Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1992 Optimization techniques for contact stress analysis. McDonald, Eric S. Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/23999 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY CA 93943-5101 . Approved for public release; distribution Optimization Techniques for is unlimited. Contact Stress Analysis by Eric S. McDonald Lieutenant, United States Navy Merchant Marine Academy, 1986 B.S., U.S. Submitted in partial fulfillment of the requirements for the degrees of MASTER OF SCIENCE IN MECHANICAL ENGINEERING and MECHANICAL ENGINEER from the NAVAL POSTGRADUATE SCHOOL December, 1992 ^ .-n i-**, Richard S. Elster Dean of Instruction nO Unclassified secuntv classification of this page REPORT DOCUMENTATION PAGE lb Restrictive Markings Report Security Classification Unclassified la 2a Secuntv Classification Authority Distribution Availability of Report 3 Approved 2b Declassification Downgrading Schedule 4 Performing Organization Report Numberis) Name r-a Naval Postgraduate School nc Address Name CA applicable ' Name of Monitoring Organization Naval Postgraduate School 34 and ZIP codei 93943-5000 7b Address Sb Office Symbol i city, state, i and ZIP code i city, if applicable 9 state, CA Monterey. Funding Sponsoring Organization of 8c Address (/ TakunJuJe 12 Personal Authorise Eric S. 1 w,nr:r>- dassi/i Procurement Instrument 10 Source of Funding Numbers 13b Time Covered 1 " 19 The views expressed IS Subject Group Abstract i No Task No Work I nit Accessioi Date of Report (year, month, day ) ! December. 1992 in this thesis are 5 Page Count 90 those of the author and do not reflect the official policy or Department of Defense or the U.S. Government. Cosati Codes Field 14 To From Supplementary Notation Project McDonald Master's Thesis Id Number OPTIMIZATION TECHNIQUES FOR CONTACT STRESS ANALYSIS , 3a Type of Report sition oi the Identification < 1 aucn and ZIP code) 93943-5000 Program Element No ii unlimited 7a city, state, i Monterey. 8a : is Monitoring Organization Report Number's) 5 no Office Symbol of Performing Organization tor public release; distribution Subgroup continue on reverse if Contact Terms i continue on reverse if necessary and Identify by block number) Stress, Optimization. necessary and identify by block number) The analysis of stresses induced by contact between two bodies is inherently difficult because the size of the contact z is unknown and constantly changing throughout loading. To overcome these difficulties, two approximation methods h been developed to determine the magnitude of contact stresses using the Rayleigh-Ritz method and the finite element met! Numerical optimization methods are employed to solve the contact problem. The solution techniques are comparec known analytical solutions and shown to yield accurate results. An application oi' this approach to solving the con problem is illustrated by examining the response of a clamped sandwich composite beam to low velocity impact. It was fcv maximum shear stress is insensitive to lamina thickness, however an increase in the contact layer thickness resu reduction in interfaeial shear stress. In addition, it was noted that a nonlinear bending stress distribution in the con layer intensified as the thickness of this layer increased. This phenomenon was found to be localized to the region of cont Finally, it was found that the compressive transverse normal stresses increased as the thickness of the contact lamina increa that the in a 21 Abstract Security Classification 2U Distribution Availability of Abstract 53 unclassified unlimited 22a Name D same as report D DTIC Unclassified users 22b Telephone of Responsible Individual YAV. Kwon 3D FORM 1473,34 i Include Area code (408) 646-2033 MAR 83 APR edition may be used until exhausted > 22c Office Svmbol 54Ss secuntv classification of this All other editions are obsolete Unclass ABSTRACT The because the to size of the contact zone To overcome loading. oped and shown The solution techniques beam to is illustrated constantly changing throughout methods have been devel- An are compared to known analytical solutions application of this approach to solving the It was found that the maximum shear however an increase reduction in interfacial shear stress. in the In addition, This phenomenon was found stress it was noted of the contact lamina increased. in in- that a nonlinear of this layer to be localized to the region of contact. was found that the compressive transverse normal is contact layer thickness resulted stress distribution in the contact layer intensified as the thickness increased. to solve the by examining the response of a clamped sandwich com- low velocity impact. sensitive to lamina thickness, it inherently difficult Numerical optimization methods are employed to yield accurate results. contact problem bending unknown and is determine the magnitude of contact stresses using the Rayleigh-Ritz method and contact problem. in a is these difficulties, two approximation the finite element method. posite two bodies analysis of stresses induced by contact between Finally, stresses increased as the thickness TABLE OF CONTENTS INTRODUCTION MOTIVATION A. B. LITERATURE SURVEY I. II. 1 I 1 FORMULATION OF THE CONTACT PROBLEM A. PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY B. CONTACT PROBLEM DESCRIPTION C. NUMERICAL OPTIMIZATION V. 8 2. Augmented Lagrange 3. Optimizer and One-Dimensional Search 14 4. Convergence 15 8 Multiplier Method 13 16 1. Background 2. Application of the Rayleigh-Ritz 3. Trial Function Selection FINITE 16 16 Method to the Contact Problem 17 17 ELEMENT APPROACH 22 1. Total Potential Derivation 22 2. Optimization and Static Condensation 26 RESULTS AND DISCUSSION A. PROCEDURE VALIDATIONS B. 6 Optimization Fundamentals APPROXIMATE SOLUTION TECHNIQUES RAYLEIGH-RITZ APPROACH A. IV. 5 1. III. B. 5 1. Rayleigh-Ritz 2. Finite Method 30 30 30 Results Element Method Results a. Two b. Roller-Foundation Contact Problem Thin Plates in Contact APPLICATION 35 35 39 43 CONCLUSIONS AND RECOMMENDATIONS IV 76 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY CA 93943-5101 A. B. C. D. E. RAYLEIGH-RITZ APPROACH 76 ELEMENT APPROACH COMMENTS ON OPTIMIZATION SANDWICH COMPOSITE MATERIAL STUDY RECOMMENDATIONS FOR FURTHER STUDY 77 FINITE 77 78 78 REFERENCES 79 INITIAL DISTRIBUTION LIST 81 LIST Table 1. Table 2. Table 3. OF TABLES RAYLEIGH-RITZ RESULTS AT CONTACT POINT FINITE ELEMENT RESULTS: TWO PLATES IN CONTACT COMPARISON OF STRESSES NEAR THE POINT OF CONTACT VI 32 39 . 43 LIST OF FIGURES Figure 1. Roller-foundation assembly 7 Figure 2. Contact zone cross section 7 Figure 3. Analytical solutions for a y and a x 9 Figure 4. Deformed contact zone 10 Figure 5. Unconstrained minimization 11 Figure 6. Constrained minimization 12 Figure 7. Rayleigh-Ritz method applied to contact problem IS Figure S. Contact zone 19 Figure 9. Finite element Figure 10. Figure 1 Figure 12. Stress contour of a y from Rayleigh-Ritz method 35 Figure 13. Stress contour of a from Rayleigh-Ritz method 36 Figure 14. Strain contour of from Rayleigh-Ritz method 37 Figure 15. Strain Figure 16. Two Figure 17. Finite element mesh for Figure 18. Finite element mesh for roller-foundation Figure 19. Deformed 1. method applied to contact problem 29 Comparison of a y in a roller contact problem 33 Comparison of a in a roller contact problem 34 x x Contour e, c y from Rayleigh-Ritz method 38 thin plates in contact finite 39 two plates 40 in contact problem 41 element solution: deformation magnified 100 times ... 44 Figure 20. Analytical solution vs. approximate a y from finite element results 45 Figure 21. Analytical solution vs. approximate o from finite element results 46 x Figure 22. Stress contour ay from finite element results (increment 0.01 GPa) .... 47 Figure 23. Stress contour a x from finite element results (increment 0.02 GPa) .... 48 from finite element results (increment 0.000035) 49 from finite element results (increment 0.00001) 50 Figure 24. Strain contour t Figure 25. Strain contour zx Figure 26. y Clamped composite beam 51 Figure 27. Finite element mesh for clamped beam model 52 Figure 28. Determination of nodal contact 55 Figure 29. Loaded beam: deformation magnified 100 times 56 Figure 30. Stress contour: j xy for clamped beam model 57 Figure 31. Stress contour: a x for clamped beam model 58 vu Figure 32. Stress contour: a y for clamped beam model (In region of contact) 59 Figure 33. Stress distribution: r xy for 0(3)-ISO(10)-0(3) laminate 60 Figure 34. Stress distribution: r xy for 0(4)-ISO(8)-0(4) laminate 61 Figure 35. Stress distribution: r ty for 0(5)-ISO(6)-0(5) laminate 62 Figure 36. Stress distribution: o y for 0(3)-ISO( 10)-0(3) laminate 63 Figure 37. Stress distribution: a y for 0(4)-ISO(8)-0(4) laminate 64 Figure 38. Stress distribution: a y for 0(5)-ISO(6)-0(5) laminate 65 Figure 39. Stress distribution: a, for 0(3)-ISO( 10)-0(3) laminate 66 Figure 40. Stress distribution: a for 0(4)-ISO(8)-0(4) laminate 67 Figure 41. Stress distribution: a for 0(5)-ISO(6)-0(5) laminate 68 x x Figure 42. Stress distribution: a x for 0(5)-ISO(6)-0(5) laminate at cross section C . 69 Figure 43. Stress distribution: z xy for 0(3)-ISO(6)-0(7) laminate 70 0(8)-ISO(6)-0(2) laminate 71 Figure 45. Stress distribution: a y for 0(3)-ISO(6)-0(7) laminate 72 Figure 46. Stress distribution: a y for 0(8)-ISO(6)-0(2) laminate 73 Figure 47. Stress distribution: a for 0(3)-ISO(6)-0(7) laminate 74 Figure 4S. Stress distribution: a for 0(8)-ISO(6)-0(2) laminate 75 Figure 44. Stress distribution: r lv for t t vui INTRODUCTION I. MOTIVATION A. Contact stresses occur when two bodies exert forces over limited contact regions. Examples of contact rollers in plate meshing gear stress include forming operations, and roller teeth, cam shaft and pushrod contact, ball bearings in contact with races, shaft and journal bearing contact, and plate-pin connections. Contact zones can be point, line, or surface in geometry. Because of the limited contact zone, the local stresses can be sufficiently high to be of major concern to the Consequently, a thorough understanding of this phenomenon designer. successful analytical solution to the contact first solved in the late 19th century by Hertz. is essential. The problem between two spheres was His solution can only be applied to simple geometries such as spheres, cylinders, and flat plates. ternate solution techniques were needed to accommodate more complex geometries and Because of these limitations, al- boundary conditions. Unfortunately, the contact problem culty is that the size of the contact zone loading. is difficult to study. unknown and its behavior. It is and numerical methods are preferred two general approaches used One class applies in the specific because of these in the significant diffi- Consequently, the problem difficulties that is approximation solution of the contact problem. There are approximate techniques to solve the contact prob- iterative equations that represent the contact The most constantly changing throughout Solutions are obtained by an iterative process. highly nonlinear in lem. is state. procedures to solve a nonlinear system of For example, the contact condition can be simulated by the introduction of additional coupling terms into the system of equations. The other functional is body constraint. The then minimized using specific numerical programming techniques. Al- class constructs a functional that includes the contact though these two classes use different procedures, there are several porate the contact condition into the problem formulation that are classes. methods to incor- common to both Examples of these include the