## Abstract

The structural performance is explored for a reticulated circular tube made from a periodic lattice: triangulated; hexagonal; Kagome; and square lattices. The finite-element (FE) method is used to determine the macroscopic bending, torsional and axial rigidities of each tube. Additional insight is obtained by examining the structural mechanics of the pin-jointed version of each topology. For all pin-jointed lattices considered, no states of self-stress exist. However, collapse mechanisms do exist for all reticulated tubes, and for the Kagome and hexagonal lattices some of these mechanisms produce macroscopic generalized strain. These strain-producing collapse modes are additional to those observed in the planar version of these lattices. Consequently, the structural rigidities of tubes with walls made from the rigid-jointed Kagome lattice or hexagonal lattice are less than those predicted from the in-plane effective properties of these two lattices. The morphing capacity of reticulated tubes is also explored by replacing a single bar with an actuator in the FE simulations. The actuation stiffness of the structure is defined by the stiffness of the reticulated tube in resisting extension by the actuated bar. The actuation stiffness is explored as a function of the type of lattice, number of unit cells around the circumference, orientation of the actuated bar and of the bar stockiness. In all cases, the macroscopic shape change of the tube can be idealized as a combination of a local rotation, axial extension, axial twist and shear displacement of the cross-section.

## 1. Introduction

Recently, lattice materials have emerged as an attractive candidate for application in shape morphing technology (Dos Santos e Lucato *et al*. 2004; Wicks & Hutchinson 2004; Symons *et al*. 2005*a*,*b*). In this study, we shall explore the morphing capability of reticulated tubes by replacing a single bar in the wall of the tube with an extensional actuator. Consequently, the structural stiffness of reticulated tubes is of direct interest. The structural rigidity of prismatic structures such as circular cylinders is traditionally calculated from the in-plane effective properties of the wall. While this is accurate for solid walls, it is unclear whether the approach remains valid for walls made from a reticulated framework, such as a hexagonal honeycomb. We shall address this unresolved issue in the paper and show that tubes with walls made from a lattice material can have a substantially degraded structural rigidity due to the activation of additional compliant modes.

As a first step, the macroscopic structural rigidity is explored for circular tubes with walls made from a number of competing topologies of lattice material: triangulated; hexagonal; Kagome; and square lattices (figure 1). The triangulated, hexagonal and Kagome lattices have isotropic, in-plane properties due to their 120° symmetries, while the square lattice is orthotropic. The structural rigidity is determined for axial extension, bending and torsion by finite-element (FE) analysis, and is then compared with the analytical prediction based on the in-plane effective properties.

The in-plane properties of a wide range of two-dimensional lattices have been explored in recent publications (Gibson & Ashby 1997; Romijn & Fleck 2007); in contrast, little is known about their behaviour in a three-dimensional structure. These effective properties scale with the relative density as summarized in table 1; the expressions are taken from Gibson & Ashby (1997) for the triangulated, hexagonal and square lattices, and from Romijn & Fleck (2007) and Srikantha Phani *et al*. (2006) for the Kagome and square lattices.

The in-plane stiffness of the lattices is sensitive to the nodal connectivity, as follows. The low nodal connectivity *Z*=3 of the hexagonal lattice endows the lattice with a compliant deviatoric response, such that the in-plane Young's modulus and shear modulus scale with . This can be traced to the fact that the response is governed by *bending* of the individual bars. The Kagome and triangulated lattices have higher nodal connectivities of 4 and 6, respectively, and their effective properties scale linearly with : these microstructures *stretch* under in-plane macroscopic loading. The square lattice stretches under axial deformation, and so its axial Young's modulus *E*^{*} scales with . In contrast, its bars bend under shear loading parallel to the principal axes, and so the shear modulus *G*^{*} scales with .

