Foundation Design Principles and Practices

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6.1 Bearing Capacity Failures 171

6

Shallow

Foundations-Bearing Capacity

When we are satisfied with the spot fIXed on for the site of the city

the foundations should be carried down to a solid bottom, if such can be found, and should be built thereon of such thickness as may be necessary for the proper support of that part of the wall which stands above the natural level of the ground They should be of the soundest workmanship and materials, and of greater thickness than the walls above If solid ground can be come to, the foundations should go down

to it and into it, according to the magnitude of the work, and the substruction to be built up as solid as possible Above the ground of the foundation, the wall should be one-half thicker than the column it is to receive so that the lower parts which carry the greatest weight, may be stronger than the upper part Nor must the mouldings of the bases of the columns project beyond the solid Thus, also, should be regulated the thickness of all walls above ground.

Marcus Vitruvius, Roman Architect and Engineer

1stcentury H.C.

as translated by Morgan (1914)

Shallow foundations must satisfy various performance requirements, as discussed in

Chapter 2 One of them is called bearing capacity, which is a geotechnical strength

re-quirement This chapter explores this requirement, and shows how to design shallowfoundations so that they do not experience bearing capacity failures

Shallow foundations transmit the applied structural loads to the near-surface soils In theprocess of doing so, they induce both compressive and shear stresses in these soils Themagnitudes of these stresses depend largely on the bearing pressure and the size of thefooting If the bearing pressure is large enough, or the footing is small enough, the shear

stresses may exceed the shear strength of the soil or rock, resulting in a bearing capacity

failure Researchers have identified three types of bearing capacity failures: general shear failure, local shear failure, and punching shear failure, as shown in Figure 6.1 A typical

load-displacement curve for each mode of failure is shown in Figure 6.2

General shear failure is the most common mode It occurs in soils that are relativelyincompressible and reasonably strong, in rock, and in saturated, normally consolidatedclays that are loaded rapidly enough that the undrained condition prevails The failuresurface is well defined and failure occurs quite suddenly, as illustrated by the load-displacement curve A clearly formed bulge appears on the ground surface adjacent to thefoundation Although bulges may appear on both sides of the foundation, ultimate failureoccurs on one side only, and it is often accompanied by rotations of the foundation.The opposite extreme is the punching shear failure It occurs in very loose sands, in

a thin crust of strong soil underlain by a very weak soil, or in weak clays loaded underslow, drained conditions The high compressibility of such soil profiles causes large set-tlements and poorly defined vertical shear surfaces Little or no bulging occurs at theground surface and failure develops gradually, as illustrated by the ever-increasing load-settlement curve

Local shear failure is an intermediate case The shear surfaces are well definedunder the foundation, and then become vague near the ground surface A small bulge mayoccur, but considerable settlement, perhaps on the order of half the foundation width, isnecessary before a clear shear surface forms near the ground Even then, a sudden failuredoes not occur, as happens in the general shear case The foundation just continues to sinkever deeper into the ground

Vesic (1973) investigated these three modes of failure by conducting load tests onmodel circular foundations in a sand These tests included both shallow and deep founda-tions The results, shown in Figure 6.3, indicate shallow foundations (D/Bless than about2) can fail in any of the three modes, depending on the relative density However, deepfoundations (DIBgreater than about 4) are always governed by punching shear Althoughthese test results apply only to circular foundations in Vesic's sand and cannot necessarily

be generalized to other soils, it does give a general relationship between the mode of ure, relative density, and theDIBratio

fail-Complete quantitative criteria have yet to be developed to determine which of thesethree modes of failure will govern in any given circumstance, but the following guidelinesare helpful:

• Shallow foundations in rock and undrained clays are governed by the general shearcase

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172 Chapter 6 Shallow Foundations-Bearing Capacity 6.2 Bearing Capacity Analyses in Soil-General Shear Case

Load

C

<IJ E

circles indicate various interpretations of

failure (Adapted from Vesic 1963).

C

<IJ E

~tl

</l

C

<IJ E

Figure 6.1 Modes of bearing capacity failure: (a) general shear failure: (b) local shear

failufe: (c)punching shear failure.

I

~

• Shallow foundations in dense sands are governed by the general shear case In this

context, a dense sand is one with a relative density, D" greater than about 67%.

• Shallow foundations on loose to medium dense sands (30% <Dr< 67%) are bly governed by local shear

proba-• Shallow foundations on very loose sand (Dr < 30%) are probably governed bypunching shear

For nearly all practical shallow foundation design problems, it is only necessary tocheck the general shear case, and then conduct settlement analyses to verify that the foun-dation will not settle excessively These settlement analyses implicitly protect againstlocal and punching shear failures

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174 Chapter 6 Shallow Foundations-Bearing Capacity

shal-Occasionally, geotechnical engineers perform more detailed bearing capacity ses using numerical methods, such as the finite element method (FEM) These analysesare more complex, and are justified only on very critical and unusual projects

analy-We will consider only limit equilibrium methods of bearing capacity analyses, cause these methods are used on the overwhelming majority of projects

be-Simple Bearing Capacity Formula

Methods of Analyzing Bearing Capacity

Figure 6.3 Modes of failure of model

cir-cular foundations in Chattahoochee Sand

(Adapted from Yesic 1963 and 1973).

