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 $\delta P=\sigma \delta A$.

Buckling is a process by which a structure cannot withstand loads so it must change its shape. Stable equilibrium is when the force (pressure) applied doesn't reach the critical load, allowing the structure to return to its original equilibrium.

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. $\epsilon ={\frac {\ell _{1}-\ell }{\ell }}={\frac {\Delta y}{R}}$ which for linear elastic material (using Hooke's law) gives ${\frac {\sigma }{E}}={\frac {\Delta y}{R}}=\epsilon$ where $\sigma$ is bending stress

Radius of curvature ${\frac {1}{R}}={\frac {\partial v^{2}}{\partial x^{2}}}$, and $\epsilon _{xx}=-y{\frac {\partial ^{2}v}{\partial x^{2}}}$.

Column

Column is a vertical beam.

Cylindrical Pipe

Axial stress

The force due to the pressure is $F=\int _{0}^{r}2p\pi rdr=2p\pi {\frac {r^{2}}{2}}=p\pi {\frac {D^{2}}{4}}$

The axial stress is $\sigma _{L}={\frac {F}{A}}={\frac {p\pi {\frac {D^{2}}{4}}}{\pi Dt}}={\frac {pD}{4t}}$

The pressure effect

The longitudinal stress and hoop (radial) stress.

$\sigma _{L}={\frac {pD}{4t}}$ 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 p} is the internal pressure, $D$ is the mean diameter of cylinder and $t$ is the wall thickness. Also, $\sigma _{H}={\frac {pD}{2t}}$.

Applying Hooke's law and the fact that $\sigma _{H}=2\sigma _{L}$ we get

$\sigma _{L}={\frac {\epsilon _{L}+v\epsilon _{H}}{1-\nu ^{2}}}E={\frac {\epsilon _{H}}{2-\nu }}E$

where $\nu$ is the Poisson's ratio and $E$ is the Young's modulus.

Buckling of cylinders

The differential equation of the classical buckling theory of a thin-walled shell. . .

Analytic solution.

Glassfiber Cansat

The stress is $\sigma ={\frac {F}{A}}$. However, the area changes (usually grows: spreads laterally) while compressing. While in Hooke's regime, we have $\sigma =E\epsilon$, where $E$ is Young's modulus and $\epsilon$ is the deformation. In this region the material deforms elastically and returns to its original shape.

The longitudinal stress (force per unit area) is $\sigma _{L}={\frac {PD}{4t}}=\epsilon E$, where $E$ is the Young's modulus and $\epsilon$ is called strain (the relative deformation, $\epsilon ={\frac {\Delta \ell }{\ell }}$)

Diameter $D=66$ mm. Thickness $t$ is to be determined. $P$ is the pressure of the can, or the force acting on the can.

Compressive stress is the capacity of a material to withstand loads tending to reduce size.

The Young's modulus

Fiber type	Tensile strength [MPa]	Compressive strength [MPa]	Young's modulus E [GPa]
E	3445	1080	76
C	3300	-	69
S-2	4890	1600	85.5

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

https://core.ac.uk/download/pdf/10851171.pdf

https://www.sciencedirect.com/science/article/abs/pii/S0263823101000660 Paywall