ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules

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which disengagement could occur as a result of progressive distortion, shall be limited to the minimum

specified yield strength at temperature,

S , or evaluated using the procedure in paragraph 5.5.7.2.

5.5.7 Ratcheting Assessment – Elastic-Plastic Stress Analysis

5.5.7.1 Overview

To evaluate protection against ratcheting using elastic-plastic analysis, an assessment is performed by

application, removal and re-application of the applied loadings. If protection against ratcheting is satisfied, it

may be assumed that progression of the stress-strain hysteresis loop along the strain axis cannot be

sustained with cycles and that the hysteresis loop will stabilize. A separate check for plastic shakedown to

alternating plasticity is not required. The following assessment procedure can be used to evaluate protection

against ratcheting using elastic-plastic analysis.

5.5.7.2 Assessment Procedure

a) STEP 1 – Develop a numerical model of the component including all relevant geometry characteristics.

The model used for analysis shall be selected to accurately represent the component geometry,

boundary conditions, and applied loads.

b) STEP 2 – Define all relevant loads and applicable load cases (see Table 5.1).

c) STEP 3 – An elastic-perfectly plastic material model shall be used in the analysis. The von Mises yield

function and associated flow rule should be utilized. The yield strength defining the plastic limit shall be

the minimum specified yield strength at temperature from Annex 3.D. The effects of non-linear geometry

shall be considered in the analysis.

d) STEP 4 – Perform an elastic-plastic analysis for the applicable loading from STEP 2 for a number of

repetitions of a loading event (see Annex 5.B), or, if more than one event is applied, of two events that

are selected so as to produce the highest likelihood of ratcheting.

e) STEP 5 – The ratcheting criteria below shall be evaluated after application of a minimum of three

complete repetitions of the cycle. Additional cycles may need to be applied to demonstrate

convergence. If any one of the following conditions is met, the ratcheting criteria are satisfied. If the

criteria shown below are not satisfied, the component configuration (i.e. thickness) shall be modified or

applied loads reduced and the analysis repeated.

1) There is no plastic action (i.e. zero plastic strains incurred) in the component.

2) There is an elastic core in the primary-load-bearing boundary of the component.

3) There is not a permanent change in the overall dimensions of the component. This can be

demonstrated by developing a plot of relevant component dimensions versus time between the last

and the next to the last cycles.

5.6 Supplemental Requirements for Stress Classification in Nozzle Necks

The following classification of stresses shall be used for stress in a nozzle neck. The classification of stress

in the shell shall be in accordance with paragraph 5.2.2.2.

a) Within the limits of reinforcement given by paragraph 4.5, whether or not nozzle reinforcement is

provided, the following classification shall be applied.

1) A

P classification is applicable to equivalent stresses resulting from pressure induced general

membrane stresses as well as stresses, other than discontinuity stresses, due to external loads and

moments including those attributable to restrained free end displacements of the attached pipe.

2) A

P classification shall be applied to local primary membrane equivalent stresses derived from

discontinuity effects plus primary bending equivalent stresses due to combined pressure and

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external loads and moments including those attributable to restrained free end displacements of the

attached pipe.

3) A

PPQ++ classification (see paragraph 5.5.2) shall apply to primary plus secondary equivalent

stresses resulting from a combination of pressure, temperature, and external loads and moments,

including those due to restrained free end displacements of the attached pipe.

b) Outside of the limits of reinforcement given in paragraph 4.5, the following classification shall be applied.

1) A

P classification is applicable to equivalent stresses resulting from pressure induced general

membrane stresses as well as the average stress across the nozzle thickness due to externally

applied nozzle axial, shear, and torsional loads other than those attributable to restrained free end

displacement of the attached pipe.

2) A

PP+ classification is applicable to the equivalent stresses resulting from adding those stresses

classified as

P to those due to externally applied bending moments except those attributable to

restrained free end displacement of the pipe.

3) A

PPQ++ classification (see paragraph 5.5.2) is applicable to equivalent stresses resulting

from all pressure, temperature, and external loads and moments, including those attributable to

restrained free end displacements of the attached pipe.

c) Beyond the limits of reinforcement, the

S limit on the range of primary plus secondary equivalent

stress may be exceeded as provided in paragraph 5.5.6.2, except that in the evaluation of the range of

primary plus secondary equivalent stress,

PPQ

+ , stresses resulting from the restrained free end

displacements of the attached pipe may also be excluded. The range of membrane plus bending

equivalent stress attributable solely to the restrained free end displacements of the attached piping shall

be less than

S .

