UNIT – III FINITE ELEMENT METHOD Introduction – Discretisation of a structure – Displacement functions – Truss element – Beam element – Plane stress and plane strain - Triangular elements

Finite Element Method Defined Many problems in engineering and applied science are governed by differential or integral equations. The solutions to these equations would provide an exact, closed-form solution to the particular problem being studied. However, complexities in the geometry, properties and in the boundary conditions that are seen in most real- world problems usually means that an exact solution cannot be obtained or obtained in a reasonable amount of time.

Finite Element Method Defined (cont.) In the FEM, a complex region defining a continuum is discretized into simple geometric shapes called elements. The properties and the governing relationships are assumed over these elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes. An assembly process is used to link the individual elements to the given system. When the effects of loads and boundary conditions are considered, a set of linear or nonlinear algebraic equations is usually obtained. Solution of these equations gives the approximate behaviour of the continuum or system.

Finite Element Method Defined (cont.) The continuum has an infinite number of degrees-of-freedom (DOF), while the discretized model has a finite number of DOF. This is the origin of the name, finite element method. The number of equations is usually rather large for most real- world applications of the FEM, and requires the computational power of the digital computer. The FEM has little practical value if the digital computer were not available. Advances in and ready availability of computers and software has brought the FEM within reach of engineers working in small industries, and even students.

Finite Element Method Defined (cont.) Two features of the finite element method are worth noting. The piecewise approximation of the physical field (continuum) on finite elements provides good precision even with simple approximating functions. Simply increasing the number of elements can achieve increasing precision. The locality of the approximation leads to sparse equation systems for a discretized problem. This helps to ease the solution of problems having very large numbers of nodal unknowns. It is not uncommon today to solve systems containing a million primary unknowns.

Origin of the Finite Element Method It is difficult to document the exact origin of the FEM, because the basic concepts have evolved over a period of 150 or more years. The term finite element was first coined by Clough in In the early 1960s, engineers used the method for approximate solution of problems in stress analysis, fluid flow, heat transfer, and other areas. The first book on the FEM by Zienkiewicz and Chung was published in In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering problems.

Origin of the Finite Element Method (cont.) The 1970s marked advances in mathematical treatments, including the development of new elements, and convergence studies. Most commercial FEM software packages originated in the 1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s (FENRIS, LARSTRAN ‘80, SESAM ‘80.) The FEM is one of the most important developments in computational methods to occur in the 20th century. In just a few decades, the method has evolved from one with applications in structural engineering to a widely utilized and richly varied computational approach for many scientific and technological areas.

How can the FEM Help the Design Engineer? The FEM offers many important advantages to the design engineer: Easily applied to complex, irregular-shaped objects composed of several different materials and having complex boundary conditions. Applicable to steady-state, time dependent and eigenvalue problems. Applicable to linear and nonlinear problems. One method can solve a wide variety of problems, including problems in solid mechanics, fluid mechanics, chemical reactions, electromagnetics, biomechanics, heat transfer and acoustics, to name a few.

How can the FEM Help the Design Engineer? (cont.) General-purpose FEM software packages are available at reasonable cost, and can be readily executed on microcomputers, including workstations and PCs. The FEM can be coupled to CAD programs to facilitate solid modeling and mesh generation. Many FEM software packages feature GUI interfaces, auto-meshers, and sophisticated postprocessors and graphics to speed the analysis and make pre and post- processing more user-friendly.

How can the FEM Help the Design Organization? Simulation using the FEM also offers important business advantages to the design organization: Reduced testing and redesign costs thereby shortening the product development time. Identify issues in designs before tooling is committed. Refine components before dependencies to other components prohibit changes. Optimize performance before prototyping. Discover design problems before litigation. Allow more time for designers to use engineering judgement, and less time “turning the crank.”

Theoretical Basis: Formulating Element Equations Several approaches can be used to transform the physical formulation of a problem to its finite element discrete analogue. If the physical formulation of the problem is described as a differential equation, then the most popular solution method is the Method of Weighted Residuals. If the physical problem can be formulated as the minimization of a functional, then the Variational Formulation is usually used.

