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## Terzaghi's Bearing Capacity Equations

### Terzaghi’s bearing capacity theory Figure 1.1:  Shear stresses based on Terzaghi’s soil bearing capacity theory

Based on Terzaghi’s bearing capacity theory, column load P is resisted by shear stresses at edges of three zones under the footing and the overburden pressure, q (=gD) above the footing.  The first term in the equation is related to cohesion of the soil.  The second term is related to the depth of the footing and overburden pressure.  The third term is related to the width of the footing and the length of shear stress area.  The bearing capacity factors, Nc, Nq, Ng, are function of internal friction angle, f.

Strip footings:

Qu = c Nc + g D Nq + 0.5 g B Ng                                            [1.1]

Square footings:

Qu = 1.3 c Nc + g D Nq + 0.4 g B Ng                                      [1.2]

Circular footings:

Qu = 1.3 c Nc + g D Nq + 0.3 g B Ng                                       [1.3]

Where:

C: Cohesion of soil, g : unit weight of soil, D: depth of footing, B: width of footing

Nc, Nq, Nr:  Terzaghi’s bearing capacity factors depend on soil friction angle, f.

Nc=cotf(Nq –1)                                                                        [1.4]

Nq=e2(3p/4-f/2)tanf / [2 cos2(45+f/2)]                                   [1.5]

Ng=(1/2) tanf( Kpr /cos2 f -1)                                                   [1.6]

Kpr=passive pressure coefficient.

(Note: from Boweles, Foundation analysis and design, "Terzaghi never explained..how he obtained Kpr used to compute Ng")

Table 1: Terzaghi’s Bearing Capacity Factors

 f Nc Nq Nr 0 5.7 1 0 5 7.3 1.6 0.5 10 9.6 2.7 1.2 15 12.9 4.4 2.5 20 17.7 7.4 5 25 25.1 12.7 9.7 30 37.2 22.5 19.7 35 57.8 41.4 42.4 40 95.7 81.3 100.4 #### Example 1: Strip footing on cohesionless soil

Given:

• Soil properties:

• Soil type: cohesionless soil.

• Cohesion: 0 (neglectable)

• Friction Angle: 30 degree

• Unit weight of soil: 100 lbs/ft3

• Expected footing dimensions:

• 3 ft wide strip footing, bottom of footing at 2 ft below ground level

• Factor of safety: 3

RequirementDetermine allowable soil bearing capacity using Terzaghi’s equation.

Solution:

From Table 1 or Figure 1, Nc = 37.2, Nq = 22.5, Nr = 19.7 for f = 30 degree

Determine ultimate soil bearing capacity using Terzaghi’s bearing capacity equation for strip footing

Qu = c Nc + g D Nq + 0.5 g B Ng

= 0 +100x2x22.5+0.5x100x6x19.7

= 10410 lbs/ft2

Allowable soil bearing capacity,

Qa = Qu / F.S. = 10410 / 3 = 3470 lbs/ft2 @ 3500 lbs/ft2

#### Example 2: Square footing on clay soil

Given:

• Soil type:  Clay

• Soil properties:

• Cohesion:2000 lbs/ft2

• Friction Angle: 0 (neglectable)

• Unit weight of soil: 120 lbs/ft3

• Expected footing dimensions:

• 6 ft by 6 ft square footing, bottom of footing at 2 ft below ground level

• Factor of safety: 3

RequirementDetermine allowable soil bearing capacity using Terzaghi’s equation.

Solution:

From Table 1 or Figure 1, Nc = 5.7, Nq = 1.0, Nr = 0 for f = 0 degree

Determine ultimate soil bearing capacity using Terzaghi’s bearing capacity equation for square footing

Qu = 1.3 c Nc + g D Nq + 0.4 g B Ng

= 1.3x1000x5.7 +120x2*1+ 0

= 7650 lbs/ft2

Allowable soil bearing capacity,

Qa = Qu / F.S. = 7650 / 3 = 2550 lbs/ft2 @ 2500 lbs/ft2

#### Example 3: Circular footing on sandy clay

Given:

• Soil properties:

• Soil type: sandy clay

• Cohesion: 500 lbs/ft2

• Friction Angle: 25 degree

• Unit weight of soil: 100 lbs/ft3

• Expected footing dimensions:

• 10 ft diameter circular footing for a circular tank, bottom of footing at 2 ft below ground level

• Factor of safety: 3

Requirement

Determine allowable soil bearing capacity using Terzaghi’s equation.

Solution:

From Table 1 or Figure 1, Nc = 17.7, Nq = 7.4, Nr = 5.0 for f = 20 degree

Determine ultimate soil bearing capacity using Terzaghi’s bearing capacity equation for circular footing

Qu = 1.3 c Nc + g D Nq + 0.3 g B Ng

= 1.3x500x17.7 +100x2x7.4+0.3x100x10x5.0

= 17985 lbs/ft2

Allowable soil bearing capacity,

Qa = Qu / F.S. = 17985/ 3 = 5995 lbs/ft2 @ 6000 lbs/ft2