University of Illinois at Urbana-Champaign College of Engineering
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1 University of Illinois at Urbana-Champaign College of Engineering CEE 570 Finite Element Methods (in Solid and Structural Mechanics) Spring Semester 03 Quiz # April 8, 03 Name: SOUTION ID#: PS.: A the pages must be stapled at all times. PS: You are allowed to use pen, pencil, eraser, and a calculator. Mobile phones are NOT allowed during the examination. Please turn your cell phones off. PS3.: This is a closed book, closed notes, open minds exam. Show A work on the exam sheets SCORE: Problem : 0 / 0 points Problem : 0 / 0 points TOTA : 00 Problem 3: 30 / 30 points Problem 4: 30 / 30 points Page of
2 Problem (0 points total) In each question below, define or describe the term given and explain its significance in relation to finite element analysis, theory, or practice. Use a few sentences, several phrases, or an outline form. You will be graded for content and logical presentation of the topics. Topic (): Boundary versus domain methods (e.g. BEM x FEM) Numerical techniques, such as boundary and domain methods, are approximate methods to solve the governing differential equations of a problem. For a linear problem, the boundary element method (BEM) requires discretization only on the boundary of the domain, e.g. 3D problems are discretized on the D surface (using plane elements), and D problems are discretized along the D contour (using line elements). The most common form of the BEM is based on the collocation method and BEM equations can be derived either using Green s theorem or the method of weighted residuals (MWR). The BEM is based on the so called fundamental solution of the problem, which in the case of elasticity, is a solution of a point load in an infinite medium (Kelvin solution). On the other hand, the finite element method (FEM) is a domain method that requires discretization both on the boundary and interior of the domain. The FEM relies on variational methods, e.g. MWR, from which the most popular is the Galerkin Method. The standard BEM leads to matrices that are full and unsymmetric, while the FEM leads to matrices that are banded and symmetric. For the same problem and similar level of accuracy, BEM matrices tend to be smaller than FEM matrices. The advantages of the BEM in comparison with FEM is that the BEM is especially suited to model infinite domain problems (without the need for infinite elements) and BEM can treat incompressible materials without any difficulty associated with ill-conditioning. However, the BEM is not as general as the FEM in the sense that BEM is not applicable to problems where the fundamental solution is unknown. Moreover, the mathematics of BEM is comparatively more difficult than the FEM. Other domain methods include the finite volume method (FVM) and the finite difference method (FDM). Finite volume refers to the small volume surrounding each node point. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. The FVM is popular in modeling computational fluid dynamics (CFD) problems. The FDM consists of approximating the solutions to differential equations using finite difference equations to approximate derivatives. The domain methods are alternatives to boundary methods. Moreover, it is not necessary to do an analysis based entirely on a single method. There are ways to couple two methods in a single analysis so that each is used to represent portions of the model to which it is best suited. Thus the best properties of each method can be explored in the solution of a problem. Page of
3 Problem (continued) Topic (): Self-Adjoint Operator The method of weighted residuals (MWR) relies on the inner product <R,W>=0 where R is the residual and W is the weight. The residual is associated to a linear operator (u). If we integrate the inner product <(u),w>=0 by parts until all derivatives in u have been eliminated, this will lead to the transposed form on the inner product. As a result, the integration by parts produces domain <*(w), u> and boundary terms S and G which are differential operators. The term S*(w) contains the w terms resulting from the initial phase of the integration, and S(u) contains the corresponding u terms. The operator * is the adjoint of. If *=, then the operator is selfadjoint. In this case, S=S* and G=G*. In addition to determining if the operator is self-adjoint, the integration by parts also generates two types of boundary conditions (BCs): the set S(u) is called essential BCs and the set G(u) is called non-essential or natural BCs. One can specify either type of BC on the boundary of the domain. The essential BCs must be enforced for the solution to be unique (e.g. FEM). We notice that self-adjointness of an operator (in the continuum sense) is similar to symmetry of a matrix (in the discrete sense). In the context of the FEM, the most popular MWR is the Galerkin method, which leads to a symmetric matrix, when the associated operator is self-adjoint. Page 3 of
