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Tresca Yield Criterion (Maximum Shear Stress Theory)

Predicts material yielding when the maximum shear stress reaches half the yield strength in uniaxial tension.

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τ_{ma x} = \frac{σ _{1} - σ _{3}}{2} = \frac{σ _{y}}{2}

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Core idea

Overview

The Tresca Yield Criterion, also known as the Maximum Shear Stress Theory, states that yielding of a ductile material begins when the maximum shear stress in the material reaches a critical value. This critical value is defined as half the yield strength ($\sigma_y$) obtained from a simple uniaxial tension test. It is expressed as $\tau_{max} = (\sigma_1 - \sigma_3)/2 = \sigma_y/2$, where $\sigma_1$ and $\sigma_3$ are the maximum and minimum principal stresses, respectively. This criterion is often used for ductile materials and provides a conservative estimate for yielding compared to the Von Mises criterion.

When to use: Use this criterion for predicting the onset of yielding in ductile materials under complex stress states, especially when a conservative design approach is preferred. It is particularly applicable when the material's behavior is dominated by shear stress, such as in torsion or thin-walled pressure vessels.

Why it matters: Predicting material yielding is crucial for ensuring structural integrity and preventing catastrophic failures in engineering components. The Tresca criterion allows engineers to design parts that can safely withstand applied loads without permanent deformation, which is vital in fields like mechanical, civil, and aerospace engineering for components ranging from shafts to pressure vessels.

Symbols

Variables

$τ_{ma x}$ = Maximum Shear Stress, $σ_{1}$ = Maximum Principal Stress, $σ_{3}$ = Minimum Principal Stress, $σ_{y}$ = Yield Strength (Uniaxial)

τ_{ma x}

Maximum Shear Stress

MPa

σ_{1}

Maximum Principal Stress

MPa

σ_{3}

Minimum Principal Stress

MPa

σ_{y}

Yield Strength (Uniaxial)

MPa

Walkthrough

Derivation

Formula: Tresca Yield Criterion

The Tresca yield criterion states that yielding occurs when the maximum shear stress in a material reaches half of its uniaxial yield strength.

The material is ductile.
The material exhibits isotropic behavior (properties are uniform in all directions).
The material's yield strength in compression is equal to its yield strength in tension.

Mohr's Circle for Shear Stress:

For any general 3D stress state, the maximum shear stress ( $τ_{ma x}$ ) occurs on planes at 45 degrees to the principal planes and is equal to half the difference between the maximum ( $σ_{1}$ ) and minimum ( $σ_{3}$ ) principal stresses. This is a fundamental result from Mohr's Circle analysis.

τ_{ma x} = \frac{σ _{1} - σ _{3}}{2}

Uniaxial Tension Test:

Consider a simple uniaxial tension test where a material yields at a stress $σ_{y}$ . In this state, the principal stresses are $σ_{1} = σ_{y}$ , $σ_{2} = 0$ , and $σ_{3} = 0$ . Applying the maximum shear stress formula from Mohr's Circle to this state gives $τ_{ma x, y i e l d} = \frac{σ _{y} - 0}{2} = \frac{σ _{y}}{2}$ .

σ_{1} = σ_{y}, σ_{2} = 0, σ_{3} = 0

Tresca Criterion Formulation:

The Tresca criterion postulates that yielding in any general stress state will occur when the maximum shear stress ( $τ_{ma x}$ ) in that state reaches the same critical value as observed during uniaxial yielding. Therefore, the condition for yielding is $\frac{σ _{1} - σ _{3}}{2} = \frac{σ _{y}}{2}$ .

τ_{ma x} = \frac{σ _{1} - σ _{3}}{2} = \frac{σ _{y}}{2}

Result

τ_{ma x} = \frac{σ _{1} - σ _{3}}{2} = \frac{σ _{y}}{2}

Source: Beer, F. P., Johnston Jr., E. R., DeWolf, J. T., & Mazurek, D. F. (2012). Mechanics of Materials (6th ed.). McGraw-Hill. Chapter 8: Theories of Failure.

Visual intuition

Graph

The graph is a straight line with a positive slope, indicating that the maximum shear stress increases steadily as the maximum principal stress increases. For an engineering student, this linear relationship means that doubling the maximum principal stress results in a proportional increase in the maximum shear stress, highlighting how stress states directly influence material failure. The most important feature of this curve is that the constant relationship between the variables remains unchanged regardless of the magnitude of the stresses, demonstrating a predictable and consistent transition toward the yield limit.

