The Shelf That Sags
A pupil designs a wall-mounted book shelf: a single 18 mm MDF board, 1200 mm long, 250 mm deep, fixed to the wall along one long edge by three 60 mm L-brackets at 0, 600 and 1200 mm. Load testing with 15 kg of books spread evenly produces a visible 9 mm sag at mid-span after 30 minutes, and the middle bracket's screws begin to pull out of the plasterboard wall (no studs were located). The pupil's specification says 'must carry 20 kg with no permanent deformation'. The workshop has 18 mm plywood, 18 mm MDF, 6 mm steel flat bar, and a pillar drill but no welder.
1. Diagnose the two separate failures and state which one must be solved first.
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There are two independent failures. (1) Bending stiffness of the shelf. A 1200 mm span of 18 mm MDF is very flexible: stiffness rises with the cube of thickness and falls with the cube of span, so doubling thickness cuts deflection eightfold, while halving span does the same. 9 mm sag at 15 kg shows the section is inadequate, and MDF also creeps under sustained load, so 'no permanent deformation' at 20 kg is unlikely. (2) Fixing failure. The screws are pulling out of plasterboard, which has almost no pull-out strength; the load path, not the shelf, is failing. Solve the fixing first: a shelf that holds its shape but falls off the wall is a safety hazard. The wall attachment must reach the studs or use proper heavy-duty anchors rated for the load, and the bracket spacing must match stud positions rather than an even 600 mm.
2. Propose a redesign using only stock available in the workshop, and justify each change against the specification.
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Fixing: relocate brackets to land on studs (locate with a detector, then mark), or add a continuous timber batten screwed into at least two studs and hang the shelf from that. This spreads load into the structure rather than plasterboard, addressing failure 2 directly. Stiffening: add a 6 mm steel flat bar as a front or rear edge stiffener, screwed and glued along the full 1200 mm. Steel has a far higher modulus than MDF, and putting stiff material at the extreme fibre of the section is exactly where it does most good - a small steel edge raises the second moment of area cheaply. Material/geometry: switch the deck to 18 mm plywood, which is stiffer and tougher than MDF and resists creep better; or reduce the span by adding a fourth bracket and a mid-support. Verification: re-test with 20 kg for 24 hours, measuring sag with a straight edge and dial gauge, and check for permanent set after unloading. If sag remains above the agreed limit (e.g. 2 mm), add a second stiffener or a centre support.