Mechanics of Materials II - Mohr's Circle

[Double A]

Homework Statement

For an element [Stress Block], determine the range of values of $$\tau_{xy}$$ for which the maximum tensile stress is equal to or less than 60 MPa.

Given in the provided figure:
$$\sigma_x$$ = -120 MPa
$$\sigma_y$$ = -60 MPa

Homework Equations

$$\sigma_{ave} = \frac{\sigma_x + \sigma_y}{2}$$

$$R = \tau_{max} = \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$$

$$\sigma_{max,min} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$$

$$\tan{2\theta_p} = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$$

The Attempt at a Solution

I have drawn a representation of Mohr's Circle using the provided data. I am confused with the statement saying "tensile stress" when the provided stresses are in compression. They represent the shear stress in the positive direction. I am also unclear about how to approch this beyond my Mohr's circle figure.

I'm not sure if this is correct but I tried this:

$$\sigma_{max} = \frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$$

Solve for $$\tau_{xy}$$ and inputing known values:

$$\tau_{xy} = \pm$$59.9 MPa

 

[LightHero]

Therefore, the range of values of \tau_{xy} for which the maximum tensile stress is equal to or less than 60 MPa would be -59.9 MPa to + 59.9 MPa.

 

[piercas]

Therefore, the range of values for \tau_{xy} is -59.9 MPa \leq \tau_{xy} \leq 59.9 MPa.

As for the statement about "tensile stress" in the problem, it is likely a mistake or typo. The provided stresses are indeed in compression, so the maximum tensile stress would actually be the minimum compressive stress, which is -120 MPa in this case. Therefore, the problem can be solved using the given stresses as is.