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Coupled instabilities in thin-walled beams: a qualitative approach
European Journal of Mechanics A/Solids 22 (2002) 139–149
Coupled instabilities in thin-walled beams: a qualitative approach Marcello Pignataro ∗ , Giuseppe C. Ruta Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy Received 12 June 2001; accepted 6 September 2002
Abstract A direct one-dimensional beam model is adopted. Kinematics is described by axis displacement, rigid rotation of the crosssection and an average measure of warping. Mechanical power is introduced as a linear functional of the kinematic descriptors and their first derivatives, hence mechanical actions naturally result as their duals. In particular, the bi-shear and bi-moment turn out to be quantities spending power on the warping and on its first derivative, respectively. Assuming as basic postulate the balance between external and internal power, local equilibrium equations for the mechanical actions are obtained. In addition to the standard inner constraint of shear indeformability, a linear relationship between twist and warping is assumed. To obtain field equations in terms of displacements, non-linear hyperelastic constitutive relations are formulated. Two coupled bifurcations for axially loaded beams are examined: in the first case no coupling occurs, in the second the beam can be sensitive to initial imperfections. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: One-dimensional model; Bi-shear; Bi-moment; Non-linear constitutive relations; Coupled bifurcations
1. Introduction The problem of the in-plane flexural bifurcation of a straight compressed beam was solved by Euler in 1743, using a direct one-dimensional continuum model with a linear elastic constitutive relation between the bending couple and the change of curvature of the axis. Euler’s pioneering work had no followers until the beginning of the twentieth century, when the demand of light-weight structures revived the interest in the subject. Among the outstanding scientists who gave basic contributions to the solution of technically meaningful problems, we should mention Lorenz (1908, 1911), Timoshenko (1910), Southwell (1913a, 1913b, 1915). Two decades later, the works by Wagner (1929) and Kappus (1937) drew attention also to the instabilities phenomena by torsion and flexure-torsion. In contrast with Euler’s approach, both Wagner and Kappus make use of onedimensional continuum models derived from three-dimensional ones. A slight different approach was followed by Vlasov (1961), who derived his results on thin-walled beams starting from a two-dimensional constrained shell model. It is common to refer to Vlasov as the milestone of the analysis of thin-walled structures. A few years later, a direct one-dimensional beam model was introduced by Epstein (1979). He suggested to describe the kinematics of the cross-sections of the beam by means of a set of directors, one for each segment, thus defining strain measures for each of them. No measure of warping is accounted for, though, and only linear elastic relations are used. In Reissner (1983) the one-dimensional beam model is derived from a three-dimensional one, assuming some non-linearities in strain measures and a standard quadratic form of the strain energy. In this way, though, strain measures and balance equations of difficult interpretation for the one-dimensional model are obtained. More recently, Simo and Vu-Quoc (1991) have suggested a direct one-dimensional beam model with a sophisticated kinematical description accounting for warping. However, in contrast with this accuracy, only simply linear elastic constitutive relations are assumed. * Corresponding author.
E-mail address: [email protected] (M. Pignataro). 0997-7538/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S0997-7538(02)00008-6
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One of the most interesting aspects of the post-buckling behaviour of structures regards the interaction of multiple buckling modes under the same critical load. This phenomenon is very important from the technical point of view, since single stable post-buckling equilibrium paths become unstable when interacting with each other. A number of papers has been written on the subject: in particular, as far as beam structures are concerned, we may quote Pignataro and Rizzi (1982) and Rizzi and Pignataro (1983), where, by using the Euler–Bernoulli beam model, it is shown that the post-buckling interaction of bending modes in frames causes an unstable behaviour. As far as the interaction between buckling modes in thin-walled beams is concerned, we should mention Grimaldi and Pignataro (1979), where, by using three-dimensional elasticity and Vlasov’s model, it is shown that the interaction between overall bending and bending-torsion modes is responsible for unstable post-buckling behaviour. Still on the basis of Vlasov’s beam model, a series of numerical results have been achieved by means of the finite strips method in Pignataro and Luongo (1985, 1987), where the interaction between local and global modes is investigated and it is shown that interacting modes cause instability. More recent works on this subject have been provided by Kołakowski and Kotełko (2000) and by Królak, Kołakowski and Kotełko (2001). In this paper, the kinematics introduced by Tatone and Rizzi (1991) and Rizzi and Tatone (1996) for a direct one-dimensional thin-walled beam model is adopted: the beam configurations are described by the position of the axis, the setting of the sections and a scalar parameter roughly accounting for warping. In contrast with the aforementioned papers, however, the basic notion after kinematics concerns the power spent on a velocity field. Thus, the dynamical actions on the beam naturally result as dual quantitities of the velocities. The balance of power is postulated as a basic axiom, following Germain (1973a, 1973b) and Di Carlo (1996). Balance equations are straightforwardly derived by standard theorems. We will investigate coupled bifurcation phenomena governed by conservative loads, on the basis of the general theory of elastic stability by Koiter (1945). The existence of an energy potential function is therefore necessary, and accordingly we postulate non-linear hyperelastic constitutive relations. By introducing the constitutive relations into balance and compatibility equations, the problems for evaluating the critical load(s) and the associated buckling mode(s) are stated. By means of adequate choices of the constitutive constants, i.e., of the stiffnesses of the beam, single or coupled buckling modes can be searched. While in Rizzi and Tatone (1996) the study of coupled bifurcations is marginal and limited to beams with doubly symmetric cross-sections, here it will constitute the core of our applications and will concern beams with generic sections. It is found that interaction can either occur or not, depending on the boundary conditions, and, when occurring, the beam is in general imperfection sensitive. We emphasize that only qualitative results will be provided.
2. One-dimensional beam model: kinematics Let E be the ordinary three-dimensional Euclidean ambient space. The vector space of translations associated with E will be denoted by U and will be equipped with the standard Euclidean norm and inner product. Consider in E a smooth curve C and a prototype plane region R. Let us build a set of copies of R, continuously attached to C. The region of E occupied by this construction will constitute our beam model. In the sequel, we shall refer to C as the beam axis and to the copies of R as the beam sections. Thus, beam transplacements will be described by the transformations of the axis and of the sections. The beam motions are one-parameter families of transplacements, τ ∈ [0, +∞[ being the scalar parameter measuring the evolution of the motion (the time, for instance). We characterise the beam transplacements by: (i) a smooth transformation of the axis; (ii) a rigid rotation of the sections; (iii) a warping superposed to it. While the transformations accomplishing (i) and (ii) will be described exactly, the warping will be roughly accounted for by a single scalar parameter. Let us consider a motion and denote by S0 the beam shape corresponding to τ = 0, which, without loss of generality, will be assumed as a reference configuration, characterised by sections with no warping (i.e., all the copies of R are plane). Chosen an origin in E , the beam axis in S0 is described by a smooth map of position vectors q : ρ ∈ [ρ1 , ρ2 ] → q(ρ) ∈ U , where ρ is a curvilinear abscissa along the axis. The unit tangent vector to the axis will thus be given by q (ρ), where a prime denotes the derivative with respect to ρ. We will denote by Sτ the shape corresponding to the present value τ of the evolution parameter. It is described by (i) a vector field p(ρ, τ ), providing the position of the point of the axis corresponding to the position q(ρ) in S0 ; (ii) a proper orthogonal tensor field R(ρ, τ ), describing the rotation of the section at p(ρ, τ ) with respect to the setting in S0 ; (iii) a scalar field α(ρ, τ ), roughly describing the warping of the section. The tangent vector to the axis is now given by p (ρ, τ ). The requirement for R(ρ, τ ) to be proper orthogonal stems from the necessity to prevent reflections of the section. The coarse description of the warping may be given by an average measure with respect to the area of the section, or by the value of warping at a suitable point of the section (the centroid, for instance). The fields p, R, α will be assumed sufficiently smooth for our purposes. In the sequel, when there is no risk of confusion, independent variables will be dropped from the notation; furthermore, in all the integrals the measure of integration will be understood and hence omitted.
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We will denote by a superimposed dot the derivative with respect to τ and by skw(U ⊗ U) the space of skew-symmetric tensors on U . The velocity fields of the beam are written as w := p˙ ∈ U,
˙ ∈ skw(U ⊗ U), W := RR
ω := α˙ ∈ R.
(1)
A rigid, or neutral, transplacement is characterised by the following properties: (i) the rotation of the sections is uniform with respect to ρ; (ii) all vectors connecting two arbitrary points in Sτ are consequently uniformly rotated: it may be proven that it is sufficient for this property to hold only for p (ρ, τ ); (iii) the warping remains uniformly zero. Hence, for a rigid transplacement R(ρ, τ ) = R(τ ),
p (ρ, τ ) = R(τ )q (ρ),
α(ρ, τ ) = 0.
(2)
Deformation is naturally defined as the difference between the considered transplacement and a rigid one. Hence, strain measures must vanish when evaluated in a rigid transplacement. A suitable choice of deformation measures with respect to S0 is thus e := R p − q ,
E := R R ,
α,
η := α .
