Finite Element Lecture-Background.pdf

BackgroundBackgroundReview :

Stresses and equilibriumStrain-displacementStress-strain relations

Solution to Complex Systems:Total Potential energyThe Rayleigh-Ritz MethodThe Galerkin (“residual”) MethodThe Saint Venant Principle

Failure Criteria:The von Mises stressThe Tresca stressThe Mohr-Coulomb stress

Why do we need to use the Finite Element method of analysis?

In Mechanics we have three types of problems:

(1) Steady-state (statics)

(2) Propagation (dynamics)

(3) Eigenvalue problems.

These problems ideally should be solved as continuous systems. Theseare usually very difficult to solve. Therefore, most problems are simplified into discrete systems, and that is where the FEM comes in.

In a continuous systems, for example, the large flow-net beneath a dam, or the stress flow in a loaded slab-column system, the response of the system is described by many variables at an infinite number of points. The system is solved by a large set of differential equations and boundary conditions.

In contrast, a discrete system will have its response described by variables at a finite number of points. This system is solved by a smaller set of algebraic equations. The finite element method isprecisely a simple system of algebraic equations, that finds thevalues of each variable at a finite number of points (the nodes)by combining the solution of a large matrix of equations throughthe use of computers.

Solution procedure of a discrete system:

Step 1: Idealize the problem into a system of elements

Step 2: Evaluate each element’s equilibrium requirements

Step 3: Assemble all the elements using matrices

Step 4: Find the system’s response through either direct or variational methods.