Environmental Engineering Reference
In-Depth Information
Table 1.
Dimensional measuring units.
Y
Parameter
Unit
α
X
kn/m³
Densities: γ
kn/m²
x
Pressures: β
m
lengths: l = β/γ
Force: l β = β
2
/γ
kn/m
α
Moments: l
2
β = β
3
/γ
2
kn
y
dn
dX
YAA
dY
dX
)
′
=+
(
=+
σσατ α
sin
−
cos
(7)
0
1
Figure 4. oblique axes (X, Y).
µσ τ
α
+
′
=
(
)
XY
+
[
σµ α
tan
+
1
cos
(8)
dY
dX
′
=+
(
)
+
τ
tan
αµ
−
]
YBB
0
1
where u is a parameter that can be the stress, σ, or
any other.
equation (11), taking into account equations
(10) and (12), adopts the form:
dm
dX
Yn C
dY
dX
(
)
′
=+
1
sin
αµ α
−
cos
=+
1
n
(9)
1
H
where:
dY
dX
∫
(13)
nFu
=
,
dX
0
=
+
cos
A
1
= σ sinα - τ cosα;
C
1
= sinα - µcosα
µσ τ
α
0
AB
0
= ;
σ
Thus, equation (8) leads to:
(10)
uUX
dY
dX
(14)
=
,
B
1
= σ(1+µ tanα) + τ(tanα - µ)
Taking parameter u from equation (14) to
equation (13) it is finally obtained:
3.2
Basic results
The final aim is to determine n (horizontal com-
ponent of the force on the wall) and m (tilting
moment acting on the foot on the wall) for X = h,
where h is the height of the wall.
H
nFX
dY
dX
∫
(15)
=
,
dX
0
in other words, n is a functional of the failure
line Y = Y(X).
in this integral the function Y(X) must be
sought, which will make it extremal, in other words
the maximum or minimum of the functional must
be found depending on whether active or passive
pressure is sought. This can be done by applying
euler´s variational method.
in this case the euler condition is simply:
H
(
)
∫
0
(11)
nAAY dX
=
+
′
1
0
The Y(X) function must be found, which defines
the wedge surface making the integral maximum
with the condition of the link XY′ = B
0
+B
1
Y′.
having obtained Y′, Y(X) and n determines m.
4
FailURe line
y
′
=
∂
F
Y
F
∂ ′
=
k
if the failure criterion is expressed parametrically
it gives:
being k a constant.
Developing the equations it is finally obtained a
second degree equation in Y′ (equation 16).
τ = τ(
u
); σ = σ(
u
)
(12)
