Wolfram Cloud Document

This notebook was created in collaboration with Enrico Babilio and Rana Nazifi Charandabi

ABSTRACT (original article): This work presents a tensegrity design approach for a small-to-medium-sized sports stadium, equipped with sun-tracking solar modules on its roof. A novel V-Expander unit is used to create a modular roof structure, with members optimized using a minimal mass strategy. The stability of the entire stadium structure and the deflections of the cantilever roof are analyzed under vertical, seismic, and wind loads through detailed finite element analysis. Electrical power production is estimated over one year for the Salerno area. The results demonstrate that the tensegrity solar roof exhibits limited deflections and remarkable solar energy harvesting capabilities, utilizing sun-tracking strategies tailored to different sectors of the grandstands. This research paves the way for the design of multi-scale sports stadiums featuring sustainable, adaptive lightweight tensegrity structures. These systems employ sun-tracking photovoltaic modules that can be tilted by simply wrapping a bus cable around a winch. CITATION (original article): Nazifi Charandabi, R., Babilio, E., Carpentieri, G., Spagnuolo,G., Amendola, A., Fraternali, F., A tensegrity structure for a solar stadium roof with sun-tracking capability, Thin-Walled Structures 210, 2025, 113033. https://doi.org/10.1016/j.tws.2025.113033

Introduction

Architects, engineers, and sports agencies have been working for over 20 years to enhance the sustainability of sports stadiums and reduce their energy consumption and carbon footprint, which are typically very high. A remarkable number of sports stadiums equipped with photo voltaic (PV) panels (solar-powered stadiums) have been designed and built in recent years. However, it appears that no examples of stadium roofs with sun-tracking PV panels have been realized to date. In

[1]

the tensegrity design of a novel solar stadium structure with sun-tracking capabilities is illustrated.The examined stadium roof structure tessellates the tensegrity unit illustrated below (graph reproduced without changes from

[1]

,under CC BY-NC-ND 4.0 license), to form the roof elements of the grandstand structures described in the following sections.

The used tensegrity unit is a variant of the original V-Expander, introduced by René Motro in his well-known textbook

[2]

as a V-shaped system composed of eight bars arranged in two distinct sets of four bars each, with equal lengths. The bars are connected by one vertical cable, along with eight horizontal and eight diagonal cables. The variant used is a class-4 tensegrity structure composed of eight bars and seven cables, featuring a vertical quadrangular section with height h and width d, formed by nodes 1, 2, 3, and 5. The system is completed by one base triangle (1, 4, 6) lying in a horizontal plane and a sloped triangle (2, 4, 6). The “arms” 1-4 and 1-6 lie on a circle with radius r, centered at node 1. The module is employed to create a cantilever roof structure by arranging an array of modules along the y-axis, which is perpendicular to the x-axis aligned with the bar 1-3. Such a module does not exhibit infinitesimal mechanisms. We set the origin of the (x,y,z) Cartesian coordinate system at node 1. It is worth noting that the V-Expander module designed for minimal mass is modeled as an independent structure with restraint conditions to simplify its interactions with adjacent parts of the grandstand structures. It is restrained by fixing node 3, constraining nodes 1, 2, and 5 in the longitudinal y-direction, and additionally constraining node 1 in the vertical z-direction aligned with the bar 1-2. The tensegrity stadium concept assumes the y-axis is parallel to one edge of the grandstands, which form a sports stadium structure with a rectangular plan.

Code to draw the V-Expander module shown above

In[]:=

Oxyz={0,0,0};height=5;span=2;armlength=12;angle=

;radbar=0.2;radnode=1.25radbar;pn[1]={0,0,0};pn[2]={0,0,height};pn[3]={-span,0,0};pn[4]=armlengthCos

angle

,-armlengthSin

angle

,0;pn[5]={-span,0,height};pn[6]=armlengthCos

angle

,armlengthSin

angle

,0;VExpander=Graphics3D[{{Sphere[{pn[1],pn[2]},radnode],Sphere[{pn[5],pn[3]},radnode],Sphere[{pn[1],pn[4]},radnode],Sphere[{pn[1],pn[6]},radnode],Sphere[{pn[2],pn[4]},radnode],Sphere[{pn[2],pn[6]},radnode],Sphere[{pn[2],pn[5]},radnode],Sphere[{pn[1],pn[3]},radnode]},{LightBrown,Tube[{pn[1],pn[2]},radbar],Tube[{pn[5],pn[3]},radbar],Tube[{pn[1],pn[4]},radbar],Tube[{pn[1],pn[6]},radbar],Tube[{pn[2],pn[4]},radbar],Tube[{pn[2],pn[6]},radbar],Tube[{pn[2],pn[5]},radbar],Tube[{pn[1],pn[3]},radbar]},{Blue,Line[{pn[2],pn[3]}],Line[{pn[3],pn[4]}],Line[{pn[4],pn[6]}],Line[{pn[6],pn[3]}],Line[{pn[4],pn[5]}],Line[{pn[5],pn[6]}],Line[{pn[5],pn[1]}]}},BoxRatiosAutomatic,BoxedFalse,ViewPoint{1.2,-2.6,1.8},ImageSize650]