(3)
The vector field e represents the difference between the tangent vector to the axis in Sτ , pulled back to S0 , and the unit tangent vector to the axis in S0 . The skew-symmetric tensor field E expresses the change in curvature of the axis when passing from S0 to Sτ . The scalar fields α and η provide a coarse description of the warping and of its rate of change, respectively, with respect to ρ. In rigid transplacements, Eqs. (2) imply the deformation measures (3) to vanish identically. It is also worth remarking that the components of the measures (3) are unaltered by a change of observer: this will be meaningful when dealing of objectivity of inner constraints and constitutive relations. With a view towards the applications which will be provided in this paper, we find it convenient to express the strain measures e, E in terms of their components with respect to a suitable basis. We will consider a beam for which S0 consists of a set of identical plane sections orthogonal to a straight axis. Let us then choose a Cartesian orthogonal coordinate frame such that the beam axis in S0 lies along the x1 -axis, that is, the curvilinear abscissa ρ coincides with x1 . Furthermore, let us fix a left-handed orthonormal basis (i1 , i2 , i3 ) for U compatible with the chosen coordinate frame: this implies that i1 = q . It is then possible to introduce the following decompositions e = ε1 i1 + ε2 i2 + ε3 i3 , E = κ1 i2 ∧ i3 + κ2 i3 ∧ i1 + κ3 i1 ∧ i2 ,
(4)
where ε1 is the axial elongation and ε2 , ε3 are the shearing strains; κ1 denotes the torsion curvature (twist) and κ2 , κ3 denote the bending curvatures; and ∧ is the wedge product between vectors, providing a skew-symmetric tensor. All the measures of deformation will be expressed in terms of the displacement field u of the points of the axis from S0 u := p − q = u1 i1 + u2 i2 + u3 i3 ,
(5)
while R will be decomposed as R = R3 R2 R1 .
(6)
In Eq. (6) R1 is a rotation of amplitude ϕ1 around i1 ; R2 is a rotation of amplitude ϕ2 around R1 i2 (i.e., the ϕ1 -transformed of i2 ); R3 is a rotation of amplitude ϕ3 around R2 R1 i3 (i.e., the ϕ2 ϕ1 -transformed of i3 ). Thus, by (3)–(6) the exact strain measures for this beam model are ε1 = (1 + u 1 ) cos ϕ2 cos ϕ3 + u 2 (cos ϕ1 sin ϕ3 + sin ϕ1 sin ϕ2 cos ϕ3 ) + u 3 (sin ϕ1 sin ϕ3 − cos ϕ1 sin ϕ2 cos ϕ3 ) − 1, ε2 = −(1 + u 1 ) cos ϕ2 sin ϕ3 + u 2 (cos ϕ1 cos ϕ3 , − sin ϕ1 sin ϕ2 sin ϕ3 ) + u 3 (sin ϕ1 cos ϕ3 + cos ϕ1 sin ϕ2 sin ϕ3 ), ε3 = (1 + u 1 ) sin ϕ2 − u 2 sin ϕ1 cos ϕ2 + u 3 cos ϕ1 cos ϕ2 , κ1 = ϕ1 cos ϕ2 cos ϕ3 + ϕ2 sin ϕ3 , κ2 = −ϕ1 cos ϕ2 sin ϕ3 + ϕ2 cos ϕ3 , κ3 = ϕ1 sin ϕ2 + ϕ3 .
(7)
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3. Balance of power and balance equations We assume that the interaction of the beam in the present shape Sτ with the environment is expressed by a linear functional of the velocities, called external power and denoted by P e : P e :=
ρ2 (b · w + B · W + βω) + |t · w + T · W + θω|ρ − + |t · w + T · W + θω|ρ + . 1
(8)
2
ρ1
In Eq. (8) the velocity fields have been defined by Eq. (1); the integral term represents the power spent by the distance actions, while the other terms represent the power spent by the contact actions. We define the vector fields b, t as the body and contact forces; the skew-symmetric tensor fields B, T as the body and contact couples; the scalar fields β, θ as the body and contact actions spending power on the warping rate ω. We furthermore assume that the internal interaction of the parts of the beam with each other is expressed by a linear functional of the velocities and of their derivatives with respect to ρ up to the first order. The last requirement, providing a so-called theory of grade one (see, e.g., Di Carlo, 1996), stems from the necessity of describing the influence that each point suffers from its neighbourhood. This linear functional will be called internal power and denoted by P i : P i := −
ρ2 (c0 · w + C0 · W + γ0 ω + c1 · w + C1 · W + γ1 ω ).
(9)
ρ1
In Eq. (9) the vector fields c0 , c1 , the skew-symmetric tensor fields C0 , C1 , and the scalar fields γ0 , γ1 represent internal actions; besides, the “minus” sign is immaterial and has been introduced only for reasons of convenience. A standard assumption is that P i ≡ 0 for any rigid transplacement; thus, by replacing Eq. (2) into Eq. (9) and letting P i = 0, one obtains the following reduced expression for P i Pi = −
ρ2
c1 · w − (p ∧ c1 ) · W + C1 · W + γ0 ω + γ1 ω ,
(10)
ρ1
i.e., the contact actions c0 , C0 cannot be present in this beam model. We assume, as a basic principle, the vanishing of the total power (Di Carlo, 1996), or, equivalently, the balance of (virtual) power (Germain, 1973a, 1973b). Thus, at each value of τ it must be P = P e + P i = 0 ⇒ P e = −P i ,
(11)
where P is the total power. As Eqs. (8) and (10) must hold for any interval [ρ1 , ρ2 ], by letting ρ1 → ρ − , ρ2 → ρ + for any ρ ∈ [ρ1 , ρ2 ], the integral term in the external power (8) and the whole of the internal power (9) vanish and the balance of power (11) reads |t · w + T · W + θω|ρ − = −|t · w + T · W + θω|ρ +
∀ρ ∈ [ρ1 , ρ2 ].
(12)
With no loss of generality, it is possible to choose separately the velocity fields in (12) so that at any τ only one does not vanish. Doing so for all of the velocities, it turns out that the contact actions on the two sides of any section are opposite, expressing for this beam model the so-called “law of action and reaction”. As a consequence, the boundary terms in (8) can be compacted into the difference of the argument evaluated at the positive faces of the sections at ρ2 and ρ1 , respectively. Thus, omitting the + superscript in ρ, the balance of power (11) reads ρ2 ρ2 ρ2 (b · w + B · W + βω) + |t · w + T · W + θω|ρ1 = c1 · w − (p ∧ c1 ) · W + C1 · W + γ0 ω + γ1 ω . ρ1
(13)
ρ1
If t, T, θ are regular enough, the fundamental theorem of calculus may be applied and the boundary terms in Eq. (13) transform into integral terms: ρ2
(t + b) · w + (T + p ∧ c1 + B) · W + (β + θ − γ0 )ω + (t − c1 ) · w + (T − C1 ) · W + (θ − γ1 )ω = 0.
(14)
ρ1
As the velocity fields and their first derivatives are arbitrary, one may operate in the same way as for Eq. (12); consequently, Eq. (14) implies
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c1 = t,
C1 = T,
t + b = 0,
T + p ∧ t + B = 0,
γ0 = β + θ ,
γ1 = θ.
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(15)
By Eqs. (15)1,2 the internal actions c1 , C1 coincide with contact force and couple, respectively; Eqs. (15)3,4 are local balance equations of forces and couples; Eqs. (15)5,6 are auxiliary equations for γ0 (which has the meaning of bi-shear) and γ1 (bi-moment). Eqs. (15) refer to the present shape, since they have been derived from the balance of power (13), written with respect to Sτ . As the present shape is in principle unknown, it is convenient to refer local balance equations to the only known shape S0 . The identification of internal actions with contact force and couple, provided by Eqs. (15)1,2 , is a result which does not depend on the considered shape. Eqs. (15)5,6 have scalar form and remain thus unchanged in passing from Sτ to S0 , while this is not true for Eqs. (15)3,4 because of their vectorial nature. By posing s = R t,
S = R TR,
a = R b,
A = R BR,
(16)
where s, S, a, A are contact and body actions in S0 , the local balance equations (15)3,4 and the expression of the internal power (10) become s + Es + a = 0, S + ES − SE + (q + e) ∧ s + A = 0, ρ2 ˙ + γ0 ω + γ1 ω ) = −P i = P e , (s · e˙ + S · E
(17)
ρ1
where e, E have been defined by (3)1,2 . With a view towards applications, in the same way as we decomposed the measures of strain, let us introduce the following decompositions for the contact action fields in the reference configuration S0 : s = Q1 i1 + Q2 i2 + Q3 i3 , (18)
S = M 1 i2 ∧ i3 + M 2 i3 ∧ i1 + M 3 i1 ∧ i2 ; Q1 is called normal force, Q2 , Q3 shearing forces; M1 is referred to as twisting couple, M2 , M3 as bending couples. By means of Eqs. (4) and (18), and assuming the body actions to vanish, the local balance of forces (17)1 reads Q 1 + κ2 Q3 − κ3 Q2 = 0,
Q 2 + κ3 Q1 − κ1 Q3 = 0,
Q 3 + κ1 Q2 − κ2 Q1 = 0
(19)
and the local balance of moments (17)2 reads M1 + κ2 M3 − κ3 M2 + ε2 Q3 − ε3 Q2 = 0, M2 + κ3 M1 − κ1 M3 + ε3 Q1 − (1 + ε1 )Q3 = 0,
(20)
M3 + κ1 M2 − κ2 M1 + (1 + ε1 )Q2 − ε2 Q1 = 0. For the purposes of this paper, we will need to express field equations in terms of displacements. Thus, it is necessary to introduce suitable constitutive relations and inner material constraints, which we will deal with in the following section.
