Universität Karlsruhe (TH) - am Institut für Baustatik

with membrane strains ε αβ , curvatures κ αβ and shear strains γ α . The second order curvatures
ρ αβ are neglected for thin structures. We organize the shell strains in a vector
ε G =[ε 11 ,ε 22 , 2ε 12 ,κ 11 ,κ 22 , 2κ 12 ,γ 1 ,γ 2 ] T , (6)
where the subscript G refers to the Green–Lagrangean strain tensor (3). The work conjugate
stress resultants are integrals of the Second Piola–Kirchhoff stress tensor and read
σ =[n 11 ,n 22 ,n 12 ,m 11 ,m 22 ,m 12 ,q 1 ,q 2 ] T (7)
with membrane forces n αβ = n βα , bending moments m αβ = m βα and shear forces q α .
The shell is loaded statically by surface loads ¯p on Ω and by boundary loads ¯t on the boundary
Γ σ . Hence the basic Hu–Washizu functional is formulated in matrix notation
∫
∫
∫
Π(v, σ, ε) = [W (ε)+σ T (ε G (v) − ε)] dA − u T ¯p dA − u T ¯t ds → stat. (8)
(Ω)
with the area element of the shell dA = jdξ 1 dξ 2 . Here, v =[u, ω] T , ε, andσ denote the
independent displacement, strain and stress fields, with u = x − X the displacement vector
and ω the vector of rotational parameters of the shell middle surface. The strain energy W
may be an arbitrary function of the independent strains. Introducing θ := [v, σ, ε] T and
δθ := [δv,δσ,δε] T the stationary condition reads
∫
δΠ := g(θ,δθ) = [δε T (∂εW − σ)+δσ T (ε G − ε)+δε T Gσ] dA
(Ω) ∫
∫
(9)
− δu T ¯p dA − δu T ¯t ds =0
(Ω)
with the virtual shell strains δε G =[δε 11 ,δε 22 , 2δε 12 ,δκ 11 ,δκ 22 , 2δκ 12 ,δγ 1 ,δγ 2 ] T
(Ω)
(Γ σ)
δε αβ = 1 2 (δx, α ·x, β +δx, β ·x, α )
(Γ σ)
δκ αβ = 1 2 (δx, α ·d, β +δx, β ·d, α +δd, α ·x, β +δd, β ·x, α )
(10)
δγ α = δx, α · d + δd · x, α .
With integration by parts and applying standard arguments of variational calculus one obtains
the associated Euler–Lagrange equations
1
(j ⎫
j nα ), α +¯p = 0 ε G − ε = 0 ⎬
1
(j in Ω (11)
j mα ), α +x, α ×n α = 0 ∂εW − σ = 0 ⎭
with n α := n αβ x, β +q α d + m αβ d, β and m α := d × m αβ x, β , where the summation convention
for repeated indices is used. The principle yields the static field equations with local form
of linear and angular momentum, the geometric field equations and the constitutive law.
Furthermore the static boundary conditions t − ¯t = 0 on Γ σ with t the boundary forces
related to n α follow. The geometric boundary conditions u − ū = 0 on Γ u have to be fulfilled
as constraints.
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