# Exact solutions for the buckling and postbuckling of a shear-deformable cantilever subjected to a follower force

## Abstract

The buckling and postbuckling of a shear-deformable cantilever is studied using Reissner’s geometrically exact relations for the planar deformation of beams. The cantilever is subjected to a compressive follower force whose line of action passes through a spatially fixed point. To study the buckling behavior, a consistent linearization of equilibrium and kinematic relations is introduced. The influence of shear deformation and extensibility on the critical loads is studied. The buckling behavior turns out to crucially depend on the ratio between the shear stiffness and the extensional stiffness of the structure. Closed-form solutions in terms of elliptic integrals for buckled configurations of the cantilever are derived in the present paper.

## 1 Introduction

The buckling of beams subjected to compressive forces is one of the classical problems in structural mechanics and stability. For centuries, scientists have studied critical loads at which structures become unstable and sought for methodologies to determine these loads. We mention Euler’s pioneering work in the appendix of his monograph [13], in which he determined the critical load of a column subjected to a compressive force and studied the buckled shapes. The treatise of Timoshenko and Gere [29] gives a detailed account of diverse stability problems of rods under different end restraints and loadings. Besides the four classical buckling problems with combinations of clamped, pinned and free ends, which have found their way into every mechanics textbook, they also present the buckling problem that is studied in the present paper. In the problem they refer to as *“Column with Load through a Fixed Point,”* which is illustrated in Fig. 1a, the line of action of the compressive force *P* is assumed to pass through the spatially fixed point *C*. Unlike classical buckling problems, in which the direction of the loading remains constant, the orientation of the force depends on the buckled configuration of the structure, which is why it can be regarded as one kind of a follower load. Still, the described problem is conservative which allows us to study the stability using static considerations only. By contrast, a follower load whose direction is determined by the orientation of the tangent to the beam’s axis at the point of application requires dynamics to be accounted for. This specific buckling problem is appealing from an experimental point of view, since the force can be realized by tensioning a cable that is mounted to the free tip of the beam and runs through the fixed point *C*. Exemplarily, such a setup was used in investigations to increase the critical loads of slender beams by means of piezoelectric transducers and feedback control [31, 32].

*elastica*, only admits flexural deformation but no shearing or stretching. For moderately thick structures, Timoshenko [27, 28] required that cross sections of the beam remain plane and undistorted in the course of deformation, but he no longer assumed that cross sections had to remain perpendicular to the beam’s axis. Timoshenko’s linear theory was generalized to arbitrarily large deformations by Reissner [25] and Antman [1]. For the three-dimensional relations governing the spatial deformation of beams, we refer to Eliseev [10, 11]. Engesser [12] and Haringx [17] obtained differing results for the critical loads of shear-deformable beams under a compressive force. The discrepancy was later resolved by Bažant [7, 8], who showed that Engesser’s and Haringx’ results do coincide once consistent tangent stiffnesses corresponding to the respective strain measures are used. Antman and Rosenfeld [2] studied the buckling of beams for general types of nonlinearly elastic material behavior and boundary conditions. The effects of shear deformation and extensibility on the buckling behavior of the classical problems, in particular, the statically indeterminate cases of clamped-hinged and clamped-clamped beams, were investigated by Humer [19].

