The bearing capacity equations developed in literature considers homogenous soil below the base of the footing. But in actual practice soil mass is non homogenous & anisotropic. Therefore, while deducing the expression of the bearing capacity in case of circular footing resting over layered deposits, one has to take into account for a layered profile of soil. The paper presents the theoretical equation for the bearing capacity of a circular footing resting on layered soil profile using punching shear failure mechanism following projected area approach. The punching mechanism has been adopted while at “ultimate load” the mechanism of punching shear failure developed in dense sand has a parabolic shape when full mobilization of shear force into failure surface is taken into consideration otherwise punching failure is the actual failure while punching in the lower layer continues to a larger extent depending upon the loading at interface. For the analysis part frustum is considered to be a linearize curve for the actual shape of failure and a bearing capacity expression is deduced adopting certain assumptions. Stresses acting on the frustum have been analyzed and after series of integration bearing capacity equations is generalized. The proposed bearing capacity equation has been derived as a function of upper and lower layer properties. Finally the parametric study is carried out. The results of the parametric study were compared with the available equations in literature for the circular footing. Further, the results were validated with the experimental results reported in literature by other investigator.

1. Ultimate Bearing Capacity of Circular footing over
Layered Soils
Vipin Chandra Joshi
13M108
Geotechnical Engineering
Supervisors:
Prof. R. K. Dutta
and
Prof. Rajnish Shrivastava

2. Contents
• Introduction
• Objective
• Historical Review
• Outcome of the Study conducted
• Proposed Bearing Capacity expression
• Parametric studies
• Comparison and Validation
• Design charts
• Conclusion

3. Introduction
Footing is required to transmit the load of superstructure to deep below ground. In general
footings may be shallow or deep depending upon the depth to width ratio. Considerations may be
taken for the safe transmission of load beneath footing such as safe against shear and settlement.
Most of the cases arise when one has an interest to find out the bearing capacity of soils beneath
the footing. For the case of several footings founded on soils it is easy when we assume soil to be
homogenous. But in actual sense, it is not the case but layered deposits may be encountered. So,
to get the actual realm of soil beneath the footing it is necessary to analyze the stresses acting on
footing for the layered case ie; how footing will behave when it is founded on layered deposits.
Over the years research had been done to found out the actual analyses of footing such as strip
footing on layered soils (dense sand over loose sand or dense sand over clay).
Analysis for the circular footing is done for the dense sand over saturated soft clay soil and failure
pattern is assumed over which stress analysis has been done to find out the actual behavior of
footing and hence determining the ultimate bearing capacity by using integration.

4. Objective
Ultimate bearing capacity of two-layered system has been a major concern among the researchers
till date due to the discrepancies between developed theoretical approaches and experimental
studies.
Determination of the ultimate bearing capacity of circular footing on layered soils using projected
area approach.

5. Historical Review
S.No Approaches Researchers Remarks
1. Classical Approaches Meyerhof, 1974; Purusothamaraj et al,
1974; Hanna, 1981b, 1982; Andrawes
et al., 1996; Hanna and Meyerhof,
1979; Georgiadis, 1985; Oda & Win,
1990; Michalowaski & Shi, 1995;
Okamura et al., 1998; Al-Shenawy, Al-
Karni, 2005; Carlos, 2004; Zhang &
Wan, 2008; Huang & Qin, 2009
(a) Semi- empirical Meyerhof, 1974; Hanna, 1981b, 1982;
Hanna and Meyerhof, 1979; Merifield
and Sloan 1999
Formulae for UBC prediction based on certain
assumptions (Vertical cylinder beneath strip footing)
(b) kinematic approach Purusothamaraj et al, 1974; Radoslaw,
Michalowski and Shi, 1995
Dependent on assumed failure mechanism observed
through experimental tests
(c) Numerical approach Georgiadis and Michalopoulos, 1985 Pressure transferred through the footing respective
of layers profiles and subsequently the number of
soil layers involved in mobilization, against the
external horizontal, vertical and shear forces, by the
forces along the failure surfaces

