CE 333
Geotechnical Engineering II
Sultan Mohammad Farooq
Sheikh Sharif Ahmed
Department of Civil Engineering
Chittagong University of Engineering & Technology
Bearing Capacity of Shallow Foundation
 The lowest part of a structure that transmits its
weight to the underlying soil or rock is the
foundation.
 Foundations can be classified into two major
categories — shallow foundations and deep
foundations.
 Individual footings (Figure 01 & 02), square or
rectangular in plan, that support columns and
strip footings that support walls and other similar
structures are generally referred to as shallow
foundations.
Bearing Capacity of Shallow Foundation
Figure 01 Individual Footing
Bearing Capacity of Shallow Foundation
Figure 02 Individual Footing
Bearing Capacity of Shallow Foundation
 When the soil located immediately below a given
structure is weak, the load of the structure may be
transmitted to a greater depth by piles and drilled
shafts, which are considered deep foundations.
Bearing Capacity of Shallow Foundation
To perform satisfactorily, shallow foundations must
have two main characteristics:
1. They have to be safe against overall shear failure
in the soil that supports them.
2. They cannot undergo excessive displacement, or
settlement. (The term excessive is relative,
because the degree of settlement allowed for a
structure depends on several considerations.)
The load per unit area of the foundation at which
shear failure in soil occurs is called the Ultimate
Bearing Capacity.
Bearing Capacity of Shallow Foundation
Three Types of Failures can occur in soil underlying a
shallow foundation1. General Shear Failure
2. Local Shear Failure
3. Punching Shear Failure
Bearing Capacity of Shallow Foundation
General Shear Failure
 Figure 03a shows a shallow foundation of width B
located at a depth of Df below the ground surface
and supported by dense sand (or stiff, clayey soil).
 If this foundation is subjected to a load Q that is
gradually increased, the load per unit area,
q = Q/A (A = area of the foundation), will increase
and the foundation will undergo increased
settlement.
 When q becomes equal to qu at foundation
settlement S = Su, the soil supporting the
foundation undergoes sudden shear failure.
Bearing Capacity of Shallow Foundation
Figure 03 General Shear Failure
Bearing Capacity of Shallow Foundation
General Shear Failure
 The failure surface in the soil is shown in Figure
03a, and the q versus S plot is shown in Figure
03b.
 This type of failure is called a General Shear
Failure, and qu is the Ultimate Bearing Capacity.
 Note that, in this type of failure, a peak value of
q = qu is clearly defined in the load-settlement
curve.
Bearing Capacity of Shallow Foundation
Figure 03 General Shear Failure
Bearing Capacity of Shallow Foundation
Local Shear Failure
 If the foundation is supported by a medium dense
sand or clayey soil of medium consistency (Figure
04a), the plot of q versus S will be as shown in
Figure 04b.
 Note that the magnitude of q increases with
settlement up to q = q′u , and this is usually
referred to as the first failure load.
 At this time, the developed failure surface in the
soil will be as shown by the solid lines in Figure
04a.
Bearing Capacity of Shallow Foundation
Figure 04 Local Shear Failure
Bearing Capacity of Shallow Foundation
Local Shear Failure
 If the load on the foundation is further increased,
the load-settlement curve becomes steeper and
more erratic with the gradual outward and upward
progress of the failure surface in the soil (shown
by the jagged line in Figure 04b) under the
foundation.
 When q becomes equal to qu (Ultimate Bearing
Capacity), the failure surface reaches the ground
surface.
Bearing Capacity of Shallow Foundation
Figure 04 Local Shear Failure
Bearing Capacity of Shallow Foundation
Local Shear Failure
 Beyond that, the plot of q versus S takes almost a
linear shape, and a peak load is never observed.
 This type of bearing capacity failure is called a
Local Shear Failure.
Bearing Capacity of Shallow Foundation
Figure 04 Local Shear Failure
Bearing Capacity of Shallow Foundation
Punching Shear Failure
 Figure 05a shows the same foundation located on
a loose sand or soft clayey soil.
 For this case, the load-settlement curve will be like
that shown in Figure 05b.
 A peak value of load per unit area q is never
observed.
Bearing Capacity of Shallow Foundation
Figure 05 Punching Shear Failure
Bearing Capacity of Shallow Foundation
Punching Shear Failure
 The ultimate bearing capacity qu is defined as the
point where ΔS/Δq becomes the largest and
remains almost constant thereafter.
 This type of failure in soil is called a Punching
Shear Failure.
 In this case the failure surface never extends up
to the ground surface.
Bearing Capacity of Shallow Foundation
Figure 05 Punching Shear Failure
Bearing Capacity of Shallow Foundation
Figure 06 Modes of failure of model footings in sand
(after Vesic, 1963)
Bearing Capacity of Shallow Foundation
 Terzaghi (1943) was the first to present a
comprehensive theory for the evaluation of the
ultimate bearing capacity of rough shallow
foundations.
