Linear Multi-Body Analysis
File: File:PVE-4472
Last Updated: Aug 23, 2010
By: DRV
FEA may be used to analyze single as well as multiple body designs. For multiple body analysis the interactions and restraints between bodies must be defined. The solver can then provide the resulting displacement, stress and contact pressure plots. Utilizing multiple bodies is typical of connection or joint analysis and allows the user to ensure proper preload and observe that joint separation does not occur. A complete engineering report of a multi body analysis typical of what is provided by Pressure Vessel Engineering is available for download below.
Interaction between multiple bodies can be defined as bonded, no interaction, or no penetration. A bonded condition forces the bodies to act as a single component. For example a head bonded to a shell would simulate a welded condition and solve the analysis as if the head and shell were a single component. A no interaction condition does not account for the interaction between multiple bodies; it allows the bodies to displace individually without any imposed restraints by the adjacent components. This condition could result in bodies interfering or overlapping each other. A no penetration condition allows multiple bodies to contact each other, but not to penetrate. This condition is useful when analyzing connections such as flanges, tri-clamps or split rings. No penetration conditions also provide contact pressure plots. These plots are useful to ensure joint separation does not occur.
Restraints between multiple bodies such as bolts may also be simulated. Bolt connectors are defined in place of solid model bolts, and their material properties and preload defined. The solver creates beams to simulate bolting where bolt connectors have been defined, and transfers the applied preload to the connection accordingly. The software can then output the resulting forces acting on each connector which can then be used to calculate stresses.
Defining appropriate restraints and interactions between bodies is critical to obtain accurate FEA results. Applying incorrect interaction conditions between components will result in incorrect results. FEA results may be interpreted as acceptable and allow for unsafe designs.
Downloads (pdf format):
Sample 21 - Flange
File: PVE-3396
Last Updated: March 18, 2009
BV
 |
This sample report illustrates how FEA is used to validate flange design. This report format may be used to justify ASME code compliance, provide stress and displacement analysis, provide cycle life estimates, complete thermal analysis, and perform design validation and optimization studies. This format is fully CRN compliant and may be applied to many applications. This level of analysis can typically be completed within a week. |
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Sample 22 - Thermal Analysis
File: PVE-3429
Last Updated: May 21, 2009
BV
 |
This sample report illustrates how FEA is used to analyze thermal loadings. This report format may be used to justify ASME code compliance, provide stress and displacement analysis, provide cycle life estimates, complete thermal analysis, and perform design validation and optimization studies for multiple components within an assembly. This format is fully CRN compliant and may be applied to many applications. This level of analysis can typically be completed within a week. |
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Sample 23 - Heat Exchanger
File: PVE-3520
Last Updated: Sept. 29, 2009
DV
Commentary:
Why use FEA on Heat Exchangers?
ASME UHX rules cover the design of tubesheets, tubes and the shell next to the tubesheet. But the rules are limited to designs with uniform hole patterns that cover the complete tubesheet. What if the hole pattern is not uniform, or in the case of this sample, the holes are not a uniform size?
 A heat exchanger with non-uniform hole sizes |
Burst testing is an economical way to validate inexpensive products. However it may be unreasonable to use destructive testing to validate costly or large items such as a heat exchanger; in this case finite element analysis (FEA) is the most logical approach.
Using FEA to Replace ASME Code Rules
The ASME code rules must be used if they are applicable. In this case the standard code rules can not be applied so FEA is allowed. The UHX rules account for three stresses in the design of an exchanger:
- the tubesheet stress caused by pressure and thermal loading
- the shell stress next to the tubesheet caused by tubesheet rotation
- the tube compression or tension loading caused by tubesheet displacement and thermal expansion.
Per UHX rules, these stresses are analyzed for the following load cases in fixed tube exchangers:
- Tube side pressure only
- Shell side pressure only
- Tube + shell side pressure
- Thermal loads only
- Tube side pressure + thermal loads
- Shell side pressure + thermal loads
- Tube side pressure + shell side pressure + thermal loads
For a finite element analysis to replace the UHX rules for a fixed tubesheet exchanger the three stresses need to be studied for the seven load cases.
FEA Results
See sample 6 for an in depth discussion of the FEA method applied to a simple pressurized object. Here the same methods are used with a more complicated model.
The illustration below shows the mesh used for this sample. A solid mesh has been applied to the tubesheet and adjacent shell and a shell mesh applied to the tubes. The mesh is reduced in size at locations of interest such as the tubesheet, the tubesheet to tube junction, and the adjacent shell.
 Close up of the mesh used in the sample stud. A fine mesh is used for the tubesheet and the shell next to the tubesheet. The tubes and outer shell are made of shell elements. |
 Load case 2 - shell side pressure only. |
The sample FEA report walks through all seven load cases and checks all three stresses for each case. Each stress is compared to the ASME allowable stress to determine pass/fail for each load case.
