Page 217 - Marks Calculation for Machine Design

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January 4, 2005
Brown.cls
Brown˙C05
PRINCIPAL STRESSES AND MOHR’S CIRCLE
U.S. Customary SI/Metric 199
Example 4. For the normal and shear stresses Example 4. For the normal and shear stresses
on theunrotatedtopstress element of Example 5 onthe unrotatedtopstress elementof Example 5
in Sec. 4.4, find the principal stresses, maximum in Sec. 4.4, find the principal stresses, maximum
and minimum shear stresses, and the special an- and minimum shear stresses, and the special an-
gles (φ p ) and (φ s ), and display these values on gles (φ p ) and (φ s ), and display these values on
appropriate rotated plane stress elements, where appropriate rotated plane stress elements, where
σ xx = 15.3 kpsi σ xx = 146.7 MPa
σ yy = 0 σ yy = 0
τ xy =−3.8 kpsi τ xy =−36.7 MPa
displayed in the following element: displayed in the following element:
0 0
3.8 36.7
15.3 15.3 146.7 146.7

3.8 36.7

0 0
solution solution
Step 1. Calculate the average normal stress Step 1. Calculate the average normal stress
(σ avg ) from Eq. (5.14) as (σ avg ) from Eq. (5.14) as
σ xx + σ yy (15.3 + 0) kpsi σ xx + σ yy (146.7 + 0) MPa
σ avg = = σ avg = =
2 2 2 2
= 7.65 kpsi = 73.35 MPa
Step 2. Calculate the maximum shear stress Step 2. Calculate the maximum shear stress
(τ max ) from Eq. (5.12) as (τ max ) from Eq. (5.12) as

2 2

σ xx − σ yy 2 σ xx − σ yy 2
τ max = + τ xy τ max = + τ xy
2 2

2 2

15.3 − 0 146.7 − 0
2
2
= + (−3.8 ) kpsi = + (−36.7 ) MPa
2 2

2
2
2
2
= (7.65) + (−3.8 ) kpsi = (73.35) + (−36.7 ) MPa

= (58.52) + (14.44 ) kpsi = (5,380.2) + (1,346.9 ) MPa

= (72.96) kpsi = 8.54 kpsi = (6,727.1) MPa = 82.02 MPa
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 = (7.65 + 8.54) kpsi σ 1 = σ avg + τ max = (73.35 + 82.02) MPa
= 16.19 kpsi = 155.37 MPa