Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1981 Shape optimization of trusses subject to strength, displacement, and frequency constraints. Felix, Jorge E. Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/20666 rihn :.•-;'' wsSmM wBWm ^• .-"." :.. "!'•' • > < : .-:• RBI USHK mm Hi *:';, i'-'l Kf^OX LIBRARY L POSTGRADUATE SCHOOL "REV, CALIF. 93940 &?G<?s (<k NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS SHAPE OPTIMIZATION OF TRUSSES SUBJECT TO STRENGTH, DISPLACEMENT, AND FREQUENCY CONSTRAINTS by Jorge E. Felix December 1981 Thesis Ac Ivisor; Garret N. Vanderplaats Approved for public release; distribution unlimited T204022 TTNrLA.S.STFTTTl SECURITY CLASSIFICATION OF THIS PAGE rWhott Dot* Bnlorod) READ INSTRUCTIONS BEFORE COMPLETING FORM REPORT DOCUMENTATION PAGE ««port numbTB I 2. GOVT ACCESSION NO TITLE (mnd Submit) 4. »E»IOO COVERED » performing o«o. report numier • CONTRACT OR GRANT NUMBER^ program element, project r*s< AREA «ORK UNIT NUMBERS io. ft Naval Postgraduate School Monterey, California 93940 II. » Felix E. PERFORMING ORGANIZATION NAME ANO AOORESS » TYPE OF «CPO«T Master's Thesis; December 1981 AuTmORi'«J Jorge , S. Shape Optimization of Trusses Subject to Strength, Displacement, and Frequency Constraints 7. <ECl»ltMT JC»T»LOC NUMBfn 1 CONTROLLING OFFICE NAME ANO AOORESS REPORT OATE 12. December 1981 Naval Postgraduate School Monterey, California 93940 NUMBER OF PAGES II. 72 pages U MONITORING AGENCY NAME * AOORESSff/ dlllonmt Itom Controlling Ollleo) SECURITY CLASS, IS. ol ini, rmporl) Unclassified ISa. DECLASSIFICATION/' DOWNGRADING SCHEDULE -L l«. DISTRIBUTION STATEMENT (ol thf Kopott) Approved for public release; distribution unlimited 17 DISTRIBUTION STATEMENT l«. SUPPLEMENTARY NOTES t». KEY »OROS (Conttnv (ol tho mbttracl on toworto •/<*• (/ onft*4 In Block 30, nfooarr "* i#mMt> *T II dltlotont tram Report) Woe* nuMoor) Structural optimization, finite elements, structural configuration, trusses, frequency constraints. ABSTRACT 20 (Continue on rtMNt •<*• II nocooomry md tdmmtltr *T »••«* mmmmor) Three-dimensional trusses are designed for minimum weight, joint subject to constraints on: member stresses, Euler buckling, static Multiple displacements and system natural frequencies. load conditions are considered. _ and The finite element displacement method of analysis is used technique. eigenvalues are calculated using the subspace iteration All gradient information is calculat ed analytically. _ DO , 'jSTjm 1473 • EDITION OF MOV S/N 0103-014- ««01 I I IS OBSOLETE UNCLASSIFIED . Bntfd) SICUHITY CLASSIFICATION OF TNIS PAOE (Whom Dot* UNCLASSIFIED fgewjwvv cluh>'C»^9ii a* rmtt »»qc"»«i— r>„« «.,•„. The design problem is cast as a multi-level numerical optimization problem. The joint coordinates are treated as system For each proposed configuration, the member sizes are variables. updated as a sub-optimization problem. This sub-problem is efficiently solved using approximation concepts in the reciprocal variable space. The multi-level approach is shown to be an effective technique which conveniently takes advantage of the most efficient methods available for the member sizing problem. The optimum Examples are presented to demonstrate the method. configuration is shown to be strongly dependent on the constraints which are imposed on the design. DD 1 U"3 Jan "?:? S/N 0102-014-6601 1 2 UNCLASSIFIED ucu«i»» cuamiuca-ho* o' *•• »*aei^»*« o* Approved for public release; distribution unlimited Shape Optimization of Trusses Subject to Strength, Displacement, and Frequency Constraints by Jorge E. Felix Lieutenant, Ecuadorian Navy B.S., Naval Postgraduate School, 1981 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL December 1981 DUDLEY KNOX UBIt/^Y NAVAL POSTGRADUATE SCHOOL ABSTRACT Three-dimensional trusses are designed for minimum weight, subject to constraints on: member stresses, Euler buckling, joint displacements and system natural frequencies Multiple static load conditions are considered. The finite element displacement method of analysis is used and eigenvalues are calculated using the subspace iteration technique. All gradient information is calculated analytically. The design problem is cast as a multi-level numerical optimization problem. system variables. The joint coordinates are treated as For each proposed configuration, the member sizes are updated as a sub-optimization problem. This sub-problem is efficiently solved using approximation concepts in the reciprocal variable space. The multi-level approach is shown to be an effective technique which con- veniently takes advantage of the most efficient methods available for the member sizing problem. Examples are presented to demonstrate the method. The optimum configuration is shown to be strongly dependent on the constraints which are imposed on the design. TABLE OF CONTENTS I. INTRODUCTION 12 II. MATHEMATICAL FORMULATION 16 A. INTRODUCTION 16 B. ANALYSIS 16 C. 1. Static Analysis 16 2. Dynamic Analysis 18 ANALYTIC GRADIENTS OF THE CONSTRAINTS 19 1. Gradient of Member Stresses with Respect to the Reciprocal of Area Variable 20 2. Gradient of Nodal Displacements with Respect 20 to the Reciprocal of Area Variable 3. Gradient of Frequencies with Respect to the Reciprocal of Area Variable 22 4. Gradient of Stresses with Respect to the Joint Coordinate Variables 23 Gradient of Displacements with Respect to the Joint Coordinate Variables 23 Gradient of the Natural Frequencies with Respect to the Joint Coordinate Variables 24 5. 6. D. APPROXIMATION CONCEPTS 25 E. OBJECTIVE FUNCTION 27 F. CONSTRAINTS 27 1. Stress 27 2. Euler Buckling 28 3 Frequency 28 . G. III. IV. V. 4. Limits on Areas 28 5. Displacement 23 6. Limits on Coordinate Variables 29 GENERAL FORMULATION OPTIMIZATION 29 31 A. INTRODUCTION 31 B. GENERAL FORMULATION 31 C. OPTIMUM GEOMETRY DESIGN 32 D. FIXED GEOMETRY DESIGN 34 NUMERICAL EXAMPLES A. INTRODUCTION B. CASE 1: 18-BAR PLANAR TRUSS 36 36 36 1. Case la 37 2. Case lb 38 3. Case lc 38 4. Case Id 38 C. CASE 2: 47-BAR PLANAR TOWER 39 D. CASE 3: 25-BAR SPACE TOWER 40 CONCLUSIONS AND RECOMMENDATIONS 42 A. CONCLUSIONS 42 B RECOMMENDATIONS 4 3 . APPENDIX A 44 LIST OF REFERENCES 70 INITIAL DISTRIBUTION LIST 72 LIST OF TABLES I. 18-Bar Planar Truss, Loads and Constants 56 II. 18-Bar Planar Truss, Design Information. Stress 57 III. 18-Bar Planar Truss, Design Information. Stress, Euler Buckling 58 18-Bar Planar Truss, Design Information. Stress, Euler Buckling, Displacement 59 V. 47 -Bar Planar Tower. Load Conditions 60 VI. 47-Bar Planar Tower. Load Conditions CI VII. 47-Bar Planar Tower. Area. IV. VIII. 47-Bar Planar Tower. Stress, Euler Buckling-62 Coordinates. Stress, Euler Buckling IX. X. XI. XII. 63 47-Bar Planar Tower. Coordinates. Buckling, Displacement 47-Bar Planar Tower. Displacement Areas. Stress, Euler 64 Stress, Euler Buckling, 65 47-Bar Planar Tower. Areas. Displacement, and Frequency Stress, Euler Buckling, Stress, Euler Coordinates. 47-Bar Planar Tower. Frequency and Displacement, Buckling, 66 67 XIII. 25-Bar Space Tower. Load Conditions 68 XIV. 25-Bar Space Tower. Design Information 69 LIST OF FIGURES A.l. A. 2. 