Tc 67/sc 4 n date: 2005-03-9 iso/wd XXXXXX ISO tc 67/sc 4/wg 6 Secretariat: Design of dynamic risers for offshore production systems Élément introductif — Élément central — Élément complémentaire Warning
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- Small versus large angle formulation
- Planar versus three-dimensional analysis
- Tension coupling
- Stroke and tensioner stiffness
- Stability
- Nominal forces and stresses
- Frequency domain analysis
- Time-domain analysis
- Integration approach
DiscretizationFinite element or finite difference techniques are typically employed to reduce the differential equilibrium equations to a set of coupled algebraic equations that can be solved numerically. The riser must be discretized carefully to avoid numerical errors from too coarse a mesh while producing a model that can be analyzed with a reasonable amount of computational effort. The level of discretization that is ultimately acceptable depends on the numerical representation of tension variation, the spatial variation in physical properties of the riser and in the magnitude of applied load, frequency content of the applied load and the accuracy of the desired results. In general, coarser meshes are acceptable for determining approximate displacement solutions to problems dominated by vessel motions, while finer meshes are essential for accurately determining stresses in the splash zone or at a stress joint for fatigue analysis. Finite element length is further controlled by the following considerations93: near a boundary, element length should not exceed:  ;
away from boundaries, element length should not exceed  ;
where w is the highest lateral frequency to be included in the analysis the ratio of lengths of successive elements should not exceed about 1:2. The "small angle" assumption is often used when formulating riser analysis methods, particularly for vertical risers. In practice, various operational constraints or stress limits for many types of risers are only met by keeping the maximum angle change along the riser below ten degrees. This just happens to be a generally recognized limit for the accuracy range of small angle beam theory. Use of small angle theory simplifies the solution through approximation of the curvature term, at the expense of limiting its applicability. However, small angle approaches will generally be sufficient for a wide variety of design applications. Note that small angle theory does not limit the approach to small displacements, as rather large displacements of a riser in moderate water depths can be achieved without exceeding an angle change of ten degrees. Note also that small angle theory is not limited to vertical risers. For example, any number of techniques (e.g., catenary equations) can be used to calculate the initial configuration of a catenary riser, whose subsequent dynamic response to environmental excitation can be calculated by small angle theory, subject to the limitation discussed above. Large rotations must be modeled, however, for certain analyses where angle changes will exceed ten degrees. This can be the case for flexible risers in extreme storms, particularly if the loading is normal to the catenary plane. For these situations, numerous approaches have been developed to accommodate large rotations.73,74 Planar versus three-dimensional analysisA common simplification for many riser analyses is the use of planar (two-dimensional) analysis, in which vessel motion, waves, current and any initial displacement of the riser are all assumed to be in the same plane. For many cases, especially for initially straight (vertical) risers, this is an adequate assumption that can significantly reduce the resources required for analysis. Planar analysis is therefore particularly useful for preliminary design work. Spread seas and non-collinear wave and current loads cannot be solved directly with two-dimensional techniques due to the coupling introduced by the drag force nonlinearity. In many cases, reasonable approximations will still permit the use of two-dimensional formulations. However, certain problems are inherently three-dimensional and therefore require a three-dimensional analysis (e.g. stresses near the seabed for a catenary riser subjected to loading normal to the plane of the riser). Tension couplingTension in the riser has a significant influence on stiffness. For some riser designs, the temporal variation in tension relative to the mean tension is naturally small and therefore will have little effect on lateral displacement. However, risers with relatively stiff tensioning systems may experience tension variations that are significant relative to the mean tension, leading to appreciable changes in lateral stiffness. Analysis of these risers must account for the nonlinear coupling between axial tension and lateral stiffness. The coupling comes from the fact that as a riser displaces laterally, it must stretch axially and/or displace vertically. The balance between axial stretch and vertical displacement is a function of the constraint typically provided by its attachment to a vessel, which is itself a function of vessel displacement. In the general case, accurate determination of tension variation requires assessment of the coupling between lateral and axial riser displacements, constraints between the riser and vessel and the associated vessel displacement. In some cases, pressure changes due to change in elevation of the free surface may also contribute to tension variation. Stroke and tensioner stiffnessCalculation of tensioner stroke is necessary if nonlinearities in the tensioner stiffness are to be accounted for in the tension calculation. Stroke calculations are also often desired for tensioner design. Stroke of direct-acting tensioners is simply the relative displacement between the tensioning ring and the vessel. Stroke of other tensioner designs depends on the particular geometry of their attachment to the tensioning ring as well. In any case, vertical riser displacement due to lateral motion and low frequency and wave frequency vessel displacements should be considered, as should nonlinear setdown of a vertically-moored vessel (TLP). In many cases where tension changes due to stroke are relatively small, modeling of the tensioner stiffness is not important. However, for detailed design of the tensioning ring or tensioning attachment hardware, forces due to stroke may be important. Also, in cases where a tensioner is relatively stiff, or in extreme cases when a tensioner is stroked out, accurate stiffness modeling of tensioners may be important. In these cases, tensioners can be modeled as inclined scalar springs with the appropriate stiffness characteristics. Accurate prediction of forces local to the tensioning ring may require considering large vessel rotations when formulating the spring (i.e., stiffness changes with instantaneous vessel position) .75 StabilityLow, or even negative, effective tension over a portion of the riser does not imply the riser is unstable, nor does it cause the riser to instantaneously experience Euler buckling. The direct consequence of low or negative effective tension is low lateral stiffness, the result of which is adequately estimated by the standard global riser analysis if changes in effective tension are accounted for.76 Nominal internal forces and stresses on the riser cross section can be calculated directly from the dynamic equilibrium equation for a riser element using the results from global analysis. Such nominal forces and stresses can then be used directly in design, where called for by Section 5, or they can be used as input to more detailed analysis of riser components as described in 6.5. Local equilibrium of a riser element leads directly to the following expression gotten by solving equation 38 for internal forces {q}:  ...(41)
