Terzaghi Shallow Foundation Bearing Capacity (q_u & Allowable q_all)

Use a defined shallow-foundation model

This worksheet demonstrates the classical Terzaghi general-shear equations for strip and square footings under centered vertical loading, with level ground and base. It is an arithmetic example, not a complete foundation design. Settlement can govern even when a shear-capacity check passes.

Assume homogeneous soil and one constant unit weight, with groundwater sufficiently deep not to alter the example. Groundwater, layered soil, eccentric or inclined loading, local/punching shear and seismic effects require additional analysis. Select soil parameters and bearing factors consistently for the drainage condition and design method.

Enter consistent units and supplied factors

Use cohesion c in kPa, unit weight γ in kN/m³, depth Df and width B in metres; pressures then come out in kPa. B2:B9 contain 20, 18, 1.5, 2, 37.16, 22.46, 19.13 and 3. These factors are supplied illustrative inputs; the sheet does not calculate friction angle or certify a factor table.

B11 requires numeric inputs, c and depth nonnegative, positive unit weight/width/Nc, Nq at least 1, Nγ nonnegative and FS greater than 1. The example limits depth to width or less. Passing these checks does not establish geotechnical suitability.

=IF(COUNT(B2:B9)<>8,"Check numeric inputs",IF(OR(B2<0,B3<=0,B4<0,B5<=0,B6<=0,B7<1,B8<0,B9<=1),"Check input ranges",IF(B4>B5,"Outside shallow example","OK")))

Calculate surcharge and strip ultimate pressure

B13 is γDf = 27 kPa. B14 adds cohesion 743.20, surcharge resistance 606.42 and self-weight resistance 344.34, giving gross ultimate pressure 1693.96 kPa. Keep results unrounded internally and format them for display.

=IF(B11<>"OK",B11,B2*B6+B13*B7+0.5*B3*B5*B8)

Calculate the square-footing variant

For the classical square variant, the cohesion multiplier is 1.3 and the coefficient on γBNγ is 0.4. B15 returns 1848.052 kPa. These coefficients already incorporate the shape adjustment; do not apply another shape-factor set on top.

=IF(B11<>"OK",B11,1.3*B2*B6+B13*B7+0.4*B3*B5*B8)

Distinguish net and gross allowable pressures

This example applies FS to net ultimate pressure: net ultimate = gross ultimate − surcharge; net allowable = net ultimate / FS; gross allowable = surcharge + net allowable. B17, B18 and B19 give 1666.96, 555.653333… and 582.653333… kPa for the strip.

Some exercises instead report gross ultimate / FS. B20 explicitly shows that different convention: 564.653333… kPa. It differs by q(1−1/FS), here 18 kPa. Do not label both numbers as interchangeable. Match the adopted convention and loading basis in the project specification.

=IF(B11<>"OK",B11,B13+B18)

Apply the same convention to a square footing

B22 uses the net-based factor of safety for the square example and returns gross allowable shear pressure 634.017333… kPa. FS=3 is illustrative, not a universal prescribed value. These shear-derived pressures still need settlement and other applicable design checks.

=IF(B11<>"OK",B11,B13+(B15-B13)/B9)

Try changes and reconcile the outputs

Set cohesion B2 to zero: strip ultimate becomes 950.76 kPa and square ultimate 881.892 kPa. Set depth B4 to zero: surcharge vanishes and the two gross-allowable conventions agree. Set FS to zero or erase a required input: all result cells show a message rather than dividing by zero.

For a controlled arithmetic scaling test, multiply both cohesion and unit weight by the same positive factor while keeping geometry and factors fixed. Every pressure should scale by that factor. This does not imply that a different real soil retains the same bearing factors.

Further reading

• UWE — Bearing capacity: strip equation, net pressure and factor of safety
• Environment Canada — Bearing capacity theory, square-footing equation 2b