Thick-Walled Cylinders (Notes,3.14)

1. Thick-Walled Cylinders(Notes,3.14) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering Thick-Wall Cylinders

2. Cylinders (3.14) Applications • pi = 0 • Submarine • Vacuum chamber • Shrink fit • Buried pipe r ri z pi po ro σl σt σr σr • po = 0 • Gun barrel • Liquid- or gas-carrying pipe • Hydraulic cylinder • Gas storage tank Thick-Wall Cylinders

3. Thin-Walled Pressure Vessels (Review) • For a thin-walled pressure vessel, ri/t > 10,so “hoop” stress (σt) variation in the radialdirection is minimal • Radial stress (σr) is equal to -p on the inner surface, zero on the outer surface, and varies in between. • σris negligible compared to σt. (hoop stress) (longitudinal stress) t ri p ro σt σr Thick-Wall Cylinders

4. Thick-Walled Cylinders (3.14) • For thick-walled pressure vessels • Maximum shear stress • If the ends of the cylinder are capped, must include longitudinal stress. t ri pi po ro σt σr Thick-Wall Cylinders

5. Thick-Walled Cylinders • Examples of closed cylinders include pressure vessels and submarines. • Examples of open cylinders include gun barrels and shrink fits. • Radial displacement of a thick-walled cylinder ur Thick-Wall Cylinders

6. Thick-Walled Cylinders (3.14) • Special case: Internal pressure only (po = 0) ri pi ro σt σr σt/pi Thick-Wall Cylinders

7. Thick-Walled Cylinders • Compare previous result with thin-walled pressure vessel case (po = 0) ri pi ro σt σr Thick-Wall Cylinders

8. Thick-Walled Cylinders • Continued… ri pi ro σt σr Thick-Wall Cylinders

9. Thick-Walled Cylinders • Special case: External pressure only (pi = 0) ri po ro σt σr Thick-Wall Cylinders

10. Example Find the tangential, radial, and longitudinal stress for a pipe with an outer diameter of 5 inches, wall thickness of 0.5 inches, and internal pressure of 4000 psi. Thick-Wall Cylinders

11. Example Find the maximum allowable internal pressure for a pipe with outer radius of 3 inches and wall thickness of 0.25 inches if the maximum allowable shear stress is 10000 psi. Thick-Wall Cylinders

12. Press and Shrink Fits(3.16) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering Press and Shrink Fits

13. Press and Shrink Fits (3.16) • Assume inner member has slightly larger outer radius than inner radius of outer member. • Interference pressure will develop upon assembly. Press together or shrink inner ri rf p uro p uri ro rf +δ Inner member (external pressure only) Outer member (internal pressure only) 14 Press and Shrink Fits

14. Press and Shrink Fits (3.16) • Once δ is known we can calculate p, or vice versa. • Typically, δ is very small, approximately 0.001 in. or less. Press and Shrink Fits

15. Press and Shrink Fits (3.16) • If the materials are the same: • E = Ei = Eo • υ = υi = υo • If the inner member is not hollow, ri = 0. Press and Shrink Fits

16. Example A solid shaft is to be press fit into a gear hub. Find the maximum stresses in the shaft and the hub. Both are made of carbon steel (E = 30x106 psi, ν = 0.3). • Solid shaft • ri = 0 in, R= 0.5 in. (nominal) • Tolerances: +2.3x10-3/+1.8x10-3 in. • Gear hub • R= 0.5 in. (nominal), ro = 1 in • Tolerances: +0.8x10-3/0 in. Press and Shrink Fits

17. Example A bronze bushing 50 mm in outer diameter and 30 mm in inner diameter is to be pressed into a hollow steel cylinder of 100 mm outer diameter. Determine the tangential stresses for the steel and bronze at the boundary between the two parts. • Eb = 105 Gpa • Es = 210 Gpa • ν= 0.5 • radial interference δ= 0.025 mm Press and Shrink Fits