## Determine the principal stresses, the maximum in-plane shear stress, and average normal stress of the element shown. Dra…

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Determine the principal stresses, the maximum in-plane shear stress, and average normal stress of the element shown. Draw Mohr’s circle and specify the orientation of the element in each case. 20 MPa 80 MPa 30 Mpa

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2 weeks 2023-03-15T10:00:11+00:00 1 Answer 0 views Teacher 0

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?_x = 20 MPa (normal stress)
?_y = 80 MPa (normal stress)
?_xy = ?_yx = 30 MPa (shear stress)

To find the principal stresses, we need to first calculate the determinant and trace of the stress matrix:

| ?_x ?_xy |
| ?_xy ?_y |

det = ?_x?_y – ?_xy?_xy = 20 MPa * 80 MPa – 30 MPa * 30 MPa = 1100 MPa^2

trace = ?_x + ?_y = 20 MPa + 80 MPa = 100 MPa

The principal stresses can then be found by solving the characteristic equation:

?^2 – trace*? + det = 0

?^2 – 100? + 1100 = 0

Solving for ?, we get:

?1 = 20 MPa
?2 = 80 MPa

Therefore, the principal stresses are ?_1 = 80 MPa and ?_2 = 20 MPa.

The maximum in-plane shear stress can be found using the formula:

?_max = (?_1 – ?_2) / 2

?_max = (80 MPa – 20 MPa) / 2 = 30 MPa

The average normal stress can be found using the formula:

?_avg = (?_x + ?_y) / 2

?_avg = (20 MPa + 80 MPa) / 2 = 50 MPa

To draw Mohr’s circle, we plot the normal stresses on the horizontal axis and the shear stresses on the vertical axis. We then plot the given point (?_x, ?_xy) and (?_y, ?_xy) and draw a circle passing through these points. The intersection of the circle with the horizontal axis gives the principal stresses ?_1 and ?_2, and the intersection of the circle with the vertical axis gives the maximum in-plane shear stress ?_max. The center of the circle gives the average normal stress ?_avg.

The orientation of the element is not specified in the question, so we cannot determine it.