PSI - Issue 22

Yinping Cao et al. / Procedia Structural Integrity 22 (2019) 84–92 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

87

4

Fig. 3 Stress distribution in the central section B-B of a curved casing

According to material mechanics, the relationship of normal stress with bending moment exerted on the casing is

z My I

σ = z

(3)

The bending moment at any position is:

1 2

1

2

( ) 2 M x qLx qx  

(4)

According to the definition of advanced mathematics curvature, the approximate differential equation of deflection can be deduced:

2

1 2

1

2

2 2 EI d y qLx qx dx  

(5)

The uniform load can be deduced with Eq. (5). The uniform load and bending moment when x is L /2 are as follows:

2 L M qL         2 8 EI q

k

(6)

8

In a plastic zone, the expression of shear force parallel to the cross-section of B-B is:

s F qL qx  

(7)

2

Given that shear stress is the ratio of shear force to cross-sectional area, it is zero at the middle of the casing. Substituting I,M, and q into Eq. (3), the relationship between the axial stress and wellbore curvature is obtained.