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Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force...
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Transcript of Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force...
Lecture #11Matrix methods
METHODS TO SOLVE INDETERMINATE PROBLEM
2
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
3
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
13
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.)”