18-Bar Planar Truss. Stress, Euler Buckling, Displacement, and Frequency Constraint 44 18-Bar Planar Truss. Stress 45 Weight vs. Iteration Number, A. 3. 18-Bar Planar Truss. Weight vs. Iteration Number. 46 Stress, Euler Buckling A. 4. 18-Bar Planar Truss. Weight vs. Iteration Number, 47 Stress, Euler Buckling, Displacement A. 5. 18-Bar Planar Truss. Weight vs. Iteration Number. Stress, Euler Buckling, Displacement, and Frequency 48 A. 6. 47-Bar Planar Tower 49 A. 7. 47-Bar Planar Tower. Stress, Euler Buckling, Displacement, Frequency 50 A. 8. 47-Bar Planar Tower. Weight vs. Iteration Number. 51 Stress, Euler Buckling A. 9. 47-Bar Planar Tower. Weight vs. Iteration Number. 52 Stress, Euler Buckling, Displacement A. 10. 47-Bar Planar Tower. Weight vs. Iteration Number. Stress, Euler Buckling, Displacement, Frequency 53 A. 11. 25-Bar Space Tower 54 A. 12. 25-Bar Space Tower. Weight vs. Iteration Number. Stress, Euler Buckling, Displacement, Frequency 55 LIST OF SYMBOLS Cross-sectional area of member A^ A:llcl.X A_. ill -LIl i i Minimum and maximum allowable area variables, respectively , . a,, b. Constants defined by Eq. c Factor defined by Eq. D Matrix of direction cosines E. Young's Modulus for member F Vector of applied loads G. (X) Constraint function of coordinate variable G. (X m 3 G, k (X m 43 31 i ) Constraint function of area variable ) Active constraints of area variable I Identity matrix K Global stiffness matrix K. Element stiffness matrix zi. L. Length of a bar M Global mass matrix M Global generalized mass matrix M_ Lumped mass matrix NAC Number of active constraints NIC Number of initial constraints S Direction vector ~G *L i Direction vector on the coordinate design space S g S m Direction vector on the area design space 9 u Nodal displacement vector u- ith nodal displacement vector ^/ Minimum and maximum allowable displacement value ^ X Vector of design variables X Vector of coordinate design variables X Vector of area design variables m 1th independent design vector X„ X, u, a, Minimum and maximum allowable coordinate variable X, , a*, v ai, a 2 Direction cosines Scalar parameter defining distance of move in the area or coordinate design space. S Objective function of direction function problem 5 Displacement constraint value 57 5. Lower and upper limits on displacement constraints 5 Kronecker delta „ 9. Push-off factor g. Stress in member a stress in member Allowable compressive c ci Allowable tensile stress in member a. a. i p. Material density (J). Eigenvector corresponding to to. ith eigenvalue $ 2 Q X i Allowable Euler buckling stress in member . bi co~, i co i to. Minimum and maximum allowable fundamental frequency Nodal matrix Expectral matrix Frequency limit 10 ACKNOWLEDGEMENT The author wishes to express his deep appreciation to his faculty advisor, Professor Garret N. Vanderplaats , for his continuous guidance, assistance and encouragement as well as his great enthusiasm during the course of this project; to Professor David Salinas who read this project, and gave of his time and engineering expertise; to the distinguished members of the faculty and the staff of the Department of Mechanical Engineering for their support in very way, my sincere appreciation. This investigation would not have been possible without the facilities and support of the W. R. Church Computer Center. Infinite thanks and grateful appreciation go to my wife, Yolanda, my son, Paulo, and daughter, Maria Gabriela, whose understanding and encouragement made the difficult time bearable. 11 I. INTRODUCTION The process of optimization of structures has undergone important changes since its development in the early 1960s. Minimum weight of elastic truss structures subject to multiple loadings has been an active area of research. Attention has been focused on the problem of least weight, when the overall layout was known in advance, and when the crosssectional area was the design variable. Some attention has been directed toward the optimum configuration of the structure. Design improvements in this area often exceed those in fixed-geometry and so shape optimization is of major interest. Pioneer work in shape optimization was conducted in 1964 [Ref. 1] by Dorn, Gomory and Greenberg. The optimal con- nectivity of nodes for truss members, subject to a single load condition, was found and minimum weight designs were achieved. In their work only planar trusses were tested, and the process was presented as a plastic design problem using linear programming. Their work was followed by Dobbs and Felton [Ref. 2] who in 1969 investigated the effect of multiple load conditions on the optimum configuration of trusses through the use of non-linear programming methods. 12 Again, only planar trusses were considered, subject to failure by stress and elastic buckling. Later, Pedersen [Ref. considered the 3] positions of the joints as continuous design variables in addition to the areas of the bars. Stress, displacement, and buckling constraints were considered. Pedersen 1 s work is significant because the optimization process is carried out by considering two separate design spaces. The optimization is achieved by successive iterations using a gradient method with move limits. The optimization process was advanced by Vanderplaats and Moses [Ref. 4] who divided the design space into two sub- spaces, separating the area variables and the joint position variables. Multiple load conditions and constraints on stresses and Euler buckling were considered. The optimiza- tion was carried out alternatively between the two spaces until convergence was achieved. Three-dimensional indeter- minate trusses were designed subject to multiple loading conditions. This work was extended by Vanderplaats [Ref. 5] to include displacement constraints. In other research, indeterminate trusses. Spillers [Ref. 6] considered statically The optimization followed an itera- tive design where the member sizes and node locations were the design variables. 13 Recently, Imai [Ref. 7] treated the sizing and con- figuration variables simultaneously for either determinate or indeterminate trusses. The optimization was achieved using the Augmented Lagrange multiplier method. The design problem considered in this study is the opti- mum configuration of three-dimensional indeterminate trusses, for multiple prescribed static load conditions. The objective is to minimize the weight of the structure, design variables are the node coordinates and the member sizes. Constraints include stress, Euler buckling, dis- placement, and the natural frequency of the system. Some approximation concepts are introduced in order to reduce computational effort and to reduce the nonlinear characteristic of some constraints. Among these, a first order Taylor Series expansion is applied to approximate the constraints. The optimization proceeds iteratively in two design spaces: the member sizing space, where the structure is optimized for a fix layout, and the coordinate space, where the geometry is allowed to vary. In both optimization processes the minimum weight of the structure was maintained, subject to the requirements that the constraints remain satisfied. The mathematical formulation is presented in Chapter II. The objective function and constraints are defined in terms 14 of the design variables. The analytic gradients are also formulated. The optimization technique is discussed in Chapter III. Several examples are presented and explained in Chapter Conclusions and recommendations for future work are presented. 