Optimal Stress Fields and Load Capacity of Structures Reuven Segev - PowerPoint PPT Presentation

Optimal Stress Fields and Load Capacity of Structures Reuven Segev ∗ & Lior Falach Department of Mechanical Engineering Ben-Gurion University ∗ Currently, on Sabbatical at MAE, UCSD Seminar Department of Mechanical and Aerospace Engineering University of California San-Diego October 2008 R. Segev & L. Falach ( B.G.U. ) 1 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Stress Analysis For a given structure geometry Ω and an t assumed loading , or a number of loading cases, Solve the equations of equilibrium with boundary conditions f div σ + b = 0, in Ω ; σ ( ν ) = t on ∂Ω . σ – stress field, b – volume force, t – boundary load, ν – unit normal Problem: the system is under-determined (statically indeterminate): 6 independent stress components and 3 equations. Solution: Use constitutive relations (e.g., Hooke’s Law) to relate the stress and the kinematics. One (hopefully) stress field σ 0 will solve the problem. Use a failure criterion Y ( τ ) � s permitted , where τ is a stress matrix. Find the maximal stress and check whether max x Y ( σ 0 ( x )) < s permitted . R. Segev & L. Falach ( B.G.U. ) 2 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Stress Analysis For a given structure geometry Ω and an t assumed loading , or a number of loading cases, Solve the equations of equilibrium with boundary conditions f div σ + b = 0, in Ω ; σ ( ν ) = t on ∂Ω . σ – stress field, b – volume force, t – boundary load, ν – unit normal Problem: the system is under-determined (statically indeterminate): 6 independent stress components and 3 equations. Solution: Use constitutive relations (e.g., Hooke’s Law) to relate the stress and the kinematics. One (hopefully) stress field σ 0 will solve the problem. Use a failure criterion Y ( τ ) � s permitted , where τ is a stress matrix. Find the maximal stress and check whether max x Y ( σ 0 ( x )) < s permitted . R. Segev & L. Falach ( B.G.U. ) 2 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Stress Analysis For a given structure geometry Ω and an t assumed loading , or a number of loading cases, Solve the equations of equilibrium with boundary conditions f div σ + b = 0, in Ω ; σ ( ν ) = t on ∂Ω . σ – stress field, b – volume force, t – boundary load, ν – unit normal Problem: the system is under-determined (statically indeterminate): 6 independent stress components and 3 equations. Solution: Use constitutive relations (e.g., Hooke’s Law) to relate the stress and the kinematics. One (hopefully) stress field σ 0 will solve the problem. Use a failure criterion Y ( τ ) � s permitted , where τ is a stress matrix. Find the maximal stress and check whether max x Y ( σ 0 ( x )) < s permitted . R. Segev & L. Falach ( B.G.U. ) 2 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Estimates for the Maximum of the Stress Field Question: For a given load on the structure, what estimates or bounds apply to the stress field on the basis of equilibrium alone? (no reference to material properties!) Signorini [1933], Grioli [1953], Truesdell & Toupin [1960], Day [1979]: Lower bounds on the maximal stress in terms of the applied load only. �� � Bound ( t ) �� � � σ ij ( x ) max x , i , j Collection of equilibrating for all equilibrating stresses. stresses for a given load Note: the bounds are not exact! max x , i , j | σ ij ( x ) | A lower bound R. Segev & L. Falach ( B.G.U. ) 3 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Greatest Lower Bounds and Optimal Stresses Notes: What is the greatest lower bound on the Collection of maximal stress equilibrating stresses for a given load components? Is the greatest lower bound attained for some stress field σ opt ? The greatest Another lower bound s t An optimal stress field A stress field for optimal max x , i , j | σ ij ( x ) | which the bound it A lower attained is optimal bound because it has the least maximum. R. Segev & L. Falach ( B.G.U. ) 4 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

The Setting for the Problem Definitions of the Main Variables Ω – a given body (bounded), Γ = ∂Ω – its boundary, Γ 0 – the part of the boundary where the body is fixed, t – a surface traction field given on Γ t ⊂ Γ , ν – the unit normal to the boundary, σ – a stress field that is in equilibrium with t , σ max – the maximal magnitude of the stress σ max = ess sup x ∈ Ω | σ ( x ) | = � σ � ∞ , | τ | = Y ( τ ) – a failure criterion function for the stress matrix τ , a norm. Remark: The treatment may be generalized to include body forces. There is a class of stress fields that are in equilibrium with t . We denote this class of stress fields by Σ t . R. Segev & L. Falach ( B.G.U. ) 5 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

