PSI - Issue 66

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

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

z

z

w

φ

h ( x )

C

y

x

u

C j

C 1

x = L

b

x = 0

Fig. 1: Beam Geometry and Configuration.

and bending moments so that all equilibrium conditions of the associated boundary-value problem are satisfied in a strong form. The effectiveness and accuracy of the formulation are numerically demonstrated through its application to benchmark problems, and the obtained results are analyzed and discussed.

2. Boundary-Value Problem

Consider a two-dimensional non-uniform beam with open cracks, as depicted in Figure 1. The beam geometry is described by its centroidal axis, denoted by C , parameterized by x ∈ [0 , L ], with L denoting the length of the beam in its reference configuration. C is decomposed into an internal part, represented by Ω= ]0 , L [, and a boundary part, identified by Γ = Γ N ∪ Γ D = { 0 , L } , where Γ N and Γ D correspond to the Neumann and Dirichlet boundaries, respectively, such that Γ N ∩ Γ D = ∅ . The cracks are denoted by C j , with j = 1 , 2 , ... . Let the beam be subjected to distributed loads defined per unit reference length, denoted by p and q , and bending moments, denoted by m , applied in Ω and assumed to depend on x , concentrated forces ¯ N and ¯ V , and a concentrated moment ¯ M , applied on Γ N ; prescribed displacements, ¯ u and ¯ w , and a prescribed rotation ¯ φ , defined on Γ D .While p , ¯ N and ¯ u represent axial quantities, q , ¯ V and ¯ w are transverse quantities. m , ¯ M and ¯ φ represent rotational quantities. The loads are assumed to act at the centroidal axis of the beam. The kinematic differential equations of the beam model under consideration are given as

ε = u ′ , γ = w ′ + φ, χ = φ ′ , in Ω

(1)

where ε and γ are the axial and shear deformations, respectively, and χ is the bending curvature of the beam; ( · ) ′ denotes the derivative of ( · ) with respect to x . The shear deformation, denoted by γ , is taken into account by means of the well-known Timoshenko’s beam theory. The Dirichlet (kinematic) boundary conditions of the problem are given as follows

u − ¯ u = 0 , w − ¯ w = 0 , φ − ¯ φ = 0 , on Γ D

(2)

where u and w are the axial and transverse displacement fields of the beam axis, respectively, and φ is the rotation of the beam’s cross-section. The equilibrium of an infinitesimal beam element can be expressed by the following set of differential equations

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

(3)

representing equilibrium of axial forces, shear forces and bending moments, respectively.