Page 212 - Marks Calculation for Machine Design

P1: Shibu/Sanjay
January 4, 2005
Brown˙C05
Brown.cls
194
U.S. Customary 14:35 STRENGTH OF MACHINES SI/Metric
Step 6. Similarly, using Eq. (5.3), calculate the Step 6. Similarly, using Eq. (5.3), calculate the
rotated stress (τ x y ) from the unrotated stresses rotated stress (τ x y ) from the unrotated stresses
given in step 4. given in step 4.
σ xx − σ yy σ xx − σ yy
τ x y =− sin 2θ + τ xy cos 2θ τ x y =− sin 2θ + τ xy cos 2θ
2 2
(4.8 − 9.6) kpsi (37.7 − 75.4) MPa
◦
◦
=− sin 2(60 ) =− sin 2(60 )
2 2
◦
◦
+(0 kpsi) cos 2(60 ) − (0MPa) cos 2(60 )
(−4.8 kpsi) (−37.7MPa)
◦
◦
=− sin (120 ) =− sin (120 )
2 2
◦
◦
+(0 kpsi) cos (120 ) − (0MPa) cos (120 )
=−(−2.4 kpsi)(0.866) + (0 kpsi) =−(−18.85 MPa)(0.866) − (0MPa)
= (2.08 + 0) kpsi = (16.32 − 0) MPa
= 2.1 kpsi = 16.3MPa
Step 7. Display the rotated stresses found in Step 7. Display the rotated stresses found in
steps 4, 5, and 6, in kpsi, on the rotated element steps 4 to 6, in MPa, on the rotated element of
of Fig. 5.2. Fig. 5.2.
2.1 8.4 16.3 66.0
6.0 47.1
60° 60°
6.0 47.1
2.1
8.4 66.0

As a check on the calculations involved in applying Eqs. (5.1) to (5.3), like that in
Example 1, the following relationship given by Eq. (5.6) must always be satisfied between
two sets of stresses at different rotation angles (θ).

σ x x + σ y y = σ xx + σ yy (5.6)

U.S. Customary SI/Metric
Example 2. Verify that the values for the un- Example 2. Verify that the values for the un-
rotated and rotated normal stresses of Example rotated and rotated normal stresses of Example
1 satisfy Eq. (5.6), where 1 satisfy Eq. (5.6), where
σ x x + σ y y = σ xx + σ yy σ x x + σ y y = σ xx + σ yy
(8.4 + 6.0 ) kpsi = (4.8 + 9.6) kpsi (66.0 + 47.1) MPa = (37.7 + 75.4) MPa
14.4 kpsi ≡ 14.4 kpsi 113.1 MPa ≡ 113.1 MPa
Clearly the stresses check. Clearly the stresses check.