The Variation Operator: Assuming u(x) is the minimizing path for a functional:I(u)= [" F(x,u,u')dx: Introducing a family of varied functions: u(x) = u(x)+ n(x): We call en(x) the variation of u(x) and write→ 0, n(a)=n(b)=0n(x)=Su(x)=Su=u-u,: The delta operator () represents a small arbitrary changein the dependent variable u for a fixed value of theindependent variable x, i.e. we do not associate a Sx witha Su.uXSu(K)N16
16 The Variation Operator • Assuming u(x) is the minimizing path for a functional: • Introducing a family of varied functions: • We call εη(x) the variation of u(x) and write • The delta operator (δ) represents a small arbitrary change in the dependent variable u for a fixed value of the independent variable x, i.e. we do not associate a δx with a δu. , , d b a I u F x u u x u x u x x x u x u u u , 0, a b 0
The difference between Su and a differential du: A differential du has a dx associated with it: Consider the variation for the derivative:ddidududSSuudxdxdxdxdx. In a similar manner:o[u(x)dx = [i(x)dx -[u(x)dx = J8u(x)dx. Consider a functional: F = F(u (x),u, (x),us (x),x). Its variation:aFaFaFSFSuSuouou2ouou. In contrast, the differential isaFaFaFaFdFdxdudu11"Ou2axauaus17
17 The difference between δu and a differential du • A differential du has a dx associated with it. • Consider the variation for the derivative: d d d d d d d d d d u u u u u u x x x x x • In a similar manner: u x x u x x u x x u x x d d d d • Consider a functional: • Its variation: F F u x u x u x x 1 2 3 , , , 1 2 3 1 2 3 F F F F u u u u u u • In contrast, the differential is 1 2 3 1 2 3 d d d d d F F F F F u u u x u u u x
Minimization of a Functional: Consider the problem of minimizing I (u)= [" F (x,u,u')dx: For a varied path, the integrand may be written asF(x,u+Su,u'+Su): Expanding the above in a Taylor series yieldsaFaFF(x,u+ Su,u'+ Su)= F(x,u,u')SuSu-auau: The first variation of the functional I is defined byaFaF81 :dxSFdxSu'Su+au'QubaFaFd aFSuSudx+QuQu'dx Ou': The minimizing process leads to Euler-Lagrange equation. Essential vs. natural BCs..18
18 Minimization of a Functional • Consider the problem of minimizing • For a varied path, the integrand may be written as , , d b a I u F x u u x F x u u u u , , • Expanding the above in a Taylor series yields 2 , , , , F F F x u u u u F x u u u u u u • The first variation of the functional I is defined by d d d d d b b a a b b a a F F I F x u u x u u F F F u x u u x u u • The minimizing process leads to Euler-Lagrange equation. • Essential vs. natural BCs
Principle of Virtual Work: A kinematically admissible displacement field is onepossessing continuous first partial derivatives in theinterior of a domain B and satisfying all displacementboundary conditions on S,: A kinematically admissible displacement variation Su(virtual displacement) is one possessing continuousfirst partial derivatives in the interior of a domain B andzero on Su: A statically admissible stress field is one that satisfiesthe equilibrium equation overthe interior of adomainBand all stress boundary conditions over S19
19 Principle of Virtual Work • A kinematically admissible displacement field is one possessing continuous first partial derivatives in the interior of a domain B and satisfying all displacement boundary conditions on Su . • A statically admissible stress field is one that satisfies the equilibrium equation over the interior of a domain B and all stress boundary conditions over St . • A kinematically admissible displacement variation δu (virtual displacement) is one possessing continuous first partial derivatives in the interior of a domain B and zero on Su
Principle of Virtual Work: Now consider a body with statically admissible stressfield and subjected to kinematically admissible virtualdisplacements.: The work done by the external loads against the virtualdisplacements isSW, = Jif,F.Sudv + T.SudS. In indicial notationSW, = Jf, F.Sudv + JJ, T.Suds = JJf, F8udV +T,Suds= JJJ, F,oud + JJJ, n,rou,ds = JJ, F,ou,dV + JJ,n,o,ou,ds: Recall that, Su = O on Su20
• Now consider a body with statically admissible stress field and subjected to kinematically admissible virtual displacements. • The work done by the external loads against the virtual displacements is d d t E V S W V S F u T u Principle of Virtual Work d d d d d d d d t t t E i i V S V i i i i V i i S j ji i j ji V S i S T u S n u uW V S F V F u V S F u V n S u F u T u • In indicial notation • Recall that, δu = 0 on Su . 20