The second focus of the paper lies on the construction of closed-form relations for the buckled configuration of shear-deformable beams. For the classical elastica, Kirchhoff noted the close resemblance between the equilibrium relations of a beam subjected to concentrated loads at its ends and the equations of motion of a pendulum [20]. Based on what is referred to as *Kirchhoff’s kinetic analogue* [21], Saalschütz presented closed-form solutions in terms of elliptic integrals [26]. We also mention the work of Frisch-Fay [14], who derived the elliptic integral relations describing the buckled configuration of a simply supported beam. Results for the clamped-hinged beam were presented in [9, 23]. For the extensible elastica, which has a finite extensional stiffness but is rigid with respect to shear, elliptic integral solutions were presented, e.g., by Pflüger [24] and Humer [18]. Magnusson et al. [22] studied the postbuckling behavior of a simply supported extensible elastica in detail. Elliptic integral solutions accounting for shear deformation were presented by Goto et al. [15]. The contribution by Humer [19] on the four classical buckling problems, in which he derived solutions for the buckled configurations of beams based on Reissner’s geometrically nonlinear theory, provides the groundwork for the present paper. An additional transformation enables closed-form solutions in terms of elliptic integrals also for the generalized elastica which exhibits both stretching and shear deformation. The family of Jacobi-elliptic functions can be regarded as inverse functions to elliptic integrals. Batista presented solutions in terms of Jacobi-elliptic functions for both the classical elastica [4, 5] and shear-deformable beams [3, 6]. The problem of a classical elastica whose tip is pulled by a cable [4] is closely related to the present problem. However, shear deformation and extensibility were neglected and the stability problem was not addressed.

We also want to mention that a large number of contributions seek for approximate solutions to equilibrium configurations of largely deformed beams in general and buckled configurations in particular. With the focus lying on analytical solutions, however, we do not further go into details here.

The present paper is organized as follows: In Sect. 2, we introduce the kinematics of shear-deformable beams and derive the equilibrium relations for the present problem of a cantilever subjected to a follower load. For the sake of generality, a non-dimensional setting is adopted subsequently. In Sect. 3, a consistent linearization of the geometrically nonlinear relations is performed. We obtain a transcendental equation whose roots determine the critical loads at which the trivial solution of a compressed beam becomes unstable. We provide both qualitative and quantitative results for the buckling behavior. By a series of transformations, closed-form relations for both the rotation of cross sections and the axial displacement and the transverse deflection in terms of elliptic integrals are derived in Sect. 4. Exemplarily, we illustrate several buckled configurations for different geometries.

## 2 Governing equations

In the present paper, we focus on the planar buckling and postbuckling of beams. In what follows, the beam’s axis is assumed to be straight in the undeformed reference configuration. The axis coincides with the *x*-axis of a global Cartesian frame \(\left\{ \varvec{e}_x, \varvec{e}_y, \varvec{e}_z\right\} \); the beam deforms in the (*x*, *y*)-plane of the global frame.

### 2.1 Geometrically exact beam theory

*z*-axis relative to the global frame,

*x*-direction, and

*y*-direction. The geometrically exact beam theory requires strain measures that can account for large deformations and are invariant to arbitrary rigid-body rotations. Based on variational arguments, Reissner introduced generalized strains for stretching, i.e., the axial force strain

*N*denotes the normal forces and

*Q*the shear forces, and the resultant moment

*M*about the

*z*-axis, which we simply refer to as bending moment (Fig. 1b). In our studies on buckling and postbuckling, we adopt the conventional linear material behavior of structural mechanics. Assuming that no physical nonlinearities are present and that the shape of the cross section allows a decoupling of the strains, the constitutive equations read

*E*and

*G*denote Young’s modules and the shear modulus, respectively, and

*k*is the shear correction factor that accounts for the non-uniform distribution of shear stresses over the cross section. The cross-sectional area and moment of inertia about the

*z*-axis are denoted by

*A*and

*I*.

### 2.2 Equilibrium relations

*N*,

*Q*and

*M*are introduced in the local frame, which is aligned with the deformed cross section and rotated by the angle \(\phi \) about the

*z*-axis. At the base of the cantilever, we find the horizontal and vertical components of the applied force

*P*, where the angle \(\alpha \) depends on the position of the beam’s free tip in the deformed configuration, see Fig. 1a. The reaction moment at the clamping is denoted by \(M_0\). Although the orientation of the applied force “follows” the deformation of the beam, we can find a potential

*V*, whose derivative with respect to the tip displacement gives the force. Let \(\varvec{v}\) denote the vector from the beam’s tip to the point

*C*(cf. Fig. 1a),

*P*and the length \(c - \Vert \varvec{v}\Vert \) as potential

*V*:

*c*in the potential

*V*, we obtain an intuitive interpretation: We assume that the load is imposed by pulling an (inextensible) rope that is attached to the beam’s tip and passes through

*C*. At the point

*C*, we can further assume the rope to be deflected into some constant direction. The distance \(c - \Vert \varvec{v}\Vert \) then represents the displacement of the rope’s end at which the force

*P*is applied. The potential

*V*therefore corresponds to the work performed by

*P*, which is the product of the force intensity and the displaced length.

