> For the complete documentation index, see [llms.txt](https://alpaca4d.gitbook.io/docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://alpaca4d.gitbook.io/docs/benchmark/simple-brick.md).

# Simple Brick

Two solids, solved by Alpaca4d with SSP bricks: a cantilever bar under an end load, set against Timoshenko beam theory, and NAFEMS FV52, a thick plate left to vibrate. Every Alpaca4d value on this page was produced by running the Grasshopper definitions you can download below, nothing typed in by hand.

## 1. Cantilever of bricks under an end load

|          |                                                                                  |
| -------- | -------------------------------------------------------------------------------- |
| Bar      | L = 2 m, 200 × 200 mm square                                                     |
| Material | [nD](/docs/basics/materials/nd.md) Elastic Isotropic: E = 2.1·10⁵ N/mm², ν = 0.3 |
| Load     | P = 10 kN, downwards, on the free end                                            |
| Elements | 20 × 4 × 4 [SSP Bricks](/docs/basics/elements/brick.md), 20 along the bar        |
| Supports | the fixed end: every node of the end face held in x, y and z                     |
| Analysis | linear static, one step                                                          |

Built from Alpaca4d components only — nD, Mesh Series to Brick, SSP Brick, Support, Point Load, Load Pattern, Assemble Model, Analysis Settings, Run Analysis and Nodal Displacements — with Grasshopper's own Mesh Plane for the cross-section, Series and Move to repeat it along the bar, and List Item and Deconstruct Mesh to pick the end faces.

The load is spread over the end face as a uniform shear, the way a beam's end shear acts: each of its 25 nodes takes its share in the proportions a bilinear face shares a uniform traction — ¼ for a corner node, ½ for an edge node and 1 for an inner node, scaled to add up to 10 kN.

### 1.1 Deflection at the tip

Bending (Euler-Bernoulli):

$$
\delta\_b = \frac{P,L^3}{3,E,I} = \frac{10 \cdot 2^3}{3 \cdot 2.1\cdot 10^{8} \cdot 1.333\cdot 10^{-4}} = 0.9524; mm
$$

The shear term (Timoshenko), with a shear area of 5/6 of the section:

$$
\delta\_s = \frac{P,L}{G,\kappa A} = \frac{10 \cdot 2}{8.08\cdot 10^{7} \cdot 0.0333} = 0.0074; mm
\qquad
\delta = \delta\_b + \delta\_s = \mathbf{0.9598}; mm
$$

### 1.2 Benchmark

| SSP bricks                               | Timoshenko | Alpaca4d  | Difference |
| ---------------------------------------- | ---------- | --------- | ---------- |
| **20 × 4 × 4, ν = 0.3 — the definition** | 0.9598 mm  | 0.9464 mm | −1.39%     |
| 40 × 8 × 8, ν = 0.3                      | 0.9598 mm  | 0.9506 mm | −0.95%     |
| 20 × 4 × 4, ν = 0                        | 0.9581 mm  | 0.9572 mm | −0.09%     |

Alpaca4d reads the deflection at the centre of the tip face.

{% hint style="info" %}
**Why the bricks are a little stiffer than the beam.** Holding every node of the end face does more than fix a beam's end: it also stops that face contracting and swelling sideways, as Poisson's ratio would have it under bending. Beam theory has no such effect; a solid does, and it stiffens the bar near the support. The last row is the proof: with ν = 0 there is nothing to restrain, and the bricks match Timoshenko to 0.09%. The restraint concentrates stress at the edges of the fixed face, which a finer mesh follows more closely, so refining the mesh softens the bar somewhat — to −0.95% at 40 × 8 × 8.
{% endhint %}

### 1.3 Download Grasshopper benchmark file

{% file src="/files/8xG9yA3hNWwy2f8sNVFL" %}

## 2. Simply supported solid square plate, natural vibration — NAFEMS FV52

|          |                                                                                                  |
| -------- | ------------------------------------------------------------------------------------------------ |
| Plate    | 10 × 10 m, 1 m thick, from z = −0.5 to 0.5                                                       |
| Material | [nD](/docs/basics/materials/nd.md) Elastic Isotropic: E = 2.0·10⁵ N/mm², ν = 0.3, ρ = 8000 kg/m³ |
| Elements | 8 × 8 × 3 [SSP Bricks](/docs/basics/elements/brick.md) — NAFEMS's own mesh                       |
| Supports | uz = 0 along the four edges of the bottom face, and nothing else                                 |
| Analysis | [Natural Vibration](/docs/basics/analysis/natural-vibration.md), 10 modes                        |

Nothing stops the plate sliding or spinning in its own plane, so its first three modes are those rigid-body motions, at zero frequency. NAFEMS counts them, and so does the benchmark: ask for 10 modes, and the plate's own start at the fourth.

### 2.1 Benchmark

| Mode   | NAFEMS         | Alpaca4d, 8 × 8 × 3           | Difference | 16 × 16 × 4 |
| ------ | -------------- | ----------------------------- | ---------- | ----------- |
| 1 to 3 | 0 — rigid body | 0 — rigid body, below 10⁻⁴ Hz | —          | —           |
| 4      | 44.092 Hz      | 44.076 Hz                     | −0.04%     | 43.75 Hz    |
| 5      | 106.66 Hz      | 106.61 Hz                     | −0.05%     | 105.43 Hz   |
| 6      | 106.66 Hz      | 106.61 Hz                     | −0.05%     | 105.43 Hz   |
| 7      | 156.23 Hz      | 156.05 Hz                     | −0.11%     | 157.15 Hz   |
| 8      | 193.58 Hz      | 193.58 Hz                     | 0.00%      | 193.59 Hz   |
| 9      | 200.13 Hz      | 200.13 Hz                     | 0.00%      | 198.45 Hz   |
| 10     | 200.13 Hz      | 200.13 Hz                     | 0.00%      | 198.82 Hz   |

On NAFEMS's mesh Alpaca4d is within 0.11% of every target, and on modes 8 to 10 it agrees to every printed digit. ASDEA's verification of OpenSees's SSP brick on the same test, supported on the mid-plane instead, reports up to 1.7%.

{% hint style="warning" %}
**This benchmark belongs to its mesh.** A support along a line is a singularity in a solid: the finer the mesh, the narrower the strip of material that holds the plate up, and the softer that support becomes, without limit. So FV52 has no mesh-independent answer, and NAFEMS's targets are for its 8 × 8 × 3 mesh — the last column shows the same plate on a finer one drifting away from them. The same is true of any model held along a line or at a point: to support a solid, hold a face.
{% endhint %}

### 2.2 Download Grasshopper benchmark file

{% file src="/files/yEjlX68SmiCSMzMap6MI" %}

## ⚙️ How the numbers were produced

Solved on 5 October 2026 with Alpaca4d 0.11.0 and OpenSees 3.8.0, in Rhino 8.26. The definitions were built, solved and read by a script running inside Rhino, so the values in the tables are Alpaca4d's own output, to the digits shown; the other meshes and the ν = 0 row are the same definitions with those inputs changed. Open the cantilever and a panel shows the range of the vertical displacement; open FV52 and a panel lists the frequencies.

## 📚 References

* S. Timoshenko, J. M. Gere, *Mechanics of Materials*, Van Nostrand, 1972.
* NAFEMS, *The Standard NAFEMS Benchmarks*, TNSB Rev. 3, 1990: FV52.
* ASDEA Software Technology, *Verification tests — STKO 2020 v1.1 and OpenSees 3.2.0*: FV52.