Reliability-based Design Optimization of a Nonlinear. Elastic Plastic Thin-Walled T-Section Beam

Transcription

1 Reliability-based Design Optimization of a Nonlinea Elastic Plastic Thin-Walled T-Section Beam by Mazen A. Ba-abbad Dissetation submitted to the Faculty of Viginia Polytechnic Institute and State Univesity in patial fulfillment of the equiements fo the degee of Docto of Philosophy in Aeospace Engineeing Rakesh K. Kapania, Chai Eic R. Johnson Michael J. Allen Robet L. West Suot Thangjitham Mach 4 Blacksbug, Viginia Keywods: Reliability-Based Optimization, Finite Element, Nonlinea Analysis, T-Beam, Elastic-plastic, Combined Loads, Flexual-Tosional Buckling, Reliability-Based Facto of Safety Copyight 4, Mazen A. Ba-abbad

2 Reliability-based Design Optimization of a Nonlinea Elastic Plastic Thin-Walled T-Section Beam Mazen A. Ba-abbad Viginia Polytechnic Institute and State Univesity Blacksbug, VA 46-3 ABSTRACT A two pat study is pefomed to investigate the application of eliability-based design optimization RBDO appoach to design elastic-plastic stiffene beams with T- section. The objectives of this study ae to evaluate the benefits of eliability-based optimization ove deteministic optimization, and to illustate though a pactical design example some of the difficulties that a design enginee may encounte while pefoming eliability-based optimization. Othe objectives ae to seach fo a computationally economic RBDO method and to utilize that method to pefom RBDO to design an elastic-plastic T-stiffene unde combined loads and with flexual-tosional buckling and local buckling failue modes. Fist, a nonlinea elastic-plastic T-beam was modeled using a simple 6 degee-of-feedom non-linea beam element. To addess the poblems of RBDO, such as the high non-lineaity and deivative discontinuity of the eliability function, and to illustate a situation whee RBDO fails to poduce a significant impovement ove the deteministic optimization, a gaphical method was developed. The method stated by obtaining a deteministic optimum design that has the lowest possible weight fo a pescibed safety facto SF, and based on that design, the method obtains an impoved optimum design that has eithe a highe eliability o a lowe weight o cost fo the same level of eliability as the deteministic design. Thee failue modes wee

3 consideed fo an elastic-plastic beam of T coss-section unde combined axial and bending loads. The failue modes ae based on the total plastic failue in a beam section, buckling, and maximum allowable deflection. The esults of the fist pat show that it is possible to get impoved optimum designs moe eliable o lighte weight using eliability-based optimization as compaed to the design given by deteministic optimization. Also, the esults show that the eliability function can be highly non-linea with espect to the design vaiables and with discontinuous deivatives. Subsequently, a moe elaboate 4-degees-of-feedom beam element was developed and used to model the global failue modes, which include the flexual-tosional and the out-of-plane buckling modes, in addition to local buckling modes. Fo this subsequent study, fou failue modes wee specified fo an elasticplastic beam of T-coss-section unde combined axial, bending, tosional and shea loads. These failue modes wee based on the maximum allowable in-plane, out-ofplane and axial otational deflections, in addition, to the web-tipping local buckling. Finally, the beam was optimized using the sequential optimization with eliabilitybased factos of safety SORFS RBDO technique, which was computationally vey economic as compaed to the widely used nested optimization loop techniques. At the same time, the SOPSF was successful in obtaining supeio designs than the deteministic optimum designs eithe up to% weight savings fo the same level of safety, o up to six digits impovement in the eliability fo the same weight fo a design with Safety Facto.5.

4 Acknowledgements The autho wishes to expess his gatitude fo the guidance and advice of his advises D. Rakesh K. Kapania and D. E. Nikolaidis, and of all the othe espected committee membes: D. Eic Johnson, D. Michael Allen, D. Robet West and D. Suot Thangjitham. Also, the autho would like to expess his gatitude to ONR STTR and Adoptech fo funding this eseach, D. Zafe Gudal, M. Mostafa Abdulla and D. Scott Ragon fo thei citique and advice. Finally, the autho would like to expess his deep appeciation and gatitude to his paents, family and fiends fo thei suppot and help. iv

5 Table of Contents List of Figues ix List of Tables....xv Chapte. Intoduction... Uncetainties in Engineeing Design..... Designing with Uncetainty: Deteministic Design and Reliability-Based Design...3 Reliability-Based Design Optimization RBDO.4.3. Concepts and Histoy Application in Stuctual Design Optimization Finite Element Analysis and Reliability..7.4 Scope of the Pesent Wok...7 Chapte. Reliability Based Stuctual Optimization: Backgound and Liteatue Review.9. Reliability Calculation Methods Simulation Based Reliability Methods: Monte-Calo Simulation.... The Diect Monte-Calo Simulation..... Monte-Calo Simulation with Impotance Sampling Othe Monte-Calo Simulation Sampling Methods Analytical Reliability Appoximation Methods.6... The Fist Ode Reliability Methods FORM The Conell Reliability index The Hasofe and Lind Reliability Index.9... Othe Analytical Reliability Calculation Methods. Reliability of Stuctual Systems and Stuctual Components with Multiple Failue Modes..3.3 Appoximations of Pefomance Functions 4.4 Reliability Based Stuctual Optimization Optimization Methods Standad Mathematical Statements of the Two RBDO v

6 Poblems and thei Classical Solutions Reliability-Based Design Optimization Methods A Modified Sequential Optimization with Reliability- Based Factos of Safety SORFS RBDO Appoach.4 Chapte 3. Nonlinea Finite Element Model of Six Degees of Feedom Elastic-Plastic Beam Intoduction Six Degees of Feedom Elastic-Plastic Beam Element.44 Chapte 4. RBDO Exploatoy Example: Elastic-Plastic T-Beam Failue Modes Deteministic Stuctual Optimization Reliability-Based Stuctual Optimization Numeical Application Discussion of Reliability-Based Optimization Results Summay and Conclusions.85 Chapte 5. Application of the Sequential Optimization with Pobabilistic Safety Factos RBDO to the Exploatoy Elastic-Plastic T-Beam Example Intoduction The Optimization Poblem Poblem Statement The Sequential Optimization with Pobabilistic Safety Factos Appoach Reliability-Based Optimization Results fo the Exploatoy Elastic-Plastic T-Beam Example using the SOPSF Appoach Discussion of the Results.93 Chapte 6. Nonlinea Finite Element Model of 4 Degees of Feedom Elastic-Plastic Beam The element loads Coss-Section Defomation Modeling Defomation field appoximation. vi

7 6.4 Element Stiffness Matix Application to an elastic-plastic T beam unde combined loads Element Integations Post-Buckling Analysis Accounting fo Plastic Defomations Code Compaisons and Veifications Compaisons with Published Data Compaisons with ABAQUS Beam Element..33 Chapte 7. Reliability Based Optimization of Nonlinea Elastic-Plastic T-Beam unde Random Axial, Flexual and Tosional Loads with Tosional and Local Buckling Failue Modes Failue Modes Examples Local Buckling Consideation Specifying the Constaints Deteministic Optimization fo Safety Factos.5,.75,.,.5 and Reliability Calculation Reliability of Deteministic Optimum Design with Safety Facto Reliability-Based Optimization Applied to the Deteministic Optimum Designs with Safety Factos.5,.75,.,.5 and RBDO fo the Deteministic Optimum Design SF RBDO fo Deteministic Optimum Design SF.75,.,.5 and Discussion of Reliability-Based Optimization Results 74 Chapte 8. Summay and Conclusions Summay Conclusions Suggestions fo Futue Wok 79 Refeences...8 vii

8 Appendix A: FORTRAN 9 Code Listing fo the 6 DOF Nonlinea Elastic-Plastic T-Beam..9 Appendix B: Backwad Eule Method to Obtain the Plastic Stesses.. Appendix C: FORTRAN 9 Code Listing fo the 6 DOF Nonlinea Elastic-Plastic T-Beam 8 Vita..3 viii

9 List of Figues Figue. Gaphical Repesentation of the Stess-Stength Intefeence. Figue. The Oiginal Pobability Distibution Function f X X and Impotance Sampling Function h v v in x and x Space.4 Figue.3 The Pobability Distibution Function f U U and the Pefomance Function in the standad u and u Space..... Figue.4 Flow Chat of the Sequential Safety-Facto Based RBDO Methods...39 Figue.5 Flow Chat of the Modified Sequential Safety-Facto Based RBDO Methods..4 Figue 3. The Example Beam unde the Applied Loads..46 Figue 3. Six Degees of Feedom Beam Element..46 Figue 3.3 The Added Moment due to the Axial Load.49 Figue 3.4 Newton-Raphson Iteations Combined with Incemental Pedicto 5 Figue 3.5 Plastic Zones Developed in a T-Beam Coss-section fo Inceasing Values of the Axial and Bending Loads.5 Figue 3.6 Mid-Span Deflection of a Simply Suppoted Beam of Rectangula Coss-section unde Lateal Load see Yang and Saigal Figue 4. The State of Stess fo a T-Beam unde the Combination of Axial Load and Bending Moment that Causes the Whole Coss-Section to Defom Plastically..55 Fig 4.a The Safe Design Zone unde the yield suface that Combines the Axial, Bending and Tosional Shea Loads..6 Fig 4..b The Pojection of the Design Point on the M-P Plane 63 Figue 4.3.a Geomety and Dimensions of the T beam.69 Figue 4.3.b The Loads Applied to the Beam.69 ix

10 Fig 4.4.a Load-Deflection Cuve fo a T Beam, Used hee unde Axial and Lateal Loads..7 Fig 4.4.b Location of the Failue Point on the Axial Load-Bending Moment Inteaction Cuve 7 Figue 4.5 Compaison Between the Results of Non-Linea FEM and the Linea Elastic Fomula Eq Figue 4.6 The Bounday Pefomance Functions GX =, GX = and G3X =, Plotted on the Random Loads Plane the loads ae nomalized by thei mean values..74 Figue 4.7.a Effects of Vaying the Design Vaiables on the Weight and Pobability of Failue of the Deteministic Optimum SF Fig 4.7.b Seaching fo the Optimum Design that has the Lowest Pobability of Failue Pf 77 Figue 4.8.a Effects of Vaying the Design Vaiables on the Weight and Pobability of Failue of the Deteministic Optimum SF Fig 4.8.b Seaching fo the Optimum Design that has the Lowest Pobability of Failue P f.78 Figue 4.9.a Effects of Vaying the Design Vaiables on the Weight and Pobability of Failue of the Deteministic Optimum SF....8 Fig 4.9.b Seaching fo the Optimum Design that has the Lowest Pobability of Failue Pf 8 Figue 4..a Effects of Vaying the Design Vaiables on the Weight and Pobability of Failue of the Deteministic Optimum SF.5 8 Fig 4..b Seaching fo the Optimum Design that has the Lowest Pobability of Failue Pf 8 Figue 4..a Effects of Vaying the Design Vaiables on the Weight and Pobability of Failue of the Deteministic Optimum SF.5 8 Fig 4..b Seaching fo the Optimum Design that has the Lowest Pobability of Failue P f.8 Figue 4..a Compaing the Reliability of the Deteministic and the Reliability-Based Optimum Designs...84 x

11 Figue 4..b Weight Saving with Reliability-Based Design ove the Deteministic Design...84 Figue 5. The Beam Loads and Dimensions 87 Figue 5. Compaing the Reliability of the Deteministic and the Reliability-Based Optimum Designs.9 Figue 5.3 Weight Saving of the Reliability-Based Design ove the Deteministic Design...9 Figue 6. A Pismatic Beam of Abitay Open Coss-Section unde the Action of a System of Loads.94 Figue 6. Beam Element Defomations unde a System of Loads..95 Figue 6.3 Displacements of a Point P in the Beam Coss-Section...96 Figue 6.4 Axial, Tangential and Nomal Components of Displacements at a Point in Beam Wall.98 Figue 6.5 Shea Stain Resulting fom Axial and Tangential Components of Displacement..99 Figue 6.6 The Longitudinal Stain.. Figue 6.7 The Beam Element Genealized Displacements and Foces..5 Figue 6.8 Resistance to Tosion Povided by an I-Beam..4 Figue 6.9 Resistance to Tosion Povided by a T-Beam...4 Figue 6. Flow Chat fo the Stess Calculation Algoithm 4 Figue 6. Flow Chat of the Stess Integation Pocedue...6 Figue 6. Simply Suppoted Rectangula Beam unde Unifom Lateal Load q o v the cental in-plane deflection..7 Figue 6.3 The Cental Deflection of a Simply Suppoted Rectangula Elastic-Plastic Beam 8 Figue 6.4 The Cantileve Beam Tested by Woolcock and Tahai 974 and analyzed by Ronagh and Badfod xi

12 Figue 6.5 The Response of Cantileve Beam Tested by Tso and Ghobaah 97 and analyzed by Ronagh and Badfod Figue 6.6 A Simply Suppoted Beam unde Bi-axial Bending and Toque Loads..3 Figue 6.7 The Response mid-span axial displacement w, and out-of-plane displacement u of a Simply Suppoted Beam unde Axial, Bi-axial Bending and Tosional Loads...33 Figue 6.8 Cantileve Beam with T Section unde Axial, Lateal and Tosional Loads...34 Figue 6.9 Compaison between the Results of the Developed Beam Element and ABAQUS B3OS Beam Element, fo Non-linea Elastic-Plastic Solution, fo the Case of Cantileve Beam unde Combined Axial, Flexual and Tosional Loads..36 Figue 6. Compaison between the Results of the Developed Beam Element and ABAQUS B3OS Beam Element, fo Non-linea Elastic-Plastic Solution fo the Case of Cantileve Beam unde Pue Toque in the Tip of the Beam.37 Figue 7. Cantileve Beam with Naow Rectangula Section unde Tip load..39 Figue 7. Buckling Mode Shapes of a Naow Rectangula Cantileve Beam unde Tip Load the second mode shape is the same as the fist, but with an opposite sign.4 Figue 7.3. Defomed Shape of a Non-linea Elastic-Plastic Cantileve Beam unde Tip Load.4 Figue 7.4 Non-linea Response of an Elastic-plastic Cantileve Beam with Naow Rectangula Section unde Tip Load U is the in-plane Defomation and U is the out-of-plane Defomation...4 Figue 7.5 A Cantileve Beam with T Section unde Tip Load 43 Figue 7.6 Buckling Mode Shapes of Cantileve Beam with T-Section unde in-plane Tip Load...44 Figue 7.7 Nonlinea Elastic Analysis of a Continuum Element Model fo a Cantileve Beam unde Vetical Tip Load, Revealing Local Stess Concentation Spots.45 Figue 7.8 Stess Contous of a Cantileve T-beam unde In-plane xii

13 Tip Load Nonlinea Elastic Analysis...46 Figue 7.9 Stess Contous fo a Nonlinea Elastic-plastic Analysis of a Cantileve T-beam unde In-plane Tip Load localized defomations have developed.47 Figue 7. Web Local Buckling Failue in a Cantileve T-section Beam unde Vetical Tip Load 48 Figue 7. Local Buckling of the Web and Flange of a Cantileve Beam of T-section unde Axial Load ABAQUS continuum element model linea buckling analysis.49 Figue 7. Local Buckling of the Web and Flange of a Cantileve Beam of T-section unde Unifom Lateal Pessue Load Applied at its Flange ABAQUS continuum element model linea buckling analysis...5 Figue 7.3 Cantileve Beam of T-section unde Toque Load at its Fee End ABAQUS shell model. 5 Figue 7.4 Compaison between the Pobabilities of Failue fo Each Failue Mode fo the Impoved Reliability Optimum Design and the Deteministic Optimum Design SF.5 in the Safety Index β Space...64 Figue 7.5 Compaison between the Pobabilities of Failue fo Each Failue Mode fo the Impoved Reliability Optimum Design and the Deteministic Optimum Design SF.75 in the Safety Index β Space...66 Figue 7.6 Compaison between the Pobabilities of Failue fo Each Failue Mode fo the Impoved Reliability Optimum Design and the Deteministic Optimum Design SF. in the Safety Index β Space.68 Figue 7.7 Compaison between the Pobabilities of Failue fo Each Failue Mode fo the Impoved Reliability Optimum Design and the Deteministic Optimum Design SF.5 in the Safety Index β Space.7 Figue 7.8 Compaison between the Pobabilities of Failue fo Each Failue Mode fo the Impoved Reliability Optimum Design and the Deteministic Optimum Design SF.5 in the Safety Index β Space.7 Figue 7.9 Weight Saving of the Reliability-Based xiii

14 Optimization ove the Deteministic Optimization fo Diffeent Safety Levels...73 Figue 7. Compaing the Reliability of the Deteministic and the Reliability-Based Optimum Designs 73 xiv

15 List of Tables Table 4. Deteministic Optimum Designs fo Vaious Safety Factos 73 Table 4. Pobabilities of Failue fo the Thee Pefomance Functions G X, G X, and G 3 X Using Monte-Calo Simulation with Impotance Sampling IS and by Using H-L Method fo the Deteministic Optimum Design with SF of Table 4.3 Pobabilities of Failue fo the Deteministic Optimum Designs SF of.5,.75,.,.5, and.5.75 Table 5. Convegence of RBDO of the T-Stiffene Example finding the least weight fo a cetain level of safety fo SF.5 using the SORFS Method...9 Table 5. Convegence of RBDO of thet-stiffene Example finding the highest eliability fo a cetain weight fo SF.5 using the SORFS Method...9 Table 7. Deteministic Optimum Designs fo Vaious Safety Factos..58 Table 7. the MPP loads fo each failue mode pesented as a multiple of the nominal loads..6 Table 7.3 Results of the RBDO fo SF.5 63 Table 7.4 Results of the RBDO fo SF Table 7.5 Results of the RBDO fo SF. 67 Table 7.6 Results of the RBDO fo SF Table 7.7 Results of the RBDO fo SF.5 7 xv

16 Chapte : Intoduction. Uncetainties in Engineeing Design In engineeing design, thee ae a numbe of uncetainties that esult fom the vaiability of applied loads and mateial popeties, in addition to that esulting fom the design modeling. Also, duing manufactuing, a numbe of uncetainties aise fom the manufactuing pocesses and fom the mateial selection. The poblem of uncetainty in estimating the loads can be clealy noticed in the aeodynamic and hydodynamic loads that ae consideed in the design of aicafts and ships. As these loads ae not known exactly, aeospace and naval designes have to conside some kind of statistical epesentation of these loads. Also, it is known that the mateial popeties documented in the mateial specification handbooks and manuals ae not the exact popeties of the actual poduct. Since these documented mateial popeties ae taken fom the aveages of the measued expeimental data. Also, the expeiments wee pefomed in laboatoy conditions that may diffe fom the situation in hand. In addition, some mateial defects, such as mico-cacks and voids, can seiously weaken the mateial and may be vey difficult to detect. Likewise, in some applications, the mateial may expeience envionmental deteioation such as coosion and abasion, that may not be quantified with cetainty. Moeove, thee ae uncetainties in manufactuing the components of engineeing systems despite the quality contol measues that ae applied. The eason is that inspection fo quality is pefomed most of the time on some andom samples of the components and not on all the components; othewise, the cost will be pohibitively high. Finally, in modeling the engineeing systems a numbe of idealizations may be made to simplify the analysis, which will esult in educing the accuacy in epesenting the eal system. Howeve,

