THEORIES OF FAILURE FOR PRESSURE EQUIPMENT

Why do parts fail?

If we keep increasing the magnitude of load applied to a part, at some point it will fail. But

the important questions are: how can we predict when the failure will occur? or what level

does the stresses in the part need to reach for it to fail?

Parts fail when the induced stresses exceed their strength.

To prevent failure, we must know what kind of stresses cause the failure: Tensile, Compressive,

Shear? Essentially, it depends on material’s relative strength in tension, compression and

shear. Also, it depends on the type loading (Static, Fatigue, Impact) and presence of the

cracks in the material.

What is Failure?

Failure doesn’t have a universal definition; it depends on the type of application. In mechanical design,

a part is said to have failed, if it becomes unsuitable for performing its desired function.

There are basically three types of mechanical failure:

1. Yielding

Yielding results in excessive permanent deformation which makes the part unfit to perform

its function satisfactorily.

This mostly occurs in ductile materials.

2. Fracture

Fracture results in breaking the component into two parts.

This mostly occurs in brittle materials.

3. Excess elastic deflection/ inadequate rigidity

In some applications, where components are desired to have sufficient rigidity, excess elastic

deflection (not permanent) could be the reason for failure to perform the desired function.

Examples: Flanges must be rigid, to effectively transmit the applied bolt load on the gasket to

compress/ squeeze it uniformly; tall distillation columns have restriction for maximum tip

deflection to ensure the liquid height on trays is uniform at all points.

This mostly occurs in ductile materials.

Ductile and Brittle materials

There is no sharp line of demarcation between ductile and brittle materials. However, a rough

guideline is that if the percentage elongation is less than 5%, then the material may be

treated as brittle and if it is more than 15% then the material is ductile.

However, there are many instances when a ductile material may fail by fracture. This may

occur if a material is subjected to: cyclic loading, long term static loading at elevated

temperature, Impact loading, work hardening, and severe quenching.

In general, ductile, isotropic materials are limited by their shear strengths. Brittle materials

are limited by their tensile strengths.

Static loads are slowly applied and remain constant with time. Dynamic loads are suddenly

applied (impact), or repeatedly varied with time (fatigue), or both. In dynamic loading, the

distinction between failure mechanisms of ductile and brittle materials blurs. Ductile materials

often fail like brittle materials in dynamic loading.

Figure 1: Tensile test specimen of ductile steel (Failure along principal shear stress plane)

Figure 2: Tensile test specimen of brittle steel (Failure along principal normal stress plane)

Figure 3: Compression test specimens (a) Ductile Steel (b) Brittle Cast Iron

Figure 4: Torsion test specimens (a) Ductile Steel (b) Brittle Cast Iron

Why do we need different failure theories?

For ductile materials failure is usually considered to occur at the onset of plastic

deformation and for brittle materials it occurs at fracture. These points are easy to define

for a uniaxial stress state, like a tensile test. They occur when the normal stress in the object

reaches the yield strength for ductile materials and the ultimate strength for brittle materials.

Figure 5: Ductile Vs Brittle failure

But for a more complex case of tri-axial stress, predicting failure is not that straightforward. In

fact it's so difficult to predict that we don't have one universally applicable method. Instead,

we have to predict failure by selecting the most suitable one of a range of different failure

theories, each of which we know works relatively well under certain circumstances, based on

experimentation. Failure theories which apply for ductile materials usually aren't applicable

for brittle materials, and vice-versa.

We have material properties from a uniaxial tensile test; whereas, in real world, the loading is

not uniaxial. Failure theories help us predict the failure in a component in case of multi-axial

loading. Theories of failure are required due to the unavailability of failure stresses under

combined loading conditions.

What does a failure theory do?

Theories of failure allow us to PREDICT FAILURE of a material by comparing the stress state in

the object with material properties like yield or ultimate strengths obtained by performing

uniaxial tensile test.

The stress state at a point can be described using the three principal stresses, so most failure

theories are defined as a function of the principal stresses and the material strength.

f(σ1, σ2, σ3) = Syt, Sut

By convention principal stresses are ordered from largest to smallest as σ1 ≥ σ2 ≥ σ3

At yield point in tension test:

The maximum principal stress is equal to the yield strength of the material, and the two other

principal stresses are equal to zero.

i.e. σ1 = Syt, σ2 = σ3 = 0.

Also, the maximum shear stress will be Syt/2.

For generally used ductile metals SyTension = SyCompression.

