Post-buckling behaviour of a slender beam in a circular tube, under axial load

Size: px

Start display at page:

Download "Post-buckling behaviour of a slender beam in a circular tube, under axial load"

Georgia Phyllis Lane
5 years ago
Views:

1 Computationa Metho and Experimenta Measurements XIII 547 Post-bucking behaviour of a sender beam in a circuar tube, under axia oad M. Gh. Munteanu & A. Barraco Transivania University of Brasov, Romania Ecoe Nationae Supérieure d'arts et Métiers ENSAM), Paris, France Abstract This paper deas with the study of the behaviour of a sender beam introduced in a cyindrica tube and subjected to an axia compressive force. The beam is very ong compared to its transversa dimensions and therefore it wi bucke to a very sma axia force. The post-bucking behaviour is examined. The study has important appications in the petroeum industry, for coied tubing in the case of driing in horizonta or incined webores. The sender beam has a constant cross-section that can have any form, athough a circuar cross-section is the most used in practice. The probem has a geometrica non-inearity to which the non-inearity caused by the friction has to be added. Rotations coud be arge and a specia isoparametric D beam finite eement is eaborated: the Euer- Rodrigues quaternion was preferred to describe the finite cross-section rotations. The paper presents ony the static case, but extending the presented approach to dynamic anaysis is quite natura. The method is very accurate and it is rapidy convergent due to the fact that the exact equations, written for the deformed configuration, are soved. The iterative Newton-Raphson method was used to sove the noninear differentia equations. Keywor: coied tubing, finite eement method, post-bucking behaviour, Euer quaternion, geometrica non-inearity. Introduction A ong sender initiay straight beam compressed and constrained within a circuar cyinder is studied in this work. This probem presents a great interest in rock engineering and petroeum production. The beam buckes within the narrow space of a dri hoe under the action of axia force and its own weight. Due to its WIT Transactions on Modeing and Simuation, Vo 46, 7 WIT Press ISSN 74-55X (on-ine) doi:.495/cmem755

2 548 Computationa Metho and Experimenta Measurements XIII ength, the beam wi first bucke according to the cassic Euer formua, a sinusoida bucking, when the dispacements are considered infinitey sma. The post-bucking behaviour is much more important because the beam ies on the cyinder and friction wi occur that diminishes the axia oad transmission. The probem is highy compicated because it invoves a geometrica non-inearity to which the friction has to be added. Severa theoretica modes were proposed, a of them based on anaytica modes, some of them compicated, [, ], or based on simpified (or even very simpe) modes using in genera a variationa approach, [ ]. Experimenta measurements were performed as we: some works are based ony on experiments, [, 4], whie in other works the experiments were meant to verify the theoretica modes, [,, 4,, ]. Amost a theoretica modes assume that post-bucking shape is a heix that is not aways true. Some of them consider even a constant pitch heix that it is obviousy not exact mainy when the friction is taken into consideration, [4, ]. In this work a numerica method based on finite eement method is proposed. A specia type of finite eement is presented, based on some previous works, [5 7]. This finite eement type is speciay conceived for geometrica noninear beam systems and uses Euer-Rodrigues quaternion and the axia strain as generaized coordinates. It aows a very exact and fast convergent numerica mode to be eaborated. The beam is initiay straight and ies aong a generatrix; it ies aways on the cyindrica surface even after deformation due to the axia force. Two boundary conditions were examined, camped-free beam and doube hinged beam; significanty different resuts were obtained for these two cases. Beam theory Because of beam ength, the Euer-Bernoui beam theory is appied. The beam is considered initiay straight and it has a constant cross-section. The first variation of the tota potentia energy has the expression: δπ Τ Τ T δu + δv EA d + d d d δε ε s δκ D κ s +. δ u p s δϕ m s () δu where U is the deformation energy, V is the potentia of the externa oad, u is the dispacement vector, ϕ represents the sma rotation vector, p and m are the distributed force vector and the distributed moment vector, respectivey, acting on the beam. Efforts and strains are inked by the reations (Hooke s ow): δv N EAε, M Dκ, () where N is the axia effort, E is the Young moduus, A the cross-section area and ε is the axia deformation of the beam centreine; the moment vector M and the curvature κ have foowing expressions in the oca reference frame: T ( M M M ) ( κ κ ) M κ (), κ WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

