Page 260 - Shigley's Mechanical Engineering Design

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Failures Resulting from Static Loading 235
Figure 5–18 B

Graph of maximum-normal- S ut
stress (MNS) theory failure
envelope for plane stress states.
Nonfailure region
–S uc S ut A

–S
uc

5–8 Maximum-Normal-Stress Theory
for Brittle Materials
The maximum-normal-stress (MNS) theory states that failure occurs whenever one of
the three principal stresses equals or exceeds the strength. Again we arrange the prin-
cipal stresses for a general stress state in the ordered form σ 1 ≥ σ 2 ≥ σ 3 . This theory
then predicts that failure occurs whenever

σ 1 ≥ S ut or σ 3 ≤−S uc (5–28)
where S ut and S uc are the ultimate tensile and compressive strengths, respectively, given
as positive quantities.
For plane stress, with the principal stresses given by Eq. (3–13), with σ A ≥ σ B ,
Eq. (5–28) can be written as

σ A ≥ S ut or σ B ≤−S uc (5–29)
which is plotted in Fig. 5–18.
As before, the failure criteria equations can be converted to design equations. We
can consider two sets of equations where σ A ≥ σ B as

S ut S uc
σ A = or σ B =− (5–30)
n n
As will be seen later, the maximum-normal-stress theory is not very good at pre-
dicting failure in the fourth quadrant of the σ A , σ B plane. Thus, we will not recommend
the theory for use. It has been included here mainly for historical reasons.

5–9 Modifications of the Mohr Theory
for Brittle Materials

We will discuss two modifications of the Mohr theory for brittle materials: the Brittle-
Coulomb-Mohr (BCM) theory and the modified Mohr (MM) theory. The equations
provided for the theories will be restricted to plane stress and be of the design type
incorporating the factor of safety.