5.7 Supplemental Requirements for Bolts

5.7.1 Design Requirements

a) The number and cross-sectional area of bolts required to resist the design pressure shall be determined

in accordance with the procedures of paragraph 4.16. The allowable bolt stress shall be obtained from

Part 3.

b) When sealing is effected by a seal weld instead of a gasket, the gasket factor,

m , and the minimum

gasket seating stress,

, may be taken as zero.

c) When gaskets are used for pre-service testing only, the design is satisfactory if the above requirements

are satisfied for

m and

factors equal to zero, and the requirements of paragraphs 5.7.1 and 5.7.2 are

satisfied when the appropriate

and

factors are used for the test gasket.

5.7.2 Service Stress Requirements

Actual service stress in bolts, such as those produced by the combination of preload, pressure, and

differential expansion, may be higher than the allowable stress values given in Annex 3.A.

a) The maximum value of service stress, averaged across the bolt cross section and neglecting stress

concentrations, shall not exceed two times the allowable stress values in paragraph 3.A.2.2 of Annex

3.A.

b) The maximum value of service stress, except as restricted by paragraph 5.7.3.1.b at the periphery of the

bolt cross section resulting from direct tension plus bending and neglecting stress concentrations, shall

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not exceed three times the allowable stress values in paragraph 3.A.2 of Annex 3.A. When the bolts are

tightened by methods other than heaters, stretchers, or other means which minimize residual torsion, the

stress measure used in the evaluation shall be the equivalent stress as defined in Equation 5.1.

5.7.3 Fatigue Assessment Of Bolts

5.7.3.1 The suitability of bolts for cyclic operation shall be determined in accordance with the following

procedures unless the vessel on which they are installed meets all the conditions of paragraph 5.5.2 (a

fatigue analysis is not required).

a) Bolts made of materials which have minimum specified tensile strengths of less than 689 MPa (100,000

psi) shall be evaluated for cyclic operation using the method in paragraph 5.5.3, using the applicable

design fatigue curves (see Annex 3.F), and, unless it can be shown by analysis or test that a lower value

is appropriate, the fatigue strength reduction factor used in the evaluation shall not be less than 4.0.

b) High strength alloy steel bolts and studs shall be evaluated for cyclic operation using the methodology in

paragraph 5.5.3 with the applicable design fatigue curve of Annex 3.F, provided all of the following are

true:

1) The material is one of the following: SA-193 Grade B7 or B16, SA-320 Grade L43, SA-540 Grades

B23 and B24, heat treated in accordance with Section 5 of SA-540.

2) The maximum value of the service stress at the periphery of the bolt cross section (resulting from

direct tension plus bending and neglecting stress concentrations) shall not exceed

2.7S , if the

higher of the two fatigue design curves for high strength bolting given in Annex 3.F is used (the

limit for direct tension is unchanged).

3) The threads shall be of a V-Type, having a minimum thread root radius no smaller than 0.076 mm

(0.003 in).

4) The fillet radii at the end of the shank shall be such that the ratio of fillet radius to shank diameter is

not less than 0.060.

5) The fatigue strength reduction factor used in the evaluation shall not be less than 4.0.

5.7.3.2 The bolts shall be acceptable for the specified cyclic operation application of loads and thermal

stresses provided the fatigue damage fraction,

D , is less than or equal to 1.0 (see paragraph 5.5.3).

5.8 Supplemental Requirements for Perforated Plates

Perforated plates may be analyzed using any of the procedures in this Part if the holes are explicitly included

in the numerical model used for the stress analysis. An elastic stress analysis option utilizing the concept of

an effective solid plate is described Annex 5.E.

5.9 Supplemental Requirements for Layered Vessels

The equations developed for solid wall cylindrical shells, spherical shells, or heads as expressed in this Part

may be applied to layered cylindrical shells, spherical shells or heads, provided that in-plane shear force on

each layer is adequately supported by the weld joint. In addition, consideration shall be given to the

construction details in the zones of load application. In order to assure solid wall equivalence for layered

cylindrical shells, spherical shells, or heads as described above, all cylindrical shells, spherical shells, or

heads subjected to radial forces and/or longitudinal bending moments due to discontinuities or externally

applied loads shall have all layers adequately bonded together to resist any longitudinal shearing forces

resulting from the radial forces and/or longitudinal bending moments acting on the sections. For example, the

use of the girth weld to bond layers together is shown in Figures 5.2, 5.3, and 5.4. The required width of the

attachment weld at the midpoint of the weld depth is given by Equation (5.85).