penalty method and the augmented Lagrange multiplier method. These methods specify the manner in which the contact boundary conditions are treated. The objective of this study has two parts. First, two approximate solution niques will be developed to obtain the solution of the general contact problem. tech- These techniques belong to the general class of methods that calculate a specific functional and then applies numerical programming methods to solve the contact problem. these solution techniques will be verified by comparison with With the solution techniques methods verified, the known Second, analytical solutions. be applied to study an actual will contact problem. In order to accomplish the method utilizes the deformation field first objective, two models will be developed. Rayleigh-Ritz method to solve the contact problem. that satisfies the boundary conditions is an expression for the system's enabling calculation of the total potential. The minimization of the first An assumed carefully selected. elasticity relationships are applied to obtain The Theory of strain energy total potential en- ergy enables the calculation of the contact stresses at any point in the body. The second model developed uses element analysis to accomplish the same objectives. finite numerical minimization technique used plier in both cases is the augmented Lagrange The multi- method. To accomplish the second objective, both methods will be verified by comparison With with the Hertz solution of an infinitely long cylinder in verification completed, the contact stresses resulting from low velocity impact between objects B. and composite sandwich materials will contact with a plane. be studied. LITERATURE SURVEY As discussed previously, there are One approach problem. The other equations. Although straints. uses two general approaches special procedures to solve are the Lagrange 1]. 2] to solve the proach is a nonlinear system of boundary con- both approaches use similar mathematical methods to different, method and the penalty method. of employing the Lagrange method Pian [Ref. to solving the contact creates a functional that includes the contact corporate the contact condition into the problem formulation. is flat in nonlinear A multibody contact problem. of these methods detailed explanation of the finite Alternately, the penalty finite element Two method element analysis was discussed by method was used by Cheng The fundamental concept of (i.e., Although both methods are of both methods. [Ref. the latter ap- the transformation of a constrained problem into an unconstrained one. done by penalizing in- This increasing) the objective function for constraint violations. effective, there Nour-Omid is substantial discussion [Ref. 3] described the positive on the limitations and negative aspects of both methods. The Lagrange method has been shown to be the more accurate method. However, its usage requires the introduction of additional unknowns thus increasing the II. FORMULATION OF THE CONTACT PROBLEM PRINCIPLE OF A. As MINIMUM TOTAL POTENTIAL ENERGY discussed in the introduction, the limited cessitated the development of solution techniques capable of handling the nonlinear be- havior of the contact problem with complicated geometry and complex boundary This study intends to develop two numerical procedures to solve the contact conditions. In short, the procedures will use different problem. of the analytical solutions ne- utility methods to obtain a functional, the system's total potential energy, and then use similar methods to obtain the equilibrium condition. Determination of the equilibrium position principle of minimum work and the Given a body principle of in minimum equilibrium, infinitesimal displacements and the it The The and is since the fact that this virtual virtual is from a system of forces. However, and work work vanishes work discussed thus SU, that results far is If each particle body in the to body if the particle forces in the in this and x, y, in equilibrium, the z directions example are referred is zero. to as virtual in referred to as the principle of virtual work. From the definition of strain energy, virtual virtual displacements can be calculated. viewed as energy against the bonds between elements, forces is can be subcategorized as virtual strain energy forces. from The work done by external tity. must be discussed. desired to describe the response of that summation of work done by external strain energy, energy potential energy simply the product of the generalized forces acting on each infinitesimal displacements nature. is particle's displacement. work must be zero The stresses by some generalized coordinates, then the work resulting from these described particle With equilibrium established, contact potential energy. infinitesimal displacements resulting is application of the In order to understand the details of this approach, the principle of can be quantified. virtual made by is is designated S XV and is SU is Since this a negative quan- simply the summation of the product of the external forces and the displacements of the generalized coordinates. Since the principle of virtual displacements is work states that the zero, SlV-SU = work done as a result of virtual Alternately, this can be expressed as, n =d(U- n where represents the system's total potential. The above equation This 330-331] This principle states that total potential is stationary. minimum This is The 12: pp. Hence, determination of a system's total potential energy enable the calculation of the equilibrium po- will the basis for the numerical techniques developed in this study. objective of the solution techniques to be developed calculating the total potential energy and minimize To accomplish dition. this, two isotropic case, the solution of dated for use in different which is models known. equal a distance of 2a. An work done by Hertz. normal in the is is shown Figure in half zones. solve a simple first manner, our models can be vali- plane as shown in Figure 1. A problem is available as a result of In developing the solution methods, only a very limited region from this Figure The reason than one half-contact zone Because of this, it is [Ref. 13] The diagram shows (a) for this is that the contact represents the analytical solution of the 3 contact problem. foundation diminish very rapidly. less means of Let the width of the contact zone 2. analytical solution of this a very local one. stresses resulting depths be created to will infinite adjacent to the contact zone will be examined. phenomenon to obtain a more complicated arrangements. cross section of the contact zone the is to determine the equilibrium con- it In this Consider contact between a cylinder and an 3 potential energy. [Ref. CONTACT PROBLEM DESCRIPTION B. at total potential of a system. condition of stable equilibrium, the system's in a and the minimization of that quantity sition. minimum condition of illustrates the the foundation of the principle of is = ll') As shown, that a z and a y diminishes the stresses becomes negligible significantly in less than reasonable to assume that displacements beyond a very limited region are negligible in the strain energy calculation of the contact problem. A number 1. of simplifying assumptions will be As discussed above, displacements beyond made for this study: a limited region are negligible in strain energy calculations. 2. The foundation is an elastic isotropic material. The cylinder (roller) is rigid. system's degrees-of-freedom and computational time. quire additional computational time, but conditions are exactly satisfied only is has been shown to result in boundary conditions. solutions in satisfying the contact choice of the penalty parameter it The penalty method does not when less re- accurate Since the contact boundary the penalty term goes to infinity, the correct the key to an acceptable solution. Guerra [Ref. 4] supported these claims. Because of the limitations o[ the Lagrange and penalty methods, Bishoff [Ref. 5] advocated the use of the augmented Lagrange multiplier method contact problems. This method is favorable since methods discussed above. The augmented Lagrange penalty method in that the objective function However, an additional multiplier term As combination of both terms. of this method is is it is element avoids the limitations of both multiplier method is similar to the penalized for constraint violations. is added so the optimum can be achieved by stated by Vanderplaats [Ref. that the penalty term to solve finite 6: p. a advantage 141], the not required to grow to infinity to achieve exact constraint satisfaction. The augmented Lagrange techniques to find the optimum of any detailed analysis of the functional. method in 6: numerical techniques. this method and Lowe [Ref. in 7] provide a applying this method. Rothert et al. [Ref. 8] used a to solve a nonlinear contact prob- will obtain the same functional. The augmented Lagrange in a similar fashion to solve be used to determine a set augmented be used as an integral part of two solution methods developed to solve the contact problem. These methods will numerical programming In this study, an existing numerical optimization routine utilizing the Lagrange multiplier method used in pp. 140-147] provided an excellent discussion on the numerical programming code based on lem. Pierre programming techniques necessary Additionally, Vanderplaats [Ref. practical usage of this method can be used multiplier each problem. will use different multiplier techniques to method will In each case, the optimization routine of design variables that describes the contact state. Following the development and verification of the numerical procedures, will investigate the One of the common correlation maximum this study response of composite sandwich materials to low velocity impact. failure mode of low velocity impact is delamination. Joshi and Sun [Ref. 9] studied the impact response of a three layer cross-ply graphite A then be was obtained between delamination cracks shear stress points determined numerically. epoxy laminate. initiated experimentally Sun and Rechak and [Ref. 10] fol- lowed up these findings and found that the introduction of adhesive layers between laminae reduced the shear stress distribution thus reducing delamination. Choi, Wang, and Chang [Ref. 11] studied the effects of laminae orientation, ply thickness, and stacking sequence on impact damage of graphite epoxy composites. that stacking sequence affects impact Much damage more than laminae It was determined thickness variations. of the previous work has focused on the behavior of the graphite epoxy laminate. However, there is currently interest in the development of turbine blades constructed of sandwich composites. It is therefore beneficial to investigate the response of composite sandwich materials to low velocity impact. Figure 1. Roller- foundation assembly Figure 2. Contact zone cross section Deformations normal to the cross-section are 3. negligible, hence a condition of plane strain exists. 