Bearing capacity is the soil pressure that causes foundation failure by shear. Terzaghi's shallow foundation equations split that resistance into three physical parts: cohesion, surcharge or embedment, and the weight of the soil involved in the failure wedge.
$$q_{ult}=q_c+q_q+q_\gamma$$
$$q=\gamma D_f$$
The term $q$ is the effective vertical stress at foundation level. It acts like a confinement pressure: a deeper footing has more overburden holding the soil in place, so it can usually resist more load.
Why the Bearing Capacity Factors Exist
The bearing capacity factors $N_c$, $N_q$, and $N_\gamma$ are multipliers that convert soil strength and geometry into ultimate pressure. They are not arbitrary constants; they come from Terzaghi's assumed failure mechanism under the footing.
$$q_c=cN_c$$
$$q_q=qN_q$$
$$q_\gamma=\frac{1}{2}\gamma BN_\gamma$$
$N_c$ multiplies cohesion because cohesion gives the soil a pressure-independent shear resistance. $N_q$ multiplies surcharge because deeper embedment confines the failure wedge. $N_\gamma$ multiplies soil unit weight and footing width because a wider, heavier wedge must move for failure to occur.
As $\phi$ increases, the soil mobilizes more friction. That is why $N_q$ and especially $N_\gamma$ rise rapidly with friction angle. A small increase in $\phi$ can make a large difference in sand bearing capacity.
Terzaghi Equations by Footing Shape
Terzaghi's original formulas change the coefficients in front of the cohesion and unit-weight terms depending on the footing shape. Shape matters because a square or circular footing has three-dimensional confinement, while a strip footing behaves closer to plane strain.
For rectangular footings, $B$ is the smaller dimension and $L$ is the longer dimension. If $B=L$, the rectangular expression becomes the square footing expression.
How the Terms Interact
The three terms add, but they do not all respond to changes the same way. Cohesion affects the $cN_c$ term only. Embedment affects the $qN_q$ term through $q=\gamma D_f$. Footing width affects the $\gamma BN_\gamma$ term, so width becomes especially powerful in frictional soils where $N_\gamma$ is large.
Groundwater lowers effective unit weight. That weakens the surcharge term if the water table is above the footing level, and weakens the footing-dimension term if the water table is within about one footing width below the base.
Water Table Corrections
Use effective unit weight where soil is submerged. The water correction mainly changes $q$ and the $\gamma B$ part of the third term.
$$\gamma'=\gamma_{sat}-\gamma_w$$
$$q=\gamma D_f \quad \text{when the soil above the base is dry or moist}$$
$$q=\gamma_1 z_1+\gamma_2' z_2 \quad \text{when the water table cuts the embedment depth}$$
$$\gamma_{eff}=\gamma \quad \text{when the water table is deeper than }B\text{ below the base}$$
If the water table is at the ground surface, both surcharge and width terms use submerged effective unit weight. If it is exactly at the footing base, surcharge is usually computed from the soil above the base, while the width term uses submerged unit weight below the base.
General Shear and Local Shear Failure
General shear failure is sudden and well-defined, usually associated with dense sand or stiff clay. Local shear failure is more gradual, with less obvious failure surfaces, commonly associated with loose sand or soft clay. The handout uses primed bearing capacity factors for local shear.
$$c'=\frac{2}{3}c \quad \text{for local shear calculations}$$
The primed factors are smaller because local shear cannot fully mobilize the ideal failure mechanism assumed in general shear.
Allowable, Gross, and Net Bearing Capacity
The ultimate value is divided by a factor of safety to get an allowable value. Gross values include the stress already present at foundation level. Net values subtract the existing overburden pressure.
$$q_{all}=\frac{q_{ult}}{FS}$$
$$q_{net,ult}=q_{ult}-q$$
$$q_{net,all}=\frac{q_{net,ult}}{FS}$$
$$Q_{all}=q_{all}A$$
Bearing Capacity Factor Values Used
The PDFs include full Terzaghi factor tables. These are common values used in the worked problems on this page.
