Finite Element Method in Mechanics

Variational Methods. Principle of stationary potential energy. Problems having many degrees of freedom (DOF). Potential energy of an elastic body. The Rayleigh-Ritz method. Galerkin and other weighted residual methods (MWR). Examples: Uniform bar, Beam dynamics. Galerkin FEM in two dimensions.
Bar and Beam Elements. Displacement-based elements. Shape functions. Stiffness matrix. Properties of stiffness matrices. Timoshenko beam element. Boundary conditions. Applied mechanical loads. Equilibrium equations. Stresses.
FEM Concepts. Elements of arbitrary orientation – local and global matrices. Assembly of elements ( assembly and structure node numbers ). Exploiting sparsity, numbering and sparsity. Solution of equations. Structural symmetry. Connecting dissimilar elements. Eccentric stiffeners. Rigid elements.
Basic Elements. Preliminaries: Strain-displacement relations, Stress-strain relations. Interpolation and shape functions. Formulas for element matrices. Linear triangle ( constant-strain triangle CST ). Quadratic triangle ( LST ). Bilinear rectangle
( Q4 ). Quadratic rectangle ( Q8, Q9 ). Rectangular solid elements. Choice of interpolation functions. Nature of a finite element solution.
Isoparametric Elements. Example- bar element. Bilinear quadrilateral ( Q4 ). Transformation. [B] matrix and stiffness matrix. Numerical integration and Gauss quadrature. One, two and three dimensions. Stiffness matrix integration. Static condensation. Stress calculation.
Analysis of axisymmetric solids. Elasticity relations. Axisymmetric solid elements. Loads without axial symmetry.
FEM in Structural Dynamics. Dynamic equation. Mass and damping matrices. Consistent and lumped (diagonal) mass matrix. Proportional damping, Eigenfrequencies (natural frequencies), eigenmodes (mode shapes) and solutions method. Reduction of the number of DOF.
Response History. Modal methods. Harmonic response. Direct integration methods-explicit or implicit. Central differences-stability conditions. Newmark family of methods.
Heat Transfer and Selected Fluid Problems. Heat transfer: introduction. Finite element formulation. Transient thermal analysis – Modal method and direct integration. Acoustics and FE formulation. Boundary absorption. Fluid - structure interaction.
Buckling. Geometric nonlinearity-Green strain. Energy considerations. Initial stress stiffness matrix (geometric stiffness matrix ). Linear buckling. Imperfection. Nonlinear buckling.
Nonlinearity. Newton-Raphson method. Arc-length method. Convergence criteria. Problems of gaps and contact.