6. 2. Finite Element method(FEM) Hanna, 1987; Yin et al, 2001;
Zhu, 2004; Szypcio and Dołžyk,
2006; Zhu, Michalowski, 2005;
Kumar and Kouzer, 2007
Modulus of elasticity, stress-
strain relationship, footing
settlement
3. Artificial Neural Network(ANN) Padmini et al. 2008; Kuo et al.
2009; Kalinli et al. 2011
Good capability for estimation
of stress-strain relationship,
settlement and classification
of soils

7. Meyerhof (1974) investigated the case of sand layer overlying clay: dense sand on soft clay and
loose sand on stiff clay. For several modes of failure analysis was done and were compared with
the results of model test on circular and strip footings.
Semi-empirical formulae were developed for computations of bearing capacity of strip, and
circular footings resting on dense sand overlying soft clay.
Theory and test results performed showed that the influence of the sand layer thickness beneath
the footing depends mainly on the bearing capacity ratio of the clay to the sand, the friction
angle ∅ of the sand, the shape and the depth of the foundation. Theory was limited to the vertical
loading and has no comments for the case of eccentric loading on footing.

8. • Where,
• C : undrained cohesion of clay,
• Nc: bearing capacity factor = 5.14,
• D : depth of embedment,
• H : thickness of top layer,
• B : footing width,
• 𝛾 : Unit weight of top layer,
• S : shape factor,
• Ks: punching shear coefficient,
• 𝑞𝑡=0.3𝛾B𝑁𝛾+ 𝛾DNq (for circular footing)

9. Meyerhof and Hanna (1978) considered the case of footings resting in a strong layer overlying
weak deposit and a weak layer overlying strong deposit. The analyses of different soil failure were
compared with the results of model tests on circular and strip footings on layered sand and clay.
They developed theories to predict the bearing capacity of layered soils under vertical load and
inclined loading conditions.
This paper is a development of the previous theory (Meyerhof 1974) ,taking into consideration
all possible cases of two different layer of subsoil, and also including the effect of inclined and
eccentric loading on the ultimate bearing capacity of strip, circular, and rectangular footings.
This theory and the failure mechanism considered are approximations of the real failure
mechanism, which depend on many factors.

10. Hanna and Meyerhof (1980) extended the previous theory to cover the case of footings resting
on subsoil consisting of a dense layer of sand overlying a soft clay deposit, and they presented the
results of this analysis in the form of design charts. It is a kind of revision of the assumptions
previously used in the punching theory of the previous papers in order to reduce their effects on
the analysis.
Hanna and Meyerhof (1981) investigated experimentally the ultimate bearing capacity of
footings subjected to axially inclined loads by conducting test on model strip and circular footing
on homogenous sand and clay. The results were analyzed to determine the inclination factors,
depth factors, and shape factors incorporated in the general bearing capacity equation for shallow
foundations. These values were compared with the recommended values given in the Canadian
Foundation Engineering Manual. The values of these factors given in the manual agree reasonably
well with the experimental ones, except for the shape and depth factors, for which the theoretical
values are on the conservative side when applied to inclined loads.

11. Development of bearing capacity equations using projected area approach(Classical approach)
has been studied by some researchers for two-layered soil system (Kenny and Andrawes, 1997;
Okamura et al., 1998; Carlos, 2004).
Projected area approach
In this approach external load is supposed to spread linearly from circumference of footing to a
larger area of sand as pressure penetrates deeply into the top layer through a constant angle
therefore the intensity of the load decreases along the depth.
 Kenny and Andrawes (1997)found that better and more reliable results can be obtained by
employing lower values of load spread angle.
 A theoretical equation is also developed by Okamura et al (1998) following projected area
method. The bearing capacity of bottom clay layer is supposed to be the same as the applied
vertical stress at the interface of two layers (at the base of sand block). Furthermore it was
reported that projected area method overestimates the bearing capacity of circular footings
since the highest magnitude of 𝛼 has been chosen over the range of 0° to 30°.

