   # During a test of an airplane wing, the strain gage readings from a 45° rosette (see figure) are as follows: gage A, 520 × l0-6; gage B. 360 × l0-6; and gage C,- 80 × 10-6. Determine the principal strains and maximum shear strains, and show them on sketches of properly oriented elements. ### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 7, Problem 7.7.19P
Textbook Problem
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## During a test of an airplane wing, the strain gage readings from a 45° rosette (see figure) are as follows: gage A, 520 × l0-6; gage B. 360 × l0-6; and gage C,-80 × 10-6. Determine the principal strains and maximum shear strains, and show them on sketches of properly oriented elements. To determine

The principal strains and maximum shear strains.

### Explanation of Solution

Strain along direction A is 520×106and strain along direction B is 80×106

The strain along direction C is 80×106.

Write the expression for the strain along x axis.

εx=εA     …… (I)

Here, the normal strain along x direction is εx.

Write the expression for the strain along y axis.

εy=εC     …… (II)

Here, the normal strain along y direction is εy.

Write the Equation for strain along x-y plane.

γxy=2εBεAεC        …… (III)

Here, the shear strain along x-y plane is γxy.

Write expression for the principal strains.

ε1,2=(εx+εy2)+(εxεy2)2+(γxy2)2        …… (IV)

Here, the maximum principal strain is ε1and the minimum principal strain is ε2.

Write the expression for the principal angle.

tan2θp=γxyεxεy        …… (V)

Here, θpis the principal angle.

Write the expression for maximum shear strain,

γmax2=(εxεy2)2+(γxy2)2        …… (VI)

Here, γmaxis the maximum shear strain.

Write expression for first shear angle

θs1=θp145°        …… (VII)

Here, θs1is the first shear angle.

Write expression for second shear angle

θs2=θs145°        …… (VIII)

Here, θs2is the second shear angle.

Write the Equation for average strain

εavg=εx+εy2     …… (IX)

Write expression for maximum shear strain.

γmin=γmax        …… (X)

Calculation:

Substitute 520×106for εAin Equation (I).

εx=520×106

Substitute 80×106for εCin Equation (II).

εy=80×106

Substitute 520×106for εA, 360×106for εBand 80×106for εCin Equation (III).

γxy=2(360×106)(520×106)(80×106)γxy=720×106520×106+80×106γxy=280×106

Substitute 520×106for εx, 80×106for εyand 280×106for γxy.in Equation (IV)

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