4. Constitutive relations. Inner constraints If the beam is made of elastic and homogeneous material, standard theorems (Truesdell and Noll, 1965) assure that the most general constitutive equation assumes the reduced form S = S(e, E, α, η),
(21)
where S stands for any of the s, S, γ0 , γ1 . We now introduce some inner constraints. First, the warping is usually assumed to depend on the other strain measures, i.e., α = α(e, ˆ E),
α(0, ˆ 0) = 0,
(22)
where αˆ is objective since e and E are. On the basis of previous works, we pose the following particular representation for αˆ α = ξ κ1 ,
ξ ∈ R,
η = ξ κ1 ,
(23)
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where ξ is a constant (Vlasov, 1961; Reissner, 1983; Simo and Vu-Quoc, 1991; Tatone and Rizzi, 1991; Rizzi and Tatone, 1996). In particular, on the basis of the paper of Simo and Vu-Quoc, a possible kinematical meaning for α is a weighted mean of the out-of-plane displacement over the beam section. We further assume the beam to be shear-undeformable, i.e., e = ε1 q = ε1 i1 .
(24)
The introduction of inner constraints implies the presence of reactive terms in contact actions, according to the relaxed principle of determinism for constrained materials (Truesdell and Noll, 1965). Thus, the contact actions will be the sum of an active and a reactive part. The former is determined by the motion according to Eq. (21), while the latter will be denoted by the subscript r and spends no power in any motion compatible with the constraints. Thus, writing the internal power (17)3 spent by the reactive part of the inner actions, one has ρ2 (sr · e˙ + Sr · E˙ + γ0r ξ κ˙ 1 + γ1r ξ κ˙ 1 ) = 0.
(25)
ρ1
Imposing the constraints (23), (24), for the generality of the components of the strain measures and of the interval [ρ1 , ρ2 ], from (25) it turns out that Q1r = 0,
Q2r , Q3r ∈ R,
M2r = M3r = 0,
γ0r ∈ R,
M1r = −ξ γ0r ,
(26)
γ1r = 0.
Hence, the normal force, the bending couples and the bi-moment are entirely determined by the motion; the shearing forces and the bi-shear are purely reactive; the twisting couple is the only contact action which has both active and reactive parts, the latter proportional to the bi-shear via the constraint (24). As we want to perform a static perturbation analysis, we need to deal with actions deriving from a potential. Hence, it is necessary to assume the material to be hyperelastic, and we adopt the following constitutive equations for the active part of contact actions, up to the second order in strain measures 1 Q1 = aε1 + dκ12 , 2 1 M2 = b2 κ2 + f2 κ12 , 2
M1 = (c + dε1 + f2 κ2 + f3 κ3 + gη)κ1 , 1 M3 = b3 κ3 + f3 κ12 , 2
1 γ1 = hη + gκ12 . 2
(27)
The coefficients a, bj , c, h stand for the axial, bending, torsion and warping stiffnesses, respectively; d, fj , g take into account the coupling between torsion and extension (Poynting’s effect, see Truesdell and Noll, 1965), torsion and bending, torsion and warping (Møllmann, 1986), respectively. Assuming, as usual, the body action β to vanish, Eqs. (15)5,6 , together with (23), (26)3,5,6 , (27), yield 1 γ1 = hξ κ1 + gκ12 , 2 γ0 = hξ κ1
+ gκ1 κ1 ,
(28)
M1 = (c + dε1 + f2 κ2 + f3 κ3 )κ1 − hξ 2 κ1
. By comparing Eq. (V.1.10)3 in Vlasov (1961) with (28)3 , under a suitable choice of the co-ordinate frame one has a = EA, fj = EIfj , bj = EIj , d = EId , c = GIc , hξ 2 = EIω , where: E, G are the Young and shear moduli; A is the area of the cross-section; Ij are the principal moments of inertia, Ic , Id the torsion moment of inertia and the polar moment of inertia with respect to the shear centre, respectively; Iω is the warping rigidity; and Ifj = R xj r 2 dA, with xj the co-ordinates of a point of the cross-section with respect to the centroid and r its distance from the shear centre. It is remarkable how, with this identification, the following results for the critical loads are qualitatively the same as those in Timoshenko (1910), Timoshenko and Gere (1961). By combining Eqs. (17)1,2 , (4), (18), (23), (24), (26)–(28), non-linear field equations in terms of displacements are obtained.
5. Coupled bifurcations for axially loaded beams In dealing with bifurcation problems, we make use of the static perturbation analysis developed in Budiansky (1974).
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Let us consider, under the hypotheses of the preceding section, a beam axially compressed by a dead force of magnitude λ. It is easy to prove that a fundamental solution, denoted by the superscript f, is given by λ uf = − ρi1 , a
Rf = I,
λ ef = − i1 , a
α f = 0, sf = −λi1 ,
Sf = 0,
Ef = 0, γ0f = 0,
ηf = 0,
(29)
γ1f = 0.
A bifurcated path, denoted by the superscript b, is described by ub = u −
λ ρi1 , a
α b = α, sb = s − λi1 ,
Rb = R + I,
λ eb = e − i1 , a Sb = S,
Eb = E,
ηb = η,
γ0b = γ0 ,
γ1b = γ1 ,
(30)
where from now on symbols of fields without superscripts will denote the difference between the values along the bifurcated and the fundamental paths, i.e., (·) := (·)b − (·)f . Compatibility, balance and constitutive equations can be written in terms of differences, which will be supposed to regularly depend on a scalar parameter σ : (·)|σ =0 = 0. We perform a power series expansion of the fields of interest and of the load multiplier λ in terms of σ near σ = 0. First-order field equations are (a bar denotes linearisation in σ ) a u¯
1 = 0, a − λ
+ λ u¯
u¯ 2 = 0, 2 ab3 a − λ
u¯
u¯ 3 = 0, 3 + λ ab 2 dλ − ac
ϕ¯1
+ ϕ¯1 = 0. ahξ 2
(31)
Remark that Eq. (31)4 coincides with Eq. (5.18) in Timoshenko and Gere (1961). Eqs. (31) plus boundary conditions represent an eigenvalue problem, providing the critical loads λci and the linearised displacement components. In the following subsections, we will consider two applications. 5.1. Beam with guided end and pin Let us first consider a beam of length l with a guided end at x1 = 0, pinned at x1 = l, Fig. 1. In this case, the boundary conditions for Eqs. (31) are 2 = Q 3 = M 1 = 0 at x1 = 0, u¯ 1 = ϕ¯2 = ϕ¯3 = α¯ = Q 1 = M 2 = M 3 = γ¯1 = 0 at x1 = l, u¯ 2 = u¯ 3 = ϕ¯ 1 = Q where warping is restrained at the guided end. The corresponding eigensolutions read
Fig. 1.
(32)
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u¯ 1 = 0, π x1 , u¯ 2 = U2 cos 2l
π x1 , 2l π x1 , ϕ¯1 = Φ1 cos 2l
u¯ 3 = U3 cos
a b3 π 2 λc2 = 1− 1− , 2 al 2
a b2 π 2 1− 1− , λc3 = 2 al 2 c hξ 2 π 2 1+ λc1 = a , d 4cl 2
(33)
where U2 , U3 , Φ1 are some real constants. In Eqs. (33) the bending buckling loads both depend on bending and axial rigidities; the torsion buckling load depends on axial, torsion, warping and coupled torsion-axial rigidities; when a → ∞, Eqs. (33)3,5 provide the usual Euler buckling loads, while there is no torsion buckling. Remark that (33)7 has the same structure as Eqs. (5.24) and (5.25) in Timoshenko and Gere (1961), being different only in the numerical constant multiplying π 2 / l 2 because of the different constraint at one end. By ignoring the trivial case of coupled bending modes (b2 = b3 ), we now consider the interaction between the torsion and one of the two bending modes u¯ 1 = u¯ 3 = 0, λc1 = λc2 = λc , π π u¯ 2 = U2 cos x1 , x1 . ϕ¯1 = Φ1 cos 2l 2l
(34)
By means of Eqs. (34), second-order perturbation field equations for u2 , ϕ1 read π π a − λc ¯
(a − 2λc )π 2 π 4 f3 2 ¯ + λ = − Φ cos U cos x x u ¯ + λ u¯¯
c c 1 2 1 , 2 2 ab3 l 2l 8b3 l 4 1 2ab3 l 2 f3 π 4 dπ 2 dλc − ac ¯
π π
¯ ¯ ϕ¯ 1 = λc ϕ¯ 1 + Φ1 cos U2 Φ1 cos x1 + x1 , 2l l ahξ 2 2ahξ 2 l 2 8hξ 2 l 4
(35)
where two superposed bars denote second-order quantities with respect to the perturbation parameter. According to Fredholm compatibility conditions, Eqs. (35) admit a solution if their right-hand sides are orthogonal to u¯ 2 , ϕ¯1 . Adding the normalization condition (U2 / l)2 + Φ12 = 1, we get a system of three non-linear algebraic equations in the unknowns λ¯ c , U2 , Φ1 , whose solutions are λ¯ c = 0, λ¯ c = 0,
U2 = ±1,
U2 = ±
U2 = 0,
Φ1 = 0, Φ1 = ±1,
dl 2 , d − (a − 2λc )l 2
(36)
Φ1 = ± −
(a − 2λc )l 2 , d − (a − 2λc )l 2
πaf3 λ¯ c = − U2 , 3dl 2
under the restrictions d a − 2λc < 2 , l
a < 2λc .
