University of Illinois at Urbana-Champaign College of Engineering CEE 570 Finite Element Methods (in Solid and Structural Mechanics) Spring Semester 2014 Quiz #1 March 3, 2014 Name: SOLUTION ID#: PS.: ALL the pages must be stapled at all times. PS2.: This is a closed book, closed notes, open minds exam. Show ALL work on the exam sheets SCORE: Problem 1: 25 / 25 points Problem 2: 20 / 20 points 100/ 100 TOTAL : Problem 3: 15 / 15 points Problem 4: 40 / 40 points Page 1 of 11
Problem 1 (25 points total): Part I [Historical Aspects] (10 points): The person shown at the left is Euler. State the relevance of his work in the context of the finite element method (FEM). Use the language of mathematics in your statement. Solution) The followings are some of his major contributions which should be included (at least two) in the statement. A. Bernoulli-Euler beam theory B. Calculus of variations - Euler-Lagrange Equation 0 C. Euler critical buckling load Page 2 of 11
Problem 1 Part II (15 points): In a three-node triangle (e.g. T3), the scalar field quantity φ can be written as φ = a 1 + a 2 x + a 3 y, where the a i (i=1,2,3) are generalized degrees of freedom (dof). For the particular shape of triangle shown below, express φ in the form φ = f 1 φ 1 + f 2 φ 2 + f 3 φ 3 where the f i (i=1,2,3) are functions of (x, y, a, b). b 1 a y 3 a 2 x Acknowledgement: This problem was assigned as part of Homework #1, Problem 1.3 3. Solution) Point 1:,, 0 Point 2:,, 0 Point 3:, 0, From equation (1.3-5): Point 1: Point 2: Point 3: Solving for,, yields: 2 2 2 2 Finally: 2 1 2 2 2 2 2 2 1 2 2 2 Page 3 of 11
Problem 2 (20 points): The stiffness matrix below is associated with an actual structure. Devise such a structure, using linear springs and rigid blocks, in the manner that you did Homework #2 (Problem 4.3-3) and also as done in class. Do a clear drawing of the structure and indicate each spring stiffness (magnitude), the nodal loads, and the actual displacement degrees of freedom (DOFs). Solution) 18 6 6 0 60 6 12 0 6 0 6 0 12 6 20 0 6 6 12 0 Optional to verify (Not needed) Π 1 2 6 6 6 6 6 60 20 Π 06 6 6 60 Π 06 6 Π 06 6 20 Π 06 6 666 6 6 0 60 6 6 6 0 6 0 6 0 6 6 6 20 0 6 6 66 0 Page 4 of 11
Problem 3 (15 points): Assume that the structure shown below has one degree of freedom (DOF) per node, and that each straight line between nodes is a two-node element. (a) Assign a node numbering that minimizes the largest semi-bandwidth (b max ). For this numbering, what are b max, profile p, and the fills generated by Gauss elimination? (b) Repeat part (a), but now try to assign an alternative numbering that maximizes b max. Solution) (a) Candidate numbering b max = 5, p= 69, fills = 29 Page 5 of 11
(b) Candidate numbering b max = 16, p= 66, fills = 29 Page 6 of 11
Problem 4 (40 points): Consider the notched plate shown below. The plate will be modeled using plane stress conditions, with a thickness of 0.02, a Young s modulus, E, of 200 and Poisson ratio, ν, of 0.3. A pressure load of 3.0/unit length is applied to the top and bottom edges. The domain is discretized using 4 quadratic quadrilateral (Q8) elements, as shown below. Fill in the 20 missing spaces in the ABAQUS input file, and answer questions (a) through (h). *HEADING Plate with slit problem - QUIZ 1 *NODE 1, 0.0000, 0.0000 2, 0.0000, 0.5000 3, 0.0000, 1.0000 4, 0.0000, 1.5000 5, 0.0000, 2.0000 6, 0.5000, 0.0000 7, 0.5000, 1.0000 8, 0.5000, 2.0000 9, 1.0000, 0.0000 10, 1.0000, 0.5000 11, 1.0000, 1.0000 12, 1.0000, 1.5000 13, 1.0000, 2.0000 14, 1.5000, 0.0000 15, 1.5000, 1.0000 16, 1.5000, 1.0000 17, 1.5000, 2.0000 18, 2.0000, 0.0000 19, 2.0000, 0.5000 20, 2.0000, 1.0000 21, 2.0000, 1.0000 22, 2.0000, 1.5000 23, 2.0000, 2.0000 *ELEMENT, TYPE=CPS8, ELSET=THINPLATE 1, 1, 9, 11, 3, 6, 10, 7, 2 2, 3, 11, 13, 5, 7, 12, 8, 4 3, 9, 18, 20, 11, 14, 19, 15, 10 4, 11, 21, 23, 13, 16, 22, 17, 12 *SOLID SECTION, ELSET=THINPLATE, MATERIAL=MAT01 0.02 *MATERIAL, NAME=MAT01 *ELASTIC 200, 0.3 *STEP, PERTURBATION 2 1 4 3 Page 7 of 11