Theoretical Basis: MWR One family of methods used to numerically solve differential equations are called the methods of weighted residuals (MWR). In the MWR, an approximate solution is substituted into the differential equation. Since the approximate solution does not identically satisfy the equation, a residual, or error term, results. Consider a differential equation Dy’’(x) + Q = 0(1) Suppose that y = h(x) is an approximate solution to (1). Substitution then givesDh’’(x) + Q = R, where R is a nonzero residual. The MWR then requires that  W i (x)R(x) = 0(2) where W i (x) are the weighting functions. The number of weighting functions equals the number of unknown coefficients in the approximate solution.

Theoretical Basis: Galerkin’s Method There are several choices for the weighting functions, W i. In the Galerkin’s method, the weighting functions are the same functions that were used in the approximating equation. The MWR is an integral solution method.

Theoretical Basis: Variational Method The variational method involves the integral of a function that produces a number. Each new function produces a new number. The function that produces the lowest number has the additional property of satisfying a specific differential equation. Consider the integral  D/2 y’’(x) - Qy]dx (1) The numerical value of  can be calculated given a specific equation y = f(x). Variational calculus shows that the particular equation y = g(x) which yields the lowest numerical value for  is the solution to the differential equation Dy’’(x) + Q = 0.(2)

Theoretical Basis: Variational Method (cont.) In solid mechanics, the so-called Rayeigh-Ritz technique uses the Theorem of Minimum Potential Energy (with the potential energy being the functional,  ) to develop the element equations. The trial solution that gives the minimum value of  is the approximate solution.

Sources of Error in the FEM The three main sources of error in a typical FEM solution are discretization errors, formulation errors and numerical errors. Discretization error results from transforming the physical system (continuum) into a finite element model, and can be related to modeling the boundary shape, the boundary conditions, etc. Discretization error due to poor geometry representation. Discretization error effectively eliminated.

Sources of Error in the FEM (cont.) Formulation error results from the use of elements that don't precisely describe the behavior of the physical problem. Elements which are used to model physical problems for which they are not suited are sometimes referred to as ill-conditioned or mathematically unsuitable elements. For example a particular finite element might be formulated on the assumption that displacements vary in a linear manner over the domain. Such an element will produce no formulation error when it is used to model a linearly varying physical problem (linear varying displacement field in this example), but would create a significant formulation error if it is used to represent a quadratic or cubic varying displacement field.

Sources of Error in the FEM (cont.) Numerical error occurs as a result of numerical calculation procedures, and includes truncation errors and round off errors. Numerical error is therefore a problem mainly concerning the FEM vendors and developers. The user can also contribute to the numerical accuracy, for example, by specifying a physical quantity, say Young’s modulus, E, to an inadequate number of decimal places.

Advantages of the Finite Element Method Can readily handle complex geometry: »The heart and power of the FEM. Can handle complex analysis types: »Vibration »Transients »Nonlinear »Heat transfer »Fluids Can handle complex loading: »Node-based loading (point loads). »Element-based loading (pressure, thermal, inertial forces). »Time or frequency dependent loading. Can handle complex restraints: »Indeterminate structures can be analyzed.

Advantages of the Finite Element Method (cont.) Can handle bodies comprised of nonhomogeneous materials: »Every element in the model could be assigned a different set of material properties. Can handle bodies comprised of nonisotropic materials: »Orthotropic »Anisotropic Special material effects are handled: »Temperature dependent properties. »Plasticity »Creep »Swelling Special geometric effects can be modeled: »Large displacements. »Large rotations. »Contact (gap) condition.

Disadvantages of the Finite Element Method A specific numerical result is obtained for a specific problem. A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced. The FEM is applied to an approximation of the mathematical model of a system (the source of so-called inherited errors.) Experience and judgment are needed in order to construct a good finite element model. A powerful computer and reliable FEM software are essential. Input and output data may be large and tedious to prepare and interpret.

Disadvantages of the Finite Element Method (cont.) Numerical problems: »Computers only carry a finite number of significant digits. »Round off and error accumulation. »Can help the situation by not attaching stiff (small) elements to flexible (large) elements. Susceptible to user-introduced modeling errors: »Poor choice of element types. »Distorted elements. »Geometry not adequately modeled. Certain effects not automatically included: »Buckling »Large deflections and rotations. »Material nonlinearities. »Other nonlinearities.

Types of Finite Elements

Spring Element

Example

Bar and Beam Elements

Example

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