Issue 73

N. Laouche et alii, Fracture and Structural Integrity, 73 (2025) 88-107; DOI: 10.3221/IGF-ESIS.73.07

Since there are two types of cracks (Fig. 3), that of steel outer layer and that of inner composite polymer concrete. The crack location is indexed with ௦ and ௖ , where ௦ and ௖ represents the crack location from the left end of the beam for the steel outer layer and the inner composite polymer concrete respectively. ௘ is the length of the mesh element. DQFEM formulation To represent our beam, we assume the shape functions take the form described in Eqn. (19) [9]:

 N

  N  N i 1

  u x L x u   

  w x L x w   

,

i

i

b

i

bi



i

1

(21)

 N

 

 

 

 

 wx Lxw wx Lxw , 

s

i

si

z

i

zi





i

i

1

1

   w bi b i w x ,

   w si s i w x and

   u i i u x ,

i L to denote the Lagrange polynomial, while

This formulation employs

   zi z i w w x represent the nodal displacements at the Gauss-Lobatto quadrature points within the differential quadrature (DQ) finite element framework of the beam. The nth-order derivative of a field variable f (x) at a discrete point i x is approximated as follows:                i n N n ij j n j x f x t A f x t i N x 1 , , 1,2,3, .., (22)

  n ij A denotes the weighting coefficient associated with the nth order derivative approximation.   n ij A is

In this context, derived as follows: if n = 1, so

 

M x

  1

i





  1,2, ,

A

i

j i j , ,

N



   j

ij

 x x M x

i

j

(23)

n



  1

  1



  1,2, ,

A

A

i

N

ii

ij

  1,

j

j i

N

N

  j





   k k i 1,

   k k i 1,

 







 x x M x ,



 x x

M x

(24)

i

i

k

j

k

 n 2 3 for a function f ( x ) defined within the

The Gauss-Lobatto quadrature rule, which possesses a degree of precision

interval [ − 1,1], is expressed as follows:

  j f x dx C f x 1 1 1     N j 

  j

(25)

2

2





 





C C

C

j

N

,

1,

(26)

 N N



N

j

1

 

2



1





 

 

 N j N N P x 1 1 

j x corresponds to the (j − 1)-th root of the first derivative of the Legendre polynomial    N 1 P x . To achieve fast convergence and high accuracy, a denser distribution of points near the boundaries is essential. Therefore, the sampling points are chosen based on the distribution of nodes in the Gauss–Lobatto grid and solved via Newton-Raphson iteration method.

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