Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction

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Jennifer Lambert
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1 Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations. It was originated from te need of solving complex elasticity and structural analysis problems in Civil, Mecanical and Aerospace engineering. In a structural simulation, FEM elps in producing stiffness and strengt visualizations. It also elps to minimize materialweigt and its cost of te structures. FEM allows for detailed visualization and indicates te distribution of stresses and strains inside te body of a structure. Many of FE software are powerful yet complex tool meant for professional engineers wit te training and education necessary to properly interpret te results. Several modern FEM packages include specific components suc as fluid, termal, electromagnetic and structural working environments. FEM allows entire designs to be constructed, refined and optimized before te design is manufactured. Tis powerful design tool as significantly improved bot te standard of engineering designs and te metodology of te design process in many industrial applications. Te use of FEM as significantly decreased te time to take products from concept to te production line. One must take te advantage of te advent of faster generation of personal computers for te analysis and design of engineering product wit precision level of accuracy...2 Background of Finite Element Analysis Te finite element analysis can be traced back to te work by Alexander Hrennikoff (9)and Ricard Courant(92). Hrenikoff introduced te framework metod, in wic a plane elastic medium was represented as collections of bars and beams.tese pioneers sare one essential caracteristic: mes discretization of a continuous domain into a set of discrete sub-domains, usually called elements. In 950s, solution of large number of simultaneous equations became possible because of te digitalcomputer. In 960, Ray W. Cloug first publised a paper using term Finite Element Metod. In 965, First conference on finite elements was eld. In 967, te first book on te Finite Element Metod was publised by Zienkiewicz and Cung. In te late 960s and early 970s, te FEM was applied to a wide variety of engineering problems.

2 2 In te 970s, most commercial FEM software packages (ABAQUS, NASTRAN, ANSYS, etc.) originated.interactive FE programs on supercomputer lead to rapid growt of CAD systems. In te 980s, algoritm on electromagnetic applications, fluid flow and termal analysis were developed wit te use of FE program. Engineers can evaluate ways to control te vibrations and extend te use of flexible, deployablestructures in space using FE and oter metods in te 990s. Trends to solve fully coupled solution of fluid flows wit structural interactions, bio-mecanics related problems wit a iger level of accuracy were observed in tis decade. Wit te development of finite element metod, togeter wit tremendous increases in computing power and convenience, today it is possible to understand structural beavior wit levels of accuracy. Tis was in fact te beyond of imagination before te computer age...3 Numerical Metods Te formulation for structural analysis is generally based on te tree fundamental relations: equilibrium, constitutive and compatibility. Tere are two major approaces to te analysis: Analytical and Numerical. Analytical approac wic leads to closed-form solutions is effective in case of simple geometry, boundary conditions, loadings and material properties. However, in reality, suc simple cases may not arise. As a result, various numerical metods are evolved for solving suc problems wic are complex in nature. For numerical approac, te solutions will be approximate wen any of tese relations are only approximately satisfied. Te numerical metod depends eavily on te processing power of computers and is more applicable to structures of arbitrary size and complexity. It is common practice to use approximate solutions of differential equations as te basis for structural analysis. Tis is usually done using numerical approximation tecniques. Few numerical metods wic are commonly used to solve solid and fluid mecanics problems are given below. Finite Difference Metod Finite Volume Metod Finite Element Metod Boundary Element Metod Mesless Metod Te application of finite difference metod for engineering problems involves replacing te governing differential equations and te boundary condition by suitable algebraic equations. For

3 3 example in te analysis of beam bending problem te differential equation is reduced to be solution of algebraic equations written at every nodal point witin te beam member. For example, te beam equation can be expressed as: d w q (..) dx EI To explain te concept of finite difference metod let us consider a displacement function variable namely w f( x) Fig... Displacement Function Now, Δ w f(x+δx) f(x) dw Δw f(x+δx)- f(x) So, Lt Lt ( w -w) Tus, dx Δx 0 Δx Δx 0 Δx dx dx i 2 dw d w 2 -w i w 2 -w -w +w i w 2-2w +wi 3 dw ( ) ( ) ( ) ( ) wi+3 -w -2w +2w 3 3 +w -wi dx 3 ( wi+3-3w +3w -wi ) (..2) (..3) (..)

4 dw ( ) wi+ - wi+3-3w i+3 +3w +3w - 3w - w + wi dx ( wi+ -w i+3 +6w -w +wi ) ( w -w +6wi -w i- +wi-2 ) (..5) Tus, eq. (..) can be expressed wit te elp of eq. (..5) and can be written in finite difference form as: q ( w EI i 2 wi + 6wi wi+ + wi+ 2 ) (..6) Fig...2 Finite difference equation at node i Tus, te displacement at node i of te beam member corresponds to uniformly distributed load can be obtained from eq. (..6) wit te elp of boundary conditions. It may be interesting to note tat, te concept of node is used in te finite difference metod. Basically, tis metod as an array of grid points and is a point wise approximation, wereas, finite element metod as an array of small interconnecting sub-regions and is a piece wise approximation. Eac metod as noteworty advantages as well as limitations. However it is possible to solve various problems by finite element metod, even wit igly complex geometry and loading conditions, wit te restriction tat tere is always some numerical errors. Terefore, effective and reliable use of tis metod requires a solid understanding of its limitations... Concepts of Elements and Nodes Any continuum/domain can be divided into a number of pieces wit very small dimensions. Tese small pieces of finite dimension are called Finite Elements (Fig...3). A field quantity in eac element is allowed to ave a simple spatial variation wic can be described by polynomial terms. Tus te original domain is considered as an assemblage of number of suc small elements. Tese elements are connected troug number of joints wic are called Nodes. Wile discretizing te structural system, it is assumed tat te elements are attaced to te adjacent elements only at te nodal points. Eac element contains te material and geometrical properties. Te material properties inside an element are assumed to be constant. Te elements may be D elements, 2D elements or 3D elements. Te pysical object can be modeled by coosing appropriate element suc as frame

5 5 element, plate element, sell element, solid element, etc. All elements are ten assembled to obtain te solution of te entire domain/structure under certain loading conditions. Nodes are assigned at a certain density trougout te continuum depending on te anticipated stress levels of a particular domain. Regions wic will receive large amounts of stress variation usually ave a iger node density tan tose wic experience little or no stress. Fig...3 Finite element discretization of a domain..5 Degrees of Freedom A structure can ave infinite number of displacements. Approximation wit a reasonable level of accuracy can be acieved by assuming a limited number of displacements. Tis finite number of displacements is te number of degrees of freedom of te structure. For example, te truss member will undergo only axial deformation. Terefore, te degrees of freedom of a truss member wit respect to its own coordinate system will be one at eac node. If a two dimension structure is modeled by truss elements, ten te deformation wit respect to structural coordinate system will be two and terefore degrees of freedom will also become two. Te degrees of freedom for various types of element are sown in Fig... for easy understanding. Here (,, ) represent displacement and rotation respectively. uvw and ( θ x, θy, θ z)

6 6 Fig... Degrees of Freedom for Various Elements

Instantaneous Rate of Change:

Instantaneous Rate of Change: Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over