Collapse Strength Under Biaxial Load
The use of the Eq. 2.156 has one disadvantage in that one cannot completely separate the pressure term from the axial load term unless p, = 0. To overcome this problem Pattilo and Huang (1982). presented the following expression:
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1_з /V, 4 |
|
1 2 av |
|
(2.158) |
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± |
|
21 a„ |
and showed that for a given set of material properties, casing may exhibit plastic collapse for zero axial load (see Fig. ‘2.20. Path I). The failure mode can. however, switch to ultimate strength collapse as the axial load is increased beyond a certain value. It is also observed that the collapse resistance decreases continuously with increasing axial load: Curve 0 — no axial load and Curve 4 — maximum axial load.
Path II depicts a more interesting collapse behavior. The initial collapse mode, Curve 0 — no axial load, is in the elastic region: collapse load remains constant and equal to the initial elastic collapse value until an axial load represented by Curve 1 is reached. From this point, collapse load decreases with increasing axial load as the mode of failure passes successively through the region of plastic collapse and ultimate strength collapse.
The API collapse formula for computing the additional effect of tensile stress is very similar to the Eq. 2.158 and is derived from the Lame equation lor plastic range and the theory of minimum distortional energy. Previously, it was shown the critical collapse pressure for plastic region is equal to:
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(2.159) |
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Pc = 2 (To.: |
{dp/t) ~ 1
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Fig. 2.20: Manifold of collapse curves in the presence of axial load. (After Pattilo and Huang, 1982; courtesy of JPT.) |
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DIAMETER:THICKNESS RATIO |
0 DIRECTION OF INCREASING AXIAL LOAD
If the initiation of yield in the pipe subjected to external pressure, p0, occurs only when (Jq.2 = cry, then Eq. 2.159 becomes;
(2.160)
where:
(2.161)
Neglecting the effect of internal pressure (assumes p, <C <ra) and substituting tangential stress, cr(, due to the external pressure, p0, by ay, Eq. 2.157 becomes:
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+ 0.5 |
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0.5 |
|
0.75 |
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(2.162) |
|
y(do/t, p0) (Ту |
where:
= the axial stress due to tension.
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£а_ Ov • Т х 29 lb/ft (t=0.408") N-80 Fig. 2.21: Collapse strength under combined loads. (After Krug. 1982; courtesy of IPE-TU Clausthal.) |
The API Bui. 5C3 (1989) defines the term on the left of Eq. 2.162 as Ypa, the yield strength of axial stress equivalent grade [casing] under a combined load. Thus:
Provided Ypa is greater than 24,000 psif it is then used in Eqs. 2.135, 2.139 and 2.140, as illustrated in Table 2.1, to determine the effective collapse pressure in yield, plastic, and transition ranges. Within the elastic range. Eq. 2.163 is not applicable because for an elastic mode of failure the collapse pressure is independent of effective yield strength and, therefore, the API minimum performance values suffice.
Equation 2.163 also ignores the effect of internal pressure on the correction of collapse pressure rating. The rating is the minimum pressure difference across the pipe-wall required for failure and, therefore, is assumed to be independent of
‘API collapse resistance formulas are not valid for the yield strength of axial stress equivalent grade (Ура) less than 24,000 psi — API Bui. 5C3, 1989.
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Fig. 2.22: Collapse behavior of casing subjected to external pressure and superimposed tension. Stress-strain plots and assumed distribution of stress over the pipe wall. |
internal pressure. However, API Bui. 5C3 (1989) defines an external pressure equivalent, p0 as:
Poeq — Po ~ [1 — 2/(d0/t)]pt (2.164)
Nara et al. (1981) and Krug and Marx (1980) have observed that the axial load does not affect the collapse strength to the extent predicted by the theory of minimum distortional energy. The test results presented by Krug and Marx (1980), see Fig. 2.2lT, clearly demonstrate that for larger values of d0/t ratio or of the yield strength there is a shift away from the theoretical minimum distort ion curve towards the y-axis.
