## ASME PRESSURE VESSELS

The scope of this presentation is to present basic information and understanding of the ASME code for the design of pressure vessels for the chemical and process industry as applicable in the United States and most of North and South America.

## FATIGUE ANALYSIS

Nearly all materials subject to cyclic loads break at stresses much lower than the rupture stresses produced by steady loads. The phenomenon is referred to as fatigue. Fatigue failures are defined as “repeated stress failures” and are results of stress repetition, rather than duration of time. Fatigue failure can also be a result of thermal variation; fatigue failure has occurred in boiler drums due to temperature variations in the shell at the feed water inlet.

Fatigue failures are a very common cause of service failure of components and therefore designing to resist fatigue failures is a major engineering concern. In fatigue service, the localized stresses at abrupt changes in section, such as at head junction or nozzle opening, misalignment, defects in construction, and thermal gradients are the significant stresses. Fatigue failures are characterized by a fracture which involves little plastic flow, and is transgranular in nature as compared to intergranular which is characteristic of stress rupture failures. The fatigue process may be divided into three main stages: crack initiation, crack propagation to critical size, and unstable rupture of the remaining section.

Many fatigue tests of metals have provided the following general observations of the behavior of metal structures that are useful in coping with the problem of design, construction and research to improve their fatigue resistance:

- Failure at much lower than the ultimate tensile stress occurs in most metals that exhibit some ductility in static tests, and the magnitude of the applied alternating stress range is the controlling fatigue life parameter.
- Failure depends on the number of repetitions of a given range of stress rather than the total time under load. The speed of loading is a factor of secondary importance except at elevated temperatures.
- Most metals have a safe range of stress, called the “endurance or fatigue limit” below which the failure does not occur irrespective of the number of stress cycles. Not all materials have a clear endurance limit.
- Notches, grooves, or other discontinuities of sections, including those associated with surface finishes, greatly decrease the stress range that can be sustained for a given number of cycles.
- The range of stress necessary to produce failure in a fixed number of cycles usually decreases as the main tension stress of the loading cycle is increased.

## Fatigue Crack Growth

Fatigue is a result of plastic deformation, crack initiation and growth; and the principles of fracture mechanics are used to predict fatigue behavior. Stress produces slip lines in the crystals of metals that develop into small cracks which subsequently grow, join others and result in a fracture exhibiting no gross plastic deformation. Accordingly, they start at points of high stress concentration where sharp notches, material defects, etc. serve as points of nucleation. Once a crack has been initiated, it advances a finite amount with each loading cycle. At the start of the loading cycle the crack tip is sharp, but during extension and creation of an advancing plastic zone, it becomes blunted. The effect is a balance between the applied stress and amount of plastic extension at the crack tip which establishes the crack growth rate. Crack growth continues until the crack becomes large enough to trigger final instability. In brittle material, this means a fast running crack; while in ductile materials, it means the remaining cross sectional area can no longer support the applied load, and a slow ductile shear-type rupture occurs.

Application of stress to a material containing a very sharp crack results in plastic deformation of the material about the crack tip. As the applied stress increases, the zone of plastically deformed material expands and the crack tip radius increases until a characteristic radius associated with the material fracture toughness (Kc) is reached. Once a plastic zone of critical size for the fracture has been developed at the crack tip, each succeeding application of stress in the crack-opening direction will cause extension of the crack and simultaneous motion of the plastic zone boundary in the direction of crack extension.

## Fatigue Strength Reduction Factor

The stress concentration or strength reduction factor at a given location must be included in the stress analysis in order to establish the fatigue life of that component. ASME VIII-2 lists a few stress concentration factors for fillet welds and nozzle penetrations due to internal pressure as shown in the table below.

Any abrupt change of a section (fillet weld tee joint, threads) along the path of the stress flow will reduce the fatigue strength. The strength reduction factor is then the ratio of the fatigue strength without a stress concentration to the fatigue strength with the given stress concentration. The theoretical strength reduction values are usually computed and modified by experimental data that include the effect of material ductility. If the experimental data are not available, the theoretical stress concentration factors are acceptable under the Code rules.

## Cumulative Damage

Behavior of metal under fatigue conditions varies significantly from normal stress-strain relationships. Damage accumulates during each cycle of loading and develops at localized regions of high stress until subsequent repetitions finally cause visible cracks to grow, join and spread. Progressive fractures develop from these discontinuities even though the stress is well below the static elastic strength of the material.

