Beam compliance from two support spans ($a=(C_2-C_1)/(L_2^3-L_1^3)$)

Beam compliance from two support spans separates a cubic span-dependent beam response from a span-independent contribution to the measured mechanical compliance. Assume the same uniform specimen, unchanged machine contribution $C_0$ and a small-deflection central-load model:Here $B$ is beam flexural rigidity and $L_1\ne L_2$. The midspan compliance of a simply supported beam contains the basic numerical fit, while the beam source derives the cubic dependence.^[ref]

An original uncertainty example shows why distinct spans can still be too close. Let the actual model have $B=10\mathrm{N}{\mathrm{m}}^2$, hence $a=1/480{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$. Assign independent compliance standard uncertainties $u(C_1)=u(C_2)=1\mu m/N$, and neglect span uncertainty for this example. Since the slope estimate is linear in the readings,Spans $0.200$ and $0.210\mathrm{m}$ give $u(a)\approx 0.00112{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, about54percent of the actual coefficient. Using $0.200$ and $0.300\mathrm{m}$ instead gives about3.57percent with the same readout uncertainties. The close-span subtraction strongly amplifies those errors. At the larger relative uncertainty, simply treating $B=1/(48a)$ as approximately linear is unreliable.

This conditioning of two-span beam compliance inference concerns measurement precision, separately from model adequacy. If shear adds a term proportional to $L$, then $C(L)=C_0+a{L}^3+bL$ has three unknown coefficients that two readings cannot identify. Changing contact or fixture compliance can likewise violate the assumed constant $C_0$. Further spans and an independent deflection readout test those assumptions; two fitted points alone do not.

Conditioning of two-span beam compliance inference describes how perturbations in two measured compliances and support spans affect the inferred bending coefficient. In the model$C_0$ represents an unchanged series contribution and $B$ is beam flexural rigidity. The cubic beam term follows from the midspan compliance of a simply supported beam.^[ref] The subtraction is sensitive when the spans are close, or when the beam-response difference is small compared with readout errors.

Let $D=L_2^3-L_1^3\ne 0$ and temporarily treat the spans as exact. The sensitivity coefficients are $\partial \hat{a}/\partial C_1=-1/D$ and $\partial \hat{a}/\partial C_2=1/D$. Consequently,This is the standard measurement uncertainty propagation rule applied to a linear combination; the covariance term matters when the readings share errors.^[ref]

For an original numerical comparison, stipulate $B=10\mathrm{N}{\mathrm{m}}^2$, so $a=1/480{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, and independent standard uncertainties $u(C_i)=10^{-6}m/N$. Spans $0.200$ and $0.210\mathrm{m}$ give $D=0.001261{\mathrm{m}}^3$ and $u(\hat{a})=0.001122{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, about 54 percent of $a$. Increasing the second span to $0.300\mathrm{m}$ gives $D=0.019{\mathrm{m}}^3$ and $u(\hat{a})=0.00007443{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, about 3.57 percent. The wider spacing makes the same compliance readout precision much more informative.

Span uncertainty enters through additional sensitivity coefficients:For uncorrelated span readings, their first-order contribution to $u^2(\hat{a})$ is $9{\hat{a}}^2[L_1^4{u}^2(L_1)+L_2^4{u}^2(L_2)]/D^2$, before any compliance/span correlations are included. An identical additive readout offset cancels from $C_2-C_1$, whereas a changing offset does not.

A practical span comparison holds the specimen, contacts, loading position and displacement channel as consistent as possible. Close spans amplify measurement noise; simply choosing distant spans can instead expose a failure of the assumed model. Shear produces a term proportional to $L$, and changing contacts can change $C_0$. A third span tests those assumptions. Finally, inversion to $B$ is strongly nonlinear near $a=0$: the 54-percent case cannot be adequately summarized by a small symmetric relative uncertainty on $B$.