i S o THE IMPROVEMENT OF RIVERS. 



If it is desired to include the effect of the friction of the earth, as is customary, the 



direction of T is changed to T (the point of application remaining at -), and T equal* 







Weight of triangle AGD X 0.643. 



This is a formula given by Trautwine, and has the advantage of simplicity, 

 besides giving results as practical as those of more complicated formulas. The line DG 

 is inclined to the vertical at half the angle of repose of the backing. 



The angle KLM of inclination of T' to the back of the wall is equal to the angle 

 of wall friction, and is given for this formula as about 33 41', or i$ to i, LM being 

 perpendicular to AD. 



If the back of the wall is vertical, the triangle AGD becomes the same as the 

 triangle JGD. 



If the normal pressure T is used, we must first find the resultant of T and P, 

 which equals 7"+ P= R, and acts at a point V which may be found by equating the 

 moments of P and R about L. 



Thus 



PX UL 



PXUL=*RXVL, or VL - 



R 



Combining next W and P 1 (if the latter exists) into their resultant W P*, or Q, 

 the factor of safety becomes 



QXCN 

 RXFC 



The force Q, if P' and W do not pass through the same point, acts on that side 

 of W opposite to f at a point N, found by equating the moments of W and Q 

 about S. Thus 



W y OS 

 QXSN-WX OS, or SN-- Q 



The result may be checked graphically, as in the case of the calculation for the 

 river wall. 



If the inclined pressure T' is used instead of T, the resultant of P and T, which 

 will pass through the point of intersection of these two forces, can be found by 

 graphics. As it will usually fall within CD, it will be found most convenient to 

 combine it graphically with Q, limiting the position of the final resultant through the 

 base CD as before indicated for the river wall. 



The assumption of an inclined earth pressure, as might be expected, affords an 

 apparently greater stability of wall than the assumption of a normal one. 



