Indeterminate Structures — Question 4

Question 4

A slack support engages only beyond a load threshold

Two vertical elastic ties connect the same load hook to a fixed support. Tie 1 has stiffness $k_1=1000\,\mathrm{N/m}$ and no slack. Tie 2 has stiffness $k_2=2000\,\mathrm{N/m}$ and slack $g=2\,\mathrm{mm}$. Their tensions are $T_1=k_1\delta$ and $T_2=k_2\max(\delta-g,0)$.

1. Find the downward load at which tie 2 begins carrying force.

2. For a load of $10\,\mathrm N$, find the hook displacement and both tensions.

Question 4 - Solution

A slack support engages only beyond a load threshold

(a) Derivation and calculation

The second tie engages when $\delta=g=.002$ m. Before engagement, $$P=k_1\delta,\qquad T_2=0.$$ Thus the threshold is $$\boxed{P_c=k_1g=1000(.002)=2.00\ \mathrm N}.$$

(b) Result and interpretation

Since $10>P_c$, both ties carry load: $$P=k_1\delta+k_2(\delta-g)\Rightarrow \delta=\frac{P+k_2g}{k_1+k_2}.$$ $$\boxed{\delta=\frac{10+2000(.002)}{3000}=4.67\ \mathrm{mm}},$$ $$\boxed{T_1=14/3=4.67\ \mathrm N},\qquad\boxed{T_2=16/3=5.33\ \mathrm N}.$$

Physical check

The tensions sum to 10 N, and the calculated displacement exceeds the 2 mm slack.