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The Brazier eﬀect for elastic pipe beams with foam cores ⁎

T

A. Luongo , D. Zulli, I. Scognamiglio International Center for Mathematics & Mechanics of Complex Systems, M&MoCS, University of L'Aquila, 67040 Monteluco di Roio, AQ, Italy

A R T I C L E I N F O

A B S T R A C T

Keywords: Pipe beam Foam core Brazier eﬀect Ovalization Instability

Pipe beams are considered under the action of bending. In case of elastic material, an analytical model of a onedimensional beam is introduced, where the internal constraint between ovalization and bending curvature is deduced from mechanical considerations. Then, the softening moment-curvature relationship, able to describe the Brazier eﬀect, is evaluated applying equilibrium equations in the nonlinear ﬁeld. The same model is extended to the case where a structural foam is present as core of the pipe. The contribution of the core is analyzed in terms of its action in preventing instability phenomena. Finally, a model of lumped ovalization is discussed.

1. Introduction Tubular thin-walled beams are essential assets in many industrial and civil applications, from gas and oil distribution networks, nuclear plants, aerospace structures, waterworks and many others. The evaluation of their bearing capacity appears as a crucial step in the design but, on the other hand, it is well-known that conventional beam models typically fail at that, due to the consequences of the usually signiﬁcant deformation of the cross section. In particular, a possible source of instability is the Brazier eﬀect [1], which is related to the ovalization of empty tubes under bending, giving rise to a nonlinear softening behavior of the structure and the occurrence of catastrophic limit points, even in elastic regime. This phenomenon is analyzed in [2], where a variational approach in small-strain nonlinear elastic regime is used to model the combined eﬀects of cross-section deformation and localized longitudinal buckling in case of pure bending of thin-walled tubes with circular cross-sections. In [3], the instability analysis under bending eﬀect is addressed to the case of very short cylinders, taking into account imperfections as well. In [4], long thin elastic tubes with possible initial curvature are considered in the framework of ﬁnite elements analysis, with the aim of comparing the triggering of both buckling and ovalization instability, as well as tracing the post-buckling paths. Critical failure in wind turbine blades is analyzed in [5], where it is shown how the Brazier pressure may have a signiﬁcant impact in the mechanical behavior of such kind of structures. In [6], bending collapse behavior of thin-walled circular tubes is addressed, after deriving the relationship between the applied moment and the bending angle, and then generating simpliﬁed tube models with diﬀerent cross-sections and materials. In [7], a one-dimensional continuum endowed with structure is proposed to analyze the pure ﬂexure problem of rods, ﬁnding

⁎

bifurcation conditions which can describe the Brazier instability of thinwalled tubes. In [8], anisotropic materials are considered in formulating a beam model of cylindrical tube, and use of the variationalasymptotic method is made to obtain asymptotically correct solutions reproducing the Brazier limit-moment instability. In [9], single- and double-walled elastic tubes are analyzed under a pure bending condition, and the modiﬁcation of the ovalization intensity and of the Brazier limit-moment due to the presence of the multiple layers is evaluated. Sometimes, the presence of a soft core, possibly made of foam materials, can be used to improve the performance of the pipes under bending instability. As an example, the presence of soft elastic cores in thin-walled cylindrical structures is considered in [10], in which attention is paid to structures as suggested by nature, where foam-like cellular cores ﬁll, e.g., plant stems or hedgehog spines, in order to obtain inspiration to increase the mechanical eﬃciency of engineering structures. In [11,12], analytical and experimental models are used to ﬁnd the optimum design of thin-walled, cylindrical shells with compliant cores subjected to uniaxial compression and bending. More speciﬁcally about the foam material properties, as well as its interaction with the skin structure, composite sandwich panels developed for marine applications with PVC foam core are analyzed in terms of ﬂexural behavior under quasi static load in [13], while polymeric foam composites for aerospace industry are considered in [14]. Very soft polyurethane foam cores are studied and characterized in [15], even in cases of diﬀerent fabrication angles. In [16] a locally deformable one-dimensional beam model was used to describe eﬀects of bending on thin-walled members, after identifying the nonlinear elastic response function from a corresponding three-dimensional ﬁber model. Drawing inspiration by [17], where distortionconstrained thin walled beam models are formulated to describe,

Corresponding author. E-mail address: [email protected] (A. Luongo).

https://doi.org/10.1016/j.tws.2017.11.053 Received 18 May 2017; Received in revised form 27 November 2017; Accepted 30 November 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.

