Buckling Behavior of Well Tubing: The Packer Effect

1982 ◽  
Vol 22 (05) ◽  
pp. 616-624 ◽  
Author(s):  
R.F. Mitchell
Keyword(s):  
Virtual Work ◽  
Beam Theory ◽  
Bending Moment ◽  
Radial Clearance ◽  
Bending Moments ◽  
Principle Of Virtual Work ◽  
Equilibrium Equations ◽  
Conventional Model ◽  
Shear Loads ◽  
Buckling Behavior

Abstract The equilibrium equations for a helically buckled tubing are developed and solved directly. The results show that the packer has a strong influence on the pitch of the helix, and that the pitch developed by the helix is different from the pitch calculated by conventional methods. In addition, the solution providesshear loads and bending moments at the packer andconstraining force exerted on the tubing by the exterior casing. This last result can be used to estimate friction effects on tubing buckling. Introduction The buckling behavior of well tuning and its effect on packer selection and installation have received much attention in the industry. The most well-known analysis of this problem is by Lubinski et al. Later analyses. such as by Hammerlindl, have extended and refined these results. There were two major contributions of this analysis:to clarity the roles of pressures, temperatures, fluid flow, pretension, and packer design in the buckling problem andto present a mechanical model of well buckling behavior that predicted the buckled well configuration as a function of applied loads. The principal results from this model were the motion of the tubing at the packer and the stresses developed in the tubing as a result of buckling. The major features of the conventional model of buckling behavior are summarized as follows.Slender beam theory is used to relate bending moment to curvature.The tubing is assumed to buckle into a helical shape.The principle of virtual work is used to relate applied buckling load to pitch of the helix.Friction between the buckled tubing and restraining casing is neglected. The geometry of the helix is described by three equations: (1) (2) and (3) where u1, u2, and u3 are tubing centerline locations in the x, y, and z coordinate directions, respectively; Theta is the angular coordinate (Fig. 1); r is the tubing-casing radial clearance: and P is pitch of the helix. The principle of virtual work relates P to the buckling force, F, through the following formula. (4) Several questions are not addressed by this analysis:What is the shape of the tubing from packer to fully developed helix?What are the resulting shear loads and moments at the packer caused by buckling?What are the forces exerted on the helically buckled tubing by the restraining casing? Solutions to Questions 2 and 3 would be particularly useful for evaluating friction effects on the tubing and the effect of induced loads on the packer elements. This information would allow better estimates of tubing movement and provide detailed load reactions at the packer for improved packer design. The solution to Question 1 could be particularly interesting because of its effect on results obtained by virtual work methods. SPEJ P. 616^

Geometrical Stiffness of Thin-Walled I-Beam Element Based on Rigid-Beam Assemblage Concept

2012 ◽  
Vol 28 (1) ◽  
pp. 97-106 ◽  
Author(s):  
J. D. Yau ◽  
S.-R. Kuo
Keyword(s):  
Stiffness Matrix ◽  
Virtual Work ◽  
Bending Moment ◽  
Beam Element ◽  
Narrow Beam ◽  
Cross Sectional ◽  
Geometric Stiffness ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Thin Walled

ABSTRACTUsing conventional virtual work method to derive geometric stiffness of a thin-walled beam element, researchers usually have to deal with nonlinear strains with high order terms and the induced moments caused by cross sectional stress results under rotations. To simplify the laborious procedure, this study decomposes an I-beam element into three narrow beam components in conjunction with geometrical hypothesis of rigid cross section. Then let us adopt Yanget al.'s simplified geometric stiffness matrix [kg]12×12of a rigid beam element as the basis of geometric stiffness of a narrow beam element. Finally, we can use rigid beam assemblage and stiffness transformation procedure to derivate the geometric stiffness matrix [kg]14×14of an I-beam element, in which two nodal warping deformations are included. From the derived [kg]14×14matrix, it can take into account the nature of various rotational moments, such as semi-tangential (ST) property for St. Venant torque and quasi-tangential (QT) property for both bending moment and warping torque. The applicability of the proposed [kg]14×14matrix to buckling problem and geometric nonlinear analysis of loaded I-shaped beam structures will be verified and compared with the results presented in existing literatures. Moreover, the post-buckling behavior of a centrally-load web-tapered I-beam with warping restraints will be investigated as well.

