Local Stress Analysis by Chris Hinnant
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Local Stress Analysis Chris Hinnant Paulin Research Group Houston TX Houston,
Course Outline
Establishing the need for stress analysis Brief review of ASME rules necessitating stress analysis Tools for stress analysis WRC vs vs. FEA Stress – comparison of solutions solutions. Fundamentals of ASME VIII-2 rules for stress analysis Finite element analysis topics Fi it element Finite l t analysis l i example l
Sudden impact load during transport resulted in a severed nozzle.
Why Local Stress Analysis?
The ASME VIII rules mainly address pressure containment. ASME VIII-1 and VIII-2 require that the user must consider all loadings acting on the equipment (UG-22, etc). However,, explicit p design g equations q for external loadings g on nozzles,, clips, and lugs, and various supports are not provided.
An ASME Technology, LLC. research project is underway to provide explicit design rules for the stress analysis of nozzles subject to external loads.
Certain constructions such as large diameter openings are not comprehensively addressed. To satisfy the ASME code requirements, we need a way to verify that components not explicitly addressed by the ASME rules are adequate for the intended service.
Typical Applications of Stress Analysis
Piping loads on nozzles. Combined loadings (pressure, temperature, etc) Fatigue design Loads on clips, clips lugs lugs, and supports supports. Thick walled expansion joints. Pressure design of unique parts:
Large di L diameter t openings i Rectangular nozzles and ducts Closely spaced nozzles
Analysis A l i off startup/shutdown / h d Thermal stresses and transients
Typical Applications of Stress Analysis
Jacketed equipment Skirt design – thermal, transients, wind/seismic loads, etc. Transport loads, hydro test conditions
Typical Applications of Stress Analysis
Although not covered in this course, there are numerous applications of FEA for piping systems.
Improved Stress Intensification Factors Accurate flexibilities Pipe supports (saddles, shoes, bends with staunchions) Piping with D/T > 100 Lateral and hillside branches Wye fittings Flange design
Typical Applications of Stress Analysis
Multiple p adjacent j nozzles,, such as this flare tip. p
Validation of New Designs Photo courtesy of Dynamic Products – Houston, TX
Stress analysis can be used to replace or compliment traditional validation methods such as proof testing, strain gauges, etc.
ASME Rules Related to Stress Analysis y
In this section, we’ll briefly cover some of the pertinent ASME rules related to stress analysis.
ASME VIII-1
ASME VIII-1 does not directly address the use of stress analysis. Users are directed to U-2(g) in most cases. When stress analysis is used, the designer must still satisfy the minimum requirements q of VIII-1 ((for instance thickness per p UG-27). ) Stress analysis can not be used to justify thinner parts than that permitted by the mandatory rules of VIII-1. When using stress analysis, analysis the basic allowable stresses from VIII-1 VIII 1 should be used (i.e. those from Section II-D Tables 1A and 2A). Do not use higher allowable stresses for VIII-2 from Section II-D Tables 5A and 5B.
Excerpts from ASME VIII-1
From the Foreword of ASME VIII-1
The Code is not a handbook – you must apply engineering judgment. The designer is responsible for understanding the limitations of the tools theyy use – don’t fall prey p y to the “black box” mentality. y
Excerpts from ASME VIII-1
U-2(g) is an important part of VIII-1 and stress analysis work.
Requires that the designer consider cases where complete rules are not given and ensure the design is “as safe as” if the rules had been provided by VIII-1.
Excerpts from ASME VIII-1
UG-22 defines the loadings to be considered
Designers often overlook this, especially in cases where their software doesn’t not address the topic. A good example – seismic loads on horizontal vessels. There is no simple way to address this condition.
ASME VIII-2
Completely rewritten, with a greater focus on technology (especially stress analysis) Design margin reduced to 2.4 on UTS (increased allowable stress) Permits a vessel to be designed g entirely y using g Part 5 ((e.g g FEA)) and need not satisfy minimum thickness equations of Part 4 (see code rules for limitations).
This can offer substantial savings g when the total equipment q p costs are well in excess of the analytical expenses.