*STATIC *BOUNDARY 1, 1 2, 1 3, 1, 2 4, 1 5, 1 *DLOAD 1, P1, -3 or P1, -150 2, P3, -3 or P3, -150 3, P1, -3 or P1, -150 4, P3, -3 or P3, -150 *NODE PRINT U, RF *EL PRINT, POSITION=AVERAGED AT NODES S, E *END STEP (a) The input file above, called plate_slit.inp has been emailed to you for access on a Linux machine. You download the file into your downloads folder located at /home/downloads/. From a command line prompt, you need to move the file to your working directory located at /home/workingdir/ and rename it to plate_slit_2.inp. The original file should not remain in the downloads folder. Write the command(s) you would type at the terminal to achieve this. The current directory is shown below using the pwd command. $ pwd /home Many solutions are possible, the following is one option: $ mv downloads/plate_slit.inp workingdir/plate_slit_2.inp (b) What is the total size of the full stiffness matrix for the static analysis of the problem? Hint: each node contains 2 degrees of freedom (DOFs). Number of DOFs = dofs per node number of nodes = Matrix size is (c) What is the maximum semi-bandwidth of the stiffness matrix in (b)? (dofs per node) (highest node number in an element lowest node number in an element - 1) = Page 8 of 11
(d) The problem is remeshed with 16 Q8 elements shown below. This is achieved by means of two surfaces, which are each meshed in PATRAN, as shown below. What is the number of finite element nodes after meshing (before equivalencing)? What is the number of finite element nodes after equivalencing? Is there any special care that need to be taken when equivalencing this domain? If so, please describe. Number of nodes after meshing = 74 Number of nodes after equivalencing = 69 In order to preserve the notch, the nodes on the notch surfaces must remain separated. Therefore the eight nodes on the notch surfaces should be excluded from equivalencing. (e) We would like to produce a fringe plot of one of the stress components on the deformed shape, as shown below. Circle the items that need to be selected in the PATRAN Results dialogue box, shown to the right, before clicking Apply. Page 9 of 11
(f) Which stress component is plotted in the figure of part (e)? (1) (2) (3) (4) (von Mises stress) (5) None of the above (g) If higher order terms are neglected the analytical stress at the tip of the notch (i.e. node 11) is given as, 2 where is the so-called stress intensity factor, is a dimensionless quantity that depends on the problem geometry and boundary conditions, is the radial distance from the tip of the notch, and is the angle measured from the notch tip, as shown below. (Hint: Think CEE471) Will the finite element solution resolve the analytical solution given above at the notch tip (i.e. 0) using the standard Q8 elements? Why or why not? What will happen if the mesh is refined? The elements can represent a quadratic displacement field exactly, thus they can represent and linear stress field exactly. The analytical solution of stress contains a singularity at the crack tip, (i.e. ), and the quadratic elements will never be able to resolve the analytical solution at this point. Even with mesh refinement, the analytical solution cannot be obtained at this point, because the richer quadratic space will never represent infinite stress at a point. Page 10 of 11
(h) Consider the case where the direction of the force is reversed as is shown in the figure to the right. When trying to model this problem with standard finite elements, what do you predict will happen? (Hint: Think CEE471) (1) The gap will close and the plate will be modeled as if the slit didn t exist (Case 1). (2) The free surfaces will come into contact with each other, but penetration is prevented causing to be continuous across the slit, but not and (Case 2). (3) The contact will not be detected and edges of the plate will penetrate each other without any interaction from the opposite side (Case 3). (4) ABAQUS/ANSYS/ETABS/SAP2000/etc. will report an error and fail to run the model. (5) Other. Explain (Case 1) (Case 2) (Case 3) Page 11 of 11