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The Bearing Capacity of a Shallow Foundation, as proposed by Vesic; The Settlement of a Shallow Foundation on Sand
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The Bearing Capacity of a Shallow Foundation, as proposed by Vesic; The Settlement of a Shallow Foundation on Sand

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Abstract: A shallow foundation must be designed not to excessively settle or reach the ultimate bearing capacity of the subsurface. Each criterion is dependent on the footing geometry and several soil …

Abstract: A shallow foundation must be designed not to excessively settle or reach the ultimate bearing capacity of the subsurface. Each criterion is dependent on the footing geometry and several soil properties, which must be accurately determined before design. Because soil properties are rather difficult to obtain, close scrutiny should be used when interpreting laboratory or in-situ tests and the lack of doing so may lead to grossly incorrect predictions. Once the soil properties are understood, the proper bearing capacity factors should be selected, or left out, to calculate an accurate bearing capacity.
Several load tests were interpreted using the ASSHTO (2008) bearing capacity equation for a shallow foundation. Results yielded a significant over prediction of bearing capacity for those footings that failed in local to punching shear. It is believed one major contributing factor to these discrepancies resided in the addition and deduction of two specific bearing capacity factors.
In sand, the plate load test is a good measure when predicting the ultimate bearing capacity of a shallow foundation, though not a great deal when predicting its total settlement. However, because obtaining non-disturbed sand samples to test in the laboratory is impractical, the plate load test needs to be reliable. In order to accurately predict the soils behavior it is crucial to correctly interpret the raw data and make logical changes. The importance of a soils modulus of elasticity was especially considered in respect to depth, load, and soil type.

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  • 1. 14.536 Soil EngineeringThe Bearing Capacity of a Shallow Foundation, as proposed by Vesic; The Settlement of a Shallow Foundation on Sand Rex RadloffAbstract: A shallow foundation must be designed not to excessively settle or reach the ultimate bearing capacity ofthe subsurface. Each criterion is dependent on the footing geometry and several soil properties, which must beaccurately determined before design. Because soil properties are rather difficult to obtain, close scrutiny should beused when interpreting laboratory or in-situ tests and the lack of doing so may lead to grossly incorrect predictions.Once the soil properties are understood, the proper bearing capacity factors should be selected, or left out, tocalculate an accurate bearing capacity.Several load tests were interpreted using the ASSHTO (2008) bearing capacity equation for a shallow foundation.Results yielded a significant over prediction of bearing capacity for those footings that failed in local to punchingshear. It is believed one major contributing factor to these discrepancies resided in the addition and deduction of twospecific bearing capacity factors. In sand, the plate load test is a good measure when predicting the ultimate bearing capacity of a shallow foundation,though not a great deal when predicting its total settlement. However, because obtaining non-disturbed sand samplesto test in the laboratory is impractical, the plate load test needs to be reliable. In order to accurately predict the soilsbehavior it is crucial to correctly interpret the raw data and make logical changes. The importance of a soils modulusof elasticity was especially considered in respect to depth, load, and soil type.Introduction Bearing Capacity of aTo design a shallow foundation, there must be an Shallow Foundationunderstanding of the underlying soil bearing the load. Modes of FailureIn general, every soil will fail in the same regard i.e.an increase in load forces settlement of the footing Depending on the depth and compressibility of theuntil shear planes develop and failure occurs. soil underlying a shallow foundation, different modesHowever because every subsurface and shallow of failure may arise. When a shallow foundation isfoundation is very particular, specific bearing