1. Structures Matrix Analysis
Introduction to Finite Element Analysis
Sina Askarinejad
1/1
7/8/2016

2. Hook’s Law..
Hook’s law is a principle of physics that states that the force (F) needed to extend or compress a spring by a distance X is proportional to that distance. F = kX where k is a constant factor characteristic of the spring (stiffness), and X is small compared to the total possible deformation of the spring.

3. Application to elastic materials
Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state, often obey Hooke's law. σ=𝐸ε σ = 𝐹 𝐴 , ε= ∆𝐿 𝐿 𝐹= 𝐸𝐴 𝐿 ∆𝐿

4. Structures and elements
In order to analyze engineering structures, we divide the structure into small sections. F
d1=?
Trusses
Beams
Plates
Shells
3-D solids
Nodes
Elements

5. Truss
Truss is a slender member (length is much larger than the cross-section).
It is a two-force member i.e. it can only support an axial load and cannot support a bending load.
The cross-sectional dimensions and elastic properties of each member are constant along its length.
The element may interconnect in a 2-D or 3-D configuration in space.
The element is mechanically equivalent to a spring, since it has no stiffness against applied loads except those acting along the axis of the member.
(Beam is a truss that can tolerate bending load with rotation)

6. Complex structures with simple elements

7. Finite element method
Step 1: Divide the system into bar/truss elements connected to each other through the nodes
Step 2: Describe the behavior of each bar/truss element (i.e. derive its stiffness matrix and load vector in local AND global coordinate system)
Step 3: Describe the behavior of the entire system by assembling their stiffness matrices and load vectors
Step 4: Apply appropriate boundary conditions and solve

8. Element stiffness matrix in local coordinates
E, A, L
Two nodes: 1, 2
Nodal displacements:
Nodal forces:
Spring constant:
Element stiffness matrix in local coordinates
Element nodal displacement vector
Element force vector
Element stiffness matrix

9. y x
At node 1:
At node 2:

10. x y
Rewrite as

11. In the global coordinate system, the vector of nodal displacements and loads
Our objective is to obtain a relation of the form
Where k is the 4x4 element stiffness matrix in global coordinate system

12. Relationship between and for the truss element
At node 1
At node 2
Putting these together

13. Putting all the pieces togetherx y
The desired relationship is
is the element stiffness matrix in the global coordinate system
Where

15. Assembly….

16. Same procedure for 3-D trusses
© 2002 Brooks/Cole Publishing / Thomson Learning™

17. Transformation matrix T relating the local and global displacement and load vectors of the truss element
Element stiffness matrix in global coordinates

18. Global k for 3-d trusses

19. Solve the following problem with your previous knowledge about springs and the method that you just learned
E2, A2 L1 L2
E1, A1 F