   # The surface of an airplane wing is subjected to plane stress with normal stresses x and cr and shear stress t x v , as shown in the figure. At a counterclockwise angle 9 = 32° from the x axis, the normal stress is 29 MPa in tension, and at an angle 8 = 46°, it is 17 MPa in compression. If the stress tr x equals 105 MPa in tension, what are the stresses cr„ and r„,? ### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 7, Problem 7.2.24P
Textbook Problem
383 views

## The surface of an airplane wing is subjected to plane stress with normal stresses x and cr and shear stress t x v, as shown in the figure. At a counterclockwise angle 9 = 32° from the x axis, the normal stress is 29 MPa in tension, and at an angle 8 = 46°, it is 17 MPa in compression. If the stress trxequals 105 MPa in tension, what are the stresses cr„ and r„,? To determine

The normal stresses acting on the element.

The shear stress acting on the element.

### Explanation of Solution

Given information The normal stress acting on the body at an angle of θ=32° in tension is 29MPa , the normal stress acting on the body at an angle of θ=46° in compression is 17MPa , the normal stress acting on the body along x-axis is 105MPa

The figure below shows the stresses acting on the element.

Figure-(1)

Write the expression for normal stress acting on the element.

σx=σx+σy2+σxσy2cos2θ+τxysin2θ ...... (I)

Here, normal stresses acting on the body along x-direction is σx , normal stress acting on the body in tension along x -direction is σx , normal stress in y-direction is σy and θ is the angle subtended in tension and compression.

Calculation:

Substitute 29MPa for σx , 105MPa for σx , and 32° for θ in tension. for Equation (I).

29MPa=105MPa+σy2+105MPaσy2cos64°+τxysin64°52.5MPa+σy2+(52.5MPa)×(0.4388)(0.2192)σy29MPa=τxy×(0.8988)52.5MPa+σy2+52.5MPa×(0.4388)(0.2192)σy29MPa0.8988τxy=(46.5145MPa+0.2808σy0.8988)

(51.7518MPa+0.3124σy)=τxy ...... (II)

Substitute 17MPa for σx , 105MPa for σx , and 46° for θ for compression in Equation (I)

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