The objective of this derivation is to obtain a formula for the ultimate bearing

ca-pacity, qui" which is the bearing pressure required to cause a bearing capacity failure By

To analyze spread foatings for bearing capacity failures and design them in a way toavoid such failures, we must understand the relationship between bearing capacity, load,footing dimensions, and soil properties Various researchers have studied these relation-ships using a variety of techniques, including:

• Assessments of the performance of real foundations, including full-scale load tests

• Load tests on model footings

• Limit equilibrium analyses

Detailed stress analyses, such as finite element method (FEM) analyses

Full-scale load tests, which consist of constructing real spread footings and loadingthem to failure, are the most precise way to evaluate bearing capacity However, suchtests are expensive, and thus are rarely, if ever, performed as a part of routine design Afew such tests have been performed for research purposes

Model footing tests have been used quite extensively, mostly because the cost ofthese tests is far below that for full-scale tests Unfortunately, model tests have their limi-tations, especially when conducted in sands, because of uncertainties in applying the

i-1

Plb

D

Figure 6.4 Bearing capacity analysis along a circular failure surface.

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obtain the following:

MA =(qultBb)(B/2) - (su'rrBb)(B) - er,oBb(B/2) quit =2 '11"Su +er:o

(6.1)(6.2)

p

j

Terzaghi's Bearing Capacity Formulas

ca-pacity formulas.

(6.3)

Iquit =Nesu +er:o I

Radial Shear Zone Figure 6.5 Geometry of failure surface for Terzaghi's bearing capacity formulas.

lin-earshear zonein which the soil shears along planar surfaces.

For square foundations:

throughout).

o The shear strength of the soil is described by the formula s=c'+er' tan <1>'.

[ quit = 1.3c' Ne +er;oNq +0.4-/BN~ I

(6.4)

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178 Chapter 6 Shallow Foundations-Bearing Capacity 6.2 Bearing Capacity Analyses in Soil-General Shear Case 179

These bearing capacity factors are also presented in tabular form in Table 6.1 Notice thatTerzaghi's Ne of 5.7 is smaller than the value of 6.28 derived from the simple bearing ca~

pacity analysis This difference the result of using a circular failure surface in the simplemethod and a more complex geometry in Terzaghi's method

Because of the shape of the failure surface, the values of e' and<1>'only need to resent the soil between the bottom of the footing and a depth B below the bottom Thesoils between the ground surface and a depth Dare treated simply as overburden

rep-Terzaghi's formulas are presented in terms of effective stresses However, they alsomay be used in a total stress analyses by substituting eT' <l>T>andaDfore', <1>',andaD" Ifsaturated undrained conditions exist, we may conduct a total stress analysis with the shearstrength defined aseT= SIIand<l>T= O In this case,Ne= 5.7,N'I=1.0,andNe=0.0.

The Terzaghi bearing capacity factors are:

Where:

quit= ultimate bearing capacitye'=effective cohesion for soil beneath foundation

<I>'= effective friction angle for soil beneath foundation

a,p' = vertical effective stress at depth Dbelow the ground surface

-y'=effective unit weight of the soil(-y=-y'if groundwater table is very deep;

see discussion later in this chapter for shallow groundwater conditions)

D= depth of foundation below ground surface

B= width (or diameter) of foundation

N" N,I' N~=Terzaghi's bearing capacity factors =!(<I>')(See Table 6.1 or Equations

6.7-6.12.)

(for use in Equations 6.4-6.6)

(for use in Equation 6.13)

N,

Nq

N~ Ne N~

0

5.7 1

6.0 2

6.3 3

6.6 4

7.0 5

7.3 6

7.7 7

8.2 8

8.6 9

9.1 10

9.6 11

10.2 3.0 12

10.8 3.3 13

11.4 3.6 14

12.1 4.0 10.4 3.6 15

12.9 4.4 11.0 3.9 16

13.7 4.9 11.6 4.3 17

14.6 5.5 12.3 4.8 18

15.5 6.0 13.1 5.3 19

16.6 6.7 13.9 5.8 20

17.7 7.4 14.8 6.4 21

18.9 8.3 15.8 7.1 22

20.3 9.2 16.9 7.8 23

21.7 6.8 18.0 8.7 24

23.4 7.9 19.'39.625

25.1 9.2 20.7 26

27.1 27

29.2 28

31.6 29

34.2 30

37.2 31

40.4 32

44.0 33

48.1 34

52.6 35

57.8 36

63.5 37

70.1 38

77.5 39

86.0 40

95.7 121.5 75.3 109.4 41

106.8 148.5 93.8 83.9 130.2

(6.8)

(6.9)(6.7)(6.6)

(6.11)

(6.10)for<1>'> 0

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