4. The 5. The roller-foundation contact An roller is subjected to a vertical distributed load. is frictionless. important restriction on the minimization problem prohibited from penetrating into the other body. however striction, a method of mathematically This will be that one body may seem stating this restriction Figure 4 shows the deformed and undeformed contact zone. flection of the roller due to an external force and of the foundation at any point like v(jCjy) Let be will an obvious re- must be discussed. represent the de- <5 represent the vertical deflection At the point of Contact A, the condition of no (xj,-). interference can be expressed as. v(0.0) At point condition can be stated B, this \{r sin 0,0) The latter condition > c) as. >d - r{\ many can be specified at as - cos 6) points as necessary to define this re- striction. C. NUMERICAL OPTIMIZATION 1. Optimization Fundamentals Before developing the models to be used in this study, one final area must be discussed. it Once the total potential energy has been calculated, a to find the equilibrium position mization technique to be used is A study of the numerical opti- required. The technique being used optimization. must be employed. method of minimizing in this Design optimization is study relies heavily on the methods of design the utilization of mathematical techniques to minimize or maximize a particular value to obtain an optimum solution. is ideally suited for design. utions. may is may have an number of sol- a matter of the designer's experience and given design task However, finding the best solution intuition. ution A The method infinite In the absence of significant experience in a particular field, reduce to examining a range of possible solutions by trial finding this sol- and error. Opti- mization routines can be utilized to find this solution mathematically. Optimization problems can be constrained or unconstrained. may be desired to determine the minimum of the parabola, F{x) = (x - For example, 5) 2 + 2. As it seen — " < n a LEGEND to U C/1 a- * - SIGMA X • - SIGMA Y M in a u ISJ § z o" a — ' 0.0 .5 1.0 l.S « 3.0 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 3. Analytical solutions for a y and <x, M 2.S 1 3 .0 Figure in 5, the minimum A constrained problem. clearly identified at point A. is constrained counterpart of Minimize: Subject As seen Figure in 6, the minimum this This is problem an example of an unis: F{x) to: ^(.v) > .5jc + 4 of the constrained problem simple illustration of constrained minimization. is at point B. The contact problem is This is a a constrained minimization problem. Figure 4. Deformed contact zone The value The parameters process is an to be minimized or maximized is referred to as the objective function. to be determined are referred to as design variables. iterative one. are varied thus obtaining a The new objective function objective function. tolerance or meets a certain convergence criteria, the is out of tolerance, the cycle repeats. 10 is evaluated. The optimization The design If the difference variables within a certain is optimum has been obtained. If it Figure 5. Unconstrained minimization 11 o \ a / / \ a X U. a ^-^\B a o 1 9.0 2.0 4 B .0 .0 X Figure 6. Constrained minimization 12 8 .0 IE 1.0 Augmented Lagrange Multiplier Method 2. The contact problem belongs in the class The technique used several techniques of solving constrained problems. belongs to a class of solution techniques tion techniques this is approach using an unconstrained minimization technique. if the design variables are varied in such a is violated (i.e., (i.e., way is that a constrained This transformation Thus, in as to enter the region 6: is problem minimized accomplished For example, where a constraint would be assessed a penalty order to minimize the objective function, the design variables remain within the region where no constraints are violated [Ref. is for constraint violations. the infeasible region), the objective function increased). study designed for the general nonlinear transformed into an unconstrained problem and the objective function by assessing a penalty to the objective function in this as sequential unconstrained minimiza- class of techniques The fundamental concept behind problem. is (SUMT). This known There are of constrained problems. (i.e.. the feasible region). pp. 121-123] There are a number of methods within differ in the way in this class of techniques. They essentially which penalties are assessed. The technique used augmented Lagrange multiplier (ALM) in this study is the method. Given the constrained inequality optimization problem; Minimize: Subject The Augmented Lagrangian is to: F{X) g (X) t < 0, / = 1 .2 n defined as, m A(X, k,p) = F[X) + £{/M + 2 s, ] + p[g/LX) + sff} where, X= ). t vector containing the design variables = Lagrange multipliers p =penalty parameter 5, =slack variables which convert inequality constraints to equality con- straints 13 The first two terms of A(X, Lagrange multipliers, mality. known X unknown. Hence, is used to select and modify / method of the that minimization of the Lagrangian represents opti- Like simple problems where X AL.M method is is it From Lagrangian. represent the X,p) is simply an additional unknown a mathematical routine based for successive iterations. [Ref. 6: to obtain, in the on constraint values pp. 140-147] Initial selection of this term can have a significant impact of the problem's convergence. As discussed above, violations. This feature is a penalty is assessed to the objective function for constraint apparent by examining the results in a positive value for g(X) thus resulting in which a scaling term that there small, convergence A term. an increase constraint violation in A. The value sequentially increased throughout optimization. is between convergence and numerical conditioning. a balance is last may occur with major constraint violations. If of/? This ensures I f /? remains p remains large, constraints will be satisfied at the expense of an ill-conditioned problem [Ref. 169]. As with is 7: p. the Lagrange term, selection of this term has a significant impact on the outcome of the problem. 3. Optimizer and One-Dimensional Search Thus far. a procedure has been defined which has transformed minimization problem into an unconstrained one. This is of the optimization process The formulation of referred to as the optimization strategy. function via the augmented Lagrange multiplier optimization process. level a constrained the modified objective method represents However, there are additional parts of a key portion of the this process that require comment. With the modified objective function and a procedure for assessing penalties in place, a procedure for minimizing the objective function of the process is carried out by the 'optimizer.' atically alter the design variables in a If X represents manner The must be defined. This portion optimizer's function X X — l <*(/_!) + *<->< where, = current iteration number 5 = search vector k = scalar representing distance traveled in direction 14 to system- that reduces the objective function rapidly. a vector containing the design variables, the following process the optimizer to alter / is S is used by In general, two processes must be accomplished S must be determined by direction steepest descent method where An example a systematic process. the direction of steepest gradient must be determined such that the objective function scalar k The possible in the search direction of the current iteration. as one-dimensional search. [Ref. The optimizer used complexity of count optimum. to find the this method, this study a discussion routine used in this study method of count of this finding the approach is chosen. is the Second, the minimized as much as is latter phase referred to is 6: Due the variable metric method. pp. 92-93]. the golden section is The one-dimensional search portion of efficient is of their formulation available in Vanderplaats [Ref. is of this phase pp. 10-12] 6: in is search First, the is A omitted. to the detailed ac- The one-dimensional search method with polynomial interpolation. the process merely represents a systematic and minimum in the A chosen search direction. again available in Vanderplaats [Ref. 6: detailed ac- pp. 26-49]. Convergence 4. The convergence. final point to be discussed relevant to optimization fundamentals Convergence There are a number of convergence terminate calculations. The most obvious is criteria are utilized to identify the optimum is solution and can be criteria that that of utilized. absolute convergence where the objective functions from two suc- cessive iterations are compared. prescribed limit, optimization is If the difference terminated. A between the values is within some second method signifying optimality for unconstrained problems is vector approximately zero, optimality has been achieved. This method is A'. If this called the value is calculation of the gradient with respect to the design variable Kuhn-Tucker conditions for unconstrained minimization. conditions are more involved for constrained problems. [Ref. 6: The Automated Design Synthesis (ADS) System used iterations is 100-101] in this study uses both Relative convergence these termination criteria as well as relative convergence. to absolute pp. Kuhn-Tucker is similar convergence except normalized versions of the difference between successive calculated. Again, if a specified tolerance minated. 15 is achieved, optimization is ter- APPROXIMATE SOLUTION TECHNIQUES III. RAYLEIGH-RITZ APPROACH A. Background 1. The Rayleigh-Ritz method tential is utilizing the theory the Since the total potential function. trial enables determination of the unknown differentiating with respect to the n' is Second, the system's total potential energy constants. equations and 'n' is a minimum The constants. unknown unknowns, the trial A total potential L with the origin at the left is terms oC un- in is deflection A end of the beam. 12: is and minimized by The result pp. 335-336 one that slope). a simply supported ofl minimization that the trial function kinematically admissible solution (i.e., method calculated in terms function constants. [Ref. As an example, consider satisfied. is this constants and equating to zero. geometric boundary conditions of the system ments need not be chosen at equilibrium, The only requirement of the Rayleigh-Ritz method kinematically admissible. minimum po of The fundamental concept behind energy to solve a given problem. that a trial function that represents the deformation field known is method of a is ] is satisfies the Other require- beam of length kinematically admissible solution to describe the beam's one dimensional deformation from vertical loading in the y direction is, y(x) where a represents the n A a n sin( ~- ) coefficient to be determined. Deflection boundary conditions at x increased = number of terms = in the trial and jc = L have been satisfied. Obviously, an function will yield a far more accurate solution. Fourier sine series would be a reasonable selection in this case. Although the Rayleigh-Ritz method does not on the trial significantly. now function, sensible choices of stipulate numerous requirements functions will increase solution accuracy trial For example, consider the same simply supported beam with the origin at the center. A kinematically admissible function / \ y[x) = • / an sin( 16 —— 27T.V , ). is, Naturally however, due to the placement of the origin in this problem, an even function a is much A more for sensible selection for a trial function. appropriate selection would be, y(x) The throughout A -^- ). trial function will be discussed method brief overview of the Method to the Contact Problem. to be developed in order. is Rayleigh-Ritz method necessitates the selection of the appropriate of unknown in this study. Application of the Rayleigh-Ritz 2. an cos( concerning sensible selection of the latter point detail = A coefficients. the desired features of the The theory of Application of the trial function in terms discussion of the physical nature of this problem as well as trial solution is required. elasticity relationships will obtain the system's strain energy in terms of be applied using the unknown coefficients. trial function to The system's total potential energy will then be minimized utilizing the optimization techniques discussed in Chapter II. The design variables are the the coefficients determined, the displacement to be determined throughout the body. unknown is Figure 7 utilized. The post-processing procedure shown using the now determined 3. known is is trial for function coefficients. all With points enabling the stress a flow chart of the procedure to be simply the calculation of the stresses coefficients. Trial Function Selection In order to choose an appropriate trial function, it is necessary to have an understanding of the physical phenomena to be modeled. Consider the roller-foundation system shown tical in 1. Two trial functions are needed to model horizontal and ver- There are a number of important characteristics that should be deformation. herent within the 1. Figure trial functions. in- These are outlined below: As seen in Figure 3, a t and a y are equal and compressive at the point of contact. Additionally, a t decays more rapidly than a y as distance from the contact point in- creases. 