capa- loaded, two distinct shear planes will develop directlycity factors are necessary when attempting to predict below the base and create a triangular zone (Fig. 1).exactly how the soil will fail. As this wedge moves downward, the adjacent soil will yield accordingly and its ultimate bearing On the other hand, the ultimate load of a capacity will be reached. If the soil is notshallow foundation may lie in the degree of compressible (not capable of filling its own voids)settlement, and predicting this measure can be then general shear failure will occur and the shearespecially difficult without the understanding of the planes illustrated in Fig. 1(a) will completely developsubsurfaces modulus of elasticity and shear modulus. to the surface. If the soil is very compressibleIf the soil is sand, the combination of the soil (capable of filling its own voids) then volume changeproperties with the addition of in-situ plate testing is promoted and punching shear failure will occur ascan set the groundwork for settlement prediction. illustrated in Fig. 1(c) and the additional shear planes In both cases, too much information is will barley develop. For scenarios in between thealways welcomed; however too little information is above modes of failure, local shear failure will occurtypically is the case. Either way, it is critical to as illustrated in Fig. 1(b) and an interpolation of theunderstand the reason behind settlement and bearing shear planes between general and punching shearcapacity to better predict soil strength. failure should be taken. 1
  • 2. 14.536 Soil Engineering vered, and the shear planes will develop to the ground level. Failure of a shallow foundation resting on incompressible soil consists of rapid settlement at its ultimate bearing capacity. Meanwhile, the footing may excessively tilt to one side and the soil at the ground level will heave. Relative Density (1) The relative density (Eq. 1) of a soil is directly related to its compressibility. Soils that exhibit a high relative density must be incompressible on the same magnitude, and the contrary for soils with a low relative density. For example, if a soil sample is in aFig.1. (a) General shear failure (b) Local shear 100% dense state (Dr = 1.0) then it would lack thefailure (c) Punching shear failure ability to be compressed. If this wasn’t the case, then relative densities of over 1.0 would capable, which is not feasible.Compressible Soil Fig. 2 demonstrates an empirical relation- ship between a soils relative density and its failureLoose, well graded soils with soft particles tend to be mode. The graph clearly demonstrates the previouslycompressible, and when semi-laterally loaded by the mentioned concepts of compressibility, relativedownward moving wedge, the soil will fill its own density, and mode of failurevoids (volume change) before applying a load to theadjacent soil. Therefore, the shear planes witnessed inloading an incompressible soil will cease to developas there is not an increase in stress of that area. Graphical failure of a shallow foundationresting on compressible soil consists of a few slightdeviations from the initial load-settlement curve(~modulus of elasticity, E) as shown in Fig 1(b) and1(c). In other words, the shallow foundation willsettle gradually until the soil acts plastically.Typically, when a shallow foundation fails in thisregard, the ultimate bearing capacity will be taken ata specified settlement.Incompressible SoilDense, poorly graded soils with hard round particles Fig.2. Probable mode of failure for a given relativetend to be incompressible as they have trouble filling density of the underlying soil and relative depth oftheir own voids under high loads. Therefore, the the shallow foundation.stress provided by the wedge will be transferredthroughout the subsurface, as its volume is perse- 2
