The use of tensegrity in structures is not all that common at this time and often surprises people who find out about tensegrity systems. Sometimes it seems that the structures using tensegrity should not even be stable as the heaviest elements almost float in space. The geometry of a tensegrity structure uses a multiplication of cells of identical composition but different sizes organized about the vertical axis. A tensegrity structure is a 3-D truss which has struts in compression that transfers the loads to tensioned cables which work together in order to transfer the loading of the structure to the ground. The tensioned cables offer support to the compressed struts of the structure to support the loads that can be applied to the system. Due to the way the struts and cables work with one another, the structure can be set spatially and be self-equilibrated. Even though the structure may have these positives, it still is not used as a typical structural system due to its flaws. By examining the tensegrity structure and the concepts behind it, one can hope the future use of tensegrity structures can be broadened and allow for more awe inspiring structures to be built.

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## Introduction

Architects and engineers have pushed the limits of what is possible for many years and have attempted to make new and awe inspiring structures at the same time. One such structural system is known as tensegrity. Tensegrity is not a new structural system as it has been around for roughly fifty years, however, it is not widely used for common structures such as buildings and bridges. "Tensegrity structures are spatial structures formed by a combination of rigid struts and elastic ties. No pair of struts touches and the end of each strut is connected to three non-coplanar ties. The struts are always in compression and the ties in tension (Crane)." These structures are complex in nature and calculating the equilibrium for these structures can be very complex and involved.

Figure - Elementary Equilibrium Element (Motro)

## 1.1 Definition and Description

The simplest of tensegrity structures, shown in Figure 1, is composed of three compressed struts of equal length and nine tensioned ties also of equal length. These lengths of the struts and ties are different, but create a golden ratio between the length of the struts and ties of 1.468. If the ratio of these lengths were to be smaller than 1.468, then the system will not be rigid making it impossible to give a shape to it. However, if the ratio was larger, then that would result in a system that would be very difficult or even impossible to assemble (Motro).

Anthony Pugh provides a useful definition of tensegrity that can be utilized in further discussing tensegrity structures. Pugh states "a tensegrity system is established when a set of discontinuous compression components interacts with a set of continuous tensile components to define a stable volume in space (Pugh)." As can be seen from Figure 1, the structure is standing on its own and conforms to the definition that Pugh provides for tensegrity structures. This definition can be modified to be even more accurate to state that "a tensegrity system is a system in a stable self-equilibrated state comprising a discontinuous set of compressed components inside a continuum of tensioned components (Motro)." The component that both definitions describe is the compressed struts and tensioned ties, which can be a combination of several simple components assembled to form a more complex structure.

## 1.2 Description based on patents

Several patents for tensegrity structures had been taken out around the same time by Richard Buckminster Fuller of America, David Georges Emmerich of France, and Kenneth Snelson of America. All three men had similar descriptions and details of these "self-stressing systems (Maculet)." Of the three men, the patent name that ultimately crafted the word "tensegrity" was Fuller's patent for "Tensile Integrity." Emmerich had applied for his patent about the same time as Fuller for "Construction de réseaux autotendants," which roughly translated is "Construction of self-stressing networks," and Snelson's patent was for "Continuous tension, discontinuous compression structures."

Figure - Emmerichs' (top left), Fullers' (top right) and Snelsons' (bottom) patents (Motro)

All three patents had described the same system and can be combined into one definition that gives a very accurate description of a tensegrity system. "Tensegrity Systems are spatial reticulate systems in a state of self-stress. All their elements have a straight middle fibre and are equivalent size. Tensioned elements have no rigidity in compression and constitute a continuous set. Compressed elements constitute a discontinuous set. Each node receives one and only one compressed element (Motro)." This definition is very similar to the definitions provided by Pugh and Motro. Based on this definition, it insists the spatiality of the system and that the components are either in compression or tension. Also, the stiffness of the structure is created by a state of self-stress, independent of all external forces and that there is a golden ratio to the lengths of the members to maintain this rigidity.

