Tamprių-plastinių prisitaikančių sistemų optimizacija su standumo ir stabilumo sąlygomis ; Optimization of elastic-plastic systems under stiffness and stability constraints at shakedown

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Sujets

• Mathematical Programming
• Optimal design
• Shakedown

Informations

• Publié par: vilnius_gediminas_technical_university
• Publié le: 01 janvier 2006
• Nombre de lectures: 23

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DovilėMERKEVIČIŪTĖ        OPTIMIZATION OF ELASTIC-PLASTIC SYSTEMS UNDER STIFFNESS AND STABILITY CONSTRAINTS AT SHAKEDOWN       Summary of Doctoral Dissertation Technological Sciences, Civil Engineering (02 T)                

                                   Vilnius

2005

1202

               

VILNIUS GEDIMINAS TECHNICAL UNIVERSITY

DovilėMERKEVIČIŪTĖ        OPTIMIZATION OF ELASTIC-PLASTIC SYSTEMS UNDER STIFFNESS AND STABILITY CONSTRAINTS AT SHAKEDOWN       Summary of Doctoral Dissertation Technological Sciences, Civil Engineering (02 T)                

                                  Vilnius

2005

Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 20012005.   Scientific Supervisor Prof Dr HabilJuozas ATKOČIŪNAS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering  02 T)   The doctoral dissertation is defended at the Council of Scientific Field of Civil Engineering at Vilnius Gediminas Technical University:   Chairman Prof Dr HabilGintaris KAKLAUSKAS Gediminas Technical (Vilnius University, Technological Sciences, Civil Engineering  02 T) Members: Prof Dr HabilVytautas STANKEVIČIUS University of (Kaunas Technology, Technological Sciences, Civil Engineering  02 T) Prof Dr HabilRomualdas MAČIULAITIS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering  02 T) Prof Dr HabilVytautas Jonas STAUSKIS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering  02 T) Prof Dr HabilAntanas ILIUKAS University of Technology, (Kaunas Technological Sciences, Mechanical Engineering  09 T) Opponents: Prof Dr HabilJonas BAREIIS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering  09 T) Prof Dr HabilRimantas KAČIANAUSKAS Gediminas Technical (Vilnius University, Technological Sciences, Civil Engineering  02 T)   The dissertation will be defended at the public meeting of the Council of Scientific Field of Civil Engineering in Senate Hall of Vilnius Gediminas Technical University at 2 p. m. on December 16, 2005. Address: Saulėtekio al. 11, LT-10223 Vilnius-40, Lithuania Tel. [phone]; fax [phone], e-mail [e-mail]   The summary of doctoral dissertation was distributed on November 16, 2005. A copy of doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saulėtekio al. 14, Vilnius).    

© D. Merkevičiūtė, 2005

VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS                

DovilėMERKEVIČIŪTĖ        TAMPRIŲ-PLASTINIŲPRISITAIKANČIŲ   SISTEMŲOPTIMIZACIJA SU STANDUMO IR STABILUMO SĄLYGOMIS       Daktaro disertacijos santrauka Technologijos mokslai, statybos ininerija (02 T)                   

                                     Vilnius

2005

Disertacija rengta 20012005 m. Vilniaus Gedimino technikos universitete.   Mokslinis vadovas prof. habil. dr.Juozas ATKOČIŪNAS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, statybos ininerija  02 T).   Disertacija ginama Vilniaus Gedimino technikos universiteto Statybos ininerijos mokslo krypties taryboje:   Pirmininkas prof. habil. dr.Gintaris KAKLAUSKAS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, statybos ininerija  02 T). Nariai: prof. habil. dr.Vytautas STANKEVIČIUS technologijos (Kauno universitetas, technologijos mokslai, statybos ininerija  02 T), prof. habil. dr.Romualdas MAČIULAITIS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, statybos ininerija  02 T), prof. habil. dr.Vytautas Jonas STAUSKIS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, statybos ininerija  02 T), prof. habil. dr.Antanas ILIUKAS technologijos universitetas, (Kauno technologijos mokslai, mechanikos ininerija  09 T). Oponentai: prof. habil. dr.Jonas BAREIIS (Kauno technologijos universitetas, technologijos mokslai, mechanikos ininerija  09 T), prof. habil. dr.Rimantas KAČIANAUSKAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, statybos ininerija  02 T).   Disertacija bus ginama vieame Statybos ininerijos mokslo krypties tarybos posėdyje 2005 m. gruodio 16 d. 14 val. Vilniaus Gedimino technikos universiteto senato posėdiųsalėje. Adresas: Saulėtekio al. 11, LT-10223 Vilnius - 40, Lietuva. Tel. [phone]; faksas [phone], el. patas. [e-mail]   Disertacijos santrauka isiuntinėta 2005 m. lapkričio 16 d. Disertacijągalima periūrėti Vilniaus Gedimino technikos universiteto bibliotekoje (Saulėtekio al. 14, Vilnius). VGTU leidyklos Technika 1202 mokslo literatūros knyga    