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1.88

⎛⎞

⎜⎟

⋅

⎝⎠

(5.85)

In Equation (5.85), the parameter

is the longitudinal bending moment per unit length of circumference

existing at the weld junction of a layered cylindrical shells, spherical shells, or head. This parameter is

determined from a stress analysis considering the pressure loading and all externally applied loads such as

Q , and

F .

5.10 Experimental Stress Analysis

Requirements for determining stresses in parts using experimental stress analysis are provided in Annex 5.F.

5.11 Fracture Mechanic Evaluations

Fracture mechanics evaluations performed to determine the MDMT in accordance with paragraph 3.11.2.8

shall be in accordance with API/ASME FFS-1. Residual stresses resulting from welding shall be considered

along with primary and secondary stresses in all fracture mechanics calculations.

5.12 Definitions

1. Bending Stress – The variable component of normal stress, the variation may or may not be linear across

the section thickness.

2. Bifurcation Buckling – The point of instability where there is a branch in the primary load versus

displacement path for a structure.

3. Event – The User’s Design Specification may include one or more events that produce fatigue damage.

Each event consists of loading components specified at a number of time points over a time period and is

repeated a specified number of times. For example, an event may be the startup, shutdown, upset

condition, or any other cyclic action. The sequence of multiple events may be specified or random.

4. Cycle – A cycle is a relationship between stress and strain that is established by the specified loading at a

location in a vessel or component. More than one stress-strain cycle may be produced at a location,

either within an event or in transition between two events, and the accumulated fatigue damage of the

stress-strain cycles determines the adequacy for the specified operation at that location. This

determination shall be made with respect to the stabilized stress-strain cycle.

5. Cyclic Loading – A service in which fatigue becomes significant due to the cyclic nature of the mechanical

and/or thermal loads. A screening criteria is provided in paragraph 5.5.2 that can be used to determine if

a fatigue analysis should be included as part of the vessel design.

6. Fatigue – The conditions leading to fracture under repeated or fluctuating stresses having a maximum

value less than the tensile strength of the material.

7. Fatigue Endurance Limit – The maximum stress below which a material can undergo

10 alternating

stress cycles without failure.

8. Fatigue Strength Reduction Factor – A stress intensification factor which accounts for the effect of a local

structural discontinuity (stress concentration) on the fatigue strength. It is the ratio of the fatigue strength

of a component without a discontinuity or weld joint to the fatigue strength of that same component with a

discontinuity or weld joint. Values for some specific cases are empirically determined (e.g. socket welds).

In the absence of experimental data, the stress intensification factor can be developed from a theoretical

stress concentration factor derived from the theory of elasticity or based on the guidance provided in

Tables 5.11 and 5.12.

9. Fracture Mechanics – an engineering discipline concerned with the behavior of cracks in materials.

Fracture mechanics models provide mathematical relationships for critical combinations of stress, crack

size and fracture toughness that lead to crack propagation. Linear Elastic Fracture Mechanics (LEFM)

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approaches apply to cases where crack propagation occurs during predominately elastic loading with

negligible plasticity. Elastic-Plastic Fracture Mechanics (EPFM) methods are suitable for materials that

undergo significant plastic deformation during crack propagation.

10. Gross Structural Discontinuity – A source of stress or strain intensification that affects a relatively large

portion of a structure and has a significant effect on the overall stress or strain pattern or on the structure

as a whole. Examples of gross structural discontinuities are head-to-shell and flange-to-shell junctions,

nozzles, and junctions between shells of different diameters or thicknesses.

11. Local Primary Membrane Stress – Cases arise in which a membrane stress produced by pressure, or

other mechanical loading associated with a primary and/or a discontinuity effect would, if not limited,

produce excessive distortion in the transfer of load to other portions of the structure. Conservatism

requires that such a stress be classified as a local primary membrane stress even though it has some

characteristics of a secondary stress.

12. Local Structural Discontinuity – A source of stress or strain intensification which affects a relatively small

volume of material and does not have a significant effect on the overall stress or strain pattern, or on the

structure as a whole. Examples are small fillet radii, small attachments, and partial penetration welds.

