One dimensional consolidation theory

Bearing capacity of shallow foundationsJ S VinodSchool of Civil, Mining and Environmental EngineeringRoom 4.36, Tel: 4221 4089, Email: vinod@.auJ S Vinod School of Civil, Mining and Environmental Engineering Room 4.36, Tel: 4221 4089, Email: vinod@.au 2Foundation Engineering•K Terzaghi at I st Conf. on soil mech. & Foundation (1936) Engineering stated•“Successful work in Soil Mech. & Found. Eng. requires not only a thorough grounding in theory combined with an open eye for possible sources of error , but also an amount of observation and measurement in the field far in excess of anything attempted by the preceding generations of engineers.”–Hence the centre of gravity of research has shifted from study and the laboratory into the construction camp , where it will remain3Shallow Foundation•A foundation is the part of the structure, whichtransmits load directly to the underlying soil. If asoil stratum near the surface is capable ofadequately supporting the structural loads, it ispossible to use either a footing or a raft foundation.Individual footing Raft foundation•In general footing and raft foundations are referred as shallow foundation•To be considered as shallow foundation D/B <1DB4Site Investigation •Knowledge of soil profile +ground water conditions across the site. Combine with local geology and local experience•Systematic description of soil in each stratum –visual observations from boring/ sampling.•In-situ or laboratory measurement of the mechanical properties of the sub-soils as relevant to the proposed structure.5 Soil exploration (soft Bangkok clay)67Types of foundation•Individual footing or Pad footing-a foundation supporting a single column•Strip footing-a foundation supporting a load bearing wall or a group of closely spaced column •Raft foundation-raft foundations are required on soils of low bearing capacity or where the structural columns are so close in both directions.–Reduces the differential settlement on variable soils or wide variations in loading between the column8Guidelines / Geotechnical categories •Highways = small embankments (compacted)•Buildings:–Houses: Shallow foundation–Multi storied structures : Pile foundation–Bridges: Bridge piers (concrete piles)•Dams: geological deposits (sound rock –improved soil)9Significant depthResults from theory of elasticityThe increase in vertical stress in the soil below the centre of a strip footing of width B is ≈20 % of the foundation pressure at a depth 3B. In the case of square footing the corresponding depth is 1.5 B.For practical purposes these depths can be normally be accepted as the limits of the zone of influence of the respective foundation and are called the significant depths10p11•The ultimate bearing capacity (qf ) is defined as thepressure which would cause shear failure of thesupporting soil immediately below and adjacent to afoundation•The allowable bearing pressure (q a ) is defined as the maximum pressure which may be applied to the soil such that an adequate factor of safety against shear failure is ensured and that settlement should not cause unacceptable damage nor interfere with the function of the structureUltimate bearing capacity12Modes of bearing capacity failureGeneral shear failure–continuous failure surface develop between the edges of the footing and the ground surface.–heaving of ground surface occurs on both sides of the footing but the final slip movement would occur only on one side, accompanied by tilting of the footing.–this type of failure is typical for soils of low compressible andultimate load is well defined13Local shear failure•Significant compression of the soil under the footing. Therefore the failure surfaces do not reach the ground surface and only aslight heaving occurs.•Tilting of the foundation would not expected.•This type of failure is expected with soils of high compressibility and characterized by the occurrence of relatively large settlements.•The ultimate load is not well defined.14Punching shear failure–when there is relatively high compression of the soil under thefooting accompanied by shearing in the vertical direction around the edges of the footing.–There is no heaving of the ground surface away from the edges and no tilting of the footing.–Relatively large settlements are also characteristics of the failure mode.