT[DHIT IF T [ALI]T [E] (2 [ALI] + [ALo]) [173] [DH]° av 31 (2.104) It may be recognized that [DH] is the strain-displacement relation given by equation 2.74 which corresponds to the linear contribution of the stiffness matrix given in equation 2.97, and the contribution of the non-linear strains results in the 2nd Piola-Kirchhoff matrix, [s], as T [S] = [r31T [ALI] [E] (2 [ALI] + [ALo]) [I3] (2.105) [r3] is given in equation 2.79. It may be shown that the matrix [s] takes the form, [ sil 1 0 101 IS [0] [0] [S] I [0] [s] [0] (2.106) The expression for the second term in the right hand side of equation 2.63 is evaluated in the same manner as the linear and nonlinear parts. On using equations 2.100 and 2.101, we obtain, [BLI]T +[BLo]T) [E] (2 [BLII+[BLo)] {d} °dv (2.107) d}T(2 IT j{6e}rTts}dv = It may be seen that by defining { (t+AF} = jV(2 [BL]r +[BLo]T )[(2[BL1 + [BLo)] {d}°dv (2.108) the expression {d}T I+A(F} represents the work done by the external loads at time t + At. Noting that { 6 i+At} {6e} (2.109) We approximate for the second term in the right hand side of equation 2.62 such that {e T {ts }odv /o j{6 +AtclT{I+ "s}Odv 24 (2.110) which represents the internal virtual work, so that the right hand side of the equilibrium equation 2.63 is the difference between the external and internal virtual work. On substitution of relations 2.97, 2.102, 2.103, 2.107, and 2.109 into 2.63 and applying the principle of virtual displacement as shown previously, we arrive at the incremental equilibrium equation 2.96, which may be solved by Newton-type methods. [Ford and Stieman, 1987] E. STRESS-STRAIN RELATIONS In this section, the stress-strain relations for a composite material, to be used in the three dimensional analysis is developed. Two approaches, one based on classical laminate theory and the other based on anisotropic material constitutive relations, are presented. 1. Classical and Higher Order Laminate Theories Typical structures composed of composite materials are built using sev- eral number of laminae, forming a laminate. Each lamina consists of, typically, uniaxial fibers embedded in a matrix, such as epoxy-resin, forming a thin plate. Figure 2.4 shows principal material axes, labelled 1 and 2 in directions parallel to and normal to the fibers, respectively. It may be noted that in each lamina, there exists a state of plane stress, as shown in Figure 2.4. Assuming elastic orthotropic material, (i.e., the lamina possesses a plane of elastic symmetry parallel to the x-y plane), the generalized Hooke's law may be written as al a2 a3 Q1 Q 12 Q13 Q22 Q2 _ a4 symm as 06 0 0 Q33 0 2Q4 L 0 0 0 0 2Qss 0 0 0 0 0 2Q66 25 E f3 f4 fs J c (2.111) x 00 4-) 0 X 00 Q) x> 0 ,, __ __ __ _ __ __ _ t -I " - x - tW4 Figure 2.4: Lamina coordinate system (2-D) 26 CM where the plane-stress elastic constants are given by: Q1I = QI - I -- E V12 21 = 1 2 E2 ' 1 - V 12 V21 = Q44 = G Q13 = Q23 = 23 12 - 3 Q22 Qss , 21 = G 1 3, Q6 = G12 (2.112) 0 The subscripts 1, 2, and 3 correspond to normal stresses or normal strains while 4, 5, and 6 correspond to shear stresses or tensorial shearing strains in yz, zx, and zy planes, respectively. The stresses in the material coordinate axes are transformed to reference coordinate axes (x, y, z) by the following equation: axz m2 n2 0 r, n2 m2 0 o, 0 0 1 Oryz or. 0 0 0 0 mn -mn rY1, 0 0 0 -2mn 2mn 0 0 0 0 m 0 -n 0 0 0 a, Or2 (2.113) 03 n 0 m 0 2 0 (m - n 2 ) 0r4 as '6J where the direction cosines m and n are given by m = cos 0 and n = sin 0. The straint, may be transformed in a similar manner. Introducing the strain transformation, along with equation 2.110 into equation 2.112 results in Q11 Q12 oQ k 0 2Q16 C. C Q22 0 0 0 2Q26 0 0 Q33 0 0 0 0 0 0 2Q44 2Q45 0 0 0 0 2Q 45 2Qss 0 Q26 0 ,z Oz~y 0 Q12 az 0 0 LQi6 0 0 2Q66 J where, Q11 = Q m 4 + 2 (Q 1 2 4 + 2Q 6 6 ) m 2 n 2 + Q 22 n 27 C2 (2.114) fxz - k 012 = (Q11 + Q22 022 = Qun 033 Q33 Q16 Q1 - 4Q6) m 2 n2 + Q12 (m4 + n 4) 4 + 2(Q12 + 2Q 3 n Q22 mn' - 6 - 4 )m 2 n 2 + Q 2 2 m (Q12 = Qjmn-3 Q6 = (QI + Q22 - 2Q2) mn 2 Q44 = Q4 4 m (Q55 - Q4) mn = Q 4 4n 2 + Q 5 5 Mn2 = + Q6 (m -n2) _ - n2) n2) 2 2 + Qs 5 n Q45 = Q5 (M2 Q 22 m 3 n + (Q12 + 2Q6) mn (m2 Q26 2 + 2Q66) mn (2.115) The stress strain relations presented correspond to kth lamina. Now, consider a laminate composed of N laminae for which, each lamina has a different orientation (0), with respect to the laminate x and y axes. For linear elastic plates, the function!L, form of the displacement may be assumed to be u(x, Y, z) = Uo(x,y) + zu 1 (X,y) v(X' Y, z) = vo(,y) + zVI(X,y) w(x, y) = WO(X,y) (2.116) and, the linear strains are given by, fi= 1(uij + u,i) so that, Ezz = fyy -= VO, 1, + ZV14 y zz UO,x + ZUi,x 0 28 (2.117) 1 fu C' + w0,11) 1(Vi 2 = 1 = 2 I(Uo, + vo+ U1 + WO ) (u, + vI) (2.118) where uo, vo and w0 are the midplane displacements, ul and v, are related to the rotations of the normals. It may be noted that the in-plane strains, 1 fo = Uo,X, EO = Vo, , (2.119) (Uo, + Vo,) zo = and the curvatures are given by = ul., UT¢ tc = vl,p, Kz 1 = (2.120) (ul, + vi) We define stress resultants for plate/shell type structures in terms of stresses and shears (see Figure 2.5) as follows for the kth layer: h QX= L 2 h ~zdz 2 /M azdz = (2.121) Similar expressions are applicable for Ny, Ny, Qy, Mv and M,,y, where h is the lamina thickness. By summing all laminae over the laminate thickness in the following manner, {} N..,,y E~i k=1 h -, f'YO } dzJ~ +~ f k_1x zdz} (2.122) which, in matrix form, may be written as {N} = [A) {o} + [B] {K} 29 (2.123) x ~co CH 4-)r CI) x 4-do Q) x 30) - where, N Ai 3 = hk (Q") k 1: - hk-I) k=1 N (h 2 - h _1 ) Bi = (2.124) k=1 with ij = 1, 2, and 6 and the moment resultants are given by {M} = [B] {co} + [D] {,} (2.125) where, Dj = E - I2.126) k=1 with i,j = 1, 2, and 6. The displacement field, as stated in equation 2.105 is linear in the thickness direction, resulting in constant shear stresses. To get better accuracy, a higher order displacement field may be used [Reddy, 1984]. In order to account for the accurate shear distribution, shape factors are used in computing shear energies. These factors are typically obtained by equating the shear energies. The procedure is outlined for linear and cubic variation of displacement fields. The shear energy due to transverse shear stresses is given by, Ua = J (ao,, (2,Ez) + oyz (2,,)) dz dA (2.127) where A is the area bounded by the lamina surface dy. On using Hooke's Law, we get = Uo~~~~ U. AL (Gxz (2cz+)2 + = dA d(Gz(~~ (2fy)2) dz (2.128) Equating this to the linear displacement field and simplifying, (1A 2= [G.. (u, + Wo,.) 