B.2 The Onset of Catenary Effects. An expression for the "critical" tension corresponding to the 

 onset of catenary behavior can be obtained from equation (B4). The result is 



He, 



mg 



^ rrit 



2/3 



EA 



1/3 



/. 



(B12) 



By requiring the cable frequency to be within 5 percent of the taut string, an approximation consistent 

 with the accuracy of the string equation, one obtains 



>.cr„=1.26. (B13) 



Since d~ s for typical cables when X^ is small, equation (B5) becomes 



L,= / (l + 8(s//)2)< 9/8/. (B14) 



The onset of slack effect occurs near the limit of s// — 0, so that a slightly conservative estimate is 



established by putting Z-^ = / to obtain 



Hcn, = Qm{W^EAyi\ (B15) 



where W = mgl is the total weight of the cable to the accuracy of the linear theory. The corresponding 



critical sag is 



W 



EA 



1/3 



/. 



(B16) 



V, = 0.134 



For the Double Armor Steel (DAS) cable, equation (815) yields 



//,„, = 200 - 300 lbs (890 - 1340 N) in air 

 and 



//„„ = 167 - 254 lbs (743 - 1130 N) in water. 

 When they are compared to the data in Figs. B4 and B5, these tensions correspond well to the regions 



where the cables begin to deviate from a taut string behavior, i.e. where /— (D^^^. It should be 



emphasized that this criterion applies only to the initiation of catenary effects in only the symmetric 



modes, since the antisymmetric modes are unaffected for si I < 1:8 and H > W. Furthermore, at // = 



//„,, the only affected mode will be the « = 1 mode. If one is interested in the onset of slack effects at 



the higher symmetric modes, then the expression becomes 



Hr, 



W^EA 



ni 



1/3 



(31 7a) 