17 these idealized engineeing system designs ae aely poduced with the exact dimensions specifications, since toleances must be given fo mateial pocessing and fabication, which adds to the level of uncetainty in the design.. Designing with Uncetainty: Deteministic Design and Reliability-Based Design A designe must deal with the existence of uncetainties in the poduct in a manne that will make the design pefom as expected in a safe and eliable way. Fo that pupose, two methods have been developed ove the yeas to quantify the uncetainties in an engineeing poduct and thei effects on the design level of safety. The fist appoach is the deteministic design, in which it is assumed that all the infomation about the design is known and a consevative assumption is made to compensate fo any unaccounted factos. Accodingly, a facto of safety o a load facto is assigned to eithe the mateial stength o the applied loads and it is the epesentation to the design level of safety. The second appoach is the eliability-based design, in which it is assumed that the infomation about the design is known to be within cetain bounds and have known distibutions of pobability. Accodingly, the pobability of suvival of the design is calculated and can be consideed to epesent the level of safety of the design. The deteministic appoach consides aveage values fo the loads and mateial popeties. It does not conside quantitatively the fequency of occuence of some paticula values of the loads and mateial popeties duing the life-span of the poduct. Instead, a subjective value is assigned to the facto of safety o the load facto on a qualitative basis. Also, the deteministic appoach becomes highly subjective in the sense that

18 facto of safety is usually estimated and modified accoding to the cost and the consequences of failue of a paticula component. In addition, the deteministic appoach does not use a quantitative method fo combining vaiable loads that ae applied to the design see Moi et al, 3. Thus, the level of safety of an engineeing design that has multiple components that diffe in thei equied level of safety will not be consistent, because the oveall level of safety of a system can not be detemined fom adding the safety factos of the individual components. As a esult, an oveestimation o undeestimation of the design level of safety will be made. Finally, it must be noted that the deteministic appoach does not identify the pats o egions of the design that may fail much ealie than othes, and also, does not identify the moe citical loads o design vaiables since all the uncetainties ae coveed by one facto thoughout the design Long and Naciso, 999. On the othe hand, the eliability of an engineeing design is calculated using statistical analysis and pobability pinciples to the samples of the expected sevice loads and the popeties of the mateial used in the design. The uncetainties ae modeled by andomly distibuted vaiables, in which the fequency of occuence of each possible value of the vaiable is consideed. Specifically, the most epeated values of a andom vaiable ae associated with the highest values in the pobability distibution function. Howeve, it must be noted that thee ae some expeience-based assumptions that ae made in detemining the type and shape of the pobability distibution of each andom vaiable, since it is impossible to pefom expeiments that cove all the possible values of a andom vaiable Hess et al, 994; Sundaaajan, 995; White et al, 995; and Melches,. Nevetheless, the accuacy of the model used to epesent the actual data inceases as the volume of the available statistical data inceases. Fo 3

19 example, flight loads specta can now be geneated with consideable accuacy Heb, 995. Also, by tightening the quality measues and by using moe accuate models, a bette assessment of the design safety may be achieved De Kiueghian, 989. Once the uncetainties ae modeled, a consistent level of safety can be obtained fo the engineeing design. In paticula, the failue of an engineeing system that has multiple components with diffeent safety levels may be defined by combining the eliabilities of the components as in the following. In the case of independent failues, the eliabilities can be combined in seial failue of one component makes the whole system fail and/o in paallel failue of all components makes the whole system fail. Also, in the case of coelated failues, joint pobabilities of failue can be calculated fo the design Ang and Tang, 975; and Melches,. Hence, the uncetainties pesent in the design can be quantified and calculations of the safety level can be pefomed in a moe coheent manne. Finally, it is appopiate to mention some othe methods of design unde uncetainty that do not conside explicit safety measues, but instead calculate estimates of the uncetain design vaiables, such as the fuzzy sets membeship functions, inteval methods see Muhanna, and Mullen, ; and Rao and Beke, 997, evidence theoy Bae et al, 3 and possibility theoy Moelle et al, Reliability-Based Design Optimization RBDO.3. Concepts and Histoy Given that one is able to detemine the eliability of a cetain engineeing design, it may be pudent to pefom optimization to seach fo the best design the safest, least cost, o least weight etc. while satisfying cetain estictions. In fact two types of eliability-based optimization 4

20 poblems ae consideed a finding the most eliable design within cetain design o cost constaints, and b finding the best design that has a constaint on the minimum value of eliability. Howeve, despite that the RBDO has a moe consistent desciption of the safety of designs; the field of design optimization is still dominated by deteministic methods. One of the easons that has slowed the acceptance of the RBDO by industy is that its computational cost is much highe as compaed to the cost of employing a deteministic appoach to the design of vaious components. In the case of the deteministic design, the analysis calculations ae pefomed fo only one set of paametes to yield the design esponse to the applied loads, and then this pocess is epeated fo a diffeent set of vaiables in the optimization iteations. In eliability-based design, on the othe hand, the analysis calculations ae epeated seveal times just to calculate the eliability of a design, and then this pocess is epeated again fo optimization. Hence, thee is a need to pefom eliability-based optimization, by using only a limited numbe of analyses. A second eason that may have stalled the use of RBDO, is that the eliability-based optimization calculations may not convege all the time. Fangopol 995 has pointed out that RBDO poblems wee solved with vaying degees of success this efeence also contains lists of efeences that cove the development of the eliability-based design fom the 973 to993. Also, this poblem was clealy addessed by Royset et. al., who explained that the eason fo this failue is that the eliability function may be highly non-linea and/o non smooth have discontinuous deivatives, and thus, poposed a pocedue to avoid this difficulty. Thei pocedue equies efomulating the eliability-optimization poblem in such a way that will make it solvable by semiinfinite optimization methods, in which the seach fo the optimum solution consides a finite domain e.g. a cicle o a sphee and then uses a semi-infinite algoithm to seach fo the optimum 5

21 solution within this finite domain see Polak, 997. Still, Royset et al. concluded that the success of thei poposed method depends on how the eliability calculations modify the optimization subpoblem. Wang et al. 995 tied to ovecome the eliability function non-lineaity and deivative discontinuity poblem by using a piece-wise appoximation of the eliability function and they poposed appoximating the design constaints using multivaiate splines and then using these splines to pefom the eliability-based optimization. Howeve, the spacing between the nodes of the splines needs to be optimized to yield the maximum computational economy. Ealie attempts did not addess the issue of non-smooth objective and constaint functions and used optimization methods that ae eithe based on the deivatives of the eliability function Nikolaidis et al, 988; Li and Yang, 994; Nikolaidis and Stoud, 996; Rajagopalan and Gandhi, 996; Pu et al, 997; Nakib, 997 o even based on andom seach of the design space Nataajan and Santhakuma, 995; and Cheng et al, 998. Thus, to pefom RBDO, a method that can be both computationally economical and handle highly non-linea functions that have deivative discontinuities should be used..3. Application in Stuctual Design Optimization Today s stuctual designe has to conside a vaiety of stuctual demands in addition to insuing the safety of the stuctue. Among these demands that concen the aeospace and ship industies ae weight and cost saving, the efficient use of mateials especially composite mateials, and the efficiency in enegy consumption. Howeve, saving stuctual weight, and at the same time, maintaining an acceptable level of safety is citical in many aeospace and naval applications. Thus, 6

22 RBDO can be applied to aeospace and naval stuctues, povided that it can poduce bette designs safe o lighte designs than those obtained by the deteministic design optimization that is cuently in use. The stuctual optimization can be caied out to find the optimum dimensions sizing, configuation e.g. the oientation of tuss membes, o fibes in composite mateials, o topology e.g. the numbe of stiffenes in a stiffened panel, o numbe of spas and ibs in a wing, see Fangopol Finite Element Analysis and Reliability The use of finite element analysis FEA has become common fo analyzing complex stuctual designs that have no closed-fom solutions. FEA has also become vey popula in analyzing stuctual elements unde complex loads. Howeve, to use the FEA in calculating the eliability of a design, it has to be fomulated based on the natue of uncetainties. Fo instance, in the case of uncetainty in the applied loads the usual deteministic FEA can be employed and the esults ae then pocessed to calculate the eliability. On the othe hand, if the mateial popeties have uncetainties in thei spatial distibution as the case of non homogenous distibution of stength o modulus of elasticity special fomulations fo FEA must be employed Liu et al, 995; Elishakoff et al 995, and 997; Melches, ; and Kapania and Goyal.4 Scope of the Pesent Wok The aim of this wok is to exploe the potential of the RBDO in poviding supeio stuctual designs when compaed with the pesent deteministic design and optimization methods. Also, to obtain a bette undestanding to the poblems that hinde the total acceptance of RBDO in the industy. Paticulaly, the aspects of computational economy i.e. the method is pefomed with as 7

23 minimum code uns as possible, obustness i.e. the optimization pocess can be caied with minimal designe intefeence, and the possible optimization limitations i.e. when is it not possible to get a design that is bette than the deteministic optimum design. Also, this wok aims to compae the state-of the at methods that ae used to pefom RBDO and ties to point out the most pomising method out of them. Fo this pupose, a pactical example a beam of T-section was devised. Fist, the beam example will be used to pefom some exploatoy study of the RBDO pocess, which addesses the above mentioned aspects. Then, one of the state of the at RBDO techniques will be modified and applied to the exploatoy beam example. Afte that, moe involved failue modes will be added to the beam. Finally, the updated beam model will be used to compae esults obtained using one of the standad RBDO methods and one of most pomising state-of the at RBDO methods. As a pepaation fo the exploatoy and advanced RBDO applications, a backgound and liteatue eview of RBDO will be pesented in Chapte. Next, an elastic-plastic six degees of feedom nonlinea beam finite element, which was used in the exploatoy study, is pesented in Chapte 3. Then, the RBDO exploatoy study is pesented in Chapte 4. Chapte 5 contains an application of the sequential optimization with eliability-based factos of safety using the coodinates of the most pobable failue points as an appoximate safety factos RBDO see Chen et al. 997; Wu et al. ; Du and Chen ; and Qu and Haftka 3 technique to the same exploatoy beam example of Chapte 4. Then in Chapte 6, we will develop an elastic-plastic, fouteen degees of feedom, nonlinea beam finite element that consides the tosional effects on beam failue. Chapte 7 pesents eliability-based optimization of the updated beam model. Finally, Chapte 8 pesents some concluding emaks and suggestions fo futue eseach. 8

24 Chapte Reliability Based Stuctual Optimization: Backgound and Liteatue Review The pocess of eliability-based optimization is concened with using a compute-based model to calculate the eliability of the design, and employing a suitable optimization method to obtain an optimum design the safest with the least cost o weight. Howeve, as we ae going to see late, it geneally equies an extensive computing effot to calculate the eliability of one design; this step is epeated seveal times to obtain an optimum design. Also, the success and efficiency of the RBDO pocess depends on the way the poblem is fomulated and suitability of the RBDO algoithms to the natue of the design. Thus, the focus is to seek methods that ae both computationally efficient and have a wide ange of applicability. In the following sections these issues will be intoduced with a focus on the methods that wee devised fo this wok. Discussing the compute-based models will be delayed until the next chapte.. Reliability Calculation Methods Since the goal is to detemine the level of safety of a design in its sevice envionment, it would be vey expensive to apply actual loads to the design and calculate the chance of failue. Instead, compute based analytical models ae used to simulate the behavio of the design unde diffeent conditions, and then, the esults ae used to calculate the eliability of the design. The numbe of simulation uns is elated to the numbe of uncetainties pesent in the design paametes o loads. Accodingly, the design is evaluated evey time with a diffeent set of values of the andom vaiables. In tun, the values of the andom vaiables that ente evey evaluation un ae 9

25 selected accoding to thei espective pobability distibution. Afte that, statistical methods ae used to evaluation the output and pedict the eliability of the design. Fo example, if we conside a simple design such as a od that has a stength R that may vay fom one od to the othe depending on some unknown manufactuing conditions. Howeve, it was known that fom pevious samples, a statistical distibution of the od s stength was obtained. Then the stength of the od can be epesented by a andomly distibuted vaiable with a known pobability density function PDF. Now, if we have a stess S that acts on the od that has a magnitude that vaies andomly, we know that the event of failue occus when: R S <. Now, to calculate the eliability of the od we need to calculate the pobability of occuence of the failue event, which can occu at any point inside the failue domain Ω that is epesented gaphically by the shaded aea in Figue.. Stess f s Stength f Failue Domain Ω Stess, Stength Figue. Gaphical Repesentation of the Stess-Stength Intefeence So, in the case that the od s stength and the applied stess values ae statistically independent, the pobability of failue P f fo this simple example can be calculated fom the following integal:

26 Ω P f = f f s d ds. Howeve, despite that the integal of equation. can have a diect analytical solution fo some special cases, o can be pefomed numeically fo some othe cases Sundaaajan and Witt, 995, in most eal life situations the integal can not be evaluated diectly. In many cases the failue domain Ω may not have an analytical expession and the poblem gets moe complicated as the numbe of andom vaiables inceases. Thus, othe methods such as the simulation-based eliability methods Monte-Calo Simulation methods, o the analytical eliability appoximation methods e.g. the fist and second ode eliability methods, and the advanced mean value method must be employed, and ae pesented in the following sub-sections... Simulation Based Reliability Methods: Monte-Calo Simulation To pefom Monte-Calo Simulation MCS to calculate eliability we need an analytical o an appoximate model of the physical system that can be defined in the compute envionment. Then, the analysis needs to be evaluated numeous times. In each analysis un, a diffeent set of values of the design andom vaiables will be used. These andom vaiables ae selected in each evaluation accoding to thei espective pobability distibution, and they ae geneated using compute built-in standad andom numbe geneatos. These, compute built-in functions geneate the andom vaiables in two steps. The fist step geneates andom numbes that ae unifomly distibuted between and, which is done eithe by using the bits and binay digits of the compute o by using some special mathematical fomulas. Once the unifomly-distibuted numbes ae geneated, the andom vaiables can be poduced by eithe the invese tansfomation method o some othe statistical methods Ayyub and Mccun 995, and Fishman 996.

27 Now, to evaluate the eliability of the design i.e. Pf fom equation. using Monte-Calo Simulation MCS, eseaches have developed a numbe of methods, such as the diect Monte-Calo Simulation MCS, o the MCS with vaiance eduction techniques. We will pesent both of these appoaches next.... The Diect Monte-Calo Simulation The diect MCS is the simplest, and at the same time, the most computationally expensive appoach to pefom a pobabilistic analysis of a component o a system. It calculates the pobability of failue Pf as the atio of the failue tials e.g. when the load exceeds the esistance of the stuctue to the total numbe of tials i.e., N f P f =.3 Ntotal The accuacy in estimating P f is epesented by the Coefficient Of Vaiation COV of P f, which can be calculated by assuming each simulation cycle to constitute a Benoulli tial and can be obtained fom: COV P = f P P N P total f f f.4 Ayyub and Mccun 995; Melches. It is appaent that as the numbe of tials inceases, the accuacy in calculating P f inceases. Howeve, this would become pohibitively expensive as the equied level of safety inceases. Theefoe, some othe techniques must be used to educe the equied numbe of tials. These techniques include the vaiance eduction methods e.g. Impotance Sampling, and Latin Hypecube Sampling Olsson and Sandbeg ; Olsson et al. 3 and

28 othe methods that depend on the physical natue of the design such as the poposed method of Dasgupta.... Monte-Calo Simulation with Impotance Sampling The impotance sampling method ISM is one of the popula methods that ae applied to educe the numbe of MCS simulation uns equied to calculate the pobability of failue by educing the vaiance of the calculated eliability value, and thus, impoving its convegence. This method modifies the sampling pocess in the diect MCS by using modified pobability distibution functions fo the andom vaiables, which is called the sampling density function o the impotance function. Then, the compute geneates the values of the andom vaiables v accoding to the defined impotance sampling function v and the analysis is pefomed accoding to these values. h v Fom the analysis, we calculate the design pefomance function GX whee X is the vecto of andom design vaiables, which defines the safe G X > and the failue G X zones in the design vaiables space. Finally, the pobability of failue is calculated fom: P f N N X j I[ G v j ] j= hv v j f v.5 whee N is the total numbe of tials, I [ ] is an indicato that is assigned a value of fo the failue event of the j th tial when G and fo the event of safe design when G >, and v j v j f X v j is the oiginal joint pobability distibution function evaluated at the impotance sampling values v j. Then the eo in estimating the pobability of failue is obtained fom see Melches, 3

29 J J COV P N f =.6.a P f whee J f X X j K = I[ G X j ] hv X j dx hv X j.6.b J f X v j = L I[ G v j ] hv v j dv.6.c h v v j Fom equations.6, we can clealy see that the choice of the impotance sampling function is impotant to the convegence of the calculation of the pobability of failue. In fact some choices may esult in slowe convegence than the diect MCS. Theefoe, Melches suggests use of a sampling pobability distibution function h v v that has its mean value centeed at the most pobable failue point; X* as shown in Fig. fo a design space that has two andom vaiables x and x. Contous of X f X x Safe Zone G X = Contous of v X* h v Unsafe Zone x Figue. The Oiginal Pobability Distibution Function f X X and the Impotance Sampling Function h v v with mean value at MPP in x and x Space 4

30 Yet, it is not easy in most cases to detemine the most pobable failue point a pioi. Howeve, it may be found by some numeical maximization techniques. Howeve, as we will see in section.., we may use a suitable FORM method to detemine the most pobable failue point....3 Othe Monte-Calo Simulation Sampling Methods As mentioned ealie, eseaches have developed othe methods to educe the equied numbe of tial points. Fo instance, the Latin Hype Cube Sampling LHS method has some populaity Ayyub and Mccun 995, Olsson and Sandbeg ; Olsson et al. 3. Howeve, as pointed out by Olsson et al. 3, its efficiency ove the diect MCS is ealized in the case that the pobability of failue is dominated by a single andom vaiable. Howeve, they popose using LHS with the impotance sampling method to incease its efficiency. Finally, othe sampling techniques exist, but they did not get much populaity among eseaches and can be found in the liteatue. Fo example, eseaches have used Statified sampling, adaptive sampling, and diectional sampling techniques see Ayyub and Mccun 995, Moony 997, and Melches fo a eview of some of these methods... Analytical Reliability Appoximation Methods: To calculate the eliability of a design we may need to use a suitable way to appoximate the integal in equation.. Howeve, depending on the types of the functions involved and the shape of the failue egion Ω, we may be able to appoximate the eliability by using fist o second ode Taylo seies appoximation to the limit state function o the pefomance function GX. In the following sub-sections we ae going to pesent one of the widely used methods and efe to othe existing methods. 5

31 ... The Fist Ode Reliability Methods FORM This method is a development of the fast pobability integation method FIP see Haskin et al, 996., and the name fist ode eliability method comes fom appoximating the pefomance function GX by a fist ode Taylo seies. Also, when we ae only consideing the fist two moments of the andom vaiables fo nomally distibuted andom vaiables, the fist moment is mean value, and the second is the vaiance, and ignoing the highe moments i.e. skewness, flatness etc., then these methods ae called the fist-ode second-moment methods FOSM. Howeve, befoe we pesent some of the FORM methods, it is appopiate to define the Conell safety o eliability index β.... The Conell Reliability index The Conell eliability index was the fist analytical appoximation method to calculate the pobability of failue, and it had paved the way fo othe methods that have a wide domain of application Ang and Tang, 975. To intoduce it in a simple way, let s ecall the simple example of the od unde load pesented in Sec.., whee we had the following simple limit state function G X = R S.7 Assuming that R and S ae statistically independent and nomally distibuted andom vaiables, we may define a new andom vaiable Z with the following popeties Ang and Tang, 975: Z = R S µ = µ µ Z Z R R S σ = σ σ S.8 6