2. Taking the origin at the point of contact as shown takes the form of an odd function. 3. The v-deformation takes the form of an even function origin 4. To and greatest satisfy the be satisfied. boundaries. in Figure (i.e. 2, the u-deformation symmetric about the at the origin). Rayleigh-Ritz method requirements, the boundary conditions must and v deformation to be zero at the far In this case, this requires u 17 f Initiialize Program 2> Define Optimization Design Variables Define Assumed Solution Compute Strain Energy , f Calculate Total Potential Alter Design Variables • Define Constraints 1 i Call Optimizer to Minimize Total Potential Figure 7. Rayleigh-Ritz method applied to contact problem 18 The characteristics outlined in item 4 satisfy the requirements of this method. However, since this study will be a stress analysis, it is also desired that the stresses also reflect the physical phenomena. Since a and a y are functions of c and c c, and c y should also exhibit certain characteristics. Referring to Figure 8 for dimensions and the coordinate system, it is desired that c and c y equal zero at x= L 2 and y= H. This will ensure that stresses are zero at the boundaries. To satisfy the requirements of item above, c, and c y should be equal and negative at the point of contact and decrease in magnitude as the distance from the point of contact in. t s t x 1 creases. Figure 8. With Contact zone the above guidelines functions chosen for this study are in mind, the trial composed of a = an {H-y) a u(x,y) v(x y) = b n (H-y) c i ( function can be selected. series The of terms of the general form: — -x) (j--x) (3- c (3.2) These expressions were carefully chosen and represent a compromise due to the culties of satisfying all boundary conditions with the physical phenomena of lem. 19 trial this diffi- prob- From the above expressions, boundary conditions x=L2 at and immediately obvious that the geometric is it y=H quirements of the Rayleigh-Ritz method. have been This satisfies the satisfied. In addition, there are a re- number of important characteristics that illustrate the advantage of this selection: and are negative thus simulating a compressive environment in the vicinity of The importance of this is obvious. If normal strains were not negative, the resulting requirement would be for the coefficient to be less than zero ix 1. £ y the point of contact. to simulate It is obvious that this would result in deformations opwas desired by the choice of the trial function. compression. posite to that which This selection for deformation fields has the important characteristic of decreasing, deformation as we move away from the point of contact. Also note that deformation is maximum at the point of contact. 2. The exponents a,b.c. and d can be varied to simulate subsurface stress fields. For example, if the analytical solution indicates a large y gradient for a, at x = 0, the: objective would be to increase the rate at which c, decreases as the distance from the point of contact increases. This would be easily simulated by raising the value: of a. If this change had a detrimental effect on the behavior of a the exponents of the vertical deformation could be varied to restore the solution. 3. , v It is important to note that this selection significant limitation of the trial functions u. Physically, However, This is it is expected the in this selection of trial is behave as an odd function as discussed above. w(.rj,') function, a positive value of contrary to the physical behavior of the problem and small, this error is Another less severe limitation is this jc = , deformation c are at least It is range is compromise is in the elastic this error. a restriction on the order of the exponents in Since a, and a y are combinations of Since the normal strains are equal to first ex derivatives, this requires that 2. worthwhile to note that most of the considerations discussed above ceed the requirements stipulated by the Rayleigh-Ritz method. to utilize trial functions that closely fort to some both normal strains must be zero at the boundaries to ensure that stresses are zero at these locations. b and will lead to the origin. 0. order to maintain zero stress at the boundaries. y w(.vj-) exists at Another consequence of believed to be limited. the existence of non-zero shear strain at i not without compromise. The most with regard to the horizontal deformation However, considering that the magnitude of and is The far ex- objective has been match the physical nature of the problem in an ef- maximize accuracy. The specific trial functions used in this study are listed below. deformation was assumed to be; 20 The horizontal rri Ya (H-yf +n \ \ - x) = u( Xxy) 2 (3.3; n n=\ where // and L represent the height and length of the bearing foundation, Summation was done for n equal and 1 4. The vertical respectively. deformation was assumed to be; m Yb = v{xy) +n) (H-yf( -j - xf n (3.4) . 1=1 As discussed to be previously, manipulation of the exponents enables the matched with the analytical ponents chosen in the above functions achieve With the deformations chosen, the in the calculation As solution. Y-2a = n function results in the results, the ex- this goal sufficiently. and stresses of the foundation strain enersv. tx be illustrated will trial (H-y) { strains can be calculated for use These values are shown below: * +r,) (-^-x) (3. n=\ n Zy = Y -2b (H-y)(j--xf j a where stress E and and v are +n) n (1 +v)(l-2v) ^(l + v,fl-2v) 1(\-v)e x +ve v ] C(1 - V) Young's modulus and Poisson's ^ +V ^ ] ratio, respectively. (3.6) (3.7a) (3.7b) For the shear strain, m |t.y-(4 + .K(II-/«(frf oy (3.S) i—i JL = Yj -(l + n)b n (H-y) 21 2 (j--xf (3.9) + f. -f. cy ox y y T where G is (3.10) xy=Gv Xy (3.11) the shear modulus. Using the above quantities, the strain energy U can be calculated. of strain energy applied to a two dimension deformation From the definition field, ii — \\ ( axi x + oy zy + z xy y xy )dydx (3.12) o Because of symmetry about the origin, strain energy can be calculated for half of the domain and doubled. With strain energy calculated, the total potential for the system can be found from. n = U-FS (3.13) where, F= = (3 external force per unit length applied to the roller vertical distance traveled The quantity FS represents B. work done by the roller on the bearing foundation. Total Potential Derivation The finite roller. ELEMENT APPROACH FINITE 1. the by the finite element method can be employed to solve the contact problem. A element mesh can be constructed to approximate the behavior of an elastic foun- dation subjected to line contact loading from an rigid between the foundation and roller roller. The enables the calculation of the foundation's strain en- ergy and the subsequent calculation of the total potential energy. optimization techniques discussed in Chapter mined. Thus The resultant interaction II, By again utilizing the the equilibrium position can be deter- the contact stresses can be calculated throughout the body. objective by application of the is to derive a finite means of calculating the total potential of the system element technique. Total potential energy 22 is defined as, n = U- F5 (3.14) where, U= internal strain energy F= = J The external force per unit length applied to the roller vertical distance traveled strain energy of the Q roller. system can be found from. U= where by the — + \o xz x represents the problem domain. oy e y + r xy y xy )JQ This can be expressed in matrix form T -T{c} {a}ciQ 3.15) as, 3.16) Q where. £) - {a} = l On i cx c y [o x a "<'xy} t } the element level. T ±{e} {a}dQ U= (3.17) Q! where The i represents analysis of the stress h i' matrix can be expressed element. as, {*} = [£]{£} where [D] represents the material property matnx. 23 (3.18) For a condition of plane strain, the stiffness matrix — 1 can be expressed V V E [0] V 1 +v)(l 1 — (3.19a) V 2v] 1 For as, - 2v a plane stress condition. = [D] 1 V V 1 3.1%) (1-v' The development of this technique 1 - V will use linear triangular elements. The method, however, can be applied to any type element. For linear triangular element, the defor- mations take the followin2 form: u = Hiii\ i"i H-^ih + H-u 3"3 (3.20) v = //.v, 111*22 ,», -r H,\ + H,\ ^ M 3'3 (3.21) + + where the shape functions H, are defined as: = C(-v ^'3 ~ x 3>'2) + CV'2 ~}'3) x + x ~ H = 2 [(-ViXi - xy3 + CV'3 -}'\) x + x ~ x3 ]y] H = [(*i>2 ~ x iy\) + Oi -y-d* //, i ) where, x ,y, = coordinates for node i ( A = element The strain matrix area [Ref. 14 ]. can be expressed as, 24 ( ( + 3 *:)>'] \ (*2 - *ilv] ~ (3. - (3-23) IA 37 >->} (3.24) 4- o o f ex w.= g c cv dx Substituting equations (3.20) and (3.21) into equation (3.25) yields, r o//, 6H2 dx ex c//3 -i "l V l c cH = { a//2 c// 3 u2 cy cy v2 °y c//, cy cH x c// : o//2 BH dH cy ex cy ex expressed as, ex In abbreviated notation, equation (3.26) is For the linear triangular element, [5] reduces >2->'3 [Z?] = 17 r3 - - x2 >'3 ,v 3 - x2 }'2 -}'2 3 ''3 3 V3 to, -yi x - x2 y, \ .V! - }'3 ~}'\ -y 2 x2 - ,v 3 x2 ~ x \ }'\ -V, i.28) ~}'2 Returning to the element strain energy calculation, equation (3.17), U= ±{c} T {o}d£l & and substituting (3.18) into (3.17), ±-{,}\D^z}dV. i 25 (3.29) Substituting equation (3.27) yields, U= \{lB-]{d])\D^B^{d)dV q: r = 4-{^ r[5] r[D][5]{c/}f l- where i = = T -T{cf} d\ [Bf[D]iBl4i{J} (3.30) - Defining the element stillness matrix unit depth. [A1 = [A. ] r [*] [D][5].-!/ 13.31) then the strain energy per element equals, U = ±-{d}tK\{d\ The element With strain energy stiiTness now (3.32) matrix can be expanded into the global stillness matrix. determined, total potential can be determined from equation (3.14). 