2 31 + G (v1 +Wo,,)2ldA) h (2.129) Introducing a higher order displacement field yields a more realistic stress distribution, but in doing so, a shape factor is introduced to yield consistent shear energy. Assuming a displacement field in which the displacements are expanded as cubic functions of the thickness coordinate, while the transverse displacement is assumed to be constant through the thickness, yields + U(z,i,Z) = UO(z 1') + ZUI(x,t) V(-,%,z) = VO(zY) + ZVI(z,Y) + W(Z,Y,Z) = Wo(XY) Z U 2 (z,y,) + Z3 U3(x,y) Z2 V2(z,y) + Z3 V3(x,y) 2 (2.130) With u0, vo, and wo being the displacements of the midplane, the tensorial shearing strains are evaluated as, 2fz, = [u,(Z,,) + 2u 2(r,y)z + 3U3 (,y)z 3 + WO] (2.131) + 2V2 (.y)z + 3U3(xy)Z 3 + = Using the condition that the transverse shear stresses vanish on the plate top and bottom surfaces, we have, OX X(, ,+ ,V Y, = a1 0 0 (2.132) or, (X, fX(V , ±h = Cy Y1h) 0 (2.133) Substituting relations 2.132 into 2.130, we obtain, U2 = U3 = V2 = 0 4 4 (woM + u1 ) - 4 V3 = (W, - 32 + vI) (2.134) The displacement field then becomes u = 34 uo+zu 1 -z(w,+ v = vo + zvI - W = w0 ) "Z34(wou + vI) 2 (2.135) 3h (2.136) and the shearing strains are given by E-z 2 -(uI = [1 +wo,) -4 h (2.137) (v+ wO.±) 1-4 (z)] V- = Yielding Ui. = (1j 1 2 G (i+w,) + Gy (vI + WO,Y)2] dA ~ [ 2] d (2.138) The shear energy ratio of the two displacement field is found to be 1, so that the correction factor for constant shear stress using cubic displacements is 8. It is clear from the discussion that the classical plate theory stiffens the plate by not taking into account the higher order terms. If we use the higher order theory, we need to introduce a correction factor to the shearing strains of the magnitude Using equation 2.113, the transverse shearing stresses for the kh layer are given by a.xzk = Q~IZ + 2 Q45k'EYZ a , k = 2Q45 ,C + 2Q 44 ,f. (2.139) and the resultants are obtained using equation 2.122 as QX Q= 2(As55Ex + A4 5 fyz) 2 (A 45C. + A 33 44 'y) (2.140) Note, that equations 2.139 and 2.140 are applicable for any displacement field. Hence, for the higher order theory presented, -15 A,3 I-zj k=1 (z)2] dz [-4 1~) k (2.141) or, A 15 ~ZQJk 3 8k=1 hk-hkl4-(h 3 3h k 2 h~3_)] (2.142) with i,j = 4, 5. In the present three dimensional solid element, for which only three translational degrees of freedom per node are defined, resultants are divided by the corresponding thicknesses to obtain the stress-strain relations with All A 12 0 A 12 A 22 0 0 0 0 [1E h -E A16 0 0 0 0 0 0 0 A26 0 0 0 0 2A 16 2A 26 0 0 0 0 2A44 2A 45 0 2A 45 2A 55 0 2A 66 0 (2.143) For the special case of isotropic material, the material stiffness matrix is given by [E] A+2G A A A+2G A 0 0 0 A 0 0 0 A A 0 0 0 0 0 0 A+2G 0 0 0 0 G 0 0 0 G 0 0 0 0 G 0 (2.144) where (2.145) A =E (I + v) (1 - 2v) 2. Three-dimensional Anisotropic Theory As an alternative to using laminate theories to obtain [E] matrix, we may use anisotropic definition of the laminates. ai = (Qi') 34 E (2.146) These relations are approximated by obtaining the Qj in the laminate x-y axes by suitable transformations and transverse properties (thickness direction) correspond to the matrix characteristics. F. CONSISTENT LOADS In this section, we consider the element nodal loads vector, due to applied loads. Using the virtual work principle, the distributed loads, such as surface loads and body forces, are converted into discrete loads applied at the element nodal points. Discretizing the distributed loads along these lines are referred to as consistent or work-equivalent loads. Consider a case where a uniform distributed load acts on a prescribed face of the element, as seen in Figure 2.6. The consistent load vector may be written as {r,} - j [N'] T {f'}ds (2.147) For uniform distributed surface loads, we have {f'} =p{l} (2.148) It is worth noting that the interpolation functions on a given surface, say t = 1 reduces to that of a plane biquadratic Lagrangian -mparametric element and are presented in Table 2.1. Invoking symmetry, we observe that the forces at nodes 1, 3, 5, and 7 are equal, and similarly, forces at nodes 2, 4, 6, and 8 are equal. On using the shape functions for the t = 1 surface, we have, r, = _ "l1 lr)+_ (1 -s)p dr ds -r 35 1 1 -- r2 - -r9 1 p (2.149) TABLE 2.1: SHAPE FUNCTIONS FOR 9 NODED BIQUADRATIC ELEMENT N1 N =1 l=(1 + r) (I1- s) - 2 N8 - l N2 N3 = 1( + r) (I + s) - I N2 -1N4 N5 Ns=X4 (1 -r) (I +s) -2IN4,2 N2 N7=X =14 (1 - r)(I1- N = s) - 21Ns- 2 1 N6- 2 ) (Ir1 + 1 (2 1(1-r)(X-S S) - 1 N 4N9 - I4N9 -1N9 4 14N9 ' 2 2) 2 N9 = 1N6 2 - 2 ( 1 -r2) (1 _ 2) Note: Node numbering is referred to Figure 2.2 where t = 1, upper plane. Figure 2.6 gives consistent element nodal loads for a single element. As a check, the total pressure loading on the surface, 2 x 2 x p = 4p, is seen to be equal to the sum of all the discz.f'.zed nodal point forces. The procedure may be extended for more than one element by summing loads at joint nodes, as illustrated in Figure 2.7 for four elements. G. INTEGRATION In this section, we summarize the Gauss method for numerical integration, including a discussion on some aspects of integration schemes. 36 cr. Q LL 4- 070 0 LLM CC\ LL Figure 2.6: Consistent loads 37 0) -- ------------------- c: 0 41-) L) Figure 2.7: Consistent loads on a mesh 38 1. Gauss Quadrature The nature of finite element matrices suggests the usage of numerical quadrature. Gauss's integration scheme is the most commonly used approach and is adopted in the present analysis. The method enables exact evaluation integrals, consisting of polynomials of any order, by using appropriate order of integration. In general, the Gauss quadrature for a function O(r, s, t), has the form I= _ZZ w ww¢kO(r,s,t) 0(r, s, t) dr ds dt (2.150) The integration limit reflects the limits of non-dimensional 'master' isoparametric elements, while O(r, s, t) represents the stiffness contribution. Figure 2.8 demonstrates the application of the method for a two dimensional biquadratic element. Using the weighting factors as given in Table 2.3, the element stiffness matrix is evaluated, for example, by using a 3rd order integration scheme as follows: 55 [K] = 9 (01 + €3 + €7 + 09) + 58 99 (02 + 04 + 06 + 08) + 88 9 05 (2.151) where O, = h [B(r, S)]T [E] [B(r, s)] I J(r,s) I (2.152) as is evaluated at Gauss point i as shown in the Figure. 2. Integration Scheme The term "full integration" refers to an integration scheme which evalu- ates the integral exactly as shown in the previous example. In the same manner, a lower order integration is referred to as 'reduced integration'. In the present analysis, 'full integration' is used to evaluate the stiffness matrices. When a crude mesh is used, a stiffer structure is obtained. In geieral, there 39 xS 0 o I ,I o II II x x Iw *.-I 4-) CK Figure 2.8: Gauss points 40 TABLE 2.2: SAMPLING POINTS AND WEIGHTS FOR GAUSS QUADRATURE OVER THE INTERVAL -1 to 1 Order n Weight Factor W Location of Sampling Point 1 0. 2. 2 ±0.57735 02691 89626 = ± 1 1. 3 ±0.77459 66692 41483 = ±v 6 0.55555 55555 55555 = 9 0. 