CE-828 Advanced Geotechnical Design
Part 2: Bearing capacity I
Spring 2021
Dr. Badee Alshameri, PhD
Head of Geotechnical Engineering Department
NUST Institute of Civil Engineering (NICE)
School of Civil & Environmental Engineering (SCEE)
National University of Sciences and Technology (NUST)
H-12, Islamabad, Pakistan
Soil bearing capacity and Shallow Foundation
What & Why
Why is the function of foundation?
transfer the load of the structure to the soil on which it is resting
What using the foundation!
To prevent the overstressing the soil. Thus prevent excessive settlement or shear failure of the soil, consequently
preservation the structure form damage
Soil bearing capacity and Shallow Foundation
Types of foundations
Common types of foundations are spread footing, mat foundation, pile foundation, and drilled shaft foundation
Difference between shallow and deep foundation
If the ratio of depth to embedment to the width of foundation is less than 4, it will call shallow foundation
however, on the deep foundation the ratio will be greater than 4.
(Das & Sobhan 2014)
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
According to Vesic (1973), there are three common types of
failure; general shear failure, local shear failure, and punching
shear failure
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Modes of foundation failure in sand (After Vesic, 1973)
Where Dr is the relative density of sand and Df is the depth of
foundation measured from the ground surface
2𝐵𝐿
𝐵 ∗ = 𝐵+𝐿
Vesic, A. S. (1973). “Analysis of Ultimate Loads of Shallow
Foundations,” Journal of Soil Mechanics and Foundations Division,
American Society of Civil Engineers, Vol. 99, No. SM1, pp. 45–73.
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Modes of foundation failure in sand (After Vesic, 1973)
Where Dr is the relative density of sand and Df is the depth of
foundation measured from the ground surface
2𝐵𝐿
𝐵 ∗ = 𝐵+𝐿
Vesic, A. S. (1973). “Analysis of Ultimate Loads of Shallow
Foundations,” Journal of Soil Mechanics and Foundations Division,
American Society of Civil Engineers, Vol. 99, No. SM1, pp. 45–73.
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Terzaghi assume the following
1.
General shear failure
2.
Homogeneous soil
3.
2-D analysis
4.
Vertical and concentric load
5.
Horizontal surface
6.
Depth of the foundation Df ≤ width of the foundation B
7.
Neglecting the shear resistance above the foundation
8.
The soil above the foundation level will be considered as surcharge
9.
For sandy soil (c = 0), if ∅ ≥ 36 so it’s purely general shear failure while if ∅ ≤ 29 so it’s purely local shear failure
10.
For ∅ - c soil, if the soil specimen shows failure below 5% it will be considered as general shear failure while if it shows failure
between 10 to 20% it will be considered as local shear failure
11.
For local shear failure, mobilized cohesion (cm) and mobilized friction angle (∅m)will be used, where cm = 2ൗ3 c & ∅m =
tan−1 2ൗ3 tan ∅
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Terzaghi’s Bearing Capacity Theory
According to Terzaghi (1943) has the following conditions:
• The shallow foundation should have depth (Df) equal or less than the width of its width (B)
• The general shear failure is used.
• The failure surface plane will not extend to the surface
• The effect of soil above the bottom of the foundation may also be assumed to be replaced by an equivalent surcharge: q =
γ Df
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow
Foundations
Terzaghi’s Bearing Capacity Theory
The failure zone under the foundation can be
separated into three parts:
1. The elastic zone I
2. The radial shear zones II,
3. The Rankine passive zones III
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Terzaghi’s Bearing Capacity Theory
In the equilibrium state:
𝑞𝑢 2𝑏 1 + 𝑊 = 2𝐶 sin ∅′ + 2𝑃𝑃
where b is the half of the foundation width, W is the weight of soil wedge = 2
1
′
𝑏𝑏
tan
∅
2
= 𝛾𝑏2 tan ∅′
C is the cohesive force acting along each face, AJ and BJ, that is equal to the unit cohesion times the length of
𝑐 ′𝑏
each face cos ∅′
Thus
2𝑏𝑞𝑢 = 2𝑐 ′ 𝑏 tan ∅′ + 2𝑃𝑃 − 𝛾𝑏2 tan ∅′
By dividing all on 2b:
𝑞𝑢 = 𝑐 ′ tan ∅′ +
𝑃𝑃
𝑏
−
𝛾𝑏
tan ∅′
2
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Terzaghi’s Bearing Capacity Theory
The passive pressure will be the sum of the contribution of the weight of soil, cohesion, and
surcharge;
𝑃𝑃 = 12𝛾 𝑏 tan ∅′ 2 𝐾𝛾 + 𝑐 ′ 𝐾𝑐 𝑏 tan ∅′ + 𝑞𝐾𝑞 𝑏 tan ∅′
where 𝐾𝛾 , 𝐾𝑐 , 𝑎𝑛𝑑𝐾𝑞 are earth-pressure coefficients that are functions of the soil friction angle
∅′ .
