How To Design a Cantilevered Staircase

Staircases are a lot of fun. That is, even though the design process is strictly regulated by the building code, it can be the perfect opportunity to flex your creative muscles. But how exactly do you design a cantilevered staircase? How do the stairs stay in place? And what do you need to take into consideration during the design phase?

We’ll tackle these questions by first having a look at the relevant building regulations, and then moving onto understanding the basic physics behind designing and building a cantilevered staircase. This is where it gets a bit more technical – but don’t worry – we won’t get carried away with complex equations. We’ll only look at deflection and the basic stresses under expected loading conditions, i.e. under every day use.

Cantilevered Staircase
FIGURE 1 Cantilevered Staircase

DIGITAL FILES DOWNLOAD

Digital files used for this tutorial are available for purchase in two options.

MANUFACTURING PACKAGE
  • CAD files for laser cutting and manufacturing drawings (shop floor drawings) for steelwork
  • Assembly drawings for timber cladding & balustrade
  • Section Detail and GA drawings (PDF & DWG)
  • 3D models of the assembly (STEP, 3D DWG, SKP)

You can also view our PREVIEW PDF for a full list of files included in the package.

CAD PACKAGE
  • 3D models only (STEP, 3D DWG, SKP)

REVISION NOTE: The design used in this article has been revised in October 2020.


BUILDING REGULATIONS

To set the scene, let’s imagine a typical London townhouse that is undergoing a substantial refurbishment, and you’ve been given the task to design a cantilevered staircase for it. Now, where would you start?

Well, before we run off to the drawing board, the first thing to do is to familiarise ourselves with the relevant building regulations. Below is a table taken from The Building Regulations 2010 K showing the maximum and minimum dimensions for rise and going. As we are working on a residential project, we only need to concern ourselves with the regulations related to private stairs.

Staircase---Regulations.png
Table 1 – Building Regulations (SOURCE: Building Regulations 2010 K)

CALCULATING MAIN DIMENSIONS

1. RISE

Rise is the vertical distance between each step. With the above regulations in mind, we can determine how many steps are required to complete our staircase. The critical dimension that we need to know first is the difference between finished floor levels – also known as floor-to-floor dimension. Floor levels are usually marked as FFL (finished floor level) on the drawings, so we just need to know what the distance is between the two finished floor levels. Figure 2 below shows the difference as 2850 mm.

Cantilevered Staircase Floor-to-Floor Dimension
FIGURE 2 – Measuring Floor-to-Floor Dimension

To work out the number of risers, we need to pick a number that falls within the specified range (150 – 220mm) and divide the floor-to-floor dimension with that number. Remember that we need a whole number here, so we’ll round it up to the nearest number if we end up with decimals. We can’t have half risers on our staircase, right? So, in this case, we’ll divide 2850mm by 190mm, resulting in 15. This means that our staircase will have 15 steps, each with a rise of 190mm.

2. GOING

Going is the horizontal distance between two stair nosings. Again, looking at the values specified by the building code, we’ve chosen to go with 280mm. This usually requires trying different values, and then checking what pitch it gives you until you’re happy with the result. If the space at landings is limited, then you may need to have a shorter going – i.e. a steeper pitch. Landings are the floor areas where the stairs end at the top and bottom that, again, are specified in the building code. In this case, we have plenty of space at both landings, so we can stick with 280mm going to give us the preferred pitch.

3. PITCH

Pitch is probably the most important planning parameter in staircase design. It determines how comfortable it is to walk up and down the stairs. As a rule of thumb, we can use something called the normal ratio to check our dimensions. This ratio states that twice the rise plus going should be between 550mm and 700mm. We can check this by using the values that we just worked out;

Normal Ratio: 2R + G = between 550mm and 700mm, so 2(190) + 280 = 660

To calculate the pitch, we can take the total rise (number of risers x rise) and divide it by total going (number of stairs x going). In other words, we’re calculating the tangent (tan = opposite/adjacent). The equation gives us the following;

(15 x 190) / (15 * 280) = 0.68

inverse tangent of 0.68 = 34 degrees.

The preferred pitch for comfort and safety is given as 30.58 – 35.26 degrees, so 34 degrees satisfies this criteria.