An explanation for the difference between the test results and those predicted from the theory of distortional energy is provided by the theory of buckling proposed by Engesser and V. Karman (Krug, 1982). Extending the theory of reduced modulus, Er, the combined effect of external pressure and axial load for infinitely long casing steel specimens is summarized in Fig. 2.22 (Krug. 1982).
In Fig. 2.22 (a), the steel specimen is subjected to an uniform external pressure, p0i, which induces a tangential stress aPoi, assumed to be constant over the wall thickness. Summation of tangential stress, crn, and the added bending stress, (7, over the cross-sectional area must remain equal to zero until the onset of collapse. Insofar as the maximum of the prevailing stresses lies below the limit of the proportionality of the material, the casing fails elastically.
The specimen in Fig. 2.22 (b) is subjected to an external pressure as well as an axial tension, A. In this case, the sum of the overall tangential stresses, which is equal to the algebraic sum of the tangential stress components due to the external pressure, crPoI, a(al and the bending stress, cf, lies within the limit of proportionality. Hence, the casing string fails elastically and the collapse strength is not unfavorably affected by the superimposed tensile force.
In Case 3 (Fig. 2.22(c)), the specimen is subjected to an external pressure and to a higher tensile load, A2i whereby the maximal overall stress exceeds the limit of proportionality. As a result of the altered stress-strain relationship, the tangent modulus replaces Young’s modulus, the stability behavior changes with increasing tensile load and the collapse strength decreases. The collapse behavior corresponds to that of the elastoplastic transition range.
Upon further increase in tensile force (Fig. 2.22(d)), the problem of instability no longer occurs. The combined stress due to external pressure and axial tension induces yielding of the material, and the collapse strength can be calculated using the distortional energy theory.
■^The axial stress, <7„, and tangential stress, rrt, induced by tensile load and collapse pressure (resp.) have been referred to the respective values of yield strength, oy, under load conditions. The boundary curve is the elliptical stress curve given by the distortion energy theorem.
Consider again the casing in Example 2-11. this time applying Pattilo and Huang’s correction using Eq. 2.163. Compute the nominal collapse pressure ratings: (i) without axial force, (ii) axial tension of Fa — 340,000 lbf and an internal pressure of pt = 5,400 psi. In both cases compute the values using the pre-API Bui. 5C3, 1989, method and the API Bui. 5C3. 1989. Compute the minimum external force required for failure.
From Example 2-10:
(i) d0/t = 20.392 and, therefore, pp — 4,754 psi
(ii) An axial tension of 340,000 lbf is equivalent to an axial stress of:
<7 a = = 25,051 psi
13.57
Thus, the total axial stress is:
oa = 25,051 + 5,400 = 30.451 psi The effective yield strength is, therefore:
|
30,451 |
/
cre = 80,000
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0.5 |
30,341
1 — 0.75
80,000
^80.000 = 60,303 psi
Using the effective value of yield stress, the values for the constants are: A = 3.006, В = 0.05675, С = 1365.4, F = 1.983 and G = 0.0374 And the failure models and d0/t ranges are:
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pe = 6,123 psi pt = 607 psi pp = 4,103 psi py = 5,624 psi |
24.390 < djt < 35.649
14.423 < djt < 24.39
djt < 14.423
(a) For a d0/t = 20.393, failure is in the plastic range where pp — 4,103 psi. However, this value is the corrected pressure differential (pe — p,) for the inservice condition. The actual collapse pressure rating is pc = 4,103 + 5,400 = 9,500 psi.
(b) To find the collapse pressure rating using Eq. 2.164 proceed as follows:
4,103 = p0 — (1 — 2/(d0/t))p,
-*Po = 4,103 + (1 — 2/20.393)5.400 = 8,973 psi
In this example, the presence of a large ‘arbitrary’ in-service internal pressure was considered. Generally, cra — f p, « cra and the effect of ignoring pt is negligible. Had one ignored p, in this case, the final collapse pressure obtained would be:
pp = 4,260 psi
a difference of 4 % for a 22 % change in axial stress.