introduction finite element method by a.vinoth jebaraj
TRANSCRIPT
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.