UNIT ‐ I:
Calculus of variation, Introduction to calculus of variations, Introduction to equilibrium equations in elasticity, Euler’s Lagrange’s equations, Principal of virtual work, virtual displacements, Principles of minimum potential energy, boundary value, initial value problems, Flexibility approach, Displacement approach, Different problems in structural analysis.
UNIT – II:
FEM Procedure, Derivation of FEM equations by variation principle polynomials, Concept of shape functions, Derivation for linear simplex element, Need for integral forms, Interpolation polynomials in global and local coordinates. Weighted residual Methods: Concept of weighted residual method, Derivation of FEM equationsnby Galerkin’s method, Solving cantilever beam problem by Galerkin’s approach, Derivation of shape functions for CST triangular elements, Shape functions for rectangular elements, Shape functions for quadrilateral elements.
UNIT – III:
Higher order Elements: Concept of iso-parametric elements, Concept of sub-parametric and super – parametric elements, Concept of Jacobin matrix. Numerical Integration: Numerical Integration, one point formula and two point formula for 2D formula, Different problems of numerical integration evaluation of element stiffness matrix, Automatic mesh generation schemes,
UNIT – IV:
Pascal’s triangle law for 2D shape functions polynomial, Pascal’s triangle law for 3D shape function polynomials, Shape function for beam elements, Hermition shape functions. Convergence: Convergence criteria, Compatibility requirements, Geometric isotropy invariance, Shape functions for iso-parametric elements, Special characteristics of stiffness matrix, Direct method for deriving shape functions using Langrage’s formula, Plane stress problems.
UNIT – V:
Analysis of structures: Truss elements, Analysis of truss problems by direct stiffness method. Analysis of frames and different problems, Different axi-symmetric truss problems.