## Abstract

Some 250 years after the systematic experiments by Musschenbroek and their rationalization by Euler, for the first time we show that it is possible to design structures (i.e. mechanical systems whose elements are governed by the equation of the elastica) exhibiting bifurcation and instability (‘buckling’) under tensile load of constant direction and point of application (‘dead’). We show both theoretically and experimentally that the behaviour is possible in elementary structures with a single degree of freedom and in more complex mechanical systems, as related to the presence of a structural junction, called ‘slider’, allowing only relative transversal displacement between the connected elements. In continuous systems where the slider connects two elastic thin rods, bifurcation occurs both in tension and in compression, and is governed by the equation of the elastica, employed here for tensile loading, so that the deformed rods take the form of the capillary curve in a liquid, which is in fact governed by the equation of the elastica under tension. Since axial load in structural elements deeply influences dynamics, our results may provide application to innovative actuators for mechanical wave control; moreover, they open a new perspective in the understanding of failure within structural elements.

## 1. Introduction

Buckling of a straight elastic column subject to *compressive* end thrust occurs at a critical load for which the straight configuration of the column becomes unstable and simultaneously ceases to be the unique solution of the elastic problem (so that instability and bifurcation are concomitant phenomena). Buckling is known from ancient times: it has been experimentally investigated in a systematic way by Pieter van Musschenbrok (1692–1761) and mathematically solved by Leonhard Euler (1707–1783), who derived the differential equation governing the behaviour of a thin elastic rod suffering a large bending, the so-called ‘elastica’ (see Love 1927).

For centuries, engineers have experimented and calculated complex structures, such as frames, plates and cylinders, manifesting instabilities and bifurcations of various forms (Timoshenko & Gere 1961), so that certain instabilities have been found involving tensile loads. For instance, there are examples classified by Ziegler (1977) as ‘buckling by tension’ where a tensile loading is applied to a system in which a compressed member is always present, so that they do not represent true bifurcations under tensile loads. Other examples given by Gajewski & Palej (1974) are all related to the complex *live* (as opposed to ‘dead’) loading system; for instance, loading through a vessel filled with a liquid, so that Zyczkowski (1991) points out that ‘With Eulerian behaviour of loading (materially fixed point of application, direction fixed in space), the bar cannot lose stability at all [⋯].’ Note finally that necking of a circular bar represents a bifurcation of a material element under tension, not of a structure.

It can be concluded that until now *structures made up of line elements (each governed by the equation of the elastica) exhibiting bifurcation and instability under tensile load of fixed direction and point of application (in other words ‘dead’) have never been found*, so that the word ‘buckling’ is commonly associated with compressive loads.

In the present article, we show that:

— simple structures can be designed evidencing bifurcation (buckling) and instability under tensile dead loading;

— the deformed shapes of these structures can be calculated using the equation of the elastica, but under tension, so that the deflection of the rod is identical to the shape of a capillary curve in a liquid, which is governed by the same equation, see figure 1 and §§3

*b*and 4;— experiments show that elastic structures buckling under tension can be realized in practice and that they closely follow theory predictions, §§2 and 4.

The above findings are complemented by a series of minor new results for which our system behaves differently from other systems made up of elastic rods, but with the usual end conditions. First, our system evidences load decrease with increase of axial displacement (the so-called ‘softening’); second, the bifurcated paths involving relative displacement at the slider terminate at an unloaded limit configuration, for both tension and compression.

We will see that the above results follow from a novel use of a junction between mechanical parts, namely, a *slider* or, in other words, a connection allowing only relative sliding (transverse displacement) between the connected pieces and therefore constraining the relative rotation and axial displacement to remain null.

Vibrations of structures are deeply influenced by axial load, so that the speed of flexural waves vanishes at bifurcation (Bigoni *et al.* 2008; Gei *et al*. 2009), a feature also evidenced by the dynamical analysis presented in §3*a*, so that, since bifurcation is shown to occur in our structures both in tension and compression, these can be used as two-way actuators for mechanical waves, where the axial force controls the speed of the waves traversing the structure. Therefore, the mechanical systems invented in the present article can immediately be generalized and employed to design complex mechanical systems exhibiting bifurcations in tension and compression, to be used, for instance, as systems with specially designed vibrational properties (a movie providing a simple illustration of the concepts exposed in this paper, together with a view of experimental results, is provided in the electronic supplementary material, see also http://www.ing.unitn.it/dims/ssmg.php).

