9a. Failure Theories (SSES Ch. 9.09.2) • Types of Failure • Max. Normal Stresses • Max. Shear Stress • Max. Distortional Energy 
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Tresca  Maximum Shear Stress
 for Plane Stress only The Tresca Yield states that a ductile materials yields (fails) when the Maximum Shear Stress exceeds the shear strength t_{y} the material yields. The Maximum InPlane Shear Stress is the average of the inplane Principal Stresses. The TWO Maximum OutofPlane Shear Stresses are: Note that the OutofPlane Principal Stress (s_{III}) for the plane stress condition is zero. Failure occurs when the maximum of the three Maximum Shear Stresses reaches the shear yield stress, t_{y}. 
The above plot is a Failure Map.

von Mises  Maximum Distortion Energy
The von Mises Yield Criterion states that a ductile material fails (yields) when the von Mises Stress s_{o} exceeds the yield strength S_{y}. The von Mises Stress (or Equivalent Stress) is defined by: When s_{o} = S_{y} the material is deemed to have yielded. For plane stress (s_{z} = t_{xz} = t_{yz} = 0) the von Mises Failure Criterion reduces to:
Using the general relationship with s_{x} = s_{y} = 0, the von Mises criterion predicts that ratio of the axial yield strength to the shear yield strength is: S_{y} = 1.732t_{y.} From the Tresca condition the predicted relationship between the yield strength and shear yield strength is: S_{y} = 2t_{y}. In general, metals tend to follow the axial yield strengthshear yield strength relationship of von Mises, making von Mises more accurate. However, von Mises is slightly harder to use, and any system that falls within the Tresca boundary, also falls within the von Mises boundary. 
von Mises Failure Surface The above plot is a Failure Map. If the Inplane Principal
Stresses lie outside the shaded zone, failure occurs. The dashed lines indicate the Tresca failure surface.

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