Annex X (Informative)
ASME Design Calculations
X.1 General
This annex describes the design analysis methodology used in the ASME Boiler and
Pressure Vessel Code, Section VIII, Pressure Vessels, Division 2, Alternative Methods,
Appendix 4, up to and including the 2004 version, as well as the ASME Code Section III,
Nuclear Power Components.
Methods are included for both elastic and elastic-plastic analysis, and for closed-form as
well as finite-element analysis methods of calculation, in accordance with the rules of
Appendix 4 of the 2004 Code, Section VIII Division 2.
API has adopted slightly different stress limits from the 2004 ASME Code. For the
purpose of this international standard, the basic stress limits are based on Sm and St,
which are defined as follows:
X.1.1
Sm, the design stress intensity
For standard materials (36K, 45K, 60K, and 75K), the design stress intensity is 2/3 of the
minimum specified yield strength Sy. For non-standard materials, the design stress
intensity is the lower of 2/3 of Sy or ½ of the ultimate tensile strength Su.
X.1.2
St, the maximum allowable general primary
membrane stress intensity at test pressure
API limits this stress to 90% of the yield strength for all materials.
X.2
Elastic Analysis
For elastic analysis stress components are calculated, combined, and compared to limits
for each category of stress based on multiples of the Design Stress Intensity, Sm, for the
material in use and for the category of stress.
Stress components are combined to find the stress intensity, which is defined as twice
the maximum shear stress. This can be calculated as the difference between the largest
and smallest of the three principal stresses.
X.2.1 Stress Categories
The following categories are used to classify stresses based on the consequences of
exceeding the yield strength in various manners:
X.2.1.1Primary Stress
The basic characteristic of primary stress is that it is not self-limiting, and failure, or at
least gross distortion, can occur from one application of the loading. Primary stress is
stress caused by the application of mechanical pressure, forces and moments. Thermal
stresses are not primary stresses.
Primary stress includes both membrane and bending stress and is linearly distributed
across the wall section.
X.2.1.1a.
Primary Membrane Stress Intensity
Primary membrane stress intensity is calculated from the average values of the stress
components through the wall of the vessel. Depending on the extent of the stress, is can
be classified as either General or Local.
 General Primary Membrane Stress Intensity, Pm: Membrane stress distributed in a
way such that load redistribution cannot occur, and loading beyond the yield
strength can proceed to failure. Pm is limited to Sm.
 Local primary Membrane Stress Intensity, Pl: The following is a direct quote from
ASME Section VIII Division 2 Appendix 4:
 “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. A stressed region may be considered as local if the distance over
which the stress intensity exceeds 1.1 Sm does not extend in the meridional
direction more than 1.0(Rt)1/2, where R is the midsurface radius of curvature
measured normal to the surface from the axis of rotation and t is the minimum
thickness in the region considered. Regions of local primary membrane stress
which exceed 1.1 Sm shall not be closer in the meridional direction than 2.5(Rt)
1/2
where R is defined as (R1 + R2)/2, and t is defined as (t1+ t2)/2, where t1 and
t2 are the minimum thicknesses at each of the regions considered, and R1 and R2
are the midsurface radii of curvature measured normal to the surface from the axis
of rotation at these regions where the membrane stress exceeds 1.1 Sm. Discrete
regions of local primary membrane stress, such as those resulting from
concentrated loads acting on brackets, where the membrane stress exceeds I. I Sm
shall be spaced so that there is no overlapping of the areas in which the membrane
stress exceeds I. I Sm.An example of a local primary membrane stress is the
membrane stress in a shell produced by external load and moment at a permanent
support or at a nozzle connection.”
 Local primary stress intensity is limited to 1.5 Sm.
X.2.1.1b.
Primary Bending Stress Intensity
The components of primary bending stress intensity Pb are calculated from the linear
primary stress component distributions that have the same net bending moment as the
actual stress component distribution. Bending stress components are defined as being
proportional to the distance from the centroid of a solid section.
When the bending stress components are combined with the membrane stress
components at each surface, the resulting stress intensities Pm+Pb are limited to 1.5 Sm.
X.2.1.2Secondary Stress
Secondary stress Q is caused by the constraint of adjacent parts or by self-constraint of
the structure, and yielding can cause the source of the stress to be eliminated. One load
cycle can cause local yielding and stress redistribution but cannot result in failure or gross
distortion.
