1. FINITE ELEMENT METHOD (INTRODUCTION)
E, I, L F x
Element 1 2 θ1 θ2 y1 y2 d y F
y(x)
θ(x)

2. Displacement vector of a single finite element having 4 DOF

3. C
Potential energy

4. This integral can be calculated using symbolic operations in Matlab

5. Element Stiffness Matrix
Example:
20 mm
2 mm
500 mm
1 N
d=0.5
E=2x1011
Tip displacement is calculated using single element

6. y1=0 Sınır Şartı Sınır Şartı θ1=0
Force is applied at node 2 in positive y direction
y1=0
Sınır Şartı
θ1=0
Sınır Şartı

7. Recalculate using two elements
Example:
20 mm
2 mm
500 mm
1 N 1 2 3
d=0.25 m x2
+6d
+6d x2

8. Displacement and slope can be calculated at mid point of the beam
y2= m
θ2= rad
y3= m
θ3= rad
Displacement and slope can be calculated at mid point of the beam
Sınır şartı
y1=0
θ1=0
Force is applied at node 3 in positive y direction

9. Analysis using SolidWorks
Steel beam having 20x2x500 mm3 cross section
Fixed
1 N
δL= m

10. Example: 1 N L=500 mm 2 mm δ 20 mm 1 2 3 Using two elements y1=0
Sınır şartı
y1=0
y3=0
θ1= rad
y2= m
θ2= rad
θ3= rad
Slope is zero at the mid point of the beam
Slope has opposite sign at the ends of the beam
Force is applied at node 2 in positive y direction