geotech3 ls7 bearing capacity

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Geotech3 LS7 Bearing Capacity

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  • THE UNIVERSITY OF ADELAIDE SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING

    GEOTECHNICAL ENGINEERING DESIGN III

    M. B. Jaksa

    BEARING CAPACITY OF SHALLOW FOUNDATIONS

    References: Bowles, J. E. (1996). Foundation Analysis and Design, 5th ed., McGraw-Hill, 1175p. Coduto, D. P. (1994). Foundation Design - Principles and Practices, Prentice Hall, 796p. Craig, R. F. (2004). Soil Mechanics, 7th ed., Spon Ltd., 464p. Das, B. M. (1995). Principles of Foundation Engineering, 3rd ed., PWS Publ. Co., 828p. Peck, R. B., Hanson, W. E. and Thornburn, T. H. (1974). Foundation Engineering, 2nd ed., Wiley, 514p. Schmertmann, J. H. (1978). Guidelines for the Cone Penetration Test - Performance and Design. Report No. FHWA-TS-78-209, U.S. Dept. Transportation, Federal Highway Administration, Washington, 145p. Tomlinson, M. J. (1986). Foundation Design and Construction, 5th ed., Longman, 842p. Whitlow, R. (1990). Basic Soil Mechanics, 2nd ed., Longman, 528p.

    1. INTRODUCTION

    An integral part of foundation, or footing, design is that the underlying soil or rock is able to support the loads imposed by the foundation. In other words, the subsurface material must not fail in shear. This section investigates the bearing capacity of soils and rocks.

    Before we investigate the various theories used to quantify the bearing capacity of geotechnical materials, it is necessary to define a few key terms:

    ultimate bearing capacity, qu , is defined as the pressure at which the material beneath the foundation fails in shear.

    net bearing pressure, qn , is defined as the pressure applied to the foundation minus the overburden pressure.

    allowable bearing pressure, qall , is defined as the ultimate bearing capacity divided by some assumed factor of safety, FS. That is:

    q qFSall

    u= (1.1)

    where: FS is usually between 2 and 3.

    The allowable bearing pressure is the pressure which can safely be applied to the foundation such that shear failure is unlikely to occur. Can you suggest reasons why geotechnical engineering applies larger factors of safety than structural engineering?

    Typical ultimate bearing capacities for various soil and rock types are given in Table 1.1. Note that these values should be used for preliminary design purposes only.

  • 2

    Table 1.1 Typical ultimate bearing capacities for various soil and rock types. (Source: Whitlow, 1990.)

    Soil and Rock Types Ultimate Bearing Capacity, qu (kPa)

    Remarks

    Rocks Hard igneous or gneissic rocks 10,000 Only sound unweathered Hard limestones and sandstones 4,000 rocks. Thinly bedded or Schists and slates 3,000 heavily jointed rocks Hard shales and mudstones; soft sandstones 2,000 must be assessed after Soft shales and mudstones 600 - 1,000 inspection. Hard sound chalk; soft limestone 600 Cohesionless soils Dense gravel or sand/gravel > 600 Providing: Medium-dense gravel or sand/gravel 200 - 600 the footing width B > 1 m Loose gravel or sand/gravel < 200 and the groundwater level Dense sand > 300 > B metres below the base Medium-dense sand 100 - 300 of the footing. Loose sand < 100 Cohesive soils Very Stiff to hard clays 300 - 600 Stiff clays 150 - 300 This group is susceptible Firm clays 75 - 150 to long-term settlement. Soft clays and silts < 75 Very soft clays and silts Not applicable Note: These values are to be used for preliminary purposes only and they are gross values with allowance for

    embedment.

    2. ULTIMATE BEARING CAPACITY OF SHALLOW FOUNDATIONS

    2.1 Lower and Upper Bound Solutions

    Consider a footing of width B and length L whose base is located at a depth D below the ground surface. The situation is shown diagramatically in Figure 2.1.

    Consider an element of soil located beneath the footing (Block 1, situated just to the left of the vertical line OY) and another element located adjacent to this one (Block 2, situated just to the right of the vertical line OY). Note that, at failure, the vertical stress applied to Block 1, 1,1 , is equal to the ultimate bearing capacity of the soil, qu . This vertical stress causes a lateral stress on Block 1 equal to 3,1 which, in turn causes a lateral stress on Block 2 of magnitude 1,2 . The vertical stress applied to Block 2, 3,2 , is equal to the overburden pressure, q D= ( being the bulk unit weight of the soil). Note that the lateral stress applied to Block 2 is given a subscript of 1, rather than 3, because its magnitude is greater than the vertical stress applied to Block 2.

