Chapter-2 Line Element

Co-ordinate system and stiffness matrix are defined and steps of finding stiffness matrix are mentioned

Chapter 1- Basic Definitions And theories

What is Finite Element Method?

The finite element method is a numerical method for solving problems of engineering and mathematical physics.
In this method the body to be solved is divided into a number of smaller bodies which are called elements and they are connected by two or more joints called nodes.


Applications of Finite Element Method(FEM)

The finite element method can solve both structural and non-structural problems.

Structural Problems include-

• Stress analysis including truss and frame analysis and stress concentration  problems typically associated with holes, fillets or other changes in geometry in a body.
• Buckling problem analysis.
• Vibration analysis.

Non-structural analysis include-

• Distribution of electric or magnetic potential.
• Fluid flow analysis.

Advantages of FEM

1. Bodies with irregularities in shapes can be easily analyzed.
2. Handle general load conditions without diffiiculty.
3. Model bodies composed of several different materials because of the element equations are evaluated individually.
4. Can analyze a number of and a kind of boundary conditions.
5. By varying the size of the elements to make it possible to use small element where necessary.
6. Alter the finite element model are relatively easy and cheap.
7. Can include dynamic effects.
8. handle non-linear behavior existing with large deformations and non-linear materials.

Computer Programs on FEM:

Two types-

 a. General purpose program(GPP)

    These programs can solve different types of problems.

 b. Special purpose program(SPP)

     These programs can solve only specific type of problem designed for.

Advantages of GPP 

1. The input is well organized and is developed with user case in mind. Users do not need special knowledge of computer software and hardware. Pre-processors are readily available to help create the finite element model.
2. The programs are large systems that often can solve many types of problems of large and small size using the same input format.
3. Many of the program can be expanded by adding new modules for new kinds of problems.
4. Many of the stepped-down programs have become attractive.

Disadvantages of GPP

1. Initial cost of developing GPP is very high
2. GPP is less efficient than SPP because the computer must make many kinds of check for a problem.
3. Many of the programs are propritory.
4. Large computer configuration are needed to handle GPP

Advantages of SPP

1. Initial cost of developing SPP is low.
2. Small computer configuration is able to run the programs.
3. Additions can be made to the programs quickly and at a low cost.
4. The programs are efficient in solving the problems they were designed for.

Disadvantages of SPP

The major disadvantages of SPP is their inability to solve different classes of problems.

Some GPP

ANSYS, ABAQUS, ALGOR, IDEAS

Some SPP 

STAAD , SAP, ETABS, MSC/PAL, DYANA, STRUDL-II, IMAGES

Compatibility of GPP and SPP

1. Element types availables such as beam, plane stress, 3-dimensional solid.
2. Type of analysis available such as static and dynamic.
3. Material behavior available such as linear and non-linear.
4. Load types available such as concentrated, distributed, thermal and displacement.
5. Data generation such as automatic generation of nodes elements and restraints.
6. Plotting capabilities such as original and deformed geometry and stress and temperature contours.
7. Different displacement behavior analysis such as small and large displacement and buckling.
8. Selective output such as at selected nodes, elements and maximum or minimum value.

Types of elements

• One dimensional element of line element- it is used to model trusses and frame structure.

• Two dimensional element- When the body is loaded by force in their own plane, in this case 2D element are used to discretized the body.

• Three dimensional element- 3D element are used to irregularly shaped three dimensional casting.

• Axisymmetric element-

 

Steps of FEM

Step-1: Discretizing body by selecting element type. The choice of elements depends on the physical make up of the body under actual loading conditions and how close to actual behavior the analize wants the result to be.

Step-2: Select a displacement function. Displacement function includes polynomial function, linear function, quadratic function and cubic function.

Step-3: Define the stress strain relationship.

Step-4: Define the element stiffness matrix and equations

Step-5: Assemble the element equations to obtain global total equations and induced boundary condition

Step-6: Solve the unknown degrees of freedom.

Step-7: Solve for the element strain and stress.

Step-8: Interprete the results.