TWO- AND THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY …

TWO- AND THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS

V. Q. Nguyen 1 and R. S. Merifield2

1 le Qui Don Technical University, Vietna111 1U11iversity of Newcastle, Australia

ABSTRACT

The abi lity to predict the ultimate bearing capacity of a foundation is one of the most important problems in foundat ion engineering. To solve this problem, geotechnical engineers routinely use a bearing capacity equation that contains a number of empirical factors to account for foundation shape, depth and inclination. In this paper finite element analysis is used to predict the undrained bearing capacity of strip, square, rectangular and circular footings embedded in clay. From these analyses, rigorous shape and depth factors have been derived and are compared with previous numerical and empirical solutions in the literature. The bearing capacity behaviour is discussed and the bearing capacity factors are given for various cases involving a range of embedment depths and footing shapes.

Keywords: Deformation; Numerical modelling; Strip footing; Square footing, Circular footing; Rectangular footing; Embedment, Bearing capacity; Bearing capacity factor; Clays; Shape factor; Depth factor.

1 INTRODUCTION The ability to predict the bearing capacity of embedded foundations is one of the most significant problems in foundation engineering and, as a consequence, extensive literature exists detailing both theoretical and experimental studies on this topic. ExperimentOal investigations and formulas of undrained bearing capacity of embedded footings were reported by Terzaghi ( 1943), Meyerhof (195 1) and Skempton ( 1951 ), whi le numerical investigations of the bearing capacity of embedded footings have been reported by Hansen ( 1969), Hu et al. ( 1998), Salgado et al.(2004) and Edwards et al. (2005).

Meyerhof ( 195 1) suggested a general bearing capacity theory with consideration for correction factors for eccentricity, load inclination and foundation depth. The influence of the shear strength of soil above and below the base of the foundation was investigated. According to Meyerhof ( 1951 ), the ultimate bearing capacity q11 for a purely cohesive clay

( ¢>,, = O) is written as:

- QI/ - A/FF DN C/11 - - C11H c cs cd + Y q A

(I)

where Q11 is the total reaction force on the footing from which the ultimate bearing capacity q11 can be calculated; A is

cross-sectional area of the footing; c,, is the undrained shear strength; Ne , N" are bearing capacity factors; f'.·s is

shape factor; Fed is depth factor, y is the soil unit weight, and D is the embedment depth. The shape and depth factors

for undrained soil are given by:

Fcs = 1 + 0.2( ~)

Fed = I + 0.2 ( ~)

(2)

(3)

Skempton ( 195 1) suggested the same shape factor given in equation (2) for undrained clay, however suggested the depth factor Fed can be expressed as:

(4)

with maximum values when DI B ~ 4 as follows: when BI l = 0, Ne = 7 .5 (i.e. strip footing) and when

B I L = 1.0, Ne = 9.0 (i.e. square or circular footing). Equation 4 is often used in practice.

Hansen ( 1969) investigated the bearing capacity of a vertically and centrally loaded strip footing placed at a depth D below a horizontal, unloaded surface in homogeneous clay in the undrained state by the means of the theory of plasticity for ideal rigid-plastic materials. Hansen assumed the soil is weightless, no contact at the vert ica l soil-footing

Australian Geomechanics Vol 47 No 2 June 2012 25

TWO- AND THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS NGUYEN & MERIFIELD

interface, and shear strength is constant within the soi l medium. The scope of the investigations is limited to relatively shallow footings (O<DIB<2). Hansen ( 1969) proposed one of the simplest possible rupture figures, consisting of a kinematically admissible displacement fie ld, two radia l zones with straight radial slip lines and two line ruptures. The

bearing capacity was expressed as a ratio of QI c11B as a function of DI B. Apply ing the upper-bound theorem, a

number of different solutions for bearing capacity in the form of ratio Q I c,, B as a function of D I B were found. The

most rigorous mechanism predicted a bearing capacity of

_£_ = ( 2 + Jr)+0.533[ J 1 + l.75D I B - 1] c

11B (5)

Salgado et al. (2004) studied the two and three-dimensional bearing capacity of strip, square, circular and rectangular foundations in clay using finite element limit analysis based on the lower- and upper-bound theorems of plast ic ity of Hill ( 1951) and Drucker et al. ( 1951 ) and Drucker and Prager ( 1952). The results of the analyses were used to propose more rigorous values of the shape and depth factors for foundat ions in clay. The shape and depth factors a re determined by computing the bearing capacities of footings of various geometries placed at various embedment depths D of footings from 0 to 5 times of footing width or diameter B .

