Twice elastic slope method

A significant problem in elastic-plastic Design By Analysis is that of defining a “plastic load” to be used as the basis for calculating the allowable static load for the vessel, in the same way that the limit load does in limit analysis. In practice, this is done by applying what is called a criterion of plastic collapse, although the phrase “plastic collapse” is in fact a misnomer, as the purpose of these criteria is to define the load at which plastic deformation becomes excessive, and not when actual physical collapse occurs. Throughout the years a large number of plastic collapse criteria have been proposed in the literature, amongst them are the tangent intersection method, the 1% Plastic strain pressure, the twice elastic slope pressure, the twice elastic deformation pressure, and the 0.2% offset strain pressure. Here it must be said that some common pressure vessel codes have removed these criteria of plastic collapse in their latest revision and replaced them by what are called direct route methods.

Limit-load analysis method

Limit-load analysis method assumes an ideal elastic-perfectly plastic (or rigid-perfectly plastic) material model and small deformation theory. When perfect plasticity and small deformation theory are assumed, the load carrying capacity of the structure is limited by equilibrium considerations.

Initially, the structural response is linear elastic but as yield is exceeded regions of plastic strain develop and the response becomes non-linear. As loading continues, equal increments of load cause increasingly greater plastic deformation. The plastic zone expands to equilibrate the internal and external stresses with the externally applied forces until a stage is reached when no further expansion of plastic zones can occur to accommodate the applied load increase. This is called the limit load.

At the limit load, the load deformation curve becomes horizontal: dP/dd=0. The structure can no longer maintain equilibrium with the external loads and unlimited plastic deformation occurs. The structure fails by loss of equilibrium at the limit load of the structure.

1)  The material model is elastic-perfectly plastic with a specified yield strength. 

2)  The strain-displacement relations are those of small displacement theory.

3)  Equilibrium is satisfied in the undeformed configuration.

Real structures, however may behave rather differently to the limit analysis model in two ways: the material may exhibit post-yield strain hardening and also large deformations may occur.

Elastic-plastic analysis method

As strain-hardening materials can support stresses greater than yield, plastic deformation can continue for loads above the theoretical limit load of the structure without violating equilibrium. Changes in structural configuration as loading progresses can also affect the load carrying capacity of the vessel. If large structural deformations occur the structural load-path may change. This can increase or decrease the load carrying capacity of the vessel. The effects of non-linear geometry shall be considered in this analysis.

The von Mises yield function and associated flow rule should be utilized if plasticity is anticipated.  A material model that includes hardening or softening, or an elastic-perfectly plastic model may be utilized.  A true stress-strain curve model that includes temperature dependent hardening behavior.  When using this material model, the hardening behavior shall be included up to the true ultimate stress and perfect plasticity behavior (i.e. the slope of the stress-strain curves is zero) beyond this limit.  The effects of non-linear geometry shall be considered in the analysis.

Direct route methods

EN13445/3 Annex B provides direct route, and The Design By Analysis (DBA) route is included in the standard  as  a  complement  to  the  common  (and  easy)  Design  By  Formulae  (DBF)  route, for cases not covered by the DBF route, but also as an allowed alternative,

• as a complement for cases where superposition (of pressure actions) with environmental actions – wind, snow, earthquake, etc. - is required,

• as a complement for fitness-for-purpose cases where (quality related) allowed manufacturing tolerances are exceeded,

• as a complement for cases where local authorities require detailed investigations, e. g. in major hazards' situations or for environmental protection reasons. 

Within this DBA route, there are two possibilities available:

• The so-called Direct Route (DR)

• the Stress Categorization Route (SCR)

The second, the SCR, is the one, well-known from many national standards and technical regulations, which  requires  categorization  of  stresses,  or  parts  of  stresses,  into  primary,  secondary,  and  peak  stresses.

Because of the "familiarity" with this route, it is included in this standard as well, despite its also well-known drawbacks and problems – the problems associated with non-uniqueness of the choice of stress classification lines,  and  the  problems  associated  with  the  categorization,  the  non-  uniqueness  of  the  determination  of primary stresses . This approach, derived for assessing the results from thin shell theory, cannot easily be applied to results from 3-D  (continuum)  Finite  Element  Analyses,  but  it  requires  only  linear-elastic  analyses,  with  the  advantages  of uniqueness in the determination of stress results and the possibility of (linear) superposition of these results for different actions. 

ASME III Appendix F – A Valuable Guide to the Operability Assessment of Piping Systems

The NRC Inspection Manual Chapter 0326 refers to ASME III Appendix F as an acceptable method for the evaluation of “a degradation or nonconformance associated with piping or pipe supports …”. Appendix F provides five alternative methods for the qualification of pressure equipment, piping, and their supports. They are: (1) elastic analysis, (2) plastic analysis, (3) limit collapse analysis, (4) plastic collapse analysis, and (5) plastic instability analysis. Each of the five methods provides a different way of approaching the evaluation, with criteria that are specifically matched to the method. In this manner Appendix F reduces the conservatism inherent to the design analysis methods for normal operating conditions.

The Level D Service Limits and design rules contained in Appendix F are intended (F‑1200) to prevent the rupture of the pressure‐retaining boundary, but are not intended to assure operability of components during or following the specified event.

Following is a brief description of each of the five Appendix F methods, and how to implement them in practice. A comprehensive evaluation will consider all five methods and determine which criterion is the limiting condition. A more detailed coverage of operability analysis and the use of Appendix F is provided by George Antaki in his training course on this subject.

Implementation Overview

These methods are best performed using finite element analysis. First, you should review the capabilities of the software to determine if it supports the features and material models required to perform these analyses. Some software may require the explicit inclusion or activation of nonlinear geometric effects in the solution, where the solution step is updated based on the deformed geometry. This is likely to be trivial for the elastic case, but is important for all cases with plasticity as the deformations may be significant.

Appendix F does not specify the source for the plastic stress‐strain curve. One good source is ASME Section VIII, Division 2, Annex 3‐D.3, which provides equations to develop the curves for commonly used materials. Extra caution should be applied to avoid mixing engineering stress and true stress without appropriate translation. Note that the stress‑strain curves input into the FEA software are likely to be true stress‑strain, and either the results or the plastic evaluation criterion of Appendix F must be adjusted accordingly.

Apply the load in small increments so that the effects of increasing the load may be seen. The shape of the stress‑strain response and displacement‑vs‑load curves provide valuable insight into the behavior and failure modes of the system.

All elements of the system must be considered for evaluation of stresses, etc. However, experienced analysts can accurately predict where the limiting conditions may be found, and can focus their results extraction efforts at those locations. It may be necessary to apply stress intensification factors (SIFs) to piping components, or the software may evaluate the ovalization of piping directly.

Elastic Method