= Midspan compliance of a simply supported beam
{title2=$C=L^3/(48EI)$}
The \b[midspan compliance of a simply supported beam] is the incremental central deflection per incremental central load, with the support geometry fixed. For a uniform, small-deflection <Euler-Bernoulli beam equation> model without axial preload,
$$
C_b=\frac{dw(L/2)}{dP}=\frac{L^3}{48B},\qquad B=EI.
$$
Its units are length per force. The <simply supported beam> contains the derivation; the experimental use is to infer <beam flexural rigidity> or, with known geometry, <Young's modulus>.<Beam Displacements by David Roylance>[Introduction and Example1, equation7]{ref}
An original two-span example separates beam compliance from an assumed constant machine contribution. Suppose the same uniform specimen is tested over spans $0.20$ and $0.30\,\mathrm m$. A machine-travel readout gives $0.060$ and $0.155\,\mathrm{mm/N}$. Assume
$$
C_m(L)=C_0+kL^3,\qquad k=\frac1{48B},
$$
where $C_0$ is unchanged between the tests. Subtracting the readings gives
$$
k=\frac{(0.155-0.060)\times10^{-3}}{0.30^3-0.20^3}
=0.005\,\mathrm{N^{-1}m^{-2}}.
$$
Hence $B=1/(48k)\approx4.167\,\mathrm{N\,m^2}$ and $C_0=0.020\,\mathrm{mm/N}$. The inferred specimen compliances are $0.040$ and $0.135\,\mathrm{mm/N}$, with ratio $3.375=(0.30/0.20)^3$. The raw readings have a different ratio because they include $C_0$.
This <beam compliance from two support spans> relies on the unchanged-series-compliance assumption. Changing supports, contact indentation or shear contributions can invalidate it; two points fit its two coefficients without independently validating the model. A third span can test the cubic dependence. An incremental slope also differs from simply dividing a total displacement, including an initial offset, by the current force.
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