Michael J. Rosenfeld, Willard A. MaxeyKiefner & Associates Inc.

Worthington, Ohio

A procedure has been developed that uses structural analysis of cased pipeline crossings to determine tolerable stress levels for casing maintenance that involves moving a pipeline.

This method follows a review of current U.S. and Canadian pipeline regulations and design codes to determine those guidelines' allowable stresses caused by pipe movement.

Part 1 of this two-part series reported on what this review discovered (OGJ, Oct. 24, p. 87).

This article sets forth the procedure and presents a sample analysis of a pipeline undergoing casing maintenance.

Both articles are intended to provide the means to engineer casing maintenance that meets all regulatory concerns.

### EXCAVATION LENGTH, LIFT HEIGHT

Inside the casing, the pipeline sits on several insulators assumed to be spaced uniformly. Generally, some clearance exists above the pipe.

The amount of movement that can be achieved at the end of the casing will depend on the pipe's bending strength and stiffness, the insulator's compressive stiffness, and the spacing between insulators.

A schematic of the problem is shown in Fig. 1.

The length of pipe that should be excavated depends on the angle of inclination of the pipe at the end of the casing and the radius of curvature which can be placed on the pipe. The shortest trench will be achieved assuming the pipeline displacement can be controlled to impose a uniform bend radius.

Because the problem is nonlinear, certain simplifying assumptions will be made in this discussion.

The basic geometric configuration of the lifted pipe is assumed to resemble what is shown in Fig. 1. The arcs of the pipe outside the casing are assumed to be of a constant bend radius (R).

The relationship between bending stress ((TB) and bend radius is shown in Equation 1 in the accompanying equations box.

At the end of the casing, the vertical dimension from the original installed position to the lifted position is h. The vertical distance between the pipe axis at the end of the casing in the lifted position to the maximum lift point is DELTA, as expressed in Equation 2.

The maximum lift from the original installed position of the pipe is H, given by Equation 3.

Dimension h and angle are boundary conditions at the end of the casing, to be addressed later in the analysis.

If it is observed that the pipe at the end of the casing is not lying in its original, installed position, an attempt should be made to quantify or estimate the amount that the pipe is out of position.

This dimension should be subtracted from h if the reference position is as-excavated.

The horizontal distance from the end of the casing to the maximum lift point is L1, expressed by Equation 4.

The horizontal dimension from the maximum lift point to the end of the excavation is L2, expressed by Equation 5.

The minimum length of pipe that must be excavated is LT = L1 + L2.

### STRESSES: INITIAL, INDUCED

In order to solve for H and LT, the longitudinal stresses already in the pipe and those induced by the movement must be considered.

Longitudinal stresses already exist on the pipeline as a result of internal line pressure, thermal expansion, and initial elastic curvature.

The initial longitudinal stresses are calculated as shown in Equation 6.

The design factor (F) is established be, the applicable pipeline design code or regulations, which were discussed in Part 1.

It may be noted that the longitudinal stress due to pressure was calculated on the assumption that the line is axially restrained. If field bends are located in the excavated portion of the pipe the line is arguably not well restrained axially in terms of pressure stresses.

In that case, calculate sp = PD/4t.

The bending stress induced by the line movement has already been described.

There is also an extensional stress which results from the pipe following an arc length longer than its original installed length. This stress is expressed approximated in Equation 7.

The stress margin available for line movement is shown by Equation 8.

If the full margin were used to establish the deflection profile, the length of excavation required would be minimized, which is a worthwhile objective. There would then be, however, no capacity available (within the limits of the allowable stress) for bending stresses due to sag between lift or support points.

In a liquid-filled pipe, this could be significant. Therefore, only half of the available bending-stress capacity will be used to establish the profile, reserving half for local bending.

Then the allowable bending stress becomes as shown in Equation 9 which is solved to obtain Equation 10.

Unfortunately, H is not independent of the bending stress because H is derived from and h which are themselves established by the bending capacity, as will be discussed presently.

