Stress Analysis • Section 9.2
The K Factor Method: Automating Efficiency Factors
Semi-empirical plastic efficiency factors and automated chart interpolation in USAF AFFDL-TR-69-42
1. What is an Efficiency Factor ($K$)?
In structural mechanics, an efficiency factor ($K$) is a non-dimensional coefficient that scales the material ultimate tensile allowable ($F_{tu}$) or yield allowable ($F_{ty}$) to determine the true plastic collapse load of a geometric feature:
The factor $1.304$ arises from the standard aerospace plastic bending shape factor for circular cross-sections and ensures that excessive permanent plastic deformation does not occur prior to ultimate load.
2. The Three USAF K Efficiency Factors
Chapter 9 of the USAF Stress Analysis Manual defines three specialized efficiency factors corresponding to the primary loading directions and failure modes:
A. Bearing / Shear-Out Efficiency ($K_{bru}$) • Section 9.2.1
Under axial tension, load from the pin presses against the bore face while shearing material forward toward the free tip. The efficiency depends on whether the lug is structurally thick or thin:
- Thick Lugs ($D/t \le 5$): Evaluated using Figure 9-2. The curve plots $K_{bru}$ vs. the edge distance ratio $e/D$. For edge ratios below $e/D = 1.5$, the capacity is adjusted by the ratio $a/D$ where $a = e - D/2$: $$P_{bru,L} = K_{bru} \cdot F_{min} \cdot D \cdot t \cdot \left(\frac{a}{D}\right) \quad \text{for } e/D < 1.5$$
- Thin Lugs ($D/t > 5$): Evaluated using Figure 9-3. As thickness $t$ decreases relative to bore $D$, out-of-plane lateral buckling, lip curling, and severe stress gradients reduce bearing efficiency.
- Grain Orientation Cutoffs: For high-strength aluminum die forgings and extrusions, cross-grain tension limits bearing capacity. USAF Section 9.2.1 applies Curve A or Curve B cutoff limits and caps $K_{bru} \le 2.00$ to prevent brittle shear-lip spalling.
B. Net-Section Tension Efficiency ($K_{nu}$) • Section 9.2.2
Tension across the net section $(W - D)t$ does not achieve $100\%$ of the material tensile allowable because of the notch effect created by the pin bore. The efficiency $K_{nu}$ is plotted across Figure 9-4 parameterized by:
- Bore-to-width ratio: $D/W$
- Yield-to-ultimate strength ratio: $F_{ty} / F_{tu}$ (discrete chart families for $1.0$, $0.8$, and $0.6$)
- Ductility parameter: $\frac{F_{tu}}{E \cdot \varepsilon_u}$, representing the ratio of ultimate strength to total elastic-plastic strain energy.
Higher ductility ($\varepsilon_u$) enables localized plastic flow at the hole edge, significantly increasing $K_{nu}$ toward $1.0$.
C. Transverse Load Efficiency ($K_{tru}$ & $K_{try}$) • Section 9.2.3
When a lug is loaded perpendicular to its longitudinal centerline ($90^\circ$ transverse load), failure occurs through bending, hoop tension on the trailing side, and shear-out on the leading edge. The capacity is determined by Figure 9-8 as a function of the normalized effective edge distance $h_{avg}/D$:
- Ultimate transverse factor: $K_{tru}$
- Yield transverse factor: $K_{try}$
3. Automated Interpolation in LugCalc™
Historically, reading these efficiency factors required engineers to squint at 1960s printed chart graphs, manually interpolating between closely spaced logarithmic curves. This process was time-consuming and error-prone.
The LugCalc™ calculation engine automates this entire interpolation process seamlessly:
| USAF Figure | Design Parameter | Digital Table Points | Interpolation Algorithm |
|---|---|---|---|
| Fig. 9-2 ($K_{bru}$) | $e/D \in [0.5, 2.4]$ | 40 discrete ratio steps | Piecewise linear interpolation on $e/D$ |
| Fig. 9-3 ($K_{br}$ thin) | $e/D \in [0.5, 2.2]$, $D/t \in [5, 50]$ | Multi-curve 2D surface grid | Bilinear interpolation with grain cutoff cap |
| Fig. 9-4 ($K_{nu}$) | $D/W \in [0, 1]$, $F_{ty}/F_{tu}$, $F_{tu}/(E \varepsilon_u)$ | Family of 21 parametric curves | Non-linear tri-linear interpolation across curves |
| Fig. 9-8 ($K_{tru}$) | $h_{avg}/D \in [0.2, 2.2]$ | Digitized ultimate & yield points | Hermite curve spline interpolation |