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C. Dumont reported a 1939 experimental study of stress around a circular hole in an aluminum-alloy plate under out-of-plane bending. The primary NACA record retains that exact initial-only name and lists an Aluminum Company of America affiliation.^[ref]

Dumont measured plate deflection and strain for two loads. The largest hole-edge stresses were tangential at the ends of the transverse diameter. The reported maximum was 1.59 times a computed net-section stress and 1.85 times the computed stress for an otherwise solid plate subjected to the same bending forces.^[ref]

Those ratios demonstrate why a stress concentration factor requires a stated denominator. For an original numerical illustration, stipulate a local peak $37\mathrm{MPa}$. Reference stresses $37/1.85=20.0\mathrm{MPa}$ and $37/1.59\approx 23.27\mathrm{MPa}$ then reproduce the two ratios for the same peak. These invented stresses illustrate normalization; they are not recovered loads or stresses from Dumont's experiment.

The bending result is also distinct from the factor three in the in-plane tensile Kirsch equations. Geometry, stress components and boundary loading differ. Dumont's comparison of strain and deflection provided complementary local and overall observations rather than assuming a universal circular-hole amplification.

For a linear-elastic model, doubling the load at fixed geometry doubles both strain and deflection. Taking measurements at more than one load can therefore test proportionality. Agreement at two loads alone would not establish fatigue life or the onset of yielding. This explains the value and the limits of the experiment without expanding the initial C. or assigning an unverified biography.

Rotating shaft bending fatigue occurs when a shaft rotates while carrying a bending moment whose direction is approximately fixed in space. A material point on the outer surface moves alternately through the tensile and compressive sides of the bent section. A constant external bending load can therefore produce cyclic local stress.

For an ideal circular shaft in elastic pure bending, with diameter $d$, second moment of area $I=\pi d^4/64$ and steady moment magnitude $M$, the nominal surface stress amplitude isA rotating surface point has $\sigma (t)={\sigma }_a\cos (\omega t+\varphi )$ in this model. The mean is zero, ${\sigma }_{\max }={\sigma }_a$, ${\sigma }_{\min }=-{\sigma }_a$ and stress ratio $R={\sigma }_{\min }/{\sigma }_{\max }=-1$. One shaft revolution produces one full stress cycle at that point. A bending moment rotating with the shaft would require a different analysis; shaft rotation alone does not guarantee this cycle.

For an original example, $M=20\mathrm{N}\mathrm{m}$ and $d=20\mathrm{mm}$ give ${\sigma }_a\approx 25.5\mathrm{MPa}$. At $1800$ revolutions per minute, one hour supplies $108000$ cycles. These values specify a nominal loading example, not a predicted life or a safe design stress. Stress concentration, surface finish, material condition and other loads affect the local response and mechanical fatigue behavior.

A motivating apparatus is four-point rotating-bending testing. A NIST/NASA benchmark used machined and polished titanium-alloy specimens, a $100\mathrm{Hz}$ test frequency and fully reversed loading with $R=-1$. Comparing differently treated specimens under a controlled arrangement helps connect material state to fatigue response.^[ref] A lifetime claim needs appropriate experimental fatigue data and treatment of scatter; the elastic stress formula does not supply one.

A stress concentration is a localized increase of stress caused by geometry or another disturbance of the load path. Holes, notches and abrupt section changes can make the largest local normal stress much greater than a stress calculated from the total force divided by a reference cross-sectional area. A stress concentration factor $K_t={\sigma }_{\max }/{\sigma }_{\mathrm{ref}}$ is meaningful only after both the stress component and its reference have been specified.

The Kirsch equations give an instructive exact result for a circular hole of radius $a$ in an infinite, isotropic linear elastic elastic plate under remote tensile stress ${\sigma }_{\infty }$ in one direction, using a plane stress model. Measure the polar angle $\theta$ from the loading direction. At the hole boundary,At $\theta =\pi /2,3\pi /2$, the tangential normal stress reaches $3{\sigma }_{\infty }$, giving $K_t=3$ relative to remote stress. At $\theta =0,\pi$, it is $-{\sigma }_{\infty }$. A traction-free hole therefore does not require every component of its stress tensor to vanish: it requires zero traction on the radial boundary.^[ref]

For a finite plate, distinguish gross and net reference areas. A plate of width $W$, thickness $t$ and central hole diameter $d$ transmits force $P$ through gross area $A_g=tW$ and minimum net area $A_n=t(W-d)$. Defining ${\sigma }_g=P/A_g$ and ${\sigma }_n=P/A_n$ gives the exact conversionThis is a change of normalization, not a solution for the finite plate's stress field. For an original numerical example, take $W=40\mathrm{mm}$, $t=2\mathrm{mm}$, $d=10\mathrm{mm}$ and $P=800\mathrm{N}$. The reference stresses are $10$ and $13.333\mathrm{MPa}$. If the local maximum is stipulated as $32\mathrm{MPa}$, the same physical maximum gives $K_{tg}=3.2$ and $K_{tn}=2.4$. Neither number changes the local stress by choosing a different denominator.

The distinction appears in real historical measurements. In 1939, C. Dumont loaded a perforated aluminum-alloy plate in out-of-plane bending and measured strain and deflection at two loads. The reported peak tangential stress was 1.59 times the computed net-section stress, but 1.85 times the computed stress for an unperforated plate under the same bending forces. These are bending results for that geometry, rather than the infinite-plate tensile factor three.^[ref]

Modern demonstrations can combine a metal plate instrumented with strain gauges and a transparent polymer viewed in a polariscope. The latter uses stress-induced birefringence; its fringes respond to differences of principal stresses. The MIT laboratory compares local and remote strain measurements with qualitative photoelasticity observations around holes and a slot.^[ref]

The elastic stress concentration factor is dimensionless. It differs from the stress intensity factor used to describe a sharp crack's singular field. Local yielding or a crack changes the mechanical problem; multiplying a remote stress by the elastic $K_t$ does not, by itself, establish a failure load.