Background and Introduction to Design by Analysis
This part of the module introduces the two most widely known and used methods of designing pressure vessel components. These methods are namely Design by formula and Design by analysis (DBA). After a short introduction on what is Design by formula the unit then elaborates more on Design by Analysis. This is actually the purpose of the module itself. Some historical ideas and methods are described to put the work in context of the present methods used in DBA.
Design by formula (DBF)
Design by formula uses formula and rules for calculating basic dimensions for pressure vessel components.
This method is widely used in pressure vessel design and the bulk of any pressure vessel code is concerned with this approach. It is simple and has been used for many years with long experience in various applications. The formulae, rules and tables have evolved over many decades and represent a safe approach to pressure vessel design, where applicable. The method ensures that the component is safe against all possible failure modes such as gross plastic deformation, collapse, ratchetting, brittle fracture and buckling.
Some formulae and rules are based on elastic analysis, some are based on shakedown concepts and others are based on limit load analysis.
Although relative simple and safe to use the DBF approach has some built in limitations. Formulae and rules are only available for geometries and rules that are covered by the respective standard. This poses some limits on the designer as non-standard geometries and loadings cannot be properly analysed. Furthermore the results obtained by DBF have a tendency to be over-conservative. This results in designs that may not be competitive and economically viable.
Design by Analysis (DBA)
Design by analysis uses stress analysis directly. The maximum allowable load for the design is determined by performing a detailed stress analysis and checking against specified design criteria.
Design by analysis can also be used for calculating the component thicknesses for pressure vessel components.
In the early days of DBA, the analysis methods were focused on linear elastic stress analysis. This is mainly so because inelastic analysis required considerable computer resources which at the time were not present. However as computers became more powerful inelastic analysis has become more popular.
The DBA procedures were developed with the assumption that Shell Discontinuity analysis would be used for the calculations. Today the Finite Element Method is the most popular approach and this can present some challenges as we shall see.
Some Definitions
Gross structural discontinuity
Is a geometrical structural or material discontinuity which affects the stress or strain distribution across the entire wall thickness over a region of significant size.
Common examples are
- nozzle to cylinder junction
- shell to head junction
An example of a gross structural discontinuity showing a head cylinder junction
Local structural discontinuity
Is a discontinuity which affects the stress or strain distribution quite locally across part of the wall thickness. The main effect of a notch is to produce a non-linearity in the stress distribution. Common examples for local stress discontinuity are welds. Normally the word local is a somewhat subjective term. The weld toe and any undercutting are also considered as local discontinuities.
An example of a local structural discontinuity showing a weld toe in a butt weld
Nominal stress
Generally, is referred to as the stress value obtained by applying standard strength of materials formulae. Nominal stress is found at a distance outside the effects of local or gross structural discontinuities.
Structural stress
Is a linearly distributed stress across the section thickness. It includes both nominal stresses and the effects of gross structural discontinuities. However it does not include the effects of local structural discontinuities.
The structural stress, ss can be broken down into two parts, membrane stress and bending stress.
Membrane stress, sm
Is the component of the structural stress that is uniformly distributed and equal to the average value of stress across the section thickness
Bending stress, sb
Is the component of the structural stress that varies linearly across the section thickness.
Structural stress components
Notch stress
Is the total stress located at the base of a notch. The notch stress combines the structural stress together with the effects of a stress raiser. Examples of notch stresses are the stress at weld toes and at local structural discontinuities.
The additional stress to the structural part that forms the notch stress is referred to as the non-linear part of the stress distribution, snlp
Notch stress conponents
Linear elastic analysis
Generally, by linear elastic we understand that stress is proportional to strain.
As stress is related to force, and displacement is related to strain then these are also proportional to each other. Therefore, if the stresses are known for a particular load value, they can then be calculated for any other load using simple proportionality.
Linear Elastic materiaL model
A useful aspect of linear elastic analysis is that it allows the use of the superposition principle.
Superposition
For a component under the action of a number of loads the combined effect of the loads may be calculated through a separate analysis. This is applicable only for cases where the stresses remain elastic and the analysis is based on small deformation theory. The stress field results from the individual analysis are then added together to obtain the resultant stress field corresponding to the case when the loads are acting together.
Note: Another stress-strain behaviour is non-linear elastic response. However this will not be discussed here. An example of such non linear elastic behaviour is the behaviour of rubber under load.
Linear-elastic plastic analysis
Plasticity can be defined as that property that enables a material to be deformed continuously and permanently without rupture during the application of stresses exceeding those necessary to cause yielding of the material.
In linear-elastic plastic analysis the stresses are proportional to strain only up to the yield point of the material. Beyond the yield point this no longer applies and plasticity effects need to be considered. At this stage the material exhibits non-linear strain hardening and permanent deformations take place. When the load is removed the unloading is assumed to take place linearly, parallel to the loading line.
Linear-Elastic Plastic material model
Generally the plastic response is conveniently simplified using idealized models. The most commonly used models are the bilinear hardening model and the perfect plastic model. Both are described below.
Bilinear hardening
The material is assumed to be linear elastic up to yield. Beyond yield the material exhibits linear plastic deformation.
Perfect plasticity
The material is assumed to be linear elastic up to yield. Beyond yield there is unlimited plastic flow and no further increase in stress takes place.