IN-PLANE NONLINEAR BEHAVIOUR OF CIRCULAR PINNED ARCHES WITH ELASTIC RESTRAINTS UNDER THERMAL LOADING

2006 ◽  
Vol 06 (02) ◽  
pp. 163-177 ◽  
Author(s):  
M. A. BRADFORD
Keyword(s):  
Closed Form ◽  
Thermal Loading ◽  
Virtual Work ◽  
Thermal Buckling ◽  
Nonlinear Behaviour ◽  
Principle Of Virtual Work ◽  
Equilibrium Equations ◽  
Structural Behaviour ◽  
Geometric Imperfection ◽  
Included Angle

This paper considers the nonlinear in-plane behaviour of a circular arch subjected to thermal loading only. The arch is pinned at its ends, with the pins being on roller supports attached to longitudinal elastic springs that model an elastic foundation, or the restraint provided by adjacent members in a structural assemblage. By using a nonlinear formulation of the strain-displacement relationship, the principle of virtual work is used to produce the differential equations of in-plane equilibrium, as well as the statical boundary conditions that govern the structural behaviour under thermal loading. These equations are solved to produce the equilibrium equations in closed form. The possibility of thermal buckling of the arch is addressed by considering an adjacent buckled equilibrium configuration, and the differential equilibrium equations for this buckled state are also derived from the principle of virtual work. It is shown that unless the arch is flat, in which case it replicates a straight column, thermal buckling of the arch in the plane of its curvature cannot occur, and the arch deflects transversely without bound in the elastic range as the temperature increases. The nonlinear behaviour of a flat arch (with a small included angle) is similar to that of a column with a small initial geometric imperfection under axial loading, while the nonlinearity and magnitude of the deflections decrease with an increase of the included angle at a given temperature. By using the closed form solutions for the problem, the influence of the stiffness of the elastic spring supports is considered, as is the attainment of temperature-dependent first yielding of a steel arch.

Inelastic instability of rectangular and nonrectangular frames

1999 ◽  
Vol 26 (3) ◽  
pp. 282-292 ◽  
Author(s):  
A R Kemp
Keyword(s):  
South African ◽  
Elastic Limit ◽  
Virtual Work ◽  
Collapse Mechanism ◽  
Principle Of Virtual Work ◽  
Equilibrium Equations ◽  
Plastic Properties ◽  
The Moment ◽  
Independent Mode ◽  
Steel Design

Proposals are described for extending the elastic amplification approach to frame instability given in the Canadian and South African codes for structural steel design to both nonrectangular frames and material inelasticity. The proposed approach to generalized nonrectangular frames is based on sway equilibrium equations which are derived by the principle of virtual work for each independent mode of sway collapse. The influence of material inelasticity is assessed by idealizing the moment-curvature relationship and the load-deflection behaviour between an effective elastic limit and the formation of the plastic collapse mechanism. A number of examples are given which demonstrate both the simplicity and accuracy of the method.Key words: codes, design, frames, inelastic properties, plastic properties, stability.

Effects of symmetric imperfections on the behavior of bistable struts

2016 ◽  
Vol 22 (12) ◽  
pp. 2240-2252 ◽  
Author(s):  
Jianguo Cai ◽  
Xiaowei Deng ◽  
Jian Feng
Keyword(s):  
Virtual Work ◽  
Variable Geometry ◽  
Equilibrium Equations ◽  
Buckling Mode ◽  
Concentrated Load ◽  
Virtual Work Principle ◽  
Shallow Arches ◽  
Buckling Behavior ◽  
Nonlinear Equilibrium ◽  
Post Buckling

The behavior of a bistable strut for variable geometry structures was investigated in this paper. A three-hinged arch subjected to a central concentrated load was used to study the effect of symmetric imperfections on the behavior of the bistable strut. Based on a nonlinear strain–displacement relationship, the virtual work principle was adopted to establish both the pre-buckling and buckling nonlinear equilibrium equations for the symmetric snap-through buckling mode. Then the critical load for symmetric snap-through buckling was obtained. The results show that the axial force is in compression before the arch is buckled, but it becomes in tension after buckling. Thus, the previous formulas cannot be used for the analysis of post-buckling behavior of three-hinged shallow arches. Then, the principle of virtual work was also used to establish the post-buckling equilibrium equations of the arch in the horizontal and vertical directions as well as the static boundary conditions, which are very important for bistable struts.

