Slide 1 : INTRODUCTION TO
FINITE ELEMENT ANALYSIS OF ENGINEERING INFRASTRUCTURE
INTRODUCTION : INTRODUCTION What is Finite Element Method?
Advantages of FEA
Applications
Methods of analysis
Procedure for FEA
FEA packages
A typical LISA static analysis example
Some modelling considerations
Conclusions
CLASSIFICATION OF METHODS OF ANALYSIS : CLASSIFICATION OF METHODS OF ANALYSIS
What is Finite Element Method? : What is Finite Element Method? The finite element method is a numerical analysis technique for obtaining approximate solutions of boundary value problems in engineering . These are also sometimes called field problems.
The field variables may include physical displacement, temperature, heat flux, fluid velocity or other dependent variables depending on the type of physical problem being analyzed.
Advantages of FEA : Advantages of FEA Many problem types can be addressed with the same code, merely by specifying the appropriate element types from the library.
Applicable in different fields of engineering: civil engineering, mechanical engineering, nuclear engineering, biomedical engineering, hydrodynamics, heat conduction and geo-mechanics.
Applicable in Sciences and Medicine.
APPLICATIONS : APPLICATIONS
Applications in Fluid Mechanics: Laminar Flow : Applications in Fluid Mechanics: Laminar Flow
Applications in Structural Engineering: Steel structures : Applications in Structural Engineering: Steel structures
Applications in Highway Engineering: Bridge Decks : Applications in Highway Engineering: Bridge Decks
Applications in Transportation Engineering: Car crashes : Applications in Transportation Engineering: Car crashes
Applications in Electromagnetic fields : Applications in Electromagnetic fields
Applications in Electrical Engineering : Circuit Board : Applications in Electrical Engineering : Circuit Board
Applications in Thermodynamics : Heating of Buildings : Applications in Thermodynamics : Heating of Buildings
Applications in Acoustics: Guitar : Applications in Acoustics: Guitar
Applications in Mechanical Engineering : Tyre : Applications in Mechanical Engineering : Tyre
Applications in Medicine: Human Skeletal System : Applications in Medicine: Human Skeletal System
Applications in Surgery : FEA for Knee : Applications in Surgery : FEA for Knee
Applications in Surgery: FEA for Artificial Knee : Applications in Surgery: FEA for Artificial Knee
Applications in Surgery: FEA of Human Skull : Applications in Surgery: FEA of Human Skull
Procedure in Formulating FEA : Procedure in Formulating FEA Discretize the Continuum
Select Interpolation Functions
Find the Element Properties
Assemble the Element Properties to Obtain the System Equations
Impose the Boundary Conditions
Solve the System Equations
Make Additional Computations
1. Discretizing the Continuum : 1. Discretizing the Continuum
2.Interpolation Functions : 2.Interpolation Functions
3. Element Properties : 3. Element Properties Stress Stiffness Strain
4. System Equations : 4. System Equations Stiffness Equation where
5. Boundary Conditions : 5. Boundary Conditions In order to solve the resulting system equations, certain boundary conditions must be specified to make the equation solvable.
6. Solving the System Equations : 6. Solving the System Equations The assembly of the element system equations into a global equation gives a set of simultaneous equations which could be solved using any convenient method.
The solution to this global equation gives the values of the dependent variables, like displacement, temperature, heat flux, fluid velocity at the nodes
Making Additional Computations : Making Additional Computations From the computed nodal variables, other derivates are determined using equations (1) and (2).
If the nodal variables are displacements, the strain and stresses at different nodes on the elements are computed.
If the nodal variables are temperatures, the element heat fluxes are computed.
If the nodal variables are fluid velocities, the element pressure heads are computed.
Manual Example of a Bar Element : Manual Example of a Bar Element Derive the stiffness for the bar element shown below.
