Acta Mech 224, 1493–1525 (2013)DOI 10.1007/s00707-013-0818-1

Alexander Humer

Exact solutions for the buckling and postbucklingof shear-deformable beams

Received: 1 October 2012 / Revised: 20 December 2012 / Published online: 23 February 2013© Springer-Verlag Wien 2013

Abstract The buckling and postbuckling of beams is revisited taking into account both the influence of axialcompressibility and shear deformation. A theory based on Reissner’s geometrically exact relations for theplane deformation of beams is adopted, in which the stress resultants depend linearly on the generalized strainmeasures. The equilibrium equation is derived in a general form that holds for the statically determinate andindeterminate combinations of boundary conditions representing the four fundamental buckling cases. Theeigenvalue problem is recovered by consistent linearization of the governing equations, the critical loads atwhich the trivial solution bifurcates are determined, and the influence of shear on the buckling behavior isinvestigated. By a series of transformations, the equilibrium equation is rearranged such that it allows a rep-resentation of the solution in terms of elliptic integrals. Additionally, closed-form relations are provided forthe displacement of the axis, from which buckled shapes are eventually obtained. Even for slender beams, forwhich shear deformation can usually be neglected, both the buckling and the postbuckling behavior turn outto be affected by shear not only quantitatively, but also qualitatively.

1 Introduction

The buckling of a column under a compressive force is certainly one of the most thoroughly studied problemsnot only in structural mechanics, but also in other fields such as bifurcation theory, for instance. Over centu-ries, scientists have investigated those critical loadings at which members of structures lose their stability andpossibly collapse subsequently. Needless to say, a detailed overview of the existing literature on the variousaspects of this classical problem is beyond the scope of the present paper. No paper on buckling, however, goeswithout mentioning Euler’s pioneering contribution in the Additamentum I of his monograph [1] entitled DeCurvis Elasticis, in which he determined the critical load of a column and studied the deformed configurationsfor several loadings and boundary conditions. Following a proposition of Daniel Bernoulli, Euler investigatedan elastic beam showing bending deformation only. More information on the developments before and afterEuler’s important contributions can be found in Truesdell’s comprehensive historical treatise [2].

In the present paper, the problem of planar buckling is considered for a beam theory featuring a morecomplex deformation than the classical elastica. Other than in Euler’s derivations, the generalized elastica canalso undergo shearing and compression (or extension) in addition to bending deformation. Among the variousaspects of the buckling problem, the present paper focuses on two issues: first, a systematic derivation of thecritical loads at which the equilibrium path bifurcates is presented for the classical types of boundary conditionsdepicted in Fig. 1a–d; subsequently, closed-form solutions in terms of elliptic integrals for the beam’s buckledconfigurations—the term postbuckling is typically adopted in this context—are constructed. To this end, a

A. Humer (B)Institute of Technical Mechanics, Johannes Kepler University Linz,Altenberger Str. 69, 4040 Linz, AustriaE-mail: alexander.humer@jku.at

1494 A. Humer

(a) (b) (c) (d)

PPPP

Fig. 1 Boundary conditions of the classical buckling problems: simply supported beam with both ends pinned (a); cantileverwith one end clamped, one end free (b); one end clamped, one end pinned (c); both ends clamped (d)

theory based on Reissner’s geometrically exact relations [3] for the plane deformation of beams undergoingpossibly large deformation is employed. Note that Antman [4] presented a more general formulation for thespatial deformation of nonlinearly elastic rods. By specializing these relations for the plane problem, Antmanindependently recovered Reissner’s theory [3]. Adopting linear constitutive equations within these theories,one obtains a formulation which can be regarded as a generalization of Timoshenko’s linear relations [5]. Whilethe buckling of the classical elastica is discussed quite extensively in the literature, a combined treatment ofboth the buckling and the postbuckling of shear-deformable beams has not been presented up to now. Theprevious contributions on the bifurcation problem are restricted to the statically determinate situations, cf.Fig. 1a–b; closed-form solutions describing the buckled states of equilibrium are not available in any of thesecases.

For Euler’s elastica, the derivation of the critical loads for the usual combinations of end restraints can befound in almost every textbook on structural stability. Among these, the book of Timoshenko and Gere [6]has to be mentioned first and foremost; more recently, Simitses and Hodges presented a thorough explanationin [7]. In order to determine the beam’s buckled configuration, an exact representation of the geometry ofdeformation is required since the structure may change its shape substantially in this event. Kirchhoff [8]recognized that the spatial equilibrium relations of an inextensible, non-shearable beam subjected to forcesapplied at its ends are similar to the equations governing the motion of a pendulum. From this analogy, whichis referred to as “Kirchhoff’s kinetic analogue” by Love [9], it becomes clear that the buckled configurations ofbeams can be represented in terms of elliptic integrals and the related Jacobi elliptic functions, since these areknown to describe the planar motion of a pendulum. The first exact solutions for Euler’s classical elastica interms of elliptic integrals date back to Saalschütz [10]. In his remarkable treatise, he carefully investigated theproperties of the equilibrium configuration of a cantilevered column, which is deformed under the combinedloading by a compressive and a transverse force applied at its free end. Later, elliptic integral relations forthe bending of a cantilever under a transverse tip force were also presented by Bisshopp and Drucker [11].Among several other problems of the classical elastica, Frisch-Fay [12] also derived analytic solutions for thebuckled configurations of a simply supported beam under a compressive force as well as a cantilever clampedat one end while being free at the other. Stern [13,14] investigated large deformations of hinged struts usingelliptic integrals. Domokos et al. [15] studied the buckling of Euler’s elastica, for which the lateral deflection isconstrained by walls parallel to its undeformed axis, both analytically and experimentally. In their paper, theyalso discussed the more involved statically indeterminate problem of a beam clamped at one end and pinnedat the other, which was later examined in more detail by Mikata [16].

As one would expect, both the eigenvalue analysis for determining the critical loads and the construction ofclosed-form solutions for the buckled configuration become more complicated as soon as shear deformation andaxial compressibility (or extensibility) are included. A discussion on the critical loads is given by Bažant [17]and Bažant and Cedolin [18], who clarified that previous, apparently differing results of Engesser [19] andHaringx [20] actually coincide once the correct tangent stiffness moduli corresponding to the respective finite

Buckling and postbuckling of shear-deformable beams 1495

strain measures are used. Antman and Rosenfeld [21] investigated the qualitative properties of buckled rods fora very general class of nonlinearly elastic constitutive relations and supports. In their comprehensive treatise,they showed that determining the critical loads and the corresponding eigenfunctions becomes particularlycomplicated in case of statically indeterminate problems. In these, the kinematic constraints give rise to reac-tion forces that depend on the state of deformation. Closed-form relations in terms of elliptic integrals for theextensible elastica, which is rigid with respect to shearing, were first derived by Pflüger [22]. Goto et al. [23]presented solutions for several problems of beams subjected to terminal forces and moments using a similarapproach. Later, they generalized their formulation to also include shear deformation in [24], where they alsodiscussed the buckled configuration of an imperfect cantilever under a compressive force. Humer [25] usedelliptic integrals for solving the problem of an extensible elastica, which may move relative to one of itssupports under a transverse load. A detailed investigation of both the eigenvalue problem and the postbucklingbehavior of an extensible, but non-shearable elastica was conducted by Magnusson et al. [26]. However, theyrestricted themselves to the case of a simply supported beam and did not provide relations for the displacementof the beam’s axis.

Over the years, several numerical schemes have been developed or adapted particularly for the approxima-tion of buckled shapes. Wang [27], for instance, studied the buckling of the classical elastica, for which one endis clamped and the other is pinned, using a perturbation method as well as asymptotic analysis in combinationwith a shooting method for the numerical integration. Vaz and Silva [28] generalized Wang’s investigationsby replacing the clamped end with a rotational spring of arbitrary stiffness. A different approach was recentlychosen by Mazzilli [29], who applied the method of multiple scales, which is best known from nonlineardynamics [30]. In addition to the conventional combinations of end restraints, he also considered a beam undereccentric loading. Reissner’s beam theory [3], which is adopted in the present investigations, also serves asbasis for several important finite element formulations for problems of large deformations. Among those, theapproach by Simo and Vu-Quoc [31,32] is perhaps most renowned. A beam element based on the absolutenodal coordinate formulation going entirely without rotational degrees of freedom was presented by Gerstmayret al. [33], in which thickness deformation of the beam can also be included. Recently, this approach was fur-ther improved by Nachbagauer et al. [34], who achieved a reduction in the number of generalized coordinatesrequired in the approximation, by which the locking behavior could also be improved.

The main intention of the present paper is to both generalize and particularize some of the contributionsmentioned above in order to derive several novel solutions related to the buckling of extensible and shear-deformable beams under a compressive force. A second key idea is to contribute—on the basis of Reissner’stheory [3]—a careful, systematic, and unified discussion of the problems of stability on the one hand, andpostbuckling on the other hand, which is still missing in the literature. In particular, the critical loads for thetwo statically indeterminate buckling cases depicted in Fig. 1c–d are determined, and exact solutions for thebuckled configurations in terms of elliptic integrals are provided for the first time. To this end, the paper isorganized as follows: first of all, the beam theory is outlined briefly, beginning with the geometry of defor-mation and introducing Reissner’s [3] generalized measures of strain. Subsequently, the stress resultants areexplained and the nonlinear equilibrium relations for a beam under a compressive force are derived. A set ofconstitutive equations typically employed in elastic problems of structural mechanics closes the problem. InSect. 3, the eigenvalue problem is obtained as a second-order functional-differential equation [21] by con-sistent linearization of the governing equations. First, the eigenfunctions and critical loads of the staticallydeterminate boundary settings are determined; afterward, the solutions of the more difficult statically indeter-minate problems are discussed. A non-dimensional setting is utilized to illustrate the influence of extensionaland shear stiffness on the critical load, at which the trivial becomes unstable. In the following section, thenonlinear problem is reformulated such that a representation of the solution in terms of elliptic integrals canbe obtained. Closed-form relations are presented for the angle of rotation of the beam’s cross-sections aswell as the components of the displacement vector of its axis. Having obtained the relations for the staticallydeterminate combinations of end restraints, these are generalized for the statically indeterminate problems,which require an additional transformation. For each of the four fundamental buckling cases, selected buckledconfigurations are provided exemplarily.

2 Beam theory

In the following derivations, the assumption of ideal beams lacking any kind of imperfections is adopted.This implies a perfectly straight undeformed configuration, perfect supports, and a perfect alignment of the

1496 A. Humer

compressive force which is applied at one end of the beam. Moreover, an elastic beam is considered, whosegeometry and material properties shall remain constant along its entire length l.

Additionally, it should be mentioned that the terms “beam,” “column,” and “rod,” which are not introducedconsistently in the literature, are used synonymously in the present paper.

2.1 Kinematic relations

For the description of the geometry of deformation, a fixed Cartesian frame (ex , ey, ez) is employed, thex-axis of which is coinciding with the beam’s axis in the undeformed straight configuration. Additionally, itis assumed that the deformation takes place in the xy-plane only, which also implies that this represents theplane of minimum flexural rigidity.