15 IV, II. A. MATHEMATICAL FORMULATION INTRODUCTION Several features are desirable when finite element methods of structural analysis are used in optimization. First, the number of analyses for the structure should be kept as low as possible. Second, the amount of gradient in- formation required during the design process should be reduced as much as possible. B. ANALYSIS Static Analysis 1. The initial layout of the truss, the member sizing and material properties (which may be different for each member), a set of external loads, and support conditions are initially specified. The analysis for the stresses and deflections must be carried out satisfying the conditions of equilibrium of forces at the nodes and compatibility of deformation. If the material of the structure behaves in a linear manner, Hooke ' s law will establish the force-deflection relationship For a truss, it is also necessary to establish the following assumptions before selecting the method of computing the internal forces. The discrete element is treated as pin-connected, and loads and reactions are supported at 16 the joints. In this study, the weight of the members is not included as loads. The Displacement (Stiffness) method [Ref. 8] considers the joint displacement components as the unknowns, and is written in the most general form using matrix notation, Ku = F (Eq. 1) where K is the global stiffness matrix, F is the vector or vectors of applied loads, and u is the vector or vectors of displacements. Equation is the set of equilibrium equa- 1 tions, and is formulated such that the compatibility is auto- matically satisfied. The generality of the method is important if either statically determinate or indeterminate trusses are analyzed. The global stiffness matrix is symmetric and sparce. These features are used to write the code for a computer solution, and the matrix K is stored in compact form for efficient numerical solution. Once the displacements at every node are known, the internal forces or stresses are calculated by applying forceThis is defined as: deflection relations. r E. *n ij = D _ T -VL \ u„ i and j - u. ^ ^ < V £j w„ where . . V kj - w. ) (Eq. 2) . are element and load condition numbers respec- tively, and k and I are node numbers associated with the 17 element and i. a. is the stress, . is the length. L. direction cosines. E. Matrix D = is the Young's modulus, f (a,y,v) contains the For brevity, hereafter the second sub- script is omitted and it is assumed that the stress or displacement corresponds to the appropriate loading condition, 2 Dynamic Analysis . When system natural frequency constraints are imposed in the design process, the corresponding dynamic analysis of the structure has to be carried out. This requires the solution of an eigen-problem to determine the natural fre- quencies and normal modes. For linear elastic structures, the finite element approach leads to the well-known equation of motion, considering free vibration conditions, £Ju + = Ky (Eq. 3) where M is the global mass matrix, and u is the linear acceleration. Assuming a solution of the form ia)t u = ^e where oo (Eq. 4) is the angular natural frequency of vibration of the structure, and $ is the corresponding eigenvector. After substitution into Eq. 3, the generalized eigenvalue problem becomes, K<j> = 2 u) (Eq. M$ Written in matrix form for several eigenvalues, Eq. 5) 5 becomes, KO = 2 tt m (Eq. 18 6) where $ is the modal matrix, and 2 1 = diag 2 (oj ) is the spectral matrix. From the static analysis, the global stiffness matrix is already calculated, then only the global mass matrix evaluation is needed. mass options are coded. Both generalized and lumped These are defined respectively as: NE & o-A.L. = .1 p i l l 1=1 ^ - 2I l 3 I-, 3 (Eq. 7) (Eq. 8) 21 and NE 1 1=1 A.L. ill L. • 6x6 6 VJhere I is the identity matrix, method of Bathe and Wilson [Ref. the sub-space iteration 9] is used to solve for a specified number of lowest eigenvalues and the associated eigenvectors. large problems. The method is economically efficient for The mass matrix may be diagonal or banded. The method is well suited for re-analysis when small changes are made in the design. C. ANALYTIC GRADIENTS OF THE CONSTRAINTS The necessity to compute gradients of the relevent functions in a design optimization process arise from the fact that efficient mathematical programming algorithms require information on derivatives. Furthermore, approxi- mate methods based on a Taylor Series expansion of functions, 19 requires determination of derivatives. Based on the static and dynamic analyses, the gradients of forces, displacements, and frequencies with respect to the reciprocal of the cross- sectional areas and the coordinate variables, are formulated. 1 . Gradient of Member Stresses with Respect to the Reciprocal of Area Variables Stress in a member is defined as F. 1 (Eq. 9) (Eq. 10) (Eq. 11) (Eq. 12) (Eq. 13) also K.ua = . The partial derivative is then K 3a. 3 ax 3X A. l ~l *i + A7 9X-e u^ The first term of the right hand side is zero, then —- — ^— "£ 3X„ 3X„ A, (u. v ~l' I This can be written in explicit form as 3£i s 5s> T E.D. L? l 2. Wt (yi) Gradient of Nodal Displacements with Respect to the Reciprocal of Area Variable Consider the equation (Eq. Ku = F 20 14) The derivative of [K u] with respect to some variable X, is 3 [KU] n In this case X = ^"(F) ax, = 1/A^ £ (Eq. 15} and the loads are constant and in- dependent of the areas (weight of the truss elements are ignored) , then (?) (u) t|- ~-(u) + ? = (Eq. 16) (Eq. 17) (Eq. 18) or § 3X Finally, then 9 3X where K -1 (u) = -k' 1 (K) u 3 ?£ is the inverse of K. It is necessary to compute the partial derivative of the global stiffness matrix K defined as: K = NE E A X -fc± i=l where D. i . [D (Eq. ±] 19) is the matrix of direction cosines defined earlier, Therefore, the partial derivative with respect to the reciprocal of the area variables is defined as: . NE E.Af (Eq. *l i=l i 21 20) . where and is the kronecker delta defined by 5 = 5 = 1 if i = l 6. In practice the full matrix, if i ^ I. Trvr-(K) *i 3 is not actually stored. created directly. are given in [Ref 3 . Instead the product, tttt-(K)u is Details of efficient gradient computations . 10] Gradient of Frequencies with Respect to the Reciprocal of Area Variable Consider the eigen-problem defined by, [K - co?M] = 0. (Eq. Taking the derivative with respect to the variable X 21) [Ref. 11] gives 4dX •£ 9 {[Ka - 1~ (Eq. = $.} -1 u)|M] 22; then - u>?M]} ^|- {[g T Pre-multiplying through by symmetry of the matrix [K - oojM] g|- to. *I « lt h »!