## September 2021

## Process Optimization

# External pressure design of large-diameter piping: Optimal analysis approach

Large-diameter piping used in the petrochemical, oil and gas, steel plant and power industries often carry toxic fluids and can pose significant safety hazards.

Large-diameter piping used in the petrochemical, oil and gas, steel plant and power industries often carry toxic fluids and can pose significant safety hazards. Their stability and reliability are important while they operate under external pressure loading due to vacuum formation or any other external load that can potentially collapse the piping system.

Major piping codes, such as the ASME B31 pressure piping series^{1}, refer pressure vessel codes for the design of piping subjected to external pressure. However, piping and pressure vessel configurations are usually different, and a pressure vessel external pressure design approach should be carefully considered when applied to piping. In the author’s experience, a design by rule (DBR) approach as stipulated in various pressure vessel codes is conservative in nature; those rules are conservatively applied to a specific piping system without due consideration of its actual geometry, pipe support location and configuration. However, these aspects have considerable impact on the external allowable pressure/loading capacity of a piping system.

However, a design by analysis (DBA) approach can address this issue effectively, considering the actual layout of the piping and support location/configuration and even optimizing the piping wall thickness requirement while meeting allowable external pressure loading criteria. This article briefly compares the various rules and approaches of the external pressure design per various international codes and intends to help practicing engineers to adopt a suitable approach for specific applications to optimize design within current pressure vessel code framework.

# RULES, PROCEDURES AND PRINCIPLES

## Classical approach for infinity long shell

External pressure causes compressive stress in piping or pressure vessels, creating stability problems and potentially leading to failure. Large-diameter and thin pipes are more susceptible under external pressure than thicker pipe; however, even thick piping may also be subjected to failure where external pressure is significant, such as in a subsea environment.

Bresse-Byran offered an allowable external pressure design formula^{2} with a factor of safety (FOS) of 3 considering two lobes for an un-stiffened portion of piping (Eq. 1):

(1)

Another commonly used formula (Eq. 2) for above-ground steel pipe from AWWA M-11^{3} is often used by engineers working on a water main of steel construction. This formula is from the original Timoshenko equation^{4} (without an FOS) for an unreinforced, infinite length:

(2)

## DBR approach for pressure vessel codes

DBRs on external pressure design are outlined in various international codes, such as ASME Boiler and Pressure Vessel Code (B & PVC) Section VIII, Div. 1, UG-28 to UG-30, and B & PVC Section VIII, Div. 2, EN-13445/PD 5500.^{5,6} However, these code rules are primarily meant for tubular structure design under only external pressure loading, but are not applicable for combined additional loading, such as axial compression or bending.

ASME B&PVC Code case 2286,^{7} first published in 1990 and reaffirmed in the 2004 edition, was intended to address the combined loading of compression and external pressure applied to a tubular structure. The 2286 code case is applicable to both Div. 1 and Div. 2 vessels and was later merged into the main code of Div. 2.

However, the methodologies and conservatism associated with each code related to external pressure design are different. ASME B & PVC Section VIII, Div. 2 offers both DBR and DBA in Parts. IV and V, respectively. B & PVC Section VIII, Div. 1 (UG-28 to UG-30) DBR is a little conservative in approach but still preferred and widely used for piping/pressure vessel design, as these calculations are less time-consuming and more straightforward.

## DBA approach for pressure vessel codes

In 2007, new code rules were introduced in ASME B & PVC Section VIII, Div. 2, and the DBA approach was included under the ASME B & PVC rewrite program. The DBA approach offers a rigorous method of analyzing pressure vessel/piping engineering problems using a numerical method.

The buckling analysis due to external pressure as well as other loading can be analyzed by three primary methods per the DBA approach, as defined in Part V of the Div. 2 code in a compressive stress field:

- Linear buckling (Eigen value buckling) analysis, or Type 1 analysis (
**FIG. 1**) - Non-linear buckling analysis with effect of non-linear geometry/material, or Type 2 analysis
- Non-linear collapse analysis with geometry/material non-linearity considering shell imperfection, or Type 3 analysis.

In general, Type 1 analysis results in the greatest overestimate of collapse load, so the largest design margin is used. Type 2 analysis results in a lower estimate in the buckling load and a lower design margin is used. Type 3 analysis results are the best estimate for collapse load as both material and geometric nonlinearity, along with shell imperfection, are included in numerical analysis and no capacity reduction factors are used.^{8,9} However, no discussion or analysis results have been produced on Type 3 buckling analysis in this article.

In the European pressure vessel code EN 13445, two approaches exist to DBA. One is the so-called direct approach, while the other is historically older and is referred to as the elastic stress categorizations approach. Both have different merits according to the complexity of the design. However, detailed discussion related to external pressure design DBA application per EN 13445 is outside the scope of this work.