  • 3. 14.536 Soil EngineeringRigidity IndexThe rigidity index is a means to analytically interpretthe compressibility of soil. Given as: (2a)In which G is the shear modulus defined as: (2c) Fig.3. Deformation of an elastic material subjected to a shear stress. (2c) where Nc, Nq, Nγ are dimensionless bearing capacity factors, ζc,ζq,ζγ, are dimensionless shape factors,The shear modulus in Eq. 2(c) (which is the more ζcd,ζqd,ζγd, are dimensionless depth factors, anddesirable equation in soil mechanics) was derived ζcc,ζqc,ζγc, are dimensionless compressibility factors.from Eq. 2(b) and shown in Fig 3. Since the shearmodulus examines a materials resistance to a shear Bearing Capacity Factorsforce, the rigidity index will yield the factor of safetythis material has against deflecting 45 degrees The following bearing capacity factors were defined(=tan( )) when subjected to a shear stress; in this by:case the soils shear stress at failure. Prandtl and Reissner: (4) Eq. 2(a) assumes a perfect elastic materialwith no volume change. However, if the soil Prandtl and Reissner: ( ) (5)undergoes a plastic deformation and a volume changelarger than 1% occurs, Eq. (2d) should be utilized Caquot and Kerisel: (6) (2d) The factors Nc and Nq do not vary much with φ where Nγ does significantly, hence choosing The significance of the rigidity index lies in the correct internal friction angle is critical whenthe response a material will have under a shear stress. calculating the ultimate bearing capacity. SelectingIf the response is minimal, i.e. Ir = large value, then the correct friction angle will be briefly addressedthe material will not deform under a stress, which shortly.much imply a lack of compression. In soil, the range Shape Factorsof the rigidity index can vary from 10 (verycompressible) to 250 and over (very incompressible). The following shape factors are defined as:Vesic (1973) Bearing Capacity Equation ( )( ) (7)Vesic refined the ultimate bearing capacity of ashallow foundation resting on a cohesive-frictional ( ) (8)(c’-φ’) soil subjected to an axial was: ( ) (9) (3) Depending on the shape of the shallow foundation, different modes of failure may occur which can be traced back the ultimate bearing capacity of a soil (De Beer) 3
  • 4. 14.536 Soil EngineeringDepth Factors The following compression factors are defined asThe following depth factors are defined as: If: ≤ For Df ≤ B, then: Then: (10) (11) If: ≥ (12) Then: (17) For Df > B, then: ,( ) (13) * +- (18) (14) Where Ir = Irr (Eq. 2d) if ΔV ≥ 1.0% ( ) (15) If: φ = 0, then: These depth factors incorporate the shearing ASSHTO (2008) Bearing Capacity Equationstrength of the overburden soil which increases itsultimate bearing capacity. However, this adjustment The American Association of State Highway &is discouraged as a shallow foundation is typically Transportation Officials (ASSHTO) has incorporatedburied with loose fill. To correct for this loss in shear the following shallow foundation bearing capacitystrength, it is suggested to use the residual frictional equation.angle. Regardless, each case should be analyzedseparately and realistically. (19)Compressibility Factors In most regards, Eq. 19 was based off of the Eq. 3. However, the ASSHTO equation does not incor-The degree a soil will compress under a given load is porate compressibility factors and does include depthdictated by the critical rigidity index, which is factors, both that were respectively encouraged anddefined as: discourages by the Vesic in Eq. (3). , *( ) ( )+- Case Study(16) Table 1 presents a series of shallow foundation loadIf the rigidity index is larger than the critical rigidity tests to failure. The load was applied axially andindex, then the soil will compress and the failure there was not a water table. The modes of failure formode will deviate away from general shear each test carried out by Muhs were determined byrespectively. interpreting the shape of the load-settlement curve and check its consistency with the depth of the foun- 4
  • 5. 14.536 Soil EngineeringTable 1. – Predicted (ASSHTO 2008) versus Measured Ultimate Bearing Capacity in c-phi soils Predicted Measured Df B L γ φ c qf qf errorSource Case (m) (m) (m) (kPa) (deg) (kPa) Failure Type (kPa) (kPa) (%) Muhs 1 0.0 0.50 2.00 15.69 37.0 6.40 General Shear 658 981 -33 Muhs 2 0.5 0.50 2.00 16.38 35.3 3.90 General Shear 878 1030 -15 Muhs 3 0.5 0.50 2.00 17.06 38.3 7.80 General Shear 1684 2158 -22 Muhs 4 0.5 1.00 1.00 17.06 38.3 7.80 General Shear 2280 2649 -14 Muhs 5 0.4 0.71 0.71 17.65 22.0 12.80 Local Shear 499 410 22 Muhs 6 0.5 0.71 0.71 17.65 25.0 14.70 Local Shear 782 550 42 Muhs 7 0.0 0.71 0.71 17.06 20.0 9.80 Punching Shear 228 220 3 Muhs 8 0.3 0.71 0.71 17.06 20.0 9.80 Punching Shear 311 260 20Demir 9 0.0 0.30 0.30 18.00 26.0 17.00 Punching Shear 600 198 203Demir 10 0.0 0.45 0.45 18.00 26.0 17.00 Punching Shear 610 226 170Demir 11 0.0 0.60 0.60 18.00 26.0 17.00 Punching Shear 620 223 178Demir 12 0.0 0.40 0.40 18.00 26.0 17.00 Punching Shear 607 250 143Demir 13 0.0 0.70 0.70 18.00 26.0 17.00 Punching Shear 627 188 