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Rational Design of Retaining Walls
dy
y¢ = = tga , (1.7)
dz
where у=у(z) – the function, which describe the geometry of the back surface of the retaining wall.
In view of (1.7) we have:
2
Ø
1 ø
Π1-
2
œ
Œ
1+ y¢
tgy +
œ
Œ
1 œ
Π1+
œ
2
Œ
1+
y¢ œ 1
l = Œ
- tga
œ
;
2
1
1+
y¢
Π1-
œ
2
Œ
1+ y¢ œ
Œ1-
gy
œ
Œ
1
1+
œ
Œ
2
1 y¢ œ
º
+ ß
1
Substituting of variable.: f = . We express out y¢ :
2
1+ y¢
f 2
= 1
2
1 + y
; 1 1 2 - f 2
¢ 2 = + y¢ ; y¢ 2 = 1 1- f 2
; y¢ = tga = –
2
2
f
f
f
2
Ø 1-
f ø
Πtgy +
œ
Π1+
f
l = - tga
œ f ;
Π1-
f œ
Œ1-
gy
œ
μ
1+
f ϧ
s
From (1.5): l = g (z + z ) , where: z=z o +z 1 – current depth (Figure 1.2); finally: l = s ( z)
o 1
z g
as a known function of the depth σ=σ(z), is permissible to write the following equation:
2
Ø 1- f ø
Πtgy
+
2 œ
Π1+
f 1-
f œ s ( z)
- f = ; (1.8)
2
Π1-
f f œ z g
Œ1-
gy
œ
μ
1+ f ϧ
s ( z)
We make one more substitution of variable, F( z) =
F 2 (z) = s ( z)
z g
z g , then:
2
Ø 1-
f ø
Πtgy
+
2 œ
Π1+
f 1-
f
2
- œ f = F ( z)
. Further, let that: k
2 = 1- f
2
Π1-
f f œ
1+ f
Œ1-
gy
œ
μ
1+
f ϧ
We express f = f (k) : 1- f = (1 + f ) k 2 ; 1- f = k 2 + f k 2 ; f k 2 + f =1- k 2 ;
2
1- k
f ( k 1) 2 + = 1- k 2 ; f =
2
1+ k ,
Given that the values of γ, z o , and φ are known, and the magnitude of the intensity of normal pressure can be represented
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