VIII-2 Part 5 has replaced the previous VIII-2 Appendix 4 and 5. In manyy aspects, p , Part 5 is the same. However,, the rules have been updated to better address technology used today (FEA). Part 5 provides explicit rules for the stress analysis:
Stress categories and their associated limits are defined. Requirements/recommendations for modeling are given.
Excerpts from ASME VIII- 2
Design by Analysis (Part 5) may be used in lieu of the design-byrules of Part 4.
If the temperature is such that the material properties are governed by time dependent behavior (creep), the use of Part 5 is permitted, but only with successful prior experience experience.
ASME VIII-2, Part 5
VIII-2, Part 5 is rearranged to address potential failure mechanisms:
5.2 – Plastic Collapse (ductile rupture, etc) 5.3 – Local Failure (local fracture) 5.4 – Buckling Failure 5.5 – Cyclic Failure (fatigue and ratcheting)
New load cases combinations are similar to those given in ASCE. Expanded fatigue design methods. Guidance on stress linearization.
M More di discussion i on VIII VIII-2 2P Partt 5 tto come…
ASME VIII-2, Part 5 $avings
As previously mentioned, VIII-2 allows you design vessels entirely by FEA. This may offer substantial savings. As a simple example of the potential savings, let us consider a basic cylindrical shell constructed of SA-516 operating at 70 deg F. The design pressure per Part 4.3.3 is 1,476 psig
ASME VIII-2, Part 5 Savings
Now, calculate the permitted pressure using Part 5 rules.
1,755 psig using the elastic stresses are easily calculated using hand methods. 1,705 psig estimate using limit analysis equations for cylinders. 1,699 psig estimated using burst equation for cylinders.
ASME VIII-2, Part 5 Savings
The previous slide estimated the plastic load by a semi-empirical method. The elastic-plastic solution indicates the plastic load is reached at 4,320 psig, allowing a design pressure of 1,800 psig
ASME VIII-2, Part 5 Savings
Even for simple parts, like a cylindrical shell, Part 5 offers significant savings – especially i ll if elastic-plastic l ti l ti analysis l i iis used. d For elastic results and lower bound limit analysis, savings is due to the difference in failure theory (Eq. 4.3.3.1 uses Tresca whereas Part 5 relies on Von Mises). Savings g with the elastic p plastic analysis y is a combination of the Von Mises theory, y, but primarily the strain hardening characteristics of the material. Savings may be realized by reducing material thickness. Reduced thickness has indirect benefits such as the opportunity to eliminate PWHT, reduced fabrication labor decreased inspection efforts labor, efforts, etc etc.
Method
Theory
Analysis
Design Pressure
Savings
Part 4
Tresca
By Eq. 4.3.3.1
1,476 psig
Part 5
Von Mises
Elastic
1,755 psig
18.9%
Part 5
Von Mises
Lower Bound Limit
, psig p g 1,705
15.5%
Part 5
Von Mises
Elasitc-Plastic
1,800 psig
22%
Tools for Stress Analysis
Next, we’ll highlight some of the more common methods used for stress analysis today.
Tools for Stress Analysis
There are a varietyy of tools available:
Welding Research Council Bulletins z
WRC-107, WRC-297, WRC-497, etc.
ASME Section III III, Appendix Y Y. PD-5500, EN-13445, and other foreign codes. API-650, Appendix P (Low wall tank nozzles) Fi it Element Finite El tA Analysis l i
WRC-107
A semi-empirical method for estimating stresses in the spherical and cylindrical shells with loaded attachments. First published in 1965, revised in 1979. Experimental work included:
Solid cylinders in spherical shells. Nozzles in spherical shells. Square and rectangular solids on unperforated cylindrical shells. Did not include nozzles on cylinders cylinders.
Design charts for various geometric ratios are provided. Loads are defined in local coordinate system at the nozzle-shell intersection point. Because the geometric range is somewhat limited, programmers must interpolate or extrapolate beyond the intended bounds. Despite p some shortcomings, g widely y used for analysis y of external loads on nozzles, clips, and lugs.