F
1 2 Displacement function in Polynomial form u= A1 + A2x Two nodes and 1 d.o.f at each node, only two polynomial constants F
Manual Example of a Bar Element : Manual Example of a Bar Element Displacement Function in Shape function form N1 = 1 – x/l where
Manual Example of a Bar Element : Manual Example of a Bar Element Strain-Nodal Displacement
Manual Example of a Bar Element : Manual Example of a Bar Element but Hence, Stress- Strain Hence,
Manual Example of a Bar Element : Manual Example of a Bar Element The bar has constant cross-section, A Stiffness of bar l Hence, Therefore,
FEA packages : FEA packages Freeware :LISA, Felt, FEMM, Impact, CalculiX, MYSTRAN, VisualFEA
Open-source : SLFFEA, FEniCS, freeFEM
Commercial : ABAQUS, ANSYS, NASTRAN and MARC
Steps in using FEA packages : Steps in using FEA packages Pre-processing/ Modeling
Analysis
Post-processing
Steps in using FEA packages : Steps in using FEA packages 1) Preprocessing/ Modeling
Define the geometric domain of the problem.
Define the element type(s) to be used.
Define the material properties of the elements.
Define the geometric properties of the elements (length, area, and the like).
Define the element connectivity (mesh the model).
Define the physical constraints (boundary conditions).
Define the loadings
Steps in using FEA packages : Steps in using FEA packages 2) Analysis
During the analysis phase, finite element software assembles the governing algebraic equations in matrix form and computes the unknown values of the primary field variable(s). The computed values are then used by back substitution to compute additional, derived variables, such as reaction forces, element stresses, and heat flow.
Steps in using FEA packages : Steps in using FEA packages 3) Post-processing
Sort element stresses in order of magnitude.
Check equilibrium.
Calculate factors of safety.
Plot deformed structural shape.
Animate dynamic model behavior.
Produce color-coded temperature plots.
LISA static analysis example : LISA static analysis example
Cantilever Beam Example : Cantilever Beam Example Problem Definition:
The cantilever beam shown below is loaded by a single point load of magnitude 100 N at it's free end and in the negative Y-direction. The beam is of length 10m, and cross-section 4mx4m. It is rigidly fixed at it's left end. The material properties are Young's modulus, E= 15000N/m2, Poisson ratio= 0.288. The stress distribution is required to be determined.
Cantilever Beam Example : Cantilever Beam Example
Cantilever Beam Example : Cantilever Beam Example Preliminary Analysis Expected Deformation Max. deflection at the end of cantilever,
y =FL3/3EI
Moment of inertia of the square cross-section =bh3/12 I = 21.333 m4
y = (100)(10**3) / [3(15000)(21.333)]
y = 0.1041m
Cantilever Beam Example : Cantilever Beam Example Preliminary Analysis Expected Stress Arbitrarily the longitudinal stress at the outermost fibre on the tensile side at a distance of 5m from the free end will be calculated for comparison with the computed values.
The bending moment 5m from free end, M = FL
M = 100 x 5 = 500 Nm
Cantilever Beam Example : Cantilever Beam Example Preliminary Analysis Expected Stress The longitudinal stress is
sx= (moment)(distance from the neutral axis)/(moment of inertia) =My/I
The neutral axis in this case passes through the centroid of the square cross-section, so it's value is 2m.
Therefore,
sx = (500)(2)/(21.333) = 46.875 N/m2
Cantilever Beam Example: Discretization : Cantilever Beam Example: Discretization
Cantilever Beam Example : Cantilever Beam Example Using the package
CONCLUSION : CONCLUSION Summary of the workshop
No hands-on experimentation
Continuation of the Workshop for Advanced applications in conjunction with Unilorin Consultancy Services from 29-31 July, 2010 on Campus
Upcoming International Conference on Sustainable Urban Water Supply in Developing Countries coming up from 26-28 March, 2010.
THANK YOU FOR YOUR ATTENTION : THANK YOU FOR YOUR ATTENTION