  • 3

    Figure 2.1 Bearing capacity approximation for a = 0 soil. (Source: Bowles, 1996.)

    Recall, from Geotechnical Engineering II, the relationship between the principal total stresses, 1 and 3 , at failure is given by:

    1 32 45

    22 45

    2= +

    + +

    tan tanc (2.1)

    Note that: tan sinsin

    2 452

    11

    +

    =

    +

    (2.2)

    Lower Bound

    For the special case of a cohesive soil, = 0, ( ) ( )tan tan45 2 45 2 12+ = + = . For Block 2: 3,2 = q , and therefore:

    1 2 1 2 1 2, ( ) ( )= + = +q c q c (2.3)

    Since 3,1 = 1,2:

    ( ) 1 1 3 1 1 2 1 2 2 4, , ( ) ( )= + = + + = +c q c c q c (2.4)

    Since 1,1 = qu , and if q = 0 :

    q cu = 4 (2.5)

    Upper Bound

    For a possible upper bound, consider the footing rotating about point O, as shown in Figure 2.1. Taking moments about O (anticlockwise moments are positive):

    q B B c B B q B Bu

    =

    22

    2 20pi (2.6)

    Pressure on footing Overburden pressure

    Shear developed on failure plane

  • 4

    Rearranging and solving for qu yields: q c qu = +2pi (2.7)

    Again, if q = 0 :

    q c cu = =2pi 6.28 (2.8)

    Hence, for a cohesive soil ( = 0), the ultimate bearing capacity, qu , lies between 4c and 6.28c. (Coincidently, the average, 5.14c (= pi + 2) is given by plasticity theory).

    2.2 Methods Currently Used to Estimate the Ultimate Bearing Capacity

    It should be emphasised from the outset that all of the methods currently in use to evaluate the ultimate bearing capacity of a foundation provide only estimates of this capacity. This is due largely to the complexity of the problem, the heterogeneity of soils and the enormous cost and difficulties of performing load tests on full-scale foundations.

    In this treatment we will examine 2 methods which are in current use for the estimation of the ultimate bearing capacity of a foundation: the Terzaghi and Brinch Hansen methods. Two other equations proposed by Meyerhof and Vesi are also used, and these are described in detail by Bowles (1996).

    2.2.1 Terzaghis Method

    Karl Terzaghi, in 1943, was the first to present a comprehensive theory for the evaluation of the ultimate bearing capacity of rough shallow foundations. (A foundation is assumed to be shallow if the depth from the ground surface to the base of the foundation, D, is no more than 3 to 4 times the width of the foundation). Terzaghi suggested that for a continuous, or strip, footing the failure surface may be assumed to be similar to that shown in Figure 2.2. The failure zone under the foundation can be separated into 3 parts:

    1. The triangular zone ACD immediately beneath the foundation; 2. The radial shear zones ADF and CDE, with the curves DE and DF logarithmic spirals; 3. Two triangular Rankine passive zones AFH and CEG.

    Figure 2.2 Bearing capacity failure in soil under a rough, rigid strip footing. (Source: Das, 1995.)

  • 5

    Using equilibrium analyses, Terzaghi expressed the ultimate bearing capacity in the form:

    Strip Footings: q cN qN BNu c q= + + 0.5 (2.9)

    where: c is the cohesion of the soil; is the bulk unit weight of the soil; q = D; Nc , Nq , N are non-dimensional bearing capacity factors which are

    dependent solely on , and are given by Eqns (2.10) to (2.12) and in Table 2.1.

    and: ( )

    ( )Ne

    c =

    +

    cotcos

    tan

    pi 2 3 4 2

    22 45 21 (2.10)

    ( )

    ( )Ne

    q =+

    2 3 4 2

    22 45 21

    pi

    tan

    cos (2.11)

    NKP

    =

    12

    12costan (2.12)

    where: KP is the passive pressure coefficient of the soil.

    Table 2.1 Terzaghis bearing capacity factors. (Source: Das, 1995.)

