γb = buoyant unit weight of soil below footing.
B = footing width.
Nγ = bearing capacity factor based on effective friction angle φ′ as per Fig. 7.3.
Factor of safety FS is determined as follows:
FS = q
7.4 BEARING CAPACITY ANALYSIS FOR COHESIVE SOIL WEAKENED BY EARTHQUAKE
Cohesive soils as well as organic soils can also be susceptible to a loss of shear strength during the earthquake. In dealing with such soils, it is often desirable to limit the stress exerted by the footing during the earthquake. The stress exerted should be less than the maximum past pressure of the cohesive or organic soils. This prevents the soil from squeezing out. It also prevents soil from deforming laterally from underneath footing.
It is often very difficult to predict the amount of earthquake induced settlement for foundations bearing on cohesive or organic soils. Consequently, in one approach adequate factor of safety against bearing capacity failure of foundation is ensured. Ultimate bearing capacity is determined as follows:
su is undrained shear strength, B is footing width and L is footing length. Factor of safety FS is determined as follows:
FS = q
There are standard guidelines in terms of undrained shear strength that should be utilized for earthquake engineering analysis (Triandafilidis, 1965). These guidelines for selection of undrained shear strength is given in subsections below.
7.4.1 Cohesive soil above groundwater table
These soils above groundwater table have negative pore pressures. This is due to capillary tension. This tends to hold soil particles together. It also provides additional strength.
Undrained shear strength should be determined by performing unconfined compression or vane shear tests under these conditions. Due to negative pore pressure, a future increase in water content will tend to decrease undrained shear strength of partially saturated cohesive soil. Consequently, possible change in water content in future should also be considered.
Ultimate strength obtained during unconfined compression test should be used in bearing capacity analysis.
7.4.2 Cohesive soil below groundwater table having low sensitivity
Sensitivity is ratio of undrained shear strength of undisturbed soil to undrained shear strength of completely remolded soil. Consequently, it represents loss of strength as cohesive soil is remolded. Earthquake also tends to shear the soil back-forth. Furthermore, it also remolds it. For low sensitivity soils (sensitivity < 4), reduction of undrained shear strength during earthquake is small. Consequently, undrained shear strength from unconfined compression or vane shear test should be used for bearing capacity analysis.
7.4.3 Cohesive soil below groundwater table having high sensitivity
For high sensitivity soils (sensitivity > 8), earthquake-induced ground shaking could lead to significant shear strength loss during earthquake shaking. The stress-strain curve from an unconfined compression test on such soils exhibits peak shear strength developed at low vertical strain. This is followed by dramatic drop-off in strength with continued straining.
Estimated reduction in undrained shear strength due to earthquake shaking should be included in analysis. Most critical condition develops when such soil is subjected to high static shear stress. If sum of static shear stress and seismic induced shear stress during earthquake shaking exceeds undrained shear strength, there is significant reduction in shear strength (Cunny and Sloan, 1961). Cohesive soils having sensitivity in between 4 and 8 tend to be intermediate case.
There are other factors also which may be considered in bearing capacity analysis. Peak ground acceleration and earthquake magnitude is such factor. Higher the peak ground acceleration and higher the magnitude of earthquake, greater the tendency for cohesive soil to be strained and remolded by earthquake shaking. Undrained shear strength, sensitivity, maximum past pressure and stress-strain behavior are other important soil behavior parameters, which should be included in the analysis. Increase in shear stress due to dynamic loading should also be included in analysis. Lightly loaded foundations tend to produce small dynamic loads. On the other hand heavy and tall buildings subject foundation to high dynamic loads due to rocking. Finally it can be concluded that since so many variables are involved, it takes considerable judgement in selection of undrained shear strength to be used in Equations (7.16) and (7.17).
Finally, based on results of settlement analysis and bearing capacity analysis for both static and dynamic conditions design recommendations such as minimum footing dimensions, embedment requirements and allowable bearing capacity values are provided. Consequently, the objective of earthquake resistant design of shallow foundations is achieved to support varied civil engineering structures.
Earthquake Resistant Design of Shallow Foundation 85
Example 7.1. At a particular site, ground surface is horizontal and the zone of liquefaction extends from a depth of 1.2 m to 6.7 m. During construction, additional 1.8 m thick cohesive soil is placed at ground surface. After that it is proposed to construct a sewage disposal plant at the site. Bottom of the footing for the plant is to be at a depth of 0.5 m below ground surface. For both existing 1.2 m thick unliquefiable cohesive soil layer and additional 1.8 m thick cohesive layer, the undrained shear strength is 60 kPa. Calculate the factor of safety of the footing using punching shear analysis for:
(a) 1m wide strip footing under total load of 60 kN/m.