« i = (Eq. 23) gives M ^"^ 3M 3£ " <J) and applying the condition of <£. - Ik + $. { ] m 2 '*i ~ 3u>. 1 3X^ » (Eq. *i 24) The left hand side of the equation is zero because of Eq. 21, and thus 3 3w? X l 3M K w _3x £ T f *1 i Mi~l * 22 ax Ji (Eq. 25) - The partial derivative of the generalized mass matrix with respect to the reciprocal of the area variable is defined as NE 3.M ~G " 3 p.AfL. ill I ?i 2I °i£ i=l I 3 3 h (Ec. 2I 26) 3 The partial derivative of the lumped mass matrix with respect to the reciprocal of the area variable is formulated following the same procedure as the generalized mass matrix. 4. Gradient of Stresses with Respect to the Joint Coordinate Variables Since stress in a nember is K.11. = a. ~i (Eq. 27) (Eq. 28) the gradient is calculated as «£ 8X„ In this case, —?i 3 1 (Jl_k. u. ) ] + _i_ k. 1 1 ~l the stiffness matrix is a function of the coordinate variables so the first term on the right hand side of Eq. 28 is not zero. 5 . Gradient of Displacements with Respect to the Joint Coordinate Variables Consider the following equation Ku = F (Eq. 29) (Eq. 30) The gradient of u with respect to X I is: (u) **i = -K -1 JL-(K)u **l 23 . The partial derivative of the global stiffness matrix K now is defined as: NE n A.E. 3A.E. 9K. (K) l) (dX l| frJ i 3 i ) [K i (Eq. where (-1) * and j ] 31) = 1 or 2 defines the sign of the gradient at that particular node. Note that the terms in the summa- tion are only evaluated for members connected to the joint defined by the variable X ? 6 . . Gradient of the Natural Frequencies with Respect to the Joint Coordinate Variables The value of the derivative of the eigenvalue co 2 found from 2**1 A u>fM]<j>. [K - _ = n (Eq . 32) (Eq. 33) is 3M 3K 3X, 9o)f CO' SX •i T <f>7 M h. I . c|). The partial derivative of the generalized mass matrix with respect to the coordinate variables are defined as: NE . 3X £ where (-1)- and ""I i=l 1 j ("I) l o A dX l li . . 6L. 2I 3 I3 I3 2I 3 (Eq. 34) l = 1 or 2 defines the sign of the gradient at the particular node. The partial derivative of the lumped mass matrix follows the same procedure. 24 D. APPROXIMATION CONCEPTS At this stage the analysis tools necessary for the design process under static and dynamic conditions may be regarded as available. It has been pointed out that the application of mathematical programming methods for structural design is the most widely used because of its great generality and its simple formulation. However, it is required that a large number of structural analyses and sensitivity analyses be performed. This has motivated the idea of formulating simple and explicit approximations for the most relevant response quantities. These approximations can only be expected to be of acceptable quality in some finite region of the design space surrounding the point about which the approximations were constructed. The total number of analyses required to find an optimum design using approximation concepts is significantly less than the number previously required. Some sources of simplification can be considered. First, the dimension of the design space can be reduced if a proper subspace can be identified. Second, linking of design variables which is imposed because of symmetry or practical considerations also reduces the dimension of the design space. The objective function for the design of trusses is a relatively simple explicit function of the design variables. 25 On the other hand nonlinear constraint functions are very complicated. From a computational standpoint, a small portion of the constraints play an active role in the opti- mization process; therefore, deletion of non-critical constraints avoids effort of evaluation of irrelevant constraints. A method to deal with complicated constraint which is very effective in reducing computational effort is to deduce simple and explicit expressions for the constraints. Linearization is directly and efficiently accomplished by Taylor series expansions. The order of the expansion selected is decided based on the degree of nonlinearity and the approximation required; and a trade-off must be made between the computational effort required for the highest order derivatives versus improvement of approximation. Application of a first order Taylor series expansion on stress constraints has been found to be sufficient. This was not the case for the natural frequency constraints, which may still be numerically unstable during the optimization. order. This suggests further expansion up to a second However, the computational effort needed to do this usually exceeds the benefits. Therefore, a first order expansion with move limits is used. First order Taylor series expansion of a function, W, of the variable, X, about a point, Xo, is written as W(p)* W(x) = W(Xo) + (X-Xo) VW(Xo) 26 The philosophy underlying the use of linear approxima- tions is not to transform the problem into a sequence of linear programs, but to replace the constraints by simple and explicit approximation functions. In this study the Taylor series expansion was used for all constraints with respect to the reciprocal of the member sizing variables, A.. The objective function, in reciprocal space, is now non- linear but still explicit and easily evaluated. E. OBJECTIVE FUNCTION The function whose least value is sought in the optimi- zation procedure is defined as the total weight of the truss which is given by NE W = V p.A.L. .£-,111 (Eq. 35) 1=1 Where NE is the total number of members, and p^ is the material density, both are prescribed constants. L. F. A^ and are the area and length of the ith member respectively. CONSTRAINTS The restrictions to be satisfied in order for the design to be acceptable are formulated explicitly; behavioral and side constraints are defined accordingly. 1. Stress O . ci < — a . l < — (Eq. cr ti 27 36) . . . where o^^ and a ^ are the allowable compressive and tensile stresses respectively for member i. Euler Buckling 2. The Euler buckling stress is formulated as cA.E. = " -q^- a bi <=*• 37 (Eq. 38) > where c is a prescribed constant. Frequency 3. W where ^ W K ^ ^K K and w K are the lower and upper bounds respectively oj on the first natural frequency of the system. 4 Limits on Areas A mm where, A mm l . , . < . < A. i . and A max A max (Eq. ^ 39) are the minimum and maximum allowable l cross-sectional areas of the ith member, and are taken to When symmetry of design is to be the same for all members. be preserved, linking of the variables is required. This is defined as Aj^ where K and 5 i = A. (Eq. 40) (Eq. 41) are symmetric members. Displacement < ll < u* are the lower and upper bounds respectively where u« and u v K K on joint displacements u . 28 , 6. Limits on Coordinate Variables X X K " K ~ XK < EcI- 42) where again X and X are the lower and upper bounds K K respectively on the Kth coordinate variable, and if symmetry must be preserved, linking of the variables is required. This is obtained by X i = a i + b l X K < where a and b^ are constants and X £ ECI- 43 > is the coordinate variable. G. GENERAL FORMULATION The inequality constraints are of the form, G (X) . < j = 1 NIC (Eq. 44) The constraints are normalized as follows: Stresses a -1*0 a -1*0 (Eq. 45) ^--1*0 (Eq. 46 °c Euler Buckling °b Note that because a and a, depend on the member area this constraint is treated as a nonlinear function. Taylor series expansion is performed on of a, is continually updated. 