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

14.536 Soil Engineering The Bearing Capacity of a Shallow Foundation, as proposed by Vesic; The Settlement of a Shallow Foundation on Sand Rex Radloff 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. Introduction Bearing Capacity of a Shallow Foundation To design a shallow foundation, there must be an understanding of the underlying soil bearing the load. Modes of Failure In 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 the until shear planes develop and failure occurs. soil underlying a shallow foundation, different modes However because every subsurface and shallow of failure may arise. When a shallow foundation is foundation is very particular, specific bearing capa-loaded, two distinct shear planes will develop directly city 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 not shallow 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 shear especially difficult without the understanding of the planes illustrated in Fig. 1(a) will completely develop subsurfaces modulus of elasticity and shear modulus. to the surface. If the soil is very compressible If the soil is sand, the combination of the soil (capable of filling its own voids) then volume change properties with the addition of in-situ plate testing is promoted and punching shear failure will occur as can set the groundwork for settlement prediction. illustrated in Fig. 1(c) and the additional shear planes will barley develop. For scenarios in between the In both cases, too much information is above modes of failure, local shear failure will occur always welcomed; however too little information is as illustrated in Fig. 1(b) and an interpolation of the typically is the case. Either way, it is critical to shear planes between general and punching shear understand the reason behind settlement and bearing failure should be taken. capacity to better predict soil strength. 1 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 a Fig.1. (a) General shear failure (b) Local shear 100% dense state (Dr = 1.0) then it would lack the ability to be compressed. If this wasn’t the case, then failure (c) Punching shear failure 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 failure Loose, well graded soils with soft particles tend to be mode. The graph clearly demonstrates the previously compressible, and when semi-laterally loaded by the mentioned concepts of compressibility, relative downward moving wedge, the soil will fill its own density, and mode of failure voids (volume change) before applying a load to the adjacent soil. Therefore, the shear planes witnessed in loading an incompressible soil will cease to develop as there is not an increase in stress of that area. Graphical failure of a shallow foundation resting on compressible soil consists of a few slight deviations from the initial load-settlement curve (~modulus of elasticity, E) as shown in Fig 1(b) and 1(c). In other words, the shallow foundation will settle gradually until the soil acts plastically. Typically, when a shallow foundation fails in this regard, the ultimate bearing capacity will be taken at a specified settlement. Incompressible Soil Fig.2. Probable mode of failure for a given relative Dense, poorly graded soils with hard round particles density of the underlying soil and relative depth of tend to be incompressible as they have trouble filling the shallow foundation. their own voids under high loads. Therefore, the stress provided by the wedge will be transferred throughout the subsurface, as its volume is perse-2 14.536 Soil Engineering Rigidity Index The rigidity index is a means to analytically interpret the 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, ζ ,ζ ,ζThe shear modulus in Eq. 2(c) (which is the more cqγ, are dimensionless shape factors, ζ ,ζ ,ζdesirable equation in soil mechanics) was derived cdqdγd, are dimensionless depth factors, and ζ ,ζ ,ζfrom Eq. 2(b) and shown in Fig 3. Since the shear ccqcγc, are dimensionless compressibility factors. modulus examines a materials resistance to a shear Bearing Capacity Factors force, the rigidity index will yield the factor of safety this 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 material with no volume change. However, if the soil Prandtl and Reissner: ( ) (5) undergoes a plastic deformation and a volume change larger than 1% occurs, Eq. (2d) should be utilized Caquot and Kerisel: (6) The factors N (2d) c and Nq do not vary much with φ where Nγ does significantly, hence choosing the correct internal friction angle is critical when The significance of the rigidity index lies in calculating the ultimate bearing capacity. Selecting the response a material will have under a shear stress. If the response is minimal, i.e. Ithe correct friction angle will be briefly addressed r = large value, then shortly. the material will not deform under a stress, which much imply a lack of compression. In soil, the range Shape Factors of the rigidity index can vary from 10 (very compressible) 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 a shallow 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 14.536 Soil Engineering Depth Factors The following compression factors are defined as The 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 Equation strength of the overburden soil which increases its ultimate bearing capacity. However, this adjustment The American Association of State Highway & is discouraged as a shallow foundation is typically Transportation Officials (ASSHTO) has incorporated buried with loose fill. To correct for this loss in shear the following shallow foundation bearing capacity strength, it is suggested to use the residual frictional equation. angle. Regardless, each