Various Theories of Failure

Theories of failure state the governing parameter causing failure (criteria) in general tri-axial

stress state and prescribe the safe limit for design from uniaxial tension test. Some of the most

popular failure criteria are listed below.

i. Maximum principal stress reaches tensile yield strength of the material

ii. Maximum shear stress reaches shear yield strength of the material

iii. Maximum tensile strain reaches yield point strain

iv. Total strain energy per unit volume absorbed by the material reaches the strain

energy limit

v. Shear strain energy (distorsion energy) per unit volume reaches its maximum limit

In a uniaxial tensile test all parameters listed above will simultaneously reach their limiting

values. But when the loading is complex, not all of them would reach their limiting values

simultaneously. Often only one of them may reach its limiting value causing failure, even if the

other parameters are well within their safe limits.

As pressure vessels are primarily constructed of ductile materials, the scope of this article is

limited to failure theories applicable/ used for pressure equipment design. Of the many

theories developed to predict elastic failure, the three most commonly used are:

1. Maximum Principal Stress theory

2. Maximum Shear Stress theory

3. Maximum Distortion Energy theory

Hydrostatic and deviatoric stresses

Before we go to specifics of any failure theory, it is required to understand the concept of

hydrostatic and deviatoric stresses. A general tri-axial stress state can be decomposed into

stresses which cause a change in volume and stresses which cause shape distortion.

Figure 6: Decomposition of general tri-axial stress state into hydrostatic stress and deviatoric stresses

Stresses that cause a change in volume are called HYDROSTATIC stresses, because that is the

type of stress acting on an object submerged in liquid. For a hydrostatic stress

configuration, the three principal stresses are always equal, and there are no shear

stresses. i.e. σ1 = σ2 = σ3

For a tri-axial stress state we can calculate the hydrostatic component as the average of the

three principal stresses.

Hydrostatic stress = σavg = (σ1 + σ2 + σ3) / 3

The mechanism that causes yielding of ductile materials is shear deformation. Since there are

no shear stresses for a state of hydrostatic stress, this component can be very large and still

not contribute to yielding.

Yielding is only caused by the stresses which cause shape distortion. These are called

DEVIATORIC stresses. The deviatoric component is calculated by subtracting the hydrostatic

component from each of the principal stresses.

The hydrostatic and deviatoric components can be expressed in matrix form, like this.

Here we have described the stress state using the principal stresses, but we could also

describe it for an arbitrary orientation of the stress element.

Any good failure theory needs to be consistent with experimental observations we can make

about how materials fail. There is one key observation that failure theories for ductile materials

need to capture, which is the fact that hydrostatic stresses do not cause yielding in ductile

materials.

1. Maximum Principal Stress theory (also known as RANKINE’S THEORY)

This theory states:

“Failure occurs when any one of the three principal stresses reach the yield or ultimate

strengths of the material determined from a uniaxial tension or compression test.”

What is principal stress?

It is defined as the normal stress calculated at an angle when shear stress is zero.

Condition for failure

Maximum principal stress > Failure stress in Tension test

Max [ |σ1| , |σ2| , |σ3| ] > Syt or Sut

Failure Envelope

For planer (biaxial) state of stress, σ3 = 0

Max [ |σ1| , |σ2| ] > Syt

For the limiting condition of failure, the above inequality results in a square envelope when

plotted on graph with principal stresses on x-y axes.

Figure 7: Failure envelope for Maximum Principal Stress Theory

Condition for safe design

Maximum principal stress ≤ Permissible stress

Permissible stress = Failure stress in Tension test / Factor of Safety

Max [ |σ1| , |σ2| , |σ3| ] ≤ Syt/ FOS or Sut/ FOS

For planer stress state, σ3 = 0

Max[σ1, σ2] ≤ Syt/ FOS

2. Maximum Shear Stress theory (also known as Tresca- Guest theory)

This is the oldest failure theory, originally proposed by the great French scientist C. A.

Coulomb. French engineer Henri Tresca modified it in 1864, and J. J. Guest validated by

experiments around 1900. For these reasons the maximum-shear-stress theory is sometimes

called the Tresca-Guest theory.

This theory states:

“A material subjected to any combination of loads will fail by yielding when the maximum

shear stress exceeds the shear stress at yielding in a uniaxial tensile test.”

“This theory states that yielding occurs when the difference between the maximum and

minimum principal stresses is equal to the yield strength of the material.”