*x*-direction, we obtain the equilibrium relation

*y*-direction gives

*X*, the distance

*c*as well as the components of the displacement vector

*u*and

*w*to the length

*l*of beam in the reference configuration,

*D*and the shear stiffness

*S*and the non-dimensional slenderness parameter \(\lambda \), which relates the extensional stiffness and the bending stiffness,

*l*/

*r*, where the radius of gyration is obtained from the cross-sectional area

*A*and its second moment of area

*I*as \(r = \sqrt{I / A}\). For notational simplicity, we introduce the square in our definition of the slenderness (25).

*B*. Alternatively, we can also introduce Euler’s load \(P_\mathrm{e}\), i.e., the (first) critical load, at which a simply supported beam buckles under a concentrated compressive force,

## 3 Linearized problem

*s*and evaluate the limit \(s \rightarrow 0\), by which we obtain

*u*does not vanish in the trivial (unbuckled) solution, since the extensional stiffness is finite. The corresponding axial displacement of the beam’s free tip in the pre-compressed state is denoted by \(u_0(1)\).

*x*-position of the beam’s tip, and

*a*and

*e*have been introduced for notational simplicity. The solution for the homogeneous problem of the linearized equation reads

*a*and

*e*, we realize that the roots of the transcendental equation are hyper-surfaces in the four-dimensional space spanned by the non-dimensional geometry parameter

*c*, the slenderness \(\lambda \), the stiffness ratio \(\nu \) and the force factor \(\chi \). For the case \(c = 1\), i.e., a force whose line of action runs through the base of the cantilever, the surfaces corresponding to the lowest four critical loads are exemplarily illustrated in Fig. 3. In Fig. 3, the critical loads of Euler’s classical elastica are indicated as light-gray planes parallel to the (\(\nu \), \(\lambda \))-plane. From (48), we recover the corresponding critical loads of the classical elastica as the limiting case \(\nu \rightarrow 0\), \(\lambda \rightarrow \infty \), which results in Timoshenko’s relation

To provide a deeper insight, we illustrate (up to) four critical loads as function of the slenderness \(\lambda \) for different values of the stiffness ratio \(\nu \) and for the three cases \(c = 1 / 2\), \(c = 1\) and \(c = 2\) in Figs. 4, 5 and 6. The dotted lines in the figures represent the critical loads of the classical elastica. Dashed lines delimit the admissible region of deformation and correspond to the limiting case of a beam being compressed to zero length, i.e., \(\varepsilon = -\,1\), which occurs for a load of \(\chi = \lambda \). The results in Figs. 4a–d show that the buckling behavior crucially depends on the stiffness ratio \(\nu \). In cases where the extensional stiffness is greater than the shear stiffness, i.e., \(\nu > 1\), the critical loads of the shear-deformable beam are smaller than the corresponding loads in the classical elastica. The critical loads increase with increasing slenderness \(\lambda \), and we obtain the critical loads of the classical elastica in the limit \(\lambda \rightarrow \infty \). We further note that, for a given slenderness \(\lambda \), the critical loads decrease if the shear stiffness is decreased compared to the extensional stiffness.

