Calculating Fear: Contextualising Skyscraper Sway and Lateral Movement With mathematics

During a recent visit to The Shard, I had an unnerving experience while standing on the open viewing gallery of the 72nd floor. I noticed that the building was swaying slightly due to high winds!  Skyscraper sway refers to the lateral movement or swaying of tall buildings due to external forces such as wind, seismic activity, or other environmental factors. As tall buildings are exposed to wind or seismic forces, they tend to move or sway slightly, which can be noticeable to occupants or people observing the building from outside. 

The degree of sway depends on various factors such as the building's height, shape, mass, stiffness, and damping, as well as the magnitude and direction of external forces. Skyscrapers are designed to withstand these forces and remain stable within acceptable limits to ensure the safety of occupants and the structural integrity of the building.

As far as I am aware (with a rudimentary knowledge of structural engineering), the sway of a skyscraper can be calculated using the following formula:

Δ = (C * F) / (2 * k)

where:

• Δ is the lateral deflection or sway of the building at the top
• C is the lateral load due to wind or seismic activity
• F is the height of the building
• k is the lateral stiffness of the building's structure

Note that this formula is a simplified version and may not take into account all the factors that can affect the sway of a skyscraper, such as the building's shape, materials, and foundations. Additionally, engineers and architects use advanced computer simulations and models (such as ETABS structural analysis software) to calculate the sway of skyscrapers more accurately.

We know that the Shard is roughly 306 meters tall. Nevertheless, to calculate the sway of The Shard using the formula above, we also need to know the lateral load due to wind or seismic activity (C) and the lateral stiffness of the building's structure (k). These values are typically determined through complex engineering calculations and simulations.

However, to provide an example calculation, let's assume the following values for The Shard based on information I can find online about similar-sized skyscrapers:

• Lateral load due to wind or seismic activity (C) = 1000 kN/m
• Height of the building (F) = 306 m
• Lateral stiffness of the building's structure (k) = 5000 kN/m

Using the formula Δ = (C * F) / (2 * k), we can calculate the hypothetical sway of the building:

Δ = (1000 kN/m * 306 m) / (2 * 5000 kN/m)

Δ = 153000 kNm / 10000 kNm

Δ = 15.3 m / 1000

Δ = 0.0153 m

Therefore, in this hypothetical example, the sway of the skyscraper would be 0.0153 meters or approximately 1.53 centimetres.

In the case of The Shard, a brief look at WSP’s website (the engineers who designed The Shard) suggests that the building was actually designed to have a maximum lateral sway of 300mm (approximately 12 inches) during an extreme wind event!

This amount of sway tolerance is apparently well within the safety limits set by building regulations. The literature from WSP also suggests the maximum sway acceleration is 0.15 m/s2 under high winds. The “acceleration” of the swaying property is probably what gives rise to the uneasy feeling that occupants feel as the building moves.

Calculating the "Acceleration" of Sway

In the calculation above, I calculated a hypothetical value for how much The Shard can move laterally when acted upon by external forces. Nevertheless, how do we calculate how fast this movement takes place under varying wind speeds? Clearly, such calculations are complex and require detailed analysis of the building's design and environmental conditions.

However, the general formula for calculating the acceleration of sway of a tall building due to wind is:

a = K * V^2 * H / m

where:

a = acceleration of sway

K = aerodynamic coefficient of the building

V = wind speed

H = building height

m = mass of the building

The aerodynamic coefficient, K, depends on the shape and size of the building and can be determined through wind tunnel testing or computer simulations.

It is important to note that the acceleration of sway is a measure of how much the building moves back and forth in response to wind forces. It is not the same as the actual speed of the building's movement, which would depend on the damping properties of the building's structure.

Here are some hypothetical values for The Shard based on skyscrapers of a similar height: 

K = 1.2 (typical for a rectangular building)

V = 30 m/s (about 67 mph)

H = 306 m

m = 2.5 x 10^8 kg (roughly the mass of a 100-story building)

Plugging these values into the formula, we get:

a = K * V^2 * H / m

a = 1.2 * (30 m/s)^2 * 306 m / (2.5 x 10^8 kg)

a = 0.00232 m/s^2

Therefore, the acceleration of sway for a  306-meter-tall building like The Shard due to wind could be approximately 0.00232 m/s^2. This value is relatively small, but it can still be significant enough to cause discomfort for building occupants and potential damage to the building's contents if the swaying is severe enough.

How do tall buildings protect their structural integrity? 

As well as a tiered use of concrete and steel vertically along the structure, WSP designed The Shard to incorporate a 'hat truss' on level 66 to increase stiffness, which involved using outrigger struts that extend diagonally from the perimeter columns to the central core. The primary goal of this design was to decrease lateral acceleration. However, due to construction constraints, the tightening of bolts on the truss was delayed until the later stages of The Shard's construction (to allow for the massive settlement and shrinkage of any buildings above 15 floors).

While it's natural to feel uneasy about a tall building's sway, it's important to note that this movement is imperceptible to most building occupants, and modern structural designs ensure that buildings remain safe and stable even during extreme weather events. In fact, some buildings are intentionally designed to sway slightly in response to wind or seismic forces, as this can help to distribute these forces evenly across the building and reduce the risk of damage or collapse. 

*It should be noted that none of these calculations have eased my fear of heights and building sway. I am just as likely to shit my pants going forward.