12. The above literature indicates that most of the studies were carried out for the strip footing resting
on layered soils using several approaches as discussed above. However, there is paucity of
analysis for the bearing capacity of circular footing resting on profiled soils using punching shear
mechanism following projected area approach.
There are many researches carried out to bring new and more accurate solutions following
classical and theoretical approaches however number of studies regarding application of finite
element method and artificial neural network is much less.
Outcome of the Study conducted so far

13. Assumed Failure Pattern

14. Forces acting on a frustum of thickness dz

15. 𝑞 𝑢𝑙𝑡= 𝑞 𝑏+𝐾𝑝 𝛾1 𝑠𝑒𝑐 𝛼 (𝑠𝑖𝑛 𝛿)
2𝐻(𝐻+2𝐷)
𝐷′+2𝐻 𝑡𝑎𝑛 𝛼
−
𝛾1H
where,
𝑞 𝑏 is the ultimate bearing capacity of
circular footing on a very thick bed of the lower
soft saturated clay layer.

16. 𝑞 𝑏 =c𝑁𝑐 𝑠𝑐 + 𝑞
where,
𝑠𝑐= 1.3 for circular footing
The non dimensional expression for “𝑞 𝑏” can be written as follows
𝑞 𝑏
𝛾1 𝐷′ = (
𝑐𝑁 𝑐 𝑠 𝑐+𝑞
𝛾1 𝐷′ )

17. Proposed Bearing Capacity Equation
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′ =
𝑐𝑁 𝑐 𝑠 𝑐+𝑞
𝛾1 𝐷′ + 2𝐾𝑝 𝐻
𝐻+2𝐷
𝐷′(𝐷′ 𝑐𝑜𝑠∝+2𝐻 𝑠𝑖𝑛∝)
𝑠𝑖𝑛 𝛿 − (
𝐻
𝐷′)
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′
Non –dimensionless parameter (Ratio of ultimate bearing capacity to the product of density and diameter
of circular footing)
𝛿 Mobilised angle of shearing resistance , degree
𝜶 Dispersion angle of the load through circular footing on upper layer
𝐷′
q
Diameter of circular footing, m
Surcharge
H Total thickness of upper layer- Depth of footing, m

18. Parametric studies
The various parameters were varied as follows
• Non Dimensional parameter (
𝑐
𝛾1 𝐷′) = 0.5 and 1.0
• The friction angle of the sand layer ( ϕ ) = 300, 350, 400 and 450
• Mobilised angle of shearing resistance (𝛿) = ϕ and 2ϕ/3
• Load spread angle (𝛼) in sand layer = 00 and 300
• Non dimensional parameter (
𝐻
𝐷′) = 0.5 to 4.0 at an interval of 0.5.

19. 0
4
8
12
16
20
24
0 0.5 1 1.5 2 2.5
ϕ =30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ -------------
𝜹 = (2ϕ)/3 . . . . .
0
4
8
12
16
20
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ -------------
𝜹 = (2ϕ)/3 . . . . .
Fig. 1 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ “ when
𝐻
𝐷′ = 0.5 and
𝐷
𝐷′ = 0 (a) ∝ =00 (b) ∝ = 300

20. Fig .2 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝐻
𝐷′ ” when δ = ϕ and
𝐷
𝐷′ = 0 (a)
𝑐
𝛾1 𝐷′ =0.5 and ∝ =00 (b)
𝑐
𝛾1 𝐷′ =1.0 and ∝ =00 (c)
𝑐
𝛾1 𝐷′ =0.5 and ∝ =300 (d)
𝑐
𝛾1 𝐷′=1.0 and ∝ =300

21. Fig. 3 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝐻
𝐷′ ” when δ =
2ϕ
3
and
𝐷
𝐷′ = 0 (a)
𝑐
𝛾1 𝐷′ =0.5 and ∝ =00 (b)
𝑐
𝛾1 𝐷′ =1.0 and ∝ =00 (c)
𝑐
𝛾1 𝐷′ = 0.5 and ∝ =300 (d)
𝑐
𝛾1 𝐷′ =1.0 and ∝ =300