The present investigations are restricted to slender and moderately thick beams, for which Timoshenko’swell-known hypothesis [5] holds. Accordingly, the cross-sections of the beam are expected to remain plane,but not necessarily perpendicular to the axis in the course of deformation, in which they are rotated by the angleφ(X) about the z-axis, cf. Fig. 2a. The unit vectors normal to and in direction of the deformed cross-section,eξ (X) and eη(X), represent a local frame, which is related to the fixed Cartesian system by

eξ = cosφex + sin φez, eη = − sin φex + cosφez . (1)

Let r0 denote the axis’ position in the deformed configuration of the beam, then the current position of amaterial point originally located at X = Xex + Y ey + Zez can be written as

r(X) = r0(X)+ Y eη(X)+ Zez . (2)

According to elementary differential geometry, differentiation of a vector field describing a curve in spacewith respect to the arc-length gives a unit vector tangential to that curve. Consequently, the derivative of theposition vector of the axis with respect to the material coordinate X , which represents the referential arc-lengthof the undeformed beam, yields

∂r0

∂X= Λ

(cosχeξ + sin χeζ

). (3)

In the above equation,Λ denotes the length ratio of a material line element of the axis in the deformed and in theundeformed configuration. It is also referred to as the axial stretch or stretch of the axis. The shear-angle, whichis the angle enclosed by the tangent to the beam’s axis and the outer normal of the deformed cross-section, is des-ignated byχ(X). Introducing the axial displacement u(X) and the lateral deflectionw(X) as the components ofthe displacement vector of the axis with respect to the Cartesian frame, u(X) = r0(X)−X = u(X)ex +w(X)ey ,the following relations are obtained by inserting Eq. (1) into Eq. (3):

1 + ∂u

∂X= Λ cos (φ + χ) , (4)

∂w

∂X= Λ sin (φ + χ) . (5)

(a) (b)

Fig. 2 Buckling of the generalized elastica: kinematics of the generalized elastica exhibiting bending, shearing, as well ascompression or extension (a); free-body diagram of a buckled beam (b)

Buckling and postbuckling of shear-deformable beams 1497

Reissner [3] postulated the existence of a set of three generalized strain measures: an axial force strain ε,a transverse force strain γ , and a bending strain κ . Using a variational formulation, he identified the strains interms of the previously introduced kinematic quantities as

ε = Λ cosχ − 1, (6)

γ = Λ sin χ, (7)

κ = ∂(φ − φ(0))

∂X, (8)

where φ(0) vanishes for an initially straight beam. In Reissner’s theory, the normal and shear force, N (X) andQ(X), which are conjugate to ε and γ , respectively, are introduced in the directions of the local frame (eξ ,eζ ), see Fig. 2a. The bending moment about the z-axis, which is conjugate to the bending strain κ , is denotedby M(X).

2.2 Derivation of the governing equations

In the present section, the equilibrium relations for a beam under a compressive force P applied at the materialpoint X = l are formulated before the material response is considered. There exist several possibilities for thederivation of these; the perhaps most straightforward way is to consider the free-body diagram of a buckledbeam intersected at some material point X of the axis, see Fig. 2b. At this point of the investigations, noparticular boundary condition at the ends is resorted to, since the governing equations should hold for all of thebuckling cases considered herein. In each of these, however, the compressive force evokes a reaction force ofthe same intensity in x-direction at the opposite support, i.e., at X = 0. Depending on the actual end restraints,also a reaction force V in y-direction, as well as a reaction moment M0 may additionally emerge, which areintroduced as depicted to Fig. 2b.

In the absence of distributed loads, balancing forces in x- and y-direction results in two equilibrium relations

P + N cosφ − Q sin φ = 0, (9)

V − N sin φ − Q cosφ = 0. (10)

The balance of moments with respect to the point of intersection reads

M − M0 + Pw + V (X + u) = 0. (11)

As the displacement of the beam’s axis at X = 0 vanishes, u(X = 0) = 0, the above relation can be writtenequivalently as

M − M0 + P

X∫

0

∂w

∂XdX + V

X∫

0

(1 + ∂u

∂X

)dX = 0. (12)

Taking the derivative with respect to X , one consequently obtains

∂M

∂X+ P

∂w

∂X+ V

(1 + ∂u

∂X

)= 0. (13)

Substituting the kinematic relations (4) and (5), expanding the trigonometric functions, and inserting the defi-nitions of the axial and shear force strain, Eqs. (6) and (7), subsequently yield the following local equilibriumrelation for the bending moment:

∂M

∂X+ (1 + ε) (P sin φ + V cosφ)+ γ (P cosφ − V sin φ) = 0. (14)

Alternatively, the equilibrium relations could also be obtained from a variational formulation analogously towhat Magnusson et al. [26] did for the extensible elastica rigid in shear. Their approach, however, is a bit moreinvolved since it requires the variations of the strain measures to be expressed in terms of the displacementderivatives.

1498 A. Humer

In order to close the problem, constitutive equations have to be specified, by which the stress resultants arelinked to the generalized strain measures. According to Reissner [3], these functions have to be determinedfrom appropriate experiments. In the present investigations, however, the conventional material behavior typ-ically used in elastic problems of structural mechanics is adopted, according to which the stress resultantsdepend linearly on the conjugate strains:

N = Dε, (15)

Q = Sγ, (16)

M = Bκ. (17)

Subsequently, the proportionality coefficients D, S, and B are referred to as extensional stiffness, shear stiff-ness, and bending stiffness or flexural rigidity. Note that the relations (15)–(17) do satisfy the monotonicitycondition in the sense of Antman and Rosenfeld [21], however, their requirement that the beam may not becompressed to zero length by a finite force is not met. As Magnusson et al. [26] pointed out for the extensibleelastica, the case of ε = −1, which means that the beam is compressed to a single point, represents the limit toa non-admissible region. For straight beams, this restriction implies that the compressive force has to satisfythe inequality

P < D. (18)

Of course, the validity of a beam theory is questionable for values close to that limit. The present mate-rial behavior can be interpreted as a physical linearization, while the exact description of the geometry ofdeformation is maintained.

In case of a homogeneous, isotropic material, the stiffnesses in the constitutive relations (15)–(17) can beexpressed in terms of Young’s modulus E and the shear modulus G as D = E A, S = ks G A, and B = E I ,where the cross-sectional area and the second moment of inertia about the z-axis are denoted by A and I ,respectively. Additionally, a shear-correction factor ks is required in Timoshenko-type beam theories in orderto account for the non-uniform distribution of shear stresses that depends on the shape of the cross-section.Recently, Irschik and Gerstmayr [35,36] presented an interpretation of the generalized strain measures andstress resultants adopted here in terms of fundamental quantities of nonlinear continuum mechanics. Theirfoundation allows to consistently incorporate material models formulated in finite strains and the respectiveconjugate stresses into the beam theory. For structures rigid in shear, i.e., χ = 0, Irschik and Gerstmayr [35]could show that the above constitutive equations for the normal force (15) and the bending moment (17) areequivalent to a linear relationship between Biot’s stress and Biot’s strain tensor. For shear-deformable beams,however, the fixed Cartesian frame (ex , ey, ez) adopted in the analysis does not coincide with the principal axesof Biot’s strain tensor any longer [36]. As a consequence, the representation of the stress resultants in termsof the conjugate stresses becomes more complicated, which is why the set of constitutive equations (15)–(17)employed in the present paper does not correspond to a linear relation between Biot’s measures of stress andstrain as opposed to the extensible elastica rigid in shear.

Suppose the cross-sections have a complex heterogeneous structure, which applies to sandwich and com-posite beams, for instance, the material constants are to be regarded as effective stiffnesses. These stiffnessesneed to be determined either by some more or less involved homogenization procedure or, alternatively, bymeans of material testing as Reissner suggested [3].

Now, proceeding with the derivations, the normal force and the shear force are expressed from the equi-librium relations (9) and (10) as

N = −P cosφ + V sin φ, (19)

Q = P sin φ + V cosφ. (20)

Comparing these with the constitutive relations (15) and (16), the axial force strain and the shear strainevaluate to

ε = −P cosφ + V sin φ

D, (21)

γ = P sin φ + V cosφ

S. (22)

Buckling and postbuckling of shear-deformable beams 1499

With the aid of the above relations and the constitutive equation for the bending moment (17), the equilibriumrelation for the bending moment (14) finally becomes

B∂2φ

∂X2 + (P sin φ + V cosφ)

[1 −

(1

D− 1

S

)(P cosφ − V sin φ)

]= 0. (23)

The equilibrium relation of the extensible elastica is recovered from the above equation by inserting S → ∞:

B∂2φ

∂X2 + (P sin φ + V cosφ)

[1 − 1

D(P cosφ − V sin φ)

]= 0. (24)

In case of the classical elastica, also the extensional stiffness becomes infinite, D → ∞, by which the governingequation is further simplified to

B∂2φ

∂X2 + P sin φ + V cosφ = 0. (25)

Equation (23) will later serve as starting point for the construction of closed-form solutions in terms of ellipticintegrals. Before, however, those critical loads need to be determined, at which the straight equilibrium con-figuration becomes unstable allowing the beam to buckle in the first place. In particular, the influence of theshear stiffness on the critical loads is of interest.

3 Solution of the linearized problem

The critical intensities of the compressive force, at which the trivial solution path bifurcates, are related to theeigenvalues of the governing equations. The generalized strain measures, Eqs. (6)–(8), together with the stressresultants, Eqs. (15)–(17), form a set of six quantities that are intrinsic to the present beam theory. As Antmanand Rosenfeld [21] pointed out, the buckling problem is therefore represented by the eigenvalue problem of asixth-order system of ordinary differential equations. Only four independent quantities remain in case of theclassical elastica, since the kinematic constraints of inextensibility and rigidity in shear reduce the order bytwo. Timoshenko and Gere [6] obtained the eigenvalue problem of the classical elastica assuming small defor-mations from the beginning. The constraints, i.e., ε = 0 and χ = 0, allow the angle to be approximated by thefirst derivative of the deflection φ ≈ ∂w/∂X as one can see from Eqs. (4) and (5). Inserting these assumptionsinto the equilibrium relation (25) and subsequent differentiation yield the well-known fourth-order differentialequation formulated in the lateral deflection,

B∂4w

∂X4 + P∂2w

∂X2 = 0, (26)

where the transverse reaction force V does not appear any longer. With the conventional boundary conditionsbeing also available as functions of w and its derivatives, the eigenfunctions and corresponding eigenvaluesare obtained more or less immediately for each of the classical buckling problems depicted in Fig. 1.

In view of the richer geometry of deformation of the generalized elastica, which is reflected in the complexstructure of the equilibrium relation (23), determining and solving the eigenvalue problem becomes more com-plicated. In his discussion on critical loads of an extensible, shear-deformable cantilever [37], Reissner deriveda second-order eigenvalue problem formulated in the shear force Q by linearization of the local equilibriumrelations about a pre-compressed state.

For statically indeterminate problems, the transverse reaction force V additionally enters the problem,which has to be linearized consistently as well. In line with the approach of Antman and Rosenfeld [21], asecond-order functional-differential equation is derived, from which the eigenfunctions and eigenvalues areobtained subsequently. However, as opposed to their investigations, in which they adopted general functionsof the strains as constitutive equations as well as nonlinear springs as boundary conditions, also quantitativeresults for the critical loads will be given in what follows.