# SAMPLE COMPARATIVE STUDY ACROSS PRESSURE VESSEL CODES

This section compares the different code-allowable external pressures of large diameter piping, with **TABLE 1** showing the properties for the varied magnitude of an un-stiffened length (support-to-support length).

For a sample study, an un-stiffened cylindrical shell with 20-mm wall thickness of material of construction 316 grade SS was considered. The varied support-to-support length and maximum allowable external pressure (MAEP) capabilities have been determined as per various codes to obtain the trend of conservativeness across the pressure vessel codes on an external pressure design approach.

## Assumed data for MAEP study and result table/plot

**TABLE 2** represents the MAEP per various pressure vessel codes vs. the varied support-to-support length. **FIG. 2** shows the plot of the same data.

The MAEP, as tabulated/plotted here, was calculated based on DBRs of various pressure vessel codes and other standards, such as AWWA M11. Additionally, the MAEP based on the DBA approach per ASME B & PVC Section VIII, Div. 2, Part V was calculated using the procedure below and compared with the DBR result set.

**FIG. 3** represents a finite element model result of a linear buckling analysis with a support-to-support length of 30 m (with a 2,130-mm diameter SS 316 shell with 20-mm thickness at 21°C) and with the same properties shown in **TABLE 1**.

Therefore (Eq. 3):^{10}

(3)

where,

Φ* _{B}* = 2/

*β*= 2/0.80 = 2.5

_{CR}*β*= 0.8 for external pressure on the shell.

_{CR}Similarly, for a varied support-to-support length starting from 10 m to 60 m, the result was tabulated in the sixth column (from left) of **TABLE 2**.

## Observations/discussion on comparative study

ASME B & PVC Section VIII, Div. 2 DBR for external pressure design^{10} is the most conservative, followed by EN-13445/PD-5500 and B & PVC Section VIII, Div. 2 design by rules per the data and plot shown in **TABLE 2 **and **FIG. 2**.

Also in **TABLE 2** and **FIG. 2**, DBA (linear buckling, Type 1) per ASME B & PVC Section VIII, Div. 2, Part V^{8} provides a non-conservative result for certain support-to-support length based on the outer diameter of the pipe (in this case, 2,130 mm), after which the result converges and the MAEP is almost independent of support-to-support length.

The AWWA – M11/Timoshenko formula (without an FOS) is basically an overestimate of the MAEP compared to other code rules and does not include the effect of support-to-support length.

## DBA: Type 1 linear buckling of actual piping configuration/layout with restrained condition

**FIG. 3** represents a piping FEA buckling result under external pressure from one piece of equipment to another (anchor-to-anchor), with the properties shown in **TABLE 1**. A linear buckling analysis (Type 1) was performed to evaluate the allowable external pressure, as per the DBA approach of ASME B & PVC, Section VIII, Div. 2, Part V.

## Piping model properties

The piping model properties have been kept the same as **TABLE 1** except that the configuration/layout was considered per **FIG. 3** with the length of each leg as 10 m, a total developed length of 30 m, and a restraint boundary condition where both ends are fixed (equipment end). Result summary: For the linear Type 1 buckling analysis, Eq. 4 is used:

(4)

where,

Φ* _{B}* = 2/

*β*= 2/0.80 = 2.5

_{CR}*β*= 0.8 for the external pressure on the shell.

_{CR}## DBA Type 2 non-linear buckling analysis of actual piping configuration with restrained condition

A non-linear Type 2 buckling analysis was performed on the same piping configuration as **FIG. 3** as per Part V of B & PVC, Div. 2.^{8} **FIG. 4** represents the non-linear FEA result plot, and the inset figure shows the load factor plot for non-linear time step analysis. Result summary: For the non-linear Type 2 buckling analysis^{11}, Eq. 5 is used:

(5)

where,

Φ* _{B}* = 1.667/

*β*= 1.667/0.8 = 2.08

_{CR}*β*= the effect of shell imperfection = 0.8 for external pressure on the shell.

_{CR}## DBA Type 1 and Type 2 analysis results comparison

**TABLE 3 **represents the summary of the buckling analysis results from Sections IV and V for actual piping configuration/layout with restrained condition.

One intriguing observation is that the non-linear Type 2 analysis MAEP without the design factor stipulated by the Div. 2 code is slightly less than that of the linear Type 1 (10.70 kgf/cm^{2} vs. 12.25 kgf/cm^{2}). However, for this particular application, the final MAEP for the Type 2 analysis is greater than Type 1 due to a significant difference in design factor for these two types of analysis.^{12,13,14,15}

## DBA linear and non-linear method adopting analysis approach

While Type 2 and Type 3 non-linear bucking analyses are more realistic and non-conservative methods compared to the linear elastic Type 1 analysis, they require a significant amount of engineering effort and data related to geometric and material imperfection. In particular, Type 3 analysis—which considers both geometric and material non-linearity as well as shell imperfections—is the least conservative. However, it is complex and requires significant computational effort, making it an unrealistic approach for regular engineering applications.

