Calculating Stress/Strain in a Tensegrity Structure

Understanding the complexity of the mechanics involved in Tensegrity structures, which appear similar to mechanical principles employed in nature, is challenging. In this post, a simple, reduced tensegrity structure is mathematically modeled to present the type and degree of complexity of the tools utilized to analyze these systems.

The underlying principle is that in a static tensegrity structure, the sum of forces at any node, where struts and cables converge, equals zero. While true for static structures, in movement these forces sum to a non-zero number, causing movement to occur at the node. Therefore, the math presented here is just an element of a larger set of equations characterizing tensegrity within animal movement dynamics.

Additionally, within mammals, millions of nodes scale1 many orders of magnitude from the cellular to systemwide. It is a characteristic of our Clade (Mammalia) that we have excelled at integrating tensegrity principles holistically into our biomechanics. The therapeutic application of tenegrity analysis to guide manipulation may still be in the intermediate future and involve the creation of new tools2.

OVERVIEW OF TENSEGRITY CALCULATION
FOR A SINGLE NODE IN A STATIC STRUCTURE

To demonstrate the equations for calculating tensile stresses in a simplified tensegrity model without directly using XYZ coordinates3, let’s consider a basic tensegrity module. This module consists of three tension cables forming a triangle at the base and a single compression strut suspended in the middle, connected to the corners of the triangle by three additional tension cables. This is a simplified model that illustrates the core principles of tensegrity structures.

Step 1: Geometry and Connectivity

  • Nodes: Identify the nodes at the corners of the triangle and the ends of the compression strut.
  • Elements: Three tension cables form the base triangle, and three additional tension cables connect the corners of the triangle to the top of the central strut.

Step 2: Equilibrium and Forces

For any node where tension cables meet, the sum of the forces in the tension cables must be balanced by the force in the strut (if applicable) or external forces (if any). Let’s focus on a corner node of the triangle base:

Force Equilibrium in Planar Structure

  • For simplicity, let’s assume the structure is in a plane and we are analyzing forces in two dimensions.
  • The forces in the tension cables can be decomposed into horizontal and vertical components. The sum of horizontal components (Fh) and the sum of vertical components (Fv​) must each equal zero for equilibrium.

Step 3: Tension Force and Stress Relationship

Given:

  • F is the force in the cable.
  • σ is the stress in the cable.
  • A is the cross-sectional area of the cable.

The relationship is: F=σ×A

Step 4: Trigonometry and Geometry

If the tension cable makes an angle θ with the horizontal, then:

  • Fh​=FCos(θ)
  • Fv​=FSin(θ)

For equilibrium at a corner of the base triangle, consider two tension cables and their components:

  • Sum of horizontal forces: F1​Cos(θ1​)+F2​Cos(θ2​)=0
  • Sum of vertical forces: F1​Sin(θ1​)+F2​Sin(θ2​)+Fstrut​=0 (assuming the strut’s force acts vertically at this node).

Step 5: Solving the System of Equations

Solving the system of equations involves finding the values of F1​ and F2​ that satisfy the equilibrium conditions. This typically requires knowledge of the angles (θ1​ and θ2​) and the force in the strut (if applicable).

Simplified Example Calculation

Let’s assume:

  • Each tension cable has the same cross-sectional area A.
  • The tension in each of the base cables is equal due to symmetry (F1​=F2​=Fbase​).
  • The angles formed by each of the base cables with the horizontal are equal (θ1​=θ2​=θ).

For horizontal equilibrium: 2Fbase​Cos(θ)=0 This implies either Fbase​=0 or Cos(θ)=0, but in a stable, non-collapsed structure, Fbase​ is not zero. Instead, this demonstrates that for purely vertical forces (like in the strut), the horizontal components cancel out.

For vertical equilibrium, assuming the strut applies a force Fstrut​ downwards: 2Fbase​Sin(θ)=Fstrut​ This can be rearranged to find Fbase​, given Fstrut​ and θ.

To calculate the stress in the base cables: σbase​=Fbase/A​​

This example simplifies the real-world complexity of tensegrity structures but illustrates the fundamental principles of calculating tensile stresses without directly resorting to a Cartesian coordinate system.

FOOTNOTES

  1. This scaling is fractal (self-similar across all scales). ↩︎
  2. The SPRIKE discussion in this blog converges on this query. ↩︎
  3. Cartesian XYZ coordinates are not selected for use because planar geometry seems more relevant for stance and movement on a horizontal plane – the ground. ↩︎