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If the stiffness matrix of the beam element is given as \(\frac{{2EI}}{L}\left[ {\begin{array}{*{20}{c}} 2&{ - 1}\\ { - 1}&2 \end{array}} \right]\)then the flexibility matrix is
1. \(\frac{L}{{6EI}}\;\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)
2. \(\frac{L}{{2EI}}\;\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)
3. \(\frac{L}{{3EI}}\;\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)
4. \(\frac{L}{{6EI}}\;\left[ {\begin{array}{*{20}{c}} { - 1}&2\\ 2&{ - 1} \end{array}} \right]\)

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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
  • 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. 
  • Elements of the flexibility matrix are displacements
  • Elements of stiffness matrix are the forces

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 \(\frac{L}{{6EI}}\left[ {\begin{array}{*{20}{c}} 2&1\\ 1&2 \end{array}} \right]\)

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