FIGURE 10. General wedge theory applied to timbered trench.force dR acting on any elemental area on the failure surface makes an angle </> with the
normal to this surface.
The general wedge theory formulated by Terzaghi postulates that the arc of failure is a
logarithmic spiral. Let V 0 denote a reference radius vector and v denote the radius vector
to a given point on the spiral. The equation of the curve is
r = r 0 e
aian<i>
(20)where r 0 = length of V 0 , r = length of v; a = angle between V 0 and v; e = base of natural
logarithms = 2.7 18
The property of this curve that commends it for use in this analysis is that at every
point the radius vector and the normal to the curve make an angle 0 with each other.
Therefore, if the failure line is defined by Eq. 20, the action line of the resultant force dR
at any point is a radius vector or, in other words, the action line passes through the center
of the spiral. Consequently, the action line of the total resultant force R also passes
through the center.
The pressure distribution on the wall departs radically from a hydrostatic one, and the
resultant thrust P is applied at a point considerably above the lower third point of the wall.
Terzaghi recommends setting the ratio BDIAB at between 0.5 and 0.6.
Perform the following construction: Using a suitable scale, draw line AB to represent
the side of the trench, and draw a line to represent the ground surface. At middepth, draw
the action line of P at an angle of 12° with the horizontal.
On a sheet of transparent paper, draw the logarithmic spiral representing Eq. 20, set-
ting cf> = 260 SO' and assigning any convenient value to r 0. Designate the center of the spi-
ral as O.
Select a point C 1 on the ground surface, and draw a line L through C 1 at an angle $
Logarithmic
spiral