Page 287 - Marks Calculation for Machine Design

P1: Shibu
14:56
January 4, 2005
Brown.cls
Brown˙C06
STATIC DESIGN AND COLUMN BUCKLING
As the right hand side came out just slightly greater than the guess, try 11.
40 kpsi 269
σ cr = √
1 + s[(0.36) σ cr ]
40 kpsi 40 kpsi 40 kpsi 40 kpsi
11 = √ = = =
1 + s[(0.36) 11] 1 + s [1.194] 1 + (2.72) 3.72
= 10.8
To get one decimal place accuracy, try as a last guess, 10.9.
40 kpsi
σ cr = √
1 + s[(0.36) σ cr ]
40 kpsi 40 kpsi 40 kpsi 40 kpsi
10.9 = √ = = =
1 + s[(0.36) 10.9] 1 + s [1.188] 1 + (2.68) 3.68
= 10.9
= σ cr
Notice that it required only four iterations to get one decimal place accuracy for the
critical stress. Also, this value of the critical stress would still predict a safe design.
SI/Metric
Example 3. Determine the critical stress (σ cr ) using the secant formula for the column in
Example 2 if there is an eccentricity (e) of 0.01 m, and where it was found that the radius
of gyration (k) was (0.0125 m) and the slenderness ratio (L/k) was (80). The yield stress
3
(S y ) was given as (270 MPa) and the modulus of elasticity (E) was (70 × 10 MPa). The
distance (c) for a circle is the radius (r), which in Example 2 is (0.025 m).
solution ec
Step 1. Calculate the eccentricity ratio as
k 2
ec (0.01 m)(0.025 m)
= = 1.6
k 2 (0.0125 m) 2
Step 2. Substitute the eccentricity ratio found in step 1 and the known values of the other
terms in Eq. (6.34) to give Eq. (6.36) as
S y
σ cr =
ec 1 L σ cr
1 + s
k 2 2 k E
270 MPa
= (6.36)

1 σ cr
1 + (1.6) s (80)
3
2 70 × 10 MPa
270 MPa
=
(80) √
1 + (1.6) s σ cr
2(264.6)
270 MPa
= √
1 + (1.6) s (0.15) σ cr