WRC-107
An example of a WRC-107 calculation sheet and design charts:
WRC-107 Limitations
Some limitations include:
Pad reinforced nozzles are not directly addressed, can only be approximated by an enlarged attachment diameter. Laterals and hillside nozzles are not included in the scope. Users should be cautious and avoid any cases where the geometric limits of the method are exceeded. Only considers stresses in the shell, nozzle is ignored. Reports stresses at a finite number of locations (in-plane and out-ofplane positions). Maximum off-axis stresses may be missed. Combined stress due to pressure and external loads is not effectively addressed. addressed
WRC 107 Limitations
WRC-107, paragraph 3.3.5 tells us that in the case of arbitrary combined loading there is no assurance the maximum stress will occur at one of the reported positions.
WRC-107 Limitations
Applicable geometric limits for spherical shells are:
WRC-107 Limitations
Applicable geometric limits for cylindrical shells are:
WRC 297
Estimates the stresses due to external loads acting on nozzles in cylindrical shells. Published in 1984 Based on thin shell theory y and work by y Steele. Improved stress results in comparison to WRC 107. Major improvement is that stresses in the nozzle neck, adjacent to the intersection are evaluated evaluated. Expanded range of applicability over WRC 107.
Large D/T values Improved solutions for small d/D values values.
WRC 297 Limitations
WRC 297 shares many of the same limitations of WRC 107:
Stresses are only reported at the in-plane and out-of-plane locations. No reliable method of combining pressure stresses and those calculated with WRC 297. Not intended for pad reinforced nozzles, laterals, or hillside nozzles. Restricted to nozzles in cylindrical shells. May give excessively conservative solutions when the nozzle neck is thinner than the shell plate.
WRC-297 Limitations
Applicable geometric limits for nozzles on cylindrical shells are:
Other Non-WRC 107/297 Cases
Other situations where the WRC 107/297 methods should not be used: z z z z z
Attachments or nozzles on cones Laterals, hillsides, or nozzles with internal projections Nozzles near the edge or knuckle regions of heads Attachments located on flat heads Nozzles near rigid components (tubesheets, stiffeners, etc)
Summary of WRC 107 & 297
WRC 107/297 are OK as a screening tool for external loads only.
For external loads only (no pressure), if the stress exceeds 50% of the allowable using these methods, use a better tool.
If the answer matters, use FEA. But WRC 107/297 are successful, right?
The upcoming comparison with FEA will provide some insight. Piping p g analysis y often over estimates the stiffness – real loads are often smaller than predicted. z
FEA is being used more often today to determine the intersection stiffness, so loads are becoming more accurate, requiring a better nozzle analysis.
Safety S f t margin i is i compromised. i d Failures occur, but source isn’t identified.
WRC 497
WRC 497 gives a calculation procedure for determining stresses at nozzle openings in cylinders with internal pressure and external nozzle loads. Intended for larger openings (d/D>0.33) Based on finite element analysis results. Although it represents a significant improvement in nozzle stress calculations, it has not received wide spread use.
WRC 497
Results are given in terms of shell membrane and bending stresses. Can easily combine pressure stress and external load stress. Unlike WRC 107/297, the maximum stress anywhere is reported. Does not include pad reinforced nozzles nozzles, laterals laterals, or hillsides hillsides. No charts to interpolate/extrapolate. Simple equations promote ease of use:
WRC 107/297 vs. FEA
Next, we will examine some comparisons between WRC 107/297 and FEA
FEA vs WRC 107 & 297
Noting the limitations of WRC 107 & 297 so far, it is important to examine the difference between FEA and WRC results. Ideally, FEA represents the “real” answer FEA does not suffer from g geometric limitations – any y geometry g y can be constructed and analyzed. Simple FEA tools exist today that are just as easy to use as a WRC calculator. Complexity p y and effort are no longer g jjustifications for not using FEA. In the following slides slides, we will examine the difference for axial axial, inin plane, and out-of-plane loadings on nozzles in cylindrical shells.
