Discussion on size effect of footing in ultimate bearing capacity of sandy soil using rigid plastic finite element method

Soils and Foundations 2016;56(1):93–103

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Discussion on size effect of footing in ultimate bearing capacity of sandy soil using rigid plastic finite element method Du L. Nguyena,b,n, S. Ohtsukac,1, T. Hoshinad,2, K. Isobee,3 b

a Department of Civil Engineering, Ho Chi Minh City University of Transport, Vietnam Department of Energy and Environmental Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japan c Department Civil and Environmental Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japan d Sato Kogyo Ltd. Company, 4-12-19 Honcho, Tokyo 103-8639, Japan e Division of Field Engineering for Environment, Hokkaido University, 13 Kita, 8 Nishi, Kita-ku, Sapporo, Hokkaido 060-8628, Japan

Received 25 December 2014; received in revised form 26 August 2015; accepted 14 October 2015 Available online 26 February 2016

Abstract Currently, many formulas are used to calculate the ultimate bearing capacity. However, these formulas have disadvantages when being applied in practice since they can only be applied for calculating simple footing shapes and uniform grounds. Most formulas do not take into account the size effect of the footing on the ultimate bearing capacity, except for the formula by the Architectural Institute of Japan. The advantage of using the finite element method (FEM) is its applicability to non-uniform grounds, for example, multi-layered and improved grounds, and to complicated footing shapes under three-dimensional conditions. FEM greatly improves the accuracy in estimating the ultimate bearing capacity. The objective of this study is to propose a rigid plastic constitutive equation using the non-linear shear strength property against the confining pressure. The constitutive equation was built based on experiments for the non-linear shear strength property against the confining pressure reported by Tatsuoka and other researchers. The results from tests on Toyoura sand and various other kinds of sand indicated that, although the internal friction angle differs among sandy soils, the normalized internal friction angle decreases with an increase in the normalized first stress invariant for various sands despite dispersion in the data. This property always holds irrespective of the reference value of the confining pressure in the normalization of the internal friction angle. The applicability of the proposed rigid plastic equation was proved by comparing it to the ultimate bearing capacity formula by the Architectural Institute of Japan, which is an experimental formula that takes into account the size effect of the footing. The results of rigid plastic finite element method (RPFEM) with the proposed constitutive equation were found to be similar to those obtained with the Architectural Institute of Japan’s formula. It is clear that RPFEM, with the use of the non-linear shear strength against the confining pressure, provides good estimations of the ultimate bearing capacity of the footing by taking account of the size effect of the footing. & 2016 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Ultimate bearing capacity; Size effect; Stress-dependent shear strength; Finite element method

n Corresponding author at: Department of Civil Engineering, Ho Chi Minh City University of Transport, Vietnam. Tel.: þ 81 25847 9633; mobile: þ 81 80 4947 7979. E-mail addresses: [email protected] (D.L. Nguyen), [email protected] (S. Ohtsuka), t.hoshina.offi[email protected] (T. Hoshina), [email protected] (K. Isobe). 1 Tel.: þ81 25847 9633. 2 Tel.: þ81 3 3661 1587. 3 Tel.: þ81 117066201. Peer review under responsibility of The Japanese Geotechnical Society.

http://dx.doi.org/10.1016/j.sandf.2016.01.007 0038-0806/& 2016 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

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1. Introduction In the design of buildings, the assessment of the ultimate bearing capacity of the footing is an important task in order to examine the stability of the building-ground system. Pioneering works were conducted by Prandtl (1921) and Reissner (1924). Prandtl considered a rigid-perfectly plastic half space loaded by a strip punch. The punch–soil interface can be frictional or smooth, and the material is set as weightless. The stress boundary condition is zero traction on the surface of the half space, except for the strip punch. Prandtl proposed bearing capacity factor Nc by analytical consideration. Reissner (1924) analyzed a similar problem, but there are two conditions different from those of Prandtl. The material is set as purely frictional (c ¼ 0), and a uniformly distributed pressure is loaded at the surface of the half space. Reissner applied hyperbolictype equations to solve the boundary value problem and introduced bearing capacity factor Nq. In the case of frictional-cohesive material, the analyzed slip-line is obtained similarly to the slip-line field. Bearing capacity factors Nq and Nc are adopted for many ultimate bearing capacity formulae. The ultimate bearing capacity formula for the footing by Terzaghi (1943) has been widely employed in practice. It takes into account the effects of cohesion, surcharge and soil weight (Terzaghi, 1943). The ultimate bearing capacity formula is typically expressed as q ¼ cN c þ 1=2γBN γ þ γDf N q

ð1Þ

where Νc, Νγ and Νq are the bearing capacity factors, which are functions of the internal friction angle of the soil, ϕ. The other indexes are as follows: γ : unit weight of soil (kN/m3), Df : depth of footing (m), and Β : footing width (m) Since this approach has been proposed, various studies on bearing capacity factors have been conducted. Bearing capacity factors Νq and Νc were provided by Prandtl (1921) and Reissner (1924) as
 π ϕ þ N q ¼ eπ tan ϕ tan 2 ð2Þ 4 2   N c ¼ N q  1 cot ϕ

