Indeterminate Structures — Question 10

Question 10

Balanced reaction readings can still falsify a support model

A rigid weightless beam has three initially level, equally spaced vertical supports. Each is claimed to have stiffness $k$. With a centered $100\,\mathrm N$ load, measured reactions from left to right are $(20,60,20)\,\mathrm N$. Assume small elastic deformation and no initial fit errors.

1. Check equilibrium, then test the equal-stiffness model using compatibility.

2. If only the center stiffness is unknown, infer it in terms of $k$.

Question 10 - Solution

Balanced reaction readings can still falsify a support model

(a) Derivation and calculation

The measured reactions satisfy equilibrium: $$20+60+20=100\ \mathrm N,\qquad a(20)-a(20)=0.$$ Equal stiffness would imply $$\delta_-=20/k,\qquad\delta_0=60/k,\qquad\delta_+=20/k.$$ But rigid-beam compatibility requires $\delta_0=(\delta_-+\delta_+)/2=20/k$, contradicting $60/k$.

(b) Result and interpretation

Keep the outer stiffnesses $k$ and denote the center stiffness by $k_0$. The equal outer deflections make the beam translate without rotation: $$\delta_0=20/k.$$ Thus $$60=k_0\frac{20}{k}\quad\Longrightarrow\quad\boxed{k_0=3k}.$$ This stiffness ratio reproduces all three measured forces with a common deflection.

Physical check

Balanced force data alone cannot validate an assumed elastic-support model.