# Geotechnical Eng CH3

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<p>3-13Design ofShallow FoundationsbyNagaratnam SivakuganJames Cook University, Townsville, AustraliaMarcus PachecoUniversidade do Estado do Rio de Janeiro, Brazil3.1 Introduction .................................................................................. 3-23.2 Stresses beneath Loaded Areas .................................................... 3-2Point and Line Loads Uniform Rectangular Loads NewmarksChart for Uniformly Loaded Irregular Areas3.3 Bearing Capacity of Shallow Foundations .................................. 3-6Historical Developments Terzaghis Bearing Capacity Equation Meyerhof s Bearing Capacity Equation Hansens Bearing CapacityEquation Vesics Bearing Capacity Equation Gross and NetPressures and Bearing Capacities Effects of the Water Table Presumptive Bearing Pressures3.4 Pressure Distribution beneath Eccentrically LoadedFootings ....................................................................................... 3-183.5 Settlement of Shallow Foundations in Cohesive Soils ............. 3-19Immediate Settlement Consolidation Settlement SecondaryCompression Settlement3.6 Settlement of Shallow Foundations in Granular Soils ............. 3-24Terzaghi and Peck Method Schmertmann et al. Method Burland and Burbidge Method Accuracy and Reliability of theSettlement Estimates and Allowable Pressures ProbabilisticApproach3.7 Raft Foundations ........................................................................ 3-32Structural Design Methods for Rafts Bearing Capacity andSettlement of Rafts3.8 Shallow Foundations under Tensile Loading ........................... 3-40Tensile Loads and Failure Modes Tensile Capacity Equations inHomogeneous Soils: Grenoble ModelAppendix A ........................................................................................ 3-49Appendix B ........................................................................................ 3-50Sample Chapter3-2 Geotechnical Engineering Handbook3.1 IntroductionA foundation is a structural element that is expected to transfer a load from a structure to theground safely. The two major classes of foundations are shallow foundations and deep founda-tions. A shallow foundation transfers the entire load at a relatively shallow depth. A commonunderstanding is that the depth of a shallow foundation (Df) must be less than the breadth (B).Breadth is the shorter of the two plan dimensions. Shallow foundations include pad footings,strip (or wall) footings, combined footings, and mat foundations, shown in Figure 3.1. Deepfoundations have a greater depth than breadth and include piles, pile groups, and piers, whichare discussed in Chapter 4. A typical building can apply 1015 kPa per floor, depending on thecolumn spacing, type of structure, and number of floors.Shallow foundations generally are designed to satisfy two criteria: bearing capacity andsettlement. The bearing capacity criterion ensures that there is adequate safety against possiblebearing capacity failure within the underlying soil. This is done through provision of anadequate factor of safety of about 3. In other words, shallow foundations are designed to carrya working load of one-third of the failure load. For raft foundations, a safety factor of 1.72.5is recommended (Bowles 1996). The settlement criterion ensures that settlement is withinacceptable limits. For example, pad and strip footings in granular soils generally are designedto settle less than 25 mm.3.2 Stresses beneath Loaded AreasIn particular for computing settlement of footings, it is necessary to be able to estimate thestress increase at a specific depth due to the foundation loading. The theories developed forcomputing settlement often assume the soil to be a homogeneous, isotropic, weightless elasticcontinuum.FIGURE 3.1 Types of shallow foundations.(a) Pad footing (b) Strip footing (c) Mat or raft foundation Sample ChapterDesign of Shallow Foundations 3-33.2.1 Point and Line LoadsBoussinesq (1885) showed that in a homogeneous,isotropic elastic half-space, the vertical stress in-crease (v) at a point within the medium, dueto a point load (Q) applied at the surface (seeFigure 3.2), is given byvQzx z +,, ]]]3211225 2( ) (3.1)where z and x are the vertical and horizontaldistance, respectively, to the point of interest fromthe applied load.Westergaard (1938) did similar research, assuming the soil to be reinforced by closelyspaced rigid sheets of infinitesimal thicknesses, and proposed a slightly different equation:vQzxz </p>