 According to this theory, a foundation is shallow, if
its depth is less than or equal to its width.
 Later investigators, however, have suggested that
foundations with equal to 3 to 4 times their width
may be defined as shallow foundations.
Bearing Capacity of Shallow Foundation
 Terzaghi suggested that for a continuous, or strip
foundation (i.e., one whose width-to-length ratio
approaches zero), the failure surface in soil at
ultimate load may be assumed to be similar to
that shown in Figure 07. (Note that this is the case
of general shear failure)
Bearing Capacity of Shallow Foundation
Figure 07 General Shear Failure Surface as assumed by
Terzaghi for a Strip Footing
Bearing Capacity of Shallow Foundation
Assumptions of Terzaghi’s Bearing Capacity Theory
Terzaghi made the following assumptions for
developing an equation for determining qu for a 𝒄 −
∅ soil  The soil is semi-infinite, homogeneous and
isotropic
 The problem is two dimensional
 The base of the footing is rough
 The failure is by general shear
 The load is vertical and symmetrical
 The ground surface is horizontal
Bearing Capacity of Shallow Foundation
Assumptions of Terzaghi’s Bearing Capacity Theory
 The overburden pressure at foundation level is
equivalent to a surcharge load 𝒒 = 𝜸𝑫𝒇 where 𝜸
is the effective unit weight of soil, and 𝑫𝒇 is the
depth of foundation less than the width B of the
foundation
 The principle of superposition is valid
 Coulomb's law is strictly valid, that is, 𝝈 = 𝒄 +
𝝈 𝐭𝐚𝐧 ∅
Bearing Capacity of Shallow Foundation
Mechanism of Failure
 The shapes of the failure surfaces under ultimate
loading conditions are given in Figure 07.
 The zones of plastic equilibrium represented in
this figure by the area gedcf may be subdivided
into
1. Zone I of Elastic Equilibrium
2. Zones II of Radial Shear State
3. Zones III of Rankine Passive State
Bearing Capacity of Shallow Foundation
Figure 07 General Shear Failure Surface as assumed by
Terzaghi for a Strip Footing
Bearing Capacity of Shallow Foundation
Mechanism of Failure
 When load qu per unit area acting on the base of
the footing of width B with a rough base is
transmitted into the soil, the tendency of the soil
located within Zone I is to spread but this is
counteracted by friction and adhesion between
the soil and the base of the footing.
Bearing Capacity of Shallow Foundation
Figure 07 General Shear Failure Surface as assumed by
Terzaghi for a Strip Footing
Bearing Capacity of Shallow Foundation
Mechanism of Failure
 Due to the existence of this resistance against
lateral spreading, the soil located immediately
beneath the base remains permanently in a state
of elastic equilibrium, and the soil located within
this central Zone I behaves as if it were a part of
the footing and sinks with the footing under the
superimposed load.
Bearing Capacity of Shallow Foundation
Figure 07 General Shear Failure Surface as assumed by
Terzaghi for a Strip Footing
Bearing Capacity of Shallow Foundation
Mechanism of Failure
 The depth of this wedge shaped body of soil abc
remains practically unchanged, yet the footing
sinks.
 This process is only conceivable if the soil located
just below point c moves vertically downwards.
 This type of movement requires that the surface
of sliding cd (Figure 07) through point c should
start from a vertical tangent.
Bearing Capacity of Shallow Foundation
Mechanism of Failure
 The boundary be of the zone of radial shear bed
(Zone II) is also the surface of sliding.
 As per the theory of plasticity, the potential
surfaces of sliding in an ideal plastic material
intersect each other in every point of the zone of
plastic equilibrium at an angle (𝟗𝟎° − ∅).
 Therefore the boundary be must rise at an angle
∅ to the horizontal provided the friction and
adhesion between the soil and the base of the
footing suffice to prevent a sliding motion at the
base.
Bearing Capacity of Shallow Foundation
Figure 07 General Shear Failure Surface as assumed by
Terzaghi for a Strip Footing
Bearing Capacity of Shallow Foundation
Mechanism of Failure
 The sinking of Zone I creates two zones of plastic
equilibrium, II and III, on either side of the footing.
 Zone II is the radial shear zone whose remote
boundaries bd and af meet the horizontal surface
at angles (𝟒𝟓° − ∅/𝟐), whereas Zone III is a
passive Rankine zone.
 The boundaries de and fg of these zones are
straight lines and they meet the surface at angles
of (𝟒𝟓° − ∅/𝟐).
 The curved parts cd and cf in Zone II are parts of
logarithmic spirals whose centers are located at b
and a respectively.