A Type 1 linear elastic buckling analysis is straightforward with comparatively lesser computational efforts. However, it produces a less conservative result than compared to pressure vessel code design by rule methods (DBR), leading to significant design optimization and material cost savings.

## Takeaway

Pipe wall thickness estimation and fixing the stiffener/support location for external pressure loading through the DBR of pressure vessel codes is easy to apply and has been a traditional approach for many years for large-diameter piping design. These rules are conservative and, in many cases, can result in overdesign of the system. In case of multiple loading, such as axial compression and bending and an external pressure loading scenario, these rules are not applicable to predict buckling.

However, for costly material and more critical applications, provisions within the code (e.g., the DBA approach) provide rigorous engineering analysis and, therefore, more optimized solutions with significant material savings. Additionally, the DBA approach is suitable to predict buckling in multiple loading scenarios.

Once the DBA approach is adopted for buckling analysis for external pressure design, an FEA model can be built with the actual layout of the piping system with exact restraint/support locations. The location of the stiffener can also be fixed based on the buckling mode shapes to obtain an optimized solution.

With applications of the FEA method gaining popularity among piping and pressure vessel engineering industries, DBA methods have become more relevant than ever. For critical applications involving expensive material and in-service scenarios, the material/geometric non-linearity effect can be accounted for and fitness for service can be ascertained through Type 2 or Type 3 buckling analysis—or even the simpler Type 1 analysis—for a new design, providing a more realistic result than the traditional code DBR approach. **HP**

**ABBREVIATIONS:**

DBR Design by rules

DBA Design by analysis

B & PVC Boiler and Pressure Vessel Code

LM Load multiplier

MAEP Maximum allowable external pressure

FOS Factor of safety

BC Boundary condition

FEA Finite element analysis*E _{s}* Modulus of elasticity

*D*Shell diameter

_{0}*t*Shell wall thickness

*v*Poisson ratio

_{s}*P*Allowable external pressure

_{allow}*P*

_{c}_{ }Collapsing pressure

*β*Shell imperfection effect, B & PVC, Section VIII, Div. 2, Part V

_{CR}*φ*Design factor, B & PVC, Section VIII, Div. 2, Part V

_{B}**LITERATURE CITED**

- American Society of Mechanical Engineers, ASME B31.3, “Process Piping,” 2018.
- Peng, L.-C. and T.-L. Peng,
*Pipe stress engineering,*American Society of Mechanical Engineers (ASME), 2009. - ANSI/AWWA Standard M11,
*Steel—A guide for design and installation,*American Water Works Association, 2016. - Timoshenko, S. P.,
*Theory of elastic stability,*2nd Ed., McGraw-Hill, 1985. - BS EN-13445, “Unfired pressure vessel,” Part 3, 2018.
- BS PD 5500, “Specification for unfired fusion welded pressure vessels,” 2018.
- American Society of Mechanical Engineers (ASME), Code Case 2286-1, “Alternative rules for determining allowable external pressure and compressive stresses for cylinders, cones, spheres and formed heads,” Section VIII, Div. 1 and 2, Cases of the ASME Boiler & Pressure Vessel Code, 2001.
- American Society of Mechanical Engineers (ASME), “ASME Boiler & Pressure Vessel Code,” Section VIII, Div. 2, 2019.
- American Society of Mechanical Engineers (ASME), PTB-1, “Criteria and commentary,” Section VIII, Div. 2, 2014.
- American Society of Mechanical Engineers (ASME), “ASME boiler and pressure vessel code,” Section VIII, Div. 1, 2019.
- DNV, DNVGL-RP-C208, “Determination of structural capacity by non-linear finite element analysis methods—Recommended practice,” 2019.
- American Society of Mechanical Engineers (ASME), “ASME boiler and pressure vessel code,” Section II, 2019.
- Miller, C. D. and K. Mokhtarian, “Proposed rules for determining allowable compressive stresses for cylinders, cones, spheres and formed heads,” Welding Research Council Bulletin 406, New York, 1995.
- Miller, C. D., “Commentary on the alternative rules for determining allowable compressive stresses for cylinders, cones, spheres and formed heads for Section VIII, Div. 1 and 2,” Welding Research Council Bulletin 462, New York, 2001.
- American Society of Mechanical Engineers (ASME), Section VIII, Div. 2, Example Problem Manual, PTB-3, 2013.

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