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1、有限单元法及程序设计（Finite element method and program design）Finite element method and program designIntroduction1. mechanical analysis method: analytic method, numerical methodThe finite element method, the actual structure shape and the load is complex, mostly by analytical method and numerical method is d
2、ifficult to develop, with the progress of computer, a new numerical analysis method which is developed.2 basic steps:(1) structural discretization: the structure is divided into finite units by lines or planes from the set.(2) element analysis: the relation between nodal displacement and nodal force
3、 (element stiffness matrix) of the derived element.(3) overall analysis: the whole structure of each element is analyzed, and the relation between the displacement of the structural point and the nodal force is derived.3 programming steps:Ask questions and work out a solutionStructural mathematical
4、modelDraw a program flow chartProgrammingCompiler debuggerTrial verification program4., according to the national standard (GB-1526-89) procedures, processes, icons, quasi symbols and regulations:EMBED PBrushThe graph represents the starting and ending points of the program flow chart;Diagrams show
5、the input and output of data information;A data set to be completed before a series of operations is performed by a graph;The chart shows the criteria;Graphs represent various processing functions, such as mathematical operations;The chart shows the path and direction of the process.Finite element m
6、ethod and program design of bar structuresThe finite element method of plane member elementBasic concepts of finite element methodThe basic idea: first after (the first unit analysis, then the overall analysis)Basic concept: whole number: node endpoint number by natural number 1, 2, 3,. (in the over
7、all coordinate system xOy)Local number: each unit is marked with I, J (in the local coordinates of the unit, EMBED, Equation.3, in the same direction as the overall coordinate system).3.Fe=, Ke, Delta, e, where: Ke = EMBED, Equation.3, element stiffness matrix, elements are stiffness coefficientsDel
8、ta e=, EMBED, Equation.3 element, rod section, displacement arrayRod end force array of Fe= EMBED Equation.3 unitK, EMBED, Equation.3, =P (1-7)K=, EMBED, Equation.3, integral stiffness matrix, EMBED, Equation.3 = EMBED, Equation.3, EMBED, Equation.3 displacement arrayP=, EMBED, Equation.3 node load
9、array3. finite element displacement analysis of continuous beams needs to be consideredStiffness integration method:(1-3) the K expansion order and the expanding element are 0, and the element contribution matrix is obtainedUnit 1: K = EMBED Equation.3 Unit II: K = = EMBED Equation.3The element cont
10、ribution matrix is superimposed to form the whole stiffness matrixK=, K + K = EMBED Equation.3The introduction of support conditions at both endsWithout considering the constraints, the overall stiffness matrix is obtained, and the main diagonal element K, EMBED, Equation.3 is changed to 1, line I,
11、column J, and the rest elements are changed to 0,The corresponding load element is also changed to 0.Treatment of non nodal loadsThe analysis is carried out by equivalent nodal load:Each node (including two nodes) is constrained to stop the rotation of the node, and the constraint moments are the su
12、m of the fixed end moments of the related units which are handed over to the node, and the clockwise is positiveRemove an additional constraint (P EMBED, Equation.3, the same as the constraint moment, in the opposite direction)The two part of the rod end bending moment is superimposedElement stiffne
13、ss matrix in second local coordinate systemsGeneral unitThe elastic modulus, the sectional moment of inertia and the sectional area of the element EQ, oac (L, e) are E, I and A, respectively. The I and j ends of the unit have three bars, end forces, EMBED, Equation.3 (i.e. axial forces, shear and be
14、nding moments) and their corresponding three bar end displacements, EMBED, Equation.3, as shown in figure 1-7. Figure EMBED Equation.3 as the element local coordinate, I point at the origin of EMBED, Equation.3 axis and the rod axis coincide, provided by I to j for EMBED positive Equation.3 axis, Eq
15、uation.3 axis by EMBED clockwise 90? EMBED Equation.3 direction. The positive direction of force and displacement is shown in figure 1-7.EMBED PBrushIn this unit, the element end force array and the rod end displacement array are respectivelyEMBED Equation.3 element rod end force arrayEMBED Equation.3 rod end displacement arrayIn order to derive the relationship between the end force of a general element and the displacement at the end of a rod, we consider the following two cases respectively.Firstly, the relationship between axial force EMBED Equation.3 a