scholarly journals Analytical solution for the resistance of composite beams subjected to bending

2020 ◽  
Vol 323 ◽  
pp. 01013
Author(s):  
Marek Lechman
Keyword(s):  
Analytical Solution ◽  
Bending Moment ◽  
Composite Beams ◽  
Bending Moments ◽  
Equilibrium Equations ◽  
Good Convergence ◽  
Linear Elastic ◽  
Eurocode 2 ◽  
Strain Relationship

The paper deals with the resistance of steel and concrete composite beams, named BH beams, subjected to bending. They are structurally connected with prefabricated or cast in situ slabs, forming floor slab system. The beams under consideration consist of the reinforced concrete (RC) rectangular core placed inside a reversed TT welded profile. The stress-strain relationship for concrete in compression of the RC core is assumed for nonlinear analysis according to Eurocode 2. For reinforcing and profile steels linear elastic – ideal plastic model is applied. The normalized ultimate bending moment determining the resistance of the BH beam is derived by integrating the equilibrium equations of the bending moments about the horizontal axis of the RC core rectangle, taking into account the physical and geometrical relationships. The presented model was verified by tests carried out on two BH beams subjected to bending. The comparisons made indicated good convergence between the analytical solution and the experimental results in ultimate bending moments.

The Statics Analysis and Verification of 3-DOF Parallel Mechanism Based on Two Methods

2013 ◽  
Vol 7 (2) ◽  
pp. 237-244 ◽  
Author(s):  
Guangda Lu ◽  
◽  
Aimei Zhang ◽  
Jing Zhou ◽  
Shigang Cui ◽  
...  
Keyword(s):  
Force Feedback ◽  
Jacobian Matrix ◽  
Virtual Work ◽  
Rehabilitation Robot ◽  
Principle Of Virtual Work ◽  
Equilibrium Equations ◽  
Vector Method ◽  
Ankle Rehabilitation ◽  
Moving Platform ◽  
Mathematical Relationships

Statics of the 3-RSS/S parallel ankle-rehabilitation robot is analyzed in this paper using two methods, i.e. the component vector method and the principle of virtual work. Static equilibrium equations based on component vector theory were established on a moving platform, and cranks of 3-RSS/S parallel Ankle-rehabilitation Robot, using this method, to obtain mathematical relationships between the external torque of moving platform and the output torque of three cranks. The velocity Jacobian matrix of the robot is calculated firstly using the principle of virtual work method, then the force Jacobian matrix is obtained based on the relationship between velocity Jacobian matrix and force Jacobian matrix. The results of the two methods are verified and found to be consistent by calculation, and the force Jacobian matrix of the robot is the basis of the force feedback control for the Ankle-rehabilitation Robot.

On a One-Dimensional Theory of Finite Torsion and Flexure of Anisotropic Elastic Plates

1981 ◽  
Vol 48 (3) ◽  
pp. 601-605 ◽  
Author(s):  
E. Reissner
Keyword(s):  
Linear Theory ◽  
Beam Theory ◽  
Bending Moment ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Dimensional Theory ◽  
Cross Sectional ◽  
One Dimensional ◽  
Slender Beams ◽  
Anisotropic Elastic

Equations for small finite displacements of shear-deformable plates are used to derive a one-dimensional theory of finite deformations of straight slender beams with one cross-sectional axis of symmetry. The equations of this beam theory are compared with the corresponding case of Kirchhoff’s equations, and with a generalization of Kirchhoff’s equations which accounts for the deformational effects of cross-sectional forces. Results of principal interest are: 1. The equilibrium equations are seven rather than six, in such a way as to account for cross-sectional warping. 2. In addition to the usual six force and moment components of beam theory, there are two further stress measures, (i) a differential plate bending moment, as in the corresponding linear theory, and (ii) a differential sheet bending moment which does not occur in linear theory. The general results are illustrated by the two specific problems of finite torsion of orthotropic beams, and of the buckling of an axially loaded cantilever, as a problem of bending-twisting instability caused by material anisotropy.