## 1.3 Russian Constructivists

Although Emmerich, Fuller, and Snelson were the first three to apply for patents for tensegrity, the history of tensegrity seems to go back even further to the Russian constructivists. An exhibition in Moscow in 1921 held a "sculpture-structure" by Karl Ioganson, Figure 3 below, that was a self-equilibrated structure and had the basic components of a tensegrity structure. Even though Ioganson's sculpture does not relate to the normal characteristics of tensegrity, the states of static equilibrium of tensengrity can be based on the compressed struts and the construction of the sculpture helps to explain the mechanisms of the system. "According to structural morphology, it illustrates the fact that several shapes can be linked to a single structure (Motro)."

Figure - Ioganson sculpture (Motro)

## Principles of Tensegrity

A tensegrity structure has a more complex set of equations and conditions that must be met in order to ensure that the system is in equilibrium. Many engineers have devoted much time and effort into developing the necessary equations to properly design a tensegrity structure like a simple bridge or building. As the equations can be very messy and take a while to explain, the basics of various conditions are presented here.

## 2.1 Self Equilibrium

Stability for a tensegrity structure cannot be validated easily compared to trusses and cable nets. A tensegrity structure is stable if the quadratic form, Q, of the tangent stiffness matrix, K, with respect to any non-trivial motion, d, is positive so that (Thompson):

If Q is equal to zero, the structure may be stable, however would require further investigation in higher-order terms of energy. Furthermore, a structure is considered to be degenerate if it can lie in a space with lower dimensions, such as a two dimensional truss on the x-y plane but placed in the x-y-z space. This is important for tensegrity because if a structure is non-degenerate, then its nodal coordinates in different directions are linearly independent (Zhang). The topology of a tensegrity structure can be modeled as a directed graph described by the connectivity matrix, C. If nodes i and j are connected by member k, then the ith and jth elements in the kth row of C are equal to 1 and -1 respectively, while all other elements in the row are zero (Kaveh).

Figure - Non-degenerate two-dimensional tensegrity structure (Zhang)

If any non-trivial displacement vector exists so that dâ‰ 0 that does not change the member lengths as:

Then d is considered a mechanism and the structure is therefore kinematically indeterminate, which is commonly the case in tensegrity structures (Pellegrino). The linear stiffness matrix, KE, and geometrical stiffness matrix, KG, are formulated using D and E, the self-equilibrium equation of the structure with respect to the nodal coordinate vectors.

Where Ç¨ is a diagonal matrix consisting of modified axial stiffness of the members, Id is an identity matrix, and âŠ- denotes tensor product (Zhang and Ohaski). If one was to consider only conventional materials that does not cause zero or negative axial stiffness, then Ç¨ is positive definite and KE is positive semidefinite. Positive-semidefinite means that the eigenvalues of the matrix are all non-negative values. If KG is positive semidefinite, then Q>0 will not be satisfied if and only if there exists at least one mechanism that lies in the null-space of KG (Zhang). This makes the need for higher order terms of energy to be considered to determine if the system is in equilibrium.

## 2.2 Affine Motion

"An affine motion is a motion that preserves colinearity and ratios of distances; i.e., all points lying on a line are transformed to points on a line, and ratios of the distances between any pairs of the points on the line are preserved (Weisstein)." By this definition, the affine motions of a tensegrity structure is some linear movement, whether it is a translation or rotation, as shown in Figure 5, or a dilation or shear, as shown in Figure 6. The rotation and translation affine motions maintain the lengths of the members, however the dilation and shear are considered to be non-trivial affine motions.

Figure - Translation and Rotation Affine Motions of 2-D Tensegrity Structure (Zhang)

Figure - Dilation and Shear Affine Motions of 2-D Tensegrity Structure (Zhang)

Translation and rotation are known as rigid body motions of a structure. Translation is simply displacing the structure a certain distance, i, in the x-, y-, or z-direction. The translation vectors in three dimensional space are as follows:

(5)

Rotation about the axes is also a rigid-body motion, however only the geometrical stiffness matrix is considered here. For the geometrical stiffness matrix, node i is rotated about the z-axis by an arbitrary angle Î¸ and has the following relationship between its old and new coordinates:

(6)

Where c=cosÎ¸ and s=sinÎ¸ (Zhang) and r is the geometrical stiffness matrix described. So, by letting X and áºŠ be the vectors for all of the nodes old and new coordinates, respectively, then the relationship between the old and new coordinates is:

(7)

Where R is an identity matrix:

(8)

Therefore, the motion d' that rotates the structure from the original configuration to the new one can then be written as:

(9)

Dilation causes the structure to expand or contract. This displacement is similar to that of translation; however as it moves in one direction while the other stays constant, the values for each matrix will be different. These motions, dx, dy and dz of a structure can be written as follows:

(10)

There is only one shear possible for the two dimensional case, as shown in Figure 6(c), but with a three-dimensional tensegrity structure, three shears are possible: dxy, dxz and dyz and are defined as follows:

(11)

From equations (10) and (11) above, it can be said that the non-trivial affine motions of dilation and shear are linearly independent of the rigid-body motions of translation and rotation from equations (5) and (9).

## 2.3 Equilibrium of properly loaded structures and structures with no external loading

The equilibrium of a tensegrity structure with properly loaded strings is a function of force densities (Î»), member node incidence matrix (M), and node vectors (p) (Masic). Force density is a scalar multiplier that when multiplied by the element vector, gi, produces the element force vector, fji, which corresponds to the contribution of the internal force of component, ei, to the balance of the forces at node, vj. The node vectors are then collected into a matrix according to the following definition:

(12)

Furthermore, fe and fc is the collection of external force vectors and constraint forces that act on a node, respectively, defined similarly to the node vectors matrix as:

(13)

The equilibrium of a tensegrity structure when the strings in the configuration p are properly loaded can then be written as:

(14)

In order to relate the properly loaded equilibrium to the rare case when there are no external forces applied to the structure, constitutive equations must be formed to relate the proper constraint. "The relationship between the force-density variables and actual structure parameters depends on the strain-stress relationship for the material used to build the elastic elements of the structure (Masic)." Using Hooke's law for linear elastic materials, the force densities at any configuration can be computed with vi, l0 and yi representing the volumes, rest lengths, and Young's modulus of cylindrical elements so that:

(15)

Using the matrices from equation (15), the force densities for no external loads can then be computed as follows:

(16)

With the new force densities calculated, the constraint in equation (14) then becomes:

(17)

So that when there is no external loading applied (i.e. fe=0), the equilibrium condition in equation (14) becomes:

(18)

## 2.4 Force equilibrium and force coefficients

Tensegrity structures biggest concern is the compressive forces sustained by the struts of the structure. So, given the tensile forces, tn, in the ties characterized by the string vector, sn, and the compressive force, fn, in the strut characterized by the bar vector, bn, the tensile force coefficient Î³n>0 and the compressive force coefficient Î»m>0 (Williamson) are defined as

(19)

Forces in a tensegrity structure are defined by the external force vector, w, compression vector, f, and tension vector, t, where

(20)

The diagonal matrices {Î“,Î›} are the tensile and compressive coefficient force matrix, respectively, such that:

(21)

Using the information from equations (20) and (21), equation (19) can be rewritten and expressed as:

(22)

## 2.5 External force application

There has been a lot of work completed until recently that considered no external loading applied to a tensegrity structure. As the members of a tensegrity structure is either a compressive strut or tensioned tie, the application of external loads need to be considered for a complete static analysis to be completed for a tensegrity structure. The external forces and moments applied to a tensegrity structure can be determined using the principles of virtual work. In the simplest form, the virtual work equation is:

(23)

Where Î´WF is the total virtual work done by the forces, Î´WM is the total virtual work done by the moments, and Î´V is the total potential energy associated with the forces. Since the struts are considered massless the term related to the potential energy in the principle of virtual work is the resultant of the elastic potential energy contributions given by the ties (Crane).

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## 2.6 Complexity of equilibrium

The equilibrium conditions and equations tend to grow exponentially in complexity the more that an analysis is applied to the structure. Each of the topics discussed thus far is an attempt to simplify what is needed to determine the equilibrium of any tensegrity structure. Many of the equations involve numerous matrices and variables to fully analyze a tensegrity structure. Figure 7 below shows a typical strut with the base constrained from movement on the horizontal axis and rotation about the longitudinal axis. This shows how complex the external loading can be when virtual work is used to determine the applied forces.

