General solution procedure
Elasticity condition (no dissipation): dψ = δW reflecting that dD = 0
(this is the result from analyzing the TD as done in class)
∂ψ ∂ψ ∂ψ
• Step 1: Express dψ (x1 , x2 ,..) = dx1 + dx2 + ... = dxi
∂x1 ∂x2 ∂xi
∂F ∂F ∂ψ
• Step 2: Express δW (ξ1 , ξ2 ,..) = dξ1 + dξ 2 + ... = dξ j
∂ξ1 ∂ξ 2 ∂ξ j
∂ψ ∂ψ
• Step 3: Solve equations dxi = dξ j ∀dxi ,∀dξ j
∂xi ∂ξ j
Collect all terms dxi and dξ j and set the entire expression to zero.
In EQ, the expression must be satisfied for all displacement changes dxi , dξ j
, Example II: Truss structure (1)
d
Problem statement: Structure of three trusses with applied force F :
Forces in each truss
1 1
ξ0
Fd δ1 , N 1 δ 2 , N 2 δ 3 , N 3
Distance L=1 between the trusses N 1 = kδ 1 Trusses
N 2 = kδ 2 behave
like
N 3 = kδ 3 springs
Goal: Calculate displacements δ i , ξ0 Fd
Elasticity condition (no dissipation): dψ = δW reflecting that dD = 0
(this is the result from analyzing the TD as done in class)
∂ψ ∂ψ ∂ψ
• Step 1: Express dψ (x1 , x2 ,..) = dx1 + dx2 + ... = dxi
∂x1 ∂x2 ∂xi
∂F ∂F ∂ψ
• Step 2: Express δW (ξ1 , ξ2 ,..) = dξ1 + dξ 2 + ... = dξ j
∂ξ1 ∂ξ 2 ∂ξ j
∂ψ ∂ψ
• Step 3: Solve equations dxi = dξ j ∀dxi ,∀dξ j
∂xi ∂ξ j
Collect all terms dxi and dξ j and set the entire expression to zero.
In EQ, the expression must be satisfied for all displacement changes dxi , dξ j
, Example II: Truss structure (1)
d
Problem statement: Structure of three trusses with applied force F :
Forces in each truss
1 1
ξ0
Fd δ1 , N 1 δ 2 , N 2 δ 3 , N 3
Distance L=1 between the trusses N 1 = kδ 1 Trusses
N 2 = kδ 2 behave
like
N 3 = kδ 3 springs
Goal: Calculate displacements δ i , ξ0 Fd