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the axis line of a uniform (and small) curvature κ about a3 (see Fig. 2-a); consistently with Saint Venant theory, a longitudinal stress occurs. In particular, the generic ﬁber, represented separately in Fig. 2-b, assumes 1 1 radius of curvature equal to κ + y ≃ κ , and is subjected to the longitudinal stress σs . Note that nonlinear eﬀects have to be considered in evaluating the contribution of σs , overtaking in some sense Saint Venant theory, namely σs is not generally directed as a1 but it is perpendicular to deformed (and still planar) cross section (or, equivalently, tangent to the deformed ﬁber, as in Fig. 2-b, since shear strain is null); moreover, in the same framework, the intensity of σs in a point of the cross section shall be evaluated as a linear function of the distance of the point itself, in the deformed conﬁguration, from a3 , which is the neutral axis. Following the Mariotte formula [20], the equilibrium of the longitudinal ﬁber under the action of σs is guaranteed by the existence of a radial pressure of intensity

among others, eﬀects of bending, here an internally constrained pipe beam under bending and able to be ﬁlled with structural foam is considered. The aim is to derive a suitable bending moment-curvature relationship, which is obtained with reference to a segment of beam in uniform bending, but which could be assumed as valid for a nonuniform bending condition too, as it is typically done in engineering applications. A constraint between the ovalization, assumed as occurring in a prescribed oval shape, and the imposed bending curvature is obtained using mechanical considerations. Then, the corresponding softening moment-curvature relationship is obtained through equilibrium equations in the nonlinear ﬁeld, in order to describe the Brazier eﬀect. The model is then extended to the case of ﬁlled pipes, where a structural foam is used as core, to increase the stiﬀness and reduce the possibility of instability occurrence. Even if structural foams can present complicated and rich response laws, showing nonlinear elastic, viscous and plastic behavior with plateau regions (see [15,18,19]), here they are assumed to remain in linear elastic ﬁeld, postponing the accounting of nonlinear material eﬀects, in particular plasticity, to future works. The paper is organized as follows: In Section 2 the analysis of empty pipes under bending is carried out; in Section 3, a foam core is considered and the model extended consistently; in Section 4 an application on a simple structure is carried out, suggesting the introduction of a simpliﬁed model. Finally, in Section 5, some conclusions are drawn.

p = σs bκ

(1)

On the other hand, the prescribed pressure is actually not present, therefore a contrary pressure of the same intensity must be imaged as applied in correspondence of the ﬁber itself (see Fig. 3-a), in order to vanish the total pressure. Generalizing the result for any ﬁbers, the contrary pressure has intensity depending on the distance of the ﬁber from a3 , as in Fig. 3-b. In the perspective of linear kinematic equations in the transverse direction, the pressure is assumed to act along a2 ; it is responsible for the ovalization of the cross section.

2. The empty pipe 2.2. Deformation of the cross section and application of the virtual work theorem

2.1. The mechanical interpretation of the Brazier eﬀect Here the Brazier eﬀect is referenced through a mechanical interpretation directly deduced by [1]. A beam with a thin annular cross section and length l is considered. The beam axis is spanned in the direction identiﬁed by the unitary vector a1. The average radius of the cross section is R, its uniform thickness is b ≪ R , and a2, a3 are orthogonal unitary vectors laying on it (see Fig. 1-a,b). The beam is constituted by linear elastic material of Young's modulus Es , where the subscript s stands for “skin”. A generic longitudinal ﬁber of the beam is identiﬁed in polar coordinates by the phase φ on the cross section (or equivalently by the abscissa c = Rφ ), therefore its initial distance from the axis a3 is y = R sin(φ) . A ﬁnite but small segment of beam of length Δs⪡l is considered and imaged under the action of uniform bending. The deformed shape of the beam segment is related to the imposition to

The cross section is supposed to displace to an assumed shape of oval, where the change in length of the semi-axes is referred to as α , considered uniform along the segment of beam (Fig. 3-c). In particular, the displacement of the trace point of the longitudinal ﬁber identiﬁed by the phase φ is

u (φ) = u (φ) at (φ) + v (φ) an (φ)

(2)

where at (φ), an (φ) are the tangent and (internal) normal unitary vectors at the abscissa c; the components u, v are considered as

u (φ) = αψt (φ)

(3)

v (φ) = αψn (φ) Fig. 1. Pipe beam: (a) cross section, (b) beam with a generic longitudinal ﬁber highlighted.

Fig. 2. Deformation of the beam segment under pure bending (a). Stress on the generic ﬁber plus radial pressure (b).

