analysis will yield the lift force record (both magnitudes and time 

 history) of each separate component pipe section, which may then be 

 integrated in the appropriate manner to determine the maximum wave- 

 induced stresses exerted on the pipeline at any critical section. 



This is important because the maximum lift forces may act upward 

 on a bottom-supported section of a pipeline, while acting downward on 

 the adjacent sections of the pipeline spanning the bottom at a small 

 clearance. Maximum values of both the positive and negative lift 

 forces acting in opposite directions could easily occur at the same 

 point in the wave cycle (under the crests and troughs) , thus exerting 

 stresses on the pipeline twice as high as would be calculated consid- 

 ering any pipe section alone, or in using some average clearance for 

 a long section of the pipeline. 



4. Extension of Model to Higher Order Theories . 



The lift force model (eq. 4) is based on linear theory, assuming 

 the lift force phenomenon is identical as either the wave crest or 

 trough passes over the pipeline. Such a symmetrical expression is not 

 flexible enough to consider slightly different kinematics under the 

 wave crests and troughs, which are expressed in higher order theories. 

 These different kinematics would, in reality, produce slightly different 

 lift forces under the crests and troughs of nonlinear waves. 



The lift force model described above was derived as a modification 

 of the traditional lift force equation using linear wave theory to 

 express the horizontal water particle velocities. Using linear wave 

 theory, the traditional lift force equation can be expressed as: 



Fl = 1/2 Cl P Au^^/ cos^ (9). (5) 



This equation was modified to make it a suitable expression for wave- 

 induced lift forces by adding the phase shift parameter, 4>, to account 

 for maximum lift forces occurring in places other than the cresl^ and- 

 trough in the wave cycle, and by adding the parameter, k, to account 

 for positive lift forces during part of the wave cycle and negative 

 forces during the rest of the cycle. This modified equation fits the 

 experimental data very well for all conditions tested in this investi- 

 gation. 



The model was developed after thorough inspection of the experimental 

 data. For a given pipe diameter and wave condition, the force record 

 followed a sinusoidal relationship of twice the frequency of the waves. 

 As the clearance increased, the maximum positive forces gradually dimin- 

 ished while continuously shifting to a- maximum of 90° from the wave crest 

 as the forces went to zero (Fig. 4). At the same time, the maximum 

 negative forces slowly grew from a minimum value of zero at a position 



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