The Optimization Problem Find the least value s opt of σ max , i.e., t s opt = inf σ ∈ Σ t { σ max } = inf σ ∈ Σ t {� σ � ∞ } . t ◮ Question : Is there an optimal stress field σ opt such that s opt = � σ opt � ∞ ? t R. Segev & L. Falach ( B.G.U. ) 6 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

The Corresponding Scalar Problem: the Junction Problem Given the flux density φ on the boundary of Ω with � ∂Ω φ d A = 0 (this constraint may be removed and we will get the optimal source distribution). Set V φ = { v : Ω → R 3 , v i , i = 0 in Ω , , v i ν i = φ on ∂Ω } —compatible velocity fields. For each v ∈ V φ , set v max = ess sup x ∈ Ω | v ( x ) | . Find the least value v opt of v max , i.e., φ v opt = inf v ∈ V φ { v max } . φ The optimal velocity field for the junction Ω . R. Segev & L. Falach ( B.G.U. ) 7 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

The Result Theorem The optimal value s opt is given by t � � � � ∂Ω t · w d A | t ( w ) | � s opt = Ω | ε ( w ) | d V = sup sup , � t � ε ( w ) � 1 w ∈ C ∞ ( Ω , R 3 ) w ∈ C ∞ ( Ω , R 3 ) | ε ( w ) | is the norm of the value of the stretching ε ( w ) = 1 2 ( ∇ w + ∇ w T ) . The optimum is attained for some σ opt ∈ Σ t . t Mathematically: s opt = � Force Functional � . t Motivation: recall the principle of virtual work: t � � Ω σ ij ε ij d V = t i w i d A . Γ t s opt λ t = λ s opt Note : , λ > 0 . for all t R. Segev & L. Falach ( B.G.U. ) 8 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Realization of an Optimal Stress Field Question: Can the optimal stress field be realized? Introduce residual stresses in the structure (e.g., prestressed beams, tree trunks), Introduce additional external loading, Limit design for elastic perfectly plastic materials . . . R. Segev & L. Falach ( B.G.U. ) 9 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

The Yield Condition and (Perfectly) Plastic Materials Use the yield function as a norm for stress matrices. Hydrostatic pressure does not cause failure. τ = τ H + τ D , where, σ Y τ H = 1 3 tr ( τ ) I . τ D – deviatoric component Stress of the stress matrix. Von Mises yield function: � � � τ D � � τ D � � � = 3 Y ( τ ) = 2 , � 2 2 = √ τ ij τ ij � � τ D � Strain � – the Euclidean norm. A Perfectly Plastic Material Yield condition: � � τ D � � = s Y . Y ( τ ) = R. Segev & L. Falach ( B.G.U. ) 10 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Yield Function and the (Semi-) Norm Induced Deviatoric projection – π D ( τ ) = τ − 1 3 τ ii I for every matrix τ . π D : R 6 − → D ⊂ R 6 , the space of traceless matrices. Yield function Y – a semi-norm on the space of matrices Y ( τ ) = | τ − 1 3 τ ii I | , |·| is a norm on the space of matrices. Yield condition – Y ( τ ) = s Y . Semi-norms – � σ � Y = � Y ◦ σ � , � σ � Y ∞ = � Y ◦ σ � ∞ are norms on the subspaces of trace-less fields. Thus, in the previous definitions of the optimal stress we have to use the semi-norms or restrict ourselves to the appropriate subspaces containing trace-less fields. R. Segev & L. Falach ( B.G.U. ) 11 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008

Limit Analysis of Plasticity Theory The limit analysis problem: Given t and s Y , find the largest multiplier of the force for which collapse will not occur, i.e., λ ∗ such that there exists σ , � σ � Y t = sup λ , ∞ � s Y , σ ∈ Σ λ t . Basic idea, the body can support any stress field σ as long as � σ � Y ∞ � s Y . Christiansen and Temam & Strang [1980’s]: � λ ∗ t = σ ij ε ( w ) ij d V sup inf σ Y t ( w )= 1 � σ � Y ∞ � s Y Ω � Stress = inf sup σ ij ε ( w ) ij d V t ( w )= 1 � σ � Y ∞ � s Y Ω Strain A Perfectly Plastic Material R. Segev & L. Falach ( B.G.U. ) 12 / 43 Optimal Stresses and Load Capacity UCSD, Oct. 2008