    Nc Nq N Nc Nq N Nc Nq N 0 5.70 1.00 0.00 17 14.60 5.45 2.18 34 52.64 36.50 38.04 1 6.00 1.10 0.01 18 15.12 6.04 2.59 35 57.75 41.44 45.41 2 6.30 1.22 0.04 19 16.56 6.70 3.07 36 63.53 47.16 54.36 3 6.62 1.35 0.06 20 17.69 7.44 3.64 37 70.01 53.80 65.27 4 6.97 1.49 0.10 21 18.92 8.26 4.31 38 77.50 61.55 78.61 5 7.34 1.64 0.14 22 20.27 9.19 5.09 39 85.97 70.61 95.03 6 7.73 1.81 0.20 23 21.75 10.23 6.00 40 95.66 81.27 115.31 7 8.15 2.00 0.27 24 23.36 11.40 7.08 41 106.81 93.85 140.51 8 8.60 2.21 0.35 25 25.13 12.72 8.34 42 119.67 108.75 171.99 9 9.09 2.44 0.44 26 27.09 14.21 9.84 43 134.58 126.50 211.56

    10 9.61 2.69 0.56 27 29.24 15.90 11.60 44 151.95 147.74 261.60 11 10.16 2.98 0.69 28 31.61 17.81 13.70 45 172.28 173.28 325.34 12 1076 3.29 0.85 29 34.24 19.98 16.18 46 196.22 204.19 407.11 13 11.41 3.63 1.04 30 37.16 22.46 19.13 47 224.55 241.80 512.84 14 12.11 4.02 1.26 31 40.41 25.28 22.65 48 258.28 287.85 650.67 15 12.86 4.45 1.52 32 44.04 28.52 26.87 49 298.71 344.63 831.99 16 13.68 4.92 1.82 33 48.09 32.23 31.94 50 347.50 415.14 1072.8

    For square and circular foundations, Terzaghi suggested the following eqautions:

    Square Footings: q cN qN BNu c q= + +1.3 0.4 (2.13)

    Circular Footings: q cN qN BNu c q= + +1.3 0.3 (2.14)

  • 6

    2.2.2 Brinch Hansens Method

    Hanson (1970) proposed the general bearing capacity case:

    ( ) ( ) ( )q cN s d i g b qN s d i g b BN s d i g bu c c c c c c q q q q q q= + + 0.5 (2.15)

    where: si are shape factors; di are depth factors; ii are inclination factors; gi are ground factors (base on slope); bi are base factors (tilted base).

    and:

    += pi

    245tan2taneNq (2.16)

    ( ) = cot1qc NN (2.17)

    ( ) ( )= 1.4tan1qNN (2.18)

    Values of si , di , ii , gi and bi are given in Table 2.2, and Nc , Nq and N in Table 2.3.

    Table 2.2 Shape, depth, inclination, ground and base factors for use in either the Brinch Hansen or Vesi bearing capacity equations. Use primed factors when = 0. (Source: Bowles, 1996.) [continued overleaf.]

  • 7

    Table 2.2 Shape, depth, inclination, ground and base factors for use in either the Brinch Hansen or Vesi bearing capacity equations. Use primed factors when = 0. (Source: Bowles, 1996.) [continued]

    Table 2.3 Bearing capacity factors for Brinch Hansen equation.

    Nc Nq N Nc Nq N Nc Nq N 0 5.14 1.00 0.00 17 12.34 4.77 1.73 34 42.16 29.44 28.77 1 5.38 1.09 0.00 18 13.10 5.26 2.08 35 46.12 33.30 33.92 2 5.63 1.20 0.01 19 13.93 5.80 2.48 36 50.59 37.75 40.05 3 5.90 1.31 0.02 20 14.83 6.40 2.95 37 55.63 42.92 47.38 4 6.19 1.43 0.05 21 15.81 7.07 3.50 38 61.35 48.93 56.17 5 6.49 1.57 0.07 22 16.88 7.82 4.13 39 67.87 55.96 66.76 6 6.81 1.72 0.11 23 18.05 8.66 4.88 40 75.31 64.20 79.54 7 7.16 1.88 0.16 24 19.32 9.60 5.75 41 83.86 73.90 95.05 8 7.53 2.06 0.22 25 20.72 10.66 6.76 42 93.71 85.37 113.96 9 7.92 2.25 0.30 26 22.25 11.85 7.94 43 105.11 99.01 137.10

    10 8.34 2.47 0.39 27 23.94 13.20 9.32 44 118.37 115.31 165.58 11 8.80 2.71 0.50 28 25.80 14.72 10.94 45 133.87 134.87 200.81 12 9.28 2.97 0.63 29 27.86 16.44 12.84 46 152.10 158.50 244.65 13 9.81 3.26 0.78 30 30.14 18.40 15.07 47 173.64 187.21 299.52 14 10.37 3.59 0.97 31 32.67 20.63 17.69 48 199.26 222.30 368.67 15 10.98 3.94 1.18 32 35.49 23.18 20.79 49 229.92 265.50 456.40 16 11.63 4.34 1.43 33 38.64 26.09 24.44 50 266.88 319.06 568.57

  • 8

    2.2.3 Which Equation to Use

    Bowles (1996) suggested the various equations be used in the following situations:

    Use Best for Terzaghi Very cohesive soils where D B 1 or for a quick estimate of q

    u to compare with other methods.