Recently, the actuation response of planar grids and of three-dimensional plates has been studied. For example, Wicks & Guest (2004) have explored the actuation behaviour of the two-dimensional triangulated, Kagome and hexagonal lattices. They extended a single bar by imposing a thermal strain, and they found that the stiffness of the surrounding structure is the highest for the triangulated lattice and is the least for the hexagonal geometry. The triangulated structure deforms predominantly by bar stretching while the hexagonal lattice undergoes bar bending. The Kagome lattice deforms by a mode in which some bars bend and others stretch near the actuated bar, and thereby has an intermediate actuation stiffness.

Likewise, the morphing capability has been determined for a Kagome double-layer grid by Symons *et al*. (2005*a*,*b*), and for a sandwich plate comprising a tetrahedral truss core, a solid face and a Kagome face by Hutchinson *et al*. (2003) and by Dos Santos e Lucato *et al*. (2004). Symons *et al*. (2005*a*,*b*) compared the actuation response of a pin-jointed Kagome double-layer grid with the rigid-jointed version. They demonstrated that the rigid-jointed version of a statically and kinematically determinate pin-jointed structure inherits a morphing capability: it combines a high passive stiffness under remote macroscopic loads with minimal resistance to actuation of one or more bars. This motivates us to determine whether pin-jointed reticulated tubes contain internal collapse mechanisms and states of self-stress, and to determine the implications of this upon their structural rigidity and morphing capability.

The scope of the present paper is as follows.

Circular lattice tubes are described with wall topology comprising triangulated, hexagonal, Kagome or square lattices. Linear algebra is used to determine the number of states of self-stress and internal collapse mechanisms in the pin-jointed versions.

The structural rigidities (axial, bending and torsional) are calculated for rigid-jointed lattice tubes using FE simulations, and are compared with the existing analytical predictions based on the in-plane effective properties of two-dimensional lattices.

The morphing capacity of reticulated tubes is explored by replacing a single bar with an extensional actuator in FE simulations. The actuation stiffness is compared with that of the infinite, planar lattice and the macroscopic mode of actuation is quantified in terms of a rotational hinge, an axial extension, axial twist and transverse shear of the cross-section.

### (a) Geometry of reticulated tubes

Consider a circular cylindrical tube of length *L* and radius *R*, generated by the wrapping of a periodic lattice: triangulated (topology *A*); hexagonal (topology *B*); Kagome (topologies *C* and *D*); or square (topology *E*) as defined in figure 1. The joints and axial bars of the tubular lattices lay on a circular cylinder, with the non-axial connecting bars deviating from this surface. All bars are straight and of length *l*, and for definiteness have a square solid cross-section of dimension *t*×*t*. The second moment of area is *I*=*t*^{4}/12 and the radius of gyration is . We shall present our results in terms of the stockiness *s*≡*g*/*l*, to increase the applicability of our findings to bars of other cross-section, as discussed by Wicks & Guest (2004). A Cartesian reference frame is defined in figure 1, with the *x*_{3}-direction along the longitudinal axis of each tube.

The triangulated, Kagome and hexagonal lattice tubes are arranged such that a portion of their bars lie along the longitudinal or circumferential directions. We explore the significance of this for the Kagome lattice: topology *C* denotes the longitudinal orientation and topology *D* denotes the circumferential orientation (figure 1*c*,*d*, respectively). For the triangulated and hexagonal lattices, we limit attention to the longitudinal orientation (see figure 1*a*,*b*, respectively).

The geometry of each tube is specified by the relative density of the parent planar lattice, and by the number of bars *p* circumnavigating the tube. Suppose we complete one full circuit of the tube along the circumferential orientation by moving from one bar to the next. The shortest such journey involves *p* bars. The examples shown in figure 1 are for *p*=8. The radius *R* of the mid-surface of the tube scales with (*l*, *p*) according to(1.1a)for topologies *A*–*C* and(1.1b)for topologies *D* and *E*. It is clear from the above two formulae that for a fixed value of length *l* of the bar member, the circumferential curvature 1/*R* of the lattice tube tends to zero as *p* tends to infinity. In this limit, the tubes become infinite planar lattices.