32 whee µ Z and σ Z ae the mean value and the standad deviation of the andom vaiable Z espectively. Then the pobability of failue can be calculated fom µ = < = Φ Z P f P[ Z ] = Φ β σ.9 Z whee Φ is the cumulative distibution function fo a standad nomal vaiable, and β is the safety index. The same concept can be genealized to the case of moe than two andom vaiables and to the case of nonlinea pefomance function and this can be done by Taylo seies expansion of the pefomance function aound the mean values of the andom vaiables as in the following: n n n g g Z = g X x i xi x L i xi x j x j. x x x i= i i= j = i j whee g X is the pefomance function evaluated at the mean values of the andom vaiables, and x i is the mean value of the andom vaiable x i. Then, if we tuncate the seies at the linea tems, the fist appoximate mean value and the vaiance of Z will be given by µ σ Z Z g X n n i= j= g x i g x j Cov x, x i j. whee Cov x i, x j is the coefficient of vaiation fo the andom vaiables x i and x j 7

33 Also, a bette estimation of the mean value of Z can be obtained fom consideing the squae tem in the Taylo seies g µ Z g X. n n Cov xi, x j i= j = xi x j Howeve, the second ode vaiance equies obtaining the highe moments of the andom vaiables, which may not be available in pactical situations Ang and Tang, 975 Finally, the safety index β, and the pobability of failue can be detemined fom Eq..9 as befoe. Also, since the limit state function is lineaized aound the mean value, the safety index method is also known as the mean value fist-ode second-moment MVFOSM. Howeve, it is impotant to know that the estimation of the pobability of failue using the safety index can only give accuate values fo special cases, paticulaly, when the pefomance function is a simple addition o multiplication of the statistically independent andom vaiables see Chang et. al., 998. Also, it gives diffeent eliability values fo the same design poblem if the fomulation of the pefomance function is changed to an equivalent fomulation i.e. this eliability calculation method lacks invaiance. Thus, thee was a need to develop some impoved methods that avoid this poblem some of them will be pesented shotly. Yet, these impoved methods ae based on the safety index idea, which fom equation.9 we can see that it gives a qualitative measue of safety, in the sense that lage values of β means safe design, and vice vesa. 8

34 ... The Hasofe and Lind Reliability Index The Hasofe and Lind H-L eliability index is one of the most widely used eliability calculation methods Madsen et al., 986; Nikolaidis and Budisso, 987; Yang, 989; Enevoldsen, and Soensen, 994; Halda and Mahadevan, 995; Nikolaidis and Stoud, 996; Baakat, et al., 999; De Kiueghian,, Melches, ; Stoud, et al.,. It is an impovement ove the Conell s safety index and it avoids its lack of invaiance poblem. Also, it can be used fo explicit and implicit pefomance functions. The H-L method fist tansfoms the andom vaiables into a standadized fom i.e. zeo mean value and unit standad deviation as in the following: U i X i µ X i =.3 σ X i Also, the pefomance function is tansfomed accodingly by using the andom vaiables tansfomation in Eq... Howeve, this tansfomation is only applied diectly to nomally distibuted andom vaiables, in the case of othe distibutions the andom vaiables ae tansfomed to equivalent nomal vaiables by using some appopiate tansfomations see Halda and Mahadevan, 995; Melches,. Then the H-L safety index β HL is defined as the minimum distance fom the oigin of the axis in the educed coodinate system to the limit state suface, which becomes an optimization poblem of the fom: 9

35 β HL s. t. T u. u n = min = min ui i= g u =.4 Hence, we ae essentially seaching fo the most pobable failue point MPFP in the standadized nomal vaiables space see Figue.3 fo the case of two andom vaiables. u G U = Safe Zone Contous of U f U U* u Unsafe Zone β HL Figue.3 The Pobability Distibution Function f U U Function in the standad u and u Space and the Pefomance The H-L eliability index gives an exact estimation fo the design eliability fo linea pefomance functions, and an acceptable appoximation fo most of the nonlinea pefomance functions as long as the adius of cuvatue of the pefomance function is lage compaed to β HL. Consequently, fo those cases quadatic appoximations fo the failue suface at the most pobable failue point may be appopiate, which is the basis of the second-ode eliability methods SORM, see Madsen et

36 al., 986; Halda and Mahadevan, 995; Melches,. Thus, it may be wothwhile to check fo the suitability of the H-L method by compaing its esults with some of the moe diect methods such as the Monte-Calo simulation Olsson et al., 3; Stoud et al.,. Howeve, it should be noted that fo designs of high eliability, it is often sufficient to calculate the eliability within a facto of to 5 on the pobability of failue. Also, moe impotantly, the pobability density function fo each andom vaiable PDF decays vey quickly with the distance fom the oigin with a ate of exp. Hence, the majo contibution comes fom the most pobable failue point and the points closest to the oigin Madsen et. al., 986, and thus, highe accuacy in appoximating the tue pefomance function may not be equied.... Othe Analytical Reliability Calculation Methods As was mentioned in the pevious sub-section, the second-ode eliability methods SORM may be used to calculate the eliability of the design in the cases whee the fist ode methods ae not suitable. The basic idea fo SORM is to appoximate the failue suface aound the MPFP by calculating the local cuvatues, which equies calculating the second deivative of the failue suface at MPFP fo each andom vaiable see Madsen et al., 986; Halda and Mahadevan, 995; Melches,. Also, anothe method that employs a highe ode appoximation of the pefomance function is the advanced mean value method AMV. The advanced mean value method calculates the cumulative distibution function of the design pobability to povide moe infomation about the eliability of the design Wu et. al., 99. It uses petubed values aound the mean values of the andom vaiables, and then expands the pefomance function using linea Taylo seies. Next, MPFP is detemined and the pefomance function value is detemined at MPFP. Finally, the values obtained fom the pevious iteation ae combined to obtain a second-ode

37 appoximation of the failue pobability o the cumulative distibution function of the andom stuctual esponse. Although, the AMV can calculate the eliability of the design Chandu and Gandhi, 995, it needs exta calculations to povide moe infomation than the FORM that is concened only with the pobability of failue. Hence, the AMV is used mainly to calculate the pobability of occuence of cetain values of stuctual esponses Riha et al., 99; Wisching, 995; Rajagopalan and Gandhi, 996; Pepin et al.,.. Reliability of Stuctual Systems and Stuctual Components with Multiple Failue Modes Fo a stuctue with multiple components o fo a component with multiple failue modes, thee ae some simple combinations of failue events such as in seies, paallel, k-out-of-n, o thei combinations. Fo othe combinations, such as pogessive failue situations cacks, fatigue etc., failue event logical tees may be constucted see Kaamchandani, et al a., 99; Kaamchandani, et al b., 99; and Xiao and Mahadevan, 994. In any case, the total pobability of stuctual failue may be expessed as P F = P F P F P F L P F P F I F P F I F I F L 3 3 I F P F 3 I F L 3.5 whee, PF i is pobability of the failue event F i. Howeve, the statistical coelation i.e. the intesection between failue events is geneally vey difficult to quantify, and hence, the effect of multiple failue events is usually quantified by bounds on the eliability. Fo the case of seies combination of failue events fo a stuctue o fo a

38 component, the fist-ode seies bound is among the widely used. This failue bound assumes statistically independent failue modes, and it is defined as: m i= m [ P Fi ] P F P F.6 i i= On the othe hand, highe ode failue bounds take into account the coelations of failue modes, and hence may esult in a educed value of the pobability of failue. Nevetheless, impoved accuacy fo failue bounds can be obtained by using Monte-Calo simulation even by using FORM. Howeve, it may be expensive to use MCS and the use of FORM is limited fo the case linea pefomance functions, since nonlinea functions may cause failue of the optimization pocess see Madsen et al., 986; Enevoldsen, and Soensen, 994; Moses, 995; Melches,. It is appopiate to note that thee ae othe methods that combine multiple failue modes into a global failue event, and assign weighted influence factos to each failue mode see Hong-Zong and De Kiueghian, 989. Also, fo some stuctues the actual failue modes may not be known a pioi, despite the fact that the possible failue modes may be known. Fo this situation, failue mode identification must be caied out befoe combining the esulting failue modes see Zimmemann, et. al., 99; Melches,..3 Appoximations of Pefomance Functions In stuctual analysis, sometimes the pefomance function GX can be obtained as a closed-fom equation, which will allow diect application of standad optimization techniques see next subsection fo a shot eview of some of the commonly used optimization methods fo eliability calculation and eliability-based optimization needed to calculate the design eliability using FORM see Nikolaidis and Budisso, 988; Yang, 989; Yang et al., 99; Zimmemann, 99; 3

39 Kaamchandani and Conell, 99; Kaamchandani et al., 99; Li and Yang, 994; Nikolaidis and Stoud; 996; Nakib, 997 Also, in some othe situations the pefomance function of the design may not be explicit, but may have just one andom vaiable and may be appoximated linealy o quadatically duing optimization see Hisada et al, 983; Liu and De Kiueghian a, 99; Enevoldsen and Soesnsen, 993; Enevoldsen and Soesnsen, 994; Nataajan and Santhakuma, 995; Pu et al., 997. Yet, it was ealized that in many othe stuctual designs the pefomance function can not be epesented by closed-fom equations no has moe than one andom vaiable. This situation occus mostly when failue is simulated point by point in finite element analysis. In those applications, a high non-lineaity and possible discontinuities may occu especially in the case of sudden failue e.g. buckling o cack popagation and in the case of multiple failue modes. Howeve, it may become computationally expensive to make a point-by-point discovey of the entie failue domain. Also, to ensue convegence of the optimization pocess that the FORM methods use to calculate eliability, a diffeentiable o if possible a closed-fom of the pefomance function should be obtained. Theefoe, it would be beneficial to appoximate the pefomance function athe than having it in point-by-point fomat. Fo this pupose, eseaches have used diffeent methods to appoximate the stuctual esponse. Fo instance, some eseaches have developed analytical gadients of the pefomance function with espect to the andom vaiables by efomulating the finite element model. This was done to facilitate the optimization pocess needed to pefom FORM and SORM, since thee wee some difficulties in numeical calculation of these gadients see Lin and De Kiueghian, 989; Liu and De Kiueghian b, 99. Yet, in geneal, it may not be easy to efomulate all othe finite element models especially those that ae a pat of a geneal 4

40 pupose code. Howeve, sensitivity deivatives can be obtained fom the stiffness matix of the finite element model, and thus, may have wide ange of application wee developed. Fo instance, Santos et al. 995 suggested using a continuum sensitivity method to calculate the needed gadients fo the FORM optimization, and thei wok was late adapted by Kleibe et al 999. Despite that, it would be pefeable to find a way to appoximate the pefomance function fo the case of multiple failue modes, which nomally esults in deivative discontinuity in the pefomance function appoximation see Wang et al; 995; Royset et. al., ; Ba-abbad et al.,. Theefoe, othe methods have been employed. These methods ange fom linea and quadatic esponse suface RS methods see Myes, 97, that was employed by many eseaches, to highe ode appoximations that take into account possible deivative discontinuity. Fo instance, Shuëlle et al., 99 used the esponse suface technique to appoximate the pefomance function and then used MCS to calculate P f fom the integal of Eq... Also, Tandjiia et al.,, Kishnamuthy and Romeo, and Gayton et al., 3 have used the esponse suface techniques to facilitate calculating the eliability of a design by appoximating the pefomance function and then using FORM o SORM to calculate the eliability. In addition, Stoud et al., devised RS to calculate the stuctual esponse and diect MCS to calculate the pobability of failue. Multi-vaiable highe ode appoximations wee poposed by Wang and Gandhi 995, who have used multivaiate splines to appoximate the pefomance function, and then used FORM to calculate P f. A subsequent wok, Gandhi and Wang 998 developed a two point adaptive nonlinea appoximation TANA to appoximate the pefomance function and also used FORM to 5

41 calculate the design eliability. Moeove, Penmetsa and Gandhi 3 developed a method that is based on the TANA developed by Gandhi and Wang 998 to appoximate the pefomance function and then adapted the Fast Fouie Tansfom FFT to estimate the pobability of failue. Howeve, it should be noted that the use of RS and simila methods gets to be computationally vey expensive as the numbe of andom vaiables inceases. Finally othe intepolation methods wee suggested, fo instance, Papadakakis and Lagaos,, have used neual netwok to appoximate the pefomance function and then used MCS to calculate the pobability of failue..4 Reliability Based Stuctual Optimization Befoe pesenting the methods used to pefom the eliability based optimization it would be appopiate to pesent a quick eview of some of the optimization methods that ae widely used in eliability calculations fo FORM and SORM and in pefoming the RBDO. Also, it would be appopiate to pesent the standad mathematical fomulations of the two poblems mentioned in Sec Optimization Methods The geneal optimization poblem is defined as to find the minimum o the maximum of the objective o the cost function FX X is the vecto of design vaiables, and usually, this is pefomed within a specified set of constaints. These constaints ae divided into: the inequality constaints gx that define the feasible and the unfeasible domains, the equality constaints hx that give some elations that the design paametes must satisfy, and the side constaints that constain the ange of vaiation of the design vaiables. The mathematical epesentation of the geneal optimization poblem is: 6

42 Min such that : l F X g X h X = i X i u i.7 whee l is the lowe limit and u is the uppe limit of the design vaiable X i. The classical closed fom solution of this poblem is obtained by minimizing the Lagangian of the optimization poblem Min: X, λ, λ = F X λ g X λ h X X, λ, λ = X i X, λ, λ = λ.8 X, λ, λ = λ λ i whee λ and λ ae the Lagange multiplies. The analytical solution must satisfy the Kuhn-Tucke optimality conditions see Vandeplaats, 999. Howeve, the Lagange multiplies appoach is only used if the objective function and the constaints can be epesented by closed fom equations, which is aely the case in most pactical situations. Thus, most optimization poblems ae solved numeically using iteative algoithms. In paticula, most of the optimization methods stat fom an initial X i point and then pefom calculations to detemine the diection S i and the distance α i to the next point, i.e. 7

43 X i = X α S.9 i i i The diection of the seach S i is along the diection that has the most impact in educing o inceasing the objective function, while the magnitude α i is chosen such that no constaints would be violated. The optimize seaches the design space until it eaches an optimum solution, at which no futhe eduction in the value of the objective function is possible without violating the constaints. Howeve, except in special cases, thee is geneally no guaantee that the optimum solution obtained is unique i.e. the global minimum, that is why the optimum solution is petubed and the optimization is stated fom diffeent stating points to check fo possible othe optima. The optimization methods that ae applied to non linea poblems can be classified accoding to thei use of the objective functions and its deivatives in Taylo seies expansion into thee goups. i. The zeo-ode methods, in which no gadients of the objective function is consideed e.g. andom seach, genetic algoithms... etc. See Kloda et al, 3 fo a eview of these methods ii. The fist-ode methods, in which the fist gadient of the objective function is consideed, on which most of popula optimization methods ae based e.g. steepest decent method, sequential linea and quadatic pogamming methods, the feasible diections method, the conjugate gadient method etc. iii. The second-ode methods, in which the second ode Taylo seies tems, ae consideed e.g. the Newton method. Also, the optimization methods can be classified based on the way the constaints ae consideed into diect methods that use the constaints explicitly, and indiect methods that augment the 8

44 constaints in the fom of added penalty functions with the objective function. Fo othe methods and moe details see Haftka and Gudal, 99, Polak 997, Vandeplaats Standad Mathematical Statements of the Two RBDO Poblems and thei Classical Solutions. The fist RBDO poblem is to find the design with the least cost o least weight, with constaints on the acceptable level of safety and some othe deteministic constaints. This RBDO poblem consists of two optimization poblems when using the analytical eliability appoximation methods: i. Optimization poblem unde eliability constaints Min : F x s. t. g k x β x, u β t. whee Fx is the objective function, x is the vecto of deteministic vaiables, x g k ae the deteministic constaints that do not affect the safety of the design e.g. packaging cleaance, β x, u is the eliability index of the design, u is the vecto of nomalized andom vaiables see Eq.., and β t is the taget eliability index the minimum acceptable eliability. ii. Calculation of the eliability index β x, u of the design using FORM Min s. t. : T d u = u u H x, u. 9

45 3 whee u d is the distance in the nomalized andom space, and, u x H is the tansfomed pefomance function see Eq..3. Hence, the classical solution fo these poblems consists of minimizing the two Lagangians Min: ], [,, = k k k t x g u x x f u x λ β β λ λ β Min:,,, u x H u d u x H H λ λ =. whee i λ ae the Lagange multiplies. The analytical optimum solutions of these optimization poblems ae given by: x i,, = = k i k k i i x g x x f u x λ β λ λ β λ β,,, = = u x u x t β β λ λ k,, = = x g u x k λ.3.a u j,, = = j H j H u H u d u x λ λ λ H,,, = = u x H u x H λ.3.b

46 . The second RBDO poblem is to find the design with the highest eliability, with constaints on the acceptable cost o weight. Again, this RBDO poblem consists of two optimization poblems when using analytical methods to calculate the eliability: i. Optimization poblem unde cost constaints Max : β x, u s. t. gk x f x f t.4 whee f t is the taget cost function ii. Calculation of the eliability index β x, u of the design using FORM Min s. t. : T d u = u u H x, u.4 whee du is the distance in the nomalized andom space, and H x, u is the tansfomed see Eq..3 pefomance function. Hence, the classical solution fo these poblems consists of minimizing the two Lagangians Min: x, u, λ = β x, u λ f [ f t f x] k λ g k k x.5 Min: x, u, λ = d u λ H x, u H H whee λ i ae the Lagange multiplies. The analytical optimum solutions of these optimization poblems ae given by: 3

47 β f g x, u,λ = λ f λk x i x x k x i i k i λ x, u, λ = f f x = t f.6.a λ x, u, λ = g x = k k d H x, u, λh = λh = u j u j u j x, u, λ H = H x, u = λ H.6.b.4.3 Reliability-Based Design Optimization Methods Stuctual RBDO methods can be classified, accoding to the way eliability and optimization calculations ae pefomed, in thee categoies: a. Nested double optimization loops, in which an inne optimization loop is used to calculate the eliability of the design using FORM o SORM and an oute optimization loop is used to obtain an optimum eliability-based design the one that has the highest eliability fo a specified weight o a 3

48 specified cost, o the one that has the least weight o the least cost, fo a specified value of eliability. b. Single oute optimization loop, in which the eliability is calculated by MCS and this oute optimization loop, is used to obtain a eliability-based optimum design. c. Combination and efomulation of the RBDO poblem, in which the objective function is a multi-objective function of eliability and cost function see Madsen and Hansen, 99. Also, some eseaches have developed a combined optimization space to pefom RBDO fomulation, in which they combine the design and eliability vaiables in a hybid optimization space Khamanda et al,, o in a design potential space a concept simila to the hybid optimization space as poposed by Tu et al, 999,, and Choi et al,. d. Sequential Optimization with Pobabilistic Safety Factos SORFS RBDO, in which the optimization and the eliability calculation ae pefomed in sequence afte appoximating the pobabilistic constaints with modified safety factos. Since the goal is to find the most computationally efficient and obust RBDO method, it would be appopiate to compae all these diffeent methods to find the most computationally efficient and the most obust method. Now consideing the nested optimization poblem fomat, we find that it is the most widely used. Fo example, Nikolaidis and Budisso 988 used nested gadient-based optimization loops to pefom RBDO fo a simple stuctue that was modeled by 33

