Bearing capacity equations
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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,
If the ground is subjected to additional surcharge load q, then
. 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 [70], Burmister [21-23], Sowers and Vesic [62], Poulos and Davis [55], and Perloff
[54] 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 [52]. 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 [14]. 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. [28] developed the following equation to estimate settlements of footings on
sand using SPT data:
Where 0 and 1 are settlement influence factors dependent on footing geometry, depth of embedment, and depth to the relative incompressible layer (Figure 31.17), p is average applied pressure under service load and M is modulus of compressibility. The correlation between M and average SPT blow count is given in Figure 31.18. Barker et al. [9] discussed the commonly used procedure for estimating settlement of footing on sand using SPT blow count developed by Terzaghi and Peck [64,65] and Bazaraa [10].
FIGURE 31.18 Correlation between modulus of compressibility and average value SPT blow count. (After DAppolonia
et al [28].)1.6 FACTOR OF SAFETYIt is the factor of ignorance about the soil under consideration. It depends on many factors such as,
1. Type of soil
2. Method of exploration
3. Level of Uncertainty in Soil Strength
4. Importance of structure and consequences of failure
5. Likelihood of design load occurrence, etc.
Assume a factor of safety F = 3, unless otherwise specified for bearing capacity problems. Table 7.5 provides the details of factors of safety to be used under different circumstances.
Table 7.5 Typical factors of safety for bearing capacity calculation in different situations
7.12 Presumptive Safe Bearing Capacity
It is the bearing capacity that can be presumed in the absence of data based on visual identification at the site. National Building Code of India (1983) lists the values of presumptive SBC in kPa for different soils as presented below.
A : Rocks
Sl NoDescriptionSBC (kPa)
1Rocks (hard) without laminations and defects. For e.g. granite, trap & diorite3240
2Laminated Rocks. For e.g. Sand stone and Lime stone in sound condition1620
3Residual deposits of shattered and broken bed rocks and hard shale cemented material880
4Soft Rock440
B : Cohesionless Soils
Sl NoDescriptionSBC (kPa)
1Gravel, sand and gravel, compact and offering resistance to penetration when excavated by tools440
2Coarse sand, compact and dry440
3Medium sand, compact and dry245
4Fine sand, silt (dry lumps easily pulverized by fingers)150
5Loose gravel or sand gravel mixture, Loose coarse to medium sand, dry245
6Fine sand, loose and dry100
C : Cohesive Soils
Sl NoDescriptionSBC (kPa)
1Soft shale, hard or stiff clay in deep bed, dry440
2Medium clay readily indented with a thumb nail245
3Moist clay and sand clay mixture which can be indented with strong thumb pressure150
4Soft clay indented with moderate thumb pressure100
5Very soft clay which can be penetrated several centimeters with the thumb50
6Black cotton soil or other shrinkable or expansive clay in dry condition (50 % saturation)130 - 160
Note :
1. Use d for all cases without water. Use sat for calculations with water. If simply density is mentioned use accordingly.
2. Fill all the available data with proper units.
3. Write down the required formula
4. If the given soil is sand, c = 0REFERENCES:1. Joseph E. Bowles, Foundation Analysis and Design, Fifth Edition, McGraw Hill, New York, pp.213-343.2. Braja M. Das, Principles of Foundation Engineering, Sixth Edition, India edition, pp.81-146._1394151183.unknown
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