The pipeline industry has long been interested in evaluating the effects of external loading due to fill and surface loads, such as excavation equipment, on buried pipes. This interest stems not only from the initial design of pipeline systems, but also from the need to evaluate changing loading conditions over the life of the pipeline. Variations in loading conditions can arise due to the construction of roads and railroads over the pipeline and one-time events in which, for example, heavy equipment must cross the pipeline.

The pipeline may also suffer corrosion or damage that requires excavation and repair. Heavy excavation equipment is often placed directly over a pipeline during repair work, as shown in Figure 1. Safety while excavating pressurized pipelines is a serious concern for operating companies. Both gas and liquid pipeline companies often specify reduced pressures while excavating and repairing in-service pipelines.

A common issue is determining what pressures are safe during excavation and repair procedures. Design codes, regulations and industry publications offer little guidance on what factors should be considered to determine safe pressures during in-service excavation activities. Surface-loading conditions and soil overburden result in stresses that should be evaluated in determining safe excavation pressures near areas of damage or corrosion. Large concentrated loads, like truck wheel loads, are of primary concern.

The ALA Guideline for the Design of Buried Steel Pipe presents design provisions for use in evaluating the integrity of buried pipelines for a range of applied loads. (ref: “Guideline for the Design of Buried Steel Pipe,” American Lifelines Alliance/ASCE/FEMA, 2001.) Its methodology offers an approach for evaluating the fill and surface-loading effects on buried pipelines. This approach utilizes the deflection of the pipe, calculated using a version of the classic Iowa Formula, in estimating the wall-bending stresses in the pipe. The wall-bending stress is then combined with other calculated stresses to calculate the overall stress in the pipe.

*Figure 2: Schematic of the deflection of a buried pipe due to vertical loading. *

Smith and Watkins pointed out that the Iowa Formula was derived to predict the ring deflection of flexible culverts, and not as a design equation to determine the wall thicknesses of pipes. (ref: Smith, G., and Watkins, R., “The Iowa Formula: Its Use and Misuse when Designing Flexible Pipe,” Proc. of Pipelines 2004 Int’l Conf., ASCE, 2004.) It is often used to estimate wall stresses, however, and determination of the total stress is important to safety calculations. In this article, the wall-bending stress calculation and some quirks in its behavior will be discussed.

Pipe materials are classified as being either flexible or rigid. A flexible pipe has been defined as being able to deflect at least 2% without structural distress. (ref: Moser, A.P. and Folkman, S., “Buried Pipe Design, 3rd Ed.,” McGraw Hill, 2008.) Materials such as steel and most plastics are considered flexible pipe. Concrete and clay pipes are considered rigid. The Iowa Formula was developed for use with flexible pipes.

Flexible pipes derive much of their load-carrying capacity from pressure induced at the sides of the pipe as they deform horizontally outward under vertical loading. Analysis of the effect of fill weight and surface loading is therefore a problem of interaction between the pipe and the soil. The Iowa Formula describes the interaction of the pipe and soil and the deflection that results from vertical loading.

*Figure 3: Effect of wall thickness ratio on the normalized wall-bending stress *

In his research of the performance of buried flexible pipes, M. G. Spangler observed that, compared to rigid pipes, flexible pipes provide little inherent stiffness and perform poorly in 3-edge bearing tests. However, flexible pipes performed better than predicted by these tests when buried. He reasoned that the source of strength of the flexible pipe is not the pipe itself, but is primarily the soil beside the pipe. (ref: “Insight into Pipe Deflection Predictions: An Interview with M.G. Spangler,” Sewer Sense No. 17, National Clay Pipe Association, 2004.)

The ability of buried flexible pipe to support vertical loads is based on support from the soil around the pipe and the restraining force induced on the sides of the pipe counter to the horizontal deflection. Coupling these concepts with ring-deflection theory led to the development of the Iowa Formula in 1941. (ref: Spangler, M.G., “The Structural Design of Flexible Pipe Culverts,” Bulletin 153. Iowa State College, Ames, Iowa, 1941.)

The Iowa Formula was developed to estimate the distortion of a buried flexible pipe under vertical loading. A sketch of a deflected pipe is shown in Figure 2. The formula for the deflection can be written as:

The formula has two terms in the denominator, the first of which depends on the pipe stiffness and the second on the modulus of soil reaction, E’. For thin-walled flexible pipes, the modulus of soil-resistance term tends to dominate the equation. This term defines the resistance of the soil to deformation. Unfortunately, E’ is not a true property of the soil, but instead depends upon a number of factors including compaction, texture, and fill depth. E’ is normally estimated from tables or by testing.

It can be shown that the maximum through-wall circumferential bending stress can be determined from Eq. 2:

This stress equation is what is used in the ALA guidelines to account for the stresses due to ovality of a buried pipe.

Along with the Iowa Formula, Spangler also derived a formula for determining the wall-bending stress in a vertically loaded pipe with internal pressure. This is often referred to as the Spangler Stress Formula. In this case, however, he did not include a term to represent the soil support that resists distortion of the pipe. Warman et al. derived a combined equation that includes the effects of lateral soil restraint and the distortion-resisting effects of internal pressure. Inclusion of the pressure term removes some of the conservatism of the Iowa equation when applied to pressurized pipes. The wall-bending stress term proposed by Warman et al. can be written as:

Consider the wall-bending stress without internal pressure, as shown in Eq. 3. If the ratio of the wall thickness to pipe diameter is set to zero, the wall-bending stress goes to zero. The wall-bending stress also approaches zero as the wall thickness ratio increases. A graph showing the calculated wall stress as a function of the wall thickness ratio is shown in Figure 3 using 500 psi for the value of E’. Note that the magnitude of the stress depends on other terms, but the shape of the curve is determined by the ratio of E’ to E.

