INTRODUCTION TO FINITE ELEMENT METHOD

Presented by

S.BASKARASST. PROF. , MECHANICAL

DEPT.SRM UNIVERSITY,

MODINAGAR

Fundamental concepts of engineering analysis

Objectives: The analyst needs certain requirements while

designing and assembling the parts of the product. Those requirements are Displacement at certain points Stress distribution Natural frequencies Vibrations Crack growth, residual strength and fatigue life

METHODS OF ENGINEERING ANALYSIS

1. Experimental methods, prototypes can be used. It needs man power and materials. It is time consuming and costly process.

2. Analytical methods, mathematical differential equations are used. It gives quick and closed solutions. It is used only for simple geometries.

3. Numerical methods, mathematical differential equations are used and it is suitable for complex structure.

What is Finite Element method?

FEM is a numerical method for solving problems of engineering and mathematical physics. In this method, a body or a structure is subdivided into smaller elements of finite dimensions called “Finite Elements”. The body is considered as an assemblage of these elements connected at a finite number of joints called “Nodes” or “Nodal points”.

Finite element method is used to solve physical problems involving complicated geometrics, loading and material properties which can’t be solved by analytical method.

Historical Background Hrennikof and McHenry formulated a 2D structural problem as an assembly of bars and beams

Courant used a variational formulation to approximate PDE’s by linear interpolation over triangular elements

Turner wrote a seminal paper on how to solve one and two dimensional problems using structural elements or triangular and rectangular elements of continuum.

Types of mechanical systems

Linear systems, are far less complex and generally do not take into account plastic deformation.

Non-linear systems, do account for plastic deformation, and many also are capable of testing a material all the way to fracture.

General steps in FEM

Discretization of structure: The art of subdividing a structure in to a

convenient no of smaller elements is known as discretizationElement types:

One dimensional element Two dimensional element Three dimensional element

Element Types

Selection of displacement function:It involves choosing a displacement function with in each element. Polynomial functions are frequently used as displacement functions in finite element formulation.

Types of polynomial functions:

Linear polynomial Quadratic polynomial Cubic polynomial

Define material behavior Derive element stiffness matrixAssemble the element equations to obtain the

global equationsApplying boundary conditionsSolution for the unknown displacementsCompute element stress and strain from nodal

displacements.

Shape Function The values of the field variable are computed

at the nodes are used to approximate the values at non-nodal points by interpolation of the nodal values.

In one dimensional problem, the basic field variable is displacement.u = ∑ Ni ui

For two noded bar element, the displacement at any point with in element is,

u = ∑ Ni ui = N1 u1 + N2 u2

N1 & N2 - shape functions

Characteristics of shape function The shape function has unit value at its own nodal point and zero value at other nodal points. The sum of shape function is equal to one. The shape functions are always polynomial functions.

Shape Function Formulation

u1 u21 2

Consider is bar element of length L with nodes 1 and 2 as shown in fig. u1 and u2 are the displacements at the respective nodes. U=a0+a1 xWhere, a0 and a1 are global co-ordinates.u = [1 x ] a0

At node 1, u = u1, x=0At node 2, u = u2, x=1

u1 = a0

u2 = a0+a1L

a1

u1

u2

= 1 0

1 l

a0

a1

u = [ 1 x ] 1/L

u = [ L-x/L x/L]

l 0-1 1

u1

u2

u1

u2

u = [N1 N2 ] u1

u2

N1 =L-x/L N2 = x/L

At node 1, x = 0 N1 = 1 N2 = 0

At node 2, x = L N1 = 0 N2 = 1

STIFFNESS MATRIX FORMULATION

FEM Applications• Static analysis of trusses, beams, frames,

plates, bridges, machine structures.• Structural analysis of aircraft wings, missile

and rocket structures, etc.• Natural frequencies and modes of structures,

linkages, gears, flywheels, and cams • Stability analysis of aircraft, rocket and

missiles.• Dynamic response of structures subjected to a

periodic loads and random loads.

• Stress analysis of pressure vessels, flywheels. Crankshafts, cams, linkages, gears, machine members, etc.

• Steady state temperature distribution in solids and fluids.

• Transient heat flow in IC engines, turbine blades, steam pipes, and rocket nozzles.

• Analysis of potential flows, free surface flows, boundary layer flows, viscous flows, and transonic aerodynamic problems.

• Analysis of earthquakes.

• Analysis of robots and computer chip.

• Analysis of casting, forming, welding and machining processes.

• Stress analysis of bones and teeth, load bearing capacity of implant and prosthetic systems, and mechanics of heart values.

Advantages of FEM:• Model complex shaped bodies quite easily

• Handle several load conditions without difficulty.

• Handle different kinds of boundary conditions.

• Model bodies composed of several different materials.

• Discretize the bodies with combination of different elements, because the element equations can be evaluated individually.

• Handle time dependent and time independent heat transfer.

• Handle steady and unsteady, compressible and incompressible, laminar and turbulent.

THANK YOU