29 a A. The and the value Displacements -1-0 — 6- A5 Frequency 1 < (Eq. 47) (Eq. 49) (Eq 50) + £-1*0 00 side constraints are \ " Ai " ° x - X K x K < X K " K " 30 ° * OPTIMIZATION III. A. INTRODUCTION The main goal of structural engineering is to design structural systems that efficiently perform specified purposes. Selection of a specific algorithm is required and this algorithm must minimize the number of times the structure has to be analyzed and the amount of specific Finally, the algorithm gradient information required. should provide reasonable assurance that it will attain an optimum or near-optimum design. The next two sections are a brief explanation of the algorithm used for this work. B. GENERAL FORMULATION The general constrained minimization design problem is defined as minimize W(X) (Eq. 1) (Eq. 2) subject to G. (X) where, W(X) ^ = 1, j . . . .m is the objective function. the set of inequality constraints. Functions . y 31 are The vector of design variables X includes member sizing variables X metric design variables X G. (X) and geo- OPTIMUM GEOMETRY DESIGN C. The procedure used here was to treat the geometric design parameters as independent design variables. The member sizing parameters are handled as dependent variables which are determined as a sub-problem. Beginning with an initial geometric design vector X the , design proceeds iteratively using the following relationship: X g+1 = q X + a* S* g g g (Eg. 4 where q is the iteration number and a is the search direction S y * to be determined. 3)' v g is the scalar parameter determining the distance of travel in the design space. For each proposed geometric vector, X , the structure was optimized with respect to the member sizing variables, X , by the sub-optimization problem defined in Section D. Assume that for the initial geometry the structure has been optimized with respect to the cross-sectional areas, and that, from this subproblem there are I active constraints of the form: G. (X) where, G.(X) = k - 1,...£ (Eq. 4) is defined as active if its value is close to 3 zero. Now, it is necessary to find the search direction, S so that by moving in this direction in the coordinate design space, the objective function is minimized. This direction is found by solving the following subproblem. 32 , y . Minimize VF(X).S (Eq . 5) (Eq. 6) (Eq. 7) Subject to: G R (X) S k = 1, < . S . S * 1 £ At this point, S, provides a search direction in the combined [X constraints. Once, kj + S space which is projected onto the active Only the S part of is used. is known it is substituted back into Eq. , and a one-dimensional search on a X. the vector, S y 3 * is performed to update The Optimum-Geometry design problem can be summarized in the following algorithm. Given an initial coordinate design vector, X 1. area design vector, A 3 o . Specify, D 2 r max and P max . ' (D , and max', maximum change in the coordinate at each iteration, and P , total number of iterations.) Solve the fixed geometry problem and calculate the 2 minimum weight (W*) for the current geometry. 3. Determine the set of active constraints. 4. Determine the search direction vector, 5. Find the move parameter S. * a, for a D change of some coordinate variable, or such that some coordinate constraints becomes active. Solve the fixed geometry problem for coordinates. 6. X g = X g and calculate the optimum weight, W(X). + a, S 1 g 33 . * Find the parameter a-# which minimizes the weight 7. using 2-point quadratic interpolation. step If a. >a. do to 9. Solve the fixed geometry design problem for coordi* -a -a+1 -a = X^ + ou S M nates X^ and calculate the optimal weight, r 8. g g 9. g Check convergence; if satisfied, terminate; otherwise, set q = q+1 D. 2 , update X and return to step 3 FIXED GEOMETRY DESIGN As stated earlier, a sub-optimization problem has to be solved for a proposed geometry design vector, X. The structure is now optimized based on the cross-sectional This is defined in general as: area subspace. Minimize, W(X) (Eq. 8) (Eq. 9) Subject to, G. (X) < j = 1, I The design proceeds iteratively. of design variables X , Given an initial vector find the X vector at the (q+1) ith iteration defined as: X<3 m * where, a m +1 = q + X* S q X m m (Eq. m - , - is a scaler multiplier, and, S m , 10) is a vector move direction in the design space. Now the problem becomes one of finding the direction, S, and the move parameter, a*. 34 Zoutendijk [Ref. 12] shows that the direction may be found by solving the following problem. Maximize 8 (Eq. 11) (Eq. 12) (Eq. 13) (Eq. 14) Subject to VF(X) G. (X m S S . S . S + . ) < + - 3 9 6 < j = 1,...NAC. 1 where the scalars 6. are named as "push-off" factors. If some of the constraints are violated, this algorithm is modified in order to find a feasible design [Ref. 13]. For the fixed geometry sub-problem, the reciprocals of the member sizes are used as design variables and approxima- tion techniques are employed [Ref. 5] . The optimization program CONMIN [Ref. 14] is used to solve this sub-problem. 35 IV. A. NUMERICAL EXAMPLES INTRODUCTION Design of planar trusses and space towers are presented here and the corresponding numerical results are summarized to demonstrate the purpose of this study. The examples begin with an 18-bar truss. In this case, it is shown how the optimum geometry is dependent on the constraints imposed. A single load condition was considered and the cross-sectional areas are linked. Next, a 4 7-bar planar tower is designed to of load conditions given in Table VI. support a set The design is subject to constraints on the member stresses, Euler buckling, dis- placement and first natural frequency. Linking was imposed for symmetry in both cross-sectional area and coordinate variables. Finally, a 25-bar space tower was designed. Constraints on stresses, Euler buckling, displacement, and frequencies were imposed. The truss was required to support two different load conditions. Symmetry for the member areas and coordinates was established. B. CASE 1: 18-BAR PLANAR TRUSS A cantilever truss, as shown in Fig. A.l, has been used previously as an example for the design of trusses of 36 a ; specified geometry [Ref . 7] This structure was designed . for optimum geometry subject to a single set of load condi- tions given in Table The allowable stresses are speci- I. fied as -20,000 < a 20,000 psi < i Young's modulus is taken as 10 p = 0.1 lb./cu. in. psi and the material density The allowable stress at which Euler buckling occurs is a. b =-SM L2 The independent coordinate variables were taken as X3, Y3, X5, Y5, X7, U7, X9 , in the following groups: Y9 The member areas were linked . A1=A4=A8=A12=A16 A3=A7=A11=A15; A5=A9=A13=A17 . ; A2=A6=A10=A14=A18 There are a total of eight independent coordinate variables and four independent area variables. 