1. INTRODUCTION TO FINITE ELEMENT METHOD The finite element method has a power tool for numerical solution for a wide range of engineering and scientific problems. Application range from deformation and stress analysis of automotive, aircraft, buildings and bridge structure to field analysis of heat flux, fluid flow, magnetic flux, seepage and flow problem. 1.1 Need of numerical methods:- It is not possible to obtain analytical solution for many engineering problems. For problems involving complex shapes, material properties and in many cases intractable to obtain analytical solution that satisfies the governing differential equation gives extreme values to governing functional. Hence for most of practical problems the engineer resort numerical method which provide approximate but acceptable solution. Need of FEM:- The three numerical methods are used to solve engineering problem. a) Functional approximation b) Finite difference method c) Finite element method Functional approximation is also called as variational method. In this method, a set of independent functions satisfying the boundary condition is chosen and a linear combination of finite number of them is taken approximate specify field variable at any point. It is needed that the assumed trial function must be continuous and satisfy all prescribe boundary conditions. This method suffers from disadvantages. That approximation function for problems are difficult to construct, as there is no simple guideline available to select such functions. In finite different method of different equation, derivatives in equations are replaced by different quotients, which involve values if solution at discrete mesh point of domain. The resulting discrete equation after imposing boundary conditions solves for solutions at mesh point. This method also suffers through disadvantages like inaccuracy of derivatives of approximated solution. The difficult in imposing boundary condition along non straight boundaries difficulty in accurately representing geometric complex domain and the inability to employ non uniformly and non rectangular meshes. To overcome above difficulties, associated to both methods, there should be such of method thus posses following feature. To overcome above difficulties, associated to both methods, there should be such of method thus posses following feature. 1. The method should have a sound mathematical s well as physical basis. 2. The method should not have limitations with regards to the geometry and physical composition of domain as well as the nature of 'loading'. 3. Formative procedure should be independent on shape of domain and the boundary condition. 4. The method should be flexible enough to allow choosing a desired degree of approximate without reformulation the entire problem. The FEM not only over come disadvantages. But also it enclosed with features of effective computational techniques listed above.

Discrete and continuos:- In many situations an adequate model is obtained using a finite number of a well defined components, such problems are term as discrete. In other the sub division (any body or domain can represented by dividing it into subdivision or sub-domain) is continued indefinitely and the problem can be defined by using a mathematical fiction of an infinitesimal. This leads to differential equations which simply infinite of elements, such systems are termed as continuos. 1.2 Basic Concepts:- The basic concept underlying the finite elements is not new. If we were to identify evidence of the earliest finite element concept we could probably trace it back to geometry approximations of pyramids by Egyptians some 5000 years ago. If we consider, for example the numeric approximation of pie or starting point of finite element we could find some interseting historical records in China, Egypt and Greece. The principle of discretization is used in most forms of human endeavor. Perhaps the necessity of discretising or dividing a thing into smaller manageable things arises from fundamental limitation of human beings. The limitations of human mind are such that it cannot grasp the behavior of its complex surrounding and creation in one operation entirely or totally. Even to see things immediately surrounding us, we must make several turns to obtain a jointed mental picture of our surrounding. In other words, we descretize the space around us into small segments and final assemblage that we visualize is one that simulates the real continuos surrounding. Any body or structure can be divided into smaller parts of finite dimensions called 'finite element'. The original body or structure is then considered as an assemblage of these elements connected at a finite number of joints called 'Nodes' or 'Nodal points'. The properties of elements are formulated and combined to obtain the solution for the entire body or structure. For example, displacement formulation widely adopted in FEA. Simple functions known as 'Shape functions' are chosen to approximate the variation of displacement within an element in terms of the displacement at the nodes of element. In this, the approximation to field variable is made be expressed in terms at the nodes of element. In this, the approximation to field variable is made be expressed level. The strains and stress within an element will also be expressed in terms of the nodal displacement, then the principle of virtual displacement of minimum potential energy is used to desire the equation of equilibrium for the element and the nodal displacement will be unknown in the equations. The equation of equilibrium for the finite structure or body are then obtained by combined the equilibrium equation of each elements such that the continuity or displacement is ensured at each node where the elements are connected. The necessary oundry conditions are imposed, and equation of equilibrium are solved for nodal displacement. Having this obtained values at nodes of each elements, the stress and strain are evaluated using elements, the stress and strain are evaluated using elements properties derived earlier. Thus instead of solving problem for entire body in one operation, this method is mainly devoted to formation of properties of constituent elements. The above procedure is same for any type of structural system.

