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1、第 1 页 共 8 页1 OverviewWe use a finite element model for a line as shown in the figure below.Figure1: Line modelThe line is divided into a series of line segments which are then modelled by straight massless model segments with a node at each end.The model segments only model the axial and torsional p
2、roperties of the line. The other properties (mass, weight, buoyancy etc.) are all lumped to the nodes, as indicated by the arrows in the figure above.Nodes and segments are numbered 1,2,3,. sequentially from End A of the line to End B. So segment n joins nodes n and (n+1).NodesEach node is effective
3、ly a short straight rod that represents the two half-segments either side of the node. The exception to this is end nodes, which have only one half-segment next to them, and so represent just one halfsegment.Each line segment is divided into two halves and the properties (mass, weight, buoyancy, dra
4、g etc.) of each halfsegment are lumped and assigned to the node at that end of the segment.Forces and moments are applied at the nodes - with the exception that weight can be applied at an offset. Where a segment pierces the sea surface, all the fluid related forces (e.g. buoyancy, added mass, drag)
5、 are calculated allowing for the varying wetted length up to the instantaneous water surface level.SegmentsEach model segment is a straight massless element that models just the axial and torsional properties of the line. A segment can be thought of as being made up of two co-axial telescoping rods
6、that are connected by axial and torsional spring+ dampers.The bending properties of the line are represented by rotational springs+ dampers at each end of the segment, between the segment and the node. The line does not have to have axial symmetry, since different bend stiffness values can be specif
7、ied for two orthogonal planes of bending.This section has given only an overview of the line model. See structural model for full details.2 Structural Model DetailsThe following figure gives greater detail of the line model, showing a single mid-line node and the segments either side of it. The figu
8、re includes the various spring+ dampers that model the structural properties of the line, and also shows the xyz-frames of reference and the angles that are used in the theory below.Figure: Detailed representation of Line modelAs shown in the diagram, there are 3 types of spring+ dampers in the mode
9、l:The axial stiffness and damping of the line are modelled by the axial spring+ damper at the centre of each segment, which applies an equal and opposite effective tension force to the nodes at each end of the segment. The bending properties are represented by rotational spring+ dampers either side
10、of the node, spanning between the nodes axial direction Nz and the segments axial direction Sz. If torsion is included (this is optional) then the lines torsional stiffness and damping are modelled by the torsional spring+ damper at the centre of each segment, which applies equal and opposite torque
11、 moments to the nodes at each end of the segment. If torsion is not included then this torsional spring+ damper is missing and the two halves of the segment are then free to twist relative to each other.3 Calculation StagesThe Program calculates the forces and moments on a mid-node in 5 stages:1. Te
12、nsion Forces.2. Bend Moments. 3. Shear Forces.4. Torsion Moments.5. Total Load.4 Tension ForcesFirstly the tensions in the segments are calculated. To do this, program calculates the distance (and its rate of change) between the nodes at the ends of the segment, and also calculates the segment axial
13、 direction Sz, which is the unit vector in the direction joining the two nodes.Linear axial stiffnessIn the case of linear axial stiffness the tension in the axial spring+damper at the centre of each segment iscalculated as follows. It is the vector in direction Sz and whose magnitude is given by:Te
14、 = EA.e + (1 -2u).(Po.Ao - Pi.Ai) + EA.e.(dL/dt)/L0whereTe = effective tensionEA = axial stiffness of line, as specified on the line types form (= effective Youngs modulus x cross-section area)e = mean axial strain = (L - L0) / (L0)L = instantaneous length of segmentl = expansion factor of segmentL0
15、 = unstretched length of segmentu = Poisson ratioPi, Po = internal pressure and external pressure respectively (see Line Pressure Effects)Ai, Ao = internal and external cross section areas respectively (see Line Pressure Effects)e = damping coefficient of the line, in seconds (this is defined below)
16、dL/dt = rate of increase of length.Note: The effective tension Te can be negative, indicating effective compression. For the relationship between effective tension and pipe wall tension see Line Pressure Effects.-This effective tension force vector is then applied (with opposite signs) to the nodes at each end of the
