Which theorem describe the relationship among the bending moments
A. Castigliano’s Theorem
B. Mohr’s Theorem
C. Clapeyron’s Theorem
D. Betti’s Theorem
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
Clapeyron’s Theorem:
Clapeyron’s Theorem or the Theorem of Three Moments describes the relationship among the bending moments at three consecutive supports of a horizontal continuous beam.
Let the bending moment at these points is MA, MB and MC and the corresponding vertical displacement of these points are ΔA, ΔB and ΔC, respectively. Let L1 and L2 be the distance between points AB and BC, respectively.
\({M_A}\left( {\frac{{{L_1}}}{{{I_1}}}} \right) + 2{M_B}\left( {\frac{{{L_1}}}{{{I_1}}} + \frac{{{L_2}}}{{{I_2}}}} \right) + {M_C}\left( {\frac{{{L_2}}}{{{I_2}}}} \right) = - \frac{{6{A_1}{X_1}}}{{{L_1}{I_1}}} - \frac{{6{A_2}{X_2}}}{{{L_2}{I_2}}} \)
\(+\; 6E\left( {\frac{{{{\rm{\Delta }}_B} - {{\rm{\Delta }}_A}}}{{{L_1}}} + \frac{{{{\rm{\Delta }}_B} - {{\rm{\Delta }}_C}}}{{{L_2}}}} \right)\)
where
A1 and A2 are the areas of BMD’s,
X̅1 and X̅2 are the centroidal distance of areas A1 and A2 measured from A and C and
I1 and I2 are the Moment of Inertia of the beam AB and BC respectively.