<p>j(, \,( + j(, \,(,,,]]]]21 22 21 22 2223 2 (3.2)Westergaards equation models anisotropic sedimentary clays with several thin seams of sandlenses interbedded with the clays. The stresses computed from the Boussinesq equationgenerally are greater than those computed from the Westergaard equation. As it is conservativeand simpler, the Boussinesq equation is more popular and will be used throughout thissection.If the point load is replaced by an infinitely long line load in Figure 3.2, the vertical stressincrease v is given by:vQzx z +,, ]]]2 1122( ) (3.3)3.2.2 Uniform Rectangular LoadsThe vertical stress increase at a depth z beneath the corner of a uniform rectangular load (seeFigure 3.3a) can be obtained by breaking the rectangular load into an infinite number of pointloads (dq Qdx dy) and integrating over the entire area. The vertical stress increase is givenbyxzQGLvFIGURE 3.2 Stress increase beneath a pointor line load.Sample Chapter3-4 Geotechnical Engineering HandbookvIq (3.4)where q is the applied pressure and the influence factor I is given by:Imn m nm n m nm nm nmn m nm n m n </p>
<p>+ ++ + +j(,\,( + ++ +j(, \,(+ + ++ 142 11212 12 22 2 2 22 22 212 22 2 2 2tan 1 +j(,\,(,,,,,,,,]]]]]]]]](3.5)Here m Bz and n Lz. Variation of I with m and n is shown in Figure 3.3b. Using theequation or Figure 3.3b, the vertical stress increase at any point within the soil, under auniformly loaded rectangular footing, can be found. This will require breaking up the loadedarea into four rectangles and applying the principle of superposition. This can be extended toT-shaped or L-shaped areas as well.At a depth z, v is the maximum directly below the center and decays with horizontaldistance. Very often, the value of v is estimated by assuming that the soil pressure appliedat the footing level is distributed through a rectangular prism, with slopes of 2 (vertical):1(horizontal) in both directions, as shown in Figure 3.4. Assuming the 2:1 spread in the load,the vertical stress at depth z below the footing becomes:FIGURE 3.3 Influence factor for stress beneath a corner of uniform rectangular load: (a) uniformlyloaded rectangle and (b) chart.(a) (b)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.01 0.1 1 10 Influence factor I n m = 0.1 m = 0.3 m = 0.5 m = 0.7 m = 1.0 m = 1.5 m = 2.0 m = m = 1.2 m = 3.0 x y z vzL B dx dy Sample ChapterDesign of Shallow Foundations 3-5vQB z L z </p>
<p>+ + ( ) ( )(3.6)In the case of strip footings, Equation 3.6 becomes:vQB z </p>
<p>+(3.7)3.2.3 Newmarks Chart for Uniformly Loaded Irregular AreasThe vertical stress increase at depth z below the center of a uniformly loaded circular footingof radius r is given by:vr zq +1112 3 2[( ) ] (3.8)The values of rz for v 0.1q, 0.2q1.0q are given in Table 3.1. Newmark (1942)developed the influence chart shown in Figure 3.5 using the values given in Table 3.1. Eachblock in the chart contributes an equal amount of vertical stress increase at any point directlybelow the center. This chart can be used to determine the vertical stress increase at depth zdirectly below any point (X) within or outside a uniformly loaded irregular area.The following steps are required for computing v at depth z below P:1. Redraw (better to use tracing paper) the plan of the loaded area to a scale where z isequal to the scale length given in the diagram.2. Place the plan on top of the influence chart such that the point of interest P on the plancoincides with the center of the chart.FIGURE 3.4 Average vertical stress increase with 2:1 distribution.TABLE 3.1 Influence Circle Radii for Newmarks Chartv q 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0rz 0.270 0.401 0.518 0.637 0.766 0.918 1.110 1.387 1.908 Q L B zB + zL + zSample Chapter3-6 Geotechnical Engineering Handbook3. Count the number of blocks (say, n) covered by the loaded area (include fractions ofthe blocks).4. Compute v as v Inq, where I is the influence value for Newmarks chart. For theone in Figure 3.5, where there are 200 blocks, I 1120 0.00833.3.3 Bearing Capacity of Shallow FoundationsSeveral researchers have studied bearing capacity of shallow foundations, analytically andusing model tests in laboratories. Lets look at some historical developments and three of themajor bearing capacity equations with corresponding correction factors.Typical pressure-settlement plots in different types of soils are shown in Figure 3.6. Threedifferent failure mechanisms, namely general shear, local shear, and punching shear, wererecognized by researchers. General shear failure is the most common mode of failure, and itoccurs in firm ground, including dense granular soils and stiff clays, where the failure load iswell defined (see Figure 3.6a). Here, the shear resistance is fully developed along the entireFIGURE 3.5 Newmarks influence chart.GL z I = 1/120scale: depth, z =Sample ChapterDesign of Shallow Foundations 3-7failure surface that