On Imaginary Cantilevers

Unraveling the mystery of pencheck stairs.

Staircases hardly seems like a topic to ignite much interest, however there is one type of stair that is particularly interesting and has a bit of a mystery about its load-path. So mysterious that no one can actually agree what to call such stairs. ‘Cantilever stairs’ is the most common name, but ‘hanging stairs’, ‘geometric stairs’, and ‘stairs with a void in the middle’ have all been tried too. In this post I am going to run with the Scottish term ‘pencheck stairs’, but before I do I am going to start with the one before.

   

‘Stair with a void’ is an important description, because until Palladio coined it in ‘Quattro Libri dell’Architettura’ during the sixteenth century, stair treads generally bridged between two walls. The concept of supporting a stair from one side only, so that light could flood into the stair well from above, was completely unknown...or at least we have no evidence of it.

Pallidio’s concept was taken to new heights by successive exponents and some of our grandest buildings are adorned by elegant stone staircases with narrow wedge like treads that seem to float from bottom to top. Then, at some point in the past, probably the early twentieth century, Pallidio’s concept fell out of fashion and somehow engineers forgot how they work. This remains the case for some engineers today.

One thing we do know is that many nineteenth century examples have been investigated and it turns out the stone treads are embedded only 4.5 inches into the supporting wall. That’s effectively the width of one brick.

This simple fact tells us something important about pencheck stairs. 4.5 inches is completely insufficient to support cantilever behaviour. Evidently pencheck stairs do not, as one would suppose, cantilever. Their load path is therefore a structural mystery. 

I know the load path is a mystery, because I have seen several examples where a worried engineer has added a steel beam below the outside edge of the treads, simultaneously ruining the stair’s aesthetic and their own engineering credibility. 

Once it has been recognised that pencheck stairs do not cantilever their secret can be understood in terms of a load path that consists of three component parts. 

The first part is quite simply load transfer through bearing. The front of each step sits on the back of the one below transferring load progressively from the top of the stair to the bottom. The second component of the load path is torsion. Since load is transferred from above into the back of a tread and is transferred out of the same tread at the front a load ‘couple’ is formed i.e. the front pushes down on the tread below and the back resists the tread above by pushing back up. This combination of forces is what generates torsion.

Torsion is of course resisted by embedment in a brick wall. All the brick need do is prevent the stone tread from rotating. This is why embedment can be so shallow. 

Together bearing and torsion are quite sufficient for a stair to stand without further assistance.

 

The third load-path component is only necessary when stair treads have a particularly slender profile; sometimes a mere wedge. It is developed by small rebates at the back of each tread, known in Scotland as penchecks. In this arrangement the toe of each tread interlocks snuggly with the pencheck below.

Given its size one might be forgiven for assuming the presence of a pencheck is somewhat superfluous, however closer examination of the load-path reveals this is far from the case. It is in fact crucial.

Let us first revisit the style of stair that has no pencheck. It is self-evident that since each tread bears on the one below the cumulative load on the final tread is the sum total of all the treads above i.e. the load carried by each tread increases the closer it is to the bottom of the stair. It follows that the torsional component is also greatest at the bottom of the stair. This is fortunate because, unless there is a door or window, the weight of masonry available to resist torsion is also greatest at the bottom of the stair.

We are now ready to consider the effect of penchecks on the same stair. Since they are interlocked, rotation of a tread will cause it to push into its neighbours above and below. While the top surface pushes against its upper neighbour the bottom surface pushes against the lower one. These pushing forces are in equal magnitude and opposite direction. In effect we have created a horizontal load ‘couple’, which acts against the tread’s desire to rotate. This effect passes from one tread to the next; both up and down the flight. At the top of the stair the cumulative push is met by the top landing. If the landing is secure, and cannot move, then the upward thrust is resisted. Similarly, if the base of the stair is secured by the bottom landing then the downward thrust is also resisted.

   

In order to simplify this part of the load path we can make a conservative assumption. All of the horizontal thrust generated by the penchecks is assumed to be resisted by the landings i.e. no benefit is taken from embedding the treads within a brick wall.

The effect that this has is profound. It creates an overall load ‘couple’ between the two landings that results in a uniform torsion over the length of the stair. This in turn reduces the torsion due to part two of the load path by half. Perhaps the best way to understand  this conclusion is to review the forces visually.

The top diagram depicts the torsional force in the treads without penchecks. It is minimal near the top of the stair and maximal at the base. In the middle diagram we can see a uniform torsion due to pushing forces generated by the penchecks and resisted at the landings. 

We can combine these two diagrams by subtracting one from the other, as shown in the bottom diagram The net clockwise torsion at the base of the stair is reduced by the same amount as the anti-clockwise torsion at the top is increased. The net torsion on the middle tread is therefore zero.

The net torsion transferred into the brick wall is also significantly reduced as the treads have now taken on much of the work themselves. It is this effect that allows stairs with more elegant and slender cross sections to be built.

It is accepted that part three of this load path has been resolved in a simplified manner. The true distribution of load between wall and tread is difficult to calculate, as it is sensitive to the relative stiffness and contact between all the components. Nevertheless, no matter the precise distribution of load, in qualitative terms the principles hold good and reveal to us the secret of the pencheck stair.

Now that we have solved the easy bit let us return to the difficult part. What should we call such stairs. I don’t like ‘cantilever stair’. I like the fact that there is a mystery about how such stairs work, but ‘cantilever’ is not mysterious; its just plain wrong. I am sympathetic to ‘stair with a void in the middle’, as it is an historic description that credits Palladio. That said, I can’t run with this, because people are no longer surprised by a stairwell with a void. Time has robbed this potential title of its power to impress. ‘Hanging stair’ is like ‘cantilever’ stair; it’s just plain wrong. ‘Geometric stair’ is bland and without any useful meaning. ‘Stairs with one sided support’ has been proposed. It is technically accurate, but somewhat wordy, and therefore unlikely to catch on. 

No, I am going to stick with the Scottish description ‘pencheck stair’. I like this for two reasons. Firstly, it recognises the crucial role played by those little rebates. Secondly, hardly anyone in Scotland, let alone anywhere else, knows what ‘pencheck’ means, thus the sense of mystery is preserved.