0.88888 88888 88888 = 9 4 ±0.86113 63115 94053 = ± [ L7 0.34785 48451 37454 = I2 -, 6r ±0.33998 10435 84856 = ± [3-]2 0.65214 51548 62546 = 1 + where r = rT. and (2r - 1) is the polinom order are two ways to soften the structure. One way is to refine the mesh and another by using 'reduced integration'. Thus, by using a 'reduced integration' scheme, a faster convergence and more cost-effective, accurate solution may be obtained. However, the method suffers such drawbacks as mesh instabilities or mechanisms, resulting in a singular element stiffness matrix. H. BUCKLING ANALYSIS 1. Introduction It is well known that thin columns or plates under axial compression tend to buckle. Elastic buckling occurs when the compressive stress is well below the material stress limit. A flat plate under axial compression shortens in the direction of the applied compressive loads. This shortening results in coupling between inplane and out-of-plane displacements. 41 As the applied compressive load increases, there is a configuration at which the plate offers no more resistance to deform, resulting in a state of neutral stability. The load corresponding to this configuration is referred to as the buckling load and constitutes a limit point on the load response curve. At this critical value, the deflection becomes very sensitive to any change in the configuration. For some structures, beyond the limit point, the load-displacement path may take any of multiple paths. The point where the plate can take any of the different paths is called the Bifurcation point and is illustrated in Figure 2.9. In analyzing for nonlinear response, the incremental load method is adopted, which may be summarized as follows: (a) the tangent stiffness matrix is formed, and solved for displacements for an incremental load. Keeping the stiffness matrix constant, corrections to the incremental displacements are obtained in an iterative manner until equilibrium is achieved, (b) total displacements for this load are obtained, (c) a new tangent stiffness matrix is formed at this new equilibrium position and steps (a) and (b) are repeated. This procedure is continued until the desired load is reached or the critical buckling load is reached. 2. Implementation In this section, the Finite-Element formulation for buckling will be pre- sented [Bathe, (1982), Kolar, et al., (1985)]. The problem of instability can be approached either by looking at the equilibrium of the structure in the deflected position and transforming all quantities to the initial configuration or by solving the system in the current configuration. The former approach, described earlier as the total Lagrangian formulation, is adopted here. By performing an incremental load analysis, using the nonlinear formulation described earlier, we may write ([tKLI + ['KNL]) {d}'+') 42 _= &t A{P - IF)(') (2.153) P l r0 I rit cn itpointa& 10. Patt:b u ck I ng Fig. 2.9; Instability and bifurcation points Figure 2.9: Instability and bifurcation points 43 where {P} represents the total load applied and t+AtA is a scalar, referred to as load parameter. The value A scales the incremental load and may be treated as a constant or variable during iterations. Buckling load is reached when displacements become large with no increase in the incremental load, i.e., the global stiffness of the structure, as illustrated for a single degree of freedom system in Figure 2.9, becomes small and [K] tend to be singular. Thus, using Newton-Raphson and modified Newton-Raphson methods, convergence difficulties are encountered as buckling load is approached. This is overcome by using arc length methods, described in the next section, where the load parameter is continuously updated to reflect the state of the structure. 3. Constant Arc Length Method [Kolar and Kamel (1985)] When using the Newton type iteration schemes, the stiffness matrix "be- comes singular as limit points are approached. In order to obtain post-buckling response, a method to overcome this singularity is needed. This is accomplished by treating the load parameter as a variable and thus have an adaptive load incrementation. This approach differs from the conventional Newton type schemes where the load level is held constant for all iterations at a given load step. Symbolically, at load step m and iteration i, equation 2.153 can be rewritten as follows [KI{d}(+ 1) = (mA + P),+ AA) {p} - {F}(') (2.154) where - [K] is the tangent stiffness matrix at load step m A'+' = P) + AA (2.155) The Arc Length Method (ALM) may easily be visualized for a single degree of freedom as shown in Figure 2.10. The displacements are updated as {x}(i+i) = {} 44 + {u} ' + Au (2.156) such that x (' + ' ) corresponds to the displacement at the (i + 1)th iteration of load step m. In the constant ALM, the radius of the arc at each load step is constant. It is clear from Figure 2.10 that the ALM is used in conjunction with the modified Newton-Raphson method, and may also be implemented with NR iteration schemes. The method allows one to obtain postbuckling response but bifurcation problems require modification that will seek out multiple paths after a limit point. 4. Convergence Criterion For a given load step, the iterations on displacements are carried out until a pre-set convergence is achieved. There are three convergence tests most commonly used, (a) Displacement Convergence, (b) Residual Force Convergence, and (c) Strain Energy Convergence. These criterion may be summarized as follows: {mul }T{AUI} - {gi}T{g} (AAI) 2 CtDISP < a.F. < aDJSPCaR.F. {p}T{p} f{Au}T{gi} A'1Ai{ul}{P} (2.157) - It may be noted that {Au'} is the incremental displacement at iZt iteration, {g'} is the residual force at it h iteration, AA'{P} is the incremental load at the first iteration, {Au'} is the incremental displacement at the first iteration, and a 's are the prescribed convergence parameters, usually in the order of 10-2 to 10- . It is further noted that the initial load {P} used to start the analysis is set arbitrarily and only the load parameter is modified automatically to go from zero load to the desired load level. 