Page 288 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
14:56
Brown.cls
Brown˙C06
STRENGTH OF MACHINES
270
where the units (MPa) have been dropped for the modulus of elasticity (E) in the square
root term as the critical stress (σ cr ) will also have units of (MPa). This is compatible with
the fact that the secant can only be evaluated for a nondimensional quantity.
As the critical stress (σ cr ) is on both sides of Eq. (6.36) it must be solved by trial and
error or some other numerical method. To show how quickly the trial-and-error method can
obtain a reasonably accurate value for the critical stress, start with an educated guess for
the critical stress, then modify this guess in successive iterations until the guess equals the
right hand side of Eq. (6.36). Stop when an appropriate level of accuracy is reached.
An excellent educated guess would be the yield stress divided by two, which would be
135 MPa. Substitute this value into the right hand side of Eq. (6.36) to give
270 MPa
σ cr = √
1 + (1.6) s (0.15) σ cr
270 MPa 270 MPa 270 MPa
135 = √ = =
1 + (1.6) s[(0.15) 135] 1 + (1.6) s [1.74] 1 + (−9.3)
270 MPa
= =−32.4
−8.3
As the right hand side came out negative, try a new guess of 70.
270 MPa
σ cr = √
1 + (1.6) s (0.15) σ cr
270 MPa 270 MPa 270 MPa
70 = √ = =
1 + (1.6) s[(0.15) 70] 1 + (1.6) s [1.25] 1 + (5.15)
270 MPa
= = 43.9
6.15
Split the difference between 70 and 43.9 and try 57.
270 MPa
σ cr = √
1 + (1.6) s (0.15) σ cr
270 MPa 270 MPa 270 MPa
57 = √ = =
1 + (1.6) s (0.15) 57 1 + (1.6) s [1.13] 1 + (3.77)
270 MPa
= = 56.6 = σ cr
4.77
Notice that it required only three iterations, compared to four iterations for the U.S.
Customary calculation, to get one decimal place accuracy for the critical stress. Also, this
value of the critical stress would still predict a safe design.

6.2.4 Short Columns
The big question is how short is short? The machine element could be so short that it can
be considered as a pure compression member, where failure is a shortening of the column
at the yield stress (S y ).
For columns having slenderness ratios between for pure compression and for one
which would mean that the secant formula would apply, the critical stress (σ cr )