The document provides an overview of the finite element method (FEM) including its history, basic steps, and direct formulation approach. It describes how FEM was developed in the 1950s for aerospace applications and became widely used after the 1960s. The key steps are discretizing the domain, deriving element equations, assembling to obtain whole problem equations, applying boundary conditions, solving, and post-processing results. Direct formulation relates applied forces to displacements using element stiffness matrices which are assembled into a global matrix to solve for unknown displacements. Examples show applying this to simple structural problems.

1. FINITE ELEMENT METHOD ( BDA 4033 ) Lecture #01

17. Direct Formulation (Contd..)- Stiffness Matrix Let us write the stiffness matrix for the first element with CSA, A 1 as and For the second element with CSA, A 2 as where k 1 =A 1 E 1 /L 1 and k 2 =A 2 E 2 /L 2 Then the force equation can be written as for element 1 and for element 2 F 1 2 1 2 3

18. Direct Formulation (Contd..)- Assembly Physical Assembly: Step 0: Nodes Matrix assembly : Global Matrix Order = No. of nodes X No. of displacements= 3 X 3 1 2 3

19. Direct Formulation (Contd..)- Assembly Step 1: Physical Assembly: Matrix assembly : Step 2: Matrix assembly : Physical Assembly: 1 2 3 1 1 2 3 1 2

20. Direct Formulation (Contd..)- Apply BCs In the given beam, u 1 =F 1 =F 2 =0 Substituting the values in the assembled equation Solution: Solving the above equation, the unknown displacements u 2 and u 3 can be found. F 1 2 1 2 3

21. Direct Formulation (Contd..)- Post Processing From the previous solution, the unknown displacements u 2 and u3 a re found. To find the element stresses: For the element 1, the strain ε 1 = ( u 2 - u 1 )/L 1 Applying Hooke’s law,  1 =E 1 ε 1 =E 1 (u 2 - u 1 )/L 1 Similarly for the element 2,  2 =E 2 ε 2 =E 2 (u 3 - u 2 )/L 2

25. Questions ?