## 2. Structural analysis of pin-jointed lattice tubes

We begin by treating the lattice tubes as *pin-jointed* and use linear algebra to determine the number of independent states of self-stress and the number of independent collapse mechanisms. Insight into the structural performance of the rigid-jointed versions is thereby obtained. Here, we are interested in the practical case of tubes of finite length and we shall investigate the dependence of the number of collapse mechanisms and states of self-stress upon the number of circumferential bars *p* and upon the number of unit cells *q* along the tube length. It is recognized that methods exist to analyse the infinitely long, pin-jointed lattice tube: a Bloch-wave analysis could be used to identify harmonic collapse modes that do not produce macroscopic strain, and the infinite wavelength limit could be used to explore macroscopic strain-producing collapse modes such as axial extension of the tube. Analyses of these types have been performed for planar two-dimensional lattices (see Hutchinson & Fleck 2006), but are not pursued further here: the main details are gleaned from an analysis of finite tubes.

A unit cell of each lattice tube is shown by dashed lines at the left-hand end of the tube (figure 1*a–e*). The unit cells shown give the smallest arrangement of bars, which can be tessellated along the longitudinal and circumferential directions in order to construct the full structure. (Smaller unit cells could be constructed by tessellation along intermediate directions.) For example, the unit cell of topology *A* (the triangulated tube) has two axial bars and four diagonal bars, while that of topology *D* (the circumferential Kagome tube) contains four circumferential bars and eight diagonal bars (compare figure 1*a*,*d*). A small number of additional bars are added to the other end of the tube in order to complete the lattice pattern for each topology. These additional bars are shown in grey-dashed lines at the right-hand end of each tube (figure 1*a–e*) in order to close the pattern rather than leave dangling bars connected by single nodes. We shall show below that these additional bars are not sufficient to eliminate local mechanisms at the end of the tube, and so-called ‘patching bars’ are also needed. Formulae for the number of the bars *b* and number of joints *j* are given as a function of (*p*, *q*) in table 2, for each topology.

The methodology of Pellegrino & Calladine (1986) and Pellegrino (1993) is now applied to construct the equilibrium matrix for each pin-jointed topology. The rank and the fundamental subspaces of the equilibrium matrix are obtained via the singular value decomposition (SVD), as implemented within the software package Matlab (MathWorks 2004 Matlab user's guide, v. 7). The rank of the equilibrium matrix is closely related to the number of states of self-stress *s* and to the number of internal mechanisms *m*, as follows. The Maxwell–Calladine relation (see Calladine 1978) reads as(2.1)where *b* is the total number of bars and *j* is total number of joints in the structure. An SVD analysis of all topologies *A*–*E* reveals that there are no states of self-stress (*s*=0) within any lattice tube. Consequently, *m* is given directly in terms of (*b*, *j*) by equation (2.1). Recall that (*b*, *j*) can be expressed in terms of (*p*, *q*) via the formulae given in table 2. Thus, *m* scales with (*p*, *q*), and these dependencies are included in the table.

We note from table 2 that the number of collapse mechanisms increases linearly with the number of unit cells *q* along the length of the tube, except for the triangulated lattice, topology *A*. This lattice has collapse mechanisms, independent of *q*. By patching additional *p*−3 suitable bars to each end of this tube (not shown in figure 1*a*), these collapse mechanisms can be eliminated, and the structure can be made *kinematically determinate* (it is already *statically determinate* since *s*=0). No such simple end-patching procedure can be applied to make the remaining topologies *kinematically determinate*.