$\phi$
$N_c$
$N_q$
$N_\gamma$
Failure mode
10°
9.61
2.69
0.56
General
15°
12.86
4.45
1.52
General
20°
17.69
7.44
3.64
General
25°
25.13
12.72
8.34
General
30°
37.16
22.46
19.13
General
10°
8.02
1.94
0.24
Local table
Some solved items use rounded or problem-specified factors, such as $N_c=9$, $N_q=2.5$, $N_\gamma=1.2$ for Problem 153 and $N_c'=7.5$, $N_q'=1.8$, $N_\gamma'=0.48$ for local shear examples.
Problem 147: Square Footing Contributions
A square footing 4 m on a side is founded 1.2 m below the surface. Soil has $\gamma=20$ kN/m3, $c=10$ kPa, and $\phi=20^\circ$. Use $N_c=17.69$, $N_q=7.44$, and $N_\gamma=3.64$. Evaluate the contributions to $q_{ult}$.
A 2.00 m square footing is founded 1.00 m into cohesionless soil with $\gamma=16$ kN/m3 and $\phi=25^\circ$. Use Terzaghi general shear, $N_q=12.7$, $N_\gamma=8.34$, and $FS=3$.
A 0.90 m square footing is embedded 1.00 m into cohesionless soil with $\gamma=18$ kN/m3 and $\phi=30^\circ$. Use $N_q=22.46$, $N_\gamma=19.13$, and $FS=3$.
Answer: embedment = about 405 kPa, dimension = about 124 kPa, safe load = about 143 kN.
Problem 150: Maximum Load with Water Correction
A 1.5 m by 1.5 m square footing carries a concentric load. Use $FS=2.8$, $\gamma=19.6$ kN/m3, $\phi=34^\circ$, $N_c=42.16$, $N_q=29.44$, and $N_\gamma=31.15$. The extracted solution uses a water correction for the footing-width term.
A 2 m square footing is founded 1.2 m below ground. Soil has $\gamma=20$ kN/m3, $c=10$ kPa, and $\phi=15^\circ$. Use $N_c=12.86$, $N_q=4.45$, and $N_\gamma=1.52$.
A 1.2 m by 1.2 m square footing has its base 1 m below ground. Use $N_c=35$, $N_q=22$, $N_\gamma=19$, $c=1605$ kg/m2, $\gamma=1846$ kg/m3, $\gamma_{sat}=1965$ kg/m3, and $FS=3$.
Calculate the safe load capacity of a rectangular footing 6 m long and 1.25 m wide. Given $c=50$ kPa, $D_f=4.5$ m, $\gamma=19.2$ kN/m3, $\phi=10^\circ$, $FS=2.5$, $N_c=9$, $N_q=2.5$, and $N_\gamma=1.2$.
A circular footing 1.5 m in diameter is constructed 1.22 m below the ground. Soil has $\phi=25^\circ$, unconfined compressive strength $q_u=96$ kPa, and $\gamma=20.12$ kN/m3. Use $N_c=24.1$, $N_q=13.1$, $N_\gamma=9.1$, and $FS=3$.
Problem 155: Net Allowable Capacity for Local Shear
A 1.5 m by 1.5 m footing has its base 2 m below ground. Soil has $\phi=10^\circ$, $\gamma=18$ kN/m3, $c=60$ kPa, and $\gamma_{sat}=20$ kN/m3. The water table is 0.80 m below ground. Use $N_c'=7.5$, $N_q'=1.8$, $N_\gamma'=0.48$, and $FS=3$.
Answer: net allowable bearing capacity is 138.08 kPa.
Problem 156: Net Ultimate Capacity for Strip Footing
For a soil with $c=50$ kPa, $\gamma=19.2$ kN/m3, and $\phi=10^\circ$, calculate net ultimate bearing capacity for a 1.25 m strip footing at depth $D_f=4.5$ m. Use local shear factors $N_c'=7.5$, $N_q'=1.8$, and $N_\gamma'=0.48$.