3 Computationa Metho and Experimenta Measurements XIII 549 M t is the torque and M, M are the bending moments; κ, κ and κ are the torsion and the two curvatures, that is the variation of anges between two oca reference frames situated at the infinitesima distance.the Hooke s ow matrix D is M t GI t κ M Dκ or M EI κ. (4) M EI κ I t, I and I are second order geometrica properties of the cross-section (I and I are the principa inertia moments); G is the transversa easticity moduus. The main difficuty in D geometrica non-inear study, in cassica incrementa formuation, is that the rotations form no onger a vector. This is why in this work the Euer-Rodrigues parameters are used to describe crosssection rotation, [5 7]. They are defined for a finite rotation of ange θ around the axis defined by the unit vector ν ( ν ) T x ν y ν z. The rotation is θ represented by the scaar cos and by the vector sin θ ν grouped in the foowing coumn matrix, the we-known Euer-Rodrigues quaternion: T θ θ θ θ cos ν x sin ν y sin ν z sin (,,, ) T. (5) In this work the finite rotation θ transforms the initia infinite sma ength eement of the initia straight beam into the fina position due to its deformation. The four Euer-Rodrigues parameters must verify the reation: T. (6) Knowing the Euer-Rodrigues parameters, the rotation matrix from the initia configuration to the actua one, for each cross-section, has the form: R ( ) ( ) ( ) + + ( ) ( ) ( ) ( ) ( ) ( ) (7) To sove the probem, the tota potentia energy () has to be minimized and usuay numerica metho are used, finite eement method being the most effective. The cassica arge dispacement finite eement approach considers as noda unknowns (noda generaized coordinates) sma increments of noda dispacements and rotations and this approach ea to an extremey timeconsuming incrementa formuation. In this work the unknowns are axia deformation ε and the four Euer- Rodrigues parameters. They are grouped into the generaized coordinate vector: ( ) T q. (8) ε WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

4 55 Computationa Metho and Experimenta Measurements XIII Curvature vector may be easiy expressed in Euer-Rodrigues parameters as: κ G, (9) where is the space derivative that is with respect to the curviinear coordinates s. The G matrix has the form: G. () The equations are written in the actua reference frame of the current crosssection and therefore the spatia derivatives are performed with respect of actua curviinear coordinate S different of the curviinear coordinate s for the initia configuration. As the sma strain hypothesis was considered, we may consider that the ength of the beam keep its ength during the deformation: ds ( + ε ). Thus the foowing symboic expression may be written:. S s In the oca reference frame, virtua sma rotation vector is expressed in Euer- Rodrigues parameters as: and in the goba reference frame: L δ ϕ Gδ () δϕ RGδ. () Finay the dispacements vector u of the current point has the form: s s ( + ) - u( s) u + ε u ε ( ) ( Rn( + ) n) + ( + ) ( + ), () where n is the unit vector perpendicuar to cross-section. Now there are a the ingredients to sove numericay the probem using the variationa formuation (). Kinematic equations In the case of the beam in contact with the tube, the deformed shape of the beam is given by the equations: ( s) x x y Rcos ϕ() s. (4) z Rsinϕ() s The foowing reations exists between anges θ, ϕ (see figure ) and the abscise x (for sma axia strain ε ): WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

5 Computationa Metho and Experimenta Measurements XIII 55 dx sinθ cosθ dϕ R or or dx sinθ, (5) dϕ cosθ R R being the radius of the tube, figure. Therefore the director parameters of an infinite sma segment of the beam centre-ine are: dx dy dz sinθ, Rcosθ sinϕ, Rcosθ cosϕ, (6) where the anges θ and ϕ resut from figure.that must coincide to the first coumn of the matrix R given by (7): + sinθ + cos θ sinϕ +. (7) cosθ cosϕ It is easy to show that the Euer-Rodrigues parameters satisfying equations (7), satisfy aso equation (6). R θ The pipe (beam) x z ϕ y dx The tube Figure : Definition of anges θ and ϕ. The system resuting from () is noninear and the iterative Newton-Raphson method wi be appied, that is at each iteration the system is soved in unknown increments. Equations (7) become in increments: - - cosθ cosϕ cosθ sinϕ δ δ δ sinθ sinϕ, (8) δ sinθ cosϕ δϕ δθ to which the incrementa form of the second equation (5) is added: WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