49 closed fom equations. Yang 989 and Yang et al. 99 used gadient-based nested RBDO to optimize aicaft wings that wee modeled by closed-fom equations. Also, Enevoldsen and Soesnsen 993 used gadient-based nested RBDO to optimize a seies system of paallel systems that also was modeled by closed fom equations. Then in a late wok, Enevoldsen and Soesnsen 994 fomulated the nested RBDO method on the basis of the classical decision theoy and they pesented some solution stategies fo component optimization and fo stuctual system optimization. In addition, Li and Yang 994 have lineaized the eliability index and used nested linea pogamming optimization loops to pefom RBDO fo a tuss. Moeove, Nataajan and Santhakuma 995 used nested RBDO appoach to optimize tansmission line towes, which employed gadient-based optimization algoithm to find the eliability, and andom seach optimization algoithm to find an optimum design fo the tansmission towe. Nikolaidis and Stoud 996 used nested optimization loops with gadient based penalty function optimization techniques to pefom RBDO fo -ba tuss. Rajagopalan and Gandhi 996 have developed eliability based stuctual optimization RELOPT code that pefoms nested gadient-based optimization to obtain the optimum design. Then, RELOPT was used by Baakat et al 999 to pefom RBDO of lateally loaded piles. Nakib 997 has used the nested gadient-based optimization to pefom RBDO of tuss bidges. Also, Pu et al 997 used nested gadient-based optimization to pefom RBDO fo the fame of a small twin hull ship. Some eseaches suggested appoximating the pefomance functions fist befoe caying on the optimization, especially when the analysis is pefomed using implicit models such as the finite element. Fo instance, Wang et al 995 have developed multivaiate splines to appoximate the design constaints and then they used nested sequential quadatic pogamming to cay out the 34

50 optimization fo diffeent stuctual poblems. Then, Gandhi and Wang 998 used the multivaiate splines which they called: two-point adaptive nonlinea appoximation TANA to appoximate the stuctual esponse and then used double optimization loops to pefom RBDO fo a fame and a tubine blade. Moeove, some eseaches suggested appoximating the deivatives of the pefomance function always nea the eliability constaint, instead of othe points in the design space, to acceleate the convegence. Fo example, Tu et al poposed the design potential method and epoted to get aound 64% eductions in the numbe of evaluations compaed to RBDO method that takes the deivatives of the pefomance function at othe points. In addition, some eseaches have used othe methods of optimization to avoid calculating the gadients of the eliability function. Fo example, Thampan and Kishnamoothy employed anothe fom of the nested optimization appoach which uses genetic optimization algoithm to optimize the configuation of tuss stuctues and gadient-based optimization to calculate the eliability of the design. Also, Ba-abbad et al employed nested optimization which uses gaphical optimization to optimize the design of an elastic-plastic T-beam and used gadient-based methods to calculate the design eliability see chapte 4. Finally, some eseaches have used vaiable complexity appoach to cay on RBDO. Fo example, Buton and Hajela devised a nested optimization appoach that uses vaiable complexity RBDO optimization stats by deteministic optimization, followed by FORM, then, the active constaints ae calculated by SORM, and they devised a neual netwok to appoximate eo between FORM and SORM. Hence, the nested optimization appoach can be used to pefom RBDO fo a vaiety of RBDO poblems povided that the gadients needed fo optimization ae calculated using an efficient and obust method see Enevoldsen and Soensen, 994, Santos et al., 995, Kleibe et al.,

51 Othewise, the nested optimization loop methods may not convege o may not be successful in obtaining an optimum design. On the othe hand, the single optimization loop methods that use MCS may not expeience difficulties in calculating the eliability of the design, but may expeience convegence poblems that may esult fom the high non-lineaity and/o deivative discontinuity of the eliability function. Howeve, to ovecome the eliability function discontinuity eseaches have used non-gadient based methods, such as the genetic algoithms, o efomulated the poblem to be semi-infinite optimization. Fo example, Papadakakis and Lagaos used genetic optimization algoithm to pefom the optimization of a 3-D fame and its eliability was calculated using MCS with impotance sampling. Also, Royset et al, poposed efomulating the RBDO to an oute deteministic semi-infinite optimization loop and a sepaate inne loop to calculate the eliability of the design by any eliability method fo semi-infinite optimization algoithms, see Polak, 997. The efficiency and obustness of the semi-infinite algoithm used by Royset et al was not compaed to othe widely used optimization methods, and thus, Royset et al concluded that the success of thei poposed method depends on how the eliability calculations modify the optimization sub-poblem. Also, moe impotantly, if the MCS was used to calculate the eliability instead of the analytical eliability methods such as the FORM/SORM, and AMV methods, it is geneally much moe computationally expensive than these methods. Altenatively, some eseaches have efomulated the RBDO techniques by combining the two optimization loops to educe the oveall numbe of the pefomance function evaluations. Nevetheless, these attempts sometimes wee not successful in achieving thei goals, o may wok 36

52 only fo special cases. Fo example, Madesen and Hansen 99 have found that a multi-objective function RBDO fomulation may need 5% moe iteations than a nested RBDO fomulation fo the same poblem. Also, the hybid optimization space appoach poposed by Khaamanda et al which was epoted to have a five fold eductions in the numbe of evaluations compaed to a nested RBDO fomulation, was in fact ceating a lage numbe of local minima. This is because the hybid space combines the optimization seach fo minimum cost that satisfies the eliability constaint with the eliability calculation optimization seach to locate the most pobable point MPP in FORM by multiplying the two objective functions togethe. Howeve, this combination may teminate the eliability calculation seach pematuely and give an oveestimation of the eliability. Finally, despite the fact that thee is aely an explicit elation between the safety factos and the eliability of a design Elishakoff and Stanes 999, eseaches have found that they may be able to deive appoximate pobabilistic safety factos fom the eliability calculations and use decoupled sequential loops fo optimization and eliability calculations. This is the main idea behind the Sequential Optimization with Pobabilistic Safety-Facto SORFS RBDO methods. Fo example, Chen et al. 997; Wu et al. ; Du and Chen have used the FORM o SORM to obtain the MPFP coodinates, and then used these coodinates to update the pobabilistic constaints in the design optimization loop. Also, Qu and Haftka 3 have used pobabilistic safety factos PSF that ae calculated fom MCS in the design optimization loop. In this way, the optimum design seach can be pefomed deteministically, instead of using the actual eliability constaints and seaching in the pobability space. Thus, the RBDO can be pefomed in sequential optimization loops instead of nested optimization loops. A substantial eduction in the numbe of pefomance function evaluation may be achieved. 37

53 In this scheme please see Figue.3, the fist optimization loop seaches deteministically fo an optimum design with the mean values of the andom vaiables implemented in the eliability constaints, and then, the second loop calculates the eliability of the design and finds its MPP. In the next iteation, the deteministic optimization loop seaches fo a new optimum design by using * the MPPs X v k fom the last eliability calculation loop to modify the new eliability constaints, and then, the eliability loop calculates the eliability of this new optimum design. The eliability of the design is impoved by shifting the pefomance functions fo the deteministic loop by an amount that equals the diffeence between the cuent safety index β k and the taget safety index β T Wu et. al. ; Du and Chen, o by scaling up the design vaiables based on the value of a calculated pobabilistic safety facto Qu and Haftka 3. The pocedue is teminated when the taget eliability is eached and no impovement in the design is possible. Figue.4 shows the flow chat fo such pocedue. 38

54 Stating with initial values o * o d, X = µ o k =, s = X X s * k = X * k Deteministic Optimization * Min : F d, X k * S. t.: Gi d, X k s. opt Get: d X * k = s Calculate the Reliability: FORM k Min : β Find: opt k S. t.: Gi d, µ X =. k * Calculate: s = β T β, X k MPFP, Wu et. al. ; and Du and Chen Calculate the Reliability: Response Suface, MCS Calculate: PSF s = ; o Scale d opt PSF Qu and Haftka 3 no Conveged Conveged β β opt βt opt β T no yes yes Stop Figue.4 Flow Chat of the Sequential Safety-Facto Based RBDO Methods 39

55 Howeve, these pocedues vay in the way of handling multiple constaints, and accodingly in the way the design impoves duing optimization. Fo example, Wu et. al., conside only the active constaint in each optimization cycle, which may cause the design to oscillate between competing constaints. Altenatively, Du and Chen, have shifted the constaints towads the taget eliability value β T using an invese MPP seach. Yet, fo an RBDO poblem with multiple failue modes, the system eliability may not be optimum. Finally, Qu and Haftka 3, ae suggesting handling multiple constaints using a pobabilistic safety facto that combines all the citical constaints, and they calculated this safety facto fom Monte-Calo simulation of a esponse suface. Also, simila to Du and Chen, in thei pocedue the system eliability was not optimized. Accodingly, thee is a need to develop an RBDO method that will optimize the system eliability..4.4 A Modified Sequential Optimization with Reliability-Based Factos of Safety SORFS RBDO Appoach In this RBDO poblem fomulation, we ae consideing the fist RBDO poblem safest design fo the least cost v Find : d Min : F d, X s. t. Psys d, X = P... P m P s,max.7 whee d,,d p ae the design vaiables which may contain the mean values of the andom vaiables, F d, X is the objective function cost o weight, P,, P m ae the failue pobabilities of the m failue modes, and P s,max is the maximum allowable system failue pobability. The constaint on system eliability can be witten in tems of the safety indexes of the modes as follows: 4

56 Φ β... Φ β P m s,max.8 whee β *,...,β * m ae the safety indices fo the system failue modes. We pefom deteministic optimization to find a good initial design and then pefom eliability analysis of the deteministic optimum. Let * X,..., X * m be the most pobable failue points fo the m failue modes and β *,...,β * m the coesponding safety indices. Let * Z,...,Z * m be the values of the educed andom vaiables at the most pobable point fo each failue mode whee X * = T Z * and i i T is the tansfomation fom the space of the educed andom vaiables to the space of the andom vaiables. This is a vecto with n functions, whee n is the numbe of andom vaiables. Now we pefom deteministic optimization consideing the taget safety indices of the m failue modes, β T,...,β Tm as design vaiables. The optimize will seek the both the optimum values of the design vaiables and the optimum taget values of the safety indices to minimize the weight, as in the following. v Find : d, β T Min : F d, X s. t. G d, T Z... G m d, T Z m.9 4

57 whee Z so that : P and β Ti i ae the pojected most pobable points Z... P >. m = Φ β... Φ β P T Tm i β = β s,max Ti * i Z * i, i =,..., m In Eq..8, we can use the values of the design vaiables and the safety indices of the deteministic optimum as initial guesses. Once we found the optimum, we pefom a new eliability analysis using a pefomance measue appoach that is we fix the value of the safety index and find the minimum value of the pefomance function and fomulate and solve the optimization poblem Eq..8 again. We epeat the pocess until convegence. Figue.5 illustates the appoach. Pefom deteministic optimization fo a given minimum safety facto o Pefom eliability analysis of deteministic optimum MPP s, PFS Solve optimization poblem Eq..9 Optimum design, optimum taget safety indices Pefom PMA based eliability analysis Convegence? MPP fo each failue mode, impotant failue modes Stop Figue.5 Flow Chat of the Modified Sequential Safety-Facto Based RBDO Method 4

58 The second RBDO poblem follows the same steps, but with intechanging the objective function and constaints, so Eq..7 becomes v Find : d Min : Psys d, X = P... Pm s. t. Weight d, X Weight max.3 We will pesent two examples fo applying this modified RBDO technique, the fist Chapte 5 will involve application to an exploatoy elastic-plastic T-beam example, and then we will investigate the application to a moe elaboate elastic-plastic T-beam model. 43

59 Chapte 3 Nonlinea Finite Element Model of Six Degees of Feedom Elastic- Plastic Beam 3. Intoduction In this pat of ou wok, we ae exploing the diffeent issues that may face the design enginee when pefoming RBDO fo pactical stuctual design situations. Fo this pupose, we selected a epesentative stuctual membe that exhibits nonlinea elastic-plastic behavio, which is known fom the liteatue to cause the most poblems fo RBDO. In paticula, we chose an example of T-stiffene beam that is acted upon by combined axial, bending and tosional shea loads. Howeve, to educe the computational costs at this investigative pat of the study, we chose to develop a simple six degees of feedom beam element that has nonlinea behavio. The concept of finite element modeling in stuctues stems fom the idea of discetizing the stuctue into a numbe of elements that have simple shapes o simple mechanical behavio onto which the stuctual esponse is appoximated. These simple elements include beam, plate, shell, and solid elements. Hence, complex stuctues and complex mechanical behavio can be geatly simplified. Fo this pupose, the applied system of loads is eplaced by a statically equivalent foces and moments applied to the nodes. In tun, the nodes ae the points in the element whee the intepolation functions ae specified, and whee the stuctual esponse is obtained. Howeve, the eal stuctual esponse is continuous between the joined potions. Theefoe, continuity between adjacent elements is ensued at the connecting nodes by enfocing continuity of displacements. Geneally, thee ae two souces fo nonlineaity in a finite element poblem: 44

60 a. Mateial nonlineaity, in which the mateial behavio does not follow a linea elastic path e.g. plastic mateial behavio b. Geometic nonlineaity, in which the geomety of the poblem affects the magnitude and/o the diection of the applied loads e.g. lage deflections of the stuctue may cause the loads to change diection o magnitude. Also, the stain-displacement elations may develop non-lineaity in the model by having coupled and/o highe ode tems see Ch. 5. Fo both these cases, the stuctual behavio is assumed to be linea within small iteative steps, and thus, iteative solution pocedues ae employed. Fo moe infomation and details about finite element methods and pocedues see Yang 986; Cisfield 99; Bathe 996; Belytschko et al ; and Doyle. 3. Six Degees of Feedom Elastic-Plastic Beam Element This nonlinea beam element calculates the in-plane and axial defomations of an elastic-plastic beam of the T-section subject to combined axial and lateal loads see Figue 3.. Fo this pupose, thee degees of feedom wee assigned to each of the two nodes of the element. The fist degee of feedom is the axial displacement u that coesponds to the axial compessive load F x. Also, the second degee of feedom is the in-plane deflection of the beam w that coesponds to the shea load w F z. Finally, the thid degee of feedom is the slope of the beam in-plane deflection θ = that x coesponds to the moment M y. Figue 3. illustates these six degees of feedom. 45

61 Z X q Y P Figue 3. The Example Beam unde the Applied Loads Z Fx u My θ X My θ Fx u Fz w Fz w Figue 3. Six Degees of Feedom Beam Element 46

62 47 Since, in most of the cases the vaiation in axial compession along the beam element is much smalle than the vaiation in the in-plane deflections; the intepolation functions that ae used to appoximate defomation ae linea fo the axial degee of feedom and cubic fo the othes. The displacements intepolation functions ae: = 4 u u N N x u = θ θ w w N N N N x w 3. note that x w = θ whee, ξ = ζ = 4 ; N N = = = = ζ ζ ζ ζ ζ ζ ζ ζ L N N L N N 3..a and L x = ζ 3..b whee L is the element length. The linea stain due to bending at any point x, z in the element is given by

63 48 x w z x u x = ε 3. which can be ewitten afte appoximation by the intepolation functions as = θ θ ε w u w u x N z x N z x N x N z x N z x N x 3.3 Also, it is customay to wite the stains in the following fomat { } q B x = ε 3.4 whee, B is vecto that contains the deivatives of the intepolation functions and { } q is the vecto of nodal displacements see Yang and Saigal, 984 Now, the elastic stain enegy of the beam element is given by: = = V x x V x x dv z x E dv U ε ε ε σ, 3.5 which we can ewite using the stain appoximation { } = V T dv q B y x E B q U, 3.6 Then, by using Castigliano s theoem we can get the standad element stiffness matix see Yang, 986 as the following [ ] = V T e dv B y x E B k, 3.7

64 Now, to conside the geometic nonlineaity the effect of the axial foce P that geneates an added moment to the beam that inceases as the in-plane deflection inceases see Figue 3.3 is consideed. dx Z P ds dw P X Figue 3.3 The Added Moment due to the Axial Load The pat of the stain enegy that epesents the shoting of the beam due to the applied axial foce is given by V dw = P dx 3.8 dx L Consideing an axial foce that is constant along the beam element and diffeentiating with espect to the six degees of feedom afte substituting with the defomation intepolation functions, we get the following matix of the geometic effects see Shames and Dym; 985 and Yang, 986: Symmetic L 4L P k = 3 g 3.9 L L L 4L 3 3 [ ] 49

65 It is clea that fo a given value of P this matix depends only on the length of the element, is a geometic paamete, and hence, this matix is called the geometic stiffness matix. Also, since thee is nonlineaity the two stiffness matices epesent the elation of the incemental loads to incemental displacements, and theefoe, constitute the tangent stiffness matix [ k ] [ k ] [ k ] t = 3. e g Accodingly, the stiffness equation can be witten as { P} [ k ] { q } = 3. t whee { P } is the vecto of nodal foces. In the case that the beam element displacements ae lage, a tansfomation matix [ T ] can be used to tansfom the loads, the stiffness matices, and the displacements fom the local to the global coodinates though angle of otationf at the element level see Yang, 986. G T { P } = [ T ] { P } G T [ k ] = [ T ] [ k ][ T ] t t 3. G T { q } = [ T ] { q } whee cosφ sinφ sinφ cosφ T = 3.3 cosφ sinφ sinφ cosφ [ ] 5

66 Finally, the element loads vecto, the stiffness matices, and displacements vecto ae assembled into thei global countepats. Fo computational efficiency, the solution pocedue is based on the Newton-Raphson iteative method applied within a pedicto/coecto appoach see Figue 3.4. This way, a quadatic convegence ate is achieved in detemining the equilibium path see Cisfield, 99. Load Incemental Pedicto Newton-Raphson Coecto Iteations Displacement Figue 3.4 Newton-Raphson Iteations Combined with Incemental Pedicto The full Newton-Raphson method equies that the intenal foces must be detemined afte each iteation and balanced with the extenal applied loads. Also, since the beam may yield and defom plastically at some potions of the coss-section, the modulus of elasticity contained in the linea stiffness matix would change. Theefoe, numeical integation must be pefomed to obtain the above mentioned quantities. Howeve, since the beam element is assumed to defom accoding to the Eule-Benoulli beam model plane sections emain plane afte defomation we can detemine the stess distibution ove the coss-section analytically. The analytic integation is moe accuate and moe efficient than a point by point quadatue. It uses the value of the stain along the depth of 5

67 the beam as obtained fom Eq. 3.3 to detemine the location of the yield zones on the coss-section, and fom this infomation the stess distibution ove the coss-section is obtained see Figue 3.5 q o P P σ σ y σ = σ y σ = σ y σ < σ y σ < σ y P M σ = σ y ε ε y ε > ε y ε > ε y Figue 3.5 Plastic Zones Developed in a T-Beam Coss-section fo Inceasing Values of the Axial and Bending Loads Step Contol Since some equilibium path non-lineaity due to the onset of buckling o plasticity is going to be encounteed, it is ecommended that some fom of displacement contol be used see Thomas and Gallaghe, 975; Batoz and Dhatt, 979; and Cisfield, 99. Accodingly, displacement contol of a specific displacement vaiable is used to pogess. In paticula, following the wok of Batoz and Dhatt 979, one would pescibe the value of the dominant displacement component δ q q to a cetain value say a faction of the atio of the beam s depth at the beginning of the 5

68 iteations i.e. i =. Then, the tangent stiffness matix, displacements, the esidual loads, and the applied extenal loads ae eoganized as the following: K K K K δq δr P = δλ δq δr P 3.4 wheeδ q = δq, [ ] q K is the tangent stiffness matix of ode n n with the qth ow and column ae suppessed duing pocessing i.e. factoization o invesion, { K } and K ae the qth column and ow of the tangent stiffness matix without Also,{ q} δ,{ δ R}, and { } ae oganized accodingly. Taking the second ow equation in Eq. 3.4, K = Kqq of dimension n. P ae the displacement, the esidual load, and the applied load vectos that K δ q K δq = δr δλ P 3.5 and splitting the displacement vecto into two components; one associated with the extenal loads vecto { P } and the othe with the unbalanced loads {δr} in the following fom { δ q} { δq a } δλ{ δq b } b Equation 3.4 can be ewitten as note that δ = = 3.6 q q K δ q λ δq K δq = δr δλ P 3.7 a b q Finally, solving fo the applied load faction δλ δλ K δq δr K δq a = q b P Kδq 3.8 the desied iteative o incemental step size is obtained as: λ = λ δλ i i i