Nomenclature: Pressure Vessel vs vs. Piping
Occasionally, different nomenclature is used in piping and vessel design for the same parts or directions:
In-plane : longitudinal direction Out-of-plane : circumferential direction Header : cylindrical shell Branch : nozzle
FEA vs. WRC Format of Comparisons
FEA and WRC 107/297 are compared on the basis of the calculated Stress Intensification Factor (SIF). The SIF is simply the ratio of stress to cause failure in a component to the peak to cause failure in a girth butt weld:
For vessel engineers
FEA vs. WRC Comparison Axial Loads
For d/D < 0.50, WRC 107 is non-conservative.
For the entire range of d/D, WRC 297 is more conservative.
Stresses are under predicted by up to 200%. Stresses are over predicted by 200% to 1500%.
These differences are expected for a wide range of D/T and d/t.
FEA vs. WRC Comparison Axial Loads
FEA vs. WRC Comparison In-Plane Bending Moments
For d/D < 0.70, WRC-107 is non-conservative.
For all d/D values, WRC-297 over predicts the stress.
Stress is under predicted by up to 250% for many cases. For most all cases, the stress is over predicted by 200% to 300%.
These differences are expected for a wide range of D/T and d/t.
FEA vs. WRC Comparison In-Plane Bending Moments
FEA vs. WRC Comparison Out-of-Plane Bending Moments
For d/D < 0.70, WRC-107 is non-conservative.
For all d/D values, WRC-297 over predicts the stress.
Stress is under predicted by up to 200% in some cases. For most all cases, the stress is over predicted by 200% to 500%.
These differences are expected for a wide range of D/T and d/t.
FEA vs. WRC Comparison Out-of-Plane Bending Moments
FEA vs. WRC Comparison Summary
WRC 107 is non-conservative in many cases examined here, especially for d/D < 0.70. WRC 297 over predicts the stress in all cases examined here. At large d/D ratios, the difference may be significant. Not all geometries examined here are necessarily within the bounds of the WRC methods. However, the limitations are routinely over looked because software offers extrapolated parameters. These comparisons highlight why WRC 107 & 297 should be replaced with FEA calculations on nozzles and branch connections.
Fundamentals of Stress Analysis with ASME VIII-2
Next, we’ll cover some of the background of ASME VIII-2, applicable to stress analysis.
ASME VIII-2, Part 5
Rules for finite element analysis are given in ASME VIII-2, Part 5. ASME VIII-2, Part 5 seeks to prevent the following failure modes:
5.2 – Plastic Collapse (ductile rupture, etc) 5.3 – Local Failure ((local fracture)) 5.4 – Buckling Failure 5.5 – Cyclic Failure (fatigue and ratcheting)
For linear elastic stress analysis, the stresses are categorized based on several requirements: 1. 2.
Character (sustained or self-relieving) Location and extent
ASME Stress Categories
When performing stress analysis per ASME VIII-2, Part 5, we must segregate stresses into one of three categories: 1. Primary Stress 2. Secondaryy Stress 3. Peak Stress The “Hopper Diagram” explains the location, source, and limits for these stress categories categories.
Hopper Diagram
Hopper Diagram
Hopper Diagram
Primary General Membrane Pm - Explained
Pm is a membrane stress acting across the entire cross section (global, not local). Examples include pressure stress or bending stress across the entire vessel cross section. Pm can be calculated by hand methods. Often not evaluated by FEA. Routinely satisfied by ASME designby-rule equations such as: PD/2T F/A M/Z Guards against g g gross collapse. p Pm is caused only by sustained loads. Pm is not self limiting.
Once yield is reached reached, only strain hardening of the material will prevent collapse.
Original intent was to limit Pm to 2/3 of yield. Today, the limit is represented by the basic allowable stress “S” S.