Combining the previous equations will result:
𝛾𝑏
′
tan
∅
2
tan ∅′
𝑞𝑢 = 𝑐 ′ tan ∅′ + 12𝛾𝑏 tan ∅′ 2 𝐾𝛾 + 𝑐 ′ 𝐾𝑐 tan ∅′ + 𝑞𝐾𝑞 tan ∅′ −
𝑞𝑢 = 𝑐 ′ tan ∅′ 1 + 𝐾𝑐 + 12𝛾𝑏 tan ∅′ tan ∅′ 𝐾𝛾 − 1 + 𝑞𝐾𝑞
Replacing Nc, Ny, and Nq
𝑞𝑢 = 𝑐 ′ 𝑁𝑐 + 𝑞𝑁𝑞 + 12𝛾𝐵𝑁𝛾 ** Terzaghi’s bearing-capacity equation
Where
𝑁𝑐 = tan ∅′ 1 + 𝐾𝑐
𝑁𝛾 = 12 tan ∅′ tan ∅′ 𝐾𝛾 − 1
𝑁𝑞 = 𝐾𝑞 tan ∅′
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Terzaghi’s Bearing Capacity Theory
For square and circular footings, Terzaghi suggested the following equations for ultimate soil-bearing capacity:
For square footing
𝑞𝑢 = 1.3𝑐 ′ 𝑁𝑐 + 𝑞𝑁𝑞 + 0.4𝛾𝐵𝑁𝛾
For circular footing
𝑞𝑢 = 1.3𝑐 ′ 𝑁𝑐 + 𝑞𝑁𝑞 + 0.3𝛾𝐵𝑁𝛾
where B is the diameter of the footing
For an undrained condition with ∅ = 0 and 𝜏𝑓 = 𝑐𝑢 , the bearing-capacity factors are Nγ= 0 and Nq = 1. Also, Nc
= 5.7.
𝑞𝑢 = 5.7𝑐 ′ 𝑁𝑐 + 𝑞 strip footing
𝑞𝑢 = 1.3 5.7𝑐𝑢 + 𝑞 = 7.41𝑐𝑢 + 𝑞
square and circular footing
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow
Foundations
Terzaghi’s Bearing Capacity Theory - Using the table
solution
By referring to Terzaghi’s bearing capacity Factors
𝑁𝑐 = tan ∅′ 1 + 𝐾𝑐
𝑁𝛾 = 12 tan ∅′ tan ∅′ 𝐾𝛾 − 1
𝑁𝑞 = 𝐾𝑞 tan ∅′
It obviously that the factors depend totally on the
friction angle, the table solution will depend on the
variation of the friction angle
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Terzaghi’s Bearing Capacity Theory -Example
Determine the gross allowable load per unit area (qall) that the footing can carry by continuous footing (using Terzaghi’s
bearing-capacity factors). Assume general shear failure. Given: 𝛾 =17.5 kN/m3, c’ = 21 kN/m2, ∅′ = 32°, Df = 1 m, B =
1.5 m, and factor of safety =3.