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Fig. 3. Pressure acting on the cross section: Analysis of the single ﬁber (a) and generalization to the whole section (b). Assumed shape of ovalization (c).

with α unknown amplitude and ψt , ψn components of an assumed mode, described by:

1 ψt (φ) = − sin(2φ) 2 ψn (φ) = −cos(2φ)

section, where δ is the variation operator, namely:

Le =

1 ⎡ du α dψ − v ⎤ = ⎡ t − ψn ⎤ = 0 ⎥ ⎥ R⎢ dφ R⎢ ⎣ ⎦ ⎣ dφ ⎦

(12)

where

dψt ⎤ 1 ⎡ d 2ψn + 2 R2 ⎢ dφ dφ ⎥ ⎣ ⎦

χ (φ) =

(13)

Concurrently to the bending curvature χ (φ) , a bending moment (still around a1) for the annular beam and indicated as m (φ) , arises. Therefore, the internal virtual work for the annular beam can be written as:

∫0

Li =

2π

m (φ) δχ (φ) Rdφ

(14)

Since the virtual bending curvature δχ is kinematically compatible with the virtual displacement δu2 , then the application of the virtual work equation, namely Li = Le ∀ δα , makes the bending moment m in equilibrium with p, providing the transversal equilibrium equation of the annular segment. The pipe is constituted by a linear elastic material, therefore for the annular beam the response function m (φ) = Es Jχ (φ) holds, where 1 J = 12 b3Δs is the second order principal area moment of the cross section of the annular beam. Then the virtual work equation becomes, using Eqs. (10), (12) and (14):

(6)

(7)

αJ

∫0

2π

χ (φ)2dφ = −bκ 2RΔs

∫0

2π

ψ2 (φ)sin(φ) dφ

(15)

Substituting in Eq. (15) the expressions (7), (4), (13) and performing the integration, one obtains:

(8)

α=

where the polar form of y and Eq. (6)-a are used. As a consequence, inserting Eq. (8) in Eq. (1), the expression of the pressure becomes:

p = Es bκ 2 (R sin(φ) + αψ2 (φ))

(11)

(φ) χ (φ) = αχ

Indicating with y := y + u2 the distance of the generic point of the deformed cross section from the axis a3 (see Fig. 3-c), the expression of the longitudinal stress is

σs = Es κy = Es κ (R sin(φ) + αψ2 (φ))

(10)

which becomes, in terms of the assumed mode (Eq. (3)):

(5)

The components of the assumed mode in the two bases (a2, a3) and (at , an) , respectively, are related to each others by the following trigonometric relationships:

ψ2 (φ) = ψt (φ)cos(φ) − ψn (φ)sin(φ) ψ3 (φ) = ψt (φ)sin(φ) + ψn (φ)cos(φ)

(−p (φ)) δu2 (φ) RdφΔs

1 ⎡ d 2v du ⎤ + 2 ⎥ R2 ⎢ dφ dφ ⎣ ⎦

χ (φ) =

This appears as a reasonable occurrence, at least at the ﬁrst order of the analysis, where the extension of the circumference lines emerges as a secondary eﬀect; nevertheless, more reﬁned analyses could be performed, adding more than one shape function in the description of the deformation of the cross-section (as in the spirit of the generalized beam theory, GBT, [21–23]), each one with its own amplitude factor, giving rise to more complicated (but surely reﬁned) models. It is worth noting that the assumed shape given by Eq. (4) is not an ellipse, which would be alternatively produced by components ψt (φ) = −sin(φ) , ψn (φ) = −cos(φ) . Although the two shapes are very similar (at least for small α ), the assumed one is referred to as an oval. Note that, in the case of an ellipse, it would be εt ≠ 0 . It is convenient to introduce the components of u(φ) in the a2, a3 directions as well, so that u (φ) = u2 (φ) a2 + u3 (φ) a3 , where

u2 (φ) = αψ2 (φ) u3 (φ) = αψ3 (φ)

2π

On the other hand, in order to evaluate the internal virtual work, one has to consider the abscissa of the cross section as the axis line of an annular beam whose cross section is rectangular, with base Δs and height b; its linear bending curvature (around a1) induced by the ovalization is deﬁned as

(4)

The particular choice of the expression given in Eq. (4) for the assumed mode is grounded on the fact that it actually describes the shape of an oval in the range φ ∈ [0, 2π ) where no linear circumferential strain εt of its axis line results, being:

εt =

∫0

R5 R5 κ2 ≃ 2 κ2 2 4 b 5κ R ⎞ b2 ⎛1 + 6b2 ⎠ ⎝ ⎜

⎟

(16)

where the approximation is related to the following estimation of the orders of magnitudes of the terms: O (R/ b) = 102 and O (κR) = 10−3, therefore O (κ 2R 4 / b2) = 10−2 . Note that the validity of the approximation in Eq. (16) induces the consideration that, (only) for the actual evaluation of the function α (κ ) , the expression of σs depending just on the distance of the point of the cross section from the axis a3 in the non

(9)

Considering the segment of pipe of length Δs along a1, the pressure acts on an inﬁnitesimal area dA = RdφΔs . It is possible to introduce the expression of the (external) virtual work spent by the pressure p on the corresponding component δu2 of virtual displacement of the cross 74

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A. Luongo et al.

deformed conﬁguration can be actually used (i.e., σs = Es κy and not σs = Es κy ). Eq. (16) represents an internal constraint between the bending curvature and the amplitude of ovalization.

