TERZAGHI’S BEARING CAPACITY THEORY DERIVATION OF EQUATION TERZAGHI’S BEARING CAPACITY THEORY TERZAGHI’S BEARING CAPACITY FACTORS Download vedio link https://youtu.be/imy61hU0_yo

1. TERZAGHI’S BEARING CAPACITY
THEORY
PREPARED
BY
ER. SANJEEV SINGH
SNJV432@GMAIL.COM

2. CONTENT
• TERZAGHI’S BEARING CAPACITY THEORY
• DERIVATION OF EQUATION
• TERZAGHI’S BEARING CAPACITY FACTORS

3. TERZAGHI’S BEARING CAPACITY
THEORY
• Terzaghi 1943 gave a general theory for the bearing capacity the soil under a
strip footing, making the following assumptions.
1. The base of footing is rough.
2. The footing is laid at a shallow depth i.e 𝐷𝑓≤B.
3. The shear strength of the soil above of the footing is neglected. The soil
above the base is replaced by a uniform surcharge γ𝐷𝑓.
4. The load on the footing is vertical and is uniformly distributed.
5. The shear strength of the soil is governed by Mohr Coulomb equation.

4. DERIVATION OF EQUATION
• As the base of footing is rough, the soil in the
wedge ABC immediately beneath the footing is
prevented from undergoing any lateral yield.
• The soil in this wedge zone I remains in a state
of elastic equilibrium. It behaves as if it were a
part of the footing itself.
• It is assumed that the angle CAB and CBA are
equals to the angle of shearing resistance ϕ’ of
the soil.
• The sloping edges AC and BC of the soil wedge
CBA bear against the radial shear zone CBD
and CAF zone II. The curves CD and CF are
arcs of a logarithmic spiral.

5. DERIVATION OF EQUATION
• Two triangular zones BDE and AFG are the
Rankine passive zones zone III. An
overburden pressure q=γ 𝐷𝑓 acts as a
surcharge on the Rankine passive zones.
• The shearing resistance of the soil located
above the base of the footing is neglected,
and the effect of soil is taken equivalent to a
surcharge. Because of this assumption.
Terzaghi’s theory is vaild only for shallow
foundation 𝐷𝑓≤B, in which the term γ𝐷𝑓 is
relatively small.
• The loading condition is similar to that on a
retaining wall under passive case.

6. DERIVATION OF EQUATION
• The failure occurs when the downward
pressure exerted by loads on the soil adjoining
the inclined surface CB and CA of the soil
wedge is equal to the upward pressure.
• The downward forces are due to in the load
( 𝑞 𝑢 * B) and the weight of the wedge (1/4
γ𝐵2
tanϕ′).
• The upward forces are the vertical components
of the resultant passive pressure (𝑝 𝑝) and the
cohesion (c’) acting along the inclined surface.
• As the resultant passive pressure is inclined at
an angle ϕ′ to the normal surface of the wedge,
it is vertical.

7. DERIVATION OF EQUATION
• Therefore , from the equilibrium equation in the
vertical direction.
1
4
γ𝐵2tanϕ’ + 𝑞 𝑢 ∗ B = 2𝑝 𝑝 + 2c’ * 𝐿𝑖 sinϕ’
Where 𝐿𝑖 = length of the inclined surface
CB[=(B/2)/cosϕ’]
Therefore 𝑞 𝑢 ∗ B = 2𝑝 𝑝 + Bc′tanϕ’ -
1
4
γ𝐵2tanϕ’
………(a)
• The resultant 𝑝 𝑝 on the surface CB & CA
constituents the following 3 components,
1. (𝑝 𝑝)γ Weight of soil in shear zone II & III ,
assuming the soil as cohesionless (c’=0) and
neglecting the surcharge q.

8. DERIVATION OF EQUATION
2. (𝑝 𝑝) 𝑐 due to cohesion (c’) soil, assuming the soil as
weightless γ=0 and neglecting the surcharge q.
3. (𝑝 𝑝) 𝑞 due to surcharge, assuming the soil as
cohesionless and weightless (c’=0 ;γ=0).
Thus the resultant passive pressure 𝑝 𝑝 is taken equal
to the sum of the components (𝑝 𝑝)γ, (𝑝 𝑝) 𝑐, (𝑝 𝑝) 𝑞.
From equation (a)
𝑞 𝑢B =2[(𝑝 𝑝)γ+(𝑝 𝑝) 𝑐 + (𝑝 𝑝) 𝑞]+Bc′tanϕ’-
1
4
γ𝐵2tanϕ’

9. DERIVATION OF EQUATION
substituting
2(𝑝 𝑝)γ-
1
4
γ𝐵2tanϕ’ = B *
1
2
Bγ𝑁γ effects of soil weight in shear zone.
And
2(𝑝 𝑝) 𝑐 + Bc′ tanϕ’ = B * c’ 𝑁𝐶 effects of cohesion.
And
2(𝑝 𝑝) 𝑞 = B * γ𝐷𝑓 𝑁𝑞 effects of surcharge.

10. DERIVATION OF EQUATION
• We get
𝑞 𝑢B = B * c’ 𝑁𝐶 + B * γ𝐷𝑓 𝑁𝑞 + B *
1
2
γB𝑁γ
𝑞 𝑢 = c’ 𝑁𝐶+ γ𝐷𝑓 𝑁𝑞+
1
2
γB𝑁γ……..(*)
𝑞 𝑢 = c’ 𝑁𝐶+ 𝑞 𝑜 𝑁𝑞+ 0.5γB𝑁γ
Where 𝑞 𝑜 is the overburden pressure = γ𝐷𝑓
Equation * is known as Terzaghi’s bearing capacity equation.
The bearing capacity factors Nc, Nq, Nγ are dimensionless numbers,
depending upon the angle of shearing resistance (ϕ’).

11. DERIVATION OF EQUATION
• The bearing capacity factors Nc, Nq, Nγ are dimensionless numbers,
depending upon the angle of shearing resistance (ϕ’).
These are defined by following equations:
𝑁c=cot ϕ’
𝑎
2
2𝑐𝑜𝑠
2
(45
0
+
ϕ
′
2
)
− 1 ………(b)
Nq=
𝑎
2
2𝑐𝑜𝑠
2
(45
0
+
ϕ
′
2
)
………(c)
Where a = 𝑒
3ᴫ
4
−
ϕ
′
2
𝑡𝑎𝑛ϕ′

12. DERIVATION OF EQUATION
And
Nγ=
1
2
𝐾 𝑝
𝑐𝑜𝑠2ϕ
′ − 1 𝑡𝑎𝑛ϕ′………(d)
𝑤ℎ𝑒𝑟𝑒 𝐾 𝑝= coefficient of passive earth pressure
Fig gives the values of bearing capacity factor.
Equation * gives the ultimate bearing capacity of a strip footing.

13. Terzaghi’s Bearing capacity Factors