## Abstract

Bifurcation of an elastic structure crucially depends on the curvature of the constraints against which the ends of the structure are prescribed to move, an effect that deserves more attention than it has received so far. In fact, we show theoretically and provide definitive experimental verification that an appropriate curvature of the constraint over which the end of a structure has to slide strongly affects buckling loads and can induce: (i) tensile buckling; (ii) decreasing- (softening), increasing- (hardening) or constant-load (null stiffness) postcritical behaviour; and (iii) multiple bifurcations, determining for instance two bifurcation loads (one tensile and one compressive) in a single-degree-of-freedom elastic system. We show how to design a constraint profile to obtain a desired postcritical behaviour and we provide the solution for the elastica constrained to slide along a circle on one end, representing the first example of an inflexional elastica developed from a buckling in tension. These results have important practical implications in the design of compliant mechanisms and may find applications in devices operating in quasi-static or dynamic conditions, even at the nanoscale.

## 1. Introduction

We begin with a simple example, by considering a one-degree-of-freedom elastic structure made up of a rigid rod connected with a rotational linear elastic spring on its left end and with a roller constrained to move on a circle (of radius *R*_{c}, centred on the rod's axis) on the right (figure 1). The structure is subject to a horizontal force, so that when this load is compressive and the circle degenerates to a line (null curvature), the structure buckles at the compressive force, *F*=−*k*/*l*. Our interest is to analyse the case, when the curvature of the constraint is not null, revealing that this curvature strongly affects the critical load, which results to be a *tensile* force^{1} in the negative curvature case (*F*_{t}=*k*/(3*l*), for ) and a compressive load for positive curvature (*F*_{c}=−*k*/(5*l*), for ).

The example shows that the curvature of the constraint at the end of a structure deeply affects its critical load^{2} , but also the shape of the curve defining the constraint influences the postcritical behaviour, which displays a rising-load (hardening) behaviour in the case of null curvature and a decreasing-load (softening) behaviour for circular profiles (for instance, when , as in the structure shown in figure 1). Moreover, the postcritical behaviour connected to the tensile (compressive) bifurcation evidences force reversal, because the tensile (compressive) force needed to buckle the structure decreases until it vanishes and becomes compressive (tensile), during continued displacement of the structure end.

Once the lesson on the curvature and the shape of the constraint is clear, it becomes easy to play with these structural elements and discover several new effects. Some of these are listed as follows.

— A constraint profile can be designed to provide a ‘hardening’, ‘softening’ or even a ‘

*neutral*’ (in which the displacement grows at constant load) postcritical behaviour. More in general, a formula will be given to determine the shape of the profile to obtain a desired postcritical behaviour, including situations in which the stability of the path changes during postcritical deformation.— A negative and a positive curvature can be combined in an ‘S-shaped constraint’ (see the inset of figure 2) to yield

*a one-degree-of-freedom structure with two buckling loads: one tensile and one compressive*.

In the case of the ‘S-shaped constraint’, imperfections suppress bifurcations, and the stability of the equilibrium path strongly depends on the *sign* of the imperfection. For tensile forces, if the imperfection has a positive sign (*ϕ*_{0}>0), then the equilibrium path of the system becomes unstable after a peak in the load is reached, whereas if the sign is negative (*ϕ*_{0}<0), then the structure remains in a metastable equilibrium configuration which asymptotically approaches an unstable configuration (figure 2).

Finally, we can appreciate the role played by the curvature of a constraint in the more interesting case of a structural element governed by the elastica, a research aspect passed unnoticed until now, but interesting for the applications in compliant mechanisms. We show that consideration of this curvature provides a generalization of the findings by Zaccaria *et al.* (2011), so that their ‘slider’ can be seen as a special case of the curved constraint introduced in this study, and the elastica developing after a tensile buckling is of inflexional type, while that investigated by Zaccaria *et al.* (2011) is non-inflexional. We fully develop the theory of the elastica constrained to slide with a rotational spring along a circle on one of its ends and we experimentally confirm the theoretical findings with experiments designed and realized by us at the Laboratory for Physical Modeling of Structures and Photoelasticity.

