Indeterminate Structures — Question 2

Question 2

An eccentric load changes which elastic supports remain active

Three identical compression-only springs support a rigid weightless beam at $-a,0,a$, with level unloaded tops. A downward load $W$ is at $x=e\ge0$. In the small-displacement model each contacting spring obeys $R=k\delta$.

1. Find the reactions assuming all three contacts remain active, and the allowed range of $e$.

2. For $e=.8a$, find the actual reactions and check separation of the left spring.

Question 2 - Solution

An eccentric load changes which elastic supports remain active

(a) Derivation and calculation

For all contacts active, set $\delta(x)=d+\phi x$. Balance gives $$3kd=W,\qquad 2ka^2\phi=We.$$ Consequently, $$\boxed{R_0=W/3},\qquad\boxed{R_\pm=W/3\pm We/(2a)}.$$ Compression-only contact requires $R_-\ge0$, hence $\boxed{0\le e\le2a/3}$ (strictly positive forces below the endpoint).

(b) Result and interpretation

At $e=.8a$, the left contact is inactive. Rebalance the two remaining supports: $$R_0+R_+=W,\qquad aR_+=.8aW.$$ Thus $$\boxed{R_-=0,\quad R_0=.2W,\quad R_+=.8W}.$$ Rigid-beam extrapolation gives $\delta_-=2\delta_0-\delta_+=-.4W/k<0$, confirming a gap at the left spring.

Physical check

A negative all-contact reaction must be replaced by a new contact regime, not merely set to zero without rebalancing.