Similar as in the classical buckling problems [19], the picture changes completely, if the shear stiffness becomes greater than the extensional stiffness, i.e., \(\nu < 1\). In this case, the critical loads at which the beam buckles are greater than in Euler’s classical elastica and approach that latter from above in the limit of an infinite slenderness, i.e., \(\lambda \rightarrow \infty \). Regarding the number of critical loads and the corresponding buckling modes, we can make the following observations: The classical elastica has an infinite number of critical loads and corresponding buckling modes. If the extensional stiffness is equal or greater than the shear stiffness, i.e., \(\nu \ge 1\), the number of buckling modes is restricted by the requirement \(\chi \le \lambda \) and therefore depends on the slenderness \(\lambda \). The dependence on \(\lambda \) is distinct if the shear stiffness is greater than the extensional stiffness, i.e., \(\nu < 1\). We note that a second critical load exists for each buckling mode, which lies, at least partially, within the admissible region. For each buckling mode, however, we can determine a slenderness, for which the two roots of the transcendental equation (48) and below which no critical load exists. In fact, the region enclosed by the dashed line and first critical load includes configurations of strongly compressed beams, in which the unbuckled equilibrium state is stable. Table 1 provides the critical loads \(\chi _\mathrm{crit}/ \pi ^2\) corresponding to the first three buckling modes (\(N=1,2,3\)) for two values of the slenderness \(\lambda \). In case two critical loads exist, we only provide the lower one; dashes indicate parameter combinations for which no critical load is found.

Increasing the distance *c*, the qualitative behavior remains unchanged. In Figs. 5 and 6, the critical loads are plotted as functions of the slenderness \(\lambda \) for the cases \(c=1\) and \(c=2\). The case \(c=1\), i.e., the line of action of the compressive force runs through the base of the cantilever has the same critical loads as a simply supported beam under a compressive force. Moving the point *C* further away from the base, we observe that the critical loads decrease. The limiting case \(c \rightarrow \infty \) corresponds to a clamped cantilever subjected to a compressive force that does not change its direction, i.e., the angle \(\alpha \) vanishes identically. Quantitative results for the critical loads are provided in Tables 2 and 3, where different values of \(\nu \) and \(\lambda \) have been considered.

## 4 Postbuckling solution

Critical loads \(\chi _\mathrm{crit}/ \pi ^2\) for \(c = 1/2\)

| \(\nu = 10\) | \(\nu = 2\) | \(\nu = 1/10\) | \(\nu = 1/100\) | ||||
---|---|---|---|---|---|---|---|---|

\(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | |

1 | 1.1241 | 1.5798 | 1.6820 | 1.8126 | 2.0774 | 1.8894 | 2.1065 | 1.8933 |

2 | 2.5503 | 4.1646 | 4.6779 | 5.5245 | – | 6.2114 | – | 6.2531 |

3 | 4.0180 | 7.1095 | 8.2919 | 10.6785 | – | 13.5167 | – | 13.7457 |

Critical loads \(\chi _\mathrm{crit}/ \pi ^2\) for \(c = 1\)

| \(\nu = 10\) | \(\nu = 2\) | \(\nu = 1/10\) | \(\nu = 1/100\) | ||||
---|---|---|---|---|---|---|---|---|

\(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | |

1 | 0.7501 | 0.9241 | 0.9550 | 0.9903 | 1.0489 | 1.0090 | 1.0543 | 1.0100 |

2 | 2.0795 | 3.1299 | 3.4221 | 3.8535 | 5.2017 | 4.1532 | 5.4524 | 4.1700 |

3 | 3.5144 | 5.9039 | 6.7510 | 8.3173 | – | 9.8643 | – | 9.9715 |

Critical loads \(\chi _\mathrm{crit}/ \pi ^2\) for \(c = 2\)

| \(\nu = 10\) | \(\nu = 2\) | \(\nu = 1/10\) | \(\nu = 1/100\) | ||||
---|---|---|---|---|---|---|---|---|

\(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | \(\lambda = 200\) | \(\lambda = 1000\) | |

1 | 0.3762 | 0.4072 | 0.4114 | 0.4159 | 0.4224 | 0.4181 | 0.4229 | 0.4182 |

2 | 1.5293 | 2.0916 | 2.2225 | 2.3937 | – | 2.4971 | 2.8042 | 2.5024 |

3 | 2.9169 | 4.6277 | 5.1772 | 6.0952 | – | 6.8555 | – | 6.9022 |

*e*as

*C*denotes a constant of integration. We can determine this constant of integration by considering the boundary conditions at the free end \(X=1\), which, in terms of \(\beta \), read

*e*vanishes, which is the case if the shear stiffness and the extensional stiffness are identical (\(\nu = 1\)), and for Euler’s classical elastica (\(\nu = 0\) and \(\lambda \rightarrow \infty \)).

