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2. BEARING CAPACITY OF SHALLOW
FOUNDATIONS
By
Engr.Bahadar Ali Khan
University of Engineering and Technology, Lahore Pakistan
For Video Click here

3. Methods of bearing capacity Determination
I. Analytical Methods
Plastic theory
• Prandtl’s Method
• Terzaghi’s Method
• Skemptons theory
• Mayerhof theory
• Fellinive’s theory
• Brinch Theory
• Vesics Theory
• Hansens Theory
( )C 
( )C 
( )C 
( )C 
( )C
( )C
( )C 
( )C 

4. Earth Pressure Theory
• Rankine theory
• Bell’s theory
• Paukers method
Elastic theory
• S. Schleicher theory
( )
( )C 
( )C
( )C

5. II. Field Method
• Plate load test
• Standard penetration test
• Static cone penetration test

6. Prandtl’s Method
2 tan
tan 1
tan 4 2
Cohession of soil
= Internal friction of soil
2.178
u
c
q e
c
e
  


  
    
  



7. Terzaaghi’s Method
• It is improvement of Prandtl theory.
• Both theories are plastic theories and valid for soils.
• Base of footing is smooth in Prandtl’s theory but in Terzaaghi’s
theory base of footing is rough.
( )C 
( )C 

8. Assumptions in Terzaghi’s theory
• Foundation is shallow i.e . It means that he considered only
base resistance and ignored side resistance.
• Base of footing is rough.
• Footing is strip footing (continuous footing). As so Terzaghi
ignored the length and his analysis only based on Depth Df and
Breadth B of footing i.e 2D analysis.
• Soil beneath foundation is homogeneous semi infinite mass
• No soil consolidation occurs; undrained condition
• Foundation is very rigid relative to the soil
1
fD
B

L B

9. • Loading is vertical and symmetrical. There is no eccentricity. It means that there
is no moment in foundation.
• He considered general shear failure & ground surface is horizontal.
• The shear resistance of the soil above the foundation level up to ground level is
ignored. It means that he considered footing as surface footing.
Surface footing: Surface footing is that footing resting at foundation level and soil
above this foundation level is converted in form of surcharge acting at
foundation level.
At time of failure plastic equilibrium state reaches only up to foundation level not
up to ground level.
fD

10. • Soil above the foundation level is replaced by uniform surcharge
over the foundation level.
• Initially he does not consider effect of water table but finally this theory is
modified for water table. In that case surcharge is
• Stress zone is extended up to the foundation level.
• Mohr-Coulomb model for soil applies
• No soil consolidation occurs; undrained condition
• Foundation is very rigid relative to the soil
• No applied moments present
2
( / )fD KN m
fD 

11. B

12. Terzaaghi Failure zone
Failure zone is further divided into three zones
• Region I
Zone of elastic equilibrium / Rankine Active Zone
• Region II
Zone of plastic equilibrium or Radial shear zone
• Region III
Rankine passive zone or linear shear zone

13. Zone I
• Active Rankine Zone
• In this zone soil is prevented from any lateral movement, because of
friction and adhesion between soil and footing surface.
• It acts as a part of footing and remain in elastic equilibrium.
• Note: In case of Prandtl’s theory zone I will also in plastic
equilibrium state as base of footing is smooth.

14. Zone II
• In this zone all particles of soil mass are stressed & remains in plastic
equilibrium state.
• The failure plane of zone II is circular in case of cohesive and long
spiral in case of soils.C 
Zone III
 The wedge pushes soil outwards, causing passive Rankine zones to
form with angles Angle of with the horizontal.45
2



15. Terzaaghi Ultimate bearing Capacity Equation
Note: for pure clay
according to Terzaughi: N 5.7
according to Prandtle: N 5.14
c
c


, , are Terzaghi bearing capacity factors. Their values depend upon of soil.c qN N N 
0.5u c qq cN qN BN  
I II III
0.5
ultimate bearing capacity
D Depth of foundation
B = width of footing
unit weight of soil (in zone II is for soil above the F.L and i
u c f q
u
f
q cN D N BN
q
  
 
  


 n zone III is for soil below the F.L)

16. (0.75 /2)tan
2
2
cos (45 / 2)
( 1)cot
tan
( 1)
2 cos
q
c q
p
e
N
N N
K
N
  








 
 
N

17. • Later on Terzaghi proposed shape factors for the first and last
terms of equation to account for the different shapes of the footings
such as circular, square, rectangular etc.
andcS S
0.5u c c f qq S cN D N S BN    

18. Terzaghi's bearing capacity Eq. has been modified for other types of
foundations by introducing the shape factors. The equations are:
1.3 0.4
1.3 0.3
1 0.2 0.5 1 0.2
u c f q
u c f q
u c f q
q cN D N BN
q cN D N BN
B B
q cN D N BN
L L



 
 
 
  
  
   
         
   
Square Footing
Cicular Footing
Rectangle Footing

21. Meyerhof Theory
• Terzaaghi considered the stress zone extended up to foundation level
but Meyerhof considered the stress zone extended up to ground
surface. It means Terzaaghi ignored side resistance of soil but
Meyerhof considered side resistance.

22. Bearing Capacity According to Meyerhof
tan 2
0.5
0.5
( ) tan (45 )
2
( 1)cot
( 1) tan1.4
u c c c f q q q
u c c c f q q q
q
c q
q
q cN S d D N S d BN S d
q cN d i D N d i BN d i
N e
N N
N N
  
  
 

 
 


  
  
 
 
 
In Case of Vertical Load
In Case of Inclined Load
0.5u c c c c c c f q q q q q qq cN S d i g b D S N d i g b BN S d i g b        
0.5u c c c c f q q q qq cN S d i D N S d i BN S d i      
tan 2
0.5
0.5
( ) tan (45 )
2
( 1)cot
( 1) tan1.4
u c c c f q q q
u c c c f q q q
q
c q
q
q cN S d D N S d BN S d
q cN d i D N d i BN d i
N e
N N
N N
  
  
 

 
 


  
  
 
 
 
In Case of Vertical Load
In Case of Inclined Load