A horizontal bar fixed at $x=0$ is 1 m long with area 100 mm². Its modulus varies as $E(x)=100e^{-x}$ GPa, where x is measured in metres. A 10 kN axial tensile load acts at its free end $x=1$. Find free-end axial displacement in mm.
Axial force is constant along the bar. Extension is $\delta=\int_0^1P/[AE(x)]dx$. In SI units, $P=10^4$ N, $A=10^{-4}$ m² and $E=10^{11}e^{-x}$ Pa. Hence $\delta=0.001\int_0^1e^xdx=0.001(e-1)$ m = 1.71828 mm. The decreasing modulus makes this larger than the 1 mm extension for constant 100 GPa.
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