Untitled

56 Robert L. Taylor, Eugenio Oñate ˜ and Pere-Andreu Ubach
4.1 Deformation of Cable
The deformation of the reinforcement cable may be expressed in terms of the Green-
Lagrange strain given by
Eij = 1
j i T j i
(x − x ) (x − x )
2 (X j − X i ) T (X j − X i
− 1 =
) 1
ji 2
∆x
2 ∆X ji
− 1 (52)
2 where ∆X ji = X j − X i . The variation of the strain is then expressed as
4.2 Material Constitution
δEij = (δxj − δx i ) T (∆x ji )
∆X ji 2
For simplicity we again assume that the material is elastic and may be represented
by a one-dimensional form of a St.Venant-Kirchhoff model expressed as
Sij = E Eij
where Sij is the constant second Piola-Kirchhoff stress in the cable and E is an
elastic modulus.
4.3 Weak Form for Reinforcement
A weak form for an individual reinforcement cable in an element may be written as
k
δΠr Π = δx M kl¨x l + C kl ˙x l
− δEij Sij Aij Lij ; k, l = i, j (55)
where Aij is the cross sectional area of the reinforcement; Lij the length of the cable
(i.e., ∆X ji ); M kl is the mass matrix; and C kl is the damping matrix.
The variation of the strain is rewritten from Eq. (53) as
⎧ ⎫
δEij = δx i,T j,T
δx
⎪
⎨⎪ ⎨
⎪
⎩⎪
− ∆xji
L2 ij
∆x ji
j
L2 ij
Equation (55) is appended to the other terms from the membrane by replacing
variations of end displacements and the rate terms by their representation in terms
of the membrane nodal parameters as given by Eq. (51). The result is:
where
δΠr Π = δ˜x α,T
⎪
⎬⎪ ⎬
⎪
⎭⎪
M αβ
r ¨˜x β + C αβ ˙˜x r
β − P α r
P α r = ∆ξ ji
α ∆ξ ji
β ˜xβ Sij Aij
Lij
(53)
(54)
(56)
(57)
; (58)