To begin with, the nonlinear equilibrium equation (23) is expanded into a series about the trivial solution.The transverse reaction force V appears as an additional unknown that can be regarded as Lagrange multipliertaking care of the kinematic constraint for the deflection at X = l, i.e.,w(l) = 0, which holds in both staticallyindeterminate problems considered in this paper, cf. Fig. 1c, d. Therefore, not only φ is replaced by φ0 + sφ,

1500 A. Humer

where s denotes some small parameter, but also V0 + sV has to be substituted for V in Eq. (23). Since thetrivial solution represents a merely compressed, but straight configuration of the beam, both the angle and thereaction force vanish, i.e., φ0 = 0 and V0 = 0. Taking the derivative with respect to s and setting s = 0, thelinearized equilibrium relation is obtained as

B∂2φ

∂X2 +[

1 −(

1

D− 1

S

)P

](Pφ + V ) = 0. (27)

In order to eliminate V from the above equation, it is expressed from the constraint w(l) = 0, which needs tobe satisfied by every solution of φ and V . By inserting the definitions of the axial force strains (6) and shearforce strain (7) into the kinematic relation (5), substituting Eqs. (21)–(22) and performing integration, thiscondition is formulated as

w(l) =l∫

0

(1 − P cosφ − V sin φ

D

)sin φ +

(P sin φ + V cosφ

S

)cosφ dX = 0. (28)

Proceeding similarly as before, its linearization about the trivial solution subsequently yields

w(l) =l∫

0

[1 −

(1

D− 1

S

)P

]φ dX + V

l∫

0

1

SdX = 0, (29)

from which V immediately follows as

V = − S

l

l∫

0

[1 −

(1

D− 1

S

)P

]φ dX. (30)

After substituting this result in Eq. (27), the consistently linearized equilibrium equation formulated in theangle φ is eventually obtained as

B∂2φ

∂X2 +[

1 −(

1

D− 1

S

)P

]Pφ =

[1 −

(1

D− 1

S

)P

]2 S

l

l∫

0

φ dX. (31)

Before solutions of the above relation are discussed, several quantities are introduced for a preferably generalrepresentation of the critical loads in terms of non-dimensional similarity parameters, on the one hand, and amore compact representation of the following derivations, on the other hand. For this purpose, the compressiveforce is scaled with the first critical load of a simply supported classical elastica, which is also referred to asEuler’s load:

Pe = π2 B

l2 . (32)

Furthermore, the ratio between the extensional stiffness and the bending stiffness is adopted as a secondgoverning parameter, which is defined by

λ = l

√D

B. (33)

Provided the beam’s material is isotropic and homogeneous, the similarity parameter reduces to a geometricquantity, i.e., the ratio between the cross-section’s area A and its second moment of inertia I . Consideringa particular shape of the cross-section, λ may also be written in terms of the height-to-length ratio, whichexplains why it is referred to as slenderness in the literature, cf. Magnussion et al. [26]. In case of rectangularcross-sections, for instance, the slenderness is given by λ = √

12l/h. The shape of the cross-section alsoenters the shear-correction factor, which additionally depends on Poisson’s ratio ν of the respective material;suppose the shape is rectangular, it is given by ks = 10(1 + ν)/(12 + 11ν). Poisson’s ratio in turn appearsin the relationship between the shear modulus and Young’s modulus. In the present paper, however, a more

Buckling and postbuckling of shear-deformable beams 1501

general representation is employed by introducing the ratio between the shear and extensional stiffness as athird governing parameter:

η = S

D. (34)

For an isotropic, linear elastic material, the stiffness ratio is given by

η = ks G

E= ks

2(1 + ν), (35)

which implies that the extensional stiffness apparently exceeds twice the shear stiffness for conventional struc-tures. Larger values of η, however, are not only of interest from a theoretical point of view, since such ratioscannot be excluded a priori for today’s engineering materials. As already indicated, the effective stiffnessesneed be understood as representative for the macroscopic behavior of such materials, while the microstructuremay show a much more complex response. Both polymer and metal foams as well as some kinds of honeycombstructures, for instance, are known to exhibit a negative (effective) Poisson ratio, see, e.g., Lakes [38] and Pralland Lakes [39], which allows ratios between the shear end the extensional stiffness of η > 0.5.

In addition to the non-dimensional quantities, the abbreviation

e = η − 1

η

P

D= η − 1

η

π2

λ2

P

Pe, (36)

is introduced for notational convenience. Inserting all of these definitions, the eigenvalue problem (31) can beequivalently rewritten as

∂2φ

∂X2 + (1 − e)π2

l2

P

Peφ = (1 − e)2

ηλ2

l3

l∫

0

φ dX. (37)

In order to obtain a solution of the linear differential equation (37), the homogeneous part is considered first.The solution is immediately found as

φ = A sin (a X)+ B cos (a X) , (38)

where A and B denote some arbitrary constants and

a = √1 − e

√P

Pe

π

l(39)

has been introduced. Consequently, every solution of the inhomogeneous problem (37) has to satisfy

φ = A sin (a X)+ B cos (a X)+ (1 − e)η

l

λ2

π2

Pe

P

l∫

0

φ dX. (40)

Now, the question is how to treat the integral in the last term of the above expression. First of all, integratingEq. (40) over the entire length of beam gives

l∫

0

φ dX =l∫

0

A sin (a X)+ B cos (a X) dX + (1 − e)ηλ2

π2

Pe

P

l∫

0

φ dX. (41)

Collecting the coefficients of the integral expression yields

[1 − (1 − e) η

λ2

π2

Pe

P

] l∫

0

φ dX = A [1 − cos (al)] + B sin(al)

a, (42)

1502 A. Humer

from which one eventually obtains the general solution for φ as

φ = A sin (a X)+ B cos (a X)+[

1

(1 − e)η

π2

λ2

P

Pe− 1

]−1A [1 − cos(al)] + B sin(al)

al. (43)

In order to determine the critical loads, the solution has to be adapted to the particular boundary conditions ofthe respective buckling problem. To begin with, the statically determinate cases are considered, for which theinhomogeneous part vanishes; afterward, the more complicated statically indeterminate problems are investi-gated.

3.1 Column with both ends pinned

Due to its practical relevance, Timoshenko and Gere [6] refer to the problem of a simply supported beam,cf. Fig. 1a, as the “fundamental case” of buckling. For this reason, it shall be considered first in the presentanalysis. From elementary statics, it is well-known that the bending moment must vanish at the supports. Dueto the simple structure of the constitutive equation (17), it therefore follows immediately that also the derivativeof φ is zero at both ends of the column:

[∂φ

∂X

]

X=0= 0,

[∂φ

∂X

]

X=l= 0. (44)

Substituting the solution of the homogeneous problem (38), the following transcendental relations are obtainedfor the constants A and B:

a A = 0, a A cos(al)− aB sin(al) = 0. (45)

Obviously, the boundary conditions are satisfied by the trivial solution A = B = 0, in which the beam remainsin a compressed but straight configuration. In order to have a non-trivial solution, B �= 0, the argument of thesine has to vanish, which implies

al = nπ, n = 1, 2, . . . . (46)

Recalling the definitions (36) and (39), one obtains a quadratic equation for the critical loads, at which thetrivial solution bifurcates,

(1 − η − 1

η

π2

λ2

Pcr

Pe

)Pcr

Pe= n2, n = 1, 2, . . . , (47)

for which the solution is given by

Pcr

Pe= 1

2

η

η − 1

λ2

π2 ±√

1

4

(η

η − 1

)2λ4

π4 − n2 η

η − 1

λ2

π2 , n = 1, 2, . . . . (48)

Upon closer inspection, one recognizes that three different cases can be distinguished in the above result. Forcompleteness’ sake, it should be mentioned here that negative values of the shear stiffness make no sense, ofcourse; likewise, a negative slenderness is of limited physical relevance. To begin with, the shear stiffness isassumed to be smaller than the extensional stiffness, i.e., η < 1, which probably represents the most commonsituation for conventional types of structures. For these values of η, the expression under the root alwaysremains greater than zero regardless of the choice of n, which is why no complex roots exist. The root, inturn, is always greater than the leading term in Eq. (48). Therefore, the positive sign has to be chosen, sincethe critical load is expected to be positive. Comparing the results to the critical loads of the classical elastica,which are obtained from Eq. (47) as the limiting case where λ → ∞ and η → ∞,

Pcr = n2 Pe, n = 1, 2, . . . , (49)

it is obvious that the generalized elastica buckles at lower intensities of the compressive force if the stiffnessratio is η < 1.

Buckling and postbuckling of shear-deformable beams 1503

(a) (b)

(c)

Fig. 3 Critical loads corresponding to the first three buckling modes of a simply supported beam against the slenderness for astiffness ratio of η = 1/2 (a), η = 2 (b), and η = 100 (c)

(a) (b)

Fig. 4 Critical loads corresponding the first three buckling modes of a simply supported beam against the stiffness ratio η

This can be seen in Fig. 3a, in which the critical loads of the first three buckling modes are plotted againstthe slenderness for a stiffness ratio of η = 1/2. The dotted lines represent the corresponding results of theclassical elastica. For a decreasing thickness of the beam, i.e., the slenderness increases, the critical loadsapproach those of Euler’s classical elastica from below. The dashed line corresponds to the condition (18) forthe admissible values of the load intensity, which can be rewritten as P/Pe < λ2/π2. As explained before, itmarks the boundary at which the beam is compressed to a single point. Consequently, the number of criticalloads is restricted if the beam’s axis is extensible. This behavior differs from Euler’s classical elastica, forwhich countably infinite buckling modes exist. The straight configuration remains stable for all those valuesof the load intensity within the region enclosed by the lowest curve corresponding the first buckling mode andthe abscissa.

1504 A. Humer

Table 1 Critical loads Pcr/Pe of a simply supported beam

n η = 1/2 η = 2 η = 100

λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20

1 0.9629 0.9902 0.9975 1.0208 1.0051 1.0013 1.0431 1.0101 1.00252 3.5078 3.8516 3.9608 4.3845 4.0834 4.0202 4.9835 4.1723 4.04043 7.0256 8.3095 8.8061 11.7712 9.4461 9.1036 – 9.9875 9.2099

Now, suppose that the shear stiffness exceeds the extensional stiffness, i.e., η > 1. Having a closer look atthe relation (48) for the critical loads, it is first noted that the roots do not necessarily remain real-valued anylonger. Therefore, the trivial equilibrium path may not exhibit any bifurcation point at all in case of relativelythick beams. In order to have real-valued solutions, the inequality

λ2

π2

η

η − 1− 4n2 ≥ 0, n = 1, 2, . . . (50)

has to be fulfilled. It is further observed that the leading term in Eq. (48) is always greater than the root.Therefore, one may find as many as two critical loads associated with each eigenmode, on the one hand; on theother hand, the critical intensities of the compressive force now approach those of the classical elastica fromabove for an increasing slenderness. The substantially different buckling behavior of the beam can also beobserved in Fig. 3b, c, in which the critical loads of the first three buckling modes are depicted for η = 2 andη = 100, respectively. Quantitative results corresponding to the chosen parameters are presented in Table 1,in which only the lower of the possibly two critical loads—provided there exists any critical load at all—isgiven.

For smaller values of η, the second critical load, which corresponds to the positive sign in Eq. (48), mostlylies within the non-admissible region. If the shear stiffness, however, significantly exceeds the extensionalstiffness, as in Fig. 3c, one can see that the trivial solution may become stable again for very large intensitiesof the compressive force just before it would be compressed to a single point. It is doubtful, to say the least, ifsuch stable states of equilibrium correctly reflect the actual physical behavior of strongly compressed beams,since the validity of the present theory is certainly limited in these situations.