1. Case la The structure was designed subject to stress constraints only. The resulting geometry is shown in Fig. la, and the design information is given in Table II. iteration history is plotted in Fig. A. 2. Weight versus The number of analyses for this design is 59, and the execution time 2.53 seconds on an IBM 30 33 computer. 37 Case lb 2. Stress and Euler buckling constraints were imposed for this case. Design information is given in Table III. Figure lb shows the final geometry layout and the weight versus iteration history is shown in Fig. A. 3. The number of analyses for this design is 78 and the execution time was 3.94 seconds. Case lc 3. This design is based on stress, Euler buckling and displacement constraints. number The latter was applied at node in the Y-direction. 1 Table IV. The results are shown in Figure lc presents the final layout and the weight versus iteration is plotted in Fig. A. 4. The number of analyses is 91 and the execution time was 3.94 seconds. 4. Case Id This final case includes all the constraints mentioned before plus a constraint that the first natural frequency be greater than or equal to 3 Hz. A non-structural mass of fixed value W = 1,000 lbs was placed at node 1. Steady convergence is achieved and results are summarized in Table V. The final geometry and weight iteration history are shown in Figures Id and A. 5, respectively. The number of analyses is 96 and the execution time was 4.7 3 seconds. When the mass is removed from the structure, the design fails to converge even when move limits are imposed. First, it is known by definition that the frequency is 38 proportional to the stiffness and is inversely proportional to the mass. Second, both the K and M matrices are functions of area and coordinate variables. Therefore, as K increases, M also increases, and the frequency is kept closed to the initial range. CASE C. 47-BAR PLANAR TOWER 2: The initial layout of the tower is shown in Fig. A. 6. Stress constraints were imposed as well as constraints on Euler buckling, displacement, and first natural frequency: -15,000 < a < i 20,000 psi i=l,47 and IO.IttEA. ° 0i ' bi = " 1^ psi l X * 12 hz. respectively. The members are assumed tubular with D/t=10. Young's modulus of 3.x 10 & = . 3 Ibs/cu. in. imposed. 7 psi, and material density Minimum allowable area of 10 — fi Symmetry about the y-axis was desired during optimization so linking of variables is necessary. 15, 16, in. was 17, 22 are kept fixed, to move in the x-axis direction and joints (Y=0) . 1 and 2 Joints allowed This gave a total of 27 area design variables and 17 coordinate variables. Nonstructural masses were attached at nodes 19 and 20, of W=500 lbs each. through XII. The results are tabulated in Tables VII The final geometry for the case where all 39 — : constraints were imposed is shown in Fig. A. iteration history is shown in Figs. A. 8 7 and the through A. 10. When all constraints are imposed, the design required 220 analysis and the execution time was 160.7 seconds. CASE D. 3: 25-BAR SPACE TOWER The 25-bar space tower shown in Fig. A. 11 was designed to support two independent load conditions given in Table The allowable stresses were specified as XIII. -40,000 < a <; i 40,000 psi i-1,25 Young's modulus was selected as 10 density p = .1 lb./cu. psi and the material The members are assumed to be in. turbular with a nominal diameter to thickness ratio of D/t = 100., so that the stress at which Euler buckling occurs is 10.1 ° bi tt EA. ~ psi - iZf Symmetry with respect to both x-z plane and the y-z plane was imposed, so linking of variables were made as follows: the member areas were grouped in the following sequence Al, A2=A3=A4=A5, A6=A7=A8=A9 A10=A11, A12=A13, A14=A15=A16=A17 , A18=A19=A20=A21, and A22=A23=A24=A25. X4 1 , Y4 and , 2 Z4, X8 , For the coordinates Y8 were considered as variables. were held fixed and joints to lie in the x-z plane. were attached at nodes 1 7 The joints through 10 were required Nonstructural masses of W=500 lbs. and 2, 40 respectively. The first , natural frequency was limited to a value A ^ 16 Hz. Results of this example are shown in Table XIV and the iteration history is shown in Fig. 41 A. 12. V. A. CONCLUSIONS AND RECOMMENDATIONS CONCLUSIONS The final layout dependency on constraints has been presented for the elastic design of trusses for optimal geometry. The truss may be planar or three-dimensional and may be indeterminate. sidered. Multiple load conditions were con- The design procedure was separated as: analysis, design for fixed geometry, design for optimum geometry. The displacement method for static analysis and the subspace iteration method for dynamic analysis were applied. The sequential optimization based on two design sub- spaces present substantial advantages in the reduced number of analyses and allow the designer to keep control of the optimization process. Several examples were considered. In every case, cant weight reduction was efficiently achieved. signifi- Also, the geometries obtained appear quite acceptable from an aesthetic as well as structural point of view. The graphs of Weight vs. Iteration number show that an acceptable design can be achieved in few iterations. 42 B. RECOMMENDATIONS The following recommendations may be of theoretical and practical value. 1. Weight due to the structure itself and other design dependent external forces can easily be taken into account and should be considered in future studies. 2. Application should be made to reasonable sized structures such as offshore towers and long span roof trusses. 3. The principles and the procedure described herein can also be used for optimal design of frames as well as other structures. 43 APPENDIX A @ Figure 1. r © 18-Dar Planar Truss. Initial Geometry. 4,780. (lbs) Hf« 4,524. (lbs) Wi= 6,430. (lbs) Wf- 5,724. (lbs) W1= 7,413. (lbs) Wf» 6,465. (lbs) HI » Figure la. Stress, Figure lb. Stress, Euler 3uckling. Figure lc. Stress, Euler 3uckling, Displacement. Wi= 13,686. (lbs) Wf= 11,528. (lbs) Figure Id. Stress, Euler Buckling, Displacement, and Frequency. Figure A.1. 18-Ear Planar Truss. Stress, Euler Buckling, Displacenent, and Frequency Constraint. 44 i a a z £3 o •r» -M •o M (V P en 0) ID a: > H •H <r z CE -J 0. M &. CE PQ M a it} <d <-t I a. GO u W u> M V CO (0 ai I ao M (V -J <N - (Sqi) %m6^©m 45 •H -1 2! 1 a a z a o •H V -01 (0 M a> p H 0) en • (0 tr (. > • ID -Q E D c o: CE z: CE -J a. c o •r- -p (0 c a CE m (3 •H •H <- 0) -< CJ* 3 t CQ (A W) W 3 M H 0) 9 CU M <d a l""J <tf i O 3 (0 rH <tf a. M M C/l CD (0 CQ 1 CO f~ en CD m <0 •V s a m s cu a s »< S s a to (0 CO (O CO s a (9 (sqi) *q6i3M 46 s 83 a in a.0 3 s a CD i>- in in 0) M 3 cr •H • M 0) .0 -Tfrl 1 a 3 Z • a o •H V (0 M in tn H o _Q E DC (L C Z c o CE -J a. "*" •p (0 c © CE +> m l-H (0 rH > J o> •H Q Ol a •H V H .