69 δ q δq δλ δq i = 3. a i i b i q = q δq i i i 3. It is impotant to note that the stiffness matix [ K t ] and the esidual foces vecto δr ae calculated fo the initial step afte updating the displacement with the pescibed iteative displacement component i.e. q o = q δq. This method was implemented in a FORTRAN-9 compute o q pogam that is documented in Appendix A. Also, the method gave exact match to the esults of a simply-suppoted elastic-plastic beam analyzed by Yang and Saigal 984, as shown in Figue 3.6. Cental Deflection of an elastic-plastic beam Yang and Saigal Non- Dimensional Cental Deflection Non-Dimensional Load Figue 3.6 Mid-Span Deflection of a Simply Suppoted Beam of Rectangula Coss-section unde Lateal Load see Yang and Saigal

70 Chapte 4 RBDO Exploatoy Example: Elastic-Plastic T-beam 4. Failue Modes Failue unde axial and bending stesses Fo a linea-elastic mateial, the axial stess can be linealy combined with the flexual stesses until the exteme fibes yield. Howeve, afte yielding the failue stesses do not combine linealy and they can be detemined fom the equilibium of foces and moments ove the cosssection. Let us conside the state of stess at a beam of T coss-section unde the combination of axial load and bending moment that causes the whole coss-section to defom plastically as shown in Figue 4. M σ = σ y P Figue Fig. 4. The The State of of Stess in fo the a T-Beam Coss-Section unde fo the Combination a of Axial of the Load Axial and Load Bending and the Moment Bending that Causes Moment the that Whole Causes Coss-Section Fully Plastic to Defomation Plastically of 55

71 Fom the equilibium of foces in the axial diection noting that the axial compessive load P is assumed to be acting at the centoid of the coss-section: y A A P = σ 4. c t which becomes afte simplifying P = σ A A 4. y c whee P is the applied axial load, σ y is the yield stess of the mateial, A c is the aea unde compession, A t is the aea unde tension, and A is the aea of the coss-section. Nomalizing the axial load by the maximum axial load on the coss-section that causes the fully plastic defomation Py =, the nomalized axial load becomes: σ y A Ac R p = 4.3 A Fom equation 4.3 solving fo A c we get R A c = A p 4.4 Fom the foce equilibium and Figue 4., it can be seen that the axial load at which the flange becomes entiely plastic unde compession is an impotant tansition load because the equilibium equations will change when the net compessive foce is distibuted entiely on the flange o potions of it, and when the net compessive foce is distibuted ove the flange and pat of the web. We will designate the fist situation, case a, and the second situation, case b. Hence, fom equilibium of foces we can get the axial load that causes the flange to become fully plastic unde compession. This load becomes afte nomalization by the maximum axial load and simplification becomes: 56

72 57 = w f f pc h t b t b t R 4.5 Also, fom the equilibium of the moment we get: y A y A M t c y = σ 4.6 whee y and y ae the distances between the centoid of the entie coss-section and the centoids of the compession and tension zones, espectively. Fom the geomety of case a, we can detemine the following elations: b A a c c = 4.7 c c a C = 4.8 b a t h t a t h b a t h t h C c f w c f c f w t = 4.9 Similaly, fo case b: f w f c c t t b t A a = 4. w f c f f c w f c f f c t t a b t t a t t a t b t C = 4.

73 C t h t f ac = 4. whee C c and C t ae the centoids of the compession and tension zones espectively measued fom the bottom edge of the web. Fo both cases, the expessions fo distances y and y ae given by the following elations: y = h t Y f C c 4.3 y = Y C t 4.4 whee Y gives the location of the centoid of the entie coss-section measued fom the bottom edge of the web. We can nomalize the moment by the maximum elastic moment, M e, that the coss-section can expeience befoe any plastic defomation in the exteme fibe sets in. This moment is given as: M e σ y Y I = 4.5 Then the nomalized moment becomes: R m Ac Y A Ac Y = y y 4.6 I I 58

74 We can now substitute the value of A c fom Eq. 4.4 into Eq. 4.6 and obtain two expessions that elate R p with R m and the coss-section dimensions fo cases a and b. These two expessions will be of the fom R m = f a R p 4.7.a R m = fb R p 4.7.b The fist expession is valid fo axial loads in the ange fom zeo to R pc Eq. 4.5; the second expession is valid fo loads highe than R pc. The obtained expessions wee applied to the example pesented by Smith and Sidebottom 965 and gave identical esults. The fist expession is valid fo axial loads in the ange fom zeo to R pc Eq. 4.5; the second expession is valid fo loads highe than R pc. The obtained expessions wee applied to the example pesented by Smith and Sidebottom 965 and gave identical esults. Thus, we have obtained the inteaction cuve between the axial foce and bending moment and now we will conside the effects of the shea loads The effect of tansvese shea stess: Tansvese shea stesses due to a bending load have elatively small contibution to the failue of the coss-section compaed to the bending o tosional shea stesses fo long and slende beams having height to length atio >.3. Howeve, we can include thei contibution to failue in case we found that they educed significantly the yield stength of the mateial. In that case, the new failue stess becomes 59

75 = σ y σ f 3τ 4.8 whee σ f is the failue stess of the mateial, σ y is its yield stength, and τ the aveage shea stess on the beam coss-section. Howeve, it is impotant to note the limitation on this eduction facto as it is only valid fo shea stesses up to 3% of the axial stesses due to flexue loads. see Hughes, 983. The effects of tosional shea stesses: Finally, we include the effect of the tosional shea stess on the total plastic failue of the beam and constuct the inteaction suface. These tosional effects include the thickness shea stesses St. Venant s shea stess and the waping axial stesses contou stesses. Howeve, the waping axial stesses have seconday effect in a thin walled section and can be ignoed see Megson, 999. Now, we need to detemine the value of the maximum toque that will defom the entie cosssection plastically. We will apply the Nadai sand heap analogy to the coss-section and obtain the following expession fo the maximum shea stess that the coss-section can withstand with no othe stesses pesent see Nadai, 95: t 3 3 f t t f b w f w tw h t t T p = τ y whee 6

76 σ y τ y = Von Mises 4. 3 We can nomalize the plastic toque by the maximum elastic toque, that is the toque afte which the exteme fibe defoms plastically. Te τ y J = tw τ y h t 3 w b t 3 f = 3 tw 4. Finally, we obtain the nomalized maximum plastic toque fo this coss-section [ t 3 f tw 6 t f b 6 tw h tw t f ] 4 [ h t 3 b t 3 ] R Tp tw T = = o Te w f 4. So fa, we have developed an inteaction cuve between the axial foce and bending moment and we have deived an expession fo the maximum tosional shea stess fo the coss-section. We still need to elate all these quantities togethe in one constaint. Howeve, we do not need a igoous mathematical fomulation to elate these loads to the limit combined load; but instead, we can use a suitable empiical fomulation. This empiical fomulation must satisfy the convexity condition Ducke's postulate and must not assume that the mateial will bea stesses that ae highe than the allowable Von Mises stesses at any point in the coss-section see Chakabaty, 987; Gebbeken, 998; and Ducke, With this in mind, let us assume that the inteaction of the tosional load with both the axial load and the bending moment can be epesented by a uled suface as shown in Figue 4..a. 6

77 Fig 4.a The Safe Design Zone unde the yield suface that Combines the Axial, Bending and Tosional Shea Loads This suface is constucted by extending staight lines fom the maximum nomalized tosional load to the nomalized inteaction cuve between the bending and the axial load. This way, the convexity of the failue suface is assued since we ae using staight lines. Also, no concavity is allowed to occu at any potion of the suface othewise the plastic wok will be negative, which means that we ae at the lowe bound of any possible inteaction cuve between the tosional shea and the othe loads. Theefoe, at no point in the coss-section the stess exceeds the yield stength of the mateial, and that fulfils the second condition that the yield suface must satisfy. Now, we need to obtain the facto of safety against failue detemined by this suface, which tells us how fa o nea we ae fom the constaint coesponding to failue of a beam section. Let us assume the pojection of any point inside o outside the failue suface onto the M-P plane to be denoted by P, and let us daw a vecto, V, fom the oigin to the pojected point see Figue 4..b. 6

78 Then let us extend the vecto V until it intesects the M-P inteaction cuve and name the new vecto V. Now, let R PM be the atio of the lengths: V R PM = 4.3 V which will allow us to obtain the safety facto γ afte solving fo it fom the failue suface equation below: γ R γ R = 4.4 PM T R M V V R p R Po R To P R T Fig Fig. 4..b 3 The The Pojection of of the the Design Point on on the the M-P Plane Beam Plastic Deflection Hee, we ae setting a limit fo the maximum deflection of the beam; in ou case % of the total depth of the beam. Moeove, we will assume that the applied loads on the beam do not change with the magnitude of the deflection. Consequently, we do not expect any change in the load 63

79 distibution ove the beam. Although we could have used a geneal-pupose finite element code to calculate the deflection, it was moe economical to wite a non-linea beam finite element code fo this poblem see Chapte 3. Out-of-Plane Buckling The out-of-plane buckling buckling in the plane of the flange was consideed to pevent the optimize fom obtaining naow beams that can buckle out of plane. We used Eule s beam buckling fomula fo this pupose. 4. Deteministic Stuctual Optimization To undestand the advantages and disadvantages of the eliability-based optimization, we will compae it with a numbe of deteministic optimum designs with diffeent safety factos. The intent hee is to compae the deteministic optimum designs' weight and safety level to those of eliabilitybased optimum designs. The Deteministic Optimization Poblem Fomulation find : dopt min : W d s. t.: G d, µ SF i x 4.5 d d L opt whee d is the vecto of design vaiables height of the web, the width of the flange, and thei thicknesses as shown in Figue3, W d is the weight of the beam, G d, µ SF ae the constaints that depend on and mean values of the andom loads lowe limits of the design vaiables. The constaints ae: i x µ x and the safety facto SF, and L d ae the 64

80 . A stength constaint that pevents failue by total plastic defomation at any coss-section of the beam: the beam can sustain the applied loads without total plastic defomation at any coss-section γ. in equation 4.4, which is specified by the pefomance function G d,. X. A constaint about out-of-plane stability that pevents failue by loss of out-of-plane stability: the beam would not buckle out-of-plane, which specifies the pefomance function G d,. X 3. A design constaint that pevents failue by excessive in-plane deflection: the in-plane deflection of the beam will not exceed % of its total depth, which specifies the pefomance function G d,. 3 X Also, the lowe limits on the design vaiables ae manufactuing constaints that pevent the thickness fom being less than 5 mm fo any potion of the beam. Fo the optimization pocess, we have used the Modified Method of Feasible Diections and the Sequential Quadatic Pogamming optimization algoithms of Visual Doc softwae Vandeplaats, 999; and Ghosh et al,. Both methods conveged to the same optimum design in the examples consideed. 4.3 Reliability-Based Stuctual Optimization Two fomulations of the eliability-based optimization poblem will be consideed, design fo maximum eliability and design fo minimum weight. Design fo maximum eliability is used if the objective is to find the design with highest eliability pobability of suvival whose weight does not exceed a maximum acceptable limit. In this fomulation, objective function is the beam eliability. On the othe hand, design fo minimum weight is used when the objective is to find the lightest beam design whose eliability is no less than a minimum acceptable value. In this fomulation, 65

81 objective function will be the beam weight. We will use both appoaches and compae the obtained designs to the deteministic optimum design. The beam eliability is a function of the failue modes consideed ealie, that ae the constaints in the deteministic optimization poblem. It is impotant to note that manufactuing constaints ae consideed as side constaints and will not affect the calculation of the design eliability. Fomulations of Reliability-Based Optimization Poblems a. Design fo minimum weight whee failue modes. find : d opt min : W d s.t. R d, X R L d d * d 4.6 * R d is the eliability of the coesponding deteministic optimum design against all the b. Design fo maximum eliability: find : d opt max : R d, X s.t. W d Wd L d d * 4.7 whee d opt is the vecto of optimal design vaiables, R d, X is the system eliability of the beam, which is the pobability of suvival of the beam unde the applied loads calculated against the peviously detemined failue modes G i d, X, W d * is the weight of the beam, and W d is the weight of the coesponding deteministic optimum design 66

82 Methods Used to Calculate the Reliability of the Beam To cay out the eliability-based optimization we need to calculate the eliability of diffeent beam designs unde vaiable loads and/o mateial popeties. Thus, fo this application it would be suitable to use a vey economical method to calculate the eliability of a design, such as the Fist Ode Second Moment Methods FOSM, and in the case of nomally distibuted vaiables we can diectly use Hasofe-Lind H-L method othewise we may need to tansfom the andom vaiables into equivalent nomal vaiables, see Melches,. The H-L method equies tansfoming the andom vaiables and limit-state functions into a new space space of educed vaiables with independent, nomal andom vaiables with zeo mean and unit standad deviation. Then, we need to find the most pobable failue point, which is the closest failue point to the oigin in the space of educed vaiables. Finding the most pobable failue point fo non-linea state functions is an optimization poblem. Thus, it would be wothy to check the accuacy of the esults by compaing them to those by anothe moe diect method. We chose Monte-Calo Simulation with Impotance Sampling, since using the Diect Monte-Calo Simulation needs aound 7 sample points to fo a eliability of.999 and a adius of confidence of %. Also, to simplify the eliability calculations we will calculate the pobability of failue fo each failue mode sepaately and then use the summation of the pobabilities of failue fo all the failue modes as an uppe bound on the system pobability of failue. Once the beam eliability at diffeent design points is calculated, we can use a suitable optimization scheme to find the eliability-based optimum design. Reliability-Based Optimization Method As was mentioned ealie, we needed to devise a simple method that demonstates the featues of RBDO and the diffeent consideations that the designe must conside in the RBDO, without being 67

83 computationally expensive. Howeve, although the weight of the beam is a simple function of the design vaiables, the eliability of the beam is an implicit function of the design vaiables and may not always be diffeentiable. Hence, one may not expect that we can use one of the standad optimization algoithms to obtain the eliability-based optimum designs. In light of this, one may think of othe optimization tools that will be both economical and can easily ovecome the discontinuity in the deivatives of the constaint and/o the objective functions. Thus, we ve tuned to the sensitivity analysis of the beam in the sense that the effect of each design vaiable on both the objective function and the constaint is studied sepaately. This analysis was used to identify two goups of impotant vaiables: those that significantly affect the weight and those that affect significantly the eliability. This allows us to petub the design fom the deteministic optimum to eithe incease the eliability of the design with the least incease in the cost, o to educe the cost with the minimal eduction in eliability. To achieve this, we can vay one design vaiable at a time and study its effect on both the eliability and the weight of the beam. To save the unnecessay effot, this analysis may be pefomed in the neighbohood of the optimum deteministic design with the goal of impoving its eliability o educe its cost. Using this infomation the optimization poblem is then changed into a simple one-dimensional seach poblem that can be solved gaphically. 4.4 Numeical Application We chose a beam of T coss-section, which is commonly used as a stiffene in aeospace, ship, and automotive industies. The geomety and dimensions of the beam ae shown in Figue 4.3.a. The loads applied to the beam ae an axial compessive load applied to the beam s centoid and a unifom lateal distibuted load with an eccenticity of 4mm, as shown in Figue 4.3.b. Both the axial load P and the distibuted lateal load Q o ae andom, having nomal pobability 68

84 distibution with mean values 5 kn and 38 kn/m, espectively and coefficients of vaiation given as 5% and 35%, espectively. The mateial is assumed to behave as elastic-plastic with yield stength of 4 MPa and Modulus of Elasticity of 7 GPa. X Y Z L=.9m b tf h tw Figue Fig. 4 The 4.3.a Geomety and Dimensions of the T beam Q o = 38 kn/m, COV=35% P P o =5 kn COV=5% e q = 4 mm mm. Fig 4.3.b The Loads Applied to the Beam 69

85 Combined Load and Deflection Analysis The following should be consideed in failue analysis of the beam in this example: i. The applied bending moment M ext will be a function of the applied loads and the esulting deflection will have the following fom: M ext z Q L = P δ Q o o z 4.8 whee δ is the beam deflection in the y-diection and L is the length of the beam. ii. If any potion of the beam coss-section yields, then the beam bending stiffness is educed, and thus, the beam may be consideed to have failed because of excessive deflection and lose its capacity to bea moe load even befoe eaching the yield suface see Figue 4.4. a. and 4.4.b.. Load-Deflection Cuve. Load Faction Deflection m Fig. Fig 6.a 4.4.a Load-Deflection Load-Deflection Cuve Cuve fo a Typical fo a T T Beam, Used of the hee Example unde Axial unde and Axial Lateal and Lateal Loads Loads. 7

86 M/M elastic Tension Axial load and bending moment plastic inteaction Point of bifucationbuckling failue Axial load and bending moment elastic inteaction Compession P/P plastic Fig b b The Location of of the Failue Point on on the the Axial Axial Load-Bending Moment Inteaction Cuve. Cuve iii.this situation may not occu fo othe bounday conditions, such as fixed end beams, since in that case the beam may continue caying the load until the yield suface has been eached. Thus, we have modified ou constaint fomulation accodingly and eplaced the limit combined load constaint with the maximum load faction that the beam can expeience befoe going into bifucation buckling. Howeve, to avoid getting singulaities in the stiffness matices we limited iteations up to 95% of stiffness eduction of the beam. That is the load at which the slope of the load-deflection cuve is less than 5% of its initial value i.e. when the beam loses 95% of its stiffness. Also, this obseved loss of stiffness occus much ealie than that pedicted by the linea elastic beam-column equation given below and the maximum deflection at obtained at the mid-span δ max is geate see Figue 4.5. δ max = Qo E I k 4 k L Q sec o L 8 E I k 4.9 7

87 k = P E I whee I is the coss-section moment of inetia and E is the mateial modulus of elasticity see Chen and Lui, 987. These esults emphasize the impotance of consideing the inelastic behavio of the beam..3.5 Load-Deflection Cuve fo a T-Beam Load Faction..5. Non-linea FEM Linea Fomula Deflection m Fig. 7 Compaison Between the Results of Non- Linea Figue 4.5 Compaison Between the Results of Non- FEM and the Linea Elastic Fomula Linea FEM 8 and the Linea Elastic Fomula Eq. 6 iv. The shea loads vay linealy along the beam and vanish at the mid-span. Deteministic Optimum Designs We have fist pefomed deteministic optimization fo safety factos SF of.5,.75,.,.5, and.5. The esults ae shown in Table 4.. Also, to check if the obtained designs fom optimization wee global optima, we petubed the values of the design vaiables of the designs and we have aived at the same optimum design evey time. 7

88 Table 4.. Deteministic Optimum Designs fo Vaious Safety Factos Facto of Safety Mass kg h mm t w mm b mm t f mm Now, befoe calculating the espective pobabilities of failue of these deteministic optimum designs, we may need to check on the suitability of Hasofe-Lind H-L method used to calculate the pobabilities of failue in ou application. Fo this pupose, we chose the design with the lowest safety level highest pobability of failue, which is the deteministic optimum design fo SF=.5. We have obtained the pobabilities of failue fo the thee pefomance functions using Monte-Calo Simulation with Impotance Sampling and by using H-L method. They wee in easonable ageement see Table 4.. Howeve, fo such lage pobabilities of failue a typical acceptable 7 pobability of failue is less than we may expect the ageement between FORM and Monte-Calo simulation not to be exact especially fo non-linea constaints. Since, FORM woks best fo linea constaints and non-linea constaints at small pobability of failue see Madsen et al. 986; Halda and Mahadevan, 995; Melches, Table 4.. Pobabilities of Failue fo the Thee Pefomance Functions G X, G X, and G 3 X Using Monte-Calo Simulation with Impotance Sampling IS and by Using H-L Method fo the Deteministic Optimum Design with SF of.5. G G G 3 IS H-L IS H-L IS H-L P f.e-.e-.e-.9e-.3e-3 4.E-3 73

