NAFEMS FV32: Thick Disk Free Vibration

Good question. FV32 is a benchmark dealing with the free vibration of a thick circular disk. Specifically, it's a problem where you find the natural frequency for out-of-plane bending using FEM and compare it with the theoretical solution $f_1 = 1.4568\,\text{Hz}$.

That's the core of it. For thick plates, the influence of shear deformation becomes significant. The Kirchhoff theory, used under the assumption of thin plates, assumes that "the cross-section remains perpendicular to the neutral surface after deformation," but this doesn't hold for thick plates. If you use Kirchhoff theory as-is, it will overestimate the natural frequency.

Under the FV32 conditions (plate thickness/radius = 0.1), the first natural frequency calculated by Kirchhoff theory comes out a few percent higher than Mindlin theory. In practice, if you think "the FEM result is higher than theory," the first thing you should suspect is whether the effect of shear deformation is being considered. That's precisely why FV32 is widely used for accuracy verification of Mindlin plate elements and solid elements.

It's a simple but profound setup.

Exactly. The only important dimensionless quantity is the ratio $t/R = 0.1$. This value is a subtle setting in the region that is "clearly a thick plate, but not extremely thick," where the difference between thin plate theory and thick plate theory becomes pronounced. For example, automotive brake disks are also roughly in the range of $t/R = 0.05\sim0.15$, so it's a practically meaningful thickness ratio.

First, the basic assumption of Kirchhoff (classical thin plate theory) is "the cross-section remains perpendicular to the neutral surface after deformation", i.e., the assumption of zero shear deformation. This uniquely determines the cross-sectional rotation angle $\theta$ from the gradient of the out-of-plane displacement $w$:

On the other hand, Mindlin (Reissner-Mindlin theory) treats the cross-sectional rotation angle as an independent variable and allows for shear strain $\gamma$:

That's exactly the image. If you try bending a thick book, the pages (cross-sections) will tilt and not remain perpendicular to the cover, right? That's the effect of shear deformation. For a single thin sheet of paper, the cross-section always remains perpendicular, but as it gets thicker, it can't be ignored.

Perfect understanding. Stiffness is overestimated → frequency comes out higher. This is a very important point in FEM practice as well; when selecting plate/shell elements, you must always check if they are "Mindlin type."

Writing the free vibration of a Mindlin plate in circular disk coordinates, for the axisymmetric mode (circumferential wave number $n = 0$), it becomes a coupled equation for out-of-plane displacement $w(r,t)$ and cross-sectional rotation angle $\psi(r,t)$:

Where:

• $D = \dfrac{Et^3}{12(1-\nu^2)}$: Plate bending rigidity
• $G = \dfrac{E}{2(1+\nu)}$: Shear modulus
• $\kappa = 5/6$: Mindlin's shear correction factor

Good observation. In Mindlin theory, the shear stress distribution through the thickness is assumed constant, but in reality, it's parabolic. $\kappa$ corrects for that difference, and for a rectangular cross-section, it is exactly $5/6$. By the way, for a circular cross-section beam, $\kappa = 6/7$ is sometimes used, but for plates, $5/6$ is conventionally used.

The reference solution for the first natural frequency published by NAFEMS is:

This corresponds to the first axisymmetric mode of out-of-plane bending (the $(0,1)$ mode, no nodal lines). It is derived from the analytical solution based on Mindlin thick plate theory.

The first natural frequency of a simply supported circular disk according to Kirchhoff thin plate theory is:

Here $\lambda_{01} \approx 4.935$ (first root for simply supported). Calculating with this formula gives approximately $1.50\,\text{Hz}$, which is about 3% higher than Mindlin theory's $1.4568\,\text{Hz}$. You might think it's only 3%, but in practice, not being able to distinguish between "a 3% error" and "a 3% theoretical difference" is critical.

Since it's free vibration, external forces are zero. Assuming displacement as $\{u\} = \{\phi\} e^{i\omega t}$, it reduces to a generalized eigenvalue problem:

Here $[K]$ is the global stiffness matrix, $[M]$ is the global mass matrix, $\omega = 2\pi f$ is the angular frequency, and $\{\phi\}$ is the mode vector.

Sharp question. Generally, the consistent mass matrix is more accurate. However, the lumped mass matrix is diagonalized, making computation lighter. For a problem size like FV32, consistent is recommended. For large-scale models (millions of DOF) where memory is tight, lumped mass can still yield practical accuracy. The actual difference is often less than 0.5%.

Both can give the correct answer, but each has different points to note.

Exactly. FV32 is precisely a problem where shear deformation is the main player, so if locking occurs, the natural frequency becomes significantly higher than the theoretical value. Full integration first-order elements (CQUAD4 full integration, C3D8 full integration) are particularly risky. Using reduced integration or the B-bar method, or choosing second-order elements from the start, is the best practice.

Let's look at typical convergence data when using Mindlin plate elements (equivalent to CQUAD8).

That's right. For vibration analysis, the cost-performance of second-order elements is overwhelmingly high. This is a lesson FV32 clearly demonstrates. In practice, the golden rule is also "first use second-order elements with a coarse mesh → check convergence → refine if necessary."