–Punching shear failure will also occur in soil of low compressibility if the foundation is located at considerable depths.15DBFailure mode of foundation in sand16Methods for calculating ultimate bearing capacity •Full scale load test•Load tests on model footing•Analytical Methods•Detailed stress analysis such as the FEM method17Overview of bearing capacity theories•Started with Pauker , a Russian military engineer(1889) proposed a theory to determine bearing capacity based on Rankine earth pressure theory. However, it is onlyapplicable for sands •Later Bell modified the theory incorporating cohesion component to it.•Terzaghi (1943) proposed a method by taking into account of the weight of the soil by the superposition principle by extending the theory of proposed by Prandtl (1921).•Prandtl (1921) –developed an equation based on the study of the penetration of a long hard metalpunch into softer material for computing ultimate bearing capacity 18Assumptions:•Soil is semi-infinite, homogeneous and isotropic•The problem is two dimensional•The base of the footing is rough•The failure is by general shear failure•The load is vertical and symmetrical•The ground surface is horizontal•The principle of superposition is valid•Coulombs law is strictly valid•The overburden pressure at foundation is equivalent to surcharge load (γD)Terzaghi bearing capacity theory19Terzaghi bearing capacity theory Failure mechanism under a strip footing By analyzing force acting on the wedge, efg wPpv cAφφq ultB= 2b Footing pressure = -weight of the wedge + cohesion + lateral pressurepult P c W b q 2sin 22++−=×φef g 20Terzaghi bearing capacity theory Failure mechanism under a strip footing φφπφtan )2/75.0(22)2/45(cos −=+=e a a a N q ⎟⎟⎠⎞⎜⎜⎝⎛−=1cos 2tan2φφγγp KN ()φcot 1−=q c N N Where N c , N q & N γare bearing capacity factors and depends only on shear strength parameter φ.γγγN B DN cN q q c f 21++=21cohesion Self wtsurcharge γγγN B DN cN q q c f 21++=22Simplified Shape factors •For Square footing s γ= 0.8; s c =1.3 and s q =1•For circular footing s γ= 0.6; s c =1.3 and s q =1γγγN B DN cN q q c f 4.03.1++=•For Square footing •For circular footing γγγN B DN cN q q c f 3.03.1++=)/2.01(5.0)/3.01(L B N B DN L B cN q q c f −+++=γγγRectangular footing: 23Simplified Inclination factors (i )''21L cB Hi c −=V Hi q 5.11−=2qi i =γL Be xe y X X YYx e B B 2'−=y e L L 2'−=B V H 24Hansen’s Bearing capacity equationsγγγγγi s N B i s DN i s cN q q q q c c c f 21++=()φcot 1−=q c N N Bearing capacity factors)245(tan 2tan φφπ+=e N q Where N c , N q & N γare bearing capacity factors;s and i are shape and inclination factors()φγtan 15.1−=q N N2526Shape factors, (s )6.0/4.01)](sin /1[/1−>−=+=+=L B s L B s LB N Ns q c qc γφLBNote:In case of inclined loads use B ′and L ′to compute shapefactors27Inclination factors (i )a c Ac Hi −−=15.0⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+−=1cot 5.01αφa q Ac V Hi ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+−=2cot 7.01αγφa Ac V H i L Be xe y X XYYxe B B 2'−=y e L L 2'−=BV H 521<<α522<<αCa = adhesion factor = 0.6 –1 times base cohesionBase area, A = B’xL’28•Net ultimate bearing capacity is the increase in the pressure at foundation level being the total foundation pressure less the weight of soil per unit area permanently removed.Factor of safetyDq q n γ−=•Factor of safety (F) with respect to shear failureDq Dq q q F f n nf γγ−−==29Effect of water table on bearing capacity •If the water table >1.5-2 B (footing width)–The soil is dry and total and effective stresses are identical and the analysis is identical to that described earlier.•The soil below the base of the footing is saturated –the effective unit weight (γ’=γsat -γw ) should be used for•If the water table is at the surface–the effective unit weight (γ’=γsat -γw ) should be used forγγγN B DN cN q q c f '5.0'++=γγγN B DN cN q q c f '5.0++=30Skempton’s Values of Nc•For saturated clays under undrained conditions (φu =0) the ultimate bearing capacity could be given byqN c q c u f +=31•For rectangular footing of dimensions BxL (B<L) the value of square footing should be multiplied by (0.84+0.16B/L)32Choice of soil parameters•Bearing capacity equation can be used with either drained or undrained soil parameter–For granular materials drained state at all stages of construction (φ ′).–Saturate cohesive soil operates undrained state during and immediately after construction (cu and φu ).However, long term stability can be checked with the assumptions that soil will be drained and relevantparameters are c ′and φ ′.33Example•A footing 2.25 x 2.25 m is located at a depth of 1.5 m in a sand, the shear strength parameters used in design being c’= 0 and φ’=38°. determine theultimate bearing capacity •(a) if the water table is well below the foundation level•(b) if the water table is at the surface.•The unit weight of the sand above the water table is 18 kN/m 3and saturated unit weight is 20kN/m 3.。