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:

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:

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.

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.

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