Answer: net ultimate bearing capacity is 324.88 kPa.
Concept: Water Table Correction in Terzaghi's Formula
Terzaghi's formula $q_{ult}=cN_c+qN_q+0.5\gamma BN_\gamma$ uses two unit weights: $q=\gamma D_f$ for overburden, and $\gamma$ in the $N_\gamma$ term for soil within the failure zone (roughly $B$ deep below the footing). A high water table reduces both by converting to effective (buoyant) unit weight $\gamma'=\gamma_{sat}-\gamma_w$.
Three practical cases for depth of water table $d_w$ below ground surface:
$d_w \leq D_f$ (at or above footing): $q$ accounts for saturated soil above footing; use $\gamma'$ in the $N_\gamma$ term.
$D_f < d_w \leq D_f+B$ (within failure zone): interpolate $\gamma_{\text{eff}} = \gamma' + \dfrac{d_w-D_f}{B}(\gamma-\gamma')$ for the $N_\gamma$ term.
$d_w > D_f+B$ (below failure zone): no correction needed; use moist unit weight throughout.
The practical impact: a water table at the footing level roughly halves the $N_\gamma$ contribution, reducing ultimate capacity by 20–40% in granular soils.
Problem: Required Footing Width — Square Footing on Soft Clay
A square footing at depth $D_f=2.0$ m is to support a column load of 800 kN. The soil is saturated clay with $c=60$ kPa, $\phi=0°$, and $\gamma_{sat}=18$ kN/m³. Use Terzaghi general-shear factors $N_c=5.7$, $N_q=1.0$, $N_\gamma=0$ and a factor of safety of 3 on net capacity. Find the required footing width $B$.
Problem: Strip Footing with Water Table at Foundation Level
A 1.5 m wide strip footing is placed at $D_f=1.2$ m. Soil above the water table has $\gamma=17.5$ kN/m³; below the water table (which is at foundation level) $\gamma_{sat}=20$ kN/m³. The soil has $c=0$, $\phi=32°$. Use $N_c=44.04$, $N_q=28.52$, $N_\gamma=26.87$, and $FS=3$. Find the net allowable bearing capacity and the safe load per meter of wall.
Water table at footing level: $q = \gamma D_f = 17.5(1.2) = 21$ kPa. Use $\gamma'=20-9.81=10.19$ kN/m³ in $N_\gamma$ term.
Answer: $q_{net,all}=261.2$ kPa; safe wall load $=391.8$ kN per meter of wall length.
Problem: Allowable Load on Circular Footing
A circular footing has diameter $D=2.0$ m and is placed at $D_f=1.5$ m in soil with $c=30$ kPa, $\phi=20°$, and $\gamma=17$ kN/m³. Use Terzaghi factors $N_c=17.69$, $N_q=7.44$, $N_\gamma=3.64$ and $FS=3$ on gross capacity. Find the allowable total column load.
For a circular footing: $q_{ult}=1.3cN_c+qN_q+0.3\gamma D N_\gamma$.
Problem: Check Factor of Safety on Existing Square Footing
An existing 1.5 m by 1.5 m square footing at $D_f=1.5$ m supports a column load of 600 kN. Soil properties: $c=25$ kPa, $\phi=20°$, $\gamma=17$ kN/m³. Terzaghi factors: $N_c=17.69$, $N_q=7.44$, $N_\gamma=3.64$. Compute the gross factor of safety against general shear failure.
Answer: $FS=3.01$ — the footing is just adequate with a safety factor slightly above 3.
Problem: Strip Footing Width from Given Load per Meter
A continuous wall footing at $D_f=1.5$ m must carry a wall load of 180 kN/m. Soil: $c=0$, $\phi=30°$, $\gamma=17.5$ kN/m³. Use $N_c=37.16$, $N_q=22.46$, $N_\gamma=19.13$ and $FS=3$ on gross capacity. Find the required footing width $B$.
