DHARM
486 GEOTECHNICAL ENGINEERING
= BE. BD
or c^2 = b.d, with the notation of Fig. 13.27 (a).
This is the Poncelet rule, which implies Rebhann’s condition automatically; hence the
validity of the construction.
The validity of the graphical method may be established in a different manner by deriv-
ing Eq. 13.33 for Pa from the geometry of Fig. 13.28 coupled with the notation of Fig. 13.27.
However, with a view to avoiding confusion, the data required for this are given separately in
Fig. 13.29.
f q
A
(+)ab
(+ )fd
H J
a
f
d c b
B
E
L
M
G
(–)fb
D
Ground
line
b
x
y
n
x yad=( – )
F
C
Fig. 13.29 Key figure for establishing the validity of Poncelet graphical method
x
c
=
sin( )
sin( )
θφ
θφψ
−
−+
...(Eq. 13.47)
by applying sine rule in the triangle BCG,
where c = bd (the geometric mean).
For similar triangles GCD and EAD,
x/a =
bc
bd
−
−
From the triangle ABE,
AE
AB =
sin( )
sin
φα
ψ
+
∴ a = AE = AB.
sin( )
sin sin
.
sin( )
sin
φα
ψα
φα
ψ
+
=
H +
∴ x =
ab c
bd
Hbc
bd
()
()sin
.
sin( )
sin
.
()
()
−
−
=
+−
α −
φα
ψ ...(Eq. 13.48)