2. Optimization and Static Condensation As with method mum will the previously developed model, the augmented Lagrange multiplier be utilized to determine the equilibrium condition via the theorem of mini- potential. The objective function ever, the design variables are the is again the total potential. nodal deformations, u and v„ t In this case, how- for non-fixed nodes. Constraint equations are developed in a similar manner as discussed in Chapter II to: ensure that one body does not violate space occupied by the other. Since the nodal deformations are represented as the optimization design variables, the number of design For a simple mesh, a However, it is direct application of this known number of design variables will equal twice the procedure number of non-fixed nodes. will likely yield accurate results. that the accuracy of the optimization routine declines as the variables increases. Hence, for complicated solution accuracy will be adversely affected by the large finite element meshes, number of design variables. Therefore a procedure must be adopted to eliminate the need for assigning design vari- 26 ables to nodes where information cedure is known is not necessary for evaluating a solution. This pro- as static condensation. Static condensation has been duce computer computational time. utilized The ganization of the global stiffness matrix. by References 2 and 3 in an idea behind static condensation A finite [KJ{u} = effort to reis the reor- element problem can be expressed {F} as, (3.33) where. [K,] It is = the stiffness matrix {«} = the deformation vector {F} = the force vector desired to reorganize this system of equations into the following: The vector m, ^11 ^12 n Kt, Ao-> of (3.34) contains the essential nodes while vector u2 contains non-essential nodes. Essential nodes are those where boundary conditions are applied and nodes that are signed optimization design variables. By matrix manipulation. = -lK22 TXK2l -]{u {u2 ) as- x (3.35) }. Therefore, the displacement vector can be expressed as, [/] u2 where I }"\-LK22 T l (3.36) - IK22 T\K2 {\ LK2l 2M] represents the identity matrix. Substituting this equation into the global counterpart of equation (3.32), ,; V =y 1 f ( ,71 M i) IK22 Y\K2 {] \K ~] fo S [^22] [^21] - By defining the reduced stiffness matrix [A ], the above reduces 27 to. (3.37) U = ±-{u As discussed at the beginning of { } T [K}{ Ul ) (3.38) this section, the signed as the horizontal and vertical deformations at all ADS design variables are as- non-fixed nodes. With the in- tegration of static condensation, design variable assignments are further restricted ten non-fixed, non-condensed nodes. The procedure is now in place for calculation of strain: energy and total potential energy. In summary, a flow chart of the solution procedure' utilized in this chapter is shown in figure 9. 28 ] ( V Initialize Program I y Define Optimization Design Variables Calculate Element Stiffness Matrix [K] Assemble Global Stiffness Matrix (KG] Reduce Degree of Freedom by Static Condensation Assign Design Variables to Calculate Strain Energy U={d} t (d) U (KG]{d) Alter Design Variables Calculate Total Potential Define Constraints Call Optimizer to Minimize Total Potential Post-Process C Figure 9. Finite element method applied End to contact 29 ) problem [ KG RESULTS AND DISCUSSION IV. PROCEDURE VALIDATIONS A. Method Rayleigh-Ritz 1. In problem Chapter III an approximation technique was developed via the Rayleigh-Ritz mated the deformation Results field method. As were selected discussed, two in terms of trial unknown to solve the contact! functions that approxicoefficients. The hori- zontal deformation was assumed to be; m n=] The vertical deformation was assumed to be; m vi Y bn(H-y) (-T-x) 2 x vv) = 0+n) j n=\ Using the above deformation theory of elasticity relationships and the definition fields, of strain energy were employed to obtain an expression for the total potential energy of the system to shown in Figure 1. Numerical minimization techniques were then employed determine the equilibrium condition and the contact To of illustrate the application this stresses. method, an isotropic material with the fol- lowing properties was selected: E = 200 = v G As stated in Chapter II, this because an analytical solution GPa 0.3 = 76.9 GPa problem was selected is for development of available as a result of the Contact stresses as well as the size of the strongly affected by the size of the contact zone. 30 technique work done be Hertz. desired to use this analytical solution to choose a roller size, load, that can be used to accurately simulates the contact this It is and problem domain phenomenon. surrounding region of influence are Naturally, as the size of the contact zone increases, the load The is distributed over a larger area and contact stresses decrease. extent of the affected subsurface zone also decreases. the contact zone size is and the diameter of the roller [ Ref. 13 Hence ]. and the defining the Figure 3. examining it o, domain beyond which is resulting stresses. Therefore the decay o^ ay . shows the ana- 3 is the limiting factor in From strain energy contributions are negligible. phenomenon can be estimated that the contact a region equal to Figure zones (a) awa\ from the contact point. As shown. stresses as a function of half-contact a y decreases more gradually than and This figure shows the decrease of the normal of the contact problem. lytical solution for a given material, the roller size defined the width of the contact zone as 2a. 2 the analytical solution, defined by the externally applied force, the material properties the external load define the contact zone size Figure From approximately accurately modeled by five half-zones (5a). Since a numerical integration technique was used to perform the double inte- gration required by Equation 3.12. the dimensions were selected for numerical convenience. Referring to Figure respectively. Due to the 8. height H and length L were selected as H and 2 meters, problem's symmetry, half the foundation was analyzed. enabled the double integration to be conducted between the limits of foundation height 1 has been set to 1 m, it is and 1. This Since the desired to have this distance equal to 5 contact zones (5a) as described above. Using the analytical solution, a load and roller ated a contact zone such that the foundation height The only additional restriction was that the roller radius combination used equal to a distance of 5a. resulting contact stresses remained within MPa. 90 Radius: 75 MN m values were obtained using the analytical solutions found in Reference latter dimension is of base dimensions unrealistic. However, as stated above, in the interest The load study are; in this Load: The H was Yield stress was assumed to be 300 the elastic range of the material. and diameter were selected that cre- it of numerical convenience. is 13. The a result of the selection This model simply repres- ents a scaled-up examination of the base material in a small region adjacent to the contact zone. Comparisons of the approximated contact along the axis of symmetry are shown in stresses with the analytical solution Figures 10 and 31 11. Both figures are normalized graphs of the stress distribution. the point of contact. Figure 10 is a As shown, comparison for ay As shown this stress As stated in Chapter the exponents to ax It is . III, match the maximum apparent that 1 shows the analytical and approximate 1 this method over approximates function this trial is it in this A stress in In the case of Figure 11, it brief expla- solution technique follows. must be understood that a and a y are both funcx vertical deformations. deformation to alter the this stress.- the ability to change analytical solutions with approximate solutions. and at x one of the benefits of While selecting exponents, occurs stress equal to a and o y at the point of contact: is Figure nation of the choice of exponents used tions of horizontal 3, This figure shows a stress distribution that closely . approximates the analytical solution. stress distribution for Figure in Therefore changing the exponent of one one direction behavior of the other. will influence the appears that a modification is required. An increase in the exponent of the y-portion of the horizontal deformation function seems appropriate increase the rate at which a x decays. fect However, of decreasing the rate of decay of a y exponents of the maintain a zero vertical deformation. stress is the at and Chapter y=H III. in ef- order to the exponents of compromise must be reached. 11. all It is on ensuring a y is Accordingly, a decision was Figure in a 2 in priority should be placed as close as possible to the analytical solution. shown have the undesirable possible to counter by decreasing the x=L Thus 2. dominant term, cept the over estimation of a x as is However, as stated boundary condition terms must be greater than or equal to believed that since a y It . this action will to made to ac- In this case, the estimation of a x conservative. is The contours of the Rayleigh-Ritz It solution are shown in Figures 12 through 15. has been determined that this method can accurately predict contact stresses resulting! from line contact between a roller and flat plane. Table 1 compares the approximated! contact stresses and the analytical results at the point of contact. Table RAYLEIGH-RITZ RESULTS AT CONTACT POINT 1. Current Model Analytical Solution a, (MPa) 285.6 289.7 o (MPa) 301.9 289. v 32 Pi ' LEGEI D * - APPRO CIMRTE en u m • - EXACT aa: a (- m a u S z a" 'I a a. 0.0 0.5 1.0 1.5 3.0 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 10. Comparison of a y in a roller contact problem 33 2.5 3.0 *• rw o LEGD in u U1 U1 B « - APPRO (inflTE • - EXACT fri in a u M _1 • _ § z o" ri a — 3.0 1.0 .5 l.S •— 3.0 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 1 1. Comparison of a x in a roller contact 34 problem 2.5 1 1 2 .0 Figure 2. Stress contour of 12. Finite «r y from Rayleigh-Ritz method Element Method Results In Chapter III, an approximation problem using the finite element method. method was developed A to solve the contact method of calculating ergy and the total potential energy was investigated. a system's strain en- In addition, the use of static condensation to improve optimization efficiency was described. As discussed, the nu- merical minimization techniques were again utilized to determine the equilibrium position. A means of employing these techniques introduced. In this section, a simple contact algorithms used condensation. problem will a. to calculate With confidence strain to evaluate the contact problem energy and will be investigated to validate the those in these algorithms, the phenomenon was used to implement static more complex roller-foundation be examined. Two Thin Plates in Contact The procedure developed was the solution of which was known. first validated In this example, 35 on a simple contact two thin bodies in problem plane stress were Figure studied. Stress contour of 13. As shown in Figure <x, 16, from Rayleigh-Ritz method one body, restrained on one edge and subjected to a horizontal load, comes in contact with a second body rigidly supported on three sides. The Finite element model developed to solve angular elements as shown in Figure problem was to solve this finite 17. The this problem is composed of objective of the fortran static tri- program developed to calculate the total potential energy of the element technique and the method of 14 linear system using the condensation. With this accomplished, the equilibrium position can then be found via the augmented Lagrange multiplier method. The Equation 3.14. expressed as: objective function for this problem is the total potential energy, Referring to Figure 16 and 17 the constraints imposed on the system are u& < 0.005 36 + uu Figure Strain contour of 14. t y from Rayleigh-Ritz method tig < 0.005 + u l2 where u represents the horizontal deformation of node As shown in Figure seventeen nodes were used to model the system. 17, Each node has been assigned horizontal and ingly, the degree in Chapter III, of freedom for static non-essential nodes. this /. system is vertical deformation variables. twice the number of nodes. Accord- As discussed condensation requires the identification of essential nodes and Essential nodes are those nodes where assigned and boundary conditions are applied. ADS design variables are Referring to Figure point of load application and must be assigned a design variable. 