Maximum Shear Stress Theory Quiz & Flashcards

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Questions and Answers

What is the maximum shear stress in a tension test specimen when it begins to yield?

Sy / 2 (correct)

Sy / 4

Sy * 2

What is the condition for yielding according to the Maximum Shear Stress Theory?

When the maximum shear stress in a stress element exceeds half the yield stress (correct)

When the maximum shear stress in a stress element is less than the yield stress

When the maximum shear stress in a stress element equals the yield stress

When the maximum shear stress in a stress element exceeds the yield stress

What is the expression for the maximum shear stress in a stress element according to the Maximum Shear Stress Theory?

t max = σ1 - σ3

t max = σ1 + σ3

t max = (σ1 - σ3) / 2 (correct)

t max = (σ1 + σ3) / 2

What is the purpose of incorporating a factor of safety in the Maximum Shear Stress Theory?

To account for uncertainties in the design Signup and view all the answers

What is the expression for the factor of safety in the Maximum Shear Stress Theory?

n = Sy / (2 * t max) Signup and view all the answers

What is the purpose of expressing τmax in terms of principal stresses in the Maximum Shear Stress Theory?

To compare with experimental data Signup and view all the answers

What is the assumption made in the Maximum Shear Stress Theory to simplify the calculation of maximum shear stress?

Plane stress state Signup and view all the answers

What is the ordering of the principal stresses in the Maximum Shear Stress Theory?

σ1 ≥ σ2 ≥ σ3 Signup and view all the answers

What is the expression for B1C1 in terms of St?

St Signup and view all the answers

What is the condition for Case 1 in the Coulomb-Mohr Theory?

σA ≥ σB ≥ 0 Signup and view all the answers

What is the condition for Case 1 in the Maximum Shear Stress Theory?

σA ≥ σB ≥ 0 Signup and view all the answers

What is the equation for Case 2 in the Maximum Shear Stress Theory?

σA - σB ≥ Sy Signup and view all the answers

What is the value of σ3 in Case 1?

0 Signup and view all the answers

What is the main idea behind the Distortion Energy Failure Theory?

Yielding occurs when the distortion strain energy per unit volume reaches a certain value Signup and view all the answers

What is the expression for B2C2 in terms of s1 and s3?

(s1 - s3)^2 Signup and view all the answers

What is the condition for Case 3 in the Coulomb-Mohr Theory?

0 ≥ σA ≥ σB Signup and view all the answers

What is a characteristic of the Maximum Shear Stress Theory?

It is commonly used for design situations Signup and view all the answers

What is the main assumption of the Coulomb-Mohr Theory?

The failure criteria can be derived from the geometry Signup and view all the answers

What is the value of σ1 in Case 3?

0 Signup and view all the answers

What is the expression for B3C3 in terms of Sc?

Sc^2 Signup and view all the answers

What is the condition for Case 3 in the Maximum Shear Stress Theory?

0 ≥ σA ≥ σB Signup and view all the answers

What is the equation number for the expression of s1 and s3 in terms of St and Sc?

Eq.(5-22) Signup and view all the answers

What is the equation for Case 3 in the Maximum Shear Stress Theory?

σB ≤ -Sy Signup and view all the answers

What is a characteristic of the Distortion Energy Failure Theory?

It is based on the idea that yielding is primarily affected by the distortion energy Signup and view all the answers

What is the main difference between the Coulomb-Mohr Theory and the Maximum Normal Stress Theory?

The Coulomb-Mohr Theory has different strengths for compression and tension. Signup and view all the answers

What is the equation for the Maximum Normal Stress Theory in terms of the principal stress, σ1, and the ultimate strength, Sut?

σ1 ³ Sut Signup and view all the answers

What is the modification made to the Coulomb-Mohr Theory to better fit the data in the 4th quadrant?

Modified Mohr criteria Signup and view all the answers

What is the equation for the Maximum Normal Stress Theory in terms of the principal stress, σA, and the ultimate strength, Sut, incorporating a design factor?

σA ³ Sut/n Signup and view all the answers

What is the main limitation of the Maximum Normal Stress Theory?

It is not recommended for use Signup and view all the answers

What is the main difference between the Coulomb-Mohr Theory and the Modified Mohr criteria?

The Modified Mohr criteria adjusts the Coulomb-Mohr Theory to better fit the data in the 4th quadrant Signup and view all the answers

What is the equation for the Maximum Normal Stress Theory in terms of the principal stress, σB, and the ultimate strength, Suc?