Primary General Membrane Pm - Examples
Stress due to pressure (PD/2T, PD/4T) Bending moment across the section (M/Z) Axial load across the section (F/A)
Primary General Membrane Pm - Failures
An example of the failure mode addressed by primary general membrane stress is shown here in the rupture of the stainless pipe. Before the test, the stainless pipe and aluminum pipe were the same diameter. The swelling is a result of stainless steel’s strong strain hardening characteristic characteristic. Stainless Steel Pipe
Aluminum Pipe
Primary Local Membrane PL - Explained
Average stress through the local wall thickness (not across entire cross section) Only caused by sustained/primary type loads. Pm is naturallyy included in PL
Limited in the extent over which it may act:
PL is sum of general and local membrane Must exceed 1.1S 1 1S and be contained within sqrt(R sqrt(R*T) T)
Original intent was to limit local membrane stresses to the yield strength “Sy”. Evolved into a limit of 1.5*Sm.
An open item in the ASME committees is evaluating the option to allow Sy in lieu of 1.5*Sm for some cases.
Primary Local Membrane PL - Examples
Average stress through the thickness due to pressure at a nozzleshell junction is a local membrane stress.
Primary Local Membrane PL - Failures
High local membrane stresses near nozzle openings combine with secondary bending to produce ruptures.
Primary Local M+B PL+Pb - Explained
The sum of membrane and bending must not exceed d th the lilimitit strength t th off th the section. ti This diagram relates the combined effects of general primary membrane and primary bending stress on a rectangular section. The diagram assumes an elastic-perfectlyplastic material model. Failure is expected when F/A reaches the yield strength strength. Failure is expected when M/Z causes a plastic hinge (1.5*Sy in rectangular section). The Code intends that a margin of 2/3 against i t gross collapse ll iis maintained. i t i d But, the margin may be less than desired for some combinations of membrane and bending stress.
Primary Local M+B PL+Pb - Examples
Bending stress at the center of a flat head is a good example of primary local bending stress. Be careful with bending stress at the edges of fixed heads. The stress may be classified as primary or secondary, depending on how the flat plate is designed. designed
Secondary Stress PL+Pb+Q - Explained
Secondary stresses are the linearly varying component through the thickness. Secondary stresses are due to sustained and operating loads. Self limiting g stress. Strains exceeding g yyield do not cause collapse. p Evaluated over the full range of stresses. Secondary stress limits serve three purposes: Prevent ratcheting and ensure shakedown to elastic action action. 2. Validate the elastic assumption of the ASME fatigue curves. 3. Provide nominal protection against excessive local distortion. Original intent is to limit to twice yield yield. Evolved into PL+Pb+Q < 3*Sm 3 Sm 1 1.
Secondary Stress PL+Pb+Q - Examples
The linearly varying stress through the thickness at a nozzle opening is secondary stresses when caused by internal pressure.
Peak Stress PL+Pb+Q+F - Explained
The peak stress is the increment added to the primary or secondary stress due to a concentration or notch. Maximum stress anywhere across the section. May occur at a notch or in plain base metal (such as the surface of a smooth th kknuckle) kl ) Only objectionable in the sense that repeated applications may cause fatigue failure. Peak stress limits are satisfied by fatigue analysis, analysis or fatigue exemption exemption. In shell solutions and linearized stresses, the peak alternating stress is related to the secondary stress range by the following relationship:
Peak Stress PL+Pb+Q+F - Examples
An example of peak stress is the concentration at a weld toe due to internal pressure.
Peak Stress PL+Pb+Q+F - Failures
Examples of fatigue failure at welds:
Ratcheting and Shakedown
Ratcheting
Progressive incremental deformation or strain. Requires the simultaneous presence of a constant membrane stress with a cyclic strain controlled bending stress, otherwise ratcheting will not occur. General consensus is that “ratcheting” implies changes in gross dimensions, not at a finite point. For example, progressive increase of the vessel diameter. diameter
Shakedown
Phenomenon that occurs when a structure experiences only elastic or elastic-plastic action after the first few cycles. cycles Progressive incremental distortion does not occur.
Ratcheting and Shakedown
Ratcheting & shakedown requirements may be satisfied by:
Elastic stress analysis z
Secondary stress range must be less than three times the average allowable stress, or for some materials twice average yield strength for the range.
El ti l ti analysis Elastic-plastic l i 1. 2. 3 3.
