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Any solution to these problems must satisfy the following conditions. These equations form the governing equations that the field variables must satisfy for equilibrium and compatibility conditions of the solid or structural system.
These differential equations can also be obtained by using the Euler-Lagrange equations of the variational principle as explained in Chapter 2. The governing functional may be the total potential energy or total complementary energy of the system. Using the calculus of variations it can be shown that the extremum or stationary value of the functional is obtained by equating the first variation of the functional to zero.
In problems of structural mechanics, the condition leads to the equilibrium or compatibility equation for the problem, and the field variable must satisfy these equations.
A brief summary of the principles used in this type of formulation is given in Chap- ter. The strain energy principle used for the analysis of beams, framed structures and plate bending problems treated in texts [2, 3, 4, 5] fall under this category. In problems where time is involved the initial values may have to be specified.
Figure 1. If the vertical deflection w at any point is taken as a field variable, it must satisfy the differential equation Eq- 1. Approximate Solutions It is not possible to obtain analytical solution for many engineering problems. An analytical solution is a mathematical expression that gives the value of the field variable at any location in the body.
For problems involving complex shapes, material properties and com- plicated boundary conditions, it is difficult and in many cases in- tractable to obtain analytical solution that satisfies the governing differential equations or gives extremum value to the governing func- tional.
Hence, for most of the practical problems the engineer resorts to numerical methods that provide approximate but acceptable so- lutions. The three methods that are used are as follows: 1.
Functional approximation 2. Finite difference method ite element method A brief description of the first two methods is given in the subse- quent sections and then the finite element method is introduced as a powerful numerical method, widely used in practice.
The unknown parameters that combine the functions are found out in such a way to achieve at best the field condition, which is represented through variational formulation. The well known classical methods such as Rayleigh-Ritz, Galerkin and collocation are based on functional ap- proximation but vary in their procedure for evaluating the unknown parameters 6.
The Rayleigh-Ritz method is briefly described below. Consider a simply supported beam, shown in Fig. In this problem if the deflected shape of the beam is known, the bending moment and shear force at any cross-section can be determined.
Consider the following approximation to the deflection curve that satisfies the boundary condition. Introduction 5. It can be seen from equation e that the total potential energy is now expressed in terms of the parameters a, and a2. Hence, for stationary value of I the following conditions must be satisfied. This error can be reduced by adding more terms to the approximate or trial function for w, i. The above procedure can be extended to the analysis of a three- dimensional solid.
In general a deformable body consists of infi- nite material points and, therefore, it has infinitely many degrees of freedom. By the Rayleigh-Ritz method such a continuous system is reduced to a system of finite degrees of freedom. For the case of three-dimensional solid, the variation of the field variables, displace- ments u,v and w can approximately be represented by the following trial functions. By this approximation the body is reduced to have 3n degrees of freedom.
Now the potential energy of the body can be expressed by a functional in terms of these parameters. Also the classical approach of arriving at the equations of the type 1. Hence, except in simple situations, this approach could not be used to solve practical problems. However, the concepts used in Rayleigh-Ritz method, i.
LT Fig. For example, in the case of plate bending problem the normal displacement w will be taken as unknown variable at each one of the nodes. The governing differential equation and the bound- ary conditions are converted to finite difference form.
In the case of isotropic thin plates, the finite difference form of the governing dif- ferential Eq. The resulting equations are then solved for the nodal values of this variable. The procedure is illustrated through an example given below. The thick- ness of the plate is 12 cm. The discretization of the plate with 25 nodal points is shown in Fig. It may be noted that the additional nodal points outside the plate boundaries 26 to 37 are required for analysis as will be shown below.
Since the loading and boundary conditions are symmetric about the two axes, only one quadrant of the plate is considered for analysis. The following boundary conditions are applied. However, in a general case a finer discretization may be necessary to get the solution closer to the theoretical values, The above example illustrated the method of solving a problem when the finite difference form of the governing differential equation is known. However, it may be noted that the finite difference for- m of the differential equations can be obtained easily for meshes of regular shapes, but it becomes quite involved to derive it for irregu- lar meshes.
Also the procedure does not lend itself to computerisation for solving many classes of problems in general. Finite Element Method A key contribution to the development of matrix methods for struc- tural analysis was made by Argyris and Kelsey .
In their contri- bution they presented matrix formulation for force and displacement methods of analysis using energy theorems of structural mechanics. It was the work of Turner, Clough, Martin and Topp  that led to the discovery of the finite element method.
Since then, tremendous advances have been made in the last 25 years both on the mathematical foundations and generalisation of the method to solve field problems in various areas of engineering analysis [10, 11, 12]. During the same period due to rapid devel- opment in computer technology, large number of package programs have been developed for finite element analysis which made it pos- sible for wider use of this technique in practice.
A brief summary of the method is given below and detailed description of the method for stress analysis is presented in subsequent Chapters.
The finite element method combines in an elegant way the best features of the two approximate methods of analysis discussed above. In particular the method can be explained through physical concept and hence is most appealing to the engineer.
And the method is amenable to systematic computer programming and offers scope for application to a wide range of analysis problems. The concept of discretization used in finite differ- ence method is adopted here. Figures 1. The properties of the elements are formulated and combined to obtain the solution for the entire body or structure. The strains and stresses within an element will also be expressed in terms of the nodal displacements.
Then the principle of virtual displacement or minimum potential energy is used to derive the equation of equilibrium for the element and the nodal displacements will be the unknowns in the equations. The equations of equilibrium for the entire structure or body are then obtained by combining the equilibrium equation of each ele- ment such that the continuity of displacement is ensured at each node where the elements are connected.
The necessary boundary conditions are imposed and the equations of equilibrium are solved for! Having thus obtained the values of dis- placements at the nodes of each element, the strains and stresses are evaluated using the element properties derived earlier. Thus, instead of solving the problem for the entire structure or body in one operation, in this method attention is mainly devoted to the formulation of properties of the constituent elements. The procedure for combining the elements, solution of equations and evaluation of element strains and stresses are the same for any type of structural system or body.
This modular structure of the program organisation is well exploited in the large number of program packages, now available for practical application to various disciplines of engineering. In the subsequent chapters of the book, the properties of various types of elements, procedures for assemblage of elements, solution technique and development of computer programs for various types of problems will be presented. Clough, H. Martin and L. K Limited, Zienkiewicz, 0.
Finite Element Analysis - Theory & Progr. - C.s. Krishnamoorthy
Finite Element Analysis : Theory and Programming
Finite Element Analysis Theory and Programming by C. S. Krishnamoorthy