Rain dancing

When Sir Christopher Wren was asked to design the Sheldonian Theatre in the 1660s, he began with visions of the great amphitheatres of Ancient Rome. Oxford having rather more rain than Rome, his design of a modern amphitheatre was in need of a roof.

How would you use timber beams to build a simple roof?

The most obvious solution span the walls with the timber beams.  But the dimensions of the Sheldonian are 70 foot by 80 foot and this simple roof design would need beams far longer than the timber beams available at the time.  To overcome this Wren planned to use internal columns to support the roof but this was vetoed by the university officials – they didn’t want columns impeding any dancing at the venue!  Today the Sheldonian is used for graduation ceremonies but back in the 17th century they had much wilder parties in mind!
 

Solving the puzzle of an unsupported roof

Wren couldn’t use internal columns to support the roof and he couldn’t span the space with individual beams.  Therefore to complete  the Sheldonian, Wren would need to build the largest unsupported roof the 17th Century had ever seen. 

Luckily Wren had studied mathematics here at Oxford, taught by John Wallis.  Wallis was the Savilian Professor of Geometry, a chair that still exists at Oxford today, currently held by Nigel Hitchens.  It was Wallis' ingenious design that provided Wren with the answer he needed.

Wren's problem was that the walls could only support the roof at one end of the timbers.   At first sight it might seem impossible to build a stable roof with beams that are only supported in one place – but if you have ever balanced on a seesaw then you’re already part of the way to the solution.  Just like a seesaw balancing on its single pivot, Wallis’ ingenious idea hinged on statics: balancing all the forces involved in so they cancel out making the seesaw, or roof truss, stable.

If the timber beams are only supported at one end by the walls, how can you arrange them so that together they provide stable, strong roof? The answer is to interlock the beams.  You can see a great video demonstrating the construction on Amy Mason's Sheldonian site.

Wallis’ solution

Wallis’ devised an ingenious pattern of interlocking beams, so that every beam was supported at both ends – either by the walls or by other beams – while every beam also supported the ends of two other beams.  So for every beam, the downward forces from those resting on it are balanced by the upward forces from the beams, or wall, supporting it.   In an impressive feat of calculation, Wallis demonstrated that his geometrical flat floor could carry loads when supported by the walls alone by solving  a set of 25x25 simultaneous equations using just pen and paper!

So Wren’s roof, inspired by Wallis’ design, not only keeps dancers (and graduates and their families) dry, it can also support significant loads.  The Oxford University Press stored books on the first floor for many years, proving that you can build a strong stable roof supported by mathematics instead of columns.

Demonstration

Props needed:
    •    3 flat sticks, about a metre long (the beams)
    •    3 maths books
    •    Laminated picture of Wallis’ design

If you have a large tour group (over 15) you’ll need to divide them into two groups, with a set of 3 beams and 3 books for each group. Identify 3 volunteers in each group and place them in the centre of the group in a triangle roughly a beam length apart.  These three volunteers will act as the walls supporting the roof, where the whole group will work together to design the roof.

Your groups can explore the principles at this site by trying to answer the questions in italics under the pictures in the explanation above:

How would you use timber beams to build a simple roof?

With the three volunteers standing about a beam’s length apart, ask them to build a simple roof.  Mostly likely they will build one using single beams to span the walls.  You can demonstrate how this design fails if the room is bigger than the length of the beams, by asking them to take a small step backwards so they are now more than a beam length apart.

If the timber beams are only supported at one end by the walls, how can you arrange them so that together they provide stable, strong roof?

The most successful approach with tour groups has been to just stand back and let the group get on with experimenting and solving this puzzle.  If they are struggling after several minutes you could suggest they need to have the beams supporting each other, interweaving them in some way.  The answer will be something like the following construction:
    •    End of first beam rests on one wall (you temporarily support other end)
    •    End of second beam rests on next wall and other end on middle of first beam
    •    End of third beam rests on last wall, other end on middle of second beam, and the centre of third beam supporting end of first beam.

To demonstrate that the structure is not only stable but can support significant weights, place several books in the centre of the arrangement (if the beams interlock close to the centre) or book at each of the three points where the beams cross.  Encourage other people in the group to try holding the ends of the beams to feel the solidity of the arrangement.