Basics of Structural Analysis

Introduction

Structural analysis using about high school level physics and maths.

Aim to calculate fiberglass cansat structure

Basic theory

Principle of superposition: linearity. Displacement at location $B$ from forces $P_{1}$ and </math>P_2</math> located at different positions is calculated as $\Delta _{B}=\Delta _{BP_{1}}+\Delta _{BP_{2}}$

The energy principle: $dW=Fd\Delta$ giving the total energy as $W=\int _{0}^{\Delta }Fd\Delta$ which is called strain energy. For linear deformation this gives $U={\tfrac {1}{2}}F\Delta$.

Virtual work principle.

Dead loads, live loads, impact loads (impact factor), wind loads.

Equilibrium.

Forces:

• Normal force and axial force (out-of-plane forces, in-plane forces)
• Shearing force ${\frac {V(x)}{x}}=-w(x)$. Thus we have $\Delta V=\int w(x)dx$. $w(x)$ is the intensity of applied (normal?) force.
• Bending moment ${\frac {dM}{dx}}=V(x)$ and thus ${\frac {d^{2}M}{dx^{2}}}=-w(x)$.
• Torsion (of a plate)
• Curvature and twist

Hooke's law hold for linear elastic material: $\epsilon ={\frac {\sigma }{E}}$, where $\sigma$ is the bending stress. The force $\delta P$ is $\delta P=\sigma \delta A$.

Plate

• Torsion (of a plate)
• Curvature and twist $T_{xy}$.

Curvature in the $x$ direction $\kappa _{x}={\frac {1}{R_{x}}}$ is the rate of change of the slope with respect to arch length, giving $\kappa _{x}={\frac {\frac {\partial ^{2}w}{\partial x^{2}}}{{\sqrt {(1+\left({\frac {\partial w}{\partial x}}\right)^{2}}}^{3}}}$

Strains in a plate $\epsilon _{xx}$.

Von Kármán strains.

Beam

Forces:

• Normal force and axial force
• Shearing force ${\frac {V(x)}{x}}=-w(x)$. Thus we have $\Delta V=\int w(x)dx$. $w(x)$ is the intensity of applied (normal?) force.
• Bending moment ${\frac {dM}{dx}}=V(x)$ and thus ${\frac {d^{2}M}{dx^{2}}}=-w(x)$.

Deflection of beams

The deflection of a beam.

The strain $\epsilon$ in the filament is due to the different lengths of filaments in bended beam. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon = \frac{\ell_1 - \ell}{\ell} = \frac{\Delta y}{R}} which for linear elastic material (using Hooke's law) gives ${\frac {\sigma }{E}}={\frac {\Delta y}{R}}=\epsilon$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} is bending stress

Radius of curvature ${\frac {1}{R}}={\frac {\partial v^{2}}{\partial x^{2}}}$, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{xx} = -y \frac{\partial^2 v}{\partial x^2}} .

Column

Column is a vertical beam.

References

https://scholarshare.temple.edu/bitstream/handle/20.500.12613/7150/Udoeyo-Textbook-2020.pdf?sequence=1

https://tiij.org/issues/issues/spring2006/12_Dues-Accepted/Dues.pdf