Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Compression
test
τ
Tension
test
σ
σ
UT
p2
A
663
THEORIES OF FAILuRE
−σ
UC
σ
UT
−σ
UC
σ
UT
σ
p1
τ
(a)
B
−σ
UC
Experimental data from
tension and compression tests.
(b)
FIGURE 15.12 Mohr’s failure criterion for an element subjected to plane stress.
compressive strength s UC , respectively, of the material. Mohr’s circles for the tension and
compression tests are shown in Figure 15.12a. Mohr’s theory suggests that failure occurs in a
material whenever Mohr’s circle for the combination of stresses at a point in a body exceeds
the “envelope” defined by the Mohr’s circles for the tensile and compressive tests.
Mohr’s failure criterion for a plane stress state may be represented on a graph of principal
stresses in the s p1 − s p2 plane (Figure 15.12b). The principal stresses for all Mohr’s
circles that have centers on the s axis and are tangent to the dashed lines in Figure 15.12a
will plot as points along the dashed lines in the s p1 − s p2 plane of Figure 15.12b.
Mohr’s failure criterion applied to a plane stress state with in-plane principal stresses
s p1 and s p2 predicts that failure will occur under the following conditions:
• If s p1 and s p2 are both positive (i.e., tension), then failure will occur if s p1 ≥ s UT .
• If s p1 and s p2 are both negative (i.e., compression), then failure will occur if
s p2 ≤ − s UC .
If the naming convention for principal stresses is followed (i.e., s p1 > s p2 ), then all
combinations of s p1 and s p2 will plot to the right of (i.e., below) line AB shown in Figure
15.12b. Stress states with s p1 > 0 and s p2 < 0 fall into the fourth quadrant of Figure 15.12b.
In these cases, Mohr’s failure criterion predicts that failure will occur for those combinations
which plot on the dashed line—in other words, under the following condition:
s p1 s p2
• If s p1 is positive and s p2 is negative, then failure will occur if − ≥ 1.
s s
If torsion-test data are available, the dashed line in the fourth quadrant may be modified to
incorporate these experimental data.
Examples 15.8 and 15.9 illustrate the application of the theories of failure in predicting
the load-carrying capacity of a member:
UT
UC
ExAmpLE 15.8
The stresses on the free surface of a machine component are illustrated on the stress
element shown. The component is made of 6061-T6 aluminum with a yield strength of
s Y = 270 MPa.
(a) What is the factor of safety predicted by the maximum-shear-stress theory of failure
for the stress state shown? According to this theory, does the component fail?
y
x
90 MPa
75 MPa