# Bearing capacity equations

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Different types of bearing capacity equations of soil

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Bearing Capacity

BEARING CAPACITY EQUATIONS IN DESIGNING SHALLOW FOUNDATIONS1.1 BEARING CAPACITY EQUATIONS:1.1.1 Terzaghis bearing Capacity TheoryTerzaghi (1943) was the first to present a comprehensive theory for evaluating the ultimate bearing capacity of shallow foundations with rough base.According to this theory a foundation is shallow if its depth Df is less than or equal to its width. Later investigators, however, have suggetsed that foundation with Df equal to 3 to 4 times their width may be defined as shallow foundations.

Fig.1.1: Typical Shallow Foundation

Fig.1.3: Terzaghis concept of Footing with five distinct failure zones in foundation soil

Ultimate bearing capacity,

. Net ultimate bearing capacity, Or,

Safe or allowable bearing capacity,

Here, F = Factor of safety (usually2-3)c = cohesion of soil, = unit weight of soil, D = Depth of foundation

q = Surcharge at the ground level, B = Width of foundationNc, Nq, N = Terzaghis bearing capacity factors depend on soil friction angle .Where:

Nc = cot(Nq 1) Nq = (e 2(3/4-/2)tan ) / 2 cos2(45+/2)N = 1/2 tan ( Kpr /cos2 -1)

Kpr = passive pressure coefficient.Table 1: Bearing capacity factors for different

NcNqNgN'cN'qN'g

05.71.00.05.71.00.0

57.31.60.56.71.40.2

109.62.71.28.01.90.5

1512.94.42.59.72.70.9

2017.77.45.011.83.91.7

2525.112.79.714.85.63.2

3037.222.519.719.08.35.7

3452.636.535.023.711.79.0

3557.841.442.425.212.610.1

4095.781.3100.434.920.518.8

45172.3173.3297.551.235.137.7

48258.3287.9780.166.850.560.4

50347.6415.11153.281.365.687.1

Effect of shape of Foundation

The following are the corrections for circular, square and rectangular footings.Circular footing,

Square footing,

Rectangular footing,

Table 2: Summary of Shape factors for different shapes of footing

Shapescsqs

Strip111

Square1.310.8

Round1.310.6

Rectangle

1

The equation for bearing capacity explained above is applicable for soil experiencing general shear failure. If a soil is relatively loose and soft, it fails in local shear failure. Such a failure is accounted in bearing capacity equation by reducing the magnitudes of strength parameters c and as follows.

Table summarizes the bearing capacity factors to be used under different situations. If is less than 360 and more than 280, it is not sure whether the failure is of general or local shear type. In such situations, linear interpolation can be made and the region is called mixed zone.

Table 3: Bearing capacity factors in zones of local, mixed and general shear conditions.

Local Shear FailureMixed ZoneGeneral Shear Failure

< 28o28o < < 36o > 36o

Nc1, Nq1, N1Ncm, Nqm, NmNc, Nq, N

1.1.2 General Bearing Capacity EquationIt is evident that Tergaghis equation is only valid for the case of general shear failure because no soil compression is allowed before the failure occurs.

Meyerhof, Hansen, and Vesic further extended Terzaghis bearing capacity equation to account for footing shape (si), footing embedment depth (d1), load inclination or eccentricity (ii), sloping ground (gi), and tilted base (bi). Chen reevaluated N factors in Terzaghis equation using the limit analysis method. These efforts resulted in significant extensions of Terzaghis bearing capacity equation. The general form of the bearing capacity equation can be expressed as:qu = c.Nc. Sc. dc. ic + q.Nq. Sq. dq. iq + 0.5.BN. S. d. i

Equations are available for shape factors (sc, sq, s), depth factors (dc, dq, d) and load inclination factors (ic, iq, i). The effects of these factors are to reduce the bearing capacity.Table 4: Bearing capacity factors for general bearing capacity equation

Note: Nc and Nq are same for all four methods; subscripts identify author for M = Meyerhof; H = Hansen; V = Vesic; C = Chen.

Vesic suggested that a flat reduction of might be too conservative in the case of local and

punching shear failure. He proposed the following equation for a reduction factor varying with

relative density Dr:

Bearing Capacity from Standard Penetration Test (SPT)The SPT is widely used to obtain the bearing capacity of soils directly .Meyerhof (1956, 1974) published equation for computing the allowable bearing capacity for a 25 mm (1 inch) settlement.