*c*on the other hand. For \(c > 1\), the number of inflection points along the axis is

*N*, where

*N*denotes the corresponding buckling mode, whereas the axis shows \(N+1\) inflection points for \(c \le 1\). Note that inflection points at end points of the beam are included in the numbers above. Exemplarily, we consider the first buckling mode (\(N=1\)), for which we find a single inflection point at the free tip if \(c > 1\), whereas an additional inflection point occurs along the axis if \(c \le 1\). To avoid a cumbersome case-by-case analysis regarding the change of signs, we exploit the periodicity of trigonometric functions, which allows \(\psi \) to be taken as a monotonically increasing function ranging from \(\psi _0 > 0\) at the clamped end to \(\psi _1 = N \pi + \pi / 2\) at the free end; under this assumption, we can distinguish the three different cases with respect to \(\psi _0\): For \(c < 1\), we obtain the interval \(0< \psi _0 < \pi /2\); we find \(\psi _0 = \pi / 2\) for \(c = 1\), and for \(c > 1\), the angle must lie in the range of \(\pi / 2< \psi _0 < \pi \).

*X*and the angle \(\psi \) yields

*F*(

*m*) and \(F(m, \psi _0)\) denote the complete and incomplete elliptic integrals of the first kind. We want to emphasize that the notation of elliptic integrals is not consistent in the literature. In what follows, we adopt the notation that is used in the symbolic software

*Mathematica*[30]. The above result relates the compressive force to the angle \(\psi _0\) and, through the transformations (61), (58), (50), to the rotation \(\phi _1 = \phi (1)\) at the tip of the cantilever. As a second unknown, relation (70) also contains the angle \(\alpha \), which depends on the equilibrium configuration of the buckled beam. We can establish a further equation by considering the reaction moment at the clamped end of the beam, whose non-dimensional representation is given by

*X*and the corresponding angle \(\psi (X)\) if we integrate (63) only over a part of the beam rather than the whole structure,

*e*in the integrands of (34)–(35), which gives the relation

*X*, at which the displacements

*u*,

*w*and the angle \(\psi \) occur. In addition to the elliptic integral of the first kind, which has already emerged in the solution of the rotation (73), further, more complicated terms emerge, which we want to briefly discuss in what follows. For further details, we refer to the table of integrals by Gröbner and Hofreiter [16]. We start with the integral of the second term on the right-hand side of the above equations, which evaluates to

*n*is also referred to as the characteristic, and

*C*is some constant of integration. The integral of the second term in the curly braces on the right-hand side of (81) and (82) gives

*X*, the angle of rotation \(\psi \), the axial displacement

*u*and the transverse deflection

*w*by means of the results (73), (86) and (88). To construct the buckled shapes, we first have to solve the nonlinear relations (70)–(72) for \(\phi _1\) and \(\alpha \) and compute the corresponding \(\psi _0\) to subsequently evaluate the elliptic integral solutions within the range \(\psi \in \left[ \psi _0, N \, \pi + \pi / 2 \right] \).

In Figs. 7, 8 and 9, several buckled configurations are illustrated for the cases \(c = 1/2\), \(c = 1\) and \(c = 2\), where the respective load intensities are provided in the figures. The illustrated shapes correspond to the first three buckling modes of the cantilever.

As for the critical loads, we obtain the solutions of the extensible elastica as the limiting case \(\nu = 0\). If we further let \(\lambda \rightarrow \infty \), the corresponding relations of Euler’s classical elastica are recovered from the above results.

