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Rational Design of Retaining Walls
y
x
δ
Δ
Δ
H
В
Fig.2.2. the current cross section of wall.
2.1):
table:
2.3. Internal Forces
Since, considered the plane bending, there are bending moment and shear force at sections of element (Fig.
q L 2
2
M (x ) =
6
Q(x ) = q 2L
2
g =
q 1
q
2
, x
x 2 [ 3 g (1 - g ) x]
+ , (2.5)
x[ 2 g + (1- g ) x ],
x
= , γ, x ˛[0;1], L – height of wall;
L
M(ξ); Q(ξ) – bending moment and shear force.
2.4. Terms of Rationalization.
The height of section of wall H(ξ) will be searched from condition [3]:
1-n
1+
n
s + s
2 + mt
2 = R +
red
, (2.7)
2 2
where the parameters ν and m correspond to different criteria, and limit states are defined by the following
Table 2.1
№ criterion ν m
(2.6)
1 Galileo-Rankine 0 4
2 Saint-Venant m * 4
3 Coulomb 1 4
R bt
4 Mohr D = Δ 4
R
b
5 Mises-Genk 1 3
*) m – Poisson's ratio
Сonsidering (2.2) (2.3) (2.4) (2.5) (2.6), normal and shear stresses are represented by the following
dependencies:
6M(
x)
s =
j 2
,
B H ( x )
(2.8)
3Q(
x ) h
t = , and B(x) = w H(x) = const .
2a
B j H(
x )
(2.9)
In addition to five criteria, presented in Table 2.1, criterion introduced in [2] is considered:
e(x) = , (2.10)
e u
where е(ξ) –- the yield value of the potential strain energy density per unit length;
[
cu tu
] 2 )eshu
e u
= 0,5c 2 e
( c e
+ 1)e -( c e
- 1)e + (1 - c , (2.11)
е и – the ultimate value of the potential strai energy density per unit length,
c 2e -e -e
e
e - e
2 1 3
= - Lode-Nadai parameter,
1
3
e
110