There is no plasticity in the component. The section of interest is elastic at the core. There are no permanent changes in the overall dimensions of the component.
Secondary Stress Why a limit of twice yield?
The elastic limit of twice yield for secondary stresses has a unique and important meaning. If the linear elastic stresses are limited to the range of twice yield, then ratcheting should not occur.
Although, some recent studies have shown this may not be true in all cases.
In the following slides, we’ll explore various responses to combinations of steady membrane and cyclic bending stress.
Ratcheting and Shakedown Responses to Cyclic Loading
In this case, the stress never exceeds the yield strength and the cyclic response is wholly elastic. Loading an unloading will occur along the line between Point 0 and Point A.
Ratcheting and Shakedown Responses to Cyclic Loading
In this case, the stress exceeds the yield strength but does not exceed twice the yield stress. As shown below, this is important since the cyclic response will be purely elastic upon subsequent loading (between C and B).
Ratcheting and Shakedown Responses to Cyclic Loading
If the elastic stress exceeds twice yield, then several responses are possible. If the bending stress is high, but the membrane stress is small, then shakedown to elastic action with alternating plasticity may occur.
Ratcheting and Shakedown Responses to Cyclic Loading
If the elastic stress exceeds twice yield, with high membrane and high bending stress, then ratcheting with two sided yielding may occur. Notice how the hysteresis loop progresses along the strain axis, a classic case of ratcheting.
Ratcheting and Shakedown Responses to Cyclic Loading
If the elastic stress exceeds twice yield, with very high membrane stress and cyclic bending stress, then ratcheting with single sided yielding may occur.
Ratcheting and Shakedown Bree Diagrams
Bree performed the original work to describe ratcheting phenomena. Developed diagrams relating the ratio of membrane and bending stress to cyclic response (shakedown or ratcheting).
Finite Element Analysis y Topics p
FEA - Methods of Analysis
Three common analysis techniques in ASME VIII-2:
Also have a choice of geometric behavior:
Linear elastic analysis Limit load analysis Elastic-plastic analysis Small displacement theory Large displacement theory
The choice of which method to use is ultimately a decision based on the needs at hand. Some considerations:
Complexity of the component being evaluated evaluated. Type of loadings Time available Need for optimization Level of experience
FEA - Methods of Analysis
Although several analytical methods exist for the various failure mechanisms, only one need be satisfied to qualify the component. This leads to some interesting questions
What if one method p passes, while one fails? May analytical methods be mixed (for instance elastic-plastic design for primary loads and linear elastic design for fatigue).
Linear Elastic Analysis
Linear elastic analysis assumes a linear relationship between stress and strain, for any magnitude of stress or strain. Easiest of the three methods for common PVP components. Is typically yp y more conservative than non-linear methods. Requires that the stresses are classified into specific categories.
Categorization may be difficult for complex or arbitrary geometries.
Non-Linear Analysis Options
Two inelastic options are available in ASME VIII-2
Lower bound limit analysis y is for p primary y ((sustained)) loads only. y
Lower bound limit analysis Elastic-plastic analysis A good alternative to nozzle reinforcement design and can offer substantial savings with little analysis effort. Individual components may be sized while the remainder of the vessel is designed by common rules. l
Elastic –plastic analysis can be used for any load combination.
Can be used to highly optimize a design. Significant reductions in minimum i i wallll thi thickness k can b be achieved. hi d 25% savings i iis routine. ti
Lower Bound Limit Analysis
Represents an idealized lower bound estimate of the actual load to cause plastic collapse in the structure. Ensures that an unrestrained plastic deformation does not occur (i.e. a plastic collapse state is not reached). Si l example Simple l off a lower l bound b d limit li it analysis l i iis a b bar with ith an axial i l ttensile il load applied. The lower bound collapse load is the load at which F/A = Sy.
Lower Bound Limit Analysis
ASME requires a margin of 2/3 against the lower bound limit load.
For design purposes purposes, the yield strength for the elastic-perfectly elastic perfectly plastic material model is approximated as 1.5*S.
