Aero Structures 1: Worked Example Walkthroughs

Abstract

This article presents a structured walkthrough of core concepts in introductory aerospace structures, emphasizing the governing equations of linear elasticity, principal stress analysis, and thin-walled beam theory. Through worked examples, we illustrate how these foundational tools apply to aircraft structural components. The material is drawn from undergraduate aeronautical engineering coursework and is intended for students and practitioners seeking clarity on the mechanics underlying wing spars, fuselage sections, and other load-bearing members.

Background

Aerospace structures must carry complex, multi-directional loads while minimizing weight. This demands a rigorous analytical foundation. The discipline rests on three coupled equation systems that together form a complete framework for predicting how structures deform and fail under load ^{[governing-equations-of-linear-elasticity]}.

Equilibrium equations enforce force and moment balance throughout the structure. Kinematic equations ensure geometric compatibility—deformations must fit together without gaps or overlaps. Constitutive equations encode the material's mechanical response, typically through Hooke's Law for linear elastic materials. Solving these three systems simultaneously, subject to boundary conditions and applied loads, yields the complete stress, strain, and displacement fields.

In practice, aircraft designers apply these equations to idealized structural members: beams, plates, and shells. Two classes of beams are particularly important:

1. Open thin-walled beams (channels, I-sections) used in secondary structure
2. Multi-cell closed beams (box sections) used in primary load-carrying members like wing spars

Understanding the behavior of these members under bending, torsion, and shear is essential for safe, efficient design.

Key Results

Principal Stresses and Failure Prediction

Real structures experience complex, multi-directional loading. At any point in a loaded body, three mutually perpendicular directions exist—the principal axes—where the stress tensor simplifies to contain only normal stresses with zero shear ^{[principal-stresses-and-strains]}.

The principal stresses are ordered as $\sigma_1 \geq \sigma_2 \geq \sigma_3$. This transformation is powerful because materials typically fail when normal stresses exceed yield limits or when shear stresses trigger slip. By rotating to principal axes, engineers expose the most dangerous stress components directly.

For ductile materials, two widely used yield criteria relate principal stresses to the material's uniaxial yield strength $\sigma_y$:

Von Mises criterion: $\sqrt{\frac{1}{2}[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2]} = \sigma_y$

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

Both criteria convert a multi-axial stress state into an equivalent uniaxial stress that can be compared directly against material strength. For aircraft structures, this is essential: a wing spar under combined bending and torsion has a complicated stress state, but finding its principal stresses and applying a failure criterion reveals whether the design is safe.

Shear Center in Open Thin-Walled Beams

An open thin-walled beam (such as a channel or I-section) has a unique geometric property called the shear center ^{[shear-center-of-open-thin-walled-beams]}. When a transverse load passes through the shear center, it produces pure bending without torsion. Any load applied away from this point induces both bending and twisting.

In symmetric sections (equal-flange I-beams), the shear center coincides with the centroid. However, asymmetric or open sections have shear centers that differ from their centroids. This distinction is critical in aircraft design because unintended torsion from off-center loading can trigger flutter, accelerate fatigue, or cause structural failure.

Multi-Cell Beams and Torsional Efficiency

Aircraft wings typically employ multi-cell box beams as main spars ^{[thin-walled-multi-cell-beams]}. These structures comprise multiple closed cells (compartments) formed by interconnected thin walls. Under torsion or transverse shear loading, shear flow circulates within and between cells.

Multi-cell beams are structurally superior to open sections for resisting torsion because closed cells prevent cross-sectional warping and distribute shear stresses more uniformly. The beam's torsional rigidity and shear resistance depend on cell geometry, wall thickness, and material properties. This efficiency makes multi-cell designs ideal for weight-critical aerospace applications, though analysis is more complex than open or single-cell beams.

Worked Examples

Example 1: Principal Stress Calculation

Problem: A point in an aircraft wing spar experiences the following stress state (in MPa): $\sigma_x = 100, \quad \sigma_y = 50, \quad \tau_{xy} = 30$

Find the principal stresses and determine whether the material (aluminum alloy with $\sigma_y = 250$ MPa) will yield using the Von Mises criterion.

Solution:

For a 2D stress state, the principal stresses are found from: $\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

Substituting: $\sigma_{1,2} = \frac{100 + 50}{2} \pm \sqrt{\left(\frac{100 - 50}{2}\right)^2 + 30^2}$ $= 75 \pm \sqrt{625 + 900} = 75 \pm \sqrt{1525} = 75 \pm 39.05$

Thus: $\sigma_1 = 114.05 \text{ MPa}, \quad \sigma_2 = 35.95 \text{ MPa}, \quad \sigma_3 = 0$

The Von Mises equivalent stress is: $\sigma_{\text{eq}} = \sqrt{\frac{1}{2}[(114.05-35.95)^2 + (35.95-0)^2 + (0-114.05)^2]}$ $= \sqrt{\frac{1}{2}[6084.2 + 1292.4 + 13007.4]} = \sqrt{10192} = 100.95 \text{ MPa}$

Since $\sigma_{\text{eq}} = 100.95$ MPa $< \sigma_y = 250$ MPa, the material does not yield. The design is safe at this point.

Example 2: Shear Center Location (Qualitative)

Problem: A channel section (C-beam) is used as a secondary structural member. Explain why loads must be applied through the shear center and what happens if they are not.

Solution:

The channel section is asymmetric: the web is centered, but the flanges are on one side only. The centroid lies on the web centerline, but the shear center is offset toward the open side ^{[shear-center-of-open-thin-walled-beams]}.

If a vertical load is applied at the centroid (not at the shear center), it creates:

• A bending moment about the neutral axis (expected)
• A torsional moment about the longitudinal axis (undesired)

The torsional moment arises because the load's line of action does not pass through the shear center. This twisting can cause:

• Increased stress in the flanges and web
• Coupling between bending and torsion, complicating analysis
• Flutter risk if the structure is part of a control surface
• Accelerated fatigue failure

By identifying the shear center location (typically via integration of shear stress distribution) and ensuring loads pass through it, engineers guarantee that wing and fuselage loads produce primarily bending rather than twisting, maintaining structural integrity and flight safety.

Example 3: Multi-Cell Beam Torsion (Conceptual)

Problem: A wing box (multi-cell beam) with two cells is subjected to a torque $T$. Explain how shear flow distributes and why this design is superior to an open section.

Solution:

In a multi-cell beam under torsion, shear flow $q$ (shear stress times thickness) circulates around each cell. The flow is governed by equilibrium and compatibility conditions. For a symmetric two-cell box, shear flow is typically distributed such that:

• Flow in the outer walls is higher than in the inner wall
• The inner wall carries shear flow in both directions (from both cells)
• Total torque is resisted by the combined effect of all cells

This distributed resistance is far more efficient than an open section because:

1. No warping: Closed cells prevent the cross-section from warping out of plane, maintaining structural stiffness.
2. Uniform stress distribution: Shear stresses are distributed over multiple load paths rather than concentrated in a single web.
3. Higher torsional rigidity: The torsional constant $GJ$ is much larger for a multi-cell box than an equivalent open section, reducing twist per unit torque.

For weight-critical aircraft, this efficiency translates to thinner walls and lower structural mass while maintaining the same torsional strength—a critical advantage in aerospace design.

References

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

AI Disclosure

This article was drafted with the assistance of an AI language model based on personal class notes and course materials from Aero Structures 1. The mathematical derivations, worked examples, and conceptual explanations are original compositions synthesizing course content. All factual claims are attributed to source notes via citation. The article has been reviewed for technical accuracy and clarity by the author.