= Conditioning of two-span beam compliance inference
\b[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
$$
C_i=C_0+aL_i^3,\qquad
\widehat a=\frac{C_2-C_1}{L_2^3-L_1^3},\qquad
B=\frac1{48a},
$$
$C_0$ 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>.<Beam Displacements by David Roylance>[Introduction; Example 1, equation 7]{ref} The subtraction is sensitive when the spans are close, or when the beam-response difference is small compared with readout errors.
Let $D=L_2^3-L_1^3\ne0$ and temporarily treat the spans as exact. The sensitivity coefficients are $\partial\widehat a/\partial C_1=-1/D$ and $\partial\widehat a/\partial C_2=1/D$. Consequently,
$$
u^2(\widehat a)=\frac{u^2(C_1)+u^2(C_2)-2\operatorname{Cov}(C_1,C_2)}{D^2}.
$$
This is the standard <measurement uncertainty> propagation rule applied to a linear combination; the <covariance> term matters when the readings share errors.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§5.1 and Appendix A, equation A-3]{ref}
For an original numerical comparison, stipulate $B=10\,\mathrm{N\,m^2}$, so $a=1/480\,\mathrm{N^{-1}m^{-2}}$, and independent <standard uncertainties> $u(C_i)=10^{-6}\,\mathrm{m/N}$. Spans $0.200$ and $0.210\,\mathrm m$ give $D=0.001261\,\mathrm{m^3}$ and $u(\widehat a)=0.001122\,\mathrm{N^{-1}m^{-2}}$, about 54 percent of $a$. Increasing the second span to $0.300\,\mathrm m$ gives $D=0.019\,\mathrm{m^3}$ and $u(\widehat a)=0.00007443\,\mathrm{N^{-1}m^{-2}}$, about 3.57 percent. The wider spacing makes the same compliance readout precision much more informative.
Span uncertainty enters through additional sensitivity coefficients:
$$
\frac{\partial\widehat a}{\partial L_1}=\frac{3\widehat aL_1^2}{D},\qquad
\frac{\partial\widehat a}{\partial L_2}=-\frac{3\widehat aL_2^2}{D}.
$$
For uncorrelated span readings, their first-order contribution to $u^2(\widehat a)$ is $9\widehat a^2[L_1^4u^2(L_1)+L_2^4u^2(L_2)]/D^2$, 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 $B$.
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