45 k=k LO kLi + NL .1U) I I P- D I I Figure~ 2.0 ~~ osatArIeghMto 46I W III. PROGRAM IMPLEMENTATION A. INTRODUCTION This chapter presents certain aspects of computer implementation of the ele- ments matrices developed in the previous chapter. As mentioned in Equation 2.25, the general problem to be solved is given by [K]{d} = {r} (3.1) = {R} (3.2) [K]iDI where equations 3.1 and 3.2 are the static equilibrium equations for the element and structural assemblage respectively. If the stiffness matrix, [K], is independent of displacements, the analysis reduces to solving a set of linear algebraic equations. In the case of the stiffness matrix being displacement independent, the structural behavior is nonlinear, and an incremental load analysis together with a suitable iterative methods has to be adopted. B. LINEAR ANALYSIS In the case of linear analysis, we assume small displacements and small strains, and the resulting force-displacement relations are solved only once. Using the displacement independent part in equation 2.61 to get the strain-displacement relations [BLo], as given by equation 2.80, equation 2.20 is used to form the element stiffness matrix. A series of Fortran subroutines was developed incorporating the element stiffness matrix for this element. The material characteristics may be either isotropic, laminate theory definitions, or anisotropic description. The subroutines are implemented in an existing computer program, FEMCOM, which is capable of 47 element assembly and subsequently calculation of the displacement solution for both linear and incremental load methods. C. NONLINEAR ANALYSIS In this research effort, geometric nonlinearities, namely, large displacements but small strains are considered. Consequently, the element consists of a linear displacement independent stiffness matrix, [KLo], and two other contributions. The first contribution is due to the linear displacement dependent stiffness matrix, [KLI], based on [BL1], as given in equation 2.81. The other contribution comes from nonlinear stiffness matrix, [KNL], as given by equation 2.95. Note that the stress-strain matrix, [E], may be used both for isotropic as well as composite materials using relations 2.131 and 2.132. D. SOLUTION PROCEDURE 1. Composite Material In order to generalize the procedure of implementing the solid element with composite materials, the plate built of solid elements may be stacked in all three directions. Figure 3.1 shows such a stack, where rows of elements are arranged in the thickness direction. For each finite element, the stress-strain matrix is computed in a subroutine separately, though it would be more efficient to compute it for the whole row of elements, taking into account the appropriate layers. In assigning a certain number of layers in each row, a constraint to be noted is that the total number of layers of all rows match the number of layers of the structure being modeled. 2. Linear Case As mentioned earlier, the matrices corresponding to the linear displace- ment independent part was coded into several subroutines and implemented into a 48 €4-1 ~Q) Q 0)) 0 U4 CUJ -Ir CeO o -. Figure 3.1: Thick composite plate-element arrangement 49 general purpose finite element program, FEMCOM. The program does automatic element assembly and yields solutions to prescribed loads. The flow chart shown in Figure 3.2 shows various steps that may be summarized as follows. 1. The material properties, model geometry, applied loads, integration scheme, boundary conditions and solution parameters are input. The material properties needed for isotropic material are Young's modulus and Poisson ratio. For composites, data needed includes the number of layers, rows of elements, fiber orientations, Young's moduli, shear modulus in three directions, and Poisson ratio. 2. Using the shape function derivatives, the coordinates transformation relations, Jacobian and the strain displacement relations are established. 3. Using the specified Gauss quadrature, the element stiffness matrix is formed in global coordinates. 4. The element global stiffness matrix is assembled. 5. Using Gauss elimination technique, the displacement vector is computed. 6. Stresses may be computed using equations 2.14 and 2.15. 3. Nonlinear Case In order to obtain nonlinear response, either for studying the extension- twist-flexure coupling or nonlinear buckling and post-buckling, the analysis procedure is termed the incremental load method, and a variation of Newton-type iteration is used. The element formulation, assembly and equation solving proceed as before, except that additional element stiffness contributions have to be taken into account. The assembly and solution to get displacements needs to be done as 50 E INPUT IR=I,,NHRON= H E ko:IDTED °d'. K D:K-" 'k FIG. :3.2; Flow chartlinear analysis Figure 3.2: Flow chart 51 - linear analysis frequently as the load steps increments and iterations continue, depending on the solution strategy selected. A typical flow chart is given in Figure 3.3. For a given load step, the incremental displacements are computed iteratively until the convergence criterion is satisfied. At that point, equilibrium is achieved and new incremental load is applied and a new tangent stiffness matrix is computed. The iterations continue until the new equilibrium position is obtained. In the modified Newton-Raphson method, the tangent stiffness matrix is kept constant for all iterations for a given load step, while, for the Newton-Raphson method, the stiffness matrix for the whole structure is formed at every iteration. By tracing the load-displacement path, critical points, characterizing buckling, and stable and unstable regions of post-buckling equilibrium states may be identified. 52 kLo kil kf B" ERCd. kHL B2DL1D LO 13' Ff ERdy I F 3.3 FIG.~ hat Nonlinear anaysi Figre3.: Lowcat-Nnieraayi RUL i-l<to53e IV. NUMERICAL EXAMPLES A. INTRODUCTION AND NOTATIONS In this chapter, selected numerical examples are used to evaluate element idiosynchrasies and demonstrate its application in solving critical structural components that use thick composites. Solutions obtained here are compared with available elasticity solutions or other numberical solutions. 1. Material Properties In all the examples to be discussed, the material characteristics used are as follows. For isotropic materials, E = 30 x 106 psi, v = 0.30 and for composite materials used for laminated plates, layer properties are given by E, = 40x106 psi E2 = 106 psi G 12 = G1 3 =0.6x 106 psi G23 = 0.5 x 106psi v = 0.25 An eight-layered symmetric laminate configuration using this material is selected. All the dimensions presented in this chapter are in inches. In the discussion on effects of numerical integration rules, L x M x N notation refers to the number of integration points in x, y, and z directions respectively. B. COLUMNS AND BARS Two simple cases have been selected as part of the element validation process. 54 1. Bars A bar clamped at one end and loaded at the other end with uniformly distributed traction (Figure 4.1a) was studied and compared to the theory of elasticity. In the numerical solution, work-equivalent loads were used. The dimensions of the bar are 10 in. x 1 in. x 1 in., and it is isotropic. The boundary conditions are given by U(0,h h) =V(0,h -h-\ = w 0, -h+) =0 (4.1) Figure 4.2 depicts the effects of reduced integration and mesh refinement on the maximum deflection. It is obvious that when the mesh is refined in the thickness direction, for instance, one element in each of x and y direction and two in z direction [1 x 1 x 2] mesh, provides a stiffer solution than for the [1 x 1 x 1] mesh. It may be noted that the full (F) and reduced (R) integration schemes converge to about 95% of the classical solution (See Appendix C). It may be noted that the classical elasticity solution does not account for transverse shear stresses. A reduced integration in the axial direction (2 x 3 x 3) gives the same results as the full integration. However, when reduced integration in the thickness directions is performed (3 x 2 x 2), the solution converges slowly. On using (2 x 2 x 3) integration scheme in the thickness direction for (12 x 1 x 1) mesh, spurious mode is observed. Table 4.1 summarizes the effects of various integration schemes and mesh sizes. 