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Determination of Ng for Rough Circular Footing
Assistant Professor, Department of Civil Engineering, and Research Scholar, Department of Civil Engineering, |
ABSTRACT
The bearing capacity factor Ng for a rough circular footing was determined in this study by using the method of stress characteristics. The failure mechanism considered in the study comprises of a curved non-plastic trapped wedge below the footing being tangential to its base at its edge and inclined at an angle (p/4-f/2) with the axis of symmetry. The chosen curved trapped wedge ensures that the angle of interface friction between the footing base and underlying soil mass remains equal to f at the footing edge. The computed values of Ng were found to be significantly smaller than those obtained with the consideration a triangular (conical) trapped wedge below the footing wedge.
Keywords: bearing capacity; failure; foundations; numerical modeling and analysis; plasticity.
INTRODUCTION
It is known that the bearing capacity factors Nc and Nq, due to the components of cohesion and surcharge, respectively, remain almost unchanged with respect to alteration in the roughness of the footing-soil interface (Meyerhof, 1951, 1955; Griffiths, 1982; and Bolton and Lau, 1993). The magnitude of Ng on the other hand is affected quite significantly with changes in the roughness of the footing-soil interface (Meyerhof, 1951, 1955; Griffiths, 1982; Bolton and Lau, 1993; Michalowski, 1997; and Erickson and Drescher, 2002). By using the method of characteristics, Kumar (2003) has recently employed the failure mechanism of Lundgren and Mortensen (1953) to compute the bearing capacity factor Ng for a strip footing. The computed values of Ng using this mechanism were found to be significantly smaller than those reported in literature for a rough footing by using a triangular trapped wedge below the footing base. The failure mechanism of Lundgren and Mortensen automatically ensures that (i) the vertical plane passing through the centre of the strip footing remains free from any shear stress at the bottommost point of the trapped wedge, and (ii) the shear resistance of the soil mass along the interface of the footing soil interface is fully mobilized along its either edge; the later condition follows from the findings of Meyerhof (1955) and Frydman and Burd (1997). The present note determines the bearing capacity factor Ng for a rough circular footing using the failure mechanism of Lundgren and Mortensen (1953). Unlike plane strain, the axi-symmetric stability problems as such cannot be simply solved only with the use of two equilibrium equations and one failure condition due to the involvement of an additional unknown stress variable sq (hoop stress). For obtaining the bearing capacity of a circular footing, it is generally assumed that the value of sq is equal to the minor principal stress (Harr and Von Karman, 1909; Cox et al., 1961; Larkin, 1968; and Bolton and Lau, 1993). This assumption was also followed in this note to generate the solution for a rough circular footing. The results obtained from the analysis were thoroughly compared with those published in literature.
GOVERNING EQUATIONS FOR AN AXI-SYMMETRIC PROBLEM
The various equations associated with the method of characteristics for an axi-symmetric problem are summarized below; the paper of Larkin (1968) and Bolton and Lau (1993) can be referred for this purpose.
