Matrix Stiffness Method 1011 - Structural Analysis IV...

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Structural Analysis IV Matrix Stiffness Method 4th Year Structural Engineering 2010/11 Dr. Colin Caprani Dr. C. Caprani 1
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Structural Analysis IV Contents 1. Introduction . ........................................................................................................ 4 1.1 Background. ..................................................................................................... 4 1.2 Basic Concepts. ................................................................................................ 5 1.3 Matlab Truss Analysis Program . ..................................................................... 7 2. Basic Approach. ................................................................................................... 9 2.1 Individual Element . ......................................................................................... 9 2.2 Assemblies of Elements. ................................................................................ 11 2.3 Example 1. ..................................................................................................... 13 2.4 General Methodology. ................................................................................... 19 2.5 Member contribution to global stiffness matrix. ........................................... 21 2.6 Interpretation of Stiffness Matrix . ................................................................. 26 2.7 Restricting a Matrix. ...................................................................................... 28 3. Plane Trusses . .................................................................................................... 31 3.1 Introduction. ................................................................................................... 31 3.2 Truss Element Stiffness Matrix. .................................................................... 34 3.3 Element Forces . ............................................................................................. 39 3.4 Example 2: Basic Truss. ................................................................................ 42 3.5 Example 3: Adding Members. ....................................................................... 51 3.6 Example 4: Using Symmetry. ........................................................................ 55 3.7 Self-Strained Structures. ................................................................................ 58 3.8 Example 5 – Truss with Differential Temperature. ....................................... 62 3.9 Example 6 – Truss with Loads & Self Strains . ............................................. 68 3.10 Problems. .................................................................................................... 73 4. Beams. ................................................................................................................. 75 4.1 Beam Element Stiffness Matrix. .................................................................... 75 4.2 Beam Element Loading . ................................................................................ 80 4.3 Example 7 – Simple Two-Span Beam. .......................................................... 82 Dr. C. Caprani 2
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Structural Analysis IV 4.4 Example 8 – Non-Prismatic Beam . ............................................................... 86 4.5 Problems . ....................................................................................................... 90 5. Plane Frames. ..................................................................................................... 92 5.1 Plane Frame Element Stiffness Matrix. ......................................................... 92 5.2 Example 9 – Simple Plane Frame . .............................................................. 101 5.3 Example 10 –Plane Frame Using Symmetry . ............................................. 106 5.4 Problems . ..................................................................................................... 112 6. Appendix . ......................................................................................................... 114 6.1 Plane Truss Element Stiffness Matrix in Global Coordinates. .................... 114 6.2 Coordinate Transformations. ....................................................................... 123 6.3 Past Exam Questions . .................................................................................. 131 7. References . ....................................................................................................... 141 Dr. C. Caprani 3
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Structural Analysis IV 1. Introduction 1.1 Background The matrix stiffness method is the basis of almost all commercial structural analysis programs. It is a specific case of the more general finite element method, and was in part responsible for the development of the finite element method. An understanding of the underlying theory, limitations and means of application of the method is therefore essential so that the user of analysis software is not just operating a ‘black box’. Such users must be able to understand any errors in the modelling of structures which usually come as obtuse warnings such as ‘zero pivot’ or ‘determinant zero: structure unstable: aborting’. Understanding the basics presented herein should hopefully lead to more fruitful use of the available software. Note: LinPro is very useful as a study aid for this topic: right click on a member and select “Stiffness Matrix” to see the stiffness matrix for any member. Dr. C. Caprani 4
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Structural Analysis IV 1.2 Basic Concepts Node The more general name for a connection between adjacent members is termed a node. For trusses and frames the terms joint and node are interchangeable. For more complex structures (e.g. plates), they are not. Element For trusses and frames element means the same as member. For more complex structures this is not the case. Degree of Freedom The number of possible directions that displacements or forces at a node can exist in is termed a degree of freedom (dof). Some examples are: Plane truss: has 2 degrees of freedom at each node: translation/forces in the x and y directions.
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Matrix Stiffness Method 1011 - Structural Analysis IV...

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