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Lecture

MECH 4101 Lecture Notes - Cylinder Stress, Residual Stress, Stress Concentration


Department
Mechanical Engineering
Course Code
MECH 4101
Professor
Shashank Pant

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MECH 4101 PROBLEM SOLUTIONS
Axi-Symmetric Deformations ......................................................................................................................................... 2
1. Internal Pressure ONLY ........................................................................................................................................... 2
open-ended case .......................................................................................................................................................... 2
closed-ended case ....................................................................................................................................................... 3
restrained ends case .................................................................................................................................................. 3
2. Rotational Effects ONLY ........................................................................................................................................... 4
maximum hoop stress .............................................................................................................................................. 4
bearing effect (restrained ends) ......................................................................................................................... 5
3. Internal Pressure with Thermal Loads ........................................................................................................... 5
open-ended case .......................................................................................................................................................... 5
closed-ended case ....................................................................................................................................................... 6
restrained ends case .................................................................................................................................................. 7
4. Rotational Effects with Thermal Effects and Blades ................................................................................ 8
Stress Functions ................................................................................................................................................................... 11
1. Fourth Order Polynomial ..................................................................................................................................... 11
(a) Assume a1, a3, a4 , and a5 are all equal to zero..................................................................................... 11
(b) Assume a2, a4 , and a5 are all equal to zero ........................................................................................... 12
2. Triangular Plate ......................................................................................................................................................... 12
3. Wedge .............................................................................................................................................................................. 13
5. (a) Knife Edge .............................................................................................................................................................. 15
6. Stress Concentration Factor of a Small Hole .............................................................................................. 16
(a) biaxial tension ..................................................................................................................................................... 16
(b) pure shear ............................................................................................................................................................. 19
Shock and Impact Loading ............................................................................................................................................. 20
1. Axial Tension ............................................................................................................................................................... 20
2. Long, Elastic Shock in a Cord .............................................................................................................................. 21
3. Uniform Rigid Bar ..................................................................................................................................................... 22
4. Simply Supported Beam ........................................................................................................................................ 23
(a) rigid supports ...................................................................................................................................................... 23
(b) spring supports .................................................................................................................................................. 24
5. Sliding Collar ............................................................................................................................................................... 25
(a) max. stress in bar ............................................................................................................................................... 25

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1
(b) max. extension .................................................................................................................................................... 26
Elasto-Plastic Analysis ..................................................................................................................................................... 27
1. Elastic-Perfectly Plastic Beam ........................................................................................................................... 27
(a) max elastic bending moment ...................................................................................................................... 27
(b) elastic-plastic moment ................................................................................................................................... 27
(c) residual stress distribution .......................................................................................................................... 27
(d) resultant stress distribution ....................................................................................................................... 27
2. Elastic-Perfectly Plastic Strip ............................................................................................................................. 28
3. Narrow Aluminum Beam (Aluminum Core) .............................................................................................. 28
4. Thick-Walled Cylinder Max Internal Pressure ......................................................................................... 29
5. Thick-Walled Cylinder Autofrettage .............................................................................................................. 29
6. Thick-Walled Cylinder Proof .............................................................................................................................. 30

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2
Axi-Symmetric Deformations
1. Internal Pressure ONLY
The steel cylinder is long, so it is a plane strain case.
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