## 2. A simple single-degree-of-freedom structure that buckles for tensile dead loading

The best way to understand how a structure can bifurcate under tensile dead loading is to consider the elementary single-degree-of-freedom structure shown in figure 2, where two rigid rods are connected through a ‘slider’ (a device which imposes the same rotation angle and axial displacement to the two connected pieces, but null shear transmission, leaving only the possibility of relative sliding).

Bifurcation load and equilibrium paths of this single-degree-of-freedom structure can be calculated by considering the bifurcation mode illustrated in figure 2 and defined by the rotation angle *ϕ*. The elongation of the system and the potential energy are, respectively,
2.1so that solutions of the equilibrium problem are
2.2for *ϕ*≠0, plus the trivial solution (*ϕ*=0,∀*F*). Analysis of the second-order derivative of the strain energy reveals that the trivial solution is stable up to the critical load
2.3while the non-trivial path, *evidencing softening*, is unstable. For an imperfect system, characterized by an initial inclination of the rods *ϕ*_{0}, we obtain
2.4so that the force–rotation relation is obtained, which is represented as two dashed lines in figure 2 for *ϕ*_{0}=1^{°} and *ϕ*_{0}=10^{°}.

The simple structure presented in figure 2, showing possibility of a bifurcation under dead load in tension and displaying an overall softening behaviour, can be realized in practice, as shown by the wooden model reported in figure 3.

## 3. Vibrations, buckling and the elastica for a structure subject to tensile (and compressive) dead loading

In order to generalize the single-degree-of-freedom system model into an elastic structure, we consider two *inextensible* elastic rods clamped at one end and jointed through a slider, identical to that used to joint the two rigid bars employed for the single-degree-of-freedom system (see the inset of figure 4). The two bars have bending stiffness *B*, length *l*^{−} (on the left) and *l*^{+} (on the right) and are subject to a load *F*, which may be tensile (*F*>0) or compressive (*F*<0).

### (a) The vibrations and critical loads

The differential equation governing the dynamics of an elastic rod subject to an axial force *F* (assumed positive if tensile) is
3.1where *ρ* is the unit-length mass density of the rod and *v* the transversal displacement, so that time-harmonic motion is based on the separate-variable representation
3.2in which *ω* is the circular frequency, *t* is the time and is the imaginary unit.

Substitution of equation (3.2) into equation (3.1) yields the equation governing time-harmonic oscillations
3.3where the function ‘sign’ (defined as sign(*α*)=|*α*|/*α* ∀*α*∈*Re* and sign(0)=0) has been used and
3.4The general solution of equation (3.3) is
3.5where
3.6Equation (3.5) holds both for the rod on the left (transversal displacement denoted with ‘−’) and on the right (transversal displacement denoted with ‘+’) shown in the inset of figure 4, so that the boundary conditions at the clamps impose
3.7while at the slider we have the two conditions
3.8expressing the vanishing of the shear force. The imposition of the six conditions (3.7) and (3.8) provides the constants as functions of the constants , so that the continuity of the rotation at the slider
3.9and the equilibrium of the slider
3.10yields finally a linear homogeneous system (with unknowns and ), whose determinant has to be set equal to zero, to obtain the frequency equation, function of *α*^{2}, *ω* and sign(*F*).

The circular frequency *ω* (normalized through multiplication by ) versus the axial force (normalized through multiplication by 4*l*^{2}/(*Bπ*^{2})) is reported in figure 4, where the first four branches are shown for a system of two rods of equal length. In this figure, the grey zones represent situations that cannot be achieved, in the sense that the axial force falls outside the interval where the straight configuration of the system is feasible (in other words, for axial loads external to the interval of first bifurcations in tension and compression, the straight configuration cannot be maintained).

The branches shown in figure 4 intersect the horizontal axis in correspondence of the bifurcation loads of the system, namely, 4*F*_{cr}*l*^{2}/(*π*^{2}*B*)=−16,−15.19,−4,−3.17,+0.58, so that there is one critical load in tension (the corresponding branch is labelled ‘1st slider mode’ in figure 4), and infinitely many bifurcation loads in compression; the first three are reported in figure 4 (bifurcations corresponding to the label ‘global mode’ do not involve relative displacement across the slider).

