Authors: Derek E. Slovenec, Ph.D., P.E., Senior Engineer I, Matthew Kaiser, E.I.T., Staff Engineer I
Introduction
Buckling is a concern for slender components subjected to compressive stresses. Industrial process environments are full of such structures, including pressure vessels, piping, storage tanks, support frames, and others. Regardless of component type, the underlying mechanics of buckling phenomena are the same: buckling is a bifurcation of equilibrium from a stable to an unstable state generally characterized by a large transverse deformation in response to small increments in the compressive load. An understanding of buckling and the factors that contribute to its occurrence are essential to many design, analysis, and fitness-for-service (FFS) applications.
Background
Nearly all components in an industrial setting are subjected to some amount of compressive loads and are, therefore, potentially at risk of buckling. Propensity to buckle increases in proportion to the slenderness of the structure under compression. Simply put, a component that is thin and long (in the direction of compressive loading) is slender. For a given geometry and loading, the load at which buckling is expected to occur, as well as the buckled shape the component is expected to take, can be obtained theoretically by solving the eigenvalue problem for the differential equation governing equilibrium of the deformed component. This is a complex process for all but the simplest of cases, and these theoretical solutions may overestimate critical loads compared to experimental results due primarily to geometric imperfections and material residual stresses. As such, many engineering codes and standards specify buckling capacity equations informed by both theory and experimental results and rely on safety factors and fabrication tolerances (which are particularly critical for buckling) to ensure adequate protection against this limit state. While powerful finite element analysis (FEA) tools are available today to solve complex buckling problems, these methods can give misleading results if the analyst does not have a thorough understanding of the underlying mechanics and relevant factors. Buckling is a universal presence in any industrial process facility, and this discussion intends to highlight some of the methods for assessing buckling, including key factors, limitations, and more.
Buckling of Pressure Vessels
Pressure vessels experience compressive stresses when subjected to external pressure events. Vertical vessels experience longitudinal compressive stresses as they support their own weight, and also in response to lateral loading such as wind or earthquake demands that cause bending. An overview of buckling evaluation methods in ASME Section VIII Divisions 1 and 2 is provided below. Note that there are other approaches for evaluating buckling of pressure vessel components, including ASME Code Case 2286 and various international standards; however, these are outside the scope of this article.
ASME VIII-1 Methods
ASME Section VIII Division 1 (ASME VIII-1) addresses two causes of buckling failure: buckling from external pressure and buckling from axial compression. The code procedure changes depending on the equipment geometry and the loading, but the underlying method to evaluate buckling is the same. This simplified approach makes it easy to evaluate buckling, but it may be difficult for inexperienced engineers to understand the assumptions used in the code procedure.
Using the example of the procedure for cylindrical structures under axial compression, ASME VIII-1 gives the user an equation for “Factor A”:
In this equation, Ro is the outside radius of the cylinder and t is the thickness of the cylinder, but without any further background, it is unclear what is happening in the equation. The code then directs the user to reference the external pressure charts in subpart 3 of ASME Section II part D. With the correct chart in hand, the user finds Factor A along the bottom axis, then draws a vertical line directly upwards until it intersects the appropriate material/temperature curve, and finally draws a horizontal line from the intersection to the right axis of the chart to find Factor B. Thankfully, the procedure explicitly tells the user that B is the maximum allowable compressive stress for the cylinder with the dimensions Ro and t. But what is this procedure actually doing? And what is Factor A a measure of?
Factor A is the critical buckling strain of the component geometry. It is based on a combination of theoretical relationships between geometry and critical buckling loads derived from elastic shell stability theory, and empirical relationships between predicted buckling loads from theory and observed buckling loads from experiment. The equation for the theoretical critical buckling stress of a cylinder under axial compression is as follows:
This equation results in a predicted critical buckling stress that is anywhere from 150% to 500% of what is observed in experiments. The reason for this discrepancy is that imperfections in the geometry of real-world cylinders will initiate buckling before the predicted buckling load is reached. Therefore, a knockdown factor is applied to the theoretical equation to account for geometric imperfections.
Buckling can happen in the elastic regime (thin-walled cylinders) or in the inelastic regime (heavy-walled cylinders), both of which are accounted for in the external pressure charts. Additionally, the external pressure charts have another safety factor baked in, which varies based on the component geometry. For axially compressed cylinders, this factor of safety is 2.
The ASME VIII-1 methods have some notable limitations. First, the allowable stress limits assume a uniform membrane stress distribution, which means these buckling procedures do not handle load combinations. Looking at the stresses in a vessel from a combination of deadweight axial loading and a lateral shear load, such as a column vessel with wind loading might experience, may result in overly conservative results since the method can only evaluate the maximum compressive stress that only occurs on one side of the equipment. Furthermore, the procedures use a very conservative factor for geometric imperfections which may be unnecessarily large for equipment designed to ASME VIII-1. Finally, the procedure does not evaluate Euler column buckling whatsoever.
ASME VIII-2 Methods
The limitations of the ASME VIII-1 method are addressed by the “Design-by-Rule” approach in part 4.4 of ASME VIII-2, which provides full consideration of elastic/inelastic buckling, column buckling, and the specific load combinations applied to the equipment (e.g., internal/external/hydrostatic pressure, axial/shear/bending loads). As such, it is a more complex procedure that uses capacity equations specific to the types of combined loads (e.g., external pressure plus overturning moment, strictly axial compression, etc.).
