Lecture #11 Matrix methods

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

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

FLOWCHART OF MATRIX METHOD 4

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

EXAMPLE OF MEMBER STIFFNESS MATRIX Stiffness relation for a rod: Stiffness matrix: 6

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

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

SOLUTION USING MATRIX METHOD - EXAMPLE 9

SOLUTION USING MATRIX METHOD - EXAMPLE 10

SOLUTION USING MATRIX METHOD - EXAMPLE 11

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

TRANSFORMATION MATRIX EXAMPLE For simplest member (rod) we get: 13

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

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

EXAMPLE FOR A TRUSS 16

EXAMPLE FOR A TRUSS 17

EXAMPLE FOR A TRUSS 18

EXAMPLE FOR A TRUSS 19

EXAMPLE FOR A TRUSS 20

EXAMPLE FOR A TRUSS 21

EXAMPLE FOR A TRUSS 22

EXAMPLE FOR A TRUSS 23

How are they implemented in matrix method THREE BASIC EQUATIONS How are they implemented in matrix method 24

… Internet is boundless … WHERE TO FIND MORE INFORMATION? Megson. Structural and Stress Analysis. 2005 Chapter 17 Megson. An Introduction to Aircraft Structural Analysis. 2010 Chapter 6. … Internet is boundless … 25

Stress state of sweptback wing 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