Lecture #11Matrix methods

METHODS TO SOLVE INDETERMINATE PROBLEM

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Displacement methods

Force method

Small degreeof statical

indeterminacy

Large degreeof statical

indeterminacy

Displacement methodin matrix formulation

Numerical 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

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Advantages:• very formalized and computer-friendly;• versatile, suitable for large problems;• applicable for both statically determinate and indeterminate problems.

FLOWCHART OF MATRIX METHOD

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Classificationof members

Stiffness matrices for members

Transformed stiffness matrices

Stiffness matrices are composed according to

member models

Stiffness matrices are transformed from local to global

coordinates

Final equationF = K · Z

Stress-strain state of structure

Unknown displacements and reaction forces are calculated

Stiffness matrices of separate members are assembled into a

single stiffness matrix K

STIFFNESS MATRIX OF STRUCTURAL MEMBER

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Stiffness matrix (K) gives the relation between vectors

of nodal forces (F) and nodal displacements (Z):

EXAMPLE OF MEMBER STIFFNESS MATRIX

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Stiffness relation for a rod:

Stiffness matrix:

i j i

EAF x x

L

ASSEMBLY OF STIFFNESS MATRICES

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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.

Let’s consider a system of two rods:

ASSEMBLY OF STIFFNESS MATRICES - EXAMPLE

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SOLUTION USING MATRIX METHOD - EXAMPLE

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SOLUTION USING MATRIX METHOD - EXAMPLE

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i j10k

SOLUTION USING MATRIX METHOD - EXAMPLE

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i j10

k

TRANSFORMATION MATRIX

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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:

1 MT T

F T F Z T Z

The transformation from local CS to global CS: T TF T F Z T Z

For simplest member (rod) we get:

TRANSFORMATION MATRIX EXAMPLE

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i

i

j

j

x

yZ

x

y

i

i

j

j

x

yZ

x

y

Z T Z

TRANSFORMATION MATRIX

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

EXAMPLE FOR A TRUSS

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The truss has three members, thus 6 degrees of freedom. The stiffness matrix will be 6x6.

EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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EXAMPLE FOR A TRUSS

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THREE BASIC EQUATIONS

Equilibriumequations

Constitutiveequations

Compatibilityequations

Taken into account when global stiffness matrix is assembled from

member matrices

Through member stiffness matrices

Taken into account when global stiffness matrix is assembled from

member matrices

How are they implemented in matrix method

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WHERE TO FIND MORE INFORMATION?

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Megson. Structural and Stress Analysis. 2005Chapter 17

Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 6.

… Internet is boundless …

TOPIC OF THE NEXT LECTURE

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Stress state of sweptback wing

All materials of our course are availableat department website k102.khai.edu

1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”