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<br />rearranging and substituting terms gives the final form of the factor of safety equations: <br /> <br />cose(~) <br />SF= l1 <br />T]'( ~~ ) +sine cos ~ <br /> <br />(Factor of Safety) <br /> <br />A t -1 cost.. <br /> <br /> <br />I-' = an [ M ] <br />-+1 <br />1'L-~ sine +sint.. <br />T] l2 <br /> <br />The stability number, 11, is defined as; <br /> <br />'t <br />T] =......2.. <br />'to <br /> <br />(Stability Number) <br /> <br />where: <br /> <br />'to = the shear stress or tractive force acting on the channel boundaries and can be <br />computed from design hydraulic conditions. <br /> <br />'to = the critical shear stress when "failure" occurs. <br /> <br />1 M + sin(t.. + ~ )) <br />1]'= N T] <br />M <br />-+1 <br />N <br /> <br />(Stability Number on a side slope) <br /> <br />where <br /> <br />M = l4FL <br />N laFo <br /> <br />The above equations can be solved by knowing 'to and 'to and the angles e and A, and <br />assuming the ratios l 11 l 2 ,l a/l4 and FL/Fo. <br /> <br />Incipient motion analysis identifies 'to as the loading which causes a single particle to begin <br />to move. Critical shear stress for sediments can be estimated based on particle size <br />diameter from relationships such as the Shields equation. Extensive research has been <br />conducted for incipient motion analysis of sediments and larger sized rocks. However, there <br />are limited test data on the performance of proprietary products such as ACB's. Therefore, <br />hydraulic testing of ACB's must be conducted before a complete design procedure can be <br /> <br />4.7 <br />