(37)
These results show that there is one bending and one torsion mode, which do not interact and exhibit symmetric postbuckling equilibrium path. Remark, however, that (37)2 is in contrast with (33)3 , where a > 2λc is expected. Thus, in general no interaction between bending and torsion modes is possible, unless the cross-section exhibits at least one axis of symmetry. Indeed, if x2 is an axis of symmetry, f3 = 0 ⇒ λ¯ c = 0, and the buckling modes amplitudes can be determined only by resorting to next order perturbation equations, as done in Rizzi and Tatone (1996). This seeming paradoxical result depends on the right-hand side of Eqs. (35), which in general is, starting from the considered fundamental path and considering the interaction between u¯ 2 and ϕ¯ 1 , a − 2λ
−2f3 (ϕ¯1
)2 + ϕ¯1 ϕ¯ 1
− 2λ¯ u¯ 2 , a
d
¯ 2f3 (ϕ¯ 1
u¯
2 + ϕ¯1 u¯
2 ) − 2λ a ϕ¯ 1 .
Thus, the general expressions of Fredholm compatibility conditions are
(38)
M. Pignataro, G.C. Ruta / European Journal of Mechanics A/Solids 22 (2002) 139–149
U f3 2 Φ12 l
l
2 a − 2λ U22 (ϕ¯1 ) + ϕ¯1 ϕ¯1
u¯ 2 + λ¯ a l2
0
U f3 2 Φ12 l
l
l
147
(u¯
2 u¯ 2 ) = 0,
0
¯d 2 (ϕ¯1
u¯
2 + ϕ¯1 u¯
2 )ϕ¯ 1 − 2λ a Φ1
0
l
(ϕ¯1
ϕ¯1 ) = 0.
(39)
0
It is apparent that, whenever the two buckling modes u¯ 2 and ϕ¯1 have the same expression, the algebraic system formed by (39) and the normalization condition does not admit real solutions apart from the two buckling modes occurring separately. This is the case of the considered beam, depending on the adopted constraints. To study a meaningful interaction case, other boundary conditions shall be considered, as it will be done in the next application. 5.2. Clamped beam with roller Let us now consider a beam of length l clamped at x1 = 0 and with a roller at x1 = l, Fig. 2. The boundary conditions in this case read as follows u¯ 1 = u¯ 2 = u¯ 3 = ϕ¯1 = ϕ¯2 = ϕ¯3 = α¯ = 0 at x1 = 0, 1 = Q 2 = M 2 = M 3 = γ¯1 = 0 at x1 = l, u¯ 3 = ϕ¯1 = Q
(40)
where warping is restrained at the clamped end. The corresponding eigensolutions are u¯ 1 = 0,
πx1 −1 , u¯ 2 = U2 cos 2l
a π 2 b3 λc2 = 1− 1− , 2 al 2
A0 A0 A0 x − A0 cos x + (l − x1 ) , u¯ 3 = U3 sin l 1 l 1 l
4A20 b2 a λc3 = 1− 1− , 2 al 2
A0 A0 A0 x1 − A0 cos x1 + (l − x1 ) , ϕ¯1 = Φ1 sin l l l A2 hξ 2 ac 1 + 20 , λc1 = d c l
(41)
where U2 , U3 , Φ1 are arbitrary constants and A0 is the smallest solution of the characteristic equation tan(x) = x (see, for instance, Timoshenko and Gere (1961)). As in the previous case, the bending buckling loads depend on both bending and axial rigidities, while the torsion buckling load depends on axial, torsion, warping and coupled torsion-axial rigidities. When a → ∞, Eqs. (39)3,5 yield the usual buckling loads provided, for instance, by Timoshenko and Gere (1961). Remark, however, that in this case no torsion buckling occurs. We shall consider interaction between the torsion mode ϕ¯1 and the bending mode u¯ 2 . Hence we have, ν1 and ν2 being non-dimensional constants,
Fig. 2.
148
M. Pignataro, G.C. Ruta / European Journal of Mechanics A/Solids 22 (2002) 139–149
Fig. 3.
λc1 = λc2 = λc , u¯ 1 = u¯ 3 = 0, πx1 b3 u¯ 2 = ν2 cos −1 , a 2l
A0 A0 A ϕ¯1 = ν1 sin x1 − A0 cos x1 + 0 (l − x1 ) . l l l
(42)
The second-order perturbation field equations for u2 , ϕ1 read π 2 ¯
a − 2λc 2f u¯¯
λ¯ c κ¯ 3 − 3 (κ¯ 1 )2 + κ¯ 1 κ¯ 1
, 2 + 2 u¯ 2 = −2 ab b3 4l 3 A20
2d 2f ϕ¯¯ 1 = − + ϕ¯¯
λ¯ c κ¯ 1 + 32 (κ¯ 1 κ¯ 3 ) . 1 2 2 l 2ahξ hξ
(43)
Fredholm compatibility conditions involving the right-hand sides of Eqs. (43) and u¯ 2 , ϕ¯ 1 , in addition to the normalization condition ν12 + ν22 = 1 yield a system of three non-linear algebraic equations in the unknowns λ¯ c , ν1 , ν2 , whose solutions are λ¯ c = 0,
U2 = ±1,
λ¯ c = 0,
U2 = 0,
ν2 = ±
Φ1 = 0,
Φ1 = ±1, √ 50.04 dl
(50.04)2 ad + 2.47(a − 2λc )b3
,
2.47(a − 2λc )b3 , (50.04)2 ad + 2.47(a − 2λc )b3 √ 56.15 b3 f3 ¯λc = ∓ √ , dl 2 (50.04)2 ad + 2.47(a − 2λc )b3 ν1 = ∓
(44)
under no restrictions, provided a > 2λc because of (41)3 . According to Bezout’s theorem, besides the two buckling modes occurring separately, there are two actual solutions, depending on the sign combinations of ν1 and ν2 . Remark that, due to the normalization condition, Eq. (44)3 can be equivalently written as a linear function in terms of ν1 . The slope of the bifurcated coupled paths is not zero and depends on many mechanical parameters, among which d and f3 play a crucial role. The post-buckling equilibrium path is thus non-symmetric and the beam is imperfection sensitive. When f3 = 0 (no coupling between bending and torsion, the cross section exhibits symmetry with respect to x2 ), λ¯ c = 0 and the bifurcated path becomes symmetric. In particular, this happens when the section has two axes of symmetry, thus confirming the results previously achieved in Grimaldi and Pignataro (1979). On the other hand, when for instance x3 is an axis of symmetry for the section, f3 = 0 and the post-buckling behaviour is asymmetric. In Fig. 3, the projection of a qualitative post-buckling equilibrium path on the plane λ − u2 is shown, both for perfect and imperfect beams. It is apparent that, in presence of initial imperfections, the bifurcation load is never attained, since the beam reaches a collapse load below the bifurcation one.
6. Conclusions In this paper a one-dimensional direct beam model suitable to qualitatively describe coupled bifurcations has been adopted. The axis displacement and the rotation of the cross-sections are described exactly, while warping is accounted for by an
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149
average scalar descriptor. Balance equations are derived by localisation of the balance of mechanical power. Suitable non-linear hyperelastic constitutive equations are introduced and the balance equations are written in terms of displacements. Hence, two coupled bifurcation problems are investigated. In the first case, no interaction occurs; in the second case, it is found that interaction between buckling in torsion and bending leads to imperfection sensitivity. Further investigations will be performed to improve the capability of the model in describing coupling phenomena, for instance reformulating the assumed constitutive relations, so to account for the different behaviour of the beam, when its axis is imagined passing through the centroids or the shear centres of the cross-sections.
