The state of plane stress shown occurs at a critical point of a steel machine component. As a result of several tensile tests, it has been found that the tensile yield strength is sY = 250 MPa for the grade of steel used. Determine the factor of safety with respect to yield, using (a) the maximum

Question

The state of plane stress shown occurs at a critical point of a steel machine component. As a result of several tensile tests, it has been found that the tensile yield strength is [latex]s_Y[/latex] = 250 MPa for the grade of steel used. Determine the factor of safety with respect to yield, using (a) the maximum-shearing-stress criterion, and (b) the maximum-distortion-energy criterion.

Accepted Answer

Mohr’s Circle. We construct Mohr’s circle for the given state of stress and find

[latex]s _{ ext {ave }}=O C=\frac{1}{2}\left( s _{x}+ s _{y} ight)=\frac{1}{2}(80-40)=20 MPa[/latex]

[latex]t _{m}=R=\sqrt{(C F)^{2}+(F X)^{2}}=\sqrt{(60)^{2}+(25)^{2}}=65 MPa[/latex]

Principal Stresses

[latex]s _{a}=O C+C A=20+65=+85 MPa[/latex]

[latex]s _{b}=O C-B C=20-65=-45 MPa[/latex]

a. Maximum-Shearing-Stress Criterion. Since for the grade of steel used the tensile strength is [latex]s _{Y}=250 MPa[/latex], the corresponding shearing stress at yield is

[latex]t _{Y}=\frac{1}{2} s _{Y}=\frac{1}{2}(250 MPa )=125 MPa[/latex]

For [latex]t _{m}=65 MPa[/latex]: [latex]F . S .=\frac{ t _{Y}}{ t _{m}}=\frac{125 MPa }{65 MPa }[/latex] [latex]F . S .=1.92[/latex]

b. Maximum-Distortion-Energy Criterion. Introducing a factor of safety into Eq. (7.26), we write

[latex]s_{a}^{2}- s _{a} s _{b}+ s _{b}^{2}< s _{Y}^{2}[/latex] (7.26)

[latex]s _{a}^{2}- s _{a} s _{b}+ s _{b}^{2}=\left(\frac{ s _{Y}}{F \cdot S .} ight)^{2}[/latex]

For [latex]s _{a}=+85 MPa , s _{b}=-45 MPa[/latex], and [latex]s_{Y}=250 MPa[/latex] , we have

[latex](85)^{2}-(85)(-45)+(45)^{2}=\left(\frac{250}{F \cdot S .} ight)^{2}[/latex]

[latex]114.3=\frac{250}{F . S}[/latex] [latex]F . S .=2.19[/latex]

Comment. For a ductile material with [latex]s _{Y}=250 MPa[/latex], we have drawn the hexagon associated with the maximum-shearing-stress criterion and the ellipse associated with the maximum-distortion-energy criterion. The given state of plane stress is represented by point H of coordinates [latex]s_{a}=85 MPa[/latex] and [latex]s _{b}=-45 MPa[/latex]. We note that the straight line drawn through points O and H intersects the hexagon at point T and the ellipse at point M. For each criterion, the value obtained for F.S. can be verified by measuring the line segments indicated and computing their ratios:

(a) [latex]F . S .=\frac{O T}{O H}=1.92[/latex] (b) [latex]F . S .=\frac{O M}{O H}=2.19[/latex]

Question 7.S-P.4: The state of plane stress shown occurs at a critical point o...

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