Aero Structures 1: Reference Tables and Quick Lookups

Abstract

This article consolidates core concepts from Aero Structures 1 into a reference guide for structural analysis of aircraft components. It covers the governing equations of linear elasticity, principal stress identification, yield failure criteria, and specialized topics in thin-walled beam theory. The material is organized for quick lookup and practical application in design and analysis workflows.

Background

Aircraft structural analysis rests on three pillars: equilibrium, kinematics, and material constitutive behavior ^{[governing-equations-of-linear-elasticity]}. Engineers must predict internal stress and strain states under flight loads, identify critical failure modes, and ensure safety margins. This requires both theoretical grounding and practical reference tools.

The notes below synthesize lecture material into lookup tables and concise definitions suitable for design calculations, failure assessments, and component analysis.

Key Results

The Three Governing Equations

Linear elastic structural analysis is governed by three coupled systems ^{[governing-equations-of-linear-elasticity]}:

Equation Set	Role	Purpose
Equilibrium	Force and moment balance	Ensures internal stresses support applied loads
Kinematic	Strain-displacement relations	Ensures deformations are geometrically compatible
Constitutive	Stress-strain relations (Hooke's Law)	Encodes material mechanical response

Together, these form a complete boundary value problem that yields stresses, strains, and displacements throughout a structure given loads and boundary conditions.

Principal Stresses and Strains

At any point in a loaded structure, there exist three orthogonal directions—the principal axes—along which shear stresses vanish and only normal stresses remain ^{[principal-stresses-and-strains]}. These principal stresses are ordered as:

$\sigma_1 \geq \sigma_2 \geq \sigma_3$

Similarly, principal strains $\varepsilon_1, \varepsilon_2, \varepsilon_3$ align with the principal stress directions. Identifying principal values is essential for failure analysis because materials typically fail along planes of maximum normal or shear stress. In aircraft wing analysis, principal stresses at stress concentrations (e.g., rivet holes, spar junctions) directly reveal the most damaging stress components.

Yield Failure Criteria

Yield criteria extend uniaxial material strength to multi-axial stress states by defining a limiting surface in stress space ^{[yield-failure-criteria]}. When the stress state reaches this surface, permanent deformation begins.

Von Mises Criterion

Yield occurs when the equivalent stress reaches the material's yield strength:

$\sqrt{\frac{1}{2}\left[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\right]} = \sigma_y$

This criterion is widely used for ductile metals (aluminum alloys, steel) in aircraft structures.

Tresca Criterion

Yield occurs when the maximum shear stress reaches half the yield strength:

$\max(|\sigma_1-\sigma_2|, |\sigma_2-\sigma_3|, |\sigma_3-\sigma_1|) = \sigma_y$

The Tresca criterion is more conservative than Von Mises and is sometimes preferred for safety-critical applications.

Both criteria allow engineers to assess failure likelihood at critical locations by computing an equivalent stress and comparing it against known material strength.

Shear Center of Open Thin-Walled Beams

For open thin-walled beams (channels, I-beams, hat sections), the shear center is the unique point through which transverse loads must pass to produce pure bending without torsion ^{[shear-center-of-open-thin-walled-beams]}.

Beam Type	Shear Center Location
Symmetric I-beam	Coincides with centroid
Unsymmetric or open section	Offset from centroid

When a load is applied away from the shear center, it induces both bending and twisting. In aircraft wings, loads applied off-center cause unwanted torsion, leading to flutter, fatigue, or structural failure. Proper identification of the shear center ensures that wing loads produce primarily bending rather than twisting.

Thin-Walled Multi-Cell Beams

Closed-section beams with multiple internal cells (box beams, wing boxes) are superior to open sections for resisting torsion and shear loads ^{[thin-walled-multi-cell-beams]}.

Key advantages:

• Closed cells prevent warping and distribute shear stress efficiently
• Torsional rigidity is much higher than open sections of equivalent weight
• Shear flow circulates around and through each cell under loading

In aircraft wings, the main spar is typically a multi-cell box beam carrying bending, torsion, and shear simultaneously. Analysis requires determining shear flow distribution across cells and calculating resulting stresses and deflections—more complex than single-cell or open beams, but necessary for accurate design.

Worked Examples

Example 1: Von Mises Failure Check

Given: A critical point in an aircraft skin experiences principal stresses $\sigma_1 = 250$ MPa, $\sigma_2 = 100$ MPa, $\sigma_3 = 0$ MPa. The aluminum alloy has yield strength $\sigma_y = 350$ MPa.

Find: Does the material yield?

Solution:

$\sigma_{\text{eq}} = \sqrt{\frac{1}{2}\left[(250-100)^2 + (100-0)^2 + (0-250)^2\right]}$

$= \sqrt{\frac{1}{2}\left[22500 + 10000 + 62500\right]} = \sqrt{47500} \approx 218 \text{ MPa}$

Since $\sigma_{\text{eq}} = 218$ MPa $< \sigma_y = 350$ MPa, the material does not yield. A safety factor of approximately $350/218 \approx 1.6$ exists.

Example 2: Shear Center Identification

Given: An open-section channel beam with flanges of unequal thickness. A vertical load $P$ is applied at the centroid.

Expected result: The load is not at the shear center, so the beam experiences both bending and torsion. To eliminate torsion, the load must be shifted horizontally to the shear center location, which can be computed from the section geometry and wall thicknesses.

References

• ^{[governing-equations-of-linear-elasticity]}
• ^{[principal-stresses-and-strains]}
• ^{[yield-failure-criteria]}
• ^{[shear-center-of-open-thin-walled-beams]}
• ^{[thin-walled-multi-cell-beams]}

AI Disclosure

This article was drafted with AI assistance from personal class notes (Zettelkasten). All factual and mathematical claims are cited to source notes. The article paraphrases rather than copies note text, and no results or claims are invented beyond the source material. The worked examples are original applications of the cited concepts.