89 Also, we have plotted the thee limit state functions G i X = on the load space to see thei degee of non-lineaity and to check fo the existence of local minima see Figue 4.6. The Pefomance Functions Q/Qn.5.5 G3 G G P/Pn Figue Fig The The Bounday Pefomance Functions G X GX =, G= X, GX = and = G, 3 X = and, Plotted G3 X on = the Random Plotted on Loads the Plane Random the loads Loads ae Plane nomalized nomalized by thei mean by values. thei mean values. These limit state functions did not show a high level of non-lineaity; no did they have a local minimum. These tends show that it would be adequate to use H-L method to calculate the eliability of the T beam. Howeve, it is inteesting to note that the deflection limit state function G 3 X = was not monotonic, which is due to the fact that cetain load atios axial/lateal poduce moe deflection befoe the buckling failue than the othes. Finally, we obtained the pobabilities of failue fo the othe deteministic optimum designs SF of.75,.,.5, and.5 fo the thee failue modes and they ae listed along with the esults of SF.5 in Table

90 Table 4.3. Pobabilities of Failue fo the Deteministic Optimum Designs SF of.5,.75,.,.5, and.5 Failue Modes Facto of Safety G G G 3 System.5.E-.9E- 4.E-3 3.6E-.75.3E-4.47E E-7.67E-3..7E-6 3.E-5.9E-3 3.E E- 3.7E-7 3.E-5 3.8E-7.5.5E-3.94E-9 3.3E-44.94E-9 Reliability-Based Optimum Designs We have pefomed sensitivity analysis aound each deteministic optimum design. Specifically, we have vaied each design vaiable aound the deteministic optimum and plotted the beam pobability of failue against the pecent incease in weight. This allowed us to detemine which design vaiables we should incease to get the maximum incease in eliability see Figs. 4.7.a 4.. a. We used this infomation to solve both eliability-based stuctual optimization poblems a. and b. design fo maximum eliability and design fo minimum weight, espectively. Initially, we planned to vay each design vaiable aound % of its value, but we late modified this ange duing calculations accoding to the contibution of each design vaiable to the eliability and weight of the design. Also, since the thicknesses of the flange and the web wee constained to be no less than 5 mm we studied the sensitivity fo values above that limit up to 5%. In addition, we used quadatic intepolation between the points. Following ae the esults of eliability-based optimization stating fom the coesponding deteministic design fo safety factos of.5,.75,. and.5. Deteministic optimum design with SF.5: Fig 4.7.a shows the eliability of the deteministic optimum design as a function of the weight when the each of the fou dimensions of the coss 75

91 section is changed -- one at a time -- while the othe thee dimensions ae fixed at thei espective values coesponding to the deteministic optimum design. The flange width, b, is the most impotant design vaiable fo -4% to % change in the design weight, while the web thickness, t w, has the least effect within that ange see Fig 4.7.a. Howeve, since the web thickness was at its lowe limit we chose the web height, h, to be the vaiable to be educed. Now, fo the design fo maximum eliability Pob. a., we can impove the eliability of the design fo the same weight by inceasing the flange width and educing the web height in a way that the weight emains the same. Howeve, the impovement was insignificant, but at least it achieved a moe equal balance between diffeent failue modes see Figue 4.7.b.. Consequently, fo the second optimization poblem Pob. b., it was unlikely to obtain a design with a weight less than the weight of the deteministic optimum design. Effects of Vaiying the Design Vaiables on the Weight and Pf SF Log Pf.5.5 tw tf b h % Change in Weight Figue 4.7.a Effects of Vaying the Design Vaiables on the Weight and Fig. 9.a The Effects of Vaying the Design Vaiables on the Design Pobability of Failue of the Deteministic Optimum SF.5 Weight and its Pobability of Failue Pf. 76

92 Seaching fo the Optimum Design with the Lowest Pf SF Log Pf System G G G3 Det SF.5 Rel-Based Optimum h m Fig Fig 4.7.b 9.b Seaching fo the Optimum Design that has the Lowest Pobability of of Failue PPf f Deteministic optimum design with SF.75 : The flange width, b, is the most impotant design vaiable fo -4% to % change in the design weight, while the web height h has the least effect within that ange see Fig 4.8.a. Fo the fist eliability-optimization poblem Pob. a. we have impoved the eliability of the design fo the same weight by inceasing the flange width and educing the web height see Fig 4.8.b.. The situation was favoable fo the second optimization poblem Pob. b. and we wee able to save about.5% of the weight without educing eliability. 77

93 Effects of Vaiying the Design Vaiables on the Weight and Pf Sf LogPf tw tf b h %Change in Weight Figue Fig. 4.8.a.a Effects The Effects of Vaying of Vaying the Design the Vaiables Design Vaiables on the Weight on and the Pobability Design Weight of Failue and of the its Pobability Deteministic of Optimum Failue SF Pf..75 Seaching fo the Optimum Design with the Lowest Pf SF Log Pf System Pf G G G3 Det SF h m Fig Fig. 4.8.b.b Seaching Seaching fo fo the the Optimum Optimum Design Design that has that the has Lowest the Pobability Lowest Pobability of Failue P of Failue f Pf 78

94 Deteministic optimum designs with SF.,.5, and.5: These thee designs showed simila tends as in the deteministic design with SF.75 see Figs. 9.a to.a and we wee able to impove thei espective values of the eliability fom to. 5. 9, fom to. 4. 6, and fom.. 9 to.. espectively. Also, we wee able to save weight by.7%, 3.6%, and 5.% espectively. Figues 4..a and 4..b summaize these findings. 79

95 Log Pf tf b h tw % Change in Weight Figue 4.9.a Effects of Vaying the Design Vaiables on the Weight and Pobability of Failue of the Deteministic Optimum SF. 3 -LogPf h m System G G G3 Det SF. Fig 4.9.b Seaching fo the Optimum Design that has the Lowest Pobability of Failue P f 8

96 Effects of Vaiying the Design Vaiables on the Weight and Pf Sf.5 -Log Pf tf b h tw % Change in Weight Figue Fig..a 4..a The Effects Effects of Vaying of Vaying the Design the Design Vaiables Vaiables on the Weight on the and Design Pobability Weight and of Failue its Pobability of the Deteministic of Failue Optimum Pf. SF.5 Seaching fo the Optimum Design with the Lowest Pf SF Log Pf System G G G3 Det SF h m Fig 4..b Seaching fo the Optimum Design that has the Lowest Pobability of Fig..b Seaching fo the Optimum Design that has the Lowest Failue P Pobability f of Failue Pf 8

97 Effects of vaiying the design vaiables on the weight and Pf Sf.5 -Log Pf % Change in Weight tf b h tw Figue 4..a Effects of Vaying the Design Vaiables on the Weight and Fig. Pobability 3.a. The of Effects Failue of Vaying the Deteministic the Design Optimum Vaiables SF on.5 the Design Weight and its Pobability of Failue Pf. Seaching fo the Optimum Design with the Lowest Pf SF.5 -Log Pf h m System G G G3 Det SF.5 Fig 4..b Seaching fo the Optimum Design that has the Lowest Pobability of Failue Fig. P f 3.b Seaching fo the Optimum Design that has the Lowest Pobability of Failue Pf 8

98 4.5 Discussion of Reliability-Based Optimization Results Fom Table 4.3, we obseve that the eliability of the beam designed deteministically fo a facto of safety of.5 is athe low <.97. This shows that the facto of safety may not be an adequate method to descibe the level of safety of a design. Figs 4.7.a to 4..a povide an insight into the behavio of the eliability and the weight of the beam as functions of each design vaiable. In addition, looking at Figs 4.7 to 4., one may notice the discontinuity of the beam eliability function and how the contibution of each design vaiable to the eliability of the beam may diffe befoe and afte this discontinuity. Moeove, looking at Figs 4.7.b-4..b we can see that the location of the deteministic optimum designs in the space of the eliability-design vaiables is located nea just one failue mode. This is because the deteministic optimize stops as soon as it eaches any of the constaints and no futhe impovement in the objective function is possible. Thus, the isk of failue may not be evenly distibuted among the failue modes and futhe impovement of the design by making it moe eliable o obtain one with less weight may be possible by edistibuting the mateial to moe equal isks of failue among the thee failue modes. Finally, as we can see fom Figs. 4..a. and 4..b. the benefits of the eliability-based optimization ove the deteministic optimization incease fo highe safety levels and the incease in benefits is monotonic as found in an ealie study on the design of aicaft wings subject to gust loads 35. This is because as the safety level of the design inceases moe mateial may be used, and thus, a bette distibution of the mateial can yield highe benefits. 83

99 4 -LogPf 8 6 Det Optimum Rel-Based Optimum 4 SF.5 SF.75 SF. SF.5 SF.5 Design Facto of Safety Figue 4..a Compaing the Reliability of the Deteministic and the Reliability- Based Optimum Designs 6.E 5.E % Weight Saving 4.E 3.E.E.E.E SF.5 SF.75 SF. SF.5 SF.5 Design Facto of Safety Figue 4..b Weight Saving with Reliability-Based Design ove the Deteministic Design 84

100 4.6 Summay and Conclusions An elastic-plastic beam design of T coss-section unde combined loads was optimized. Thee failue modes wee consideed: failue due to yielding of the entie beam coss section unde the combined action of axial and lateal loads, buckling in the plane of the flange, and excessive deflection in the plane of the web. A load inteaction suface was constucted to detemine the limit load that combines axial, bending, and shea loads. Also, a non-linea finite element pocedue was used to detemine the deflection of the beam in the plastic ange. Next, deteministic optimization was caied out to find the lightest beam design that can suppot the applied loads fo a given safety facto. The deteministic optimization was epeated to obtain optimum designs fo a numbe of diffeent safety factos. Then, the pobability of failue was calculated fo each deteministic optimum design using a Monte-Calo simulation with impotance sampling and b fist-ode second-moment methods. Finally, eliability-based optimization was pefomed to find a the lightest design that has same eliability as the deteministic design and b the most eliable design that has same weight as the deteministic design. The esults show that the eliability-based optimization can obtain designs with lowe pobability of failue than the designs obtained by the deteministic optimization. Consequently, this advantage can be used to obtain safe designs that have the same weight as the deteministic optimum designs, o designs that have less weight and the same pobability of failue as the deteministic ones. Also, these esults show we wee able to deal with the poblem of high non-lineaity and deivative discontinuity of the eliability function in such a way that allowed us to obtain an impoved design ove the deteministic design in most cases. 85

101 Chapte 5 Application of the Sequential Optimization with Pobabilistic Safety Factos RBDO to the Exploatoy Elastic-Plastic T-Beam Example 5. Intoduction Befoe consideing a moe involved example, it would be appopiate to use the exploatoy elastic-plastic T-beam example that we have consideed in Chaptes 3 and 4 to investigate the SORFS RBDO technique see Sec.4.4. In paticula, we will compae the esults of the gaphical method used ealie see Ch. 4 to the esults of SORFS RBDO method. Also, we will examine the ability of the SORFS method to ovecome the poblems of high computational costs and the high non-lineaity and deivative discontinuity of the eliability function. Finally, we will study the convegence of the SORFS RBDO method. 5. The Optimization Poblem 5.. Poblem Statement The following is a bief pesentation of the beam optimization poblem that we have consideed in Chapte 4. The beam loads and dimensions ae shown in Figue 5.. It is equied to find the beam design that has the minimum weight and satisfies the following constaints. The beam can sustain the applied load without failue G. The beam must not expeience out-of-plane buckling G 3. The beam must not have an in-plane defomation that exceed % of its total depth G 3 86

102 4. Design Constaint: no thickness less than 5 mm The Design Vaiables ae: the web depth h, web thickness tw, flange width b and flange thickness tf P o = 5 kn COV= 5% L=.9m b t f h t w Figue 5. The Beam Loads and Dimensions Also, it is equied to obtain thee optimum designs: a. A deteministic optimum design that is obtained by using deteministic safety 87

103 facto. b. A eliability-based optimum design that has the same level of safety of the deteministic optimum design but with the least possible weight. c. A eliability-based optimum design that has the same weight as the deteministic optimum design but with the highest eliability. 5.3 The Sequential Optimization with Reliability-Based Factos of Safety Appoach To compae the esults of the SORFS technique with the gaphical appoach that we have used befoe, we will pefom the optimization fo the five levels of safety consideed in Chapte 4 safety factos of.5,.75,.,.5 and.5. Accodingly we will follow the following steps to obtain the two eliability-based optimum designs fo each facto of safety see Ch. fo moe details:. Pefom deteministic optimization fo the given deteministic safety facto. Calculate the eliability of the deteministic optimum design using FORM o any othe eliability calculation method, and we set the maximum allowable pobability of failue to equal the calculated eliability of this deteministic optimum design. Also, we identify the active and the nea active constaints having pobability of failue that is lage than % of the active constaints.. 3. Pefom semi-pobabilistic optimization using the values of the safety indices β T of i the deteministic optimum design as initial guesses see Eq..8 and.9, and fixing the coodinates of the most pobable failue points MPP X see Eq..9 88

104 4. Pefom pefomance measue analysis PMA to find coected most pobable failue points MPP X that coespond to the optimum safety indices β obtained in step 3, and use them as the lowe bounds on the β T fo the next optimization iteation. i 5. Calculate the eliability of the new optimum design Repeat steps 3-5 until getting a design with the least weigh that has a eliability that is equal o bette than the eliability of the coesponding optimum design is obtained. Ti Finally, to obtain a eliability-based optimum design that has the highest pobability fo a given weight we use the same steps above, but with the system eliability as the objective function and the weight as a constaint. Dissetation 5.4 Reliability-Based Optimization Results fo the Exploatoy Elastic-Plastic T- Beam Example using the SORFS Appoach Fo this example we found that the constaints G and G wee competing constaints see Figue 4.6. Since G depends only on the value of the axial load, and to impove the design against the coesponding out-of-plane failue, the flange width has to be inceased. On the othe hand, G depends moe on the flexual stength of the beam, which equies deepe web moe than a wide flange. The esults wee in excellent ageement with the esults that we have obtained ealie in Chapte 4 compae Figues 5. and 5.3 to Figues 4..a and 4..b. In addition, we wee able to achieve an aveage eduction of calculations of about 8% compaed to the gaphical method used in Chapte 4. 89

105 4 -LogPf 8 6 Deteministic Reliability-Based 4 SF.5 SF.75 SF. SF.5 SF.5 Safety Levels Figue 5. Compaing the Reliability of the Deteministic and the Reliability-Based Optimum Designs 9

106 6 5 Weight Saving % 4 3 SF.5 SF.75 SF. SF.5 SF.5 Safety Levels Figue 5.3 Weight Saving of the Reliability-Based Design ove the Deteministic Design. Finally, to check the convegence of the method, we have pefomed convegence analysis fo both the RBDO poblems fo SF.5, and the esults ae shown in Table 5. and Table 5. Simila convegence esults wee found fo SF.5 9

107 Table 5. Convegence of RBDO T-Stiffene Example finding the least weight fo a cetain level of safety fo SF.5 using the SORFS Method Deteministic Opt RBDO Cycle Cycle Cycle 3 Cycle 4 h=.34 t w =.5 b=.9 t f =.5 Weight= 4.5 G Xp = Xq =.786 β T = P f = E-3 G Xp = Xq =. β T = P f =.94E-9 h =.9657 t w =.5 b =.989 t f =.5668 β T = β T = 6.67 β T3 = 5. Weight= 3.57 G Xp =.7 Xq =.5 β T = 5.93 P f =.536E-7 G Xp =.5444 Xq =. β T = P f = 3.85E- h = t w =.5 b = t f =.5 Weight= G Xp=.4 Xq =.5 β T = 6.38 P f = 9.996E- G Xp =.568 Xq =. β T = 6.73 P f = E- h =.3576 tw =.5 b = tf =.5 β T = β T = β T3 = 5. Weight= 3.57 G Xp=.36 Xq = β T = P f = 7.9E- PMA Xp =.969 PMA Xq =.676 G Xp =.638 Xq =. β T = 6.53 P f = 3.78E- PMA Xp =.584 PMA Xq =. h =.3576 tw =.5 b = tf =.5 No change G Xp=.36 Xq = β T = P f = 7.9E- No change G Xp =.638 Xq =. β T = 6.53 P f = 3.78E- No change G 3 Out of ange G 3 Out of ange G 3 Out of ange G 3 Out of ange G 3 No change Table 5. Convegence of T-Stiffene Example RBDO finding the highest eliability fo a cetain weight fo SF.5 using the SORFS Method Deteministic Opt RBDO Cycle Cycle Cycle 3 Cycle 4 h=.34 t w =.5 b=.9 t f =.5 Weight= 4.5 G Xp = Xq =.786 β T = P f = E-3 G Xp = Xq =. β T = P f =.94E-9 h = t w =.5 b =.5763 t f =.5 Weight= 4.5 β T = β T = β T3 =5. G Xp =.6 Xq =.7 β T = P f = 7.3E- G Xp = 3.93 Xq =. β T = 8.36 P f = E-7 h =.3345 t w =.5 b =.9746 t f =.5 Weight= 4.5 G Xp =.55 Xq =.875 β T = P f = E-3 G Xp =.735 Xq =. β T = P f =.769E- h = t w =.5 b =.397 t f =.5 β T = β T = 7.46 β T3 =5. Weight= 4.5 G Xp =.9834 Xq = β T = P f =.9E- PMA Xp =.84 PMA Xq =.934 G Xp = Xq =. β T = P f = 6.573E-4 PMA Xp =.8 PMA Xq =. h = t w =.5 b =.397 t f =.5 Weight= 4.5 G Xp =.9834 Xq = β T = P f =.937E- G Xp = Xq =. β T = 7.47 P f = 6.573E-4 G 3 Out of Range G 3 Out of Range G 3 Out of Range G 3 Out of Range G 3 Out of Range 9

108 5.5 Discussion of the Results Fom Figues 5. and 5.3, we may ealize that we wee geneally able to obtain impoved optimum designs that has less weight than the coesponding deteministic optimum design fo the same eliability o that has highe eliability fo the same weight of the deteministic optimum design. Howeve, fo the same easons that wee explained in Chapte 4, we could not get a substantial impovement fo the deteministic optimum design with safety facto.5. Yet, the optimization pocess did not encounte any difficulty that is elated to the eliability function, since the optimization is pefomed deteministically without using an actual eliability objective function o constaints in the optimization pocess. Moeove, using the SORFS RBDO method we wee able to achieve lage eductions in calculations an aveage of 8% eduction. Finally, the convegence analysis as pesented in Tables 5. and 5. indicates that the method conveged afte the second semi-pobabilistic optimization cycle. 93

109 Chapte 6 Nonlinea Finite Element Model of 4 Degees of Feedom Elastic-Plastic Beam 6. The element loads We conside a beam of abitay open constant coss-section unde the action of a system of loads see Figue 6.. These loads that ae applied extenally to the beam sufaces may include an axial load N, shea foces in the X and Y diections V x and V y, moments aound the X and Y axis M x and M y, tosion aound the shea cente T, and distibuted lateal loads in the X and Y diection w x and w y. Y V y w y M y X w x C N G T SC M x V x Z Figue 6. A Pismatic Beam of Abitay Open Coss- Section unde the Action of a System of Loads 94

