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In 1943, Karl Terzaghi expanded on Prantl's 1921 study on the penetration of hard bodies on softer materials, the plastic failure theory. Terzaghi used this theory to determine the bearing capacity of soils for shallow strip footing.

Calculation

Inputs

Footing Geometry

Geometry



Footing shape
:Square​



B
:1m​



Df
:1m​



D
:0m​


Soil Properties



γ
:18.11kN/m3​



Saturated soil unit weight
:19kN/m3​



ϕ
:30deg​



Water Table Cases




Case
:II​


Output



Factor of safety, FoS
:2​


General Shear Failure



(General) - qu
:1240kPa​



(General) - qu(net)
:620kPa​


Local Shear Failure



(Local) - qu
:426kPa​



(Local) - qu(net)
:213kPa​


Explanation

According to Terzaghi, a foundation is considered shallow if the footing depth, D, is less than or equal to its width, B. Additionally, he assumed that a uniform surcharge, q = γDf can replace the weight of soil above the base of the footing.

Terzaghi's bearing capacity equations are based on the following assumptions:
  1. The width of the foundation is equal or greater than its depth (i.e. B > D)
  2. No sliding between foundation and soil
  3. The soil beneath the foundation is a homogenous, semi-infinite mass
  4. The general shear failure governs
  5. No applied moments
  6. The applied load is compressive and is vertically applied to the foundation centroid
Terzaghi's equation consists of three components:
  1. the cohesion of the soil edge,
  2. the surcharge,
  3. the friction along the soil edge.
Combining these three together, gives the general equation:

qu=KccNc+KqqNq+Kγγc BNγ where:qu = ultimate bearing capacityγc =unit weight of soilB = width of the footing c = soil cohesionNγ = factor for unit weight of soilNc = factor of soil cohesionNq = factor of overburden pressureq = overburden pressure (effective stress)Kc,Kq,Kγ = constants q_u = K_ccN_c +K_qqN_q + {K_\gamma}{\gamma_c\ }BN_{\gamma}\ \\\text{where:}\\q_u\ =\text{\ ultimate\ bearing\ capacity}\\\gamma_c\ = \text{unit\ weight\ of\ soil} \\B\ =\ \text{width\ of\ the\ footing}\ \\c\ =\ \text{soil\ cohesion}\\N_\gamma\ =\ \text{factor\ for\ unit\ weight\ of\ soil}\\N_c\ =\ \text{factor\ of\ soil\ cohesion}\\N_q\ =\ \text{factor\ of\ overburden\ pressure}\\q\ =\ \text{overburden\ pressure\ (effective\ stress)}\\K_c,K_q,K_\gamma\ =\ \text{constants}

General Shear Failure

For general shear failure, the ultimate bearing capacity is dependent on the footing shape:
Strip footing

qu=cNc+qNq+0.5γc BNγq_u = cN_c +qN_q + 0.5{\gamma_c\ }BN_{\gamma}
Square footing

qu=1.3cNc+qNq+0.4γc BNγq_u = 1.3cN_c +qN_q + 0.4{\gamma_c\ }BN_{\gamma}
Circular Footing

qu=1.3cNc+qNq+0.3γc BNγq_u = 1.3cN_c +qN_q + 0.3{\gamma_c\ }BN_{\gamma}


Local Shear Failure

For local shear failure, it is assumed that the soil cohesion is reduced:

cˉ = 23c\bar{c}\ =\ \frac{2}{3}c
Similar to 'general shear failure', the ultimate bearing capacity is dependent on the footing shape:
Strip footing

qu =cˉNc′ + qNq′ +0.5γcBNγ′q_u\ =\bar{c}N'_c\ +\ qN'_q\ +0.5\gamma_cBN'_{\gamma}
Square footing

qu = 1.3cˉNc′ + qNq′ + 0.4γc BNγ′q_u\ =\ 1.3\bar{c}N'_c\ +\ qN'_q\ +\ 0.4\gamma_c\ BN'_{\gamma}
Circular footing

qu=1.3cNc′+qNq′+0.3γc BNγ′q_u = 1.3cN'_c +qN'_q + 0.3{\gamma_c\ }BN'_{\gamma}

Allowable Bearing Capacity

A factor of safety (FoS) is incorporated into the footing design, which reduces the ultimate bearing capacity. FoS generally varies anywhere from 1.5 ~ 3, depending on the uncertainty in soil properties. The allowable bearing capacity is calculated as:

qallowable =quFoSq_{allowable}\ =\frac{q_u}{FoS}

Effect of Water Table

The equations above assume the groundwater table depth is much greater than the footing width B. However, in other cases wherein the water table is near or above the footing, the subsoil becomes saturated and the unit weight of the submerged soil is reduced, resulting in a decrease in the soil's ultimate bearing capacity.
Case 1: Water table is above the footing base

 q = γ(Df−D) + γ′D γc = γ′ = γsat−γw\ q\ =\ \gamma(D_f-D)\ +\ \gamma'D\ \\\gamma_c\ =\ \gamma'\ =\ \gamma_{sat}-\gamma_w
Case 2: Water table is at the footing base

q = γ Dfq\ =\ \gamma\ D_f
Additionally, γ in the third term of the bearing capacity equations should be replaced by γ'.
Case 3: Water table is below the footing base

For D≤B:     γ = γave =1B[γ D + γ′(B−D)]For D > B:     γ = unchanged\text{For\ D}\leq\text{B:\ }\ \ \ \ \gamma\ =\ \gamma_{ave}\ =\frac{1}{B}[\gamma\ D\ +\ \gamma'(B-D)]\\\text{For\ D\ >\ B:\ }\ \ \ \ \gamma\ =\ \text{unchanged}


References

  1. Foundations of Geotechnical Engineering by DIT Gillesania
  2. Principles of Geotechnical Engineering 7th Edition by Braja M. Das

Related Resources

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