Questions and Answers
What is the maximum shear stress in a tension test specimen when it begins to yield?
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
What is the expression for the maximum shear stress in a stress element according to the Maximum Shear Stress Theory?
t max = (σ1 - σ3) / 2
What is the purpose of incorporating a factor of safety in the Maximum Shear Stress Theory?
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What is the expression for the factor of safety in the Maximum Shear Stress Theory?
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What is the purpose of expressing τmax in terms of principal stresses in the Maximum Shear Stress Theory?
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What is the assumption made in the Maximum Shear Stress Theory to simplify the calculation of maximum shear stress?
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What is the ordering of the principal stresses in the Maximum Shear Stress Theory?
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What is the expression for B1C1 in terms of St?
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What is the condition for Case 1 in the Coulomb-Mohr Theory?
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What is the condition for Case 1 in the Maximum Shear Stress Theory?
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What is the equation for Case 2 in the Maximum Shear Stress Theory?
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What is the value of σ3 in Case 1?
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What is the main idea behind the Distortion Energy Failure Theory?
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What is the expression for B2C2 in terms of s1 and s3?
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What is the condition for Case 3 in the Coulomb-Mohr Theory?
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What is a characteristic of the Maximum Shear Stress Theory?
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What is the main assumption of the Coulomb-Mohr Theory?
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What is the value of σ1 in Case 3?
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What is the expression for B3C3 in terms of Sc?
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What is the condition for Case 3 in the Maximum Shear Stress Theory?
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What is the equation number for the expression of s1 and s3 in terms of St and Sc?
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What is the equation for Case 3 in the Maximum Shear Stress Theory?
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What is a characteristic of the Distortion Energy Failure Theory?
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What is the main difference between the Coulomb-Mohr Theory and the Maximum Normal Stress Theory?
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What is the equation for the Maximum Normal Stress Theory in terms of the principal stress, σ1, and the ultimate strength, Sut?
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What is the modification made to the Coulomb-Mohr Theory to better fit the data in the 4th quadrant?
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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?
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What is the main limitation of the Maximum Normal Stress Theory?
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What is the main difference between the Coulomb-Mohr Theory and the Modified Mohr criteria?
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What is the equation for the Maximum Normal Stress Theory in terms of the principal stress, σB, and the ultimate strength, Suc?
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What is the main advantage of the Coulomb-Mohr Theory?
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What is the condition for the Modified-Mohr Theory when σA ≥ σB ≥ 0?
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What is the expression for sB when sA ≥ 0 ≥ sB?
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What is the condition for the Modified-Mohr Theory when 0 ≥ σA ≥ σB?
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What is the expression for sA when sA ≥ 0 ≥ sB and £1?
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What is the expression for sA when sA ≥ 0 ≥ sB and > 1?
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What is the expression for sB when sA ≥ 0 ≥ sB and £1?
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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.