Out[]=

Code to draw the soccer field and the roof panels

Geometric data

In[]:=

d=span;h=height;r=armlength;αpaper=angle;ell=rCos

angle

;p=

ell

;L$=2rSin

angle

;evec[i_]:=IdentityMatrix[3]〚i〛;

PV panels data

Soccer field measurements

The roof panels

The soccer field

A view of the soccer field and and the roof panels

Code to draw movable roof panels

Rodrigues’ formula

Major rotation axes (one-axis tracking)

Minor rotation axes (two-axes tracking ready)

Rotated vectors and panels

The role of master to four triangles is assigned

A view of the soccer field and and the roof panels with “manually” operated motion

The movement of the roof pitches can be explored by moving the slider controls.

Sun data

As in [4], the sun path and solar irradiance data for the examined days were obtained using PVLib v0 .10.3, a Python toolbox designed to simulate the performance of photovoltaic (PV) systems [5].
Here, however, only Mathematica commands are used.

Decimal latitude and longitude (Salerno area) and example day (summer solstice)

Sun path data

Maximization

Centroids and unit normal vectors

Computation of the optimal tilting angle

For the one-axis tracking, φ is held fixed, while θ is optimized.

Results

The variation of the secondary (i.e. additional) tilting angles θ during the daylight hours is shown according the following color coding:
(i) N grandstand roof: solid blue line;
(ii) E grandstand roof: dot-dashed black line;
(iii) S grandstand roof: solid red line;
(iv) W grandstand roof: dashed gray line.

Roof pitches motion

Resampling to reduce the number of animation frames

Current configuration of the roof panels

Current unit normal vectors

Position of the sun in the sky

Some movie editing

The movie shows the tracking of the sun during daylight hours, from sunrise to sunset.
As the sun rises, the roof tilts from the resting configuration, maintained during the night, to the optimal configuration.
The latter is adjusted during daylight hours.
At sunset, the roof returns to the rest configuration.

Optimized motion during daylight hours

Sunrise adjustments

Sunset adjustments

The motion of the roof pitches from sunrise to sunset

Conclusions

This Mathematica notebook reports part of the code used in the study of a tensegrity roof of a small-to-medium-scale stadium with a capacity of nearly 10 000 seats. The roof is designed to be employed for solar energy harvesting by exploiting sun-tracking capabilities. In this study, the roof panels of the various grandstand sections are designed to tilt at distinct angles, maintaining uniform tilting within each section (one-axis tracking). With such a strategy a remarkable peak electrical power output can be achieved, reaching up to 510 kW when using thin-film solar panels with 18% power conversion efficiency. An additional, under investigation, improvement considers further optimization by allowing the individual modules forming the roofs of the grandstand sections to exhibit different tilting angles, incorporating two-axis tracking.

References

[1] R. Nazifi Charandabi, E. Babilio, G. Carpentieri, G. Spagnuolo, A. Amendola, F. Fraternali, “A tensegrity structure for a solar stadium roof with sun-tracking capability”. Thin-Walled Structures, 210, 2025, 113033. https://doi.org/10.1016/j.tws.2025.113033

[3] J. S. Dai, “Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections”. Mechanism and Machine Theory 2015, 92, 144–152. https://doi.org/10.1016/j.mechmachtheory.2015.03.004

[4] F. Fraternali, E. Babilio, R. Nazifi Charandabi, G. Germano, R. Luciano, G. Spagnuolo, “Dynamic origami solar eyes with tensegrity architecture for energy harvesting Mashrabiyas”, Applications in Engineering Science, 19, 2024,100190. https://doi.org/10.1016/j.apples.2024.100190

[5] W. Holmgren, K. Anderson, C. Hansen, Robwandrews, M. Mikofski, A.R. Jensen, A. Lorenzo, U. Krien, Bmu, A. Driesse, C. Stark, DaCoEx, M.S. de León Peque, T. Transue, E. Luis, Kt, N. Priyadarshi, Mayudong, Heliolytics, E. Miller, M.A. Anoma, V. Guo, L. Boeman, J. Stein, S. Jane, W. Vining, Jforbess, T. Lunel, C. Leroy, A. Morgan, “pvlib/pvlib-python: v0 .10.3”, 2023, Zenodo. http://dx.doi.org/10.5281/zenodo.10412885

CITE THIS NOTEBOOK

A tensegrity structure for a solar stadium roof with sun-tracking capability
by Rana Nazifi Charandabi, Enrico Babilio, Gerardo Carpentieri, Giovanni Spagnuolo, Ada Amendola & Fernando Fraternali
Wolfram Community, STAFF PICKS, February 19, 2025
https://community.wolfram.com/groups/-/m/t/3396614