Again using a value of 500 psi for E’ and the Young’s modulus of steel (2.9×106 psi), the maximum stress occurs at t/D of about 0.01. This roughly corresponds to a 48-inch pipe with a 0.5 inch wall thickness. At thicknesses greater than the critical thickness, the stress equation predicts that the wall stress gets lower as the wall thickness is increased. However, for thickness ratios below the critical value, using a thicker wall results in a higher wall-bending stress. This is interesting, in that the thinner the wall is made, the lower the stress becomes.

Looking at the Iowa Formula in Eq. 1, it can be seen that reducing the wall thickness to zero results in a finite value of deflection. At this point, the wall-bending stress is zero, so the soil is carrying the entire vertical load. It appears that for gravity fed flows, that a hole in the soil does not need a pipe at all! Unfortunately, there are reasons that we can’t get rid of the pipe. The Iowa Formula assumed that the pipe transfers the vertical load to the side walls, so without the pipe, the formula doesn’t work. Also, ignoring the wall-bending stress, the vertical load itself will cause the pipe to fail due to buckling or crushing if the walls get too thin.

*Figure 4: Example minimum wall thickness ratio calculation using the wall-bending stress formula without a pressure term. *

A maximum value of the circumferential stress can be determined by adding the hoop stress and the wall-bending stress. If the circumferential stress and its Poisson contribution to the longitudinal stress are used to calculate the Von Mises stress, the resulting equations can be solved to determine the minimum acceptable wall thickness ratio as a function of internal pipe pressure.

As an example, consider a steel pipe 48 inches in diameter. Assume some combination of burial depth, lag factor, and live-loading to achieve a vertical pressure at the pipe of about 13 psi. This is a high value of vertical loading, but it was chosen to illustrate peculiar behavior of the wall-bending stress equation. In this example, we also assume a modulus of soil reaction E’ of 500 psi and constants based on a bedding angle of 30º. Using the wall-bending stress equation given in Eq. 3, the minimum required wall thickness as a function of internal pressure was calculated. The results are graphed in Figure 4. Wall thickness ratios above and to the left of the curve are acceptable, below and to the right are beyond the acceptable stress limit (in this case taken to be 0.6 times the ultimate tensile strength).

The results show that, for pressures between roughly 160-180 psi internal pressure, an S-curve occurs where a range of wall thicknesses are not acceptable while thinner walls are. In our example, at 170 psi internal pressure, a wall thickness ratio of 0.85% is not acceptable, but a wall thickness ratio of 0.45% is okay. Based on the pipe diameter of 48 inches, these correspond to wall thicknesses of 0.408 inches and 0.216 inches, respectively. This behavior certainly appears unrealistic.

Next we repeat the example, but using the wall-bending stress determined with Eq. 4, which incorporates the pressure term. Again, the pipe has a diameter of 48 inches and a vertical-loading pressure of 13 psi.

Figure 5 shows the results based on the Iowa Formula with the pressure term included and the previous results without the pressure term. The minimum required wall thickness calculated using the hoop stress only (neglecting the wall-bending stress entirely) is also shown for comparison. The results for the calculation using the Iowa Formula with the pressure term included do not show the strange behavior observed when the pressure term is not included.

*Figure 5: Comparison of minimum wall thickness ratio calculations using several approaches. *

Comparing the results shows that not including the pressure effects in the wall-bending stress equation leads to calculated wall thicknesses that become extremely conservative as pressure increases. On the other hand, comparing the results based on the Iowa Formula with the pressure term included to the results for hoop stress alone shows that the effect of the wall-bending stress is not insignificant, and the results become increasingly different as pressure increases. This indicates that neglecting the effects of wall-bending stress could result in non-conservative stress calculations and wall thickness values.

The chart in Figure 5 also shows the calculated wall thickness ratio at the buckling limit with no internal pressure. This is a performance limit based on the pipe walls buckling under the vertical load. Since internal pressure puts the walls into tension rather than compression, this effect will be observed for unpressurized pipes or pipes under vacuum. However, since all pipelines will be unpressurized at some time, this performance limit must be satisfied for pipes that normally operate pressurized as well.

The buckling limit wall thickness ratio for the example pipe is about 0.48%. Therefore, for any design pressure less than about 360 psi, the buckling limit will determine the minimum required wall thickness for the example pipe. Note that the buckling limit is a function of the vertical load pressure, which was specifically chosen in this example to be high.

**Conclusions**

Smith and Watkins pointed out that the Iowa Formula was derived to predict the ring deflection of flexible culverts, and not as a design equation to determine the wall thicknesses of pipes. It is, however, widely used in stress calculations, and is part of the methodology used to predict the stresses in pipelines due to vertical loading in the ALA Guideline for the Design of Buried Steel Pipe. The use of the Iowa Formula to calculate the wall-bending stresses in a pressurized buried pipe is generally unrealistically conservative, and can, under certain circumstances, lead to results that behave strangely, particularly for high vertical loading.

Inclusion of a pressure-stiffening term in the stress equation appears to improve the behavior and remove some of the excessive conservatism inherent in the Iowa Formula. At high vertical-loading pressures and low internal pressures, the wall buckling limit may be the dominant factor in the minimum allowable wall thickness.

**Author Lawrence Matta, Ph.D.**, is a staff consultant at Stress Engineering, Houston. He received his Ph.D. from Georgia Tech. He is registered in Texas as a Professional Engineer and is a certified fire and explosion investigator (CFEI). His experience includes investigation of pipeline failures and the resulting explosions and fires. He can be reached at lmatta@stress.com, 281-955-2900.