Yield and Failure Criteria – Tresca, von Mises & Rankine

Engineering components are frequently subjected to combined normal and shear stresses rather than simple uniaxial loading. A yield or failure criterion is therefore required to compare a multiaxial stress state with material strength data obtained from standard tests.

Need help choosing a mechanics or stress analysis method? See the Mechanics and Stress Analysis Guide for guidance on forces, motion, loading, stress, strain, yielding, buckling, pressure vessels, rotating components, torsion and plate analysis.

This page summarises the classical maximum principal stress, maximum shear stress, distortion energy and maximum strain energy criteria. The appropriate criterion depends on the material behaviour and the type of failure being assessed.

Introduction
Principal Stresses
Maximum Principal Stress Theory – Rankine
Maximum Shear Stress Theory – Tresca
Distortion Energy Theory – von Mises
Maximum Strain Energy Theory
Comparison of Criteria

Introduction

Under a simple uniaxial tensile test, yielding or fracture can be related directly to measured material properties such as yield strength, proof strength and ultimate tensile strength. Under a multiaxial state of stress, failure cannot generally be assessed by considering only one Cartesian stress component.

Failure theories reduce the combined stress state to a criterion that can be compared with an appropriate material strength obtained from a simpler test.

Ductile and Brittle Behaviour

Metals are often broadly classified as ductile or brittle according to their deformation and fracture behaviour.

Ductile metals normally undergo appreciable plastic deformation before fracture. Some exhibit a distinct yield point, while others are characterised using a specified proof strength. A common engineering value is the 0.2% proof stress.

Brittle materials generally exhibit relatively little plastic deformation before fracture. Cast irons, ceramics and some high-strength or low-temperature metal conditions can display brittle behaviour, although actual behaviour depends on material condition, stress state, temperature, loading rate, geometry and environment.

Terminology: For ductile materials, the usual design event considered by Tresca and von Mises is the onset of yielding, not ultimate fracture. For brittle materials, fracture criteria based on principal stresses or other brittle-fracture models may be more appropriate.

Common Classical Criteria

Maximum principal stress theory (Rankine) – principally of historical and practical relevance to brittle failure where normal stress is dominant.
Maximum shear stress theory (Tresca-Guest) – widely used yield criterion for ductile materials and generally conservative relative to von Mises.
Distortion energy theory (von Mises-Hencky) – widely used yield criterion for isotropic ductile metals.
Maximum strain energy theory – classical historical criterion based on total elastic strain energy; less commonly used for modern ductile-metal design.

Principal Stresses

At a point in a stressed body there are three mutually perpendicular principal planes on which the shear stress is zero. The corresponding normal stresses are the principal stresses:

σ_{1} \geq σ_{2} \geq σ_{3}

For plane stress with Cartesian stresses σ_x, σ_y and τ_xy, the two in-plane principal stresses are:

σ_{1, 2} = \frac{σ_{x} + σ_{y}}{2} \pm \sqrt{{(\frac{σ_{x} - σ_{y}}{2})}^{2} + τ_{x y}^{2}}

For plane stress, the third principal stress is zero:

σ_{3} = 0

when the two calculated in-plane values are labelled separately from the out-of-plane stress. For use with three-dimensional criteria, all three values must then be reordered so that σ₁ ≥ σ₂ ≥ σ₃.

See the RoyMech Mohr's Circle reference for stress transformation and graphical determination of principal stresses.

Maximum Principal Stress Theory – Rankine

The maximum principal stress theory is commonly associated with Rankine. In its simplest form, failure is predicted when a principal normal stress reaches the corresponding limiting stress measured under uniaxial loading.

For a material having different tensile and compressive strengths, the principal stresses should be checked against the appropriate tensile and compressive limits:

σ_{max} \leq \frac{S_{t}}{n}

| σ_{min} | \leq \frac{S_{c}}{n}

where:

S_t = allowable reference tensile strength;
S_c = allowable reference compressive strength;
n = factor of safety.

For a material whose tensile limit governs, the corresponding factor of safety may be written:

n = \frac{S_{t}}{σ_{max}}

Application: Rankine's criterion is not normally preferred for predicting yielding of ductile metals under general multiaxial stress. It is more relevant as a simple normal-stress criterion for brittle behaviour, but real brittle fracture can require more sophisticated criteria, particularly where tensile and compressive strengths differ substantially.

Maximum Shear Stress Theory – Tresca

The maximum shear stress theory is associated with Tresca and Guest and is widely used for predicting the onset of yielding in ductile materials.

In a uniaxial tensile test at yield:

σ_{1} = S_{y}, σ_{2} = σ_{3} = 0

The maximum shear stress at yield is therefore:

τ_{max,y} = \frac{S_{y}}{2}

For a general three-dimensional stress state with:

σ_{1} \geq σ_{2} \geq σ_{3}

the maximum shear stress is:

τ_{max} = \frac{σ_{1} - σ_{3}}{2}

Tresca predicts yielding when:

σ_{1} - σ_{3} \geq S_{y}

The Tresca equivalent stress can therefore be defined as:

σ_{Tresca} = \max (| σ_{1} - σ_{2} |, | σ_{2} - σ_{3} |, | σ_{3} - σ_{1} |)

and the factor of safety against yielding is:

n = \frac{S_{y}}{σ_{Tresca}}

Combined Direct Stress and Shear Stress

For the common plane-stress case consisting of a direct stress σ_x combined with shear stress τ_xy, with σ_y = 0, the principal stresses are:

σ_{1, 2} = \frac{σ_{x}}{2} \pm \sqrt{{(\frac{σ_{x}}{2})}^{2} + τ_{x y}^{2}}

For this stress state, the Tresca equivalent stress becomes:

σ_{Tresca} = \sqrt{σ_{x}^{2} + 4 τ_{x y}^{2}}

and:

n = \frac{S_{y}}{\sqrt{σ_{x}^{2} + 4 τ_{x y}^{2}}}

Engineering note: Tresca is often convenient for hand calculations and is conservative relative to von Mises for many multiaxial ductile-metal stress states. Both criteria give the same result for simple uniaxial tension.

