{ PRESTUBE.PDE }
{ A Tube with an internal pressure
- from "Fields of Physics on the PC" by Gunnar Backstrom }
title
' Tube With Internal Pressure'
variables
u
v
definitions
mm = 1e-3
r1 = 3*mm r2 = 10*mm q21= r2/r1
mu = 0.3
E = 200e9 {Steel}
c = E/(1-mu**2) G = E/2/(1+mu)
dabs= sqrt(u**2+ v**2)
ex= dx(u) ey= dy(v) exy= dx(v)+ dy(u)
sx= c*(ex+ mu*ey) sy= c*(mu*ex+ ey) sxy= G*exy
p1= 1e8 { the internal pressure }
{ Exact expressions }
rad= sqrt(x**2+ y**2)
sr_ex= -p1*((r2/rad)**2 - 1)/(q21**2 - 1)
st_ex= p1*((r2/rad)**2 + 1)/(q21**2 - 1)
dabs_ex= abs( rad/E*(st_ex- mu*sr_ex))
initial values
equations { Constant temperature, no volume forces }
u: dx( c*(dx(u) + mu*dy(v)) ) + dy( G*(dx(v)+ dy(u)) )= 0
v: dx( G*( dx(v)+ dy(u)) )+ dy( c*(dy(v) + mu*dx(u)) )= 0
constraints { Since all boundaries are free, it is necessary
to apply constraints to eliminate
rigid-body motions }
integral(u) = 0
integral(v) = 0
integral(dx(v)-dy(u)) = 0
boundaries
region 1
start (r2,0)
load(u)= 0 { Outer boundary is free }
load(v)= 0
arc to (0,r2) to (-r2,0) to (0,-r2) to close
start (r1,0) { Cut-out }
load(u)= p1*x/r1 { Normal component of x-stress }
load(v)= p1*y/r1 { Normal component of y-stress }
arc to (0,-r1) to (-r1,0) to (0,r1) to close
monitors
contour(dabs)
plots
grid(x+200*u, y+200*v)
elevation(sx, sr_ex) from (r1,0) to (r2,0)
elevation(sy, st_ex) from (r1,0) to (r2,0)
contour(dabs) contour[(dabs-dabs_ex)/dabs_ex]
contour(u) contour(v)
vector(u,v) vector(u/dabs, v/dabs)
contour(sx) contour(sy) contour(sxy)
end