2. Beams The next example considered is a clamped, cantilever beam loaded at the free end by a shear load. Using the clamped boundary conditions, dimensions and material as the previous example, solutions using full and reduced (R) integration are compared with the elasticity solution in Figure 4.3. The comparisons also include the solution obtained using eight noded first order solid element of 'GIFTS' software. 55 (at "Nem) Fig. 4.1; Bar Sample problems Figure 4.1: Bar Sample Problems 56 Fig. 4.2; Clamped Bar Under Uniaxial Load w 130 . 120-... 1 ~9 1 ... . 0 ..... ....... F Ituegraion Rule 80-S -.-~-RR - -au- 70 . ..... R (2x 3 x3) (3x2x2) -F - Thickness Mesh Elmsticiiy 60............... 50 0 1 100 r 2C0 300 400 500 600 #d. o. f Figure 4.2: Clamped Bar Under Uniaxial Load 57 700 TABLE 4.1: EFFECTS OF REDUCED INTEGRATION AND REFINED MESH, ON THE MAXIMUM DEFLECTION OF CLAMPED ISOTROPIC CANTILEVER BAR UNDER UNIAXIAL LOAD # d.o.f. Mesh Integration Rule u-, 5.7 F(3x3x3) R (3 x 3 x 3) RR (2 x 2 x 3) 2-2x x 111 F R 3 = 3 x Ix 1 4 = 4x Ix 1 6 = 6 x 1x 1 12 = 12 x 1 x 1 99 165 219 327 651 129.56 99.43 99.84 RR 2 = 1 x 1 x2 97.49 98.17 115.98 F R 90.77 91.20 RR 91.83 F R 100.40 100.81 RR 112.64 F R 101.04 101.59 RR 111.39 F R 101.83 102.80 RR 110.74 F R 102.74 105.00 RR Umax 11,500.00 Uma AE Uelasticity pl 58 The reduced integration shows a better convergence than both the full integration and the first order solid element. It may be noted that the eight noded solid element converges more rapidly than the full integration scheme of the present element up to about 300 degrees of freedom (d.o.f.). It can be seen from Table 4.2 that mesh refinement in the thickness direction results in reduced performance and one element in the thickness direction consistently yields good results. In Figure 4.4, the effect of transverse shear deformation is studied for a 12 x 1 x 1 mesh using reduced integration scheme, and compared to the theory of elasticity solution (See Appendix C). The results are summarized in Table 4.3 versus the aspect () ratio. It is clear that for thin beams where the elasticity solution is adequate, the present element gives stiff solutions, whereas for thick beams (- < 10), better solutions are predicted. The reason for these effects may be attributed to the transverse shear stresses. In the case of thin bars, or beams, the element aspect ratio is very large and the parasitic shear strains appear at Gauss points, resulting in a phenomena called 'shear locking' [Cook, 1989, and Hughes, 1978]. When the beam is thick and the aspect ratio is of the order of 1/10, the transverse shear stresses start to become significant, whereas in the elasticity solution, they are taken into account only to a limited degree together with the restrictions of Saint-Venant's principle. C. CLAMPED PLATES An isotropic clamped plate of dimensions 20 in. x 20 in. x 1 in. under a central concentrated load is shown in Figure 4.5a. This problem is studied for mesh sensitivity and the effects of different integration rules. The present solution 59 Fig. 4.3; End Loaded Beam Bending 110 . . 100--- 90P-( 7 0 ...... ~Z E 50 ........ 40 ....... 30 -P~- 3 0---- ..... 0 1I (F) ...... "GIFTS" 8solid (F) ............. w 100 let Prcscni Elcrncni (R) 20 0 Prcscti 200 300 _E lasticit 400 500 #d.o.f Figure 4.3: End Loaded Beam Bending 60 600 700 TABLE 4.2: EFFECTS OF REDUCED INTEGRATION AND MESH CONFIGURATION ON THE MAXIMUM DEFLECTION OF CLAMPED ISOTROPIC CANTILEVER BEAM LOADED AT THE END Mesh * # d.o.f. (27 solid) f --Wma F p1 27 solid R 100 8 solid t F I = 1x I x 1 57 5.7 26.7 9.4 3= 3xI x 1 165 37.0 54.0 47.2 4 = 4x I x 1 219 56.0 70.3 60.5 9 = 9 x 1x 1 489 91.3 95.4 85.8 12 = 12 x 1 x 1 651 95.0 97.3 89.8 3 = I x 1 x3 147 5.5 4=2xl xl 135 16.9 6 = 2 x 1x 3 273 17.0 6 =3x 1x2 285 36.0 9=3xI x3 399 39.1 w~eaotirit = 101.0 F: Full integration e 3 x 3 x 3 for 27 solid * 2x2x2for8solid R: Reduced integration * 2 x 3 x 3 for 27 solid f 8 solid is generated in "GIFTS". * Mesh configuration for 8 solid is twice of 27 solid in each direction, i.e., 2 x 1 x 3 for 27 solid is 4 x 2 x 6 for 8 solid. 61 TABLE 4.3: CENTER DEFLECTION VS. ASPECT RATIO (i) OF AN ISOTROPIC CANTILEVER CLAMPED BEAM LOADED AT ONE END ___ We, 124.38 110.07 .5 103.90 99.49 10 100.98 97.07 50 100.04 53.43 100 100.01 17.85 2 h, lh) 3E1 2 w1I, w P 12 w(1, h, jh) Eh10 w +e 3 [+(1 See Appendix C. 62 +V) h] 102 Fig 4.4; End Loaded Beam Deflection 130j 120 .... 2 100-................80...............................s... 70 S 0-- -4 Elasticity *- Precnt Elemcmi (R)..............V 20 ............................ 10 111 Lenght/Thickness [L/h] Figure 4.4: End Loaded Beam Deflection 63 is compared with the elasticity solution (See Appendix C for details on elasticity solutions). Invoking symmetry conditions of the problem, a quarter of the plate is modeled with the following boundary conditions imposed: = *,Y,±u (X, , ±)2 V 1Y ) V(X, 0 ± W0 ,±h = (X, a, (4.2) =02 (4.3) and the symmetry conditions. u (, y, z) =v (X, a, z) =0 (4.4) The load was taken as one quarter of the total load. Figure 4.6 shows the comparison of a mesh composed of elements arranged in one row of elements (N x N x 1) vs. a mesh of the type (2 x 2 x M), composed of M rows of elements arranged in the thickness direction with 2 x 2 elements in each row. Full integration is employed in the computations. Table 4.4 summarizes the resultant deflection and mesh sizes. It is clea- from this and the previous examples that one row of elements in the thickness direction is adequate to predict the response of the structures. Figure 4.7 presents the convergence characteristics of three integration schemes. It may be noted that reduced integration (3 x 3 x 2) in the thickness direction yields very close results to that of the full integration scheme. Reduced integration produces good results by compensating for the estimation of finite element approximation. The in-plane reduced integration scheme (3 x 2 x 2) shows divergence in the computed response. It may be mentioned that using one element to model quarter plate resulted in much higher deflection than expected. This implies that a one element model contains spurious modes and a one-element modeling of plate/shell problem should be avoided. On examining the convergence plot, with less than 600 d.o.f., the 64 CC >C' Fig. 4. 