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- # problem parameters for
- # 12"-inch schedule 100
- b = 323.8/2 # external radius [ mm ]
- a = b-21.5 # internal radius [ mm ]
- l = 2*(b-a) # axial length [ mm ]
- n = 4 # number of elements through thickness
-
- E = 200e3 # Young modulus [ MPa ]
- nu = 0.3 # Poisson's ratio [ non-dimensional ]
- p = 10 # internal pressure [ MPa ]
-
-
- # definition of analytical solutions for comparison from
- # <http://eprints.whiterose.ac.uk/110536/1/art%253A10.1007%252Fs00707-016-1762-7.pdf>
- ur(x,y,z) = (p*a**2*sqrt(y**2+z**2))/(E*(b**2-a**2)) * ((1-2*nu)*(1+nu) + (1+nu)*b**2/(y**2+z**2))
- sigmal(x,y,z) = 2*nu*p*a**2/(b**2-a**2)
- sigmar(x,y,z) = p*a**2/(b**2-a**2) * (1 - b**2/(y**2+z**2))
- sigmatheta(x,y,z) = p*a**2/(b**2-a**2) * (1 + b**2/(y**2+z**2))
-
-
- # analytical results in [ MPa ]
- M_tresca = 69.9467
- M_vonmises = 61.7785
- M_1 = 65.3023
- M_2 = 18.1974
- M_3 = -4.64439
-
- MB_tresca = 79.9062
- MB_vonmises = 70.2562
- MB_1 = 70.2821
- MB_2 = 18.1974
- MB_3 = -9.62415
-
-
- ur1(x) = ur(0,x,0)
- sigmal1(x) = sigmal(0,x,0)
- sigmar1(x) = sigmar(0,x,0)
- sigmatheta1(x) = sigmatheta(0,x,0)
-
- delta = 0.05
-
- set preamble "\usepackage{amsmath}"
-
- set width 14*unit(cm)
-
- set axis x arrow nomirrored
- set axis y arrow nomirrored
-
-
- set grid
- set terminal pdf
-
-
- set output "ur.pdf"
- set xrange [a:b]
- set yrange [*:*]
- set key top right
- set xlabel "$r$ [mm]"
- set ylabel "$u_r(r)$ [mm]"
- plot ur1(x) w l lw 3 lt 1 color olivegreen ti "Analytical solution $u_r(r)$",\
- "pipe-dist-1-3.dat" u 2:4 w l lw 1 lt 2 color maroon ti "",\
- "pipe-dist-1-3.dat" u 2:4 w p pt 18 ps 1.5 color maroon every (100)/(1*3) ti "$n=3$ first order",\
- "pipe-dist-1-6.dat" u 2:4 w l lw 1 lt 4 color tan ti "",\
- "pipe-dist-1-6.dat" u 2:4 w p pt 16 ps 1.5 color tan every (100)/(1*6) ti "$n=6$ first order",\
- "pipe-dist-2-2.dat" u 2:4 w l lw 2 lt 3 color cerulean ti "",\
- "pipe-dist-2-2.dat" u 2:4 w p pt 17 ps 1.5 color cerulean every (100)/(2*2) ti "$n=2$ second order"
-
- set output "sigmatheta.pdf"
- set xrange [a:b]
- set yrange [*:*]
- set key top right
- set xlabel "$r$ [mm]"
- set ylabel "$\sigma_\theta(r)$ [mm]"
- plot sigmatheta1(x) w l lw 3 lt 1 color springgreen ti "Analytical solution $\sigma_\theta(r)$",\
- "pipe-dist-1-3.dat" u 2:5 w l lw 1 lt 2 color melon ti "",\
- "pipe-dist-1-3.dat" u 2:5 w p pt 18 ps 1.5 color melon every (100)/(1*3) ti "$n=3$ first order",\
- "pipe-dist-1-6.dat" u 2:5 w l lw 1 lt 2 color redorange ti "",\
- "pipe-dist-1-6.dat" u 2:5 w p pt 16 ps 1.5 color redorange every (100)/(1*6) ti "$n=6$ first order",\
- "pipe-dist-2-2.dat" u 2:5 w l lw 2 lt 3 color turquoise ti "",\
- "pipe-dist-2-2.dat" u 2:5 w p pt 17 ps 1.5 color turquoise every (100)/(2*2) ti "$n=2$ second order"
-
- set output "sigmal.pdf"
- set xrange [a:b]
- set yrange [*:*]
- set key top right
- set xlabel "$r$ [mm]"
- set ylabel "$\sigma_l(r)$ [mm]"
- plot sigmal1(x) w l lw 3 lt 1 color forestgreen ti "Analytical solution $\sigma_l(r)$",\
- "pipe-dist-1-3.dat" u 2:6 w l lw 1 lt 2 color peach ti "",\
- "pipe-dist-1-3.dat" u 2:6 w p pt 18 ps 1.5 color peach every (100)/(1*3) ti "$n=3$ first order",\
- "pipe-dist-1-6.dat" u 2:6 w l lw 1 lt 2 color dandelion ti "",\
- "pipe-dist-1-6.dat" u 2:6 w p pt 16 ps 1.5 color dandelion every (100)/(1*6) ti "$n=6$ first order",\
- "pipe-dist-2-3.dat" u 2:6 w l lw 2 lt 3 color periwinkle ti "",\
- "pipe-dist-2-3.dat" u 2:6 w p pt 17 ps 1.5 color periwinkle every (100)/(2*2) ti "$n=2$ second order"
-
- set output "sigmar.pdf"
- set xrange [a:b]
- set yrange [*:*]
- set key top left
- set xlabel "$r$ [mm]"
- set ylabel "$\sigma_r(r)$ [mm]"
- plot sigmar1(x) w l lw 3 lt 1 color emerald ti "Analytical solution $\sigma_r(r)$",\
- "pipe-dist-1-3.dat" u 2:7 w l lw 1 lt 2 color bittersweet ti "",\
- "pipe-dist-1-3.dat" u 2:7 w p pt 18 ps 1.5 color bittersweet every (100)/(1*3) ti "$n=3$ first order",\
- "pipe-dist-1-6.dat" u 2:7 w l lw 1 lt 2 color orange ti "",\
- "pipe-dist-1-6.dat" u 2:7 w p pt 16 ps 1.5 color orange every (100)/(1*6) ti "$n=6$ first order",\
- "pipe-dist-2-3.dat" u 2:7 w l lw 2 lt 3 color royalblue ti "",\
- "pipe-dist-2-3.dat" u 2:7 w p pt 17 ps 1.5 color royalblue every (100)/(2*2) ti "$n=2$ second order"
-
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