References Budiansky, B., 1974. In: Yih, C.S. (Ed.), Theory of buckling and postbuckling behavior of elastic structures. In: Adv. Appl. Mech., Vol. 14. Academic Press, New York. Di Carlo, A., 1996. A non-standard format for continuum mechanics. In: Batra, R.C., Beatty, M.F. (Eds.), Contemporary Research in the Mechanics and Mathematics of Materials, CIMNE, Barcelona. Epstein, M., 1979. Thin-walled beams as directed curves. Acta Mech. 33, 229–242. Germain, P., 1973a. La méthode des puissance virtuelles en mécanique des milieux continus, 1ère partie: la théorie du second gradient. J. Mécanique 12, 235–274. Germain, P., 1973b. The method of virtual power in continuum mechanics. Part II. Application to continuous media with microstructure. SIAM J. Appl. Math. 23, 556–575. Grimaldi, A., Pignataro, M., 1979. Postbuckling behavior of thin-walled open cross-section compression members. J. Structural Mech. 7/2, 143–159. Kappus, R., 1937. Drillknicken zentrich gedrückter Stäbe mit offenem Profil im elastischen Bereich. Translated in NACA TM 851 (1938). Koiter, W.T., Over de stabiliteit van het elastidch evenwicht. Thesis, Delft. Translated as NASA TT F-10, 833 (1967) and AFFDL Report TR 70-25 (1970). Kołakowski, Z., Kotełko, M., 2000. Coupled instabilty related collapse behaviour of channel-section beam-column. In: Proceedings of CIMS 2000, Lisbon, Portugal, pp. 205–212. Królak, M., Kołakowski, Z., Kotełko, M., 2001. Influence of load-nonuniformity and eccentricity no the stability and load carrying of orthotropic tubular columns of regular hexagonal cross-section. Thin-Walled Structures 39, 483–498. Lorenz, R., 1908. Achsensymmetrische Verzerrungen in dünnwandigen Hohlzylindern. VDI-Zeitschrift 52 (43), 1706–1713. Lorenz, R., 1911. Die nicht-achsensymmetrische Knickung dünnwandiger Hohlzylindern. Phys. Z. 12 (7), 241–260. Møllmann, H., 1986. Theory of thin-walled beams with finite displacements. In: Pietraszkiewicz, W. (Ed.), Proceedings of the EUROMECH Colloquium 197. Springer-Verlag, New York, pp. 195–209. Pignataro, M., Luongo, A., 1985. Simultaneous buckling modes and imperfection sensitivity of channels in compression. In: Proceedings EUROMECH 200. Pignataro, M., Luongo, A., 1987. Asymmetric interaction buckling of thin-walled columns with initial imperfections. Thin-Walled Structures 5, 365–386. Pignataro, M., Rizzi, N., 1982. The effect of multiple buckling modes on the postbuckling beavior of plane elastic frames. Part I. Symmetric frames. J. Structural Mech. 10/4, 437–458. Reissner, E., 1983. On a simple variational analysis of small finite deformations of prismatical beams. Z. Angew. Math. Phys. 34, 642–648. Rizzi, N., Pignataro, M., 1983. The effect of multiple buckling modes on the postbuckling beavior of plane elastic frames. Part II. Asymmetric frames. J. Structural Mech. 10/4, 459–474. Rizzi, N., Tatone, A., 1996. Nonstandard models for thin-walled beams with a view to applications. J. Appl. Mech. 63, 399–403. Simo, J.C., Vu-Quoc, L., 1991. A geometrically exact rod model incorporating shear and torsion-warping deformation. Int. J. Solids Structures 27, 371–393. Southwell, R., 1913a. On the collapse of tubes by external pressure I. Philos. Magazine 25 (149), 687–697. Southwell, R., 1913b. On the collapse of tubes by external pressure II. Philos. Magazine 26 (153), 502–510. Southwell, R., 1915. On the collapse of tubes by external pressure III. Philos. Magazine 29, 67–76. Tatone, A., Rizzi, N., 1991. A one-dimensional model for thin-walled beams. In: Schneider, W., Troger, H., Ziegler, F. (Eds.), Trends in Applications of Mathematics to Mechanics. Longman, Avon, pp. 312–320. Timoshenko, S.P., 1910. Einige Stabilitätsprobleme der Elastizitätstheorie. Z. Angew. Math. Phys. 28 (4), 357–385. Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability. McGraw-Hill, New York. Truesdell, C., Noll, W., 1965. The non-linear field theories of mechanics. In: Handbuch der Physik, Vol. III/3. Springer-Verlag, New York. Vlasov, V.Z., 1961. Thin-Walled Elastic Beams. Monson, Jerusalem. Wagner, H. 1929. Verdrehung und Knickung von offenen Profilen. Translated in NACA TM 807 (1936).
Coupled instabilities in thin-walled beams: a qualitative approach Marcello Pignataro ∗ , Giuseppe C. Ruta Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy Received 12 June 2001; accepted 6 September 2002
Abstract A direct one-dimensional beam model is adopted. Kinematics is described by axis displacement, rigid rotation of the crosssection and an average measure of warping. Mechanical power is introduced as a linear functional of the kinematic descriptors and their first derivatives, hence mechanical actions naturally result as their duals. In particular, the bi-shear and bi-moment turn out to be quantities spending power on the warping and on its first derivative, respectively. Assuming as basic postulate the balance between external and internal power, local equilibrium equations for the mechanical actions are obtained. In addition to the standard inner constraint of shear indeformability, a linear relationship between twist and warping is assumed. To obtain field equations in terms of displacements, non-linear hyperelastic constitutive relations are formulated. Two coupled bifurcations for axially loaded beams are examined: in the first case no coupling occurs, in the second the beam can be sensitive to initial imperfections. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: One-dimensional model; Bi-shear; Bi-moment; Non-linear constitutive relations; Coupled bifurcations
1. Introduction The problem of the in-plane flexural bifurcation of a straight compressed beam was solved by Euler in 1743, using a direct one-dimensional continuum model with a linear elastic constitutive relation between the bending couple and the change of curvature of the axis. Euler’s pioneering work had no followers until the beginning of the twentieth century, when the demand of light-weight structures revived the interest in the subject. Among the outstanding scientists who gave basic contributions to the solution of technically meaningful problems, we should mention Lorenz (1908, 1911), Timoshenko (1910), Southwell (1913a, 1913b, 1915). Two decades later, the works by Wagner (1929) and Kappus (1937) drew attention also to the instabilities phenomena by torsion and flexure-torsion. In contrast with Euler’s approach, both Wagner and Kappus make use of onedimensional continuum models derived from three-dimensional ones. A slight different approach was followed by Vlasov (1961), who derived his results on thin-walled beams starting from a two-dimensional constrained shell model. It is common to refer to Vlasov as the milestone of the analysis of thin-walled structures. A few years later, a direct one-dimensional beam model was introduced by Epstein (1979). He suggested to describe the kinematics of the cross-sections of the beam by means of a set of directors, one for each segment, thus defining strain measures for each of them. No measure of warping is accounted for, though, and only linear elastic relations are used. In Reissner (1983) the one-dimensional beam model is derived from a three-dimensional one, assuming some non-linearities in strain measures and a standard quadratic form of the strain energy. In this way, though, strain measures and balance equations of difficult interpretation for the one-dimensional model are obtained. More recently, Simo and Vu-Quoc (1991) have suggested a direct one-dimensional beam model with a sophisticated kinematical description accounting for warping. However, in contrast with this accuracy, only simply linear elastic constitutive relations are assumed. * Corresponding author.
E-mail address: [email protected] (M. Pignataro). 0997-7538/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S0997-7538(02)00008-6
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One of the most interesting aspects of the post-buckling behaviour of structures regards the interaction of multiple buckling modes under the same critical load. This phenomenon is very important from the technical point of view, since single stable post-buckling equilibrium paths become unstable when interacting with each other. A number of papers has been written on the subject: in particular, as far as beam structures are concerned, we may quote Pignataro and Rizzi (1982) and Rizzi and Pignataro (1983), where, by using the Euler–Bernoulli beam model, it is shown that the post-buckling interaction of bending modes in frames causes an unstable behaviour. As far as the interaction between buckling modes in thin-walled beams is concerned, we should mention Grimaldi and Pignataro (1979), where, by using three-dimensional elasticity and Vlasov’s model, it is shown that the interaction between overall bending and bending-torsion modes is responsible for unstable post-buckling behaviour. Still on the basis of Vlasov’s beam model, a series of numerical results have been achieved by means of the finite strips method in Pignataro and Luongo (1985, 1987), where the interaction between local and global modes is investigated and it is shown that interacting modes cause instability. More recent works on this subject have been provided by Kołakowski and Kotełko (2000) and by Królak, Kołakowski and Kotełko (2001). In this paper, the kinematics introduced by Tatone and Rizzi (1991) and Rizzi and Tatone (1996) for a direct one-dimensional thin-walled beam model is adopted: the beam configurations are described by the position of the axis, the setting of the sections and a scalar parameter roughly accounting for warping. In contrast with the aforementioned papers, however, the basic notion after kinematics concerns the power spent on a velocity field. Thus, the dynamical actions on the beam naturally result as dual quantitities of the velocities. The balance of power is postulated as a basic axiom, following Germain (1973a, 1973b) and Di Carlo (1996). Balance equations are straightforwardly derived by standard theorems. We will investigate coupled bifurcation phenomena governed by conservative loads, on the basis of the general theory of elastic stability by Koiter (1945). The existence of an energy potential function is therefore necessary, and accordingly we postulate non-linear hyperelastic constitutive relations. By introducing the constitutive relations into balance and compatibility equations, the problems for evaluating the critical load(s) and the associated buckling mode(s) are stated. By means of adequate choices of the constitutive constants, i.e., of the stiffnesses of the beam, single or coupled buckling modes can be searched. While in Rizzi and Tatone (1996) the study of coupled bifurcations is marginal and limited to beams with doubly symmetric cross-sections, here it will constitute the core of our applications and will concern beams with generic sections. It is found that interaction can either occur or not, depending on the boundary conditions, and, when occurring, the beam is in general imperfection sensitive. We emphasize that only qualitative results will be provided.
2. One-dimensional beam model: kinematics Let E be the ordinary three-dimensional Euclidean ambient space. The vector space of translations associated with E will be denoted by U and will be equipped with the standard Euclidean norm and inner product. Consider in E a smooth curve C and a prototype plane region R. Let us build a set of copies of R, continuously attached to C. The region of E occupied by this construction will constitute our beam model. In the sequel, we shall refer to C as the beam axis and to the copies of R as the beam sections. Thus, beam transplacements will be described by the transformations of the axis and of the sections. The beam motions are one-parameter families of transplacements, τ ∈ [0, +∞[ being the scalar parameter measuring the evolution of the motion (the time, for instance). We characterise the beam transplacements by: (i) a smooth transformation of the axis; (ii) a rigid rotation of the sections; (iii) a warping superposed to it. While the transformations accomplishing (i) and (ii) will be described exactly, the warping will be roughly accounted for by a single scalar parameter. Let us consider a motion and denote by S0 the beam shape corresponding to τ = 0, which, without loss of generality, will be assumed as a reference configuration, characterised by sections with no warping (i.e., all the copies of R are plane). Chosen an origin in E , the beam axis in S0 is described by a smooth map of position vectors q : ρ ∈ [ρ1 , ρ2 ] → q(ρ) ∈ U , where ρ is a curvilinear abscissa along the axis. The unit tangent vector to the axis will thus be given by q (ρ), where a prime denotes the derivative with respect to ρ. We will denote by Sτ the shape corresponding to the present value τ of the evolution parameter. It is described by (i) a vector field p(ρ, τ ), providing the position of the point of the axis corresponding to the position q(ρ) in S0 ; (ii) a proper orthogonal tensor field R(ρ, τ ), describing the rotation of the section at p(ρ, τ ) with respect to the setting in S0 ; (iii) a scalar field α(ρ, τ ), roughly describing the warping of the section. The tangent vector to the axis is now given by p (ρ, τ ). The requirement for R(ρ, τ ) to be proper orthogonal stems from the necessity to prevent reflections of the section. The coarse description of the warping may be given by an average measure with respect to the area of the section, or by the value of warping at a suitable point of the section (the centroid, for instance). The fields p, R, α will be assumed sufficiently smooth for our purposes. In the sequel, when there is no risk of confusion, independent variables will be dropped from the notation; furthermore, in all the integrals the measure of integration will be understood and hence omitted.