## 3. Structural rigidity of lattice tubes

The macroscopic structural rigidity of reticulated tubes has been explored by FE simulations (the commercial FE software Abaqus (Abaqus standard user's mannual, 2004 v. 6.4) is used), and the calculated values have been compared with analytical predictions using the in-plane effective properties of the two-dimensional lattices. The five types of lattice tubes shown in figure 1 were subjected to axial tension, axial torsion and bending about the *x*_{1}-axis (figure 2*b*), and the associated structural rigidities were obtained. In all cases, the tube length *L* is much larger than the bar member length *l*; for topologies *A*, *B*, *C* and *E*, the tube length is *L*=100*l*, while for topology *D*, *L*=84.1*l*. Each bar is simulated by a single two-noded Euler–Bernoulli beam element. These linear elastic elements have cubic interpolation formulae and are of element type *B33* in Abaqus notation. Appropriate displacements are prescribed on the ends of each tube in order to generate axial tension, transverse bending and axial torsion, as sketched in figure 2*a–c*, respectively. The FE method is used to obtain the work-conjugate resultant loads. All simulations assume infinitesimal displacements.

### (a) Imposed loading on tubes

#### (i) Axial tension

The two ends of the reticulated tube are extended by a relative displacement of 2*u* along the *x*_{3}-direction, as shown in figure 2*a*. The *effective axial rigidity in tension* (EA)_{eff} of the tube is defined in terms of the computed axial resultant force *T* as(3.1)It is instructive to compare the axial rigidity of the lattice tube with that of a solid-walled tube of equal radius *R* and wall thickness *t* to that of the lattice tube. The solid wall is endowed with the effective in-plane moduli (*E*^{*}, *G*^{*}) of the two-dimensional lattice plate (given in table 1). The cross-sectional area *A*_{s}, second moment of area *I*_{s} and polar moment of area *J*_{s} of the solid-walled tube are given by the usual thin-wall approximations(3.2)The effective axial rigidity (EA)_{eff} of the lattice tube is normalized by the axial stiffness *E*^{*}*A*_{s} of the equivalent continuum tube to obtain(3.3)

#### (ii) Transverse bending

The lattice tube is subjected to transverse bending by rotating the cross-section of the tube by +*ϕ* at the right-hand end and −*ϕ* at the left-hand end. This rotation is imposed by a distribution of axial displacement, which is linear in the *x*_{2}-direction. The FE calculation gives us the imposed moment *M* on the end cross-section. The imposed curvature on the tube is *κ*=2*ϕ*/*L* and the non-dimensional bending rigidity of the tube is(3.4)

#### (iii) Axial torsion

The two ends of the lattice tube are given a relative rotation 2*ω* about the longitudinal, *x*_{3}-axis. This is achieved by imposing a circumferential displacement to each node of the end faces. The tube responds with an axial torque *Q*, and the non-dimensional torsional rigidity of the tube is(3.5)

### (b) Predictions of structural rigidity for each type of lattice tube

FE calculations have been performed on each topology of figure 1 in order to obtain the dependence of (, , ) upon for the particular choice *p*=8. The effect of the number of circumferential bars *p* upon the response is explored subsequently for the longitudinal Kagome tube in torsion.

The non-dimensional structural rigidities (, , ) are approximately equal to unity for the triangulated tube (topology *A*) and for the square-lattice tube (topology *E*), for all . (There is no need to show these results graphically.) The triangulated tube has a stretching-dominated response under axial tension, torsion and bending; in contrast, the bars of the square-lattice tube stretch when the tube is subjected to axial tension or bending, while the bars bend when the tube is twisted.

Next, consider the hexagonal tube, topology *B*, with *p*=8. The normalized structural rigidities are almost independent of (figure 3*a*). Recall that the two-dimensional hexagonal lattice deforms by bar bending under in-plane deviatoric loading (Gibson & Ashby 1997). The hexagonal tube under axial torsion has the same deformation mode as that of the unwrapped hexagonal plate under shear: the axial bars deform by bending and is close to unity. Under macroscopic tension and macroscopic bending, the hexagonal tube is somewhat more compliant than that given by the in-plane effective stiffness of the two-dimensional lattice. This is traced to the fact that the bars aligned with the axis of the tube undergo local bending, while they remain straight in the two-dimensional planar lattice.

