Toy-Brick Stability Lab (2D)

Independent educational simulation; not affiliated with any toy or film brand. The bricks are generic (2x4 bricks, 31.8 x 15.8 x 9.6 mm, 2.5 g). Everything below is a static model of a stack seen from the side; there is no impact dynamics.

Centre of mass and contact

The piece above a seam (all bricks from that course upward) stays put against gravity only while its centre of mass x_cm = sum(m_i x_i) / sum(m_i) lies over the contact interval between the two bricks at that seam (the cyan bar). The distance from x_cm to the nearest contact edge is the lever arm b of the restoring moment W b.

Wind, shake and tipping at every seam

Wind pushes with F_w = 1/2 rho C_d A v^2 (rho = 1.225 kg/m3, C_d = 1.2, A = height above the seam x brick depth) acting at half the height of the piece; a shake table adds the inertial force m a at the centre of mass. The piece tips about the downwind edge of its contact when M_over = F_w h_cp + m a h_cm exceeds M_res = W b + n_edge f_stud w_c; the last term is the clutch of the upwind stud row (pull-off f_stud = 2 N per stud, typical 1-3 N, a model value). It slides when the horizontal load exceeds mu W + n_studs f_shear (f_shear = 3 N). The safety factor of a seam is the smaller of the tipping and sliding ratios; the page tests all seams and reports the weakest.

Critical shake of a free stack

For a stack of width b and height H without clutch, tipping starts at a_crit = g b / H: three 2x4 bricks on their 15.8 mm side give 5.38 m/s2. With clutch on, small shakes move nothing, so the shake table is most interesting with clutch off.

Harmonic overhang

Without clutch, n bricks of length L stacked so that each overhangs the next by L / (2 j) (counting from the top) reach D_n = (L/2)(1 + 1/2 + ... + 1/n); for n = 10 and L = 31.8 mm that is 46.57 mm, and it grows like ln n. The page's seam solver reproduces this: it keeps the stack standing exactly up to scale factor 1 of the harmonic offsets.

Honest simplifications

Two-dimensional side view of a straight stack; no collapse dynamics (the preview rotates the unstable piece rigidly about its edge); constants are model values, not measurements; wind is steady and uniform, applied on the end face; real bricks have tolerances that change clutch strength a lot. The 3D builder uses the same constants and formulas on a voxel grid of real bricks.

• "Clutch off: wooden blocks" with the shake table: three bricks tip at 5.38 m/s2 across the width, but slide earlier with low friction.
• "Harmonic overhang": try the extra shift slider on the top brick and watch seam safety factors turn red.
• Raise the wind on the 14-brick tower and compare the critical wind with and without clutch in the second graph.
• Press "Collapse preview" on an unstable build.