Contributions to internal force by inertia loads are represented in the mass times acceleration term. Similarly damping contributions are represented in the damping term, and curvature is the stiffness term. Note also that contributions by internal and external pressure terms are also in the stiffness term via the effective tension, as is the contribution of axial tension to shear and moment. Frequency domain analysisFrequency domain analysis is appropriate when the effects of tension coupling are known to be small, and there are no other nonlinearities significantly affecting responses. Frequency domain analysis is often used for fatigue analysis with the objective of obtaining estimates of root-mean-square (rms) axial and bending stresses. Frequency domain analysis is also useful to estimate rms stresses for use in strength calculations of certain components as well as estimating clearance between risers (see 6.6). The principal advantage of frequency domain analysis is a reduction in computational effort for linear systems, coupled with very simple unambiguous output. Analysis of linear systems is well understood, and the application of frequency domain results to design criteria is straightforward. The limitation of frequency domain analysis is the difficulties and added complexities associated with modeling nonlinear behavior. Wave and current forces as modeled by Morison's equation are nonlinear in velocity. This is typically a significant force term for risers, but it can be successfully linearized as described in 6.3.436,37,38,98. With a linearized drag term, the frequency domain equilibrium equations become:  ...(42)
where [A] = -4p2 ¦2 [M] + i2p¦[C] + [K] {RL} = ½r½jei(2p¦+fj)
In addition to properly linearizing the drag force, careful selection of analysis frequencies is essential to adequately model riser response. Frequencies used in the analysis should result in adequate definition of 1) the wave energy spectrum, 2) vessel response characteristics and 3) natural frequencies of the riser. Time-domain analysisTime-domain analysis is typically used when accurate representation of nonlinear behavior is important to meeting the analysis objective. Nonlinear effects encountered for some riser analyses such as tension coupling, large rotations, nonlinear loading or foundation stiffness can be directly modeled in the time-domain.83,84 In addition, time-domain analysis is used to analyze transient events such as disconnecting with overpull on a drilling riser or loss of station keeping ability. Finally, time-domain analysis can be used to assess the relative accuracy of equivalent frequency domain analyses and calibrate them for use in design. Time-domain analysis requires defining the environment as a function of time, typically simulating wave time histories. Vessel motions may be calculated from the simulated waves and vessel frequency response functions (uncoupled vessel/riser analysis) or they may be calculated in the time-domain along with riser response (coupled vessel/riser analysis). Time-domain analysis is essentially satisfying static equilibrium, including inertial, damping and applied loads, at particular points in time. Integration approachDirect integration methods integrate the equilibrium equation in time, with the objective of satisfying dynamic equilibrium at discrete times. The general form for such methods is:  ...(43)
for explicit integration methods such as central difference or for implicit methods, such as the Newmark method.  ...(44)
where A = coefficient matrix, U = function of mass, stiffness and damping, R = function of load, mass, stiffness, damping and the solution at previous time steps. The primary difference between the two classes of integration methods is that explicit methods are derived from the equilibrium equation at time t and implicit methods at t+dt. This has implications on the numerical effort required to perform the integration. Explicit methods typically require fewer computations per time step but often require shorter time steps to achieve an accurate solution. Implicit methods often require substantial numerical effort at each time step (like decomposition of the coefficient matrix) but can often utilize larger time steps. Selecting a method is usually a tradeoff between accuracy and economy. A variety of methods are available from the literature.85.86 Stability and accuracy of the time integration should be carefully considered when setting up the time-domain analysis run. Most popular methods are conditionally stable, meaning that the time step size must be below a certain threshold for the analysis to yield meaningful results. One of the most popular methods is the Newmark Constant Average Acceleration method, which is stable for any time step (i.e., unconditionally stable). However, all methods require that the time step be small enough to accurately reflect important frequencies in the load or response. This is analogous to proper spatial discretization of the model and careful selection of frequencies in the frequency domain method. Large time steps may result in a quicker analysis that is accurate for the frequencies represented but may miss important higher frequency contributions. All methods have some degree of integration error that is associated with frequency and amplitude of the integrated response. In certain situations, slight errors in frequency alone can accumulate and lead to numerical "beating" of the response. It is important to recognize and understand these errors when performing time-domain analysis, particularly for the purpose of simulating long time histories and developing statistics for extremes. Directory: standards standards -> Request for proposal contract form for the transit industry standards -> Indiana Academic Standards Resource Guide Grade 8 United States History – Growth and Development (to 1877) Updated April 2016 standards -> Us history Dictionary Based on the sc state Standards standards -> Integrating Standards Education into the Business School Curriculum standards -> Use of Sprint U301 3G/4g mobile Broadband usb device Installation standards -> International organisation for standardisation organisation internationale de normalisation standards -> Standards of Excellence standards -> Common Core State Standards for Mathematics (ccssm) Download 2.34 Mb. Share with your friends:
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