Indeterminate Structures — Question 8

Question 8

Measured support deflections identify hidden stiffness differences

A weightless rigid beam rests on initially level springs at $x=-1,0,+1\,\mathrm m$. Measured downward deflections are $1,2,3\,\mathrm{mm}$ and upward reactions are $40,80,180\,\mathrm N$, in the same order. One downward point load acts on the beam.

1. Infer each spring stiffness and verify rigid-beam compatibility.

2. Infer the load magnitude and position.

Question 8 - Solution

Measured support deflections identify hidden stiffness differences

(a) Derivation and calculation

Infer each stiffness from its measured force and displacement: $$k_-=\frac{40}{.001}=40\ \mathrm{kN/m},\quad k_0=\frac{80}{.002}=40\ \mathrm{kN/m},$$ $$\boxed{k_+=\frac{180}{.003}=60\ \mathrm{kN/m}}.$$ The equal spacing requires $\delta_0=(\delta_-+\delta_+)/2=(1+3)/2=2$ mm, exactly as measured.

(b) Result and interpretation

The load magnitude is $$\boxed{W=40+80+180=300\ \mathrm N}.$$ Taking moments about the center, $$We=(-1)(40)+(1)(180)=140\ \mathrm{N\,m}.$$ Thus $$\boxed{e=140/300=7/15=.467\ \mathrm m\ \text{right}}.$$

Physical check

The positive load offset agrees with the larger right reaction, while the deflections remain collinear.