Finite Element Analysis of Pressure Vessels
As a fabricator responsible for the design of a pressure vessel there may come a time when performing a finite element analysis is necessary to meet the requirements of the ASME Boiler & Pressure Vessel Code. You may come to this decision on your own or it may have been specifically required by your customer. If you are unfamiliar with finite element analysis with respect to pressure vessels this article should help.
What is Finite Element Analysis?
A finite element analysis (FEA) is a type of analysis that uses a numerical technique known as the finite element method to determine the response of an object subjected to loading by representing it with an assembly of simple shapes. The representation of the object is referred to as a model, and the simple shapes making up the model are called elements. Adjacent elements are connected to each other at locations known as nodes and the assembly of elements is referred to as a mesh. The response of an individual element due to an applied load is dependent on the magnitude and type of load, material properties assigned to the element, and the element’s size and shape. While the response of each element may be different, adjacent elements sharing nodes will have the same response at these locations. The response at each node can be described with a mathematical equation. In a structural analysis conducted using FEA the equation will relate the load on the element at that particular node with the stiffness of the element and displacement at the node. Each of these nodes has the ability to translate (and possibly rotate) from its original position. This ability to move is known as a degree of freedom. Each node in a finite element model will have an equation for each degree of freedom. The number of nodes and number of degrees of freedom in an FEA model will determine how many equations the computer has to solve. It is not unusual for finite element models to have tens of thousands or hundreds of thousands of nodes. As a result, solving finite element analyses requires substantial computing power that is often beyond the capabilities of your typical office computer.
When is FEA Typically Used in the Design of Pressure Vessels?
When a pressure vessel has complex geometry or loading conditions such that traditional methods (i.e. hand calculations) are inadequate in accurately evaluating the vessel, finite element analysis can be used to ensure the design is acceptable.
Qualifications of the Analyst
Finite element analysis software is readily available and its use is becoming fairly common in engineering design. However, one should not underestimate the qualifications necessary to correctly evaluate pressure vessels using finite element analysis. The engineer using FEA to evaluate a pressure vessel should be experienced with finite element analysis in general, know how to correctly use the particular FEA software chosen, and be well-versed in the requirements of the ASME Code. Even a cursory look at the requirements for evaluating a pressure vessel using FEA in Part 5 of Section VIII, Division 2 of the Code reveals the methods and acceptance criteria are significantly different than those in Section VIII, Division 1. If an engineer within your company does not have the experience and expertise to independently evaluate pressure vessels using FEA, enlisting the services of a pressure vessel consultant experienced in FEA may be prudent.
CAD Model Simplifications
In creating a finite element model of a pressure vessel, the analyst will decide which geometric features to include or exclude in the model. While including all of the geometric features in the model is possible, doing so usually results in an unnecessarily large number of elements leading to longer computer run times. The simplifications to a model of a pressure vessel usually include the removal of small geometric features where the magnitude of stress will not be significant. An engineer experienced in performing finite element analyses of pressure vessels will know which features can be excluded without adversely affecting the analysis results.
Creation of the Finite Element Model
Once the model of the pressure vessel has been simplified, the element mesh will be created. Finite element models of pressure vessels are created using either solid elements or shell elements. There are a number of factors that will determine whether shell elements or solid elements should be used. These include the overall size of the vessel, whether thermal gradients exist through the thickness of the shell, and if the plates comprising the vessel are thick. Often the use of either shell or solid elements can be justified and it becomes a matter of which element type is more advantageous in performing the analysis. An engineer experienced in evaluating pressure vessels using FEA will be able to make these decisions. The element mesh for the model should be such that it will accurately predict the stresses in the pressure vessel. There will be a change in stress in areas of structural discontinuities such as at locations of nozzles. The size of elements at and near structural discontinuities will often be smaller than elsewhere in the model to accurately predict the change in stresses in these areas. If using solid elements, the number of elements through the thickness of the components is important in accurately predicting bending stresses at structural discontinuities.
