FEATool Multiphysics
v1.14
Finite Element Analysis Toolbox

A long cylindrical thick walled pressure vessel is here subjected to both a high internal pressure and thermal expansion. The model and solution is symmetric and also suitable for modeling with the twodimensional plane strain approximation. This problem allows for an analytical solution which is used to validate and test the accuracy of the simulation [1].
This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Structural Mechanics > Cylindrical Pressure Vessel from the File menu. Or alternatively, follow the stepbystep instructions below.
To decrease the time and cost of the simulation, symmetry and and the plane strain approximation are used so that only a quarter of a 2D slice has to be modeled.
The geometry can be created by first subtracting a smaller inner from a larger outer cylinder. And the overlaying a rectangle and using the intersection operation to get the resulting quarter geometry section.
0 0
into the center edit field.0.95
into the x_{radius} edit field.0.95
into the y_{radius} edit field.0 0
into the center edit field.0.2
into the x_{radius} edit field.0.2
into the y_{radius} edit field.0
into the x_{min} edit field.1
into the x_{max} edit field.0
into the y_{min} edit field.1
into the y_{max} edit field.0.02
into the Grid Size edit field.0.33
for the Poisson's ratio and 2e11
for the Modulus of elasticity, 1.3e5
for the Thermal expansion coefficient, and the expanded expression for the temperature field (TiTo)/log(ro/ri)*log(ro/r)+To. The other coefficients can be left to their default values.(60030)/log(0.95/0.2)*log(0.95/sqrt(x^2+y^2))+30
into the Temperature edit field.To account for the symmetry, apply fix left vertical boundary by setting the displacement in the xdirection there to zero, and similarly fix the ydisplacement on the lower horizontal boundary.
Apply the normal pressure 6e7
on the inner boundary and 2.5e7
on the outer one. Note that these loads are scaled with the outward pointing unit normal vectors nx
and ny
in order to apply the forces normal to the boundaries.
nx*6e7
into the Displacement/load, xcomponent edit field.ny*6e7
into the Displacement/load, ycomponent edit field.nx*2.5e7
into the Displacement/load, xcomponent edit field.ny*2.5e7
into the Displacement/load, ycomponent edit field.The analytical solution for the displacement and stresses in the radial direction are as follows
r = 0.2  r = 0.95  

rdisplacement  0.00068  0.00302 
rstress  6e7  2.5e7 
thstress  1.59e9  0.59e9 
zstress  2.1e9  0.107e9 
To compare it is easier to either look at the horizontal xaxis (where the xdirection quantities correspond to the radial ones) or alternatively the vertical yaxis. Click anywhere on the surface plot to evaluate the expression in the chosen point. For a more clear representation one can use the Point/Line Evaluation functionality to study and plot quantities at points and lines.
0.2:0.01:0.95
into the Evaluation coordinates in xdirection edit field.0
into the Evaluation coordinates in ydirection edit field.The cylindrical pressure vessel structural mechanics model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB mscript text file (available as the example ex_planestrain2 script file), or GUI script (.fes) file.
[1] Barber JR. Solid Mechanics and Its Applications, Intermediate Mechanics of Materials. vol. 175, 2nd ed. Springer, 2011.