17, Nodes node 8, 9, 3 is 11, the and 12 are assigned design variables in order to define the constraint equations described above. After eliminating all fixed only candidates for condensation. nodes from consideration, nodes As discussed in Chapter 37 III, 2, 5, and 6 are the the global stillness ma- Figure trix is Strain Contour 15. et from Rayleigh-Ritz method rearranged according to Equation 3.34. the non-essential nodal information. u2 {w2 } where u, and v\ Strain energy 3.14. The r is , ={w2 For this problem, the vector containing arranged as follows; u5 v2 v5 u6 v6 } represent the horizontal and vertical deformation of node and total potential energy scribed in Chapter The results obtained de- compared with the II. from Y.W. Kvvon and place to solve the method multiplier solution obtained from this simple problem were The solutions were respectively. were calculated according to Equations 3.38 and was minimized using the augmented Lagrange latter /, in agreement. It J.E. Akin [ Ref. 15 was concluded that a ] in Table satisfactory procedure more complex roller-foundation problem. 38 and are shown 2. was in V/ '/_ W/y/////////////zzzv7//, Figure Table Tno 16. 2. FINITE Load (N) thin plates in contact ELEMENT RESULTS: TWO PLATES l.OxlO No structed to Contact .503x10-' "12 4 .336X10" .319x10^ "9 .702xlO- 2 .734x10^ 1 finite No Contact .503xlO- 2 Roller- Foundation Contact A Reference 15 "9 un b. CONTACT Current Model Deformation 1.0x10 s 1.0x10° IN .201xlO~ 2 .235xlO" 2 Problem element grid composed of 512 linear triangular elements was con- model the roller-foundation assembly shown in Figure symmetry of the problem, one-half of the foundation was modeled. constructed in the vicinity of the point of contact. The mesh is 1. A Because of the refined shown in mesh was Figure 18 where the origin represents the point of contact. The domain dimensions are similar 39 to Figure Finite element 17. mesh for two plates in contact those chosen in the Rayleigh-Ritz method discussed in Part radius was chosen as 75 meters A of this chapter. The and the half-domain dimensions are roller 1.40 x 1.40 meters, j As discussed in Part A, these dimensions represent an analysis of the region immediately adjacent to the contact zone and are a result of the local nature of the contact problem. Constraint Chapter • II Boundary' conditions were imposed in the the discussion of following manner: Horizontal and vertical deformations were prohibited on the remote mesh boundaries • Part B. equations were constructed according to (i.e., m( 1.40, y) = v( 1.40, y) = and u(jr,1.40) = v<x,1.40) = 0). Horizontal deformation was prohibited on the axis of symmetry 40 (i.e., u(0,y) = 0). *• n I— U (C Z o CJ u. o "" o iz E-o" E o a: u. u IS1— to E * tae u >• a" o 1 1 3.0 1 0.4 0.2 0.8 1 0.8 1 1.0 1 1.2 HORTIZONTflL D I STANCE FROM POINT OF CONTACT i Figure 18. Finite element mesh for roller-foundation 41 problem i 1 Since the system has 289 nodes, the resulting 578 degrees of freedom necessitated the utilization of static condensation. Referring to the contact surface, the deformation variables that correspond to nodes on straint equations. all subject to The nodes this surface are required for con- that comprise the other three borders of the boundary conditions. Consequently, the nodes that are candidates 1 for static condensation. interior domain arc: nodes of the mesh are the In this model, all interior nodes were condensed. The original system was reduced from 578 to 128 degrees of freedom. After; application of the boundary conditions, there were 46 possible deformations, a number of ciently small design variable was used to represent the distance of travel by the needed to calculate the As One design variables for the optimization algorithm. work done by the before. Equation 3.38 roller was used suffi- additional This value roller. is' on the bearing foundation. to calculate the system's strain energy. Following calculation and the subsequent minimization of the total potential energy, a post-processing procedure was followed to determine the contact stresses. of the optimization routine represents the nonzero components of the The output {«,} vector. In order to calculate stresses throughout the body, the remaining deformations contained within the condensed vector using Equation 3.35. must be determined. {;/,} is calculated directly With deformations known throughout the domain, determined by applying Equation 3.27 to each element. Equation 3.18 enables determination of To This vector illustrate the capability stress for strains can be The subsequent application of each element. of this method, an isotropic material with the following properties was chosen: E = 240 v = GPa 0.3 G = 92.3 GPa Load = 90.0 MPa Figure 19 shows the deformation resulting from the loading. For clarity, the defor- mations have been magnified 100 times their original values. Comparisons of the distributions with the analytical solution along the axis of ures 20 and 21. These figures are similar to Figures 10 and versions of contact stresses. mation of the As shown stress distribution in the symmetry are shown 1 in these figures, this foundation of a loaded 42 1 stress in Fig- and represent normalized method is a good approxi- roller bearing. If the mesh was more refined near the contact zone and the domain extended further, the between the numerical and analytical solutions would be figures 22 and 23 rep- resent normal stress contours of this problem. Table 3 shows a comparison of the better, show normal Figures 24 and 25 results of this agreement strains. model and the analytical solution at a selected element in the region of contact. Table COMPARISON OF STRESSES NEAR THE POINT OF CONTACT 3. FEM Analytical Solution Solution a, (MPa) at x = 0.0137. y = 0.0273 281.3 272.0 ex, (MPa) at x = 0.0137, y = 0.0273 315.7 313.8 APPLICATION B. The preceding lated using the show how this section illustrated that the contact problem can be accurately simu- methods developed in Chapter approach can be applied III. It is the objective of this section to problem to a contact composite plate sub- in a jected to low-velocity impact. A multi-ply laminate model has been constructed to investigate the response of composite materials to low velocity impact. subject to impact damage commonly fail It due has been found that composite bodies currently being considered for use as turbine blades. It an understanding of the behavior of sandwich materials In order to accomplish this task, a picted in Figure 26 has been modeled. cm. Sandwich composites are to delamination. would be to impact clamped composite beam The beam length is 25 cm. beneficial to acquire damage. similar to the Beam thickness is Because of symmetry, half the beam was modeled with 256 bilinear elements. shown in Figure 27, the mesh is The major assumptions of strain and that the dynamic this effect model are that the beam condition of plane of the impact can be neglected. Reference 10 ap- cordingly, the stress distributions obtained is maximum normal and from this time of the response of the body at the instant of finite element program developed to solve alter the material stiffness As in a the peak load will be examined to study the The 2.5 refined near the point of contact, the origin of the mesh. proximated the loading resulting from low velocity impact as a sinusoid. in one de- matrix [D] shown 43 in In this study, shear stresses. Ac- study will represent a 'snap-shot' maximum this loading. problem is sufficiently flexible to Equation 3.18 during construction of the 1— cc o U o Z o £o a: U. LJ 83- o _] " d- > o " — , .. d . i i 0.0 i 0.4 0.2 -i— 0.8 -i 0.B i 1.0 i \ 1.2 HORTIZONTflL 01 STANCE FROM POINT OF CONTACT Figure 19. Deformed finite element solution: deformation magnified 100 times 44 ri LEGE* Y0 m U x - APPRO K I MATE X • - aa: o- x (— en a U EXACT N. x > x (si >_ L IE O z X X X *• o" ri o" o 0.0 0.S 1.5 1.0 2.S 2.0 3.0 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 20. Analytical solution vs. approximate a y from 45 finite element results rt o LEGEIO en U a3= O « - APPRO • - EXACT (I HATE 1— in o u CE O z o" n\ <<• X X X o X - • 0.0 O.S 1.0 X •— r- 3.0 l.S 2.S • 3.0 NORMALIZED DEPTH BELOW CONTACT SLRFACE Figure 21. Analytical solution vs. approximate 46 <x x from finite element results 0.4 0.2 0.8 0.S 1.0 HORIZONTAL DISTANCE FROM POINT OF CONTACT Figure 22. Stress contour a from finite element results (increment 0.01 GPa) 47 0.0 0.2 0.4 0.B 0.0 1.0 1.2 HORIZONTAL DISTANCE FROM POINT OF CONTACT Figure 23. Stress contour <x, from finite element results (increment 0.02 GPa) 48 Figure 24. Strain contour sy from finite element results (increment 0.000035) 49 Figure 25. Strain contour £, from finite element results (increment 0.00001) 50 Figure 26. Clamped composite beam 51 fl B C FINITE ELEMENT MESH 5.0 2.5 5 7.S 10. HORIZONTAL DISTANCE FROM POINT OP CONTACT 4 o H 0= Figure 27. finite Finite element element global mesh for stiffness matrix. clamped beam model Therefore by defining the layup for the laminate, the lamina stiffness matrices can be varied from element to element to accurately model the behavior of the body. This enables a variety of laminate layups and lamina thick- nesses to be studied. The sandwich materials used in this study are stiffness matrix is given by Equation 3.19a. orthotropic exterior laminae, the material stiffness matrix [Ref. 16 isotropic interior Since a condition of plane strain was as- material and orthotropic exterior laminae. sumed, the isotropic material composed of an ]. 52 is For the given by Equation 4.1 1 — V23V32 v i E2 E3 T [Z>] v = \2+ + 2 V 32 V I3 £,£ 3r v 32 v 13 1 " v 13 v 3 1 (4.1) E E3 T £,£ r 3 { r 12 where. r= v 1 :: v :, - V3VJ,- the r* . = - 2l\,v 3: v 13 £,£2 £3 £ = Young's Modulus v v 31 v 13 in /" direction Poisson's ratio for lateral contraction in/'" direction resulting from loading in direction The derivation of done using the total potential energy calculation in Chapter III. Part B Since bilinear elements were used linear triangular elements. calculation of the element stiffness matrix was more computationally in this wise, the calculation of total potential energy Chapter was model, Indi- intensive. vidual entries of the element stiffness matrix were obtained from Reference 17. was Other- identical to the procedure outlined in III. There were two groups of boundary conditions applied to the problem. clamped edge, deformation was prohibited. As prohibited along the axis of symmetry. In addition, horizontal deformation the was before, the application of static condensation requires the identification of essential nodes the information of which the {w,} vector. Along is contained within In addition to the essential nodes associated with the above boundary- conditions, the nodes along the contact surface are needed for the constraint equations. All other nodes were To condensed out. illustrate this application of problem solving, sandwich material with the lowing properties were studied: Exterior Laminae £u = 170 Ex = 11.8 Gu = 5.2 v i2 = GPa GPa GPa 0.33 53 fol- Isotropic Core E = 2.24 GPa G = 0.84 GPa v = 0.35 The beam was loaded by contact with was 250 N. Some unexpected mined by the optimizer. a 10 cm The transmitted radius ball. force trends were observed in the equilibrium position deter-| By examining the deformations along the gradually decreasing trend in deformation moving away from interrupted in lamina of significantly decreased stiffness. It is axis of symmetry, a was the point of contact believed that this difficulty resulted from an inability of the optimization routine to approximate deformations through regions containing very different orders of strain energy. ficulties, some beginning of In spite of