2. FUNDAMENTAL TERMS IN FEM 2.1 Equilibrium Condition:- Consider the case or a deferrable body, which is in equilibrium. The body is subjected to external force and set a internal force (i.e stress) developed due to deformation of the solid. The external force may due to i) Body forces due to weight of the body and expressed as force per unit volume, and . ii) Surface forces, which act on the boundary of the body, expressed as force per unit area of the surface. Consider a differential element of sides d?, dy and dz within the boundaries of deformed body and general state of the stress, and the variation from point A to A are also shown there in fig. (2.1a). In firg. (2.2) the force acting on the element parallel to the ? axis, are shown as ? f? = 0 for equilibrium, we get ?? x/? x ? d? dy dz + ??yz/?y ? dx dy dz + ?? z x/?z ? dx dy dz + Xb dx dy dz = 0 ? ?? x/?x + ?? yz /? y + ? zx / ? z + Xb = 0 Similarly equation can be written for y and z direction. ?? xy /? x + ?? y / ? y + ?? z y/ ? z + yb = 0 ? xz / ? x + ?? yz / ? y + ?? z/ ? z + Zb = 0 Where Xb ,Yb, Zb are body force components in respective direction. The three remaining equation of static's for moments about the axes x,y,z are ?Mx = 0 and ? Mz = 0 Now taking moments of forces about the x, y, and z axes we get. ? xy = ? yx, ? xz = ? zx, ? yz = ? zy thus the state of stress at any point can be defined by six components of stress ?x, ?y, ?z, ?xy, ?yz and ? zx. It may be represented as {?} = [? x ? y ? z ? xy ? yz ? zx ]. 2.2 Finite element types:- In FEM there are several types of elements are used some typical elements for various types of structure are discussed. Triangular or quadrilateral elements are employed for plane stress or plane strain or plate bending problems. Shell structures are discretized with either flat or curved elements. Axisymmetric solids are discretized using ring type element for three dimensional problems hexahedral o tetrahedral element are used. 2.3 Degree of Freedom:- Degree of freedom can be defined as an independent (unknown) displacement that can occur at a point. For the problem of one-dimensional column, there is only one way in which a point is free to move, that is in the uniaxial direction, then point has one degree of freedom. For a two dimensional problems, if deformations can occur only in the plane of the body ( and bending effects are ignored) a point in free to move only in two independent co-ordinate direction, thus a point has two degree of freedom.

2.4 Shape functions (Interpolation Function):- In finite element analysis using the displacement modes the variation of displacement within an element is assumed since the true variation of displacement is known. In general, in higher mathematics, it is necessary in many situations to deal with functions whose analytical form is either totally unknown or else is of a nature that the function can not easily be subjected to such operations as may be required in either case it is desirable to replace the given function by another functions which can be handled more easily is called as 'shape' , 'interpolation', or' basis function'. 2.5 Global or Local Co-Ordinates:- Let us consider an example, here it is need to define ( a point of land ) an area and relate it to a standard or global point A. It can be done by establishing the distance of each point in that area from point A. let, point A is far away from the plot area, and it is difficult to established a direct relation with A. A local point B is available, and its distance from A is known Xa point B is accessible from all points in the plot if the distance of any point p from b is XB, then its distance from point A is. XA = YB + XAB Here the measurement w.r.t. point B called as local or natural coordinates, and those from point A, global coordinates can be the measurement of plot w.r.t. point B, is much simpler than those w.r.t. point A. there can be number of possible points w.r.t. which define local coordinate system. Local coordinates are in non-dimensional form, and their values are expressed as number, and often lie between simplicity to the subsequent derivations. 2.6 Isoparametric Elements:- For analysis of structural problem of complex shapes involving curved boundaries or surfaces, simple triangular or rectangular elements are no longer sufficient. This has led to the development of elements of more arbitrary shape and are called isoparmetric elements. These elements are widely used in two and three dimensional stress analysis and, plates and shell problems. The concept of isoparametric elements is based on transformation of the parent element in local and natural coordinate system to an arbitrary shape in the cartesian system as shown in figure. A convenient way of expressing the transformation is to make use of the shape functions of the rectilinear elements in their natural coordinates system and the nodal values of the coordinates thus the cartesian correlates of a point is an element may be expressed as. x = N'1 x1 +N'2 x2 + ------+ N'n xn y = N'1 y1 + N'2 y2 + ------+ N'n yn z = N'1 z1 + N'2 z2 + ------+ N' nzn. Or in Matrix form {x} = [N'] {xn}