ð3Þ

With regard to the Νγ factor, several formulations have been proposed, but no formula is totally accurate. For example, the formula by Meyerhof (1963) is expressed in the following way:   N γ ¼ N q  1 tan ð1:4ϕÞ ð4Þ Meyerhof (1951, 1963) introduced other factors, such as semi-empirical inclination factors ic, iγ and iq. The ultimate bearing capacity formula is described as follows: q ¼ ic cN c þ 1=2iγ γ 1 BN γ þ iq γ 2 Df N q

ð5Þ


 θ 2 ic ¼ iq ¼ 1  3 90

ð6Þ


 θ 2 iγ ¼ 1  ϕ

ð7Þ

where θ is the inclination angle of the load with respect to the vertical plane. The Architectural Institute of Japan (AIJ, 1988, 2001) developed the ultimate bearing capacity formula and it is now widely used in Japan. It was developed semi-experimentally. By using factors Νc and Νq , given by Prandtl, and Νγ, described by Meyerhof, the ultimate bearing capacity formula is expressed as follows: q ¼ ic αcN c þ iγ γ 1 βBηN γ þ iq γ 2 Df N q

ð8Þ

In the above equation, α and β express shape coefficients for which α ¼ 1 and β¼ 0.5 are recommended by De Beer (1970). η is the size effect factor defined in the following:
m B η¼ ð9Þ B0 where B0 is the reference value in the footing width, m is the coefficient determined from the experiment and m ¼  1/3 is recommended in practice. The ultimate bearing capacity formula by AIJ successfully takes into account the size effect of the footing which has not been considered in past formulae employing the Mohr– Coulomb criteria for soil strength. Since the past formulae overestimated the ultimate bearing capacity with the increase in footing width, this effect needs to be examined. Ueno et al. (1998) reported that the size effect on the ultimate bearing capacity was mainly attributed to the stress level effect on the shear strength of soils. Their research indicated that the mean stress ranged from 2γB to 10γB beneath the footing and caused changes in the internal friction angle of the ground mainly due to the mean stress. This study attempts to discuss the size effect on the ultimate bearing capacity by using a finite element analysis with the rigid plastic constitutive equation, which simulates the non-linear shear strength property of sandy soil against the confining pressure. In recent years, the finite element method (FEM) has become widely accepted as one of the well-established and convenient techniques for solving complex problems in various fields of engineering and mathematical physics. The latest four decades have observed a growing use of the finite element method in geotechnical engineering. FEM has been applied to estimate the bearing capacity of strip footings on cohesionless soils, such as Sloan and Randolph (1982), Griffiths (1982) and Frydman and Burd (1997). The rigidplastic finite element method (RPFEM) was developed for geotechnical engineering by Tamura et al. (1984) and Tamura et al. (1987a, 1987b). In this method, the limit load is calculated without the assumption on the potential failure mode. The method is effective in calculating the ultimate bearing capacity of a footing against three-dimensional boundary value problems where the soil conditions are varied as a multi-layered ground. Although RPFEM was originally developed based on the upper bound theorem in plasticity, Tamura et al. proved that it could be derived directly using the

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103

95

rigid plastic constitutive equation. The advantage of the rigid plastic constitutive equation is the scalability for considering the material property of soils as the non-associated flow rule. This study improves RPFEM by using the non-linear shear strength property of soils and introduces the rigid plastic constitutive equation of the parabolic yield function for the confining pressure. Tatsuoka et al. (1986) and other researchers (Hettler and Gudehus, 1988) have reported the effects of confining pressure on the internal friction angle for sandy soils by experiments. The obtained results from tests on Toyoura sand, Degebo sand, Eastern Scheldt sand and Darmstadt sand indicated that although the internal friction angles are different for the different soils, the normalized internal friction angle shows the same trend for all case studies. In this study, the non-linear shear strength property against confining pressure is introduced into RPFEM in order to assess the ultimate bearing capacity of sandy soils by taking account of the size effect of the footing. The agreement in ultimate bearing capacity between RPFEM and the AIJ formula shows the applicability of RPFEM. The size effect of the footing in the ultimate bearing capacity can be observed for not only uniform grounds, but also multi-layered grounds. Since the ultimate bearing capacity formula was developed for uniform grounds, the applicability of the method is severely limited in design practice. The results in both the ultimate bearing capacity and the failure mode are shown to have been appropriately obtained for the prescribed footing width. Through an examination on the computed results, the developed rigid plastic FEM has been proved effective for rational assessments of problems in which the ultimate bearing capacity is difficult to assess with the current bearing capacity formulas.

Drucker–Prager yield function is expressed as follows: pffiffiffiffiffi f ðsÞ ¼ aI 1 þ J 2  b ¼ 0 ð10Þ   where I 1 ¼ tr sij is first stress invariant, J 2 ¼ 12 sij sij is second invariant of deviator stress where is Kronecker’s operator. tan ϕ 3c ffi and b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi express The coefficients a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

2. Rigid plastic constitutive equation for the finite element method

The first term expresses the stress component, uniquely determined for the yield function, and the second term expresses the indeterminate stress component along the yield function. The indeterminate stress parameter, 3β, remains unknown until the boundary value problem with Eq. (12) is solved. In this study, the constraint condition on the strain rate is directly introduced into the constitutive equation using the penalty method (Hoshina et al., 2011). 0 1

The rigid plastic finite element method (RPFEM) was basically developed based on the upper bound theorem in the limit analysis. It is widely employed for stability assessments of soil structures in geotechnical engineering. Reissner (1924) derived the rigid plastic constitutive equation and proved that FEM with the rigid plastic constitutive equation matches RPFEM developed by the upper bound theorem. The advantage of the rigid plastic constitutive equation exists in its extensibility to more complicated material properties, such as the non-associated flow rule. In this chapter, the rigid plastic constitutive equation for the Drucker–Prager yield function is exhibited. Hoshina et al. (2011) derived the rigid plastic constitutive equation by introducing the dilatancy condition explicitly modeled by using the penalty method. 2.1. Rigid plastic constitutive equation for the Drucker– Prager yield function Tamura (1991) developed the rigid plastic constitutive equation for frictional material (Tamura et al., 1987). The

9 þ 12tan ϕ

9 þ 12 tan ϕ

the soil constants corresponding to the internal friction angle and cohesion, respectively. The volumetric strain rate is expressed as follows:




 ∂f ðrÞ s 3a ¼ tr λ αI þ pffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ė ε̇v ¼ tr ðε̇Þ ¼ tr λ ∂s 2 J2 3a2 þ1=2

ð11Þ where λ is the plastic multiplier and e_ is the norm of the strain rate. I and s express the unit and the deviatoric stress tensors, respectively. Strain rate ε_ , which is a purely plastic component, should satisfy the volumetric constraint condition which is derived by Eq. (11), as follows: 3a hðε̇Þ ¼ ε̇v  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ė ¼ ε̇v  η3ė ¼ 0 3a2 þ 1=2

ð12Þ

Any strain rate which is compatible with the Drucker–Prager yield criterion must satisfy the kinematical constraint conditions of Eq. (12). 3η is a coefficient determined by Eq. (12) which is one of the dilation characteristics. The rigid plastic constitutive equation is expressed by the Lagragian method after Tamura (1991), as follows: 0 1 b ε̇ 3a ε̇C B r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ β3@I  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ė 3a2 þ 12 3a2 þ 12 ė

b ε̇ 3a ε̇C B r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ κ ðε̇v  η3ėÞ@I  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ė 3a2 þ 12 3a2 þ 12 ė

ð13Þ

ð14Þ

where κ is a penalty constant. This technique makes the computation faster and more stable. With this constitutive equation and FEM, the same formulation of the upper bound theorem in plasticity (Tamura et al., 1987) is provided. This method is called RPFEM in the present study. In RPFEM, the occurrence of zero energy modes has been pointed out and some numerical techniques to avoid it have been introduced into FEM. However, zero energy modes have not been observed in the computation with the rigid plastic constitutive equation using the penalty method.

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2.2. Ultimate bearing capacity of a footing under plane strain conditions In this study, the input parameters for the ultimate bearing capacity analysis under plane strain conditions are derived from triaxial compression tests in the same way as with conventional methods. If the computed results show a good agreement between the RPFEM and the conventional formulas, it indicates that RPFEM can provide a good estimation of the ultimate bearing capacity since the conventional formulas are developed semi-empirically. In this study, the ultimate bearing capacity of a strip footing subjected to a uniform vertical load is investigated by RPFEM. The load is applied at the center of the footing with width B. This footing is modeled by a solid element, the strength of which is set large to be rigid. The typical finite element mesh and the boundary conditions employed for RPFEM are shown in Fig. 1. The ultimate bearing capacity is computed for B ¼ 10 m and ϕ ¼ 301. The obtained velocity field is shown in Fig. 2, which indicates the typical failure mode of the ground. The norm of the strain rate, e_ , is presented by contour lines. It is illustrated by the range between ėmax and 0ðėmin Þ, since it is basically indeterminate and the relative magnitude in e_ affects the magnitude of the ultimate bearing capacity. The slip-line assumed in the conventional bearing capacity formula is also plotted in the figure. The failure mode that is inferred by the computation results is similar to the slip-line assumed in the conventional formula. It is difficult to determine the slip-line by RPFEM since FEM is based on the continuum theory. However, it can be seen to provide a similar slip-line, although it is slightly smaller than that of the conventional formula. In the case of a rigid footing, stress concentration is widely known to generate at the edge of the footing. This causes a problem of singularity in the stress distribution of the ground. Since the finite element analysis is based on a continuous function for the shape function, it is not able to analyze the singularity problem directly. Thus, it analyzes the problem approximately. In sandy soil, the shear strength at the edge of the footing is affected by the free stress conditions of the ground surface outside the footing. The degree of singularity in the stress distribution is, therefore, comparatively moderate in the case of sandy soil, since the shear strength depends on the confining stress. In this study, no special numerical technique is employed to analyze the ultimate bearing capacity as in the

past references (Ukritchon et al., 2003; Lyamin and Sloan, 2002). As shown in Fig. 2, the velocity field of the ground at the edge of the footing is obtained greatly from the viewpoint of total balance in the velocity field. This seems to reflect the above-mentioned problem, but it is due to the limitation of the regular finite element method. This problem is partly resolved by using finer finite elements. The applicability of the rigid plastic finite element method is examined through a comparison with the past bearing capacity formulas and the finite element analysis. Fig. 3 expresses a comparison of bearing capacity factor Nγ among the various methods for changes in the internal friction angle. It proves that the rigid plastic finite element method gives a good estimation of the ultimate bearing capacity, although there is a defect in the treatment of singularity problems. The ultimate bearing capacity is computed for various footing widths from 1 m to 100 m at internal friction angles of 201 and 301. The results are presented in Fig. 4a and b. The larger the footing width, the higher will be the ultimate bearing capacity. The values obtained from RPFEM with the Drucker– Prager (DP) yield function coincide with the results from the formulas of Meyerhof and Euro-code 7 when the footing width is less than 30 m. Since the Euro-code formula employs different concepts, regarding the bearing capacity factor, it leads to ultimate bearing capacity values in a different way than the other formulas. Thus, the discrepancies among them become larger at the footing width of 100 m. This width seems too large in practice, but it is considered clearly to discuss the size effect of the footing on the ultimate bearing capacity.

Fig. 2. Deformation diagrams of the Drucker–Prager yield function with B¼ 10 m in case ϕ ¼301.

Fig. 1. Typical finite element mesh and boundary condition in case of B ¼ 10 m.

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103

97

120

100

Meyerhof

RPFEM Eurocode

80

Ukritchon et al. (2003)

60

40

20

0 0

5

10

15

20

25

30

35

40

Internal friction angle φ (°)

Ultimate bearing capacity q[kPa]

Bearing capacity factor Nγ

4000 Meyerhof

Fig. 3. Comparison of bearing capacity factor Nγ among the various methods.

RPFEM (DP) 3000

Eurocode AIJ

2000

1000

0 1

10

100

Footing width B[m] = 20deg 4000

20000

RPFEM (DP) 3000

Meyerhof

Eurocode

Ultimate bearing capacity q[kPa]

Ultimate bearing capacity q[kPa]

Meyerhof

2000

1000

0 1

10

RPFEM (DP) Eurocode

15000

AIJ 10000

5000

100

Footing width B[m]

0 1

= 20deg

10

100

Footing width B[m] = 30deg

20000

Fig. 5. Effect of footing width on ultimate bearing capacity for vertical load application.

Meyerhof

Bearing capacity q[kPa]

RPFEM (DP) 15000

Eurocode

10000

5000

0 1

10

100

Width of foundation B[m]

= 30deg. Fig. 4. Ultimate bearing capacity for vertical load application in case (a) ϕ ¼201 and (b) ϕ ¼301.

In the preliminary analysis, the effect of mesh size on the ultimate bearing capacity was investigated by comparing the bearing capacities computed for 1640 and 3423 element meshes which produces ultimate bearing capacities of 201.9 kPa, 504.9 kPa, 1530.7 kPa, 3822.1 kPa and 13691.2 kPa. The finite element meshes in this study produce ultimate bearing capacities of 201.8 kPa, 503.8 kPa, 1528.8 kPa, 3821.7 kPa and 13685.4 kPa with footing widths of 1 m, 3 m, 10 m, 30 m and 100 m, respectively. The obtained results almost coincide for all cases where the footing width is varied from 1 m to 100 m. Thus, the employed finite element meshes provide good estimations for various cases in this study. The AIJ formula takes into account the size effect of the footing on the ultimate bearing capacity. Fig. 5 indicates a comparison in the ultimate bearing capacity among the AIJ

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103

3. Rigid plastic constitutive equation for sandy soils 3.1. Strength tests on Toyoura sand by Tatsuoka et al.

Void Ratio e = 0.65 Void Ratio e = 0.75 Void Ratio e = 0.85 45

40

35

30 0

1000

2000

3000

First stress invariant I1[kPa] Fig. 7. Relationship between internal friction angle and first stress invariant for Toyoura sand.

1.00

Void Ratio e = 0.65 0.99

Void Ratio e = 0.75 Void Ratio e = 0.85

0.98

/

As mentioned above, the effect of the confining pressure on the shear strength is clearly presented in Fig. 6 through the experiments by Tatsuoka et al. on Toyoura sand. This figure shows that the internal friction angle decreases with the increase in confining pressure for a constant void ratio. In this study, in order to estimate the influence of the pressure level on ϕ under triaxial compression, the relationship between the internal friction angle and the first stress invariant is arranged in the normalization form. The general property in the internal friction angle is surveyed against the confining pressure. Fig. 6 indicates that internal friction angle ϕ can be inferred by the confining pressure for various void ratios. Fig. 7 demonstrates the relationship between internal friction angle ϕ and the first stress invariant I1 at failure. In reality, the friction angle

50

(deg)

formula and others. The results from the AIJ formula are smaller than those from other formulas that do not consider the size effect of the footing. A great discrepancy can be seen in the ultimate bearing capacity at the footing width of 100 m. Since the AIJ formula was developed semi-experimentally, it implies that RPFEM needs to take into account the size effect of the footing when assessing the ultimate bearing capacity.

Internal friction angle

98

0.97

0.96

0.95 0

2

4

6

8

10

I1 /I10 Fig. 8. Relationship between normalized internal friction angle ϕ/ϕ0 and normalized first stress invariant I 1 =I 10 for Toyoura sand.

Fig. 6. Experimental result of Toyoura sand (Tatsuoka et al., 1986).

decreases with an increase in the first stress variant in a logarithmic function. The range of the first stress variant is chosen according to the test results. The secant friction angle corresponds to the peak of each first stress variant and was larger than the approximated value obtained from the Mohr– Coulomb approach. Although the relationship is different, depending on the void ratio, the figure shows that the internal friction angle decreased with an increase in the first stress invariant, irrespective of the void ratio. Fig. 8 indicates the relationship between the normalized internal friction angle and the normalized first stress invariant. ϕ0 and I10 are the reference values for the internal friction angle and the first stress invariant, respectively. The figure shows that the normalized internal friction angles display a similar trend irrespective of the void ratio, which means that the obtained relationship exhibits the common property of Toyoura sand.

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103

ϕ ¼ arcsin  ζ s2 s20

sin ϕ  

ζ  þ sin ϕ 1 ss202

1.05

Eastern Scheldt sand Darmstadt Sand Toyoura sand 0.95

0.90

ð15Þ

where s2 is lateral stress, ζ estimated from triaxial tests; ϕ* is internal friction angle for reference lateral stress . Hettler and Gudehus (1988) also indicated that ζ is close to 0.1 and remains unchanged for various sands and densities, as seen in Table 1. Regarding Fig. 9, references I10 and ϕ0 can be chosen according to the examiner in the laboratory. However, the property of the normalization between the internal friction angle and the first stress invariant always holds irrespective of the reference value of the confining pressure in the standardization of the internal friction angle. Tatsuoka et al. (1986a, 1986b) and (Ueno et al., 1998) indicated that the effect of the confining pressure is considerable. Therefore, this study improves the rigid plastic finite element method by introducing the non-linear shear strength property against the confining pressure.

Degebo Sand

1.00

/

Hettler and Gudehus (1988) used three different types of sands, namely, Degebo sand, Eastern Scheldt sand and Darmstadt sand. The normalized internal friction angle, ϕ=ϕ0 , and the first stress invariant, I 1 =I 10 , for all types of soils show the same trends. It proves that the obtained relationship in the figure can be applied not only to Toyoura sand, but also to various kinds of sands. Hettler and Gudehus (1988) proposed a formula showing the relationship between internal friction angle ϕ and ϕ*, as follows:

99

0.85 0

2

4

6

8

10

I1/I10 Fig. 9. Relationship between ϕ/ϕ0 and I 1 =I 10 for various kinds of sand.

3.2. Proposal of the rigid plastic constitutive equation for the non-linear strength property In this study, the higher order hyperbolic function is introduced into the yield function of sandy soils, as follows: f ðrÞ ¼ aI 1 þ ðJ 2 Þ  b ¼ 0

ð16Þ

n

where a and b are soil constants. Index n expresses the degree of non-linearity in the shear strength against the first stress invariant. Eq. (16) is identical to the Drucker–Prager yield function in the case of n¼ 1/2. The non-linear parameters, a, b and n, are identified by the testing data. In the figure, the results by triaxial compression tests are plotted for various confining stresses. Fig. 10 shows an example of how parameter n in Eq. (16) influences the internal friction angle for the confining pressure. This figure indicates that parameter n greatly affects the non-linear property in the shear strength of soils. This means that parameter n increases when the Table 1 Data for different sands (Hettler and Gudehus, 1988).

Fig. 10. Non-linear parameter n affects the non-linear property in shear strength of soils in case ϕ0 ¼ 301.

internal friction angle decreases at various levels of confining pressure. Based on the associated flow rule, the strain rate is obtained as follows for the yield function of Eq. (16): ε̇ ¼ λ

  ∂f ðrÞ ∂ ¼ λ ðaI 1 þ ðJ 2 Þn  bÞ ¼ λ aIþ nJ n2  1 s ∂r ∂r

ð17Þ

In the above equation, λ is the plastic multiplier. The volumetric strain rate is expressed as    3a ε̇v ¼ trε̇ ¼ tr λ aI þ nJ n2  1 s ¼ 3aλ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ė 3a2 þ 2n2 ðb  aI 1 Þ2  1=n

ð18Þ

Sand

ϕ* (deg)

r20 ðkPaÞ

ζ

Toyoura Degebo Eastern Scheldt Darmstadt

41 40 38 43.8

10 50 50 50

0.1 0.1 0.08 0.1

First stress invariant I1 is identified from Eq. (16) to Eq. (18) as the following equation: ( "
#)2n n 1  b 1 1 ė 2 2 I1 ¼  3a  3a ð19Þ a a 2n2 ε̇v

100

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103

In this study, the non-linear rigid plastic constitutive equation for confining pressure is finally obtained as follows: r¼

( "
#)2n1 n1  3a 1 ė 2 ε̇ 2 3a  3a n 2n2 ε̇v ε̇v 0 " #2n n 1 " #2n1 n1 1
2
2 b 1 1 ė a 1 ė AI þ@  3a  3a2  3a  3a2 3a 3a 2n2 ε̇v n 2n2 ε̇v

ð20Þ In this equation, the stress is uniquely determined for the plastic strain rate and is different from Eq. (14) for the Drucker–Prager yield function.

Fig. 11. Deformation diagram of the non-linear shear strength with B ¼ 10 m.

4. Discussion on the size effect of the footing on the ultimate bearing capacity

Ultimate bearing capacity q[kPa]

600

AIJ 500

RPFEM (NL) 400

300

200

100

0 1

10

100

Footing width B[m] o

= 20deg

3200

Ultimate bearing capacity q[kPa]

The conventional RPFEM with the Drucker–Prager function does not take into account the size effect on the ultimate bearing capacity, which is considered in the AIJ formula, because RPFEM is based on the same framework as the other conventional ultimate bearing capacity formulae. This study improves RPFEM by using the non-linear shear strength property of soils and introduces the rigid plastic constitutive equation for the parabolic yield function for the confining pressure. This study has shown that the internal friction angle is not constant, but decreases with the increase in confining pressure in sandy soils. This implies that the confiningpressure dependency in the soil shear strength may be one of the most important factors influencing the size effect of the footing. In bearing capacity problems, the larger the footing width is, the higher the confining pressure will be. This leads to a decrease in the internal friction angle, as discussed above. It is necessary, therefore, to apply the non-linear shear strength property against the confining pressure to take into account the size effect of the footing on the ultimate bearing capacity. On the other hand, the internal friction angle is set to be constant in RPFEM in the case of the Drucker–Prager yield function. Therefore, the ultimate bearing capacity calculated with the non-linear rigid-plastic constitutive equation becomes smaller than that obtained from the Drucker–Prager yield function. This means that the size effect of the footing is properly taken into account in the computation. The non-linear yield function (Eq. (16)) is defined by parameters a, b and n, which are derived from the experiment. In this study, a series of numerical simulations are conducted for Toyoura sand based on the experiment in Tatsuoka (1986a, 1986b). Through the case studies, the non-linear shear strength parameters of Toyoura sand are set as a ¼ 0.24, b¼ 2.4 (kPa) and n ¼ 0.56. Fig. 11 shows the deformation of the ground at the limit state computed by multiplying the arbitrary time increment to the velocity field obtained by RPFEM for B ¼ 10 m.The obtained failure mode of the ground is similar to that in Fig. 2 for the linear shear strength of the Drucker–Prager yield function. However, the deformation area in the case of linear shear strength is obtained as a larger value than that in the case

2800

AIJ

2400

RPFEM (NL)

2000 1600 1200 800 400 0 1

10

100

Footing width B[m] o

= 30deg.

Fig. 12. Ultimate bearing capacity with non-linear shear strength in case (a) ϕo ¼201 and (b) ϕo ¼301.

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103

of non-linear shear strength, especially around the edge of the footing. Fig. 12 shows the results of RPFEM with non-linear shear strength in the case internal friction angles of 201 and 301. In the figure, these results are clearly identical to those of AIJ. This means that the results obtained by employing the non-linear shear strength property are rational and that they show that the size effect of the footing in the ultimate bearing capacity can be expressed well by considering the non-linear shear strength against the confining pressure. The computed results are utilized to determine the bearing capacity factor Nγ for various internal friction angles from 01 to 401. The obtained bearing capacity factor Nγ is compared with these factors defined by the empirical methods by Meyerhof (1963-Semi-empirical) and Muhs and Weiss (1969-Eurocode7, Semi-empirical). Although the cohesion of soils (c ¼ 1 kN/m2) is introduced into the analysis to stabilize the computation process, it does not affect the ultimate bearing capacity too much. Therefore, Eqs. (21) and (22) are applied to approximately define Nγ. The bearing capacity factor Nγ of RPFEM for Drucker– Prager is calculated by the following equation: N DP γ ¼

2qDP γ1B

ð21Þ

On the other hand, the bearing capacity factor Nγ for nonlinear shear strength is determined by N NL γ ¼

2qNL γ1B

ð22Þ

The bearing capacity factor Nγ was compared among the bearing capacity formulas of AIJ, Euro-code 7 and Meyerhof with RPFEM. Fig. 13 shows a comparison of bearing capacity factors derived by changing the internal friction angle from 01 to 401. As shown in the figure, the bearing capacity factor by RPFEM, employing non-linear shear strength against the confining pressure, matches that by the AIJ formula for a wide range of internal friction angles. It is obtained as a smaller value than that by the formulas of Euro-code 7 and Meyerhof. When the internal friction angle is less than 301, there is not much difference in the bearing capacity factor 110

Meyerhof

Bearing capacity factor N

100 90

RPFEM Result (DP)

80

Eurocode

70

AIJ

60

RPFEM Result (NL)

50

N DP γ

N γNL

40 30 20 10

101

among them. However, the difference becomes greater at the internal friction angle of 401.

5. Conclusions Terzaghi (1943) and others (e.g. Meyerhof, 1951, 1963) have proposed many formulas to evaluate the ultimate bearing capacity. However, the application of these formulas is limited due to their disadvantages. The rigid plastic finite element method (RPFEM) is effective for solving complex problems, such as multi-layered soil and the footing shape under threedimensional conditions. Moreover, it is possible to conduct a limit state analysis without assuming potential failure modes. In this study, RPFEM has been employed to assess the ultimate bearing capacity. The applicability of the method has been presented through a comparison with that of semiexperimental ultimate bearing capacity formulas. The size effect of the footing has been observed in the ultimate bearing capacity, but basically it has not been taken into account in the ultimate bearing capacity formulas. In this study, a discussion on the size effect has been conducted in the case of a uniform sandy ground. A rigid plastic constitutive equation has been proposed for sandy soils by considering the experiments, where the secant internal friction angle decreased with the increase in confining pressure. This equation was expressed by the higher order parabolic function and was easily applied to RPFEM. The obtained ultimate bearing capacity showed a good agreement with that of the ultimate bearing capacity formula by the Architectural Institute of Japan (AIJ, 1998, 2001), which takes into account the size effect of the footing. It is clear that RPFEM, with the use of the proposed constitutive equation, provides a good estimation in ultimate bearing capacity assessments by considering the size effect of the footing. On the other hand, all the numerical calculations were for the vertical loading cases of rigid flat footing under plane strain conditions. When the inclined load was considered, the vertical load at failure decreased with the increase in inclination angle. This caused a decrease in confining pressure and changes in the internal friction angle in the ground. Therefore, the limit state in vertical and horizontal load spaces is not as simple as seen in the previous work by Meyerhof due to the variance in the internal friction angle. The assessment of the ultimate bearing capacity for inclined loads is a subject for future study, but the analytical method will provide reliable computation results to this problem. Through the case studies for various footing widths, changes in both the ultimate bearing capacity and the failure mode due to the footing width were shown to have been properly simulated. The obtained conclusions are summarized as follows:

0 0

10

20

Internal friction angle

30

40

()

Fig. 13. Relationship between bearing capacity factor Nϕ and internal friction angle ϕ.

(1) For sandy soils, the size effect of the footing in the ultimate bearing capacity was well simulated by RPFEM with the use of the proposed constitutive equation. It was proved by a comparison in the ultimate bearing capacity between the

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semi-experimental bearing capacity formulas of AIJ and RPFEM. (2) A rigid plastic constitutive equation was proposed for sandy soils based on the experiments by Tatsuoka and other researchers for various soils. The relationship between the secant internal friction angle and first stress invariant was uniquely expressed in normalized form, although some scatter existed. The yield function was modeled into the higher order parabolic function regarding the first stress invariant. (3) Bearing capacity factor Nγ was compared among the bearing capacity formulas of AIJ, Euro-code 7 and Meyerhof with RPFEM by changing the internal friction angle from 01 to 401. The bearing capacity factor by RPFEM, employing non-linear shear strength against the confining pressure, matched that by AIJ formula in the wide range of internal friction angles. It was obtained as a smaller value than that by the formulas of Euro-code 7 and Meyerhof. The difference in bearing capacity factors was shown to be greater at the internal friction angle of 401. (4) The wide applicability of the developed RPFEM to the assessment of the ultimate bearing capacity was shown through the case studies. References AIJ (1988, 2001), Architectural Institute of Japan. Recommendations for design of building foundations, 430p. De Beer, E.E., 1970. Experimental dertermination of the shape factors and the ultimate bearing capacity factors of sand. Geotechnique 20 (4), 387–411. Frydman, S., Burd, H.J., 1997. Numerical studies of ultimate bearing capacity factor, Nγ. J. Geotech. Geoenviron. Eng. 123 (1), 20–29. Griffiths, D.V., 1982. Computation of bearing capacity on layered soil. In: Proceedings of the 4th International Conference on Numerical Methods in Geomechanics, Edmonton, Alberta, Canada, Balkema, Rotterdam, the Netherlands, May, vol. 1, pp. 163–170. Hettler, A., Gudehus, G., 1988. Influence of the foundation width on the ultimate bearing capacity factor. Soils Found. 28 (4), 81–92. Hoshina, T., Ohtsuka, S., Isobe, K., 2011. Ultimate bearing capacity of ground by rigid plastic finite element method taking account of stress dependent non-linear strength property. J. Appl. Mech. 6, 191–200 (in Japanese). Lyamin, A.V., Sloan, S.W., 2002. Upper bound limit analysis using linear finite elements and non-linear programming. Int. J. Numer. Anal. Methods Geomech. 26, 181–216. Meyerhof, G.G., 1951. Ultimate bearing capacity of foundations. Geotechnique 2, 301–332. Meyerhof, G.G., 1963. Some recent research on the ultimate bearing capacity of foundations. Can. Geotech. J. 1 (1), 243–256. Prandtl, L., 1921. Über die Eindringungsfestigkeit(Härte) Plastischer Baustoffe und die Festigkeit von Schneiden. Z. Angew. Math. Mech. 1, 15–20. Reissner, H., 1924. Zumerddruckproblem. In: Proceedings of the 1st International Congress of Applied Mechanics, Delft, The Netherlands, pp. 295– 311. Sloan, S.W., Randolph, M.F., 1982. Numerical prediction of collapse loads using finite element methods. Int. J. Numer. Anal. Methods Geomech. 6, 47–76. Tamura, T., Kobayashi, S., Sumi, T., 1987. Rigid plastic finite element method for frictional materials. Soils Found. 27 (3), 1–12. Tamura, T., Kobayashi, S., Sumi, T., 1987. Limit analysis of soil structure by rigid plastic finite element method. Soils Found. 24 (1), 34–42. Tatsuoka, F., Goto, S., Sakamoto, M., 1986c. Effects of some factors on strength and deformation characteristics of sand at low pressures. Soils Found. 26 (4), 79–97.

Tatsuoka, F., Sakamoto, M., Kawamura, T., Fukushima, S., 1986. Strength and deformation characteristics of sand in plane strain compression at extremely low pressures. Soils Found. 26 (1), 65–84. Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wiley and Sons Ltd., 510. Ueno, K., Miura, K., Maeda, Y., 1998. Prediction of ultimate bearing capacity of surface footings with regard to size effects. Soils Found. 38 (3), 165–178. Ukritchon, B., Whittle, A.W., Klangvijit, C., 2003. Calculation of bearing capacity factor Nγ using numerical limit analysis. J. Geotech. Geoenviron. Eng., ASCE 129 (6), 468–474.