The study is organized as follows. We begin presenting a generalization of the one-degree-of-freedom structure shown in figure 1, to highlight: (i) the effects of the curvature of the constraint, (ii) the multiplicity of bifurcation loads, (iii) the behaviour of the imperfect system, and (iv) the possibility of designing a constraint profile to obtain a given postcritical behaviour. Later, we analyse a continuous system, made up of an inextensible beam governed by the Euler elastica and we solve the critical loads and the nonlinear postcritical large-deformation behaviour, through explicit integration of the elastica. We systematically complement theoretical results with experiments confirming all our findings for discrete and continuous elastic systems. An electronic supplementary material is given, with movie S1 providing a simple illustration of the concepts exposed in this study, together with a view of experimental results (see also http://ssmg.unitn.it/).

## 2. Effect of the constraint's curvature on a one-degree-of-freedom elastic structure

Bifurcation load and equilibrium paths of the one-degree-of-freedom structure illustrated in figure 3 (where the constraint is assumed smooth and described in the *x*_{1}−*x*_{2} reference system as *x*_{2}=*lf*(*ψ*), with *ψ*=*x*_{1}/*l*∈[0,1] and *f*′(0)=0) can be calculated by considering a deformed mode defined by the rotation angle, *ϕ*. Assuming a possible imperfection in terms of an initial inclination *ϕ*_{0}, the elongation of the system and the potential energy are, respectively,
2.1
and
2.2
so that solutions of the equilibrium problem are governed by
2.3
where *f*′=∂*f*/∂*ψ*, so that the critical load for the perfect system, *ϕ*_{0}=0, is
2.4
where, because *f*′(0)=0, is the signed curvature at *ϕ*=0. Stability can be judged based on the sign of the second derivative of the potential energy
2.5
showing that the trivial configuration of the perfect system is always unstable beyond the critical load.

In the case when the profile of the constraint is a circle^{3} of dimensionless radius as in figure 1, the non-trivial equilibrium configurations are given by
2.6
and result to be stable when
2.7
Equations (2.6) and (2.7) have been used to solve the special case of figure 1 (), with an ‘S-shaped’ constraint (so that is discontinuous at *ϕ*=0), to obtain the results plotted in figure 2.

### (a) The design of the postcritical behaviour

It is important to emphasize that *the shape of the profile on which one end of the structure has to slide can be designed to obtain ‘desired postcritical behaviours’*. Let us assume that we want to obtain a certain force–displacement *F*/*δ* postcritical behaviour for the structure sketched in figure 3. Because
2.8
to assume a certain *F*/*δ* relation is equivalent to assume a given dependence of *F* on *ψ*; therefore, we introduce the dimensionless function
2.9
Employing equation (2.3), we obtain the condition
2.10
satisfying *f*(0)=1 and *f*′(0)=0.

Three different profiles designed to obtain particular force *F* versus rotation *ϕ* postcritical behaviours (a sinusoidal, a circular and a constant) are sketched in figure 4. An interesting case is that of the neutral (or constant) postcritical behaviour, in which the rotation *ϕ* (and therefore also the displacement *δ*) can grow at constant load^{4} that can be obtained employing the constraint profile expressed as
2.11

### (b) Experiments on one-degree-of-freedom elastic systems: multiple buckling and neutral postcritical response

The behaviours obtained employing the simple one-degree-of-freedom structures are not a mathematical curiosity, but can be realized in practice. In particular, we have realized the ‘S-shaped’ circular constraint shown in the inset of figure 2 and the profile illustrated in figure 4*b* (labelled ‘constant’), the latter to show a ‘neutral’ or, in other words, ‘constant-force’ response. The experimental apparatuses are shown in figures 5 and 6 (the former relative to the ‘S-shaped’ semicircular profile; the latter relative to the profile providing the neutral postcritical response), where the grooves have been laser cut (by HTR Laser & Water cut, BZ, Italy) in a 2 mm thick plate of AISI 304 steel and the roller has been realized with a (17 mm diameter) steel cylinder mounted with two roller bearings (SKF-61801-2Z). The rigid bar (600 mm×50 mm×20 mm) has been machined from an aluminium bar and lightened with longitudinal grooves (see appendix A), so that its final mass is 820 g. The hinge with rotational spring has been realized with three identical rotational springs, which have been designed using eqns (32) of Brown (1981) and realized in music wire (4 mm diameter; ASTM A228; see appendix A for further details).

