MEE 825 YIELD CRITERIA
1. As an analyst you are interested in predicting whether the film under this biaxial stress state
will yield according to two different yield criteria: maximum normal stress (also called
Rankine criterion); Tresca criterion; and von Mises or J2-flow criterion. Here, explicitly
consider that the through thickness stress σzz = 0 as one of our three principal stresses. Assume
the yield strength of the Al film in uniaxial tension and compression σy = 100 MPa.
(a)
Determine whether the Al film will yield using each of the three yield criteria mentioned
above. Given these set principal normal stresses σx=σ1 =100.3 MPa; σyy =σ2 =19.7MPa,
and σzz =σ3 = 0MPa (plane-stress conditions).
(b)
Plot the yield locus for each of the three different yield criterion on a graph of σ2/σy vs. σ1/
σ y, where σ3=0. Which of the three criteria is the most conservative in predicting the stress
state required for yielding? Which, if any, of the criteria is not suitable for the analysis of
ductile materials such as aluminum?
2. A thin-walled circular cylinder of wall thickness t, radius D/2 is subjected to internal pressure
p and axial loading by the hydraulic ram arrangement shown in Fig. 1. For total ram area Ao
to cylinder area A, of (a) 2, (b) 5, determine the pressure p required to cause yielding of the
cylinder if the yield stress in simple radial stresses in the wall can be neglected. What ratio of
Ao/A results in plane strain yield conditions in the cylinder?
Fig, 1
Fig. 2
3. Figure 2 shows a thin-walled circular cylinder of wall thickness t and radius r with closed ends
and with the cylindrical surface enclosed by an outer cylinder. The inner cylinder with
pressure p1, acting internally and a lower pressure p2 acts externally on the cylindrical
surface. Sliding seals between the inner and outer cylinders ensure free relative axial
movement.
If the inner cylinder has yield stress Y in uniaxial tension, show that the pressure P1 necessary to
cause yield according to the Mises criterion is
1
2𝑡𝑌
𝑝1 =
(4𝑥2 − 6𝑥 + 3)−2
𝑟
Notes
a.
The Von Mises Yield criteria in terms of Hoop Stress, axial stress and radial stress is given as
(𝜎𝜃 − 𝜎𝑟 )2 + (𝜎𝑟 − 𝜎𝑧 )2 + (𝜎𝑧 − 𝜎𝜃 )2 = 2𝑌 2
Where σϴ= hoop stress, σr = radial stress, σz = axial stress
b. Study how to plot yield criteria is essential for question 1b