Page 214 - Marks Calculation for Machine Design

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January 4, 2005
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Brown˙C05
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STRENGTH OF MACHINES
One value of (φ s ) gives the maximum shear stress (τ max ), and the other value of (φ s ) gives
(τ min ). These two values of (φ s ) are 90 degrees apart, like the directions for the principal
stresses, (σ 1 ) and (σ 2 ), which are also 90 degrees apart.
Subsituting one of the values for the angle (φ s ) defined by Eq. (5.10) in Eq. (5.3) gives
the maximum shear stress (τ max ), given by Eq. (5.12),

2

σ xx − σ yy
2
τ max = + τ xy (5.12)
2
and substituting the other value for the angle (φ s ) defined by Eq. (5.10) in Eq. (5.3) gives
the minimum shear stress (τ min ), given by Eq. (5.13),

2
σ xx − σ yy 2
τ min =− + τ xy =−τ max (5.13)
2
which is just the negative of the maximum shear stress (τ max ).
On the rotated elements associated with the maximum and minimum shear stresses, the
normal stresses will be equal, and also equal to the average stress (σ avg ) given by Eq. (5.14).
σ xx + σ yy
σ avg = (5.14)
2
Noting that the first terms in both Eqs. (5.7) and (5.8) are the average stress (σ avg ) given
by Eq. (5.14), and that the magnitude of the second terms are the maximum shear stress
(τ max ), Eqs. (5.7) and (5.8) for the principal stresses (σ 1 ) and (σ 2 ) can be rewritten in the
following forms.
σ 1 = σ avg + τ max (5.15)
σ 2 = σ avg − τ max (5.16)
Similar to the relationship in Eq. (5.6), the values found for the principal stresses from
Eqs. (5.15) and (5.16) must satisfy the relationship in Eq. (5.17).
σ 1 + σ 2 = σ xx + σ yy (5.17)
Before going to the next topic where the principal stresses (σ 1 ) and (σ 2 ), the maxium
and minimum shear stresses (τ max ) and (τ min ), and the associated angles (φ p ) and (φ s ),
will be determined graphically using Mohr’s circle, consider the following examples, which
hopefully will provide an appreciation for the usefulness of the graphical approach called
Mohr’s circle.

U.S. Customary SI/Metric
Example 3. For the normal and shear stresses Example 3. For the normal and shear stresses
on the unrotated stress element of Example 1, on the unrotated stress element of Example 1,
find the principal stresses, maximum and mini- find the principal stresses, maximum and mini-
mum shear stresses, and the special angles (φ p ) mum shear stresses, and the special angles (φ p )
and (φ s ), and display these values on appropri- and (φ s ), and display these values on appropri-
ate rotated plane stress elements, where ate rotated plane stress elements, where
σ xx = σ axial = 4.8 kpsi σ xx = σ axial = 37.7MPa
σ yy = σ hoop = 9.6 kpsi σ yy = σ hoop = 75.4MPa
τ xy = 0 τ xy = 0
displayed on the following element. displayed on the following element.