Validation Case: Infinite Cylinder Subjected to a Volume Load
This validation case belongs to solid mechanics. In this case, an infinite cylinder is subjected to a voluminal load. The parameter being validated is:
Volume load
SimScale simulation results were compared to the analytical solution presented in [SSLA100]\(^1\).
Mesh and Element Types: The mesh used in this case is a second-order mesh created in SimScale with the standard algorithm.
Table 2 contains details to the resulting mesh:
Mesh Type
Nodes
Element Type
Second-order standard
991445
Standard
Table 2: Second-order standard mesh characteristics
Figure 2 shows the standard mesh used for this case:
Figure 2: Discretization obtained with a second-order standard mesh
Simulation Setup
Material:
Material behavior: Linear elastic
Young’s modulus \(E\) = 10 \(Pa\)
Poisson’s ratio \(\nu\) = 0.3
Density \(\rho\) = 1 \(kg/m³\)
Boundary Conditions:
The boundary conditions will be defined based on the following nomenclature:
Figure 3: Nomenclature for the assignment of the boundary conditions
Constraints
\(d_z\) = 0 on both sides
Isotropic elastic support applied to the inner wall, with a spring constant \(k\) = 0.001 \(N/m\). The purpose of this boundary condition is to avoid rigid body motion.
Surface loads
Pressure of 1 \(Pa\) applied to theinner wall
Volume load applied in the radial direction, with a magnitude of \(\alpha r^2\ (\frac {N}{m^3})\), where \(\alpha = 1\ \frac {N}{m^5}\) and \(r\) is any given radius, in meters.
Figure 4: Volume load applied in a radial direction. The magnitude of the load is radius squared.
Note
Since the volume load is in cylindrical coordinates\(^1\), it was necessary to convert it to Cartesian coordinates using trigonometric functions. The resulting input formulae are:
$$Load_x = (x^2+y^2)cos(atan2(y,x)) \tag {1}$$
$$Load_y = (x^2+y^2)sin(atan2(y,x)) \tag {2}$$
$$Load_z = 0 \tag {3}$$
In equations 1 and 2, the \((x^2+y^2)\) term represents radius squared. The second part of the equations, consisting of trigonometric functions, is responsible for the conversion from cylindrical to Cartesian coordinates. As a result, the loads will be applied in a radial direction.
Result Comparison
The analytical solution for the displacements of the inner and outer wall are presented in [SSLA100]\(^1\). Find below a comparison between SimScale results and the analytical solution:
Radius \([m]\)
Analytical displacement\(^1\)\([m]\)
Displacement – SimScale \([m]\)
Error [%]
1
0.521309
0.521259
-0.0096
1.4
0.442031
0.441986
-0.0102
Table 3: Comparison of SimScale results with the analytical solution
In Figure 5, we can see the resulting displacement contours. SimScale results show great agreement with the analytical solution.
Figure 5: Displacement contours obtained with SimScale
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