Load–displacement curves are reported in figure 7 for the ‘S-shaped’ circular profile and in figure 8 for the profile giving the neutral response, as obtained from experiments, and directly compared with the theoretical predictions.

We note a good agreement, with buckling detected prior to the attainment of the theoretical value, in agreement with the known effect of imperfections. Friction at the roller–profile contact has induced some irrelevant load oscillation, minimized by hand-polishing the edges of the groove and using Aero Lubricant AS 100 (from Rivolta s.p.a, Milano, Italy). Finally, we may comment that the experiments confirm the possibility of practically realizing mechanical systems behaving as the theoretical modelling predicts.

## 3. The buckling and postcritical behaviour of an elastic rod with a circular constraint on one end

We consider an *inextensible* elastic rod (of bending stiffness *B* and length *l*), with a movable clamp at one end, and having a rotational elastic spring (of stiffness *k*) on the other, which can slide on a circle centred on the axis of the rod, see the inset of figure 9. The rod is subject to an axial load *F* which may be tensile (*F*>0) or compressive (*F*<0).

### (a) The critical loads

The differential equilibrium equation of an elastic rod subject to an axial force *F*, linearized near the rectilinear configuration, is
3.1
where *v* is the transversal displacement, ‘sgn’ is defined as sgn(*α*)=|*α*|/*α* ∀*α*∈Re−{0}, sgn(0)=0, and
3.2
The general solution of equation (3.1) is
3.3
as well as the boundary conditions (the third involving the rotational spring stiffness *k*) are
3.4
plus the kinematic compatibility condition defining *ϕ*
3.5
involving the signed, dimensionless curvature of the circle.

Imposing conditions (3.4)–(3.5), the solution (3.3) provides the condition for the critical loads 3.6 corresponding in the two limits and to a pinned and clamped constraint on the right end, respectively.

Buckling loads (made dimensionless through multiplication by *l*^{2}/(*π*^{2}*B*)) are reported in figures 9–11, as functions of the signed radius of curvature of the constraint.

Results reported in figures 10 and 11 (where the negative signs denote compressive loads) are given in terms of effective length factor *ξ* defined as
3.7

We note from figures 9 to 11 that for certain curvatures of the constraint there is one buckling load in tension, whereas there are always infinite bifurcations in compression (hence we can comment that the bifurcation problem remains a Sturm–Liouville problem). The results reveal the strong effect of the constraint curvature, hence for instance for (for ), there is a buckling load in tension much smaller (much higher) than that in compression, taken in absolute value. Moreover, not only for , but also for all positive curvatures , there is no tensile bifurcation.

### (b) The elastica

The shape of the constraint has a strong effect on the postcritical behaviour, as will be shown below with reference to the case of the circular profile. This effect can be exploited for the design of compliant mechanisms, hence the solution of this problem is not only of academic interest. Therefore, we derive the solution for an elastic rod clamped to the left and constrained on the right to slide with a rotational spring (of stiffness *k*_{r}) on a ‘S-shaped’ bi-circular profile, as sketched in figure 12, where the local reference system to be used in the analysis is also indicated.

The elastic line problem is governed by the following equations.

— A condition of kinematic compatibility can be obtained by observing from figure 12 that the coordinates of the elastica evaluated at

*s*=*l*, namely*x*_{1}(*l*) and*x*_{2}(*l*), are related to the angle of rotation of the local reference system*ϕ*and to the radius*R*_{c}of the constraint via 3.8 where*ϕ*is assumed positive if anticlockwise; note that in equation (3.8) the sign ‘−’ (‘+’) holds for the case of the rotational spring lying on the left (right) half-circle.— The curved constraint transmits to the rod a moment and a force pointing the centre of the circle, in other words, parallel to

*x*_{1}and assumed positive when opposite to the direction of the*x*_{1}-axis, so that for 0≤*ϕ*<*π*/2 (*π*/2<*ϕ*≤*π*) it corresponds to a positive tensile (negative compressive) dead force*F*applied to the structure defined by 3.9— Through introduction of the curvilinear coordinate