Now consider the longitudinal Kagome tube (topology *C*; figure 3*b*). The non-dimensional axial and bending rigidities are approximately equal to unity, . The interpretation is that the longitudinal Kagome tube under macroscopic tension (and bending) has the same stretching-dominated response as that of the planar Kagome plate. In contrast, the normalized torsional rigidity scales with (figure 3*b*). Similarly, all the non-dimensional structural rigidities of topology *D* (the circumferential Kagome) scale with (figure 3*c*). The dependence of these structural rigidities upon implies that the deformation mode of the tubes involves bar bending, whereas the planar response is stretching governed.

To gain additional insight into the above compliant modes of the Kagome tubes, the sensitivity of the structural rigidity to the number *p* of bars on the tube circumference is explored for the longitudinal Kagome tube in axial torsion. The normalized torsional rigidity of topology *C* is shown in figure 4, as a function of *p* for selected relative densities. increases to unity as *p* tends to infinity: at large *p* the circumferential curvature of the tube diminishes and the planar response is recovered.

Recall from §2 that the Kagome tubes possess a number of collapse mechanisms and some of these can generate macroscopic strain. These strain-producing mechanisms in the pin-jointed limit become local bar-bending modes of the rigid-jointed Kagome. We proceed to examine the local bar-bending modes in the deformed FE mesh of the rigid-jointed Kagome tube at low relative density; significant beam curvatures develop near the joints, thereby indicating the collapse mechanism in the pin-jointed variant.

### (c) Macroscopic strain-producing mechanisms of pin-jointed Kagome tubes

The deformed shape of a rigid-jointed longitudinal Kagome tube, of relative density , has been determined by FE analysis, for the choice *p*=8, *q*=50. To aid identification of the collapse mode, the cylindrical tube in the undeformed and deformed states is unwrapped to eliminate its circumferential curvature. The undeformed Kagome lattice is shown as dashed lines in the unwrapped state, whereas the deformed lattice is shown as solid lines (figure 5). In order to indicate the radial component of nodal displacement (normal to the initial cylindrical surface), additional labelling of the nodes is required: nodes denoted by filled circles move inwards, those denoted by double circles move outwards while those denoted by open circles move radially by less than 5 per cent of the maximum radial displacement of any node. It is clear from figure 5 that the torsional collapse mode involves a combination of circumferential torsion (affine deformation) and a superimposed short-wavelength twist of neighbouring triangular elements about an axis within the cylindrical surface. This short-wavelength twisting motion generates additional macroscopic torsion of the tube and significantly degrades the torsional rigidity of the rigid-jointed tube.

A similar explanation can be invoked for the low torsional rigidity of the circumferential Kagome tube. The deformed lattice for the rigid-jointed version at , *p*=8, *q*=25 is shown in the unwrapped state in figure 6*a*. We note again that the deformed state comprises an affine torsional motion within the surface of the initial cylinder, and an additional superimposed short-wavelength twist of neighbouring triangular elements. The twisting moves joints in the radial direction such that they move out of the plane of the unwrapped configuration. This additional compliant mode reduces the torsional rigidity of the tube. The structural rigidity in tension and in bending of the circumferential Kagome tube is also degraded by the presence of additional collapse mechanisms. A side view of the rigid-jointed circumferential Kagome lattice (, *p*=8, *q*=25) in the initial and deformed states is shown in figure 6*b* for macroscopic tension and in figure 6*c* for macroscopic bending. In the initial configuration, adjacent bars are not co-directional. Consequently, under macroscopic tension of the tube, the bars undergo short-wavelength relative rotations to become more co-directional with their neighbours (figure 6*b*). Likewise, under macroscopic bending of the tube, those bars under tension are pulled into a straighter configuration while those bars under compression rotate away from each other into a more angular configuration (figure 6*c*).

## 4. Single-bar actuation in reticulated tubes

The actuation stiffness of the lattice tubes is now explored. For each tube, a single bar at mid-length of the tube is replaced by an extensional actuator. The actuated bar lies along the axial direction, labelled (I), along the diagonal direction (II) or along the circumferential direction (III), as defined in figure 1. The actuation motion is achieved by deleting the relevant bar and by imposing a small relative extension *u*_{a} of the joints at the ends of the removed bar. The above FE procedure is used to compute the work-conjugate force *f*_{a} on these two joints. The actuation stiffness *k* of the structure is the ratio(4.1)This stiffness is normalized by the axial stiffness *Et*^{2}/*l* of a single bar to obtain(4.2)where *t*^{2} is the cross-sectional area of one bar and *E* is Young's modulus of the parent material.