From the results of the limit analyses, Salgado et al. (2004) proposed an equation for depth factor of square, circular and rectangular footings:

Fed = 1+0.271%

And they also proposed an equation for the shape factor:

Fcs = I + c, ( ~) + C2 J% (6)

(7)

Where C1 and C2 are regression constants depending on the foot ing shape. They concluded that the shape factors are

not constant with depth. This is in contrast to the assumption of independence of shape and depth factors implied by traditional expressions.

2 PROBLEM DEFINITION The two and three dimensional bearing capacity problems to be considered are illustrated in Figure I. The model inc ludes a footing placed at an embedment depth D from the surface and the clay soil has an undrained shear strength e11 and infinite depth. Strip footings have a width B, square footings have a side length B, rectangular footings have

width B and length L and circular foot ings have a diameter B. In this study equation ( I) is used to obtain the bearing capacity factors. Symmetry has been exploited for the problems. Only o ne quarter of the problem domain has been modelled for three-dimensional analyses and one hal f for two-dimensional analyses. The soil was modelled as an isotropic e lasto-perfectly plastic continuum with yie ld ing described by the Mohr-Coulomb yield c riterion. The elastic behaviour was described by a Poisson's ratio v = 0.48 and a ratio of Young's modulus to shear strength of

E I e11

= 300 . The soil is treated as weightless. The interactions be tween the soil and the bottom of the footings are

characterized as rough. The undrained bearing capacity of embedded footings will be a function of the ra tio DI B and/or B IL. In this work, solutions have been computed for problems where D I B ranges from 0 to 5 and in the

rectangular case, the additional parameter BIL varies from 0.2 to 0.5. These ranges cover most problems of practical inte rest.

The purpose of this study is to propose bearing capacity, depth and shape factors for strip, square, circular and rectangular footings at some depth in clay using finite e lement method. To do this, the input files were created and run using commercial software package ABAQUS. The output of the analyses is the total reaction force Q,, on the footi ng,

from which the ultimate bearing capacity q11

can be calculated. In addition, the depth factor F;_.d and the shape factor

Fe, are obtained via equation ( I).

3 ABAQUS MODELS The ABAQUS model consisted of two parts, namely the footing and the soil as presented in Figure I. Typical meshes for these problems are shown in Figure 2. For the strip footing case, the mesh (Figure 2a) consisted of 8-node p lane strain quadrilateral reduced integration elements because this e lement was found to provide the best solution convergence. For the three-dimensional case of square, rectangular (Figure 2c) and circula r footings (Figure 2d), the I 0-node modified quadra tic te trahedron e lement (C3D 1 OM) was adopted for similar reasons. These elements are specific to

26 Australian Geomechanics Vol 47 No 2 June 2012


ABAQUS and have been constructed to reduce "node locking" and to have an unambiguous sign of the contact normal stress that is usually associated with second order e lements in contact analyses.

Footing

D [

B/2

Soil

L. (a) Strip footing

D

Footing

I I I I I I

Soil I P6 I

I 12

I ---l -I.. 3 '

----"\..[5';,.... 1

- ' ' ' p ",3

Elements in planes P2 and P7 have 1i 1 = ur2 = 11r3 = 0

Elements in planes P3, P4 andP5 have :

U. 1 = Uz = U3 = 111'1 = 111'2 = U/'3 = 0 Footing has u 11 = 0, u21 = - o, u31 = 0

I I I

Soil l I 2 I I I 3 .-I.. I - , 1 --"" -- \ p' -- ...... 5-"' - - '

' ' '"P '

Eleme11ts in planes P1 have 1i3 = 111·1 = 11r2 = 0

Eleme11ts i11planesP2 have u1 = 11r 2 = ur 3 = 0 Elements in planes P 3, P4 and P 5 have :

lt1 = u2 = u3 = 11r1 = ur2 = 11r3 = 0 Footing has u11 = 0, " 'lf = - o, u31 = 0

J11te1face of footing sides and soil P 6 just has vertical displacement only

(b) Square and rectangular footing (c) Qrcular footing

Figure I : Models and boundary conditions of strip, square, rectangular and circular embedded footings



The overall mesh dimensions were selected to ensure that the zones of plastic shearing and the observed displacement fields were contained within the model boundaries at all times. The underside of all footings was modelled as perfectly rough by specifying a tied contact constraint at the footing/soil interface.