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Effect of Groundwater Table
𝑞𝑢 = 𝑐 ′ 𝑁𝑐 + 𝑞𝑁𝑞 + 12𝛾𝐵𝑁𝛾
Three different conditions can arise regarding the location of the groundwater table with respect to
the bottom of the foundation
Case I: If the groundwater table is located at a distance D above the bottom of the foundation, the
magnitude of surcharge q at second term will be:
𝑞 = 𝛾 𝐷𝑓 − 𝐷 + 𝛾 ′ 𝐷
and the unit weight of soil 𝛾 at third term will be 𝛾 ′
Case II: If the groundwater table coincides with the bottom of the foundation:
𝑞 = 𝛾𝐷𝑓
and the unit weight of soil 𝛾 at third term will be 𝛾 ′
Case III: When the groundwater table is at a depth D below the bottom of the foundation:
𝑞 = 𝛾𝐷𝑓
and the unit weight of soil 𝛾 at third term will be:
1
𝛾𝑎𝑣 = 𝐵 𝛾𝐷 + 𝛾 ′ 𝐵 − 𝐷
𝛾𝑎𝑣 = 𝛾
For D > 𝐵
For D ≤ 𝐵
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Effect of Groundwater Table- Example
Determine the gross allowable load per unit area (Qall) that the footing can carry by square footing (using Terzaghi’s
bearing-capacity factors). Assume general shear failure. Given: 𝛾 =16 kN/m3, : 𝛾𝑠𝑎𝑡 =18.9 kN/m3, c’ = 17 kN/m2, ∅′ =
32°, Df = 1.2 m, h = 0.9 m, B = 1.75 m, and factor of safety = 3.5.
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Factor of Safety
Generally, a factor of safety, Fs, of about 3 or more is applied to the ultimate soil-bearing
capacity to arrive at the value of the allowable bearing capacity. An Fs of 3 or more is
not considered too conservative
The gross allowable bearing capacity is is the allowable load per unit area to which the
soil under the foundation should be subjected to avoid any chance of bearing capacity
failure
𝑞𝑎𝑙𝑙 =
𝑞𝑢
𝐹𝑠
=
𝑊 𝐷+𝐿 +𝑊𝑠 +𝑊𝐹
𝐴
The net allowable bearing capacity is the allowable load per unit area of the foundation
in excess of the existing vertical effective stress at the level of the foundation
The vertical effective stress at the foundation level is equal to 𝑞 = 𝛾𝐷𝑓 =
𝑞𝑢(𝑛𝑒𝑡) = 𝑞𝑢 − 𝑞
𝑞𝑎𝑙𝑙(𝑛𝑒𝑡) =
𝑞𝑢 − 𝑞 𝑊 𝐷+𝐿
=
𝐹𝑠
𝐴
𝑊𝑠 +𝑊𝐹
𝐴
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
General Bearing Capacity Equation
Hansen (1970) proposed a general expression for bearing capacity considering the effects of shape, depth of foundation
and inclination of applied load
1
𝑞𝑢 = 𝑐 ′ 𝐹𝑐𝑠 𝐹𝑐𝑑 𝐹𝑐𝑖 𝑁𝑐 + 𝑞𝐹𝑞𝑠 𝐹𝑞𝑑 𝐹𝑞𝑖 𝑁𝑞 + 2 𝐵𝛾𝐹𝛾𝑠 𝐹𝛾𝑑 𝐹𝛾𝑖 𝑁𝛾
Where
𝐹𝑐𝑠 , 𝐹𝑞𝑠, 𝑎𝑛𝑑 𝐹𝛾𝑠 is the shape factors, developed by De Beer (1970)
𝐹𝑐𝑑 , 𝐹𝑞𝑑, 𝑎𝑛𝑑 𝐹𝛾𝑑 is the depth factors, developed by Hansen (1970)
𝐹𝑐𝑖 , 𝐹𝑞𝑖, 𝑎𝑛𝑑 𝐹γ𝑖 is the inclination factors, developed by Meyerhof (1963)
In Terzaghi’s bearing capacity the following factors are not considered:
1-Length of the footing
2- The ratio footing depth to width
3- inclination of the load on the foundation with respect to the vertical
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
General Bearing Capacity Equation
1
𝑞𝑢 = 𝑐 ′ 𝐹𝑐𝑠 𝐹𝑐𝑑 𝐹𝑐𝑖 𝑁𝑐 + 𝑞𝐹𝑞𝑠 𝐹𝑞𝑑 𝐹𝑞𝑖 𝑁𝑞 + 2 𝐵𝛾𝐹𝛾𝑠 𝐹𝛾𝑑 𝐹𝛾𝑖 𝑁𝛾
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
General Bearing Capacity Equation
There are several solution for the bearing capacity factors which can be refer from many texts and references
Group assignment 1 from 8-3-2021 to 22-3-2021
Review the empirical correlation to predict the friction angle using geotechnical field tests
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
3.3 Notes in bearing capacity and settlement effective zone
• The effective zone for considering the bearing capacity can be roughly calculated until depth below the foundation equal
to B (B is the width of the foundation).
