Page 221 - Marks Calculation for Machine Design

P1: Shibu/Sanjay
January 4, 2005
14:35
Brown.cls
Brown˙C05
PRINCIPAL STRESSES AND MOHR’S CIRCLE
U.S. Customary SI/Metric 203
and the minimum shear stress (τ min ) is and the minimum shear stress (τ min ) is
τ min =−τ max =−7.5 kpsi τ min =−τ max =−58 kpsi
Step 3. Using the average normal stress (σ avg ) Step 3. Using the average normal stress (σ avg )
found in step 1 and the maximum shear stress found in step 1 and the maximum shear stress
(τ max ) found in step 2, calculate the maximum (τ max ) found in step 2, calculate the maximum
principal stress (σ 1 ) from Eq. (5.15) as principal stress (σ 1 ) from Eq. (5.15) as
σ 1 = σ avg + τ max = (3.5 + 7.5) kpsi σ 1 = σ avg + τ max = (25 + 58) MPa
= 11 kpsi = 83 MPa
and use Eq. (5.16) to calculate the minimum and use Eq. (5.16) to calculate the minimum
principal stress (σ 2 ) as principal stress (σ 2 ) as
σ 2 = σ avg − τ max = (3.5 − 7.5) kpsi σ 2 = σ avg − τ max = (25 − 58) MPa
=−4 kpsi =−33 MPa
Step 4. Before going further, check that the Step 4. Before going further, check that the
values for the principal stresses (σ 1 ) and (σ 2 ) values for the principal stresses (σ 1 ) and (σ 2 )
satisfy Eq. (5.17) satisfy Eq. (5.17)
σ 1 + σ 2 = σ xx + σ yy σ 1 + σ 2 = σ xx + σ yy
[11 + (−4)] kpsi = [10 + (−3)] kpsi [83 + (−33)]MPa = [75 + (−25)]MPa
7 kpsi ≡ 7 kpsi 50 MPa ≡ 50 MPa

and they do. and they do.
Step 5. Using Eq. (5.9), calculate the rotation Step 5. Using Eq. (5.9), calculate the rotation
angle (φ p ) for maximum and minimum princi- angle (φ p ) for maximum and minimum princi-
pal stresses as pal stresses as
2τ xy 2τ xy
tan 2φ p = tan 2φ p =
σ xx − σ yy σ xx − σ yy
2 (−4 kpsi) 2 (−30 MPa)
= =
[10 − (−3)] kpsi [75 − (−25)]MPa
−8 kpsi −60 MPa
tan 2φ p = =−0.615 tan 2φ p = =−0.600
13 kpsi 100 MPa
2 φ p =−31.6 ◦ 2 φ p =−31.0 ◦
φ p =−15.8 ◦ φ p =−15.5 ◦
Step 6. Without the benefit of the graphical Step 6. Without the benefit of the graphical
picture of Mohr’s circle, the only way to tell picture of Mohr’s circle, the only way to tell
which principal stress this value of the rotation which principal stress this value of the rotation
angle (φ p ) is associated with is to substitute angle (φ p ) is associated with is to substitute
this angle in Eq. (5.1) and see which stress is this angle in Eq. (5.1) and see which stress is