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Shallow Foundations Allowable Bearing Capacity and Settlement Introduction Apart from bearing pressure, the other major design consideration for shallow footings is settlement Excessive settlement (primarily differential settlement) can cause a number of problems Settlement • Calculate vertical stress increase • Calculate settlement due to the stress increase • Bearing capacity based on settlement criterion Components of Settlement • Initial or elastic settlement, Se • Primary consolidation settlement, Sc • Secondary compression or creep, Sc(s) It occurs at constant effective stress (i.e. no drainage of pore water occurs) and is irreversible. • The total settlement, S = Se + Sc + Sc(s) Elastic Settlement Q E H E Generalized stress z and strain field Se = H /E = H.z Se = z .dz 0 Stress Due to a Concentrated Load • Boussinesq solution P 3P 5/ 2 R z r 2 2 z 1 2 z z where r r x2 y 2 x y Stress due to a Circularly Loaded Area For the incremental area shown settlement at the centre can be calculated as: Load on incremental area load, q q o rd dr d 5/2 dr r 2 2 z 1 2 z d After the double integration a r At Centre : z 1 q 1 2 3/ 2 B z 1 2z Distribution of Stress (from a vertical line load) The stresses at point X due to a line load of Q per unit length on the surface are as followings: Q/m 2Q Z3 z (x2 Z 2 )2 2 2Q xZ z x ( x 2 Z 2 )2 z 2Q xZ 2 x xy x ( x 2 Z 2 )2 Square B/2 Strip B/2 1.5B 1B 1B 2B 3B .9 0.9 0 0.7 0.5 1B 0.5 0.3 1B 0.3 2B 0.2 0.2 3B 0.1 2B 0.1 4B 5B 0.05 3B 6B 0.05 7B Stress below a Rectangular Area • Solution after Newmark for stresses under the corner of a L uniformly loaded flexible rectangular area: B • Define m = B/z and n = L/z • Solution by charts or numerically z = qo.I (It must be emphasised that the chart gives the stresses under the corner of z uniformly load flexible rectangular area.) 2 2 1/2 I = 1 2mn(m2+n2+1)1/2 . m2+n2+2 + tan-1 2mn(m +n +1) 4 m2+n2-m2n2+1 m2+n2+1 m2+n2-m2n2+1 Stress Influence Chart 0 .2 6 I m = oc 0 .2 4 A rea covered m = 2.0 m = 3 .0 m = 2.5 The influence w ith u nifo rm m = 1 .8 0 .2 2 n orm al load, q m = 1.6 m = 1 .4 m = 1 .2 factor is nz 0 .2 0 y mz z x m = 1.0 m = 0.9 shown in 0 .1 8 z = q.I m = 0.8 m = 0 .7 Table 5.2 (Pgs 0 .1 6 z m = 0 .6 208, 209) N ote: m and n are interch ang eable m = 0 .5 0 .1 4 0 .1 2 m = 0.4 0 .1 0 m = 0.3 0 .0 8 m = 0.2 0 .0 6 0 .0 4 m = 0 .1 0 .0 2 m = 0 .0 0 0.0 1 2 3 4 5 0 .1 2 3 4 5 1.0 2 3 45 10 n V E R T IC AL S T R E S S BE LO W A C O R N E R O F A U N IF O R M LY L O A D E D F L E X IB L E R E C TA N G U LA R AR E A . 2:1 method qo BL ( B z )( L z ) Average Vertical Stress Increases Due to a Rectangularly Loaded Area qo I a I a f (m2 , n2 ) B m2 H L n2 H Average Vertical Stress Increases Due to a Rectangularly Loaded Area qo I a I a f (m2 , n2 ) B m2 H L n2 H Average Vertical Stress Increases in a given layer H 2 I a ( H 2 ) H1 I a ( H1 ) qo H 2 H1 Elastic settlement Based on the Theory of Elasticity 1. Determine vertical strains: Q 2. Integrate strains: z = 1 [z - (x + y )] E R z Se = z .dz 0 z r x y Elastic Settlement of Rectangular footings )(1-s2)IsIf/Es Se = qo (B’ where qo = net applied pressure on the foundation s = Poisson’ ratio of soil s B’= B/2 for center of foundation =B for the corner of foundation Is = shape factor = F1 + (1 -2s)(1-s)F2 (see Tables 5.4 and 5.5) If = depth factor (see Fig. 5.15; pg 227) Elastic Settlement on Saturated Clay The average vertical immediate displacement under a flexible area carrying a uniform pressure q is given by q0 B se A1 A2 Es where A1 depends on the shape of the loaded area A2 depends on the depth of footing The above solutions for vertical displacement are used mainly to estimate the immediate settlement of foundation on saturated clays; such settlement occurs s under undrained conditions, the appropriate value of Poisson’ ratio being 0.5. The value of the undrained modulus Es is therefore required. A2 A1 Fig. 5.17; Pg. 231 Multi-layer systems q Se = Se(H1,E1) + Se(H1+H2,E2) - Se(H1,E2) B H1 E1 H2 E2 “Rigid” Flexible vs Rigid F F stress stress deflection deflection centre 0.93 centre RF = 0.93 Settlement in Sandy Soil (Schmertmann, 1970) The method was developed as a means to computing the settlement of spread footings on sandy soils It has a physical base and calibrates with empirical data CPT results are often used with this method Schmertmann conducted extensive research on the distribution of vertical strain below the spread footings He found that the greatest strains do not occur immediately below the