5; Plate Sample problems Figure 4.5: Plate Sample Problems 65 TABLE 4.4: MESH COMPARISON OF AN ALL EDGES CLAMPED RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD QP =1000 lb.) # d.o.f. w 1= 1 x 1 x 1 37 13.8529 4 = 2x 2x1 145 4.2230 9 = 3 x 3x 1 325 5.0690 16 = 4 x 4 x 1 577 5.6592 8 = 2 x 2x 2 275 4.0133 12 = 2 x 2 x 3 405 3.9955 16 = 2 x 2 x 4 535 3.9597 18 = 3 x 3 x 2 591 5.0647 Mesh h) Eh 3 w W 2 2 2 pa 100 evaluated deflection is within 90% of the elasticity solution. Table 4.5 summarizes the effects of various integration schemes and mesh sizes. D. SIMPLY SUPPORTED PLATES Bending of a simply supported rectangular plate under uniformly distributed force is presented herein. (See Figure 4.5b) Both isotropic and laminated plates are investigated using one quarter of the plate, as discussed previously. The results are 66 Fig. 4.6; Clamped Plate Under Central Load 7.06.5-...................... 4 .0-- - --- - - - -- - - - --- - - - .5 35............. 30. . ........... . ......... ... .... ... .... ... ..... .... ..-... .. .. .. . .......... 2 x . ...... . . . . . 2.5-CT 2.0 0 1 i 100 200 300 400 500 #d.o.f Figure 4.6: Clamped Plate Under Central Load 67 600 Fig. 4.7; Clamped Plate Integration Rules 9.08.5 ..... .......... 8 .0 .-.... ~ * .- .................... 7.0 -............... . . . . . . .. . .. . . . . . . . . . . . . . . . S 6.5-.................................... 6.0 5 .5 -. (F).3...3...3 ...... 2.5.......... 2.00.5 Type I1 100 200 30 40 #do RR 2f 00 Figure 4.7 Clamped.. ..... Plate..tegratin.Rule 3 .0 .............. c68 60 TABLE 4.5: INTEGRATION RULES COMPARISON OF AN ALL EDGES CLAMPED RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD QP = 1000) Mesh #d.o.f. w for various integration rules (3x3x3) (3x3x2) (2x2x3) 4= 2 x2 x 1 145 4.2230 4.2575 6.9918 9= 3x 3 x 1 325 5.0690 5.0853 7.6576 16 = 4 x 4 x 1 577 5.6592 5.6763 8.1128 w w(~ = S, 2 h) Eh3 2 Pa 69 100 compared with Classical Plate Theory (CPT) as given in Appendix C and higher order shear-deformation (HSDT) plate theories [Reddy, 1985, Lo et al., 1977]. The following boundary conditions are imposed, W (0, Y, ±-. = W(X, 0, )= 0 (4.5) and the symmetry conditions are as given in equation 4.4. Reduced integration (3 x 3 x 2) is adapted throughout all the computations presented in this section. The convergence characteristics of the element and comparison to CPT is depicted in Figures 4.8 and 4.9 for both isotropic and laminated plates. In the isotropic case, the present element shows convergence within 90% of the elasticity solution for less than 200 d.o.f. In the case of laminated plates, (Figure 4.9), the classical .;olution [Vinson, 1987] gives a more flexible solution than the present element. Table 4.6 summarizes the deflections of the isotropic and laminated plates and mesh sizes. It may be noted that the classical solution uses laminate theory, which neglects the transverse shear stresses, and hence the contribution of these stresses is not taken into account. This assumes more significance for thick plates (a < 10 to 15). Furthermore, when the plate stiffness in the thickness direction is significantly lower than its stiffness in the in-plane direction and when the shear modulus in the thickness direction is significant, the classical laminate theory does not predict the response of the structure accurately. In the present example, a = 20 and - = 40, G1 3 - G12 . It may be concluded that using laminate theory for bending of thick plates yields a nonconservative estimate of deflections and special attention should be given to the stiffness ratio [] and shear modulus ratio [-] in determining the of such plates. In Figures 4.10 and 4.11, the maximum deflection is presented for different aspect ratios, (S), of the plate. Both isotropic and laminated plates are analyzed and compared to CPT. In addition, the solution of the laminated plate is 70 also compared -o Higher Order Shear Deformation Theory [Reddy, 1985], as shown in Table 4.7. As in the beam bending c; e, shear locking is observed for thin isotropic plates. For thick plates, (say, = 4), the computed deflections become significantly larger than predicted by CPT, as expected. Examining Figure 4.11, it may be deduced that even the HSDT [Reddy, 1985] underpredicts the deflections. For thin laminated plates, the shear locking effect is not as significant as observed for isotropic plates. This may be attributed to the fact the laminated plate has more flexible transverse material stiffness in bending than the coefficients than the isotrop,i': plate, so that shear locking is expected to develop only for thin isotropic plates. 71 TABLE 4.6: ALL EDGES SIMPLY SUPPORTED RECTANGULAR PLATE, UNDER UNIFORMLY DISTRIBUTED LOAD # d.o.f. Mesh Isotropic w Composite w 4 = 2 x 2x 1 186 3.2826 0.3871 9 = 3 x 3 x1 386 3.6335 0.3878 16 = 4 x 4 x 1 658 4.0541 0.4053 4.4335 *0.3634 CPT * Neglects G1 3 and G 23 . See Appendix C. The uniformly distributed load is taken as the total consistent load over the area of the quarter plate. q F = (!2 The maximum deflection is taken at upper surface. w (z, 2, h) E2 h3 100 4 w = q a 72 TABLE 4.7: CENTER DEFLECTION VS. ASPECT RATIO OF SIMPLY SUPPORTED RECTANGULAR PLATE UNDER UNIFORMLY DISTRIBUTED LOAD Isotropic Orthotropic Reference* w w w 4 9.8275 5.1324 1.6340 10 4.9581 0.8221 0.5904 20 4.0541 0.4053 0.4336 100 1.0902 0.2406 0.3769 CPT 4.4335 0.3634 *Reference: Reddy, 1985. 73 Fig,. 4.8-, Simply SUlplpiwdt Isoltitpic P1h11C 3 41 ' ... . . ... . .. . . :.. =2 -IV or I(H4) 2004 4044 30(1 6004 5400 700 (1-.... o -15 . ....... Simply Supported Isotropic Plate . . Figure 4.8: Fig. 4.9; Sinmply Suppotrted Lanmi mlcd Iluc o050 -- - - - -- .. ------ ------- ----------------------------....... --- LLJ 0.35 . .... . ..... u.. a .3 5 ... : 0.25 0.20o 0 ... . .. ... .. ... .... ... . 100 200 3(o( 40) 5 600 Skitlhllickuic>s ku/i Figure 4.9: Simply Supported Laminated Plate 74 700 Inii!. 4. 10 s();( )I ic Pla~te Delflcclions vs. Aspect Ratio 10 9 0 .~ 7 . 2 -,-- ~ !r....... ..... -- - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - Side/Ilickness (;Aj Figure 4.10: Isotropic Plate Deflections vs. Aspect Ratio Fig. 4.11; Laminated H'awc Dclcctions vs. Aspect Ratio 4 ........ i . ....... -0.