How do I calculate the weight threshold for cabinet carcasses based on the dimensions, material (e.g., plywood, mdf), and material thickness?
Has someone already worked out a nice formula that I can use?
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The sagulator will give deflection values for horizontal shelf spans using various materials and thicknesses.
Unless you are working with extremely heavy loads (more than the weight of a countertop with people dancing on it), 3/4" material for the (vertical) cabinet walls should be more than adequate. As you make the walls thinner, they will still remain surprisingly strong as long as your cabinet design prevents racking.
Even 1/2" particleboard, which is not known for its strength, should hold up well to vertical compression/buckling, as you may know if you've ever had to pack up books that were stored on a cheap particleboard bookcase.