A STUDY ON BEARING CAPACITY OF STRIP AND SQUARE FOOTINGS IN SAND FROM N AND φ
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1 Proceedings of Indian Geotechnical Conference December 22-24,2013, Roorkee A STUDY ON BEARING CAPACITY OF STRIP AND SQUARE FOOTINGS IN SAND FROM N AND φ S.V. Abhishek, PG Student, A.U. College of Engineering, Visakhapatnam, svabhi.92@gmail.com C.N.V. Satyanarayana Reddy, Professor, A.U. College of Engg., Visakhapatnam, cnvsnreddy@rediffmail.com ABSTRACT: The allowable bearing capacity of shallow foundations in cohesionless soils such as sand, is commonly estimated based on results obtained from in-situ penetration tests, most notably, the standard penetration resistance. However, estimation of safe bearing capacity from the angle of internal friction obtained based on the standard penetration resistance, has not been investigated. Hence the present paper deals with an analytical parametric study performed to compare the net safe bearing capacity estimated from the standard penetration resistance using Teng s equation, with that obtained from the angle of internal friction according to IS:6403, for strip and square footings in sand. Likewise, the net safe settlement pressure estimated from IS:8009 (Part 1) based on the standard penetration resistance, is compared with that attained from the semi-empirical approach of Schmertmann et al. The results obtained, indicate that Teng s equation and the equation specified by IS:8009 (Part 1) are appropriate for estimating the net safe bearing capacity and net safe settlement pressure respectively, of strip and square footings in sand. INTRODUCTION Penetration tests such as the standard penetration test (SPT) are conducted for estimating the density, strength and compressibility of cohesionless soils such as sand, due to difficulty in obtaining undisturbed (UD) samples. The allowable bearing capacity is an important parameter that governs the design of shallow foundations and is taken as the least of safe bearing capacity and safe settlement pressure. Correlations of net safe bearing capacity and net safe settlement pressure with the standard penetration resistance (N) have been developed by past researchers, and the same are widely used for estimating the allowable bearing capacity of shallow foundations in sand. Indian standard code IS:6403 [1] specifies a chart relating the standard penetration resistance with the angle of internal friction ( φ ) corresponding to different states of soil denseness. However, no attempt has been made to study and compare the net safe bearing capacity computed from φ (obtained based on N ) with that attained directly from N. Hence in the present paper, the net safe bearing capacity (q ns ) determined from N using Teng s equation [2], is compared with that estimated from φ from IS:6403 by using the N vs. φ chart of IS:6403, for strip and square footings in sand. Likewise, the net safe settlement pressure (q nssp ) obtained based on N from IS:8009 (Part 1) [3] is compared with that attained from the semiempirical approach of Schmertmann et al. [4] through an analytical parametric study. The parametric study is performed by varying the size, shape and depth of foundation for different standard penetration resistance ranging from 10 to 40 in increments of 5. From the results of the parametric study, interactive charts are prepared for net safe bearing capacity and net safe settlement pressure in terms of N and φ (obtained based on N ) for strip and square footings in sand. LITERATURE REVIEW Several researchers have developed equations for estimating the net safe bearing capacity and net safe settlement pressure of footings in sand from the standard penetration resistance and angle of internal friction. The equations used in the present study for estimation of net safe bearing capacity and net safe settlement pressure of strip and square footings in sand based on N and φ are illustrated below. Page 1 of 10
2 S.V. Abhishek, C.N.V. Satyanarayana Reddy Net Safe Bearing Capacity from N Empirical equations [2] developed for determination of net ultimate bearing capacity of strip and square footings based on the standard penetration resistance are presented below. The net ultimate bearing capacity (q nu ) of a strip footing in kpa is given by, q nu = [ 3N BR w2 + 5(100 + N )Df R w1] (1) 6 The net ultimate bearing capacity of a square footing in kpa is given by, q nu = [ 2N BR w2 + 6(100 + N )Df R w1] (2) 6 where N = standard penetration resistance (blows/30 cm); B = size of footing (m); D f = depth of foundation (m); R w1 and R w2 are water table correction factors defined by, R w1 z w1 = D (3) f z w2 R w2 = (4) B where z w1 = depth of water table below ground surface (m); and z w2 = depth of water table below base of footing (m). Further, the depth of foundation (D f ) should be restricted to the size of the footing (B) when D f exceeds B [2]. The net safe bearing capacity is calculated by dividing the net ultimate bearing capacity with a factor of safety. Net Safe Bearing Capacity from φ The equation specified for estimating the net ultimate bearing capacity (kpa) of footings in general shear failure ( φ > 36 0 ) is given by [1], q nu = cn s d i + q(n 1)s d i + 0.5γBN s d i W (5) c c c c q q q q where c = cohesion of soil (kpa); γ = bulk unit weight of soil (kn/m 3 ); q = initial effective overburden pressure at base of footing (kpa); B = size of footing (m); N c, N q and N γ are bearing capacity factors; s c, s q and s γ are shape factors; d c, d q and d γ are depth factors; i c, i q and i γ are inclination factors; W' is a water table correction factor. γ γ γ γ ' The bearing capacity factors, N c, N q and N γ depend on the angle of internal friction of the soil and are calculated using the following expressions proposed by Vesic [5]. 2 φ π.tan φ N q = tan e (6) 2 N c = (N q 1) cot φ (7) N = 2(N q + 1) tan φ (8) γ The shape, depth and inclination factors have been considered as per IS:6403. The expression for the water table correction factor is similar to that of Eq. 4. The equation specified for estimating the net ultimate bearing capacity (kpa) of footings in local shear failure ( φ < 28 0 ) is given by [1], 2 ' ' ' ' q nu = cn cs cd ci c + q(n q 1)s qd qi q + 0.5γBN γs γd γi γ W (9) 3 ' where N ' c, N q and ' N γ are Vesic s bearing capacity factors corresponding to local shear failure, obtained by substituting the mobilized angle of internal friction ( φ m ) in the place of φ in Eqs The mobilized angle of internal friction can be evaluated using the expression [6], 2 φ = Tan 1 m tan φ (10) 3 where φ = friction angle of soil (degrees). For intermediate values of friction angle between 28 0 and 36 0, the bearing capacity factors can be obtained by linear interpolation between the local shear failure and the general shear failure. The net safe bearing capacity is calculated by dividing the net ultimate bearing capacity with a factor of safety. Net Safe Settlement Pressure from N The equation specified for computing the net safe settlement pressure (kpa) from the standard penetration resistance is given by [3], q nssp B = 1.385(N 3) R w2. sa (11) 2B 2 Page 2 of 10
3 where B = size of footing (m); R w2 = water table correction factor (similar to Eq. 4); and s a = allowable settlement (50 mm for isolated footings in sand [7]). Schmertmann et al. [4] proposed a semi-empirical equation for estimating the net safe settlement pressure of footings on granular soils based on the results of SPT. The soil on which the footing rests, is divided into several layers based on the standard penetration resistance profile, each layer having a constant value of strain and soil modulus. The net safe settlement pressure (kpa) is calculated by summing up the influence of all the layers using the expression, sa q nssp = + 0.5q (12) 2B,4B Iz C z 0 Es where s a = allowable settlement (50 mm for isolated footings in sand [7]); q = initial effective overburden pressure at foundation level (kpa); I z = vertical strain influence factor; E s = modulus of elasticity of soil (kpa); Δz = thickness of elemental soil layer; C = correction factor to take into account creep in soil = 1+0.2log 10 (t / 0.1) where t is the time in years for which period the settlement is required. A study on bearing capacity of strip and square footings in sand from N and φ influence factor diagram can be obtained by interpolation. The modulus of elasticity of sand E s (kpa) can be determined from the standard penetration resistance using the following expression [8], E s = 766N (13) ANALYTICAL WORK Interactive charts are developed for net safe bearing capacity and net safe settlement pressure in terms of the standard penetration resistance, using the equations illustrated earlier, for strip and square footings in medium dense to dense sand, by performing an analytical parametric study. The parametric study is performed for N values of 10, 15, 20, 25, 30, 35 and 40 by varying the, 1. size of footing (1.0, 1.5 and 2.0 m for strip footings and 1.5, 2.0, 2.5, 3.0 m for square footings) keeping depth of foundation constant. 2. depth of foundation (1.5, 2.0, 2.5 and 3.0 m) keeping size constant. The N vs. φ chart developed by Peck et al. [9] and adopted by IS:6403 is shown in Fig. 2. Fig. 1 Vertical strain influence factor diagram (after Schmertmann et al. [4]) The vertical strain influence factor diagrams [4] for square footing (L/B = 1) and strip footing (L/B 10 ) are presented in Fig. 1. For intermediate values of L/B between 1 and 10, the vertical strain Fig. 2 Variation of standard penetration resistance with angle of internal friction (after Peck et al. [9]) Referring to Fig. 2, the values of angle of internal friction corresponding to different standard penetration resistance are presented in Table 1. The Page 3 of 10
4 S.V. Abhishek and C.N.V. Satyanarayana Reddy possible mode of shear failure in the foundation soil is also mentioned. Table 1 Values of angle of internal friction for different standard penetration resistance N φ Possible Mode of Failure Local Shear Failure * Transition Transition Transition General Shear Failure General Shear Failure General Shear Failure * Since the standard penetration resistance is 10, the soil is considered to exist in loose condition and hence local shear failure is assumed to take place. Before commencement of the parametric study, the following considerations have been made. 1. The soil is homogeneous, isotropic and semiinfinite. 2. The ground water table is located at a depth greater than the width of the footing, measured below the footing base. 3. The load transmitted to the footing is vertical and uniformly distributed. 4. The modulus of elasticity of sand is equal to 766N at the base of the footing and increases linearly with depth. 5. The bulk unit weight of sand (γ) is 18 kn/m The factor of safety against risk of shear failure is The time period for settlement analysis is 50 years. RESULTS AND DISCUSSION Net Safe Bearing Capacity of Strip Footing Effect of Size of Footing for Given Depth of Foundation Figure 3 shows the variation of the net safe bearing capacity with the standard penetration resistance for a strip footing of varying width (B = 1.0, 1.5, 2.0 m) installed at a given depth (D f ). Fig. 3 Influence of size of strip footing on net safe bearing capacity for a given depth of foundation Page 4 of 10
5 The net safe bearing capacity computed from both Teng s equation and IS:6403 increases non-linearly with the standard penetration resistance for a given size and depth of footing. For D f /B 1.0, the net safe bearing capacity computed from Teng s equation is in fair agreement with that computed from IS:6403. However for D f /B>1.0, it is observed that Teng s equation yields conservative value of net safe bearing capacity due to restriction of depth of footing (D f ) to size (B). A study on bearing capacity of strip and square footings in sand from N and φ Effect of Depth of Foundation on Strip Footing of Given Size Figure 4 depicts the variation of the net safe bearing capacity with the standard penetration resistance for a strip footing installed at varying depth (D f = 1.5, 2.0, 2.5, 3.0 m) corresponding to a given size (B). Similar to the results of Fig. 3, the net safe bearing capacity estimated from Teng s equation compares well with IS:6403 for D f /B 1.0. However for D f /B>1.0, Teng s equation is conservative compared to IS:6403, due to the restriction of the value of D f to B. Therefore Teng s equation is to be preferred for estimating the net safe bearing capacity of strip footings in sand for D f /B>1.0 as it yields conservative values. It can also be observed that the net safe bearing capacity estimated from IS:6403 increases with increased depth of foundation for a given standard penetration resistance and size of footing. But referring to Fig. 3, the equation of IS:6403 fails to reflect the increase in net safe bearing capacity with increased size of footing especially for D f 2.5m and thus contradicts the fundamental concept that bearing capacity should increase with the size of the footing. Fig. 4 Effect of depth of foundation on net safe bearing capacity of strip footing of a given size Net Safe Settlement Pressure of Strip Footing Effect of Size of Footing for Given Depth of Foundation Figure 5 shows the variation of the net safe