6 55 Computationa Metho and Experimenta Measurements XIII s s sinθδθ cosθ δϕ ( + ε ) + δε. (9) R R Equations (8) and (9) represent the reations between the increments of Euer- Rodrigues parameters and the increments of the two anges θ and ϕ. There are 4 equations, inking 7 unknown functions, ε,,,,, θ, ϕ, imposing the deformed beam to ie on the interna surface of a tube of radius R. Therefore ony independent generaized coordinates were chosen: q ˆ ε θ. () 4 Friction forces ( ) T Between the beam and the atera surface of the tube there are reaction forces. Moment equiibrium equations for a sice of the beam, written in the oca reference frame of each cross-section, is: M + n R s or ( κ ) D + κ ( D κ ) + n R. () s After each iteration, the generaized coordinates qˆ are known and the Euer- Rodrigues parameters are known as we; that aow the curvatures κ to be found. The axia effort N and the two shear efforts Q and Q (the components of the vector R) are computed using equation (). Finay, from the force equiibrium equation written in the cross section oca reference frame: R + κ R + p () s it resuts the contact pressure p between the beam and the tube. Distributed force p wi be represented in the cyindrica reference frame. Axes of the cyindrica reference frame are: x direction of the goba frame (see figure ), - norma direction and tangent direction in the yz goba pane. The norma distributed force is p. The friction force cannot exceed the vaue µ p, where µ is the friction coefficient. If the tangentia force is smaer than µ p, there is no reative motion between the points in contact beonging to the beam and to the tube. In any point the foowing condition must be respected: ~ p + ~ p µ ~ p, () otherwise the ink is broken and a reative motion is aowed. At each oad step few iterations are needed in order to estabish which among a nodes are moving and which are not because of the friction. The tube is compressed by an axia force with the condition that the tube aways ies on the surface of the tube. In the absence of the friction the method is not incrementa and the soution may be find directy for the fina oad of the oad. But in the presence of the friction the approach must be incrementa, the resut depending on the history of the force variation. WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

7 Computationa Metho and Experimenta Measurements XIII 55 Figure : Two-node and three-node finite eements. 5 Specia finite eement To sove the probem, finite eement method was appied. Isoparametric twonode and three-node finite eements were used, figure. This finite eement is described in detai in [5 7]. Each node has five degrees of freedom per node accordingy to (8). For the three-node finite eement the shape function is: q s s s s s s) k,i- + ( + ), + ( + ),,...5 q qk i qk i+ k. (4) h h h h h k ( The beam is divided in severa finite eements and the oad is appied incrementay. Finay a non-inear differentia system is obtained from functiona () using a procedure simiar to the standard one: E q K q q Q, (5) ( ) ( ) where K is the secant stiffness matrix that is function of q, obtained by assembing eementa stiffness matrices; q is the vector of the generaized noda unknown (8) for the whoe structure and Q is the vector of noda oa. System (5) incudes the equation (6) for each node. It is interesting to note that a eements of secant stiffness matrix K are poynomias in Euer-Rodrigues and strain ε. To sove the non-inear system, the iterative Newton-Raphson method is appied. At each step, at each iteration, equations (8) and (9) are used in order to repace noda degrees of freedom (8) with (), to impose the beam to ie on the cyindrica surface of the tube. In the oad step i and for the k-th iteration it can be written: ˆ ( k ) ( ˆ ) ˆ ( ) K δq k + + k i E, (6) where: t T q Cqˆ, K C Kˆ C, E CEˆ, (7) t C being a square matrix provided by appying of equations (8) and (9) for a ( k+ ) ( k+ ) ( k ) nodes of the discretization and δ qˆ i qˆ i qˆ i. K t is the tangent stiffness moduus, in fact the Jacobean J: t WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

8 554 Computationa Metho and Experimenta Measurements XIII E J K t. (8) q The approximate evauation of the Jacobean may be used, athough in the paper the exact computing were performed: Q K t K +. (9) q Often an under-reaxation factor β coud improve the convergence: ( k+ ) ( k ) ( k+ ) q qˆ qˆ ˆ βδ. () i+ i+ + i+ Ange θ [ o ] θ z y Axia force [N] a) Rotation ange θ of the mid node versus axia force. b) Beam (pipe) view aong x axis, the beam axis. Figure : Post-bucking diagram. 6 Numerica exampe To iustrate the method a MATLAB program was eaborate and the foowing exampe was anayzed, [6, 7, ]: doube hinged beam (pipe), ength L4 mm, outer pipe diameter D.7 mm, inner pipe diameter d6.5 mm, diameter of the tube D 48.6 mm, Young moduus E.7e5 MPa, Poisson coefficient ν., -node finite eements, 6 nodes. Figure shows the dependence ange θ versus axia force, dependence found by the computer code. The Euer bucking force is 7.45 N (see figure a) and much higher force vaues were studied. Figure 4 presents the deformation configuration of the pipe under an axia force of 8 N. One can see that the shape is not exacty a heix: the symmetry about the pane perpendicuar on the beam, on the mid-node, has to be ensured. On the contrary, figure 5 shows the unsymmetrica deformed configuration of a camped-free beam. WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

Computationa Metho and Experimenta Measurements XIII 555 a) Isometric view (view direction: [;.6;.]) Figure 4: b) View aong x axis, the hoe axis.

In the case of compex structura probems, numerica metho, especiay those based on finite eement method, may describe much more exacty the rea phenomenon than anaytica metho.

A specia finite eement type, mainy conceived for geometrica non-inearity of beam systems, was proposed which proved to be fast convergent and exact.