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We will denote by a superimposed dot the derivative with respect to τ and by skw(U ⊗ U) the space of skew-symmetric tensors on U . The velocity fields of the beam are written as w := p˙ ∈ U,
˙ ∈ skw(U ⊗ U), W := RR
ω := α˙ ∈ R.
(1)
A rigid, or neutral, transplacement is characterised by the following properties: (i) the rotation of the sections is uniform with respect to ρ; (ii) all vectors connecting two arbitrary points in Sτ are consequently uniformly rotated: it may be proven that it is sufficient for this property to hold only for p (ρ, τ ); (iii) the warping remains uniformly zero. Hence, for a rigid transplacement R(ρ, τ ) = R(τ ),
p (ρ, τ ) = R(τ )q (ρ),
α(ρ, τ ) = 0.
(2)
Deformation is naturally defined as the difference between the considered transplacement and a rigid one. Hence, strain measures must vanish when evaluated in a rigid transplacement. A suitable choice of deformation measures with respect to S0 is thus e := R p − q ,
E := R R ,
α,
η := α .
(3)
The vector field e represents the difference between the tangent vector to the axis in Sτ , pulled back to S0 , and the unit tangent vector to the axis in S0 . The skew-symmetric tensor field E expresses the change in curvature of the axis when passing from S0 to Sτ . The scalar fields α and η provide a coarse description of the warping and of its rate of change, respectively, with respect to ρ. In rigid transplacements, Eqs. (2) imply the deformation measures (3) to vanish identically. It is also worth remarking that the components of the measures (3) are unaltered by a change of observer: this will be meaningful when dealing of objectivity of inner constraints and constitutive relations. With a view towards the applications which will be provided in this paper, we find it convenient to express the strain measures e, E in terms of their components with respect to a suitable basis. We will consider a beam for which S0 consists of a set of identical plane sections orthogonal to a straight axis. Let us then choose a Cartesian orthogonal coordinate frame such that the beam axis in S0 lies along the x1 -axis, that is, the curvilinear abscissa ρ coincides with x1 . Furthermore, let us fix a left-handed orthonormal basis (i1 , i2 , i3 ) for U compatible with the chosen coordinate frame: this implies that i1 = q . It is then possible to introduce the following decompositions e = ε1 i1 + ε2 i2 + ε3 i3 , E = κ1 i2 ∧ i3 + κ2 i3 ∧ i1 + κ3 i1 ∧ i2 ,
(4)
where ε1 is the axial elongation and ε2 , ε3 are the shearing strains; κ1 denotes the torsion curvature (twist) and κ2 , κ3 denote the bending curvatures; and ∧ is the wedge product between vectors, providing a skew-symmetric tensor. All the measures of deformation will be expressed in terms of the displacement field u of the points of the axis from S0 u := p − q = u1 i1 + u2 i2 + u3 i3 ,
(5)
while R will be decomposed as R = R3 R2 R1 .
(6)
In Eq. (6) R1 is a rotation of amplitude ϕ1 around i1 ; R2 is a rotation of amplitude ϕ2 around R1 i2 (i.e., the ϕ1 -transformed of i2 ); R3 is a rotation of amplitude ϕ3 around R2 R1 i3 (i.e., the ϕ2 ϕ1 -transformed of i3 ). Thus, by (3)–(6) the exact strain measures for this beam model are ε1 = (1 + u 1 ) cos ϕ2 cos ϕ3 + u 2 (cos ϕ1 sin ϕ3 + sin ϕ1 sin ϕ2 cos ϕ3 ) + u 3 (sin ϕ1 sin ϕ3 − cos ϕ1 sin ϕ2 cos ϕ3 ) − 1, ε2 = −(1 + u 1 ) cos ϕ2 sin ϕ3 + u 2 (cos ϕ1 cos ϕ3 , − sin ϕ1 sin ϕ2 sin ϕ3 ) + u 3 (sin ϕ1 cos ϕ3 + cos ϕ1 sin ϕ2 sin ϕ3 ), ε3 = (1 + u 1 ) sin ϕ2 − u 2 sin ϕ1 cos ϕ2 + u 3 cos ϕ1 cos ϕ2 , κ1 = ϕ1 cos ϕ2 cos ϕ3 + ϕ2 sin ϕ3 , κ2 = −ϕ1 cos ϕ2 sin ϕ3 + ϕ2 cos ϕ3 , κ3 = ϕ1 sin ϕ2 + ϕ3 .
(7)
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3. Balance of power and balance equations We assume that the interaction of the beam in the present shape Sτ with the environment is expressed by a linear functional of the velocities, called external power and denoted by P e : P e :=
ρ2 (b · w + B · W + βω) + |t · w + T · W + θω|ρ − + |t · w + T · W + θω|ρ + . 1
(8)
2
ρ1
In Eq. (8) the velocity fields have been defined by Eq. (1); the integral term represents the power spent by the distance actions, while the other terms represent the power spent by the contact actions. We define the vector fields b, t as the body and contact forces; the skew-symmetric tensor fields B, T as the body and contact couples; the scalar fields β, θ as the body and contact actions spending power on the warping rate ω. We furthermore assume that the internal interaction of the parts of the beam with each other is expressed by a linear functional of the velocities and of their derivatives with respect to ρ up to the first order. The last requirement, providing a so-called theory of grade one (see, e.g., Di Carlo, 1996), stems from the necessity of describing the influence that each point suffers from its neighbourhood. This linear functional will be called internal power and denoted by P i : P i := −
ρ2 (c0 · w + C0 · W + γ0 ω + c1 · w + C1 · W + γ1 ω ).
(9)
ρ1
In Eq. (9) the vector fields c0 , c1 , the skew-symmetric tensor fields C0 , C1 , and the scalar fields γ0 , γ1 represent internal actions; besides, the “minus” sign is immaterial and has been introduced only for reasons of convenience. A standard assumption is that P i ≡ 0 for any rigid transplacement; thus, by replacing Eq. (2) into Eq. (9) and letting P i = 0, one obtains the following reduced expression for P i Pi = −
ρ2
c1 · w − (p ∧ c1 ) · W + C1 · W + γ0 ω + γ1 ω ,
(10)
ρ1
i.e., the contact actions c0 , C0 cannot be present in this beam model. We assume, as a basic principle, the vanishing of the total power (Di Carlo, 1996), or, equivalently, the balance of (virtual) power (Germain, 1973a, 1973b). Thus, at each value of τ it must be P = P e + P i = 0 ⇒ P e = −P i ,
(11)
where P is the total power. As Eqs. (8) and (10) must hold for any interval [ρ1 , ρ2 ], by letting ρ1 → ρ − , ρ2 → ρ + for any ρ ∈ [ρ1 , ρ2 ], the integral term in the external power (8) and the whole of the internal power (9) vanish and the balance of power (11) reads |t · w + T · W + θω|ρ − = −|t · w + T · W + θω|ρ +
∀ρ ∈ [ρ1 , ρ2 ].
(12)
With no loss of generality, it is possible to choose separately the velocity fields in (12) so that at any τ only one does not vanish. Doing so for all of the velocities, it turns out that the contact actions on the two sides of any section are opposite, expressing for this beam model the so-called “law of action and reaction”. As a consequence, the boundary terms in (8) can be compacted into the difference of the argument evaluated at the positive faces of the sections at ρ2 and ρ1 , respectively. Thus, omitting the + superscript in ρ, the balance of power (11) reads ρ2 ρ2 ρ2 (b · w + B · W + βω) + |t · w + T · W + θω|ρ1 = c1 · w − (p ∧ c1 ) · W + C1 · W + γ0 ω + γ1 ω . ρ1
(13)
ρ1
If t, T, θ are regular enough, the fundamental theorem of calculus may be applied and the boundary terms in Eq. (13) transform into integral terms: ρ2
(t + b) · w + (T + p ∧ c1 + B) · W + (β + θ − γ0 )ω + (t − c1 ) · w + (T − C1 ) · W + (θ − γ1 )ω = 0.
(14)
ρ1
As the velocity fields and their first derivatives are arbitrary, one may operate in the same way as for Eq. (12); consequently, Eq. (14) implies
M. Pignataro, G.C. Ruta / European Journal of Mechanics A/Solids 22 (2002) 139–149
c1 = t,
C1 = T,
t + b = 0,
T + p ∧ t + B = 0,
γ0 = β + θ ,
γ1 = θ.
