Need your help please show all work and thank you Large cali

Need your help please, show all work and thank you

Large caliber gun barrels can be constructed by press fit of a liner tube into an outer tube as shown in Figure 1.a. The diametral interference must be closely controlled to avoid damages to both tubes in the press fitting process. For a liner tube having an inside diameter 3.000 in and outside diameter 4.003 in and an outer tube of inside diameter 3.997 in and outside diameter of 6.000 in, calculate the maximum tangential and radial stresses in (a) the liner tube and (b) the outer tube. Assume both tubes are made from alloy steel.

Solution

For alloy steel, Modulus of Elasticity E = 200 GPa = 29*10⁶ psi

Radial interferece delta = (4.003 - 3.997)/2 = 0.003\"

Average Radius at interface R = (4.003 + 3.997)/2/2 = 2\"

Outer radius r_o = 6/2 = 3\"

Inner radius r_i = 3/2 = 1.5\"

Interface pressure p = (E*delta / R) [(r_o² - R²) (R² - r_i²) / (2R² (r_o² - r_i²))]

= (29*10⁶ * 0.003 / 4) [(3² - 2²)(2² - 1.5²) / (2*1.5² (3² - 1.5²))]

= 6265.4 psi

a)

In liner tube, max tangential stress (occurs at its Outer Dia) = -p*(R² + r_i²) / (R² - r_i²)

= -6265.4 * (2² + 1.5²) / (2² - 1.5²)

= -22376.4 psi

In liner, Max. radial stress = -p = -6265.4 psi

b)

In outer tube, max tangential stress (occurs at its Inner Dia) = p*(r_o² + R²) / (r_o² - R²)

= 6265.4*(3² + 2²) / (3² - 2²)

= 16290 psi

In outer tube, max radial stress = -p = -6265.4 psi