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

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.

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

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

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)

Complex structures with simple elements

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

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

y x At node 1: At node 2:

x y Rewrite as

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

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

Putting all the pieces together x y The desired relationship is is the element stiffness matrix in the global coordinate system Where

Assembly….

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

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

Global k for 3-d trusses

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