Eventually, the remaining case, in which shear and extensional stiffness coincide exactly, is considered.For η = 1, the quadratic term in Eq. (47) vanishes. Consequently, the buckling behavior is similar to Euler’sclassical elastica, cf. (50), although shear and axial deformation are taken into account. With the beam beingextensible, however, the non-admissible region still has to be considered.

The dependence of the first three critical loads on the stiffness ratio is depicted in Fig. 4a, b for a slender-ness of λ = 5π and of λ = 10π , respectively. For structures that are very soft in shear, the critical loads aresignificantly reduced compared to the classical elastica. One can also see that the critical loads of the highermodes are much more sensitive to the stiffness ratio than the lower ones.

The present result for the critical loads (48) coincides with that obtained by Reissner [37], which he referredto as “modified Engesser result.” Reissner’s solution, in turn, can be regarded as a generalization of Timoshenkoand Gere’s “modified shear equation” [6], in which the axial compressibility is not taken into account, i.e.,D → ∞. The difference between these results and Engesser’s original expression for the critical loads [19]originates in the distinct definitions of the shear strain and the shear force. Other than in the present beam theory,Engesser, who was among the first to investigate the influence of shear on the buckling behavior, introducedthe shear force Q perpendicular to the deformed axis and proportional to the angle φ − χ . Engesser’s result,however, proved to incorrectly predict the critical loads of structures for which shear deformation is particularlysignificant, such as helical springs, for instance. For very low values of the slenderness, experiments showedthat helical springs do not buckle at all. To circumvent this shortcoming, Haringx [20] derived a relation forshort springs, in which—unlike in Engesser’s solution—the critical load becomes infinite as the slendernesstends to zero. In Haringx’ derivation, the stress resultants were introduced similar as in the present beamtheory, see Timoshenko and Gere [6] as well as Bažant and Cedolin [18] for further details. For this reason,Reissner’s result was also denoted “modified Haringx formula” by Attard and Hunt [40], although Reissnerhimself referred to Engesser. The explanation for the apparently conflicting results was given by Bažant [17]and Bažant and Cedolin [18], who showed that Engesser’s and Haringx’ relations in fact do not contradict eachother. At a state of finite deformation, in which the compressed column may well be, the different definitions ofthe shear force and the shear strain require a different tangent stiffness to be used. The correct transformation

Buckling and postbuckling of shear-deformable beams 1505

Table 2 Critical loads Pcr/Pe of a cantilever

n η = 1/2 η = 2 η = 100

λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20

1 0.2476 0.2494 0.2498 0.2513 0.2503 0.2501 0.2525 0.2506 0.25022 2.0774 2.2015 2.2375 2.3615 2.2759 2.2564 2.4969 2.3025 2.26273 5.1777 5.9017 6.1553 7.3223 6.4586 6.2996 11.3636 6.6936 6.3498

of the tangent stiffness moduli corresponding to the respective strain measure was presented by Bažant andCedolin [18]. Once these are inserted in Engesser’s and Haringx’ formulations, their results can be transformedinto each other.

3.2 Column with one end clamped, one end free

For a cantilever with one end clamped and the other free, the critical loads are obtained analogously to thesimply supported beam. At the clamped end, the rotation of the cross-section is prohibited, while the bendingmoment is zero at the free end:

φ(X = 0) = 0,

[∂φ

∂X

]

X=l= 0. (51)

Inserting the solution of the homogenous problem (38) into these boundary conditions yields the equations

B = 0, a A cos(al)− aB sin(al) = 0, (52)

from which the condition

al = (2n − 1)π

2, n = 1, 2, . . . . (53)

for the existence of a non-trivial solution follows immediately. The critical load, at which buckling occurs, isthen obtained as

Pcr

Pe= 1

2

λ2

π2

η

η − 1±

√λ4

4π4

(η

η − 1

)2

− (2n − 1)2

4

λ2

π2

η

η − 1, n = 1, 2, . . . . (54)

Qualitatively, the buckling behavior of a cantilever corresponds to that of a column with both ends pinned. Ifthe shear stiffness is lower than the extensional stiffness, the critical loads are below those of Euler’s elastica.The reverse behavior is observed if the shear stiffness is higher than the extensional stiffness, i.e., the criticalloads are increased then. In the case η = 1, the critical load of the n-th buckling mode exactly equals that ofthe classical elastica, which is given by

Pcr = (2n − 1)2

4Pe, n = 1, 2, . . . . (55)

In Table 2, numerical results are collected for the parameters utilized in Fig. 5a–c.

3.3 Column with one end clamped, one end pinned

In statics, a structure is referred to as statically indeterminate if the reaction forces and moments at the supportscannot be determined from the equilibrium relations alone. A beam, which is clamped at one end while theother one is pinned, represents an example for a structure of this type. As soon as it buckles from its straightconfiguration due to a compressive force, a transverse reaction force V emerges at the supports that dependson the actual state of deformation. At the pinned end, the bending moment must vanish, which is why theboundary conditions remain the same as for the cantilever:

φ(X = 0) = 0,

[∂φ

∂X

]

X=l= 0. (56)

1506 A. Humer

(a) (b)

(c)

Fig. 5 Critical loads corresponding to the first three buckling modes of a cantilever against the slenderness for a stiffness ratio ofη = 1/2 (a), η = 2 (b), and η = 100 (c)

In contrast to a cantilever, however, the kinematic constraint for the deflection at the pinned end gives rise toa reaction force V in transverse direction. Consequently, the general solution of the inhomogeneous problem(43) has to be inserted into the boundary conditions, by which the following set of equations is obtained:

B +[

1

(1 − e)η

π2

λ2

P

Pe− 1

]−1A [1 − cos(al)] + B sin(al)

al= 0, a A cos(al)− aB sin(al) = 0. (57)

Solving the second equation for A and substituting the result in the first equation gives the relation

B

{

1 +[

1

(1 − e)η

π2

λ2

P

Pe− 1

]−1tan(al)

al

}

= 0, (58)

from which the following requirement for the existence of a non-trivial solution is obtained:

tan(al)

al= 1 − 1

(1 − e)η

π2

λ2

P

Pe. (59)

Recalling the definitions of e and a, cf. Eqs. (36) and (39), the above relation is reformulated as

tan

√

π2 P

Pe

(1 − η − 1

η

π2

λ2

P

Pe

)=

√

π2 P

Pe

(1 − η − 1

η

π2

λ2

P

Pe

)[

1 − 1

η − (η − 1) π2

λ2PPe

π2

λ2

P

Pe

]

. (60)

In order to obtain the critical loads from this nonlinear equation, some numerical procedure such as Newton’smethod, for instance, has to be employed. In Fig. 6a–c, the first three critical loads are depicted as a functionof the slenderness for the same three values of η as in the previous cases. Obviously, the qualitative behavior

Buckling and postbuckling of shear-deformable beams 1507

(a) (b)

(c)

Fig. 6 Critical loads corresponding to the first three buckling modes of a clamped-pinned column against the slenderness for astiffness ratio of η = 1/2 (a), η = 2 (b), and η = 100 (c)

Table 3 Critical loads Pcr/Pe of a clamped-pinned beam

n η = 1/2 η = 2 η = 100

λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20

1 1.8728 1.9976 2.0333 2.1268 2.0649 2.0505 2.2455 2.0889 2.05622 4.9621 5.6977 5.9521 6.9828 6.2344 6.0916 10.0263 6.4598 6.14013 8.7604 10.8259 11.6931 18.8404 12.8589 12.2309 – 13.9822 12.4294

remains quite similar to the statically determinate problems; for η = 2, however, the critical loads exhibit amaximum value outside the non-admissible region, which has not been observed before. Selected quantitativeresults corresponding to the graphs are summarized in Table 3.

3.4 Column with both ends clamped

As the last of the fundamental problems, the buckling of a column with both ends clamped is discussed. In thiscase, two different situations have to be distinguished in the first place: If the buckled shape exhibits a plane ofsymmetry at the structure’s midpoint X = l/2, no reaction force in transverse direction emerges, i.e., V = 0.Therefore, the homogeneous part of the general solution (43) has to be considered in order to determine thecritical loads. In case the buckled configuration is antisymmetric, the reaction moments at the clamped ends areoriented in the same direction. Similar as in the previous problem, a transverse reaction force must be presentif the structure is in equilibrium. In either case, however, the same boundary conditions have to be satisfied,which trivially read

φ(X = 0) = 0, φ(X = l) = 0. (61)

1508 A. Humer

Assuming first that the buckled shape is symmetric, it is further recognized that φ must also vanish at themidpoint, i.e., φ(X = l/2) = 0. By substituting the solution of the homogenous problem (38), one obtainsthe relations

B = 0, A sin

(al

2

)+ B cos

(al

2

)= 0, (62)

from which the conditional

2= nπ, n = 1, 2, . . . (63)

is obtained, which has to be fulfilled in order to have a solution different from the trivial one. The critical loadsare consequently found as

Pcr

Pe= 1

2

λ2

π2

η

η − 1±

√λ4

4π4

(η

η − 1

)2

− 4n2 λ2

π2

η

η − 1, n = 1, 2, . . . . (64)

If the buckled shape of the beam is antisymmetric, the solution of the inhomogeneous problem (43) has to beinserted into the boundary conditions (61). Thereby, one obtains the following equations for the constants Aand B:

B +[

1

(1 − e)η

π2

λ2

P

Pe− 1

]−1A [1 − cos(al)] + B sin(al)

al= 0, (65)

A sin(al)+ B cos(al)+[

1

(1 − e)η

π2

λ2

P

Pe− 1

]−1A [1 − cos(al)] + B sin(al)

al= 0, (66)

whose combination yields

B

{

1 + 2

[1

(1 − e)η

π2

λ2

P

Pe− 1

]−1 tan( al

2

)

al

}

= 0, (67)

from which the transcendental equation for the critical loads is eventually determined as

tan

√π2

4

P

Pe

(1 − η − 1

η

π2

λ2

P

Pe

)= 1

2

√

π2 P

Pe

(1 − η − 1

η

π2

λ2

P

Pe

)[

1 − 1

η − (η − 1) π2

λ2PPe

π2

λ2

P

Pe

]

. (68)

Similar as for the clamped-pinned beam, this nonlinear relation can only be solved with the aid of some numer-ical method. Figs. 7a–c illustrate the values of the first three critical loads as a function of the slendernessof the column. Both the lower and the top curve correspond to a symmetric buckling pattern, in which notransverse reaction force shows up, while the middle curve belongs to the first antisymmetric buckling mode.For relatively thick beams (λ/π < 5) and a conventional shear stiffness, cf. Fig. 7a, it is recognized that thecritical load intensity of the antisymmetric shape is actually closer to the first critical load of the classicalelastica than to its second one. Several numerical results are presented in Table 4 for the family of parametersthat has repeatedly been used in the present investigations.

4 Solution of the postbuckling problem

Having obtained the loads at which buckling occurs, the actual configuration of the beam after buckling is tobe determined in what follows. To this end, one has to resort to the nonlinear equilibrium relation (23), whichis first rewritten in terms of the quantities (32)–(34) that have been introduced for the sake of generality:

∂2φ

∂X2 + π2

l2

(P

Pesin φ + V

Pecosφ

) [1 − η − 1

η

π2

λ2

(P

Pecosφ − V

Pesin φ

)]= 0. (69)

Solving the statically indeterminate cases, in which the transverse reaction force does not vanish, unsurprisinglyis a bit more involved. Therefore, it appears reasonable to consider those boundary settings at first, for whichno transverse force emerges in the column’s buckled configuration. These results are subsequently generalizedfor the remaining statically indeterminate cases, for which V �= 0 holds.