settlement pressure with the standard penetration resistance for a strip footing of varying width (B = 1.0, 1.5, 2.0 m) installed at a given depth (D f ). The net safe settlement pressure computed from both IS:8009 (Part 1) and the semi-empirical equation of Schmertmann et al. increases almost linearly with the standard penetration resistance for a given size and depth of footing. The net safe settlement pressure decreases with increased size of footing due to increase in the size of the pressure bulb. Further, the net safe settlement pressure estimated from Schmertmann et al. decreases with increased size of footing due to increase in the value of the strain influence factor. It can be observed that the net safe settlement pressure estimated from IS:8009 (Part 1) is conservative compared to Schmertmann et al. This is because no depth factor is present in the equation specified by IS:8009 (Part 1). If a depth factor is incorporated, Page 5 of 10
6 S.V. Abhishek and C.N.V. Satyanarayana Reddy the resulting net safe settlement pressure could be nearer to that estimated from Schmertmann et al. Effect of Depth of Foundation on Strip Footing of Given Size Figure 6 depicts the variation of the net safe settlement pressure with the standard penetration resistance for a strip footing installed at varying depth (D f = 1.5, 2.0, 2.5, 3.0 m) corresponding to a given size (B). The net safe settlement pressure estimated from Schmertmann et al. decreases with increased depth of foundation due to reduction in modulus of elasticity of sand at a given depth below the ground surface. This is because the modulus of elasticity is considered to increase linearly with depth, possessing a value of 766N at the base of the footing. Fig. 5 Influence of size of strip footing on net safe settlement pressure for a given depth of foundation Fig. 6 Effect of depth of foundation on net safe settlement pressure of strip footing of a given size Page 6 of 10
7 It can be observed that the equation of Schmertmann et al. overestimates the net safe settlement pressure by times that of IS:8009 (Part 1). The suitability of the methods may be ascertained by obtaining data of plate load tests conducted in sands of different relative densities. But for design purposes, the equation of IS:8009 (Part 1) may be preferred as it is found to be conservative. Net Safe Bearing Capacity of Square Footing A study on bearing capacity of strip and square footings in sand from N and φ Effect of Size of Footing for Given Depth of Foundation Figure 7 illustrates the variation of the net safe bearing capacity with the standard penetration resistance for a square footing of varying width (B = 1.5, 2.0, 2.5, 3.0 m) installed at a given depth (D f ). As expected, the net safe bearing capacity computed from both Teng s equation and IS:6403 increases non-linearly with the standard penetration resistance for a given size and depth of footing. Similar to the results obtained for strip footing, the net safe bearing capacity estimated from Teng s equation is in good agreement with IS:6403 for D f /B However for D f /B>1.0, Teng s equation yields conservative value of net safe bearing capacity due to restriction of depth of footing (D f ) to size (B). Fig. 7 Influence of size of square footing on net safe bearing capacity for given depth of foundation Effect of Depth of Foundation on Square Footing of Given Size Figure 8 shows the variation of net safe bearing capacity with the standard penetration resistance for a square footing installed at varying depth (D f = 1.5, 2.0, 2.5, 3.0 m) corresponding to a given size (B). Similar to the results of Fig. 7, the net safe bearing capacity computed from Teng s equation compares well with IS:6403 for D f /B 1. 0 and for D f /B>1.0, Teng s equation [2] yields conservative values. It can also be observed that the net safe bearing capacity estimated from IS:6403 increases with increased depth of foundation for a given standard penetration resistance and size of footing. But referring to Fig. 7, the equation of IS:6403 fails to reflect the increase in net safe bearing capacity with increased size of footing especially for D f 2.5m. A similar behaviour is observed in the earlier case of strip footing (Fig. 3) and such a tendency contradicts the fundamental concept that bearing capacity should increase with the size of the footing. Page 7 of 10