4 Problem (0 points) Consider the differential equation: u, xx + 4u = in the range 0<x<, with essential boundary conditions u=3 at x=0 and u= at x=. There are no natural boundary conditions. An admissible one-parameter approximating polynomial is u* = 3 - x + a(x -x). (a) Determine the single parameter a by the east Squares Method. (b) Determine the Residual and the Error at x=0.6 and compare the quantities. The exact solution is given by u (exact ) = sin (x) [Acknowledgement: this question is based on Homework #4, Problem 5.-3 of Cook et al. 00] Solution) Part (a) Exact solution at: at x 0.6 u 3.5sin( 0.6) 0.48 Residual methods: exact u 3 xa( x x) ; u, x a(x ) ; u, xx a R 4 ax (4a 8) x a R a 4x 4x east Squares Method: 0 R R dx 0 a [4 ax (4a 8) x a](4x 4x ) dx 0 0 yields: 0 a and 0 34 u x x Thus R x x Part (b) Rx ( 0.6).885 ux ( 0.6).43 Error: u u exact % Error = u u exact exact u % 0.48 Page 4 of
5 Problem 3 (30 points total) - Part I (0 points) Consider a uniformly distributed transverse loading q acting on the beam element shown in the figure below. Determine the consistent nodal load vector using the equivalent nodal load concept of FEM. q The beam shape-functions are (0 < x < ): N = 3x / + x 3 / 3 ; N = x x / + x 3 / N 3 = 3x / x 3 / 3 ; N 4 = x / + x 3 / [ Hint: Draw a FBD to verify that your answer in indeed correct! ] [ Acknowledgment: This question is similar to the one in the Practice Quiz #.] Solution) Equivalent oad oads: N T T T N Wext ue Peq ue N dx u u u3 u4 qdx N 0 3 N 4 N N N4 Peqq dx N x x Ndx x x x x N dx x x N3 dx x x N dx Page 5 of
6 Peq q q q q q Fy q q q q q M () Page 6 of
7 Problem 3 (30 points total) - Part II (0 points) continued Draw the quadratic strain triangle (QST) and determine its corner node shape functions using area coordinates (,, 3 ). Please do a clear drawing. [ Hint: Use the concept of SHAPE FUNCTION MAGIC! Note: This problem was discussed in class and was suggested as an optional HW. ] Solution) Recall the triangle: x x y xy y x xyxy y 3 3 N c(3)(3 ) At Node :, N c c N (3)(3) Ni i(3i )(3i ) ( i,,3) N4 c(3 ) At Node 4:,, N4 c c N4 (3 ) N0 c 3 At Node 0:,, 3 N N5 (3 ) N6 3(3 ) N7 3(33 ) N8 3(33) N 3(3), N0 c c Page 7 of
8 Problem 4 (30 points total) [Patran/ABAQUS] [Acknowledgement: This question is related to your project, and is intended to assess your knowledge of FEM to solve practical engineering problems. It is also a continuation of the last problem of Quiz #]. We want to conduct finite element analysis of the -D plate with a hole, shown in the first figure below, using linear triangular elements, i.e. T3 (you programmed this element in HW#5). The bar has a unit thickness, and is subjected to an axial stress applied over the top and bottom (complete model). A plane-stress idealization is adopted for the -D model. A quarter-symmetric FE model is shown in the second figure below along with the corresponding finite FE mesh. y 3 R= (0,0) x C A B Figure. Geometry of a plate with a hole Figure. Finite element mesh Table : FEM Stress distribution along the A-B cross section. Nodal position Averaged stress values X Y σ x σ y τ xy Page 8 of
9 (a) Do a clear sketch of the quarter-symmetric model and indicate the displacement boundary conditions to be applied to the nodes along the lines Y=0 and X=0. y (0, x (b) Describe the load boundary conditions (*COAD) that should be applied to the nodes along the line Y=. The six element edges at Y= have the same length *COAD,, 0.5,, 0.5 3,, 0.5 4,, 0.5 5,, 0.5 6,, 0.5 7,, 0.5 (c) The averaged nodal stresses across the section A-B are listed in Table. Using those values, check equilibrium along the Y-direction (i.e. quantify how well equilibrium is satisfied). Use the trapezoidal rule to obtain the net force along the Y=0. ( )( ) / ( )( ) / ( )(.83.5) ( )(.5.70) / (.5.5)(.70.) / (.375.5)(. 0.) (.5.375)(0. 0.7) /.5.5 Page of
10 (d) In general, how can the FEM solution for stresses be improved? h -, p -, r -, s - refinements or a combination (e) Estimate quantitatively the quality of the solution at point B. Note that point B is located at a traction-free boundary. (Hint: Recall Cauchy relation from CEE47). n n x x xy x n n 0.43 y y xy xy The solution quality at point B is not very good since it does not satisfy the traction-free boundary (f) From a mechanics point of view (CEE47 is a pre-requisite for this class), what are the stresses σ Y and τ XY at point C? What do you expect the FEM solution at point C to be and why? σ Y and τ XY should be zero because of the traction-free boundary condition at point C. However FEM solution may show non-zero results since results of stress are obtained numerically. (g) Based on your knowledge of CEE47 and using the stress values provided in Table, what are the principal stresses at (X,Y) = (.375,0)? - Mohr s circle, , Stress tensor Eigenvalues of the stress tensor are,.068, Page 0 of
11 (h) The stress concentration factor K for flat bars with circular holes is shown in the below figure. In the figure below, circle the value of the stress concentration factor (K) of the bar with a hole (see Figure ). max P K, nom, t thickness nom ct 3.8 P b c/ d P.6 c/ K d/b Figure Stress concentration factor K for flat bars with circular holes. (Ref. Roark, R.J., and Young, W. C. (8). Formulars for stress and strain, 6th Ed., Mcgraw- Hill Book co., Inc., Newyork.) (i) Calculate the stress concentration factor (K) from the actual finite element analysis results. nom P 3, 4.5 max 4.5, K=.5 ct.5 Page of
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