If we do not know the direction of the force, we must assume a direction. If the assumption is correct, the answer will have a positive sign; if the assumption is incorrect, the answer will have a negative sign. Although in this case it is possible to g uess the sign of the reactions, it is not always so. Choosing "all posative"- up or to the right- minimizes possible confusion over the meaning of resulting signs.
Use the three equations of equilibrium to solve for the reactions.
Sum Fx = 0 = 4/5(40k) + AxThe first and third equations can be solved immediately.
Sum Fy = 0 = -3/5(40k) + Ay + B
Sum MA = 0 = 3/5(40k)(6ft) - B(20ft)
Ax = -32kNow use B = 7.2k in the second equation to solve for Ay.
B = 7.2k
Sum Fy = 0 = -24k + 7.2k + AyA can be found using the Pythagorean Theorem.
Ay = 16.8k
A = SQRT (-32k^2 + 16.8k^2)The direction is up and to the right. The angle above the x axis equals
A = 36k
ø = arctan (-16.8 / 32) = 27.7 degrees
In summary, the reactions of the beam are as follows:
at A a force of 36 kips acting up and to the right at an angle of 27.7 degrees and
at B a force of 7.2k acting up.