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Module 1 / Topic 2
THE “EFFECTIVE STRESS PRINCIPLE” – A THEORETICAL BUILDING
BLOCK
The ‘effective stress principle’ was enunciated by Terzaghi (1925) in his celebrated
book ‘Erdbaumechanik’ which was the first seminal publication heralding the birth of
modern Soil Mechanics.
To the extent effective stress controls the mechanics of saturated soils (soils
whose pore space is filled with water), it can be called a determinant of the engineering
behaviour of soils (Gulhati and Datta, 2005).
However, what is interesting is the fact that effective stress is not a real quantity –
in the sense of being a quantity that can be physically measured – but a rather fictitious
quantity, dwelling in the realm of concepts.
2.1 Definition
Effective stress or (pressure) p is defined as the difference between two quantities
which can be measured or determined, namely total stress (or pressure) and pore water
pressure, or simply pore pressure, which is the pressure of water existing in the pore
space.
Referring to Fig. 2.1 which depict a simple static condition, at level A-A the total
pressure due to overburden = 𝛾𝑠𝑎𝑡 x h, where 𝛾𝑠𝑎𝑡 is the saturated unit weight of the soil.
The soil being saturated, there is a continuous body of water running through the pore
space in the soil. Hence the pore water pressure at level A-A due to the head of water
above = 𝛾𝑤 x h. (h can be determined by a piezometer if it is different from the static
head.) If we call effective pressure, p, as per the above definition we can state,
p = p – u, where u is the pore pressure due to the head h.
Hence,
p = 𝛾𝑠𝑎𝑡 x h - 𝛾𝑤 x h
= (𝛾𝑠𝑎𝑡 - 𝛾𝑤 ) h
= 𝛾𝑠𝑢𝑏 x h,
(2.1)
where 𝛾𝑠𝑢𝑏 is the submerged unit weight of the soil. (Please see Sec.4.5.7 explaining
how the submerged unit weight is obtained as the difference between the saturated unit
weight of the soil and unit weight of water).
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Just as water is a continuous body running through the pore space, the soil
skeleton is also continuous thanks to the mechanical contact between the grains which
constitute the solid phase of the soil. Hence the total pressure at any depth is sustained
together by the soil grains and the pore water. Therefore the effective pressure (or
stress), in a physical sense, can be looked upon as the stress transmitted from grain to
grain at their points of contact, and in that sense it is called the ‘intergranular pressure.’
But a closer examination, which follows, will reveal that, in a real sense it is not the same
as the physical quantity described above, but only a conceptual quantity, defined as the
difference between two real quantities, viz. the total pressure and the pore water
pressure.
2.2 Examination of the nature of the effective stress
Let us assume the solid phase of the soil medium as consisting of small spherical
balls or beads of identical size place one above the other as shown in Fig.2.2. Let us
also leave some distance between the columns of spheres so that the pore space is filled
with water and forms a continuous body.
At level A-A running through the points of contact between the spherical balls, if the total
pressure is p, the total force.
P = p x A, where A is the total area over which p acts.
If the balls are perfectly rigid, the contact between the balls is a point, which theoretically
has no area (Fig. 2.2a). However, if the balls are of some softer material, a small area
can be assumed over which contact exists. (Note that if the area of contact is 0, the
contact stress would be ∞; even if one has a small positive value for the area of contact,
the contact stress would be less than ∞, but will still have a very high value.)
Let us call this small area A’.
A = A’ + Aw
(Fig.2.2b) where, Aw is the area occupied by
water.
Hence we can state,
P = P’ + u Aw ,
Dividing by A,
P
=
A
P'
A
+
where P’ is the part of P transmitted
through the solid phase.
u.Aw
A
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Since 𝐴𝑤 ≃ A, we can state,
p
=
P'
A
+u
= p + u, from which
p=p–u
(2.2)
From the above one notes that p is not P' divided by A' , but the full area A over which
p
acts. It is this fact which gives p its fictitious attribute.
Let us now look at Fig. 2.3 which depicts the actual situation in a saturated soil.
The plane C-C passing through the actual points of contact between the soil particles is
wavy, and its area is slightly higher than A which is actually its projected area on a
horizontal plane. If this area is treated as equal to A' , the same situation as in Eq. (2.2)
will repeat here also.
Thus p is not the actual ‘intergranular’ pressure in so far as it relates not to the
small area A' which is the actual area of contact, but A the total area. (It may still be called
‘intergranular’ pressure in a literary sense, but not in a quantitative sense.)
It is indeed amazing to note that a whole body of knowledge in the field of modern
geotechnical engineering has been built on such a seemingly innocuous concept!
P.S.: The above picture has a parallel in ‘permeability’ (Topic 6), where k the coefficient
of permeability is defined in relation to the total cross sectional area of the soil and not
the actual cross sectional area of the pore space through which water flows.
References
1. Gulhati, S. K. and Datta, M. (2005), Geotechnical Engineering, New Delhi, Tata
McGraw-Hill, xxviii + 738 pp.
2. Terzaghi, K. (1925), Erdbaumechanik auf bodenphysicalischer Grunlage, Franz
Deuticke, Leipzig und Wien.