* • 00 W a M M 3 H O a <D .H M <tJ 6 (0 CD •H k •P H i y <V t &. a a v <u 4-» H q: -M « W w (V CU M U CO <0 cq • M GO 1— L § m in M § S s in § IV. in IS. IN. to to (sqi) imBism 47 CO in M US <0 u (V -Q iM a 9 z t3 a (0 * a o •H V 21 ID U U) -M a <U a (V o <v (0 •M (H H a, 01 if) • 81 ID H QH CE 8 Z CE e © •«-» JQ -fiS •H C • +> to i. c 4* H I W £ 3 £ O CE PQ W > Q C •H r-l V X 9 33 W) w 9 M H M a M 0) H3 W «1 (d w • r-» (V >i M M o a V) a) o« GO •H M (0 » 1 00 ^ in i i*- a IO a IV. in en CO in cu CO a Q CO 1 . 1 in 3 IN. in CU cu <sqn a in cu cu ii|6lom 48 o s 8 CU —»3 1 o in IN. *4 a in •* in cu ^i 4 M 3 H • M 3 O! a> Q) M -a Cu Synim. 600" ? { ! I? 7 IS 14 © Top Sub -structure /JJ [ iy 3 ®. t A/^ / ©« xX \ ©- 420" 360" 35 \ 43 / Yy\ 45 G>; -*| 60" Figure A. 6. 4 1" 60" |«— 47-Bar Planar Tower, 43 240" - 120' 40 41/N. /^ - V^ \,47 44 Sub - s true ture 38 \42 39 480" WX 33 ©1 \37 3 Bottom 10, 29 34 540" Figure A. 7. 47-Ear Planar lower. Stress, Euler Buckling, Displacement, Frequency. 50 a t81 3 » SB £3 o •H 91 «J M M • • > 21 en JQ 01 £ D C C O 8 +» J3 tr> a> H JC t BQ •H 3 M <u * e-t fl c ID 9 V +> w a •H M « a 3 U a> H9 W % f-i <w cu M M tn (0 • 1 u r- (V * A - s s in CVI s V 63 M 8 5 CO cu 2 .0 63 63 ru ru cu (sqi) imBiom 51 M 9 •H 0S i a a Tl 2 (3 61 O H 4-> <0 M 91 <y H a o (0 f-l a. «i •H M > O U O H 55 M J3 D> H D C 01 cm 8 i. c -p a (V • CO •H m u a> •p o> •H rH .* C o Q a U 3 M a> H H 9 » M (0 a <TJ iH 04 M * W W <1> M M cn (0 CU 1 (9 63 Q CD CO ID CO IO 53 O IV. 53 53 * 53 Q *4 T (sqi) "*• 53 53 00 co a in eg 53 53 CM 53 cn CO CO cu 53 m6i9M 52 f^ V J* -U .3 53 53 53 53 CO cu CO CU • M 0) M 9 o> •H 1 a zz z 81 a o £3 •H 0) (d M 0) 91 H t > frl &. © JQ 21 E D C P JS o* •H 0) 3 a u :0 H a. •H Q % cr> c •H H .* U 3 t 01? la I c o M 0) 0) H •^» +> 8 <0 &. o •p 9 n H W3 M (0 a (d % W U] • d) >i H cu M M en (d o V - M r- 0) M cr -U cu (V M 9 cr (sqi) imBiqm 53 u* i 2 .0 t u a 0> 3 •H fa <i> Figure A. 11. 25-Bar Space Tower, 54 • >1 C u c 1*1 r a a <D 3 3 * CP •P <D (3 u O Cm K *. 4J on u o C c <U a) a e 0) 0) o u (0 (0 .H rH 0) en •H •H & Q h- u u CE EL tn tP Oi H M O 3 o 3 .q J3 -Q M U U a; ai J3 O 3 < > 1 - J 8 C O J t < > • i c -9 3 „ < > I : : 01 1 01 ' 1 a) n < > ! t i +> (0 C © W Ul E D C M J5 a» •H 'if s rtJ Ha. Q " - > en CVJ a> a •H H .* U 3 CO • Ul (V s 3 /F 1 4J w 3 M ui in > < r-i W 01 * u> c a) o . •H a> • 0) > 1 ) < c 0) • a u Ul •H -H .* i m • -01 :: c 01 <V +> •H r^ » m M :: a «. 3 -P (D a a, u a: CE •H -21 • +J a a> s u 0) H 3 H w cu % u w <0 </> t a* a> >1 en H o u -P en <w 3 <d < aj t > 1 — -2 in CN CN —^La - § (0 o in in 63 in I s in T ! a a T L 1 s co in en -J «£_1_ a a 29 in 3J CO ru ru (sqD ^q6^9M 55 _i a m — 3 M cn •H • M <v A a» a» M En TABLE I 18-Bar Elanar Truss. Loads and Constants LCAE CONDITION Jcint (lbs .) Ex Ey 1 0. 2 0. -20,000 -20,000 4 0. -20,000 6 0. -20,000 a 0. -20,000 TAELE OF CONSTANTS Young's Modulus E= Allowable Stress 5" = Density P Buckling Coefficient = 1EC8 psi. (.69E08 KN/m 0.2E05 psi. (.138E06 KN/m ) lbs/cu.in (.276E04 Kg/m ) (.276E09 KN/m ) 0. 0. c= 56 1 .4E08 psi. ) TABLE II 18-Bar Flanar Truss, resign Inf or nation. AREA* Member A1=A4=A8=A12=A16 A2=A6=A10=A14=A18 A3=A7=A1 1=A 15 A4=A9=A13=A17 (sg.in.) Initial Final 10.0 1 15.0 15.07 5.0 a. 54 7.07 5.33 COORDINATES 1.05 (in.) Initial Joint * Stress Final X y X 3 000 0.0 99 5 750 0.0 745.88 7 500 0.0 494.61 34.538 9 250 0.0 249.54 23.609 1. y 17 19.686 15. 12 Optimum Weight for Initial Geometry = 4,780.5 Optimum Weight for Final Geometry = 4,524.7 (lbs.) (Its.) Areas ars the optimum values fcr the initial and final geometry. 57 TAELE III Stress, Euler 18-Bar Planar Truss. Design Information. Buckling. AREA* Kember A1=A4=A8=A12=A16 A2=A6=A10=A14=A18 A3=A7=A11=A15 A4=A9=A13=A17 (sg.in.) Initial Final 10.00 11.34 21.65 19.28 12.5 1C.97 7.071 CCOBDIMTZS (in.) Initial Joint z * 5.306 Final X y y 3 1000 0.0 994.57 162.31 5 747.36 102.92 750 0.0 7 500 0.0 482.90 32.962 9 250 0.0 221.71 17. 105 Optimum Weight fcr Initial Geometry = 6,430.7 (Its.) Optimum Weight for Final Geometry = 5,713.C (lbs.) Areas are tie optimum values for the initial and final geometry. TABLE IV Stress, Euler 18-Bar Planar Truss. Design Information. Buckling, Displacement. AREA* Member (sg.in.) Initial Final A1=A4=A8=A12=A16 A2=A6=A10=A14=A18 A3=A7=A11=A15 14.18 16.27 21.66 20.06 12.5 11.18 A4=A9=A13=A17 10.32 COORDINATES Joint 7.863 (in.) Initial x Final x y 5 750 0.0 962.19 703.95 7 500 0.0 452.7 9 250 0.0 208.48 3 1000 0.0 Optimum Weight for Initial Geometry Optimum Weight for Final Geometry = = 7,413.1 6,466.9 y 109.93 55.309 27.307 - 2.884 (lbs.) (lbs.) * Areas are the optimum values fcr the initial and final gcemetry. 59 TABLE V 18-Bar Planar Truss. Design Information. Stress, Euler Euckling, Displacement, Frequency. AREA* Hember A1=A4=A8=A12=A16 A2=A6=A10=A14=A18 A3=A7=A11=A15 A4=A9=A13=A17 (sg.in.) Initial Final 31.66 2S.66 40.58 35.29 12.5 13.03 21. 10 16.09 CCCEDIHATES Initial Jcint * (in.) Final X X 1 000 0.0 972.52 88.0 750 0.0 717.08 27.8 4 500' 0.0 250 0.0 468.75 200.98 y - 3.91 -19.29 Cptimun Weight fcr Initial Geometry = 13,686.0 (Its.) Optimum Height for Final Geometry = 11, 528. C (lbs.) Areas are the optimum values for the initial and final geometry. 60 TABLE VI 47-Bar Planar Tower. Load Conditions LOAD CONDITION 1 Fy JOINT Fx 17 6, COO. 22 (lbs.) -14,000. 0. 0. LOAD CONDITION 2 (lbs.) Fy Joint Fx 17 0. 0. 22 6,000. -14,000. LOAD CONDITION 3 Fx Joint (Its.) Fy 17 6, COO. -14,000. 22 6,000. -14,000. 61 TAELE VII 47-3ar Planar Tcwer. Heater Area. Stress, Euler Buckling. Initial irea * Final (sg.in.) Area * 3 3.764 2.727 It 3.315 2.468 5 0.787 0.727 7 0.864 0.2 8 0.856 0.938 10 1.754 1.076 12 1.691 14 2.087 1.188 0.695 15 1 .52S 1.058 18 2.087 1.412 20 0.648 0.263 22 0.843 0.811 24 1.700 1.060 26 1.700 1.052 27 1.354 0.820 28 0.847 0.302 30 3.609 2.766 31 1.435 0.657 33 0.638 0.207 35 2.842 2.697 36 0.676 0.266 38 1.596 1.408 40 3.686 3.429 13 41 1.526 0.991 43 45 0.677 4.486 0.170 3.650 46 1.532 1.005 * Optimum areas for initial and 62 (sg.in.) final geometry. % (3 O "1 (N O O o CN » o 03 <n <T in •H O • o\ «— o f r» ft CN ^ vO » c* in <J\ CN o vO o O CN OO o .0 03 M <9 * o •P j» % en * CN ^» II l •3 a - ti -a M o o u CN X m CO <x> rn r* m C7» in r» o •• m CN J id 4T •» o> ^ * o o <J\ o ^ o in r» o • CN CN o> *— -p CO M M a> a o r^ - >% id C3 Dm •H (/) w -3 fO a CD •H •a H M o o 3 — a t •H ** >• o o o o o o o O o o a o o o o a o O o o 04 .» CN CO r» r» •» o o o o * (N en » * m m -O « SO W » O b-t M o CJ r^ * o o O o o o o o O o O jS vO vO ro en m m <J> o in C7» f» «CN CN CN •i-l P a CO fi4 J-l o n a o •-» a •i-j X, U o M O <M <H +» •H -c 3 G 03 a> en s * o ro a H 01 1-t - M •H J3 o> « s •a <D u a* 'D P >i +> 03 O i- CN r- I 63 lO «- O CN a 3 a v o> o r-t <a a •H V o. o o a 00 CM vn f*1 •H 1* r-l M o O 3 CN m <X> (V o as o o as tN m p* »" O o o o 03 -o 03 o w <* M 10 a 0) iH 3 •O U O o W W 00 as 00 o 0"» 0"> o ao o o o in o o ut CN m a o • t . . a? o in a* o> •" o\ •" J* a o <v u M P H CO X w J a Q) (1) a a •H U M o o o a O o O o o o O cj CN » vO CN ao ;T r* o o o o *" CM m » » m in VO o o CD • O <d »-i C3 •rt H u o w w -a en (0 •H •H •H a a, M O O a 0> 10 (/) <d (!) •H J c M M V a •ri V a M O O u o o O o o o m • o c C7> M (0 C (0 CU Q0 N » O *" r" f— (0 m » 64 XI f» <(N CN CN W o 'M J ja Ol •H (V 3 a 9 a 9 a a •H •H •fJ O 61 it. 3 Q. rr) o (0 V a. o TABLE X 47-3ar Planar Tcuer. Areas. Stress, Euler Buckling, Displacement. Hemter Initial Area * (sg.in.) Final Area * 15.28 4.478 4 16.55 3.671 5 7 3.066 2.691 0.561 8 2.435 1.156 10 3.706 1. 