Determine the resultant of the loads:

A system of loads acting on beam is shown in the figure given below. Determine the resultant of the loads.
Sol.: Let R be resultant of given system. And   ∑H and   ∑V be the horizontal and vertical component of resultant. And resultant makes an angle of θ with the horizontal.

Resolving all the forces horizontally

∑H = 20 cos 60º

∑H = 10KN                                                                                                       ...(i)

By resolving all the forces vertically

∑V = 20 + 30 + 20 sin 60º

∑V = 67.32KN                                                                                  ...(ii)

Since,  ∑R = √(∑H)² + (∑V)² √(10)²  + (67.32)²

R = 68.05KN                                                                           .......ANS

Let     θ= Angle makes by the resultant

tanθ=  ∑V/∑H = 67.32/10 = 81.55º For position of the resultant

Let, d = Distance between Point A and the line of action of the resultant force. Now apply varignon's theorem

R.d = 20 × 2 + 30 × 4 + 20 sin 30º × 7

68.05.d = 281.2

d = 4.132 m                                                                            .......ANS