A Practical Guide to Tensegrity Design

Table of Contents

6 Double-Layer Tensegrity Domes

Chapter 7

Tensegrity Member Force Analysis

7.1 Force Analysis: Introduction

A method for ascertaining the forces in the various members of a tensegrity structure is useful to the builder. It allows the builder to make a sensible choice of materials for the different members which will meet the requirements of the loads the members will have to bear. In early design stages, force analysis will point up any overloaded members in the structure as well as situations where a member is bearing no load or a load which is not appropriate to it (for instance when calculations show a tensile member is bearing a compressive load). Force analysis aids the formulation of an assembly strategy: it is easier to install the tighter members earlier when they bear less of their full load.

The gross analysis^{1}
of forces in a tensegrity structure is comparatively
simple due to the flexible interconnection of the
members. Shear forces can be neglected, and
only the axial tensile and compressive forces need to be taken
into account.^{2}
However, a detailed analysis of a tensegrity, for example of the various
parts of a hub, may require attention to shear forces. Since only
axial forces are considered in the analyses here, in the interest
of simplicity, the terminology used takes a small
freedom: sometimes when a "force," technically a vector-valued
quantity, is discussed in the chapters of this book, what is actually meant is
a signed magnitude — a scalar value — corresponding to the force.
This seems
permissible since the force always coincides with the direction of the member,
and if the magnitude is known, the corresponding vector-valued force can
easily be computed. When forces at hubs are summed, the analysis will require
the vector-valued force to be computed explicitly; but in many places, just
referring to the magnitude is very sufficient, and the context should make
it clear when a scalar is being referred to and when a vector is being
referred to.

In most non-tensegrity trusses, the forces in the members of the truss are
only due to the propagation through the structure of external loads
**exogenous** to the structure
such as the force of gravity and the foundation of the structure
pressing up against it. However, tensegrity structures are prestressed so
that an additional portion (and, in some applications, the total portion) of
the force in a member can be attributed to the structure
itself. This is due to the fact that a tensegrity structure
relies on the isometric straining of the inwardly pulling tensile
members against the outwardly pushing compression members to
create a stable structural system. The geometry of the structure
determines the relative magnitudes of the member
forces due to these **endogenous** factors.

So, in analyzing the forces in a tensegrity structure,
both **exogenous** and **endogenous** factors must be taken into account.
The analysis of the endogenous forces is derived directly
from the model used for computing tendon lengths and
is discussed first. The analysis of exogenous forces
is discussed second since it presumes the analysis of endogenous
forces has already been done. Stress, that is force divided by the
cross-sectional area of a member, is completely neglected since
once a force has been computed, it is very simple to reinterpret
it as a stress by dividing by the appropriate cross-sectional area.

^{1}
Recall from Section 3.1.6
that in the gross analysis, where the structure is considered in a more
abstract way, the details of strut-tendon connections are omitted,
and the hubs are considered to be simple points. A more detailed
analysis would take into account the details of the structure of the hub
where struts and tendons are connected. Such a detailed analysis will
be undertaken in
Section 7.3.5
to more accurately model the effects of exogenous forces.

^{2}
For example, see *Chajes83*,
pp. 36-37. "Axial" means the direction of the force
coincides with the direction of the member.