The normalized actuation stiffness is closely related to the ‘internal resistance to actuation’ as defined by Wicks & Guest (2004) and Wicks & Hutchinson (2004). They define as the elastic energy stored in the lattice and actuated bar, normalized by the elastic energy stored in an actuated bar, if the remainder of the structure is rigid. Elementary algebra reveals that for small .

### (a) The actuation stiffness of each type of lattice tube

FE simulations have been performed on rigid-jointed tubes of length *L*=100*l* for topologies *A*, *B*, *C* and *E*, and *L*=84.1*l* for topology *D*. (Preliminary numerical experiments confirmed that the chosen length of lattice tube is adequate to eliminate the effect of tube length upon the actuation stiffness.) The non-dimensional actuation stiffness is determined as a function of the bar stockiness *s*=*g*/*l* within the practical range of 0.005–0.045. First, for the three topologies *A*–*C* (with *p*=8 circumferential bars) is plotted as a function of the bar stockiness *s* in figure 7*a* for actuation of an axial member and in figure 7*b* for actuation of a diagonal member. Likewise, for the circumferential Kagome lattice (*D*) and the square lattice (*E*) (with *p*=8) is given in figure 8. Second, the sensitivity of to the value of *p* is explored for the single case of the longitudinal Kagome tube (figure 9).

In order to aid interpretation of the results for the lattice tubes, additional calculations have been performed for the actuation stiffness for two-dimensional planar lattices. In the FE simulations, the actuator is located at the centre of the lattice, and is parallel to the longer dimension of the structure. The planar lattice is modelled as a rectangle of width *W*=100*l*; the height is for the triangulated, Kagome and hexagonal lattices, whereas *H*=60*l* for the square lattice. Wicks & Guest (2004) have demonstrated that these dimensions are sufficiently large to represent an infinite plate for the triangulated, Kagome and hexagonal topologies (for *s*≥0.005). The normalized actuation stiffnesses of the planar lattices are included in figures 7–9 as solid lines for reference purposes. We note in passing that our results for the planar lattices *A*–*D* are in excellent agreement with those of Wicks & Guest (2004).

#### (i) Results

First, consider axial actuation of the tubular lattices. scales with *s* to the power of 1/2, 1, 1 and 2 for the triangulated, longitudinal Kagome, square and hexagonal topologies, respectively (figures 7*a* and 8). These values lay below those for the corresponding planar lattice by a factor of approximately 4.

Second, consider for diagonal actuation of the tubular lattices. scales with *s* to the power of 3/2, 2, 2 and 2 for the triangulated, longitudinal Kagome, circumferential Kagome and hexagonal topologies, respectively (figures 7*b* and 8). For the hexagonal lattice tube, is below that for the two-dimensional planar lattice by a factor of approximately 4, as already noted for the case of axial actuation. In contrast, there is a much larger drop in stiffness upon switching from a two-dimensional lattice to a tubular lattice for the triangulated, longitudinal Kagome and circumferential Kagome topologies. Recall that scales with *s* to the power of 0, 1 and 1 for the planar triangulated, longitudinal Kagome and circumferential Kagome topologies, respectively.

Third, consider for circumferential actuation of the tubular lattices with square and circumferential Kagome topologies (figure 8). Again, the tubes are much more compliant than the two-dimensional plates. For both lattices, scales with *s* to the power of unity for the two-dimensional plate and to the power of 2 for the tube.