The “Design-by-Analysis” methods in ASME VIII-2 part 5.4 provide a means for performing fully generalized buckling analyses using powerful FEA software. There is no limit to the geometry nor load combinations that can be analyzed using FEA. Three buckling analysis types are outlined in Part 5.4:
- Type 1: Linear-Elastic Bifurcation Buckling Analysis
- Type 2: Elastic-Plastic Bifurcation Buckling Analysis
- Type 3: Collapse Analysis
The Type 1 and Type 2 procedures are extremely similar in practice. The Type 1 procedure uses elastic material properties, does not consider geometric non-linearity in the preload step, and uses a design factor of 2. The Type 2 procedure uses elastic-plastic material properties, considers geometric non-linearity in the preload step, and uses a lower design factor of 1.667. Each procedure solves an eigenvalue problem created from the finite element geometry of the base state. In essence, it solves the same elastic instability equations that were solved by hand as the basis of the ASME VIII-1 procedures. The lowest eigenvalue calculated from the analysis is the critical buckling load for the “perfect” geometry. Corrections for geometric imperfections are provided in the procedure; the appropriate correction depends on the component geometry and applied loading.
The Type 3 analysis is a full elastic-plastic collapse analysis in accordance with the load factors described in part 5.2.4. The geometric imperfections must be explicitly modeled in the FEA geometry, which means no capacity reduction factors are required. One challenge of this approach is that the required load factors are based on plastic collapse, which has a higher required safety factor than buckling. Convergence of the FEA at the required load factors for protection against plastic collapse may not be achieved if the structure is slender and prone to buckling, requiring Type 1 or 2 analysis to assess buckling and a limit load analysis for plastic collapse (effectively decoupling these limit states). Obtaining appropriate data and incorporating it into FEA is also challenging. Thankfully, E²G is experienced in this procedure and has successfully guided clients and contractors through the data acquisition process to complete complex Level 3 FFS assessments on buckled equipment.
Steel Structures
Beams and columns that form the backbone of any industrial process facility are subject to their own buckling phenomena. Steel structures are commonly used to support piping, process equipment, and to frame walk/working surfaces and buildings. The design of steel structures is governed by AISC 360 in the United States, which includes provisions to assess buckling capacity depending on the type of member and loading application.
Columns
Columns are generally controlled by their compressive load-carrying capacity calculated per Chapter E of AISC 360. This capacity is a function of the slenderness of the member, defined as its effective unbraced length divided by the radius of gyration about the buckling axis. Buckling capacity reduces as this slenderness ratio increases. An approximate illustration of this relationship is provided in Figure 4.
For a very low slenderness ratio, the column is capable of reaching its so-called “squash” load where capacity is limited by yield of the cross section rather than buckling instability. For larger slenderness ratios, capacity is calculated based on the Euler buckling stress modified by various factors based on experimental test results to account for fabrication tolerances (out-of-straightness) and other factors that reduce capacity relative to the theoretical solution. Various buckling modes must be considered depending on the cross section, including flexural buckling, torsional buckling, and flexural-torsional buckling. The applicability of various buckling modes and capacity calculations are detailed in Chapter E of AISC 360.
One additional note for compression members, as well as flexural members, is that the slenderness of individual elements of the cross section, such as the web or flanges of W-shapes, must be considered as these elements can buckle locally, thus limiting the load-carrying capacity of the entire section. The necessary checks are included in AISC 360 Tables B4.1a and b for compression and flexure, respectively.
Beams
A beam section subjected to bending has compressive stresses on one side of its neutral axis. Similar to columns in compression, beams have flexural load-carrying capacity that is a function of slenderness. Various buckling modes are relevant for beams depending on their cross section and bending axis. The most common buckling mode affecting beams, including W-shapes (so-called I-beams) bent about their strong axis, is lateral torsional buckling (LTB). The flexural capacity of a member limited by LTB as a function of slenderness is shown in Figure 5.
LTB is a mode that is characterized by the beam section “rolling over,” or simultaneously displacing laterally while also twisting (torsion). Non-slender beams (in the leftmost region of the plot) are able to achieve the full plastic strength of the section in flexure without buckling. As the unbraced length (i.e., slenderness) increases, the member is expected to undergo inelastic LTB after some material in the cross section yields but before reaching the full plastic section capacity. Highly slender beams will undergo elastic LTB prior to the development of any plastic strain. Beams are rarely designed in this elastic LTB region. Lateral bracing can be added to beams to reduce the unbraced length and, thus, increase flexural capacity. Bracing is attached to the compression flange of the beam to restrain its tendency to buckle laterally and twist. Strength and stiffness requirements for effective lateral bracing are detailed in Appendix 6 of AISC 360.
Conclusions
Buckling is a complex phenomenon that manifests in a myriad of ways in industrial process facilities. While the full breadth and depth of this topic cannot be covered in an article of this length, the authors hope that this overview provided a basic primer on the subject so the reader can recognize the key factors that contribute to buckling and analysis approaches available for various common structures. If you have any questions regarding this (or any other) topic, please contact us by submitting the form below:
