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CHAPTER EIGHT UPLIFT CAPACITY OF SHALLOW FOUNDATIONS 8.1 INTRODUCTION Foundations and other structures may be subjected to uplift forces under special circumstances. For those foundations, during the design process it is desirable to apply sufficient factor of safety against failure by uplift. During the last thirty or so years, several theories have been developed to estimate the ultimate uplift capacity of foundations embedded in sand and clay soils, and some of those theories are detailed in this chapter. The chapter is divided into two major parts: foundations in granular soil and foundations in saturated clay soil (! = 0). Figure 8.1 shows a shallow foundation of width B. The depth of embed- ment is Df . The ultimate uplift capacity of the foundation Qu can be expressed as Qu = frictional resistance of soil along the failure surface + weight of soil in the failure zone and the foundation (8.1) If the foundation is subjected to an uplift load of Qu , the failure surface in the soil for relatively small Df /B values will be of the type shown in Fig. 8.1. The intersection of the failure surface at the ground level will make an angle " with the horizontal. However, the magnitude of " will vary with the relative density of compaction in the case of sand and with the consistency in the case of clay soils. FIGURE 8.1 Shallow foundation subjected to uplift © 1999 by CRC Press LLC When the failure surface in soil extends up to the ground surface at ultimate load, it is defined as a shallow foundation under uplift. For larger values of Df /B, failure takes place around the foundation and the failure sur- face does not extend to the ground surface. These are called deep foundations under uplift. The embedment ratio, Df /B, at which a foundation changes from shallow to deep condition is referred to as the critical embedment ratio, (Df /B)cr. In sand the magnitude of (Df /B)cr can vary from 3 to about 11 and, in saturated clay, it can vary from 3 to about 7. FOUNDATIONS IN SAND During the last thirty years, several theoretical and semiempirical methods have been developed to predict the net ultimate uplifting load of continuous, circular, and rectangular foundations embedded in sand. Some of these theories are briefly described in the following sections. 8.2 BALLA’S THEORY Based on results of several model and field tests conducted in dense soil, Balla [1] established that, for shallow circular foundations, the failure surface in soil will be as shown in Fig. 8.2. Note from the figure that aa! and bb! are arcs of a circle. The angle " is equal to 45! !/2. The radius of the circle, of which aa! and bb! are arcs, is equal to FIGURE 8.2 Balla’s theory for shallow circular foundations © 1999 by CRC Press LLC Df r= (8.2) φ sin 45 + 2 As mentioned before, the ultimate uplift capacity of the foundation is the sum of two components: (a) the weight of the soil and the foundation in the failure zone and (b) the shearing resistance developed along the failure surface. Thus, assuming that the unit weight of soil and the foundation material are approxi- mately the same Df D Q u = D 3 γ F1 φ, f + F3 φ, f B (8.3) B where # = unit weight of soil ! = soil friction angle B = diameter of the circular foundation The sums of the functions F1(!, Df /B) and F3(!, Df /B) developed by Balla [1] are plotted in Fig. 8.3 for various values of the soil friction angle ! and the embedment ratio, Df /B. In general, Balla’s theory is in good agreement with the uplift capacity of shallow foundations embedded in dense sand at an embedment ratio of Df /B " 5. However for foundations located in loose and medium sand, the theory overestimates the ultimate uplift capacity. The main reason Balla’s theory overestimates the ultimate uplift capacity for Df /B > about 5 even in dense sand is because it is essentially deep foundation condition, and the failure surface does not extend to the ground