For several loads with different alternating stress amplitudes Salt a linear damage relationship is assumed and cumulative damage is computed as follows:

n1/N1 + n2/N2 + …. + ni/Ni ≤ 1

where,

ni = number of stress cycles during the operating life of the vessel due to the load Wi.

Ni = maximum permissible number of cycles for the alternating calculated stress intensity Salt(Sa) due to the load Wi.

The ratio n/N is called the cycle ratio since it represents this fraction of the total life which each stress value uses up. The value of N is determined from Sa-N curves for the material. If the sum of these cycle ratios is less than unity, the structure is presumed safe. This is particularly important in designing an economic and safe structure which experiences only a relatively few cycles at high stress level and the major number at relatively low stress level.

As an example, a vessel subjected to 500 cycles at a stress of 55,000 psi, 2000 cycles at 42,000 psi, and 10,000 cycles at 31,000 psi, and fabricated of material with allowable fatigue strength properties given by Sa-N curve of Figure 1, would be considered safe fatigue-wise because the sum of the cycle ratios is less than 1.0.

There are exceptions to this linear cumulative damage rule. For instance, it does not take into account the order in which the stress cycles are encountered, and it has been demonstrated that if the higher stresses are applied early in life, the cumulative usage factor at failure will be less than unity, while if they are applied later in life the factor will be greater than unity.

## Figure 1: Fatigue Evaluation

## Fatigue Curves

Fatigue curves are typically in two forms: fatigue curves that are based on smooth bar test specimens and fatigue curves that are based on test specimens that include weld details of quality consistent with fabrication and inspection requirements of the Code.

- Smooth bar fatigue curves may be used for components with or without welds. The welded joints curves shall only be used for welded joints.
- Smooth bar fatigue curves are applicable to the maximum number of cycles given on the curves. The welded joint fatigue curves do not exhibit an endurance limit and are acceptable for all cycles.
- If welded joint fatigue curves are used in the evaluation, and if thermal transients result in a through-thickness stress difference at any time that is greater than the steady-state difference, the number of design cycles shall be determined as the smaller of the number of cycles for the base metal established using either paragraphs 5.5.3 or 5.5.4, and for the weld established in accordance with paragraph 5.5.5.

Stresses and strains produced by any load or thermal condition that does not vary during the cycle need not be considered in a fatigue analysis if the fatigue curves utilized in the evaluation are adjusted for mean stresses and strains. The design fatigue curves referenced in paragraphs 5.5.3 and 5.5.4 of ASME VIII-2 are based on smooth bar test specimens and are adjusted for the maximum possible effect of mean stress and strain; therefore an adjustment for mean stress effects is not required. The fatigue curves referenced in paragraph 5.5.5 of ASME VIII-2 are based on welded test specimens and include explicit adjustments for thickness and mean stress effects.

The VIII-2 allowable stress cycling data are expressed by the design fatigue strength curves Sa-N (see Figure 2). Sa is the maximum allowable stress amplitude (Salt in Figure 3) of the alternating stress range, Sr. It is plotted against the permissible number of operating cycles N.

## ASME Section VIII Requirements for Fatigue Analysis

Presently, ASME VIII-1 does not list any rules for fatigue evaluation of components. When fatigue evaluation of a component is required in accordance with UG-22 or U-2(g) of ASME VIII-1, the general practice is to use the ASME VIII-2 fatigue criteria as guidance up to the temperature limits of VIII-2. At temperatures higher than those given in VIII-2, the rules of III-H are used for guidance. Other fatigue criteria, such as those given in other international codes and ASME B31.3, may also be considered as long as the requirements of U-2(g) of VIII-1 are met. This article will concentrate on the guidance provided in VIII-2.

VIII-2 contains detailed rules regarding fatigue. Part 5 of VIII-2 deals with “Design by Analysis Requirements” and Section 5.5 specifically addresses “Protection against Failure from Cyclic Loading”. Reader is advised to review this section for requirements relating to Fatigue, as the Code has been extensively revised in the 2007 edition.

