Based on the sinusoidal walls of Eladio Dieste, this study aims to analyze and compare the benefits gained by creating walls (and other structures) with varying degrees of sinusoidal profiles.

Background

The Church of Christ the Worker, located in Atlantida, Uraguay, was constructed between 1958 and 1960 from designs by architect/engineer Eladio Dieste.  The most striking feature of this church is the sinusoidal side walls — which are based on cosine waves mirrored across the center aisle, as shown in the schematic below.

Goal

The goal of this ongoing study is to create models based off of these sinusoidal walls and to compare them under a variety of loading scenarios.  Comparing models with varying degrees of “curviness” — which is mathematically controlled by increasing/decreasing the amplitude of the cosine waves — will provide insight into why Dieste would choose to use walls of this type.  Such double-curved designs recur in his works, and as an engineer, he informed these decisions based, at least in part, on the advantages that these shapes provided with respect to loading.

Description

I started this study by creating a small variety of shapes to “play around with” in SolidWorks.   Once I determined how I would proceed with the study, I went back and normalized all of the parameters so that valid comparisons could be drawn.  I first generated a surface by using two juxtaposed cosine waves as the top and bottom limits — the peaks of one cosine correlated with the troughs of the other, and vice-versa.  This surface was then thickened.  Six shapes were generated from this template — with cosine amplitudes of 0 (a flat, planar wall), 0.5, 1, 1.5, 2, and 2.5.  They are all 24 inches tall and slightly more than 25 inches wide (8 pi, to be exact, since each has a period of 2 pi repeated 4 times).

I then applied a fixed set of conditions using COSMOSWorks.  For each model, I applied three different sets of restraints and pressure loads, all with a fixed value of 10psi compression: (1) bottom fixed and loaded from the top, (2) bottom fixed and loaded from the sides, and (3) bottom fixed and loaded from the top and the sides.  For those familiar with the software, this is done with a basic static analysis, using PVC as the material.  Each design scenario produces five graphs: stress, strain, deformation, displacement, and factor of safety.  Due to the comparitive nature of these analyses, the graph of interest is that of the stress analysis.

As part of the ongoing study, with excellent feedback from several instructors and professionals, more loading scenarios will be included, including changing the restraints (i.e. “real life” wall restraints on all sides) and the directions of the pressure loads.  There will also be analyses of several more shapes based on Dieste’s designs, including those with simple extruded cosine profiles instead of juxtaposed ones, and with different materials.

Completed Analysis Images

As it exists now, there is already a fairly large set of images generated.  I caution you to note that the scale of each image is slightly different (”blue” on one graph is not necessarily equal to “blue” on another).  The usefulness of these graphs as images is that they provide insight into where the shapes experience the highest levels of stress — mechanically, these areas are where the structures are the weakest from a load-carrying perspective.  The basis of the comparison lies in looking at the maximum and minimum stress values for each iteration, which will be catalogued for comparison at a future date.

I will begin with the planar case, and work in increasing increments.  For all cases other than the planar one, I included screenshots of both sides of the models — because the shapes are based on cosines, there are more full peaks on one side than on the other, thus creating a “preference” for bending in that direction.  In each image, the green arrows indicate which portion is fixed, and the red arrows indicate on which face the pressure load is being applied.  Blue represents the lowest stress levels, green the intermediate stress levels, and red the highest stress levels, as per the scale on the right.  The scale is difficult to read at the required image resolution, but it varies for each model.

Iteration 1: Planar

Iteration 2: Amplitude of 0.5

Iteration 3: Amplitude of 1

Iteration 4: Amplitude of 1.5

Iteration 5: Amplitude of 2

Iteration 6: Amplitude of 2.5

A Short Note

You can already begin to see, just visually, the effect that adding curvature has on the stress distribution.  Rather than being evenly distributed throughout the body (as in the planar case), the curvature concentrates the stress in the central, neraly planar area of the body.  For example, under the third loading scenario (bottom fixed, pressure from the top and from the sides), the planar body experiences a max stress of 18.4 psi, whereas the most curved model (iteration 6) experiences a max stress of 347 psi under identical conditions.  The planar body has an average stress of 9.7 psi, whereas the curved body has an average stress of 62.3 psi.  However, the curved body also has more places wherein the stress is equal to or lower than that of the planar body, as confirmed by probe sampling of those areas.  Will planar surfaces hold up better than curved ones to loads on the largest faces?  Discovering these kinds of trade-offs are the end goal of these studies.

Study by Gerard Delatour II originally posted on core.form-ula and was produced under the guidance of Neil Katz + Ajmal Aqtash in the Stevens PAE Skyscraper Design Studio