The reported results can also be further used to provide design optimization and cycle life results.
 Deformation plot with the displacements magnified 100x. The rotation of the tubesheet with the adjacent shell bending it causes can be seen. |
 Deformation plot with the displacements magnified 100x. The rotation of the tubesheet with the adjacent shell bending it causes can be seen. |
Summary:
FEA can be used to address ASME code rules where calculations cannot be applied. It is an excellent, and in some cases the only option to validate a design. It can be cost effective, reduce lead time and expedite registration.
We have successfully used this FEA method to provide reports justifying heat exchanger designs reviewed by Authorized Inspectors and review engineers.
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ASME VIII-2 Stress Classifications and Limits
File: File:PVE-4495
Last Updated: Aug 31, 2010
By: DRV
For pressure equipment FEA is typically completed in accordance with ASME VIII-2 Part 5 Design by Analysis. The user creates a solid model, applies an appropriate mesh, material properties, loads and boundary conditions. The FEA then provides corresponding displacement and stress results. The difficult part however is analyzing these results to ensure a safe design or code compliance.
 A Stress Plot of a Valve under pressure |
The ASME code provides guidance and acceptance criteria for FEA results. It categorizes stresses into two categories based on the failure mode they are expected to induce and provides allowable stress limits for each. The first "primary stresses" are stresses located in areas which plastic collapse is expected to occur if loadings exceed design, such as a pipe under internal pressure. The second "secondary stresses" are caused by geometrical discontinuities and are in addition to primary stresses. These stresses are in proportion with primary stresses until the yield point. After reaching yield secondary stress regions are expected to be self-limiting, supported by adjacent material, and result in localized yielding opposed to plastic collapse. The hub of a flange of the valve below is a good example of this. |
Primary stresses that exceed the yield limit by some margin will result in failure and as such are limited to 1X the allowable ASME stress limit.
Pressure Vessel Engineering has over 8 years of experience analyzing components to the rules of ASME VIII-2 using FEA. A complete engineering report is available below representing a typical analysis and report done in compliance with ASME VIII-2.
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Sample 25 - Cycle Life Analysis
File: PVE-3380
Last Updated: October 20, 2009
DV
Commentary:
When is a Cycle Life Analysis required?
VIII-1 does not provide a hard and fast guideline for when fatigue analysis is required. The older VIII-2 code specified that it would not be required if the number of cycles is less than 1000 (VIII-2 2001 AD-160.2 Condition A). The new VIII-2 states that fatigue life analysis is required above 100,000 cycles and often required at lower cycles (VIII-2 2008 5.5.2.1(c)).
From our experience, we know that a vessel designed to VIII-1 has maximum stresses between 2x and 4x the allowable design stress Sa. These maximum stresses are found in "local" areas such as flange necks, knuckles or heads, nozzle to shell junctions or other changes in shape. Some of this stress is caused by flange seating loads or support loads that do not fluctuate so do not affect the cycle life. Other stresses change with each applied pressure cycle and need to be considered.
The difference in stress between unpressurized and pressurized conditions depends on the design features used in the vessel - what type of head, how the nozzles are attached, weld efficiency and more. For a SA-516 70 vessel at ambient temperature the maximum allowable stress Sa = 20,000 psi. The expected maximum stress is between 40,000 and 80,000 psi. This corresponds to an expected cycle life of 1,100 to 8,500 cycles.
Many pressure vessels are not expected to experience 1,100 cycles. Many other factors will affect the actual fatigue life: corner joint details; operating at less than the full design pressure; temperature difference effects; design of nozzle details; and type and quality of welding or other factors that change the surface finish and more.
Cycle Life Sample
For this sample filter vessel the operating pressure range cycles between a full vacuum and a pressure of 230 psi. The design has flanged connections. The flange stresses are part seating and part operating. To get an accurate fatigue analysis two runs are required - one at the pressure case, and the other at the vacuum case. When the two cases are subtracted (according to the rules of VIII-2 5.5.3.2) the operating stress range is obtained without flange seating effects. This stress difference is not the same as simply taking the peak stress from one the higher pressure case.
Step by Step
See sample #6 for an introduction to FEA on a simple shape. This cycle life study uses the same modeling and analysis methods on a more complex shape.