A fatigue evaluation is generally performed if the component is subject to cyclic operation, and is based on the number of applied cycles of a stress or strain range at a point in the component. The screening criteria to determine if fatigue analysis is required as a part of a design are provided in paragraph 5.5.2. If the component does not satisfy the screening criteria, a fatigue analysis is required. Also if the specified number of cycles is greater than 106, then the screening criteria are not applicable and a fatigue analysis is required. Fatigue analysis can be performed using the techniques in paragraphs 5.5.3, 5.5.4 or 5.5.5.

Screening Options

If any one of the screening options in paragraph 5.5.2 is satisfied, then a fatigue analysis is not required as part of the vessel design. The fatigue analysis exemption in accordance with this paragraph is performed on a component or a part basis. One component (integral) may be exempt, while another component (non-integral) is not exempt. If anyone component is not exempt, then a fatigue analysis shall be performed for that component.

First Screening Option

The first screening option is based on experience with comparable equipment. If successful experience over sufficient time-frame is obtained and documented with comparable equipment subject to similar loading, then a fatigue analysis is not required.

Second Screening Option

The second screening option applies to materials with a specified minimum tensile strength that is less than or equal to 80,000 psi. Paragraph 5.5.2.3 provides step-by-step instructions on determining:

- Expected number of full-range pressure cycles including startup and shutdown
- Expected number of operating pressure cycles in which the range of pressure variation exceeds 20% of design pressure for integral construction, and 15% of the design pressure for non-integral construction. Pressure cycles in which the pressure variation does not exceed these percentages of the design pressure and pressure cycles caused by fluctuations in the atmospheric conditions do not need to be considered in this evaluation.
- Effective number of changes in metal temperature difference between any two adjacent points. The Code provides for the definition of the adjacent points, and for the effective number of metal temperature differences.
- Number of temperature cycles for components involving welds between materials having different coefficients of thermal expansion that causes the value (α1-α2)ΔT to exceed 0.00034

If the number of cycles obtained by adding 1, 2, 3 and 4 satisfy the criterion in the table below, then a fatigue analysis is not required.

Third Screening Option

The third screening option can be used for all materials. Paragraph 5.5.2.4 provides step-by-step instructions for this option.

a. Determine the fatigue screening criteria factors, C1 and C2, based on the type of construction in accordance with the table below.

b. Determine the design number of full range pressure cycles including startup and shutdown, NΔFP. If the following equation is satisfied, proceed to the next step, otherwise a detailed fatigue analysis is required.

c. Determine the maximum range of pressure fluctuations during normal operation, excluding startups and shutdowns, ΔPN, and the corresponding number of significant cycles, NΔP. Significant pressure fluctuation cycles are defined as cycles where the pressure range exceeds Sas/3S times the design pressure. If the following equation is satisfied, proceed to the next step, otherwise a detailed fatigue analysis is required.

d. Determine the maximum temperature difference between any two adjacent points of the vessel during normal operation, and during startup and shutdown, ΔTN, and the corresponding number of cycles, NΔTN. If the following equation is satisfied, proceed to the next step, otherwise a detailed fatigue analysis is required.

e. Determine the maximum range of temperature difference fluctuation, ΔTR, between any two adjacent points of the vessel during normal operation, excluding startups and shutdowns, and the corresponding number of significant cycles, NΔTR. Significant temperature difference fluctuation cycles are defined as cycles where the temperature range exceeds Sas/2Eymα. If the following equation is satisfied, proceed to the next step, otherwise a detailed fatigue analysis is required.

f. Determine the range of temperature difference fluctuation between any two adjacent points for components fabricated from different materials of construction during normal operation, ΔTM, and the corresponding number of significant cycles, NΔTM. Significant temperature difference fluctuation cycles for this step are defined as cycles where the temperature range exceeds Sas/[2(Ey1α1- Ey2α2)]. If the following equation is satisfied, proceed to the next step, otherwise a detailed fatigue analysis is required.

g. Determine the equivalent stress range computed from the specified full range of mechanical loads, excluding pressure but including piping reactions, ΔSML, and the corresponding number of significant cycles, NΔS. Significant mechanical load range cycles for this step are defined as cycles where the stress range exceeds Sas. If the total specified number of significant load fluctuations exceeds the maximum number of cycles defined on the applicable fatigue curve, the Sas value corresponding to the maximum number of cycles defined on the fatigue curve shall be used. If the following equation is satisfied, proceed to the next step, otherwise a detailed fatigue analysis is required.

