Indeterminate Structures — Question 7

Question 7

Bonded members can carry stress with zero external axial load

Two equal-length parallel rods are joined by rigid end plates free of external axial load. Rod 1 would freely acquire strain $\epsilon_1^*=.001$; rod 2 has $\epsilon_2^*=0$. Their axial rigidities are $E_1A_1=2\times10^6\,\mathrm N$ and $E_2A_2=10^6\,\mathrm N$. Both share strain $\epsilon$ and obey $N_i=E_iA_i(\epsilon-\epsilon_i^*)$.

1. Find the common strain and both internal forces.

2. Explain the force signs and check external equilibrium.

Question 7 - Solution

Bonded members can carry stress with zero external axial load

(a) Derivation and calculation

External axial equilibrium requires $$N_1+N_2=0,$$ $$(2\times10^6)(\epsilon-.001)+(10^6)\epsilon=0.$$ Hence $$\boxed{\epsilon=\frac{2000}{3\times10^6}=6.67\times10^{-4}}.$$

(b) Result and interpretation

The resulting internal forces are $$\boxed{N_1=(2\times10^6)(.0006667-.001)\simeq-667\ \mathrm N},$$ $$\boxed{N_2=(10^6)(.0006667)\simeq+667\ \mathrm N}.$$ Rod 1 is compressed relative to its preferred expanded length; rod 2 is stretched. Their exact forces are $\mp2000/3$ N, so $N_1+N_2=0$.

Physical check

The common strain lies between the two free strains, weighted toward the axially stiffer rod.