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3 FAILURE THEORIES (From Shigley s Mechanical Engineering Design)

4 MSS theory is an acceptable but conservative predictor of failure; and since engineers are conservative by nature, it is quite often used. For a general state of stress, three principal stresses can be determined and ordered such that σ1 σ2 σ3.

Equation reduces to a yield condition of Case 2: σa 0 σb.

5 Assuming that σa σb, there are three cases to consider for plane stress: Case 1: σa σb 0. For this case, σ1 = σa and σ3 = 0. Equation reduces to a yield condition of Case 2: σa 0 σb. Here, σ1 = σa and σ3 = σb, and Eq. becomes Case 3: 0 σa σb. For this case, σ1 = 0 and σ3 = σb, and Eq. gives

6 The distortion-energy (DE) theory originated from the observation that ductile materials stressed hydrostatically exhibited yield strengths greatly in excess of the values given by the simple tension test.

7 For the element showed the strain energy per unit volume is The strain energy for producing only volume change Then the distortion energy is obtained by subtracting from u

8 For the simple tensile test, at yield, σ1 = Sy and σ2 = σ3 = 0, so the distortion energy is Before we saw that So for the general state of stress given, yield is predicted if equals or exceeds This gives

A single, equivalent, or effective stress for the entire general state of stress given by σ1, σ2, and σ3 is usually called the von Mises stress, σ, named

9 A single, equivalent, or effective stress for the entire general state of stress given by σ1, σ2, and σ3 is usually called the von Mises stress, σ, named after Dr. R. von Mises, who contributed to the theory. For plane stress, let σa and σb be the two nonzero principal stresses then the von Mises stress is given by:

10 The equations given allow the most complicated stress situation to be represented by a single quantity, the von Mises stress, which then can be compared against the yield strength of the material through The distortion-energy theory predicts no failure under hydrostatic stress and agrees well with all data for ductile behavior. Hence, it is the most widely used theory for ductile materials and is recommended for design problems unless otherwise specified.

11 The shear yield strength predicted by the distortion-energy theory is which as stated earlier, is about 15 percent greater than the 0.5 Sy predicted by the MSS theory.

12 Not all materials have compressive strengths equal to their corresponding tensile values. For example, the yield strength of magnesium alloys in compression may be as little as 50 percent of their yield strength in tension. Coulomb-Mohr theory or the internal-friction theory

14 For design equations, incorporating the factor of safety n, divide all strengths by n.

15 Shah (2009)

16 FAILURE THEORIES (From Shigley s Mechanical Engineering Design)

18 The maximum-normal-stress (MNS) theory states that failure occurs whenever one of the three principal stresses equals or exceeds the strength. The principal stresses for a general stress state in the ordered form σ1 σ2 σ3. This theory then predicts that failure occurs whenever where Sut and Suc are the ultimate tensile and compressive strengths, respectively, given as positive quantities. For plane stress, with σa σb, σa Sut or σb Suc

19 Graph of maximum-normal stress (MNS) theory of failure for plane stress states. Stress states that plot inside the failure locus are safe.

23 Esfuerzos principales

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