MechanicalStaticseasyFE ExamCoursework

Overhanging Beam Support Reactions

A beam $ABC$ rests on a pin support at $A$ ( $x = 0$ ) and a roller at $B$ ( $x = 6\ \text{m}$ ). It overhangs to point $C$ at $x = 8\ \text{m}$ .

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Find the vertical reaction at $A$ .

Given

Hint 1

Convert the distributed load into a single resultant force before writing any equilibrium equation.

Hint 2

That resultant acts at the centroid of the loaded length — which is not the middle of the beam here.

Hint 3

Take moments about B: it removes R_B from the equation and leaves R_A alone.

Worked solution — try the problem first

Replace the UDL by its resultant. Total $= 10 \times 6 = 60\ \text{kN}$ , acting at the centroid of the load, $x = 3\ \text{m}$ .

Moments about A (this eliminates $R_A$ , leaving one unknown):

$\sum M_A = 0: \quad R_B(6) - 60(3) - 15(8) = 0$ $6R_B = 180 + 120 = 300$ $R_B = 50\ \text{kN}$

Vertical equilibrium:

$\sum F_y = 0: \quad R_A + R_B = 60 + 15 = 75$ $R_A = 75 - 50 = 25\ \text{kN}$

Check — moments about B should also vanish:

$-R_A(6) + 60(3) - 15(2) = -150 + 180 - 30 = 0 \ \checkmark$

$\boxed{R_A = 25.0\ \text{kN upward}}$

Concepts:Static equilibriumDistributed loadsSupport reactions

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