calfem 3.4 有限元程序.pdf

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C A L F E M - A I F I N T E E L E M E N T T O O L B O X V e r s i o n 3 . 4 C A L F E M A FINITE ELEMENT TOOLBOX Version 3.4 P-E AUSTRELL, O DAHLBLOM, J LINDEMANN, A OLSSON, K-G OLSSON, K PERSSON, H PETERSSON, M RISTINMAA, G SANDBERG, P-A WERNBERG

C A L F E M A FINITE ELEMENT TOOLBOX Version 3.4 P-E AUSTRELL, O DAHLBLOM, J LINDEMANN, A OLSSON, K-G OLSSON, K PERSSON, H PETERSSON, M RISTINMAA, G SANDBERG, P-A WERNBERG Copyright © 2004 by Structural Mechanics, LTH, Sweden. ISBN: 91-8855823-1

The software described in this document is furnished under a license agreement. The software may be used or copied only under terms in the license agreement. No part of this manual may be photocopied or reproduced in any form without the prior written consent by the Division of Structural Mechanics. c Copyright 1992–2004 by the Division of Structural Mechanics at Lund University. All rights reserved. CALFEM is the trademark of the Division of Structural Mechanics, Lund University. Matlab is the trademark of The MathWorks, Inc. E-mail address: calfem@byggmek.lth.se Homepage: http://www.byggmek.lth.se/Calfem Contacts: The Division of Structural Mechanics Lund University PO Box 118 SE–221 00 Lund SWEDEN Phone: +46 46 222 0000 Fax: +46 46 222 4420

Preface R is an interactive computer program for teaching the ﬁnite element method CALFEM (FEM). The name CALFEM is an abbreviation of ”Computer Aided Learning of the Finite Element Method”. The program can be used for diﬀerent types of structural mechanics problems and ﬁeld problems. CALFEM, the program and its built-in philosophy have been developed at the Division of Structural Mechanics, Lund University, starting in the late 70’s. Many coworkers, former and present, have been engaged in the development at diﬀerent stages. This release represents the latest development of CALFEM. The functions for ﬁnite element applications are all MATLAB functions (.m-ﬁles) as described in the MATLAB manual. We believe that this environment increases the versatility and handling of the program and, above all, the ease of teaching the ﬁnite element method. CALFEM also works with Octave, presently with exception for some graphical functions. Lund, October 7, 2004 The authors

Contents 1 Introduction 2 General purpose functions 3 Matrix functions 4 Material functions 5 Element functions 1 – 1 2 – 1 3 – 1 4 – 1 5.1 – 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 – 1 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 – 1 5.2 Spring element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 – 1 5.3 Bar elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 – 1 5.4 Heat ﬂow elements 5.5 Solid elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 – 1 5.6 Beam elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 – 1 5.7 Plate element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 – 1 6 System functions 6.1 – 1 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 – 1 6.2 Static system functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 – 1 6.3 Dynamic system functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 – 1 7 Statements and macros 8 Graphics functions 9 User’s Manual, examples 7 – 1 8 – 1 9.1 – 1 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 – 1 9.2 MATLAB introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 – 1 9.3 Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 – 1 9.4 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 – 1 9.5 Nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 – 1

1 Introduction The computer program CALFEM is a MATLAB toolbox for ﬁnite element applications. This manual concerns mainly the ﬁnite element functions, but it also contains descriptions of some often used MATLAB functions. The ﬁnite element analysis can be carried out either interactively or in a batch oriented fashion. In the interactive mode the functions are evaluated one by one in the MATLAB command window. In the batch oriented mode a sequence of functions are written in a ﬁle named .m-ﬁle, and evaluated by writing the ﬁle name in the command window. The batch oriented mode is a more ﬂexible way of performing ﬁnite element analysis because the .m-ﬁle can be written in an ordinary editor. This way of using CALFEM is recommended because it gives a structured organization of the functions. Changes and reruns are also easily executed in the batch oriented mode. A command line consists typically of functions for vector and matrix operations, calls to functions in the CALFEM ﬁnite element library or commands for workspace operations. An example of a command line for a matrix operation is C = A + B where two matrices A and B’ are added together and the result is stored in matrix C . The matrix B’ is the transpose of B. An example of a call to the element library is Ke = bar1e(k) where the two-by-two element stiﬀness matrix Ke is computed for a spring element with spring stiﬀness k, and is stored in the variable Ke. The input argument is given within parentheses ( ) after the name of the function. Some functions have multiple input argu- ments and/or multiple output arguments. For example [lambda, X] = eigen(K, M) computes the eigenvalues and eigenvectors to a pair of matrices K and M. The output variables - the eigenvalues stored in the vector lambda and the corresponding eigenvectors stored in the matrix X - are surrounded by brackets [ ] and separated by commas. The input arguments are given inside the parentheses and also separated by commas. The statement help function provides information about purpose and syntax for the speciﬁed function. 1 – 1 INTRODUCTION

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