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as a child all of us have built a tower out of Lego. we built it as high as we could but then eventually it topples over.

is there a formula to calculate how wide and heavy the base has to be for the tower not to topple

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Yes there is. Lets say you have a tall building with multiple stories and several unsymmetrical mass protrusions at different floors.

To verify that your building is not going to topple:

  • You find the center of mass of the buliding.

  • You drop a vertical line from this center of mass to the footprint of the building, which is usually the exterior extents of its foundation.

If the vertical line fall within the footprint the building will not topple. If it falls outside of the footprint it is not stable and will topple on the side where it falls outside of.

This is called static overturning stability.

Many building codes require the vertical line to fall within an area cantered around the center of the geometry of that surface with sides parallel to footprint and 1/3 each side length.

In siesmic areas this rule changes to more conservative set of rules considering the dynamic behavior of the building and its tendency to potential rotational vibration or possibly resonance of buildings vibration with the ground motions spectra.

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If your building is very tall and "slender", then the rigidity also comes into play in a set of phenomenon called slender column buckling. For a Lego tower, the tl;dr; is that at some point, the tower will start bending over and then topple due to the kind of situation that kamran described.

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