   # Solve the preceding problem if the plate is made of aluminum with E = 72 GPa and Poisson’s ratio v = 0.33. The plate is loaded in biaxial stress with normal stress s x = 79 MPa, angle Ø = 18°, and the strain measured by the gage is e = 925 × 10-6. ### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 7, Problem 7.7.16P
Textbook Problem
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## Solve the preceding problem if the plate is made of aluminum with E = 72 GPa and Poisson’s ratio v = 0.33. The plate is loaded in biaxial stress with normal stress sx= 79 MPa, angle Ø = 18°, and the strain measured by the gage is e = 925 × 10-6. To determine

The maximum shear stress and strain in x-y plane.

The maximum shear strain in x-z plane and y-z plane.

### Explanation of Solution

Given information:

Modulus of elasticity is 72GPaand Poissons ratio is 0.33direct stress in x direction is 79MPaand strain in x direction at an orientation of 18°is 925×106.

Write the Equations to obtain maximum in-plane shear stress.

εx'=εx+εy2+εxεy2cos2θ+γxy2sin2θ(I)

Here, the normal strain along x direction is εx', the normal strain along x direction is εxand the normal strain along the y direction is εyand the shear strain along x-y lane is γxy.

Write the expression for the strain along x direction.

εx=1E(σxνσy) (II)

Here, direct stress along x direction is σx, the Poissons ratio is νand the modulus of elasticity is E.

Write the expression for strain along y direction.

εy=1E(σyνσx) (III)

Here, direct stress in y direction is σy.

Substitute value of direct stresses of x and y plane in Equation (I).

εx'=12E(σxνσy+σyνσx+cos2θσxνcos2θσycos2θσy+νcos2θσx)(IV)

Write the expression for stress in x-y direction

(τmax)xy=(σxσy2)2+τxy2 (V)

Here, τxyis stress in x-y plane

Write the expression for strain in x-y direction

(γmax)xy=2(1+ν)×(τmax)xyE(VI)

Write the expression for the maximum shear stress in plane x-z.

(τmax)xz=σx2 (VII)

Write the expression for the maximum shear strain in x-z direction.

(γmax)xz=2(1+ν)×(τmax)xzE(VIII)

Write the expression for the maximum shear stress in y z direction.

(τmax)yz=σy2 (IX)Similarly write the expression for the maximum strain in y-z direction.

(γmax)yz=2(1+ν)×(τmax)yzE (X)

Calculation:

Substitute 79MPafor σx, 0.33for νand 18°for θin Equation (IV).

εx'=12E((79MPa)(0.33)σy+σy((0.33)(79MPa))+cos36°×(79MPa)(0.33)×cos(2×18°)×σycos(2×18°)×σy+(0.33)×cos(2×18°)×σx)

2εx'=12E((79MPa)(10.33+0.8090+0.2669)+σy(0.33+10.26690.8090))

Substitute 925×106for εx, 72GPafor Ein above Equation.

925×106×2=(1((72GPa))(1.7459((79MPa))+(0.4059)σy

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