Es ⪢Ec . As a consequence, the pressure p is considered as applied just to the skin ﬁbers, as in Fig. 3-a, and still assumes the expression in Eq. (1) where σs = Es κy , consistently with the evaluation of the order of magnitude of the terms given in Section 2.2. (3) The displacement causing the ovalization of the cross section still generates an oval shape as shown in Fig. 3-c, which now is extended as follows:

2.3. The moment-curvature relationship In order to obtain the moment-curvature relationship, the equivalence condition between the bending moment and the resultant of the longitudinal stress is considered at the extremal cross section of the beam segment. In particular, considering that the deformation does not produce any modiﬁcation of the element of area of the cross section, being the circumferential element inextensible (Eq. (5)), the condition reads:

M=

∫0

2π

y σs bRdφ

r sin(2φ) 2R r ψn (r , φ) = − cos(2φ) R ψt (r , φ) = −

where internal points corresponding to the core are considered (where r < R ). Consequently, the displacement at the generic point at radius r and phase φ has components

(17)

u (r , φ) = αψt (r , φ) v (r , φ) = αψn (r , φ)

where M is the bending moment acting on the beam segment (around the a3 axis) able to guarantee the existence of the curvature κ . Note that in Eq. (17) the arm of the couple produced by the longitudinal stress σs is evaluated in the ovalized conﬁguration. Substituting Eq. (8) in Eq. (17), the latter becomes

M = Es κbR

∫0

2π

y 2 dφ

(18)

Le =

Inserting Eq. (16) in Eq. (19), the moment-curvature relationship becomes ⎜

⎟

(20)

18 − 2 31 b b ≃ 0.524 2 5 R2 R

(−p (φ)) δu2 (R, φ) RdφΔs

(25)

v 1 ∂u − =0 r ∂φ r ∂v εn (r , φ) = − ∂r ∂u 1 ∂v u + − γtn (r , φ) = ∂r r ∂φ r

(26)

where εt is the (null) circumferential strain, εn is the (internal) radial strain and γtn is the shear strain. Consequently, the corresponding stress components in the core are

σt (r , φ) = λ c εn σn (r , φ) = (λ c + 2μc ) εn

(21)

τtn (r , φ) = μc γtn

where the limit moment is

4 1497 + 217 31 Es πb2R ≃ 0.333Es πb2R Ml = 625

(22)

A foam core of elastic material identiﬁed by Lamé constants λ c , μc 3λ c + 2μc λ c + μc

(27)

(6) In evaluating the internal virtual work, besides the contribution due to the bending moment and curvature of the annular beam corresponding to the skin, which has an expression very similar to Eq. (14), a new contribution due to the foam core must be considered. In particular, the internal virtual work now assumes the following expression:

3. The pipe with foam core

(and Young's modulus Ec =

2π

εt (r , φ) =

Note that, contrary to what happens for Eq. (16), it is crucial to consider here the dependence of the longitudinal stress on the deformed distance of the point from the axis a3 , namely σs = Es κy , in order to have a consistent nonlinear dependence of the bending moment M on the curvature κ and amplitude of ovalization α . The softening nature of the relationship is evident in Eq. (20), due to the negative cubic term in κ . The Brazier eﬀect occurs at the limit point corresponding to the triggering of the instability, i.e. at the (smallest) dM (κ ) value of curvature corresponding to dκ l = 0 , namely

κl =

∫0

(5) The annular beam described by the skin abscissa still presents bending curvature having expression (12), where Eq. (13) is valid, and bending moment m (φ) = Es Jχ (φ) . On the other hand, if a segment of pipe of length Δs along a1 is taken, the foam is assumed in planar deformation regime, and its (linear) strain measures become [24]:

(19)

3R7 3 5R11 5 ⎞ κ + κ M = Es π ⎛bR3κ − 2b 8b3 ⎠ ⎝

(24)

along tangent and normal directions to an abscissa at radius r, respectively. Moreover, u2 (r , φ) = αψ2 (r , φ) is the displacement component along a2 , and u3 (r , φ) = αψ3 (r , φ) along a3 , where relationships among ψ2 , ψ3 and ψt , ψn are still like in Eq. (7). (4) Being the pressure applied just to the skin, the external virtual work assumes an expression very similar to Eq. (10), namely:

Using the expression of the component ψ2 (φ) of the assumed deformation of the cross section in Eq. (17) and performing the integration, one obtains:

3 5 M = Es bκπ ⎛R3 − R2α + Rα 2 ⎞ 2 8 ⎝ ⎠

(23)

μc , where the subscript c stands for

“core”) is considered now in the pipe, under the hypothesis of perfect gluing between the skin and core at the interface. The formulas obtained in Section 2 now modify as follows.