*s*, the fully nonlinear equation of the elastica-governing deflections of the rod is 3.10 where*θ*is the rotation angle (assumed positive if clockwise) of the normal at each point of the elastica, hence with the symbols introduced in figure 12, we find the condition 3.11

Integration of equation (3.10) from 0 to *s*, after multiplication by d*θ*/d*s*, leads to
3.12
where
3.13
in which the term *θ*(0)*k*_{r} corresponds to the moment evaluated at *s*=0. The introduction of the change of variable
3.14
where H denotes the Heaviside step function, allows to rewrite equation (3.12) as
3.15
so that a second change of variable yields
3.16

Restricting the treatment to the case ‘+’, which corresponds to *θ*(0)≥0, and as *β*=*β*(0) at *u*=0, equation (3.16) provides the following solution for *β*
3.17
where ‘am’ and ‘F’ are the Jacobi elliptic function amplitude and the incomplete elliptic integral of the first kind of modulus *k*, respectively (Byrd & Friedman 1971). Keeping into account that and , an integration provides the two coordinates *x*_{1} and *x*_{2} of the elastica expressed in terms of *u* as
3.18
in which the constants of integration are chosen, so that *x*_{1} and *x*_{2} vanish at *s*=0. In equation (3.18), ‘dn’ is the Jacobi elliptic function delta-amplitude of modulus *k*, where ‘E’ is the incomplete elliptic integral of the second kind (Byrd & Friedman 1971). Equation (3.18) generalizes the expressions derived by Zaccaria *et al.* (2011, their eqns (3.23) and (3.24)), which are recovered when *θ*(0)=0.

The horizontal displacement *δ* of the clamp on the left of the structure (assumed positive for a lengthening of the system) is given in the form
3.19
where, as for equation (3.8), the sign ‘−’ (‘+’) holds for the case of the rotational spring lying on the left (right) half-circle.

The axial load *F* can be obtained as a function of the rotation *ϕ*, or as a function of the end displacement *δ*, through the following steps.

A value for

*θ*(0) is fixed, hence*k*can be expressed using equation (3.13) as a function of*R*;the expressions (3.18) for the coordinates of the elastica and equation (3.17), evaluated at

*s*=*l*, become functions of*R*only;equation (3.11) provides

*ϕ*, so that equation (3.8) becomes a nonlinear equation in the variable*R*, which can be numerically solved (we have used the function FindRoot of Mathematica 6.0); andonce

*R*is known,*F*,*ϕ*and*δ*can be, respectively, obtained from equations (3.9), (3.11) and (3.19).

The postcritical behaviour (corresponding to the first modes branching from both tensile and compressive critical loads) of the structure is reported in figure 13 in terms of dimensionless axial load 4*Fl*^{2}/(*Bπ*^{2}) versus dimensionless displacement *δ*/*R*_{c}, for the particular case of a roller sliding on the profile, *k*_{r}=0.

We note that the elastica obtained in this case is *inflexional* and therefore different from that reported by Zaccaria *et al.* (2011), moreover, the postcritical behaviour is always unstable, evidencing decrease of the load with increasing edge displacement (‘softening’). Special features of the postcritical behaviour (already present in the one-degree-of-freedom system) are (i) that there is a transition from a tensile (a compressive) to a compressive (to a tensile) elastica when the constraint reaches the points denoted with ‘2’ and ‘5’ in the graph, and that (ii) the postcritical branches emanating from the critical loads are the same, but horizontally shifted.

### (c) Experiments on the elastica

We have tested the behaviour of an elastic rod by employing the same experimental set-up used for testing the one-degree-of-freedom structures in §2*b*, but with the rigid system replaced by elastic rods realized with two (250 mm×25 mm×4 mm) C72 carbon-steel strips (Young modulus 200 GPa, mass 968 g), see appendix A for details. The experimental set-up with photos taken during the tests is shown in figure 14. These experiments represent *the practical realization of a designed compliant mechanism*.

Experimental results are reported in figure 15 in terms of theoretical (dashed line) versus experimental force–end-displacement data.

Moreover, the photos reported in figure 16, which are details of the photos shown in figure 14*a*,*b*,*d*,*e* are compared with the theoretical shape of the elastica (shown with a white dashed line and obtained from equation (3.18)) at two different end angles (45° and 90° for tension and compression).

