Structure Analysis 2

1. Government Engineering College, Bhavnagar.
Civil Engineering Department

2. 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

3. 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.

4. 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

5. 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):

6. EXAMPLE OF MEMBER STIFFNESS MATRIX
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Stiffness relation for a rod:
Stiffness matrix:
( )i j i
EA
F x x
L
= − ⋅ −

7. 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.

8. Let’s consider a system of two rods:
ASSEMBLY OF STIFFNESS MATRICES - EXAMPLE
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9. SOLUTION USING MATRIX METHOD - EXAMPLE
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10. SOLUTION USING MATRIX METHOD - EXAMPLE
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i j
1 0
k

11. SOLUTION USING MATRIX METHOD - EXAMPLE
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i j
1 0
k

12. 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 T
F T F Z T Z= ⋅ = ⋅

13. For simplest member (rod) we get:
TRANSFORMATION MATRIX EXAMPLE
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i
i
j
j
x
y
Z
x
y
 
 
 =
 
 
 
i
i
j
j
x
y
Z
x
y
 
 
 =
 
 
  Z T Z= ×

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

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

16. EXAMPLE FOR A TRUSS
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17. EXAMPLE FOR A TRUSS
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18. EXAMPLE FOR A TRUSS
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19. EXAMPLE FOR A TRUSS
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20. EXAMPLE FOR A TRUSS
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21. EXAMPLE FOR A TRUSS
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22. EXAMPLE FOR A TRUSS
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23. EXAMPLE FOR A TRUSS
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24. 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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