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Chapter 5

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Failures Resulting from Static Loading

Lecture 9

The McGraw-Hill Companies © 2012

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Distortion Energy (DE) Failure Theory For ductile materials

  • Also known as:
    • Octahedral Shear Stress
    • Shear Energy
    • Von Mises
    • Von Mises – Hencky

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Distortion Energy (DE) Failure Theory

  • Originated from observation that ductile materials stressed hydrostatically (equal principal stresses) exhibited yield strengths greatly in excess of expected values.
  • Theorizes that if strain energy is divided into hydrostatic volume changing energy and angular distortion energy, the yielding is primarily affected by the distortion energy.

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Distortion Energy (DE) Failure Theory

  • Theory: 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.

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Deriving the Distortion Energy

  • Hydrostatic stress is average of principal stresses

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  • Strain energy per unit volume,
  • Substituting Eq. (3–19) for principal strains into strain energy equation,

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Deriving the Distortion Energy

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  • Strain energy for producing only volume change is obtained by substituting σav for σ1, σ2, and σ3

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  • Substituting σav from Eq. (a),

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  • Obtain distortion energy by subtracting volume changing energy, Eq. (5–7), from total strain energy, Eq. (b)

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Deriving the Distortion Energy

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  • Tension test specimen at yield has σ1 = Sy and σ2 = σ3 =0
  • Applying to Eq. (5–8), distortion energy for tension test specimen is

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  • DE theory predicts failure when distortion energy, Eq. (5–8), exceeds distortion energy of tension test specimen, Eq. (5–9)

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Von Mises Stress

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  • Left hand side is defined as von Mises stress

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  • For plane stress, simplifies to

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  • In terms of xyz components, in three dimensions

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  • In terms of xyz components, for plane stress

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Distortion Energy Theory With Von Mises Stress

  • Von Mises Stress can be thought of as a single, equivalent, or effective stress for the entire general state of stress in a stress element.
  • Distortion Energy failure theory simply compares von Mises stress to yield strength.

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  • Introducing a design factor,

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  • Expressing as factor of safety,

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Failure Theory in Terms of von Mises Stress

  • Equation is identical to Eq. (5–10) from Distortion Energy approach
  • Identical conclusion for:
    • Distortion Energy
    • Octahedral Shear Stress
    • Shear Energy
    • Von Mises
    • Von Mises – Hencky

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DE Theory Compared to Experimental Data

  • Plot von Mises stress on principal stress axes to compare to experimental data (and to other failure theories)
  • DE curve is typical of data
  • Note that typical equates to a 50% reliability from a design perspective
  • Commonly used for analysis situations
  • MSS theory useful for design situations where higher reliability is desired

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Shear Strength Predictions

  • For pure shear loading, Mohr’s circle shows that σA = −σB = τ
  • Plotting this equation on principal stress axes gives load line for pure shear case
  • Intersection of pure shear load line with failure curve indicates shear strength has been reached
  • Each failure theory predicts shear strength to be some fraction of normal strength

Shigley’s Mechanical Engineering Design

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Shear Strength Predictions

  • For MSS theory, intersecting pure shear load line with failure line [Eq. (5–5)] results in

Shigley’s Mechanical Engineering Design

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Shear Strength Predictions

  • For DE theory, intersection pure shear load line with failure curve [Eq. (5–11)] gives

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  • Therefore, DE theory predicts shear strength as

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Example 5-1

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Example 5-1

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Example 5-1

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Example 5-1

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Example 5-1

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Example 5-1

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