Structural Theory

This page details the structural analysis methods and assumptions used in lightaero.

Finite Element Model (FEM)

lightaero implements a 1D finite-element model of the wing spar based on Euler-Bernoulli beam theory.

Euler-Bernoulli Beam

The model uses standard Hermite cubic shape functions for transverse displacement and slope.

Element Stiffness Matrix ($K_e$)

The $4 \times 4$ element stiffness matrix for a beam element with length $L$, Young's modulus $E$, and moment of inertia $I$ is:

\[ \mathbf{K}_e = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix} \]

Consistent Mass Matrix ($M_e$)

To account for inertial effects in dynamic analysis, a consistent mass matrix is derived using the same Hermite cubic shape functions as the stiffness matrix (Reddy, 1993):

\[ \mathbf{M}_e = \frac{\rho A L}{420} \begin{bmatrix} 156 & 22L & 54 & -13L \\ 22L & 4L^2 & 13L & -3L^2 \\ 54 & 13L & 156 & -22L \\ -13L & -3L^2 & -22L & 4L^2 \end{bmatrix} \]

where $\rho$ is the material density and $A$ is the cross-sectional area.

Implementation Note: Static Analysis Only

Although the theoretical mass matrix formulas are provided above, the current structural discipline implementation in lightaero is limited to static analysis. Dynamic effects such as flutter, gust response, and modal analysis are not yet supported.

Thin-Walled Section Properties

Wingbox section properties ($I_{xx}$, $I_{zz}$, $J$) are computed using numerical integration over a closed single-cell box outline (typically 15% to 65% chord).

Shear Center

The chordwise shear center position ($x_{sc}$) is determined via the Bredt-Batho method for a unit vertical shear.

Torsional Constant

The torsional constant ($J$) for the closed single-cell box is calculated as:

\[ J = \frac{4 A_e^2}{\oint \frac{1}{t} ds} \]

where $A_e$ is the enclosed area and $t$ is the skin thickness.

Stress Analysis

Bending Stress

Bending stress is calculated from the vertical ($M_x$) and in-plane ($M_z$) bending moments:

\[ \sigma_{bending} = \frac{|M_x| y}{I_{xx}} + \frac{|M_z| x}{I_{zz}} \]

Torsional Shear Stress

Torsional shear stress ($\tau$) is computed using the Bredt-Batho formula for closed sections:

\[ \tau = \frac{T}{2 A_e t} \]

where $T$ is the local torque.

Von Mises Stress

The combined stress state is evaluated using the Von Mises criterion: $$ \sigma_{vm} = \sqrt{\sigma_{bending}^2 + 3\tau^2} $$

Weight Estimation

Structural Mass

Wing structural mass is obtained by spanwise integration of the material density ($\rho$) over the cross-sectional area ($A_{cross}$), adjusted by a structural factor ($k_{struct}$):

\[ m_{wing} = 2 \int_0^{b/2} \rho A_{cross} k_{struct} dy \]

Operating Empty Weight (OEW)

OEW is estimated using the Roskam log-linear regression for transport aircraft (Roskam, 1985):

\[ \log_{10}(OEW) = -0.0833 + 0.9647 \log_{10}(MTOW) \]

References

Reddy, J. N. (1993). An Introduction to the Finite Element Method (2nd ed.). McGraw-Hill.
Roskam, J. (1985). Airplane Design Part V: Component Weight Estimation. DARcorporation.