Geometrical and Material Non-Linear Formulation for Bend Stiffeners

2004 ◽  
Author(s):  
Murilo Augusto Vaz ◽  
Carlos Alberto Duarte de Lemos
Keyword(s):  
Differential Equations ◽  
Constitutive Relations ◽  
Mathematical Formulation ◽  
Power Series Expansion ◽  
Constitutive Models ◽  
Bending Moment ◽  
Accurate Analysis ◽  
Linear Ordinary Differential Equations ◽  
Linear Elastic ◽  
Non Linear

A mathematical formulation and a numerical solution for the geometrical and material non-linear analysis of bend stiffeners — employed to protect the upper terminations of flexible risers and subsea umbilical cables — are presented in this paper. The differential equations governing the problem result from geometrical compatibility, equilibrium of forces and moments and material constitutive relations, which can be linear elastic symmetric or non-linear elastic asymmetric. In this latter case, the bending moment versus curvature for each cross-section is calculated and then expressed by a polynomial power series expansion. Hence, a set of four first order non-linear ordinary differential equations is written and boundary conditions are defined at both ends. A one-parameter shooting method is employed and results are presented for a case study where linear elastic symmetric and non-linear elastic asymmetric constitutive models are compared and discussed. It is shown that an accurate analysis of bend stiffeners depends on a precise assessment of the material constitutive property.

A Nonlinear Analysis Formulation for Bend Stiffeners

2007 ◽  
Vol 51 (03) ◽  
pp. 250-258 ◽  
Author(s):  
M. A. Vaz ◽  
C. A. D. de Lemos ◽  
M. Caire
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Nonlinear Analysis ◽  
Cross Section ◽  
Contact Force ◽  
Oil And Gas ◽  
Constitutive Relations ◽  
Mathematical Formulation ◽  
Power Series Expansion ◽  
Gas Production

Bend stiffeners are polymeric structures with a conical shape designed to limit the curvature of flexible risers and umbilical cables at their uppermost connections, protecting them against overbending and from accumulation of fatigue damage. Thus, they are of vital importance to deep water oil and gas production systems. This work develops a mathematical formulation and a numerical solution procedure for the geometrical and material nonlinear analysis of the riser/bend stiffener system considered as a beam bending model. The structures are separately modeled, which allows the numerical calculation of the contact force along the system arc length. The governing differential equations are derived considering geometrical compatibility, equilibrium of forces and moments, and nonlinear asymmetric material constitutive relations, which leads to a shift in the neutral axis position from the cross-section centroid. The eccentricity and the bending moment versus curvature relation for each cross section are numerically calculated and then expressed by a polynomial power series expansion. A set of four first-order nonlinear ordinary differential equations is written and four boundary conditions are specified at both ends. Once the global problem is solved, the contact force may be promptly calculated. A finite difference method is implemented in Fortran code to obtain the numerical solution. A case study is carried out where linear elastic symmetric and nonlinear elastic asymmetric constitutive models are compared and discussed. The results are presented for the riser/bend stiffener deflected configuration, angle, curvature, and contact force distribution. The results demonstrate that an accurate structural analysis of bend stiffeners depends on a precise assessment of the nonlinear asymmetric polyurethane property.

scholarly journals THERMAL POST-BUCKLING OF SLENDER ELASTIC RODS WITH DIFFERENT BOUNDARY CONDITIONS

2006 ◽  
Vol 5 (2) ◽  
pp. 50
Author(s):  
R. F. Solano ◽  
M. A. Vaz
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Mathematical Formulation ◽  
Different Boundary Conditions ◽  
Linear Ordinary Differential Equations ◽  
Linear Elastic ◽  
Buckling Behavior ◽  
Non Linear ◽  
Post Buckling ◽  
Complex Boundary

This paper presents mathematical formulation, critical buckling temperature and analytical and numerical solutions for the thermal post-buckling behavior of slender rods subjected to uniform thermal load. The material is assumed to be linear elastic, homogeneous and isotropic. Furthermore, large displacements are considered hence the formulation is geometrically non-linear. Three different boundary conditions are assumed: (i) double-hinged non-movable, (ii) hinged non-movable at one end, whereas at the other end longitudinal displacement is constrained by a linear spring, and (iii) double-fixed non-movable. The governing equations are derived from geometrical compatibility, equilibrium of forces and moments, constitutive equations and strain-displacement relation, yielding a set of six first-order non-linear ordinary differential equations with boundary conditions specified at both ends, which constitutes a complex boundary value problem. The buckling and post-buckling solutions are respectively accomplished assuming infinitesimal and finite rotations. The results are presented in non-dimensional graphs for a range of temperature gradients and different values of slenderness ratios, and it is shown that this parameter governs the rod post-buckling response. The influence of the boundary conditions is evaluated through graphic results for deformed configuration, maximum deflection, maximum inclination angle and maximum curvature in the rod.