170 12 5.263 2.318 14 4.048 0.669 15 5.077 1 3 (sq.in.) 1.037 .385 18 4.780 1. 169 20 1.650 0.562 22 2.231 1.295 24 4.665 1.526 26 4.816 1.511 27 3.693 1.036 28 2.923 0.566 30 17.91 5.231 31 7.091 0.949 33 2.566 0.558 35 17.59 6.175 36 6.640 1.217 38 3.385 1.076 40 28.11 7.730 41 6.792 1.570 43 2.701 0.555 45 37. 11 9.878 46 7.363 2.153 * Optimum ares for initial and final geometry. 65 ) TABLE XI 47-Bar Planar Tower. Areas. Stress, Euler Buckling, Displacement, and Frequency. Henter Initial Area * (sg.in.) 3 16.59 4.825 4 15.37 4.605 5 1.321 0.S38 7 1.871 0.348 8 1.356 1.081 10 5.152 1.089 12 8.593 3.025 14 5.988 1.088 15 8.468 1.691 18 8.593 1.656 20 1.416 0.635 22 1.836 1.724 24 8.470 2.480 26 8.470 2.186 27 5.999 1.342 28 1.949 0.452 30 14.54 4.833 31 7.096 0.855 33 1.709 0.307 35 36 9.710 1.414 4.579 0.368 38 1.385 1.439 40 10.43 4.3 96 41 1.484 1.126 43 1.465 0.270 45 11.17 4.365 46 1.482 1.664 Final Area * (sg. in. * Optimum areas for initial and final geometry. 66 en rt O <7\ •H * H .* iT» r-1 (N f* U 3 r*" * m * in CM CM T- o O in (N m » 03 r» •=r ro in <T> *— in O o r* m "> (N r» o m o oa rn o •o lO o r* _ u • • </) ji r-l (U in CM M 0) <H Itf o 3 •«-» » a • * </] w a> H H X W 0) u +j en o* M O o * o © 9 o O o * in o CM o o o -3 m o o on o o oo 0} r» m m 0% »— 43 <7s cr» (J X <T> 0"l 43 o o in o> *— 00 0"> r-l 10 a Dm - u t M <D X (J H « •P ^^ • c •H ^^ » >i O o o o O o o o o C3 o o o o o o o o O o o :N * SO CN CO » r* o o o T~ CO » CM in •» m H3 o ^o a >« a a o +J <u M HI (9 a o r-l O r-l •H a a •H M U o IX* u a >*-i (t> a a JS M — (V u •H -*i •rt CO V <V H O o a rH o< » T •O W u •H fl U +' a 3 s a a a 3 a in r— •H •H • x o o o o o o a o o o m a* o o *o V0 m rn rn CT> •P a. o id c (0 Oj © o n t- i 67 CN t- 3- «- lO I- cn o M M M +> •H (0 M O o II >< id ' T3 <d cs II % CD ft* -p •H "3 ro •r» 0) <P r» r" • M • -u e-t u a 3 )-) «»: >i % <N O «- CM CN CM CM V O o -o.no o .0000 VI a o • t Jl 4 J3 •H o o o o o -M l»1 in •H CN G O u H J o M fc« ,, o o o o CJ o% « O o m CN • CM i u 1 o M ft H a z (J O • CN o o o o o o« o * ID o «» 1 <y cq H o M M Q 3 O o H a «* a •4 o o j hJ o (0 a. to o o o o o o % % in o © CN o o o o o o o o o« « • *" *" * % in i i M CO 03 t ir> -t-> a S s o n Cm 68 (tl SlH A. TAELE XIV 25-Ear Space lower. Design Information COORDINATES (in. Initial Strs-6c)c. X 37.5 21 .487 1 37.5 48 .271 602 .83 47.271 z 100. 1C0.27 1 .26 128.77 X 100. 22 .146 47. 83 107.56 1 100. 96 .353 140 .53 110.077 Joint AREAS • STRESS- BOCKl INITIAL FINAL EENBER 6 0.9E-3 0.782 0.754 0.013 0.414 0.842 10 0. 1E-3 0.033 12 0. 130 14 0.558 0.982 0.801 1 2 18 22 Height ) (lbs.) I )isplaceaent 33. 512 14 Frequency 64.9 13 (sq.in.) DISPLACEMENT INITIAL FINAL 0.668 0. .029 PREQUENCI INITIAL FINAL 0.54 b 0.07U 0, .450 2.533 ,219 2.635 1.209 0.533 0. .015 0.547 0.054 0.101 0.549 0. 118 C.496 0.206 0.121 0.668 0. ,084 0.533 0.350 0.739 1.592 0, .751 1.449 0.720 0, .668 3.36 3 1. 2. 0.554 Stress-Buckl. Pinal Initial 128.55 229.52 032 3. 100 1, 2.6S3 . Displacement Initial Fianal 169.9 565.2 0.73a 130 Frequency Initial Final 587.73 261.52 Iteration 12 44 9 Tiie 5.38 37.14 21.09 * (sec.) Optimua areas for initial and final geoaetry. 69 LIST OF REFERENCES 1. Dorn, W. S., Gomory, R. E., and Greenberg, H. J., Automatic Design of Optimal Structures Journal de Mechanique, Vol. 3, March 1964, pp 25-52. , 2. Dobbs, M. W. and Felton, L. P., Optimization of Truss Geometry Journal of the Structural Division, ASCE, Vol. 95, No. ST10, Proc. Paper 6797, October 1969, pp 2105-2118. , , 3. Pedersen, P., Optimum Joint Positions for Space Trusses Journal of Structural Division, ASCE, Vol. 99, No. ST12, December 1973, pp. 2459-2475. 4. Vanderplaats, G. N. and Moses, F. Automated Design of Trusses for Optimum Geometry Journal of Structural Division, ASCE, Vol. 98, No. ST3, March 1972, pp. 671- , , , , 690. 5. Vanderplaats, G. N. Design of Structures for Optimum Geometry Technical Memorandum TM-X-62,462, National Aeronautics and Space Administration, August 1975. , , 6. Spillers, W. R. Iterative Design for Optimal Geometry Journal of Structural Division, ASCE, Vol. 101, No. ST7, July 1975, pp. 1435-1442. 7. Imas K. and Schmit, L. A., Jr., Configuration Optimization of Trusses Journal of the Structural Division, ASCE, Vol. 107, No. ST5, Proc. Paper 16251, May 1981, pp 745-756. , , , , , 8. and Topp, Martin, H. C. Turner, M. J., Clough, R. W. Stiffness and Deflection Analysis of Complex L. J. Structures J. Aeronaut. Sci. Vol. 23, No. 9, pp 805824, September 1956. , , : , , 9. Large Eigenvalue Bathe, K, J., and Wilson, E. L. of the Engineering Journal Problems in Dynamics Analysis Mechanics Division, ASCE, Vol. 98, June 1973, pp 213-226. , , 10. Aurora, J. S. and Haug, E. J., Methods of Design Sensitivity Analysis in Structural Optimization AIAA Journal, Vol. 17, September 1979, pp 970-974. , 11. Nelson, R. B., Simplified Calculation of Eigenvectors Derivatives AIAA Journal, Vol. 14, No. 9, September 1976, pp 1201-1205. , 70 12. Zoutendijk, K. G. Methods of Feasible Directions Elsevier Publishing Co., Amsterdam, Netherlands, , , 1960. 13. Vanderplaats, G. N. and Moses, F., Structural Optimization by Method of Feasible Directions, Computers and Structures Vol. 3, Pergamon Press, 1973, pp 739, , 755. 14. Vanderplaats, G. N., CONMIN - A FORTRAN Program for Constrained Function Minimization - User's Manual NASA TMX-62,282, August 1973. , 15. Martin, H. C. Truss Analysis by Stiffness Considerations, Trans. Am. Soc. Civil Engrs., Vol. 123, pp 1182, 1194, 1958. 71 . INITIAL DISTRIBUTION LIST No. Copies 1. Defense Technical Information Center Cameron Station Alexandria, Virginia 22314 2 2. Library, Code 0142 Naval Postgraduate School Monterey, California 93940 2 3. Professor G. N. Vanderplaats Code 67Vn Department of Mechanical Engineering Naval Postgraduate School Monterey, California 93940 3 4. Professor David Salinas, Code 69Zc Department of Mechanical Engineering Naval Postgraduate School Monterey, .California 93940 1 5 Naval Attache Embassy of Ecuador 2535 15th Street NW Washington, DC 20009 1 6. LT Jorge E. Felix Box 5147 Guayaquil, Ecuador, SOUTH AMERICA 2 7. LT David J. Mulholland, USN COMNAVSHIPYD PEARL HARBOR Pearl Harbor, HI 96782 1 8. Dr. Richard H. Gallegher Dean of College of Engineering University of Arizona Tucson, Arizona 85702 1 9. Department Chairman, Code 69 Department of Mechanical Engineering Naval Postgraduate School Monterey, California 93940 1 , 72 Felix Shape lV55k optimization eS S " b e J st re„ gth dl « straints. Thesis F2533 c.l Felix Shape optimization of trusses subject to strength, displacement, and frequency constraints. to thesF2533 Shape optimization of trusses subject to 3 2768 002 06513 8 DUDLEY KNOX LIBRARY HP M ggafljBB *" II '' ; "-''VL\ ;' ,), :V.,,', , t)3w3 SHShsS 11 r' ' ••.>' ' i HP : m

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