ladil123: Here is how you compute the stress on an interference fit, press fit, or shrink fit for a shaft and collar.
(1) Description of parameters.
d1 = shaft inside diameter at reference temperature T1o (zero for solid shaft).
d2 = shaft outside diameter at reference temperature T1o.
d3 = collar inside diameter at reference temperature T2o.
d4 = collar outside diameter at reference temperature T2o.
d = resultant contact interface diameter.
E1 = shaft modulus of elasticity.
E2 = collar modulus of elasticity.
nu1 = shaft Poisson ratio.
nu2 = collar Poisson ratio.
alpha1 = shaft coefficient of thermal expansion (CTE).
alpha2 = collar CTE.
T1 = shaft temperature.
T2 = collar temperature.
T1o = reference temperature at which shaft dimensions d1 and d2 are measured.
T2o = reference temperature at which collar dimensions d3 and d4 are measured.
sd = surface roughness deformation = 1.2(Ra1 + Ra2).
Ra1 = shaft outside diameter surface roughness, Ra value.
Ra2 = collar inside diameter surface roughness, Ra value.
pc = contact pressure between shaft and collar.
sigma_t1 = hoop stress at shaft inside diameter.
sigma_t2 = hoop stress at shaft outside diameter.
sigma_t3 = hoop stress at collar inside diameter.
(2) Apply temperatures T1 and T2.
d1 = d1*[1 + alpha1*(T1 - T1o)]
d2 = d2*[1 + alpha1*(T1 - T1o)]
d3 = d3*[1 + alpha2*(T2 - T2o)]
d4 = d4*[1 + alpha2*(T2 - T2o)]
(3) Initially estimate d as follows.
d = 0.5(d2 + d3)
(4) Compute sd if surface roughnesses are known; or you can set sd = 0 if you want to neglect yielding of surface roughness asperities.
sd = 1.2(Ra1 + Ra2)
(5) Compute contact interface pressure.
pc = [(d2 - d3 - sd)/d]/{[(d^2 + d1^2)/(E1*(d^2 - d1^2))] + [(d4^2 + d^2)/(E2*(d4^2 - d^2))] - (nu1/E1) + (nu2/E2)}
(6) Refine d, and repeat step 5 one time. You can skip this step if the shaft or collar is not relatively flexible compared to the other member.
d = d2*[1 - (pc/E1)(1 - nu1)]
(7) Compute shaft and collar hoop stress.
sigma_t1 = pc*2(d^2)/(d1^2 - d^2)
sigma_t2 = pc*(d1^2 + d^2)/(d1^2 - d^2)
sigma_t3 = pc*(d4^2 + d^2)/(d4^2 - d^2)
Hoop stress at the collar outside diameter is not given, above, because it is not maximum. Radial stress at the shaft outside diameter and collar inside diameter is sigma_r2 = sigma_r3 = -pc. Radial stress at the shaft inside diameter (or center) and collar outside diameter is sigma_r1 = sigma_r4 = 0. Combine the radial and hoop stress using an appropriate failure theory. For ductile materials, use von Mises theory. The foregoing analysis is applicable only if the combined stress is below the material yield strength. Also, the foregoing analysis and formulas are applicable only if pc in item 5 is positive.
For example, for your given problem in post 7, if T1o = T2o = 20 C, T2 = 30 C, T1 = 80 C, and sd = 1.2(0.7 micrometer + 1.0 micrometer) = 2.04 micrometer, we have the following.
d1 = 0 mm, d2 = 87 mm, E1 = 205 GPa, nu1 = 0.30, alpha1 = 11.5e-6/C
d3 = 87 mm, d4 = 142 mm, E2 = 540 GPa, nu2 = 0.24, alpha2 = 5.6e-6/C
d1 = (0.000 m)[1 + (11.5e-6/C)(80 C - 20 C)] = 0 mm
d2 = (0.087 m)[1 + (11.5e-6/C)(80 C - 20 C)] = 87.060 030 mm
d3 = (0.087 m)[1 + (5.6e-6/C)(30 C - 20 C)] = 87.004 872 mm
d4 = (0.142 m)[1 + (5.6e-6/C)(30 C - 20 C)] = 142.007 952 mm
d = 0.5(d2 + d3) = 87.032 451 mm
(5) Compute contact interface pressure.
pc = [(0.087 060 030 - 0.087 004 872 - 2.04e-6)/0.087 032 451]/{[(0.087 032 451^2 + 0^2)/((205e9 Pa)*(0.087 032 451^2 - 0^2))] + [(0.142 007 952^2 + 0.087 032 451^2)/((540e9 Pa)*(0.142 007 952^2 - 0.087 032 451^2))] - (0.30/(205e9 Pa)) + (0.24/(540e9 Pa))} = 76.8773 MPa
(6) Refine d, and repeat step 5 one time. (This step is not expressly required in your particular problem, but I will perform it anyway.)
d = 0.087 060 030{1 - [(76 877 259 Pa)/(205e9 Pa)](1 - 0.30)} = 87.037 176 mm
pc = 76.8693 MPa
(7) Compute shaft and collar hoop stress.
sigma_t1 = (76 869 334 Pa)*2*0.087 037 176^2/(0^2 - 0.087 037 176^2) = -153.7 MPa
sigma_t2 = (76 869 334 Pa)(0^2 + 0.087 037 176^2)/(0^2 - 0.087 037 176^2) = -76.88 MPa
sigma_t3 = (76 869 334 Pa)(0.142 007 952^2 + 0.087 037 176^2)/(0.142 007 952^2 - 0.087 037 176^2) = 169.4 MPa