(a) Strip footing

(c) Square, rectangular footing

(b) 8-node plane strain quadrilateral, hybrid, linear pressure, reduced integration element (CPE8RH)

(e) 10-node modified quadratic tetrahedron ele

ment (C3D10M)

2

3--( 1

(d) Circular footing

Figure 2: Typical meshing of the finite element models of strip, square, rectangular and circular embedded footings

To determine the collapse load of the footing, displacement defined ana lyses were performed where the footing was considered as being perfectly rigid. That is, a uni form vertical prescribed displacement was applied to all those nodes on the footing. The total displacement was applied over a number of substeps and the nodal contact forces along the footing were summed to compute the equivalent bearing capacity. The number of displacement increments is automatically determined by ABAQUS, within initial, minimum and maximum values prescribed by users.



A di stinct advantage of using the finite element method is that it provides the complete load de formation response. This can provide insight into general footing behaviour, particularly in regard to the development of the collapse mechanism and deformation issues. By observing the load displacement response, a check can be made to ensure that the ultimate bearing capacity has been reached and that overall collapse has in fact occurred (i.e. the load-displacement plot reaches a plateau).

4 RESULT AND DISCUSSION All the ABAQUS results are presented in Table 2 and Figure 3. Also shown are the results of lower and upper bound limit analysis of Salgado et al. (2004) and those of Skemp ton ( 1951) using equations (2) and ( 4). The normalised undrained bearing capacity from the current fini te element study compare well to those calculated us ing the numerical solutions of Salgado et al. (2004) In general the current results plot close to the upper bound for strip, circular and square foundations and lie between the bounds solutions of Salgado for rectangular foundat ions. The results using of Skempton's ( 195 1) equations are always conservative.

It is clear from Table I that deeper foundations mobil ise larger volumes of soil and dissipate more plastic energy, therefore the larger bearing capacities are at larger DI B ratios. Samples of the load-displacement behaviour of embedded footings is presented in Figure 4 to Figure 6 in terms of the dimensionless parameters 0£

11 I Be

11, where o is

the displacement of the footing. The plateaus shown in Figure 4 to Figure 6 indicate that collapse is clearly defined for each embedment ratio DI B and footing shape. For footings at relatively small embedment ratios DIBS.0.6 collapse is very clearly defined because the load-displacement plot reaches a plateau at small normalised displacements.

The failure mechanism can be seen in displacement vectors and contours in Figure 7 to Figure 9. For embedment depths of DIBS.0.6, the modes of failure can best be described as being general shear, as illustrated in Figure 7. As can be seen from Figure 7 the modes of failure are similar to those of sur face footings on undrained soil. The soil elements with greatest displacement are underneath and along the centre line of the footing. When DIB>0.6, the fa ilure mechanism is similar to that of a local shear failure of a surface footing. The failure zone is deeper and is located more in the vicinity of the footing as the ratio of DIB increases. For values of D>0.6, more soil elements move horizontally away from the footing instead of moving to the surface. It is like ly that when the ratio of DIB rises to a infi ni te depth, soil displacements will no longer be observed at the surface. In that case, the shear failure mechan ism will be contained around the footing toe.

For deeper footings (DIB>0.6) the di splacement at collapse are very large. The plateaux shown in Figure 3 to Figure 6 also indicate that, for the same footing depth at collapse, the value of the footing displacement o £11 I Be11 increases as

the footing shape changes. For example, for DIB=0.6, at collapse, the value of 0£11

I Be11

=65.5, 67.32, 80.23, 95.44,

I 04.09 and 132. 12 as the footing shape changes from strip, to circular, to square, and to rectangular Bll=0.5 , B/L=0.33 , B/l=0.25 and BIL=0.20.

For the same value of DIB, the bearing capacity increases as the ratio of BIL rises from 0 (for strip footings) to I (for square or circular footings). Al the same depth, if DI B ~ 3, the bearing capacity of circular footings is larger than of square footings by about ?-4% and if D I B > 3 bearing capacity of circular footings is smaller than those of square footings by about 1-3%.

Equation ( I) assumes implicitly that shape factors F;_'S and depth factors Fed are independent from each other. The shape factors will be a function of Bil and can be calculated us ing the bearing capaci ty of surface footings (DIB=O) for strip, square, circular and rectangular footings. These shape fact; rs do not depend on the footing depth. The depth factors Fe,/ are functions of both parameters Bil and DIB and are calculated from (I ) when the shape factors have been determined beforehand. Both the shape and depth factors will be presented in the following sections.

4.1 SHAPE FACTORS

The shape factor Fcs can be calculated using the values of undra ined bearing capacity for embedded footings q11

I c,, in

Table I in conj unction with equation ( I). The shape factor Fcs is determined by divid ing the bearing capacity factor

Ne for any surface footing by Ne fo r strip surface footing. In the present work, rectangular footi ng dimensions of

B I L = 0.5, 0.33, 0.25 and 0.2 were considered and B I L = I applies to square and circular foot ings. For strip footing, shape factor f',.

5 = I .

For a square footing, shape factor is determined as:

Australian Geomechanics Vol 47 No 2 June 201 2 29


Table l : Normalised undrained bearing capacity q,/c,, for embedded strip, square, ci rcular and four selected rectangular footings of FEM-ABAQUS, compared to result of Salgado et al. (2004)and Skemptom ( 1951) in equation (2)

Strip footing Square footing Circular foot ing Rectangle footing

DIE BIL=0.5

AB LB UB SK AB LB UB SK AB LB UB SK AB LB 0.00 5.240 5.1 32 5.203 5.140 5.950 5.523 6.227 6.168 6.048 5.856 6.227 6. 168 5.730 5.359

0.01 5.250 5.164 5.259 5.258 6.2 11 5.610 6.503 6.310 6.265 5.962 6.503 6.310 5.822 5.424

0.05 5.480 5.293 5.384 5.405 6.659 5.886 6.840 6.486 6.85 1 6.295 6.840 6.486 6.186 5.640

0.10 5.670 5.448 5.548 5.5 14 7.063 6. 171 7. 140 6.617 7.324 6.49 1 7. 140 6.6 17 6.596 5.860

0.20 5.860 5.696 5.806 5.669 7.539 6.590 7.523 6.803 7.843 6.897 7.523 6.803 6.947 6. 197

0.40 6.270 6.029 6.133 5.888 8. 123 7.194 8. 104 7.066 8.334 7.303 8.104 7.066 7.448 6.680

0.60 6.530 6.240 6.341 6.057 8.548 7.671 8.608 7.268 8.774 7.866 8.608 7.268 7.835 7.082

0.80 6.771 6.4 11 6.509 6.198 8.92 1 8.068 9.034 7.438 9. 183 8.370 9.034 7.438 8. 185 7.427

1.00 6.871 6.526 6.657 6.323 9.27 1 8.429 9.429 7.588 9.574 8.77 1 9.429 7.588 8.482 7.729

2.00 7.520 7.130 7.227 6.813 10.607 9.752 11 .008 8. 176 11.00 1 9.973 11.008 8. 176 9.692 8.968

3.00 7.93 1 7.547 7.652 7.190 12. 108 10.532 12. 140 8.627 12.08 1 10.686 12. 140 8.627 10.593 9.860 4.00 8. 166 7.885 7.994 7.500 13.075 10.94 1 13.030 9.000 12.978 10.954 13.030 9.000 11.380 10.5 13

5.00 8.498 8.168 8.284 7.500 13.848 11 .206 13.743 9.000 13.700 10.998 13.743 9.000 12.043 10.880

Rectangle foot ing Rectangle footing BIL=0.33 Rectangle footing BIL=0.25 Rectangle foot ing B/L=0.2

DIB BIL=0.5

UB

0.00 6.022 0.01 6.249

0.05 6.503

0. 10 6.756

0.20 7. 11 6

0.40 7.574

0.60 7.993

0.80 8.377

1.00 8.724

2.00 10.055

3.00 11.076 4.00 11.878

5.00 12.545

30

SK AB LB UB SK AB LB UB SK AB

5.654 5.613 5.256 5.886 5.479 5.540 5.20 1 5.820 5.397 5.513

5.784 5.669 5.3 11 6.085 5.605 5.572 5.253 6.006 5.521 5.572

5.945 5.983 5.503 6.300 5.761 5.920 5.430 6.203 5.675 5.839

6.066 6. 154 5.697 6.533 5.878 6.122 5.6 14 6.4 13 5.790 6.136

6.236 6.630 5.997 6.867 6.043 6.523 5.895 6.73 1 5.953 6.529

6.477 7. 11 2 6.408 7.271 6.277 6.965 6.272 7. 11 3 6.183 6.950

6.662 7.417 6.740 7.608 6.456 7.239 6.567 7.412 6.359 7.209

6.8 18 7.709 7.030 7.936 6.607 7.485 6.7 17 7.705 6.508 7.42 1

6.956 7.994 7.297 8.240 6.74 1 7.726 7.048 7.976 6.639 7.6 19

7.495 9. 13 1 8.447 9.476 7.263 8.7 19 8. 109 9.086 7. 154 8.525

7.909 9.987 9.296 I 0.473 7.664 9.545 8.920 10.026 7.549 9.309

8.257 10.698 10.0 18 11 .242 8.002 10.259 9.594 10.769 7.882 9.961

8.565 11.347 10.464 11 .887 8.300 10.894 10.11 7 11.408 8.175 10.58 1

Note: AB is res11lts ofABAQUS in this st11dy

LB is lower bo1111d and VB is upper bound result of Salgado et al. (2004)