Where (N') are the shape functions of the parents rectilinear elements and (xn) are the nodal coordinates of the elements. The shape functions will be expressed through the natural coordinates system r,s, and t. The shape functions (N') used in the above transformation thus help us to geometry of the element in the Cartesian coordinate system. If these shape functions (N') are the same as the shape functions N used to represent the variation of displacement in the element, these element are called Isoparametric element. i.e. {x} = [N] {xn} And in cases where the geometry of the element is defined by shape functions of order higher than that for representing the variation of displacement, the element are called 'super parametric'. Similarly if more nodes are used to define displacement composed to the nodes used to represent t the geometry of the elements then they would be refereed to as ' sub parametric' elements.

3. STEPS IN FEM 3.1 Introduction:- Formulation and application of FEM are considered to consists of eight basic steps. The main aim of general description is to prepare for complete and detailed consideration each step. 3.2 General Idea: - The nature of distribution of effects (such as deformations, stresses, temp. fluid press, fluid relocates caused by forces such as applied loads a pressures and thermal and fluid fluxes) in body depends upon characteristics of force system and of the body itself put aim is to find this distribution of effects. For convenience we shall often use displacement or deformation 'u' in place of effects. But it is difficult to find the distribution u by using conventional methods and decide to use finite element method, which is based on concept of 'discretization' i.e. the body is divided into a number of smaller regions called 'finite elements'. 3.3 Steps:- The various steps adopted in FEA are described as bellow in detail: Step 1 : Discretize and select element configuration This step involves subdividing the body into a suitable number of small bodies called 'finite elements'. The intersection of sides of elements are called nodes or nodal points, and the interfaces between the elements are called nodal lines or nodal planes. The elements used will depends upon characteristics of continuum and the indentilization, for e.g. if structure of body is indentilized as a one- dimensional line the elements we use is a 'line' element. For two-dimensional bodies, we use triangular of quadrilateral, for three-dimensional identilization a hexahedron with different specialization can be used. Although we could subdivide body into regular shaped elements in interior, we have to make special provision in boundary is irregular. Step 2 : Select approximation model or function In this step, we choose or pattern a shape for distribution of the unknown quantity that can be a displacement and / or stress of stress- deformation problem, temp in heat - flow problem, and both temp and displacement for coupled problems involving effects of both flow and deformation. The nodal points of the elements provide strategic point for writing mathematical functions to describe shape of distribution of unknown quantity over domain of the element. Mathematical functions such as polynomials and trigonometric series can be used for this purpose, especially polynomial because of ease and simplification. If we denote 'u' as unknown, the polynomial interpolation function can be expressed as u = N1u1 + n2u2 + -------- + Nnun Here, u1,u2,u3 ---- un = values of unknown at nodal points N1,N2,N3 ---- Nn = interpolation function. The solution obtained will be in terms of the unknown only at nodal points. The discretization should be such a that composed solution is as close as possible to the exact solution i.e. error is a minimum. Step 3 : Define strain (Gradient) - displacement (unknown) and stress - strain (constitutive) Relationship. The displacement at a point of a deferrable body can be described by the components u, v, and w parallel to the Cartesian coordinates axes. These components can in general, be represented as functions of x,y, and z. the strain in the deformed body can be expressed as partial derivatives of the displacement u, ,v and w. For example, in case of deformation occurring only in one direction y, strain ? y assumed to be small is given by: ? y = [dv/dy] where, v is deformation in y direction. In addition to strain must be define an addition quantity, the stress, usually this is done by expressing its relationship with strain. Such a relation is called as 'stress - strain law'. In general it is a constitutive law and describe the response or effect (displacement , strain) in a system due to applied cause (force). The stress-strain law is one of the most vital part in FEM. As an elementary illustration considers Hook's law, which defines stress and strain in solid body ? y = E ? yd where, ? y = stress in vertical direction. E = Young's