It’s still the holiday season, so no apologies for doodling on about gingerbread, which, as it turns out, can be pretty strong stuff – if a bit bendy.
Cue my wife Erin’s first attempt at a gingerbread house (above). Pretty good, huh? The heat from the incandescent fairy lights has kept it from turning mushy, and nicely spiced up the room at the same time. The house is only eight inches tall, but prompts the obvious question: “How high can you build with gingerbread?”
A structural analysis of a full-on house with walls, windows and doors is too tall an order, even with finite element techniques, so I settled on calculating a ballpark maximum height based on standard engineering equations for a free-standing gingerbread column.
There’s no wind blowing through our lounge, so we can ignore sideways forces and focus on the two likely failure modes a column of gingerbread might suffer – just because of its own weight as it gets taller, i.e.:
(b) the column can buckle, which is more related to the material’s elastic, or tensile properties.
The heights at which these two failures occur can be found from, respectively:
where =column height at compressive failure (m), is the failure pressure (N/m2) = compressive strength of the gingerbread, g=gravity (9.8 ms-2), and is gingerbread density (kg m-3). And for buckling: is the critical height, E is Young’s Modulus of elasticity calculated as tensile stress/strain, I is the Area Moment of Inertia3, and is a factor called a Bessel function, used to solve this type of equation (Ref.2)
Using published gingerbread properties data1 (amazingly, there actually are some) for compressive strength and tensile stress/strain, I calculated values of:
(Workings in box below if you’re interested.)
which essentially means a gingerbread column will start to lean over and buckle sideways long before the gingerbread breaks up through compression under its own weight (I used an arbitrary but realistic 20 cm column diameter). You might think there’s no reason why a uniform, vertical, column would start to lean, but in real life the weight distribution is never uniform and, if the column is sufficiently slim, a turning moment will establish and drive a progressive buckle.
So if you’re going to build a gingerbread house out of free-standing columns, better stop at 3 metres.
Buckling is clearly the limiting factor, but the 3 metre figure is based on a relatively small 0.2m column diameter, and buckling is particularly sensitive to cross-sectional area (whereas compressive fracture of a column under its own weight is independent of area). Also, most real buildings are more complex than a bunch of pillars, and I’d expect the right combination of interconnecting members building up from a broad foundation could reduce buckling potential, making a full-size gingerbread house a reality.
Indeed, the Guinness Book of Records ‘Worlds largest gingerbread house’ is 18.28m (60ft) on a 13.86m by 10.8m base; but closer inspection shows it’s built around a steel frame that presumably keeps incipient buckling in check. But then it’s more of a gingerbread and steel house – a bit of a con really.
Anyhow, our room’s about 3 metres high, so nothing stopping a more ambitious project next year: Empire State Building or Cathédrale Notre Dame ?
1. ‘Building with gingerbread: Engineering students put holiday delight to the test’ refers to ‘Structural Analysis of Gingerbread. Engineering Design Project Term 2’ by Mercedes Duifhuis and Sean Heisler (pdf)
2. The Shape of the Tallest Column. Steven J.Cox, C.Maeve McCarthy, Society for Industrial and Applied Mathematics. Vol29,No.3. pp.547-554. (Also see Wikipedia page on buckling.)
3. Engineering Fundamentals efunda.com/math)
Of related interest
How to create the perfect sand castle Nature Scientific Reports 2, Article number:549, doi:10.1038/srep00549