1. Matrix methods

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

3. ADVANTAGES AND DISADVANTAGES OF MATRIX METHODS
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 METHOD4

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:
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 - EXAMPLE9

10. SOLUTION USING MATRIX METHOD - EXAMPLE10

11. SOLUTION USING MATRIX METHOD - EXAMPLE11

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

13. TRANSFORMATION MATRIX EXAMPLE
For simplest member (rod) we get: 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. How are they implemented in matrix method
THREE BASIC EQUATIONS
How are they implemented in matrix method 24