Modulus of elasticity. If we substitute E from equation 1 into above equation we have expression for stress in terms of displacement as ? y = E [? v/ ? y]. Step 4 : Derive Element Equations:- By using the available laws and principles, we obtain equations governing the behavior of the element. Here the equations are obtained in general terms and hence can be used for all elements in discretized body. We can derive the elementary equation by two methods i.e. energy methods and residual method. Element Equation We use of either of two methods will leads to equation describing behavior of an element which are commonly expressed as [k]{q} = {Q} [k] = Element property matrix. {q} = Vector of unknown at element nodes. {Q} = Vector of element nodal forcing parameters. For a specific problem of stress analysis [k] = Stiffness to matrix. {q} = Vector of nodal displacements. {Q} = Vector of nodal forces. Step 5 : Assemble Element Equations to obtain Global or Assemblage Equations and introduce to boundary conditions. Our final aim is to obtain equations, for entire body that defines approximately the behavior of the entire body of structure. Once the element equation are establish in step 4, for a generic element, we are generates the equations for other elements again and again. Then we add them together to find global equations. This assembling process is based on law of compatibility or continuity. Which requires the body remains continuous, i.e. neighboring points should remains in neighborhood of each other after load applied. In other words, displacement of two adjacent or consecutive points must be identical values for eq. For deformations occurring in a plane, it may be sufficient to enforce continuity of displacement only. Finally, we obtain the assemblage equations, which are expressed in matrix notation. [k] {r} = {R} [k] = assemblage property matrix. {r} = assemblage vector of nodal unknowns. {R} = assemblage vector of nodal forcing parameters. Boundary conditions Until now we considers only properties i.e. capabilities of the body to withstand applied forces. It is just like saying that one in on engineer. How he will be perform his engineering duties will depends on thew surroundings and problem he faces; these aspects are called 'constraints'. In case of engineering bodies the surrounding or constraints are the boundary conditions. When we introduce these conditions, we can decide how body can perform. Boundary conditions care physical constraints are commonly specified in terms of known values of the unknowns on a part of surface or boundaries S1, and / or gradients or derivatives of unknowns on S2. In case of simply supported beams, the boundary S1 is two end points where displacements are given. This type of constraints expressed in terms of displacement is often called the essential, forced, or geometric boundary conditions. At the simple support of the beam bending moment is zero; i.e. the second derivative of displacement vanishes. This type is often called or natural boundary conditions. To reflect the boundary conditions in finite approximation of the body , it is usually necessary to modify these equation (i.e. [f] {r}= {R} only for geometrical boundary conditions. So final modified assemblage equations are expressed by inserting overbears as [k] {r} = {R} Step 6: Solve for primary unknowns:- The above equation is set of linear ( or non-linear) simultaneous algebraic equations, which can be written in standard formulas form as : K11 r1 + k21 r2 +…………+ k1n rn =R1 K21 r1 + k22 r2 +…………+ k12 rn =R2 | kn1 r1 + kn2 r2 +…………+ knn rn =Rn These equations can be solved by using the well-known Gaussion elimination or iterative methods. These are called primary unknowns because they appears as the first quantities. For e.g. if the problem is formulated by using stresses as unknown , the stresses will be called primary quantities. For flow problem, the primary quantity can be fluid a velocity head or potential. Step 7 : Solve for derived or Secondary Quantities:- Very often additional or secondary quantities must be computed from the primary quantities. In case of stress deformation problem such quantities can be strain, stresses moments, and shear forces. If is relatively straightforward to find the secondary quantities once primary quantities are known since we can make use of relations between strain and displacement and stress and strain. Step 8: Interpretation of Results:- The final and important aim is to reduce result from use of finite elements procedures to a form that can be readily used for analysis and design. The results are usually obtained in form of printed output from computer. We then select critical sections of body and plot the values of displacement and stresses along them, or we can tabulate the results. It is often very convenient and less time consuming to use routines and ask the computer to plot or tabulate the results.