    Brinch Hansen, Meyerhof, Vesi

    Any situation which applies, depending on user preference or familiarity with a particular method.

    Brinch Hansen, Vesi When base is tilted; when footing is on a slope; or when D B 1 .

    2.2.4 Choice of Factor of Safety

    The factor of safety, FS, to be used to determine the allowable bearing pressure, qall , should be chosen with consideration given to the following:

    the type of structure; the quality of the soil exploration program; the likelihood of the maxiimum load occurring; and the consequences of failure.

    High factors of safety (4 to 6) are used where limited soil exploration is undertaken, the soil conditions are very variable, the consequences of failure are disastrous, and the maximum design load is likely to occur often.

    Low factors of safety (2 to 2.5) are adopted where thorough and complete site investigations are undertaken, the soil conditions are relatively uniform, and where the maximum design load is unlikely to occur.

    2.3 SPECIAL CONSIDERATIONS

    2.3.1 Effect of Water Table on Bearing Capacity

    The ultimate bearing capacity equations presented thus far have inherently assumed that the groundwater table is located well below the base of the foundation. However, if the water table is located close to the foundation, some modifications of the bearing capacity equations are necessary, depending on the depth of the water table, Dw, below ground surface. Three cases are examined and these are shown diagrammatically in Figure 2.3.

    Case 1. 0 Dw D (The water table is located at or above the base of the footing)

    Firstly, the effective surcharge pressure, q , needs to be modified to account for the effective unit weight, ' (= sat w), of the soil; that is:

    ( )q D D Dw w= + ' (2.19)

    Secondly, the parameter , in the last term (...BN) is replaced by '.

    Case 2. D < Dw D+B (The water table is located below the base of the footing and above D+B)

    The parameter , in the last term (...BN) is replaced by $ , such that:

  • 9

    $ =

    w

    wD DB

    1 (2.20)

    Case 3. Dw > D+B (The water table is located below D+B beneath the ground surface)

    No modifications are necessary.

    Figure 2.3 The three groundwater cases which influence bearing capacity. (Source Coduto, 1994.)

    2.3.2 Bearing Capacity of Footings on Layered Soils

    The ultimate bearing capacity equations examined thus far have treated the soil beneath the footing as being a single, homogeneous deposit. In some instances, the subsoil may be stratified into thin layers. Bowles (1996) suggests a number of methods for handling layered soils located beneath the footing. A useful technique for treating c soils is to evaluate average values, cav and av , and to substitute these into the various bearing capacity models, such that:

    cc H c H c H c H

    Hav

    n n

    ii

    n=

    + + + +

    =

    1 1 2 2 3 3

    1

    L

    (2.21)

    av n ni

    i

    n

    H H H H

    H=

    + + + +

    =

    tan

    tan tan tan tan1 1 1 2 2 3 3

    1

    L

    (2.22)

    where: ci is the cohesion of layer i; Hi is the thickness of layer i; i is the internal angle of friction of layer i ( may equal zero); Hi is the effective shear depth and is limited to 0.5Btan(45+/2).

  • 10

    2.3.3 Short-Term and Long-Term Bearing Capacity

    Except in special cases, it is advisable to analyse the stability of a foundation against bearing capacity failure for both short-term and long-term stability to ensure that the foundation performs satisfactorily throughout its design life.

    Short-Term Case (Undrained): This case considers the situation just after construction or soon after rapid loading. Excess pore

    pressures, created by the loading, have not had time to dissipate. In the various bearing capacity equations, use the undrained strength parameters cu and u , and

    use total stresses, that is, use the bulk unit weight, .

    Long-Term Case (Drained): This case considers the situation a long time after construction or after very slow loading. As a

    consequence, all excess pore pressures have had time to dissipate. In the various bearing capacity equations, use the effective strength parameters c' and ', and use

    effective stresses, that is, substitute ' for .

    2.3.4 Bearing Capacity from CPT

    Schmertmann (1978) suggested that Nq and N can be estimated from:

    N N qq c 1.25 (2.23)

    where: qc is the measured cone tip resistance averaged over the depth interval from about B/2 above to 1.1B below the footing base.

    The values of Nq and N can then be substituted into the various bearing capacity equations.

    Geotech3_LS7_Bearing Capacity.doc, 2006, M. B. Jaksa