g. Calculation of Forces and Moments on Groups of Vertical Cylindrical 

 Piles. To find the maximum horizontal force and the moment around the mud 

 line for a group of piles supporting a structure, the approach presented in 

 Section III,l,b must be generalized. Figure 7-86 shows an example group of 

 piles subjected to wave action. The design wave concept assumes a two- 

 dimensional (long-crested) wave; hence the x-direction is chosen as the 

 direction of wave propagation. If a reference pile located at x = is 

 chosen, the x-coordinate of each pile in the group may be determined from 



x = Z cos a (7-56) 



n n n 



where the subscript n refers to a particular pile and i and a are as 

 defined in Figure 7-86. If the distance between any two adjacent piles is 

 large enough, the forces on a single pile will be unaffected by the presence 

 of the other piles. The problem is simply one of finding the maximum force on 

 a series of piles. 



In Section III,l,b, the force variation in a single vertical pile as a 

 function of time was found. If the design wave is assumed to be a wave of 

 permanent form (i.e., one that does not change form as it propagates), the 

 variation of force at a particular point with time is the same as the 

 variation of force with distance at an instant in time. By introducing the 

 phase angle 



e=2ix_2ut (^_3^) 



where L is wavelength, the formulas given in Section III,l,c (eqs. (7-25) 

 and (7-26)) for a pile located at x = may be written in general form by 

 introducing 9 , defined by 2irx/L - 2irt/T in place of -2irt/T . 



Using tables (Skjelbreia et al., 1960, and Dean, 1974), it is possible to 

 calculate the total horizontal force F(x) and moment around the mud line 

 M(x) as a function of distance from the wave crest x . By choosing the 

 location of the reference pile at a certain position x = x relative to the 

 design wave crest the total force, or moment around the mud line, is obtained 

 by summation 



N - 1 

 F , = Z f(x + X 1 (7-58) 



Total 



n = 



N - 1 



M , = Z Mfx + X 1 (7-59) 



where 



Total n = " 



N = total number of piles in the group 



X = from equation (7-56) 



X = location of reference pile relative to wave crest 



7-150 