4. FINITE ELEMENT ANALYSIS SOFTWARE 4.1 Introduction:- One of the reason for used application of FEM is due to the available of number of package programs. The FEA programming may divided in three basic step i.e., pre-processing, solution and post processing. This is shown by schematic chart. Start Data input module Pre-processor Data processing module Processor Results/Output module Post processor Stop

Simplified Schematic Structure Of Finite Element Program 4.2 Pre-Processor:- The pre-processor stage involves the following section: specifying the title, that is the name of the problem. Setting the preference, this is the type of filtering to be used, e.g., structural, fluid, thermal or electromagnetic, defining the element type, this may be 2D or 3D in the structure element types and these are many types. This is possibly the most crucial port of on analysis is a highly accurate set of result is required. Defining the material properties, i.e., the Young's modules, Positions ratio, the density and if applicable, the coefficient of expansion friction, thermal conductivity, damping effect specific heat etc. creating the model in appropriate dimensions. This is where the actual model is drawn in 2D or 3D space in the appropriate unit (M, mm, in. etc.). defining the mesh density. This may be done by manually defining the number of element along the lines of the model, thus customizing the number of elements. In complex cores, the mesh density may be generated by specification the element edge length, model automatically on the command using the edge length specified. Solution :- Here , the loading and boundary conditions are applied to the model. The boundary conditions are the second must critical stage at the analysis. The boundary condition usually are in the term of zero displacement can be placed on nodes, key points, areas or an lines. The one on lines can be in the form of symmetric or unit-symmetric boundary conditions are allowing in plane rotation and out of plane translation, the other allowing in plane translation and out of plane rotation for a given line. The loading may be in the form of a point load, a pressure of a displacement, again the values should be in the same units as the model drawn and the material properties given in the pre-processor section of the analysis. The solution of the problem done automatically. The package then proceed, to form the element stiffen matrix for the problem, followed by solving for the matrix and then updating the displacement value for each node within the components or continuum. Pre - Processor :- Here the result of the analysis can be read. They can be in the form of table, a contour plot, deformed shape of the components or the mode shapes and natural frequencies if frequency analysis is involved. The contour plots can be complete for the displacement, stress or strain according to the main theories of failure other information such as the strain energy. Plastic strain and creep strain may be obtained, which may otherwise not be available due to the complexity of the geometry, loading and for boundary condition. 4.3 FEA Software Packages :- The rapid advances made in computer hardware and software led to significant developed in FEA software. Finite element programming has emerged as a specialized discipline, which requires knowledge of and experience in the diverse areas such as finite element technology including foundations of mechanics and numeric analysis on one hand the computational skill in the areas of software technology including programming techniques, data base management and computer graphics on the other hand. It requires several man- years to develop a general purpose FEA software with processing capability and facility for the user to have a wide choice of several types of elements, analysis of different types of problems. ADINA, ANSYS, ASKA, NASTRAN, NONSAP, NISA, SAP, SAFE, STRUOL, Are some FEA software packages. Above listed software packages has different capability and application. But the basic procedure or steps adopted while solving any problem remains same.

