PPT: Matrix Method Notes | EduRev

Structural Analysis

GATE : PPT: Matrix Method Notes | EduRev

 Page 1


METHODS TO SOLVE INDETERMINATE PROBLEM
2
Displacement methods
Force method
Small degree
of statical 
indeterminacy
Large degree
of statical 
indeterminacy
Displacement method
in matrix formulation
Numerical methods
Page 2


METHODS TO SOLVE INDETERMINATE PROBLEM
2
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
3
Advantages:
•
 very formalized and computer-friendly;
•
 versatile, suitable for large problems;
•
 applicable for both statically determinate and 
indeterminate problems.
Page 3


METHODS TO SOLVE INDETERMINATE PROBLEM
2
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
3
Advantages:
•
 very formalized and computer-friendly;
•
 versatile, suitable for large problems;
•
 applicable for both statically determinate and 
indeterminate problems.
FLOWCHART OF MATRIX METHOD
4
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
Page 4


METHODS TO SOLVE INDETERMINATE PROBLEM
2
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
3
Advantages:
•
 very formalized and computer-friendly;
•
 versatile, suitable for large problems;
•
 applicable for both statically determinate and 
indeterminate problems.
FLOWCHART OF MATRIX METHOD
4
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
5
Stiffness matrix (K) gives the relation between vectors 
of nodal forces (F) and nodal displacements (Z):
Page 5


METHODS TO SOLVE INDETERMINATE PROBLEM
2
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
3
Advantages:
•
 very formalized and computer-friendly;
•
 versatile, suitable for large problems;
•
 applicable for both statically determinate and 
indeterminate problems.
FLOWCHART OF MATRIX METHOD
4
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
5
Stiffness matrix (K) gives the relation between vectors 
of nodal forces (F) and nodal displacements (Z):
EXAMPLE OF MEMBER STIFFNESS MATRIX
6
Stiffness relation for a rod:
Stiffness matrix:
 
( )
i j i
EA
F x x
L
= - · -
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