Question 17.23: A simple supported beam supports a UDL and a moment as shown...
A simple supported beam supports a UDL and a moment as shown in Fig. 17.26. Using the principle of virtual work, determine support reactions.
The beam is turned for virtual angle \delta \theta about A as shown in Fig. 17.26 (a). The weight of UDL is 12 kN and acts at mid–point ‘L’. The moment 20 kNm is acting clockwise thus virtual angle for this will be (–\delta \theta ) as beam is turned in anticlockwise direction.
Using principle of virtual work,
R _{ D } \cdot(+\delta y)_{ D }+20 \cdot(-\delta \theta)+0 \cdot(\delta y)_{ B }+12(-\delta y)_{ L }=0
R _{ D ^{*}}(\delta y)_{ D }-20 . \delta \theta-12 \delta y_{ L } ….. (1)
From right angle triangles,
\delta \theta=\frac{(\delta y)_{D}}{5}=\frac{(\delta y)_{B}}{3}=\frac{(\delta y)_{L}}{1.5}
(\delta y)_{D}=5 . \delta \theta,(\delta y)_{B}=3 . \delta \theta,(\delta y)_{L}=1.5 \delta \theta
Substituting virtual distances in equation (1),
R _{ D }(5 . \delta \theta)-20 \delta \theta-12 .(1.5 \delta \theta)=0
R _{ D }=7.6 kN
Since, R _{ A }+ R _{ D }=12 kN , R _{ A }=12-7.6=4.4 kN
