10/21/2015 Introduction to PV Design by Analysis http://personal.strath.ac.uk/j.wood/CCOPPS_DBA/Notes/dba_intro_content_1.htm 1/17 CONTENT: Introduction to PV Design by Analysis Introduction Navigator: Background and Introduction to Design by Analysis Some Definitions Linear Elastic Analysis Linearelastic Plastic Analysis Yield Criteria Limit and Plastic Collapse loads 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.
CONTENT: Introduction to PV Design by AnalysisIntroduction
Navigator:
Background and Introduction to Design by Analysis
Some Definitions
Linear Elastic Analysis
Linearelastic Plastic Analysis
Yield Criteria
Limit and Plastic Collapse loads
Background and Introduction to Design by Analysis
This part of the module introduces the two most widely known and used methods ofdesigning pressure vessel components. These methods are namely Design byformula and Design by analysis (DBA). After a short introduction on what is Designby formula the unit then elaborates more on Design by Analysis. This is actually thepurpose of the module itself. Some historical ideas and methods are described to putthe 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 forpressure vessel components.
This method is widely used in pressure vessel design and the bulk of any pressurevessel code is concerned with this approach. It is simple and has been used formany years with long experience in various applications. The formulae, rules andtables have evolved over many decades and represent a safe approach to pressurevessel design, where applicable. The method ensures that the component is safeagainst 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 onshakedown concepts and others are based on limit load analysis.
Although relative simple and safe to use the DBF approach has some built inlimitations. Formulae and rules are only available for geometries and rules that arecovered by the respective standard. This poses some limits on the designer as nonstandard geometries and loadings cannot be properly analysed. Furthermore theresults obtained by DBF have a tendency to be overconservative. This results indesigns that may not be competitive and economically viable.
Design by Analysis (DBA)
Design by analysis uses stress analysis directly. The maximum allowable loadfor the design is determined by performing a detailed stress analysis andchecking against specified design criteria.
Design by analysis can also be used for calculating the component thicknessesfor pressure vessel components.
In the early days of DBA, the analysis methods were focused on linear elastic stressanalysis. This is mainly so because inelastic analysis required considerablecomputer resources which at the time were not present. However as computersbecame more powerful inelastic analysis has become more popular.
The DBA procedures were developed with the assumption that Shell Discontinuityanalysis would be used for the calculations. Today the Finite Element Method is themost popular approach and this can present some challenges as we shall see.
Return to top
Some Definitions
Gross structural discontinuity
Is a geometrical structural or material discontinuity which affects the stress or straindistribution across the entire wall thickness over a region of significant size.
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 acrosspart of the wall thickness. The main effect of a notch is to produce a nonlinearity inthe stress distribution. Common examples for local stress discontinuity are welds.Normally the word local is a somewhat subjective term. The weld toe and anyundercutting 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 ofmaterials formulae. Nominal stress is found at a distance outside the effects of localor gross structural discontinuities.
Is a linearly distributed stress across the section thickness. It includes both nominalstresses and the effects of gross structural discontinuities. However it does notinclude the effects of local structural discontinuities.
The structural stress, s can be broken down into two parts, membrane stress andbending stress.
Membrane stress, m
Is the component of the structural stress that is uniformlydistributed and equal to the average value of stress across thesection thickness
Bending stress, b
Is the component of the structural stress that varies linearlyacross the section thickness.
Structural stress components
Notch stress
Is the total stress located at the base of a notch. The notch stress combines thestructural stress together with the effects of a stress raiser. Examples of notchstresses 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 asthe nonlinear part of the stress distribution, nlp
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 alsoproportional to each other. Therefore, if the stresses are known for a particular loadvalue, they can then be calculated for any other load using simple proportionality.
A useful aspect of linear elastic analysis is that it allows the use of the superpositionprinciple.
Superposition
For a component under the action of a number of loads the combined effect ofthe loads may be calculated through a separate analysis. This is applicable onlyfor cases where the stresses remain elastic and the analysis is based on smalldeformation theory. The stress field results from the individual analysis are thenadded together to obtain the resultant stress field corresponding to the casewhen the loads are acting together.
Note: Another stressstrain behaviour is nonlinear elastic response. However thiswill not be discussed here. An example of such non linear elastic behaviour is thebehaviour of rubber under load.
Plasticity can be defined as that property that enables a material to be deformedcontinuously and permanently without rupture during the application of stressesexceeding those necessary to cause yielding of the material.
In linearelastic plastic analysis the stresses are proportional to strain only up to theyield point of the material. Beyond the yield point this no longer applies and plasticityeffects need to be considered. At this stage the material exhibits nonlinear strainhardening and permanent deformations take place. When the load is removed theunloading is assumed to take place linearly, parallel to the loading line.
LinearElastic Plastic material model
Generally the plastic response is conveniently simplified using idealized models. Themost commonly used models are the bilinear hardening model and the perfect plasticmodel. Both are described below.
The material is assumed to be linear elastic up to yield. Beyond yield the materialexhibits linear plastic deformation.
Perfect plasticity
The material is assumed to be linear elastic up to yield. Beyond yield there isunlimited plastic flow and no further increase in stress takes place.
It may be argued that this highly idealised model may seem unrealistic. However it isregarded by most pressure vessel code committees as a useful conservative modelfor design purposes.
