CE 57000 – Advanced Structural Mechanics

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: Studies of stress and strain, failure theories, and yield criteria; flexure and torsion theories for solid and thin-walled members; and energy methods.
  • Prerequisites: CE 27000 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:
    • solve simple boundary problems in linear elasticity
    • estimate stresses and strains in three-dimensional bodies
    • set-up weighted residual forms of governing equations for numerical analysis
    • find numerical solutions for elasticity problems
    • assess accuracy and convergence of numerical solutions
    • write Matlab programs for numerical analysis of elasticity problems
    • communicate analysis methods and discuss results in a technical paper
  •  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:

  • VECTOR OPERATIONS
    • Index notation, dot product, cross product, tensor product
  • TENSORS, TENSOR OPERATIONS
    • Tensor operations on vectors and tensors, calculating tensor components, tensor and vector calculus
  • CHANGE OF BASIS, TENSOR INVARIANTS
    • Transformation matrix, transforming vector and tensor components, invariants
  • STRETCH AND STRAIN, CHANGE OF AREA AND VOLUME
    • Definition of stretch and strain in 3D, different measures of strain, linearized strains.
  • EIGENVALUE PROBLEMS
    • Tensor eigenvectors and eigenvalues, characteristic equation, orthogonality of eigenvectors, Cayley-Hamilton theorem
  • PRINCIPAL STRETCH, SPECTRAL DEC
    • Principal directions and values of stretch and strain, spectral decomposition theorem
  • TRACTIONS, STRESS TENSOR
    • Traction vector, Newton’s law, tractions on arbitrary surfaces, the Cauchy stress tensor
  • PRINCIPAL STRESS
    • Normal and shear stresses, maximum normal stress, stress tensor eigenvectors and eigenvalues
  • EQUILIBRIUM, ALTERNATIVE REPRESENTATIONS OF STRESS
    • Governing differential equations for forces and moments, first and second Piola-Kirchhoff stress tensors
  • CONSTITUTIVE THEORY
    • Hyper-elasticity, isotropy, stress-strain relationships for linear elastic materials, definition of material moduli: Lamé parameters, Young’s modulus and Poisson’s ration, bulk and shear modulus
  • BOUNDARY VALUE PROBLEMS, CLASSICAL SOLUTION
    • Boundary-value problems in linear elasticity, natural and essential boundary conditions, well-posed problems and unique solutions, torsion
  • LITTLE BOUNDARY VALUE PROBLEMS
    • 1-D elasticity equations, classical solutions to 1-D problems
  • VIRTUAL WORK
    • Weighted residuals for 1-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
  • BEAM THEORY (IF TIME PERMITS)
    • Beam theory equations in 3D, 2D beam theory as a subset of 3D elasticity