Figure - Free Body Diagram for an Arbitrary Strut Modeled with a Universal Joint (Crane)

The equations to determine the force applied externally to a tensegrity structure has a tendency to get quite lengthy as each item in the simple equation is a function of the free length (LS), each individual top (T), bottom (B), and lateral (L) tie, the rotations about the x (Îµj) and y (Î²j) axis and horizontal displacements, aj and bj. The equation can only be solved iteratively by using a proper set of values from Yin's work with unloaded position structures. Because of the complexity of the equilibrium equations it is essential to verify the answers obtained. An independent validation of the results can be accomplished using Newton's Third Law. If there are no external moments acting on an isolated strut, it is sufficient to perform the summation of moments with respect to the lower end of the strut. If there are external moments the verification process involves additional steps (Crane). As an example of how much complexity is involved with the external loading applied to a tensegrity structure, the virtual work done by the external forces, Î´WF, of equation (23) expands as follows with all the forces of Figure 7 considered:

(24)

The virtual work done by the external moments and potential energy equations expand similarly to the external forces and has many parameters that are needed in order to properly calculate the external load applied to the tensegrity structure.

## 3. Stability and Rigidity of Tensegrity

Much like the equilibrium equations required for tensegrity structures, the need to ensure that the stability and rigidity of the structure is proper so that the structure does not just fall apart when assembled. There are several approaches to determining the both the stability and rigidity as well as certain conditions that are required to maintain the proper structure.

Figure - Extended Tensegrity Framework Rigidity and Stability Hierarchy (Juan)

## 3.1 Static analysis

There are three primary approaches used to ensure that the stability and rigidity of a tensegrity structure either constrains the possible motions or the member stresses to reach a given configuration. The first approach involves the motion of the structure. Rigidity means the absence of relative motion between the members of a structure, which implies that the lengths of the members linking the vertices of the framework are kept constant (Juan) such that:

(25)

Where pi and pj are the placement in the i and j directions and cij is the squared length of the {ij} edge. From a motion point of view, a tensegrity framework is said to be rigid if all the neighbor configurations, qi, of a given configuration p are congruent (Juan). If this not the case, the tensegrity structure is flexible and has a set of non-rigid motions. Since equation (25) only deals with non-linearities rigidity conditions, it is useful to use the first derivative of equation (25) to ensure the velocities, pi'(0) and pj'(0), at time t=0 satisfy the conditions for it and remains true for higher order derivatives:

(26)

From this derivative, a tensegrity structure can be determined if it is first order rigid structure or if it is an infinitesimal flex structure. Higher order derivatives of equation (25) determine the higher levels of rigidity as according to Figure 8 (Juan).

The second approach that can be used is the force approach. This is an extension of the structures equilibrium as the tensegrity framework must be in equilibrium at each of its vertices in order for it to be a rigid configuration. Using dij0 and Tij as the rest length and internal tension for each adjacent member, respectively, the equation for the forces applied is:

(27)

For any external equilibrium force, Feq, ext, and proper stress, Ï‰, and using force density coefficients introduced by Schek, which represent the force per unit of length that vertex j applies to vertex i, then the equilibrium condition for the framework can then be stated as:

(28)

If all of the equilibrium forces are resolvable, then the framework is statically rigid. "Any tensegrity framework can compensate for an external force in two different ways: keeping the initial configuration but modifying the stress present in each member, or directly modifying the initial configuration. Only the external forces verifying the former condition are considered resolvable for a given tensegrity framework, since, in the second case, the framework itself is modified in order to withstand the external load (Juan)." Static rigidity is only preserved for affine transformations, not projective transformations or orthogonal projections. It has been proven that a tensegrity framework is statically rigid if and only if it is infinitesimally rigid.