surface. The simplest procedure to determine the embedment ratio at which deep foundation condition is reached may be determined by plotting the nondimen- sional breakout factor Fq against Df /B as shown in Fig. 8.4. The breakout factor is derived as Qu Fq = (8.4) γAD f where A = area of the foundation. * The breakout factor increases with Df /B up to a maximum value of Fq" Fq at Df /B" (Df /B)cr . For Df /B > (Df /B)cr the breakout factor remains practically * constant (that is, Fq). © 1999 by CRC Press LLC FIGURE 8.3 Variation of F1 + F3 [Eq. (8.3)] © 1999 by CRC Press LLC FIGURE 8.4 Nature of variation of Fq with Df /B FIGURE 8.5 Continuous foundation subjected to uplift 8.3 THEORY OF MEYERHOF AND ADAMS One of the most rational methods for estimating the ultimate uplift capacity of a shallow foundation was proposed by Meyerhof and Adams [2], and it is des- cribed in detail in this section. Figure 8.5 shows a continuous foundation of width B subjected to an uplifting force. The ultimate uplift capacity per unit length of the foundation is equal to Qu . At ultimate load the failure surface in © 1999 by CRC Press LLC soil makes an angle " with the horizontal. The magnitude of " depends on several factors, such as the relative density of compaction and the angle of friction of the soil, and it varies between 90#! a! to 90#! b!. Let us consider the free body diagram of the zone abcd. For stability consideration, the following forces per unit length of the foundation need to be considered. a. the weight of the soil and concrete, W, and b. the passive force P´ per unit length along the faces ad and bc. The p force P´ is inclined at an angle $ to the horizontal. For an average p value of "" 90! !/2, the magnitude of $ is about b!. If we assume that the unit weights of soil and concrete are approximately the same, then W = #Df B Ph′ 1 1 Pp′ = = ( K γ D ) 2 (8.5) cos δ 2 cos δ ph f where P´ = horizontal component of the passive force h Kph = horizontal component of the passive earth pressure coefficient Now, for equilibrium, summing the vertical components of all forces $Fv = 0 Qu = W + 2P´ sin$ p Qu = W + 2(P´ cos$)tan$ p Qu = W + 2P´ tan$ h or 1 Q u = W + 2 K ph γD 2 tan δ = W + K ph γD 2 tan δ f f (8.6) 2 The passive earth pressure coefficient based on the curved failure surface for $ = b! can be obtained from Caquot and Kerisel [3]. Furthermore, it is convenient to express Kphtan$ in the form Ku tan! = Kphtan$ (8.7) Combining Eqs. (8.6) and (8.7) Qu = W + Ku #Df² tan! (8.8) © 1999 by CRC Press LLC FIGURE 8.6 Variation of Ku where Ku = nominal uplift coefficient The variation of the nominal uplift coefficient Ku with the soil friction angle ! is shown in Fig. 8.6. It falls within a narrow range and may be taken as equal to 0.95 for all values of ! varying from 30# to about 48#. The ulti- mate uplift capacity can now be expressed in a nondimensional form (that is, the breakout factor, Fq ) as defined in Eq. (8.4) [4]. Thus, for a continuous foundation, the breakout factor per unit length is Qu Fs = γBD f or W + K u γD 2 tan φ D Fq = f = 1 + Ku f B tan φ (8.9) W For circular foundations, Eq. (8.8) can be modified to the form % Qu = W + – SF #BDf²Ku tan! (8.10) 2 % W% – B2Df # (8.11) 4 © 1999 by CRC Press LLC where SF = shape factor B = diameter of the foundation The shape factor can be expressed as D S F = 1 + m f B (8.12) where m = coefficient which is a function of the soil friction angle ! Thus, combining Eqs. (8.10), (8.11), and (8.12), we obtain π 2 π Df Qu = B D f γ + 1 + m B γ BD 2 K u tan φ f (8.13) 4 2 The breakout factor Fq can be given as π 2 π D B D f γ + 1 + m f γ BD 2 K u tan φ B f Qu 4 2 Fq = = γ AD f π γ B 2 D f 4 D D f = 1 + 2 1 + m f B B K u tan φ (8.14) For rectangular foundations having dimensions of B × L, the ultimate capacity can also be expressed as Qu = W + #D2f (2SFB + L & B)Ku tan! (8.15) The preceding equation was derived with the assumption that the two end por- tions of length B/2 are governed by the shape factor SF , while the passive pressure along the central portion of length L & B is the same as the continuous foundation. In Eq. (8.15) W % #BLDf (8.16) and D S F = 1 + m f B (8.17) © 1999 by CRC Press LLC Thus Df Q u = γBLD f + γD 2 2 1 + m B + L − B K u tan φ (8.18) f B The breakout factor Fq can now be determined as Qu Fq = (8.19) γBLD f Combining Eqs. (8.18) and (8.19), we obtain [4] D B D f Fq = 1 + 1 + 2 m f + 1 K u tan φ L B (8.20) B The coefficient m given in Eq. (8.12) was determined from experimental observations [2] and its values are given in Table 8.1. In Fig. 8.7, m is also plotted as a function of the soil friction angle !. TABLE 8.1 Variation of m [Eq. (8.12)] Soil friction angle, ! m 20 0.05 25 0.1 30 0.15 35 0.25 40 0.35 45 0.5 48 0.6 As shown in Fig. 8.4, the breakout factor Fq increases with Df /B to a maximum value of Fq* at (Df /B)cr and remains constant thereafter. Based on experimental observations, Meyerhof and Adams [2] recommended the variation of (Df /B)cr for square and circular foundations with soil friction angle ! and this is shown in Fig. 8.8. Thus, for a given value of ! for square (B = L) and circular (diameter = B) foundations, we can substitute m (Table 1) into Eqs. (8.14) and (8.20) and © 1999 by CRC Press LLC FIGURE 8.7 Variation of m FIGURE 8.8 Variation of (Df /B)cr for square and circular foundations © 1999 by CRC Press LLC calculate the breakout factor (Fq) variation with embedment ratio (Df /B). The * maximum value of Fq = F q will be attained at Df /B = (Df /B)cr . For Df /B > * (Df /B)cr , the breakout factor will remain constant as F q . The variation of Fq with Df /B for various values of ! made in this manner is shown in Fig. 8.9. * The variation of the maximum breakout factor F q for deep square and circular foundations with the soil friction angle ! is shown in Fig. 8.10. Laboratory experimental observations have shown that the critical embed- ment ratio (for a given soil friction angle !) increases with the L/B ratio. Meyerhof [5] indicated that, for a given value of !, FIGURE 8.9 Plot of Fq [Eqs. (8.14) and (8.20)] for square and circular foundations © 1999 by CRC Press LLC FIGURE 8.10 Fq* for deep square and circular foundations Df B cr -continuous ≈ 1.5 (8.21) Df B cr -square Based on laboratory model test results, Das and Jones [6] gave an empirical relationship for the critical embedment ratio of rectangular foundations in the form Df D L D = f 0.133 + 0. 867 ≤ 1.4 f B B B B (8.22) cr - R cr -S cr -S © 1999 by CRC Press LLC D where f B = criticalembedmentratio of a rectangular foundation cr-R having dimensions of L × B Df = criticalembedmentratio of a square foundation having B cr-S dimensions of B × B Using Eq. (8.22) and the (Df /B)cr-S values given in Fig. 8.8, the magnitude of (Df /B)cr-R for a rectangular foundation can be estimated. These values of (Df /B)cr-R can be substituted into Eq. (8.20) to determine the variation of Fq = F *with the soil friction angle !. 8.4 THEORY OF VESIC Vesic [7] studied the problem of an explosive point charge expanding a spherical cavity close to the surface of a semi-infinite, homogeneous, isotropic solid (in this case, the soil). Now, referring to Fig. 8.11, it can be seen that, if the distance Df is small enough, there will be an ultimate pressure po that will shear away the soil located above the cavity. At that time, the diameter of the spherical cavity is equal to B. The slip surfaces ab and cd will be tangent to the spherical cavity at a and c. At points b and d they make an angle " = 45!!/2. For equilibrium, summing the components of forces in the vertical direction we can determine the ultimate pressure po in the cavity. Forces that will be involved are: 1. Vertical component of the force inside the cavity, PV ; 2. Effective self-weight of the soil, W = W1 + W2 ; and 3. Vertical component of the resultant of internal forces, FV . FIGURE 8.11 Vesic’s theory of expansion of cavities © 1999 by CRC Press LLC For a c–! soil, we can thus determine that ! po = c!c + #Df Fq F (8.23) 2 B D D where Fq = 1.0 − + A1 f + A2 f 2 2 (8.24) 3 Df B B 2 2 D D Fc = A3 f + A4 f (8.25) B B 2 2 where A1 , A2 , A3 , A4 = functions of the soil friction ! For granular soils c = 0, so ! po = #Df Fq (8.26) Vesic [8] applied the preceding concept to determine the ultimate uplift capacity of shallow circular foundations. In Fig. 8.12 consider that the circular foundation ab having a diameter B is located at a depth Df below the ground FIGURE 8.12 Cavity expansion theory applied to circular foundation uplift © 1999 by CRC Press LLC surface. Assuming that the unit weight of the soil and the unit weight of the foundation are approximately the same, if the hemispherical cavity above the foundation (that is, ab) is filled with soil, it will have a weight of 3 2 B W3 = π γ (8.27) 3 2 This weight of soil will increase the pressure by p1 , or 3 2 B π γ 2 B = 2 = γ W3 3 2 p1 = 3 2 2 B B π π 2 2 If the foundation is embedded in a cohesionless soil (c = 0), then the pressure p1 should be added to Eq. (8.26) to obtain the force per unit area of the anchor, qu , needed for complete pullout. Thus Qu Qu 2 B qu = = = po + p1 = γD f Fq + γ A π 3 2 ( B) 2 2 2 B = γD f Fq + 3 2 (8.28) Df or 2 D D Qu + A2 f = γD f Fq qu = = γD f 1 + A1 f (8.29) A B B 2 2 ↑ Breakout factor The variation of the breakout factor Fq for shallow circular foundations is given in Table 8.2 and Fig. 8.13. In a similar manner, using the analogy of © 1999 by CRC Press LLC TABLE 8.2 Vesic’s Breakout Factor, Fq , for Circular Foundations Soil friction Df /B angle, ! (deg) 0.5 1.0 1.5 2.5 5.0 0 1.0 1.0 1.0 1.0 1.0 10 1.18 1.37 1.59 2.08 3.67 20 1.36 1.75 2.20 3.25 6.71 30 1.52 2.11 2.79 4.41 9.89 40 1.65 2.41 3.30 5.43 13.0 50 1.73 2.61 3.56 6.27 15.7 FIGURE 8.13 Vesic’s breakout factor, Fq , for shallow circular foundations © 1999 by CRC Press LLC expansion of long cylindrical cavities, Vesic determined the variation of the breakout factor Fq for shallow continuous foundations. These values are given in Table 8.3 and are also plotted in Fig. 8.14. TABLE 8.3 Vesic’s Breakout Factor, Fq , for Continuous Foundations Soil friction Df /B angle, ! (deg) 0.5 1.0 1.5 2.5 5.0 0 1.0 1.0 1.0 1.0 1.0 10 1.09 1.16 1.25 1.42 1.83 20 1.17 1.33 1.49 1.83 2.65 30 1.24 1.47 1.71 2.19 3.38 40 1.30 1.58 1.87 2.46 3.91 50 1.32 1.64 2.04 2.60 4.20 FIGURE 8.14 Vesic’s breakout factor, Fq , for shallow continuous foundation © 1999 by CRC Press LLC 8.5 SAEEDY’S THEORY A theory for the ultimate uplift capacity of circular foundations embedded in sand was proposed by Saeedy [9] in which the trace of the failure surface was assumed to be an arc of a logarithmic spiral. According to this solution, for shallow foundations the failure surface extends to the ground surface. How- ever, for deep foundations [that is, Df > Df (cr)] the failure surface extends only to a distance of Df (cr) above the foundation. Based on this analysis, Saeedy [9] proposed the ultimate uplift capacity in a nondimensional form (Qu /!B2Df ) for various values of " and the Df /B ratio. The author converted the solution into a plot of breakout factor Fq = Qu /!ADf (A = area of the foundation) versus the soil friction angle " as shown in Fig. 8.15. According to Saeedy, during the foundation uplift the soil located above the anchor gradually becomes com- pacted, in turn increasing the shear strength of the soil and, hence, the ultimate uplift capacity. For that reason, he introduced an empirical compaction factor µ, which is given in the form µ = 1.044Dr + 0.44 (8.30) where Dr = relative density of sand Thus, the actual ultimate capacity can be expressed as Qu (actual) = (Fq !ADf ) µ (8.31) 8.6 DISCUSSION OF VARIOUS THEORIES Based on the various theories presented in the preceding sections, we can make some general observations: 1. The only theory that addresses the problem of rectangular foundations is that given by Meyerhof and Adams [3]. 2. Most theories assume that shallow foundation conditions exist for Df /B ! 5. Meyerhof and Adams’ theory provides a critical embedment ratio (Df /B)cr for square and circular foundations as a function of the soil friction angle. 3. Experimental observations generally tend to show that, for shallow foundation in loose sand, Balla’s theory [1] overestimates the ultimate uplift capacity. Better agreement, however, is obtained for foundations in dense soil. 4. Vesic’s theory [8] is, in general, fairly accurate in estimating the ulti- mate uplift capacity of shallow foundations in loose sand. However, laboratory experimental observations have shown that, for shallow foundations in dense sand, this theory can underestimate the actual uplift capacity by as much as 100% or more. © 1999 by CRC Press LLC FIGURE 8.15 Plot of Fq based on Saeedy’s theory Figure 8.16 shows a comparison of some published laboratory experi- mental results for the ultimate uplift capacity of circular foundations with the theories of Balla, Vesic, and Meyerhof and Adams. Table 8.4 gives the refer- ences to the laboratory experimental curves shown in Fig. 8.16. In developing the theoretical plots for " = 30" (loose sand condition) and " = 45" (dense sand condition) the following procedures were used. 1. According to Balla’s theory [1], from Eq. (8.3) for circular founda- tions Qu = D 3 γ ( F1 + F3 ) f ! "## $ Fig. 8.3 © 1999 by CRC Press LLC FIGURE 8.16 Comparison of theories with laboratory experimental results for circular foundations © 1999 by CRC Press LLC TABLE 8.4 References to Laboratory Experimental Curves Shown in Fig. 8.16 Circular foundation Curv Reference diameter, B Soil properties e 1 Baker and Kondner [10] 25.4 " = 42"; ! = 17.61 kN / m3 2 Baker and Kondner [10] 38.1 " = 42"; ! = 17.61 kN / m3 3 Baker and Kondner [10] 50.8 " = 42"; ! = 17.61 kN / m3 4 Baker and Kondner [10] 76.2 " = 42"; ! = 17.61 kN / m3 5 Sutherland [11] 38.1–152.4 " = 45" 6 Sutherland [11] 38.1–152.4 " = 31" 7 Esquivel-Diaz [12] 76.2 8 Esquivel-Diaz [12] 76.2 " # 43"; ! = 14.81 kN / m3–15.14 kN / m3 9 Balla [1] 61–119.4 " = 33"; ! = 12.73 kN / m3–12.89 kN / m3 Dense sand So 2 π 2 π B Qu B Qu 4 D f F1 + F3 = u3 = Q 4 = γD f 3 π 2 γD f A γD f B 4 or Qu F1 + F3 Fq = = (8.32) γ AD f 2 π B 4 D f So, for a given soil friction angle the sum of F1 + F3 was obtained from Fig. 8.3, and the breakout factor was calculated for various values of Df /B. These values are plotted in Fig. 8.16. 2. For Vesic’s theory [8] the variations of Fq versus Df /B for circular foundations are given in Table 8.2. These values of Fq are also plotted in Fig. 8.16. 3. The breakout factor relationship for circular foundations based on Meyerhof and Adams’ theory [3] is given in Eq. (8.14). Using Ku # 0.95, the variations of Fq with Df /B were calculated, and they are also plotted in Fig. 8.16. © 1999 by CRC Press LLC Based on the comparison between the theories and the laboratory experi- mental results shown in Fig. 8.16, it appears that Meyerhof and Adams’ theory [3] is more applicable to a wide range of foundations and provides as good an estimate as any for the ultimate uplift capacity. So this theory is recommended for use. However, it needs to be kept in mind that the majority of the experi- mental results presently available in the literature for comparison with the theory are from laboratory model tests. When applying these results to the design of an actual foundation, the scale effect needs to be taken into consideration. For that reason, a judicious choice is necessary in selecting the value of the soil friction angle ". EXAMPLE 8.1 Consider a circular foundation in sand. Given for the foundation: diameter, B = 1.5 m; depth of embedment, Df = 1.5 m. Given for the sand: unit weight, ! = 17.4 kN / m3; friction angle, " = 35". Using Balla’s theory, calculate the ultimate uplift capacity. Solution From Eq. (8.3) Qu = D3 !(F1 + F3) f From Fig. 8.3, for " = 35" and Df /B = 1.5/1.5 = 1, the magnitude of F1 + F3 # 2.4. so Qu = (1.5)3(17.4)(2.4) = 140.9 kN !! EXAMPLE 8.2 Redo Example Problem 8.1 using Vesic’s theory. Solution From Eq. (8.29) Qu = A! D3 Fq f From Fig. 8.13, for " = 35" and Df /B = 1, Fq is about 2.2. So π Qu = (1.5) 2 (17.4)(1.5)(2.2) = 101.5 kN 4 !! © 1999 by CRC Press LLC EXAMPLE 8.3 Redo Example Problem 8.1 using Meyerhof and Adams’ theory. Solution From Eq. (8.14) Df D f Fq = 1 + 2 1 + m B B K u tan φ For " = 35", m = 0.25 (Table 8.1). So Fq = 1+2[1 + (0.25)(1)](1)(0.95)(tan35) = 2.66 So π Qu = Fq γAD f = (2.66 )(17.4) (1.5) 2 (1.5) = 122.7 kN 4 !! 8.7 EFFECT OF BACKFILL ON UPLIFT CAPACITY Spread foundations constructed for electric transmission towers are subjected to uplifting force. The uplift capacity of such foundations can be estimated by using the same relationship described in the preceding sections. During the construction of such foundations, the embedment ratio Df /B is usually kept at 3 or less. For foundation construction, the native soil is first excavated. Once the foundation construction is finished, the excavation is backfilled and com- pacted. The degree of compaction of the backfill material plays an important role in the actual ultimate uplift capacity of the foundation. Kulhawy, Trautman, and Nicolaides [13] conducted several laboratory model tests to ob- serve the effect of the degree of compaction of the backfill compared to the native soil. According to their observations, failure in soil in most cases takes place by side shear as shown in Fig. 8.17. However, wedge or combined shear failure occurs for foundations with Df /B < about 2 in medium to dense native soil where the backfill is at least 85 percent as dense as the native soil (Fig. 8.18). Figure 8.19 shows the effect of backfill compaction on the breakout factor Fq when the native soil is loose. Similarly, Fig. 8.20 is for the case where the native soil is dense. Based on the observations of Kulhawy et al. [13], this study shows that the compaction of the backfill has a great influence on the breakout factor of the foundation, and the ultimate uplift capacity greatly increases with the degree of backfill compaction. © 1999 by CRC Press LLC FIGURE 8.17 Failure by side shear FIGURE 8.18 Wedge or combined shear failure © 1999 by CRC Press LLC FIGURE 8.19 Effect of backfill on breakout factor — square foundation with loose native soil (after Kulhawy et al. [13]) FIGURE 8.20 Effect of backfill on breakout factor — square foundation with dense native soil (after Kulhawy et al. [13]) © 1999 by CRC Press LLC FIGURE 8.21 Shallow foundation in saturated clay subjected to uplift FOUNDATIONS IN SATURATED CLAY ! (! = 0 CONDITION) 8.8 ULTIMATE UPLIFT CAPACITY—GENERAL Theoretical and experimental research results presently available for deter- mining the ultimate uplift capacity of foundations embedded in saturated clay soil are rather limited. In the following sections, the results of some of the existing studies are reviewed. Figure 8.21 shows a shallow foundation in a saturated clay. The depth of the foundation is Df , and the width of the foundation is B. The undrained shear strength and the unit weight of the soil are cu and γ, respectively. If we assume that the unit weight of the foundation material and the clay are approx- imately the same, then the ultimate uplift capacity can be expressed as [8] Qu = A("Df + cu Fc) (8.33) where A = area of the foundation Fc = breakout factor " = saturated unit weight of the soil 8.9 VESIC’S THEORY Using the analogy of the expansion of cavities, Vesic [8] presented the theoretical variation of the breakout factor Fc (for ! = 0 condition) with © 1999 by CRC Press LLC ! TABLE 8.5 Variation of Fc (! = 0 Condition) Df /B Foundation type 0.5 1.0 1.5 2.5 5.0 Circular (diameter = B) 1.76 3.80 6.12 11.6 30.3 Continuous (width = B) 0.81 1.61 2.42 4.04 8.07 FIGURE 8.22 Vesic’s breakout factor Fc the embedment ratio Df /B, and these values are given in Table 8.5. A plot of these same values of Fc against Df /B is also shown in Fig. 8.22. Based on the laboratory model test results available at the present time, it appears that Vesic’s theory gives a closer estimate only for shallow foundations embedded in softer clay. In general, the breakout factor increases with embedment ratio up to a maximum value and remains constant thereafter as shown in Fig. 8.23. The maximum value of Fc = Fc* is reached at Df /B = (Df /B)cr . Foundations located at Df /B > (Df /B)cr are referred to as deep foundations for uplift capacity consid- eration. For these foundations at ultimate uplift load, local shear failure in soil © 1999 by CRC Press LLC FIGURE 8.23 Nature of variation of Fc with Df /B located around the foundation takes place. Foundations located at Df /B! (Df /B)cr are shallow foundations for uplift capacity consideration. 8.10 MEYERHOF’S THEORY Based on several experimental results, Meyerhof [5] proposed the following relationship Qu = A("Df + Fc cu) For circular and square foundations Df Fc = 1.2 B ≤9 (8.34) and, for strip foundations Df Fc = 0.6 B ≤8 (8.35) The preceding two equations imply that the critical embedment ratio (Df /B)cr is about 7.5 for square and circular foundations and about 13.5 for strip foundations. © 1999 by CRC Press LLC 8.11 MODIFICATIONS TO MEYERHOF’S THEORY Das [14] compiled a number of laboratory model test results on circular foundations in saturated clay with cu varying from 5.18 kN / m2 to about 172.5 kN / m2. Figure 8.24 shows the average plots of Fc versus Df /B obtained from these studies, along with the critical embedment ratios. From Fig. 8.24 it can be seen that, for shallow foundations D Fc ≈ n f B ≤ 8 to 9 (8.36) where n = a constant The magnitude of n varies between 5.9 to 2.0 and is a function of the un- drained cohesion. Since n is a function of cu and Fc = Fc* is about 8 to 9 in all cases, it is obvious that the critical embedment ratio (Df /B)cr will be a function of cu . Das [14] also reported some model test results with square and rectangular foundations. Based on these tests, it was proposed that Df = 0.107 c u + 2.5 ≤ 7 (8.37) B cr −S D where f B = critical embedment ratio of square foundations cr −S (or circular foundation) cu = undrained cohesion, in kN/m 2 It was also observed by Das [19] that Df D L D = f 0 .73 + 0 . 27 ≤ 1 . 55 f B B B B (8.38) cr − R cr − S cr −S D where f B = critical embedment ratio of rectangula r foundation s cr − R L = length of foundation © 1999 by CRC Press LLC FIGURE 8.24 Variation of Fc with Df/B from various experimental observations—circular foundations; diameter = B © 1999 by CRC Press LLC Based on the above findings, Das [19] proposed an empirical procedure to obtain the breakout factors for shallow and deep foundations. According to this procedure, #" and $" are two nondimensional factors defined as Df α′ = B (8.39) Df B cr and Fc β′ = (8.40) Fc* For a given foundation, the critical embedment ratio can be calculated by using Eqs. (8.37) and (8.38). The magnitude of Fc* can be given by the following empirical relationship B Fc*−R = 7.56 + 1.44 (8.41) L * where Fc-R = breakout factor for deep rectangular foundations Figure 8.25 shows the experimentally derived plots (upper limit, lower limit, and average of $" and #". Following is a step-by-step procedure to esti- mate the ultimate uplift capacity. 1. Determine the representative value of the undrained cohesion, cu . 2. Determine the critical embedment ratio using Eqs. (8.37) and (8.38). 3. Determine the Df /B ratio for the foundation. 4. If Df /B > (Df /B)cr as determined in Step 2, it is a deep foundation. However, if Df /B ! (Df /B)cr , it is a shallow foundation. 5. For Df /B > (Df /B)cr B Fc = Fc* = 7.56 + 1.44 L Thus B Qu = A 7.56 + 1.44 cu + γD f (8.42) L where A = area of the foundation © 1999 by CRC Press LLC FIGURE 8.25 Plot of $" versus #" 6. For Df /B! (Df /B)cr B Qu = A(β′ Fc* cu + γD f ) = A β′ 7.56 + 1.44 cu + γD f (8.43) L The value of $" can be obtained from the average curve of Fig. 8.25. The procedure outlined above gives fairly good results in estimating the net ultimate capacity of foundations. EXAMPLE 8.4 A rectangular foundation in a saturated clay measures 1.5 m × 3 m. Given: Df = 1.8 m; cu = 52 kN / m2; " = 18.9 kN / m3. Estimate the ultimate uplift capacity. Solution From Eq. (8.37) Df = 0.107cu + 2.5 = ( 0.107)(52) + 2.5 = 8.06 B cr −S © 1999 by CRC Press LLC So use (Df /B)cr-S = 7. Again, from Eq. (8.38) Df D L = f 0 .73 + 0.27 B B B cr − R cr − S 3 = 7 0 .73 + 0.27 = 8.89 1 .5 D Check : 1.55 f B = (1.55)( 7) = 10.85 cr −S So use (Df /B)cr-R = 8.89. The actual embedment ratio is Df /B = 1.8/1.5 = 1.2. Hence this is a shallow foundation. Df B α′ = = 1.2 = 0.13 Df 8.89 B cr Referring to the average curve of Fig. 8.25, for #" = 0.13 the magnitude of $" = 0.2. From Eq. (8.43) B Qu = Aβ′7.56 + 1.44 cu + γD f L 1.5 = (1.5)(3)(0.2)7.56 + 1.44 (52) + (18.9)(1.8) = 540.6 kN 3 !! 8.12 FACTOR OF SAFETY In most cases of foundation design, it is recommended that a minimum factor of safety of 2 to 2.5 be used to arrive at the allowable ultimate uplift capacity. REFERENCES 1. Balla, A., The resistance to breaking out of mushroom foundations for pylons, in Proc., V Int. Conf. Soil Mech. Found. Eng., Paris, France, 1, 1961,569. © 1999 by CRC Press LLC 2. Meyerhof, G. G., and Adams, J. I., The ultimate uplift capacity of foundations, Canadian Geotech. J., 5(4)225, 1968. 3. Caquot, A., and Kerisel, J., Tables for Calculation of Passive Pressure, Active Pressure, and Bearing Capacity of Foundations, Gauthier- Villars, Paris, 1949. 4. Das, B. M., and Seeley, G. R., Breakout resistance of horizontal anchors, J. Geotech. Eng. Div., ASCE, 101(9), 999, 1975. 5. Meyerhof, G. G., Uplift resistance of inclined anchors and piles, in Proc., VIII Int. Conf. Soil Mech. Found. Eng., Moscow, USSR, 2.1, 1973, 167. 6. Das, B. M., and Jones, A. D., Uplift capacity of rectangular founda- tions in sand, Trans. Res. Rec. 884, National Research Council, Washington, DC, 54, 1982. 7. Vesic, A. S., Cratering by explosives as an earth pressure problem, in Proc., VI Int. Conf. Soil Mech. Found. Eng., Montreal, Canada, 2, 1965, 427. 8. Vesic, A. S., Breakout resistance of objects embedded in ocean bottom, J. Soil Mech. Found. Div., ASCE, 97(9), 1183, 1971. 9. Saeedy, H. S., Stability of circular vertical earth anchors, Canadian Geotech. J., 24(3), 452, 1987. 10. Baker, W. H., and Kondner, R. L., Pullout load capacity of a circular earth anchor buried in sand, Highway Res. Rec.108, National Research Council, Washington, DC, 1, 1966. 11. Sutherland, H. B., Model studies for shaft raising through cohesionless soils, in Proc., VI Int. Conf. Soil Mech. Found. Eng., Montreal Canada, 2, 1965, 410 12. Esquivel-Diaz, R. F., Pullout Resistance of Deeply Buried Anchors in Sand, M.S. Thesis, Duke University, Durham, NC, USA, 1967. 13. Kulhawy, F. H., Trautman, C. H., and Nicolaides, C. N., Spread foundations in uplift: experimental study, Found. for Transmission Towers, Geotech. Spec. Pub. 8, ASCE, 110, 1987. 14. Das, B. M., Model tests for uplift capacity of foundations in clay, Soils and Foundations, Japan, 18(2), 17, 1978. 15. Ali, M., Pullout Resistance of Anchor Plates in Soft Bentonite Clay, M.S. Thesis, Duke University, Durham, NC, USA, 1968. 16. Kupferman, M., The Vertical Holding Capacity of Marine Anchors in Clay Subjected to Static and Dynamic Loading, M.S. Thesis, Univer- sity of Massachusetts, Amherst, , USA, 1971. 17. Adams, J. K., and Hayes, D. C., The uplift capacity of shallow foundations, Ontario Hydro. Res. Quarterly, 19(1), 1, 1967. 18. Bhatnagar, R. S., Pullout Resistance of Anchors in Silty Clay, M.S. Thesis, Duke University, Durham, NC, USA, 1969. 19. Das, B. M., A procedure for estimation of ultimate uplift capacity of foundations in clay, Soils and Foundations, Japan, 20(1), 77, 1980. © 1999 by CRC Press LLC