Finally, consider the sensitivity of to the number of circumferential bars *p* for the longitudinal Kagome tube (figure 9). When an axial bar is actuated, is proportional to *s* for all values of *p*, and tends towards the two-dimensional Kagome plate limit with increasing *p* (figure 9*a*). In contrast, is highly sensitive to *p* for the case of actuation of a diagonal bar (figure 9*b*). For low values of *p*, scales as *s*^{2} whereas for large *p*, increases linearly with *s*.

#### (ii) Interpretation of the actuation stiffness in terms of the deformed shape of tubes

Insight into the actuation stiffness is obtained by examining the deformed shape of lattice tubes at low bar stockiness. We consider each topology in turn. The triangulated tube actuates in a more compliant manner than that of the planar lattice due to the radial *bending* of arrays of adjacent diagonal bars (figure 10). The loading is reminiscent of shear-lag, with the progressive decay in axial force of a line of axial bars by the bending of the neighbouring diagonal bars.

The hexagonal tube has a slightly lower actuation stiffness than that of the hexagonal plate, but the overall mode of deformation is the same. The deformation mode involves the bending of bars near to the actuated bar, and is omitted here for the sake of brevity; the mode for the hexagonal plate is shown in fig. 6 of Wicks & Guest (2004).

The longitudinal Kagome tube under actuation of an axial bar deforms in a similar manner to that of the planar Kagome plate (compare figure 11*a* with fig. 7 of Wicks & Guest 2004). In contrast, diagonal actuation of the longitudinal Kagome tube generates a local deformation mode involving the bending of a small number of bars in the radial direction (figure 11*b*). This is a more compliant and more local mode than that observed for actuation of an axial bar of the longitudinal Kagome tube. Likewise, actuation of the circumferential Kagome tube by extension of a circumferential bar or a diagonal bar generates a local compliant bending mode, as shown in figure 11*c*. The actuation stiffness of the circumferential Kagome tube is almost identical to that for diagonal actuation of the longitudinal Kagome tube (compare figures 7*b* and 8).

The square-lattice tube under axial actuation has a similar deformation mode to that of the longitudinal Kagome tube: a long wavelength shear-lag mode exists, with negligible radial motion of the nodes. An unwrapped view of the deformed square lattice is given in figure 12*a*. Wicks & Guest (2004) have developed a simple analysis of this deformation mode for the planar Kagome lattice; we modify this analysis in appendix A for the case of the two-dimensional square lattice and thereby obtain expressions for the actuation stiffness and the size of deformation zone. The actuation stiffness of the square-lattice tube is slightly below that of the Kagome plate (figure 8). The square-lattice tube under circumferential actuation has a similar deformation mode to that of circumferential actuation of the circumferential Kagome tube; it involves bar bending near the actuated bar (compare figures 11*c* and 12*b*). The actuation stiffnesses for these two topologies are almost identical (figure 8).

### (b) The generalized hinge at the actuated section of a reticulated tube

The FE simulations reveal that actuation of a bar in the reticulated tube gives rise to a *generalized hinge* at the actuated cross-section of the tube. This hinge involves an axial extension Δ*u*_{3}, two shearing displacements (Δ*u*_{1}, Δ*u*_{2}), an axial twist Δ*ω*_{3} and two bending rotations (Δ*ω*_{1}, Δ*ω*_{2}) (figure 13). The magnitudes of (Δ*u*_{i}, Δ*ω*_{i}) are obtained by the following protocol. At each end of the actuated lattice tube, the average nodal displacement and rotation are calculated over the cross-section. Denote the average values at the left-hand end by A and the values at the right-hand end by B. Then, we obtain(4.3a)and(4.3b)(4.3c)(4.3d)The hinge values (Δ*u*_{i}, Δ*ω*_{i}) are then normalized by the actuator displacement *u*_{a} to obtain the performance metrics(4.4)

#### (i) Results

The FE calculations reveal that (, ) are almost independent of the stockiness *s* for each type of lattice tubes with *p*=8 circumferential bars. Thus, there is no need to plot the results graphically, and instead values are reported in table 3 for each topology, with the choice *p*=8, *s*=0.01.