Li =

∫0

2π

m (φ) δχ (φ) Rdφ +

2π

∫0 ∫0

R

(σn δεn + τtn δγtn ) rdrdφΔs (28)

where rdrdφ is the area element of the core. (7) Application of the virtual work equation Li = Le ∀ α , i.e. by equating Eqs. (25) and (28), leads now to the following expression of the constraint between the amplitude of ovalization and the bending curvature, obtained after performing the relevant integration:

(1) A generic point of the cross section, which is now the whole circle and not only the circumference, is identiﬁed by polar coordinates r , φ , where r ∈ [0, R], and points at r = R correspond to the skin. (2) Just for the evaluation of the pressure p, the longitudinal stress σc in correspondence of the generic longitudinal ﬁber of the foam is considered vanishing with respect to that of the skin σs , being 75

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A. Luongo et al.

α=

R5 κ2 3 λ + 2μ R 2 c c 2 b (1 + ) Es 3 b3

applications. Here, linear kinematics and equilibrium equations are assumed for the beam, considered as Euler-Bernoulli type, while nonlinear eﬀects are related only to the moment-curvature relationship. The kinematic boundary value problem reads:

(29)

It is worth noting how the constraint Eq. (29) now depends on the ratio between the elastic coeﬃcients of the two materials constituting the pipe, and that if λ c , μc → 0 , i.e. no foam is considered, Eq. (29) degenerates to Eq. (16). (8) The bending moment-curvature relationship for a pipe with skin and core is obtained from the equivalence of the applied bending moment of the segment with that due to the emerging longitudinal stress:

M=

∫0

2π

y σs bRdφ +

2π

∫0 ∫0

R

y σc rdrdφ

κ (s ) w (0) w′ (0) w′ (l)

(30)

(31)

3 5 M = Es − bR2α + bRα 2 ⎞ κ 2 8 ⎝ ⎠ 5R2α 2 ⎞ R4 R3α ⎛ + Ec π − + κ 2 16 ⎠ ⎝ 4

3R7 (6w″w‴ 2 + 3w″2 w⁗) Es π ⎛⎜bR3w⁗ − 2b ⎝

π ⎛bR3

+

⎟

(32)

Finally, substituting Eq. (29) in Eq. (32), the moment-curvature relationship becomes

+

πR7 (3Es b + Ec R) κ3 2 R3 λ c + 6μc ⎞ 2b2 ⎛1 + Es 3 b3 ⎝ ⎠

5πR11 (2Es b + Ec R) κ5 2 2 R3 λ c + 6μc ⎞ 16b4 ⎛1 + 3 b3 Es ⎝ ⎠ ⎜

(35)

where V (s ) indicates the (reactive) shear force of the beam. Using Eq. (20) and writing the equilibrium equations in terms of displacement (displacements method) after static condensation of the shear force, the fundamental problem reads:

i.e., as it is done in the empty pipe case, the evaluation of the moment-curvature relationship requires the full expressions for σs and σc , that is their dependence on y . Substituting Eq. (31) in Eq. (30), and performing the integration, one obtains

⎜

(34)

V ′ (s ) = βf M ′ (s ) + V (s ) = 0 V (l) = 0

σs = Es κy = Es κ (R sin(φ) + αψ2 (R, φ)) σc = Ec κy = Ec κ (r sin(φ) + αψ2 (r , φ))

1 M = π ⎛Es bR3 + Ec R 4 ⎞ κ − 4 ⎝ ⎠

w″ (s ) 0 0 0

where w (s ) is the vertical displacement (along a2 ) of the axis line and the prime indicates derivative with respect to the abscissa s (Fig. 2-b). The boundary value problem related to the equilibrium is:

where

⎜

= = = =

5R11 (20w″3 w‴ 2 + 5w″4 w⁗) ⎞⎟ = −βf 8b3 ⎠

w (0) = 0,

w′ (0) = 0

w′ (l) = 0,

Es πw‴ (l) ⎛bR3 − ⎝ ⎜

9R7w″ (l)2 25R11w″4 (l) ⎞ + =0 2b 8b3 ⎠ ⎟

(36)

However, in case of ﬁlled pipe, the b.v.p. reads:

