EQUATIONS SHORTEN PIPE COLLAPSE CALCULATIONS

Vladimir A. Avakov
Halliburton Energy Services
Duncan, Okla.

Operating and service company engineers can substitute two pipe collapse pressure equations for the 12 API equations now in general use.' As shown in Figs. 1 (80630 bytes), 2 (44338 bytes), and 3 (81054 bytes) the shorthand results are almost the same as those from the API equations.

The shorthand method has the additional advantage of allowing units from any measurement system. The API equations restrict calculations to U.S. units only.

The equation box lists the API (Equations 1-12) and the shorthand (Equations 13-14) equations (68066 bytes). The API equations are based on work started shortly after the turn of the century.2-6

COLLAPSE EQUATIONS

The API suggests collapse pressure equations for long, perfectly round, steel oil field casing, tubing, drill pipe, and line pipe.

Yield strength collapse pressure (Equation 1) is a part of the Lame equation when tubing is loaded by external pressure only.7 The equation helps ensure that stress on the inner pipe surface is below the yield strength. API recognizes that it is unsafe to apply external pressure above the limit that causes a material to yield.

Equation 3 (68066 bytes), API plastic collapse pressure, has a strong empirical basis. It was derived by statistical regression of 2,488 collapse tests performed on three grades of seamless casing of various ct = D/t.3 The statistical regression was based on the plastic collapse pressure equation originally developed by R.T. Stewart.5

Equation 3 provides statistically minimum collapsing pressure based on one-sided tolerance developed using methods described in References 8 and 9. There is a 95% probability or confidence level that the actual collapsing pressure will exceed the minimum pressure, defined by Equation 3, with no more than 0.5% failures.1

Equation 5, transition collapse pressure, is derived on an arbitrary basis. When the curves for minimum collapse pressure (Equation 3) are extended to higher D/t values, they fall below the minimum elastic collapse pressure curves without intersecting them. To overcome this anomaly, a plastic/elastic transition collapse pressure equation was derived.1 The transition collapse pressure equation defines minimum collapse pressure between its tangency to the minimum elastic collapse pressure curve and its intersection with the minimum plastic pressure curve.

Equation 7, elastic collapse pressure, is based on the classical equation of collapsing pressure. It was derived by M. Levy,10 who assumed that outside pressure is acting over the tubing mean diameter, D-t. In 1939, W.O. Clinedinst redeveloped the equation showing that the pressure is acting over the outside diameter, D.2 The API adopted the new elastic collapsing pressure equation as Equation 15, assuming E = 30 x 10 6 psi and m = 0.30.

In 1939, the API adopted an average collapse pressure equation correction factor of KE = 0.95. For the minimum elastic collapse pressure equation adopted in 1968, the correction factor is KE = 0-95 x 0.75 = 0.7125.

The dashed line in Fig. 1, at a yield strength uv=80 kpsi, plots the API collapse pressure as a function of the D/t ratio. In practical applications, the API collapse pressure should be divided by a suitable safety factor. The magnitude of this safety factor is beyond the scope of this article.

Since the beginning of the 1900s, API equations have provided a reliable method for evaluating collapse pressure. Equations were updated whenever new data became available. But some features of the equations could be improved, simplified, and changed so that all consistent measurement units can be input.

The API defines four collapse modes: yield, plastic, transition, and elastic. These are calculated by Equations 1, 3, 5, and 7, respectively. Each collapse pressure equation has its own (x = D/t domain. Boundary a ratios should be obtained with Equations. 2, 4, and 6 before determining the equation to apply). The API provides boundary ratio tables for various specific tubing grades. In general, boundary ratios are functions of five intermittent factors, A, B, C, F, G, and the tubing yield strength in psi.1

The equations for the five factors, three boundary ratios, and two collapse pressure equations (plastic and transition) do not allow units other than psi. Even such multiples as kpsi cannot be used, because they distort the results unless corrections to the equations are applied.

Classical elastic and yield strength are rational expressions that provide results in units consistent with input modulus of elasticity, E, and yield strength, oy, respectively. That is, yield and elastic collapse pressure equations do not have problems due to the selected measurement unit. Nevertheless, to avoid confusion with units when various collapse modes are under consideration, API nomenclature uses only U.S. units. Therefore, pressure and stress are expressed only in psi.

SHORTHAND METHOD

A simple collapse pressure equation from the API collapse data is feasible. Such a solution can be based on collapse pressure, P, expressed as a function of yield strength collapse pressure, Py, and elastic collapse pressure, PE In Equation 15, exponent a was made a positive constant. Each tubing has positive parameters of Py and PE. Then function P, as defined by Equation 15, is always less than either component Py or PE. Furthermore, when one component, such as PE, is greater than the second component, PY, and the ratio PE/PY 'S increasing, then P approaches the value of the smaller component, Py. This tendency is more pronounced when exponent a is greater.

In Equation 15, exponent a should be chosen such that the resulting pressure, P, has a minimum deviation, AP, from the API collapse pressure, PAPI, in the practical range of tubing yield strength. Equation 16 defines this deviation.

Fig. 2A at a yield strength 80. kpsi, plots AP(a) for three exponents: a = 1, 2, and 3. In Fig. 2B, exponent a is plotted at three yield strengths: (oy = 60, 80, and 100 kpsi. From -these graphs, we have selected a = 2 as an acceptable value. Then, Equation 16 becomes Equation 13 or 14.

Fig. 1 at oy = 80 kpsi, comes the collapse pressure from equation 13, shown as a continuous line, and the API collapse pressure, the dashed lines. Fig. 3 compares collapse pressure, P, at eight yield strengths with the API collapse pressure at cry = 60 and 120 kpsi.

Equation 14 can accommodate any measurement units defined by the input units of E and oy. It also can determine required yield strength, oy, when outside pressure, P, is given.

REFERENCES

1. Formulas and Calculations for Casing, Tubing, Drill Pipe and Line Pipe Properties, API Bulletin 5C3, Sixth Edition, Oct. 1, 1994.

2. Clinedinst, W.O., "A Rational Expression for the Critical Collapsing Pressure of Pipe under External Pressure," API Annual meeting, Chicago, 1939.

3. Clinedinst, W.O., 'Development of Collapse Pressure Formulas," API, Dallas, 1963.

4. Holmquist, J.L., and Nadai, A., "A Theoretical and Experimental Approach to the Problem of Collapse of Deep-Well Casing," 20th API Annual Meeting, Chicago, Nov. 16,1939.

5. Stewart, R.T., "Collapsing Pressure of Bessemer-Steel Lapwelded Tubes," Trans. ASME, Vol. 27, 1906.

6. Timoshenko, S., 'Working Stresses for Columns and Thin-Walled Structures," Trans. ASME, Applied Mechanics, Vol. 1, 1933, pp. 173-83.

7. Lame, Gabriel, Lecons Sur La Theorie Mathematique De L'Elasicite Des Corps Solides. Paris, Bachelier, 1852.

8. Hald, A., Statistical Theory with Engineering Applications, Wiley, N.Y., 1952.

9. Snedecor, G.W., and Cochran, W.G., Statistical Methods. Eighth Ed., Iowa State Univ. Press, Ames, 1989.

10. Levy, M., journal d. Math. pure et Appl. (Iiouville), Ser. 3, Vol. 10, 1884, p. 5.

11. Sturm, R.G., A Study of the Collapsing Pressure of Thin-Walled Cylinders. Doctorate Thesis, University of Illinois, Eng. Exp. Sta. Bull., No. 12, Nov. 11, 1941.