these dif- information was obtained from this program. As discussed critical this study, one of the greatest difficulties termination of the size of the contact zone. of the contact problem and the distance between the (r) can be determined node distance r' to the i if m a node is is r' is greater than r, is By com- and the node - (r illustrated in Figure 28. ), it The given by the equation: r' If ball center This condition in contact. the de- is Fortunately, the size of the contact zone can be readily determined by examining the output from the constraint equations. paring the ball radius in the the node is =v not (r in - + vf + <5 x? contact with the body. With the extent of the contact zone known, the solution to this tained by applying the contact boundary condition to a direct finite Since the validity of the optimization program was in question, this problem was obelement program. method was applied using a 0-90-0 layup similar to one used in a study conducted by Sun and Rechak [Ref. The solution obtained from the current approach was very 10]. close to the other re- sults. With confidence model in the to various layups. procedure, The objective be used for meaningful research. it was desired was to illustrate concern direction, the is examine the behavior of how this solution this technique can The study conducted by Sun and Rechak analyzed methods of reducing the likelihood of composite ticular to failure due to delamination. the magnitude of shear stress distribution two predominant causes of delamination 54 and failure. Of par- tensile stress in the y- Using materials with the B z 8 o SHEAR STRESS /— a. n- o - 0.0002 ' tr. H"! in 1 o fcB#5TO / 3.0 a. 5 ^T" 0.0002 5.0 7.S 10.0 ia.s HORIZONTAL DISTANCE FROM POINT OF CONTACT o fa: U Figure 30. Stress contour: x xy for clamped beam model thickness of the exterior layer increases, the However, this stress is always compressive the magnitudes of the shear appear that if by high shear 41. value of this stress increases. with the core. at the interface and transverse normal delamination was to occur at By comparing stresses at the interface, this interface, it is more likely to stresses at cross section A are shown in be caused These figures show that as the thickness of the exterior layer increases, indicate that lamina. would Figures 39, 40, linear stress distribution intensifies in the layer closest to the contact surface. would it stresses. Comparisons of the bending and maximum beam theory is This nonlinear behavior a non- This trend unsuitable for estimating bending stress through this is local to the contact zone. Figure 42 shows the counterpart for Figure 41 at cross section C. The stress distribution in the contact layer is approximately bending linear. stress graphs. Another As significant observation can be made by examining the the thickness of the exterior layer opposite to the contact layer increases, the stress distribution within this layer transforms 57 from purely tensile 8 STRESS-X S.S 5.0 7.5 10.0 HORIZONTAL DISTANCE FROM POINT OF CONTACT Figure 31. Stress contour: <t x for clamped beam model behavior to compressivc-tensile behavior. by initiated tensile stresses tends to is significant because a bending crack propagate to the core interface and cause delami- The presence of compressive nation. This stress within this layer will tend to slow the growth of this crack toward the core. Thus far symmetric layups have been studied. metric layups respond to contact loading, two To analyze how beams beams with were studied: 0(3)-ISO(6)-0(7) and 0(8)-ISO(6)-0(2). lamina closest to the contact surface. utions for these two layups. As The Figures 43 and 44 before, the maximum with asym- the following designations designated layer first show is the the shear stress distrib- shear stress is relatively unaffected by the different layups. As was the case for the symmetric beams, a thicker exterior layer close to the contact zone results in a core. The As seen same as in result is more gradual transition of shear stress into the a lower shear stress at the interface for the 0(8)-ISO(6)-0(2) case. Figures 45 and 46, the trends for the transverse normal stress, a y were the , those found in the svmmetnc beams. 58 Deflection was less for the ~ xi I r^y 6 vi t t rrr / -v ^ r r' Figure 28. Determination of nodal contact properties outlined above, a sandwich composite degrees was beams Since the studied. first finite studied will be described with the element model number of is finite outer fibers aligned to composed of element layers For example, a 0(3)-ISO( 10)-0(3) beam following the layer description. 10 isotropic core layers within beam with 3 16 layers, the in parenthesis composed of is layers of material with the fibers oriented at degrees on the top and bottom of the beam. Three symmetric layups of varying core thickness were have the following designations: mations magnified 100 times. and stress ites, is As is 32. 0(3)-ISO( 10)-0(3), Stress contours for this beam The beams are beam shown in with deforFigures 30, Figure 30 shows the shear stress contour for the loaded condition. of particular concern since delamination, a commonly initiated by high shear beam increases failure mode stresses or tensile transverse the figure shows, a very high stress gradient distance along the common away from is and 0(4)-ISO(8)-0(4), Figure 29 shows the deformed 0(3)-ISO(lO)-0(3) 0(5)-ISO(6)-0(5). 31, initially studied. This for compos- normal stresses. present near the contact zone. As the the contact zone, the magnitude of the gradient decreases until the cross sectional shear stress distribution becomes parabolic. The transverse normal stress Figures 33, 34, and 35 three symmetric is also concentrated show around the contact zone. the cross sectional shear stress distributions for the beams described above. Three separate cross sections are shown on 55 s — a'S Figure 29. ' 7 S ia-5 10 -° POINT OF CONTACT HWIZDKIfL DISTBNDE FROM magnified 100 times Loaded beam: deformation "™*^^ Ftgure 27. The of which are indicated in each oraph the locat.ons she, that interfaces. These figures lamma the identify gr Ph onea'ch . <h«ir stress "o? is a,, stgntfi an c- However, to the core thickness. relatively insensitive stress transtuon tncreases, the shear to the contact surface ea a reduction ,n the s core. The resuit is the tnterfa'c, w,th the that lamma. it ts also noteworthy w.th th.c.er extenor lavups for interface atthe sect.on A. sect.on B vice cross stress occurs at cross shear maxtmum lh e same normal stress. ., for the for the transverse graphs are 38 and F.sures 36 37 nes the th s show an increase in .. as layups. These graphs stronger a produces these layers The increased th.ckness of , o lavers increases 1 : -est mlg li « I;: J— str esses increase. Is, . of* At cross sect.on C, some stress stated. tens„e transverse is stress tens.le transverse ,s euden, of delam.nat.on. a potentia, source 56 A. As P« STRESS-Y: ZOOM OF CONTfCT ZONE i I i 3.0 1.0 3.0 I 4.0 5.0 HORIZONTAL DISTANCE FROM POINT OF CONTACT 2 Stress contour: o y for clamped Figure 32. 0(8)-ISO(6)-0(2) case. As seen in the resulting smaller contact symmetric cases, an increase contact zone resulted stress, The in beam model an increase seen at cross section C. in the (In region of contact) zone lead in the thickness magnitude of the However, the stress at the to higher contact stresses. of the layer closest to the maximum tensile transverse laminate interface was always compressive. With regard to bending stresses for these layups, Figures 47 and 48 clearly show the nonlinear behavior as the contact layer thickness increases. In addition, the thickness of the exterior layer opposite to the contact surface shows similar results as the symmetric cases. As the thickness of this layer decreases, the bending crack that propagates into the interface. 59 beam is more susceptible to a 0.2 0.4 0.S 0-8 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 33. Stress distribution: zxy for 0(3)-ISO(10)-0(3) laminate 60 ^3* LEGEND GROSS SECTION (ROSS SECTION laDss-SECikm 0.4 0.2 0.8 O.S NORMAL I ZED DEPTH BELOW CONTACT SURFACE Figure 34. Stress distribution: t for 0(4)-ISO(8)-0(4) laminate 61 o i • '-i m •- i S- » *-~~^M a o / a. co — I >-?" J X 1 cs m J a j T" LEGEND o«- 1 a • - ( ROSS SECTION A C ross Section B r Rrrvi rfttthn r i o 0.0 0.2 0.4 0.8 0.0 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 35. Stress distribution: r for 0(5)-ISO(6)-0(5) laminate 62 t .0 CROSS SECTION CROSS SECTION [ROSS SECTION a.o 0.2 0.4 O.B 0.8 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 36. Stress distribution: a y for 0(3)-ISO(10)-0(3) laminate 63 1.0 o __J 1 a a T // / <E a. r: en / o T i a njFkin ' o- c ROSS SECTION * - ( ROSS SECTION a • - [ ROSS SECTION c a 0.0 0.2 0.4 O.S 0.8 NORMALIZED OEPTH BELOW CONTACT SURFACE Figure 37. Stress distribution: <r y for 0(4)-ISO(8)-0(4) laminate 64 1.0 en a. a - [ROSS X - SECTION (Ross Section (ROSS SECTION 0.0 0.2 0.4 0.8 0.8 NORMAL I ZED DEPTH BELQU CONTACT SURFACE Figure 38. Stress distribution: <t for 0(5)-ISO(6)-0(5) laminate 65 1.0 J / V / / LEGENO Q- 0.0 0.4 o.a [ ROSS SECTION o.s in o.a NORMALIZED DEPTH BELOW CONTRCT SURFACE Figure 39. Stress distribution: <j x for 0(3)-ISO(10)-0(3) laminate 66 i.o t o J to x ' ca in / o LEGEND 1 a- ' [ I ROSS SECTilQN n o 0.0 0.2 0.4 0.8 0.8 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 40. Stress distribution: a x for 0(4)-ISO(8)-0(4) laminate 67 1.0 / o m a » e • a Co X C3 ' O LEGENO 1 Q- ( RQSS SECTION A o N 0.0 0.2 0.4 O.t 0.8 NORnflLIZED DEPTH BELflH CONTACT SURFACE Figure 41. Stress distribution: <r, for 0(5)-ISO(6)-0(5) laminate 68 1.0 / o E Q_ X CJ ' o i LEGI • - c ROSS SECTION c o w. 1 0.0 0.2 0.4 0.8 o.a 1.0 NORMALIZED 0EPTH BELOW CONTACT SURFACE Figure 42. Stress distribution: a x for 0(5)-ISO(6)-0(5) laminate at cross section 69 C o 1 \ V^/ o a ri" a r LEEENO ROSS SECTION n ROSS SECTION B o i- r RRSS SFTTrnN in o 0.0 0.4 0.2 3.8 O.S NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 43. Stress distribution: t,, for 0(3)-ISO(6)-0(7) laminate 70 l.a * A ^^-*c I — i w i U U- — — — d£z* —• * v^ ^S^ a EC Q. E w a J >< C3 en a T LEGENO o*o • - C ROSS SECTION R c ROSS SECTION B r Rnqq SFTTTriN n 1 0.0 0.4 0.2 0.8 0.8 NORMALIZED DEPTH BELOW CONTRCT SURFACE Figure 44. Stress distribution: x xy for 0(8)-ISO(6)-0(2) laminate 71 1.0 o -** •^ ^ i a i o f a. 9 hit >- in o T i a rerun oi- [ ROSS SECTION a *!" 1 ROSS SECTION B • ( ROSS SECTION C 1- i a 0.0 0.2 0.4 0.8 0.8 NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 45. Stress distribution: a y for 0(3)-ISO(6)-0(7) laminate 72 1.0 1 w o n •— 1 • 1 *f o r o T O T i o«O • - n;rwn c ROSS SECTION R C ROSS SECTION B c ROSS SECTION C 1 o ri 0.0 0.2 0.4 3.8 0.8 NORMAL I ZED DEPTH BELOW CONTRCT SURFACE Figure 46. Stress distribution: <r y for 0(8)-ISO(6)-0(2) laminate 73 L.O » e LEGEND GROSS SECTION o.o 0.2 0.4 0.8 0,8 NORMAL I ZED DEPTH BELOW CONTRCT SURFACE Figure 47. Stress distribution: <r x for 0(3)-ISO(6)-0(7) laminate 74 1.0 / o in" X C9 ^ in" ' <n o LEGEND t a- [ ROSS SECTION n o H. 1 0.0 0.2 0.4 O.B o.a NORMALIZED DEPTH BELOW CONTACT SURFACE Figure 48. Stress distribution: a x for 0(8)-ISO(6)-0(2) laminate 75 i.o V. CONCLUSIONS AND RECOMMENDATIONS This study has developed two methods for approximating contact stresses using the augmented Lagrange multiplier method. methods accurately approximate the and plane surface. As illustrated in Part stresses that result This study has also illustrated how A of Chapter IV, these from contact between this a cylinder approach can be applied to understand the behavior of an actual contact problem by examining the response of a composite plate to low velocity impact. RAYLEIGH-RITZ APPROACH A. In the process of developing these methods, a number of comments can be made regarding the application of the Rayleigh-Ritz method to solving contact stress problems. 