Basics of Structural Analysis

Introduction

Structural analysis using about high school level physics and maths.

The terms used

• $E$ Young's modulus $\sigma =E\epsilon$ is to be compared to $F=kx$
• $\sigma$ is the stress (force) in the material. It is defined as $\sigma ={\frac {F}{A}}$
• $\sigma _{cr}$ Critical stress.
• $\sigma _{cr}$
• $\sigma _{cr}$
• $\sigma _{cr}$

• $\epsilon$ is the strain or deformation (displacement) in the object (perhaps in the dimension of length ratio): $\epsilon ={\frac {\partial }{\partial X}}(x-X)=F'-I$ where $X$ is the reference position of material. For a uniform bar $\epsilon ={\frac {\Delta L}{L}}={\frac {\sigma }{E}}={\frac {F}{AE}}$.
• $\lambda$ is the stretch ratio: $\lambda ={\frac {L+\Delta L}{L}}$
• $\nu$ Poisson ratio is a measure of deformation of material. $\nu =-{\frac {d\epsilon _{y}}{d\epsilon _{x}}}$
• ${\displaystyle }$

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 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 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 $\delta P=\sigma \delta A$. Yield point (strength) is when Hooke's law is not acceptable anymore; the limit of elastic behavior. It is often used to determine the maximum allowable load in the component.

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 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 \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

Cylinder stress

• Axial (Longitudinal) stress $\sigma _{a}$
• Tangential (hoop or circumferential) stress $\sigma _{t}$
• Radial stress $\sigma _{r}$

Axial stress

The axial stress is uniform across the cylinder wall. The stress is found by equilibrium calculations (see below). See the difference between external and internal pressure.

External pressure

The critical stress is $\sigma _{cr}={\frac {F}{t}}={\frac {RE}{S(1-\nu ^{2})}}$, where $E$ is the Young's modulus, $\nu$ is the Poisson ratio, $R$ is the radius of the shell and . . .[Not ok. . . , see https://core.ac.uk/download/pdf/10851171.pdf page 28. ]

Internal pressure

The force $p_{i}A_{i}$ from the internal pressure is equilibrated by the wall stress $\sigma _{a}A_{a}=\sigma _{a}(A_{o}-A_{i})$ where $A_{i}$ is internal area and $A_{o}$ is the outer area.

The force on the direction of axis (from center to the loop) due to the internal 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 $p$ 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.

Fiberglass (Glassfiber) Cansat

Also impact strength should be considered as the explosion is hard.

The stress is in general defined by $\sigma ={\frac {F}{A}}$. 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 strain (deformation) $\epsilon ={\frac {\Delta \ell }{\ell }}$. In this region the material deforms elastically and returns to its original shape.

The longitudinal stress of cylindrical pipe is $\sigma _{L}={\frac {F}{4t}}$, where $F$ is the applied force, $E$ is the Young's modulus.

The diameter of the cansat can is $D=66$ mm. Thickness $t$ is to be determined. The force $F$ acting on the soda can when the payload is ejected from the rocket is CHECK this $F=700$ N.

Thus, we will have for the thickness $t={\frac {F}{4\sigma _{L}}}=frac{700N}{4}$.

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

Fibreglass is fibre-reinforced plastic, in which the fibres originates from glass, carbon, aramid or basalt, usually. The polymer is usually an epoxy, vinyl ester or resin (hartsi in Finnish, vaik in Estonian). The polyester resin is liquid, which will solidify when the hardener is added. The hardener (Methyl Ethyl Ketone Peroxide).

However, below is structural properties of glass fiber. Normal glass fiber is E-glass, which is alumino-borosilicate glass with less than 1% alkali oxides. See more details at https://kevra.fi/?s=lasikuitu or https://composite24.ee/tooted/kangas-ja-kiud

The Young's modulus

	Tensile strength [MPa]	Compressive strength [MPa]	Yield strength [Mpa]	Ultimate strength [Mpa]	Young's modulus E [GPa]	Density g/cm3
E glass	3445	1080	-	5650	76	2.58
Unsaturated polyester	90	55	3.23			1.35
Epoxy		-	26-85		3	1.35
S-2	4890	1600	85.5

The fiber is usually made to be a fabric or mat (or more).

• Thickness of the fabric is between 0.04 mm and 0.23, usually.

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

Mechanical Properties of Glass Fiber Reinforced Polyester Composites

https://www.engineeringtoolbox.com/young-modulus-d_417.html#gsc.tab=0 The table for strengths.

http://k-mac-plastics.com/data-sheets/fiberglass_technical_data.htm