• The effective zone for considering the settlement can be roughly calculated until depth below the foundation equal to
2B.
• The soil below the level where the stress reach 10% of the actual applied stress, has no effect on the distribution of the
bearing capacity
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Calculate the suitable dimension of foundation-Example
Example 1
A sandy soil with dry and saturated unit weight of 16.5 kN/m3 and 18.5 kN/m3 respectively, has friction angle 340. A gross
allowable load equal to 668 kN need to applied on square foundation placed on depth of 1.2 m where the depth of the water
is 0.6 m and using factor of safety of FS. Determine the size of the foundation?
Example 2
A square column foundation is to be constructed on a sand deposit (unit weight of 18 kN/m3 ) at depth 0.7 m and width of
1.25 m. The allowable load Q will be inclined at an angle β = 20° with the vertical and the friction angle of the soil is 30°.
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Meyerhof’s Method
The bearing capacity equation was modified by Meyerhof
(1963) to extend the surface of failure to the ground surface
Meyerhof bearing capacity equation for strip footing is similar
in the form of Terzaghi but the values of Nc, Nq, and Ny are
different.
The method generally overestimates the bearing capacity for
the sandy soil because footings may fail from the settlement
consideration much before the failure reaches the ground
surface
The method gives a good results for the clay soil
Induvial assignment (Self-study)
Compare Terzaghi and Meyerhof bearing capacity (advantage,
limitation, recommendations)
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Vesic’s Method
The failure surface at Vesic (1963) is similar to Terzaghi with the exception that zone I below the footing will be in
∅
active Rankine state with the inclined faces of 45 + 2 to the horizontal
Group assignment 2 (8-3-2021 to 22-3-2021)
Compare the reliability of Vesic (1963) to Terzaghi’s bearing capacity?
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Effect of Soil Compressibility
The mode of bearing-capacity failure such as general shear failure, local shear failure and punching shear failure. The
change of failure mode is due to soil compressibility, to account for which Vesic (1973) proposed the following
modification of Eq.
1
𝑞𝑢 = 𝑐 ′ 𝐹𝑐𝑠 𝐹𝑐𝑑 𝐹𝑐𝑐 𝑁𝑐 + 𝑞𝐹𝑞𝑠 𝐹𝑞𝑑 𝐹𝑞𝑐 𝑁𝑞 + 2 𝐵𝛾𝐹𝛾𝑠 𝐹𝛾𝑑 𝐹𝛾𝑐 𝑁𝛾
Where Fcc, Fqc, and Fγc are soil compressibility factors
According to that theory, in order to calculate the soil compressibility factors, the following steps should be taken:
Step 1: Calculate the rigidity index, Ir, of the soil at a depth approximately B/2 below the bottom of the foundation:
𝐼𝑟 =
𝐺𝑠
𝑐 ′ +𝑞′ tan ∅′
Where Gs is the shear modulus of the soil and q’ is the effective overburden pressure at a depth of (Df + B/2)
Step 2: The critical rigidity index, Ir(cr), can be calculated from the offered table in the next table
Step 3: if 𝐼𝑟 ≥ 𝐼𝑟
𝑐𝑟
However if 𝐼𝑟 < 𝐼𝑟
, than Fcc= Fqc, = Fγc =1
𝑐𝑟
, use the offered tables in the next slide
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for Shallow Foundations
Effect of Soil Compressibility
Step 2: The critical rigidity index, Ir(cr), can be calculated from the offered table in the next table
Variation of Ir(cr) with ø’ and B/L (Das 2016)
Soil bearing capacity and Shallow Foundation
Ultimate Soil-Bearing Capacity for
Shallow Foundations
Effect of Soil Compressibility
Step 3: if 𝐼𝑟 ≥ 𝐼𝑟
𝑐𝑟
Then, Fcc= Fqc, = Fγc =1
However if 𝐼𝑟 < 𝐼𝑟 𝑐𝑟 , use the graphical
solutions for Fqc, and Fγc
For ∅′ = 0,
𝐵
𝐹𝑐𝑐 = 0.32 + 0.12 𝐿 + 0.6 log 𝐼𝑟
For ∅′ > 0
1−𝐹𝑞𝑐
𝐹𝑐𝑐 = 𝐹𝑞𝑐 − 𝑁
𝑞
tan ∅′