It may be argued that this highly idealised model may seem unrealistic. However it is regarded by most pressure vessel code committees as a useful conservative model for design purposes.
Bilinear hardening and perfect plasticity material models
Yield Criteria
The yield stress sy, the point at which the stress-strain relationship is no longer proportional is generally obtained from test specimens under uniaxial loading.
Real structures and components are usually in a state of multiaxial stress systems. To be able to determine the maximum load that can be applied before the onset of plasticity it is therefore necessary to have a means of relating the multiaxial stresses to the one-dimensional yield stress value. This is achieved by using what are called multiaxial yield criteria.
The multiaxial stress tensor in the x,y,z components and the principal stresses
Two multiaxial yield criteria commonly used for ductile materials are the Tresca criterion and the von Mises criterion. These are also frequently used in pressure vessels codes of practice.
Yielding criteria are usually conveniently expressed in terms of principal stresses, since they completely determine the general state of stress.
Tresca yield criterion
This theory is based on the assumption that yielding is governed by the maximum principal shear stress. For a general three dimensional stress system the yield criterion is represented by the following equation.
but from strength of materials principles,
Rearranging gives, the Tresca yield criterion as,
Therefore in this case the material behaviour will remain elastic provided that the maximum difference between any two principal stresses is less than the yield stress, sy.
Below is Tresca’s yield locus for a two dimensional stress system. The blue line is referred to as the yield surface. Stress states inside the yield surface are elastic and stress states outside are plastic thus resulting in strain hardening and permanent deformation. In a three dimensional stress system the Tresca yield criterion is a prism whose cross section has the shape of a regular hexagon and whose main axis is the line (locus) given by;
In elastic perfectly plastic models the plastic stresses will lie on the yield surface. As such if a model does not include strain hardening, then the resultant stresses cannot be larger than the yield stress.
Tresca yield criterion
In the ASME Boiler and pressure vessel code the term 
is referred to as the Stress Intensity, S
Stress intensity,
On the other hand in the European Code EN13445, the term is referred to as the Equivalent stress, seq
Equivalent stress,
Though the two codes use different names the definition is the same.
Therefore the Tresca criteria becomes;
(according to the code of standard being used)
von Mises yield criterion
This theory is based on the assumption that yielding is governed by the maximum shear strain energy component. For a general three dimensional stress system the yield criterion is represented by the following equation.
but since
then rearranging gives,
Below is the locus of the von Mises criteria for a two dimensional stress system. The blue line is referred to as the yield surface. Stress states inside the yield surface are elastic and stress states outside are plastic thus resulting in strain hardening and permanent deformation.
In elastic perfectly plastic models the plastic stresses will lie on the yield surface. As such if a model does not include strain hardening and so the resultant stresses cannot be larger than the yield stress. In a three dimensional stress system the von Mises yield criterion is a circle whose main axis is the line (locus) given by;
von Mises yield criterion
Similar to the case of the Tresca criterion, the equivalent stress can also be expressed in terms of the von Mises criterion
This definition of the equivalent stress is used in the European code EN13445.
Rearranging the above equation becomes
And the von Mises criterion becomes;
Tresca’s versus von Mises’ Yield criteria
The figure shown below compares the Tresca and the von Mises yield criteria plotted on the same axis for a two dimensional state of stress. The Tresca’s yield surface lies inside the von Mises one. Therefore it can be said that the Tresca’s yield criterion is conservative when compared to the von Mises yield criterion.
The maximum difference between the two criteria is a factor of 
.
Comparison of Tresca and von Mises loci
For most ductile steels the von Mises criterion fits the experimental data more closely than Tresca’s, but usually Tresca’s yield criterion is simpler to use in elementary/manual calculations. The von Mises yield criterion lends itself more useful for computer programming because it is a mathematically continuous curve. Because of this reason most commercially available finite element software use the von Mises yield criterion and associated flow rule to solve elastic plastic problems.
Limit and plastic collapse loads
Gross plastic deformation
Gross plastic deformation is associated with excessive plastic deformation of the vessel under the application of a load. Unless the load is limited, this ultimately leads to plastic collapse or rupture of the vessel.
The form of plastic collapse mechanism differs between structural configurations. In some cases, the entire volume of the structure experiences plastic deformation at failure. In other cases only local regions of the body experience plastic straining with the rest remaining elastic.
This is demostrated in the next two video clips. Press the play button to view.
According to the design codes, two types of stress analysis may be used to guard against gross plastic deformation: elastic analysis and elastic-plastic analysis.
- When elastic analysis is used, the allowable load is calculated indirectly by partitioning the elastic stress into primary, secondary and peak categories and limiting the primary stress to a specified allowable value.
- When design is based on elastic-plastic analysis, the allowable load is determined directly from the elastic-plastic response of the vessel. Two types of inelastic analysis may be used; limit analysis and plastic analysis.
Limit analysis
Limit analysis 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. A plot of applied load against resulting deformation for a hypothetical limit analysis is shown in the below figure.
Load against deformation - limit load
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.
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.
Plastic analysis
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.
In Design by Analysis terminology, elastic-plastic analysis including strain hardening and large deformation effects is called plastic analysis. A hypothetical plastic analysis load-deformation curve is compared with a limit analysis curve for the same vessel in the figure shown below.
Load against deformation - limit load and strain hardening
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.