case should be analyzed separately 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 depth dictated by the critical rigidity index, which is factors, both that were respectively encouraged and defined as: discourages by the Vesic in Eq. (3). , *( ) ( )+- Case Study (16) Table 1 presents a series of shallow foundation load If the rigidity index is larger than the critical rigidity tests to failure. The load was applied axially and index, then the soil will compress and the failure there was not a water table. The modes of failure for mode will deviate away from general shear each test carried out by Muhs were determined by respectively. interpreting the shape of the load-settlement curve and check its consistency with the depth of the foun- 4 14.536 Soil Engineering Table 1. – Predicted (ASSHTO 2008) versus Measured Ultimate Bearing Capacity in c-phi soilsPredicted Measured Dγφ f B L c qf qf error Source 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 20 Demir 9 0.0 0.30 0.30 18.00 26.0 17.00 Punching Shear 600 198 203 Demir 10 0.0 0.45 0.45 18.00 26.0 17.00 Punching Shear 610 226 170 Demir 11 0.0 0.60 0.60 18.00 26.0 17.00 Punching Shear 620 223 178 Demir 12 0.0 0.40 0.40 18.00 26.0 17.00 Punching Shear 607 250 143 Demir 13 0.0 0.70 0.70 18.00 26.0 17.00 Punching Shear 627 188 234 Demir 14 0.0 1.00 1.00 18.00 26.0 17.00 Punching Shear 648 168 285 dation and the soils internal friction angle. For It is now possible to interpret the accuracy example, the load settlement curve for case 2 of the ASSHTO (2008) bearing capacity equation (in exhibited very little settlement with an increase in respect to other soil properties) on cases 1-4. Because load and when the ultimate bearing capacity was these tests failed in general shear, the suggested reached 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 35.3 degrees. For the load tests carried out by Demir, Table 2. – Predicted (ASSHTO 2008) versus every failure mode was visually verified as punching Measured Ultimate Bearing Capacity in c-phi soils shear after each test. without a depth factor. Results show an over prediction of bearing Case Predicted Measured Error capacity for soils that fail in punching to local shear. 2 737 1030 -32 It is speculated that this is because the ASSHTO 3 1427 2158 -35 (2008) bearing capacity equation does not incor-4 2088 2649 3 porate compressibility factors, which influences the 5 421 410 3 failure mechanism based on the soils likelihood to 6 640 550 16 8 272 260 5 compress. However, this assessment cannot be verified because none of the tests gave the soils modulus of elasticity and shear modulus, and without It should be noted, that the compressibility these soil properties the compressibility factors factors, at worse, is a conservative reduction of the cannot 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 could nated from the ASSHTO (2008) equation and for the have had shear strength. applicable cases (seen in Table 2) there was a decrease in predicted bearing capacity. For cases 5, 6, and 8 the error seems to approach zero, while the remaining cases yielded a larger under prediction. 5 14.536 Soil Engineering Settlement 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 = depth dependent pattern of settlement will occur which can of interest; R = radius of load area, and complete be broken up and labeled as the initial, primary, and vertical stress increase as secondary compression. The region of initial compression is dictated by the theory of elasticity, Δσv = (Δσvc - 2νΔσhc) (3) and is not relatively time dependent. The region of secondary compression is a function of the rate at where Δσv = Eq. 2(a) and Δσh = Eq. 2(b) (both which 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 any vary in respect to time. As the pore water dissipates, point throughout the entire system. However, total the rate of settlement will coincide with the theory of strain or more importantly total settlement cannot yet elasticity. Finally, secondary compression will take be determined. It is possible to estimate the into effect and the soil will completely settle. settlement to a specified location, but this measure is However, this region is negligible and the following not accurate as the stress changes with depth. Also will 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. To pressure will immediately dissipate and the primary consider these conditions, the following integral of consolidation cannot be witnessed. With the lack of strain has been taken: this range the soil will solely undergo initial cons-ρ=∫ zεolidation until the final settlement has been reached. 0vdz→ΔqsR/E*2(1-ν2) (4a) Therefore, the rate of consolidation for sand (or any where: granular material) is not time dependent and is a function 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 material A 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 and degree 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 Test large in each direction then the stress will dissipate accordingly. Fig. 1 and 2 illustrate how the developed In sand, the total settlement of a plate can be vertical 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 or two figures are derived using the following stress poisson’s ratio can be determined with an educated increase equations. assumption of either/or through the back calculation of Eq. 4(a). However, when dealing with soil, this is (2a) 6 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 medium underlying a uniform circular load. Change in Modulus due to Soil Type and Load an overwhelming assumption as these parameters can vary under several conditions. The following demon-Hard round uniformly granular material will maintain strates 