BUITEMS

BUITEMS
Quality & Excellence in Education
Rational Design of Retaining Walls
2
( 1+
k 2 )-
k( 1+
k ) 1
2
(1 g y k)( k ) 1
2
2
Ø tgy
ø - k
1+ k 2 tgy - k 1-
k
tgy - k
Œ
= F ( z)
2
1
œ = F ( z)
; = F(z)
2 2
º - - ß Ł + k ł
y k 1-
k ( 1+
k ) (1 - gy
k)
tgy - k = F( z) ( 1- gy k) ; tgy - k = F(z ) - F(z) tgy k ; F(z) tgy k - k = F(z ) -tgy ;
k ( F( z ) tgy - 1 ) = F( z) - tgy ;
k F ( z)
- tgy
=
F(
z)
tgy
-1
;
2
1- f 2k
y¢ = = –
2
f 1-
k
k
y = – 2 dz
1- k
2 ;
Where:
– ;
Ø F ( z)
- tgy ø
k 2 = Œ
( ) 1
œ
º F z tgy
- ß
2
[ F(
z)
- tgy
]
[ F(
z)
tgy
-1]
2
2
;
; ( )( )
2
F(
z)
tgy
1 F(z
1- k = 1-
=
2 2
F ( z ) tgy
1
2
F 2 (z) tg y - 2F(
z)
(1 - g )( ) ( )
[ - ] 2 - [ ) - tgy ] 2
[ - ]
tgy
+ 1-
F
=
2
( F ( z ) tgy
- 1)
2
(z) + 2F(z)
( 2 ) ( 2 ) ( 2 2
( z)
-1
- F ( z)
-1
F ( z)
-1)( =
( F(
z)
tgy
-1) 2 F ( z ) tg
2
-1)
= tg y F
tg y
y -1
2( F ( z) - tgy )( F( z) tgy -1) 2
2k
=
2
1- k F z tgy F z y
Finally:
2
2
( ( ) -1)( ( ) -1)( tg -1)
2( F ( z) - tgy )( F( z) tgy -1)
( ) 2
;
=
2
tgy - tg y
=
;
( )
2k
=
;
2
2
2
1- k ( F ( z ) -1)( tg y -1)
s ( z)
Given that: F 2 ( z) = (1.11)
z g
and differentiating the left and right side of equation (1.11), we define dz:
1
2F( z ) dF(
z)
=
g
2
2F ( z ) z
dz =
g dF(z)
z
Ł
-
y = –
( z) z ( z)
ds -s
2
z
ds
( z)
s ( z) dz ł
2( F( z) - tgy
)( F(
z)
tgy
-1)
;
dz 1
=
g
dz z
Ł
2F(
z)
z
2
( F
2 ( z) -1)( tg y -1) ds
( z)
z
Ł
Consider the special case s ( z) = const =s , Then F
dz
ds
dz
z
2
g
- s
( z)
2
( z)
-s
ł
s
(z) = :
z g
( z)
ł
dF( z)
2
2F( z)
z 2 g dF(
z)
g s 2F(
z)
z g dF(
z)
g s 2 F(
z)
s
2 s
dz = -
= -
= - dF(
z)
= - dF ( z)
4
3
s g s
s g s F ( z)
g F ( z)
g
2( F( z)
-tgy
)( F(
z)
tgy
-1)
2 s
2
y = m dF( z)
2
3
F 2 . We give:tg y = m , we obtain:
z - tg y -1
F z g
( ( ) 1)( ) ( )
;
(1.12)
;
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