PSI - Issue 48

Udaya B Sathuvalli et al. / Procedia Structural Integrity 48 (2023) 207–214

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Sathuvalli and Suryanarayana/ Structural Integrity Procedia 00 (2019) 000–000

The second term on the right hand side of Eq. (6) is a correction term to maintain the perimeter constant during bending of the median surface. The curvature of the deformed surface can be shown to be (Paslay et al. 2006) ( ) ( ) 2 2 3 3 cos2 ; cos2 o o o o u u R R κ θ θ κ θ θ ∆ = ∆ = . (7) The hoop strain ( ε θ ) and its increment ( ∆ ε θ ), in the cylinder cross section are given by (Paslay et al. 2006) ( ) ( ) 2 2 2 9 , 3 cos 16 o o o o o r R u u r R R θ ε θ θ − =− + 2 (8) ( ) ( ) 2 2 9 , 3 cos 8 o o o o o o r R u u u r R R θ ε θ θ − ∆ ∆ ∆ =− + 2 (9) where r is the radial distance from the cylinder axis and θ is the circumferential angle. A numerical solution (see Appendix B) determines the displacement of the median surface as a function of the load (pressure and axial load or strain). The solution considers the effects of end conditions (constant load or displacement at the ends of the pipe), manufacturing imperfections such as initial ovality in the pipe, and the effect of material stress strain curve. Fig. 6 shows the results of these calculations for two pipe sizes. The red horizontal lines represent the Timoshenko collapse pressure, calculated from elastic stability theory, (Timoshenko and Gere, 1989, section 7.4) ( ) ( ) , 3 2 2 1 1 c e E p ν γ = − − . (10) The American Petroleum Institute (API TR 5C3 2008) recommends the following equation to calculate the collapse resistance of the thin-walled casings- [ ] ( ) ( ) , 2 2 2 0.95 0.70 1 1 c API E p ν γ γ     = × ×   − −   . (11) The collapse pressure predicted by Eq. (11) is indicated by the red arrows. The Timoshenko and the API collapse formulae are independent of axial stress.

Fig. 6. Collapse pressure of thin-walled pipe as a function of axial stress

4. Conclusions An understanding of the structural limit of wellbore casings to displacement controlled axial load in the presence of external pressure is important. In the past finite element methods have been used to study the mechanical response of casings in the situations described