Cantilevered Staircase Assembly Dimensions
FIGURE 3 – Measuring Floor-to-Floor Dimension

DESIGN PRINCIPLES

Now that we have the main dimensions worked out, we can start designing our staircase. In Fig. 4 below we can see the full staircase is made up of two sections – top and bottom flight assembly. So although at a first glance the staircase looks like it’s made from solid timber treads, there’s actually a steel structure underneath it all. This steel structure is made up of two large 10mm stringer profiles to which all the tread supports are welded onto. The idea is to laser cut all the steel profiles so that they fit together like flat pack furniture. This method eliminates the need for measuring things out during fabrication, as the profiles only fit where they’re suppose to. Now let’s take a look at each component that make up the assembly.

Exploded Staircase Assembly
FIGURE 4 – Exploded Staircase Assembly
STRINGER

The stringer is the backbone of the steel structure as it supports the cantilevered treads and acts as a fixing plate for the whole staircase. Therefore it needs to be strong enough to withstand the stresses and rotational forces that are introduced while the stairs are being used. This is why there are four fixing points at the centreline of each tread. By using multiple fixings per tread, the forces are shared evenly which keeps the stringer nice and rigid.

STAIR TREAD ASSEMBLY

In the exploded visual we can also see that each tread is made up of four support profiles, one end profile and two cover plates. The support profiles locate into laser cut slots in the stringer where they are welded to provide the main structural support for the treads. At the end of these supports there is an end profile, which has tapped holes for attaching balustrade standoffs or fixing pins. Finally, the two cover plates are welded onto top and bottom of the tread to provide additional stiffness.

Cantilevered Staircase Fixing Detail
FIGURE 5 – Cantilevered Staircase Fixing Detail
FIXINGS

The fixing detail in Fig.5 above shows us how the end of the staircase is fixed to the wall. This is what the word cantilever means – fixed at one end only. You may have heard the term floating stairs thrown around, which is another term to describe cantilevered stairs. This ‘floating’ effect is achieved by hiding the fixings and support structure from sight, making it appear as if the stairs are floating. Here the stringer and fixings are hidden behind a plasterboard finish which is offset from the concrete wall. There are four lengths of M16 studding resin anchored at the centreline of each tread. We’ll see why we need this many fixings in the structural analysis a bit later, but first let’s familiarise ourselves on how resin anchoring actually works.

RESIN ANCHORS

Resin anchoring basically means drilling a hole into the concrete wall with a masonry drill, filling it with resin, and then inserting the piece of studding into the hole. Once the resin goes off, the stair assembly can be bolted onto these pieces of studding and secured with lock nuts. In this example we’ll look at Sika AnchorFix-1 resin – see imperial or metric data sheet. There are of course other brands available, but always remember to check that the tensile strength of the resin is able to withstand the forces acting on the fixing points. To demonstrate the installation method of a resin anchor, I’ve included a link to an instruction video from the manufacturer below (outside content).

STRUCTURAL ANALYSIS

As the stairs are fixed only at one end, the weight of a person walking on the stairs will introduce significant asymmetrical loading to the wall and the fixings. It is therefore absolutely crucial that these forces are taken into account. Usually, this is where a structural engineer would come in and perform calculations to check the integrity of the design and the wall structure before manufacture. If not, then you run the risk of making a staircase feels bouncy when you walk on it, or even worse, something that ends up crumbling down under your feet!

Typical Staircase Loading Conditions
FIGURE 6 – Typical Staircase Loading Conditions
AIM OF THE ANALYSIS

The aim of this structural analysis is to a) to make sure there is no deflection (displacement) on the stair treads and b) to make sure the steel structure is strong enough to handle the load. Meaning we want to simulate real life loading conditions – as demonstrated in Fig.6 above. Every time a person walks on the stairs, they are applying a load to the structure. So to validate the integrity of our design we need to simulate this user load. To keep things simple, we will only consider the steel tread part of the structure as this is where the main structural support comes from. Because if this part of the assembly isn’t structurally sound, well, then the rest of the project would go horribly pear-shaped.

