1. GLE/CEE 330: Soil Mechanics Bearing Capacity of Shallow Footings
Geological Engineering
University of Wisconsin-Madison

2. Learning Objectives Discuss failure mechanisms
Describe limit equilibrium analysis approach
Learn basic design approach for shallow footings (strength limit state)

3. Foundation Design Philosophy
Limit State = “condition beyond which a component/member of a foundation or other structure ceases to satisfy the provisions for which the component/member was designed”
Strength Limit State (bearing capacity analysis)
Service Limit State (settlement analysis)
Extreme Event Limit State
Fatigue Limit State

4. Failure of Shallow Footings
Dense soil
“Brittle” response
General Shear Failure
Local Shear Failure
Loose soil
“Ductile” response
Punching Failure
(Coduto)

5. Practice is to check general shear case
Then do settlement analysis
(implicitly checks local and punching)
(Coduto)

6. Limit Equilibrium Approach
1) Select plausible failure mechanism (failure surface)
2) Determine forces on failure surface
3) Use static equilibrium to determine failure load
Strip Footing on Saturated Clay (Undrained Analysis)
Semi-circular failure surface
For refined geometry:

7. Terzaghi’s Bearing Capacity Equation
Shallow Footing (B X L)
Embedment depth (Df): accounts for frost, environmental effects, etc., provides surcharge
General Shear Failure Surface
Ultimate Bearing Capacity

8. More general form… (Vesic, 1973)
qult = ultimate bearing capacity (stress)
c’ = cohesion (or su for undrained load andf = 0; short term analysis for clay)
q = overburden stress (q = gDf)
B = footing width (or diameter)
g = total unit weight of soil (need to correct for water table)
Nc, Nq, Nf = “Bearing capacity factors” = f(f)
fc, fq, fg = correction factors (shape, depth, load inclination, slope, etc…)

9. Bearing Capacity Factors (Vesic, 1973)

10. Correction Factors
Shape
Depth
Inclination P b Df L B

11. Ultimate, Net, and Allowable Capacity
Subtract pressure from excavated soil above footing
F.S. = factor of safety (F.S. = 2 – 4)

12. Effect of Water Table Case 1 (Shallow water table):
If Dw < Df, then use g’ in 1/2BgNg term , where (g’ = g – gw)
If Dw = 0, then use g’ in qNq term and 1/2BgNg term
Case 2 (Intermediate Water Table):
If Df < Dw < Df + B, then use “average g” in 1/2BgNg term
Case 3: (Deep Water Table):
If Dw > Df +B, then no effect
Case 1
Case 2
Case 3

13. Example: Sand P Find Pall for FS = 3.0 Df = 5’ L = 10’
Df = 5’
L = 10’
Check water table:
Dw (13) > Df + B (11) so no effect
B = 6’
Sand (c = 0)
g = 110 pcf
f = 33
Dw = 13’
From Vesic (1973): for f = 33 deg.
Nq = 26.1
Ng = 35.2
Correction factors:

14. P
Find Pall for FS = 3.0
Df = 5’
L = 10’
B = 6’
Sand (c = 0)
g = 110 pcf
f = 33
Dw = 13’

15. Example: Clay t s (psi) P Dw = 0 Find Pall for FS = 3.0
Assume undrained (rapid) loading (f = 0, su)
This is the critical case (weakest soil)
Df = 6’
L = 10’
Clay
g = 100 pcf
B = 6’ t
s (psi)
Undrained Shear Strength
su = 8.7 psi = 1250 psf
Results from UU tests on Clay:
Test
s3f (psi)
s1f (psi) A 10 27 B 20 38 C 40 57

16. P
Dw = 0
From Vesic (1973), for f = 0:
Nq = 1.0
Nc = 5.14
Ng = 0
Df = 6’
L = 10’
No need to check water table (Ng = 0)
Clay
g = 100 pcf
B = 6’
Correction Factors:

17. P
Dw = 0
Df = 6’
L = 10’
Clay
g = 100 pcf
B = 6’