234Demir 14 0.0 1.00 1.00 18.00 26.0 17.00 Punching Shear 648 168 285dation and the soils internal friction angle. For It is now possible to interpret the accuracyexample, the load settlement curve for case 2 of the ASSHTO (2008) bearing capacity equation (inexhibited very little settlement with an increase in respect to other soil properties) on cases 1-4. Becauseload and when the ultimate bearing capacity was these tests failed in general shear, the suggestedreached there was a significant amount of settlement. compressibility factors would not influence the soil,This must indicate general shear failure which as it is relatively incompressible.coexists with the internal friction angle of 35.3degrees. For the load tests carried out by Demir, Table 2. – Predicted (ASSHTO 2008) versusevery failure mode was visually verified as punching Measured Ultimate Bearing Capacity in c-phi soilsshear after each test. without a depth factor. Results show an over prediction of bearing Case Predicted Measured Errorcapacity for soils that fail in punching to local shear. 2 737 1030 -32It is speculated that this is because the ASSHTO 3 1427 2158 -35(2008) bearing capacity equation does not incor- 4 2088 2649 3porate compressibility factors, which influences the 5 421 410 3failure mechanism based on the soils likelihood to 6 640 550 16 8 272 260 5compress. However, this assessment cannot beverified because none of the tests gave the soilsmodulus of elasticity and shear modulus, and without It should be noted, that the compressibilitythese soil properties the compressibility factors factors, at worse, is a conservative reduction of thecannot be determined. projected ultimate bearing capacity. Also, the elimination of the depth factor in table 2 cannot be Meanwhile, the depth factors were elimi- completely justified as the overburden soil couldnated from the ASSHTO (2008) equation and for the have had shear strength.applicable cases (seen in Table 2) there was adecrease in predicted bearing capacity. For cases 5, 6,and 8 the error seems to approach zero, while theremaining cases yielded a larger under prediction. 5
  • 6. 14.536 Soil EngineeringSettlement of a Shallow Foundation [ ] (2b)on Sand (2c)Principle of Consolidation √( )When soil is subjected to an axial load a time where q = applied stress; ν = poisons ratio; z = depthdependent pattern of settlement will occur which can of interest; R = radius of load area, and completebe broken up and labeled as the initial, primary, and vertical stress increase assecondary compression. The region of initialcompression is dictated by the theory of elasticity, Δσv = (Δσvc - 2νΔσhc) (3)and is not relatively time dependent. The region ofsecondary compression is a function of the rate at where Δσv = Eq. 2(a) and Δσh = Eq. 2(b) (bothwhich excess pore water pressure will dissipate, and conditions are for circular loads only)depending on the soils permeability this range can Relative strain can now be analyzed at anyvary in respect to time. As the pore water dissipates, point throughout the entire system. However, totalthe rate of settlement will coincide with the theory of strain or more importantly total settlement cannot yetelasticity. Finally, secondary compression will take be determined. It is possible to estimate theinto effect and the soil will completely settle. settlement to a specified location, but this measure isHowever, this region is negligible and the following not accurate as the stress changes with depth. Alsowill not consider this range. this method does not take into account the additional When sand is loaded any excess pore water settlement that may occur at a further location. Topressure will immediately dissipate and the primary consider these conditions, the following integral ofconsolidation cannot be witnessed. With the lack of strain has been taken:this range the soil will solely undergo initial cons- ρ=∫0zεvdz→ΔqsR/E*2(1-ν2) (4a)olidation until the final settlement has been reached.Therefore, the rate of consolidation for sand (or any where:granular material) is not time dependent and is afunction of the theory of elasticity. εv = 1/E (Δσvc - 2νΔσhc) (4b)Theory of Elasticity and ρ = total settlement. The elastic theory has been transformed to yield the total settlement of a materialA perfectly elastic material will strain when subjected underlying the center of a uniformly circular plate.to a stress and fully rebound when removed. The Fig. 3(a) and 3(b) show the