Fatigue Assessment

*Elastic Stress Analysis and Equivalent Stresses*

Effective total equivalent stress amplitude is used to evaluate the fatigue damage for results obtained from a linear elastic stress analysis. The controlling stress for the fatigue evaluation is the effective total equivalent stress amplitude, defined as one-half of the effective total equivalent stress range (PL + Pb + Q + F) calculated for each cycle in the loading histogram.

The primary plus secondary plus peak equivalent stress (see Figure 4) is the equivalent stress, derived from the highest value across the thickness of a section, of the combination of all primary, secondary and peak stresses produced by specified operating pressures and other mechanical loads and by general and local thermal effects and including the effects of gross and local structural discontinuities.

The procedure to evaluate protection against failure due to cyclic loading based on the effective total equivalent stress amplitude is given in paragraph 5.5.3.2. The procedure makes use of fatigue strength reduction factors for welds that are provided in the Tables 5.11 and 5.12 of the Code.

*Elastic-Plastic Stress Analysis and Equivalent Strains*

The effective Strain range is used to evaluate the fatigue damage for results obtained from an elastic-plastic stress analysis. The Effective Strain range is calculated for each cycle in the loading histogram using either cycle-by-cycle analysis or the Twice Yield Method. For the cycle-by-cycle analysis, a cyclic plasticity algorithm with kinematic hardening shall be used. Twice Yield Method is an elastic-plastic stress analysis performed in a single loading step, based on a specified stabilized cyclic stress range-strain range curve and a specified load range representing a cycle. Stress and strain ranges are direct output from this analysis. The Twice Yield Method can be used with an analysis program without cyclic plasticity capability.

The stress range and strain range of a cycle at a point in the component can be calculated using a stabilized cyclic stress-strain curve and other material properties. The cyclic curve may be either obtained by test for the material, or from paragraph 3.D.4 of Annex 3.D which also provides such curves for certain materials.

The procedure to evaluate protection against failure due to cyclic loading based on elastic-plastic stress analysis is given in paragraph 5.5.4.2.

*Elastic Analysis and Structural Stress*

An equivalent structural stress range parameter is used to evaluate the fatigue damage for results obtained from a linear elastic stress analysis. The controlling stress for fatigue evaluation is the structural stress that is a function of the membrane and bending stresses normal to the hypothetical crack plane. This method is recommended for evaluation of welded joints that have not been machined to a smooth profile. Weld joints with controlled smooth profiles may be evaluated using paragraphs 5.5.3 or 5.5.4.

Fatigue cracks at pressure welds are typically located at the toe of a weld. For as-welded and weld joints subject to post weld heat treatment, the expected orientation of a fatigue crack is along the weld toe in the through-thickness direction, and the structural stress normal to the expected is the stress measure used to correlate fatigue life data. For fillet-welded components, fatigue cracking may occur at the toe of the fillet weld or the weld throat, and both locations shall be considered in the assessment. It is difficult to predict accurate fatigue life at the weld throat due to variability in throat dimension, which is a function of the depth of the weld penetration. It is recommended to perform sensitivity analysis where the weld throat dimension is varied.

This fatigue method may only be used when approved by the owner/ user. The procedure to evaluate protection against failure due to cyclic loading using the equivalent structural stress range is given in paragraph 5.5.5.2.

Supplemental Requirements

Additional supplemental requirements for nozzle necks, bolts, perforated plates and layered vessels are provided in paragraphs 5.6, 5.7, 5.8 and 5.9 respectively. Requirements for determining stresses in parts using experimental stress analysis are provided in Annex 5.F. The requirements for Fracture Mechanic Evaluations are provided in paragraph 5.11.

For the fatigue analysis, only the stresses that vary during the cycle have to be considered. The stresses due to steady loads, which do not vary during an operating cycle, need not be considered since they are taken as mean stresses and their effect was included in the fatigue design curves by a modifying technique.

### Sources:

1. Bednar, Henry H., Pressure Vessel Design Handbook

2. Farr, James R. and Jawad, Maan H., Guidebook for the Design ASME Section VIII Pressure Vessels

3. Harvey, John F., Theory and Design of Modern Pressure Vessels

4. ASME Boiler and Pressure Vessel Code, Section VIII, Div. 2

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