Lecture #9 Matrix methods

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1 Lecture #9 Matrix methods

2 METHODS TO SOLVE INDETERMINATE PROBLEM Small degree of statical indeterminacy Force method Displacement methods Displacement method in matrix formulation Large degree of statical indeterminacy Numerical methods 2

3 ADVANTAGES AND DISADVANTAGES OF MATRIX METHODS Advantages: very formalized and computer-friendly; versatile, suitable for large problems; applicable for both statically determinate and indeterminate problems. Disadvantages: bulky calculations (not for hand calculations); structural members should have some certain number of unknown nodal forces and nodal displacements; for complex members such as curved beams and arbitrary solids this requires some discretization, so no analytical solution is possible. 3

4 FLOWCHART OF MATRIX METHOD Classification of members Stiffness matrices for members Transformed stiffness matrices Final equation F = K Z Stiffness matrices are composed according to member models Stiffness matrices are transformed from local to global coordinates Stiffness matrices of separate members are assembled into a single stiffness matrix K Stress-strain state of structure Unknown displacements and reaction forces are calculated 4

5 STIFFNESS MATRIX OF STRUCTURAL MEMBER Stiffness matrix (K) gives the relation between vectors of nodal forces (F) and nodal displacements (Z): 5

6 EXAMPLE OF MEMBER STIFFNESS MATRIX Stiffness relation for a rod: F EA = L ( x x ) i j i Stiffness matrix: 6

7 ASSEMBLY OF STIFFNESS MATRICES To assemble stiffness matrices of separate members into a single matrix for the whole structure, we should simply add terms for corresponding displacements. Physically, this procedure represent the usage of compatibility and equilibrium equations. 7

8 ASSEMBLY OF STIFFNESS MATRICES - EXAMPLE Let s consider a system of two rods: 8

9 SOLUTION USING MATRIX METHOD - EXAMPLE 9

10 SOLUTION USING MATRIX METHOD - EXAMPLE i j 10 k 10

11 SOLUTION USING MATRIX METHOD - EXAMPLE 10 i j k 11

12 TRANSFORMATION MATRIX Transformation matrix is used to transform nodal displacements and forces from local to global coordinate system (CS) and vice versa: F = T F Z = T Z Transformation matrix is always orthogonal, thus, the inverse matrix is equal to transposed matrix: T 1 = T M The transformation from local CS to global CS: T F = T F Z = T Z T 12

13 TRANSFORMATION MATRIX EXAMPLE For simplest member (rod) we get: x x i i y y i Z = Z = i x x j j y y j j Z = T Z 13

14 TRANSFORMATION MATRIX To transform the stiffness matrix from local CS to global CS, the following formula is used: 14

15 EXAMPLE FOR A TRUSS The truss has three members, thus 6 degrees of freedom. The stiffness matrix will be 6x6. 15

16 EXAMPLE FOR A TRUSS 16

17 EXAMPLE FOR A TRUSS 17

18 EXAMPLE FOR A TRUSS 18

19 EXAMPLE FOR A TRUSS 19

20 EXAMPLE FOR A TRUSS 20

21 EXAMPLE FOR A TRUSS 21

22 EXAMPLE FOR A TRUSS 22

23 EXAMPLE FOR A TRUSS 23

24 THREE BASIC EQUATIONS How are they implemented in matrix method Equilibrium equations Taken into account when global stiffness matrix is assembled from member matrices Constitutive equations Through member stiffness matrices Compatibility equations Taken into account when global stiffness matrix is assembled from member matrices 24

25 WHERE TO FIND MORE INFORMATION? Megson. Structural and Stress Analysis Chapter 17 Megson. An Introduction to Aircraft Structural Analysis Chapter 6. Internet is boundless 25

26 TOPIC OF THE NEXT LECTURE Stress state of sweptback wing All materials of our course are available at department website k102.khai.edu 1. Go to the page Библиотека 2. Press Structural Mechanics (lecturer Vakulenko S.V.) 26

ME 345: Modeling & Simulation. Introduction to Finite Element Method

ME 345: Modeling & Simulation. Introduction to Finite Element Method ME 345: Modeling & Simulation Introduction to Finite Element Method Examples Aircraft 2D plate Crashworthiness 2 Human Heart Gears Structure Human Spine 3 F.T. Fisher, PhD Dissertation, 2002 Fluid Flow