Beside the possibility of bifurcation in tension, an interesting and novel effect related to the presence of the slider is that a tensile (compressive) axial force yields a decrease (increase) of the frequency of the system, while an opposite effect is achieved when ‘global modes’ are activated.

Quasi-static solutions of the system and related bifurcations can be obtained in the limit of the *frequency equation*, which yields
3.11In the particular case of rods of equal length *l*, equation (3.11) simplifies to
3.12

Equation (3.12) shows clearly that *there is only one bifurcation load in tension* (branch labelled ‘1st slider mode’ in figure 4), but there are bifurcation loads in compression (the first three branches are reported in figure 4). In compression, the bifurcation condition , providing solutions, yields the critical loads of a doubly clamped beam of length 2*l* and defines what we have labelled ‘global modes’ in figure 4.

Bifurcation loads, normalized through multiplication by (*l*^{+}+*l*^{−})^{2}/(*π*^{2}*B*), are reported in figure 5 as functions of the ratio *l*^{+}/*l*^{−} between the lengths of the two rods.

Note that the graph is plotted in a semi-logarithmic scale, which enforces symmetry about the vertical axis. In the graph, the first two buckling loads in compression are reported: the first corresponds to a mode involving sliding, while the second does not involve any sliding (and when *l*^{+}=*l*^{−} corresponds to the first mode of a doubly clamped rod of length 2*l*). Used as an optimization parameter, *l*^{+}=*l*^{−} corresponds to the lower bifurcation load in tension (+0.58), near five times smaller (in absolute value) that the buckling load in compression (−3.17).

### (b) The elastica

The determination of the non-trivial configurations at large deflections of the mechanical system requires a careful use of Euler’s elastica. It is instrumental to employ the reference systems shown in figure 6 and impose one kinematic compatibility condition and three equilibrium conditions. These are as follows:

— The kinematic compatibility condition can be directly obtained from figure 6 noting that the jump in displacement across the slider (measured orthogonally to the line of the elastica),

**Δ**_{s}, can be related to the angle of rotation of the slider*Φ*_{s}, a condition that assuming the local reference systems shown in figure 6 becomes 3.13where*x*_{1}(*s*) and*x*_{2}(*s*) are the coordinates of the elastica and the index minus (plus) denotes that the quantities are referred to the rod on the left (on the right). Note that*Φ*_{s}is assumed positive when anticlockwise and**Δ**_{s}is not restricted in sign (negative in the case of figure 6).— Since the slider can only transmit a moment and a force

*R*orthogonal to it, equilibrium requires that (see the inset in figure 6) 3.14where*F*is the axial force providing the load to the rod, assumed positive (negative) when tensile (compressive), so that since*Φ*_{s}∈[−*π*/2,*π*/2],*R*is positive (negative) for tensile (compressive) load. Note that with the above definitions we have 3.15— Equilibrium of the slider requires that 3.16where

*B*is the bending stiffness of the rod and*κ*^{±}_{s}is the curvature on the left (−) or on the right (+) of the slider. Note that*B*is always positive, but*R*,*κ*^{±}_{s}and**Δ**_{s}can take any sign.— For both rods (left and right) rotational equilibrium of the element of rod singled out at curvilinear coordinate

*s*requires 3.17where*θ*is the rotation of the normal at each point of the elastica, assumed positive when anticlockwise, with the superscript − (+) added to denote the rod on the left (on the right).

Equation (3.17) is usually (see for instance Love 1927, his eqn (8) at §262) written with a sign ‘+’ replacing the sign ‘−’ and *R* is assumed positive when compressive; the same equation describes the motion of a simple pendulum (see for instance Temme 1996). The ‘+’ sign originates from the fact that the elastica has been analysed until now only for deformations originating from compressive loads. However, an equation with the ‘−’ sign and with *R*/*B* replaced by the ratio between unit weight density and surface tension of a fluid—thus equal to equation (3.17)—determines *the shape of the capillary curve of a liquid* (Lamb 1928), which therefore results in being identical to the deflection of a rod under tensile load.

