To understand the how and when a pressure vessel or in general a machine component fails, one should have insight into the different failure theories which describe how material fail and what should be the design criteria’s so as to avoid failure.
Failure can be thought of the as the condition where the part fails to do the intended purpose. In present day terminology failure theories are better known as Yield Theories. Yield theories are criteria which signify when the material is going to yield or lose its elasticity and enter the plastic deformation stage.
Below mentioned are the failure theories that are usually used in the design of pressure vessel.
Maximum Principle Stress Theory
Maximum principle stress theory was proposed by Rankine and is utilized by ASME section 8 Div 1 and ASME Section 1. This theory states that failure will occur when the maximum principle stress exceeds the value corresponding to yield point of the material in uniaxial tension. It is considered that the value of the yield point in compression is about the same as that in the tensile condition.
Above figure gives a plot for the bi axial state of stress. 1st quadrant shows both the stresses being positive and the plot gives the failure locus. There will be no failure of the component for which the produced stresses lie inside the plot.
Similarly the 2nd, 3rd, and 4th quadrant gives the plot that defines the failure locus based on the criteria that any of stresses would not increase beyond the yield point stress for the material.
The failure locus plot can be understood from below mentioned relationships.
σ1 > σ2 then σ2 < σyp for safe design and when σ2 > σ1 then σ2 < σyp for safe design. The dotted line inside the safe area shows the margin imposed by ASME for vessel design. In case of pressure vessels a bi axial state of stresses is considered for a thin walled pressure vessel. The two stresses being the Circumferential or Hoop Stress & the Longitudinal or Axial stress. This theory is more accurately predict failure in brittle material, but in case of ductile materials there have been experimental evidence for showing that this theory cannot always predict failure accurately, the most common case being the failure of ductile material along planes 45° to the application of load, due to shear stress. However this theory is utilized by ASME Sec 8 Div 1 & ASME Section 1, because it is simple and reduces the requirement of detailed analysis. A large factor of safety is used for this application. For a cylindrical shell component:
σh: Hoop Stress or Circumferential Stress =( P*R)/t
σa: Axial or Longitudinal Stress = (P*R)/(2t)
Where: P = Internal design pressure, R = Inside Radius, t = thickness of the cylindrical shell Clearly σh > σa
Therefore σh < σyp (should be) for a safe design of the cylindrical shell component. Where σyp is the stress value corresponding to the yield point of the material. This criteria can also be considered for a triaxial state of stress where the principle stresses are in the order σ1>σ2>σ3. Here σ1 < σyp (should be) for a safe design.
Maximum Shear Stress Theory
This theory was proposed by French engineer Henri Tresca, which states that failure will occur when the shear stress in a component exceed the maximum shear stress in case of a uniaxial tension test. The value of maximum shear stress in a uniaxial tension test is equal to (σyp/2), i.e. One half of the Yield stress for the material.
This theory was utilized by ASME Section 8, Div 2 before the 2007 Edition, further to which the code addresses the design using the Distortion Energy Theory.
However Maximum shear stress theory gives good results for ductile materials and has shown sufficient experimental proofs for the same. Also this theory is the second most widely used design theories. The first being the Distortion energy or the Von Mises criterion.
For a cylindrical shell component:
σh: Hoop Stress or Circumferential Stress =( P*R)/t
σa: Axial or Longitudinal Stress = (P*R)/(2t)
Where: P = Internal design pressure, R = Inside Radius, t = thickness of the cylindrical shell
Clearly σh > σa, and are the principle stresses acting on the planes.
Shear stress for such a state is given by the equation:
τmax = (σh – σa)/2 > (σyp/2) for the component to not fail.
Above plot represents the failure locus for the safe area in case of a component designed as per this theory.
Considering the case when both the principle stresses are tensile or compressive, the failure locus will represent the same safe zone as in case on the Maximum stress theory. However in case one of the principle stresses is tensile and the other compressive as represented in the 2nd and 4th quadrant, considering the point B where σ1 = – σ2, then τmax = σ1 < (σyp/2) , which is half the value for the allowable stress compared to the Maximum principle stress theory. Therefore the plot for the failure locus changes in the 2nd and 4th quadrant.