## 5 Conclusion

In the present paper, we have discussed the buckling and postbuckling of a cantilever subjected to a follower load. Adopting Reissner’s theory for large deformations of beams, we have included the effects of shear deformation and extensibility of the beam’s axis. Linearizing the governing equations about the unbuckled configuration, we have obtained an inhomogeneous eigenvalue problem, from which the critical loads at which the trivial solution becomes unstable have been determined. The critical loads depend on both the slenderness of the structure and, more importantly, on the ratio between the shear stiffness and the extensional stiffness. While the buckling behavior of a shear-deformable beam qualitatively follows the classical elastica if the shear stiffness is smaller than the extensional stiffness, we find a different behavior if the shear stiffness exceeds the extensional stiffness. In the latter case, up to two bifurcation points exist per buckling mode. Moreover, we have observed admissible regions of the slenderness, in which no critical loads are found, i.e., buckling does not occur. Subsequently, we have determined exact solutions for buckled configurations of the beam in terms of elliptic integrals. An additional nonlinear equation related to the orientation of the force has to be solved before the closed-form relations for the rotation of the cross sections and the displacement of the beam’s axis can be evaluated. We have presented examples of buckled shapes for several locations of the fixed point, through which the follower force passes.

## Notes

### Acknowledgements

Open access funding provided by Johannes Kepler University Linz. This work has been supported by the COMET-K2 Center of the Linz Center of Mechatronics (LCM) funded by the Austrian federal government and the federal state of Upper Austria.