FINITE ELEMENT ANALYSIS

To simulate these conditions, we’ll be using finite element analysis. FEA is basically a computer simulation that uses a numerical method to predict how parts behave under given load conditions. For this experiment, let’s assume a person that weights 85kg. To convert this weight into force, we simply apply Newton’s second law of motion (Force = mass x acceleration). We’ll round the answer up to 850 N, or 0.85 kN. This force, which we’ll call Force-1, is then applied to the tip of the stair as a point load – as shown below.

Forces on Cantilevered Tread
FIGURE 7 – Forces on Cantilevered Tread

Before diving into the FEA results, let’s think about the types of forces we can expect to see acting on a cantilevered structure. By applying a point load to the tread, we can expect to see few things happening. The forces are in general the shear force, bending moment and rotational forces acting on the stringer and fixing points. Seeing as the treads are only fixed at one end, this is where we can expect to see the highest stress values. And naturally the highest displacement values are expected to show up at the free end of the tread.

RESULTS
Material Properties & Loads and Fixtures
TABLE 2 – Material Properties & Loads and Fixtures
Stress & Displacement FEA Results
TABLE 3 – Stress & Displacement FEA Results

After running the simulation, we’re presented with this magical rainbow-coloured visuals of the steel structure. The two results shown are for stress (von Mises stress) and displacement. The colours represent the area of deflection and stresses in the structure – red being the highest values and blue being the lowest. So these look pretty much what we predicted – the highest deflection at the free end and the highest stress at the fixed end. Now let’s take a look at these values and find out what they’re saying about our design.

A) DISPLACEMENT

The maximum displacement worked out to be 1.06 mm – located at the very end of the tread. This does exaggerate the load a little bit, as normally you would step somewhere towards the middle of the tread. So this is a result we can be happy with – knowing there will be no bounce on the stairs. One thing to bear in mind though, is that the simulation assumes no movement at the fixing points. Which highlights the importance of having a strong stringer with multiple fixing points, as well as having an adequately reinforced wall structure to make sure no such movement occurs.

Stress & Displacement Values
FIGURE 8 – Stress & Displacement Values
B) STRESS

Now what about the stress values, what do all these numbers mean? As mentioned, the stress calculations are based on the von Mises stress criterion, so by understanding this theory a little bit will help us understand the results. Without going too deep into a physics rabbit hole, we can simplify the theory into the following; a ductile material (steel in this case) will start to yield at a location when the von Mises stress becomes equal to the stress limit. In other words, we want all the values to be less than the stress limit – which is the ultimate yield strength of steel.

The yield strength is given in Material Properties in Table 2 as 2.827 x 108 N/m2. Now, when we look at the maximum stress value in Table 3 given as 3.191 x 107 N/m2, it tells us that we are well below the stress limit – by almost a ninefold. So – theoretically speaking – we would need nine people weighting at 85kg each standing at the very end of the stair tread to start seeing any yield in the steelwork. If the stress values were close to the yield strength then we would need change the material thickness to something more substantial, or tweak the design in other ways. But this result proves that the steel assembly is structurally sound to withstand any loads under normal conditions.

CONCLUSION

So back to the original question – how to design a cantilevered staircase? Well, as we’ve recently discovered, it all starts with understanding the relevant building regulations then taking the floor-to-floor dimension for working out the rise, going and pitch. With these dimensions the supporting structure can be designed. Once the design is complete, you need to perform a structural analysis to make sure the structure can handle the stresses under normal loading conditions and no deflection or excessive stresses are present. Then all that is left is to build the staircase.

When it comes to installation and setting out the staircase, the first task at hand would be to drill holes to the masonry wall for the resin anchors. But measuring the hole distances by hand would be a lengthy, painstaking task, not to mention prone to dimensional errors.

To make life easier, you’d be better off using a CNC cut plywood template that gives you the exact hole locations to drill. By fixing the template to the wall, all you have to do is drill through the template holes. Once all holes are drilled, simply remove the template and start resin anchoring the studding. Once all studding is in place, you can start fixing the stair assemblies to place. We have included this plywood template in our staircase package, all you have to do is send the DWG or DXF to a supplier that have a CNC router to cut the templates for you.

REFERENCES & FURTHER READING

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