Condition for failure

Maximum shear stress > Yield strength in shear at yield point in tension test

Yield strength in shear in tensile test = Syt/ 2

τmax > Syt/ 2

For tri-axial state of stress,

Max [ | τ12 , τ23 , τ31 ] > Syt / 2

Max [ | (𝜎1−𝜎2)

2 | , |

(𝜎2−𝜎3)

2 | , |

(𝜎3−𝜎1)

2 | ] >

𝑆𝑦𝑡

2

Max [ | (σ1-σ2) | , | (σ2-σ3) | , | (σ3-σ1) | ] > Syt

Failure Envelope

For planer (biaxial) state of stress, σ3 = 0

Max [ | (σ1-σ2) | , | σ2 | , | σ1 | ] > Syt

For the limiting condition of failure, the above inequality results in a hexagonal envelope when

plotted on graph with principal stresses on x-y axes.

Figure 8: Failure envelope for Maximum Shear Stress Theory

Condition for safe design

Maximum shear stress ≤ Permissible shear stress (τper)

Permissible shear stress = Yield strength in shear at yield point in tension test / Factor of

safety

τper = Syt / (2*FOS)

τmax ≤ Syt / (2*FOS)

For tri-axial state of stress,

Max [ | (σ1-σ2) | , | (σ2-σ3) | , | (σ3-σ1) | ] ≤ Syt / FOS

When σ1 ≥ σ2 ≥ σ3,

|σ1- σ3| ≤ Syt / FOS

For planer (biaxial) state of stress, σ3 = 0

Max [ | (σ1-σ2) | , | σ2 | , | σ1 | ] ≤ Syt / FOS

| σ1 | ≤ Syt / FOS, when σ1 and σ2 are like in nature

| σ1 - σ2| ≤ Syt / FOS, when σ1 and σ2 are unlike in nature

3. Maximum Distortion Energy (Shear Strain Energy) theory (also known as von

Mises, Huber and Hencky’s Theory)

It was initially developed by the Austrian scientist Richard von Mises, but a number of others

were involved in refining it, so it is sometimes called the Maxwell–Huber–Hencky–von Mises

theory.

This theory states:

“It states that yielding occurs when the maximum distortion energy in a material is equal to

the distortion energy at yielding in a uniaxial tensile test.”

What is the distortion energy?

It is essentially the portion of strain energy in a stressed element corresponding to the effect

of the deviatoric/ shear stresses i.e. stress which cause shape distortion.

Condition for failure

Max. Distortion Energy per unit volume > Distortion Energy per unit volume at Yield Point in

Tension Test

Ud > Ud Y.P.T.T

The Maximum Distortion Energy per unit volume (Ud) can be derived as follows:

Total strain energy (UTotal) is the sum of strain energy corresponding to hydrostatic/ volumetric

stresses (UVol) and strain energy corresponding to deviatoric/ shear stresses (Ud).

UTotal = Uvol + Ud

Ud = UTotal - UVol

Strain energy is the area under the stress-strain curve.

Figure 9: Strain energy = Area under stress-strain curve

Strain energy = ½ * Stress * Strain

Strain energy due to principal stress σ1 = Uσ1

= 1

2 * σ1 * 1

= 1

2 ∗ 𝜎1 ∗ [

𝜎1

𝐸 −

µ∗𝜎2

𝐸 −

µ∗𝜎3

𝐸]

= [σ12 − µ∗σ1∗σ2 − µ∗σ1∗σ3])

2E

Total Strain energy due to all principal stresses = UTotal

= Uσ1 + Uσ2 + Uσ3

= [σ12 − µ∗σ1∗σ2 − µ∗σ1∗σ3])

2E +

[σ22 − µ∗σ2∗σ3 − µ∗σ2∗σ1])

2E +

[σ32 − µ∗σ3∗σ2 − µ∗σ3∗σ1])

2E

UTotal = [𝜎12 + 𝜎22 + 𝜎32 − 2µ(𝜎1∗𝜎2 + 𝜎2∗𝜎3 + 𝜎3∗𝜎1) ]

2𝐸

Uvol = ½* Volumetric stress * Volumetric strain

Uvol = 1

2 (

𝜎1 + 𝜎2 + 𝜎3

3)[(

1−2µ

𝐸)( σ1 + σ2 + σ3) ]

Uvol = (𝟏−𝟐µ

𝟔𝑬)( σ1 + σ2 + σ3)^2

As, Ud = UTotal - UVol

Ud = 𝟏+µ

𝟔𝑬 [ (σ1 – σ2)2 + (σ2 – σ3)2 + (σ3 – σ1)2 ]

