# Terzaghis Method

Courses > Foundation Analysis and Design > Bearing Capacity of Shallow Foundations > Terzaghis Method

## Introduction on Terzaghi's Method :

In geotechnical engineering, bearing capacity is the capacity of soil to support the loads applied to the ground. The bearing capacity of soil is the maximum average contact pressure between the foundation and the soil which should not produce shear failure in the soil. Ultimate bearing capacity (qf) is the theoretical maximum pressure which can be supported without failure; allowable bearing capacity (qa) is the ultimate bearing capacity divided by a factor of safety. Sometimes, on soft soil sites, large settlements may occur under loaded foundations without actual shear failure occurring; in such cases, the allowable bearing capacity is based on the maximum allowable settlement.

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There are three modes of failure that limit bearing capacity: general shear failure, local shear failure, and punching shear failure.

Terzaghi's bearing capacity method is the earliest method proposed in 1943. There are more advanced and accurate methods proposed later by Meyerhof (1951, 1963) and Brinch and Hansen (1970).

## Concepts and Formulas of Terzaghi's Method:

### Failure Mechanism:

Figure 1.  Shear stresses based on Terzaghi’s soil bearing capacity theory.

Zone I: A relatively undeformed wedge of soil below the foundation forms an active Rankine zone with angles (45º + f'/2).

Zone II: The transition zones take the form of log spiral fans.

Zone III: The wedge pushes soil outwards, causing passive Rankine zones to form with angles (45º - f'/2).

### Terzaghi’s 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.

### Terzaghi's Bearing capacity equations:

Strip footings:

Qu = c Ncg D Nq + 0.5 g B Ng

Square footings:

Qu = 1.3 c Ncg D Nq + 0.4 g B Ng

Circular footings:

Qu = 1.3 c Ncg D Nq + 0.3 g B Ng

Where:

C: Cohesion of soil (apparent cohesion intercept); g: unit weight of soil; D: depth of footing (depth of embedment); B: width/breadth of footing; Nc, Nq, Nr:  Terzaghi’s bearing capacity factors depend on soil friction angle, f:

$Nq = \frac{e^{(0.75\pi-\phi/2)tan\phi}}{cos^{2}(45+\phi/2)}$

$Nc = (Nq-1)cot\phi$

$N\gamma =\frac{tan\phi}{2}(\frac{K_{p}}{cos^{2}\phi}-1)$

Kp=passive pressure coefficient.

(Note: from Bowles' Foundation analysis and design book, "Terzaghi never explained..how he obtained Kp used to compute Ng")

Table 1. Terzaghi’s Bearing Capacity Factors.

 f Nc Nq Ng 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 45 172.3 173.3 297.5 48 258.3 287.9 780.1

Figure 2. Terzaghi’s bearing capacity factors.

## Solved sample problems of Terzaghi's Method:

### Example 1: Strip footing on cohesionless soil (British units)

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 (British units)

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 (British units)

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

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