Page 281 - Marks Calculation for Machine Design

P1: Shibu
14:56
January 4, 2005
Brown˙C06
Brown.cls
STATIC DESIGN AND COLUMN BUCKLING
U.S. Customary SI/Metric 263
Step 2. Calculate the slenderness ratio (L/k) Step 2. Calculate the slenderness ratio (L/k)
L 72 in L 2m
= = 248 = = 278
k 0.29 in k 0.0072 m
so the Euler’s Buckling formula applies. so the Euler’s Buckling formula applies.
Step 3. Using Eq. (6.27), calculate the critical Step 3. Using Eq. (6.27), calculate the critical
stress (σ cr ) as stress (σ cr ) as
2
2
C ends π E C ends π E
σ cr = σ cr =
2 2
L L
k k
2
6
2
9
2
2
(1)π (30 × 10 lb/in ) (1)π (207 × 10 N/m )
= =
(248) 2 (278) 2
6
9
296 × 10 lb/in 2 2,043 × 10 N/m 2
= =
61,504 77,284
2
2
= 4,813 lb/in = 4.8 kpsi = 26,435,000 N/m = 26.4MPa
Step 4. Using Eq. (6.28), calculate the axial Step 4. Using Eq. (6.28), calculate the axial
stress (σ axial ) as stress (σ axial ) as
P 24,000 lb P 108,000 N
σ axial = = σ axial = =
A (1in)(3in) A (0.025 m)(0.075 m)
2
2
= 8,000 lb/in = 8.0 kpsi = 57,600,000 N/m = 57.6MPa
Step 5. Comparing the critical stress found in Step 5. Comparing the critical stress found in
step 3 with the axial stress found in step 4, it is step 3 with the axial stress found in step 4, it is
clear the design is unsafe as clear the design is unsafe as
σ axial >σ cr σ axial >σ cr
6.2.2 Parabolic Formula
For columns where the slenderness ratio (L/k) is less than a certain value, the Euler formula
does not accurately predict column buckling. As mentioned earlier, the Euler formula given
in Eq. (6.27) states that the critical stress (σ cr ) is inversely proportional to the square of the
slenderness ratio (L/k). This inverse relationship is presented graphically in Fig. 6.21 as the
curve from point A to point B.
The lower limit of the slenderness ratio for which the Euler formula is appropriate is
indicated by point D, denoted (L/k) D , where the critical stress is set equal to the yield
stress divided by two (S y /2). The value of the slenderness ratio at this point is given in
Eq. (6.29).
1/2

2
L 2π CE
= (6.29)
k
D S y
Also shown in Fig. 6.21 is point C, also on the Euler curve, that defines a slenderness
ratio, denoted (L/k) C , where the critical stress has been set equal to the yield stress (S y ).