Question 3.23: A beam 8 m long is hinged at A and supported on rollers over...
A beam 8 m long is hinged at A and supported on rollers over a smooth surface inclined at 30° to the horizontal at B as shown in Fig. 3.35. Determine support reactions.
(UPTU 2003)
Consider FBD of beam as showing Fig. 3.35 (a)
\sum Y=0, \qquad R_{A Y}+R_{B} \cdot \cos 30^{\circ}=10+8 \sin 45^{\circ}+10
=20+4 \sqrt{2} ….. (1)
\sum M_{A}=0, \qquad R_{B} \cos 30^{\circ} \times 8=10 \times 7+8 \sin 45^{\circ} \times 4+10 \times 2
R_{B}=16.25 kN
Substituting value of R_{B} in equation (1),
R_{A Y}=11.58 kN
\sum X=0, \quad R_{A X}+R_{B} \cdot \sin 30^{\circ}=8 \cos 45^{\circ}
R_{A X}+16.25 \sin 30^{\circ}=8 \cos 45^{\circ}
R_{A X}=-2.47 kN
R_{A}=\sqrt{R_{A X^{2}}+R_{A Y^{2}}}
=\sqrt{(-2.47)^{2}+(11.58)^{2}}
R_{A}=11.84 kN
\tan \alpha=\frac{R_{A Y}}{R_{A X}}=\frac{11.58}{2.47}
\alpha=77.96^{\circ}
negative value shows that hinge reaction is acting on right side as shown in Fig. 3.35(b)
