Computer Applications: Finite Elements in Solids and Structures [until 2009]

Lecturer(s)

PD. Dr. Andrei A. Gusev

Hour(s) of lecture(s), exercise(s) and credit points

2L
2E
4CP

Teaching goals

To introduce the finite element method to the students with only a general interest in the topic

Summary and outline

In many engineering and research situations, an adequate, approximate discrete model of a structure or a process can be obtained using a finite number of well-defined components. With the advent of digital computers, such discrete problems can generally be solved readily even if the number of discrete elements is very large. It is the object of this course to present a view of the finite element method as a general discretization procedure for solving structural and continuum problems posed by mathematically defined statements. The lecture topics include:

Introduction: historical development, solid and structural analysis using finite elements.

Preliminaries: the elastic continuum, deformation, strain, stress, stress-strain relation¬ship, thermal effects, Bernoulli-Euler beam theory, Kirchhoff’s thin plate theory.

Energy formulations: variational approach, principle of virtual work, principle of station¬ary total potential, strain energy, strain energy of beams and plates, potential energy of external loads, total energy, the Rayleigh-Ritz method.

Displacement finite elements: element definition, element shape functions, strain-dis¬placement matrix, element stiffness matrix, equivalent nodal forces, thermal effects, equiv¬alent thermal loads, equivalent nodal forces, total energy of an element, assembly, minimization and solution.

Solutions to the finite element equations: direct and indirect methods, Gaussian elimi¬nation, LU methods, Choleski factorization, banded and profile (‘skyline’) solvers, effects of degree of freedom numbering.

Linear elements: linear triangles for plane-stress and plane-strain analysis, subdivision into elements, equivalent nodal forces, thermal loads, body forces, surface tractions, axi¬symmetric analysis, the linear tetrahedron.

Convergence, compatibility and completeness: compatibility and completeness crite¬ria, non-conforming elements, the patch test.

Higher order elements: higher-order interpolation, inter-element compatibility, higher-order triangular elements, isoparametric formulation, curved boundaries, convergence and completeness, Gauss-Legendre integration, Gauss quadrature.

Beam and frame elements: simple beam element, effects of shear deformation, the Timoshenko beam element, frame elements in two and three dimensions.

Plate and shell elements: Kirchhoff displacement elements, non-confirming triangles, mindlin plate elements, shell elements, accuracy and convergence.

Dynamics and vibrations: dynamic equations, internal loads, element mass matrix, lumped and diagonal masses, normal mode analysis, direct integration, explicit and implicit schemes.

Generalization of the FE concepts: Integral statements, weak form statements, the gen¬eral quasi-harmonic equation, variational formulations.

Literature

• Astley R.J. Finite Elements in Solids and Structures,
  Chapman & Hill, 1992;
• Zienkiewicz O. C. & Taylor R.L. The Finite Element Method,
  Volume 1: The Basis, 5th edn, Butterworth-Heinemann 2000;
  FE packages: ANSYS & Palmyra.