Course Instructors:
Prof. Dr. Saeed Asiri
- Office: 24E38
- Website: https://www.asiri.net
- WhatsApp: +966565555275
- Email: saeed@asiri.net
- X: @profsaeedasiri
- Facebook: saeedasiri
Download: Calendar
Text: Reddy, J. N., (2002) Energy Principles and Variational Methods in Applied Mechanics, 2d
Edition, John Wiley, New York, NY.
Catalog Description: Derivation, interpretation, and application to engineering problems of the
principles of virtual work and complementary virtual work. Related theorems such as the principles of the
stationary value of the total potential and complementary energy, Castigliano’s Theorems, theorem of least
work and unit force and displacement theorems. Introduction to generalized, extended, mixed, and hybrid
principles. Variational methods of approximation, Hamilton’s principle, and Lagrange’s equations of
motion. Approximate solution to problems in structural mechanics by use of variational theorems.
Course Philosophy: Energy principles form the basis for many methods (including finite element
methods) that are widely used in structural mechanics. Understanding these principles requires an in-depth
look at the principles. We will attempt to take the “scenic route”, to encourage you to consider the
principles and methods in depth.
Probable Coverage: (as time permits)
1. Introduction (Ch. 2 and appendix I of Reference 5)
• A few historical notes
• Extrema of functions and constrained functions
• functionals defined
• Extrema of functionals and constrained functionals
2. Mathematical Preliminary (Ch. 2):
• Vector
• Tensors
3. Review of Equation of Solid Mechanics (Ch.3):
• Stress
• Strain
• Constitutive Laws
4. Work, Energy and Variational Calculus (Ch.4):
• Concept of Work and Energy
• Strain Energy and Complementary Strain Energy
• Virtual Work
• Calculus of Variations
5. Energy Principles of Structural Mechanics (Ch.5):
• Principles of Virtual Work
• Principles of Complementary Virtual Work
• Principles of Stationary and Minimum Potential Energy
• Principle of Stationary Complementary Energy
• Castigliano’s Theorems
6. Dynamical Systems: Hamilton’s Principles (Ch.6):
• Extension of Virtual Work to Dynamic Systems
• Generalized Hamilton’s Principle
• Applications
7. Finite Element Method (Ch.7):
• Spring Element
• Bar Element
• Beam Element
Recommended References:-
- Weinstock, R., (1952), Calculus of Variations, McGraw Hill, New York, NY (Available from
Dover Publications). - Gelfand, I. M., and Fomin, S. V., (1963) Calculus of Variations, Prentice Hall, Englewood Cliffs
NJ (Available from Dover publications) - Przemieniecki, J. S., (1968), Theory of Matrix Structural Analysis, McGraw Hill, New York, NY.
- Langhaar, H. L., (1962) Energy Methods in Applied Mechanics, John Wiley, New York, NY
- Shames, I. H., and Dym, C. L., (2003), Energy and Finite Element Methods in Structural Mechanics, Taylor and Francis, New York, NY.
- Washizu, K., (1975) Variational Methods in Elasticity and Plasticity, 2d Edition, Pergamon Press,
Oxford.
How to Succeed:-
- Take responsibility for your learning and engage actively with the material.
- Attend lectures and participate in class activities to enhance your understanding.
- Review the topic in advance using available resources, so you can contribute effectively during discussions.
- Utilize external resources (internet, classmates, professors, and the library) for additional support when needed.
- Practice problem-solving regularly to strengthen your grasp of key concepts.
Course Learning Outcomes (CLOs):
By the end of the semester, the student will be able to:
- Understand the fundamental concepts of variational calculus and apply them to engineering mechanics.
- Solve engineering problems using variational principles such as the principle of virtual work and potential energy.
- Apply Castigliano’s theorem and the complementary energy principle to analyze mechanical systems.
- Use approximation methods like the Ritz method, Petrov-Galerkin method, and Least Squares method to solve complex problems.
- Understand and apply the basics of the finite element method to solve real-world mechanical engineering problems.
- Analyze the stability and monotonicity of potential energy functions in mechanical systems.
Grading Policy:
MT | 20% |
Collaborative Learning Activities | 40% |
Final Opportunity To Shine | 40% |
- Note: 75% attendance is required. No makeup for any exams. Student must attend the presentaions to pass the course.