Buckling and postbuckling of shear-deformable beams 1509

(a) (b)

(c)

Fig. 7 Critical loads corresponding to the first three buckling modes of a clamped-clamped column against the slenderness for astiffness ratio of η = 1/2 (a), η = 2 (b), and η = 100 (c)

Table 4 Critical loads Pcr/Pe of a beam with both ends clamped

n η = 1/2 η = 2 η = 100

λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20 λ/π = 5 λ/π = 10 λ/π = 20

1 (sym) 3.5078 3.8516 3.9608 4.3845 4.0834 4.0202 4.9835 4.1723 4.04041 (asym) 6.1522 7.4911 7.9904 9.8693 8.5072 8.2597 – 8.9807 8.35562 (sym) 11.0850 14.0312 15.4066 – 17.5379 16.3335 – 19.9338 16.6894

4.1 Column with both ends pinned

To begin with, the fundamental problem of a simply supported beam is investigated. The equilibrium equationdealt with for the moment is obtained by setting V = 0 in Eq. (69), which then simplifies to

∂2φ

∂X2 + π2

l2

P

Pesin φ

(1 − η − 1

η

π2

λ2

P

Pecosφ

)= 0. (70)

After multiplying the above relation by ∂φ/∂X , one spatial derivative can be pulled out such that the expressionreads

∂

∂X

[1

2

(∂φ

∂X

)2

− π2

l2

P

Pecosφ + 1

2

η − 1

η

π2

λ2

π2

l2

(P

Pe

)2

cos2 φ

]

= 0. (71)

The first integral of the equilibrium relation is therefore obtained as

1

2

(∂φ

∂X

)2

− π2

l2

P

Pecosφ + 1

2

η − 1

η

π2

λ2

π2

l2

(P

Pe

)2

cos2 φ = C, (72)

1510 A. Humer

where C denotes a constant of integration. Regardless of the particular boundary conditions, there alwaysexists at least one material point in the buckled configuration at which the bending strain vanishes, i.e., κ = 0.These points may either represent pinned or free ends, or inflection points of the beam’s axis. For a simplysupported beam, this holds for both of the hinged ends, which has already been made use of in the bucklinganalysis, cf. Eq. (44). Supposing that φ(X = 0) = φ0, the constant of integration can be identified as

C = −π2

l2

P

Pecosφ0 + 1

2

η − 1

η

π2

λ2

π2

l2

(P

Pe

)2

cos2 φ0. (73)

By Inserting this result into Eq. (72), one obtains

1

2

(∂φ

∂X

)2

= π2

l2

P

Pe

[(cosφ − cosφ0)− 1

2

η − 1

η

π2

λ2

P

Pe

(cos2 φ − cos2 φ0

)]. (74)

Next, the trigonometric identities

cosφ = 1 − 2 sin2 φ

2, cos2 φ = 1 − 4 sin2 φ

2+ 4 sin4 φ

2(75)

are utilized, with the aid of which one arrives at

∂φ

∂X= ±2

π

l

√P

Pe

√

(1 − e)

(sin2 φ0

2− sin2 φ

2

)+ e

(sin4 φ0

2− sin4 φ

2

), (76)

where the parameter e, cf. Eq. (36), has been introduced again for the sake of brevity. The sign on the righthand side of Eq. (76) may change several times along the beam’s axis depending on the position X , at whichit is evaluated as well as the considered buckling mode. Moreover, it has to be chosen in accordance with thesign of the angle φ0.

The corresponding relations for the limiting case of the extensible elastica rigid in shear are recovered bysetting

limη→∞ e = π2

λ2

P

Pe. (77)

For the classical elastica showing bending deformation only, the parameter becomes

limη,λ→∞ e = 0. (78)

By separation of variables and subsequent integration, a relationship between the axial coordinate X and theangle φ can be established. For a representation in terms of elliptic integrals in the canonical Legendre form,however, a change in variables has to be performed before. Following Magnusson et al. [26] and Humer [25],two transformations are introduced consecutively in order to keep things short and simple. To begin with, anew variable θ is introduced, which is related to φ by

sinφ

2= q sin θ, (79)

where q = sin(φ0/2) has been introduced as an abbreviation. In case of a simply supported beam, φ is knownto vary from φ0 to ±φ0 in the buckled configuration. Along the axis, it exhibits n roots and attains a total ofn + 1 extreme values at the inflection points. By the transformation (79), the range of the angle is altered;the new angle θ varies from π/2 at one end to ±π/2 at the other instead. Both the positions of the roots andthe number of extreme values, however, remain unchanged by the substitution of variables. Mathematicallyspeaking, the transformation represents a bijective mapping for values φ ∈ [−φ0, φ0] and θ ∈ [−π/2, π/2].Now, inserting the relation (79) and its Jacobian,

∂φ

∂θ= 2q cos θ

√1 − q2 sin2 θ

, (80)

Buckling and postbuckling of shear-deformable beams 1511

the equilibrium equation (76) is rewritten in terms of the new variable as

∂θ

∂X= ±π

l

√P

Pe

√1 − q2 sin2 θ

√1 − e + eq2

(1 + sin2 θ

). (81)

In case of the classical elastica, for which e = 0 holds, the equilibrium relation already has the requestedstructure, since the second root becomes unity then. Otherwise, a second transformation is required, whichreplaces θ with another variable denoted by ψ according to the relation

sin θ =√

1 − e + eq2 sinψ√

1 − e + 2eq2 − eq2 sin2 ψ. (82)

Forψ ∈ [−π/2, π/2], the above mapping is bijective, and the positions of the roots and the number of extremevalues are preserved as with the first transformation. Recalling the restriction (18) for the admissible intensityof the compressive force, it is immediately understood that the roots in the above relation remain real-valuedregardless of the choice of parameters.

Again, the differential relation between the two angles,

∂θ

∂ψ=

√1 − e + eq2

√1 − e + 2eq2

1 − e + 2eq2 − eq2 sin2 ψ, (83)

is needed to rewrite Eq. (81) in terms ofψ , and after some rearrangements, one eventually arrives at the desiredrepresentation of the equilibrium relation, which reads

∂ψ

∂X= ±π

l

√P

Pe

√1 − e + 2eq2 − q2(1 + eq2) sin2 ψ. (84)

In order to avoid a cumbersome case-by-case analysis in what follows, only the positive sign in the above rela-tion is kept from now on. By this assumption, the angle ψ is monotonically increasing from ψ(X = 0) = π/2to ψ(X = l) = nπ + π/2 along the beam’s axis. Of course, the transformation (82) does not represent aone-to-one correspondence any longer. Given the solution for ψ , however, the corresponding values of θ andφ are recovered correctly using Eqs. (79) and (82) due to the periodicity of the trigonometric functions.

After separation of variables, Eq. (84) can be integrated over the length of the beam, by which one obtains

π

l

√P

Pe

l∫

0

dX =nπ+π/2∫

π/2

dψ√

1 − e + 2eq2 − q2(1 + eq2) sin2 ψ. (85)

The periodicity of sin2 ψ allows the above equation to be equivalently rewritten as

π

√P

Pe= 2n

π/2∫

0

dψ√

1 − e + 2eq2 − q2(1 + eq2) sin2 ψ= 2n

√1 − e + 2eq2

F(m), (86)

where F(m) denotes the complete elliptic integral of the first kind. The elliptic integral takes the parameter mas its argument, which is given by

m = q2 1 + eq2

1 − e + 2eq2 . (87)

Among the various notational variants, that of Abramowitz and Stegun [41] is adopted in the present paper.With the compressive force given, the only unknown in Eq. (86) is the parameter q and thereby the angle φ0.Due to the nonlinearity of the equations involved, some numerical method has to be employed in order to finda solution for φ0. Once φ0 is available, however, a closed-form relation between ψ and the material coordinateX is established by performing integration not over the entire column, but only over some portion,

X

l= 1

π

√Pe

P

1√

1 − e + 2eq2[K (ψ |m)− F(m)] . (88)

1512 A. Humer

The function K (ψ |m) represents the incomplete elliptic integral of the first kind, which takes the so-calledamplitudeψ as additional argument. Note that the modulus k used in other notations is related to the parameterm by k2 = m. Suppose that ψ should be represented as a function of X , then the inverse of the elliptic inte-grals—the so-called Jacobi elliptic functions—show up. For the sake of brevity, however, a detailed discussionof the resulting relation is skipped in the present paper.

In the analysis of the postbuckling behavior, the components of the displacement vector are what oneis actually interested in. For the lateral deflection, the equilibrium of moments (13) is recalled, whichsimplifies to

∂M

∂X+ P

∂w

∂X= 0, (89)

if the transverse reaction force is absent, i.e., V = 0. Inserting the constitutive relation for the bending moment(17) and subsequent integration immediately yield

w

l= − B

Pl

∂φ

∂X+ C. (90)

The constant of integration is determined by evaluating the above relation at X = 0,

C = B

Pl

[∂φ

∂X

]

X=0= 0, (91)

where the boundary conditions for the deflection, w(X = 0) = 0, which holds for all of the classical bucklingcases considered here, as well as those for the angle (44) have been taken into account. Using the transforma-tions (79) and (82), the deflection can be represented as a function of ψ ,

w

l= ± 2

π

√Pe

P

q(1 − e + 2eq2)√

1 − e + eq2 cosψ

1 − e + 2eq2 − eq2 sin2 ψ, (92)

where a scaling with the length of the column has been introduced. The sign in the above equation depends onthe direction into which the structure buckles from its straight configuration and has to be chosen in accordancewith the angle φ0.

