BENDING MOMENT IN PIPES-2

FAILURE MODES

As pointed out in the previous section the ultimate moment capacity is highly dependent on the amount of longitudinal force

and pressure loads and for cases with high external pressure also initial out-of-roundness. To clarify the approach used in the

development of the analytical equations and to give a better understanding of the obtained results, characteristics of the

ultimate strength for pipes subjected to single loads and combined loads are discussed below.

The cross sectional deformations just before failure of pipes subjected to single loads are shown in Figure 2.

Figure 2: Pipe cross sectional deformation of pipes subjected

to single loads.

Pure Bending

A pipe subjected to increasing pure bending will fail as a result of increased ovalisation of the cross section and reduced slope in

the stress-strain curve. Up to a certain level of ovalisation, the decrease in moment of inertia will be counterbalanced by

increased pipe wall stresses due to strain hardening. When the loss in moment of inertia can no longer be compensated for by

the strain hardening, the moment capacity has been reached and catastrophic cross sectional collapse will occur if additional

bending is applied. For low D/t, the failure will be initiated on the tensile side of the pipe due to stresses at the outer fibres

exceeding the limiting longitudinal stress. For D/t higher than approximately 30-35, the hoop strength of the pipe will be so low

compared to the tensile strength that the failure mode will be an inward buckling on the compressive side of the pipe. The

geometrical imperfections (excluding corrosion) that are normally allowed in pipeline design will not significantly influence the

moment capacity for pure bending, and the capacity can be calculated as, SUPERB (1996):

M p=(1.05−0. 0015⋅Dt )⋅SMYS⋅D2⋅t( 1 )

where D is the average pipe diameter, t the wall thickness and SMYS the Specified Minimum Yield Strength.

(1 .05−0 .0015⋅D / t )⋅SMYS represents the average longitudinal cross sectional stress at failure as a function of the diameter

over wall thickness ratio. The average pipe diameter is conservatively used in here while SUPERB used the outer diameter.

Pure External Pressure

Theoretically, a circular pipe without imperfections should continue being circular when subjected to increasing uniform

external pressure. However, due to material and/or geometrical imperfections, there will always be a flattening of the pipe,

which with increased external pressure will end with a total collapse of the cross section. The change in out-of-roundness,

caused by the external pressure, introduces circumferential bending stresses, where the highest stresses occur respectively at

the top/bottom and two sides of the flattened cross-section. For low D/t ratios, material softening will occur at these points and

the points will behave as a kind of hinge at collapse. The average hoop stress at failure due to external pressure changes with

the D/t ratio. For small D/t ratios, the failure is governed by yielding of the cross section, while for larger D/t ratios it is

governed by elastic buckling. By elastic buckling is meant that the collapse occurs before the average hoop stress over the cross

section has reached the yield stress. At D/t ratios in-between, the failure is a combination of yielding and elastic collapse.

Several formulations have been proposed for estimating the external collapse pressure, but in this paper, only Timoshenko’s

and Haagsma’s equations are described. Timoshenko’s equation, which gives the pressure at beginning yield in the extreme

fibres, will in general represent a lower bound, while Haagsma’s equation, using a fully plastic yielding condition, will represent

an upper bound for the collapse pressure. The collapse pressure of pipes is very dependent on geometrical imperfections and

here in special initial out-of-roundness. Both Timoshenko’s and Haagsma’s collapse equation account for initial out-of-

roundness inside the range that is normally allowed in pipeline design.

Timoshenko’s equation giving the pressure causing yield at the extreme pipe fibre:

pc2−[ pp+(1+1 .5⋅f 0⋅Dt )⋅pel]⋅pc+ p p⋅pel=0

( 2 )

where:

pel =

2⋅E(1−ν2 )

⋅( tD )3

( 3 )

pp =2⋅SMYS⋅ t

D ( 4 )

and:

pc = Characteristic collapse pressure

f0 = Initial out-of-roundness, (Dmax-Dmin)/D

D = Average diameter

t = Wall thickness

SMYS = Specified Minimum Yield Strength, hoop direction

E = Young’s Module

= Poisson’s ratio

It should be noted that the pressure ‘pc’ determined in accordance to Eq. (2) is lower than the actual collapse pressure of the

pipe and it becomes equal to the latter only in the case of a perfectly round pipe. Hence, by using ‘pc’ calculated from Eq. (2) as

the ultimate value of pressure, the results will normally be on the safe side (Timoshenko and Gere, 1961).

Haagsma’s equation giving the pressure at which fully plastic yielding over the wall thickness occurs can be expressed as:

pc3−pel⋅pc

2−( pp2+ pel⋅pp⋅f 0⋅Dt )⋅pc+ pel⋅p p2=0( 5 )

and represent the theoretical upper bound for the collapse pressure. For low D/t, the collapse pressure will be closer to the

collapse pressure calculated by Haagsma’s equation than that calculated by Timoshenko’s equation (Haagsma and Schaap,

1981).

The use of Timoshenko’s and Haagsma’s equations relates specifically to pipes with initially linear elastic material properties

where the elastic collapse pressure can be derived from classical analysis. This would be appropriate for seamless pipes or for

pipes that have been subjected to an annealing process. However, for pipes fabricated using the UO, TRB or UOE method there

are significant non-linearity’s in the material properties in the hoop direction, due to residual strains and the Bauschinger effect.

These effects may be accounted for by introducing a strength reduction factor to the plastic collapse pressure term given by Eq.

(4). In this study no attempt has been given to this reduction factor, but according to DNV 2000 the plastic collapse pressure is

to be reduced with 7% for UO and TRB pipes and with 15% for UOE pipes.