typical stress increase anddegree of this tendency can be defined as settlement curve of a medium under the center of a (1) circular plate. It should be noted, that up to this point the subsurface was assumed to be a homogeneous,where E = modulus of elasticity; σ = stress; and ε = isotopic, perfectly elastic material.strain (ΔL/L). If the material being loaded is infinitely Settlement – Plate Load Testlarge in each direction then the stress will dissipateaccordingly. Fig. 1 and 2 illustrate how the developed In sand, the total settlement of a plate can bevertical and lateral stresses will dissipate throughout immediately recorded with an increase in stress.a medium underlying a uniform circular load. These Theoretically the absolute modulus of elasticity ortwo figures are derived using the following stress poisson’s ratio can be determined with an educatedincrease equations. assumption of either/or through the back calculation of Eq. 4(a). However, when dealing with soil, this is (2a) 6
  • 7. 14.536 Soil Engineering Fig.2. Lateral stress increase throughout a medium underlying a uniform circular load. Change in Modulus with Initial Loading The above principle applies; however the modulus is a function of the present load and not depth.Fig.1. Vertical stress increase throughout a mediumunderlying a uniform circular load. Change in Modulus due to Soil Type and Loadan overwhelming assumption as these parameters canvary under several conditions. The following demon- Hard round uniformly granular material will maintainstrates how these values can deviate. its modulus best with an increase in load. Because its strength and lack to fill its own void, these soils areChange in Modulus with Depth not as susceptible to a transforming modulus.With an increase in vertical stress, soil particles will Soft angular residual soils are highlyfill its voids and become denser as consolidation vulnerable to a change in modulus with thetakes place. Over time as soil (sedimentary) builds up application of a load. As a stress is increased, soonon itself, the underlying subsurface will mimic this crushing will occur due the lack of strength and lowbehavior and again fills its own voids. This process cross sectional area of the particle to particle contact.will maintain the cross section of its macrostructure, This crushing will induce a large shift in the soilbut the net cross section of its microstructure has yielding a poor modulus at that load. If the loading isincreased. Therefore, the soil requires a larger load to increased the modulus will again increase untilreach the same stress to yield the same deflection. crushing between the already fractured particlesFor example, the modulus of elasticity of quartz is happens again.around 12-14 x 106 psi, yet when it is grounded andcompacted this value will drop because it is Change in Modulus with Cyclic Loadingimpossible to fill every void and the net crosssectional area is less, making a specific stress easier Hard round uniformly granular materials is notto reach with a lesser load. It is important to note, that susceptible to cyclic loading and will yield a fairlythe absolute soil particle modulus does not change,rather the macrostructure of the specimen. 7
  • 8. 14.536 Soil Engineering loading, the theory of elasticity should be re- evaluated to take these parameters into effect. Eq. 6 with the use of Eq. 2(a) and Eq. 2(b) models the in- situ settlement as ∫ (6) This equation can now approach a more realistic prediction of how a circular shallow foundation will settle under any load. If a rectangular shallow foundation were to be evaluated, Eq. 2(a) and Eq. 2(b) would have to be re-derived to take these dimensions into consideration. (a) (b) Methods to determine varied parametersFig.3. (a) Vertical and horizontal stress increasethroughout a medium directly under the center of a In-situ methods: The pressuremeter and dilator-uniformly loaded circular plate. (b) example of meter test can measure the lateral stress ratio at anyincremental and total settlement of a soil underlying a depth, while the CPT can measure both thisuniform circular load. parameter and the modulus of elasticity which is dependent on K0 within this method.large rebound as the modulus is maintained. Softangular material will continue to crush and very little Empirical methods: A popular empirical equationrebound will occur as the particles lost its stored by Jaky (1994) is suggested asenergy through