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

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COMbINEd LOAdS
Similarly, Equation (15.4) can also be used to compute the Mises equivalent stress:
2
sM =
2
⎡⎣ ( sx − s y ) 2 + ( s y − sz ) 2 + ( sx − σz ) 2 + 6( τxy 2 + τyz 2 + τxz
2 ) ⎤ 1/2
⎦ (15.7)
For the case of plane stress, the Mises equivalent stress can be expressed from Equation
(15.5) as
2
σ = [ σ − σ σ + σ 2 ]
1/2
(15.8)
M p1
p1 p2 p2
or it can be found from Equation (15.4) by setting σ z = τ yz = τ xz = 0 to give
2 2 2 1/2
σ = [ σ − σσ + σ + 3 τ ]
(15.9)
M x x y y xy
To use the Mises equivalent stress, σ M is calculated for the state of stress acting at any
specific point in the component. The resulting value of σ M is then compared with the tensile
yield stress σ Y , and if σ M > σ Y , then the material is predicted to fail according to the maximumdistortion-energy
theory. The utility of the Mises equivalent stress has led to its widespread
use in tabulated stress analysis results and in the form of color-coded stress contour plots
common in finite element analysis results.
B
σ
− U
σ
σ
U
p2
Experimental data from tension test.
FIGURE 15.11 Failure
diagram for maximum-normalstress
theory for an element
subjected to plane stress.
If the naming convention for
principal stresses is followed
(i.e., σ p1 > σ p2 ), then all
combinations of σ p1 and σ p2 will
plot to the right of or below line
AB shown in Figure 15.11.
σ
− U
σ
U
A
σ
p1
Brittle materials
Unlike ductile materials, brittle materials tend to fail suddenly by fracture, with little evidence
of yielding; therefore, the limiting stress appropriate for brittle materials is the fracture
stress (or the ultimate strength) rather than the yield strength. Furthermore, the tensile
strength of a brittle material is often different from its compressive strength.
maximum-Normal-Stress Theory. 6 The maximum-normal-stress theory predicts
that failure will occur in a specimen that is subjected to any combination of loads when the
maximum normal stress at any point in the specimen reaches the axial failure stress determined
from an axial tension or compression test of the same material.
The maximum-normal-stress theory is presented graphically in Figure 15.11 for an
element subjected to biaxial principal stresses in the p1 and p2 directions. The limiting
stress σ U is the failure stress for the material the element is made of when it is loaded
axially. According to this theory, any combination of biaxial principal stresses σ p1 and σ p2
represented by a point inside the square of Figure 15.11 is safe whereas any combination
of stresses represented by a point outside of the square will cause failure of the element.
mohr’s Failure criterion. For many brittle materials, the ultimate tension and compression
strengths are different, and in such cases, the maximum-normal-stress theory should
not be used. An alternative failure theory, proposed by the German engineer Otto Mohr, 7 is
called Mohr’s failure criterion. To use this criterion, a uniaxial tension test and a uniaxial
compression test are performed to establish the ultimate tensile strength σ UT and ultimate
6
Often called Rankine’s theory after W. J. M. Rankine (1820–1872), an eminent engineering educator at Glasgow
University in Scotland.
7
The same Otto Mohr of Mohr circle fame.