# -18 through 7.3-22 An element in plane stress (see figure) is subjected to stresses o, a., and (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element.

### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347

### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 7, Problem 7.3.20P
Textbook Problem
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## -18 through 7.3-22 An element in plane stress (see figure) is subjected to stresses o, a., and (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element.

(a)

To determine

The principal stresses and sketch the orientation of elements.

### Explanation of Solution

Given information:

The normal stress along x-axis is 16.5MPa , the normal stress along y-axis is 91MPa ,and the shear stress along x-y direction is 39MPa .

Write the expression for the maximum principal stress.

σ1=σx+σy2+σxσy2cos(2θp1)+τxysin(2θp1) .....(I)

Here, the maximum principal stress is σ1 , the normal stress along x-direction is σx , the normal stress along y-direction is σy , the angle of inclination of principal stresses are θp1 and θp2 ,and the shear stress is τxy .

Write the expression for the minimum principal stress.

σ2=σx+σy2+σxσy2cos(2θp2)+τxysin(2θp2) .....(II)

Here, the minimum principal stress is σ2 .

Write the expression for the angle of inclination of principal stress.

tan2θ=2τxyσxσy .....(III)

θp1=θθp2=θ+90 .....(IV)

Calculation:

Substitute 16.5MPa for σx , 91MPa for σy , and 39MPa for τxy , in Equation (III).

tan2θ=2×(39MPa)16.5MPa(91MPa)=78MPa107.5MPa=0.725

2θ=tan1(0.725)=35

(b)

To determine

The maximum shear stress.

The associated normal stress.

Sketch the orientation of the elements.

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