Bearing Capacity of Shallow Foundation
Figure 07 General Shear Failure Surface as assumed by
Terzaghi for a Strip Footing
Bearing Capacity of Shallow Foundation
Strip or Continuous Footings
Terzaghi developed his bearing capacity equation for
strip footings by analyzing the forces acting on the
wedge abc in Figure 07. The equation for the ultimate
bearing capacity qu is-
Where,
𝟏
𝒒𝒖 = 𝒄𝑵𝒄 + 𝒒𝑵𝒒 + 𝜸𝑩𝑵𝜸
𝟐
𝑐 = 𝑐𝑜ℎ𝑒𝑠𝑖𝑜𝑛 𝑜𝑓 𝑠𝑜𝑖𝑙
𝛾 = 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑜𝑖𝑙
𝑞 = 𝛾𝐷𝑓 = 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑂𝑣𝑒𝑟𝑏𝑢𝑟𝑑𝑒𝑛 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑁𝑐 , 𝑁𝑞 , 𝑁𝛾 = 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒
𝑛𝑜𝑛𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑎𝑛𝑑 𝑎𝑟𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑜𝑛𝑙𝑦 𝑜𝑓 𝑡ℎ𝑒
𝑠𝑜𝑖𝑙 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑎𝑛𝑔𝑙𝑒 ∅
Bearing Capacity of Shallow Foundation
The bearing capacity factors 𝑵𝒄 , 𝑵𝒒 𝑎𝑛𝑑 𝑵𝜸 are
defined by𝑵𝒄 = (𝑵𝒒 −𝟏) 𝒄𝒐𝒕 ∅
𝑵𝒒 =
𝟑𝝅 ∅
𝟐 𝟒 −𝟐 𝐭𝐚𝐧 ∅
𝒆
𝟐𝒄𝒐𝒔𝟐 𝟒𝟓 +
∅
𝟐
𝟏
𝐭𝐚𝐧 ∅ 𝐭𝐚𝐧 ∅
𝑵𝜸 = 𝑲𝒑𝜸
−
𝟐
𝒄𝒐𝒔𝟐 ∅
𝟐
Where, 𝐾𝑝𝛾 = 𝑃𝑎𝑠𝑠𝑖𝑣𝑒 𝐸𝑎𝑟𝑡ℎ 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
Bearing Capacity of Shallow Foundation
∅
Nc
Nq
Ny
∅
Nc
Nq
Ny
∅
Nc
Nq
Ny
0°
5.70
1.00
0.00
17°
14.56
5.45
3.63
34°
52.64
36.50
35.23
1°
6.00
1.10
0.08
18°
15.52
6.04
4.13
35°
57.75
41.44
41.08
2°
6.30
1.22
0.18
19°
16.56
6.70
4.70
36°
63.53
47.16
48.11
3°
6.62
1.35
0.28
20°
17.69
7.44
5.34
37°
70.07
53.80
56.62
4°
6.97
1.49
0.39
21°
18.92
8.26
6.07
38°
77.50
61.55
67.00
5°
7.34
1.64
0.51
22°
20.27
9.19
6.89
39°
85.97
70.61
79.77
6°
7.73
1.81
0.65
23°
21.75
10.23
7.83
40°
95.66
81.27
95.61
7°
8.15
2.00
0.80
24°
23.36
11.40
8.90
41°
106.81
93.85
115.47
8°
8.60
2.21
0.96
25°
25.13
12.72
10.12
42°
119.67 108.75 140.65
9°
9.09
2.44
1.15
26°
27.09
14.21
11.53
43°
134.58 126.50 172.99
10°
9.60
2.69
1.35
27°
29.24
15.90
13.15
44°
151.95 147.74 215.16
11°
10.16
2.98
1.58
28°
31.61
17.81
15.03
45°
172.29 173.29 271.07
12°
10.76
3.29
1.84
29°
34.24
19.98
17.21
46°
196.22 204.19 346.67
13°
11.41
3.63
2.12
30°
37.16
22.46
19.75
47°
224.55 241.80 451.29
14°
12.11
4.02
2.43
31°
40.41
25.28
22.72
48°
258.29 287.86 600.15
15°
12.86
4.45
2.79
32°
44.04
28.52
26.21
49°
298.72 344.64 819.32
16°
13.68
4.92
3.19
33°
48.09
32.23
30.33
50°
347.51 415.15 1155.97
Bearing Capacity of Shallow Foundation
Terzaghi’s Bearing Capacity Equations for Different
Foundations
Square Foundation
𝒒𝒖 = 𝟏. 𝟑𝒄𝑵𝒄 + 𝒒𝑵𝒒 + 𝟎. 𝟒𝜸𝑩𝑵𝜸
Circular Foundation
𝒒𝒖 = 𝟏. 𝟑𝒄𝑵𝒄 + 𝒒𝑵𝒒 + 𝟎. 𝟑𝜸𝑩𝑵𝜸
Rectangular Foundation
𝒒𝒖 = 𝒄𝑵𝒄 𝟏 + 𝟎. 𝟑
𝑩
𝟏
𝑩
+ 𝒒𝑵𝒒 + 𝜸𝑩𝑵𝜸 𝟏 − 𝟎. 𝟐
𝑳
𝟐
𝑳