110 6. Coss-Section Defomation Modeling In the following we ae going to model the coss-section defomations in a way simila to that of Tahai 993. Let s define the beam element tanslations in the X, Y, Z diections the oigin is at the centoid by u, v, w espectively. We will follow the Eule-Benoulli beam assumptions, and hence, the otations aound the X and Y axes ae given by fo small otations aound the X and Y axes: dv dz du v and u, whee a pime indicates deivatives with espect to the beam axis, and the dz counte clock-wise is taken to be the positive diection. Also, the ate of change of twist of the beam aound the shea cente will be denoted by φ see Figue 6.., which is also positive if in the counte clock-wise. w CG v u Y X u φ φ SC v Z Figue 6. Beam Element Defomations unde a System of Loads Fo a long pismatic beam unde the given load the coss-section expeiences shea cente displacements u S, v S, w S paallel to the oiginal X, Y, Z axes that displaces S o to S. Also, a twist otation φ aound the displaced shea cente S. Theefoe a point P o in the coss-section will 95

111 expeience a displacement to P paallel to S o -S and will otate aound S to P see Figue 6.3 a- c. Y P v P P u, v φ S v s u P P o x, y S o x o, y o y o X u s x o a In the X-Y Plane Z Y Z S Z x-x o - y-y o φ v P, y-y o x-x o φ -u P, - u {x-x o - y-y o φ } S -v {y-y o x-x o φ } X Z b In the X-Z Plane c In the Y-Z Plane Figue 6.3. Displacements of a Point P in the Beam Coss-Section 96

112 We ae assuming that the coss-section defomations will be modeately small, and that the otations will be modeately lage. Theefoe, we ae using small angle appoximation sin φ φ, cosφ we can wite the displacements of point P as u v = u y y φ 6. P S o P = v x x φ 6. S o The otation of the line S P in the X-Z plane is denoted by u and its coesponding deflection in that plane makes a displacement in the axial diection Z of: u wp = u { x x φ} o y yo 6.3 Similaly, the otation of the line S P in the Y-Z plane is denoted by v, and its coesponding deflection in that plane makes a displacement in the axial diection Z of: v wp = v { y y φ} o x xo 6.4 Now, we need to conside the axial St.Venant waping displacement of point P due to twist aound the shea cente. Fo that we need to conside the two types of waping that an open-section thinwalled beam may expeience; the pimay waping along the mid-thickness of the section and the seconday waping acoss the thickness see Figue 6.4. To detemine the pimay waping of a thin-walled beam, let s look at the components of displacement of a point in a thin walled open section beam. 97

113 Y s,v t n,v n w sc f Z ρ sc X Figue 6.4 Axial, Tangential and Nomal Components of Displacements at a Point in Beam Wall whee v t is the tangential component of displacement at a point v n is the nomal component of displacement at a point w is the axial component of displacement at a point f is the otation angle of the coss-section sc is the shea cente of the coss-section ρ sc is the nomal distance between the shea cente and the tangent to the midthickness of the coss-section at a point s is a dimension that is measued aound the coss-section of the beam tangent to the mid-thickness Now, let s look at the shea stains at an element δ z δs of the beam wall as shown in Figue

114 δw δs δz δv t Figue 6.5 Shea Stain Resulting fom Axial and Tangential Components of Displacement We can expess the shea stain as: zs w vt = s z γ 6.5 Also, we can expess the tangent component of displacement as Then, we can wite v = ρ n φ 6.6 t sc w γ = ρ n φ sc s zs 6.7 The seconday waping distibution is assumed to be that of a naow ectangula stip, so that w = [ n s ω s] φ 6.8 ω whee ω s is the Vlasov s sectoial coodinate. 99

115 The shea stain is given by ω s γ = [n ρ ] φ SC s zs 6.9 Fo open coss-section the shea flow at the mid-thickness at n = is identically zeo which gives us at the mid-thickness t / t / Gγ ds = 6. zs ω s = ρsc s 6. Let s define the waping function whee the limits of integation ae chosen such that Then the shea stain educes to Also, we can wite the axial displacement as s ω s = ρsc ds 6. ω s t ds = 6.3 γ = n φ P 6.4 w P = w S u n s ω φ { x x y y φ} v { y y x x φ} o o o o 6.5 So at the centoid C whee x = and y = C = w x u v φ y v u φ n s ω φ w 6.6 S o o C C C whee ω C = c ρ SC ds, and n C and s C ae the n and s coodinates of the centoid.

116 Let, φ ω = C C C C s n w w 6.7 Then, u v y v u x w w o o S = φ φ 6.8 and φ φ φ ω u y xv s n y v xu w w P = 6.9 Now, we can detemine the longitudinal and shea stains. The longitudinal stain at point P of the coss-section is obtained as the following see Figue 6.6. Z P Z P P Z Z Z P v u w δ δ δ δ δ ε = 6. Using the binomial theoem assuming v w P << and u P P P P v u w ε 6. o, φ φ φ φ φ φ φ ω ε = o o P x x v y y u yu yu xv xv s n y v xu w 6. Afte eaanging φ φ φ φ ω φ φ ε = o o o o P y y x x u y v x s n u v y v u x v u w 6.3

117 ε P δ Z Y δ Z w P δ Z u P δ Z v P δ Z Z X 6.3 Defomation field appoximation We will use Hemite polynomials to appoximate the element defomations along the element. We will need a fist ode polynomial to appoximate the axial defomation w, and cubic polynomials to appoximate the lateal u, v and twist φ defomations. Thus, fo the axial defomation, we can wite = = L o L o w w N N w L z w L z w, 6.4 whee w o and w L ae the axial displacements at the element ends at z = and z = L. Similaly, fo the lateal defomations, we can wite = = L L o o L L o o u u u u H H H H u L z L z u L z L z u L z L z z u L z L z u whee u o and o u ae the lateal displacement and its ate of change along the z-diection espectively, and u L and L u ae those at the othe end. Also, we can wite Figue 6.6 The Longitudinal Stain

118 3 = L L o o v v v v H v 6.6 and = L L o o H φ φ φ φ φ 6.7 Finally, we can wite the defomations vecto in the fom of { } T c q q q T v u w 4,,, ] = [ K φ 6.8 whee q = w o and q 8 = w L ae the axial displacements that coespond to axial loads N o and N L, espectively. q = u o and q 9 = u L ae the tansvese displacements that coespond to shea loads V xo and V xl, espectively. q 3 = v o and q = v L ae the tansvese displacements that coespond to shea loads V yo and V yl, espectively. q 4 = φ o and q = φ L ae the twist otations that coespond to the unifom tosion T o and T L, espectively.

119 4 q 5 = u o and q = u L ae the beam end otations that coespond to the end moments M yo and M yl, espectively. q 6 = v o and q 3 = v L ae the beam end otations that coespond to the element end moments M xo and M xl, espectively. q 7 = φ o and q 4 = φ L ae the ates of change of twists that coespond to the ends bi-moments B o and B L, espectively. And [ ] = = H H H H H H H H H H H H N N T T T T T v u w c φ 6.9 See Figue 6. 7 fo the beam element genealized displacements and foces

120 q 3 q 4 M yl B L q q 8 q q 9 q V yl N L T L V xl M xl q 6 q 3 q q 4 q q 7 q 5 Y M yo V yo N o T o V xo B o M xo Z X Figue 6.7 The Beam Element Genealized Displacements and Foces. 6.4 Element Stiffness Matix The stain enegy is expessed as: U = σ P ε P τ Pγ P dv = Eε P Gγ P dv 6.3 V V U T = ε P γ γ V P E ε P dv G P 6.3 5

121 o symbolically U T 6.3 = {}[ ε D]{}dV ε V T whee { ε } ε P = γ P We can wite {} ε = [ G x, y] [ Γ z] 6.33 Whee x y n s ω x x y y o o [ G x, y ] = n 6.34 And T { Γ z } = [ w φ φ φ φ φ φ φ ] u v x v y u u v v u o o 6.35 Now, we can expess the enegy equation as U l T T = { } [ G] [ D] [ G]{ } da dz Γ Γ 6.36 A which can be witten as U = l T T {} Γ [ G] [ D][ G] da {}dz Γ A 6.37 Let s define 6

122 7 [ ] [ ] [ ][ ]da G D G C A T = 6.38 whee,6,5,4,3,, = = = = = = = C EI da y y x x E C da s n E C da y E C da x E C EA C Po A o o A A A ω,6,5,4,3, = = = = = = C da y y x x x E C da s n x E C EI da y x E C I E C o o A A xy A y ω 3,6 3,5 3,4 3,3 = = = = C da y y x x y E C da s n y E C I E C o o A A x ω d A s n E C A =,4 4 ω 4,6 4,5 = = C da y y x x s n E C o o A ω 4,5 5 P A o o I da y y x x E C = =

123 C 5,6 = C6,6 = G 4 n A da so we can wite U = l } T { Γ} [ C x, y]{ Γ dz 6.39 Taking the fist vaiation of the enegy whee l T δ U = {δγ} [ C x, y]{ Γ} dz 6.4 { δγ} δw u δu v δv x o δv φ v δφ y δu δv φ v δφ δv δu φ u δφ = δφ φ δφ δφ o δu φ u δφ 6.4 Now, we will substitute in { δγ} the fou defomations {, u, v,φ} of two matices { δ } = [ A] { δ d} w and wite{ δγ} as a multiplication Γ 6.4 8

124 9 whee [ ] = φ φ φ φ φ v x u y u v x v y u A o o o o T 6.43 and { } { } δφ δφ δφ δ δ δ δ δ δ = v v u u w d T ' 6.44 Then, we can use the intepolation functions and expess { } d δ in tems of the element defomations { } [ ] { } q Z d c δ δ = 6.45 Whee { } { } q q δ δ = 6.46 And fom equation 6.9. [ ] = φ φ φ T T T T T T T T Z v v u u w c 6.47 Fo convenience, we will define the matix [B]

125 [ B] [ A] [ ] = 6.48 Z c Hence, we can wite { δ } = [ B] { δq} Γ 6.49 So, at the equilibuim path l T { δq} [ B] [ C x, y]{ Γ} dz { δq}{ F } = δ { U V} = in 6.5 whee { F in } is the intenal foce vecto To obtain the stiffness matices, we can take the fist vaiation of the intenal foce vecto, which will be: o l T T [ δb] [ C x, y]{ Γ} dz [ B] [ C x, y]{ Γ { δ F } = δ } dz 6.5 in l l l { q} T T { δ F } = [ δb] [ C x, y]{ Γ} dz [ B] [ C x, y] [ B] dz δ 6.5 in Taking the vaiation of the [A] matix

126 [ ] = δφ δ δ δ δ δφ δφ δ δφ δφ δ δ v x u y u v x v y u A o o o o T 6.53 We can wite [ ] [ ][ ] Z c B δa δ = 6.54 Howeve, we need to multiply the matices in the fist tem of equilibium equation to be able to extact the stiffness matix. Let s define { } [ ][ ] Γ = Ω, y x C 6.55 and let s define the vecto { } [ ] { } Ω = T A S δ 6.56 {} Ω Ω Ω Ω Ω Ω Ω Ω = δφ δ δ δ δ δφ δφ δ δφ δφ δ v x u y u v x v y u S o o o o 6.57 that can be split into

127 {S}= [S q ] { } d δ 6.58 Whee [ ] Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω = o o o o q x y x y S 6.59 Substituting fo { } d δ by its intepolation functions Eq. 6.4 in the { } F in δ expession equation 6.5, we get { } [ ] [ ][ ] [ ] [ ] [ ] { } q B dz y x C B dz Z S Z F l T l c q T c in δ δ =, 6.6 Now, we define the element tangent stiffness matix as [ ] [ ] [ ][ ] [ ] [ ][ ] dz B y x C B dz Z S Z K T l c q T c l t, = 6.6 in which, [ ] [ ] [ ][ ] dz Z S Z K l c q T c = σ 6.6 is the element stability matix that is known as the geometic stiffness matix, and [ ] [ ][ ][ ] dz B y x C B K T l e =, 6.63

128 is the element stiffness matix that contains linea and non-linea tems. 6.5 Application to an elastic-plastic T beam unde combined loads Fo a T section with the oigin at the centoid we have C = C = C = C 6.64,,3,4,3 = In addition, since we have a mono-symmetic T section, we have x =. Also, since the shea cente of the T section is at the intesection of its banches, then ρ =. This means that ω =, and thus, the pimay axial waping igidity is educed. Howeve, the esistance to tosion is also povided by the esistance to the seconday waping mechanism that acts acoss the thickness, which is esisted by the igidities in [C] and G n da = GJ 6.65 A o sc E A n s da 6.66 Also, the tosion is esisted by the othe coss-igidities that esist the twist φ combined with the othe lateal displacements u and v. To have a bette view, we can compae the diffeence between the esistances to tosion fo an I-section and T-section see Figs. 6.8 and 6.9. In the case of an I-beam ω, and thus both the pimay and seconday waping mechanisms can esist tosion if the beam was constained against waping. On the othe hand, fo a T-section and unde the same constaint conditions only the seconday waping mechanism is esisting tosion along with the othe coss igidities in [C]. 3

129 T T SC Pimay waping: tosion is esisted by bi-moment acting on the flanges in the pesence Seconday waping: tosion is esisted by waping though the thickness its effect is much Figue 6.8. Resistance to Tosion Povided by an I-Beam T SC Figue 6.9 Resistance to Tosion Povided by a T-Beam 4

130 6.6 Element Integations Since the axial defomation is assumed to change linealy along the beam element, we will use Gauss quadatue of two points fo the integation along the element. Howeve, the integations ove the coss-section to calculate the intenal foces and the element igidities need special consideations. Fist, we need to integate ove the coss-section using a suitable numbe of integation points. Fo that, we will use Lobatto quadatue since it povides integation points on the edges of the coss-section whee we expect the stesses to be the highest. Second, we need to devise a method to detemine the state of stess on the coss-section fom which we can calculate the intenal foces and the igidities. This method must take into account thee dimensional state of stess that is pesent in the coss-section sinceσ, τ, τ. Also, it must be known that as the z loads ae inceased, some potions of the coss-section may develop plastic egions which may gow plastically o unload elastically. This means that we must be able to detemine when the stess at an integation point eaches the yield suface, and to be able to etun to the yield suface if we have depated fom it duing iteation. Also, we need to contol the iteation step size to limit the divegence eos. Finally, we will use a consistent tangent modula matix to impove the convegence of the method. We will use the genealized Von Mises yield citeion. y [ σ σ σ σ σ σ 6 τ τ τ ] σ = 6.67 x y y z x z xy yz xz xz yz Also, we will devise a backwad Eule etun pocedue to ensue that ou stesses do not depat fom the yield suface. Moeove, we will use the stains to update the stesses as in the following module: Solution Method 5

131 We have used the full Newton-Raphson method fo bette convegence, and since the tangent stiffness matix duing iteations does not elate to an equilibium state, then we may not be able to use the line seach technique to optimize the iteations step length. Instead, we will use the full amount of displacements duing iteations. Step contol Since we ae going to encounte some path non-lineaity due to the onset of buckling o plasticity, it is ecommended that we use some fom of displacement contol. Hence, the same step contol method used in Ch.3 will be used hee. Howeve, since the equilibium cuve has a vaying slope, we have implemented a scheme ove the displacement incements to educe the step size when thee is a lage vaiation in the equilibium cuve slope. 6.7 Post-Buckling Analysis Afte the onset of buckling, the beam may expeience eithe a limit-point type of buckling in which case the equilibium path is unique and may aise o dop; o it may expeience type of bifucation buckling in which case moe than one equilibium path could exist. In ou study, we may expect to have a bifucation buckling, since the beam in ou poblem has multiple failue modes. Hence, the stability of the equilibium path is what decides the post-buckling behavio of the beam. Howeve, the analysis of the second vaiation of the potential enegy gives us infomation about the citical points and the states of stability along the diffeent equilibium paths. Paticulaly, we know that when the beam is citically stable, the tangent stiffness matix is singula, which means that at least one eigenvalue of the stiffness matix is zeo. 6

132 The eigenvecto that coesponds to the vanishing eigenvalue gives diection of defomation of the post-buckling state of the beam see Thomas and Gallaghe, 975; De Bost, 987; Riks et.al., 996; Ronagh and Badfod 999; Riks, ; and Bazant and Cedolin 3 Based on the above, eseaches have poposed a numbe of ways to switch to the stable equilibium banch afte the onset of buckling. Fo instance, Ronagh and Badfod 999 employed a method in which they constain the solution fom conveging to the pimay equilibium path by supeimposing a displacement vecto that does not contain the pimay path displacements ove the displacements that exist at the initiation of buckling. In that, method they augmented the total enegy potential equation with a Lagangian multiplie constaint that used a matix to assign a zeo value to the component of displacement in the diection of the pimay solution; othewise it gives a non-zeo value. This value depends on the angle between the pimay solution vecto and the new vecto, and is popotional to the Eigenvecto at the onset of buckling Eigenvecto that coesponds to the vanishing eigenvalue at the citical point. Finally, iteations ae pefomed to convege to the displacement vecto that satisfies the constaints. Howeve, since we ae using displacement contol, we can use a moe diect method that was poposed by De Bost 987, which is simila to the one used by Riks et al 996 and Riks, in which they used displacement contol and modified the post-buckling displacement { δ q PB } vecto by using a weighted and nomalized buckling eigenvecto {} ν in the following fom: { } { } T { } { } { } T δ q v δ q δ q { δ q }{ v δ q } PB T { q} { q} T δ δ δ q v = 6.68 { } { } It must be noted that the tangent stiffness matix that we ae using is composed of two pats; the element stiffness matix [ K e] that came fom the fist vaiation of the stain enegy see equations 7

133 6.5 and 6.63, and the element stability matix [ K σ ] that we obtained by consideing the second vaiation of the stain enegy see equations 6.5 and 6.6. Thus, we may lineaize the Eigenvalue poblem nea the citical point and conside the tangent stiffness matix to be linea with espect to the applied loads faction λ. This is expessed in the fom: [ K ]{ q} = [ K ] δλ [ K ]{ δq} = { } t δ σ 6.69 e Then, we can multiply equation 6.69 by the invese of the element stiffness matix [ ] [] I [ Ke] [ K ]{ δq} = { } δλ σ 6.7 whee [ I ] is the identity matix. Also, we can put the Eigenvalue poblem in the fom K to get e [ ] [ K ] = [] I K e σ δλ 6.7 Finally, we can detemine the citical Eigenvecto { ν } to be the one coesponding to the maximum Eigenvalue of [ K ] e [ ] K σ see Ronagh and Badfod 999; Bazant and Cedolin Accounting fo Plastic Defomations Assumptions and Consideations Since the beam may expeience consideable plastic defomations befoe its total collapse, we must account fo loads that exceed the beam elastic limit and cause the beam to defom plastically. Fo this pupose, we must conside the state of the stess at each integation point and update the displacements accodingly. Howeve, elastic unloading may occu duing defomation 8

134 due to possible occuences of diffeent buckling modes, which may add some complication to the convegence of the poblem. Since this elastic unloading may cause some potions of the beam to unload elastically, and thus, would not follow the same tangent stiffness matix as the othe potions of the beam that ae unde inceasing plastic loads Cisfield, 99, Bathe 996, Belytschko et al,. Also, we must conside the new elations that will exist between the stesses and the stains at the onset of plastic yielding. In paticula, we must obtain the stiffness matix that elates the applied extenal loads to the beam defomation afte the onset of plasticity, and we must be able to calculate the intenal foces that must be in equilibium with the extenal foces fom integation of the stesses ove the beam s coss-section. Howeve, the state of stess in ou case only consides the axial and the though-thickness shea stesses see Eq This is because the contou shea stesses ae contibuting to the axial stess at evey point though the waping function see Eq. 6.9 and 6.5. Theefoe, we can use the following Von Mises flow ole f = σ P 3τ P σ 6.7 y whee f is the yield function, σ P the axial stess at a point P of the coss-section, τ P the shea stess at the point P, and σ y is the yield stength of the beam mateial. The tangent modula elastic-plastic matix As was mentioned befoe, we need to detemine the stesses at each integation point of the element span-wise and at the coss-section given the stains. 9

