PSI - Issue 18

Yaroslav Dubyk et al. / Procedia Structural Integrity 18 (2019) 630–638

631

2

Yaroslav Dubyk et al. / Structural Integrity Procedia 00 (2019) 000–000

Nowadays, mitered bends are analyzed using FEM solutions, but in this study we have tried to used approximate (simplified) shell theory, based on the terms of Long (which slowly decay in axial direction) and Short (which quickly decay in axial direction) solutions. Long solution is, in fact, a semi-membrane Vlasov’s solution, when the derivative of any geometrical or force function in axial direction is much smaller than in the circumferential one. Short solution is inverse – the derivative of any in circumferential direction is much smaller than in the axial one, its main goal is to give the local behavior of stress in the vicinity of the oblique weld. In fact, such approach lead to that instead of considering a complicated general thin shell differential equation of 8 th order, we have a set of equations of 4 th order, which can be easily solved. Despite the sufficient amount of publications Zhu et al. (2010) and Zheng et al. (2016) devoted to the study and analysis of the stress state of pipes with defects in geometry such as angular misalignment, there is lack in simple analytical equations. Ideally, to understand mechanical relationships it would be best to develop a closed-form analytical model of the situation. Expressions for determining the stress concentration at sectoral bending in modern standards are often replaced by corresponding expressions for smooth bending of pipes ASME B31.8 (2014). The work is a continuation of our previous studies Orynyak et al. (2016) and improves the application of the concepts of short and long solutions in the part of the exact solution of the system of differential equations of the fourth degree, and complete coupling between the solutions.

Nomenclature , , L R h

length, radius and thickness of the shell Young’s modulus and Poisson ratio axial and circumferential coordinate axial, circumferential and radial displacements axial and circumferential normal forces; axial and circumferential bending forces

, E  , x  , , u v w , x N N  , x Q Q  , x L M  , x M M 

tangential force and moment axial and circumferential moments

2. Main Equations Equilibrium equations for the thin shell are well known Orynyak and Dubyk (2018), and are written according to Donell-Mushtari theory:

N

Q

Q N      



N

Q





L

L









0;

0;

0

(1)

 

  

x

x

x R

R x R   

x R R   







Also we will need equation for the bending forces through the displacements:

3

3 w w 

2

3

2 w v 

3

2

  

  

  

  

(1 ) 

(1 ) 

v

w

v



 





 

;

(2)

Q D 

Q D 











x



3 x R x 2

2

2

3 3 R R  

3 2

2  

2

2

2

R

x

R x

R x 



 



  







Eq. (1) is usually solved using Fourier series expansion:

 

  





cos or n

sin   n

  

 

(3)

n

n

0

1

n

n