The third method of determining the rigidity of a tensegrity structure is the energy approach. This takes into consideration the previous methods, but expressed in terms of energy instead of force or motion. It should be noted that the energy in a cable (tie) increase when stretched, energy of a strut increase when shortened and the energy of a bar increases under a length change, similar to what is shown in Figure 9 below (Juan). In terms of energy, the local minimum of the energy function associated to the tensegrity framework corresponds to a different configuration. The framework is said to be globally rigid if there is only one minimum in the configuration. The primary function used for determining the energy in a tensegrity member is:

(29)

Similarly to the force and motion approaches, the derivatives of equation (29) have to satisfy certain conditions to ensure a proper self-stress for the tensegrity framework. The energy approach does deal with the physical stiffness of the framework and leads to many of the aspects of the force and motion approaches.

Figure - Energy Functions for Cables, Bars, and Struts (Juan)

## 3.2 Conditions

A necessary condition for non-degenerate tensegrity structures is based on positive definiteness of the tangent stiffness matrix along with two other conditions will ensure that the tensegrity structure is stable. Considering the non-trivial affine motion, d, in three-dimensional space is a linear combination of the six non-trivial affine motions due to the fact that they are linearly independent (Zhang)

(30)

Since the affine motion, d, multiplied by the geometrical stiffness matrix is equal to zero, the quadratic form of the tangent stiffness matrix can be reduced:

(31)

When the tensegrity structure is created of conventional materials, then Q is either positive or zero. If there is a non-trivial affine motion that satisfies DTd=0, then the member lengths are invariant by the motion of equation (31). Since Q cannot have a negative value due to the linear and geometrical stiffness matrices being positive semidefinite, then the force density matrix is positive semidefinite, which is one of the stability conditions. If the affine motions span the whole null-space of the geometrical stiffness matrix, then the force density matrix has at minimum rank deficiency of d+1, another stability condition. Also, since no non-trivial affine motion in the geometrical stiffness matrix leads to Q=0, then the rank of the geometry matrix, G, is d(d+1)/2, the third stability condition (Zhang). Similarly, the third condition can be replaced by one by Connelly which states that the member directions do not lie on the same conic at infinity (Connelly).

## 4. Current Uses of Tensegrity

Tensegrity currently is not used widely in structures such as buildings and bridges. Although uncommon in such applications, it still has been used for the Kurilpa Bridge in Australia (Figure 10). The Kurilpa Bridge is a pedestrian tensegrity cable stay bridge spanning 1,540 feet over the Brisbane River. Although this is just a pedestrian bridge, the application of tensegrity in the design is still apparent. The masts that reach above the deck are in compression while the cables connected to the deck are in tension. Tensegrity is also sometimes used as artwork much like the Ioganson sculpture of Figure 3 as well as other artistic uses such as the Needle Tower by Kenneth Snelson.

http://upload.wikimedia.org/wikipedia/commons/d/dd/KurilpaBridge1.JPG

Figure - Kurlipa Bridge (Source: http://en.wikipedia.org/wiki/Kurilpa_Bridge)

Although the Kurlipa Bridge is a rare example of the application of a permanent tensegrity structure, tensegrity is more commonly used for deployable structures such as antennas, retractable roofs and towers. A tensegrity tower can have three to ten stages composing the tower, see Figure 11. A tensegrity tower has three bars forming a triangular base, attached by a ball and socket joint to the base and is attached similarly at the top (Sultan). In a three stage tower, the bars have a total of thirty-three ties connecting the structure together.

Figure - Three Stage Tensegrity Tower (Sultan)

Another deployable tensegrity structure is known as a SVD tensegrity structure (see Figure 12). The term SVD describes the directions that the ties run; saddle (S) runs along the edge at an incline connecting two stages together (A32 to B11 in Figure 12), vertical (V) runs vertically from one level to the next (A31 to B21), and diagonal (D) runs diagonally from one bar to the next (A11 to A32). This is in effect a slightly different type of tower that is possible compared to the tensegrity tower.

Figure - Two Stage SVD Tensegrity Structure (Sultan)

## 5. Conclusion

Tensegrity structures are unique in their design and fundamental principles as they have the necessity to maintain self-equilibrium and have only tension forces in the cables and compressive forces in the struts. Even though there is a fair amount of information and equations associated with tensegrity at this time, the uses of it are still fairly limited to deployable objects and artistic uses. Even though tensegrity has been so limited in its uses to this point, the desire to use tensegrity for far greater uses, much like the Kurlipa Bridge, is being pushed for by architects and engineers alike.

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