8 S.V. Abhishek and C.N.V. Satyanarayana Reddy Net Safe Settlement Pressure of Square Footing Effect of Size of Footing for Given Depth of Foundation Figure 9 shows the variation of the net safe settlement pressure with the standard penetration resistance for a square footing of varying width (B = 1.5, 2.0, 2.5, 3.0 m) installed at a given depth (D f ). The net safe settlement pressure computed from both IS:8009 (Part 1) and Schmertmann et al., increases almost linearly with the standard penetration resistance for a given size and depth of footing. Schmertmann et al. s equation considerably overestimates the net safe settlement pressure by times that of IS:8009 (Part 1) due to the absence of a depth factor in the equation specified by the latter. Fig. 8 Effect of depth of foundation on net safe bearing capacity of square footing of a given size Page 8 of 10
9 A study on bearing capacity of strip and square footings in sand from N and φ Fig. 9 Influence of size of square footing on net safe settlement pressure for a given depth of foundation Effect of Depth of Foundation on Square Footing of Given Size Figure 10 depicts the variation of net safe settlement pressure with the standard penetration resistance for a square footing installed at varying depth (D f = 1.5, 2.0, 2.5, 3.0 m) corresponding to a given size (B). Similar to the results obtained for strip footing, the net safe settlement pressure estimated from the equation proposed by Schmertmann et al. decreases with increased depth of foundation due to reduction in the modulus of elasticity of sand at a given depth below the ground surface. This is because the modulus of elasticity has been considered to increase linearly with depth, possessing a value of 766N at the base of the footing. The equation of IS:8009 (Part 1) is independent of the depth of foundation and hence yields much conservative values of net safe settlement pressure compared to Schmertmann et al. [4]. As a result, the equation specified by IS:8009 (Part 1) is suggested for estimating the net safe settlement pressure of square footings in sand. Fig. 10 Effect of depth foundation on net safe settlement pressure of square footing of given size Page 9 of 10
10 S.V. Abhishek and C.N.V. Satyanarayana Reddy CONCLUSIONS Based on the analytical parametric study on bearing capacity of strip and square footings in sand from N and φ, the following conclusions are drawn: 1. The empirical equation of Teng (1962) based on N, is advocated for estimating the net safe bearing capacity of strip and square footings in sand for any D f /B ratio. 2. For D f /B 1. 0, the empirical equation of Teng (1962) and the equation given by IS: based on φ, are found to be equally suitable for estimating the net safe bearing capacity of strip and square footings in sand. 3. The equation specified by IS:8009 (Part 1) based on N, is suggested for estimating the net safe settlement pressure of strip and square footings in sand. 4. It is worthwhile to consider revising the equation given by IS:8009 (Part 1)-1976 by conducting large scale plate load tests in sands of different relative densities, as it is much conservative compared to the equation proposed by Schmertmann et al. (1978). ACKNOWLEDGMENT The authors gratefully acknowledge the advice offered by Prof. M.R. Madhav, Professor Emeritus, JNTU and Visiting Professor, IIT, Hyderabad, during the course of the study. REFERENCES 1. IS: , Code of practice for determination of bearing capacity of shallow foundations, BIS, New Delhi. 2. Teng, W.C. (1962), Foundation Design, Wiley, New York. 3. IS:8009 (Part 1)-1976, Code of practice for calculation of settlement of foundations (shallow foundations subjected to symmetrical static vertical loads), BIS, New Delhi. 4. Schmertmann, J.H., Hartman, J.P. and Brown, P.R. (1978), Improved strain influence factor diagrams, Journal of Geotechnical Engineering Division, ASCE, 104(GT8), Vesic, A.S. (1973), Analysis of ultimate loads of shallow foundations, J. Soil Mech. and Foundations Division, ASCE, 99(SM1), Terzaghi, K. (1943), Theoretical Soil Mechanics, John Wiley & Sons, New York. 7. IS: , Code of practice for design and construction of foundations in soils: general requirements, BIS, New Delhi. 8. Schmertmann, J.H. (1970), Static cone to compute static settlement over sand, Journal of Soil Mechanics and Foundations Division, ASCE, 96(SM3), Peck, R.B., Hanson, W.E. and Thornburn, T.H. (1974), Foundation Engineering, Wiley, NY. Page 10 of 10
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