The ultimate bearing capacity of shallow strip footings using slip-line method

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The ultimate bearing capacity of shallow strip footings using slip-line method Ming-xiang Peng a,⇑, Hong-xi Peng b a

China Energy Engineering Group Guangdong Electric Power Design Institute Co., Ltd., Guangzhou 510663, China b College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China Received 20 April 2017; received in revised form 30 September 2018; accepted 21 January 2019 Available online 27 March 2019

Abstract Based on the limit equilibrium theory, an accurate approach is proposed to solve the ultimate bearing capacity of shallow strip footings under general conditions. The foundation soil is considered to be an ideal elastic-plastic material, which obeys the Mohr-Coulomb yield criterion, and is assumed to be an ideal continuous medium which is isotropic, homogeneous and incompressible or non-expansive. Through analyzing the relative motion and interaction between the footing and soil, the problem of the ultimate bearing capacity of shallow strip footings is divided into two categories. A minimum model with the total vertical ultimate bearing capacity as its objective function is established to solve the ultimate bearing capacity using the slip-line method with no need to make any assumptions on the plastic zone and non-plastic wedge in advance. A convenient and practical simplified method is also proposed for practical engineering purposes. Furthermore, the first category of the problem in the case of the same uniform surcharges on both sides of footings is the focus of the study: the applicable conditions of Terzaghi’s ultimate bearing capacity equation as well as the theoretical exact solutions to its three bearing capacity factors are derived, and a new bearing capacity equation is put forward as a replacement for Terzaghi’s equation. The geometric and mechanical similarity principle is proposed by a dimensionless analysis. The results show that for perfectly smooth footings, the total vertical ultimate bearing capacity obtained by the present method is in good agreement with those by existing methods, whereas the existing methods underestimate the ultimate bearing capacity in the case of perfectly rough footings. The classic Prandtl mechanism is not the plastic failure mechanism of the ultimate bearing capacity problem of perfectly smooth footings on weightless soil. Ó 2019 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: Bearing capacity; Footing–soil interaction; Limit equilibrium theory; Minimum principle; Shallow foundation; Slip-line method

1. Introduction The ultimate bearing capacity of foundations is one of the three classic subjects relating to limit equilibrium theory in soil mechanics, and is thus far the most extensively studied area. Prandtl (1920) was the first to solve the ultimate bearing capacity of strip footings on weightless soil using the limit equilibrium theory in the case of a perfectly smooth footing, a central vertical load and zero ground

Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author. E-mail address: [email protected] (M.-x. Peng).

load. He derived that the ultimate pressure on the footing base was equal everywhere, with the closed-form solution as follows: quv ¼ cN c

ð1Þ

where c is the cohesion of soil; N c is the bearing capacity factor due to soil cohesion, which can be calculated by Eq. (27) given in this paper. Later, Reissner (1924) extended the method to relate to cases in which there are the same uniform surcharges on both sides of the footing and obtained the following Prandtl-Reissner solution: quv ¼ cN c þ qN q

https://doi.org/10.1016/j.sandf.2019.01.008 0038-0806/Ó 2019 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.

This is an open access article under CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

ð2Þ

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List of symbols footing width effective width of footing width of footing base OA1 or OA optimal solution of variable b cohesion of soil adhesion between footing and soil eccentricity of line load P node number in first b slip-line of difference mesh mean vertical bearing capacity factor Nv N vM vertical bearing capacity factor at point M N v0 ,N v1 vertical bearing capacity factors at points A and O N c ,N q ,N c bearing capacity factors due to soil cohesion, uniform surcharge and soil unit weight n total number of b slip-lines of difference mesh P line load on footing normalized mean stress at point M p0M total horizontal ultimate bearing capacity Quh Quv total vertical ultimate bearing capacity Quv1 ,Quv2 total vertical ultimate bearing capacities acting on footing bases OA1 and OA2 qu ultimate bearing capacity distribution along footing base qu1 ,qu2 ultimate bearing capacity distributions along footing bases OA1 and OA2 B B0 b b c cw e m

where q is the uniform surcharge, and N q is the bearing capacity factor due to surcharge. N c and N q are also determined by Eq. (27). However, if the soil self-weight and footing roughness are taken into account, the problem will be much more complicated. For a fully rough shallow strip footing with equal uniform surcharges on both sides, Terzaghi (1943) assumed the non-plastic wedge under the footing to be an isosceles triangle and the slip surface was a composite curve comprised of a logarithmic spiral curve and a line segment. He then used the superposition principle and limit equilibrium method to deduce an ultimate bearing capacity equation in which the effect of the soil weight is considered, famously known as the Terzaghi’s equation: Quv 1 ¼ cN c þ qN q þ cBN c 2 B

ð3Þ

where Quv is the total vertical ultimate bearing capacity, B is the footing width, c is the unit weight of soil, and N c is the bearing capacity factor due to the contribution of soil unit weight. N c , N q and N c are purely dependent on the soil internal friction angle /. Among them N c and N q are calculated by Eq. (29), while N c can be obtained through chart look-up. Over the past 70-odd years, a lot of theoretical studies on various problems of the ultimate bearing capacity of

vertical ultimate bearing capacity distribution along footing base quv1 ,quv2 vertical ultimate bearing capacity distributions along footing bases OA1 and OA2 q uniform surcharge uniform surcharges on both sides of footing q1 ,q2 critical value of q1 q1cr R0M ,R0w normalized radii of stress circle at point M t width of wedge base V 1 ,V 2 vertical downward tensions on sides of nonplastic wedge x,y Cartesian coordinates normalized x-coordinate x0 v variable defined in Eq. (14) c unit weight of soil d friction angle between footing and soil normalized parameters g1 ,g2 k footing roughness w base angle of isosceles triangle non-plastic wedge x inclination angle of line load P / internal friction angle of soil X1 ,X2 plastic zones quv

foundations have been conducted using Terzaghi’s equation. The calculation methods include the limit equilibrium method (Meyerhof, 1951; Sarma and Iossifelis, 1990; Kumbhojkar, 1993; Dewaikar and Mohapatra, 2003; Silvestri, 2003; Zhu et al., 2003; Choudhury and Subba Rao, 2005; Tsuchida and Athapaththu, 2014; Van Baars, 2014; Chen and Abu-Farsakh, 2015); the slip-line method (Sokolovskii, 1965; Davis and Booker, 1971; Bolton and Lau, 1993; Ueno et al., 2001; Zhu et al., 2001; Kumar and Mohan Rao, 2002; Kumar, 2003; Cheng and Au, 2005; Martin, 2004, 2005; Lau and Bolton, 2011a); the limit analysis method (Chen, 1975; Michalowski, 1997; Soubra, 1999; Zhu, 2000; Wang et al., 2001; Kumar, 2004); the numerical limit analysis method (Sloan, 1988, 1989; Ukritchon et al., 2003; Hjiaj et al., 2005; Lyamin et al., 2007); and the finite-element and finite-difference methods (Griffiths, 1982; Manoharan and Dasgupta, 1995; Frydman and Burd, 1997; Yin et al., 2001; Loukidis and Salgado, 2009; Nguyen et al., 2016). It is generally believed that the slip-line method can be expected to give a good estimate of the correct solution (Chen, 1975). Sokolovskii (1965) used the slip-line method to research various limit equilibrium problems, including foundation stability, slope stability and earth pressure on retaining walls. He tried to give a general method for obtaining the numerical solution of a slip-line field, but his method is usually applicable only

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to cohesionless soil. Bolton and Lau (1993) studied the bearing capacity problems of strip and circular footings using the slip-line method, and analyzed the influence of smooth and fully rough footings on bearing capacity coefficients. Kumar and Mohan Rao (2002) analyzed the influence of horizontal seismic forces on the ultimate bearing capacity of strip footings using the slip-line method. However, they failed to consider the inherent friction characteristic of the footing–soil interface, and simply assumed that the ratio of the shear stresses to the normal stresses equals the horizontal seismic coefficient everywhere on the footing base. Martin (2004) developed software ABC for calculating the ultimate bearing capacity of strip and circular footings based on the slip-line method. The program takes the influence of linear variation of soil cohesion c with depth into account, but only applies to the cases of smooth and fully rough footings with the same uniform surcharge on both sides. Cheng and Au (2005) studied the ultimate bearing capacity problem of smooth strip footings on sloping ground using the slip-line method and single-side failure mechanism. For the case of cohesionless granular soil, some researchers (Ueno et al., 2001; Zhu et al., 2001; Lau and Bolton, 2011a, 2011b) considered the effects of stressdependent internal friction angle / of soil on ultimate bearing capacity. They used the slip-line method to approximately solve the ultimate bearing capacity of strip and circular footings, and carried out test verification. However, the angle of internal friction varies with mean stress, which makes the mathematical solution of the limit equilibrium problem more complicated and difficult, and the governing equations with variable / have not been rigorously derived so far. Although great progress has been made in the study of the ultimate bearing capacity of shallow strip footings, a complete theoretical solution to the problem has not yet been obtained, and the well-known problem of how to evaluate N c accurately has not been well solved either. According to the authors, the reasons lie in the lack of understanding about the following problems: (1) The genuine solution to the limit equilibrium problem. Chen (1975) made such comments: ‘‘The slipline solution is not necessarily the true solution. If the associated flow rule is employed and the resulting stress-strain rate equations can be integrated to yield a kinematically admissible velocity field, the slip-line solution is an upper-bound solution. If, in addition, the slip-line stress field can be extended over the entire half-space of the soil domain such that the equilibrium equations, the stress boundary conditions and the yield condition are satisfied, the slip-line solution is also a lower bound and is hence the true solution.” Quite a few researchers (Frydman and Burd, 1997; Michalowski, 1997; Martin, 2004,2005; Hjiaj et al., 2005) have quoted this view. However, as the limit earth pressure on retaining walls under general

603

conditions has been successfully solved using the slip-line method (Peng, 2011; Peng and Chen, 2013), there are reasons to believe that the slip-line solution represents the genuine solution to the limit equilibrium problem in theory. (2) The stress boundary conditions on the footing base. The difficulty of determining the boundary conditions lies in that the relative motion and interaction between the footing and soil remain unclear (Bolton and Lau, 1993). (3) The non-plastic wedge below the footing base. Few studies have been carried out on this aspect and it remains unknown how the non-plastic wedge forms theoretically, and also what its existence conditions are. (4) Terzaghi’s (1943) ultimate bearing capacity equation. While Terzaghi’s equation is used in nearly all current research, the problems relating to its very essence remain open for further study. For instance, is it only a conservative approximate equation based on the superposition assumption (Davis and Booker, 1971; Griffiths, 1982; Bolton and Lau, 1993; Zhu et al., 2003), or an exact theoretical one? What are its strict applicable conditions? What are the basic parameters influencing its three bearing capacity factors? Can the theoretical exact solutions to these factors be obtained? In this paper, based on the limit equilibrium theory, first, the problem of the ultimate bearing capacity of shallow strip footings is divided into two categories based on the analysis of relative motion and interaction between the footing and soil. Next, a minimum model is established for solving the ultimate bearing capacity in general cases using the slip-line method, without having to make any assumptions on the plastic zone and non-plastic wedge in advance. Finally, the first category of problem in the case of the same uniform surcharges on both sides of the footing is studied, and the applicable conditions of Terzaghi’s equation as well as the theoretical exact solutions to its three bearing capacity factors are presented. A new bearing capacity equation is put forward to replace Terzaghi’s equation. Furthermore, the geometric and mechanical similarity principle is also proposed by means of dimensionless analysis, which is expected to provide theoretical guidance for the experimental study on the foundation bearing capacity. 2. Calculation models and solutions 2.1. Basic assumptions Fig. 1 shows the calculation models of the ultimate bearing capacity of a shallow strip footing. The adhesion and the friction angle between the rigid footing and the soil are cw and d respectively. A line load P from the upper

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Fig. 1. Calculation models: (a) First category of problem with no nonplastic wedge; (b) First category of problem with non-plastic wedge; (c) Second category of problem.

structure acts on the footing centre, and the inclination angle of load P is x. The uniform surcharges q1 and q2 apply respectively on the ground at both sides of the footing. The basic assumptions are as follows: (1) The problem is a plane-strain problem. (2) The foundation soil is an ideal elastic-plastic material that obeys the Mohr-Coulomb yield criterion, and is an ideal continuous medium that is isotropic, homogeneous and incompressible or non-expansive, represented by three constant parameters c, c and /. (3) The interface between the rigid footing and the soil satisfies Coulomb’s friction law, represented by two constant parameters cw and d. (4) The shear strength of the soil above the level of the footing base and the resistance on the footing lateral sides are not considered. The soil above the footing base level is replaced by the uniform surcharges q1 and q2 respectively.

2.2. Two categories of problem According to whether or not the footing slides horizontally along the footing–soil interface under the upper line load P , the problem of the ultimate bearing capacity of shallow strip footings can be classified into the following two categories. 2.2.1. First category of problem As shown in Fig. 1(a), the horizontal component of load P is too small to lead to any horizontal slip of the footing

along the footing–soil interface towards the positive direction of the x-axis. When the footing is loaded, the soil near both footing edges A1 and A2 usually passes into the plastic flow state first, forming two local plastic zones X1 and X2 . As the load continues to increase, the two plastic zones keep expanding before finally intersecting at point O. The first category of problem is characterized by the facts that no horizontal slip of footing occurs while the plastic soil is pressured to move towards both sides of the footing – or ‘‘soil moves while footing not”, and that the friction force of footing on soil always impedes the plastic flow of soil. It can be inferred that the interaction force always exists between the footing and plastic flow soil, and that the total vertical pressure applied by plastic flow soil on footing base is the total vertical ultimate bearing capacity Quv . Generally, assuming the ultimate bearing capacities on the footing bases OA1 and OA2 to be qu1 and qu2 , and the corresponding vertical ultimate bearing capacities to be quv1 and quv2 , then Quv can be determined by Z b Z 0 Quv ¼ quv2 dx þ quv1 dx B

Z ¼

b

b B

Z qu2 dx þ



0 b

qu1 dx

cos2 d

ð4Þ

where b is the width of footing base OA1. In some cases, however, the plastic zones X1 and X2 are tangent at point E under the footing base, and forming a non-plastic wedge O1EO2, as shown in Fig. 1(b). Assuming the wedge and the footing to be an integral whole, which constitutes a new footing that can be referred to as a ‘‘wedge footing”, then Quv can be obtained as Z bt Z 0 Quv ¼ quv2 dx þ quv1 dx  V 1  V 2 ð5Þ B

b

where V 1 and V 2 respectively represent the vertical downward tensions on sides O1E and O2E of the non-plastic wedge by plastic flow soil; t denotes the width of the base O1O2 of the wedge. If t ¼0, then V 1 ¼ V 2 ¼0, and Eq. (5) degenerates into Eq. (4). 2.2.2. Second category of problem As shown in Fig. 1(c), the horizontal component of load P is large enough to cause the footing to slide horizontally along the footing–soil interface towards the positive direction of the x-axis. For instance, the ultimate bearing capacity of strip footings under seismic actions (Sarma and Iossifelis, 1990; Soubra, 1999; Kumar and Mohan Rao, 2002; Choudhury and Subba Rao, 2005), and that of gravity dams and rigid retaining walls mostly fall into this category. The characteristics of the second category of problem can be described as ‘‘footing moves prior to soil”, and that the friction force of soil on footing always impedes the horizontal movement of footing. However, the friction force of footing on soil plays a different role: for the footing base OA1, the direction of friction force is the same as that of the soil plastic flow, hence the friction

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force accelerates the plastic flow of soil; for the footing base OA2, the direction of friction force is opposite to that of the soil plastic flow, hence the friction force impedes the plastic flow. The progressive forming process of the plastic zones for the second category of problem is similar to that for the first category. The total vertical ultimate bearing capacity Quv is calculated by Eq. (4) and the total horizontal ultimate bearing capacity Quh is calculated by Z b  Z 0 Quh ¼ cw B þ quv2 dx þ quv1 dx tand Z ¼ cw B þ

B

b B

Z qu2 dx þ

b 0

b

 qu1 dx sindcosd

ð6Þ

By combining Eqs. (4) and (6), the following formula can be obtained: Quh ¼ cw B þ Quv tand

ð7Þ

where Quh and Quv satisfy the linear relationship. Once Quv has been solved, Quh can be calculated by the formula given above. In engineering applications, the category of a problem is entirely dependent on the magnitude and inclination of the upper load P , the footing width B as well as the interface friction characteristics cw and d. It can be derived that the horizontal slip of a footing along the footing–soil interface towards the positive direction of the x-axis will not occur when the following condition is satisfied: P sinðd  xÞ þ cw Bcosd > 0

ð8Þ

The problem satisfying the above condition falls into the first category; otherwise the second one. In the case of completely smooth footings, namely that when cw ¼ 0 and d ¼ 0, the calculation results of the two categories are exactly the same. If the footing load P is an eccentric load and its eccentricity is e, then the footing width should be taken as the effective width B0 ¼ B  2e (Meyerhof, 1953). Note that the footing loads in this paper are assumed to be a central load unless otherwise stated. 2.3. Solution method It can be seen from Eq. (4) or Eq. (5) that when the other parameters keep unchanged, Quv is a function of b. The value of Quv varies with that of b, and the minimum value Quv ðb Þ is the total vertical ultimate bearing capacity to be solved. The mathematical model for this optimization problem is b : Quv ðb Þ ¼ Minimize Quv ðbÞ Subject to 0 6 b 6 B

ð9Þ

where b and Quv ðb Þ are the optimal solutions to the problem. The problem can be solved by one-dimensional search methods such as the fraction method, the golden section method and the parabolic method. In order to get an objec-

Fig. 2. Basic boundary value problem.

tive function value Quv ðbÞ, two slip-line stress fields X1 and X2 need to be solved in advance. The solution methods of the two are the same, which can be come down to solving the basic boundary value problem as illustrated in Fig. 2. For the first category of problem, the calculation process is described as follows: (1) Assign a value to b and let q ¼ q1 ; then solve the basic boundary value problem to derive the slip-line stress field X1 . (2) Determine t by iteration. If the non-plastic wedge is non-existent, then t ¼0. (3) Replace b with B  b  t and let q ¼ q2 ; then solve the basic boundary value problem again, and carry out a mirror transformation about the axis x ¼ B=2 to derive the slip-line stress field X2 . (4) Calculate the objective function value Quv ðbÞ using Eq. (5). Particularly, when t ¼0, Quv ðbÞ can be calculated directly by Eq. (4). For the second category of problem, Quv ðbÞ can be determined in a similar way. But because the direction of the friction force of footing on soil is always the same as that of soil plastic flow, the values of cw and d should be negative when solving X1 . In addition, the value of t remains zero. 3. Basic boundary value problem As shown in Fig. 2, the basic boundary value problem can be treated as a special case of the problem of passive earth pressure on a retaining wall, where the inclination of wall is p, the height of wall is 0, the length of wall–soil interface is b, and the ground is horizontal. The limit equilibrium problem can be solved by the slip-line method. The basic equations, boundary conditions, finite-difference method and dimensionless analysis relating to this problem were already presented by Peng (2011) and are therefore not repeated here. The slip-line stress filed obtained generally consists of active zone ADO, radial shear zone ACD and passive zone ABC.

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The normalized slip-line stress field for the basic boundary value problem is determined by /, d, g1 and g2 , among which the dimensionless parameters g1 and g2 are defined by g1 ¼ cþqtan/ cb g2 ¼ cw þqtand cb

ð10Þ

Once the normalized slip-line stress field has been solved, the ultimate bearing capacity at an arbitrary point M on the footing base OA can be determined by qu ¼ cbNcosvM2 dþq quv ¼ cbN vM þ q

ð11Þ

where N vM represents the normalized vertical ultimate bearing capacity at point M, also referred to as the vertical bearing capacity factor, which can be calculated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 ð12Þ N vM ¼ p0M  R0w sind þ cosd R02 M  Rw where p0M is the normalized mean stress at point M; R0M and R0w are the normalized radii of stress circle at point M, that is, R0M ¼ p0M sin/ þ g1 cos/ and R0w ¼ p0M sind þ g2 cosd. In general, the distribution of the vertical ultimate bearing capacity quv is non-linear along the footing base OA. But for convenience, it is usually assumed to be linear (see the dashed line in Fig. 2) based on the principle of equal total bearing capacity in engineering practice, so that only the vertical ultimate bearing capacities at points A and O need to be determined. Let the vertical ultimate bearing capacities at points A and O be quv ð0Þ and quv ðbÞ, then: quv ð0Þ ¼ cbN v0 þ q quv ðbÞ ¼ cbN v1 þ q

ð13Þ

where N v0 and N v1 respectively represent the vertical bearing capacity factors at points A and O, which depend on /, d, g1 and g2 . To facilitate engineering applications, the authors have prepared some tables of N v0 and N v1 containing frequently-used values of / and d, which are freely available to readers as a PDF. The internal friction angle / of soil includes 5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50° and 55°. For each / value, the value of d is selected as 0, /=4, /=3, /=2, 2/=3, 3/=4 and /, respectively. Thus according to /, d, g1 and g2 =g1 , the corresponding values of N v0 and N v1 can be determined through looking up tables and using linear interpolation. Table 1 gives N v0 and N v1 for several different values of / and d. From the data in Table 1, the authors discovered and proved a very useful relation, that is, N v0 ¼ g1 f ð/; d; g2 =g1 Þ. However, the function f has no explicit expression under general conditions; only when Eq. (15) or Eq. (22) given later is satisfied, does the equation f ¼ N c constantly exist, where N c is determined by Eq. (24). Note also that N v0 can generally be obtained by a simple iteration, while N v1 can only be determined after the slip-line stress field is solved.

4. Plastic failures 4.1. Plastic zones When the load on a footing increases constantly, the soil near the footing gradually enters into the plastic failure state. The shape and size of the plastic zone completely depend on the category and relevant parameters of the problem, and can be automatically determined by solving the optimization problem described above, with no need to make any assumptions on them in advance. For example, consider the firstcategory problem for a strip footing with parameters B ¼1 m, c ¼20 kN/m3, c ¼10 kPa, / ¼25°, cw ¼5 kPa and d ¼10°. If q2 ¼30 kPa remains unchanged but q1 varies, then different plastic failures can occur. When q1 63.781 kPa, the right-side plastic failure occurs, and the total vertical ultimate bearing capacity Quv increases with q1 , as shown in Fig. 3(a); when 3.781 kPa< q1 <58.723 kPa, the both-side plastic failure occurs, and Quv increases with q1 , as shown in Fig. 3(b) and (c); when q1 P58.723 kPa, the left-side plastic failure occurs, and Quv does not change with q1 , as shown in Fig. 3(d). In addition, the plastic failures in several special cases are also given as follows: Case 1: The first category of problem for a strip footing on a purely cohesive soil d ¼ / ¼0, quv is equal everywhere along the footing base. When q1 < q2 , the optimal solution b ¼ B can be obtained, namely that the right-side plastic failure occurs. When q1 > q2 , then b ¼0, and the leftside plastic failure occurs. When q1 ¼ q2 , the optimization problem has countless solutions b 2[0,B], and the plastic failure is indeterminate. Case 2: The second category of problem for a strip footing on a purely cohesive soil d ¼ / ¼0, quv is equal everywhere along the footing base. If q2 is a fixed value, then there always exists a critical value q1cr of q1 , that is, q1cr ¼ q2 þ 2csin1 ðcw =cÞ. When q1 < q1cr , then b ¼ B, and the right-side plastic failure occurs. When q1 > q1cr , then b ¼0, and the left-side plastic failure occurs. When q1 ¼ q1cr , then b 2[0,B], and the plastic failure is indeterminate. Case 3: The first category of problem for a strip footing on weightless soil c ¼0, quv is equal everywhere along the footing base. When q1 < q2 , then b ¼ B, and the rightside plastic failure occurs. When q1 > q2 , then b ¼0, and the left-side plastic failure occurs. When q1 ¼ q2 , then b 2[0,B], and the plastic failure is indeterminate. It should be noted that Prandtl (1920), Reissner (1924) and Hill (1950) first studied the situation of a horizontal smooth footing on weightless soil (i.e., c ¼0, cw ¼0, d ¼0 and q1 ¼ q2 ), and gave the same ultimate load expressed by Eq. (2), but they used different failure mechanisms. As a matter of fact, the Hill mechanism (Hill, 1950) was only a plastic failure corresponding to b ¼ B=2. However, the Prandtl mechanism (Prandtl, 1920; Reissner, 1924) was falsely based on the case where the left-side and right-side

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Table 1 Vertical bearing capacity factors N v0 and N v1 . / ¼25° d ¼10°

/ ¼25° d ¼10°

g2 =g1

1

g1

N v0

N v1

N v0

0.5 N v1

N v0

N v1

N v0

N v1

N v0

N v1

N v0

N v1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

0 2.5517 5.1035 7.6552 10.207 12.759 15.310 17.862 20.414 22.966 25.517 28.069 30.621 33.173

3.4229 7.4421 10.365 13.117 15.797 18.439 21.059 23.663 26.256 28.843 31.423 33.999 36.572 39.142

0 2.7981 5.5962 8.3943 11.192 13.991 16.789 19.587 22.385 25.183 27.981 30.779 33.577 36.375

3.4229 7.9451 11.187 14.228 17.183 20.093 22.975 25.839 28.689 31.531 34.365 37.194 40.019 42.840

0 3.0133 6.0266 9.0399 12.053 15.067 18.080 21.093 24.106 27.120 30.133 33.146 36.160 39.173

3.4229 8.4079 11.936 15.234 18.433 21.579 24.693 27.785 30.861 33.927 36.984 40.034 43.080 46.121

0 5.2255 10.451 15.676 20.902 26.127 31.353 36.578 41.804 47.029 52.255 57.480 62.706 67.931

11.330 21.239 27.988 34.131 39.992 45.692 51.288 56.813 62.286 67.720 73.123 78.503 83.864 89.208

0 5.3274 10.655 15.982 21.309 26.637 31.964 37.292 42.619 47.946 53.274 58.601 63.928 69.256

11.330 21.515 28.426 34.711 40.703 46.527 52.243 57.884 63.471 69.018 74.533 80.023 85.493 90.946

0 5.4223 10.845 16.267 21.689 27.111 32.534 37.956 43.378 48.800 54.223 59.645 65.067 70.489

11.330 21.784 28.851 35.270 41.385 47.327 53.156 58.907 64.602 70.254 75.873 81.466 87.039 92.593

0

0

/ ¼30° d ¼0

0.5

1

/ ¼30° d ¼0

g2 =g1

1

g1

N v0

N v1

N v0

0.5 N v1

N v0

N v1

N v0

N v1

N v0

N v1

N v0

N v1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

0 5.6607 11.321 16.982 22.643 28.304 33.964 39.625 45.286 50.947 56.607 62.268 67.929 73.590

15.311 26.100 33.481 40.186 46.573 52.776 58.862 64.866 70.811 76.711 82.577 88.416 94.232 100.03

0 5.8498 11.700 17.549 23.399 29.249 35.099 40.949 46.798 52.648 58.498 64.348 70.197 76.047

15.311 26.553 34.216 41.169 47.789 54.216 60.518 66.734 72.887 78.994 85.063 91.103 97.120 103.12

0 6.0279 12.056 18.084 24.112 30.140 36.168 42.195 48.223 54.251 60.279 66.307 72.335 78.363

15.311 26.993 34.928 42.119 48.961 55.601 62.109 68.526 74.877 81.178 87.440 93.671 99.877 106.06

0 6.0279 12.056 18.084 24.112 30.140 36.168 42.195 48.223 54.251 60.279 66.307 72.335 78.363

15.311 26.993 34.928 42.119 48.961 55.601 62.109 68.526 74.877 81.178 87.440 93.671 99.877 106.06

0 6.1966 12.393 18.590 24.787 30.983 37.180 43.376 49.573 55.770 61.966 68.163 74.360 80.556

15.311 27.422 35.619 43.039 50.094 56.936 63.640 70.249 76.789 83.275 89.721 96.134 102.52 108.89

0 6.3570 12.714 19.071 25.428 31.785 38.142 44.499 50.856 57.213 63.570 69.927 76.284 82.641

15.311 27.841 36.291 43.931 51.190 58.227 65.120 71.912 78.631 85.295 91.916 98.502 105.06 111.60

g2 =g1

1

0

0

/ ¼30° d ¼30°

0.5

1

/ ¼30° d ¼30° 0.5

0

0

0.5

1

g1

N v0

N v1

N v0

N v1

N v0

N v1

N v0

N v1

N v0

N v1

N v0

N v1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

0 0.6048 1.2095 1.8143 2.4191 3.0238 3.6286 4.2333 4.8381 5.4429 6.0476 6.6524 7.2572 7.8619

0.4330 1.0378 1.6425 2.2473 2.8521 3.4568 4.0616 4.6664 5.2711 5.8759 6.4807 7.0854 7.6902 8.2949

0 1.2577 2.5154 3.7731 5.0308 6.2885 7.5462 8.8039 10.062 11.319 12.577 13.835 15.092 16.350

0.4330 2.2781 3.6235 4.9236 6.2068 7.4816 8.7516 10.019 11.284 12.547 13.810 15.071 16.332 17.593

0 1.5689 3.1379 4.7068 6.2758 7.8447 9.4137 10.983 12.552 14.120 15.689 17.258 18.827 20.396

0.4330 2.8837 4.5982 6.2401 7.8539 9.4534 11.045 12.631 14.213 15.793 17.371 18.947 20.523 22.097

0 10.317 20.634 30.950 41.267 51.584 61.901 72.218 82.534 92.851 103.17 113.48 123.80 134.12

36.165 51.659 60.705 67.957 74.284 80.353 86.135 91.710 97.130 102.43 107.64 112.77 117.83 122.84

0 10.328 20.655 30.983 41.310 51.638 61.965 72.293 82.621 92.948 103.28 113.60 123.93 134.26

36.165 51.632 60.612 67.790 74.242 80.300 86.071 91.635 97.045 102.34 107.53 112.65 117.70 122.70

0 10.333 20.666 30.999 41.333 51.666 61.999 72.332 82.665 92.998 103.33 113.66 124.00 134.33

36.165 51.607 60.531 67.738 74.220 80.273 86.038 91.596 97.000 102.29 107.48 112.59 117.64 122.63

failure mechanisms were simply overlapped, not representing the plastic failure of this limit equilibrium problem. Case 4: The second category of problem for a strip footing on weightless soil c ¼0, quv is equal everywhere along the footing base. If the q2 value keeps unchanged, then

the critical value q1cr of q1 can always be determined by iteration. When q1 < q1cr , then b ¼ B, and the right-side plastic failure occurs. When q1 > q1cr , then b ¼0, and the left-side plastic failure occurs. When q1 ¼ q1cr , then b 2[0,B], and the plastic failure is indeterminate.

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M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

Fig. 3. Variation of plastic failures with q1 : (a) q1 63.781 kPa; (b) q1 ¼30 kPa; (c) q1 ¼40 kPa; (d) q1 P58.723 kPa.

Fig. 4. Non-plastic wedges for perfectly rough footing: (a) isosceles triangle wedge; (b) Kumar’s (2003) wedge; (c) wedge used in this paper.

4.2. Non-plastic wedges It is generally believed that in the first category of problem for a perfectly rough footing with equal uniform surcharges on both sides (i.e., cw ¼ c, d ¼ / and q1 ¼ q2 ¼ q), there will be a non-plastic wedge below the footing base. However, unanimous understanding has not been reached by far regarding the shape and size of the wedge. As illustrated in Fig. 4(a), most current studies assumed the non-plastic wedge to be an isosceles triangle with base angles (i) w ¼ /(Terzaghi, 1943; Kumbhojkar, 1993); (ii) w ¼ p=4 þ /=2 (Meyerhof, 1951; Bolton and Lau, 1993; Dewaikar and Mohapatra, 2003; Zhu et al., 2003); (iii) w corresponding to the minimum value of N c (Zhu, 2000; Silvestri, 2003). The size of non-plastic wedge in all these cases increases with /; when / ¼0, the wedge does not exist except in case (ii). As shown in Fig. 4(b), Kumar (2003) assumed that the friction angle d along the

footing–soil interface increases gradually from zero at the footing centre to / at either edge of footing. He then proposed a curved-edge non-plastic wedge bounded by two slip-lines. Each slip-line was tangent to the footing base at the footing edges, and inclined at an angle p=4  /=2 with the footing centerline. However, he believed that one of the curved edges of the wedge was the a slip-line while the other was the b slip-line. Some researchers (Davis and Booker, 1971; Martin, 2004, 2005) used a non-plastic wedge bounded by two a slip-lines. The non-plastic wedges mentioned above share a common point, namely that the underlying soil within the full width of footing base is assumed to be in a non-plastic state; however, this does not accord with the results predicted by the limit equilibrium theory. It can be proved that as the load on a footing constantly increases, the equilibrium state of soil can always be changed from the elastic to the plastic state gradually, except for the soil near the footing centre that might still be in the elastic state. This has also been verified by the numerical calculation results of the finite-element and finite-difference methods (Frydman and Burd, 1997; Yin et al., 2001; Loukidis and Salgado, 2009). Fig. 4(c) shows a non-plastic wedge for a perfectly rough footing that is brought forward for the first time by this paper. The wedge is a curved-edge isosceles triangle O1EO2 formed by the fact that the plastic zones X1 and X2 are tangent at point E under the footing base. The two curved edges of the wedge are both b slip-lines, each forming a horizontal inclination p=2  / at the footing base. The size of the non-plastic wedge increases with /; when / ¼ 0, the wedge does not exist. For the first category of the ultimate bearing capacity problem under general conditions, when the foundation soil undergoes the single-side plastic failure, the nonplastic wedge does not exist; when the both-side plastic failure occurs, the necessary and sufficient condition for the non-existence of the non-plastic wedge is

M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

v¼/þdþ

  0   0  1 R R p 6 sin1 0w1 þ sin1 0w2 RM1 RM2 2 2

ð14Þ

where R0M1 , R0w1 , R0M2 and R0w2 represent the normalized radii of stress circle at point O in Fig. 1(a). Particularly, if parameters /, d, c and cw satisfy the following equation: cw tan/ ¼ ctand then Eq. (14) can be simplified as   sind v ¼ / þ d þ sin1 sin/ 6 p2 ð/ > 0Þ

v ¼ sin1 ccw 6 p2 ðd ¼ / ¼ 0Þ

ð15Þ

ð16Þ

Fig. 5 shows the plastic failures of cohesionless soil for different values of d, with parameters B ¼2 m, c ¼20 kN/ m3, cw ¼ c ¼0, / ¼30°, q1 ¼20 kPa and q2 ¼10 kPa. Eq. (15) is always satisfied. When d 619.1066°, Eq. (16) is met, the plastic zones X1 and X2 intersect at point O on the footing base as shown in Fig. 5(a) and (b), or are tangent at point O as shown in Fig. 5(c); therefore the nonplastic wedge is non-existent. When d >19.1066°, Eq. (16) is not satisfied, the plastic zones X1 and X2 are tangent at point E below the footing base, the non-plastic wedge O1EO2 exists as shown in Fig. 5(d) and (e). The authors

609

also noticed that Kumar (2004) gave a similar result via the upper-bound limit analysis, that is, for a given value of /, there always exists a corresponding value of dR . When d 6 dR , the non-plastic wedge does not exist; when d > dR , however, he concluded that the footing base can be assumed to be fully rough. It can be proved that in the second category of problem, the non-plastic wedge does not exist in any case. 5. New bearing capacity equation Now turning to the study of the first category of the ultimate bearing capacity problem where q1 ¼ q2 ¼ q. As shown in Fig. 6, when Eq. (14) is satisfied, the nonplastic wedge does not exist below the footing base, the distribution of the vertical ultimate bearing capacity along the footing base is continuous and symmetrical about the centerline of the footing. The total vertical ultimate bearing capacity can be deduced as follows: Z 0 Z 0 quv1 dx ¼ 2  ðcbN vM þ qÞdx Quv ¼ 2  B2

B2

1 ¼ cB2 N v þ qB 2

ð17Þ

where N v denotes the mean vertical bearing capacity factor, which can be determined by Z 0 N vM dx0 ð18Þ Nv ¼ 1

where x0 is a normalized coordinate (i.e., x0 ¼ x=b, b ¼ B=2); N v is a function of /, d, g1 and g2 , and increases with these parameters. The mean vertical ultimate bearing capacity on the footing base can be expressed as Quv 1 ¼ cBN v þ q B 2

ð19Þ

Eq. (19) is the new bearing capacity equation put forward in this paper. It also applies to the case of weightless soil, just let c in the preceding equations equal any nonzero value, such as the water unit weight, and the equation N v ¼ N v0 constantly exists. In engineering practice, the distributions of the vertical ultimate bearing capacity along the footing bases OA1 and OA2 are typically assumed to be linear for convenience, hence N v can be calculated by

Fig. 5. Plastic failures of cohesionless soil with / ¼30°: (a) d ¼0; (b) d ¼10°; (c) d ¼19.1066°; (d) d ¼25°; (e) d ¼30°.

Fig. 6. First category of problem without non-plastic wedge.

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M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

the radius ratio R0w =R0M of the stress circles at any point M on the footing base is independent of the mean stress. Therefore, Terzaghi’s equation is only applicable to the first category of problem satisfying q1 ¼ q2 ¼ q, Eq. (14) and Eq. (22) simultaneously. k¼

Fig. 7. First category of problem with non-plastic wedge: (a) plastic failure; (b) zoomed view around footing.

Nv ¼

N v1 þ N v0 2

ð20Þ

As shown in Fig. 7, when Eq. (14) is not met, a nonplastic wedge O1EO2 exists under the footing base, leading to the discontinuous distribution of the vertical ultimate bearing capacity along the footing base. From Eq. (5), the mean vertical ultimate bearing capacity on the footing base can be obtained as ! Z 0 Quv 2 ¼ quv1 dx  V 1 ð21Þ B B Bt 2 Eq. (21) can also be expressed in the form of the new bearing capacity equation. Based on the principle of equal total bearing capacity, the distribution of the vertical ultimate bearing capacity along the footing base is assumed to be symmetrical linear continuous, then N v can still be determined by Eq. (20). According to the dimensionless analysis (Peng, 2011), it is not hard to derive the geometric and mechanical similarity principle on the first category of problem where q1 ¼ q2 ¼ q, / >0 and c >0, which can be expressed as follows: If parameters /, d, g1 and g2 of two problems are equal, then the slip-line fields are geometrically similar, with a similarity coefficient as the B ratio of the two; the mean vertical ultimate bearing capacities on the footing base after subtracting q are similar, with a similarity coefficient as the cB ratio of the two; the total vertical ultimate bearing capacities after subtracting qB are similar, with a similarity coefficient as the cB2 ratio of the two. Theoretically, Terzaghi’s equation is not applicable to general situations, but only to the case where the vertical ultimate bearing capacity along the footing base has a symmetrical continuous distribution and that at the footing edges can be decomposed into the sum of cN c and qN q . The first condition means that there are the same uniform surcharges on both sides of the footing and no non-plastic wedge below the footing base, and the second means that

R0w  Constant R0M

ð22Þ

where k indirectly reflects the degree of mobilization of the soil shear strength along the footing base, which can be referred to as ‘‘footing roughness”, with 06 k 61. If k 0, then d ¼0 and cw ¼0, suggesting that the shear strength along the footing–soil interface is zero, namely that the footing base is perfectly smooth. If k 1, then d ¼ / and cw ¼ c, meaning that the soil shear strength is fully mobilized, namely that the footing base is perfectly rough. It is not difficult to prove that if and only if Eq. (15) is met, Eq. (22) is true and can be simplified to ð/ > 0Þ

sind k  sin/

k

ð23Þ

ðd ¼ / ¼ 0Þ

cw c

If Eq. (22) is satisfied everywhere on the footing base, then the vertical ultimate bearing capacity at the footing edges quv ð0Þ or quv ðBÞ can be expressed as cN c þ qN q , and the theoretical exact solutions to N c and N q are qffiffiffiffiffiffiffiffiffiffiffiffi

Nq ¼

1þsin/ 1sin/

1k2 1k2 sin2 /

  exp sin1 ðksin/Þ þ sin1 k þ p tan/ Nc ¼

N q 1 tan/

ð/ > 0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi N c ¼ 1  k2 þ sin1 k þ p þ 1

ð24Þ

ðd ¼ / ¼ 0Þ

where N c and N q are dependent on / and k, and increase with these parameters. On the other hand, no commonly accepted value of N c in general cases has been found so far. Usually different researchers suggested varied results, some of which even present great difference (Sieffert and Bay-Gress, 2000; Ukritchon et al., 2003; Diaz-Segura, 2013; Motra et al., 2016). However, it can be proved that for the first category of problem satisfying q1 ¼ q2 ¼ q, Eq. (14) and Eq. (22), Terzaghi’s equation is always equivalent to the new bearing capacity equation, and the theoretical exact solution to N c is N c ¼ N v  N v0 ¼ N v  g1 N c

ð25Þ

where N c is a function of /, d, g1 and g2 , and increases with these parameters. It can thus be deduced that N c decreases as B increases (Griffiths, 1982; Ueno et al., 2001), but increases with c=cB and q=cB (Michalowski, 1997), and ðq þ ccot/Þ=cB (Davis and Booker, 1971; Zhu et al., 2003). Obviously, the traditional approach assuming c ¼0 and q ¼0 to solve N c is inappropriate. Eq. (25) provides the long-sought theoretical exact solution to N c , which can typically be solved through numerical integration. Under the assumption of symmetrical linear

M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

distribution of the vertical ultimate bearing capacity along the footing base, N c can be calculated by Nc ¼

N v1  N v0 2

ð26Þ

It can be further deduced from the preceding analysis that Terzaghi’s equation is applicable only to the first category of problem where q1 ¼ q2 ¼ q, Eqs. (15) and (16) are satisfied simultaneously. The following special cases are going to be discussed. Case 1: Perfectly smooth footing, i.e., d ¼0, cw ¼0 and k 0. In such case, both Eq. (15) and Eq. (16) are satisfied, hence Terzaghi’s equation is applicable, and Eq. (24) can be simplified as expðptan/Þ N q ¼ 1þsin/ 1sin/ Nc ¼

N q 1 tan/

Nc ¼ p þ 2

ð/ > 0Þ

ð27Þ

ðd ¼ / ¼ 0Þ

Case 2: Perfectly rough footing, i.e., d ¼ /, cw ¼ c and k 1. Eq. (16) is not satisfied except for the case of d ¼ / ¼0, therefore Terzaghi’s equation is inapplicable. But because Eq. (22) is still met, Eq. (24) can be simplified to 

N q ¼ ð1 þ sin/Þexp 32 p þ / tan/ Nc ¼

N q 1 tan/

Nc ¼

3 p 2

þ1

ð/ > 0Þ

ð28Þ

ðd ¼ / ¼ 0Þ

However, for perfectly rough footing, Terzaghi (1943) assumed that the non-plastic wedge was an isosceles triangle with w ¼ / as shown in Fig. 4(a), and derived the analytical solutions to N c and N q as follows: 

1 N q ¼ 1sin/ exp 32 p  / tan/ Nc ¼

N q 1 tan/

N c ¼ 32 p þ 1

ð/ > 0Þ

ð29Þ

ðd ¼ / ¼ 0Þ

Terzaghi (1943) also assumed w ¼ p=4 þ /=2, the obtained N c and N q are exactly the same with that of perfectly smooth footing. Case 3: Footing on cohesionless soil, i.e., cw ¼ c ¼0, k  sind=sin/ and / >0. In such case, Eq. (15) is always satisfied but Eq. (16) is not always met, hence Terzaghi’s equation is not always applicable in this case. Case 4: Footing on purely cohesive soil, i.e., d ¼ / ¼0, k  cw =c and c >0. In such case, both conditions are satisfied; therefore, Terzaghi’s equation is applicable, and N c ¼0 and N q ¼1 constantly exist. Finally, it should also be pointed out that, compared to Terzaghi’s equation, the new bearing capacity equation presents obvious advantages such as definite physical meaning, wider application scope and concise expression form; therefore it can be more helpful to theoretical analysis, experimental research and engineering application. The calculation results in two special cases of cohesionless soil

611

are given below and compared with those reported in literature.

6. Comparisons 6.1. Perfectly smooth footings (k 0) Solve the first category of the ultimate bearing capacity problem for a strip footing with parameters B ¼1 m, c ¼20 kN/m3, cw ¼ c ¼0, d ¼0 and q1 ¼ q2 ¼ q. Terzaghi’s equation is applicable to this case, and N q is determined by Eq. (27). The calculation results for different values of q and / are given in Table 2 and Fig. 8. When q ¼ 0, the results of Quv and N c obtained by Martin (2004) using the slip-line method are almost the same as those presented in this paper; the results by Bolton and Lau (1993) using the slip-line method are 1.10%6.51% slightly higher than the present results. However, all these N c values only represent the exact values in the case where d ¼0 and g1 ¼ g2 ¼ 0, and do not apply to other general situations. The results by Chen (1975) using the upper-bound limit analysis are 55.03%70.49% higher than the present results. The results by Sokolovskii (1965) using the slip-line method are about twice as large as those in this paper, the reason is that his N c is determined by a single-side failure mechanism, while the present N c by a symmetrical one. It should also be noted that in order to avoid any floating error, Ueno et al. (2001), Kumar and Mohan Rao (2002), Kumar (2003) and Martin (2004,2005) kept q as a minimum possible finite value, whereas in this paper, through increasing the mesh density near the stress singularity, the dimensionless number q=cB can be an arbitrarily small non-zero value such as 10-8, 10-16 or 10-99. When q ¼20 kPa, the results of Quv calculated by Martin’s (2004) software ABC are almost the same with those in this paper. The Quv values obtained by Bolton and Lau (1993) are 3.60%12.00% lower than those by the present method, and the relative error increases with /. The Quv values by Chen (1975) are relatively consistent with the present results, with a maximum error 2.84%. For / 615°, the Quv values by Sokolovskii (1965) are 0.12%1.21% lower than those presented in this paper, with the error decreasing as / increases; for 15°< / 6 40°, the results by Sokolovskii (1965) are 1.22%9.70% higher than the present results, with the error increasing with /. When q ¼40 kPa, the results of Quv calculated by Martin’s (2004) ABC are almost the same with those in this paper. The Quv values obtained by Bolton and Lau (1993) are 2.08%8.56% lower than those by the present method, and the relative error increases with /. The Quv values by Chen (1975) are 0.17%1.97% lower than the present results. For / 615°, the Quv values by Sokolovskii (1965) are 0.58%0.94% lower than those presented in this paper, with the error decreasing as / increases; for 15°< / 640°,

612

Table 2 Comparison of results for perfectly smooth footings. /(°)

Present study

Martin (2004)

Bolton and Lau (1993)

Chen (1975)

Sokolovskii (1965)

Nc

Quv (kN)

Nc

Quv (kN)

Error (%)

Nc

Quv (kN)

Error (%)

Nc

Quv (kN)

Error (%)

Nc

Quv (kN)

Error (%)

0.085 0.281 0.699 1.578 3.461 7.655 17.599 43.293

0.085 0.281 0.699 1.578 3.461 7.655 17.599 43.293

0.85 2.81 6.99 15.78 34.61 76.55 175.99 432.93

0.084 0.281 0.699 1.579 3.461 7.653 17.577 43.187

0.84 2.81 6.99 15.79 34.61 76.53 175.77 431.87

0.05 0.07 0.03 0.06 0.01 0.03 0.13 0.24

0.09 0.29 0.71 1.60 3.51 7.74 17.80 44.00

0.90 2.90 7.10 16.00 35.10 77.40 178.00 440.00

6.51 3.28 1.60 1.39 1.43 1.10 1.14 1.63

0.131 0.461 1.16 2.68 5.90 12.70 28.60 71.60

1.31 4.61 11.60 26.80 59.00 127.00 286.00 716.00

55.03 64.17 66.00 69.83 70.49 65.89 62.51 65.38

0.17 0.56 1.40 3.16 6.92 15.30 35.20 86.50

1.70 5.60 14.00 31.60 69.20 153.00 352.00 865.00

101.18 99.43 100.34 100.25 99.97 99.86 100.01 99.80

q ¼20 kPa 5 10 15 20 25 30 35 40

1.346 3.562 7.293 13.767 25.462 47.723 93.059 193.890

0.210 0.619 1.411 2.968 6.138 12.921 28.467 67.499

33.46 55.62 92.93 157.67 274.62 497.23 950.59 1958.90

0.210* 0.619* 1.411* 2.968* 6.138* 12.921* 28.467* 67.496*

33.46 55.62 92.93 157.67 274.62 497.23 950.59 1958.86

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.09 0.29 0.71 1.60 3.51 7.74 17.80 44.00

32.25 52.33 85.92 143.99 248.34 445.42 843.92 1723.90

3.60 5.92 7.54 8.68 9.57 10.42 11.22 12.00

0.131 0.461 1.16 2.68 5.90 12.70 28.60 71.60

32.66 54.04 90.42 154.79 272.24 495.02 951.92 1999.90

2.38 2.84 2.70 1.83 0.87 0.44 0.14 2.09

0.17 0.56 1.40 3.16 6.92 15.30 35.20 86.50

33.05 55.03 92.82 159.59 282.44 521.02 1017.92 2148.90

1.21 1.06 0.12 1.22 2.85 4.78 7.08 9.70

q ¼40 kPa 5 10 15 20 25 30 35 40

2.496 6.545 13.264 24.748 45.164 83.336 159.508 324.929

0.225 0.660 1.499 3.151 6.515 13.732 30.324 72.148

64.96 105.45 172.64 287.48 491.64 873.36 1635.08 3289.29

0.225* 0.660* 1.499* 3.151* 6.515* 13.732* 30.322* 72.142*

64.96 105.45 172.64 287.48 491.64 873.36 1635.06 3289.23

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.09 0.29 0.71 1.60 3.51 7.74 17.80 44.00

63.61 101.76 164.75 271.98 461.58 813.44 1509.84 3007.81

2.08 3.50 4.57 5.39 6.11 6.86 7.66 8.56

0.131 0.461 1.16 2.68 5.90 12.70 28.60 71.60

64.02 103.47 169.25 282.78 485.49 863.04 1617.84 3283.81

1.45 1.88 1.97 1.64 1.25 1.18 1.05 0.17

0.17 0.56 1.40 3.16 6.92 15.30 35.20 86.50

64.41 104.46 171.65 287.58 495.68 889.04 1683.84 3432.81

0.85 0.94 0.58 0.03 0.82 1.80 2.98 4.36

*

indicates that based on Terzaghi’s equation the N c value is converted from Quv calculated by ABC.

M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

Nv q ¼0 5 10 15 20 25 30 35 40

M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

613

14.55% lower than the present results, and the error increases with /. It can be seen from the comparisons that other methods underestimate the ultimate bearing capacity of perfectly rough footings, mainly because they adopted a sharper wedge footing. It is worth pointing out that although Terzaghi’s equation used in other methods is not strictly applicable here, the application of Terzaghi’s equation in practical engineering is still safe and effective because the results given by this equation are conservative. 7. Numerical examples Fig. 8. Calculation results for perfectly smooth footings.

the results by Sokolovskii (1965) are 0.03%4.36% higher than the present results, with the error increasing with /. The results in Table 2 also show that the present N c and Martin’s (2004) N c increase with q and /; whereas other researchers’ N c is independent with q, but only increases with /. 6.2. Perfectly rough footings (k 1) Solve the first category of the ultimate bearing capacity problem for a strip footing with parameters B ¼1 m, c ¼20 kN/m3, cw ¼ c ¼0, d ¼ /–0 and q1 ¼ q2 ¼ q. Since Eq. (16) is not satisfied in such a case, Terzaghi’s equation is not strictly – but only approximately applicable. Martin (2004), Bolton and Lau (1993) and Chen (1975) calculated N q by Eq. (27), while Terzaghi (1943) by Eq. (29). The calculation results for different values of q and / are given in Table 3 and Fig. 9. When q ¼ 0, the Quv values calculated by Martin (2004) using the slip-line method are 0.81%23.21% lower than the present results, with the error increasing with /. The Quv values by Bolton and Lau (1993) using the slip-line method are 8.58%442.43% higher than those obtained by the present method, with the error decreasing as / increases. The Quv values by Chen (1975) using the upperbound limit analysis are 31.92%234.21% higher than the present results, with the error decreasing as / increases. For / 630°, the Quv values by Terzaghi (1943) using the limit equilibrium method are 8.95%337.45% higher than those presented in this paper, with the error decreasing as / increases; for 30°< / 640°, the results by Terzaghi (1943) are 2.92%9.90% lower than the present results, with the error increasing with /. Note here that the value of q=cB is selected to be 10-9 to avoid the floating error. When q ¼20 kPa, the results of Quv obtained by Martin (2004), Bolton and Lau (1993), Chen (1975) and Terzaghi (1943) are respectively 3.03%21.59%, 5.71%22.75%, 2.86%14.94% and 2.81%18.55% lower than those presented in this paper, and the error increases with /. When q ¼40 kPa, the results of Quv are respectively 3.77% 21.33%, 0.87%24.13%, 4.29%18.91% and 0.72%

Consider a strip footing with parameters B ¼2 m, q1 ¼40 kPa, q2 ¼20 kPa, c ¼19 kN/m3, c ¼15 kPa, / ¼25°, cw ¼10 kPa and d ¼10°, and respectively solve the ultimate bearing capacity for the first and second categories of problem. (1) For the first category of problem: The total number of b slip-lines of the difference mesh is n ¼66, the node number in the first b slip-line is m ¼86, and the total number of nodes is 9966. The golden section method is adopted as the optimization method, and when the one-dimensional search interval diminishes to smaller than B10-8, the iteration terminates. The calculation results are as follows: v ¼59.972° <90°; the width of OA1 b ¼0.580 m; the total vertical ultimate bearing capacity Quv ðb Þ ¼1941.150 kN, including Quv1 ¼613.439 kN and Quv2 ¼1327.711 kN. The slip-line fields and the distribution of the vertical ultimate bearing capacity are illustrated in Fig. 10(a). (2) For the second category of problem: The calculation results are as follows: the width of OA1 b ¼1.886 m; the total vertical ultimate bearing capacity Quv ðb Þ ¼1246.608 kN, including Quv1 ¼ 1167.603 kN and Quv2 ¼ 79.005 kN; the total horizontal ultimate bearing capacity Quh ðb Þ ¼ 239.811 kN. The slip-line fields and the distribution of the vertical ultimate bearing capacity are shown in Fig. 10(b). It can be seen that the shape and size of the plastic zones in this category of problem are of differ considerably from those in the first category, and the total vertical ultimate bearing capacity is reduced by 35.780%. (3) In engineering practice, the ultimate bearing capacity of shallow strip footings can be also determined through a simplified method. Take the first category of problem as an example, and the calculation steps are described as follows: (i) Let b ¼ B=2 and q ¼ q1 , and calculate g1 ¼1.771174, g2 ¼0.897530 and g2 =g1 ¼ 0.506743. According to Table 1, the vertical bearing capacity factors N v0 ¼47.190 and N v1 ¼68.235 are determined by linear interpolation, and then quv1 ð0Þ ¼ 936.601 kPa and quv1 ðbÞ ¼ 1336.465 kPa can be obtained.

614

Table 3 Comparison of results for perfectly rough footings. /(°)

Present study

Martin (2004)

Bolton and Lau (1993)

Chen (1975)

Terzaghi (1943)

Quv (kN)

Nc

Quv (kN)

Error (%)

Nc

Quv (kN)

Error (%)

Nc

Quv (kN)

Error (%)

Nc

Quv (kN)

Error (%)

q ¼0 5 10 15 20 25 30 35 40

0.114 0.447 1.267 3.183 7.618 18.083 43.677 111.435

1.14 4.47 12.67 31.83 76.18 180.83 436.77 1114.35

0.113 0.433 1.181 2.839 6.491 14.754 34.476 85.566

1.13 4.33 11.81 28.39 64.91 147.54 344.76 855.66

0.81 3.16 6.76 10.80 14.79 18.41 21.07 –23.21

0.62 1.71 3.17 5.97 11.60 23.60 51.00 121.00

6.20 17.10 31.70 59.70 116.00 236.00 510.00 1210.00

442.43 282.29 150.28 87.58 52.28 30.51 16.77 8.58

0.382 1.16 2.73 5.87 12.40 26.70 60.20 147.00

3.82 11.60 27.30 58.70 124.00 267.00 602.00 1470.00

234.21 159.33 115.54 84.44 62.78 47.66 37.83 31.92

0.50 1.20 2.50 5.00 9.70 19.70 42.40 100.40

5.00 12.00 25.00 50.00 97.00 197.00 424.00 1004.00

337.45 168.28 97.38 57.10 27.34 8.95 2.92 9.90

q ¼20 kPa 5 10 15 20 25 30 35 40

1.621 4.516 9.722 19.187 36.951 72.205 147.066 320.843

36.21 65.16 117.22 211.87 389.51 742.05 1490.66 3228.43

0.376* 1.122* 2.579* 5.457* 11.326* 23.887* 52.651* 124.75*

35.11 60.65 104.61 182.56 326.50 606.89 1192.43 2531.45

3.03 6.92 10.76 13.83 16.18 18.21 20.01 21.59

0.62 1.71 3.17 5.97 11.60 23.60 51.00 121.00

37.55 66.53 110.52 187.69 329.24 604.02 1175.92 2493.90

3.71 2.10 5.71 11.41 15.47 18.60 21.11 –22.75

0.382 1.16 2.73 5.87 12.40 26.70 60.20 147.00

35.17 61.03 106.12 186.69 337.24 635.02 1267.92 2753.90

2.86 6.34 9.47 11.89 13.42 14.42 14.94 14.70

0.50 1.20 2.50 5.00 9.70 19.70 42.40 100.40

37.84 65.87 113.92 198.77 351.41 646.11 1252.79 2629.42

4.49 1.09 2.81 6.18 9.78 12.93 15.96 18.55

q ¼40 kPa 5 10 15 20 25 30 35 40

2.951 8.061 16.902 32.560 61.424 117.337 233.012 493.947

69.51 120.61 209.02 365.60 654.24 1213.37 2370.12 4979.47

0.418* 1.233* 2.814* 5.926* 12.262* 25.824* 56.911* 134.96*

66.89 111.19 185.78 315.23 549.10 994.29 1900.95 3917.45

3.77 7.81 11.12 13.78 16.07 18.06 19.80 21.33

0.62 1.71 3.17 5.97 11.60 23.60 51.00 121.00

68.91 115.96 189.35 315.68 542.49 972.04 1841.84 3777.81

0.87 3.86 9.41 13.66 17.08 19.89 –22.29 24.13

0.382 1.16 2.73 5.87 12.40 26.70 60.20 147.00

66.53 110.46 184.95 314.68 550.49 1003.04 1933.84 4037.81

4.29 8.42 11.52 13.93 15.86 17.33 18.41 18.91

0.50 1.20 2.50 5.00 9.70 19.70 42.40 100.40

70.67 119.74 202.85 347.55 605.82 1095.23 2081.59 4254.83

1.68 0.72 2.95 4.94 7.40 9.74 12.17 14.55

*

indicates that based on Terzaghi’s equation the N c value is converted from Quv calculated by ABC.

M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

Nv

M.-x. Peng, H.-x. Peng / Soils and Foundations 59 (2019) 601–616

615

The total vertical ultimate bearing capacity Quv ðb Þ ¼1950.728 kN can be further obtained, which is 0.493% slightly higher than the slipline solution given above. (4) The second category of problem can be also solved using the above simplified method, as shown in Fig. 10(d). The calculation results are: b ¼1.839 m, Quv ðb Þ ¼1254.689 kN and Quh ðb Þ ¼241.236 kN. The approximate solutions to the ultimate bearing capacity are 0.648% and 0.594% greater than the corresponding exact solutions, respectively. 8. Conclusions The following can be concluded from this study: Fig. 9. Calculation results for perfectly rough footings.

Fig. 10. Calculation results: (a) slip-line solution to first category of problem; (b) slip-line solution to second category of problem; (c) approximate solution to first category of problem; (d) approximate solution to second category of problem.

Let b ¼ B=2 and q ¼ q2 ; then quv2 ðBÞ ¼ 669.267 kPa and quv2 ðbÞ ¼ 1057.701 kPa can be obtained by following the same steps as above. (iii) As shown in Fig. 10(c), assume the distributions of the vertical ultimate bearing capacities quv1 and quv2 to be linear respectively, then the width of OA1 b ¼0.646 m and the vertical ultimate bearing capacity at point O quv ðb Þ ¼ 1195.062 kPa can be determined by solving the intersection of the two straight lines.

(ii)

(1) Depending on the relative motion and interaction between the footing and soil, the problem of the ultimate bearing capacity of shallow strip footings can be divided into two categories. A minimum model with the total vertical ultimate bearing capacity as its objective function is established to reveal the intrinsic mechanical nature of the ultimate bearing capacity, and a general method for accurately solving this problem is proposed. (2) The plastic failure mechanism depends on the category and relevant parameters of the ultimate bearing capacity problem, and can be determined automatically through solving the optimization problem. The Prandtl mechanism does not represent the failure mechanism of the ultimate bearing capacity problem of perfectly smooth footings on weightless soil. (3) In engineering practice, the ultimate bearing capacity of shallow strip footings can be calculated approximately using the proposed simplified method, which is both convenient and reliable. (4) The new bearing capacity equation is applicable to the first category of problem with q1 ¼ q2 ¼ q, whereas Terzaghi’s equation only applies to the first category of problem satisfying q1 ¼ q2 ¼ q, Eq. (15) and Eq. (16) simultaneously. Terzaghi’s equation is an exact theoretical equation, and can be completely replaced by the new one. The theoretical exact solutions to N v , N c , N c and N q are obtained. Among them N v and N c increase with /, d, g1 and g2 , while N c and N q increase with / and k. (5) For perfectly smooth footings, the total vertical ultimate bearing capacity obtained by the proposed method is consistent with those by existing methods. For perfectly rough footings, the differences between the two are larger but no more than 25%, and the existing methods underestimate the ultimate bearing capacity. The present theoretical solution has yet to be compared with the results obtained by the finiteelement method.

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(6) In order to facilitate the application of the present study in practical engineering, preparing a handbook of tables on N v0 and N v1 available for engineers to use, and carrying out various laboratory and on-site test researches to verify are expected to be the primary tasks in the future studies on this subject.

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