Strength • Section 9.2.2
Lug Net-Section Tension Efficiency ($K_{nu}$)
AFFDL-TR-69-42 Section 9.2.2 tension allowable, Figure 9-4 families, and multi-variable plasticity interpolation
1. Fundamentals of Net-Section Tensile Failure
Under axial tension, the material on either side of the pin hole resists hoop tension. The nominal cross-sectional area across the hole centerline is:
Classical elementary strength of materials assumes that failure occurs when the average stress across $A_{net}$ reaches $F_{tu}$. In reality, the hole introduces severe elastic stress concentrations ($K_t \approx 3.0$). In ductile aerospace alloys, local yielding allows plastic redistribution across the net ligament before rupture, resulting in an effective efficiency factor $K_{nu}$ that often differs significantly from 1.0 (ranging from $0.6$ up to $1.2+$).
2. USAF Mathematical Formulation (Equation 9-4)
Per USAF AFFDL-TR-69-42 Section 9.2.2, the ultimate net-section tension capacity ($P_{nu,L}$) is:
where:
- $W = h_0 + D + h_2$ is the true transverse width across the hole centerline (equal to $w$ for straight lugs).
- $D$ is the hole diameter and $t$ is the lug plate thickness.
- $F_{min} = \min\left(F_{tu},\; 1.304 \cdot F_{ty}\right)$ is the governing allowable material stress.
- $K_{nu}$ is the dimensionless net-tension efficiency factor.
3. The Governing Parameters of USAF Figure 9-4
Figure 9-4 in the Air Force manual is divided into three multi-curve chart families (Figures 9-4b, 9-4c, and 9-4d). The efficiency factor $K_{nu}$ is parameterized simultaneously across three physical variables:
| Parameter | Definition | Physical Role |
|---|---|---|
| $D/W$ | Hole diameter / Net transverse width | Horizontal axis of all Figure 9-4 charts ($0.0 \le D/W \le 0.8$). Controls ligament aspect ratio. |
| $\frac{F_{ty}}{F_{tu}}$ | Yield strength / Ultimate tensile strength | Differentiates chart families: Fig 9-4b ($0.85 \le \text{ratio} \le 1.0$), Fig 9-4c ($0.75 \le \text{ratio} \le 0.85$), and Fig 9-4d ($\text{ratio} \le 0.75$). |
| $\frac{F_{tu}}{E \cdot \varepsilon_u}$ | Tensile strength / (Modulus × Ultimate Elongation) | Strain parameter curve family. High elongation alloys have smaller values and achieve higher plastic stress redistribution ($K_{nu}$). |
4. Automated Interpolation Engine in LugCalc™
In manual calculations, engineers previously had to visually pick an approximate curve, read values off millimeter grid paper, and manually interpolate between separate charts.
The LugCalc™ calculation engine automates this multi-dimensional interpolation completely:
- Evaluates the material's yield-to-ultimate ratio $\frac{F_{ty}}{F_{tu}}$ and strain index $\frac{F_{tu}}{E \cdot \varepsilon_u}$.
- Identifies the bounding curve families across the relevant sub-figures of Figure 9-4.
- Performs cubic and bi-linear spline interpolation along the exact $D/W$ coordinate.
- Renders the interpolated curve in the Tension Efficiency interactive SVG plot with your live operating point highlighted.