The pipe radius in the bending plane is reduced as the cross section becomes ovalized, and the formula no longer is valid. As a tube bends, its cross section tends to ovalize rather than remain circular. It is necessary to remember the assumptions of this formula before using it, especially with tubes. Young's modulus usually is in units of psi, kPa, or MPa, and the bending stress is in the same units. In SI units, both measures should be in meters. In USC units, the radius of the pipe usually is in inches and the radius of curvature of the borehole path usually is in feet, so they must be converted to the same unit (it does not matter which). It is important that the units used are consistent. R Оє = radius of curvature of borehole path R = radius of pipe where stress is determined (i.e., inside or outside) ОЈ b = bending stress, (+) for tension, (в€’) for compression The bending deflections are small in comparison to the length, so that the radius of the tube remains constant in all directions.

The tube radius is small compared to the length. The central longitudinal axis (neutral axis) experiences no axial strain ДЄll cross sections normal to the longitudinal axis before bending remain normal to the axis after bending.The tube cross section is symmetric about the central longitudinal axis. For the case of tubes, such as casing, these are the typical assumptions:

No theory really is involved, but merely a set of a priori assumptions about the way a beam deforms in bending that allows for an analytic solution to the more complicated boundary-value problem. These ad hoc assumptions are known variously as Euler-Bernoulli beam theory, planar beam theory, or simply just beam-bending theory. Even the planar bending problem is a two-dimensional elasticity boundary-value problem, and several assumptions usually are adopted so that a simple solution may be obtained from a one-dimensional ordinary differential equation, a derivation of which is given in Appendix C. That may seem to avoid the specific, but for certain, we can say that it becomes meaningless if the yield point is exceeded.Д¬alculating bending stresses can be a formidable undertaking in general. For short-radius wells, it should be used with caution, and again, it would depend on the pipe diameter and wall thickness. For long and medium radius of curvature wells, it seems to work well for all but larger-diameter, thin-wall pipe. That makes the formula possibly a bit conservative in casing design. Since there is no easy way to determine the point at which the shape is too ovalized to use of the formula, the tendency is to ignore it, since it will overpredict the maximum bending stress when the pipe is slightly ovalized.

2, kPa, or MPa, and the bending stress is in the same units. Young's modulus usually is in units of lbf/in. In oil-field units, the radius of the pipe usually is in inches and the radius of curvature of the bore-hole path usually is in feet, so they must be converted to the same unit (it does not matter which). R = radius of pipe where stress is determined (i.e. ОЈ b = bending stress, (+) for tension, (-) for compression

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