Carlos Sarria
MANE6940
Homework 6
3) Consider a long hollow steel cylinder subjected to uniform pressures on its inner and outer surfaces
and with the following characteristics: Inner radius a =1 m; outer radius b=1.05 m; E =
2e11 Pa; ν = 0.3; P = 104 N, pin = 105 Pa , pout = 106 Pa.
a) Assume the cylinder ends are stress-free (plane stress condition) and obtain exact values for
the displacement, strain and stress fields.
Please refer to the HW06_#3_Sarria.pdf file lines (3) through (6) for the expressions of the stress,
displacement and strain fields. Figure 1 shows a plot of the displacement field.
Figure 1: Displacement field
Figure 2 shows the radial stress field. Notice that the stress equals to the applied pressures at the
corresponding edges.
Figure 2: Radial stress field
Figure 3 shows the hoop stress field.
Figure 3: Hoop Stress Field
Figure 4 shows the radial strain field plot.
Figure 4: Radial strain field
Figure 5 shows the hoop strain field.
Figure 5: Hoop strain field
b) Use the finite element method to solve the problem approximately and compare with the
exact solution.
The FEA results are similar to the analytical solutions. For instance, Figure 6 shows the radial
displacement using ANSYS. Notice that the maximum displacement occurs at the inner wall with a value
of-9.72e-5 m. The exact solution yields a displacement of -9.715e-5m (line (7) in the Maple file).
Figure 6: Radial displacement plot from ANSYS
For the radial stress, the FEA answer was the same order of magnitude, but it was a little off (Figure 7),
with a radial stress of around 1.8e5 vs the 1e5 Pa from the analytical solution (line (8)). A more refined
mesh will probably help converge the stress.
Figure 7: Radial stress plot from ANSYS
Figure 8 shows the Hoop stress plot from ANSYS. It shows a maximum stress at the ID of 1.94e7 Pa
compressive. This is very similar to the analytical solution (line (10)) with a value of 1.946e7
compressive.
Figure 8: Hoop stress plot from ANSYS
Figure 9 shows the radial strain from ANSYS with a max value of 2.82e-5. It is very similar to the
analytical result of 2.87e-5 (line (12)).
Figure 9: Radial Strain from ANSYS
Figure 10 shows the hoop strain plot using FEA. The max value is -9.69e-5 at the ID, which is very similar
to the -9.71e-5 from the analytical solution (line (14)).
Figure 10: Hoop strain from ANSYS
4) Consider a steel disk of uniform thickness h = 0.025 m radius b = 1 m that rotates with angular
velocity ω = 6000 rpm.
a) Assume axis-symmetric conditions and that the stress-stress function (F) relationship is of the form
σrr = (1/r) F ; σΘΘ = F,r + ρ ω^2 r^2 and obtain the differential equation that must be satisfied by the
stress function to satisfy the condition of mechanical equilibrium.
Please refer to the HW6_Sarria.pdf file for the derivation of the expression below:
𝑟2 ∗
𝑑2 𝐹
𝑑𝐹
+𝑟∗
− 𝐹 + (3 + 𝜈) ∗ 𝜌 ∗ 𝜔2 ∗ 𝑟 3 = 0
2
𝑑𝑟
𝑑𝑟
b) Solve the resulting equation for F and obtain exact expressions for the stress, strain and
displacement fields in the disk.
Please refer to the HW06#4_Sarria.pdf file for the solution of the differential equation (line (2)).
Expressions for the stress, strain and displacement fields can be found on lines (8) thru (11). Figure 11
shows the radial stress field.
Figure 11: Radial stress for the rotating disk
Figure 12 shows the tangential stress field.
Figure 12: Hoop stress for the rotating disk
Figure 13 shows the radial displacement field.
Figure 13: Radial displacement field
c) Use the finite element method to solve the problem approximately and compare with the
exact solution.
Using COMSOL we obtained very similar plots compared to the analytical solution. Figure 14
shows the radial stress field obtained using FEA. Notice that the maximum stress is 1.28e9 Pa,
while the analytical solution yields around 1.3e9 (see Figure 11).
Figure 14: Radial stress along the disk's cross section
Once again, COMSOL gets a very similar solution (Figure 15) of 1.28e9Pa. The analytical solution can be
found on Figure 12.
Figure 15: Hoop stress along the disk's cross section
Displacement wise, they’re very similar too. Figure 16 shows COMSOL’s radial displacement plot, which
is very close to
Figure 16: Radial displacement field along the disks cross section