An element in plane stress is subjected to stresses ?? = -8400 psi, ??y = 1100 psi, and
(see figure). The material is aluminum with modulus of elasticity E = 10,000 ksi and Poisson’s ratio v = 0.33.
Determine the following quantities: (a) the strains for an element oriented at an angle 0 = 30°, (b) the principal strains, and (C) the maximum shear strains. Show the results on sketches of properly oriented elements.
(a)
The strains for an element oriented at
Answer to Problem 7.7.17P
The normal strain in x direction is
The normal strain along y direction is
The shear strain is
Explanation of Solution
Given:
Stress along x direction is
Write the Equation for strain along x axis
Here, the normal strain along x direction is
Write the Equation for strain along y axis
Here, the normal strain along the y direction is
Write the expression for the shear strain ix-y plane
Here, modulus of rigidity is
Write the expression for strain along x direction
Write the expression for strain along y direction
Write the expression for shear strain
Here, the shear strain along x-y plane is
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Figure (1)
Conclusion:
The normal strain along x direction is
The normal strain along y direction is
The shear strain along x-y plane is
(b)
The principal strains.
Answer to Problem 7.7.17P
The maximum principal strain is
Explanation of Solution
Write expression for the principal strains.
Here, the maximum principal strain is
Write the expression for the principal angle.
Here,
Calculation:
Substitute
While taking positive sign you get maximum principal stress.
While taking negative sign you get minimum principal stress.
Substitute
In below figure principal strains and principal angle are shown.
--> Figure (2)
Conclusion:
The maximum principal strain is
(c)
The maximum shear strain.
Answer to Problem 7.7.17P
The maximum shear strain is
Explanation of Solution
Write the expression for maximum shear strain,
Here,
Write expression for shear angle
Here,
Write the Equation for average strain
Write expression for maximum shear strain.
Calculation:
Substitute
Substitute
Substitute
Substitute
In below figure average strain and minimum strain are shown.
Figure-(3)
Conclusion:
The maximum shear strain is
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Chapter 7 Solutions
Mechanics of Materials (MindTap Course List)
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