5. ANSYS 5.1 Introduction:- ANSYS is an engineering analysis system developed by Swanson Analysis system. ANSYS has excellent graphical power. Model generation is easy. In ANSYS it is easy call or import model from another software i.e. IDEA Auto cad, for solving the problem no need to come out from working menu correction or changes can be done easily. 5.2 Step In ANSYS Step 1. Model Generation :- In order to build a model we have to follow the procedure listed below. 1 a). define the elements type There is element library which includes element such as 20 Quadrilateral, 2D triangle, 3D brick 3D tetrahedron, 3D shell, etc. the user have option to use above element. 1 b). specify material properties and for real constant material properties may be isotropic, orthotropic or antisotropic. We have to specify Young's Modulus for a static analysis, density for mass calculation, Poissions Ratio, etc. Real constants may defined optional thickness for 2D elements. 1 c). Define the model geometry Using one or the technique from various techniques available in FEA package can create the model. Modeling is much similar to CAD system.

Step 2. Meshing:- Meshing is nothing but generation of nodes and elements, which define complete geometry of model. Meshing can be done automatically. It is not need to have some mesh size for all structure, it may have more than one meshing for different regions for a complex structure.

Step 3. Applying Boundary conditions and constrains:- These are applied for sliding surfaces, displacement, to define rigid support point etc. displacement boundary conditions applied to all nodes or an element edge or face will constrain the deformation along the edge or face. Displacement boundary condition are not applied only at mid side no on an edge.

Step 4. Applying loads:- There are several types of loads such as displacement, forces, pressure, temperature, gravity. Centrifugal, etc. the required load can be applied to the solid model (key point, lines, areas, etc.) or to the finite element model (nodes and elements) except for inertia loads (gravity, rotational velocity etc. ) which are independent of model.

Step 5. Checking of element parameters:- It is necessary to check whether initially elements are destroyed or not before going for solution. It they are deformed or not before going for solution. If they are deformed then are removed or corrected from the model. All above steps i.e. to 5 are come under the category of pre- processing stage of FEA.

Step 6. Solve:- It is the processing stage. In this using input data in pre-processing step to obtain the solution or out put results performs mathematical computation. We have to give command for solving in the step and the processor module solves the problems automatically.

Step 7. Review the Results:- In this step, we get results in the different forms such as tabular, pictures, graphs etc. i.e. list of reaction forces and moments, minimum and for max. Stress and strain induced. In x,y and z directions deformed shape of structure sue to application of load, combine deformed and deformed (i.e. shape before load application) can be also seen.

6. Case Study 6.1 Problem:- A rectangular steel plate having dimensions 50 ? 24 CMS is drilled with a hole of diameter 4cm as shown fig 6.1 consider plate having uniform thickness 1 cm. Determine stress distribution and deformation for plate when 10 Nsm2 pressure is applied take E = 2.08 e 11, ? ? = 0.3

Solution By Using ANSYS:- As we are interested in the deformation of body, the symmetry of the geometry and symmetry of loading can be use effectively. Let x and y represents the axes of symmetry. The points along the x-axis involves along x direction and are constrained in the y direction and vise-versa. Hence one quarter part of the full area, with the loading and boundary constrain as shown in fig 6.2 is sufficient to solve problem for deformation and stress. Here p method is used for analysis, which obtains results such as displacement, stress or strain to a user-specified degree to accuracy. To calculate those results, the p method manipulates the polynomial level (p level) of the finite elements shape functions which are used to approximate the real solution. p method is used for analysis because of its benefits for linear structure static analysis, that are not available with the more traditional h method. p method has ability to obtain good result to a desired level of accuracy without regroups user define meshing control. For new user it provides advantage of accurate mesh. Adaptive refinement of mesh can be done in p method. Following procedure is adapted for solving the problem Step 1 Set the Analysis File: - PWH Step 2 Select p method : - Step 3 Define the element type 2D quadrilateral 145 Step 4 Define the real constant Enter the thickness 0.01m Step 5 Define material properties Isotropic, enter Young's Modulus Ex = 2.08 e11 and Poissions Ratio = 0.3. Step 6 Create plate with hole a) Create rectangle, Refer figure 6.3 b) Create circle, Refer figure 6.4 c) Substract i.e. (rectangle area - circle area), Refer figure 6.5 Step 7 Mesh the area Smart mesh size and free mesh, refine mesh, see figure 6.6

Step 8 Define symmetry boundary conditions To constrain the left side and bottom of the nodes see figure 6.7,6.8. Step 9 Define pressure load of 10 N/m2 Pressure tensile negative, compressive positive in ANSYS Force tensile positive, compressive negative in ANSYS Step 10 Static Analysis Step 11 Solve the problem Then we will get on screen the figure 6.10. Step 12 Review results and exit ANSYS We get results such as stress along X direction along Y direction, max displacement, deform shape with deform shape etc.