From the figures, we can observe the following facts.

— The experiments definitely substantiate theoretical findings.

— The comparison between the deformed beam during a test and the predictions of the elastica, shown in figure 16, reveals a very tight agreement between theory and experiments.

As for the one-degree-of-freedom systems, we can again conclude that the experiments confirm the possibility of practically realizing elastic systems behaving in strict agreement with theoretical predictions.

## 4. Conclusions

Effects related to the curvature and the shape of the constraint profile on which an end of a structure has to slide have been shown to be important on bifurcation and instability. In particular, we have found possibility of buckling both in tension and compression and multiple buckling loads, as for instance in the case of a one-degree-of-freedom structure evidencing two critical loads. Our experiments have confirmed that these effects can be designed to occur in real structural prototypes, so that new possibilities are opened in exploiting simple deformational mechanisms to obtain flexible mechanical systems.

## Acknowledgements

D.B. and G.N. gratefully acknowledge financial support from Italian Prin 2009 (prot. no. 2009XWLFKW-002); D.B. also acknowledges support from grant no. PIAP-GA-2011-286110.

## Appendix A. Details on the experimental set-up

Experiments reported in this study have been performed at the Laboratory for Physical Modeling of Structures and Photoelasticity of the University of Trento (managed by D.B.). A Midi 10 (10 kN maximum force, from Messphysik Materials Testing) electromechanical testing machine has been employed to impose displacements (velocity 0.2 mm s^{−1}) at the ends of the structures. Loads and displacements have been measured with the loading cell and the displacement transducer mounted on the Midi 10 machine, and, independently, with a MT 1041 (0.5 kN maximum load) load cell (from Mettler-Toledo) and a potentiometric displacement transducer Gefran LTM-900-S IP65.

The rotational springs employed for the one-degree-of-freedom systems have been designed to provide a stiffness equal to 211.5 Nm by employing eqns (32) of Brown (1981). After machining, the springs have been tested and found to correspond to a stiffness equal to 169.5 Nm, the value which has been used to compare experiments with theoretical results.

An IEPE accelerometer (PCB Piezotronics Inc., model 333B50) has been attached at one end of the structure to precisely detect the instant of buckling. This has been observed in all tests to correspond to an acceleration peak ranging between 0.15 and 0.2 g, while before buckling and during postcritical behaviour the acceleration did not exceed the value 0.003 g.

Data from the load cell MT 1041, the displacement transducer Gefran LTM-900-S IP65, and the accelerometer PCB 333B50 have been acquired with a system NI CompactDAQ, interfaced with Labview 8.5.1 (National Instruments), whereas acquisition of the data from the Midi 10 has been obtained from a Doli EDC 222 controller.

Temperature near the testing machine has been monitored with a thermocouple connected to a Xplorer GLX Pasco and has been found to lie around 22°C, without sensible oscillations during tests.

Photos have been taken with a Nikon D200 digital camera, equipped with AF Nikkor (18–35 mm 1:3.5–4.5 D) lens (Nikon Corporation) and movies have been recorded during the tests with a Sony handycam (model HDR-XR550VE). The testing set-up is shown in figure 17. Additional material can be found at http://ssmg.unitn.it/.

## Footnotes

↵1 Tensile buckling of an elastic structure, governed by the elastica in which all elements are strictly subject to tension, has been recently discovered by Zaccaria

*et al.*(2011).↵2 The fact that the curvature influences the critical load was observed in different terms already by Timoshenko & Gere (1961), who analysed the case of the so-called ‘load through a fixed point’. However, they did not generalize the problem enough to discover that: tensile buckling, multiple bifurcations and inflexional tensile elastica during the postcritical behaviour can be obtained, which is the topic attacked in this study.

↵3 Note that in the case of a circle, the dimensionless-signed curvature is , where

*l*is the length of the rigid bar, and*R*_{c}is the radius of the circle.↵4 A neutral postcritical behaviour has also been found by Gáspár (1984), employing a structural model completely different from that we considered.

- Received December 16, 2011.
- Accepted February 8, 2012.

- This journal is © 2012 The Royal Society