It can be deduced directly from the *symmetry* of each lattice that certain components of and vanish. For example, axial actuation of tubes *A*, *B* and *E* induces no lateral motion, and , and no twist, (table 3). Likewise, circumferential actuation of the tube *D* induces . The square-lattice tube *E* gives rise to a hinging motion (, , ) upon actuation of an axial bar. In contrast, actuation of a circumferential bar does not generate any hinging motion.

It is instructive to compare the actuation performance of the competing lattice tubes according to various metrics. The ideal topology will possess: (i) a low actuation stiffness , (ii) a high ‘displacement efficiency of hinging’ and , and (iii) a high ‘force efficiency of hinging’, as characterized by and . The values of these metrics are listed in table 4. While it would be useful to give a detailed study of the practical values needed for the performance metrics, it is beyond the scope of the present study. This issue has been addressed for the case of the Kagome double-layer grid (see Symons *et al*. 2005*a*).

We consider each topology in turn. The triangulated lattice is statically and kinematically determinate with suitable end patching of the pin-jointed version by a small number of additional bars. Thus, it has a high structural rigidity in the rigid-jointed form, and can also be actuated with only moderate actuation stiffness (1–12% of that of a single bar). It is a promising topology for morphing application.

The Kagome lattice has moderate values of actuation stiffness and high displacement efficiency. Consequently, its force efficiency is intermediate between that of the triangulated and hexagonal lattices. Recall that the longitudinal Kagome lattice has structural rigidities in tension and bending, which are comparable with those of the triangulated tube. However, its torsional rigidity is low (recall figure 3).

The hexagonal lattice has the lowest actuation stiffness and has a high value of displacement efficiency. Thus, it has the highest value of force efficiency. However, the hexagonal lattice has a low structural rigidity in macroscopic tension, bending and torsion.

Finally, the square lattice does not develop a generalized hinge when actuated by a circumferential bar; although an axial actuator generates a moderate macroscopic displacement, the actuation stiffness is large compared with that of the Kagome lattice. Recall that the square-lattice tube has similar structural rigidities to those of the longitudinal Kagome tube: high axial and bending rigidities, and low torsional rigidity. Consequently, the square lattice is an inferior choice for actuation purposes.

## 5. Concluding remarks

In this paper, the structural performance of several reticulated tubes with walls made from periodic lattices has been explored by matrix analysis of the pin-jointed trusses and FE analysis of the rigid-jointed versions. Some main conclusions are drawn as follows.

The pin-jointed topologies of all lattice tubes considered have no states of self-stress and are therefore statically determinate. The pin-jointed triangulated tube, with a nodal connectivity of 6, is also kinematically determinate (no internal mechanisms) upon suitable patching of a small number of additional bars at the ends of the tube. These properties of the pin-jointed triangulated-lattice tube endow the rigid-jointed version with high structural rigidity and useful morphing capacity. In contrast, the remaining pin-jointed lattice tubes contain internal collapse mechanisms and the rigid-jointed versions have lower structural rigidities.

It is commonplace to use the in-plane effective properties of a thin-walled tube in order to derive its structural rigidity for axial tension, flexural bending and torsion. While this approach is adequate for the triangulated, hexagonal and square lattices, it is inadequate for the Kagome lattice: additional compliant modes of deformation arise in the Kagome lattice and these severely degrade its structural rigidity. Similarly, it is misleading to use the actuation stiffness of a two-dimensional planar lattice in order to predict the actuation stiffness of the corresponding three-dimensional tube: radial modes of local deformation can occur for the tube, and these reduce the actuation stiffness.

## Acknowledgments

This work is a part of the ‘Advanced Hybrid Mechatronic Materials for ultra precise and high performance machining systems design’ (HyMM) project, funded by the European Commission under the NMP priority (NMP3-CT-2003-505206). Financial support has also been provided by the EPSRC, grant number EP/D055806/1. S.P.M. wishes to thank the Cambridge Overseas Trust for its support.

## Footnotes

- Received August 11, 2008.
- Accepted October 20, 2008.

- © 2008 The Royal Society