[Paper Review] Parametric Design of Minimal Mass Tensegrity Bridges Under Yielding and Buckling Constraints

Gerardo Carpentieri, Robert E. Skelton+1 more|arXiv (Cornell University)|Nov 28, 2014

Structural Analysis and Optimization5 references3 citations

TL;DR

This paper presents a parametric optimization framework for minimal-mass tensegrity bridges using cables (tension) and bars (compression), subject to yielding and buckling constraints. It shows that the minimal mass solution under buckling matches Michell's 1904 classic, and reveals that optimal complexity tends to infinity without joint mass, but remains finite when joint mass is included, with the lowest-complexity substructure bridge yielding the lightest design for practical materials.

ABSTRACT

This paper investigates the use of the most fundamental elements; cables for tension and bars for compression, in the search for the most efficient bridges. Stable arrangements of these elements are called tensegrity structures. We show herein the minimal mass arrangement of these basic elements to satisfy both yielding and buckling constraints. We show that the minimal mass solution for a simply-supported bridge subject to buckling constraints matches Michell's 1904 paper which treats the case of only yield constraints, even though our boundary conditions differ. The necessary and sufficient condition is given for the minimal mass bridge to lie totally above (or below) deck. Furthermore this condition depends only on material properties. If one ignores joint mass, and considers only bridges above deck level, the optimal complexity (number of elements in the bridge) tends toward infinity (producing a material continuum). If joint mass is considered then the optimal complexity is finite. The optimal (minimal mass) bridge below deck has the smallest possible complexity (and therefore cheaper to build), and under reasonable material choices, yields the smallest mass bridge.

Motivation & Objective

• To develop a parametric design method for minimal-mass tensegrity bridges using only cables and bars.
• To analyze the impact of yielding and buckling constraints on optimal bridge topology and mass.
• To determine the optimal complexity (number of elements) for minimal mass, considering both structural and joint mass effects.
• To compare substructure (below deck) and superstructure (above deck) configurations for mass efficiency.
• To establish a necessary and sufficient condition for minimal mass bridges to lie entirely above or below deck, based solely on material properties.

Proposed method

• Uses a fractal-based self-similar subdivision strategy to generate tensegrity modules of increasing complexity (n, p, q).
• Applies form-finding techniques to determine equilibrium configurations under axial forces and prestress.
• Derives analytical expressions for maximum stress in cables and bars to enforce yielding and buckling constraints.
• Introduces a mass minimization problem with constraints on material yield strength and Euler buckling for compression members.
• Incorporates deck and joint masses as penalty terms to model real-world cost and fabrication constraints.
• Solves the optimization problem numerically for various configurations and material properties (e.g., steel cables and bars).

Experimental results

Research questions

• RQ1 What is the minimal mass configuration of a tensegrity bridge composed of only cables and bars under yielding and buckling constraints?
• RQ2 Does the minimal mass solution under buckling constraints coincide with Michell’s 1904 solution for a simply-supported beam?
• RQ3 What is the optimal complexity (number of elements) for minimal mass when joint mass is included versus ignored?
• RQ4 Under what material properties does a substructure (below deck) bridge yield the lightest design?
• RQ5 What condition ensures that the minimal mass bridge lies entirely above or below the deck level?

Key findings

• The minimal mass solution for a simply-supported tensegrity bridge under buckling constraints matches Michell’s 1904 solution, despite differing boundary conditions.
• When joint mass is ignored, the optimal complexity tends toward infinity, approaching a continuous material distribution.
• When joint mass is included, the optimal complexity is finite, and the minimal-mass bridge is a substructure with the smallest possible complexity (n=1, p=1, q=0).
• For reasonable material choices, the substructure bridge with minimal complexity yields the lightest overall mass.
• The necessary and sufficient condition for a minimal mass bridge to lie entirely above or below deck depends only on material properties, not geometry or loading.
• Numerical results show that mass decreases with increasing complexity, approaching asymptotic values (e.g., μ* ≈ 45.00 for n=1, p=50, q=0 under combined constraints), indicating convergence to a continuous limit.