Bilinear hardening and perfect plasticity material models
Return to top
Yield Criteria
The yield stress y, the point at which the stressstrain relationship is no longerproportional 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 ofplasticity it is therefore necessary to have a means of relating the multiaxial stressesto the onedimensional yield stress value. This is achieved by using what are calledmultiaxial 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 Trescacriterion and the von Mises criterion. These are also frequently used in pressurevessels 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 maximumprincipal shear stress. For a general three dimensional stress system the yieldcriterion 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 themaximum difference between any two principal stresses is less than the yield stress,y.
Below is Tresca’s yield locus for a two dimensional stress system. The blue line isreferred to as the yield surface. Stress states inside the yield surface are elastic andstress states outside are plastic thus resulting in strain hardening and permanentdeformation. In a three dimensional stress system the Tresca yield criterion is aprism whose cross section has the shape of a regular hexagon and whose main axisis the line (locus) given by;
In elastic perfectly plastic models the plastic stresses will lie on the yield surface. Assuch if a model does not include strain hardening, then the resultant stresses cannotbe 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
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 maximumshear strain energy component. For a general three dimensional stress system theyield 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. Theblue line is referred to as the yield surface. Stress states inside the yield surface areelastic and stress states outside are plastic thus resulting in strain hardening andpermanent deformation.
In elastic perfectly plastic models the plastic stresses will lie on the yield surface. Assuch if a model does not include strain hardening and so the resultant stressescannot be larger than the yield stress. In a three dimensional stress system the vonMises yield criterion is a circle whose main axis is the line (locus) given by;
Similar to the case of the Tresca criterion, the equivalent stress can also beexpressed 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 plottedon the same axis for a two dimensional state of stress. The Tresca’s yield surfacelies inside the von Mises one. Therefore it can be said that the Tresca’s yield criterionis conservative when compared to the von Mises yield criterion.
The maximum difference between the two criteria is a factor of .
For most ductile steels the von Mises criterion fits the experimental data more closelythan Tresca’s, but usually Tresca’s yield criterion is simpler to use inelementary/manual calculations. The von Mises yield criterion lends itself moreuseful for computer programming because it is a mathematically continuous curve.Because of this reason most commercially available finite element software use thevon Mises yield criterion and associated flow rule to solve elastic plastic problems.
Return to top
Limit and plastic collapse loads
Gross plastic deformation
Gross plastic deformation is associated with excessive plastic deformation of thevessel under the application of a load. Unless the load is limited, this ultimately leadsto plastic collapse or rupture of the vessel.
The form of plastic collapse mechanism differs between structural configurations. Insome cases, the entire volume of the structure experiences plastic deformation atfailure. In other cases only local regions of the body experience plastic straining withthe 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 guardagainst gross plastic deformation: elastic analysis and elasticplastic analysis.
When elastic analysis is used, the allowable load is calculated indirectly bypartitioning the elastic stress into primary, secondary and peak categories andlimiting the primary stress to a specified allowable value.When design is based on elasticplastic analysis, the allowable load isdetermined directly from the elasticplastic response of the vessel. Two types ofinelastic analysis may be used; limit analysis and plastic analysis.
Limit analysis assumes an ideal elasticperfectly plastic (or rigidperfectly plastic)material model and small deformation theory. When perfect plasticity and smalldeformation theory are assumed, the load carrying capacity of the structure is limitedby equilibrium considerations. A plot of applied load against resulting deformation fora 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 ofplastic strain develop and the response becomes nonlinear. As loading continues,equal increments of load cause increasingly greater plastic deformation. The plasticzone expands to equilibrate the internal and external stresses with the externallyapplied forces until a stage is reached when no further expansion of plastic zonescan occur to accommodate the applied load increase. This is called the limit load.
At the limit load, the load deformation curve becomes horizontal: dP/d=0. Thestructure can no longer maintain equilibrium with the external loads and unlimitedplastic deformation occurs. The structure fails by loss of equilibrium at the limit loadof the structure.
Real structures, however may behave rather differently to the limit analysis model intwo ways: the material may exhibit postyield strain hardening and also largedeformations may occur.
As strainhardening materials can support stresses greater than yield, plasticdeformation can continue for loads above the theoretical limit load of the structurewithout violating equilibrium. Changes in structural configuration as loadingprogresses can also affect the load carrying capacity of the vessel. If large structuraldeformations occur the structural loadpath may change. This can increase ordecrease the load carrying capacity of the vessel.
In Design by Analysis terminology, elasticplastic analysis including strain hardeningand large deformation effects is called plastic analysis. A hypothetical plastic analysisloaddeformation curve is compared with a limit analysis curve for the same vessel inthe figure shown below.
Load against deformation limit load and strain hardening
A significant problem in elasticplastic Design By Analysis is that of defining a “plasticload” to be used as the basis for calculating the allowable static load for the vessel, inthe same way that the limit load does in limit analysis. In practice, this is done byapplying what is called a criterion of plastic collapse, although the phrase “plasticcollapse” is in fact a misnomer, as the purpose of these criteria is to define the loadat which plastic deformation becomes excessive, and not when actual physicalcollapse occurs. Throughout the years a large number of plastic collapse criteriahave been proposed in the literature, amongst them are the tangent intersectionmethod, 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 besaid that some common pressure vessel codes have removed these criteria of plasticcollapse in their latest revision and replaced them by what are called direct routemethods.
Return to top
Introduction to Design by Analysis | Buckling | Cyclic loading |