ON THE INFLUENCE OF MATERIAL COUPLINGS ON THE LINEAR AND BUCKLING BEHAVIOR OF I-SECTION COMPOSITE COLUMNS

2007 ◽  
Vol 07 (02) ◽  
pp. 243-272 ◽  
Author(s):  
N. FREITAS SILVA ◽  
N. SILVESTRE
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Plane Deformation ◽  
Beam Theory ◽  
Laminated Plates ◽  
Element Formulation ◽  
Equilibrium Equations ◽  
Composite Columns ◽  
Buckling Behavior ◽  
Section Analysis

This paper presents the incorporation of shear deformation effects into a Generalized Beam Theory (GBT) developed to analyze the structural behavior of composite thin-walled columns made of laminated plates and displaying arbitrary orthotropy. Unlike other existing beam theories, the present GBT formulation incorporates in a unified fashion (i) elastic coupling effects, (ii) warping effects, (iii) cross-section in-plane deformation and (iv) shear deformation. The main concepts and procedures involved in the available GBT are adapted/modified to account for the specific aspects related to the member shear deformation. In particular, the GBT fundamental equilibrium equations are presented and their terms are physically interpreted. An I-section is used to illustrate the performance of GBT cross-section analysis and the mechanical properties are explained in detail. With the purpose of solving the GBT system of differential equilibrium equations, a finite element formulation is briefly presented. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, the linear (first-order) and stability behavior of three composite I-section members displaying non-aligned orthotropy are analyzed and the results obtained are thoroughly discussed and compared with estimates available in the literature.

Measurement of Residual Stress-Induced Bending Moment of P+ Silicon Films

1991 ◽  
Vol 239 ◽  
Author(s):  
W. H. Chu ◽  
M. Mehregany ◽  
X. Ning ◽  
P. Pirouz
Keyword(s):  
Residual Stress ◽  
Stress Distribution ◽  
Beam Theory ◽  
Bending Moment ◽  
Surface Region ◽  
Residual Stress Distribution ◽  
Silicon Films ◽  
Bending Moments ◽  
Free Standing ◽  
Near Surface

AbstractThis paper presents results from measurements of residual stress-induced bending moment of heavily-boron-doped (p+) silicon films. Microfabricated free-standing cantilever beams of p+ silicon were fabricated by using anisotropie etching of (100) silicon wafers in ethylene-diamine and pyrocatechol. The p+ etch stops forming the cantilevers were created by diffusion from a solid source at 1125°C for one and two hour time durations. The nonuniform residual stress distribution through the thickness of the p+ silicon cantilevers resulted in a deflection of the beams. The as-diffused p+ silicon films had a residual stress distribution through the film thickness which resulted in negative bending moments. Thermal oxidation subsequent to the diffusion step modified the residual stresses near the top surface or, perhaps, plastically deformed the near surface region of the p+ thin film. As a result, thermally oxidized p+ silicon films exhibited a positive bending moment. Measurements of the deflection curves of the beams in conjunction with beam theory were used to calculate the residual stress-induced bending moments.

A Finger Model with Constant Tendon Moment Arms

1993 ◽  
Vol 37 (10) ◽  
pp. 710-714 ◽  
Author(s):  
Koo-Hyoung Lee ◽  
Karl H.E. Kroemer
Keyword(s):  
Sagittal Plane ◽  
Virtual Work ◽  
Static Equilibrium ◽  
Principle Of Virtual Work ◽  
Finger Movements ◽  
Equilibrium Equations ◽  
Finger Joints ◽  
Moment Arms ◽  
Finger Model

A kinematic finger model was developed with the assumption that the tendon moment arms at the finger joints were constant, and that the finger moved in the sagittal plane. Equations of static equilibrium for the model, derived using the principle of virtual work, were indeterminate. The number of variables was reduced based on the muscular activities in finger movements. The finger strengths were computed from the equilibrium equations, and mathematically expressed as functions of finger positions, tendon moment arms, and lengths of phalanges. Experiments were performed to measure finger strengths, and the measured finger strengths were compared to the computed results.

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