03
May
06

Torsion Beam

Here is an exaggerated example of a beam being twisted. The following code is based on Stephen Timoshenko’s analytical solution. The color represents the equivalent (Von Mises) stress where red is high stress and blue is low stress.
(* runtime: 12 seconds *)
a = b = 2.0; L = 10; T = G = 1; J = 16/3a^3b(1 - 196a/(b Pi^5)Sum[n^-5Tanh[n Pi b/(2a)], {n, 1, 200, 2}]); k = T/(G J);
phi = 32G k a/Pi^3Sum[1/n^3(-1)^((n - 1)/2)(1 - Cosh[n Pi y/(2a)]/Cosh[n Pi b/(2a)])Cos[n Pi x/(2a)], {n, 1, 10, 2}];
tauyz = -D[phi, x]; tauxz = D[phi, y]; tauxy = sx = sy = sz = 0;
f[x_, y_, z_] := Module[{}, {s1, s2, s3} = Eigenvalues[{{sx, tauxy, tauxz}, {tauxy, sy, tauyz}, {tauxz, tauyz, sz}}]; sv = Sqrt[((s1 - s2)^2 + (s2 - s3)^2 + (s3 - s1)^2)/2]; {x - k z y, y + k z x, z, {EdgeForm[], SurfaceColor[Hue[2(1 - sv/0.029)/3]]}}];
<< Graphics`Master`;
DisplayTogether[Map[{ParametricPlot3D[f[#a/2, y, z], {y, -b/2, b/2}, {z, 0,L}, Compiled -> False], ParametricPlot3D[f[x, #b/2, z], {x, -a/2, a/2}, {z, 0, L}, Compiled -> False]} &, {-1, 1}], ParametricPlot3D[f[x, y, L], {x, -a/2, a/2}, {y, -b/2, b/2}, Compiled -> False]];

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