API Collapse Formula
The collapse strength for the yield range (yield strength collapse) is calculated using the Lame equation. In this equation, critical external pressure is defined with reference to a state at which the tangential stress reaches the value of yield strength at the internal surface for the casing subjected to the maximal stress. Although the results reported by Krug (1982) have shown that the real values of collapse pressure are in fact higher, the onset of yield in the casing material is considered to be the decisive factor. Nevertheless, no further correction factor has been introduced into the following formulas to take into account the geometrical deviations from the nominal data:
where:
Yv — yield strength as defined by the API, lbf/in.2
For determining the elastic collapse strength. p’f. an equation proposed by Clinedinst (1939) is utilized:
p1 = —^————————— (2 136)
Though the formula for p’e is very similar to the Bresse equation (Eq. 2.88) it results in higher calculated collapse values, especially for smaller <7,// ratios. Test results presented by Krug (1982) show that the equation for p’e provides a good approximation only for the upper scatter range of the results. The ultimate formula has been specified by introducing a correction factor which decreases the value of the external pressure to 71.25 9c of the theoretical value. For values
of Young’s modulus, E — 30 x 106 lbf/in.2. and Poisson’s ratio, i/ = 0.3. the
numerical equation is:
46’95 X 106 (2.137)
{d0/0 {{d0/t) — l]2
The equation for plastic transition zone (plastic range) has been derived empirically from the results of almost 2,500 collapse tests on casing specimens of grades
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Table 2.1: API minimal collapse resistance formulas. (After API Bui. 5C3, 1989.) |
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1 — 0.75 ( — |
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Y — rn — |
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0.5 |
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= yield strength of axial stress equivalent grade, psi |
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= (Ty for aa — 0 |
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Failure model |
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Applicable d0/t range |
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1. Elastic |
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46.95 x 106 |
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da ^ 2 + B/A 1 — 1В/Л |
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d0/t (d0/t — l)2 2. Transition F |
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Ypa(A-F) <d1<2 + B/A |
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Pt da/t |
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С + Ypa (B — G) — t — 3 B/A [(A — 2)2 + 8 (g + СIYpa)Y12 + (.4 — 2) 2(B + С/ Ypa) < d, < Ypa (.4 — F) |
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3. Plastic Pp = Ipi 4. Yield py = 2 Yr |
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B-C |
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d0/t |
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t ~ C + Ypa(B-G) (A — 2)2 + 8(5 + C/Ypr,)]*/’2 + (.4 — 2) |
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(djt)-l Pa (d0/t)2 |
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< |
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2(B + СI Гр?) |
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where: A = В С |
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2.8762 + 0.10679 x 10"5 Ypa + 0.21301 x Ю“10 УД -0.53132 x 10~16 Yl |
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= 0.026233 + 0.50609 x 10“6 Ypc = — 465.93 + 0.030867 Ypa — 0.10483 x 10"7 УД + 0.36989 x 10~13 УД |
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46.95 x 10« |
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F = |
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3 В/А V lYB/A) |
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Y ( SB/A pa 2 + B/A |
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G = (F x B)/A |
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Table 2.2: Empirical parameters used for collapse pressure calculation — for zero axial load, i. e., aa = 0. (After API Bui. 5C3, 1989.)
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* Grades indicated without letter designation are not API grades but are grades in use or grades being considered for use and are shown for information purposes. |
K-55, N-80 and P-110. The formula for average collapse strength. pPar. has been determined by means of regression analysis:
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A d0/t |
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(2.138) |
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В |
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Pp., = Yp |
The parameters A and В are dependent on the respective yield point. In order to take into account the effect of tolerance limits, a constant pressure С has subsequently been calculated for each steel grade. Thus, minimum plastic collapse is obtained by subtracting the factor С from the average collapse strength. pPut:
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Table 2.3: Ranges of da/t ratios for various collapse pressure regions when axial stress is zero, i. e., aa = 0. (After API Bui. 5C3, 1989.) |^- Yield —• [ — Plastic — ] — Transition ] — Elastic —| Grade* Collapse Collapse Collapse Collapse
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* Grades indicated without letter designation are not API grades but are grades in use or grades being considered for use and are shown for information purposes. |
The introduction of the parameter С and the associated generalized decrease of the critical external pressure gives rise to an anomaly: the line corresponding to the plastic collapse, which depends on the respective value of the yield strength, no longer intersects the curve for elastic collapse (Fig. 2.14). Consequently, it is no longer possible to take elastic collapse behavior into consideration.
The discontinuity problem has been mathematically resolved by the creation of an artificial fourth collapse range: the transition collapse. Determination of the collapse strength in this range is accomplished by means of a functional equation. The associated curve begins at the intersection of the curve corresponding to the equation for average plastic collapse strength with the d0/t coordinate axis. (Ppav = 0): *s tangent to the curve for elastic collapse, and subsequently intersects the curve for plastic collapse:
л_Ч(<7о-с] (iuo)
where:
pt = transition collapse pressure.
The constants F and G are dependent on the respective parameters A and В in Eq. 2.139. In Fig. 2.14, the development and behavior of the collapse strength for the individual collapse ranges for steel grade N-80 are presented. Table 2.1 provides a survey of the individual equations for collapse, as well as the formulas for calculating the individual parameters. Tables 2.2 and 2.3 show the values of empirical parameters used for calculating collapse pressure and the range of d0/i ratios for various collapse pressure regions, respectively.
Using data from Table 2.2 and the API formulas from Table 2.1. calculate values of collapse resistance for N-80. 9| in.. 47 lb/ft casing in the. elastic, transition, plastic, and yield ranges. By calculating the d0jt range determine what value is applicable to this sample casing. Assume zero axial stress.
Calculate the d0/t ratio.
9 625
^ = Й72 = 2°’:№
From Table 2.2:
A = 3.071, В = 0.0667, С = 1955, F = 1.998 and G = 0.0434
Substituting these values into the formulas in Table 2.1 gives the results in Table 2.4. Thus, for our sample casing of N-80 with d0/t = 20.392. collapse failure occurs in the plastic range, i. e., pc = pp = 4.760 psi (API rounds-up figures to the nearest 10 psi).
Assuming a zero axial stress is of a rather limited practical application because it applies only to the neutral point. A more general approach to the calculation of collapse pressure is presented in the section on Biaxial Loading on page 80.