1. The is an extremely challenging process. If it is dedetermine the deformation in a contact problem, the proper stress field must be first satisfied. Because of this, the selection of possible trial functions is limited. For example, when selecting a trial function for the vertical deformation of the foundation of Figure 1. a suitable selection is given by the following selection of the trial function sired to equation: v{xy) =AX ) cos ( -fff ) This equation exhibits the favorable characteristics of maximum deformation at the contact surface and diminishing deformation as the distance from the contact surface increases. If contact stresses are to be modeled, this trial function is inappropriate. Calculation of t y is as follows: ~ dy- = -/Wl7T sin (ltf ) This function exhibits zero strain at the point of contact increasing to strain at the lower boundary. 2. 3. maximum Since the selection of trial functions is difficult, the task is further impeded by complicated geometries. Furthermore, selection of a trial function necessitates that some knowledge of the deformation field exists. Without a sensible selection of trial functions, an accurate approximation is unlikely. This method assumes the trial function in the form of an infinite series. Solution accuracy theoretically should improve with an increased number of terms. However, precautions must be taken to ensure the solution is numerically stable as the number of terms increases. Since the strain energy calculations require integration, there are choices of trial functions that will increase without bound as the number of terms is increased. This problem can be controlled by normalizing dimensions or limiting the choice of trial functions. 76 4. An increase in accuracy was observed as the number of constraints was increased and the distance between consecutive constraints was decreased. It is believed that the improved accuracy results from a better definition of the contact surface. FINITE B. The ELEMENT APPROACH results in ment method Chapter III illustrated that this to contact stress analysis is effective. approach of applying the A number finite ele- of comments can be made regarding this approach to problem solving. 1. this method accurately approximated the isotropic However, some difficulties were encountered during the bearing problem. modeling of the multi-ply composite. In this model, smooth trends of decreasing deformations were often interrupted by spurious deformations or groups of deformations. These interruptions occurred within layers of significantly reduced stiffness. It is believed that the optimization routine had difficulty approximating the deformations through these layers because of their very small contribution to strain energy. As stated in the results, the contact boundary conditions were obtained from the optimization program and applied to a direct finite element program to solve the problem. The above difficulty is recognized as a limitation of this As illustrated in the results, roller approach. 2. 3. This approach is much more flexible for complicated geometries than Rayleigh-Ritz approach. In addition, detailed knowledge of the deformation is not needed as required by the Rayleigh-Ritz approach. the field The application of static condensation is crucial to the successful implementation method. Every effort should be made to reduce the number of design variables to improve optimization efficiency. of this COMMENTS ON OPTIMIZATION C. A number of observations were Design Synthesis System and the made specific regarding the general use of the Automated usage of the augmented Lagrange multiplier method. 1. The global optimum was more was normalized. likely to be determined when the objective function 2. For both the Rayleigh-Ritz approach and the finite element approach, the initial choice of design variables had a significant affect on the possibility of obtaining the global optimum. For the Rayleigh-Ritz method, initial selections of design variables can result in largely dissimilar values of strain energy and external work, the two components of the objective function. It was determined that convergence was more likely when optimization commenced with these two terms on the same order of magnitude. With regard to the finite element approach, sensible choices of the initial design variable vector was necessary for convergence to the global optimum. This was accomplished by intuitive selection of design variables to model the likely deformation. 3. Solution accuracy can be improved by scaling constraint equations. It has been stated that in some circumstances, some constraints change more rapidly than 77 others and can influence the solution excessively while others have [Ref. 6: p. 136]. 4. little influence With regard to the usage of the augmented Lagrange multiplier method, it was frequently necessary to tune' the optimization algorithm to a specific problem. This was done by varying the initial penalty term p and the initial Lagrange multiplier term / As stated by Vanderplaats, commencing optimization with a small value of p should theoretically suffice for most problems [Ref. 6: pp. 137-138]. However, it was frequently necessary to select an initial value for p due to convergence to unrealistic solutions. Similarly an initial choice of the Lagrange multiplier term can effect the solution. Commencement with a small value is again recommended. [Ref. 6: p. 141]. This need to tune' the problem is a significant drawback to using this optimization method. The ideal way to overcome this limitation is to first tune the optimization routine using a known solution. With this accomplished, this approach can be used for meaningful data collection. . , SANDWICH COMPOSITE MATERIAL STUDY D. The behavior of sandwich composite materials to low velocity impact loading was successfully investigated by the application of the finite element approach. of observations can be 1. is observed. Tensile transverse normal stresses exist at some cross sections away from the contact zone. However, this stress is always compressive at the interface. Compressive transverse stresses increase in beams with smaller cores due to reduced deflection and contact zone 3. the results. The maximum shear stress is relatively insensitive to layer thicknesses. However, as the thickness of the contact layer increases, a reduction of the interface shear stress 2. made from examining A number size. As the thickness of the layer closest to the contact zone increases, a nonlinear distribution of bending stress within this layer intensifies. This phenomenon is local- ized to the region of contact. 4. As the thickness of the layer opposite to the contact zone increases, bending crack propagation toward the core is less likely due to increased compressive bending stresses within the layer. E. RECOMMENDATIONS FOR FURTHER STUDY The methods developed can grow. Chapter II A in this reasonable direction Part B. frictionless surface is study offer a basis from which additional research the relaxation of For example, relaxation of the some of the assumptions rigid roller assumption would provide challenging research. plex geometry could be created. made in assumption and the Models with com- For example, a model of a pin loaded bolt connection could be created with rigid or non-rigid pins. Implementation of these changes would provide a versatile and highly applicable model for contact stress analysis. 78 . . REFERENCES 1. and Kubomura. K.. "Formulation of Contact Problems by Assumed Hybrid Elements,' pp. 49-59: Wunderlich, W., et.al., Fds., Xonlinear Finite Element Analysis in Structural Mechanics: Proceedings of the Europe-L'.S. WorkPian, T. H. H.. - Stress shop, Ruhr-L niversitat at Bochum, Germany, Julv 28-31, 1980. Berlin, Springer, 1981. 2. Cheng. W. Q., Zhu. F. W., and Luo, J. W., "Computational Finite Element Analand Optimal Design for Multibody Contact System," Computer Methods in Applied Mechanics and E/igineering, v. 71, pp. 31-39. November 1986. ysis 3. 4. Nour-Omid, B.. and Wriggers, P.. "A Two-Level Iteration Method for Solution of Contact Problems." Computer Methods in Applied Mechanics and Engineering, v. 54. pp. 131-144, February 19S6. Guerra. F. \F, and Browning. R. V., "Comparison of Two Slideline Methods L'sing Structures, v. 17, pp. 819-834, June 1983. ADINA." Computers and 5. Bischoff, D., "Indirect Optimization Algorithms for Nonlinear Contact Problems," pp. 533-545: Conference on the Mathematics of Finite Elements and Applications (5th: 1984: Brunei University), London, Academic Press, 1985. 6. Yanderplaats. G. N., Xumerical Optimization Techniques for Engineering Design with Applications, / S. McGraw-Hill Book Company, D. A., and Lowe, M. J., Mathematical Programming Lagrangians, Addison-Wesley Publishing Company, Inc., 1975. Pierre. 10. \'ia Augmented Rothert, H.. Idelberger. H., Jacobi. W., and Niemann, L., "On Geometrically Nonlinear Contact Problems with Friction," Computer Methods in Applied Mechanics and Engineering, 9. 1984. v. 51, pp. 149-154, September 1985. and Sun, C. T., "Impact Induced Fracture in a Laminated Composite," Journal of Composite Materials, v. 19, pp. 51-66, January 1985. Joshi, S. P., Sun, C. T., and Rechak, S., "Effects of Adhesive Layers on Impact Damage in Testing and Design Composite Laminates," pp. 97-123, Composite Materials: ASTM STP 972, J. D. Whitcomb, Ed., American Society for ( Eighth Conference) Testing and Materials, Philadelphia, 1988. , 1 1 Choi, H. Y., Wang, H. S., and Chang, F. K., "Effect of Laminate Configuration and Impactor's Mass on the Initial Impact Damage of Graphite Epoxy Composite Plates due to Line-Loading Impact," Journal of Composite Materials, v. 26. No. 6, pp. 804-827, 1992. A. C, and Fenster, S. K., Advanced Strength Elsevier Science Publishing Co., Inc., 1987. 12. L'gural, and Applied Elasticity, 2d ed., 13. Shislev, J. E., and Mishke, C. R., Mechanical Engineering Design, 5th ed., pp. 71-74. McGraw-Hill Publishing Company, 1989. 14. Burnett, D. S., Finite Element Analysis from Concept Addison-Wesley Publishing Company, 1987. 79 to Applications, p. 568, 15. Kwon, Y. W., and Akin, J. E., "Materially Non-Linear Analysis of Body Contact Using a Finite Element Method," Engineering Compulations, v. 3, pp. 317-322, December 1986. 16. Jones, R. M., Stechanics of Composite Materials, pp. 37-41, Hemisphere Publishing Corporation, 1975. 17. Babiloglu, E., A Numerical Study of Dynamic Crack Propagation in Composites, Master's Thesis, Naval Postgraduate School, Monterev, California, September 1992. 80 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Technical Information Center 2 Cameron Station Alexandria. VA 22304-6145 2. Library, Code 52 2 Naval Postgraduate School Monterey. CA 93943-5002 3. Department Chairman. Code ME Department of Mechanical Engineering Naval Postgraduate School Monterey. CA 1 93943-5000 4. Naval Engineering Curncular Office. Code 34 Department of Mechanical Engineering Naval Postgraduate School Monterey. CA 93943-5000 5. Professor Young \V. Kwon 1 2 Department of Mechanical Engineering Naval Postgraduate School Monterey, 6. CA 93943-5000 Dr. Rembert F. Jones, Jr. Submarine Structures Division Code 172, Bldg 19, 1 Room A236B Naval Surface Warfare Center, Carderock Division Bethesda, Maryland 20084-5000 7. Lieutenant Eric Route 2 Box 174 Kenbndge, VA S. McDonald, 23944 USN 1 DUDLLY KNOX uBRARY NAVAL POSTGRADUATE SCHOOL GAYLORD S

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