© D. Merkevičiūtė, 2005

General Characteristics of the Dissertation Field and need of research.Optimization problems (to which is dedicated this dissertation) of structural mechanics are introductory stage of structure optimum design based on principles of solid deformable body mechanics, mathematical programming theory, its methods and their mechanical interpretation. In order to base calculation on real operating conditions of structure, it is necessary evaluate as exact as possible structure material properties, external effects and other factors in mathematical models of optimization problems. Partially it is achieved by including plastic properties of material. Calculation and design of the structures, taking in to account plastic strains, allows to use their bearing capacity more efficiently and make more economic project (in this dissertation research is developed on the basis of perfect plasticity theory). From the other side, real effect for structure are often cyclic (variable repeated load character is also evaluated in this work). In the dissertation it is assumed that load is quasistatic and is characterised by load variation bounds (deterministic formulation of problems is considered). Under repeated loading a structure can lose its serviceability because of its progressive plastic failure or because of alternating strain (usually both cases are called cyclicplastic collapse). But, if residual forces together with variable part that do not violate the admissible bounds appear in the initial stage of loading, the structure adapts to existing load and further behaves elastically. This phenomenon is considered in shakedown theory. Applying the classical Melan and Koiter theorems, it is possible to analyse only simple systems at shakedown by elementary methods. Meanwhile for civil engineering, calculation of any complexity elastic plastic structures subjected by variable repeated load is relevant. That had influence on the choice of research aspect of the dissertation:optimization of elasticplastic systems, subjected by variable repeated load, under stiffness and stability constraints, applying extremum energy principles, theory of mathematical programming, numerical methods and computer technologies.Growing number of scientific works dedicated to adapted structure calculation shows importance of these researches. But there is especially small number of works concerning optimisation of adapted structures under stiffness and stability constraints. Solution of structure optimization problems at shakedown is difficult as stressstrain state of dissipative systems depends on loading history. Not only general point of adapted structure analysis and optimization theory is solved in dissertation, but also new methods and algorithms of barstructure optimization are presented. That has also practical value in civil engineering. Main objective.Further development of theory of elasticplastic adapted system optimization, creation of new calculation methods and algorithms. Main tasks:1) to review calculation methods of systems at shakedown; 2) to formulate extremum energy principles for perfectly elasticplastic discrete systems at shakedown; 3) to construct general nonlinear mathematical models of analysis

 

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and optimization problems; 4) to bring out connection between extremum problems of adapted system optimization and theory of nonlinear mathematical programming; 5) to perform optimization of barstructures taking in to account stiffness and stability conditions according to requirements for the second group of limit states: mathematical models of optimum design and load oriented problems, solution methods and numerical experiments; 6) to create algorithm to solve structure optimization problems, which do not have full initial information; 7) to create technique of incremental analysis for unloading phenomenon fixation. Scientific novelty. 1) potential, which is provided by connections New between mathematical programming and extremum energy principles, are shown for formulation of analysis and optimization problems of shakedown theory and their numerical solution. 2) New nonlinear mathematical models of load oriented and optimum design problems with constraints ensuring shakedown and serviceability of structure are created. 3) It is determined that HaarKármán principle is suitable only for systems with holonomic behaviour. 4) It is proved that residual displacements are varying nonmonotonically. Dual problems of mathematical programming can not be applied for analysis of stressstrain state of unloading system. 5) It is proved that complementary slackness condition simulates only holonomic deformation process during shakedown of structure. 6) It is shown application possibilities of strain compatibility equations for formulation of new mathematical models of residual displacement variation bound calculation problems. 7) Technique of barstructure optimization is implemented: new non linear mathematical models for frames (under strength and stiffness constraints) and trusses (with stability conditions (EC3)) optimization problems, solution algorithms, numerical experiments are carried out. 8) Optimization technique is created for trusses subjected by moving load. 9) Technique of incremental analysis for unloading phenomenon fixation is created; numerical experiment is carried out for bending plate with nonlinear von Mises yield conditions. Approbation and publications.The main results of this work were submitted in 10 scientific conferences. Thirteen papers were published on the topic of dissertation: 6 of them were published in the acknowledged editions, 5  in proceedings of international conferences, 2  in proceedings of republican conferences. The scope of the thesis. Lithuanian written thesis consists of The introduction, five main chapters, conclusions and a list of references. The total scope of the dissertation  131 pages, 36 pictures and 10 tables. Content of the Work 1. Review of Calculation Methods of the Structures at Shakedown This chapter reviews methods of adapted system calculation. The important part is comparative analysis of displacement calculation methods after shakedown.    