Definition of Boundary Conditions
The extents of the finite element model and the constraints applied there are referred to as boundary conditions. With respect to pressure vessels, this would include how the vessel supports are constrained, and any constraints or loads at connections to piping systems. Care must be taken in applying boundary conditions. Over-constraining or under-constraining the model may produce erroneous results.
Applying Loads
There are numerous types of loading conditions that can be analyzed using FEA. These include: internal pressure, external pressure, deadweight, thermal loads, cyclic loads, impact and shock, seismic loads, wind loads, vibratory loads, and external nozzle loads. One consideration in evaluating pressure vessels using FEA is whether the loads are static or dynamic. If the loads vary with respect to time, a transient analysis may be required. Typically, the loads on pressure vessels can be assumed to be static. Noted exceptions are: thermal transients and thermal shock, and mechanical shock such as impact. Since pressure vessels are typically stout (rigid), structural dynamic loads can usually be modeled as static loads. An example of this is evaluating a pressure vessel for seismic loads using a static acceleration.
Analysis Method
Another aspect of evaluating pressure vessels using FEA is the type of analysis method used. Section VIII, Division 2 of the ASME Boiler & Pressure Vessel Code allows the use of three different analysis methods: elastic stress analysis method, elastic-plastic stress analysis method, and limit load analysis method. The structural integrity of a pressure vessel for various loading scenarios can be determined using one of the these methods. Each of these methods have their own requirements and acceptance criteria. The engineer performing the analysis should be knowledgeable in the method he chooses to analyze and evaluate the pressure vessel.
Evaluation of Results
After the analyses have been performed, the results are reviewed. The results of interest may be displacements, stresses, strains, temperatures, or mode shapes corresponding to natural frequencies in the case of evaluating a pressure vessel for vibration. While the acceptance criteria for the ASME Code is largely concerned with stresses and strains, other results such as displacements or mode shapes may be of interest in understanding how the pressure vessel responds to applied loads. It is not unusual for design changes to be made based on predicted displacements, temperature profiles, or mode shapes.
Part of evaluating the results of an analysis is determining if the results are accurate. There are several different methods typically used in validating a finite element model and analysis results. One is basically just using common sense. Are the results reasonable? Excessive displacements or stresses may indicate improper boundary conditions or modeling and warrant further investigation. Another method of validating a model and analysis is comparing the FEA results with hand calculations. An example of this is confirming the FEA calculated hoop stress away from discontinuities when internal pressure is the only applied load agrees with hand calculations. Another example is to compare the reactions at the boundary conditions to the applied loads. The aforementioned methods are typically used in verifying the model is properly constrained and the loads have been applied correctly. However, even if the model is properly constrained and loads applied correctly the predicted stresses may not be accurate in areas of structural discontinuities (e.g. at a nozzle). The magnitude of the predicted stresses are dependent on the element mesh. If the size of the elements are too large in areas of high stress, the magnitude of the stress may be under-predicted. A common method for determining if the element mesh is adequate is to perform a mesh sensitivity study. A mesh sensitivity study involves modifying the element mesh by reducing the size of the elements in areas of high stress and seeing how much the stress results change. One area of caution concerning this approach is the presence of locations where mathematical singularities exist. An example of this is a sharp corner at the junction of a nozzle and the shell when the weld is not modeled. The stress concentration factor associated with a sharp corner approaches infinity. Increasing the element mesh at a sharp corner will produce ever increasingly larger stresses, resulting in a stress magnitude that is fictitiously high. An engineer experienced in evaluating pressure vessels using FEA will know how to properly evaluate the stresses in these areas. After the results of the analysis have been determined to be sufficiently accurate they are compared to the Code acceptance criteria.
Documentation
Lastly, after the analyses and evaluation have been completed the work is documented. Documentation is necessary to demonstrate the design of the pressure vessel meets the ASME Code. The extent of the documentation can vary from a series of stress plots and summary tables, to a comprehensive report detailing every aspect of the analysis, results, and evaluation. The extent of the documentation will largely depend on contractual obligations between the fabricator and owner, and the fabricator’s in-house procedures for documenting engineering work.