13. Membrane Stress – The component of normal stress that is uniformly distributed and equal to the

average value of stress across the thickness of the section under consideration.

14. Normal Stress – The component of stress normal to the plane of reference. Usually the distribution of

normal stress is not uniform through the thickness of a part.

15. Operational Cycle – An operational cycle is defined as the initiation and establishment of new conditions

followed by a return to the conditions that prevailed at the beginning of the cycle. Three types of

operational cycles are considered: the startup-shutdown cycle, defined as any cycle which has

atmospheric temperature and/or pressure as one of its extremes and normal operating conditions as its

other extreme; the initiation of, and recovery from, any emergency or upset condition or pressure test

condition that shall be considered in the design; and the normal operating cycle, defined as any cycle

between startup and shutdown which is required for the vessel to perform its intended purpose.

16. Peak Stress – The basic characteristic of a peak stress is that it does not cause any noticeable distortion

and is objectionable only as a possible source of a fatigue crack or a brittle fracture. A stress that is not

highly localized falls into this category if it is of a type that cannot cause noticeable distortion. Examples

of peak stress are: the thermal stress in the austenitic steel cladding of a carbon steel vessel, the thermal

stress in the wall of a vessel or pipe caused by a rapid change in temperature of the contained fluid, and

the stress at a local structural discontinuity.

17. Primary Stress – A normal or shear stress developed by the imposed loading which is necessary to

satisfy the laws of equilibrium of external and internal forces and moments. The basic characteristic of a

primary stress is that it is not self-limiting. Primary stresses which considerably exceed the yield strength

will result in failure or at least in gross distortion. A thermal stress is not classified as a primary stress.

Primary membrane stress is divided into general and local categories. A general primary membrane

stress is one that is distributed in the structure such that no redistribution of load occurs as a result of

yielding. Examples of primary stress are general membrane stress in a circular cylindrical or a spherical

shell due to internal pressure or to distributed live loads and the bending stress in the central portion of a

flat head due to pressure. Cases arise in which a membrane stress produced by pressure or other

mechanical loading and associated with a primary and/or a discontinuity effect would, if not limited,

produce excessive distortion in the transfer of load to other portions of the structure. Conservatism

requires that such a stress be classified as a local primary membrane stress even though it has some

characteristics of a secondary stress. Finally a primary bending stress can be defined as a bending

stress developed by the imposed loading which is necessary to satisfy the laws of equilibrium of external

and internal forces and moments.

18.

Ratcheting – A progressive incremental inelastic deformation or strain that can occur in a component

subjected to variations of mechanical stress, thermal stress, or both (thermal stress ratcheting is partly or

wholly caused by thermal stress). Ratcheting is produced by a sustained load acting over the full cross

section of a component, in combination with a strain controlled cyclic load or temperature distribution that

is alternately applied and removed. Ratcheting causes cyclic straining of the material, which can result in

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failure by fatigue and at the same time produces cyclic incremental growth of a structure, which could

ultimately lead to collapse.

19. Secondary Stress – A normal stress or a shear stress developed by the constraint of adjacent parts or by

self-constraint of a structure. The basic characteristic of a secondary stress is that it is self-limiting. Local

yielding and minor distortions can satisfy the conditions that cause the stress to occur and failure from

one application of the stress is not to be expected. Examples of secondary stress are a general thermal

stress and the bending stress at a gross structural discontinuity.

20. Shakedown – Caused by cyclic loads or cyclic temperature distributions which produce plastic

deformations in some regions of the component when the loading or temperature distribution is applied,

but upon removal of the loading or temperature distribution, only elastic primary and secondary stresses

are developed in the component, except in small areas associated with local stress (strain)

concentrations. These small areas shall exhibit a stable hysteresis loop, with no indication of progressive

deformation. Further loading and unloading, or applications and removals of the temperature distribution

shall produce only elastic primary and secondary stresses.

21. Shear Stress – The component of stress tangent to the plane of reference.

22. Stress Concentration Factor – The ratio of the maximum stress to the average section stress or bending

stress.

23. Stress Cycle – A stress cycle is a condition in which the alternating stress difference goes from an initial

value through an algebraic maximum value and an algebraic minimum value and then returns to the initial

value. A single operational cycle may result in one or more stress cycles.

