Lecture #11Matrix methods

METHODS TO SOLVE INDETERMINATE PROBLEM*Displacement methodsForce methodSmall degreeof statical indeterminacyLarge degreeof statical indeterminacyDisplacement methodin matrix formulationNumerical methods

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.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.

FLOWCHART OF MATRIX METHOD*

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

EXAMPLE OF MEMBER STIFFNESS MATRIX*Stiffness relation for a rod:

Stiffness matrix:

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.

Lets consider a system of two rods:

ASSEMBLY OF STIFFNESS MATRICES - EXAMPLE*

SOLUTION USING MATRIX METHOD - EXAMPLE*

SOLUTION USING MATRIX METHOD - EXAMPLE*

SOLUTION USING MATRIX METHOD - EXAMPLE*

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:

For simplest member (rod) we get:

TRANSFORMATION MATRIX EXAMPLE*

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

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

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

EXAMPLE FOR A TRUSS*

THREE BASIC EQUATIONSHow are they implemented in matrix method*

WHERE TO FIND MORE INFORMATION?*Megson. Structural and Stress Analysis. 2005Chapter 17

Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 6. Internet is boundless

TOPIC OF THE NEXT LECTURE*Stress state of sweptback wingAll materials of our course are availableat department website k102.khai.edu1. Go to the page 2. Press Structural Mechanics (lecturer Vakulenko S.V.)