How Do You Calculate Principal Stresses and Orientations in a 2D Stress State?

[rock.freak667]

Homework Statement

In an element of material, subjected to general two-dimensional stress one axial stress is 66N/mm² (tensile) and the shear stress is 48N/mm². Calculate the values and directions of the principal stresses and the normal and shear stresses on planes equally inclined to the axes if the other axial stress of 22N/mm² is a) tensile and b)compressive.

Homework Equations

[tex]\sigma_n=\frac{1}{2}(\sigma_x+\sigma_y)+\frac{1}{2}(\sigma_x-\sigma_y)cos2\theta+\tau_{xy}sin2\theta[/tex]

[tex]\tau_n= \frac{1}{2}(\sigma_x-\sigma_y)sin 2\theta-\tau_{xy}cos2\theta[/tex]

The Attempt at a Solution

Well I used the fact that the principal stresses are given by

[tex]\sigma_1,\sigma_2=\frac{1}{2}(\sigma_x+\sigma_y) \pm \sqrt{(\sigma_x-\sigma_y)^2+4\tau_{xy}^2[/tex]

and got the values to be 96.8 N/mm² and -8.80N/mm²

Then I used the fact that

[tex]tan2\theta=\frac{2\tau_{xy}}{\sigma_x-\sigma_y}[/tex]

and got [itex]\theta=32.69[/itex]. That part I got out, but I don't know how to get the normal and shear stresses on planes equally inclined to the axes.

I am not sure about what the question means by equally inclinded to the axes.

EDIT:solved.

 

[AvroArrow]

\sigma_n=\frac{1}{2}(\sigma_x+\sigma_y)+\frac{1}{2}(\sigma_x-\sigma_y)cos2\theta+\tau_{xy}sin2\theta\tau_n= \frac{1}{2}(\sigma_x-\sigma_y)sin 2\theta-\tau_{xy}cos2\thetaFor a tensile stress of 22N/mm2,\sigma_n=74.8 N/mm2\tau_n=58.4 N/mm2For a compressive stress of 22N/mm2,\sigma_n=53.2 N/mm2\tau_n=-38.4 N/mm2

 

[nlightner]

To find the normal and shear stresses on planes equally inclined to the axes, you can use the equations provided in the homework statement. Substitute the values for the principal stresses and the angle \theta that you calculated to find the normal and shear stresses for both cases (tensile and compressive).

For example, for the case where the other axial stress is tensile, you can use the equation \sigma_n=\frac{1}{2}(\sigma_x+\sigma_y)+\frac{1}{2}(\sigma_x-\sigma_y)cos2\theta+\tau_{xy}sin2\theta. Substitute the values for \sigma_x, \sigma_y, and \tau_{xy}, along with the value of \theta that you calculated, to find the normal stress on a plane equally inclined to the axes.

Similarly, for the shear stress, you can use the equation \tau_n= \frac{1}{2}(\sigma_x-\sigma_y)sin 2\theta-\tau_{xy}cos2\theta and substitute the values to find the shear stress on a plane equally inclined to the axes.

Repeat this process for the compressive case to find the values for the normal and shear stresses on planes equally inclined to the axes.