We are visiting the American University of Cairo for an educational and research collaboration on hygroscopic surfaces. In Old Cairo, we had the surprise of running into convertible textile umbrellas in front of the Al Hussein Mosque, Cairo, Egypt designed and built by SL Rasch GmbH Special and Lightweight Structures in 2000. These umbrellas are similar to the large retractable umbrellas in front of the Prophet´s Holy Mosque in Medina, Saudi-Arabia. I have always been a fan of the way the seam patterns in this doubly curved prestressed membrane are key to the design of the canopy and how the patterns fit into the local context.

These adaptive umbrellas shade the floors in front of the mosque when needed and create a comfortable microclimate throughout the year. Conceptually, the conic membrane form carries tensile forces through a series of horizontal rings and radial lines. For me, these umbrellas are one of the archetypal prestressed membrane forms. Therefore I would like to use them as an example to better understand the relationship between form and force in pre-stressed membranes.

The umbrellas do not have a simple cone shape. Since they have anticlastic curvature, finding the optimal form of these umbrellas is more complex. The surface shape of these conic membranes is determined by the ratio of stresses in the textile’s two perpendicular directions. When the textile is woven, the weft is the term for the thread or yarn which is drawn through the warp yarns to create the textile. Warp is the lengthwise or longitudinal thread in a roll, while weft is the transverse thread.

In the conic membrane, the warp direction is represented by radial lines while the weft direction can be represented conceptually by the horizontal rings.

In the following parametric study, the effect of changing the geometry of the umbrella is studied on the stresses in the warp and weft directions. We approximate the base as a circular base of 8.75m radius for simplification. Additionally, the opening where the mast of the umbrella is ignored.

Using the equations of equilibrium for general surfaces of revolution, the tensile forces and radii of curvature in each direction depend upon the normal pressure, p:

*p= T1 / R1 + T2 / R2*

Where T1 and T2 are tensile forces and R1 and R2 are radii of curvature in the warp and weft directions, respectively.

For this example we will call the warp direction D1 and the weft direction D2. In the form finding process we assume that no permanent external pressure acts upon the membrane (thus p=0). We are interested in finding the shape under a set of pre-stress forces in the warp and weft directions. Thus when the normal pressure for these umbrellas is equated to zero, the relationship between stresses in opposing directions is easy to find.

In this analysis, the ratio of these stresses will be examined. The mosque umbrellas have a height of 5.2m and an approximated radius of 8.75m at their widest horizontal ring.

CASE 1:

*T1 / R1 + T2 / R2 = 0 , where T1 = T2 *

*T / R1 + T / 8.75 = 0 , so R1 = -8.75 m*

This case uses the minimum surface area of fabric. In Case 2, the stresses in the weft direction is reduced to half of those in the warp direction.

CASE 2:

*T1 / R1 + T2 / R2 = 0 , where T1 = 2T2 *

*T / R1 + 2T / 8.75 = 0 , so R1 = -17.5 m*

This case creates a ‘flatter’ curve for the membrane which requires higher stresses in the warp direction to maintain its form. Comparing Cases 1 and 2, it can be observed that the stress and radius ratios are directly related.

When the warp stress is k times as large as the weft, the warp radius is k times larger than the weft radius (see table below). Therefore, as k increases, the material stress increases, the warp radius increases, and the curvature of the cone decreases.

Cairo is without a doubt full of architectural gems. I am very grateful that my host, Prof. Sherif Abdelmohsen (American University of Cairo), and the excellent local guide Tarek showed me some of them.

##### Author: Sigrid Adriaenssens

##### Contributions: Hiba Abdel-Jaber

##### Editor: Emre Robbe