Some Examples of Static Equilibrium — Question 10

Question 10

A three-legged support tests the full horizontal gravity projection

A rigid platform of weight $600\,\mathrm N$ stands on upward-only legs at $A=(0,0)$, $B=(4,0)$ and $C=(0,3)\,\mathrm m$ in plan view. Its gravity projection is initially $(1,1)\,\mathrm m$.

1. Find all three vertical reactions.

2. If the gravity projection moves to $(3,1)$ while weight stays fixed, can all three legs remain in contact?

Question 10 - Solution

A three-legged support tests the full horizontal gravity projection

(a) Derivation and calculation

Vertical force and two independent moment equations are $$R_A+R_B+R_C=600,\qquad4R_B=600x_G,\qquad3R_C=600y_G.$$ For $(x_G,y_G)=(1,1)$ m, $$\boxed{R_B=150\ \mathrm N},\quad\boxed{R_C=200\ \mathrm N},\quad\boxed{R_A=250\ \mathrm N}.$$

(b) Result and interpretation

For $(3,1)$ m, the formal solution is $$R_B=450,\quad R_C=200,\quad R_A=-50\quad(\mathrm N).$$ The required negative contact force is impossible. Equivalently, the support triangle requires $$x_G\ge0,\quad y_G\ge0,\quad\frac{x_G}{4}+\frac{y_G}{3}\le1.$$ Here $3/4+1/3=13/12>1$, so the platform tips about edge $BC$.

Physical check

The first center lies inside the support polygon; the second lies beyond its opposite edge.