To get Ud Y.P.T.T, substitute σ1 = Syt, σ2 = σ3 = 0 in equation above

Ud Y.P.T.T = [ 𝟏+µ

𝟑𝐄 ] * Syt2

For the limiting condition of failure, equating Ud and Ud Y.P.T.T we get,

[ (σ1 – σ2)2 + (σ2 – σ3)2 + (σ3 – σ1)2 ] > 2*Syt2

Failure Envelope

For planer (biaxial) state of stress, σ3 = 0

σ12 + σ2

2 – σ1*σ2 > Syt2

For the limiting condition of failure, above inequality becomes an equation of an ellipse with

Semi-major axis = √2*Syt and Semi-minor axis = √2/3*Syt

Figure 10: Failure envelope for Maximum Distorsion Energy Theory

Condition for safe design

Maximum Distortion Energy/ volume ≤ Distortion energy/ volume at yield point in tension test

[ (σ1 – σ2)2 + (σ2 – σ3)2 + (σ3 – σ1)2 ] ≤ 2*(𝑺𝒚𝒕

𝑭𝑶𝑺)𝟐

√𝟏

𝟐[ (𝛔𝟏 – 𝛔𝟐)^𝟐 + (𝛔𝟐 – 𝛔𝟑)^𝟐 + (𝛔𝟑 – 𝛔𝟏)^𝟐 ] = Syt = σeq

The term on the left is often called the equivalent von Mises stress. If it is larger than the

yield strength of the material, yielding is predicted to have occurred. The equivalent von Mises

stress is a common output from stress analysis performed using the finite element method.

Contour plots are typically generated to show the distribution of the von Mises equivalent

stress within a component, as these allow areas at risk of yielding to be identified.

For planer (biaxial) state of stress, σ3 = 0

σ12 + σ2

2 – σ1*σ2 ≤ (𝑺𝒚𝒕

𝑭𝑶𝑺)^2

Comparison of failure theories

For comparing failure theories, the failure envelope (region of safety) of each theory is plotted

on a single graph. A failure envelope is the representation of a failure theory in the principal

stress space. Yielding is considered to have occurred when the stress state reaches the

boundary of envelope.

Figure 11: Superimposed failure envelopes of three theories

Figure 12: Comparison of failure criteria with experimental results

From the above figure it is clear that the Maximum Principal Stress theory has large areas

where its use is potentially unsafe. So we will compare Tresca and von Mises. Both agree well

with experimental observations, although von Mises agrees slightly better. The Tresca yield

surface lies entirely inside the von Mises surface, meaning that Tresca is more conservative.

The difference between the Tresca and von Mises theories is largest for the three

configurations indicated in figure below. The maximum difference between the two theories

can be calculated to be 15.5%.

Theory

[Researchers]

Ease of

use

Failure

envelope

τy/ σyt

on shear

diagonal

(τ=σ1=-σ2)

Accuracy/ Experimental correlation ASME Code Sec

VIII

Max principal

stress (MPST)

[Rankine]

Simplest Square 1

Potentially unsafe, when σ1 and σ2 are unlike in nature.

Correlates well for brittle materials under all loading conditions because

brittle materials are weak in tension.

But for ductile materials, it doesn’t agree because ductile materials are

weak in shear.

This theory can be applied for ductile materials when state of stress

condition such that maximum shear stress is less than or equal to maximum

principal stress i.e.

• Uniaxial state of stress(τmax=σ2)

• Biaxial loading when principal stresses are like in nature (τmax=σ2)

• Under hydrostatic stress condition (shear stress in all planes is 0).

Adopted in Div-1

Max shear

stress (MSST)

[Henri Tresca,

Guest]

Easier to

apply than

von Mises

Hexagon 0.5

Gives safe but uneconomic design.

More conservative for ductile materials.

This theory can be applied for ductile materials when uniaxial state of stress

and biaxial state of stress when principal stresses are like in nature.

It is not suitable for hydrostatic loading.

Was adopted in

Div-2,

before edition 2007

Max distorsion

energy (MDET)

[Richard von

Mises, Maxwell,

Huber, Hencky]

Relatively

complex for

manual

calculations

Ellipse 1/√3 = 0.58

Gives safe and economic design.

Agrees better with experimental data for ductile materials.

It cannot be applied materials under hydrostatic stress condition.

Adopted in Div-2,

edition 2019