24. Thermal Stress – A self-balancing stress produced by a non-uniform distribution of temperature or by

differing thermal coefficients of expansion. Thermal stress is developed in a solid body whenever a

volume of material is prevented from assuming the size and shape that it normally should under a change

in temperature. For the purpose of establishing allowable stresses, two types of thermal stress are

recognized, depending on the volume or area in which distortion takes place. A general thermal stress

that is associated with distortion of the structure in which it occurs. If a stress of this type, neglecting

stress concentrations, exceeds twice the yield strength of the material, the elastic analysis may be invalid

and successive thermal cycles may produce incremental distortion. Therefore this type is classified as a

secondary stress. Examples of general thermal stress are: the stress produced by an axial temperature

distribution in a cylindrical shell, the stress produced by the temperature difference between a nozzle and

the shell to which It is attached, and the equivalent linear stress produced by the radial temperature

distribution in a cylindrical shell. A Local thermal stress is associated with almost complete suppression

of the differential expansion and thus produces no significant distortion. Such stresses shall be

considered only from the fatigue standpoint and are therefore classified as local stresses. Examples of

local thermal stresses are the stress in a small hot spot in a vessel wall, the difference between the non-

linear portion of a through-wall temperature gradient in a cylindrical shell, and the thermal stress in a

cladding material that has a coefficient of expansion different from that of the base metal.

5.13 Nomenclature

a radius of hot spot or heated area within a plate or the depth of a flaw at a weld toe, as

applicable.

thermal expansion coefficient of the material at the mean temperature of two adjacent points,

the thermal expansion coefficient of material evaluated at the mean temperature of the cycle,

or the cone angle, as applicable.

thermal expansion coefficient of material 1 evaluated at the mean temperature of the cycle.

thermal expansion coefficient of material 2 evaluated at the mean temperature of the cycle.

material factor for the multiaxial strain limit.

capacity reduction factor.

C factor for a fatigue analysis screening based on Method B.

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D is cumulative fatigue damage.

D is fatigue damage for the

k cycle.

cumulative strain limit damage.

orm

strain limit damage from forming.

strain limit damage for the

k loading condition.

,ij k

eΔ change in total strain range components minus the free thermal strain at the point under

evaluation for the

k cycle.

Δ local nonlinear structural strain range at the point under evaluation for the

cycle.

Δ elastically calculated structural strain range at the point under evaluation for the

k cycle.

()

,tk

Δ equivalent strain range for the

cycle, computed from elastic-plastic analysis, using the

total strain less the free thermal strain.

()

,tk

equivalent strain range for the

k cycle, computed from elastic analysis, using the total strain

less the free thermal strain.

,ij k

Δ component strain range for the

cycle, computed using the total strain less the free

thermal strain

,peq k

Δ equivalent plastic strain range for the

k loading condition or cycle.

,eff k

Δ Effective Strain Range for the

cycle.

,ij k

Δ change in plastic strain range components at the point under evaluation for the

k loading

condition or cycle.

PΔ maximum design range of pressure associated with

,nk

SΔ

primary plus secondary equivalent stress range.

SΔ range of primary plus secondary plus peak equivalent stress for the

k cycle.

,LT k

SΔ local thermal equivalent stress for the

k cycle.

,ess k

SΔ equivalent structural stress range parameter for the

cycle.

SΔ equivalent stress range computed from the specified full range of mechanical loads, excluding

pressure but including piping reactions.

QΔ

range of secondary equivalent stress.

TΔ operating temperature range.

TΔ effective number of changes in metal temperature between any two adjacent points.

TΔ temperature difference between any two adjacent points of the vessel during normal

operation, and during startup and shutdown operation with

TΔ temperature difference between any two adjacent points of the vessel during normal

operation, and during startup and shutdown operation with

TΔ temperature difference between any two adjacent points of the vessel during normal

operation, and during startup and shutdown operation with

Δ stress range associated with the principal stress in the

i direction.

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Δ stress tensor range.

Δ local nonlinear structural stress range at the point under evaluation for the

k cycle.

Δ elastically calculated structural stress range at the point under evaluation for the

k cycle.

Δ elastically calculated structural bending stress range at the point under evaluation for the

cycle.

,ij k

Δ stress tensor range at the point under evaluation for the

k cycle.

Δ elastically calculated structural membrane stress range at the point under evaluation for the

k cycle.

Δ structural shear stress range at the point under evaluation for the

cycle.

elastically calculated bending component of the structural shear stress range at the point

under evaluation for the

cycle.

elastically calculated membrane component of the structural shear stress range at the point

under evaluation for the

cycle.

out-of-phase angle between

and

for the

k cycle.

