PSI - Issue 18

Yaroslav Dubyk et al. / Procedia Structural Integrity 18 (2019) 622–629 Yaroslav Dubyk and Iryna Seliverstova / Structural Integrity Procedia 00 (2019) 000–000

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2

addressed using FEA only (Li and Dang, 2017; Wu et al., 2016). Finite Element Analysis is a powerful tool, but it does not give understanding of the problem, as being empirical in nature. Ideally, to understand mechanical relationships it would be best to develop a closed-form analytical model of the situation. Modern regulatory documents regulate only the permissible depth of such defects, neglecting other geometrical parameters and the load on the defect. Based on the methods of the theory of shells and the method of equivalent loads, a simple engineering approach to assess of the local stress state of the pipeline in the presence of dents has been developed, it will allow to conduct the risk damage analysis.

Nomenclature , R h

radius and thickness of the shell axial and circumferential coordinate

, x 

axial, circumferential and radial displacements axial and circumferential normal forces; axial and circumferential moments change curves in two main directions

, , u v w , x N N  , x M M  , xx   

x  

change curvature at torsion

dent dimensions in axial and circumferential directions

1 2 , l l

2. Main Equations The method of equivalent loads is approximative, but sufficiently accurate, proposed by Calladine (1972) to describe the stress state of shells with shape imperfections. According to this method, the stress-strain state of a shell with shape defects is equal to the sum of the stress field of the ideal shell under the action of the initially applied load and the stress field of the ideal shell under the action of an equivalent system of loads caused by shape defects:

2 x xx x x p N N N          

(1)

Here , and x x N N N    membrane forces, that arise from the action of the initial load on the curved shell. Equation (1) is standard in the solution of non-ideal shell problems and is used by Godoy (1996). The shape of the dent is given in an analytic smooth function:

   

exp        

   

2

2

0        

0   x     x

1 2

1 2

( , ) R x R

exp   

 



(2)

Here 0 x  length of the dent in the axial direction, 0   is the length in circumferential direction and   dent depth. As a base for our analysis, a harmonic shell is considered, because for such imperfections displacements can be easily calculated, therefore, forces and moments can be easily found:

  1 cos cos m n l     

  





w

x



(3)

Here, the parameters and m n depend on the size of the dent, that is 1 2 / 2 n R l   . Since the nonideal profile can be expressed by expansion in the Fourier series, the equivalent function of the load can also be expressed in the form of a Fourier series. Thus, the problem of finding the stress-strain state of the shell with the dent can be solved by Fourier series, and study of the dent of the "harmonic profile" is the basis for further analysis. 2 and l l , where 1 2 m  and