143
(15)
By Eqs. (15)1,2 the internal actions c1 , C1 coincide with contact force and couple, respectively; Eqs. (15)3,4 are local balance equations of forces and couples; Eqs. (15)5,6 are auxiliary equations for γ0 (which has the meaning of bi-shear) and γ1 (bi-moment). Eqs. (15) refer to the present shape, since they have been derived from the balance of power (13), written with respect to Sτ . As the present shape is in principle unknown, it is convenient to refer local balance equations to the only known shape S0 . The identification of internal actions with contact force and couple, provided by Eqs. (15)1,2 , is a result which does not depend on the considered shape. Eqs. (15)5,6 have scalar form and remain thus unchanged in passing from Sτ to S0 , while this is not true for Eqs. (15)3,4 because of their vectorial nature. By posing s = R t,
S = R TR,
a = R b,
A = R BR,
(16)
where s, S, a, A are contact and body actions in S0 , the local balance equations (15)3,4 and the expression of the internal power (10) become s + Es + a = 0, S + ES − SE + (q + e) ∧ s + A = 0, ρ2 ˙ + γ0 ω + γ1 ω ) = −P i = P e , (s · e˙ + S · E
(17)
ρ1
where e, E have been defined by (3)1,2 . With a view towards applications, in the same way as we decomposed the measures of strain, let us introduce the following decompositions for the contact action fields in the reference configuration S0 : s = Q1 i1 + Q2 i2 + Q3 i3 , (18)
S = M 1 i2 ∧ i3 + M 2 i3 ∧ i1 + M 3 i1 ∧ i2 ; Q1 is called normal force, Q2 , Q3 shearing forces; M1 is referred to as twisting couple, M2 , M3 as bending couples. By means of Eqs. (4) and (18), and assuming the body actions to vanish, the local balance of forces (17)1 reads Q 1 + κ2 Q3 − κ3 Q2 = 0,
Q 2 + κ3 Q1 − κ1 Q3 = 0,
Q 3 + κ1 Q2 − κ2 Q1 = 0
(19)
and the local balance of moments (17)2 reads M1 + κ2 M3 − κ3 M2 + ε2 Q3 − ε3 Q2 = 0, M2 + κ3 M1 − κ1 M3 + ε3 Q1 − (1 + ε1 )Q3 = 0,
(20)
M3 + κ1 M2 − κ2 M1 + (1 + ε1 )Q2 − ε2 Q1 = 0. For the purposes of this paper, we will need to express field equations in terms of displacements. Thus, it is necessary to introduce suitable constitutive relations and inner material constraints, which we will deal with in the following section.
4. Constitutive relations. Inner constraints If the beam is made of elastic and homogeneous material, standard theorems (Truesdell and Noll, 1965) assure that the most general constitutive equation assumes the reduced form S = S(e, E, α, η),
(21)
where S stands for any of the s, S, γ0 , γ1 . We now introduce some inner constraints. First, the warping is usually assumed to depend on the other strain measures, i.e., α = α(e, ˆ E),
α(0, ˆ 0) = 0,
(22)
where αˆ is objective since e and E are. On the basis of previous works, we pose the following particular representation for αˆ α = ξ κ1 ,
ξ ∈ R,
η = ξ κ1 ,
(23)
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where ξ is a constant (Vlasov, 1961; Reissner, 1983; Simo and Vu-Quoc, 1991; Tatone and Rizzi, 1991; Rizzi and Tatone, 1996). In particular, on the basis of the paper of Simo and Vu-Quoc, a possible kinematical meaning for α is a weighted mean of the out-of-plane displacement over the beam section. We further assume the beam to be shear-undeformable, i.e., e = ε1 q = ε1 i1 .
(24)
The introduction of inner constraints implies the presence of reactive terms in contact actions, according to the relaxed principle of determinism for constrained materials (Truesdell and Noll, 1965). Thus, the contact actions will be the sum of an active and a reactive part. The former is determined by the motion according to Eq. (21), while the latter will be denoted by the subscript r and spends no power in any motion compatible with the constraints. Thus, writing the internal power (17)3 spent by the reactive part of the inner actions, one has ρ2 (sr · e˙ + Sr · E˙ + γ0r ξ κ˙ 1 + γ1r ξ κ˙ 1 ) = 0.
(25)
ρ1
Imposing the constraints (23), (24), for the generality of the components of the strain measures and of the interval [ρ1 , ρ2 ], from (25) it turns out that Q1r = 0,
Q2r , Q3r ∈ R,
M2r = M3r = 0,
γ0r ∈ R,
M1r = −ξ γ0r ,
(26)
γ1r = 0.
Hence, the normal force, the bending couples and the bi-moment are entirely determined by the motion; the shearing forces and the bi-shear are purely reactive; the twisting couple is the only contact action which has both active and reactive parts, the latter proportional to the bi-shear via the constraint (24). As we want to perform a static perturbation analysis, we need to deal with actions deriving from a potential. Hence, it is necessary to assume the material to be hyperelastic, and we adopt the following constitutive equations for the active part of contact actions, up to the second order in strain measures 1 Q1 = aε1 + dκ12 , 2 1 M2 = b2 κ2 + f2 κ12 , 2
M1 = (c + dε1 + f2 κ2 + f3 κ3 + gη)κ1 , 1 M3 = b3 κ3 + f3 κ12 , 2
1 γ1 = hη + gκ12 . 2
(27)
The coefficients a, bj , c, h stand for the axial, bending, torsion and warping stiffnesses, respectively; d, fj , g take into account the coupling between torsion and extension (Poynting’s effect, see Truesdell and Noll, 1965), torsion and bending, torsion and warping (Møllmann, 1986), respectively. Assuming, as usual, the body action β to vanish, Eqs. (15)5,6 , together with (23), (26)3,5,6 , (27), yield 1 γ1 = hξ κ1 + gκ12 , 2 γ0 = hξ κ1
+ gκ1 κ1 ,
(28)
M1 = (c + dε1 + f2 κ2 + f3 κ3 )κ1 − hξ 2 κ1
. By comparing Eq. (V.1.10)3 in Vlasov (1961) with (28)3 , under a suitable choice of the co-ordinate frame one has a = EA, fj = EIfj , bj = EIj , d = EId , c = GIc , hξ 2 = EIω , where: E, G are the Young and shear moduli; A is the area of the cross-section; Ij are the principal moments of inertia, Ic , Id the torsion moment of inertia and the polar moment of inertia with respect to the shear centre, respectively; Iω is the warping rigidity; and Ifj = R xj r 2 dA, with xj the co-ordinates of a point of the cross-section with respect to the centroid and r its distance from the shear centre. It is remarkable how, with this identification, the following results for the critical loads are qualitatively the same as those in Timoshenko (1910), Timoshenko and Gere (1961). By combining Eqs. (17)1,2 , (4), (18), (23), (24), (26)–(28), non-linear field equations in terms of displacements are obtained.
5. Coupled bifurcations for axially loaded beams In dealing with bifurcation problems, we make use of the static perturbation analysis developed in Budiansky (1974).
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Let us consider, under the hypotheses of the preceding section, a beam axially compressed by a dead force of magnitude λ. It is easy to prove that a fundamental solution, denoted by the superscript f, is given by λ uf = − ρi1 , a
Rf = I,
λ ef = − i1 , a
α f = 0, sf = −λi1 ,
Sf = 0,
Ef = 0, γ0f = 0,
ηf = 0,
(29)
γ1f = 0.
A bifurcated path, denoted by the superscript b, is described by ub = u −
λ ρi1 , a
α b = α, sb = s − λi1 ,
Rb = R + I,
λ eb = e − i1 , a Sb = S,
Eb = E,
ηb = η,
γ0b = γ0 ,
γ1b = γ1 ,
(30)
where from now on symbols of fields without superscripts will denote the difference between the values along the bifurcated and the fundamental paths, i.e., (·) := (·)b − (·)f . Compatibility, balance and constitutive equations can be written in terms of differences, which will be supposed to regularly depend on a scalar parameter σ : (·)|σ =0 = 0. We perform a power series expansion of the fields of interest and of the load multiplier λ in terms of σ near σ = 0. First-order field equations are (a bar denotes linearisation in σ ) a u¯
1 = 0, a − λ
+ λ u¯
u¯ 2 = 0, 2 ab3 a − λ
u¯
u¯ 3 = 0, 3 + λ ab 2 dλ − ac
ϕ¯1
+ ϕ¯1 = 0. ahξ 2
(31)
Remark that Eq. (31)4 coincides with Eq. (5.18) in Timoshenko and Gere (1961). Eqs. (31) plus boundary conditions represent an eigenvalue problem, providing the critical loads λci and the linearised displacement components. In the following subsections, we will consider two applications. 5.1. Beam with guided end and pin Let us first consider a beam of length l with a guided end at x1 = 0, pinned at x1 = l, Fig. 1. In this case, the boundary conditions for Eqs. (31) are 2 = Q 3 = M 1 = 0 at x1 = 0, u¯ 1 = ϕ¯2 = ϕ¯3 = α¯ = Q 1 = M 2 = M 3 = γ¯1 = 0 at x1 = l, u¯ 2 = u¯ 3 = ϕ¯ 1 = Q where warping is restrained at the guided end. The corresponding eigensolutions read
Fig. 1.