Distortion Energy Theory – von Mises

The distortion energy theory, commonly called the von Mises or von Mises-Hencky criterion, is widely used to predict yielding of homogeneous isotropic ductile metals.

Elastic strain energy can be separated into a volumetric component associated with change in volume and a distortional component associated with change in shape. The von Mises criterion relates yielding to the distortional component of the stress state.

Principal-Stress Form

The von Mises equivalent stress in terms of the three principal stresses is:

σ_{vm} = \sqrt{\frac{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2}}{2}}

Yielding is predicted when:

σ_{vm} \geq S_{y}

and the corresponding factor of safety against initial yielding is:

n = \frac{S_{y}}{σ_{vm}}

Three-Dimensional Cartesian Stress Form

For normal stresses σ_x, σ_y, σ_z and shear stresses τ_xy, τ_yz and τ_zx:

σ_{vm}^{2} = \frac{1}{2} [{(σ_{x} - σ_{y})}^{2} + {(σ_{y} - σ_{z})}^{2} + {(σ_{z} - σ_{x})}^{2} + 6 (τ_{x y}^{2} + τ_{y z}^{2} + τ_{z x}^{2})]

Plane Stress

For plane stress, where σ_z = τ_yz = τ_zx = 0:

σ_{vm} = \sqrt{σ_{x}^{2} - σ_{x} σ_{y} + σ_{y}^{2} + 3 τ_{x y}^{2}}

Combined Direct Stress and Shear

For the common case:

σ_{y} = 0

with direct stress σ_x and shear stress τ_xy:

σ_{vm} = \sqrt{σ_{x}^{2} + 3 τ_{x y}^{2}}

and:

n = \frac{S_{y}}{\sqrt{σ_{x}^{2} + 3 τ_{x y}^{2}}}

Pure Shear

For a state of pure shear τ, the von Mises equivalent stress is:

σ_{vm} = \sqrt{3} τ

Therefore von Mises predicts shear yielding at:

τ_{y} = \frac{S_{y}}{\sqrt{3}} \approx 0.577 S_{y}

By comparison, Tresca predicts:

τ_{y} = \frac{S_{y}}{2} = 0.5 S_{y}

This illustrates the greater conservatism of Tresca for this stress state.

Maximum Strain Energy Theory

The maximum strain energy theory is a classical elastic failure criterion associated with total elastic strain energy. It assumes that failure occurs when the total strain energy per unit volume reaches the value corresponding to failure or yielding in a uniaxial tensile test.

For an isotropic linear-elastic material subjected to principal stresses σ₁, σ₂ and σ₃, the total elastic strain energy per unit volume is:

U = \frac{1}{2 E} (σ_{1}^{2} + σ_{2}^{2} + σ_{3}^{2} - 2 ν (σ_{1} σ_{2} + σ_{2} σ_{3} + σ_{3} σ_{1}))

For uniaxial yielding at stress S_y:

U_{y} = \frac{S_{y}^{2}}{2 E}

The maximum strain energy criterion therefore predicts yielding when:

σ_{1}^{2} + σ_{2}^{2} + σ_{3}^{2} - 2 ν (σ_{1} σ_{2} + σ_{2} σ_{3} + σ_{3} σ_{1}) \geq S_{y}^{2}

Historical criterion: The maximum strain energy theory is important in the development of classical failure theories, but it is generally not preferred for predicting yielding of ductile metals. Tresca and particularly von Mises are more commonly used for that purpose.

Comparison of Yield and Failure Criteria

Criterion	Primary quantity	Typical use	Comment
Maximum principal stress (Rankine)	Largest principal normal stress	Brittle-material assessment	Simple criterion; tensile and compressive strength should be considered separately where appropriate.
Maximum shear stress (Tresca)	Maximum difference between principal stresses	Ductile-metal yielding	Simple and conservative; convenient for hand calculations.
Distortion energy (von Mises)	Distortional component of stress state	Ductile-metal yielding	Widely used and generally agrees well with yielding of isotropic ductile metals.
Maximum strain energy	Total elastic strain energy	Classical / historical analysis	Less commonly used for modern ductile-metal yield assessment.

Criterion selection: No single failure theory describes every material and failure mechanism. Yield criteria such as Tresca and von Mises should not be used as universal fracture, fatigue, buckling, creep or brittle-fracture criteria. The governing failure mechanism and appropriate material data must be identified for the application.

Related RoyMech Mechanics References

Component Loading – loading modes, stress, strain and component response.
Stress and Strain – direct and shear stress, strain and elastic constants.
Mohr's Circle – principal stresses, maximum shear stress and stress transformation.
Stress-Strain Curve – yield strength, proof strength and tensile behaviour.
Buckling Notes – instability of compression members.
Thick Cylinders – multiaxial stresses in thick-walled pressure vessels.
Rotating Discs and Cylinders – radial and circumferential stresses produced by rotation.
Torsion – torsional shear stress and deformation.
Loaded Flat Plates – plate stresses and deflections.

Other Mechanics References

External Failure-Criteria References

Failure Criteria – eFunda – engineering reference covering common material failure criteria.
Chapter 15: Theories of Yielding – Victoria University – reference material on classical yield theories.
Solid Mechanics – Wikibooks – general solid-mechanics reference.

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Yield and Failure Criteria for Multiaxial Stress

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