DHARMLATERAL EARTH PRESSURE AND STABILITY OF RETAINING WALLS 487But the ratio()
()bc
bd−
−may be transformed as follows:()
()bc
bd−
−=bbd
bddb
db db−
−= −
−=
+1
11
1/
/ /...(Eq. 13.49)From triangles ABE and ABD, by the application of the sine rule, one may obtain d/b as
follows:
d/b = d
ABAB
b. sin( )
sin
.sin( )
sin( )sin( ).sin( )
sin( ).sin( )= +−
+= +−
−+φδ
ψφβ
αβφδ φβ
αδ αβ...(Eq. 13.50)Hence, substituting this in Eq. 13.49.
()
()bc
bd−
− = –11 ++−
−+sin( ).sin( )
sin( ).sin( )φδ φβ
αδ αβ...(Eq. 13.51)Substituting this in Eq. 13.48,x =H
sin.sin( )
sin( ).
sin( ).sin( )
sin( ).sin( )αφα
αδ φδ φβ
αδ αβ+
−
+ +−
−+11...(Eq. 13.52)Since Pa =1
2γψx^2 .sin from Eq. 13.46, one obtainsPa =1
2112
22
22γαδ
αφα
αδ φδ φβ
αδ αβ.sin( ).
sin.sin ( )
sin ( ).
sin( ).sin( )
sin( ).sin( )− +
−
++−
−+L
N
M
M
M
M
MO
Q
P
P
P
P
PHor Pa =1
2
1222.. 2
sin ( )sin .sin( )sin( ).sin( )
sin( ).sin( )γ αφααδφδ φβ
αδ αβH +−++−
−+L
N
M
MO
Q
P
P
which is the same as Eq. 13.33 obtained previously from Coulomb’s theory.
Another form for Pa is as follows (Taylor, 1948):Pa =1
2γ^2 ααφ
αδ φδ φβ
αβHcos .sin( )sin( ) sin( ).sin( )
sin( )ec −−+ +−
+L
N
M
M
M
M
MO
Q
P
P
P
P
P...(Eq. 13.53)It is interesting to note that when α = 90° and δ = φ, both Eqs. 13.33 and 13.53 reduce to
the corresponding value obtained by using Eq. 13.17 of Rankine’s theory. (Also the form for Pa
expressed in Eq. 13.53 may be derived by considering the equality of the side ratios x/a and
CD/AD, and those in the similar triangles BJC and BCD, JG being parallel to CD, and substi-
tuting in Eq. 13.46 for Pa)