introduction finite element method by a.vinoth jebaraj

Dr.A.Vinoth Jebaraj, SMECVIT University, Vellore

DIFFERENT FAILURES OF MATERIALS?

So, On what basis we

have to design a machine

component?

Methods to solve any Engineering problem

Experimental Analytical Numerical

Time consuming & needs experimental setup

Atleast 3 to 5 prototypes must be tested

Applicable only if physical model is available

Approximate solution

Applicable if physical model is not available

Real life complicated problems

100% accurate result

Applicable only for simple problems

=

y ?

Is this equation is correct for the above beam?

Area = l × b

Area = ?

Error Solution

FEA is a numerical method to find the location and

magnitude of max stress and deflection in a structure.

Solid Plate - Theoreticalsolution is possible

Plate with Holes – No theoreticalsolution available

Load

Load

Challenge lies in representing the exact geometry of the structure, especially, the curves.

Coarser mesh

Fine mesh

Regions where geometry is complex (curves,

notches, holes, etc.) require increased

number of elements to accurately represent

the shape.

Atomic Structure Finite Element model

Infinite to Finite

Degrees of Freedom ?

Why do we carry out MESHING?

Machine component

Types of Finite elements

1D (line) element 2D (plane) element 3D solid element

Truss, beam, spring, pipe etc.

Membrane, plate, shell etc. 3D fields

Traditional Design cycle Vs. FEA

FE Model & BC’sFinite Element ModelCAD Model

Max Stress

Max Displacement

Simple Bracket

FEA Replacement for costly and Time consuming Testing

Pre-processing or modeling the structure

Post processing

Stresses vs. Resisting Area’s(Fundamentals of stress analysis)

For Direct loading or Axial loading

For transverse loading

For tangential loading or twisting

Where I and J Resistance properties of cross sectional area

I Area moment of inertia of the cross section about the axes lying on the section (i.e. xx and yy)

J Polar moment of inertia about the axis perpendicular to the section

Plane of Bending

X – Plane

Y - Plane Z - Plane

Under what basis Ixx, Iyy and Izz have to be selected in bending

equation?

Bending

Bending Twisting

Stress Tensor

Planar Assumptions

All real world structures are three dimensional.

For planar to be valid both the geometry and the loads must be constant across the thickness.

When using plane strain, we assume that the depth is infinite. Thus the effects from end conditions may be ignored.

Plane Stress

All stresses act on the one plane – normally the XYplane.

Due to Poisson effect there will be strain in the Zdirection. But We assume that there is no stress inthe Z – direction.

σx, τxz, τyz will all be zero.

All strains act on the one plane – normally the XYplane. And hence there is no strain in the z-direction.

σz will not equal to zero. Stress induced to preventdisplacement in z – direction.

εx, εxz, εyz will all be zero.

Plane Strain

A thin planar structure with constant thickness and loading within the plane of thestructure (xy plane).

A long structure with uniform cross section and transverse loading along its length (z –direction).

Stiffness

Axial stiffness =

; Bending stiffness =

; Torsional stiffness =

Types of Analysis

One dimensional analysis

Two dimensional analysis

Three dimensional analysis

Uniaxial Loading Plane Loading Multiaxial Loading

Axial stress Nodal displacement

FE Model Nodal displacement

Axially loaded Bar Element (Tension – Compression only)

Transverse loading Beam Element (Bending)

Nodal displacement

Bending stress

FE Model

Why I – section is better?

Beam Element (Torsion)

Shear stress

Shear stress

11.02 MPa

11.3 MPa

89.9 MPa

Plane Element (In plane loading)

Uy = 0

Ux = 0

Shell Element (plate bending)

“Membrane forces + bending moment”

Example: car body and tank containers

Quadratic Element Vs. Triangular Element

Quadratic element is moreaccurate than triangularelement (due to betterinterpolation function)

Tria element is stiffer than quad,results in lesser stress anddisplacement if used in criticallocations.