Equations of statical equilibrium:
 | (1) |
 | (2) |
In the above equations, for a given point r is the radial distance measured from the centre of the footing and z is the vertical distance from the ground surface as shown in Figure 1(a). The stresses sr, sz and trz are defined in Figures 1(b), 2(a) and 2(b). Compressive magnitudes of sr, sz and sq were taken as positive and the direction for the positive shear stress trz is indicated in Figure 2(b). Following Sokolovski (1960), by making use of the Mohr-Coulomb failure criterion, for a cohesionless soil deposit, the three stress components (sr, sz and trz) can be expressed in terms of two independent variables, namely, s and q; where s is the distance on the Mohr stress diagram, between the centre of the Mohr circle and a point where the Coulomb’s linear failure envelope joins with the t-axis (Figure 2a), and q is an angle made by the direction of the major principal stress (s1) with the positive z-axis as shown in Figure 2(c); the sign of q corresponding to the counter-clockwise rotation of s1 with respect to the positive z-axis was taken as positive,
Figure 1. Failure mechanism for the footing and the freebody diagram of soil element below the footing base
 | (3) |
 | (4) |
 | (5) |
Following the hypothesis of Harr and Von Karman (1909), the value of sq was assumed to be equal to the minor principal stress s3; this assumption was also made previously by Larkin (1968) and Bolton and Lau (1993) to obtain the bearing capacity value for a smooth and rough circular footing, respectively. Therefore, by using st = s3, it can be indicated (Sokolovski, 1960) that
 | (6) |
By substituting the values of sr, sz, and trz from equations (3)-(6) in the equilibrium equations (1) and (2), and then on simplification, the following expressions applicable along two different families of characteristics can be derived:
 | (7) |
 | (8) |
The upper and lower signs in the above equations correspond to (p/2 + f/2 ) and (p/2 - f/2 ) characteristics, respectively; the textbook of Sokolovski (1960) and the paper of Larkin (1968) can be referred for the derivations of the above equations.
ANALYSIS
For a perfectly rough footing it is known that at failure a soil wedge just below the footing base moves vertically downward as if it was a part of the footing itself. It was indicated by Lundgren and Mortensen (1953) that this wedge must start tangentially to the footing base at its either edge and intersect the vertical axis of the footing with an inclination of m as shown in Figure 1(a); where m = p/4 - f/2. Such a geometry of the trapped wedge ensures that (i) the shear resistance at the interface of soil media and footing base gets fully mobilized along its either edge (Meyerhof, 1951; and Frydman and Burd, 1997), and (ii) the vertical plane, for a strip footing, passing through the centre of the foundation remains free from any shear stress at the bottommost point of the trapped wedge. The outer boundary of the trapped wedge becomes one of the characteristics starting tangentially from the edge of the footing base. In order that the characteristics start tangentially from the footing base, the magnitude of q becomes equal to -p/2 + m along the footing base at its right edge. The ground surface was taken horizontal with uniform surcharge pressure (q). The magnitude of q along the ground surface, on the right side, will, therefore, become equal to p/2. On account of the difference in the value of q over the ground surface from that along the footing base, the edge of the footing becomes a singular point. Starting from the ground surface, all the q + m characteristics converge about the right edge (OR) before intersecting the last q - m characteristics emerging from OR (refer Fig. 1a). From the known boundary conditions along the ground surface (q=p/2 and sz = q), by using the equations applicable along two different families of characteristics, the solution can be numerically established gradually towards the non-plastic trapped curved wedge below the footing. The computations were continued until the values of q along the last q+m characteristics (ORQ) emerging from the point OR as well as the values of r simultaneously become equal to 0. Computations were performed by using the finite difference procedure framed by Sokolovski (1960). All the computations were carried out twice; first by using the forward difference technique and later on, by using the central difference method the accuracy of the computations was increased. Using a series of computational runs, the geometry of the trapped wedge along with the variation of the boundary stresses along its periphery ORQ was established. The complete computational procedure for solving the stability problems using the method of characteristics has been described in the classical textbook of Sokolovski (1960) and in the paper of Kumar and Mohan Rao (2002).
DETERMINATION OF THE FAILURE LOAD
After knowing the state of stress everywhere along the boundary of the trapped curved wedge, the overall vertical equilibrium of the trapped soil wedge, OLQOR (refer Figure 1a), was then considered to compute the failure load Pu for the footing. The total vertical component (V) of the shear resistance available along the outer periphery of the wedge against the downward movement of the footing was determined by using the following expression:
 |
(9) |
where n is the number of segments chosen to divide the arc QOR (refer to Figure 1a); sz and trz are the average values of the stresses along a small discrete soil element cd; and, daz and dar are the area projections of the annular element cd along horizontal and vertical planes, respectively (refer Figure 1b). It can be noted that daz = 2prdr, and dar = 2p(r+0.5dr)dz; where dr and dz are horizontal and vertical width of the element cd, and r is the horizontal distance between the middle point of the element and the vertical axis of the footing.
