Revision of Solid Mechanics through
Worked Examples
3D Stress, Equivalent Stress and more…
About
This resource serves as a refresher for key concepts in the mechanics of solids, essential for successfully completing this course. It is strongly recom-mended that you have a solid understanding of these concepts, typically gained through completing prerequisite courses (Mechanics of Solids 3 and Mechanical Design 1 ).
If you have not successfully completed these prerequisites, please consider dropping this course and fulfilling the requirements first, or consult with the teaching staff for guidance.
For those who have met the prerequisites, this document offers a concise review, complementing the revision videos available on Canvas. It provides additional information and practical applications of concepts through simple analysis problems. The problems are organised by topic and progressively increase in complexity.
Instructions
To make the most of this resource, follow these steps:
1. Attempt: Solve each problem independently, without consulting the solutions.
2. Identify Gaps: Determine which concepts you struggle with or need to review.
3. Relearn: Revisit these concepts using relevant literature and attempt the problems again.
4. Check Solutions (Only after steps 1-3): Compare your solutions to the provided ones.
5. Further Revision: Note areas where your understanding can be im-proved.
6. Revise Again: Focus your studies on these areas to solidify your knowledge.
7. Optional Challenge (Advanced): Create and solve variations of the problems you found most challenging.
Questions
Please attempt them first without consulting the solutions.
1. A pressure vessel is designed to operate at a maximum design stress of 180 MPa. The vessel is made of a material with a yield strength of 240 MPa. Determine the factor of safety against yielding.
2. A cylindrical pressure vessel has an internal diameter of 0.5 m and a wall thickness of 10 mm. If the internal pressure is 2 MPa, determine the hoop stress and longitudinal stress in the vessel.
3. A cylindrical pressure vessel made of steel (yield strength Sy = 250 MPa) has an internal diameter of 0.8 m and a wall thickness of 15 mm. Determine the maximum allowable internal pressure using the Tresca yield criterion.
4. Repeat Question 3 using the von Mises yield criterion.
5. The state of stress at a point in a cylindrical pressure vessel is given by: σl = 10 MPa, σθ = 20 MPa, τrθ = 5 MPa. Determine the principal stresses and their orientations.
6. A thin-walled cylindrical pressure vessel has a radius of 0.2 m and a wall thickness of 5 mm. It is subjected to an internal pressure of 3 MPa and a torque of 10 kNm. Determine the principal stresses and the maximum shear stress in the vessel.
7. A cylindrical pressure vessel is subjected to combined loading: internal pressure of 4 MPa, axial load of 200 kN, and torque of 50 kNm. The vessel has a mean radius of 0.3 m and a wall thickness of 10 mm. Determine the equivalent stress using the von Mises and Tresca criteria.
8. A thick-walled cylindrical pressure vessel has an inner radius of 0.1 m and an outer radius of 0.15 m. The internal pressure is 5 MPa. Determine the radial and hoop stresses at the inner and outer surfaces of the vessel.
9. A cylindrical pressure vessel has an internal diameter of 1 m and a wall thickness of 20 mm. The internal pressure fluctuates between 2 MPa and 6 MPa. Determine the mean stress and stress amplitude.
10. A cylindrical pressure vessel made of a material with a fatigue strength of 200 MPa is subjected to fluctuating stresses as described in Question 9. Determine the fatigue life of the vessel using the Soderberg criterion. (Assume Sy = 350 MPa)