# Slenderness ratio

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111 West 57 St is the world's most slender skyscraper

In architecture, the slenderness ratio, or simply slenderness, is an aspect ratio, the quotient between the height and the width of a building.

In structural engineering, slenderness is used to calculate the propensity of a column to buckle. It is defined as ${\displaystyle l/k}$ where ${\displaystyle l}$ is the effective length of the column and ${\displaystyle k}$ is the least radius of gyration, the latter defined by ${\displaystyle k^{2}=I/A}$ where ${\displaystyle A}$ is the area of the cross-section of the column and ${\displaystyle I}$ is the second moment of area of the cross-section. The effective length is calculated from the actual length of the member considering the rotational and relative translational boundary conditions at the ends. Slenderness captures the influence on buckling of all the geometric aspects of the column, namely its length, area, and second moment of area. The influence of the material is represented separately by the material's modulus of elasticity ${\displaystyle E}$.

Structural engineers generally consider a skyscraper as slender if the height:width ratio exceeds 10:1 or 12:1. Slim towers require the adoption of specific measures to counter the high strengths of the wind in the vertical cantilever, like including additional structures to endow greater rigidity to the building or diverse types of tuned mass dampers to avoid unwanted swinging.[1]

Tall buildings with high slenderness ratio are sometime referred to as pencil towers.[2]

## Examples

Building Location Floors Height (m) Slenderness Year
111 West 57th Street New York, NY 82 438 24:1 2018
Highcliff Happy Valley, Hong Kong 73 252 20:1 2003
150 North Riverside Chicago, Illinois 54 228 20:1 at base 2017
Collins House (Melbourne) Melbourne, Australia 61 190 16.25:1 2019
432 Park Avenue New York, NY 85 425.5 15:1 2015
One Madison Park New York, NY 50 188 12:1 2016
Sky House New York, NY 55 179 Between 12:1 and 20:1[3] 2008
Icon New York, NY 42 158 Between 15:1 and 18:1[4] 2009

## References

1. ^ Willis, Carol. "The Skyscraper Museum: SKY HIGH & the logic of luxury WALKTHROUGH". Retrieved 2016-01-24.
2. ^ Wainwright, Oliver (5 February 2019). "Super-tall, super-skinny, super-expensive: the 'pencil towers' of New York's super-rich". The Guardian. Retrieved 1 December 2020.
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