In the following, we derive equations holding along both rods ‘+’ and ‘−’, so that these indices will be dropped for simplicity. Multiplication of equation (3.17) by *dθ*/*ds* and integration from 0 to *s* yields
3.18where, using the Heaviside step function *H*, we have
3.19Equation (3.18) can be re-written as
3.20so that the change of variable yields
3.21The analysis will be restricted for simplicity to the case ‘+’ in the following. At *u*=0 it is *θ*=0, so that equation (3.21) gives the solution
3.22where am and dn are, respectively, the Jacobi elliptic functions amplitude and delta-amplitude and *K* is the complete elliptic integral of the first kind (Byrd & Friedman 1971). Since in the local reference system we have and , an integration gives the coordinates *x*_{1} and *x*_{2} of the elastica expressed in terms of *u*,
3.23for tensile axial loads, while for compressive axial loads
3.24in which the constants of integration are chosen so that *x*_{1} and *x*_{2} vanish at *s*=0. In equations (3.23) and (3.24) sn and cn are, respectively, the Jacobi elliptic functions sine-amplitude and cosine-amplitude and E is the incomplete elliptic integral of the second kind (Byrd & Friedman 1971).

Equation (3.24) differs from eqn (16) reported by Love (1927, §263) only in a translation of the coordinate *x*_{2}, while equation (3.23), holding for tensile axial force, is new.

Finally, with reference to figure 6, we note that the horizontal displacement **Δ**_{c} of the right clamp can be written in the form
3.25

To find the axial load *F* as a function of the slider rotation *Φ*_{s}, or as a function of the end displacement **Δ**_{c}, we have now to proceed as follows:

— values for

*κ*^{−}_{s}and*κ*^{+}_{s}are fixed (as a function of the selected mode, for instance, , to analyse the bifurcation mode in tension);—

*k*can be expressed using equation (3.19)_{2}as a function of ;— the equations for the coordinates of the elastica, equation (3.23) for tensile load, or equation (3.24) for compressive load, and equation (3.22)

_{1}, evaluated at*l*^{−}and*l*^{+}, become functions of only ;— equations (3.15)

_{2}and (3.16) provide*Φ*_{s}and**Δ**_{s}, so that equation (3.13) becomes a nonlinear equation in the variable , which can be numerically solved (we have used the function FindRoot of Mathematica 6.0);— when is known,

*R*and*F*can be obtained from equations (3.19)_{1}and (3.14);— finally,

*Φ*_{s}and**Δ**_{c}are calculated using equations (3.15)_{2}and (3.25).

Results are shown in figure 7 for tensile loads and in figure 8 for compressive loads, in terms of dimensionless axial load 4*Fl*^{2}/(*Bπ*^{2}) versus slider rotation *Φ*_{s} (*a*) and dimensionless end displacement **Δ**_{c}/(2*l*) (*b*).

Note that, while there is only one bifurcation in tension, there are infinite bifurcations in compression, so that we have limited results to the initial three modes in compression. Two of these modes involve slider rotation (labelled ‘slider mode’), while an intermediate mode (labelled ‘global mode’) does not.

The load/displacement curve shown in figure 7 on the right is plotted until extremely large displacements, namely **Δ**_{c}=20*l* (a detail at moderate displacement is reported in the inset). It displays a *descending, in other words softening and unstable, post-critical behaviour*, which contrasts with the usual post-critical behaviour of the elastica under various end conditions, in which the load rises with displacement. In compression, the post-critical behaviour evidences another novel behaviour, so that the first and the second slider modes present an initial part where the load/displacement rises, followed by a softening behaviour. Finally, it is important to note that the curve load versus *Φ*_{s} in figures 7 and 8, both *for tension and compression intersect each other at null loading* at the extreme rotation *Φ*_{s}=90^{°}, which means that two unloaded configurations (in addition to the initial configuration) exist. These peculiarities, never observed before in simple elastic structures, are all related to the presence of the slider.

Deformed elastic lines are reported in figure 9, both for tension and compression, the latter corresponding to the first three slider modes (the global mode is not reported since it corresponds to the first mode of a doubly clamped rod).

## 4. Experimental

The structure sketched in figure 6 has been realized with two carbon steel AISI 1095 strips (250×25×1 mm; the Young modulus 200 GPa) and the slider with two linear bearings (type Easy Rail SN22-80-500-610, purchased from Rollon), commonly used in machine design applications, see the inset of figure 10.