E unmodified Young's modulus for plate material.

value of modulus of elasticity on the fatigue curve being utilized.

,ya k

E value of modulus of elasticity of the material at the point under consideration, evaluated at the

mean temperature of the

cycle.

E Young’s Modulus of material 1 evaluated at the mean temperature of the cycle.

E Young’s Modulus of material 2 evaluated at the mean temperature of the cycle.

Young’s Modulus of the material evaluated at the mean temperature of the cycle.

cold forming strain.

uniaxial strain limit.

,Lk

limiting triaxial strain.

mean stress correction factor for the

k cycle.

additional stress produced by the stress concentration over and above the nominal stress

level resulting from operating loadings.

F externally applied axial force.

()F

a fatigue modification factor based on the out-of-phase angle between

Δ and

Δ .

correction factor used in the structural stress evaluation.

correction factor used in the structural shear stress evaluation.

css

K material parameter for the cyclic stress-strain curve model.

,ek

K fatigue penalty factor for the

cycle.

,vk

K plastic Poisson’s ratio adjustment for local thermal and thermal bending stresses for the

cycle.

fatigue strength reduction factor used to compute the cyclic stress amplitude or range .

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K equivalent stress load factor.

K ratio of peak stress in reduced ligament to the peak stress in normal ligament.

total number of stress ranges at a point derived from the cycle counting procedure.

longitudinal bending moment per unit length of circumference existing at the weld junction of

layered spherical shells or heads due to discontinuity or external loads.

externally applied bending moment.

m material constant used for the fatigue knock-down factor used in the simplified elastic-plastic

analysis

m mechanical strain tensor, mechanical strain is defined as the total strain minus the free

thermal strain.

m exponent used in a fatigue analysis based on the structural stress.

n material constant used for the fatigue knock-down factor used in the simplified elastic-plastic

analysis

n actual number of repetitions of the

cycle.

css

n material parameter for the cyclic stress-strain curve model.

N permissible number of cycles for the

k cycle.

()

NCS number of cycles from the applicable design fatigue curve (see Annex 3.F, paragraph 3.F.1.2)

evaluated at a stress amplitude of

CS.

()

number of cycles from the applicable design fatigue curve (see Annex 3.F, paragraph 3.F.1.2)

evaluated at a stress amplitude of

S .

design number of full-range pressure cycles including startup and shutdown.

number of significant cycles associated with

expected number of operating pressure cycles in which the range of pressure variation

exceeds 20% of the design pressure for integral construction or 15% of the design pressure

for non-integral construction.

number of significant cycles associated with

, significant cycles are those for which the

range in temperature exceeds

S .

number of cycles associated with

number of significant cycles associated with

number of temperature cycles for components involving welds between materials having

different coefficients of expansion.

Poisson’s ratio.

specified design pressure.

P primary bending equivalent stress.

P general primary membrane equivalent stress.

P local primary membrane equivalent stress.

Φ design factor for buckling.

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Q secondary equivalent stress resulting from operating loadings.

Q externally applied shear force.

inside radius measured normal to the surface from the mid-wall of the shell to the axis of

revolution , or the ratio of the minimum stress in the

k cycle to the maximum stress in the

k cycle, as applicable.

stress ratio for the

k cycle.

,bk

ratio of the bending stress to the membrane plus bending stress.

,bk

ratio of the bending component of the shear stress to the membrane plus bending component

of the shear stress.

SF computed remaining strength factor.

mid-surface radius of curvature of region 1 where the local primary membrane stress exceeds

1.1S .

mid-surface radius of curvature of region 2 where the local primary membrane stress exceeds

1.1S .

allowable stress based on the material of construction and design temperature.

S alternating stress obtained from a fatigue curve for the specified number of operating cycles.

S stress amplitude from the applicable design fatigue curve (see Annex 3.F, paragraph 3.F.1.2)

evaluated at 1E6 cycles.

S computed equivalent stress.

S allowable limit on the secondary stress range.

S allowable limit on the primary plus secondary stress range (see paragraph 5.5.6.)

minimum specified yield strength at the design temperature.

specified plastic limit for limit-load analysis.

,ak

value of alternating stress obtained from the applicable design fatigue curve for the specified

number of cycles of the

k cycle.

,alt k

S alternating equivalent stress for the

k cycle.

,yk

yield strength of the material evaluated at the mean temperature of the

k cycle.

()