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 are Change 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 highly fill its voids and become denser as consolidation vulnerable to a change in modulus with the takes place. Over time as soil (sedimentary) builds up application of a load. As a stress is increased, soon on itself, the underlying subsurface will mimic this crushing will occur due the lack of strength and low behavior 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 soil but the net cross section of its microstructure has yielding a poor modulus at that load. If the loading is increased. Therefore, the soil requires a larger load to increased the modulus will again increase until reach the same stress to yield the same deflection. crushing between the already fractured particles For example, the modulus of elasticity of quartz is happens again. around 12-14 x 106 psi, yet when it is grounded and compacted this value will drop because it is Change in Modulus with Cyclic Loading impossible to fill every void and the net cross sectional area is less, making a specific stress easier Hard round uniformly granular materials is not to reach with a lesser load. It is important to note, that susceptible to cyclic loading and will yield a fairly the absolute soil particle modulus does not change, rather the macrostructure of the specimen. 7 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 parameters Fig.3. (a) Vertical and horizontal stress increase throughout 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 any incremental and total settlement of a soil underlying a depth, while the CPT can measure both this uniform circular load. parameter and the modulus of elasticity which is dependent on K0 within this method. large rebound as the modulus is maintained. Soft angular material will continue to crush and very little Empirical methods: A popular empirical equation rebound will occur as the particles lost its stored by Jaky (1994) is suggested as energy through fracturing. For the granular soil that is found in between this range, interpolated results are (7) witnessed. This relationship is very valuable due to our good Change in Poisson’s Ratio with Depth understanding of acquiring the friction angle (ф) from standard penetration tests. Because this is an Poisson’s ratio is a function (given in Eq. 5) of the empirical equation, further insight to its origin should lateral stress ratio (Kbe investigated and a factor of safety assigned 0) which is a function of several soil parameters. accordingly. (5) Additional Notes The secant modulus of the soil determined by the Because the lateral stress ratio is influenced by the plate load test is not adequate to use in predicting the type of particles and initial arrangement, it can also settlement of a shallow foundation. Because the be a function of depth and stress these properties are; depth of influence is much less than the shallow similar to the modulus. Therefore, by appropriately foundation, the stress increase at any point will selecting the K0 in regards to depth, poisson’s ratio deviate by the same magnitude. For example, by becomes 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. This established as a function of its depth and current difference allows higher stresses which can change 8 14.536 Soil Engineering the modulus between the plate and shallow References foundation. Also, the depth of influence for the shallow foundation is greater than the plate, allowing ASSHTO (2008). LRFD Bridge Design deeper and possibly weaker layers to be affected by a Specifications Section 10: Foundations, p. 3.20– 3.23 stress increase. ASTM (2003), "Specification for Plate Load The effects of plate rigidity and the location of Testing," bedrock were left out of the evaluation. Bowles, J.E., (1996). “Foundation Analysis and Conclusion Design – Fifth Edition” A properly designed shallow foundation should not Das, B.M., (2007). “Principles of Foundation excessively settle or reach the ultimate bearing Engineering – Sixth Edition” capacity of the subsurface. These measures may seem Das, D.M., (2006). “Principles of Geotechnical simple, but it involves the complete understanding of Engineering – Sixth Edition” a soils tendency to, frankly, do anything under an applied 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 International capacity, it is important to predict the mode of failure Conference on Soil Mechanics and Foundation under the giving footing. This can be determined by Engineering, Vol. 2A, p.13-16 making use of the soils rigidity index, which is a function 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 foundations assumed that the soil will fail in general shear, then on soft soils” the cohesion and internal friction angle of the soil would be of primary concern. On the other hand, if Lambe, T.W. , and R.V. Whitman, (1969). “Soil local or punching shear is projected to take place at Mechanics”, p.198-199, M.I.T its ultimate bearing capacity, then the compressibility of 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”, Journal on 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.417-427 properties should be determined at varied depth to produce the most accurate results when calculating Milovic D.M. (1965). “Comparison between the settlement. In clay, relatively undisturbed samples Calculated and Experimental Values of the Ultimate can be taken for laboratory testing, but in the case of Bearing Capacity”,p.142-144 sand, this is not realistic. The use of in situ tests, such as the plate load test, standard penetration test, pressure meter, and cone penetration test, can be very useful in determining every soil property needed to predict total settlement. 9