 The solid model used for the fatigue analysis |
 Mesh with refined areas where higher stresses are located. This model has over 1,000,000 nodes. |
|
 230 psi loading deformation results (deformation x100) |
 Full vacuum loading deformation results (deformation x200) Note that the vessel deformation shape is different than the 230 psi case especially note the deformed shapes of the heads) |
|
 Stress plot for the 230 psi case. The peak stress indicated is not the final maximum alternating stress used. |
Once the vacuum case is subtracted from the 230 psi case, the maximum stress range is found to be 42,000 psi. This is the difference in stress between the full vacuum and the 230 psi operating case without any flange seating effects. (It is also less than the 62,000 psi peak shown in the picture above which is caused by a combination of flange seating and operating loads). The alternating stress is computed to be 21,000 psi for a predicted cycle life of 49,000 cycles.
Fatigue life curve for the vessel - expected life is 49,000 cycles.
Conclusion
Cycle life analysis is often required for products subject to cyclic processes and as a requirement for provincial CRN registration for many classes of pressurized components. We have successfully used this FEA method to provide reports calculating pressure component fatigue life's accepted by many Authorized Inspectors and review engineers.
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Thermal Analysis
File: File:PVE-4437
Last Updated: Aug 23 2010
By: BTV
Transient and Steady State Analyses
Thermal analyses are used to study thermal loadings and their resulting temperatures, heat transfer rates, displacements and stresses. These analyses are broken into two main types, steady state and transient. Steady state analysis will determine the energy balanced state at an infinite period in time without any detail on what happens while progressing to this point. Transient thermal analysis is able to analyze the heat flow through a body on a step by step basis allowing temperature effects to be observed over time.
Steady State Analysis
Steady state analysis is used to observe the effects of thermal loadings once the object in question has reached a constant, or steady state. This is useful to determine sustained temperatures, heat transfer rates, displacements and stresses. Steady state analysis is also useful to determine thermal loads and material properties to obtain a final desired result. As a steady state analysis only provides a final continuous result it only requires a single computation making it a very efficient solver.
Transient Analysis
 Transient Temperature Graph |
A transient analysis is used to observe the effects of thermal loadings over time. It allows the user view the changing temperature gradient through a component from initial though to a steady state condition. Transient thermal impacts are important to analyze as thermal loadings may result in peak stresses prior to reaching a steady state. It is also useful to answer questions such as how long will a component take to reach a desired temperature. As a transient analysis provides solutions for a defined number of time steps many computations are required resulting in a much more complex analysis. |
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Non-Linear (Elastic-Plastic) Analysis
File: File:PVE-4433
Last Updated: Aug. 12, 2010
By: DRV
Non-linear analysis is an elastic-plastic analysis method to determine if plastic collapse will occur. It is a more accurate assessment than a linear elastic analysis as it accounts for yielding and the redistribution of stress that occurs in a component as a result of plastic deformation. This example provides a comparison of a linear and non-linear analysis. Advantages and disadvantages of each are discussed and recommendations for their use provided. A complete engineering report is available below representing a typical non-linear analysis and report provided by Pressure Vessel Engineering.
 Elastic-Plastic Material Curve |
This graph compares the modulus of elasticity used for a linear and non-linear analysis of the same material. A linear analysis assumes that the material is perfectly elastic. As a result the analysis does not account for the effects of yielding and reports much higher stresses past the yield limit opposed to a non-linear study. A non-linear (elastic-plastic) analysis also assumes the material to be perfectly elastic up to the yield point, providing results identical to a linear solver. However past the yield point the redistribution of stress as a result of plastic deformation as well as the effects of strain hardening are accounted for. This results in higher deformation and lower reported stresses than a linear analysis.
 Linear Material Model - A maximum displacement of 0.018" is observed (displacements are magnified) |
 Elastic Plastic Model - A much higher, more realistic displacement of 0.044" is observed (displacements are magnified) |
 A maximum stress of 41,698 psi is observed |
 A lower, more realistic maximum stress of 36,011 psi is observed. |
So why not always use a non-linear solver if it is more accurate? A linear analysis reports the instantaneous effects of the applied boundary conditions and loadings on the component. As a result only a single computation is required to obtain the results. To analyze the effects of plastic deformation a non-linear solver must run numerous iterations. Past the yield point the component starts to plastically deform and permanently change shape. The non-linear solver accounts for this and runs as many iterations as required to accurately update the displacement of the component and the effects this has on the applied loads. For this reason a non-linear study requires tens, hundreds, even thousands of times the computations a linear study would require.
Results
As a linear analysis assumes the material to be perfectly elastic it will "overestimate" stresses in excess of the yield point. To overcome this issue allowable stresses are established for different regions of the component. For example ASME VIII-2 will only allow 1X the listed material stress for general areas where plastic collapse would be expected (center of a flat head), but up to 3X (well above the yield and approaching the tensile strength) for local regions which are expected to plastically deform and "self-limit" themselves.
Recommendations
For most analysis a linear solver is adequate. It reports identical displacements and stresses to that of a non-linear solver up to the yield limit and is much more efficient in doing so. If stresses are in excess of the yield point allowables can be established to provide acceptance criteria.
