1. Reference problem#
1.1. Geometry#
Straight beam of length \(L\), with direction vector \((\mathrm{4,}\mathrm{3,}0)\).
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Pipe section |
Tubular section with outer radius \(a\mathrm{=}\mathrm{0.04m}\), inner radius \(b\mathrm{=}\mathrm{0.032m}\), inner radius, and thickness \(e\mathrm{=}0.008m\).
1.2. Material properties#
The material used has elastic behavior. The material parameters take the following values:
Young’s module \(E=2.{10}^{11}\mathrm{Pa}\),
Poisson’s ratio \(\nu =0.3\),
Density \(\rho =7800\mathrm{kg}/{m}^{3}\),
Thermal expansion coefficient \(\alpha ={10}^{-5}\).
1.3. Boundary conditions and loads#
Embedding in \(O\)
6 basic loads at the \(B\) end
in coordinate \((x,y,z)\) linked to the beam:
\({F}_{x}\mathrm{=}{5.10}^{2}N\) |
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\({F}_{y}\mathrm{=}{5.10}^{2}N\) |
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\({F}_{z}\mathrm{=}{5.10}^{2}N\) |
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or, in the global frame of reference \((X,Y,Z)\):
1 traction load: \({F}_{X}={4.10}^{2}N\) and \({F}_{Y}={3.10}^{2}N\)
2 sharp efforts: in the \((\mathrm{oxy})\) \({F}_{X}=–3.102N\) and \({F}_{Y}={4.10}^{2}N\) plan and in the \((\mathrm{oyz})\) \({F}_{Z}={5.10}^{2}N\) plan
1 torsional moment: \({M}_{X}={4.10}^{2}\mathrm{Nm}\) and \({M}_{Y}={3.10}^{2}\mathrm{Nm}\)
2 sharp efforts: in the \((\mathrm{oxy})\) \({M}_{X}\mathrm{=}–{3.10}^{2}\mathit{Nm}\) and \({M}_{Y}\mathrm{=}{4.10}^{2}\mathit{Nm}\) plan and in the \((\mathrm{oyz})\) \({M}_{Z}\mathrm{=}{5.10}^{2}\mathit{Nm}\) plan
Internal pressure: \(P={10}^{7}\mathrm{Pa}\)
Gravity, with \(g=\mathrm{10m}/{s}^{2}\), in the \(-Z\) direction
Line load, \(\mathrm{Fz}=-141.146N/m\) (which corresponds to the load due to gravity: \(\mathrm{Fz}=\mathrm{mg}\))
Thermal expansion: \(\mathrm{Temp}=100°C\)
1.4. Notation of cross-sectional characteristics#
The geometric characteristics of the cross sections are noted:
\(S\): area of the section
\({I}_{y},{I}_{z}\): geometric moments of inertia in relation to the main axes of inertia of the section
\({J}_{x}\): torsional constant