After determining the value of V, the magnitude of the ultimate load was determined by considering the vertical equilibrium of the trapped wedge (OLQOR) below the footing, that is,
| Pu = V-W | (10) |
where W is the weight of the trapped wedge, OLQOR.
The above methodology for the determination of the failure load is similar to that followed by Kumar (2003) for the case of a strip footing.
RESULTS AND COMPARISON
Bearing Capacity Factor
For computing the unit weight component of the bearing capacity, a certain minimum magnitude of surcharge pressure (q) is always needed to avoid the computational difficulties. However, the contribution of the chosen surcharge pressure from the total ultimate load can be deducted if the bearing capacity factor Nq due to the surcharge component is separately known. It is understood that the bearing capacity factor Nq remains unaffected with changes in the roughness of the footing (Bolton and Lau, 1993). The magnitude of Nq was computed by extending the characteristics up to the entire footing-soil interface; the value of f along the footing-soil interface was taken equal to zero (smooth footing for which the direction of the major principal stress becomes vertical). The obtained values of Nq from the present computations along with the results from Bolton and Lau (1993) are given in Table 1. It can be seen the computed values of Nq match very closely with those of Bolton and Lau (1993).
After establishing the values of Nq, the magnitude of Ng was determined by using the following expression (Terzaghi, 1943):
 |
(11) |
In the above expression, A is the area of the footing, A = pD2/4; wherein D is the diameter of the footing. It can be seen that that the minimum chosen value of surcharge (qmin) needed to avoid the computational difficulties, increases continuously with the decrease in f. The obtained variation of qmin/(gD) with f is similar to that observed earlier by Kumar (2003) for a strip footing.
The obtained values of Ng with f on a semi-log plot are shown in Figure 3; the total number of (q - m) characteristics at the singular point (OR) were taken equal to 2000 for all values of f and the number of (q + m) characteristics were varied from 8586 for of f = 5o to 95598 for of f = 50o. The calculated values of Ng were compared with the those reported for a rough circular footing by (i) Bolton and Lau (1993) using the method of characteristics but with the employment of a triangular trapped wedge having horizontal inclination, a = 45 + f/2, (ii) Erickson and Drescher (2002) using the FLAC with an associated flow rule, and (iii) Manoharan and Dasgupta (1995) using the finite element method with an associated flow rule. The comparison of the results is shown in Table 2. It can be seen that the present Ng values are significantly smaller than those obtained by Bolton and Lau (1993) and the difference between the two theories increases continuously with increase in f. The finite element results of Manoharan and Dasgupta (1995) are found to be a little higher than the present values. The Ng values of Erickson and Drescher (2002) are found to be lower than the values obtained in the present analysis.
Figure 2. Definitions of various stress parameters
Figure 3. The variation of the bearing capacity factor Ng with f
Failure Patterns
The failure patterns were drawn by keeping the scale same for the horizontal and vertical axes. The failure patterns for f = 30o, 40o and 50o are shown in Figure 4. The total number of characteristics used for obtaining Ng were much higher than those shown in drawing the failure patterns; the ratio of the number of characteristics used for computing Ng to those chosen in drawing the failure patterns was approximately varying from 900 for f = 30o to 3600 for f = 50o. It can be noted that the size of the plastic zone relative to that of the trapped non-plastic wedge increases with increase in f. This observation is similar to that made earlier by Kumar (2003) for a strip footing.
Figure 4. The geometry of the failure patterns for (a) f = 30o; (b) f = 40o; and (c) f = 50o
Pressure Distribution Along The Footing Base
Since the boundary of the plastic domain has not been extended up to the base of the footing (excluding the edge of the footing) the variation of the vertical pressure along the base of the footing cannot be simply generated. However, if it is assumed that for a chosen infinitesimal annular vertical element of the trapped wedge the resultant of the shear forces on the vertical faces ad and bc of the element abcd (Figure 1b) does not have any component in the vertical direction, then an approximate variation of the base pressure can be generated from the vertical equilibrium of the soil wedge abcd; the horizontal lines ab and cd of the wedge lie along the footing base and the outer periphery of the trapped wedge, respectively. The expression for obtaining the footing pressure at any point is given below:
 |
(12) |
dW is the weight of the element abcd (refer Figure 1b); and the parameters sz, trz, daz and dar have already been defined in paragraph following Equation (9). The obtained variation of (p/gD) on the basis of Equation (12) along the footing base for of f = 30o, 40o and 50o is shown in Figure 5. It can be seen that the base pressure decreases with the increase in the distance from centre of the footing. It should be mentioned although the computed base pressure on the footing can not be said to be perfectly correct, however, the obtained value of Ng will not involve any error due to the fact that the overall vertical equilibrium of the soil wedge OLQOR does not involve any shear force along the vertical axis CQ.
Figure 5. The variation of p/(gD) over the base of the footing for different values of f
DISCUSSION
As per the theorem of limit analysis, for an associated flow rule material, the method of characteristics provides a lower bound estimate with reference to the magnitude of the failure load (Lysmer, 1970). This method does not take into account the kinematics of the problem, and therefore, the solution obtained with this method cannot be said to be correct. At present much better numerical methods are available in literature which consider simultaneously the determination of lower and upper bound solution with the help of finite element method and linear programming (Sloan et al. 1982; and Sloan, 1988). However, these methods require much more computational efforts as compared to the method of characteristics.
CONCLUSIONS
By employing a curved non-plastic trapped soil wedge below the footing base, the bearing capacity factor Ng was computed for a rough circular footing on the basis of the method of characteristics. The chosen curved trapped wedge ensures that along the footing-soil interface the friction angle becomes equal to f at the edge of the footing. The computed magnitudes of Ng were found to be significantly smaller than those reported for a rough circular footing with the choice of a triangular trapped wedge. It implies that the assumption of the triangular trapped wedge below the footing base, which is often made to avoid the computational complexities associated with the curved wedge, will lead to an overestimation of the bearing capacity of the foundation. The size of the plastic zone relative to that of the trapped wedge was found to increase with increase in the friction angle of the soil mass.
Table 1. The values of Nq using the method of characteristics
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Table 2. A comparison of Ng values for rough circular footing
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REFERENCES
1. Bolton, M. D., and C. K. Lau (1993) “Vertical Bearing Capacity Factors for Circular and Strip Footings on Mohr-Coulomb Soil,” Canadian Geotechnical Journal, Vol. 30, pp. 1024-1033.
2. Cox, A. D., G. Eason, and H. G. Hopkins (1961) “Axially Symmetric Plastic Deformation in Soils,” Philosophical Transactions of the Royal Society of London, Series A, Vol. 254, pp. 1-45.
3. Erickson, H. L., and A. Drescher (2002) “Bearing Capacity of Circular Footings,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 128, No. 1, pp. 38-43.
4. Frydman, S., and H. J. Burd (1997) “Numerical Studies of Bearing Capacity Factor Ng,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, No. 1, pp. 20-29.
5. Griffiths, D. V. (1982) “Computations of Bearing Capacity Factors Using Finite Elements,” Géotechnique, Vol. 32, No. 3, pp. 195-202.
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8. Kumar, J., and V. B. K. Mohan Rao (2002) “Seismic Bearing Capacity Factors for Spread Foundations,” Géotechnique, Vol. 52, No. 2, pp. 79-88.
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14. Michalowski, R. L. (1997) “An Estimate of the Influence of Soil Weight on Bearing Capacity Using Limit Analysis,” Soils and Foundations, Vol. 37, No. 4, pp. 57-64.
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17. Sokolovski, V. (1960) Statics of Soil Media, Butterworths Publications, London.
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