The slider is certified by the producer to have a low friction coefficient, equal to 0.01. Tensile force on the structure has been provided by imposing displacement with a load frame Digital Tritest 50 (ELE International Ltd), the load measured with a load cell Gefran OC-K2D-C3 (Gefran Spa), and the displacement with a potentiometric transducer Gefran PY-2-F-100 (Gefran Spa). Data have been acquired with system NI CompactDAQ, interfaced with Labview 8.5.1 (National Instruments). Photos have been taken with a Nikon D200 digital camera, equipped with a AF-S micro Nikkor lens (105 mm 1:2.8G ED) and movies with a Sony Handycam HDR-XR550. Tensile and compressive tests were run at a velocity of 2.5 mm s^{−1}.

Photos taken at different slider rotations (and thus load levels) are shown in figure 11 for tension (*Φ*_{s}=0^{°},10^{°},20^{°},30^{°}) and in figure 12 for compression (*Φ*_{s}=0^{°},5^{°},10^{°},20^{°}). A comparison between theoretical predictions and experiments is reported in the lower parts of the figures where photos are superimposed to the line of the elastica, shown in red and plotted using equation (3.23) for tensile load and equation (3.24) for compression.

These experiments show clearly, the existence of the bifurcation in tension and provide an excellent comparison with theoretical results obtained through integration of the elastica both in tension and in compression. A further quantitative comparison between theoretical results and experiments is provided in figure 10, where the axial load in the structure (positive for tension and negative for compression) is plotted versus the end displacement **Δ**_{c}. The experimental result is compared with theoretical results (marked red) expressed by equation (3.25), and used in the way detailed at the end of §3*b*.

The theoretical result marked in red with a continuous curve has been calculated assuming an initial length of the rods (25 cm) measured from the end of the clamps to the middle of the slider. However, the slider and the junctions to the metal strips are 58 mm thick, so that the system is stiffer in reality. Therefore, we have plotted a dashed line for the theoretical results obtained employing an ‘effective’ initial length of the rods reduced of 10 per cent (so that the effective length of the system has been taken equal to 45 cm). The experimental curve evidences oscillations of ±1 N for tensile loads and ±5 N for compressive loads. These oscillations are due to friction within the slider, so that it is obvious that the oscillations are higher in compression than in tension, since in the former case the load is higher. Except for these oscillations, the friction (which is very low) has been found not to influence the tests.

The fact that experimentally the bifurcations initiate before the theoretical values are attained represents the well-known effect of imperfections, so that we may conclude that the agreement between theory and experiments is excellent.

To provide experimental evidence to the fact that the elastica in tension corresponds to the shape of the free surface of a liquid in a capillary channel, we note that a meniscus in a capillary channel satisfies (by symmetry) a null-rotation condition at the centre of the channel, so that it corresponds to a clamped edge of a rod. If the tangent to the meniscus at the contact with the channel wall is taken to correspond to the rotation of the non-clamped edge of the rod and the width of the channel is calculated employing the elastica, the elastic deflection of the rod scales with the free surface of the liquid. Therefore, we have performed an experiment in which we have taken a photo (with a Nikon SMZ800 stereo-zoom microscope equipped with Nikon Plan Apo 0.5× objective and a Nikon DD-FI1 high definition colour camera head) of a water meniscus in a polycarbonate channel. We proceeded as follows. First, we have observed that the contact angle between a water surface in air and polycarbonate (at a temperature of 20^{°}C) is 70^{°}. Second, we have taken a photo of the meniscus formed in a polycarbonate ‘V-shaped’ channel with walls inclined at 10^{°} with the vertical, so that the angle between the horizontal direction and the free surface results to be 30^{°} and the distance between the walls results 6 mm. This photo has been compared with a photo taken (with a Nikon D200 digital camera, and shown in figure 11) during buckling in tension when the elastic rods form the same angle of 30^{°}. The result is shown in figure 1, together with the theoretical solution shown red.

## 5. Conclusions

We have theoretically proven and fully experimentally confirmed that elastic structures can be designed and practically realized in which bifurcation can occur with tensile dead loading. In these structures no parts subject to compression are present. The finding is directly linked to the presence of a junction allowing only for relative sliding between two parts of the mechanical system. Our findings open completely new and unexpected perspectives, related for instance to the control of the propagation of mechanical waves and to the understanding of certain failure modes in material elements.

## Acknowledgements

The authors gratefully acknowledge financial support from PRIN grant no. 2007YZ3B24.

- Received October 1, 2010.
- Accepted December 6, 2010.

- This journal is © 2011 The Royal Society