Concepts & Filtering Criteria

Project Overview

Calculations of moment of inertia $I$ and torsion constant $J$ will occur for each concept. Once finished, the following considerations will be evaluated (relative to the circular tube baseline) in a conclusion. Emphasis is placed on how each concept influences bending and torsional behavior, structural efficiency, and practical implementation:

  1. The extent to which it increases second moment of inertia $I$, which governs tip deflection.
  2. The extent to which it increases polar moment of inertia $J$, which governs angle of twist.
  3. Whether compliance is introduced, and where structural flexibility or failure modes originate.
  4. The fabrication complexity and cost implied by the design.
    1. Whether material is distributed farther from the boom axis without a drastic increase in mass.

In tandem, these considerations identify which structural concept most effectively increases stiffness per unit mass while remaining practical to fabricate.

This section outlines the rigorous evaluation, mathematical derivation, and analytical filtration of candidate structural geometries for the 5-foot camera crane boom. To isolate cross-sectional shape as the primary independent variable and ensure a physically meaningful comparison, all non-geometric variables [material properties ($E, G$), span ($L$), loading ($P, w$), and wall thickness ($t$)] are held strictly constant across uniform profile concepts.

All candidate profiles are evaluated as thin-walled members where the uniform wall thickness is negligible relative to cross-sectional dimensions ($t \ll R, B, D$). Thin-wall expressions assume uniform thickness and negligible warping.

Design Concept 1

Single Tube:

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A closed, hollow circular profile of constant radius $R$. This is the baseline design serving as our analytical benchmark due to its axisymmetry and optimized shear flow loop.

For $I_{\text{single}}$, the exact equation for a hollow circle is:

$$ I_{\text{single}} = \frac{\pi}{4}\left(R_{\text{outer}}^4 - R_{\text{inner}}^4\right)

$$

Expanding via $R_{\text{outer}} = R + \frac{t}{2}$ and $R_{\text{inner}} = R - \frac{t}{2}$:

$$ I_{\text{single}} = \frac{\pi}{4}\left[\left(R + \frac{t}{2}\right)^4 - \left(R - \frac{t}{2}\right)^4\right] $$

Utilizing the binomial expansion and dropping higher-order terms ($t^2, t^3, t^4 \to 0$) yields the thin-wall approximation: $I_{\text{single}} = \pi R^3 t$

$J_{\text{single}}$ is modeled using Saint-Venant torsion for thin-walled closed sections where shear flow is continuous. Evaluated via Bredt’s formula where $A_m = \pi R^2$ is the area enclosed by the wall median line and $l = 2\pi R$ is the median perimeter: