- Open Access
The bearing capacity factor N γ of strip footings on c–ϕ–γ soil using the method of characteristics
© The Author(s) 2016
- Received: 3 June 2016
- Accepted: 15 August 2016
- Published: 5 September 2016
The method of characteristics (also called as the slip-line method) is used to calculate the bearing capacity of strip footings on ponderable soil. The soil is assumed to be a rigid plastic that conforms to the Mohr–Coulomb criterion. The solution procedures proposed in this paper is implemented using a finite difference method and suitable for both smooth and rough footings. By accounting for the influence of the cohesion c, the friction angle ϕ and the unit weight γ of the soil in one failure mechanism, the solution can strictly satisfy the required boundary conditions.
The numerical solution of N γ are consistent with published complete solutions based on cohesionless soil with no surcharge load. The relationship of N γ between smooth and rough foundations is discussed which indicates that the value of N γ for a smooth footing is only half or more of that for a rough footing. The influence of λ (λ = (q + ccot ϕ)/γB) on N γ is studied. Finally, a curve-fitting formula that simultaneously considers both ϕ and λ is proposed and is used to produce a series of N γ versus λ curves.
The surcharge ratio λ and roughness of the footing base both have significant impacts on N γ . The formula for the bearing capacity on c–ϕ–γ soil can be still expressed by Terzaghi’s equation except that the bearing capacity factor N γ depends on the surcharge ratio λ in addition to the friction angle ϕ. Comparisons with the exact solutions obtained from numerical results indicate that the proposed formula is able to provide an accurate approximation with an error of no more than ±2 %.
- Bearing capacity
- Strip footing
- The method of characteristics
- Numerical analysis
- Shallow foundation
When the bearing capacity is computed on general c–ϕ–γ soil without superposition and the result is still written in the form of Eq. (1), some researchers have found that the value of N γ relates to not only the soil friction angle ϕ but also to other parameters, such as q, c, γ and B. Cox (1962) revealed that the parameters associated with stress characteristic equations are ϕ and a dimensionless parameter G(G = γB/2c) for a smooth footing without surcharge. Chen (1975) introduced a foundation depth and width ratio, D/B, and computed the changes in N γ with the D/B for different internal friction angles. Xiao et al. (1998) calculated the bearing capacity using the method of characteristics(MOC) and revealed that q, c, γ and B all affect N γ , and N γ is only affected by ϕ and γB/(c + qtan ϕ) when the load is vertical. Michalowski (1997) and Silvestri (2003) studied the influence of c/γB and q/γB on N γ using the limit analysis method and the limit equilibrium method, respectively. Their research demonstrated that for a given ϕ, the value of N γ significantly changes with c/γB or q/γB. Zhu et al. (2003) showed that N γ is not only related to the friction angle ϕ but also to the surcharge ratio λ (λ = (q + ccot ϕ)/γB). Sun et al. (2013) noted that there are two types of failure mechanisms for rough footings, and whether the trapped non-plastic wedge traverses the footing edge depends on the surcharge ratio λ. Sun et al. (2013) also studied the variation of N γ with ϕ when λ equals the critical surcharge ratio λ c. The researchers above studied different factors influencing N γ , but none of these studies proposed a formula to calculate N γ .
The MOC is one of main methods applied in the bearing capacity issue which has been discussed by many researchers (Bolton and Lau 1993; Lundgren and Mortensen 1953; Martin 2003; Sokolovskii 1965). The classical bearing capacity factor N γ when q = 0, c = 0, γ ≠ 0 by MOC is calculated to a high degree of precision and is proven to be exact by checking for coincident lower and upper bounds, and by extending the lower bound stress field throughout the semi-infinite soil domain (Martin 2005; Smith 2005).The bearing capacity on general c–ϕ–γ soil is calculated in one failure mechanism and can be therefore treated as exact solution. In this paper, the MOC is employed to calculate the bearing capacity of strip footings on general c–ϕ–γ soil and is implemented with a self-coded finite difference method program. The computation of the bearing capacity is carried out with one failure mechanism instead of using superposition approximation, which avoids assuming the shape of the slip lines. The present procedures for computing both smooth and rough footings are unified, which satisfies all of the requirements of the boundary conditions and the symmetric conditions of the surface footings. The numerical results of N γ are compared with other published results, and the sources of errors in the other results are discussed. The bearing capacity factor N γ on general c–ϕ–γ soil is found to be a function of not only the friction angle ϕ but also the surcharge ratio λ. Then, the effects of the footing base roughness and the surcharge ratio λ on N γ are investigated. Finally, a curve-fitting formula for N γ that considers ϕ and the surcharge ratio λ is proposed based on the numerical results.
Characteristic equations at any point in the limit equilibrium state
Derivation of the finite difference equation
The state of the field is determined point by point, and the final point exists at the bottom of the footing, for which y and η are known. As a result, x and σ at the final point can be directly solved using Eq. (7) without iteration.
However, for a rough footing, η equals 3π/4 + ϕ/2 if the characteristic exists at the footing base. It is impossible for all of the soil under the footing base to be in the yield state because of the roughness. Therefore, the critical problem for rough foundations is determining the boundary of the non-plastic wedge and the yield zone. The α characteristics are assumed to progress to the footing at the beginning, and all the α characteristics thus start from the free boundary and end at the footing base. According to the symmetry requirements, there is no shear stress at the center of the foundation. If the last α characteristic and the centerline of the foundation intersect at point I, the point I will have the properties of x = −B/2 and η = π/2 at the same time. The β characteristic noted as FI in Fig. 4b is the boundary of the non-plastic wedge and the yield zone. The area between FI and the footing is the non-plastic wedge in which the dotted lines in the figure represent nonexistent characteristics. The region FIAO is the yield zone, and the characteristics in this region are real.
From the construction of the characteristic field above, the computational procedure for both smooth and rough footings can be unified. The computation initiates with α characteristics progressing from the free surface to the footing and terminates at the point in which x = −B/2 and η = π/2 simultaneously. The area enclosed by the β characteristic passing through the terminal point and the footing base is the non-plastic zone. The smooth footing is simply the special case presented in Fig. 4b in which the terminal point I coincides with point F and the middle point E of the footing base, which indicates that there is no non-plastic wedge, as shown in Fig. 4a.
Martin (2003) and Sun et al. (2013) found that there are two types of failure mechanisms for rough footings when constructing the stress field. In one type, no α characteristics progress to the footing base; therefore, a complete non-plastic wedge exists under the footing base. Whereas in the other type, the α characteristics enter the region beneath the footing and result in a partial non-plastic wedge. Sun et al. (2013) stated that the type of failure mechanism depends on ϕ and λ. The proposed construction method of the characteristics in this paper satisfies all the boundary requirements, and the type of failure mechanism is automatically determined using the computation. Moreover, the computation of the bearing capacity of smooth and rough footings is unified with the same termination condition.
Equivalent solution of the bearing capacity problem
When the bearing capacity is computed on general c–ϕ–γ soil without superposition and the result is still written in the form of Eq. (1), the bearing capacity factor N γ is not the value that computed by superposition method.
Equation (18) is a general solution of the bearing capacity of strip footings that is equivalent to Eq. (1). The N γ values deduced by exact bearing capacity equal to those by superposition method only when λ = 0. To obtain an exact solution of N γ on general c–ϕ–γ soil, it is necessary to calculate the bearing capacity in the real failure mechanism using a method other than the superposition approximation proposed by Terzaghi. Zhu et al. (2003) computed the bearing capacity factor N γ of rough strip foundations using the critical slip field method. p u and λ are defined as the normalized bearing capacity and the surcharge ratio, respectively. The value of N γ was found to be influenced not only by ϕ but also by the surcharge ratio λ. However, Zhu et al. (2003) assumed that the inclined angle of the active wedge underneath the footing was π/4 + ϕ/2 with respect to the horizontal line, leading to discrepancies between the calculations and the exact solutions. The proposed method in this paper avoids this assumption and results in better numerical results. Moreover, the improved computation extends the application to both a smooth footing and a rough footing. This approach is helpful in attaining a better fitting formula for N γ based on the exact numerical results.
Shield (1954) studied the bearing capacity of strip footings using plastic theory and reached the conclusion that the well-known, closed-form expressions of N q and N c given by Reissner (1924) and Prandtl (1921) are exact solutions for weightless soil regardless of the footing roughness. It is most straightforward to use the closed form solutions for N q and N c derived for weightless soil and use N γ to account for all the effects of self weight and its interaction with q and c, by using the surcharge ratio λ.
Comparisons of Nγ with other known results
Comparison of the bearing capacity factor N γ for a smooth footing
Bolton and Lau (1993)
Frydman and Burd (1997)
Woodward and Griffiths (1998)
Hjiaj et al. (2005)
The present values when λ is 0 in Table 1 are equivalent to complete solutions given by Martin (2005) and Smith (2005), which indicates that the results by present method can be treated as exact solutions. The computations provided by Bolton and Lau (1993) or Kumar (2009) using the MOC have little difference compared to present results. Moreover, the results determined by the proposed method are between the upper and lower bounds given by Hjiaj et al. (2005). The N γ values determined by Woodward and Griffiths (1998) using the finite element method are consistent with those obtained using the MOC. Compared to the calculations given by Hjiaj et al. (2005) and Smith (2005), the results from Frydman and Burd (1997) using fast lagrangian analysis of continua(FLAC) exceed the upper bound when ϕ equals 35° and are below the lower bound when ϕ equals 40° or 45°, although the errors are small compared with the calculations in this paper.
The calculations of N γ in Table 1 when equals 104 have errors of no more than 0.1 % compared to the solutions by Eq. (21). When the weight of soil decreases to 0, the surcharge ratio λ will approach ∞. In this case, the failure surface computed by MOC is consistent with the Hill mechanism. So the upper bound of N γ in Eq. (21) can be treated as the exact theoretical solution when λ = ∞.
Comparison of the bearing capacity factor N γ for a rough footing
Bolton and Lau (1993)
Zhu et al. (2001)
Hjiaj et al. (2005)
The present values of N γ when λ = 0 are basically treated as exact solutions because they are equal to complete solutions computed by Martin (2005) and Smith (2005). The N γ results obtained by Bolton and Lau (1993) are much greater than the present values and even exceed the maximum values corresponding to λ = 104 when ϕ equals 5° or 10°. The large errors are mainly ascribed to the assumption that the trapped wedge beneath the foundation has a base angle of π/4 + ϕ/2. Kumar (2009) abandoned this assumption and determined the partly trapped wedge by computation and, consequently, obtained better results. Following Terzaghi’s assumptions, Kumbhojkar (1993) achieved a numerical solution for N γ , and the results are in agreement with Terzaghi’s calculations. Zhu et al. (2001) determined the base angle of the active wedge when N γ is a minimum using the method of triangular slices, and the corresponding N γ results are better than those determined by Kumbhojkar (1993). Hjiaj et al. (2005) meshed fine finite elements to determine the yield zones instead using an arbitrary assumption. The errors do not exceed 3.42 % between the rigorous lower and upper bound solutions, and the results are in good agreement with the present calculations.
When λ equals to 0.1, 1, 10, 100 and 104, the values of N γ are also given in Table 2. Similar to smooth footings, the value of N γ approaches the upper bound in the Prandtl mechanism when λ equals to 104. The exact theoretical solution of the upper bound is also given by Chen (1975), which is twice of the value calculated with Eq. (21). The values of N γ in Table 2 when λ equals 104 are basically equal to the theoretical solutions with the errors less than 0.1 %.
Ratio of Nγ for smooth and rough footings
The numerical calculations of N γ for smooth footings and rough footings reveal that the N γ value for a smooth footing is only half or more than half of that for a rough footing. Figure 5 also demonstrates that R N becomes less sensitive to ϕ as λ increases. Equivalent to the solution for a granular soil with zero surcharge, the numerical result of N γ when λ equals 0 is a minimum solution with a determined ϕ. For a rough foundation, the collapsed surface when λ = ∞ is the same as that in the Prandtl mechanism, and the computational result of N γ equals the closed-form solution deduced in the Prandtl mechanism. Similarly, the N γ for a smooth footing is identical to the theoretical expression in Hill’s failure mechanism. As stated by Chen (1975), the N γ in the Prandtl mechanism is exactly twice the value in the Hill mechanism, i.e., R N = 0.5. The relationship of R N and ϕ when λ = ∞ in Fig. 5 verifies Chen’s judgment.
Influence of the surcharge ratio on Nγ
Proposed formula of Nγ
As mentioned previously, the N γ,max of a foundation can be exactly calculated by theoretical solutions. However, the exact closed form solution of N γ,min is not available when λ = 0, although plenty of empirical formulas are given by different researchers. The N γ,min value is proposed to be calculated by the solution of N γ,max times K N based on Eqs. (25) and (26).
The curves of N γ versus λ for smooth and rough footings are plotted in Figs. 7 and 8, respectively, based on the fitting formula (27). For both smooth and rough footings, the approximate results agree well with the numerical results within errors of ±2 %. Therefore, Eq. (27) is able to estimate N γ with adequate accuracy.
The N γ data computed by Zhu et al. (2003) are given in Fig. 8 as well and are much greater than the results of the proposed method when λ is less than 10. As mentioned above, the discrepancies in the results are mainly attributed to the assumption that the base angle of the active wedge underneath the footing base equals 45° + ϕ/2. The error resulting from that assumption rapidly decreases with increasing λ. As seen in Fig. 8, there is little difference between the present values and the results provided by Zhu et al. (2003) when λ is greater than 10. Moreover, the theoretical solution of N γ given by Zhu et al. (2003) is the same as the present value in the case of λ = ∞. Similar to the pattern of the present results, the calculations by Zhu et al. (2003) also have an “S” shape for a fixed ϕ, which implies that the results can be estimated by expression (27) as well, except N γ,min, A 0 and p differ from the values in this paper.
It should be noted the proposed approximate formula of N γ in Eq. (27) is limited to the classic issue on the bearing capacity of strip footings that the soil is treated as a rigid plastic and obeys Mohr–Coulomb criterion. If the soil beneath the strip footing does not flow Mohr–Coulomb criterion, the proposed method may be no longer applicable. Because the suggested approximate formula is based on the conclusion that the bearing capacity factor N γ depends on the surcharge ratio λ in addition to the friction angle ϕ. This conclusion is only valid for Mohr–Coulomb soil. Further research is required whether the conclusion is suitable when the soil meets other yield criteria other than Mohr–Coulomb criterion. A comprehensive research is also needed whether the conclusions and proposed formula in this paper can be extended to circular or rectangular footings.
In the case of no overload, the computed N γ values in this paper for a granular soil are treated as exact solutions because the values are consistent with complete solutions given by Martin (2005) and Smith (2005). Some researchers assume failure surfaces or mechanisms that are not the same as the real state; therefore, their results have considerable errors compared with the exact solutions.
The roughness of the footing base has a significant impact on N γ . The ratio of the bearing capacity factor N γ for smooth foundations and rough foundations, which is R N, indicates that the value of N γ for a smooth footing is only half or more of that for a rough footing. The curve of R N versus ϕ with λ = 0 has good agreement with the results given by Hjiaj et al. (2005). A value of R N equal to 0.5 when λ = ∞ supports Chen’s statement that the N γ in the Prandtl mechanism is exactly twice the value in the Hill mechanism.
The surcharge ratio λ also significantly affects N γ , and the ratio K N defined by N γ,min/N γ,max can be approximately evaluated using a polynomial expression when λ = 0 and λ = ∞. When λ is sufficiently large, the solution of N γ is demonstrated to approach the upper bound that deduced by Chen (1975) in a closed-form solution. Therefore, N γ,max is obtained by exact theoretical formula, and thus, N γ,min can be accurately estimated using K N and N γ,max.
The present N γ value in the case of λ = ∞ is exactly the same as the theoretical solution of N γ given by Zhu et al. (2003). However, the calculations of Zhu et al. (2003) have obvious errors compared with the present results when λ is less than 10 primarily due to the assumption that the base angle of the active wedge underneath the footing base equals 45° + ϕ/2. The values of N γ can be calculated by the approximate formula (27) containing two factors: ϕ and λ. The discrepancies between the approximate results and the numerical solutions are less than ±2 % for both smooth and rough foundations. Formula (27) is demonstrated to be suitable for evaluating N γ when considering the factor λ.
DH wrote the numerical program, proposed the approximate formula and drafted the manuscript. XX supervised the study and reviewed the manuscript. LZ tested the program and calculated the numerical results. LH determined the value of the approximate parameters. All authors read and approved the final manuscript.
This research was supported by the Fund for Science and Technology Innovation Team of Ningbo (Grant No. 2011B81005). The authors wish to express their gratitude to the above for financial support.
The authors declare that they have no competing interests.
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