DHARM556 GEOTECHNICAL ENGINEERING(σi + qult) BC—→
or r 0 (σi + qult)
Moment, M 0 , of this force about B isr 0 (σi + qult) ×r 0
2
Substituting for σi,M 0 = r^02
2(c cot φ + qult), counterclockwise ...(Eq. 14.43)The passive resistance Pp on the face BD is given byPp = σi · Nφ · BD—→
...(Eq. 14.44)where Nφ = tan^2 (45° + φ/2) =1
1+
−sin
sinφ
φ
This is because σi, due to cohesion alone is transmitted by the wedge BDE.
Its moment about B, Mr, is,Mr = Pp ·BD—→2= σiNφ ·()—
BD^2
2→
= cot φ · Nφ ·1
2r 02 eπ tan φ ...(Eq. 14.45)
For equilibrium of the plastic zone, equating M 0 and Mr, and rearranging,
qult = c cot φ (Nφ · eπ tan φ – 1) ...(Eq. 14.46)
This is Prandtl’s expression for ultimate bearing capacity of a c – φ soil.
Apparently this leads one to the conclusion that if c = 0, qult = 0. This is ridiculous since
it is well known that even cohesionless soils have bearing capcity. This anomaly arises chiefly
owing to the assumption that the soil is weightless. This was later rectified by Terzaghi and
Taylor.
For purely cohesive soils, φ = 0 and the logarithmic spiral becomes a circle and Prandtl’s
analysis for this special case leads to an indeterminate quantity. But, by applying L’ Hospital’s
rule, for taking limit one finds that
qult = (π + 2)c = 5.14c ...(Eq. 14.47)
Interestingly, this agrees reasonably with the Fellenius’ solution for this case.Terzaghi’s correction
Terzaghi proposed a correction to the bearing capacity expression of Prandtl with a view to
removing the anomaly that the bearing capacity is zero when cohesion is zero. He suggested
that the weight of the soil involved be considered by adding a factor c′ to the original quantity
c in Prandtl’s equation.
c′ = γH 1 tan φ ...(Eq. 14.48)
where H 1 = equivalent height of soil material= Area of wedges and sector¼
length CDE
and γ = unit weight of soil.
The area of wedges and sector obviously means one-half of the system; the idea is that a
soil mass of equivalent height, H 1 moves during shear and offers frictional resistance.