- Row)I i is i i 01 I o" Io, 102 Figure 4.11: Laminated Plate Deflection vs. Aspect Ratio 75 V. CONCLUSIONS AND SCOPE FOR FUTURE RESEARCH A. CONCLUSIONS This study suggests a three-dimensional higher-order finite element to be in- corporated in the analysis of thick plates composed of both isotropic and laminated composites. By using a tri-quadratic Lagrangian twenty seven noded solid element, no assumptions on transverse shear strains are introduced in the formulation. The formulation, based on the principle of virtual work, is presented for both linear and nonlinear analysis. The material constitutive relations for linear isotropic and composite materials are presented. For composites, both laminate theory and three dimensional anisotropic adaptations are described. Several numerical examples using linear analysis are given for bars/beams and plates using both isotropic and composite materials. Three dimensional anisotropic relations are adapted for composites. The results show that the present element is effective for analysis of thick beams and plates, but exhibits shear locking for thin beam and plates. Spurious modes are revealed for single element usage in plate modeling, as is the case for some other finite elements. Reduced Integration in the thickness direction for beams and plates gives satisfactory results. An interesting outcome is that one element is sufficient to capture transverse deformation for thick laminated structures and mesh refinement in the other two directions yields convergent solutions. 76 B. SCOPE FOR FUTURE RESEARCH More numerical experiments need to be performed to compare the present soution to closed-form solutions [Pagano, 1969] to evaluate the efficacy of this element. Implementation of buckling analysis using the nonlinear element matrices presented herein is another task that may prove useful in predicting buckling response of thick composite cylinders subject to external pressure. By incorporating the centrifugal force in the external virtual work done by body forces, this element may be used in modeling rotor blades. 77 APPENDIX A Shape Functions and Derivatives for Solid Element Shape Functions for Solid Element Mid-edge nodes: Mid-plane nodes: N2 = N4 = N6 =-I r(1+r)(1-s 2) t (I+t) (1 - r 2) s (1 + a) t (1 + t) r(1- r) (I1-s 2) t ( I + t ) N gs-- (1 - r ) s ( 1 - s) t ( N io0= - _r(1+ r) 8(1- I+ t) ) (1 - t ) r (1 + r) s(1+s) (1- t2 ) N 14 = -- r (1 - r) s (1 + s) (1 - t 2 ) Nis = r (1l- r) s(1 -s) (I - t") N2o=--r(l+r)(1-s 2) t(I - t) N2 2 = -1 (1 - r 2 ) 8 (1 + s) t (1- t) N12 = N 24 = ,- (I - r) (1 - N 26 = r(- 2) t (I - t) N 9 = .1(1 -r 2 )( N 2 7 - (1- r 2) Nil= r (1 +r) N 15 (I _r) N13 (I- r ) 1 -- s 2) t(l+t) (1- 22)) t( 1(I-- t 2t)) (1- s _ (1-_8 ( + 8 1 ) ( 1 - t 2) ) (1 - t ) r2) s (1- s) (1- t 2 ) N17 Center node: N 13 r 2 ) (I _ 82) (1 _ t 2 ) 2 r ) s(-s)t(1-t) Corner nodes: N, = 1 81 (1+,-)(1 -,s) N= N 1 +1)( 1 (I+t) - 2(N2 +#Ns+ ) 1+t 8 = ( N7 =-= (1 + r) (1-s) (1 - s) (1 (I - t)t) (1-r) N=-- + N4 + N12) - 1 (Nil + N13 + N9) - r) ( ) (1 + t)- (+ + )+ 1+ (No + N6+ N 1 ) - -4(t+NTN) (N 5 + N 1 7+ (N16+NsN 1 N) ~ I 1 tA'1 Nl N25 = N25 = (1- r)(1-a)(-t)-(N24+N26++N16)-1(Ns+N17+N27)- + ) (1- f-- -(N22 + N 24 (2+2+1) 78 4 + N 14 ) - -(N 1 3 +N 4 NsNsN Nis 1 82 I -(I 8+ (-)l = - (I2 + r)(1 + s) (1 1- t) - 1 (N2o + N22 + N12) - 1I (Ni I + N13 + N27) 1 .- N 23 1~ 111,, 1 N N21 (N2 Nio) - 4 (Nl +.N,7 + N9) - j#1, 8 1 + N 27 )- -NI - 8 I Nis Shape Function Derivatives for Solid Element Mid-plane nodes: Mid-edge nodes: N2' N 4 ,, = t (1+ 2r) (1 -s 2 )(1 +t) -1rst (1 +s8) (1 + t) Ng,, ( Ns 61' N2 0,, = t N 20,, = - N22,, = 24,= N 26, = Nl,, N3,r = I(1- N7r N 2 3,, = a(1 2r) + N 25 ,, = 2 _ 2 2 r( de(:-t Cnte (2 _ -1 +1N_ +) N,,) 4 ,+N 2 (Ns, +)1 7t2) , + -( ~( 1 , , - (N2,, + N8,, + N 1 6,,) - a) (1-+i) - (N 2 ,, + N, + N 1 ,) -1(Nl,, - (N4,, + N, + N 1 ,) (N6,, + N,3, + N 1 r)4 7 - + a) (1 + t) S ( 1+)( - 20r+N6 ~~ ~(1+8)( 81(1 '1 )-(2, , 9 , + N 1 ,, + N ,)7 + N, N1, (N1 3 ,, +IO)-1 sr - j(N3,, + N 1 ,, + N ,,) 7 - 1, - Nr - + N)- N,3. 8 N18,r N15,, +27,r) 1 (N 2 0,, + N 2 2,, + N 1 2,,) -t 1 ) 1 (Nil,, + N 1 7 ,, + N9,,) 1 N21,r= )(1 _t2 ) S2) (I t 2) 8) (1 + t) 1 -) 2 (1 - 2r) (1 Cntr node:+s)(1_2 _ t) I(1+a (1 + 3) (1- t)N, -(1 ) (1 +t) s2) (It) (+ 2r) (I_ = (1 s) (1 - t2) sa (1 + ar)(1 _ t)( t (1 + 2r) (1 - 82) (I _I1t ((I = 2r) (1 + 0) - (-(I -Irst (1 + s) (1 - t) = N 1,, - &(I + 2r) (1 N212,, = N ( Ist = Nil:: t 2 (1 -8 =-rt N 27 ,r = N8 ,r = _t (- 2r) (I-s2 )(I+t) N8, r direction - 24 1,)-1 2, - (N1 ,r + N1 ,, + N 27 ,r) N5,+N7,+N7, ,8 79 - -Nis, ,~ Shape Function Derivatives for Solid Element - 8 direction Mid-plane nodes: Mid-edge nodes: N 2 ,. - -Irst (1 + r) (1 + t) Ng,, rsi (1 - r) (1 + t) Nil, t(1-r2) (1 - 2 s ) (1 + t) r (1 + r) (1- 2s) (1- t 2 ) Ni 5,, N6,. = N, = N l o ,, = - lr(I+r)(1+2s)(I-t2) N 12 ,. = N 14 ,, = - r (1 - r) (1 + 2s) (1 - t 2 )) N 16,.- r (1 -r) (1-2s) = (1- t - -st (1 - r 2 ) (1 +t) r) (1-_t2 ) -rs (I+ rs (1- r)(1-t 1(l-r N 1 3 ,, = N 1 7, -2(1 -r 2 ) ) 2s ) (1 - t 2 ) 2 )(1-2s)(1-t 2 ) 2 (1 + t rst (1 + r) (1 - t) N 20 ,, = N 22 . = -It (1 - r2) (1 + 28) (1 - t) N 24 ', = - t rst (I - r) (I - t) t(1-r2) (1 - 2s) (1 - t) Ns,t= Center node: t2) Nis,, = -2s(I - r2 ) (1 Corner nodes: 1 (1 N 1 ,, = N 3 ,. = (1 + r) (1 + t) - = N7 ,, = N 9,s = -I (1- r) ( + t) - (N 6,. + Ns,. + N 16,.) - = I N 23 ,. N 2 5,& = - I r) ( - (N 1 3 ,. + NI5 ,. + N 9 ,.) - N 1 3 ,. N, 1 11 Nis, + N 17 ,. + N 9 ,) I (Ni,, + N 17 ,. + N 2 7 ,.) Ns, - Nls 11 (I + r) (1 - t) - I (N 2 0 ,. + N 22 ,. + N 1 2 ,.) r) (I - t) ( -) - I (Nis, 1 - 1 - (N 2o,, + N 26 ,. + N 1o,.) - + r) (1 - t) - + Ng,,) 4 (Nil,. + 1 (N 4 ,, + N 6,. + N 14 ,.) 11 -I(I - (I - r) (1 + t) - 1 N 2 1,, (N 2 ,, + N4,. + N 1 2 ,.) 1 1 N 5 ,. (N 11 ,, + N 17 ,, + Ng,,) - (N 2 ,. + N,,. + N 1o,.) - - 1 1 _1 ,(+ (1 + r) (1 + t) - I (N 22 ,. + N 24 ,. + N 14 ,.) 11r (N 24,. + N 26 ,. + N16 .,) 80 (Nil, + - N83,. + N27,.) - (N 1 3,. + N 1s,. + N 27 ,,) 11 I (N 5 ,. + N1 7 ,. + N 27 ,,) Nis, - NIs,. - Nl,. Shape Function Derivatives for Solid Element 8 (1 -r2) (I+ 8)(1 +2t) .... r (1 - r) (I - 82) (1+ 2t) = = N16 ,t N20 ,, = = N 24 ,: = 1r N26,s,= (I-r) fs(l- N1 , 1 Irst (lI- r) (1+ 8) -Irst (1 - r) (I - s) -Ir (I + r) (1 - s2) (1 - 2t) 2 ) (1 (I- r 2 ) (I1_ )(-) -2t) - 1(1 -r N2 , ,-) ) ru=I (1 Ns N 14.9= t direction Mid-plane nodes: Mid-edge nodes: N4,, N6,, - = 2 ) (1-_a 2 )(1 -2t) -ri (I +r) (1 - 82) 2 ,=s(- (+) 2 Center node: Nlg,t = -2t( - r 2 ) (I - 82 ) Corner nodes: I (N 2 ,9 + Nt3,, + N1 0 ,t) - 1 (Nil,, + N 17 ,, + N 9 ,,) N1 ,, = I (I1+ r) (1 -8s)- N 3.8 = I(1+7r) (1 +8) - I(N 2 ,t + N4,, + N12,,) - I(N 1 1 ,, N5,,t = I (I- I (N4,t + N6,, + N 1 4,0 - (N 1 3,t N7. , = - r) (I- + + r) (1 - ) - I (N2 ,, + N 2 ,, + N1 ,) 8) - N1 = N 25 ,9 = )-I(~~ -~ (1-+r) (I1+,a) - - ( (1 a) - - - 2, 81 i6,t) N, it - 8 4 1)(Nlt+N7t+N70-1N i~) I (N 2 2 ,t + N 2 2,, + N12.0) -~ + N 26 ,, +- 8t -I + N15,, + N,) - (N,, + N 17 ,, + N ,)7 - 2 8 - 1 + Nigt= + N13 ,, + N9 ,,) N,, - - (Nil,, +~N 1 3,, + N 2 7 ,0) I (N, 5,, + N 17 ,, + N 27 ,,) - , st lt APPENDIX B Jacobian Matrix Jacobian matrix elements: 27 Jl-=x, Ni, xi "27 y, J = E N,,, y j=1 27 'Ni,,z i J13--Z,r j=1 27 x,. J1= = E Ni,. x, j=1 27 J2 = Y,. = 1: N,,, Y/ j=1 27 J23 = z,s = 1_ Nis zi j= 1 27 J3 1 = X,t = E Ni,t x, j=1 27 J3 2 = y,t = E Ni,t yi j=1 27 = Zt J3 = E Ni,t z, j=1 Elements of the inverse Jacobian matrix: = (J22 J3 - J2 J32) (J13 J32 - J1 2 J33) r,2 = r21 = (J12J23 - J1 3 J22 ) (23 8 J31 -2J2 J33) 82 "722 = r23 = r3l = r'32 = Pl IJ33 - J 3 J 3 1) j(J 1 J 2 - J1J23) ~(J 2 1 J 3 2 -J j(J 2 1 (Al F3= 3 22 J3 1) J3 -J11J32) J22 -J 2 1 J12 ) Jacobian matrix determinant: J det [J] J1 J(J 22 J33 -J 23 J32 ) -J + 12 (J21 J33 -J 23 J3 1) J13 (J 21 J32 - J22 J131) 83 APPENDIX C Theories A. THEORY OF ELASTICITY SOLUTIONS 1. Cantilevered bar under traction PL AE " P = Total load " L = Bar length * A = Cross section area " E = Young modulus 2. Cantilevered Beam under end load PL 3 - +_____ 3EI 21G 3[ +(l+V) Reference: Timoshenko, 1951. B. CLASSICAL PLATE THEORY (CPT) 1. All edges clamped rectangular isotropic plate under central load Pa 2 D D a Eh= 12 (1- V2) = 0.00560 for v = 0.3 Reference: Timoshenko, 1959. 