Page 261 - Shigley's Mechanical Engineering Design

bud29281_ch05_212-264.qxd 12/9/09 6:53PM Page 236 ntt G4 Mac OS 9.2:Desktop Folder:MHDQ196/Budynas:

236 Mechanical Engineering Design
Figure 5–19 B , MPa
300
Biaxial fracture data of gray
cast iron compared with max. normal S ut
various failure criteria.
mod. Mohr
(Dowling, N. E., Mechanical
Behavior of Materials, 2nd ed., Coulomb - Mohr
–S S
1999, p. 261. Reprinted by uc ut A , MPa
–700 –300 0 300
permission of Pearson
Education, Inc., Upper Saddle
River, New Jersey.)
–S ut
–300 Torsion
Gray cast-iron data

–S uc
–700

The Coulomb-Mohr theory was discussed earlier in Sec. 5–6 with Eqs. (5–23) to
(5–25). Written as design equations for a brittle material, they are:

Brittle-Coulomb-Mohr

S ut
σ A = σ A ≥ σ B ≥ 0 (5–31a)
n
σ A σ B 1
− = σ A ≥ 0 ≥ σ B (5–31b)
S ut S uc n
S uc
σ B =− 0 ≥ σ A ≥ σ B (5–31c)
n
On the basis of observed data for the fourth quadrant, the modified Mohr theory
expands the fourth quadrant with the solid lines shown in the second and fourth quad-
rants of Fig. 5–19.

Modified Mohr

S ut
σ A = σ A ≥ σ B ≥ 0
n
(5–32a)

σ B
and ≤ 1
σ A ≥ 0 ≥ σ B
σ A

(S uc − S ut ) σ A σ B 1 σ B
− = σ A ≥ 0 ≥ σ B and > 1 (5–32b)
S uc S ut S uc n σ A
S uc
σ B =− 0 ≥ σ A ≥ σ B (5–32c)
n
Data are still outside this extended region. The straight line introduced by the modified
Mohr theory, for σ A ≥ 0 ≥ σ B and |σ B /σ A | > 1, can be replaced by a parabolic relation