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RSMSSB JE Civil Degree Dec 2020 Official Paper

Option 4 : 3 - Moment

Building Material & Concrete Technology

15652

20 Questions
20 Marks
25 Mins

**The theorem of three moments is a relationship among the bending moments at three consecutive supports of a horizontal beam. It is developed by Calpeyron’s.**

Consider the 3 points on beam marked as A, B, C as shown in figure.

Let w1, w’ be the load per unit length of these segments AB and BC.

Let ΔA, ΔB, ΔC is deflection at A, B, C.

Let MA, MB, MC are bar’s at A, B, C

As per three-moment equation:

\({M_A}\left( {\frac{{{L_1}}}{{{I_1}}}} \right) + 2{M_B}\left( {\frac{4}{{{I_1}}} + \frac{{{L_2}}}{{{I_2}}}} \right) + {M_C}\left( {\frac{{{L_2}}}{{{I_2}}}} \right)\)

\(= - \frac{{6{A_1}\overline {{x_1}} }}{{{I_1}{L_1}}} - \frac{{6{A_2}\overline {{x_2}} }}{{{I_2}{L_2}}} + 6E\left[ {\frac{{{{\rm{\Delta }}_B} - {{\rm{\Delta }}_A}}}{{{L_1}}} + \frac{{{{\rm{\Delta }}_B} - {{\rm{\Delta }}_C}}}{{{L_2}}}} \right]\)

Where A1 A2 are areas of BMD of span AB and BC considering applied loading acting as simply supported beams.

x̅1 and x̅2 are centroids of BMD of span AB, BC.

Sign:

MA, MB, MC are +ve If they are sagging.

A1, A2 are +ve If it is sagging moment.

ΔA, ΔB, ΔC are +ve If measured downward.