Issue 73

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

                      



 

T

T I Q A CA Q (1) (1)

                                  CC CC CC CC CC CC e e e e e e K K K K K K 11 12 13 14 22 23       K

1

CC

   

   

T

T I Q A CA Q (1) (2)

2

CC

T

T I Q A CA Q (1) (2)

3

CC

 

T

T I Q A CQ (1)

8

CC

 

T

T I Q A CA Q (2) (2)

5

CC

     CC e K

(33)

 

T

T I Q A CA Q (2) (2)

4

CC

 

 

T

T I Q A CQ (2)

e

9

24

CC

CC



 

T

T

       K

T

T

(2)

(2)

(1)

(1)

 I QA CAQI QACAQ

e

6

11

33

CC

CC

CC



 

T

T

      CC e K 34       CC e K 

T

T

(2)

(1)

(1)

 I QA CQI QACAQ

10

11

CC

CC

 

T

T

T

(1)

(1)

 I QCQI QA CAQ

7

11

44

CC

CC

The interval [-1, 1] is the only valid range for all kinds of node distributions in differentiation and quadrature. Therefore, the differential and quadrature matrices need to be adjusted as follows in order to put them into practice.

  l C C A A A A l l 1 1 2 2 2 2 4 , , 2          e

(34)

e

e

e l designate the beam element length. The cracked element is introduced during the assembly step of the total strain matrix, so as to replace the strain matrix of the uncracked element with that of the cracked element. This substitution makes it possible to obtain the total strain matrix of the system as showed in Fig. 4.

Figure 4: Global strain matrix with and without cracks

The global equation is as follow:

97