In order to obtain the axial displacement, the kinematic relation (4) is first expanded into

1 + ∂u

∂X= Λ cosφ cosχ −Λ sin φ sin χ. (93)

Combining the constitutive equations for the normal and the shear force, Eqs. (15) and (16), with the equilibriumrelations (19) and (20) gives

Λ cosχ = 1 − P

Dcosφ, Λ sin χ = P

Ssin φ. (94)

By inserting these expressions into Eq. (93), the derivative of the axial position is expressed as

1 + ∂u

∂X=

(1 − P

Dcosφ

)cosφ − P

S

(1 − cos2 φ

). (95)

Again, the trigonometric identity (75) is utilized along with the definition of Euler’s load (32) to rewrite theabove equation as

1 + ∂u

∂X= 1 − 1

η

π2

λ2

P

Pe− 2 sin2 φ

2− η − 1

η

π2

λ2

P

Pe

(1 − 2 sin2 φ

2

)2

. (96)

Proceeding in the same way as before, φ is first substituted by θ according to the transformation (79):

1 + ∂u

∂X= 1 − e

η − 1− 2q2 sin2 θ − e

(1 − 2q2 sin2 θ

)2. (97)

Buckling and postbuckling of shear-deformable beams 1513

Afterward, the derivative with respect to the material coordinate X is replaced by the derivative with respectto θ using Eq. (81),

∂u

∂θ= − l

π

√Pe

P

eη−1 + 2q2 sin2 θ + e(1 − 2q2 sin2 θ)2

√1 − q2 sin2 θ

√1 − e + eq2(1 + sin2 θ)

, (98)

which can be rearranged as

∂u

∂θ= − l

π

√Pe

P

2 + eη−1 − 2(1 − q2 sin2 θ)+ e

[1 − 4(1 − q2 sin2 θ)+ 4(1 − q2 sin2 θ)2

]

√1 − q2 sin2 θ

√1 − e + eq2(1 + sin2 θ)

. (99)

Except for some coefficients, the above equation is similar to what Humer [25] obtained for the deflection ofcantilever under a transverse tip force. Adapting this result for the present problem, the solution for the axialdisplacement is finally obtained as

u

l= − 1

π

√Pe

P

√1 − e + 2eq2

{(

1 + 1 + eη−1

1 − e + 2eq2

)

[K (ψ |m)− F(m)]

−2 [E(ψ |m)− E(m)] + 2eq2 sinψ cosψ√

1 − m sin2 ψ

1 − e + 2eq2 − eq2 sin2 ψ

}

, (100)

where also complete and incomplete elliptic integrals of the second kind appear, which are denoted by E(m)and E(ψ |m), respectively. The corresponding results for the classical elastica are recovered by setting e = 0 inthe above relations, which implies that the variablesψ and θ coincide identically, cf. Eq. (82). The relationshipbetween the material coordinate X and the angle ψ then reduces to

X

l= 1

π

√Pe

P

[K (ψ |q2)− F(q2)

] ; (101)

the components of the displacement vector turn into

we

l= ± 2

π

√Pe

Pq cosψ,

ue

l= − 2

π

√Pe

P

[K (ψ |q2)− F(q2)− E(ψ |q2)+ E(q2)

]. (102)

Compared to these, the results for the angle (88) and the lateral displacement (92) of the generalized elasticadiffer only in some coefficients as well as the parameters of the elliptic integrals. The axial displacement (100),in contrast, varies qualitatively since it exhibits an additional term that vanishes if both shear deformation andaxial compressibility are neglected.

The maximum deflection occurs at those points where φ = 0 and—in view of the transformations (79)and (82)—also ψ = 0 holds. From Eq. (92), one consequently finds

wmax

l= 2

π

√Pe

Pq√

1 − e + eq2, (103)

which looks the same as the results of Magnusson et al. [26] for the extensible elastica; note, however, that ealso involves η, the ratio between shear and extensional stiffness, in the present considerations, cf. Eq. (36).The beam’s end X = l, at which the compressive force is applied, undergoes the maximum displacement inaxial direction. It is obtained by inserting ψ = nπ + π/2 in Eq. (100):

umax

l= −2n

π

√Pe

P

√1 − e + 2eq2

[(

1 + 1 + eη−1

1 − e + 2eq2

)

F(m)− 2E(m)

]

. (104)

In Fig. 8, the maximum deflection is plotted against the compressive force. On the left hand side, Fig. 8a,the load-deflection curves of the first four buckling modes are given, where the shear stiffness is chosen halfthe extensional stiffness, η = 1/2, and the slenderness is λ = 10π . Except for buckling occurring at lowerintensities of the compressive force, the qualitative behavior is very similar to that of the classical elastica,cf. [26], for example.

1514 A. Humer

(a) (b)

Fig. 8 Load-deflection curves of a simply supported beam

However, if the shear stiffness exceeds the extensional stiffness, the postbuckling behavior changes quite abit. A ratio of η = 100 is used in the load-deflection curve on right hand side, Fig. 8b. As previously indicated,two critical loads related to the first buckling mode exist for this choice of parameters, cf. Eq. (48). While thelower curve of the first buckling mode looks similar as before, a second path originates close to the dottedline, which indicates the non-admissible region. The equilibrium path corresponding to the second mode alsodiffers significantly, since it is observed that two branches start from the same bifurcation point. The reasonfor the observed behavior is that the critical load corresponding to n = 2 is a double root of Eq. (48) for aslenderness of λ = 6

√11π/5. Moreover, equilibrium configurations showing the shape of the third buckling

mode can be found although no real-valued solution for critical load exists. The load-deflection curves of suchnon-bifurcating branches do not originate from singular points of the trivial solution.

Magnusson et al. [26] presented similar results for the extensible elastica lacking shear deformation. Theyalso showed that the buckling behavior may be a sub-critical bifurcation for simple roots of Eq. (48), whichimplies that the derivative ∂|wmax|/∂P becomes negative in the vicinity of the critical load for some values ofthe slenderness. In sub-critical bifurcation problems, a sudden change of the equilibrium configuration similarto a snap-through problem may take place immediately after buckling. In order to obtain those values of theslenderness at which the behavior changes, Magnusson et al. [26] expanded the nonlinear equilibrium relation(70) into a Taylor series about the trivial solution. Except for the additional coefficient (η − 1)/η, the samerelation is obtained for the present, more general problem:

∂2φ

∂X2 + π2

l2

P

Pe

[(1 − η − 1

η

π2

λ2

P

Pe

)φ −

(1

6− 2

3

η − 1

η

π2

λ2

P

Pe

)φ3

]+ O(φ5) = 0. (105)

Magnusson et al. [26] showed that the transition from super- to sub-critical bifurcation is associated with achange of sign of the cubic term in that series. Without going into detail, one can conclude from their resultthat no sub-critical bifurcation exists if the shear stiffness is lower than the extensional stiffness, since the signof the coefficient does not change for λ > 0.

Now, the solutions for axial and the lateral displacement, Eqs. (92) and (100), are utilized to determineseveral equilibrium configurations after buckling has occurred. In Figs. 9a–c, the shapes corresponding to thefirst three buckling modes are depicted, where η = 1/2 and a slenderness of λ = 10π have been chosen onceagain. For each buckling mode, the intensity of the compressive force is increased starting from the criticalvalue of the classical elastica (49) such that the ends of the column eventually pass by each other. Beyondthe point at which the ends coincide, the applied force actually no longer is compressive but rather representsa tensile force. With the buckling loads being reduced by the influence of shear, the beam’s deformation isalready fairly large at the critical intensities Euler obtained for the classical elastica. One can further see thatthe shapes of the higher modes are obtained by combining the shape of the first mode appropriately at thepoints of inflection, which is where the term “undulating elastica” originates, cf. [12].

It must be remembered, however, that the buckled configurations do not necessarily represent stable states ofequilibrium. Frequently, the second variation of the strain energy functional is evaluated in order to determine

Buckling and postbuckling of shear-deformable beams 1515

(a) (b) (c)

Fig. 9 Buckled configurations of a simply supported beam

the stability of an equilibrium configuration. For the classical elastica, Kuznetsov and Levyakov [42,43]showed how to derive an auxiliary Sturm-Liouville problem from the second variation of the strain energy,whose eigenvalues indicate the stability of the structure. In order to solve the auxiliary problem, they convertedthe boundary value problem into an initial value problem for which an approximate solution was constructedusing a numerical procedure. As Antman and Rosenfeld [21] explained, however, the Sturmian theory is notapplicable to the more general functional-differential equation that would be obtained in the statically inde-terminate problems from the present formulation. A detailed discussion of the stability of the buckled shapesis therefore deferred to future investigations.

4.2 Column with one end clamped, one end free

In view of the symmetry of the problem, the buckled configuration of a cantilever is obtained more or lessimmediately having the solution of the hinged beam at hand. Evidently, each half of a simply supported columnrepresents a cantilever half as long. Therefore, only minor modifications have to be made in the previouslyderived relations.

With the beam being clamped at X = 0, φ0 now represents the angle at its free end, φ(X = l) = φ0.Along the beam’s axis, φ exhibits n roots and as many extreme values. From the first transformation (79), onerecognizes that θ then varies from θ(X = 0) = 0 to θ(X = l) = π/2 at the free end. Therefore, ψ is assumedto increase from ψ(X = 0) = π at the clamped end to ψ(X = l) = nπ +π/2 at the opposite side. Due to thedifferent range of ψ , the nonlinear equation (86) for φ0 has to be modified for the cantilever as follows:

π

√P

Pe= 2n − 1

√1 − e + 2eq2

F(m). (106)

The changed domain of integration also has to be regarded in the relationship between ψ and the materialcoordinate X , Eq. (88), which consequently turns into

X

l= 1

π

√Pe

P

1√

1 − e + 2eq2[K (ψ |m)− K (π |m)] = 1

π

√Pe

P

1√

1 − e + 2eq2[K (ψ |m)− 2F(m)]. (107)

For the lateral deflection, Eq. (90) is reconsidered at X = 0, where the constant of integration is determinedby inserting φ(X = 0) = 0 into Eq. (76):

C = ±2Pe

P

√P

Pe

√

(1 − e) sin2 φ0

2+ e sin4 φ0

2= ± 2

π

√Pe

Pq√

1 − e + eq2. (108)

1516 A. Humer

(a) (b) (c)

Fig. 10 Buckled configurations of a cantilever

Combining the above relation with the result for the simply supported beam (92) subsequently yields

w

l= ± 2

π

√Pe

Pq√

1 − e + eq2

[(1 − e + 2eq2) cosψ

1 − e + 2eq2 − eq2 sin2 ψ+ 1

]. (109)

In the relation for the axial displacement (100), only the lower bound of integration has to be adapted to thedifferent range of ψ ,

u

l= − 1

π

√Pe

P

√1 − e + 2eq2

{(

1 + 1 + eη−1

1 − e + 2eq2

)

[K (ψ |m)− 2F(m)]

−2 [E(ψ |m)− 2E(m)] + 2eq2 sinψ cosψ√

1 − m sin2 ψ

1 − e + 2eq2 − eq2 sin2 ψ

}

, (110)

where K (π |m) = 2F(m) and E(π |m) = 2E(m) has been utilized. Given the above results, equilibriumconfigurations of a buckled cantilever can be constructed subsequently. For the exemplary configurationsillustrated in Figs. 10a–c, the same parameters as for the simply supported beam are chosen, i.e., λ = 10π aswell as η = 1/2.

4.3 Column with one end clamped, one end pinned

Although the equilibrium equation (69) appears more complicated than the one (70) considered before, thesolution is expected to have a very similar structure as before. For considerations on the uniqueness of solu-tions, Antman and Rosenfeld [21] introduced a new set of variables by which the structure of the equilibriumrelation of the statically determinate problems (70) can be recovered by an affine transformation of the angleφ. For the classical elastica, Domokos et al. [15] and Wang [27] adopted a similar approach which turns outto be applicable here as well. Introducing the transformation

φ = φ + α, α = arctanV

P(111)

together with

P =√

P2 + V 2 = P secα, (112)

the equilibrium equation (70) indeed takes the shape

∂2φ

∂X2 + π2

l2

P

Pesin φ

(1 − η − 1

η

π2

λ2

P

Pecos φ

)= 0, (113)

Buckling and postbuckling of shear-deformable beams 1517

which coincides exactly with the relation considered before, except that the quantities with an over-bar substitutethose without.

As the bending moment must vanish at the pinned end, the beam’s axis exhibits at least one inflection pointin the buckled configuration. Let φ0 denote the actual angle at the first of these points and φ0 = φ0 + α thecorresponding value of the transformed coordinate, integrating the above equation immediately gives

∂φ

∂X= ±2

π

l

√P

Pe

√

(1 − e)

(sin2 φ0

2− sin2 φ

2

)+ e

(sin4 φ0

2− sin4 φ

2

), (114)

where

e = η − 1

η

π2

λ2

P

Pe= e secα, (115)

has further been introduced. In view of Eq. (114), a geometric interpretation of the affine transformation (111)is recognized: The actual angle φ is shifted by α such that φ = ±φ0 holds at each inflection point along thebeam’s axis.