1 ⎛bEs R3 + Ec R 4 ⎞ πw⁗ − 4 ⎝ ⎠

⎟

πR7 (3Es b + Ec R) (6w″w‴ 2 + 3w″2 w⁗) 2 R3 λ c + 6μc ⎞ 2b2 ⎛1 + Es 3 b3 ⎝ ⎠ ⎜

⎟

5πR11 (2Es b + Ec R) (20w″3 w‴ 2 + 5w″4 w⁗) = −βf 3 λ + 6μ 2 R 2 c c 4 ⎞ 16b ⎛1 + Es 3 b3 ⎝ ⎠ w (0) = 0, w′ (0) = 0, w′ (l) = 0

⎟

+

(33)

⎜

4. Numerical example

(bE R s

3

⎟

1

)

+ 4 Ec R 4 πw‴ (l) −

4.1. The complete model

+

As a practical application, a structure constituted by a pipe beam of length l, clamped at one end (section A), and with a sliding clamp at the other end (section B), is considered (Fig. 4). The external constraints are assumed to limit rotations and/or displacements, but to keep free the ovalization of the cross section. A uniform vertical load of intensity βf is applied, where β is a control parameter and f a ﬁxed value of force per unit length. When the pipe is empty, the moment-curvature relationship of Eq. (20) is used; on the other hand, once the pipe has a foam core, the relevant moment-curvature relationship in Eq. (33) is retained. Note that those relationships have been obtained for a segment of beam under uniform bending, and they are now used for a (complete) beam under nonuniform bending, as it is typically done in engineering

πR7 (3Es b + Ec R) 3w″2 (l) w‴ (l) 2 R3 λ c + 6μc ⎞ 2b2 ⎜⎛1 + ⎟ 3 E 3 b s ⎝ ⎠

5πR11 (2Es b + Ec R) 4 2 5w″ (l ) w‴ (l ) 2 R3 λ c + 6μc ⎞ 4 16b ⎜⎛1 + ⎟ 3 Es 3b ⎝

=0

⎠

(37) The following values for the mechanical and geometrical paraN meters are considered: for the skin material, Es = 6.4 × 1010 2 ; for the m 2 4 λ c = 5.769 × 10 N/m μc = 3.846 × 10 4 N/m2 and foam, 2 5 (Ec = 1.0 × 10 N/m ); for the cross section, R = 0.15 m , b = 1 mm ; the beam is assumed l = 2 m long. The boundary value problems in Eqs. (36) and (37) are numerically solved using the software AUTO [25], considering f = 10 4 N/m and increasing β from zero to positive values. The law α (κ ) in case of empty pipe is shown in solid blue line in Fig. 5-a, where its parabolic evolution is evident; a plot of the relevant moment-curvature relationship, coming from Eq. (20), is shown in Fig. 5-b still in solid blue line, conﬁrming its softening nature. The limit point (highlighted by the ﬁlled circle) occurs for κl = 0.023 m−1 and Ml = 10039 Nm . In case of ﬁlled pipe, the dependence of α on κ (Eq. (29)) is superimposed in dashed orange line to that of the empty pipe case, in Fig. 5-a, where still a parabolic evolution is found but with signiﬁcantly lower amplitudes; the moment-curvature relationship of Eq. (33) is shown in Fig. 5-b, in dashed orange line, superimposed to the corresponding one previously obtained in case of empty pipe. It is

Fig. 4. Practical application: clamped structure under increasing distributed load.

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A. Luongo et al.

Fig. 5. Empty pipe (solid blue line) and with foam core (dashed orange line): (a) Dependence of the amplitude of ovalization on the curvature; (b) moment-curvature relationship. [κ ] = m−1, [α ] = m , [M ] = Nm . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

boundary eﬀects, mainly due to the diﬃculty of ﬁnely modelling the clamp with free ovalization (see Section Appendix A).