MHD STAGNATION POINT FLOW OF HYPERBOLIC TANGENT FLUID WITH VISCOUS DISSIPATION AND CHEMICAL REACTION

2021 ◽  
Vol 21 (2) ◽  
pp. 569-588
Author(s):  
KINZA ARSHAD ◽  
MUHAMMAD ASHRAF
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stagnation Point ◽  
Viscous Dissipation ◽  
Chemical Reaction ◽  
Normal Velocity ◽  
Hyperbolic Tangent ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

In the present work, two dimensional flow of a hyperbolic tangent fluid with chemical reaction and viscous dissipation near a stagnation point is discussed numerically. The analysis is performed in the presence of magnetic field. The governing partial differential equations are converted into non-linear ordinary differential equations by using appropriate transformation. The resulting higher order non-linear ordinary differential equations are discretized by finite difference method and then solved by SOR (Successive over Relaxation parameter) method. The impact of the relevant parameters is scrutinized by plotting graphs and discussed in details. The main conclusion is that the large value of magnetic field parameter and wiessenberg numbers decrease the streamwise and normal velocity while increase the temperature distribution. Also higher value of the Eckert number Ec results in increases in temperature profile.

Convective transport of thermal and solutal energy in unsteady MHD Oldroyd-B nanofluid flow

2021 ◽  
Author(s):  
Muhammad Yasir ◽  
Masood Khan ◽  
Awais Ahmed ◽  
Malik Zaka Ullah
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Ordinary Differential Equations ◽  
Heat Generation ◽  
Axisymmetric Flow ◽  
Convective Transport ◽  
Buongiorno’S Model ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

Abstract In this work, an analysis is presented for the unsteady axisymmetric flow of Oldroyd-B nanofluid generated by an impermeable stretching cylinder with heat and mass transport under the influence of heat generation/absorption, thermal radiation and first-order chemical reaction. Additionally, thermal and solutal performances of nanofluid are studied using an interpretation of the well-known Buongiorno's model, which helps us to determine the attractive characteristics of Brownian motion and thermophoretic diffusion. Firstly, the governing unsteady boundary layer equation's (PDEs) are established and then converted into highly non-linear ordinary differential equations (ODEs) by using the suitable similarity transformations. For the governing non-linear ordinary differential equations, numerical integration in domain [0, ∞) is carried out using the BVP Midrich scheme in Maple software. For the velocity, temperature and concentration distributions, reliable results are prepared for different physical flow constraints. According to the results, for increasing values of Deborah numbers, the temperature and concentration distribution are higher in terms of relaxation time while these are decline in terms of retardation time. Moreover, thermal radiation and heat generation/absorption are increased the temperature distribution and corresponding boundary layer thickness. With previously stated numerical values, the acquired solutions have an excellent accuracy.

Entropy Generation on Maxwell Fluid Flow Past an Inclined Stretching Plate with Slip and Convective Surface Conditon: Darcy-Forchheimer Model

2019 ◽  
Vol 26 ◽  
pp. 62-83
Author(s):  
Tunde Abdulkadir Yusuf ◽  
Jacob Abiodun Gbadeyan
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Entropy Generation ◽  
Mhd Flow ◽  
Maxwell Fluid ◽  
Entropy Generation Rate ◽  
Physical Parameters ◽  
Convective Boundary ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

In this study the effect of entropy generation on two dimensional magnetohydrodynamic (MHD) flow of a Maxwell fluid over an inclined stretching sheet embedded in a non-Darcian porous medium with velocity slip and convective boundary condition is investigated. Darcy-Forchheimer based model was employed to describe the flow in the porous medium. The non-linear thermal radiation is also taken into account. Similarity transformation is used to convert the non-linear partial differential equations to a system of non-linear ordinary differential equations. The resulting transformed equations are then solved using the Homotopy analysis method (HAM). Influence of various physical parameters on the dimensionless velocity profile, temperature profile and entropy generation are shown graphically and discussed in detail while the effects of these physical parameters on velocity gradient and temperature gradient are aided with the help of Table. Furthermore, comparison of some limiting cases of this model was made with existing results. The results obtained are found to be in good agreement with previously published results. Moreover, increase in local inertial coefficient parameter is found to decrease the entropy generation rate.

Radiative Unsteady Rarefied Gaseous Flow Over a Stretching Sheet with Velocity Slip and Temperature Jump Effects

2020 ◽  
Vol 87 (3-4) ◽  
pp. 261
Author(s):  
Ram Prakash Sharma ◽  
N. Indumathi ◽  
S. Saranya ◽  
B. Ganga ◽  
A. K. Abdul Hakeem
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Stretching Sheet ◽  
Velocity Slip ◽  
Skin Friction Coefficient ◽  
Gaseous Flow ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

In this study a mathematical analysis has been carried out to scrutinize the unsteady boundary layer flow of an incompressible, rarefied gaseous flow over a vertical stretching sheet with velocity slip and thermal jump boundary conditions in the presence of thermal radiation. Using boundary layer approach and suitable similarity transformations, the governing partial differential equations with the boundary conditions are reduced to a system of non-linear ordinary differential equations. The resulting non-linear ordinary differential equations are solved with the help of fourth order Runge-Kutta method with shooting technique. The results obtained for the velocity profile, temperature profile, skin friction coefficient and the reduced Nusselt number are described through graphs. It is predicted that the velocity and temperature profiles are lower for unsteady flow and has an opposite effect for steady flow.

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