Previous investigations on the buckling behavior of the classical elastica [16,27] showed that the equilib-rium path of a buckled clamped-pinned beam exhibits a secondary loss of stability, at which a snap-throughoccurs if the compressive force is increased beyond a certain limit. In the course of the snap-through, theshape of the buckled beam may change qualitatively. Immediately after buckling, the configuration of the axisclosely resembles that of the corresponding eigenfunction. For the first mode, for instance, this implies that thebuckled shape exhibits one inflection point in between the supports. After the snap-through, however, in whichthe beam’s pinned end is displaced beyond the clamped end, no such point necessarily appears any longeras will be shown subsequently. From static considerations, it is immediately clear that the inflection pointdisappears as soon as the pinned end passes by the clamped end. In the limiting case of both ends coinciding,the clamped end can be regarded as the first inflection point, since the bending moment at this point mustvanish identically, i.e., M(X = 0) = 0, which implies that φ(X = 0) = φ0, cf. Eq. (114). From the boundarycondition φ(X = 0) = 0 and the transformation (111), one recognizes that φ0 = α and φ0 = 0 must holdthen.

On the equilibrium path corresponding to the n-th critical load, either ninfl = n or ninfl = n + 1 inflectionpoints are found along the axis in the buckled configuration depending on the position of the pinned end. Theshifted angle φ therefore ranges from φ(X = 0) = α at the clamped end to ±φ0 at the opposite side withninfl − 1 roots in between the supports.

Proceeding in the same way as for the statically determinate problems, two transformations are performedin order to obtain the desired representation in terms of elliptic integrals. Adopting the first one (79) for φinstead of φ, i.e.,

sinφ

2= q sin θ, (116)

where q = sin(φ0/2), Eq. (114) can be rewritten as

∂θ

∂X= ±π

l

√P

Pe

√1 − q2 sin2 θ

√1 − e + eq2

(1 + sin2 θ

). (117)

The value at the clamped end is denoted by θα , where

θα = θ(X = 0) = arcsin

(1

qsin

α

2

). (118)

At the inflection points, and therefore also at the pinned end, θ = ±π/2 holds; similar as before, the numberof local extrema and roots remains unchanged by the transformation.

For the transition to the angle ψ , e is simply replaced by e in the second transformation (82), by whichone eventually obtains the equilibrium equation in terms of ψ as

∂ψ

∂X= π

l

√P

Pe

√1 − e + 2eq2 − q2(1 + eq2) sin2 ψ. (119)

1518 A. Humer

In order to get rid of the sign, ψ is again assumed to be monotonically increasing. Along the axis, it thereforevaries from ψ(X = 0) = ψα , where

ψα = arcsin

√1 − e + 2eq2 sin θα√

1 − e + eq2 + e sin2 θα, (120)

to ψ(X = l) = ninflπ + π/2 at the beam’s pinned end. The relation between the material coordinate X andthe angle ψ is obtained by separation of variables and subsequent integration,

X

l= 1

π

√Pe

P

1√

1 − e + 2eq2[K (ψ |m)− K (ψα|m)] , (121)

with the parameter of the elliptic integrals now being

m = q2 1 + eq2

1 − e + 2eq2 . (122)

Evaluating Eq. (121) at the beam’s pinned end, at which ψ(X = l) = ninflπ + π/2 is known, yields thenonlinear relation

1 = 1

π

√Pe

P

1√

1 − e + 2eq2[(2ninfl + 1)F(m)− K (ψα|m)] . (123)

In the statically determinate problems, it has been sufficient to solve the corresponding equation (86) for φ0,before the solutions for the components of the displacement vector have been constructed, see Sects. 4.1 and4.2 above. Here, however, the unknown angle α of the affine transformation (111) is also involved in thenonlinear relation (123) through Eqs. (118) and (120). For this reason, an additional equation that closes theproblem is required. Considering the difference between the present problem and a cantilever with a free end,it is clear that this equation represents the constraint for the lateral deflection at the pinned end,w(X = l) = 0.The kinematic relation (5) is first rewritten as

∂w

∂X= Λ sin φ cosχ +Λ cosφ sin χ. (124)

For V �= 0, the constitutive relations (15) and (16) together with the equilibrium equations (19) and (20) yield

Λ cosχ = 1 − P cosφ − V sin φ

D, Λ sin χ = P sin φ + V cosφ

S. (125)

By inserting these expressions into (124), the following relation for the derivative of the deflection is obtained,

∂w

∂X=

(1 − P cosφ − V sin φ

D

)sin φ + P sin φ + V cosφ

Scosφ, (126)

which has to be modified somewhat before it can be integrated. To begin with, substituting φ with φ accordingto the transformation (111) and recalling the definition (115) of e gives

∂w

∂X= e

η − 1sin α + sin φ cosα − cos φ sin α − e sin φ cos φ cosα + e cos2 φ sin α. (127)

Afterward, the identities (75) are used to further rearrange the above equation as follows:

∂w

∂X= e

η − 1sin α + 2 sin

φ

2

√

1 − sin2 φ

2cosα −

(1 − 2 sin2 φ

2

)sin α

− 2e sinφ

2

√

1 − sin2 φ

2

(1 − 2 sin2 φ

2

)cosα + e

(1 − 2 sin2 φ

2

)2

sin α. (128)

Buckling and postbuckling of shear-deformable beams 1519

Subsequently, the transformation (116) can be applied in order to substitute φ with θ :

∂w

∂X= 2q cosα sin θ

√1 − q2 sin2 θ

(1 − e + 2eq2 sin2 θ

) + e

η − 1sin α

+ sin α − 2(1 − q2 sin2 θ

)sin α + e

[1 − 4

(1 − q2 sin2 θ

) + 4(1 − q2 sin2 θ

)2]

sin α. (129)

Eventually, the jacobian of the transformation (116) is utilized to obtain the derivative of the deflection withrespect to the angle θ instead of the material coordinate X :

∂w

∂θ= ± l

π

√Pe

P

{cosα

2q sin θ(1 − e + 2eq2 sin2 θ)√

1 − e + eq2(1 + sin2 θ)

+ sin α1 + e

η−1 − 2(1 − q2 sin2 θ

) + e[1 − 4(1 − q2 sin2 θ)+ 4(1 − q2 sin2 θ)2

]

√1 − q2 sin2 θ

√1 − e + eq2(1 + sin2 θ)

}. (130)

The second fraction in the above equation is similar to the expression found before for the derivative of theaxial displacement (99) in the statically determinate problems. Therefore, only the first one has to be inspectedmore closely. Separation of variables, subsequent integration, and application of the transformation (82) yield

∫sin θ(1 − e + 2eq2 sin2 θ)√

1 − e + eq2(1 + sin2 θ)dθ =

∫(1 − e + 2eq2)

√1 − e + eq2(1 − e + eq2 sin2 ψ) sinψ

(1 − e + 2eq2 + eq2 sin2 ψ

)2 dψ,

= − (1 − e + 2eq2)√

1 − e + eq2 cosψ

1 − e + 2eq2 + eq2 sin2 ψ+ C. (131)

For a detailed derivation, see the table of integrals by Gröbner and Hofreiter [44]. Inserting this result into Eq.(130) along with the corresponding relation for the remaining part, cf. Eq. (100), the lateral deflection of thebeam’s axis is obtained as

w

l= ± 1

π

√Pe

P

(−2q cosα

[(1 − e + 2eq2)

√1 − e + eq2 cosψ

1 − e + 2eq2 + eq2 sin2 ψ− (1 − e + 2eq2)

√1 − e + eq2 cosψα

1 − e + 2eq2 + eq2 sin2 ψα

]

+ sin α√

1 − e + 2eq2

{(

1 +eη−1

1 − e + 2eq2

)

[K (ψ |m)− K (ψα|m)] − 2 [E(ψ |m)− E(ψα|m)]

+2eq2 sinψ cosψ√

1 − m sin2 ψ

1 − e + 2eq2 − eq2 sin2 ψα− 2eq2 sinψα cosψα

√1 − m sin2 ψα

1 − e + 2eq2 − eq2 sin2 ψα

}). (132)

The constant of integration has been determined from the boundary condition w(X = 0) = 0. The additionalconstraint equation closing the problem, w(X = l) = 0, is given by setting ψ = ψ(X = l) = ninflπ + π/2 inthe above result, which then reads

0 = ± 1

π

√Pe

P

(cosα

2q(1 − e + 2eq2)√

1 − e + eq2 cosψα1 − e + 2eq2 − eq2 sin2 ψα

+ sin α√

1 − e + 2eq2

{(

1 +eη−1

1 − e + 2eq2

)

[(2ninfl + 1)F(m)− K (ψα|m)]

−2 [(2ninfl + 1)E(m)− E(ψα|m)] − 2eq2 sinψα cosψα√

1 − m sin2 ψα

1 − e + 2eq2 − eq2 sin2 ψα

}). (133)

Before the two nonlinear equations (123) and (133) are solved for φ0 and α with the aid of some numericalprocedure, those intensities of the axial force at which the beam’s ends coincide need to be determined. Aftersubstituting φ0 = 0 as well as ninfl = n, Eqs. (123) and (133) can be solved for the values of α and P ,where the number of inflection points changes. For the set of parameters used throughout the discussion ofthe postbuckling problem, λ/π = 10 and η = 1/2, the equilibrium path originating at the first critical loadloses the inflection point at P/Pe = −1.2761. On the second and third path, the beam’s ends coincide for

1520 A. Humer

P/Pe = −3.6618 and P/Pe = −4.5018, respectively. At this point, the negative results may appear a bitsurprising, but they will become clear in the discussion of the postbuckling behavior below.

Having determined those points, at which the buckled shape changes qualitatively, one can proceed withsolving the nonlinear equations (123) and (133) for φ0 and α with the aid of Newton’s method, for instance.

