DHARM274 GEOTECHNICAL ENGINEERINGAgain from ∆DCGsin φ = DC/GC = DC/(GM + MC) =()/
cot ( ) /σσ
φσ σ13
133
2−
c ++=()
cot ( )σσ
φσ σ13(^213)
−
c ++
∴ (σ 1 – σ 3 ) = 2c cos φ + (σ 1 + σ 3 ) sin φ ...(Eq. 8.35)
or σ 1 (1 – sin φ) = σ 3 (1 + sin φ) + 2c. cos φ
∴σ 1 =
σφ
φ
φ
φ
31
1
2
1
(sin)
(sin)
cos
(sin)
- −
−
c
or σ 1 = σ 3 tan^2 (45° + φ/2) + 2c tan(45° + φ/2) ...(Eq. 8.36)
or σ 1 = σ 3 tan^2 α + 2c tan α ...(Eq. 8.37)
This is also written as
σ 1 = σ 3 Nφ + 2c Nφ ...(Eq. 8.38)
where, Nφ = tan^2 α = tan^2 (45° + φ/2) ...(Eq. 8.39)
Equation 8.36 or Eqs. 8.38 and 8.39 define the relationship between the principal stresses
at failure. This state of stress is defined as ‘Plastic equilibrium condition’, when failure is
imminent.
From one test, a set of σ 1 and σ 3 is known; however, it can be seen from Eq. 8.36, that at
least two such sets are necessary to evaluate the parameters c and φ. Conventionally, three or
more such sets are used from a corresponding number of tests.
Strength envelope (best common tangents)
sc1 sc2 sc3 s
f
c
t
Fig. 8.14 Mohr’s circles for triaxial tests with different
cell pressures and strength envelope
The usual procedure is to plot the Mohr’s circles for a number of tests and take the best
common tangent to the circles as the strength envelope. A small curvature occurs in the strength
envelope of most soils, but since this effect is slight, the envelope for all practical purposes,
may be taken as a straight line. The intercept of the strength envelope on the τ-axis gives the
cohesion and the angle of slope of this line with σ-axis gives the angle of internal friction, as
shown in Fig. 8.14.
Lambe and Whitman (1969) advocate a modified procedure to obtain the failure enve-
lope, as a function of (σ 1 + σ 3 )/2 and (σ 1 – σ 3 )/2.
Equation 8.35 may be rewritten as follows :
(σ 1 – σ 3 )/2 = d +
()σσ 13
2
. tan ψ ...(Eq. 8.40)