σB ³ -Suc Signup and view all the answers

What is the main advantage of the Coulomb-Mohr Theory?

It has different strengths for compression and tension Signup and view all the answers

What is the condition for the Modified-Mohr Theory when σA ≥ σB ≥ 0?

sA = Sut/(n*sA) Signup and view all the answers

What is the expression for sB when sA ≥ 0 ≥ sB?

sB = - Suc/(n*sA) Signup and view all the answers

What is the condition for the Modified-Mohr Theory when 0 ≥ σA ≥ σB?

sB = - Suc/(n*sA) Signup and view all the answers

What is the expression for sA when sA ≥ 0 ≥ sB and £1?

sA = Sut/(n*sA) Signup and view all the answers

What is the expression for sA when sA ≥ 0 ≥ sB and > 1?

sA = Sut/(n*(sA - sB)) Signup and view all the answers

What is the expression for sB when sA ≥ 0 ≥ sB and £1?

sB = Suc/(n*sA) Signup and view all the answers

Study Notes

Maximum Shear Stress Theory (MSS)

Yielding begins when the maximum shear stress in a stress element exceeds Sy/2.
For a tension test specimen, the maximum shear stress is s1/2.
To find the maximum shear stress, use Mohr's circle and compare to Sy/2.
The maximum shear stress can be expressed as: τmax = (s1 - s3)/2 or τmax = Sy/2.
Incorporating a factor of safety (n), the equation becomes: τmax = Sy/2n or τmax = (s1 - s3)/2n.

Graphical Representation of MSS

The maximum shear stress can be plotted on principal stress axes.
Three cases are considered: σA ≥ σB ≥ 0, σA ≥ 0 ≥ σB, and 0 ≥ σA ≥ σB.
The equations for each case are:
- Case 1: σA ≥ σB ≥ 0, σA ≥ Sy.
- Case 2: σA ≥ 0 ≥ σB, σA - σB ≥ Sy.
- Case 3: 0 ≥ σA ≥ σB, σB ≤ -Sy.
The plot forms an envelope, with the inside area being the predicted safe zone.

Comparison to Experimental Data

The MSS theory is conservative in all quadrants.
It is commonly used for design situations.

Distortion Energy (Von Mises) Failure Theory

The theory suggests that yielding occurs when the distortion strain energy per unit volume reaches the distortion strain energy per unit volume for yield in simple tension or compression of the same material.
The theory is based on the observation that ductile materials stressed hydrostatically (equal principal stresses) exhibited yield strengths greatly in excess of expected values.

Coulomb-Mohr Theory

The theory is based on the geometry of the failure criteria.
The failure criteria can be expressed as: (s1 - s3)/2 - St/2 = ±(Sc/2)^2.
The equation can be plotted on principal stress axes, with three cases considered:
- Case 1: σA ≥ σB ≥ 0, sA ≥ St.
- Case 2: σA ≥ 0 ≥ σB, sA - sB ≥ Sc/St.
- Case 3: 0 ≥ σA ≥ σB, sB ≤ -Sc.
The theory is similar to MSS, but with different strengths for compression and tension.

Maximum Normal Stress Theory

The theory states that failure occurs when the maximum principal stress in a stress element exceeds the strength.
The failure criteria can be expressed as: s1 ≥ Sut or s3 ≤ -Suc.
Incorporating a design factor (n), the equation becomes: sA = Sut/n or sB = -Suc/n.
The theory is not recommended for use due to being unsafe in part of the fourth quadrant.

Modified Mohr Criteria

The Coulomb-Mohr theory is conservative in the fourth quadrant.
The Modified Mohr criteria adjust to better fit the data in the fourth quadrant.
The failure criteria can be expressed as:
- Quadrant condition: σA ≥ σB ≥ 0, failure criteria: sA = Sut/n.
- Quadrant condition: σA ≥ 0 ≥ σB, failure criteria: sA = (Sut - Suc) / n.
- Quadrant condition: 0 ≥ σA ≥ σB, failure criteria: sB = -Suc/n.

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Description

Understand the Maximum Shear Stress Theory (MSS) in mechanics of materials, including its definition, calculation, and graphical representation. Learn how to apply the theory to predict yielding in materials.

Mechanics of Materials: Maximum Shear Stress Theory

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