With the results for both the angle (121) and the lateral deflection (132) readily available, the axial dis-placement remains to be considered. Similar as for the deflection, Eqs. (125) are inserted into the kinematicrelation (93), by which one obtains

1 + ∂u

∂X=

(1 − P cosφ − V sin φ

D

)cosφ − P sin φ + V cosφ

Ssin φ. (134)

The transformation (111) along with substituting e subsequently yields

1 + ∂u

∂X= − e

η − 1cosα + cos φ cosα + sin φ cosα − e cos2 φ cosα − e cos φ sin φ sin α. (135)

Except for some signs and interchanged sines and cosines of α, the structure of the above relation closelyresembles that of the lateral deflection (127). Repeating the same steps as before, the following relation isobtained for the derivative of the axial displacement with respect to X :

1 + ∂u

∂X= 2q sin α sin θ

√1 − q2 sin2 θ

(1 − e + 2eq2 sin2 θ

) − e

η − 1cosα − cosα

+ 2(1 − q2 sin2 θ

)cosα − e[1 − 4

(1 − q2 sin2 θ

) + (1 − q2 sin2 θ

)2] cosα. (136)

In view of the previous result for the deflection (132), integration immediately yields

u

l= − 1

π

√Pe

P

(2q sin α

[(1 − e + 2eq2)

√1 − e + eq2 cosψ

1 − e + 2eq2 + eq2 sin2 ψ− (1 − e + 2eq2)

√1 − e + eq2 cosψα

1 − e + 2eq2 + eq2 sin2 ψα

]

+ cosα√

1 − e + 2eq2

{(

1 + secα + eη−1

1 − e + 2eq2

)

[K (ψ |m)− K (ψα|m)] − 2 [E(ψ |m)− E(ψα|m)]

+2eq2 sinψ cosψ√

1 − m sin2 ψ

1 − e + 2eq2 − eq2 sin2 ψα− 2eq2 sinψα cosψα

√1 − m sin2 ψα

1 − e + 2eq2 − eq2 sin2 ψα

}), (137)

for the axial displacement.In order to obtain a better understanding for the postbuckling behavior of the present problem, the axial

displacement of the pinned end u(X = l) is inspected in detail. It is obtained by inserting ψ = ninflπ + π/2into the above equation, which gives

u(X = l)

l= − 1

π

√Pe

P

(

−2q sin α(1 − e + 2eq2)

√1 − e + eq2 cosψα

1 − e + 2eq2 + eq2 sin2 ψα

+ cosα√

1 − e + 2eq2

{(

1 + secα + eη−1

1 − e + 2eq2

)

[(2ninfl + 1) F(m)− K (ψα|m)]

−2 [(2ninfl + 1) E(m)− E(ψα|m)] − 2eq2 sinψα cosψα√

1 − m sin2 ψα

1 − e + 2eq2 − eq2 sin2 ψα

})

. (138)

In Fig. 11, the load-displacement curves of the first two equilibrium paths are illustrated. As expected fromthe linear constitutive equation for the normal force (15), the displacement of the trivial solution increaseslinearly with the applied compressive force. After the beam buckles from its straight configuration, the axialdisplacement—or rather its absolute value—initially increases significantly with the compressive force beingraised. Shortly after, however, the equilibrium paths exhibit local maxima beyond which the compressive forcedecreases if the pinned end is further displaced toward the clamped end. The maxima represent points of

Buckling and postbuckling of shear-deformable beams 1521

Fig. 11 Load-displacement curve of a clamped-pinned beam

(a) (b) (c)

Fig. 12 Buckled configuration of a clamped-pinned beam

secondary loss of stability, at which a snap-through occurs if the load is increased further. The same appliesfor the local minima when moving along the equilibrium paths in the reverse direction. As soon as the forceis decreased further—in fact, it is increased in the opposite direction—, the beam’s configuration suddenlychanges and it takes a stretched, straight shape again. The segments of the paths in between the local max-ima and minima therefore represent unstable states of equilibrium that cannot be reached in a force-drivendeformation process. Interestingly, also negative values of the applied force appear in the load-displacementcurves, which implies that the force has to be applied in the opposite direction in these states of equilibrium.This also clarifies the negative values that have been obtained before for the loads at which the two ends of thebeam coincide. Another question is whether the beam remains on the same equilibrium path in the course ofthe snap-through. To provide an answer, one certainly has to consider the dynamic behavior and the loadinghistory, which, however, is beyond the scope of the present paper.

With the aid of the results for the lateral deflection (132) and the axial displacement (137), several buckledconfigurations are constructed subsequently, where the parameters λ/π = 10 and η = 1/2 have been adoptedagain. An example for an unstable configuration on the first equilibrium path is illustrated in Fig. 12a, in whichtwo shapes are given for a load intensity of P/Pe = 2 with only one of them being stable, cf. Fig. 11. Inthe third configuration, the pinned end is displaced much further than in the other two, although the force issmaller. In Figs. 12b–c, buckled configurations on the second and third equilibrium path are depicted.

1522 A. Humer

4.4 Columns with both ends clamped

The last of the classical buckling problems to be considered here is that of a beam for which both ends areclamped. Similar to the analysis of the linear problem, in which the critical loads have been determined, one hasto distinguish whether the buckling pattern of the beam is symmetric or antisymmetric. If the buckled shape issymmetric, no reaction force in transverse direction emerges at the supports. In this case, the nonlinear relation(70) describes the deformation of the beam. Otherwise, one has to consider the equilibrium equation (69),since V does not vanish if the buckled shape is antisymmetric. In either situation, however, the solutions ofthe previously studied problems can be adapted to the present one more or less immediately.

Starting with the symmetric shape, it is first noted that the buckled configuration of a beam with both endsclamped can be constructed by combining the shapes of buckled cantilevers. Recalling the previous results,one only has to modify the range of the angle φ in order to obtain the solution for the present boundaries.Including the clamped ends, at which a rotation of the cross-sections is prohibited, φ exhibits 2n + 1 rootsalong the beam’s axis, while it attains 2n extreme values ±φ0. Consequently, ψ ranges from ψ(X = 0) = πto ψ(X = l) = (2n + 1)π , which is reflected in the nonlinear equation for φ0:

π

√P

Pe= 4n

√1 − e + 2eq2

F(m). (139)

Given the solution for φ0, the remaining results coincide exactly with those for the cantilever. The relation-ship between the material coordinate X and ψ is given by Eq. (106). For the lateral deflection and the axialdisplacement, Eqs. (109) and (110) remain valid for the present problem, too.

In case the buckled shape is antisymmetric, one recognizes that the equilibrium configuration can be brokendown into two clamped-pinned columns that have to be put together appropriately. Similar as before, onlythe range of ψ has to be adjusted, which now is twice that of the clamped-pinned beam. Along the axis, ψconsequently varies from ψ(X = 0) = ψα at the left hand side to ψ(X = l) = (2ninfl − 1)π −ψα at the rightend. The nonlinear relations for φ0 and α therefore read

1 = 1

π

√Pe

P

1√

1 − e + 2eq2

[K

((2ninfl + 1)π − ψα|m

) − K (ψα|m)], (140)

as well as

0 = ± 1

π

√Pe

P

(cosα

4q(1 − e + 2eq2)√

1 − e + eq2 cosψα1 − e + 2eq2 − eq2 sin2 ψα

+ sin α√

1 − e + 2eq2

{(

1 +eη−1

1 − e + 2eq2

)[K

((2ninfl + 1)π − ψα|m

) − K (ψα|m)]

−2[E

((2ninfl + 1)π − ψα|m

) − E(ψα|m)] − 4eq2 sinψα cosψα

√1 − m sin2 ψα

1 − e + 2eq2 − eq2 sin2 ψα

}). (141)

For the present boundaries, the asymmetric shape exhibits 2n + 1 inflection points immediately after buck-ling, which reduce to n once the two ends of the beam coincide. Just like for the clamped-pinned beam, thecorresponding intensity of the external force is obtained by inserting φ0 = 0 as well as ninfl = n into the aboveequations and solving them for P and α. Afterward, one can revert to Eq. (121) for the relation between Xand ψ , to Eq. (132) for the lateral deflection, and to Eq. (137) for the axial displacement.

Among the buckled configurations depicted in Figs. 13a–c, the shapes in the first and the third figureare symmetric, while those in the second obviously are antisymmetric. In Fig. 13b, two configurations for aload intensity of P/Pe = 8 are illustrated. Here, however, it is impossible to determine whether one of theserepresents a stable state of equilibrium without further investigation.

5 Conclusion

Up to now, the literature has been lacking a consistent discussion of the fundamental problems of buckling andpostbuckling of beams that may also exhibit axial and shear deformation in addition to bending. The main aim

Buckling and postbuckling of shear-deformable beams 1523

(a) (b) (c)

Fig. 13 Buckled configurations of a beam with both ends clamped

of the present paper is to close this gap by providing novel results within the analysis of the bifurcation problem,on the one hand, and by constructing exact solutions for the large deformation problem of postbuckling, onthe other hand. For this purpose, Reissner’s theory [3] for the plane deformation of beams has been adoptedalong with constitutive relations, by which the stress resultants are related linearly to the generalized strainmeasures introduced in the formulation. Considering the static equilibrium of a finite element of the beam,the nonlinear equation governing the problem of a column under a compressive force has been derived. Inthe present paper, all combinations of boundary conditions representing the four classical buckling situationshave been considered. Among these, the statically indeterminate problems require particular attention in thederivation of the eigenvalue problem.

A second-order functional-differential equation has been obtained by consistent linearization of the non-linear relation, from which the critical loads of both the statically determinate and the indeterminate caseshave been obtained subsequently. The results have shown that the ratio between shear and extensional stiffnesssignificantly influences the buckling behavior of the columns not only quantitatively but also qualitatively.As long as the shear stiffness is lower than the extensional stiffness, the properties have turned out the besimilar to Euler’s classical elastica, which is inextensible and rigid in shear. Despite the critical loads beingreduced, not more than one bifurcation point per buckling mode exists on the fundamental solution path foradmissible values of the compressive force. As soon as the shear stiffness exceeds the extensional stiffness, areverse behavior has been observed. On the one hand, buckling then occurs at higher loads compared to Euler’selastica—provided there exists a bifurcation point at all—, on the other hand, one may find as many as twocritical loads per buckling mode.

Having determined those intensities of the critical force at which the trivial solution becomes unstable,the actual configurations the beam occupies after buckling have been investigated looking upon the nonlinearequilibrium relation again. To begin with, the two statically determinate problems of a simply supported beamand a cantilever are considered in the derivations. By extending previous results of the classical and the exten-sible elastica rigid in shear, novel closed-form solutions in terms of elliptic integrals of the first and secondkind have been constructed for the angle of rotation of the beam’s cross-sections to begin with. Subsequently,exact relations for both the lateral and the axial displacement have been provided, which unsurprisingly turnout to be more complicated than for the classical elastica. Given these results, load-deflection curves havebeen presented for a simply supported beam, which further underline the change of the buckling behavioronce the shear stiffness is larger than the extensional stiffness. In this case, sub-critical bifurcation points mayexist at which a snap-through takes place immediately after the trivial solution becomes unstable. Moreover,non-bifurcating equilibrium paths have been found which do not originate from the fundamental path. If theshear stiffness is lower, none of these phenomena exist since buckling always remains super-critical in thiscase. The solution for a cantilever under a compressive force has been obtained simply by adjusting the rangeof the involved variables. Afterward, it has been shown that the statically indeterminate problem of a clamped-

1524 A. Humer

pinned beam can be rewritten such that the equilibrium relation attains the same structure as in the staticallydeterminate cases. Before the buckled shapes are obtained, however, an additional nonlinear relation has tobe solved. Looking at the load-displacement curves, it has been recognized that the equilibrium paths of theclamped-pinned beam exhibit an unstable region, which cannot be reached in a force-driven process. Duringthe snap-through that takes place once the applied force reaches these points of secondary loss of stability, theshape of the column may change qualitatively, since inflection points of the axis may disappear. Eventually, theconfiguration of a clamped-clamped beam has been discussed, which either represents a statically determinateor an indeterminate problem depending on the symmetry of the buckled shape. For all of the four fundamentalbuckling problems considered in the present paper, exemplary buckled configurations belonging to the firstthree equilibrium paths have been illustrated for different intensities of the external force. Although the buckledshapes look similar as for the classical elastica, the magnitude of the deformation is considerably influencedby the presence of shear deformation as well as axial compression and extension, respectively.

Acknowledgments The support of the Comet K2 Austrian Center of Competence in Mechatronics (ACCM) is gratefullyacknowledged.

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