84 2. All edges simply-supported, rectangular plate under uniformly distributed load -qa . a . 4 00406 for v =0.3 =0 omposite x =W ,, qa 8 I H r8.0n m=1,3,5... n=1,3,5... D = 4 2 D m 4 + 2 (D, 2 + 2D 6 ) (ma) + D 22 n 85 TABLE C-i SAMPLE COMPOSITE MATERIAL DATA Table C-1; Sam~ple Composite Material Data INPUT DATA; LAMINA; 8 7 6 5 4 3 2 1 THNES ;0.12500 ;0.12500 ;0.12500 ;0.12500 ;0.12500 ;0.12500 ;0.12500 0.12500 ; THETA ; 0.0 ; ; ;45.0 ;-45.0 E2 El 0.40000E+08 0.40000E+08 ;0.40000E+08 ;0.40000E+08 ;0.10000E+07 ;0.10000E+07 ;0.10000E+07 ;0.10000E+07 ;0.10000E+07 ;90.0 ;90.0 ;0.40000E+08 ;-45.0 ;0.40000E+08 ;0.10000E+07 ;45.0 ;0.40000E+08 ;0.10000E+07 ; 0.0 ;0.40000E+08 ;0.lOOOOE+07 ;Viz2 G12 ;0.25 ;0.25 ;0.60000E+06 ;0.25 ;0.60000E+06 ;0.60000E+06 ;0.60000E+06 ;0.60000E+06 ;0.25 ;0.60000E+06 ;0.25 ;0.60000E+06 ;0.25 ;0.60000E+06 ;0.25 ;0.25 OUTPUT DATA; 0.12773E+08 0.55940E+07-0.81226E+06 0.55940E+07 0.17603E+08-0.30995E+07 -0.8l226E.06-0.30995E+07 0.59436E+07 0.OOOOOE+00-0.27344E-01 0.OOOOOE+00 -0.27344E-01-0.16406E+00 0.OOOOOE+00 0.OOOOOE+00 0.OOOOOE+00-0.62500E-01 D( i, j )-MATRIX 0.20743E+07 0.28699E+06 0.72461E+05 0.28699E+06 0.81544E+06 0.17998E+06 0.72461E+05 0.17998E+06 0.31612E+06 Note; A,B and D matrices are evaluated --- contribution i.e. G13-G23-0 ,neglecting Transverse Shear Usinq Navier's solution with n-m-200, i.e. 100 terms for each direction, as given in the above, we have, Wmax - 0.052328 3 Wmax*E *h 2 *100 -0.3634 q*a Where q-90 ; a=20 86 LIST OF REFERENCES 1. Allen, D. H. and Haisler, W. E., Introduction to Aerospace Structural Analysis, John Wiley & Sons, Inc., 1985. 2. Arnold, R. R., and Mayers, J., Buckling, Postbuckling, and Crippling of Materially Nonlinear Laminated Composite Plates, Stanford University, 1983. 3. Bathe, K. J., Finite Element Procedures in Engineering Analysis, PrenticeHall, Inc., 1989. 4. CASA/GIFTS, Inc., Computer Aided Structural Analysis/GraphicalInteractive Finite Element Total System - Users Reference and Primer Manuals, 1987. 5. Cook, M. P., Concepts and Applications of Finite Element Analysis, 3rd ed., John Wiley & Sons, Inc., 1989. 6. Eisley, J. G., Mechanics of Elastic Structures, Prentice-Hall, Inc., 1989. 7. Ford, B. W. R. and Stiemer, S. F., "Improved Arc Length Orthogonality Methods For Nonlinear Finite Element Analysis," Journal of Computers J Structures, Vol. 27, No. 5, pp. 345-353, 1987. 8. Gajbir, S. and Rao, S. Y. V. M., "Buckling of Composite Plates Using Simple Shear Flexible Finite Elements," Composite Structures, Vol. 11., #4, pp. 293-308, 1989. 9. Hoskin, B. C. and Baker, A. A., Composite Materialsfor Aircraft Structures, AIAA, 1986. 10. Hughes, T. J. R. and Cohen, M., "The 'Hetrosis' Finite Element For Plate Bending," Journal of Computers & Structures, Vol. 9, pp. 445-450, 1978. 11. Kolar, R. and Karmel, H. A., "On Some Efficient Solution Algorithms for Nonlinear Finite Element Analysis," International Conference on Nonlinear Mechanics, Trondheim, Norway, 1985. 12. Lo, K. H., Christensen, R. M. and Wu, E. M., "A High-Order Theory of Plate Deformation, Part 1: Homogeneous Plates," Journal of Applied Mechanics, Vol. 44, pp. 662-668, 1977. 13. Lo, K. H., Christensen, R. M. and Wu, E. M., "A High-Order Theory of Plate Deformation, Part 2: Laminated Plates," Journalof Applied Mechanics, Vol. 44, pp. 669-676, 1977. 87 14. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Inc., 1969. 15. Natsiavas, S., Babcock, C. D., and Knauss, W. G., Postbuckling Delamination of a Stiffened Composite Panel Using Finite Element Methods, NASACR-182803, August 1987. 16. Pagano, N. J., "Exact Solutions for Composites, Laminates in Cylindrical Bending," Journal of Composite Materials, Vol. 3, pp. 398-411, 1969. 17. Ramm, E. and Stegmuller, H., The Displacement Finite Element Method in Nonlinear Buckling Analysis of Shells, Springer, Berlin, Heidelberg, New York, 1982. 18. Reddy, J. N., "A Refined Nonlinear Theory of Plates With Transverse Shear Deformation," InternationalJournalSolids and Structures, Vol. 20, pp. 293301, 1984. 19. Reddy, J. N. and Chandrashekhara, K., Nonlinear Analysis of Laminated Shells Including Transverse Shear Strains, American Institute of Aeronautics and Astronautics, Inc., 1984. 20. Reddy, J. N. and Phan, N. D., "Analysis of Laminated Composite Plates Using A Higher-Order Shear Deformation Theory," International Journal For Numerical Methods in Engineering, Vol. 21, pp. 78-91, 1985. 21. Reddy, J. N., "A Generalization of Two-Dimensional Theories of Laminated Composite Plates," Communication in Applied Numerical Methods, Vol. 3, pp. 212-220, 1987. 22. Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw-Hill, Co., 1951. 23. Timoshenko, S. P. and Krieger, S. W., Theory of Plates and Shells, 2nd ed., McGraw-Hill, Co., 1959. 24. Tsai, S. W. and Pagano, N. J., "Invariant Properties of Composite Materiais," Composite Materials Workshop, Tsai, S. W. et al., eds., Techonomic Publishing Co., Stanford, CT, pp. 233-253, 1968. 25. Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials, Martinus Nighoff, 1987. 26. Yang, T. Y., Finite Element Structural Analysis, Prentice-Hall, Inc., 1986. 27. Zienkiewicz, 0. C., Talor, R. L. and Too, J. M., "Reduced Integration Technique in General Analysis of Plates and Shells," International Journal for Numerical Methods in Engineering, Vol. 3, pp. 575-586, 1971. 88 INITIAL DISTRIBUTION LIST No. of Copies 2 1. Defense Technical Information Center Cameron Station Alexandria, VA 22304-6145 2. Library, Code 52 Naval Postgraduate School Monterey, CA 93943-5002 2 3. Superintendent Naval Postgraduate School Chairman, Code AA Department of Aeronautics and Astronautics Monterey, CA 93943 1 4. Superintendent Naval Postgraduate School Attn: Dr. Ramesh Kolar Code AA/Kj Monterey, CA 93943 6 5. Dr. Rembert M. Jones Code 1823 David Taylor Research Center Bethesda, Maryland 20084 1 6. Dr. Raymond Kvaternik Rotorcraft Structural Dynamics NASA Langley Research Center Hampton, VA 23665 1 7. Captain Alon Yair Yerushalaim St. 44/B Apartment #5 BAT-YAM 59392 ISRAEL 2 89

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