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.

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^

Characterization and Modeling of Thermal Buckling in Eccentrically Loaded Microfabricated Nickel Beams for Adaptive Cooling

2005 ◽  
Author(s):  
Matthew McCarthy ◽  
Nicholas Tiliakos ◽  
Vijay Modi ◽  
Luc Freche´tte
Keyword(s):  
Closed Form ◽  
Analytical Model ◽  
Thermal Loading ◽  
Thermal Buckling ◽  
Design Parameters ◽  
Experimental Measurements ◽  
Test Results ◽  
Actuation Mechanism ◽  
Highly Nonlinear ◽  
Form Model

The design, fabrication and testing of micromachined nickel beams buckling under thermal loading will be presented in this paper. The focus will be on characterizing key design parameters important to the implementation of electroplated nickel beams as the actuation mechanism in a thermally adaptive microvalve. An analytical model of the thermal buckling phenomena has been developed and validated with test results from electroplated nickel beams with slight eccentricities. Highly nonlinear deflection versus temperature curves were predicted by the closed form model and match well with experimental measurements. Buckling deflections of more than 50μm were achieved at actuation temperatures under 100°C. The nickel beam fabrication process will be presented, as well as various fabrication related issues impacting the actuation capabilities of the beams.

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.

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.

NONLINEAR ELASTO-DYNAMIC ANALYSIS OF I-BEAMS CURVED IN-PLAN

2009 ◽  
Vol 09 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. EMRE ERKMEN ◽  
MARK A. BRADFORD
Keyword(s):  
Dynamic Equilibrium ◽  
Virtual Work ◽  
Numerical Models ◽  
Time History ◽  
Element Formulation ◽  
Nonlinear Behaviour ◽  
Equilibrium Equations ◽  
Geometric Nonlinearities ◽  
Initial Curvature ◽  
Straight Beam

The elastic response of curved beams subjected to moving vertical loads and dynamic loads is investigated. Incremental dynamic equilibrium equations are derived by using the principle of virtual work. Newmark's step-by-step procedure is adopted to discretise the dynamic equilibrium equations and obtain the time history response. Geometric nonlinearities due to large deflections and rotations are taken into account. A total Lagrangian finite element formulation is developed. The numerical models are compared with the existing analytical solutions and employed to show the effects of geometric nonlinearities as well as the initial curvature on the dynamic behaviour of curved I-beams. It is shown that the geometric nonlinearities are significant even for service loads. The nonlinear behaviour of a curved beam is substantially different from the nonlinear behaviour of a straight beam when the initial curvature is not small.

Symbolic Closed-Form Modeling and Linearization of Multibody Systems Subject to Control

1991 ◽  
Vol 113 (2) ◽  
pp. 124-132 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz-ul Haque
Keyword(s):  
Closed Form ◽  
Multibody Systems ◽  
Space Form ◽  
Virtual Work ◽  
Dynamic Equations ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Active Suspensions ◽  
Linear Dynamic ◽  
Curved Track

Symbolic closed-form equation formulation and linearization for constrained multibody systems subject to control are presented. The formulation is based on the principle of virtual work. The algorithm is recursive, automatically eliminates the constraint forces and redundant coordinates, and generates the nonlinear or linear dynamic equations in closed-form. It is derived with respect to principal body coordinates and a moving reference frame that allows one to generate the dynamic equations for multibody systems moving along curved track or road. The output equations may be either in syntactically correct FORTRAN form or in the form as derived by hand. A procedure that simplifies the trigonometric expressions, linearizes the geometric nonlinearities, and converts the linearized equations in state-space form is included. Several examples have been used to validate the procedure. Included is a simulation using a seven-DOF automobile ride model with active suspensions.

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.

scholarly journals Spectral solution to a problem on the axisymmetric nonlinear deformation of a cylindrical membrane shell due to pressure and edges convergence

2021 ◽  
Vol 5 (7 (113)) ◽  
pp. 6-13
Author(s):  
Vitalii Myntiuk
Keyword(s):  
Exponential Convergence ◽  
Virtual Work ◽  
High Accuracy ◽  
Zero Order ◽  
Principle Of Virtual Work ◽  
Equilibrium Equations ◽  
Membrane Shell ◽  
Spectral Solution ◽  
Hyper Elastic ◽  
Fabric Material

A geometrically and physically nonlinear model of a membrane cylindrical shell, which has been built and tested, describes the behavior of a airbag made of fabric material. Based on the geometrically accurate relations of "strain-displacement", it has been shown that the equilibrium equations of the shell, written in terms of Biot stresses, together with boundary conditions acquire a natural physical meaning and are the consequences of the principle of virtual work. The physical properties of the shell were described by Fung’s hyper-elastic biological material because its behavior is similar to that of textiles. For comparison, simpler hyper-elastic non-compressible Varga and Neo-Hookean materials, the zero-, first-, and second-order materials were also considered. The shell was loaded with internal pressure and convergence of edges. The approximate solution was constructed by an spectral method; the exponential convergence and high accuracy of the equilibrium equations inherent in this method have been demonstrated. Since the error does not exceed 1 % when keeping ten terms in the approximations of displacement functions, the solution can be considered almost accurate. Similar calculations were performed using a finite element method implemented in ANSYS WB in order to verify the results. Differences in determining the displacements have been shown to not exceed 0.2 %, stresses – 4 %. The study result has established that the use of Fung, Varga, Neo-Hookean materials, as well as a zero-order material, lead to similar values of displacements and stresses, from which displacements of shells from the materials of the first and second orders significantly differ. This finding makes it possible, instead of the Fung material whose setting requires a significant amount of experimental data, to use simpler ones – a zero-order material and the Varga material

On the Mechanical Behavior of the Orthotropic Cord-Rubber Composite

1976 ◽  
Vol 4 (4) ◽  
pp. 219-232 ◽  
Author(s):  
Ö. Pósfalvi
Keyword(s):  
Mechanical Behavior ◽  
Uniaxial Tension ◽  
Elastic Properties ◽  
Large Deformations ◽  
Virtual Work ◽  
Poisson's Ratio ◽  
Principle Of Virtual Work ◽  
Effective Elastic Properties ◽  
Rubber Composite ◽  
Mooney Equation

Abstract The effective elastic properties of the cord-rubber composite are deduced from the principle of virtual work. Such a composite must be compliant in the noncord directions and therefore undergo large deformations. The Rivlin-Mooney equation is used to derive the effective Poisson's ratio and Young's modulus of the composite and as a basis for their measurement in uniaxial tension.

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