Which of the following correctly represents the stiffness matrix
Which of the following correctly represents the stiffness matrix for two-span continuous beams as shown in the figure. The co-ordinate 1 and 2 represents the rotation at A and B respectively. The flexural rigidity for each span is constant and equal to EI.
A. <span class="math-tex">\(\left[ {\begin{array}{*{20}{c}} {2{\rm{EI}}/{\rm{L}}}&{4{\rm{EI}}/{\rm{L}}}\\ {4{\rm{EI}}/{\rm{L}}}&{6{\rm{EI}}/{\rm{L}}} \end{array}} \right]\)</span>
B. <span class="math-tex">\(\left[ {\begin{array}{*{20}{c}} {4{\rm{EI}}/{\rm{L}}}&{2{\rm{EI}}/{\rm{L}}}\\ {2{\rm{EI}}/{\rm{L}}}&{8{\rm{EI}}/{\rm{L}}} \end{array}} \right]\)</span>
C. <span class="math-tex">\(\left[ {\begin{array}{*{20}{c}} {4{\rm{EI}}/{\rm{L}}}&{2{\rm{EI}}/{\rm{L}}}\\ {2{\rm{EI}}/{\rm{L}}}&{6{\rm{EI}}/{\rm{L}}} \end{array}} \right]\)</span>
D. <span class="math-tex">\(\left[ {\begin{array}{*{20}{c}} {2{\rm{EI}}/{\rm{L}}}&{4{\rm{EI}}/{\rm{L}}}\\ {2{\rm{EI}}/{\rm{L}}}&{6{\rm{EI}}/{\rm{L}}} \end{array}} \right]\)</span>
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Right Answer is: C
SOLUTION
Concept:
The stiffness matrix for the given system is shown below:
\(\left[ {\begin{array}{*{20}{c}} {{{\rm{k}}_{11}}}&{{{\rm{k}}_{12}}}\\ {{{\rm{k}}_{21}}}&{{{\rm{k}}_{22}}} \end{array}} \right]\)
To get the first column of stiffness matrix (k11 and k21), fix all coordinates and apply unit rotation at A and find the moments developed at coordinates 1 and 2.
To get the 2nd column of stiffness matrix (k12 and k22), fix all coordinates and apply unit rotation at B and find the moments developed at coordinates 1 and 2.
Calculation:
Restrained structure: (Fix all the coordinators)
1st Column
k11 = Moment at co-ordinate (1) due to unit rotation at (1)
k21 = Moment at co-ordinate at (2) due to unit rotation at (1)
\({{\rm{k}}_{11}} = \frac{{4{\rm{EI}}}}{{\rm{L}}}\) (It behaves like for end fixed)
\({{\rm{k}}_{21}} = \frac{{2{\rm{EI}}}}{{\rm{L}}}\) (as carryover factor is ½ when the far end is fixed)
2nd Column:
k22 = Moment at (2) due to unit rotation at (2)
k21 = Moment at (1) due to unit rotation at (2)
\({{\rm{k}}_{22}} = \left( {{{\rm{M}}_{{\rm{BA}}}}} \right) + \left( {{{\rm{M}}_{{\rm{BC}}}}} \right) = \frac{{4{\rm{EI}}}}{{\rm{L}}} + \frac{{4{\rm{EI}}}}{{2{\rm{L}}}} = \frac{{6{\rm{EI}}}}{{\rm{L}}}\)
\({{\rm{k}}_{12}} = \frac{1}{2}\left( {{{\rm{M}}_{{\rm{BA}}}}} \right) = \frac{{2{\rm{EI}}}}{{\rm{L}}}\) …. (As C.O.F when the far end is fixed is 1/2)
\(\therefore {\rm{Stiffness\;Matrix\;}}\left( {\rm{k}} \right) = \left[ {\begin{array}{*{20}{c}} {\frac{{4{\rm{EI}}}}{{\rm{L}}}}&{\frac{{2{\rm{EI}}}}{{\rm{L}}}}\\ {\frac{{2{\rm{EI}}}}{{\rm{L}}}}&{\frac{{6{\rm{EI}}}}{{\rm{L}}}} \end{array}} \right]\)