footing but at a depth of ½ B to B depends on the footing shape The distribution of vertical strain is idealized as two straight lines Idealized Distribution of Z Peak Vertical Strain Influence Factor Izp q q q q I zp 0.5 0.1 vp Settlement in Sandy Soil (Schmertmann, 1970) layer n Iz Se C1C2 (q q ) z layer 1 Es q C1 1 0.5 q q t C 2 1 0.2 log10 0.1 t 0.1 (in years) q stress at the level of the foundation q = Df Range of Material Parameters for Computing Elastic Settlement • Elastic modulus can be determined: – Triaxial tests – Unconfined compression tests – In-situ tests • SPT: Es = N60 • CPT, Es = 2.5 qc (Square footing), 3.5 qc (Wall footing) • PMT, DMT,… • Es (sand): 10 ~ 55 MPa • Es (clay): 4 ~ 100 MPa Elastic modulus can also be estimated using Table 5.8 Primary Consolidation • A phenomenon occurs in both sands and clays • Expulsion of water from soils accompanied by increase in effective stress and strength • Amount can be reasonably estimated based on lab data, but rate is often poorly estimated Consolidation Settlement This method makes use of the results of the conventional oedometer test where the consolidation parameters of the soil are measured To compute the stress changes within the soil mass. The stress changes are computed using a Boussinesq- type approach assuming elasticity The important parameter for consolidation settlement calculation is the net effective stress change in the soil Usually the settlements are calculated for the soil divided into a number of sub-layers and the final total settlement is the sum of individual sub-layer settlements Use Compression Index The compression index (Cc) is used to calculate the settlement for net stress change beyond the preconsolidation pressure, 'c The recompression index (Cs) is used to calculate the settlement for net stress change from initial stress to the preconsolidation pressure Cs c Cc o av Sc( p ) log log Hc 1 eo o 1 eo c Secondary Consolidation • At the end of primary settlement, settlement may continue to develop due to the plastic deformation (creep) of the soil • The stage of consolidation is called secondary consolidation Secondary Compression Index Empirical relation: C 0.04 Cc for inorganic clays and silts C 0.05 Cc for organic clays and silts C 0.075 Cc for peats Field Load Test • Ultimate load, allowable load, and settlement can be determined from a field load test (ASTM D-1194-72) • The test plates are either square (12” or 18”) or circular (12” to 30”) Allowable Bearing Capacity for Sandy Soil based on Settlement • Meyerhof proposed a correlation for the net allowable bearing capacity for foundations with SPT N value Se qnet 20 N 60 Fd ( ) B 1.22 m 25mm B 0.3 2 Se qnet 12.5 N 60 Fd ( ) B 1.22 m B 25mm Df Fd 1 0.33 B Allowable Bearing Capacity for Sandy Soil based on Settlement • Meyerhof proposed a correlation for the net allowable bearing capacity for foundations with CPT qc value qc Se qnet ( ) B 1.22 m 15 25mm qc B 0.3 2 Se qnet ( ) B 1.22 m 25 B 25mm Reliability of Settlement Computations • The predication is quite satisfactorily in general • The predication is better for inorganic, insensitive clays than for others • The time rate of consolidation settlement is not well-predicted s Example of Schmertmann’ method A footing 2.5 m square carries a net foundation pressure of 150 kN/m2 at a depth of 1 m in a deep deposit of fine sand. The water table is at a depth of 4 m. Above the water table the unit weight of the sand is 17 kN/m3 and below the water table the saturated unit weight is 20 kN/m3. The variation of cone penetration resistance with depth is given in following table. Depth (m) 1-1.9 1.9-2.4 2.4-8 qc (MN/m2) 2.3 3.6 5 Estimate the settlement of the footing using s Schmertmann’ method. 2B Iz S = C1C2 Z 0 E • Estimate the maximum Izp • Plot influence factor curve on top of the qc/z plot • Divide the soil profile into layers assuming qc is constant for each sub-layer • Estimate the E based on qc for each layer • Determine Iz at the center of each layer • Calculate the ratio Iz*z/E • Estimate C1 and C2 • Calculate settlement B = 2.5 m Square footing 2B(5 m) Depth of influence is from 1m to 6m Izp at B/2 (1.25m) Izp = 0.5 + 0.1 ' v p = 150 kPa v’ = 2.25 x 17 = 38.25 p Izp = 0.698 Plot qc versus z to obtain sub-layers Layer z qc E Iz Iz*z/E 1 0.9 2.3 5.75 0.316 0.0495 2 0.35 3.6 9 0.601 0.0234 3 0.15 3.6 9 0.684 0.0114 4 3.6 5 12.5 0.335 0.0965 0.181 C1 = 1 – 0.5*(1*17)/150 = 0.943; The correction factor for creep will be taken as unity since there is no time given. S = 0.943 * 150 * 0.181 = 25.57 ~ 26 mm