## References

- 1.Antman, S.S.: The theory of rods. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2. Springer, Berlin (1972)Google Scholar
- 2.Antman, S.S., Rosenfeld, G.: Global behavior of buckled states of nonlinearly elastic rods. SIAM Rev.
**20**(3), 513–566 (1978)MathSciNetCrossRefGoogle Scholar - 3.Batista, M.: Large deflections of shear-deformable cantilever beam subject to a tip follower force. Int. J. Mech. Sci.
**75**, 388–395 (2013). https://doi.org/10.1016/j.ijmecsci.2013.08.006 CrossRefGoogle Scholar - 4.Batista, M.: Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions. Int. J. Solids Struct.
**51**(13), 2308–2326 (2014). https://doi.org/10.1016/j.ijsolstr.2014.02.036 CrossRefGoogle Scholar - 5.Batista, M.: Large deflection of cantilever rod pulled by cable. Appl. Math. Model.
**39**(10–11), 3175–3182 (2015). https://doi.org/10.1016/j.apm.2014.10.073 MathSciNetCrossRefGoogle Scholar - 6.Batista, M.: A closed-form solution for Reissner planar finite-strain beam using Jacobi elliptic functions. Int. J. Solids Struct.
**87**, 153–166 (2016). https://doi.org/10.1016/j.ijsolstr.2016.02.020 CrossRefGoogle Scholar - 7.Bažant, Z.P.: A correlation study of formulations of incremental deformation and stability of continuous bodies. J. Appl. Mech.
**38**(197), 9–19 (1971)zbMATHGoogle Scholar - 8.Bažant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. Oxford University Press, Oxford (1991)zbMATHGoogle Scholar
- 9.Domokos, G., Holmes, P., Royce, B.: Constrained Euler buckling. J. Nonlinear Sci.
**7**(3), 281–314 (1997)MathSciNetCrossRefGoogle Scholar - 10.Eliseev, V.V.: The non-linear dynamics of elastic rods. J. Appl. Math. Mech. (PMM)
**52**, 493–498 (1988)MathSciNetCrossRefGoogle Scholar - 11.Eliseev, V.V.: Mechanics of deformable solid bodies (in Russian), St. Petersburg State Polytechnical University Publishing House (2006)Google Scholar
- 12.Engesser, F.: Die Knickfestigkeit gerader Stäbe. Centralblatt Bauverwalt.
**11**(49), 483–486 (1891)zbMATHGoogle Scholar - 13.Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti. Marc-Michel Bousquet et Soc., Lausanne (1744)Google Scholar
- 14.Frisch-Fay, R.: Flexible Bars. Butterworths, London (1962)zbMATHGoogle Scholar
- 15.Goto, Y., Yoshimitsu, T., Obata, M.: Elliptic integral solutions of plane elastica with axial and shear deformations. Int. J. Solids Struct.
**26**(4), 375–390 (1990). https://doi.org/10.1016/0020-7683(90)90063-2 MathSciNetCrossRefzbMATHGoogle Scholar - 16.Gröbner, W., Hofreiter, N.: Integraltafel. 1. Teil: Unbestimmte Integrale, 5th edn. Springer, New York (1975)Google Scholar
- 17.Haringx, J.A.: On the buckling and lateral rigidity of helical springs. I., II. Proc. Konink. Ned. Akad. Wet.
**45**, 533–539 (1942)MathSciNetzbMATHGoogle Scholar - 18.Humer, A.: Elliptic integral solution of the extensible elastica with a variable length under a concentrated force. Acta Mech.
**222**(3–4), 209–223 (2011). https://doi.org/10.1007/s00707-011-0520-0 CrossRefzbMATHGoogle Scholar - 19.Humer, A.: Exact solutions for the buckling and postbuckling of shear-deformable beams. Acta Mech.
**224**(7), 1493–1525 (2013). https://doi.org/10.1007/s00707-013-0818-1 MathSciNetCrossRefzbMATHGoogle Scholar - 20.Kirchhoff, G.: Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. J. Reine Angew. Math.
**56**, 285–313 (1859). https://doi.org/10.1515/crll.1859.56.285 MathSciNetCrossRefGoogle Scholar - 21.Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover Publications Inc, New York (1944)zbMATHGoogle Scholar
- 22.Magnusson, A., Ristinmaa, M., Ljung, C.: Behaviour of the extensible elastica solution. Int. J. Solids Struct. (2001). https://doi.org/10.1016/S0020-7683(01)00089-0 CrossRefGoogle Scholar
- 23.Mikata, Y.: Complete solution of elastica for a clamped-hinged beam, and its applications to a carbon nanotube. Acta Mech. (2007). https://doi.org/10.1007/s00707-006-0402-z CrossRefGoogle Scholar
- 24.Pflüger, A.: Stabilitätsprobleme der Elastostatik. Springer, Berlin (1964)CrossRefGoogle Scholar
- 25.Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. Z. Angew. Math. Phys. (ZAMP) (1972). https://doi.org/10.1007/BF01602645 CrossRefGoogle Scholar
- 26.Saalschütz, L.: Der belastete Stab unter Einwirkung einer seitlichen Kraft: Auf Grundlage des strengen Ausdrucks für den Krümmungsradius. Teubner (1880)Google Scholar
- 27.Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**41**(245), 744–746 (1921). https://doi.org/10.1080/14786442108636264 CrossRefGoogle Scholar - 28.Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**43**(253), 125–131 (1922). https://doi.org/10.1080/14786442208633855 CrossRefGoogle Scholar - 29.Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability, Dover Civil and Mechanical Engineering, 2nd edn. Dover Publications, Mineola (2009)Google Scholar
- 30.Wolfram Research, Inc.: Mathematica, Version 11.3. Champaign, IL (2018)Google Scholar
- 31.Zenz, G., Humer, A.: Enhancement of the stability of beams with piezoelectric transducers. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**227**(10), 744–751 (2013). https://doi.org/10.1177/0959651813503461 CrossRefGoogle Scholar - 32.Zenz, G., Humer, A.: Stability enhancement of beam-type structures by piezoelectric transducers: theoretical, numerical and experimental investigations. Acta Mech.
**226**(12), 3961–3976 (2015). https://doi.org/10.1007/s00707-015-1445-9 MathSciNetCrossRefGoogle Scholar

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