The permitted design load is achieved using the specified load case combinations in VIII-2 (essentially multiply the loads by 1.5 and substitute 1.5Sm for the yield stress in the FEA model).
Using 1.5S ensures that the limit of 2/3 on yield is achieved, but also considers the safety factor of 2.4 on UTS to ensure that high yield-to-tensile ratio materials are safely f l employed l d iin th the d designs. i
Additional non-limit load analyses are required to satisfy secondary loads.
Lower Bound Limit Analysis
Requirements for lower bound limit analysis
An elastic-perfectly plastic material model must be used. Analysis may not include geometric nonlinearity (small displacement theory must be used) Only applied to primary loads, such as pressure and weight. Thermal loads are not valid and should not be analyzed since they are strain limited and the lower bound limit analysis is invalid. Should not be used in cases where the geometry may become unstable or weaken in the deformed state. Some cases not appropriate: z z z
Compressive loads (external pressure, axial loads, etc) Closing moments on elbows and bends Out-of-plane loads on nozzles and intersections
Elastic-Plastic Analysis
Elastic-plastic analysis attempts to predict the actual collapse load of the structure by taking into account the true stress-strain behavior of the materials of construction. There is no rigorous mathematical proof that elastic-plastic analysis can predict the collapse load load. In contrast contrast, such proof does exist for the lower bound limit analysis and limit load.
Elastic-Plastic Analysis
Elastic-plastic analysis is more complex than limit analysis, but does offer some advantages:
Primary and secondary loads can be analyzed together. Large displacement theory may be used. Includes effects of strain hardening hardening.
In ASME VIII-2, a safety factor of 2.4 against the collapse load is required and included in the load combinations. This is consistent with the margin on the UTS for Part 4 design-by-rules. g y ASME VIII-2 Part 3 defines the stress-strain curves to be used for elasticplastic analyses.
Why use Non-Linear Analysis?
Simplifies the code compliance and post-processing work since the results are a “Go” or “No-go” solution. Stress categorization is difficult to perform or invalid.
Reduces cost of construction
Non-linear analyses do not require that we classify primary and secondary stresses. Limits on PL and PL+Pb+Q need not be satisfied. These more accurate analysis Th l i techniques h i allow ll you to reduce d the h required i d thickness and carry larger loads, while maintaining a consistent design margin.
Fitness for Service
Reduce the conservatism of a linear elastic analysis and allow equipment to be used longer, in more severe operating conditions, and sometime eliminate repair/replacement cost.
Non-Linear Analysis Some Important Points
You must still satisfy requirements for ratcheting, fatigue analysis, and local strain limits. Any design margins that are used should meet the intent of the governing code of construction.
For instance, if you are designing a component for an VIII-1 vessel, you should use a limit of 3.5 in the elastic-plastic methods instead of 2.4 given in the load case combination tables.
Small\Large Displacement
Small Displacement Theory
Assumes that no change in stiffness occurs as the components deform. Acceptable for most FEA work in the PVP industry. Most common method used.
Large Displacement Theory
Important when deformation affects the stiffness of the geometry. The name is a misnomer. Large g displacement p theory y isn’t used only y when the displacements are large. A more appropriate name is “Geometric Non-Linearity” Classic cases include deflection of flat plates (strengthen) and out-ofplane bending on some nozzles (weaken) Should be considered when displacements approach 10% of the structure size. This is intentionally vague since a comprehensive rule can not be established – each case is unique. unique
Geometric Effects
The effect of small & large displacement theory on a flat head test is shown below. Small displacement theory over predicts the displacement but under predicts the stress in the attached shell.
Element Types
Commonly used elements type are 8 node quadratic shells, 8 node quadratic axisymmetric, 8 & 20 node hexahedral solids Axisymmetric and solid elements allow evaluation of thru thickness distribution.
Stress Singularities
Occur at re-entrant corners of the model, changes in materials properties, and at application site of some loads. Strain energy is infinite at the singularity. Increasing g mesh density y leads to increasing g stress values. Convergence can not be achieved. In shell models, the stress at these points would normally be ignored since the shell surfaces are inside the weld volume (recall we seek stress results at the toe of the weld). In axisymmetric and solid models, we can eliminate the influence of the singularity by employing stress linearization techniques techniques.