22. COMPARISON AND VALIDATION
Fig. 4 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′ ” with “
𝐻
𝐷′”when
𝑐
𝛾1 𝐷′ =1.0, ∝ = 300, δ = ϕ and
𝐷
𝐷′ = 0 (a) ϕ =300
(b) ϕ =350
(c)
ϕ =400
(d) ϕ =450

23. Fig.5 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′ ” with “
𝐻
𝐷′” when
𝑐
𝛾1 𝐷′ =1.0, ∝ = 00, δ = ϕ and
𝐷
𝐷′ = 0 (a) ϕ =300
(b) ϕ =350
(c) ϕ
=400
(d) ϕ =450

24. Design Charts
0
4
8
12
16
20
24
0 0.5 1 1.5 2 2.5
ϕ =30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ -------------
𝜹 = (2ϕ)/3 . . . . .
0
4
8
12
16
20
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ -------------
𝜹 = (2ϕ)/3 . . . . .
Fig 6 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ " when
𝐻
𝐷′ = 0.5 and
𝐷
𝐷′ = 0 (a) ∝ =00 (b) ∝ =300

25. 0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = 2ϕ/3 . . . . . .
𝜹 = ϕ ------------
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = (2ϕ)/3 . . . . .
𝜹 = ϕ ---------
Fig. 7 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ " when
𝐻
𝐷′ = 1.0 and
𝐷
𝐷′ = 0 (a) ∝ =00 (b) ∝ =300

26. 0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5
ϕ =30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ -------------
𝜹 = (2ϕ)/3 . . . . .
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ -------------
𝜹 = (2ϕ)/3 . . . . .
Fig. 8 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ " when
𝐻
𝐷′ = 0.5 and
𝐷
𝐷′ = 0.5 (a) ∝= 00 (b) ∝= 300

27. 0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = 2ϕ/3 . . . . .
.
𝜹 = ϕ ---------
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = 2ϕ/3 . . . . .
.
𝜹 = ϕ -------
Fig. 9 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ " when
𝐻
𝐷′ = 1 and
𝐷
𝐷′ = 0.5 (a) ∝ = 00 (b) ∝ = 300

28. 0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2 2.5
ϕ =30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ ------
𝜹 = (2ϕ)/3 . . .
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = ϕ ------
𝜹 = (2ϕ)/3 . . . . .
Fig. 10 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ " when
𝐻
𝐷′ = 0.5 and
𝐷
𝐷′ = 1.0 (a) ∝ = 00 (b) ∝ = 300

29. 0
10
20
30
40
50
60
70
80
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = 2ϕ/3 . . . . . .
𝜹 = ϕ ---------
0
5
10
15
20
25
30
35
40
45
50
0 0.5 1 1.5 2 2.5
ϕ=30
ϕ=35
ϕ=40
ϕ=45
𝜹 = 2ϕ/3 . . . . . .
𝜹 = ϕ --------
Fig. 11 Plot depicting the variation of “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” with “
𝑐
𝛾1 𝐷′ " when
𝐻
𝐷′ = 1.0 and
𝐷
𝐷′ = 1.0 (a) ∝ = 00 (b) ∝ = 300

30. Conclusions
• The analysis indicate that in general with the increase in “
𝐻
𝐷′” there is an increase
in non dimensional parameter “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′ ” .
• With the increase in load spread angle there is decrease in non dimensional
parameter “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′ ”
• With the decrease in “𝜹” from ϕ to
2ϕ
3
there is a marginal decrease in non
dimensional parameter “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” .
• With the increase in friction angle of soil there is an increase in non dimensional
parameter “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” .
• With the increase in non dimensional parameter (
𝑐
𝛾1 𝐷′) there is an increase in non
dimensional parameter “
𝑞 𝑢𝑙𝑡
𝛾1 𝐷′” .

31. Paper Accepted
Joshi, V.C., Dutta R.K. and Shrivastava, R. , “Ultimate bearing capacity of
circular footing on layered soils”, Journal of GeoEngineering.

32. Thank
You