Geotechnical Engineering

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)