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 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, 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):

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

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

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

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.

Note

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.

1.3 Download Grasshopper benchmark file

GHAlpaca4d_Benchmark_BrickCantilever.ghGrasshopper file · download

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 Elastic Isotropic: E = 2.0·10⁵ N/mm², ν = 0.3, ρ = 8000 kg/m³
Elements	8 × 8 × 3 SSP Bricks — NAFEMS's own mesh
Supports	uz = 0 along the four edges of the bottom face, and nothing else
Analysis	Natural Vibration, 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%.

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.

2.2 Download Grasshopper benchmark file

GHAlpaca4d_Benchmark_SolidPlateVibration_FV52.ghGrasshopper file · download

⚙️ 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.