 

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2. General Mathematical Models of Optimization Problems The main definitions and dependencies of discretized systems.Discrete model of the structure, which degree of freedom ism (, consists of k=1, 2,...,s , k∈K ) finite elements. Eachkelement has s k nodal points ( l=1, 2,...,s k , l∈L ). The total number of design sections isζ ≤s×s k . Nodal internal forces of element compound one n vector of discrete model forces S=S 1 ,S 2 ,...,S v ,...,S ζ T =S z ) T  and strains  n vectorΘ= (Θ 1 ,Θ 2 ,...,Θ v , ...,Θ ζ T = (Θ z ) T , v1, 2,...,ζ ( v∈V ), z=1, 2,...,n . LoadF ( t ) is characterized by time,t, independent variation bounds F sup =F 1,sup ,F 2,sup , ..., F m,sup T , F inf =F 1,inf ,F 2,inf ,...,F m,inf T  ( F inf    ≤    F(t)    ≤    F sup ). Equilibrium A S=F  and geometrical A T u−D S=0  equations do not depend on characteristics of system material and can be written via residual quantities as follow:                                  A S r =0 ,A T u r =Θ r , Θ r =D S r +Θ p . (2.1)    Elastic displacements u e (t) and forces S e (t) of the structure are determined using influence matrixes of displacements and forces, β andα, respectively:     u e (t) =βF(t),S e (t) =αF(t),=A T −1    α=KA T β , K=D −1 . (2.2) βAK, The number of all possible combinations F j of load bounds F sup , F inf isp=2 m   ( F inf    ≤    F j   ≤    F sup ): S 2 α 2 F 2,sup S e1 S e2 α 1 F 1,sup S 1 S e4 α 2 F 2,inf α 1 F 1,inf S e3 Fi 1.Locus of elastic forces

 

      S ej =αF j ,j=1, 2,...,p, (j∈J). (2.3) In the case of two loads F 1 , F 2 , domain of elastic force variation (locus) is shown in Fig 1. Plasticity constantC elastic of plastic structure relates to dimensions and material yield limit y  (limit force S 0 ) of discrete model. Limit force S 0k (k∈K) is assumed as constant in whole finite element. Nonlinear yield conditions = −0          ϕ kl,j C k f kl,j S ekl,j +S rkl ≥ ,                k∈K,l∈L, j∈J  (2.4)

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are verified in each nodal pointl of the k th finite element for all load combinations j∈J . For the entire structure the nonlinear yield conditions f(S) ≤C can be written as follows:                                           f j S ej +S r ≤C ,j∈J. (2.5) In that case, direct analysis of loading history is avoided. Statically admissible residual forces S r satisfy equilibrium equations (2.1) and yield conditions (2.4). Kinematically possible residual displacementsu r satisfy geometrical equations (2.1): A T u r =D S r +Θ p . Components of vector Θ p =Θ pkl T are calculated according to formula:     Θ pkl = ∑   ∇ϕ kl,j (S ekl,j +S rkl ) T λ kl,j , λ kl,j ≥ 0 ; k∈K , l∈L ,j∈J. (2.6) j Residual strains Θ r =D S r +Θ p  displacements and u r  of structure at shakedown can be nonunique: they depend on the particular loading history F(t) . If load is defined only by variation bounds F inf , F sup , the calculation of exact values of residual displacements becomes problematical because of unloading phenomenon appearing at crosssections: then displacements u r are varying non monotonically, it is possible to determine only their lower u r,inf and upper u r,sup   variation bounds t ( u r,inf    ≤    u r ( )≤   u r,sup ). Extremum energy principles and analysis problem   Static formulation of analysis problem of minimum (principle complementary energy) reads: find ~  min F(x)= 1 2 ∑ S r T k D k S rk =a ∗ , (2.7) k subject to                        ∑ k A   k S rk =0 ,S rk =S rk1 ,S rk2 ,...,S rkl ,...,S ks k r T , (2.8)   ϕ kl,j =C k −f kl,j S ekl,j +S rkl ≥0 C=S 0k 2 , k∈K , l∈L , j∈J . (2.9) , k Vectors of limit forces S 0 = (S 01 ,S 02 , ..., S 0k ,...,S 0η T  quasielastic forces and S ej  are known. As functions ϕ kl,j ≥  convex and matrix 0 are D k  positively is defined, the optimal solutionS r ∗  is  (2.9) nonlinear analysis problem (2.7) of global.