(b) 2m wide square spread footing under total load of 600 kN.
Solution:
(a) For strip footing, using Eq. (7.1),
T = 1.8 + 1.2 – 0.5 = 2.5 m,
τf = undrained shear strength of cohesive soil = 60 kPa.
P = 60 kN/m.
Substituting the values in Eq. (7.1):
FS = 5.0
(b) For square spread footing, using Eq. (7.2),
T = 1.8 + 1.2 – 0.5 = 2.5 m,
τf = undrained shear strength of cohesive soil = 60 kPa, P = 600 kN
and B = L = 2 m.
Substituting the values in Eq. (7.2):
FS = 2.0
Example 7.2. Perform total stress analysis using Terzaghi equations for general and local shear failure to find out factor of safety for 1m wide strip footing. Use data from Example 7.1.
Solution: From Example 7.1, P = 60 kN/m for 1 m wide strip footing, T = 1.8 + 1.2 – 0.5 = 2.5 m. c1 = su = 60 kPa = 60 kN/m2 & c2 = 0. i.e. T/B = 2.5/1 = 2.5 and c2/c1= 0. For these two values and using Fig. 7.1, Nc = 5.5. Consequently, using Eq. (7.6), qult
= (60)(5.5) = 330 kN/m2 for strip footing. Hence Qult for 1 m wide strip footing = qult B
= (330)(1) = 330 kN/m. Using Eq. (7.8), factor of safety = FS = 5.5.
Example 7.3. Use data from Example 7.1. Assume that apart from vertical loads, the strip and the spread footing is subjected to earthquake induced moment equal to 5 kN.m/
m and 150 kN.m which act in single (B) direction. Determine factor of safety using Eq.
(7.11).
Solution:
(a) For 1 m wide strip footing, Q = P = 60 kN/m, e = M/Q = 5/60 = 0.0833 m, for middle one-third of footing, e can not exceed 0.17 m, and therefore e is within
middle one-third of footing. q′= 89.988 kN/m2 from Eq. (7.9). qult in Eq. (7.11) is determined using Eq. (7.6) which makes use of Fig. 7.1. Hence qult = 330 kN/m2. Consequently, Factor of safety = FS = 3.667.
(b) For 2 m wide square spread footing, Q = P = 600 kN, e = M/Q = 150/600
= 0.25 m. For middle one-third of footing, e can not exceed 0.33 m, and therefore e is within middle one-third of footing. Q = 600/2 = 300 kN/m for use in Eq. (7.9).
q′ = 262.5 kN/m2 from Eq. (7.9) for 2m wide square spread footing. Furthermore, T = 1.8 + 1.2 – 0.5 = 2.5m. c2 = 0 and c1 = 60 kN/m2. T/B = 2.5/2 = 1.25m and c2/c1 = 0. Using these and from Fig. 7.1, Nc = 3.2. Hence qult = 249.6 kN/
m2 from Eq. (7.7) with B = L = 2 m. Consequently, Factor of safety = FS = 0.95.
Home Work Problems
1. Solve Example 7.1 assuming that both the existing 1.2 m thick and additional 1.8m thick unliquefiable soil layer is cohesionless with effective friction angle equal to 31°. Coefficient of earth pressure at rest is equal to 0.5. Total unit weight of soil above water table is 18.3 kN/m3 and buoyant unit weight of soil below water table is 9.7 kN/m3. Water table is at a depth of 1.2m below existing ground surface. (Ans. (a) FS = 0.8 (b) FS = 0.32) 2. Perform total stress analysis using Terzaghi equations for general and local shear failure to find out factor of safety for 2 m wide square spread footing. Use data from Example 7.1.
(Ans. FS = 1.664) 3. Use data from Example 7.1. Assume that apart from vertical loads, the strip and the spread footing is subjected to earthquake induced moment equal to 5 kN.m/m and 150 kN.m which act in single (B) direction. Determine factor of safety using Eq. (7.12) (Ans. (a) FS = 4.58 (b) FS = 1.176.
4. A site consists of a sand deposit with a fluctuating groundwater table. The expected depth of footing will be 0.5 to 1 m. Assume that groundwater table can rise to a level close to footing base. Buoyant unit weight of sand is 9.65 kN/m3, effective friction angle for sand = 32° and pore water pressure ratio = 0.2. Using factor of safety of 5, determine allowable bearing capacity for:
(a) 1.5m wide strip footing.
(b) 2.5m wide square spread footing.
(Ans. (a) 24.318 kPa (b) 32.424 kPa)) 5. What are the guidelines to calculate undrained shear strength in the bearing capacity analysis
for cohesive soil weakened by earthquake?