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 equations by 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 quadrila1teral 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 axis-symmetric truss problems.