evident the contribution of the foam, which signiﬁcantly reduces the ovalization, increases the stiﬀness of the beam and moves forward the occurrence of the limit point and the consequent instability phenomenon, now attained at κ c , highlighted by the diamond symbol. Note that the limit point in the ﬁlled case occurs for curvatures which involve longitudinal strain for the skin that is larger than the actual yielding limit for the considered material, which in particular occurs at κ y . It means that, if one would consider curvatures larger than κ y , an elasticplastic response function should be used, at least for the skin material. Since that is not the scope of the present paper (however it will be considered in a forthcoming paper), only curvatures less than κ y are considered as meaningful. As in Fig. 5, in the following ones, solid blue lines refer to the empty pipe while dashed orange lines to the ﬁlled pipe. In Fig. 6-a, the evolution of the bending curvature κA at the clamp of section A is shown as a function of the load control parameter β (curvature and moment are drawn on the side of the ﬁbers in traction, as usual in applications): for the empty pipe, once the curvature reaches the value κl corresponding to the limit point of the moment-curvature relationship (see also Fig. 5b), attained for βl = 0.766, the tangent of the blue curve tends to be vertical and the numerical procedure stops. It means that, in section A, the bending moment has reached its limit value, and the tangent stiﬀness operator of the structure becomes singular. For the ﬁlled pipe, the limit point is signiﬁcantly moved to the right, and the problem is essentially linear up to values β ≃ 2 . In Fig. 6-b-e, displacement w, curvature κ , amplitude of ovalization α and moment M are shown, respectively, as functions of s for the last step of the numerical procedure for the empty pipe, i.e. at βl ; Furthermore, in Fig. 6-f, the shape of the ovalized shape of the cross section at abscissa s = 2 m is shown, superimposed to its initial shape in black. The results in terms of displacement w and amplitude of ovalization α are compared with those provided by a ﬁnite element model, shown in dotted blue line for the empty pipe and dash-dotted orange line for the ﬁlled pipe in Fig. 6-b,d,f (see Section Appendix A for details on the FE model). It is evident the crisis happening at section A for the empty pipe, while the foam generally increases the stiﬀness of the structure, slightly reducing the absolute values of the displacement, but signiﬁcantly reducing the curvature and the ovalization around the section A. In particular, in Fig. 6e, it is evident how the moment law along the beam is the same for the two considered cases; furthermore, in line with dots, the separate contribution of the sole skin is reported, showing that it provides the greatest part of the bearing capacity of the pipe, except for a small region close to the section A, where the contribution of the core becomes deﬁnitely not vanishing. All the results are in good agreement with the outcomes of the FE model, where the limit load multiplier for the empty pipe is found as the one at which convergence is deﬁnitely lost, occurring at the value βl = 0.75; the larger mismatches in terms of w and α (of about 10%) are found in correspondence of the abscissa s = 0 , i.e. in section A, where the FE model produces signiﬁcant

4.2. The lumped ovalization model As a general comment arising from observing the solid blue lines of Fig. 6, i.e. those related to the empty pipe case, the contribution of the nonlinear terms related to the occurrence of ovalization of the cross section and to the softening behavior of the moment-curvature relationship is substantially localized close to the section A. For such a reason, limiting the following analysis to the empty pipe case, an alternative lumped ovalization model of the structure is proposed, where the beam is completely linear (i.e., even in its moment-curvature relationship), while the nonlinear contribution is lumped to a rotational spring, located in A, where an external hinge substitutes the clamp (see Fig. 7). In particular, the spring has a nonlinear moment-rotation relationship ( μ (ϑ) ) arising from Eq. (20), where the curvature is expressed as a function of the rotation ϑ of the cross section, namely κ = ϑ/ d , with d⪡l a characteristic length, i.e.

ϑ 3R7 ⎡ ϑ ⎤3 5R11 ⎡ ϑ ⎤5 ⎞ + μ = Es π ⎛⎜bR3 − ⎟ d 2b ⎣ d ⎦ 8b3 ⎣ d ⎦ ⎠ ⎝

(38)

In this way, the equilibrium boundary value problem becomes Es πbR3w⁗ = −βf w (0) = 0, w′ (l) = 0,

w′ (0) 3R7 ⎡ w′ (0) ⎤3 5R11 ⎡ w′ (0) ⎤5 ⎞ − + Es πbR3w″ (0) = Es π ⎛⎜bR3 ⎟ d 2b ⎣ d ⎦ 8b3 ⎣ d ⎦ ⎠ ⎝ w ‴ (l ) = 0

(39) where Eq. (39)-c states that the moment of the beam in A is equal to the moment produced by the spring due to the rotation ϑ = w′ (0) . Certainly, the solution of Eq. (39) in terms of w (s ) is aﬀected by the choice of the length d, as it happens in damage theory where mesh-dependent outcomes can occur (see e.g. [26]); furthermore, as d is reduced, the initial tangent stiﬀness of the spring increases, anyway having Ml as a limit value for the moment. The relevant moment-rotation relationship is shown in Fig. 8-a and the solution in terms of w (s ) at β = βl is shown in Fig. 8-b; in both the Figures, diﬀerent values of d are used, namely d = R (green line), d = 0.5R (magenta line), d = 0.15R (cyan line), d = 0.1R (yellow line), d = 0.005R (red line), while in blue line is reported the solution for the complete model (clamped case). From Fig. 8-b, the outcome for d = 0.15R seems to better ﬁt the clamped case in terms of displacement w, while smaller values of d tend to produce solutions converging to smaller values of the maximum displacement, i.e. determined by an excessively stiﬀ spring. As a matter of fact, the parameter d may assume the physical meaning of longitudinal dimension of the boundary layer, identifying the length scale of the region inﬂuenced by the boundary condition in A. Further studies need to be carried out to get to an a-priori determination of the value of d. 77

