The Requirements of Equilibrium — Question 10

Question 10

An equilibrium solution can remain nonunique after every balance equation is used

A weightless horizontal beam is held by two ideal planar pins at $x=0$ and $x=L$. A vertical load $W$ acts at the midpoint. Each pin can provide horizontal and vertical forces but no couple.

1. Find the vertical reactions and the relation between horizontal reactions.

2. Can rigid-body equilibrium determine each horizontal reaction separately?

Question 10 - Solution

An equilibrium solution can remain nonunique after every balance equation is used

(a) Derivation and calculation

About the left pin, $$B_yL-WL/2=0\quad\Longrightarrow\quad B_y=W/2.$$ Force balance then gives $$\boxed{A_y=B_y=W/2},\qquad\boxed{A_x+B_x=0}.$$

(b) Result and interpretation

All horizontal reaction pairs $$\boxed{A_x=H,\qquad B_x=-H}$$ satisfy the balance equations because both horizontal forces act along the beam line. The arbitrary parameter $H$ requires deformation, fit, or prestress information. There are four reaction components but only three independent planar equilibrium equations.

Physical check

Equal opposite collinear horizontal forces contribute neither a resultant nor a couple.