extends to the ground level, and a clearly formed heave appears at theground level near the footing. The other extreme is punching shear failure, which occurs inweak, compressible soils such as very loose sands, where the failure surface does not extend tothe ground level and the failure load is not well defined, with no noticeable heave at the groundlevel (Figure 3.6c). In between these two modes, there is local shear failure (Figure 3.6b), whichoccurs in soils of intermediate compressibility such as medium-dense sands, where only slightheave occurs at the ground level near the footing.In reality, the ground conditions are always improved through compaction before placingthe footing. For shallow foundations in granular soils with Dr > 70% and stiff clays, the failurewill occur in the general shear mode (Vesic 1973). Therefore, it is reasonable to assume thatthe general shear failure mode applies in most situations.From bearing capacity considerations, the allowable bearing capacity (qall) is defined asqqFallult (3.9)where qult is the ultimate bearing capacity, which is the average contact pressure at the soil-footing interface when the bearing capacity failure occurs, and F is the factor of safety, whichtypically is taken as 3 for the bearing capacity of shallow foundations.3.3.1 Historical DevelopmentsPrandtl (1921) modeled a narrow metal tool bearing against the surface of a block of smoothsofter metal, which later was extended by Reissner (1924) to include a bearing area locatedbelow the surface of the softer metal. The Prandtl-Reissner plastic limit equilibrium planestrain analysis of a hard object penetrating a softer material later was extended by Terzaghi(1943) to develop the first rational bearing capacity equation for strip footings embedded insoils. Terzaghi assumed the soil to be a semi-infinite, isotropic, homogeneous, weightless, rigidplastic material; the footing to be rigid; and the base of the footing to be sufficiently rough toensure there is no separation between the footing and the underlying soil. It also was assumedthat the failure occurs in the general shear mode (Figure 3.7).3.3.2 Terzaghis Bearing Capacity EquationAssuming that the bearing capacity failure occurs in the general shear mode, Terzaghi ex-pressed his first bearing capacity equation for a strip footing as:FIGURE 3.6 Failure modes of a shallow foundation.Settlement Applied pressure (a) General shear Settlement Applied pressure (c) Punching shear Settlement Applied pressure (b) Local shear Sample Chapter3-8 Geotechnical Engineering Handbookq cN D N B Nc f q ult + + 1 20 5 . (3.10)Here, c is the cohesion and 1 and 2 are the unit weights of the soil above and below,respectively, the footing level. Nc, Nq, and N are the bearing capacity factors, which arefunctions of the friction angle. The ultimate bearing capacity is derived from three distinctcomponents. The first term in Equation 3.10 reflects the contribution of cohesion to theultimate bearing capacity, and the second term reflects the frictional contribution of theoverburden pressure or surcharge. The last term reflects the frictional contribution of the self-weight of the soil in the failure zone.For square and circular footings, the ultimate bearing capacities are given by Equations 3.11and 3.12, respectively:q cN D N B Nc f q ult + + 1 2 0 41 2. . (3.11)q cN D N B Nc f q ult + + 1 2 0 31 2. . (3.12)It must be remembered that the bearing capacity factors in Equations 3.11 and 3.12 are still forstrip footings. For local shear failure, where the failure surface is not fully developed and thusthe friction and cohesion are not fully mobilized, Terzaghi reduced the values of the frictionangle and cohesion by one-third to: tan ( . )10 67 (3.13) c c 0 67 . (3.14)Terzaghi neglected the shear resistance provided by the overburden soil, which was treatedas a surcharge (see Figure 3.7). Also, he assumed that in Figure 3.7. Subsequent studiesby several others show that 45 + 2 (Vesic 1973), which makes the bearing capacityfactors different than what were originally proposed by Terzaghi. With 45 + 2, thebearing capacity factors Nq and Nc become:N eq +j(, \,( tantan2452(3.15)FIGURE 3.7 Assumed failure surfaces within the soil during bearing capacity failure.Df (Surcharge) Q (Line load)log spiralpassive Strip footing Failure surfaceradial shearBactivepassive radial shearDf 45 /2 45 /221Sample ChapterDesign of Shallow Foundations 3-9N Nc q ( ) cot 1 (3.16)The above expression for Nc is the same as the one originally proposed by Prandtl (1921), andthe expression for Nq is the same as the one given by Reissner (1924). While there is generalconsensus about Equations 3.15 and 3.16, various expressions for N have been proposed int...</p>