Page 284 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
Brown˙C06
Brown.cls
266
U.S. Customary 14:56 STRENGTH OF MACHINES SI/Metric
As the slenderness ratio calculated in step 2 As the slenderness ratio calculated in step 2
is less than the minimum value found in step is less than the minimum value found in step
3 for the Euler formula, the parabolic formula 3 for the Euler formula, the parabolic formula
applies. applies.
Step 4. Using Eq. (6.31), calculate the critical Step 4. Using Eq. (6.31), calculate the critical
stress (σ cr ) as stress (σ cr ) as
2 2

1 S y L 1 S y L
σ cr = S y − σ cr = S y −
CE 2π k CE 2π k
1 1
= (40 kpsi) − = (270 MPa) −
3
3
(1.5)(10 × 10 kpsi) (1.5)(70 × 10 MPa)
2 2
40 kpsi 270 MPa

× (72) × (80)
2π 2π
1 1
= (40 kpsi) − = (270 MPa) − 3
3
(15 × 10 kpsi) (105 × 10 MPa)
× (458 kpsi) 2 × (3,438 MPa) 2
3
4
21 × 10 kpsi 2 11,820 × 10 MPa 2
= (40 kpsi) − = (270 MPa) − 3
3
15 × 10 kpsi 105 × 10 MPa
= (40 kpsi) − (14 kpsi) = (270 MPa) − (113 MPa)
= 26 kpsi = 157 MPa
Step 5. Using Eq. (6.28), calculate the axial Step 5. Using Eq. (6.28), calculate the axial
stress (σ axial ) as stress (σ axial ) as
P 12,000 lb P 54,000 N
σ axial = = σ axial = =
A π(1in) 2 A π(0.025 m) 2
2
2
= 3,820 lb/in = 3.8 kpsi = 27,500,000 N/m = 27.5MPa
Step 6. Comparing the critical stress found in Step 6. Comparing the critical stress found in
step 4 with the axial stress found in step 5, the step 4 with the axial stress found in step 5, the
design is safe as design is safe as
σ axial <σ cr σ axial <σ cr
6.2.3 Secant Formula
The Euler and parabolic formulas are based on a column that is perfectly straight and is
loaded directly along the axis of the column. However, if the column has eccentricities,
either produced during manufacture or assembly, or an eccentricity in the application of the
compressive load, the column can fail at a critical stress (σ cr ) value lower than predicted
by either the Euler or parabolic formulas.
Without providing the details of its development, the appropriate formula for columns
with an eccentricity, called the secant formula, is given in Eq. (6.34) as
P cr S y
σ cr = = (6.34)
A ec 1 L σ cr
1 + 2 s
k 2 k E