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Fig. 6. Results for the empty pipe case: (a) curvature at section A as a function of the load parameter; (b) displacement along the beam for βl ; (c) curvature along the beam for βl ; (d) amplitude of ovalization along the beam for βl ; (e) moment along the beam for βl ; (f) deformed shape of the mid-span cross section of the beam for βl . Solid blue line: empty pipe; dashed orange line: ﬁlled pipe; dotted blue line: fem of empty pipe; dot-dashed orange line: fem of ﬁlled pipe. [κ ] = m−1, [s] = m , [w] = m , [α ] = m , [M ] = Nm . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

beam realized by the cross section of the pipe is evaluated. Then, in order to deduce the nonlinear elastic constitutive law, the equilibrium equation between the resultant of the longitudinal stress and the bending moment at the extremal cross section of a segment of pipe is evaluated in the ovalized shape. The obtained law reproduces the Brazier eﬀect for an empty pipe, giving rise to the well-known softening behavior, the occurrence of a limit point and the consequent instability phenomenon. When an elastic foam is included as ﬁlling of the pipe, the relevant constitutive law shows a shifting of the instability limit to larger curvatures, indicating a beneﬁcial mechanical contribution of the foam. An application is presented, consisting of a structure realized by a clamped pipe beam under uniform external load. In particular, for increasing amplitude of the load, the empty pipe shows the failure of the bearing capacity and the triggering of the instability phenomenon when the curvature of one of the cross sections, the clamped one, induces a limit value for the ovalization: for this value, the longitudinal stress is no more able to balance the bending moment. On the other hand, in case of a ﬁlled pipe, the structure essentially behaves as in the linear ﬁeld in the considered load range, while the instability phenomenon is shifted to signiﬁcantly higher values of the load. Good agreement is

Fig. 7. The reduced structure: linear beam with nonlinear lumped stiﬀness.

5. Conclusions A pipe beam with a deformable cross section, possibly ﬁlled with structural foam, is considered as bent in the elastic regime. First, following mechanical deductions, an internal constraint equation between the bending curvature and the amplitude of ovalization of the cross section is obtained. In particular, to get such a constraint, two steps are performed: 1) an assumed modal shape of oval is used to represent the deformed cross section, leaving the amplitude of ovalization as unknown; 2) the weak form of the balance equation between the pressure responsible for ovalization and the bending moment of the annular 78

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Fig. 8. Reduced model: (a) moment-rotation relationship for diﬀerent values of d; (b) displacement along the beam for βl . [ϑ] = rad , [s] = m , [w] = m , [μ] = Nm . (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article).

spring. In particular, the nonlinear moment-rotation relationship of the spring is deduced in terms of a characteristic length, which assumes the physical meaning of longitudinal dimension of the boundary layer. In the latter case, calibration of the value of the characteristic length is required in order to get a reliable solution.

obtained in terms of displacements, ovalization and deformed shape of the cross section with a FE model realized within commercial software. Finally, for the same case-study, a reduced model with a nonlinear lumped stiﬀness is proposed, where the beam is considered as linear and the clamp is substituted by a hinge with a nonlinear softening Appendix A. The FE model

A few details on the FE model are described here. It is realized within the commercial software ADINA [27], and used to compare the results and evaluate the reliability of the analytical model. In the case of the empty pipe (see Fig. A.1-a), shell elements made of linear elastic material have been used, in large displacement and small strain regime. The same model is retained for the skin of the composite pipe, whereas the core is realized using 3D elements (see Fig. A.1-b), with linear elastic material as well. In the latter case, a more reﬁned (and much more computational-time consuming) mesh is required in order to guarantee meaningful results. The clamp with free ovalization, located at section A, is realized by restraining the vertical displacements (along the Y axis) of the nodes at angles φ = 0, π , and the horizontal displacement (along the Z axis) of the nodes at angles φ = π /2, 3/2π . The load is applied as uniform mass proportional, directed as − Y and distributed in the skin material, with a multiplier factor increasing with step Δβ = 0.05. The displacement of the axis line along Y is evaluated as the semi-sum of the displacements of the top and bottom external node lines (i.e., those at Z= 0 ), respectively, while the ovalization is the displacement along Z of the left (or right) external node line (i.e., those at Y= 0 ).

Fig. A.1. Finite element model: (a) empty pipe; (b) ﬁlled pipe.

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