Geometry • Section 9.2
Effective Edge Lengths ($h_0 \dots h_4$) & Harmonic Mean ($h_{avg}$)
Radial edge distance definitions and the USAF harmonic average formulation for transverse capacity
1. What are Effective Edge Lengths?
When a pinned lug is subjected to a transverse load (acting perpendicular to the lug's longitudinal axis), failure does not occur across a single flat plane. Instead, the pin tends to rotate and push through the lug wall along curved shear-out trajectories.
To evaluate this complex stress field analytically, USAF AFFDL-TR-69-42 (Section 9.2) defines five critical radial edge dimensions, designated $h_0, h_1, h_2, h_3,$ and $h_4$, measured from the hole boundary to the external perimeter of the lug:
| Segment | Orientation / Angle | Description & Role |
|---|---|---|
| $h_0$ | $+90^\circ$ (Upper side) | Material thickness on the upper side perpendicular to the centerline (away from transverse load). Controls upper width. |
| $h_1$ | $-45^\circ$ (Forward lower) | Material distance directed at $45^\circ$ forward toward the transverse load side. Has the largest statistical weighting ($3\times$) in resistance. |
| $h_2$ | $-90^\circ$ (Direct lower) | Direct transverse edge distance in line with the transverse force vector $P_{tr}$. |
| $h_3$ | $-180^\circ$ (Trailing lower) | Rearward/trailing material distance behind the pin hole resisting rotation and rear blow-out. |
| $h_4$ | $+45^\circ$ (Forward upper) | Radial distance directed at $45^\circ$ forward on the upper side opposite the transverse load. |
2. The USAF Harmonic Mean Formulation ($h_{avg}$)
Instead of using a simple arithmetic mean, the Air Force Stress Analysis Manual establishes a weighted harmonic mean to determine the effective transverse dimension $h_{avg}$:
Why a weighted harmonic mean?
- Defect Sensitivity: The harmonic mean is governed by the smallest values in the set. If any ligament ($h_1, h_2, h_3,$ or $h_4$) is disproportionately small (e.g., due to an aggressive side taper or offset hole), $h_{avg}$ drops sharply, penalizing allowable strength accordingly.
- Shear-Out Path Weighting: Experimental photoelastic tests showed that approximately 50% of the transverse shear resistance is concentrated along the $45^\circ$ forward diagonal ($h_1$). Hence, $h_1$ receives a weight of 3 out of 6 in the denominator.
3. How $h_{avg}$ Governs Transverse Load Capacity ($P_{tru}$)
Once $h_{avg}$ is computed by the geometric solver, it is non-dimensionalized by dividing by the pin bore diameter $D$:
This ratio is then looked up on USAF Figure 9-8 to extract the ultimate transverse efficiency factor $K_{tru}$ and yield transverse efficiency factor $K_{try}$. The ultimate transverse load capacity of the lug is directly proportional to $h_{avg}$: