Question 17.20: A simple supported beam of span 10 m is carrying a point loa...
A simple supported beam of span 10 m is carrying a point load at its mid points as shown in Fig. 17.23. Using virtual work, determine reaction of hinge and roller bearing.
Let the beam is kept fixed about A and virtually turned by angle ‘\delta \theta ’ which causes virtual displacement (\delta y)_{c} for reaction R_{c} and (\delta y)_{B} for point load 30 kN at mid–point as shown in Fig. 17.23(a).
ΣY = 0,
R _{ A }+ R _{ C }=30 kN ….. (1)
Using principle of virtual work,
R _{ C } \cdot(+\delta y)_{ c }+30 \cdot(-\delta y)_{ B }=0
i.e., R _{ C } \cdot(+\delta y)_{ c }-30 \cdot(\delta y)_{ B }=0 ….. (2)
(\delta y)_{B} is taken as negative as it is opposite to the direction of 30 kN.
From right angle triangles,
\tan \delta \theta=\delta \theta=\frac{(\delta y)_{C}}{10}=\frac{(\delta y)_{B}}{5}
(\delta y)_{C}=10 . \delta \theta \text { and }(\delta y)_{B}=5 . \delta \theta
Substituting the values in equation (2),
R _{ C } \cdot(10 \delta \theta)-30(5 . \delta \theta)=0
R _{ C } =15 kN
Using equation (1),
R _{ A } =15 kN
