Geotechnical Engineering

DHARM

LATERAL EARTH PRESSURE AND STABILITY OF RETAINING WALLS 483

∴ Pa =

γbx ψ

bkx

ax

2( )

.( )sin

−

−

For the value of Pa to be a maximum, ∂

∂

P =

x

a 0,

since x is the only value which varies with the orientation of the failure plane.

∴ ∂

∂

P =− − + −

x

a ()( )()bkxa x kxax 2 = 0

(a – x) (b – kx + kx) – x(b – kx) = 0

b(a – x) = cx ...(Eq. 13.42)

Multiplying throughout by^12 sinψ,

1

2 ba sin ψ –

1

2 bx sin ψ =

1

2 cx sin ψ

or ∆ABD – ∆BCD = ∆BCG

or ∆ABC = ∆BCG ...(Eq. 13.43)

This equation signifies that for EC to be the failure plane the requirement is that the

area of the failure wedge ABC be equal to the area of the triangle BCG.

This is known as “Rebhann’s condition”, since it was demonstrated first by Rebhann is

1871.

The triangles ABC and BCG which are equal have a common base BC; hence their

altitudes on to BC should be equal;

or AJ. sin ∠AJB = CG. sin ∠BCG But ∠AJB = ∠BCG

as CG is parallel to AJ. This leads to CG = AJ = x; and

JE = a – x

Triangles DAE and DCG are similar.

Hence

()

()

.

bd

bc

xa

−

−

=

Also, triangles BCG and BJE are similar.

Consequently, d

c

.xax=−

Subtracting one from the another,

x

bd

bc

d

c

−

−

−

F

HG

I

KJ

= x

Simplifying, c^2 = bd

or c = bd ...(Eq. 13.44)
Thus if c is known, the position of G and hence that of the most dangerous rupture p
surface, BC, can be determined and the weight of the sliding wedge, W, and the active thrust,
Pa, can be calculated.