Page 69 - Pipelines and Risers

42 Chapter 3

Under the action of external uniform pressure ‘pe), there will be an additional flattening of the
pipe, and the corresponding additional radial displacement ‘w’ is calculated using the
differential equation:

The decrease in the initial curvature as a consequence of the external pressure will introduce a
positive bending moment in section AB and CD and a negative bending moment in section
AD and BC. At points A, B, C and D the bending moment is zero, and the actions between the
parts are represented by the forces ‘S’ tangential to the dotted circle representing the ideal
circular shape.

The circle can be considered as a funicular curve for the external pressure ‘pe’ and the
compressive force along this curve remains constant and equal to ‘S’. Thus, the bending
moment at any cross section is obtained by multiplying S by the total radial displacement ‘wi
+ w’ at the cross section. Then:

M = (w + w, COS@)) (3.3)

Substituting in Equation (3.3):

de Et (3.4)

or

(3.5)

The solution of this equation satisfying the conditions of continuity at the points A, B, C and
D is

in which ‘pe,cl is the critical value of the uniform pressure given by equation:

It is seen that at the points A, B, C and D, ‘w’ and ‘d2wfd02’ are zero. Hence the bending
moment at these points are zero as assumed above. The maximum moment occurs at 0 = 0
and at 0 = IC, where