(32)
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u¯ 1 = 0, π x1 , u¯ 2 = U2 cos 2l
π x1 , 2l π x1 , ϕ¯1 = Φ1 cos 2l
u¯ 3 = U3 cos
a b3 π 2 λc2 = 1− 1− , 2 al 2
a b2 π 2 1− 1− , λc3 = 2 al 2 c hξ 2 π 2 1+ λc1 = a , d 4cl 2
(33)
where U2 , U3 , Φ1 are some real constants. In Eqs. (33) the bending buckling loads both depend on bending and axial rigidities; the torsion buckling load depends on axial, torsion, warping and coupled torsion-axial rigidities; when a → ∞, Eqs. (33)3,5 provide the usual Euler buckling loads, while there is no torsion buckling. Remark that (33)7 has the same structure as Eqs. (5.24) and (5.25) in Timoshenko and Gere (1961), being different only in the numerical constant multiplying π 2 / l 2 because of the different constraint at one end. By ignoring the trivial case of coupled bending modes (b2 = b3 ), we now consider the interaction between the torsion and one of the two bending modes u¯ 1 = u¯ 3 = 0, λc1 = λc2 = λc , π π u¯ 2 = U2 cos x1 , x1 . ϕ¯1 = Φ1 cos 2l 2l
(34)
By means of Eqs. (34), second-order perturbation field equations for u2 , ϕ1 read π π a − λc ¯
(a − 2λc )π 2 π 4 f3 2 ¯ + λ = − Φ cos U cos x x u ¯ + λ u¯¯
c c 1 2 1 , 2 2 ab3 l 2l 8b3 l 4 1 2ab3 l 2 f3 π 4 dπ 2 dλc − ac ¯
π π
¯ ¯ ϕ¯ 1 = λc ϕ¯ 1 + Φ1 cos U2 Φ1 cos x1 + x1 , 2l l ahξ 2 2ahξ 2 l 2 8hξ 2 l 4
(35)
where two superposed bars denote second-order quantities with respect to the perturbation parameter. According to Fredholm compatibility conditions, Eqs. (35) admit a solution if their right-hand sides are orthogonal to u¯ 2 , ϕ¯1 . Adding the normalization condition (U2 / l)2 + Φ12 = 1, we get a system of three non-linear algebraic equations in the unknowns λ¯ c , U2 , Φ1 , whose solutions are λ¯ c = 0, λ¯ c = 0,
U2 = ±1,
U2 = ±
U2 = 0,
Φ1 = 0, Φ1 = ±1,
dl 2 , d − (a − 2λc )l 2
(36)
Φ1 = ± −
(a − 2λc )l 2 , d − (a − 2λc )l 2
πaf3 λ¯ c = − U2 , 3dl 2
under the restrictions d a − 2λc < 2 , l
a < 2λc .
(37)
These results show that there is one bending and one torsion mode, which do not interact and exhibit symmetric postbuckling equilibrium path. Remark, however, that (37)2 is in contrast with (33)3 , where a > 2λc is expected. Thus, in general no interaction between bending and torsion modes is possible, unless the cross-section exhibits at least one axis of symmetry. Indeed, if x2 is an axis of symmetry, f3 = 0 ⇒ λ¯ c = 0, and the buckling modes amplitudes can be determined only by resorting to next order perturbation equations, as done in Rizzi and Tatone (1996). This seeming paradoxical result depends on the right-hand side of Eqs. (35), which in general is, starting from the considered fundamental path and considering the interaction between u¯ 2 and ϕ¯ 1 , a − 2λ
−2f3 (ϕ¯1
)2 + ϕ¯1 ϕ¯ 1
− 2λ¯ u¯ 2 , a
d
¯ 2f3 (ϕ¯ 1
u¯
2 + ϕ¯1 u¯
2 ) − 2λ a ϕ¯ 1 .
Thus, the general expressions of Fredholm compatibility conditions are
(38)
M. Pignataro, G.C. Ruta / European Journal of Mechanics A/Solids 22 (2002) 139–149
U f3 2 Φ12 l
l
2 a − 2λ U22 (ϕ¯1 ) + ϕ¯1 ϕ¯1
u¯ 2 + λ¯ a l2
0
U f3 2 Φ12 l
l
l
147
(u¯
2 u¯ 2 ) = 0,
0
¯d 2 (ϕ¯1
u¯
2 + ϕ¯1 u¯
2 )ϕ¯ 1 − 2λ a Φ1
0
l
(ϕ¯1
ϕ¯1 ) = 0.
(39)
0
It is apparent that, whenever the two buckling modes u¯ 2 and ϕ¯1 have the same expression, the algebraic system formed by (39) and the normalization condition does not admit real solutions apart from the two buckling modes occurring separately. This is the case of the considered beam, depending on the adopted constraints. To study a meaningful interaction case, other boundary conditions shall be considered, as it will be done in the next application. 5.2. Clamped beam with roller Let us now consider a beam of length l clamped at x1 = 0 and with a roller at x1 = l, Fig. 2. The boundary conditions in this case read as follows u¯ 1 = u¯ 2 = u¯ 3 = ϕ¯1 = ϕ¯2 = ϕ¯3 = α¯ = 0 at x1 = 0, 1 = Q 2 = M 2 = M 3 = γ¯1 = 0 at x1 = l, u¯ 3 = ϕ¯1 = Q
(40)
where warping is restrained at the clamped end. The corresponding eigensolutions are u¯ 1 = 0,
πx1 −1 , u¯ 2 = U2 cos 2l
a π 2 b3 λc2 = 1− 1− , 2 al 2
A0 A0 A0 x − A0 cos x + (l − x1 ) , u¯ 3 = U3 sin l 1 l 1 l
4A20 b2 a λc3 = 1− 1− , 2 al 2
A0 A0 A0 x1 − A0 cos x1 + (l − x1 ) , ϕ¯1 = Φ1 sin l l l A2 hξ 2 ac 1 + 20 , λc1 = d c l
(41)
where U2 , U3 , Φ1 are arbitrary constants and A0 is the smallest solution of the characteristic equation tan(x) = x (see, for instance, Timoshenko and Gere (1961)). As in the previous case, the bending buckling loads depend on both bending and axial rigidities, while the torsion buckling load depends on axial, torsion, warping and coupled torsion-axial rigidities. When a → ∞, Eqs. (39)3,5 yield the usual buckling loads provided, for instance, by Timoshenko and Gere (1961). Remark, however, that in this case no torsion buckling occurs. We shall consider interaction between the torsion mode ϕ¯1 and the bending mode u¯ 2 . Hence we have, ν1 and ν2 being non-dimensional constants,
Fig. 2.
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Fig. 3.
λc1 = λc2 = λc , u¯ 1 = u¯ 3 = 0, πx1 b3 u¯ 2 = ν2 cos −1 , a 2l
A0 A0 A ϕ¯1 = ν1 sin x1 − A0 cos x1 + 0 (l − x1 ) . l l l
(42)
The second-order perturbation field equations for u2 , ϕ1 read π 2 ¯
a − 2λc 2f u¯¯
λ¯ c κ¯ 3 − 3 (κ¯ 1 )2 + κ¯ 1 κ¯ 1
, 2 + 2 u¯ 2 = −2 ab b3 4l 3 A20
2d 2f ϕ¯¯ 1 = − + ϕ¯¯
λ¯ c κ¯ 1 + 32 (κ¯ 1 κ¯ 3 ) . 1 2 2 l 2ahξ hξ
(43)
Fredholm compatibility conditions involving the right-hand sides of Eqs. (43) and u¯ 2 , ϕ¯ 1 , in addition to the normalization condition ν12 + ν22 = 1 yield a system of three non-linear algebraic equations in the unknowns λ¯ c , ν1 , ν2 , whose solutions are λ¯ c = 0,
U2 = ±1,
λ¯ c = 0,
U2 = 0,
ν2 = ±
Φ1 = 0,
Φ1 = ±1, √ 50.04 dl
(50.04)2 ad + 2.47(a − 2λc )b3
,
2.47(a − 2λc )b3 , (50.04)2 ad + 2.47(a − 2λc )b3 √ 56.15 b3 f3 ¯λc = ∓ √ , dl 2 (50.04)2 ad + 2.47(a − 2λc )b3 ν1 = ∓
(44)
under no restrictions, provided a > 2λc because of (41)3 . According to Bezout’s theorem, besides the two buckling modes occurring separately, there are two actual solutions, depending on the sign combinations of ν1 and ν2 . Remark that, due to the normalization condition, Eq. (44)3 can be equivalently written as a linear function in terms of ν1 . The slope of the bifurcated coupled paths is not zero and depends on many mechanical parameters, among which d and f3 play a crucial role. The post-buckling equilibrium path is thus non-symmetric and the beam is imperfection sensitive. When f3 = 0 (no coupling between bending and torsion, the cross section exhibits symmetry with respect to x2 ), λ¯ c = 0 and the bifurcated path becomes symmetric. In particular, this happens when the section has two axes of symmetry, thus confirming the results previously achieved in Grimaldi and Pignataro (1979). On the other hand, when for instance x3 is an axis of symmetry for the section, f3 = 0 and the post-buckling behaviour is asymmetric. In Fig. 3, the projection of a qualitative post-buckling equilibrium path on the plane λ − u2 is shown, both for perfect and imperfect beams. It is apparent that, in presence of initial imperfections, the bifurcation load is never attained, since the beam reaches a collapse load below the bifurcation one.
6. Conclusions In this paper a one-dimensional direct beam model suitable to qualitatively describe coupled bifurcations has been adopted. The axis displacement and the rotation of the cross-sections are described exactly, while warping is accounted for by an
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149
average scalar descriptor. Balance equations are derived by localisation of the balance of mechanical power. Suitable non-linear hyperelastic constitutive equations are introduced and the balance equations are written in terms of displacements. Hence, two coupled bifurcation problems are investigated. In the first case, no interaction occurs; in the second case, it is found that interaction between buckling in torsion and bending leads to imperfection sensitivity. Further investigations will be performed to improve the capability of the model in describing coupling phenomena, for instance reformulating the assumed constitutive relations, so to account for the different behaviour of the beam, when its axis is imagined passing through the centroids or the shear centres of the cross-sections.
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