9 Computationa Metho and Experimenta Measurements XIII 555 a) Isometric view (view direction: [;.6;.]) Figure 4: b) View aong x axis, the hoe axis. Post-bucking shape of a doube hinged beam; compression force 8 N. Free end Camped end View direction: [;.5;.5]. Figure 5: 7 Concusions Post bucking configuration for a compressed camped-free beam. In the case of compex structura probems, numerica metho, especiay those based on finite eement method, may describe much more exacty the rea phenomenon than anaytica metho. This is aso the case for ong and sender beam compressed in a cyinder tube, the topic of the present work. A specia finite eement type, mainy conceived for geometrica non-inearity of beam systems, was proposed which proved to be fast convergent and exact. Equiibrium equations are exact and written on the deformed configuration of the beam: this is possibe ony because the generaized noda coordinates are the eements of Euer-Rodrigues quaternion and the axia strain. In this way the boundary conditions can be described better and a significant infuence of boundary conditions was found even for quite ong beams (pipes). References [] Sorenson, K.G., Post bucking behavior of a circuar rod constrained within circuar cyinder, PhD thesis, Rice University, Houston, Texas, 984. WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

10 556 Computationa Metho and Experimenta Measurements XIII [] Chen, Yu-Che, Post bucking behavior of a circuar rod constrained within an incined hoe, PhD thesis, Rice University, Houston, Texas, 988. [] Miska, A & Cunha, J.C., An anaysis of heica Bucking of Tubuars Subjected to axia and torsiona Loading in incined webores, Society of Petroeum Engineers, SPE 946, pp. 7-8, 995. [4] Tan, X.C. & Forsman, B., Bucking of sender string in cyindrica tube under axia oad: experiments and theoretica anaysis, Experimenta Mechanics, 5(), pp. 55-6, 995. [5] Jang Wu & Juvkam-Wod, H.C., Coied tubing bucking impication in driing and competing horizonta wes, SPE Driing & Competion, March, pp. 6-, 995. [6] Miska, S., Qiu, W., Vok L. & Cuhna, J.C., An improved anaysis of axia force aong coied tubing in incined, horizonta webores, Society of Petroeum Engineers, SPE 756, pp. 7-4, 996. [7] Kuru, E., Martinez, A. & Miska, S., The bucking behavior of pipes and its infuence on the axia force transfer in directiona wes, Society of Petroeum Engineers, SPE 584, pp. -9, 999. [8] Qiu, W, Force transmission of coied tubing in horizonta wes, Society of Petroeum Engineers, SPE 54584, pp. -7, 999. [9] Mitche, R.F., Exact anaytic soutions for pipe bucking in vertica and horizonta wes, SPE Journa, December, pp. 5-5,. [] Mitche, R.F., Latera Bucking of pipes with connectors in horizonta wes, SPE Journa, June, pp. 4-5,. [] Gao, De-Li, Gao, Bao-Kui, A method for cacuating tubing behavior in HPHT wes, Journa of Petroeum Science & Engineering, 4, pp. 8-88, 4. [] Sun, C. & Lukasiewicz, S., A new method on the bucking of a rod in tubing, Journa of Petroeum Science & Engineering, 5, pp.78-8, 6. [] Martinez, A., Miska, S., Kuru, E. & Sorem, J., Experimenta Evauation of the Latera Contact Force in Horizonta Wes, Journa of Energy Resources Technoogy, (), pp. -8,. [4] Kuru, E. Martinez, A., Miska, S. & Qiu, W., The bucking behavior of pipes and its infuence on the axia force transfer in directiona wes, Journa of Energy Resources Technoogy, (), pp. 9-5,. [5] Barraco, A. & Munteanu, M.Gh., A specia finite eement for static and dynamic study of mechanica systems under arge motion, part, Revue européenne des ééments finis, Vo. (6), pp.77-79,. [6] Munteanu, M.Gh. & Barraco, A., A specia finite eement for static and dynamic study of mechanica systems under arge motion, part, Revue européenne des ééments finis, Vo. (6), pp.79-84,. [7] Barraco, A. & Munteanu, M.Gh., Dynamic study of eastic beam systems under arge motion, Proc. of European Congress on Computationa Metho in Appied Sciences and Engineering, ECCOMAS, Finand, 4-8 Juy, pubished in extensor on CD, 4. WIT Transactions on Modeing and Simuation, Vo 46, ISSN 74-55X (on-ine) 7 WIT Press

Nonlinear Analysis of Spatial Trusses

Nonlinear Analysis of Spatial Trusses Noninear Anaysis of Spatia Trusses João Barrigó October 14 Abstract The present work addresses the noninear behavior of space trusses A formuation for geometrica noninear anaysis is presented, which incudes