Collapse Behavior in the Elastoplastic Transition Range
The collapse behavior of casing specimens, which fail in the elastoplastic transition range, represents a problem of instability, as does elastic collapse behavior. The prediction of critical external pressure, however, can no longer be based on Young’s modulus because the bending stiffness now depends on the local slope of the stress-versus-strain curve (Heise and Esztergar, 1970). Young’s modulus. E, is, therefore, replaced by the tangent modulus. Et (see Fig. 2.12(a)) in Eq. 2.88. Thus, the equation for transition collapse, pctt. is:
where:
Pctt — critical external pressure for collapse in transition range based on Et, tangent modulus.
Calculation of collapse pressure using Eq. 2.119 yields values which are lower than experimentally derived results. Heise and Esztergar (1970) introduced the concept of a ‘reduced modulus’ which results in higher calculated collapse values.
The reduced modulus, ET. is based on the theory of buckling according to En — gesser and Von Karman (Szabo, 1977).
The following assumptions are made in the development of the theory (Bleich. 1952):
1. The displacements are very small in comparison to the cross-sectional dimensions of the pipe.
2. Plane cross-sections remain plane and normal to the center-line after bending.
3. The relationship between stress and strain in any longitudinal fiber is given by the stress-strain diagram. Fig. 2.12(a).
4. The plane of bending is a plane of symmetry of the pipe section.
Consider that the section in Fig. 2.12(b) is compressed by an axial load. Fa. such that cr = Fa/A exceeds the limit of proportionality. Upon further increase in Fa, the pipe reaches a condition of unstable equilibrium at which point it is deflected slightly. In every cross-section there will be an axis n — n (Fig. 2.12(c)) perpendicular to the plane of bending in which the cross-sectional stress prior to
bending, гг, remains unchanged. On one side of the n — n plane, the longitudinal compressive stresses will be increased by bending at a rate proportional to deride = Eu whereas on the other side of n — n. there will be a reduction in the longitudinal stresses due to the superimposed bending stresses associated with strain reversal.
In the case of the stress reduction, Hooke’s Law. a = E ex, is applicable because the reversal only relieves the elastic portion of the strain. In the stress diagram, Fig. 2.12(c), the concave (stress relief) side is bounded by N A and the convex (stress increase) side by NB.
Referring again to Fig. 2.12(c), equilibrium between the internal stresses and the external load, Fa, requires that:
jkl sxdA — jk2dA = 0 (2.120)
Jo Jo
and,
/ — e)dA — f ^2 (~2 + e)dA — Fay — M
Jo Jo
The deflection у is taken with respect to the centroid axis as illustrated in Fig. 2.12(b). From Fig. 2.12(c) one can infer:
5i — — r~ a. nd s2 — — z2 h i h2
Similarly:
Adx = h2dO = ^ E
|
(2.121) (2.122) |
Thus, it follows that: d9 <r2 ах
For small deformations:
TOC o "1-5" h z d9 d2y dx dx2
Thus, combining Eq. 2.122 with Eq. 2.122 yields: n, dly, d2y
a2 — bh2 — r-г and ax = thx-—
dx1 dx2
Substituting the above expressions for <X] and a2 into Eq. 2.120 yields:
^2У [hl j a r^d2y fk2
EtSl-ESi = 0 (2.123)
where:
Si and S2 — statical moments of the cross-sectional areas to the left
and right of the axis n — n. respectively.
In order to represent the pipe section as a rectangular cross-section, pipe wall thickness, t, is considered as height, h. and the unit length. 1. as base (Refer to Fig 2-12(c)). Using this notation. Eq. 2.123 reduces to:
|
(2.124) |
E h{ = Et hi
As shown in Fig 2-12(c), h = hl + h2• Thus, the changes in cross-sectional areas
(hi x 1) and (h2 x 1) from the neutral axis are given by:
_ h y/Et
1 ^Ё+^/Et “J)
and
, hyjE
h’ = 7WwW, l2J26>
The moment of inertia of the deformed sections can be given as:
«1 = щ = w
3 3 12
Combining Eqs. 2.125 and 2.126 and substituting for the moment of inertia, one
can define an additional parameter, ET (reduced modulus):
4 E ■ Et
ET = —1=——- =— (2.12/)
(n/£ + V^)2
Hence, the collapse pressure for elasto-plastic transition range can be determined by means of the equation:
where:
pctr — critical external pressure for collapse in transition range based on Er, reduced modulus.
The average tangential stress is obtained using the following equation:
= (2Л29)
where:
atEr = average tangential stress for a particular value of ET.
In contrast to Young’s modulus. ET. is not a constant, but depends on the particular value of the stress. Exact knowledge of the stress-strain behavior of the material is, therefore, necessary for the determination of the collapse pressure and the calculation must be performed by means of an iterative procedure.
Sturm (1941) proposed using the tangential modulus as the effective modulus in order for the results to be conservative and to simplify the calculations in determining the collapse pressure. His general equation for collapse strength, for which the stresses exceed the limit of proportionality, is given by:
p* = K’Et(t/d0)3 (2.130)
where:
p* = collapse pressure for stresses above the elastic limit (Sturm. 1941).
A’* denotes the collapse coefficient, which becomes equal to:
/С = (ГЬ) <2Л31)
for infinitely long casing steel specimens.
The stress-strain relationship is presented in Fig. 2.13. The curve of tangential modulus has been approximated by a single straight line, resulting in three distinct cases:
|
STRESS |
|
TENSILE STRENGTH |
YOUNG’S
MODULUS
|
STRAIN £ |
TANGENT MODULUS Et
Fig. 2.13: Relationship between stress, strain and the tangent modulus. (After Krug, 1982; courtesy of ITE-TU Clausthal.)
1. If the average nominal stress. an. lies between the limit of elasticity. <j£. and the yield limit, ay. the following equation applies:
The parameter <f denotes the ratio of Young’s modulus to the tangent modulus at the yield point, <7y.
2. If the average nominal stress. an. lies between the yield point. and the tensile stress, era, the equation becomes:
3. If the average nominal stress lies below the limit of proportionality, ap. whereas the maximal total stress. сгтях. lies above the limit of elasticity because of eccentricity, the experimentally determined formula applies:
For the calculation of the collapse pressure in the elastoplastic transition range according to the methods described, accurate knowledge of stress-strain relationships for each material is required. Furthermore, the equations do not take into
|
Fig. 2.14: Critical collapse pressure according to API. |
account the fact that Poisson’s ratio for steel varies from г/ = 0.3 in the elastic range to и — 0.5 in the plastic range. Moreover, imperfections may occur in the pipe body, which can influence the collapse strength. For these reasons, it appears both sensible and expedient to describe the collapse behavior by simple empirical formulas from the start. In practice, these simplifications are made for oilfield tubular goods because their standardized dimensions lie. for the most part, in this range (Krug, 1982).