Lecture #9

Matrix methods

METHODS TO SOLVE INDETERMINATE PROBLEM

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Displacement methods

Force method

Small degree

of statical

indeterminacy

Large degree

of statical

indeterminacy

Displacement method

in 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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Classification

of 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 equation

F = 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 j

10

k

SOLUTION USING MATRIX METHOD - EXAMPLE

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i j

10

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 M

T 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

Equilibrium

equations

Constitutive

equations

Compatibility

equations

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. 2005

Chapter 17

Megson. An Introduction to Aircraft Structural Analysis. 2010

Chapter 6.

/ Internet is boundless /

TOPIC OF THE NEXT LECTURE

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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.)”