IN FPS SYSTEM:

qall = but (1+0.33)1.33 and B 4.0 ft.qall = but (1+0.33)1.33 and B > 4.0 ft.

Where: qall = Allowable bearing pressure in ksf, for H = 1inch settlement.D = Depth of foundation (ft)

B = Width of foundation (ft).IN SI UNIT:

qall = but (1+0.33)1.33 and B 1.2 m.qall = but (1+0.33)1.33 and B > 1.2 m.Where: qall = Allowable bearing pressure in kpa, for H = 25 mm settlement.

D = Depth of foundation (m)

B = Width of foundation (m).The Standard blow count can be computed from the measured N as follows:

= CN.N.

EMBED Equation.3

EMBED Equation.3 Where: CN=and

= Effective overburden pressure in (kpa).Hammer for = 1.14 (normally)Rod length correction = 1.00 when rod length >10.0m = 0.95 when rod length 6-10m

= 0.85 when rod length 4-6m

Sampler correction = 1.00 without liner.Borehole diameter correction= 1.00 for 60 mm-120 mm = 1.05 for 150 mm.The allowable soil pressure for any settlement Hj is =. Where Ho = 25 mm or 1 inch and Hj = settlement that can be tolerated in mm or inch.Parry (1977) proposed computing the allowable bearing capacity of cohesion less soils as:

qa = 30N55 (kpa) for DBWhere, N55 is the average SPT value at a depth about 0.75B below the proposed base of the footing. The allowable bearing pressure qa is computed for settlement checking as qa = (kpa) for a Ho = 20 mmAngle of internal friction can be calculated by using SPT value as:

Here, is the effective overburden pressure at the location of the average N55 count.

N/avg. is an average value of the SPT blow counts, which is determined within the range of depths

from footing base to 1.5B below the footing. In very fine or silty saturated sand, the measured SPT blow count (N) is corrected for submergence effect as follows:

BEARING CAPACITY FROM PLATE LOAD TESTFor clay soil qult is independent of footing size, giving qult,foundation = qult, load test and for (c-) soil qult,foundation = M+ N where M includes the Nc and Nq terms and N is the term. Practically for sand use the following relation qult,foundation = qult, load test.The use of this equation is not recommended unless the is not much more atan about 3.0.Housels method for bearing capacity from plate load testHousels (1929) and Williams (1929) both gave an equation for using at least two plate load tests to obtain an allowable load Ps for some settlement as Ps = Aq1 + pq2 (kpa or ksf)Where A = area of plate used for the load test, m2 or ft2.P = perimeter of the load test plate, m or ft.

q1= bearig pressure of interior zone of plateq2= edge shear of plate31.5.2 Layered SystemsWestergaard , Burmister [21-23], Sowers and Vesic , Poulos and Davis , and Perloff

 discussed the solutions to stress distributions for layered soil strata. The reality of interlayer

shear is very complicated due to in situ nonlinearity and material inhomogeneity [37,54]. Either

zero (frictionless) or with perfect fixity is assumed for the interlayer shear to obtain possible

FIGURE 31.9 Pressure bulbs based on the Bossinesq equation for square and long footings. (After NAVFAC 7.01,

1986].)

solutions. The Westergaard method assumed that the soil being loaded is constrained by closed

spaced horizontal layers that prevent horizontal displacement . Figures 31.10 through 31.12 by the Westergaard method can be used for calculating vertical stresses in soils consisting of alternative layers of soft (loose) and stiff (dense) materials.

31.5.3 Simplified Method (2:1 Method)

Assuming a loaded area increasing systemically with depth, a commonly used approach for computing the stress distribution beneath a square or rectangle footing is to use the 2:1 slope method as shown in Fig. below. Sometimes a 60 distribution angle (1.73to1 slope) may be assumed.The pressure increase .q at a depth z beneath the loaded area due to base load P is

FIGURE 31.10 Vertical stress contours for square and strip footings [Westerqaard Case].(After NAVFAC 7.01, 1986.) Where symbols are referred to Figure 31.14. The solutions by this method compare very well with those of more theoretical equations from depth z from B to about 4B but should not be used for depth z from 0 to B . A comparison between the approximate distribution of stress calculated by a theoretical method and the 2:1 method is illustrated in Figure 31.15.

31.6.2 Settlement of Shallow Foundations on Sand

SPT Method

DAppolonio et al.  developed the following equation to estimate settlements of footings on

sand using SPT data:

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