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 analysis1 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.