CE 59500 – Finite Elements in Elasticity
Credits and contact hours:
- 3 credits
- Lecture meets 2 times per week for 75 minutes per meeting for 15 weeks
Specific course information:
- Catalog description: Fundamentals of theory of elasticity; variational principles; one-, two-, and three-dimensional elasticity finite elements; interpolation methods; numerical integration; convergence criteria; stress interpretation.
- Prerequisites: CE 47400 or equivalent, or graduate standing
- Course status: Elective course
Specific Goals for the course:
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Student learning outcomes - Upon successful completion of this course the student shall be able to:
- have a good command of 3D elasticity governing equations
- estimate stresses and strains in three-dimensional bodies
- set-up weighted residual forms of governing equations for problems in linear elasticity
- have a fundamental understanding of finite element technology in 1D, 2D, and 3D
- find numerical solutions for elasticity problems
- identify numerical issues (locking, spurious energy modes)
- assess accuracy and convergence of finite element solutions
- interpret and assess stress estimates
- write Matlab programs for numerical analysis of elasticity problems
- use commercial finite element software (ABAQUS)
- communicate analysis methods and discuss results in technical papers
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Relationship of course to program outcomes
- Outcome 1: An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.
- Outcome 3: An ability to communicate effectively with a range of audiences.
- Outcome 5: An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives.
Topics:
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REVIEW OF 3D LINEAR ELASTICITY
- Stresses and strains in 3D, equilibrium, constitutive equations, boundary value problems
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2D ELASTICITY
- Plane stress and plane strain conditions, constitutive equations in 2D
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VARIATIONAL FORMULATIONS
- Weighted residuals for 1-D, 2-D and 3-D elasticity, divergence theorem, virtual work statement, essential and natural boundary conditions, the Fundamental Theorem of the Calculus of Variations
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DISCRETIZATION VIA THE RITZ METHOD
- The Ritz method of approximation, discretized governing equations, error and convergence, Lagrangian shape functions, finite element approximations
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FINITE ELEMENTS IN 1D ELASTCITY
- Finite element approximation, error and convergence
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NUMERICAL IMPLEMENTATION
- Assembly procedure, boundary conditions, solution of systems of linear equations, numerical integration using the Gauss quadrature
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FINITE ELEMENTS IN 2D/3D ELASTCITY
- Element technology in 2D (T3, T6, Q4, Q8, Q9, ..), and 3D
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STRESS INTERPRETATION
- Stress discontinuity, averaging techniques, interpolation/extrapolation.
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NUMERICAL ISSUES
- Locking, reduced integration, spurious energy modes.
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COMMERCIAL SOFTWARE
- Abaqus tutorial
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ADVANCED ELEMENT TECHNOLOGY
- Higher-order elements, “node-less” degrees of freedom
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SPECIAL TOPICS (if time permits)
- Large deformations, material nonlinearity, plate and shell finite element formulations, other problems (heat transfer)