The Direct Method

ByS. Ziaei Rad

Model Based Simulation

Types of Finite Elements1-D (Line) Element

2-D (Plane) Element

Types of Finite Elements3-D (Solid) Element

(3-D fields - temperature, displacement, stress, flow velocity)

Spring ElementOne Spring Element

Spring force-displacement relationship:

Spring ElementConsider the equilibrium of forces for the spring. At node i,we have

and at node j,

In matrix form,

or,

Spring Element

where

k = (element) stiffness matrix

u = (element nodal) displacement vector

f = (element nodal) force vector

Note that k is symmetric. Is k singular or nonsingular? That is, can we solve the equation? If not, why?

Spring System

For element 1,

For element 2,

Assemble the stiffness matrix for the whole system

where is the (internal) force acting on local node i of Element m (i = 1, 2).

Consider the equilibrium of forces at node 1,

at node 2,

and node 3,

Spring System (Assembly)That is,

In matrix form,

or

K is the stiffness matrix (structure matrix) for the spring system.

An alternative way of assembling the whole stiffness matrix

“Enlarging”the stiffness matrices for elements 1 and 2, we have

An alternative way of assembling the whole stiffness matrixAdding the two matrix equations (superposition), we have

This is the same equation we derived by using the force equilibrium concept.

Boundary and load conditionsAssuming,

we have

which reduces to

and

SolutionsUnknowns are and the reaction force F1

Solving the equations, we obtain the displacements

and the reaction force

Checking the Results·Deformed shape of the structure·Balance of the external forces·Order of magnitudes of the numbers

Notes About the Spring Elements

Suitable for stiffness analysisNot suitable for stress analysis of the spring itselfCan have spring elements with stiffness in the lateral direction, spring elements for torsion, etc.

Example 1.1

Given:For the spring system shown above,

Find:(a) the global stiffness matrix(b) displacements of nodes 2 and 3(c) the reaction forces at nodes 1 and 4(d) the force in the spring 2

Example 1.1 : Solution(a) The element stiffness matrices are

Example 1.1 : SolutionApplying the superposition concept, we obtain the global stiffnessmatrix for the spring system as

Example 1.1 : Solutionor

which is symmetric and banded.Equilibrium (FE) equation for the whole system is

(*)

Example 1.1 : Solution(b) Applying the BC

or deleting the 1st and 4th rows and columns, we have

Solving, we obtain

(c) From the 1st and 4th equations in (*), we get the reaction forces

Example 1.1 : Solution(d) The FE equation for spring (element) 2 is

Here i = 2, j = 3 for element 2. Thus we can calculate the springforce as

Example 1.2Problem: For the spring system with arbitrarily numbered nodesand elements, as shown above, find the global stiffness matrix.

Example 1.2 : SolutionFirst we construct the following

which specifies the global node numbers corresponding to thelocal node numbers for each element.

Then we can write the element stiffness matrices as follows

Example 1.2 : Solution

Finally, applying the superposition method, we obtain the Global stiffness matrix as follows

Example 1.2 : Solution

The matrix is symmetric, banded, but singular. Why?