fracturing. For the granular soil that is (7)found in between this range, interpolated results arewitnessed. This relationship is very valuable due to our goodChange in Poisson’s Ratio with Depth understanding of acquiring the friction angle (ф) from standard penetration tests. Because this is anPoisson’s ratio is a function (given in Eq. 5) of the empirical equation, further insight to its origin shouldlateral stress ratio (K0) which is a function of several be investigated and a factor of safety assignedsoil parameters. accordingly. (5) Additional Notes The secant modulus of the soil determined by theBecause the lateral stress ratio is influenced by the plate load test is not adequate to use in predicting thetype of particles and initial arrangement, it can also settlement of a shallow foundation. Because thebe a function of depth and stress these properties are; depth of influence is much less than the shallowsimilar to the modulus. Therefore, by appropriately foundation, the stress increase at any point willselecting the K0 in regards to depth, poisson’s ratio deviate by the same magnitude. For example, bybecomes a function of depth. using fig. 1 the stress increase 1 ft. under a 2 ft.Re-evaluation of the Elastic Theory diameter plate yielding 100 lb/ft2 will be 65 lb/ft2. The stress under the same conditions of a 6 ft.Now that the modulus and poisson’s ratio has been diameter shallow foundation will be 95 lb/ft2. Thisestablished as a function of its depth and current difference allows higher stresses which can change 8
  • 9. 14.536 Soil Engineeringthe modulus between the plate and shallow Referencesfoundation. Also, the depth of influence for theshallow foundation is greater than the plate, allowing ASSHTO (2008). LRFD Bridge Designdeeper and possibly weaker layers to be affected by a Specifications Section 10: Foundations, p. 3.20– 3.23stress increase. ASTM (2003), "Specification for Plate LoadThe effects of plate rigidity and the location of Testing,"bedrock were left out of the evaluation. Bowles, J.E., (1996). “Foundation Analysis andConclusion Design – Fifth Edition”A properly designed shallow foundation should not Das, B.M., (2007). “Principles of Foundationexcessively settle or reach the ultimate bearing Engineering – Sixth Edition”capacity of the subsurface. These measures may seem Das, D.M., (2006). “Principles of Geotechnicalsimple, but it involves the complete understanding of Engineering – Sixth Edition”a soils tendency to, frankly, do anything under anapplied stress. De Beer, E.E., (1965). “The Scale Effect on the Phenomenon of Progressive Rupture in Cohesionless When calculating the soils ultimate bearing soils”, Proceedings of the Sixth Internationalcapacity, it is important to predict the mode of failure Conference on Soil Mechanics and Foundationunder the giving footing. This can be determined by Engineering, Vol. 2A, p.13-16making use of the soils rigidity index, which is afunction of the soils shear modulus and modulus of Demir A., M. Ornek., M. Laman., A. Yildiz., and G.elasticity. If, by in-situ or laboratory testing, it is Misir. (2009). “Model studies of circular foundationsassumed that the soil will fail in general shear, then on soft soils”the cohesion and internal friction angle of the soilwould be of primary concern. On the other hand, if Lambe, T.W. , and R.V. Whitman, (1969). “Soillocal or punching shear is projected to take place at Mechanics”, p.198-199, M.I.Tits ultimate bearing capacity, then the compressibilityof the soil needs to be taken into consideration. Lee J., J. Eun., and M. Prezzi, (2008). “Strain Influence Diagrams for Settlement Estimation of The degree of settlement a load can induce Both Isolated and Multiple Footings in sand”, Journalon the subsurface is also a function of the soils shear of Geotech and Geoenv Engineering, April 2008,modulus and modulus of elasticity. However, these p.417-427properties should be determined at varied depth toproduce the most accurate results when calculating Milovic D.M. (1965). “Comparison between thesettlement. In clay, relatively undisturbed samples Calculated and Experimental Values of the Ultimatecan be taken for laboratory testing, but in the case of Bearing Capacity”,p.142-144sand, this is not realistic. The use of in situ tests, suchas the plate load test, standard penetration test,pressure meter, and cone penetration test, can be veryuseful in determining every soil property needed topredict total settlement. 9