PSI - Issue 66

H.A.F.A. Santos et al. / Procedia Structural Integrity 66 (2024) 195–204

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6

H.A.F.A. Santos et al. / Structural Integrity Procedia 00 (2025) 000–000

with U s being the statically admissible space defined as

U s = { ( N , M , V ) ∈H 1 ( Ω ) ×H 1 ( Ω ) ×H 1 ( Ω ) |

N ′ + p ( x ) = 0 , − M ′ + V + m ( x ) = 0 , V ′ + q ( x ) = 0 in Ω ;

nN − ¯ N = 0 , nM + ¯ M = 0 , nV − ¯ V = 0on Γ N }

c if, and only if, the following condition holds

It can also be shown that ( N , M , V ) is a stationary point of Π

c = 0 , ∀ ( δ N ,δ M ,δ V ) ∈V s

(17)

δ Π

where V s is the pure homogeneous statically admissible space defined as

V s = { ( δ N ,δ M ,δ V ) ∈H 1 ( Ω ) ×H 1 ( Ω ) ×H 1 ( Ω ) | δ N ′ = 0 , − δ M ′ + δ V = 0 ,

δ V ′ = 0 in Ω ; n δ N = 0 , n δ M = 0 , n δ V = 0 , on Γ N }

4. Finite Element Formulation

A finite element approximation of Eq. (17) consists of seeking ( N h , M h , V h ) ∈U h

s such that Eq. (17) holds for all

( δ N h ,δ M h ,δ V h ) ∈V h

s , where U h

s ⊂U s and V h

s ⊂V s are the discretized statically admissible spaces.

Let us assume that the entire domain Ω is partitioned in subdomains Ω e ⊂ e , where n e is the number of beam elements. If the inter-element equilibrium conditions and the Neumann boundary conditions are relaxed within the framework of the complementary energy principle, then the following augmented Lagrangian (or hybrid complementary energy) must be considered Ω , such that Ω = ∪ n e e = 1 Ω

n e e = 1

n int i = 1

i

λ N

+ λ V

+ λ M

(18)

i [[ N ]] Γ i

i [[ V ]] Γ i

i [[ M ]] Γ

L c

=

Π

+

c , e

i is the inter-element boundary i . [[( · )]] denotes the jump

where n int is the number of inter-element boundaries and Γ

N i

, λ V i

and λ M i

of ( · ) on Γ

i . λ

are the Lagrange multipliers, defined on Γ

i , that are energy-conjugate of N , V and M ,

respectively. The solutions to the equilibrium differential equations are as follows

1 − p ( x ) dx 2 − q ( x ) dx

N h ( x ) = c

(19a)

V h ( x ) = c

(19b)

+ V h ( x ) dx + m ( x ) dx

M h ( x ) = c 3

(19c)

where c 1 , c 2 and c 3 are constants. These expressions are herein taken as the trial finite element approximations and a Galerkin approach is adopted, i.e. , the problem is numerically approached assuming the same trial and test approxi mation function spaces within the framework of the augmented Lagrangian given by Eq. (18).