135 Now, to deive the modula matix we will adapt the method that can be found in Cisfield 99, Bathe 996, and Belytschko et al,, which equies obtaining the gadient of the yield function with espect to each of the stesses: f σ P = f τ P σ P σ P 3τ P τ P σ P 3τ P 6.73 The elation between plastic stess ates and plastic stain ates fo isotopic elastic mateial ignoing the dynamic effects of the stain ates is given by: δσ P = δτ P δε δγ e δε δε δγ t δγ P e t P [ C ] = [ C ] e e 6.74 whee E C e [ ] = G 6.75 δε e and δγ e ae the vaiations in the elastic longitudinal and shea stain ates espectively, δε t and δγ t ae the total vaiations in longitudinal and shea stain ates espectively, and δε and δγ P P ae the vaiations in the plastic longitudinal and shea stain ates espectively, and. The vaiations in the plastic stain ates ae given by: δε P = δλp δγ P f σ f τ 6.76

136 whee P δλ is a plastic stain ate multiplie, which we need to detemine. Now, when the plastic flow occus, the ate of change of stess will be tangent to the yield suface and othogonal to the gadient of the yield function at that point: = P P T f f δτ δσ τ σ 6.77 Hence, we can find the plastic stain ate multiplie P λ & by solving the following equation: [ ] = τ σ δλ δγ δε τ σ f f C f f P t t e T 6.78 which yields fo elastic-plastic behavio P P P P P P P t P t P G E G E τ σ τ σ τ σ τ δγ σ δε δλ = 6.79 The plastic stain ate multiplie P δλ detemines the state of stess at each integation point since positive sign means plastic defomation and negative sign means elastic defomation a value of zeo is not likely to occu because of the numeical eos in computations. Substituting Eq fo the plastic stains of Eq 6.76 and then in Eq. 6.74, we get the equied tangent modula elastic-plastic matix that elates the change in the stess vecto to the change in the stain vecto as: [ ] = t t tm P P C δγ δε δτ δσ 6.8

137 9E Gτ 3 P E Gτ P σ P [ C ] tm = Eσ P 9Gτ P E Gσ P 3E Gτ P σ P whee [ Ctm] is the tangent modula matix that elates the ates of stess to the ates of stain in the plastic egion. Howeve, fo the case of elastic defomation the stess and stain ates ae elated though the elastic matix [ C e] see Eq Stess Integation and Updating So fa, we have obtained the elastic-plastic elation between the ates of stesses and the ates of stains at each integation point. Yet, we still need to integate these ates assuming the ates to be deivatives in pseudo time to obtain the values of stesses and stains at each integation point. One simple way of caying on the ates integation is to use the fowad Eule method at each integation point. Howeve, the fowad Eule method will cause an unsafe dift fom the yield suface at each integation point, which can be coected by employing a suitable algoithm to etun to the yield suface o an algoithm to scale the step size. Nevetheless, a backwad Eule method can be used in pedicto/coecto scheme without the need to scale the step size o to use any technique to etun to the yield suface see Cisfield, 99, Bathe 996, and Belytschko et al,. Stess integation using backwad Eule method We stat by incementing the stains at evey integation point by ε, ε = ε ε n n 6.8

138 whee, n denotes the pevious conveged step, and n is the new pedicto incement. Howeve, we will incement the stess by an elastic pedicto σ tial n and ou aim is to find the plastic coecto component of the tial stess k σ σ k tial = σ n n σ k 6.8 whee, k denotes the iteation numbe. Howeve, it is impotant to note that duing the elastic pedicto step the plastic stain is unchanged and duing the plastic coecto steps the total stain elastic plastic is fixed. Accodingly, the plastic component of the tial stess is given by σ k = P P P [ C ] ε = [ C ] ε ε = λp [ C ] e n e n n n e f σ n 6.83 E C e whee [ ] = is the matix of elastic axial and shea modulus, P ε G n is the plastic stain, λ p is the plastic stain ate at incement n, and n f σ n is the gadient of the flow ule Von Mises. Accodingly, it is equied to calculate the plastic stain ate at incement n, and this is pefomed using Newton s method see details at appendix B, which is adapted fom Belytschko et al,. Now, we can obtain the stesses accoding to the following algoithm shown in Figue 6. 3

139 4 At the end of the last incemental step n: set = = = p k λ σ Check convegence Tol f p and Tol f n P n P n k n σ λ ε ε σ Calculate k k and σ δλ Update k k k k k k k k P k P C σ σ σ δλ λ λ σ ε ε = = = Set k k Incement the stains and calculate the tial stesses P n n n n C ε ε σ ε ε ε = = Figue 6.. Flow Chat fo the Stess Calculation Algoithm

140 Plasticity consideation fo the stiffness matix and the intenal foces Having calculated the stesses at the plastic ange, we need to detemine the tangent stiffness matix that elates the displacements to the unbalanced foces, and we need to detemine the intenal foces on each element. Accodingly, we will integate the stesses ove the beam element coss-section and obtain the axial, bending, shea, and bi-moment loads vecto { Ω } see Eq Then, we update the initial stess matix [ K σ ] see Eq , and the intenal element foces F int T { F } [ B] { Ω}dV int = V 6.84 whee [ B ] is a matix that elates the element stains to the genealized displacements as defined in Eq Finally, we will substitute the tangent modula matix [ C tm] Eq. 6.8 in the place of the matix [ D ] Eq. 6.3 and update the element stiffness matix [ ] the flowchat of the pocedue. K accodingly. Figue 6. shows e 5

141 6 Incement the displacements pedicto loop {} {} { } q q q K K = : K incement Calculate the incemental stains at each Gauss integation point points span-wise [ ] [ ] [ ] Γ Γ = Γ K i K i K i : i Gauss point Obtain the incemental stains at each integation point i, j Eq [ ] [ ] [ ], j j i j K i y x G z Γ = ε : j Lobbato Point Obtain the updated stesses at each integation point using backwad Eule { } { } { } [ ] j K i K i K i σ σ σ = Obtain the stesses at Gauss integation points { } [ ] { } = Ω da y x G K j i j j K i,, σ Update the matices { } { } = Ω = = l T C q C K l K i K in l T T tm K e dz Z S Z K dz B F dz B da G C G B K ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ σ Obtain the tangent modula matix at each integation point [ ] = P P P P P P P P tm E G E G E G E G G E C σ σ τ σ τ τ τ σ Use the above matices in the displacement coecto iteative loop Solve { } [ ] { } { } int F F K q ex t = δ Figue 6. Flow Chat of the Stess Integation Pocedue

142 6.9 Code Compaisons and Veifications 6.9. Compaisons with Published Data The esults of the code wee compaed and veified against othe published woks, as in the following fou examples. Example : In-plane plastic bending unde unifom lateal load A simply suppoted ectangula beam unde lateal load was investigated using the pesent finite element model see Figue 6., and compaed to the example found in Yang and Saigal 98 see Ch. 3. In the analysis, 6 elements wee used and the displacement was contolled fom the midspan in plane displacement degee of feedom q see Figue 6.7. The esults wee in excellent ageement with the esults pesented by Yang and Saigal see Figue 6.3 q o b t L/ v L Figue 6. Simply Suppoted Rectangula Beam unde Unifom Lateal Load q o v the cental in-plane deflection 7

143 . Load Faction Pesent study code Yang and Saigal Centeal Deflection v non-dimensional Cental Deflection v nomalized Figue 6.3 The Cental Deflection of a Simply Suppoted Rectangula Elastic-Plastic Beam six elements wee used, and step contol fom the in plane mid-span DOF q Example : Detemining the Citical Flexual-Tosional Buckling Load fo a Cantileve Beam with a Tip Load. The citical flexual-tosional buckling load fo a cantileve beam with a tip load see Figue 6.4 was investigated using the pesent finite element model, and compaed to the example found in Ronagh and Badfod 999, and Woolcock and Tahai 974. In this analysis, 6 elements wee used and the displacement was contolled fom the beam tip in-plane displacement degee of feedom q see Figue 6.7. Also, the citical load was obtained by detemining the load faction at which the stiffness matix becomes singula. The esults wee in good ageement with the esults pesented the pedicted citical load was.9% less than the citical load obtained by Woolcock and Tahai 974, and 3.38% less than the citical load pedicted by Ronagh and Badfod

144 Y E = 93 kip/ in ν =.5 P Z P C = 7.8 lb Woolcock and Tahai, 974 P C = 7.39 lb Ronagh and Badfod, 999 P C = 7.4 lb Pesent Code Figue 6.4 The Cantileve Beam Tested by Woolcock and Tahai 974 and analyzed by Ronagh and Badfod 999. Example 3: Detemining the non-linea Tosional esponse of a Cantileve Beam with a Tip Toque Load. The non-linea esponse of a cantileve beam with a tip toque load see Figue 6.5.a was investigated using the pesent finite element model, and compaed to the example found in Ronagh and Badfod 999, which was expeimentally tested by Tso and Ghobaah 97. In this analysis, 6 elements wee used and the displacement was contolled using the beam tip axial 9

145 otational degee of feedom q see Figue 6.7. The esults wee in excellent ageement with the esults pedicted by Ronagh and Badfod 999 as in Figue 6.5 b. Y T γ E I = 3 L ω 5.8 mm Z 6.35 mm 3.75 mm.944 m a I-Beam Subject to End Toque E = GPa ν = mm Load facto Cuent study Code Tso and Ghobaah End twist ad b Non-Linea Toque-Twist Response the esults of Ronagh and Badfod finite element ae ovelapped by the cuent code Figue 6.5 The Response of Cantileve Beam Tested by Tso and Ghobaah 97 and analyzed by Ronagh and Badfod 999 six elements wee used, and step contol fom the tip axial otation DOF q 3

146 Example 4: Detemining the Response of a Beam unde Axial, Bi-axial Bending and Tosional loads The non-linea esponse of a simply suppoted beam unde axial, bi-axial bending and toque loads see Figue 6.6 was investigated using the pesent finite element model, and compaed to the example found in El-Khenfas and Nethecot 989. In this analysis, elements wee used and the displacement was contolled fom the mid-span out-of-plane displacement degee of feedom q 9 see Figue 6.7 The esults wee in excellent ageement with the esults pedicted by El-Khenfas and Nethecot 989 see Figue

147 σ = 8ksi y 6 E = psi 6 G =. psi Y,v Z, w X, u T M y M x A L/ P y B L = 9 P y = 448 lb T = 3 kip.in M y = P y e x M x = P y e y e x =. 8. e y = End A End B.56 Figue 6.6 A Simply Suppoted Beam unde Bi-axial Bending and Toque Loads 3

148 Load Faction Pesent study Code w Pesent study Code u El-Khenfas and Nethecot w El-Khenfas and Nethecot u Displacements Nomalized non-dimensional Displacements Figue 6.7 The Response mid-span axial displacement w, and out-of-plane displacement u of a Simply Fig. 5.X The Response mid-span axial displacement w, and out-of-plane Suppoted displacement Beam unde u Axial, of a Simply Bi-axial Suppoted Bending Beam and unde Tosional Axial, Bi-axial Loads Bending elements wee used and the displacement and Tosional was contolled Loads. fom the mid-span out-of-plane displacement degee of feedom q Compaisons with ABAQUS Beam Element Example: Non-linea Elastic-Plastic Cantileve Beam with T-section Section unde Axial, Lateal and Tosional Loads A non-linea elastic-plastic cantileve beam unde axial, lateal and tosional loads see Figue 6.8 was analyzed using ABAQUS 4 DOF thee nodes quadatic open section shea flexible beam element B3OS fo moe infomation about this element efe to ABAQUS Standad Use s Manual, and the esults wee compaed to the pesent code see Figue

149 Y Q=38 kn/m 6 mm P=5 kn Z 3 mm mm mm mm Lateal Load Eccenticity= 8 mm E = 7. GPa ν =.3 s y = 4 MPa Figue 6.8 Cantileve Beam with T Section unde Axial, Lateal and Tosional Loads Load Faction Code This study Abaqus Beam Element Tip Axial Displacement m 34

150 .3.5 Load Faction..5. Code This study Abaqus Beam Element Lateal Defomation m Load Faction Code This study Abaqus Beam Element Out-of-Plane Defomation m 35

151 .3.5 Load Faction..5. Code This study Abaqus Beam Element Axial Rotation ad Figue 6.9 Compaison between the Results of the Developed Beam Element and ABAQUS B3OS Beam Element, fo Non-linea Elastic-Plastic Solution, fo the Case of Cantileve Beam unde Combined Axial, Flexual and Tosional Loads 36 elements wee used, and the displacement was contolled mid-span in plane displacement degee of feedom q. It can be seen fom the figues that the ABAQUS solution is moe flexible when compaed to the code and this is may be due to the diffeence between the Eule-Benoulli beam model that is used in this fomulation and the shea flexible Timoshenko beam model of the B3OS ABAQUS element. Also, this diffeence may be due to the fact that the ABAQUS beam models have no though thickness integation points and consides a constant tosional igidity of the coss-section ABAQUS Standad Use s Manual. Thus, the ABAQUS beam model does not conside mateial non-lineaity unde pue tosional loads as shown in Figue 6. 36

152 .6.4. Load Faction.8.6 Code This study Abaqus Beam Element Axial otation ad Figue 6. Compaison between the Results of the Developed Beam Element and ABAQUS B3OS Beam Element, fo Non-linea Elastic-Plastic Solution fo the Case of Cantileve Beam unde Pue Toque in the Tip of the Beam 36 elements wee used and the displacement was contolled fom the tip axial otation degee of feedom q 37

153 Chapte 7 Reliability Based Optimization of T-Beam unde Random Axial, Flexual and Tosional Loads with Tosional and Local Buckling Failue Modes This is the same beam example that was consideed in Chapte 4, howeve, this time tosional and local buckling effects will be consideed, also a moe economic RBDO method will be used. 7. Failue Modes Even though, moe degees of feedom wee added to the beam model, but still thee is no guaantee that the actual beam will behave accoding to the pedictions of the moe advanced model. Fo example, the 4 degees-of-feedom beam model can not pedict local buckling modes and localized failues that may occu. Theefoe, highe ode elements such as a shell o a continuum element must be used to exploe the existence of local failues. Fo this pupose, examples of cantileve beams unde simple in-plane tip load ae analyzed using shell and continuum elements and pesented in the following sub-section. 7.. Examples Example : Continuum Element Analysis of Non-linea Elastic-Plastic Cantileve Beam with Naow Rectangula Section unde in-plane Tip Load To get familia with the continuum 3-D element analysis and to futhe test the capabilities of the FE code developed hee, a non-linea elastic-plastic cantileve beam with naow ectangula 38

154 section unde tip load see Figue 7. was analyzed using the ABAQUS continuum nodes quadatic continuum hexahedal bick element C3DR. Fist, a linea buckling analysis was pefomed fo the ABAQUS continuum model to detemine the buckling mode shapes. In the analysis the tip load. lb was distibuted ove the fou cones of the ectangula coss-section and a small out-of-plane load. lb was added to tigge the buckling instability. The linea buckling analysis evealed no local buckling modes fo the fist thee buckling modes see Figue 7. a and b. Next, a nonlinea elastic-plastic analysis was pefomed fo the continuum model to investigate the collapse of the beam. Duing the analysis, localized effects stess hot spots have occued at the cones of the fixed end of the beam see Figue 7.3. Howeve, since these hot spots wee esticted to the fixed end of the beam, they had only a limited influence on the equilibium path. Accodingly, thee was a good ageement between the analysis esults of the beam model and the continuum model see Figue 7.4 and the expeimental esults found in Woolcock and Tahai 974 Y E = 93 kip/ in ν =.3 s y = 3 kip/in P = lb Z Figue 7. Cantileve Beam with Naow Rectangula Section unde Tip load 39

155 Figue 7. Buckling Mode Shapes of a Naow Rectangula Cantileve Beam unde Tip Load the second mode shape is the same as the fist, but with an opposite sign. ABAQUS quadatic continuum hexahedal bick element C3DR 4

156 Figue 7.3 The Defomed Shape of a Non-linea Elastic-Plastic Cantileve Beam unde Tip Load, ABAQUS quadatic continuum hexahedal bick element C3DR. 4

157 8 7 6 Load Faction Code This study U U Abaqus 3D Element U Code This study U U Abaqus 3D Element U Displacement in Figue 7.4 The Non-linea Response of an Elastic-plastic Cantileve Beam with Naow Rectangula Section unde Tip Load U is the in-plane Defomation and U is the out-of-plane Defomation. Example : Continuum Element Analysis of a Non-linea Elastic-Plastic Cantileve Beam having T- Section and unde in-plane Tip Load ABAQUS continuum element model was used to investigate the failue modes of a cantileve beam having T-section and subject to an in-plane tip load see Figue 7.5. The aim is to detemine what types of failue modes ae pesent in such a situation. A linea elastic buckling analysis was caied out fist to detemine the citical buckling loads and the coesponding mode shapes as 4

158 shown in Figs. 7.6 a and b. The buckling analysis evealed that local buckling modes ae pesent. Y P=. kn. mm Z 75. mm. mm mm. mm E = 7. GPa ν =.3 s y = 4 MPa Figue 7.5 Cantileve Beam with T Section unde Tip load 43

159 Figue 7.6 Buckling Mode Shapes of a Cantileve Beam with T-Section unde in-plane Tip Load see Figue

160 Then, a nonlinea elastic analysis was pefomed and localized stess concentation spots stated developing see Figue 7.7. These localized stesses have developed a localized buckling defomation as shown in Figue 7.8. Simila esults wee obtained fo nonlinea elastic-plastic analysis, but with a slight localized defomation in addition to the local buckling. Theses localized defomations wee moe ponounced as the web depth is inceased as shown in Figue 7.9 Figue 7.7 Nonlinea Elastic Analysis of a Continuum Element Model fo a Cantileve Beam unde Vetical Tip Load, Revealing Local Stess Concentation Spots 45

161 Figue 7.8 Stess Contous of a Cantileve T-beam unde In-plane Tip Load Nonlinea Elastic Analysis 46

162 Figue 7.9 Stess Contous fo a Nonlinea Elastic-plastic Analysis of a Cantileve T-beam unde In-plane Tip Load localized defomations have developed Also, the same beam cantileve beam with vetical tip load was modeled and analyzed by a shell model and the beam expeienced local web buckling, which is shown in Figue

163 Figue 7. Local Defomation of the Web of a Cantileve T-section Beam unde Vetical Tip Load see Figue 6.5 Thus, one may conclude that local buckling and localized effects can occu fo a cantileve elastic-plastic beam unde athe simple loading conditions such as the vetical tip load, and theefoe, we must constaint the loads fom exceeding the values of the local buckling loads. In the next section local buckling constaints will be povided to constaint any local buckling behavio of the beam. 48

164 7. Local Buckling Consideation As was demonstated in the pevious section, a beam with a T-section is susceptible to local buckling failues that may have diffeent equilibium paths than the global buckling o the plastic hinge failues. The main contibutos to the local buckling failues of the web and flange plates ae the axial, shea and bending loads as shown in Figue 7. and Figue 7.. The pue tosional load may not induce local buckling as can be seen fom Figue 7.3, which shows an ABAQUS shell model of a cantileve T-beam unde pue toque applied at its fee end. Figue 7.3 shows that the beam has developed a significant plastic zone without poducing any local buckling due to toque. Figue 7. Local Buckling of the Web and Flange of a Cantileve Beam of T-section unde Axial Load ABAQUS continuum element model linea buckling analysis 49