Assignment 2 Simulation and modeling, Spring 2010

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1 Assignment 2 Simulation and modeling, Spring Background The consultant company B&W (Besser & Wisser) AB has decided to develop an in-house FEcode. With this code the company intends to offer their customers standalone applications for stress and strain analysis. The applications are intended to operate as problem specific solvers with a user-friendly graphical user interface (GUI), requiring a minimum of knowledge in the Finite element method. The applications will be parameter ruled with respect to certain key geometrical and loading condition parameters, which are provided by the user through the GUI. The company believes that there is a great demand for such a product, particularly among small and medium sized manufacturing companies, which do not have sufficient knowledge, or need, for a general purpose FE-code like ABAQUS or ANSYS. Also, many companies do not need many of the capabilities offered by the abovementioned codes, whence they cannot motivate the great cost for licensing such codes. However the need for accurate stress analysis still remains. The development of the project began with one engineer assigned to carry out the numerical programming. MATLAB was chosen as programming language during the development phase (although later versions preferably should be ported to Fortran). After some time of development work, however, he resigned from the firm shortly after an unavailing wage negotiation. At this time, a great deal of work had been carried out on the code, but some parts were still missing in order to complete the code. Now he is working as a well-paid investment analyst at a stockbroker firm located close to Stureplan in Stockholm, and he is not particularly keen to answer any questions related to the development work from his old employer. Meanwhile, the sales department has managed to attract interest for the product from the industry. A manufacturer of warehouse shelf systems for heavy goods has commissioned B&W to develop a FEM application specifically designed to analyze shelf consoles. Considering the fact that this is the first customer for the product it is extremely important that the customer stays satisfied with the application, and that it is delivered within reasonable time. The customer has also agreed to act as a reference case, i.e. as a so called Success story, in B&W s marketing of the product provided, of course, that everything turns out to their satisfaction. First the FE-code must be finished, and to speed up the development work, two of the company s most skilled engineers are assigned to complete the task. The first goal is to get the code running for one specific console configuration and to compare the result to that obtained from ABAQUS. 2 Problem formulation The geometry of the console is shown in figure 1 below. The worst case loading condition is applied to the console, i.e. the entire load is applied as a line load along the outer edge of the console as shown in the figure. The total load is assumed to be N. The inner surface is supposed to be clamped, i.e. all displacement components are set to zero. The console is made of steel with Young s modulus E = MPa and Poisson s ratio.

2 10 Figure 1 To simplify things the structure is analyzed as a plane structure under plain strain conditions (see figure 2). The thickness is 10 mm. 300 F= N Figure 2

3 3 Status of software development What can be found from the previous development is a folder containing a number of MATLAB files. The program package is built around a main program (the_ultimate_fem_prog.m), which calls a number of subprograms (functions). The first subprogram (readabainp.m) reads ABAQUS input files (although for very specific type of problems). The output from this function is described below: nodemat = contains the nodal coordinates. The first row is the x and y coordinates of node no. 1. The second row is the x and y coordinates of node no. 2, and so on, down to the last row which is the x and y coordinates of node no. n, where n is the total number of nodes in the model. elemmat = contains the element nodes for each element in the model (see figure below). The code presumes that bilinear elements are used in the analysis. The first row lists the element nodes for element no. 1. The second row lists the element nodes for element no. 2, and so on, until the last row, which lists the element nodes for element no. m, where m is the total number of elements in the model. no3 no4 no2 no1 Figure 3, Node numbering for bilinear plane strain element (CPE4)

4 ubound = ( no dof ) 1 ( no dof ) 2 M no dof ( ) nu contain all degree of freedoms (DOFs) with prescribed displacements in the model. The first row lists the node and the DOF corresponding to the first constrained DOF. The second row lists the node and the DOF corresponding to the second constrained DOF, and so on, down to the last row which lists the node and the DOF corresponding to the last constrained DOF. The total number of prescribed DOF s in the model is nu. Note: The node numbers are arranged in ascending order and for each node the DOFs are arranged in ascending order. For instance ubound = means that the following DOFs are constrained: x-displacement of node 2, y-displacement of node 3, x- and y-displacements of node 7 and x-displacement of node 8. umagn = contains the magnitudes of the prescribed DOF s corresponding to Ubound. The first component is the magnitude of the first prescribed DOF and so on. fbound = contain all degree of freedoms (DOFs) with prescribed forces in the model. The first row lists the first node/dof with a prescribed force. The second row lists the second node/dof with a prescribed force, and so on, down to the last row, which lists the last node/dof with a prescribed force. The total number of prescribed forces in the model is nf. The DOF s are arranged in ascending order in the same way as ubound. fmagn = contains the magnitudes of the prescribed forces corresponding to fbound. The first component is the magnitude of the first prescribed force, and so on. emod: The Young s modulus.

5 ny: The Poisson s ratio t: Thickness of the 2D model. The meshplot.m function plots the mesh and the boundary conditions applied to it. The next function that is called is updategeom.m, which updates the node coordinates with respect to the global displacement vector d. Obviously there is something missing between the call to meshplot.m and the call to updategeom.m, namely a function which computes the unknown displacement vector d. An inspection of the updategeom.m function reveals however that d should be a column vector with 2n components, where n is the number of nodes in the model. Specifically d = where is the x-displacement of node 1, is the y-displacement of node 1, is the x-displacement of node 2, is the y-displacement of node 2, and so on. The last components are, which is the x-displacement of the last node and, which is the y-displacement of the last node. This is the normal way in which the DOF s are numbered. After the call to updategeom.m, the meshplot.m function is called again in order to plot the deformed mesh. The last two functions that are called from the main program are routines for stress plotting. Let s simply assume that they work satisfactory once d is computed. Strangely enough, although most of the core of the program is missing there is still a function, which computes the element stiffness matrix for a plane bilinear element (bilinear2dke.m). The description of input and output parameters to this function is reasonably well explained in the head of the MATLAB file containing that function. 4 Procedure The following procedure is recommended: 1. Perform the ABAQUS analysis of the problem. The ABAQUS input file can be created from the Job manager from which the ABAQUS analysis is launched. Simply press the Write Input button. Review the results from ABAQUS. 2. Save all the MATLAB files (These can be found in a zipped archive file at kurstorget ) in one directory along with the ABAQUS input file created in step Enter MATLAB and run the main MATLAB program: the_ultimate_fem_prog.m. Check that the model and the boundary conditions are correct. Terminate the analysis. 4. Code the missing parts of the FE-code and run the program until it works. 5 Report The assignment is presented in a written report containing key verification results along with the complementary MATLAB program files.