Stress Singularities Some examples Singularities at Sharp Corner
No Singularity at Inside Corner
Stress Classification Lines
For elastic analysis, the ASME code requires that we determine the membrane, bending, and peak stresses to satisfy:
Pm – Primary General Membrane Stress PL+Pb – Primary Local Membrane plus Bending Stress PL+Pb+Q – Secondary stress PL+Pb+Q+F – Peak stress
Unfortunately, in volumetric models (e.g. axisymmetric) the results are only given in terms of raw stress components or total stress. stress Therefore, we need a way to convert raw stress results into shell-like results (membrane and bending). Th solution The l i is i to linearize li i the h stresses along l S Stress Cl Classification ifi i Lines (SCL’s). SCL’s are also effective means of eliminating stress singularities.
Stress Classification Lines
A graphical demonstration of stress linearization along an SCL
B
Membrane + Bending Membrane M b Stress Stres ss
A
Raw Stress
A
B Distance Thru Thickness
FEA Models – Weld Treatment
For external loads on nozzles and attachments, the maximum stress occurs along the weld toes. Stresses should not be calculated within weld volume when using shell elements since these elements are not capable of representing the complex through thickness stress distributions. Stresses should be calculated along the weld toe. For fatigue g design, g , failures typically yp y originate g from the weld toe in full penetration nozzle and attachment welds. Fatigue failure due to cyclic internal pressure often occurs at inside corner of nozzle opening. p g WRC 429, WRC 335, and ASME VIII-2 provided similar guidance.
FEA Models – Weld Treatment
When processing results for primary and secondary stresses, results within the weld volume are always discarded. The goal is achieve a discontinuity style solution with membrane and bending stresses. These shell like behaviors are not defined within the weld volume.
FEA Model – Weld Treatment
An example of shell models using PRG software where stresses inside the volume of the weld are discarded. Similar results are available via PVElite’s connection to NozzlePRO.
FEA Model – Weld Treatment
In shell models, tapered thickness elements can be used to replicate the local stiffness of weld zones. Defining the appropriate thickness and distribution around the thickness requires experience. PRG has h d developed l d proprietary i t methods th d b based d on extensive t i lit literature t reviews and experimental testing.
Applying Loads
Apply loads to the end of the nozzle along stiffened elements or rigid spars. For resting support lugs, apply loads along stiffener plate edges. In reality, the flat plates will elastically deform and the load transfer path is into the orthogonal plates plates.
Loads on stiffened ring
Not on Flat Plate
Apply loads on stiff edges
FEA Mesh Density
Use higher order elements when available (8 node quadratic shell elements, etc). As good practice, place at least 4 elements within sqrt(R*T) of an opening. Elements should be well shaped, corner angles near 90 degrees when possible. Structured meshing g is often p preferred when available. The re-entrant corners of plate attachments will cause singularities. Increasing mesh density at these locations will not lead to convergence. g
Stress Analysis Validation
Validation is critical in FEA work
Evaluate the mesh (density, quality, unmerged nodes) Stress patterns should be smooth without abrupt changes across element edges Validate stresses in simple parts using manual calculations Material properties Thickness and other dimensions Boundary conditions loads Displaced shapes are reasonable and expected Check reaction loads at boundary conditions Use simple initial models to validate complex problems Verify that the solution is converged When the mesh density is increased to study convergence, the increase should meaningful enough to provoke stress change.
Stress Analysis Validation Improved mesh & solution but not converged
Dense Mesh & Converged Solution
Bad Mesh & Poor Solution
Finite Element y Demonstration Analysis
Finite Element Analysis Demonstration
Pressure design g of an unreinforced nozzle using g linear elastic analysis:
Model without a weld included says allowable pressure is 260 psig. Model with a ¼” ¼ weld included says allowable pressure is 446 psig psig. Burst test says 1812 psig / 3.5 = 518 psig.
Thanks! Questions?
Chris Hinnant Paulin Research Group Houston TX Houston,
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