 

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Matrix B ⎢ ⎡ = ⎣ −A ′′ T A ′ T −1 ,I ⎥ ⎤ ⎦ allows analysis problem (2.7)  (2.9) rewrite as follows: find  min 2 1 k ∑ S r ′ T B k D k B k T S r = ′′ min 1 2 k ∑ S r ′ T D ~ k S r ′′ , (2.10) subject to     ϕ kl,j =C k −f kl,j S ekl,j +B k T l S ′ r ≥ ′ )0 , C k = (S 0k ) 2 , k∈K , l∈L ,j∈J. (2.11) Optimal solutions S r ′ ∗   (2.11) are unknowns of the force of the problem (2.10) method. Kinematic formulation of analysis problem(principle of minimum total potential energy) reads: find ⎧  max− ⎨ ⎪ ⎪ ⎩ 1 2 ∑ k S r T k D k S rk − ∑ k ∑ j λ k, T j ∇ϕ k,j S k,j S rk − k ∑ l ∑ j ∑ λ kl,j C k −f kl,j S k,j ⎬ ⎪ ⎫ ⎪ ⎭ ,  (2.12) subject to                        D k S rk + ∑ ∇ϕ k,j S k,j T λ k,j −A k T u r =0, λ k,j ≥0 , (2.13) j                            S k,j =S ek,j +S rk ,k∈K,l∈L, j∈J . (2.14) Unknowns of the problem (2.12)  (2.14), which is dual to (2.7)  (2.9), are residual forces S r , displacementsu r and plasticity multipliers λ j ,j∈J.   Complementary slackness conditions of the mathematical programming               λ kl,j C k −f kl,j S k,j =0 ,λ kl,j ≥0 , k∈K , l∈L , j∈J , (2.15) which are included in to objective function of the problem (2.12)  (2.14), do not allow direct evaluation of unloading phenomenon (it appears when existsλ i >0 and ϕ i >0 ,i= ...,1, 2,ζ,i∈I during deformation process). Optimal solution S r ∗ , u r ∗ ,λ ∗ j ( j∈J ) of the problem (2.12)  (2.14) is obtained without consideration of   loading history. Nevertheless, particular loading history exists F(t) ( F inf    ≤    F(t)≤    F sup ), which leads structure to shakedown with S r ∗ , u r ∗ and λ j ∗ . It becomes obvious, that the mathematical model of analysis problem (2.7)  (2.9) of structure at shakedown can be obtained according to HaarKármán principle (structure with holonomic behaviour).

 

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Residual strain compatibility equations are obtained after elimination of residual displacements u r from geometrical equations (2.13):                                                          −BΘ p =B r S r , (2.16)     Θ pkl = ∑   ∇ϕ kl,j (S ekl,j +S rkl ) T λ kl,j , λ kl,j ≥ 0 ; k∈K , l∈L , j∈J . (2.17) j HereB r A ′′ T A ′ T −1 D ′ + D ′′ . Unknowns of the problem (2.12)  (2.16), (2.17) = − become residual forces S r and vectors of plasticity multipliers λ j , j∈J . Problem of optimal design.This work is more oriented to frames and beams with Ishape section or plates with threelayered, sandwich type crosssection. Residual displacementsu r  forces andS r  are related to vector of plasticity multipliersλ(Θ p =Φ T λ) by influence matrixes andG:                                     u r =HΦ T λ=Hλ , S r =GΦ T λ=Gλ , (2.18) H AKA T −1 AK   G=KA T H−K  , .                         = HereΦ matrix of peacewise linearized yield conditions ΦS r +S ej ≤S 0 . Structure optimal design problem is stated as follows: when load variation bounds F inf , F sup are prescribed, the vector of structure limit forces S 0 , satisfying optimality criterion minψ(S 0 ) , strength and stiffness conditions, are to be found.   Mathematical models of structure optimal design problem are presented in Table 1: (2.19)  (2.22) has linear, (2.23)  (2.26) has nonlinear yield conditions. In case of ideal crosssectional form and homogenous yield condition, mini-  Table 1.Mathematical models of structure optimal design problem Linear yield conditions Nonlinear yield conditions find find      min(S 0 ) =minL T S 0 , (2.19) minL T S 0 , (2.23) subject to subject to j =S 0 −ΦGλ+S ej ≥0 2.2 2(2.)0           ,     minF ~ (S r ) =n 2 1 miS r T D S r , (2.24)             λ j T ϕ j =0 , λ j ≥0 , ( 1) A S r =0 , j =C−f j S r +S ej ≥0 , j∈J ,              λ= ∑ λ j , j∈J ,                  C=C S 0 ) , S 0 ≥0 , (2.25)    j u r,min ≤u r,inf ,u r,sup u r,max .(2.22) u r,min ≤u r,inf ,u r,sup ≤u r,max . (2.26)

 

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