Thus we do not yet have a closed-form solution.

The analysis can be simplified, however, by recognizing that in clearing casings (as contrasted with line relocation), H should be on the order of several inches compared to L, on the order of dozens of feet, so that H/L2

It is then reasonable to simplify the calculation for L2 as shown in Equation 11, noting that the bending stress induced by the deformation is rendered by Equation 12 (observing elastic modulus and stress in units of psi).

The exposed length of pipe will be lifted at one or more points, giving it the configuration of a multi-span, uniformly loaded beam.

The peak bending moment will be estimated from that for a four-span beam, M = 0.11WL S2, W being the total weight per unit length of the pipe, coating, and contents.

The available bending stress capacity is SB, so that the maximum unsupported span is given by Equation 13.

The pipe between the end of the casing and the maximum lift point, if it were unsupported, would have a bending moment of approximately 0.04W (b + a + L1)2 in addition to the bending moment due to the deflection.

If L1

To close the solution loop, a further simplifying assumption is made: the pipe is supported so that the end of the casing is near an inflection point in the bending-moment profile for the uniform weight loading.

Then the bending moment used in order to calculate , h, and subsequently, H, L1, and L2, will be Mo 2SBZ.

The multiple support condition of the pipeline within the casing means that the rotational flexibility of the pipe is greater than if it were fully restrained or built in.

It will be assumed that the bending moment beyond the fourth span is negligible and the flexibility saturates with increasing N spans greater than four (Fig. 2).

### ROTATIONS, MOMENTS, REACTIONS

With reference to Fig. 2, the rotations at Node b, due to a moment Mo at Node a, are written as shown in Equation 14.

Analogous equations involving M1,2,3 may be written for the rotations at Nodes c, d, and e.

The equations are solved simultaneously, assuming that M4 = 0, to obtain the moments, rotations, and reactions as a function of Mo shown in Table 1.

(Note that the problem could also have been solved with 0e = 0 and M4 = 0, with only a negligible difference in results, particularly at Node a.)

A beam with a built-in end at Node b would have an end rotation at Node a equal to 0 = 0.25Mob/EI, so that the multi-span beam is about 15% more flexible. Consider that for N 4, the rotational stiffness tends toward 0.28Mob/EI.

### PIPE DEFLECTION

The pipe deflection at the end of the casing will result from a stack-up of deflections and rotations due to pipe flexibility, insulator flexibility, and clearances. These are developed in the following discussion.

With reference to Fig. 3, the average reaction between the two end insulators is shown in Equation 15 so that the rotation due to insulator flexibility is shown in Equation 16.

Assuming the typical insulator width (w) and depth (d) to be 2 in. each yields Equation 17.

Typical E values are shown in Table 2.

The spacer deflection actually affects the development and distribution of moments and reactions, requiring an iterative solution. To simplify the problem, however, it is assumed that these secondary effects are negligible.

Rotation due to total clearance (c) between the insulators and casing is c/b, as shown in Fig. 4.

Under the influence of moment Mo, the net rotation () and displacement (h) of the pipe at the end of the casing are given by Equations 18 and 19, as shown in Fig. 5.

These are used subsequently to obtain H and L as described.

In the clearing operation, movement of the pipeline may not be restricted to the vertical upward direction.

The displacement in any one direction should be limited to the value calculated for the vertical movement. This is somewhat conservative because deadweight bending stresses are not directly additive to those induced by displacement except in the vertical plane.

It is assumed that the insulators will be reinstalled and the pipe repositioned very close to its original installed position. Furthermore, it is assumed that the pipe will be fully supported along its entire length by restoration of the ditch profile and compaction.

In those cases, the stresses in the pipe following completion of the operation should be no more than when the pipe was first installed, and stresses due to displacement of the pipe and intermittent support are only temporary.

*Copyright 1994 Oil & Gas Journal. All Rights Reserved.*

* Issue date: 11/07/94*