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:

  • 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
  •  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:

  • REVIEW OF 3D LINEAR ELASTICITY
    • Stresses and strains in 3D, equilibrium, constitutive equations, boundary value problems
  • 2D ELASTICITY
    • Plane stress and plane strain conditions, constitutive equations in 2D
  • 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
  • DISCRETIZATION VIA THE RITZ METHOD
    • The Ritz method of approximation, discretized governing equations, error and convergence, Lagrangian shape functions, finite element approximations
  • FINITE ELEMENTS IN 1D ELASTCITY
    • Finite element approximation, error and convergence
  • NUMERICAL IMPLEMENTATION
    • Assembly procedure, boundary conditions, solution of systems of linear equations, numerical integration using the Gauss quadrature
  • FINITE ELEMENTS IN 2D/3D ELASTCITY
    • Element technology in 2D (T3, T6, Q4, Q8, Q9, ..), and 3D
  • STRESS INTERPRETATION
    • Stress discontinuity, averaging techniques, interpolation/extrapolation.
  • NUMERICAL ISSUES
    • Locking, reduced integration, spurious energy modes.
  • COMMERCIAL SOFTWARE
    • Abaqus tutorial
  • ADVANCED ELEMENT TECHNOLOGY
    • Higher-order elements, “node-less” degrees of freedom
  • SPECIAL TOPICS (if time permits)
    • Large deformations, material nonlinearity, plate and shell finite element formulations, other problems (heat transfer)