Check whether a beam or shelf holds a load: bending stress and deflection.

Closed-form Euler–Bernoulli check. "point" puts the whole load at the centre (simply supported) or the free end (cantilever); "udl" spreads magnitude_n evenly along the span. Give the section either directly (section_modulus_m3 and inertia_m4) OR as a solid rectangle (width_m and height_m).

Args: span_m: clear span / length, metres. support_type: "simply_supported" (held both ends) or "cantilever" (one end). load_type: "point" or "udl" (uniformly distributed). magnitude_n: total load in newtons (1 kg ≈ 9.81 N). e_pa: Young's modulus of the material, pascals (steel ≈ 2.0e11, pine ≈ 9e9). allowable_stress_pa: allowable bending stress of the material, pascals. section_modulus_m3: Z, if known. Else give width_m and height_m. inertia_m4: I, if known. Else give width_m and height_m. width_m, height_m: for a solid rectangle, used to compute Z and I. deflection_limit_ratio: optional, e.g. 250 means limit deflection to span/250.

Returns: max_moment_nm, max_stress_pa, max_deflection_m, stress utilisation, pass/fail, governing check, and a short explanation.

Check whether a collection of weights will tip over its support footprint.

Computes the centre of mass and tests whether its vertical projection falls inside the support base (the polygon where the object touches the ground). If it falls outside, the object tips. Use this for shelving, stacked loads, machinery on legs, vehicles, or anything that could topple.

Args: masses: list of point masses, each {"x": metres, "y": metres, "mass": kg}. x/y are top-down (plan-view) positions. Height does not affect whether it tips on level ground, only the horizontal centre of mass does. base_polygon: the support footprint as ordered vertices [{"x", "y"}, ...] in metres (e.g. the four feet of a shelf, or the contact outline).

Returns: center_of_mass, is_stable (bool), tipping_margin_m (positive = inside the base with this much clearance; negative = already outside / tipping), tipping_direction (unit vector toward the closest base edge), and a short human-readable explanation.

Compute how much vertical force each support (leg, bracket, foot) carries.

Models a rigid object resting on point supports of equal stiffness and solves static equilibrium so the reactions balance every applied load in force and moment. Handles any number of supports. Flags supports that exceed their rated capacity, and supports with a negative reaction (the object is lifting off / would tip rather than rest evenly).

Args: supports: [{"id": str, "x": m, "y": m, "capacity_n": N (optional)}]. loads: downward point loads [{"x": m, "y": m, "magnitude_n": N}]. self_weight_n: optional self-weight of the object (N), applied at the centroid of the supports.

Returns: per-support reaction forces, over-capacity / lift-off flags, total load, max utilisation, any warnings, and a short explanation.