7. APPLICATION ADVANTAGES AND LIMITATIONS 7.1 Applications:- FEM has wide range of application as described below: 1. Finite element method (FEM) is used in analyzing housing of machine tools gear boxes, automobile radiator fan blabs, fabricated gear wheel, bearing pedestals, storage vessels with or without partition, blade disc, pressure vessels, bus bodies and rail coaches. 2. Single and multiunit cutting tools, spiral bevel and hypoid gear teeth and object of comparable length and breadth and depth are obtain by FEM. 3. Rotating equipment's like kilns, kiln types tumbling mills, cylindrical or conical container also obtain by FEM. 4. The behavior of super-structure made of frame to support boilers, motors and other industrial equipment's deformation is calculated by FEM. Industrial Applications of FEM:- 1. Multi layer pressure vessel stress analysis The deformation and stress experienced by multi layer pressure vessel is obtain by potter scheme. 2. Stress analysis main housing of hydraulic press Stress analysis of C frame of open front hydraulic of 40 tones capacity is solved by using potter scheme (cut-off-case solution) 3. Rigidity and stress analysis of two stage gear box The average stress values is found by using FEM at each of notes which has resulted in the material saving in gear box. 4. Earthquake analysis of circuit breaks The high voltage circuit breaks is modeling as a space elements by applying FEM only. 5. Determination of critical speed including the Gyroscopic effect is obtain by FEM. 6. Eigen pairs of boiler frame are computed by FEM. 7. Similarly Eigen value of compressor disc is also computationally found. 8. Static analysis and under cyclic symmetric loading is done using skyline approach and under generalized loading. 9. Analysis of the fabricated gear wheel, centrifugal fan impellers, Eigen values of impellers is an approach of FEM. 7.2 Advantages of FEM:- The main advantages of FEM is 1. The method can efficiently be applied to cater irregular geometry. 2. It can take care of any type of boundary. 3. Material anisotropy and inhomogenity can be treated without much difficulty. 4. Any type of loading can be handled. 5. The physical problems that were so far intractable and complex for any closed boundary solution can be analyzed. 6. Mechanical components of any complicated and complex shape can be analyzed by this method. 7. The technique can be easily applied in case of a component made of non- homogenous or composite materials. 8. Solution is obtain at faster speed which reduces the design lead time. 9. The behavior of component is tested before its manufacturing, which eliminates field testing. 10. The technique can be integrated with a joint CAD/CAM system, which includes design and manufacturing the components. 7.3 Limitations: 1. There are many types of problems where same other method of analysis may prove efficient then FEM. 2. For vibration and stability problem in many cases the cost of analysis by FEM may prohibits. 3. This method is cost involved in solution of problem (too much). 4. Stress values may vary by 25% from fine mesh analysis to average mesh analysis. 5. There are other trouble spots such as "aspect ratio" (ratio of longer to smaller dimensions of elements) which may effect final results. However, there are easier to guard against then problem of proper mesh size and distribution. 6. For vibration and stability analysis of simpler structure. 'finite strip' method and other semi analytic methods are more efficient and involves less low then FEM.

BIBLIOGRAPHY 1. Krishanamoorthy C.S., Finite Element Analysis Theory & Programming, Second Edition TATA Mc GRAW HILL New Delhi. 2. C.S. Desai, Elementery, Finite Element Method Practice - Hall, Inc. Englewood Cliffs. 3. S. Rajasekaran. Finite Element Analysis in Engineering Design. 4. Modeling And Meshing Guide, Release 5.4 Internet Sites Referred ? www.Google.com ? www.findarticles.com ? www.ansys.com