Correct Answer - Option 1 :
\(\frac{L}{{6EI}}\;\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)
Concept:
Matrix method of structural analysis:
- It is one of the methods of solving statically indeterminate beams and frames.
There are two methods of matrix analysis:
Flexibility matrix method |
Stiffness matrix method |
- Elements are determined by applying unit force in the direction of anyone co-ordinate and calculate the displacement of the desired co-ordinate
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- In this, we identify the unknown joint displacement and fix them and later permit deflection in the co-ordinate direction and calculate force developed in various co-ordinate direction.
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- Elements of the flexibility matrix are displacements
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- Elements of stiffness matrix are the forces
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Calculation:
Given:
\(\left[ A \right] = \left[ {\begin{array}{*{20}{c}} 2&{ - 1}\\ { - 1}&2 \end{array}} \right]\)
Flexibility matrix is inverse of the stiffness matrix
So, A-1 is the stiffness matrix of A
A-1 = \(\frac{{{\rm{Adj}}\left( {\rm{A}} \right)}}{{\left| A \right|}}\)
Adj(A) = \(\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)
\(\left| A \right|\) = \(\frac{{2EI}}{L}\)× (4-1)
= \(\frac{{2EI}}{L}\)× 3
So, A-1 = \(\frac{L}{{6EI}}\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)
∴The flexibility matrix of given stiffness matrix is \(\frac{L}{{6EI}}\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)