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Sample 28 - Addition of a Nozzle to an Existing Vessel
File: Samples/Sample 1
Last Updated: June 16, 2010
Laurence Brundrett
A Simple Calculation And Drawing Set
This is one of the simplest pressure vessel drawing and calculation sets possible - the addition of a new nozzle to an existing vessel - a tower in this case. The calculation set re-calculates the tower shell at the location of the nozzle and provides the nozzle reinforcing calculations. Depending on the condition of the vessel and the corrosion allowance, field thickness testing might be required to provide the wall thickness used in the calculations.
The Catch
The catch with alterations to existing National Board registered vessels is that the National Board code must be met as well as the ASME code. For this sample, the tower is assumed to be built to the ASME VIII-I pressure vessel code, 1992 edition, 1994 addenda and NB registered. The National Board code states that "all required design information, applicable drawings, design calculations, specifications and instructions are to be prepared, obtained, controlled and interpreted to provide the basis for an alteration in accordance with the original code of construction" (AISI NB-23 2001 RC-3020 emphasis added).
This alteration is designed to the 2001 code rules, but the 1994 allowable material stress levels are used. The allowable stresses prior to 1999 were lower than the current levels. This calculation set uses these lower stress levels. The shell was recalculated to determine the minimum thicknesses required for the nozzle calculations. The nozzle was calculated using the standard area replacement calculations. No nozzle loads are calculated, so hangers are required to support the pipe from the nozzle.
The Software We Use at PVEng
Three complete sets of calculations are shown here: calculations done in Advanced Pressure Vessel; PVElite; and our in house spreadsheets. We use these different methods to provide flexibility when solving difficult problems. For a real job we would only provide one set of calculations, preferably Advanced Pressure Vessel our favorite. Click on the links to the left to view the different calculation methods and to see the drawing.
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Buckling Analysis
File: File:PVE-4474
Last Updated: Aug 27, 2010
By: DRV
Buckling is defined as the sudden failure of a structural member under compressive load, or failure due to elastic instability. It is often associated with items under external pressure or compressive load such as vessels for vacuum processes and supports. In simple cases classical calculations can be completed to determine the critical load or buckling point. For more complex geometry and loadings FEA is utilized to complete buckling analysis. A complete engineering report of a buckling analysis typical of what is provided by Pressure Vessel Engineering is available for download below.
 Vessel with a supported flat head |
For more complex geometry and load cases a linear-elastic analysis is typically completed to validate the design. The displacements and stresses from the analysis are reviewed and acceptability determined. The results of a linear-elastic analysis however are based on the material being perfectly elastic and as a result do not account for failure due to elastic instability or buckling. As a linear-elastic analysis will not report buckling it is the designer’s responsibility to determine if it is a potential failure mechanism. For components subject to vacuum or under direct compressive load such as legs it is fairly obvious that buckling should be investigated. For other components it may be less obvious and neglected. For instance supports on a flat head subjected to internal or external pressure may buckle. A side load is imposed on the supports as the flat head balloons out under pressure. It is quite possible the flat head will meet code calculations and be shown as acceptable with a linear-elastic analysis but in reality it may fail prematurely due to buckling of the supports. |
 A view showing buckling of the flat head supports |
To ensure buckling does not occur Pressure Vessel Engineering completes a buckling analysis. All material properties, loads, and boundary conditions from a linear-elastic analysis are imported into a buckling solver. The buckling solver then provides several possible buckling failure modes and the respective safety factor of each. Results less than 1 indicate the design will fail. For ASME code compliance a safety factor of 3 is typically required. |
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Frequency / Vibration Analysis
File: File:PVE-4492
Last Updated: Sept 2, 2010
By: DRV
FEA is used for frequency and vibration analysis to determine the natural frequency of objects which cannot be obtained from classical calculations. The natural frequency can then be compared to the system resonance to ensure large amplitude oscillations will not occur. The natural frequency may also be used with building codes to determine a base shear force which can then be input to a stress analysis to validate designs subject to seismic or other oscillating conditions. A complete engineering report is available below representing a typical seismic analysis and report completed by Pressure Vessel Engineering.
 Seismic Analysis of Pressure Equipment |
To find the natural frequency two inputs are required, mass and stiffness. The object to be analyzed in modeled and its density (mass) and modulus of elasticity (stiffness) defined. The FEA solver then determines several displacement modes and their corresponding natural frequencies. Pressure Vessel Engineering then uses the natural frequency provided by the FEA to determine a base shear force in accordance with building codes such as NBC, IBC, and UBC. The base shear force is then converted to an acceleration (F = ma, both force and mass are known) and input into a stress analysis to investigate seismic loadings on the object. These stress results then either validate the design or provide insight as to how it should be revised. |