Secondary stresses are membrane plus bending stresses that can occur at gross structural
discontinuities, from general thermal stress, from mechanical preload conditions, or from
combinations of these sources.
The secondary stress variation, for any sequence of test or operating conditions, is limited
to 3 Sm.
X.2.1.3
Peak Stress
Peak stress is the increment of stress added by a stress concentration or other source that
does not cause noticeable distortion. Such sources include thermal stress in a cladding
material with a different coefficient of expansion from the base material; by transient
thermal stress, or by the non-linear portion of a thermal stress distribution. The only
concern with peak stress is that it may cause the initiation of a fatigue crack or brittle
fracture.
The total stress, including peak stress, may be used in fatigue analysis, which is beyond
the scope of this annex.
X.3
X.3.1
Special stress limits
Bearing Stress
The average bearing stress from primary and secondary loads is limited to Sy. In the
cases where the distance to a free edge is greater than the distance over which the bearing
load is applied, the bearing allowable stress may be increased by a factor of 1.5. allowed
to exceed the yield strength of the material provided that the other stresses in the vicinity
of the bearing load are within acceptable limits. When bearing loads are applied to parts
having free edges, the possibility of a shear failure shall be considered. In the case of load
stress plus secondary stress, the average shear stress shall not exceed 0.5 Sy at all
temperatures.
X.3.2
Pure Shear Stress
The average primary shear stress across a section loaded under design conditions in pure
shear (for example, keys, shear rings, screw threads) shall be limited to 0.6 Sm. The
maximum primary shear under design conditions, exclusive of stress concentration at the
periphery of a solid section in torsion, shall be limited to 0.8 Sm. [Higher limits on shear
allowable stress need to be justified].
X.3.3
Progressive distortion of nonintegral connections
Screwed-on caps, screwed-in plugs, shear ring closures, breech lock closures, clamps and
unions are examples of nonintegral connections which are subject to failure by bellmouthing or other types of progressive deformation.
If any combination of loading produces yielding, such joints are subject to ratcheting
because the mating members may slip at the end of each complete cycle, and start the
next cycle in a new relationship with one another. Additional distortion may occur at
each subsequent cycle so that interlocking parts like threads may lose engagement.
Therefore, primary plus secondary stress intensities which could produce slippage shall
be limited to Sy.
X.4
Non-linear analysis
X.4.1 General
Finite-element methods may be used that consider the yielding of the material. The first
of these is called limit analysis, and the second, elastic-plastic analysis.
X.4.2 Limit analysis
Limit analysis assumes elastic-perfectly plastic material properties, and may be based on
small-displacement analysis. As a practical matter, the stress-strain curve that is used is
actually a bi-linear representation. This curve, for stress less than the yield strength has a
slope equal to the elastic modulus of the material and above that point, a slope as near
zero as practical, since a zero slope would force the finite-element program to divide by
zero and immediately stop as soon as the first element reached yield strength.
The yield strength to be used is 1.5 Sm, which for non-standard materials may be less than
the actual specified minimum yield strength. Loading is incrementally increased until the
model diverges, which is the collapse load. Actual rated load capacity can be no more
than 2/3 of the limit analysis collapse loading.
Limit analysis may be used to justify high primary stresses but not secondary stresses. In
addition, limit analysis cannot be used to justify a wall thickness thinner than that
calculated on an elastic basis.
X.4.3 Plastic Analysis
Plastic analysis is a method of structural analysis by which the structural behavior under
given loads is computed by considering the actual material stress-strain relationship and
and stress redistribution, and it may include either strain hardening, and largedisplacement change in geometry, or both.
Plastic analysis can be used to justify high primary and secondary stresses. However,
limits for bearing stress, triaxial stress, and buckling shall be calculated elastically.
The design is acceptable if shakedown occurs. That is, after successive applications of the
design loading, there is no progressive distortion or stress ratcheting. In addition the
deformations which occur prior to shakedown shall not exceed specified limits.
[I think that a better approach is to consider 2/3 of the ultimate capacity, where the
ultimate capacity is set by a limit on strain (ISO 13628-7 Annex D) or last converged
load].
X.5
Triaxial Stresses
The algebraic sum of the three prmary principal stresses (σ1+σ2+σ3) shall not exceed
four times the design stress intensity Sm.