Further reading Aysen, A., 2002. Soil Mechanics – Basic Concepts and Engineering Applications. Balkema AA Publisher413–419. Baglioni, V.P. , Chow, G.S., Endley, S.N., 1982. Jack-up foundation stability in stratified soil profiles. In: Proceedings of the 14th Offshore Technology Conference, vol. 4, pp. 363–369. Bolton, M.D., Lau, C.K., 1993. Vertical ultimate bearing capacity factors for circular and strip footings on Mohr–Coulomb soil. Can. Geotech. J. 30, 1024–1033. Clark, Jack I., 1998. The settlement and ultimate bearing capacity of very large foundations on strong soils. Can. Geotech. J. 35, 131–145. Drucker D.C., Greenberg H.J., Lee E.H., Prager, W., 1951. On plastic rigid solutions and limit design theorems for elastic plastic bodies. 1st US NCAM, pp. 533–538. Du, N.L., Ohtsuka, S., Hoshina, T., Isobe, K., Kaneda, K., 2013. Ultimate bearing capacity analysis of ground against inclined load by taking account of non-linear properties of shear strength. Int. J. GEOMATE 5 (2), 678–684. Edgar, G.D., 2013. Assessment of the range of variation of Nϕ from 60 estimation methods for footings on sand. Can. Geotech. J. 50, 793–800. Fukushima, S., Tatsuoka, F., 1984. Strength and deformation characteristics of saturated sand at extremely low pressure. Soils Found. 24 (4), 30–48. Hanna, A.,M., Meyerhof, G.G., 1980. Design charts for ultimate bearing capacity of foundations on sand overlying soft clay. Can. Geotech. J. 17, 300–303. Hanna, A.,M., Meyerhof, G.G., 1981. Experimental evaluation of ultimate bearing capacity of footings subjected to inclined loads. Can. Geotech. J. 18 (4), 599–603. Hettler, A., Vardoulakis, I., 1984. Behaviour of dry sand tested in a large triaxial apparatus. Géotechnique 34 (2), 183–198. Hettler, A., Gedehus, G., 1985. A pressure-dependent correction for displacement results from 1 g model tests with sand. Géotechnique 35 (4), 497–510. Hjiaj, M., Lyamin, A.V., Sloan, S.W., 2004. Ultimate bearing capacity of cohension-frictional soil under non-eccentric inclined loading 31, 491–516. Hjiaj, M., Lyamin, A.V., Sloan, S.W., 2005. Numerical limit analysis solutions for the ultimate bearing capacity factor Nγ. Int. J. Solids Struct. 42, 1681–1704. Houlsby, G.T., Milligan, G.W.E., Jewell, R.A., Burd, H.J., 1989. A new approach to the design of unpaved roads, Part 1. Ground Eng. 22 (3), 25–29. Kraft, L.M., Helfrich, S.C., 1983. Bearing capacity of shallow footing, sand over clay. Can. Geotech. J. 20 (1), 182–195. Meyerhof, G.G., 1974. Ultimate bearing capacity of footings on sand layer overlying clay. Can. Geotech. J. 11 (2), 223–229. Okamura, M., Takemura, J., Kimura, T., 1997. Centrifuge model tests on ultimate bearing capacity and deformation of sand layer overlying clay. Soils Found. 38 (1), 181–194. Okamura, M., Takemura, J., Kimura, T., 1997. Ultimate bearing capacity predictions of sand overlying clay based on limit equilibrium methods. Soils Found. 37 (1), 73–87. Shiraishi, S., 1990. Variation in ultimate bearing capacity factors of dense sand assessed by mode loading tests. Soils Found. 30 (1), 17–26.

D.L. Nguyen et al. / Soils and Foundations 56 (2016) 93–103 Siddiquee, M.S.A., Tatsuoka, F., Tanaka, T., Tani, K., Yoshida, K., Morimoto, T., 2001. Model tests and FEM simulation of some factors affecting the ultimate bearing capacity of a footing on sand. Soils Found. 41 (2), 53–76. Tamura, T., 1990. Rigid-plastic finite element method in geotechnical engineering. Soc. Mater. Sci., Jpn. 7, 135–164. Tamura, T., 1992. Simulation of strain localization by means of rigid-plastic finite element method. Adv. Micromech. Granul. Mater., 183–192. Terzaghi, K., Peck, R.B., 1948. Soil Mechanics in Engineering Practice first ed. Wiley, New York, 592.

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Yamaguchi, H., Terashi, M., 1971. Ultimate bearing capacity of multi-layered ground. In: Proceedings of the 4th Asian Regional Conference on SMFE, vol. 1, pp. 99–105. Yamamoto, K., Otani, J., 2002. Ultimate bearing capacity and failure mechanism of reinforced foundations based on rigid-plastic finite element formulation. Geotext. Geomembr., 367–393. Yamamoto, N., Randolph, M.F., Einav, I., 2009. Numerical study of the effect of foundation size for a wide range of sands. J. Geotech. Geoenviron. Eng., ASCE 135, 37–45.