Graph type: linear

Why it behaves this way

Intuition

The Tresca criterion visualizes material yielding as occurring when the radius of the largest Mohr's circle (representing the maximum shear stress)

τ_{ma x}

Maximum shear stress experienced within the material

This is the largest 'cutting' or 'sliding' force per unit area acting internally. When it exceeds a critical value, the material yields.

σ_{1}

Maximum principal stress

The largest normal stress (tension or compression) acting on a plane where shear stress is zero. It represents the most extreme pulling or pushing force within the material.

σ_{3}

Minimum principal stress

The smallest normal stress (tension or compression) acting on a plane where shear stress is zero. It represents the least extreme pulling or pushing force within the material.

σ_{y}

Yield strength of the material from a uniaxial tension test

The stress level at which a material begins to deform plastically (permanently) when pulled in one direction. It serves as a benchmark for the material's strength limit before permanent deformation.

Signs and relationships

(\sigma_1 - \sigma_3): This difference represents the diameter of the largest Mohr's circle for the given stress state. A larger difference indicates a greater range of normal stresses, which directly corresponds to a larger maximum shear
/2: Dividing the difference between the maximum and minimum principal stresses by two yields the radius of the largest Mohr's circle, which is precisely the maximum shear stress.

Free study cues

Insight

Canonical usage

All terms in the Tresca criterion represent stress and must be expressed in consistent units of force per unit area to maintain dimensional homogeneity.

Common confusion

Forgetting that the Tresca criterion compares the maximum shear stress (half the difference of principal stresses) to half the uniaxial yield strength, or mixing units like MPa and GPa in the same calculation.

Dimension note

This equation is not dimensionless; it is a relationship between stress quantities.

Unit systems

$s i g m a_{1}, s i g m a_{3}$ MPa or ksi - Principal stresses must be ordered such that sigma_1 is the maximum and sigma_3 is the minimum algebraic stress.

$s i g m a_{y}$ MPa or ksi - The yield strength is a material property determined from a uniaxial tension test.

$t a u_{m} a x$ MPa or ksi - Represents the maximum shear stress acting on any plane at a point.

Ballpark figures

Quantity:

Study smarter

Tips

Always identify the principal stresses ( $σ_{1}, σ_{2}, σ_{3}$ ) correctly and ensure $σ_{1} \geq σ_{2} \geq σ_{3}$ for accurate calculation of maximum shear stress.
The Tresca criterion is generally more conservative than the Von Mises criterion, meaning it predicts yielding at lower stress levels.
Remember that $σ_{y}$ is the yield strength from a uniaxial tension test.
Ensure consistent units for all stress values.

Common questions

Frequently Asked Questions

The Tresca yield criterion states that yielding occurs when the maximum shear stress in a material reaches half of its uniaxial yield strength.

Use this criterion for predicting the onset of yielding in ductile materials under complex stress states, especially when a conservative design approach is preferred. It is particularly applicable when the material's behavior is dominated by shear stress, such as in torsion or thin-walled pressure vessels.

Predicting material yielding is crucial for ensuring structural integrity and preventing catastrophic failures in engineering components. The Tresca criterion allows engineers to design parts that can safely withstand applied loads without permanent deformation, which is vital in fields like mechanical, civil, and aerospace engineering for components ranging from shafts to pressure vessels.

Always identify the principal stresses ($\sigma_1, \sigma_2, \sigma_3$) correctly and ensure $\sigma_1 \ge \sigma_2 \ge \sigma_3$ for accurate calculation of maximum shear stress. The Tresca criterion is generally more conservative than the Von Mises criterion, meaning it predicts yielding at lower stress levels. Remember that $\sigma_y$ is the yield strength from a uniaxial tension test. Ensure consistent units for all stress values.

References

Sources

Mechanics of Materials by Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, and David F. Mazurek
Mechanics of Materials by R. C. Hibbeler
Wikipedia: Tresca criterion
Shigley's Mechanical Engineering Design
Mechanics of Materials (Hibbeler)
Wikipedia: Tresca yield criterion
Mechanics of Materials by Beer, Johnston, DeWolf, Mazurek
Fundamentals of Machine Component Design by Juvinall and Marshek