Considering a tri axial state of stress, where the principle stress are σx > σy > σz. Tresca criterion can be applied as
τmax = (σx – σz)/2 > (σyp/2) for the component to not fail.
Explained in detail in the book Strength of materials Part 1, by S.Timoshenko, page 62 using Mohrs circle.
Maximum shear stress theory or Tresca’s criterion gives better optimized results compares to the Maximum principle stress theory but is conservative when compared to the Distortion energy theory also known as Von Mises criterion.
Distortion Energy Criterion/Von Mises Criterion
Understanding a few important parameters first.
Poisson Ratio (µ) = Unit lateral contraction/Unit axial elongation; This ratio is constant for a material with same elastic properties in all directions. µ can be taken as 0.3 for structural steel.
Knowing µ the change in volume of a bar in tension can be calculated.
Strain in case of biaxial stress state:
Consider a rectangular bar in shape of a parallelepiped under stress in two perpendicular directions, let
σx be the stress in x-direction and σy be the stress in the y-direction.
Unit elongation in x-direction = σx/E; due to σx
Unit lateral contraction in y-direction = µ(σy/E)
If both stresses act simultaneously the unit elongation in x-direction
(ϵx) = σx/E – µ(σy/E)
Similarly for Y direction (ϵy) = σy/E – µ(σx/E)
Therefore for a tri axial state of stress the respective strain in the three directions would be given by.
Modulus of rigidity:
G = E/2(1+ µ); where E is the youngs Modulus and µ is the Poissons Ratio.
Elastic strain energy
Consider a prismatic bar under tensile load, which cause an elongation within the bar as the load is gradually increased. Elongation of the bar within the elastic limit means work is done on the bar and the work is converted completely into potential energy of strain and stored within the bar, and can be recovered when the load is removed.
Total energy stored in the bar during strain, is given by the area under the curve represented by the graph below between load P and deflection.
U = (1/2)*P*δ
In terms of applied load P, and deflection δ.
For practical application strain energy per unit volume is often of importance
also written as = (σ * ∈)/2, also 
can also be defined as (Uo)
Therefore the maximum value of strain energy per unit volume would be
w = (σyp*σyp/2E)
Therefore the strain energy per unit volume for an element in state of tri axial stress, or considering the stress acting on the element are the principle stresses.
Uo or 
Substituting values of strains ϵx, ϵy, ϵz for tri axial state of stress as described above.
Total strain energy theory:
This theory states that when the total strain energy for a component exceeds the total strain energy for the component in a uniaxial tension test, failure or yielding occurs.
Therefore,
or
where σx,y,z = σ1,2,3
This theory gives good results for ductile materials.
Total strain energy theory was further developed for Distortion energy theory or the
Von Mises criterion.
Distortion energy theory is also referred to as the Maxwell–Huber–Hencky–von Mises theory, in credit to the minds which worked on this theory.
The strain energy density at a point on a solid is can be divided into two parts:
- Dilatational strain energy density (w1 or Uh) – responsible for change in volume
- Distortional state of energy density (w2 or Ud) – responsible for change in shape
Total strain energy density= Dilatational strain energy density + Distortion strain energy density
Principle stresses σx, σy, σz can also be defined as σ1, σ2, σ3
In a tri axial state of stress the stress due to dilatational component is defined by
σm or σh = (σx+σy+σz)/3 is an average stress which is compression in all three directions.
Dilatational strain energy density (w1 or Uh) can be found by substituting σx,y&z as σh in the equation for “w” above.
Now, σh = (σx+σy+σz)/3 where, σx, σy, σz can also be defined as σ1, σ2, σ3
or
Uh = 
Distortion strain energy density can be defined as,
Ud = Uo-Uh
or
In terms of G, the modulus of rigidity
It is customary to write Ud or w2 in terms of an equivalent stress called Von Mises stress σvm
Where,
= 
Distortion Energy Criterion:
This theory states that when the strain energy density due to distortion (Ud or w2) for a component exceeds the strain energy density due to distortion for the component in a uniaxial tension test, failure or yielding occurs.
Therefore for a uniaxial state of stress σy, σz do not appear and w2 or Ud can be defined as below from the equation for w2,
Therefore Distortion energy or Von Mises criterion can be expressed as,
Material yields when the Von Mises stress exceeds the Yield stress in uniaxial tension.
Pressure Vessel guide 