SK is the rernlts ofSkempton (1951) in equation (2)

Australian Geomechanics Vol 47 No 2 June 2012

LB UB SK

5.1 69 5.776 5.346

5.218 5.949 5.469

5.389 6.126 5.62 1

5.565 6.32 1 5.735

5.836 6.637 5.896

6. 190 7.003 6. 124

6.465 7.299 6.299

6.695 7.570 6.446

6.904 7.8 19 6.576

7.860 8.835 7.086

8.607 9.696 7.477

9.249 I 0.403 7.807

9.796 11.030 8.097

TWO· AND THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS

~ c.

lfu

c.

NGUYEN & MERIFIELD

9 ,0

8.5 Strip

8.0

7.5

7.0

6,5

6.0

55

5.0 0

D/8 14

13 Circular

12

11 -10

9

0 2 D/8 3

Rectangular B/L=0.33 11

/ /

10 /

/ /

/

8 // / /,,, /

/ /

11/~~--

6 '/

D/8


10

/

0

/

/ /

/

D/8

-

14

13 Square

12 A

"' 11 /'. / --/.

<Ju 10 /, -~ / c / u

9 /

/.-7 I 6

0 2 3 D/8

14

13 Rectangular B/L=0.5

12

11 /

/'

q., 10 /' /

/ c / • 9 / /,

fi / ,If/

n_,? _..

5 0 2

D/8 12


10 / /'

/ / /

9 / / '111 /

/ / c. / /

8 / / /

/, / 7 /,fi ./. ...-"//

0

D/8

Abaqus Salgado LB Salgado UB Skempton 1951

Figure 3: Normalised bearing capacity of embedded footin gs


5

-

31

TWO- ANO THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS

32

NGUYEN & MERIFIELD

7

~ - " I

;; v

/

f/ v I D/ B = 0, 0.01 ,0.05, 0. l,0.2, 0.4,0.6 I

II! I

1;

6

5

4 ~ Be,,

3

2

8

6

~ Be,,

4

2

L--- -~

/ ,_.

~ 7 f I I ID/ B = 0.8, 1,2,3,4,5

0 0 40 80

0£11 /Be,, 120

0 0 100 200 300 400 500

8

~ 6

Be,,

4

2

0

10

8

6 ~ Be,,

4

2

0

0£ 11 / Be,,

Figure 4 : Load-displacement behaviour of embedded strip footings

--- 11 14

1--- I

A ~ I

I/ J

I

12

10

~ I

~ v---- ........ v f v ,,.,,,.- I r

0

/

f_ {(_ IY

0

~ ......._

ID/ B = 0, 0.0 l,0.05,0. l, 0.2, 0.4,0.6 I ~ Be,, 8

6

1....--

~ 7

v ID/ B = 0.8, 1,2,3,4,5 I

4

2

20 40 60 80 100 120 0

0 30 60 90 120 150 180 210 o~/~ o~/~

Figure 5: Load-displacement behaviour of embedded square footi ngs

14 /

i..-- 11

tr-:: I /

I

I I

.......

1.....---L---' L---

v L--i-- I/ ~ r:-: L--

~·~ 7

v 7

12

10

I D/ B =0,0.0 l,0.05,0. 1,0.2,0.4,0.6, 0.8 A: r D/ B - 1,2, 3, 4, 5 i ~ 8

Be,,

60 120 180 240 300

6

4

2

0 0 80 160

oE,, /Be,, oE,,/Be,, Figure 6: Load-disp lacement behaviour of embedded c ircular foot ings


1 I

240 320


DIB=0.01 D/B=0.05 DIB=O. I

D/8=0.2 D/B=0.4 DIB=0.6

D/8=1.0 D/8=2.0 D/8=4.0

Figure 7: Typical contours of total displacement for strip foot ings (not to scale).


TWO· AND THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS NGUYEN & MERIFIELD

DIB=0.01 D/8=0.01

..

' . ' DIB=0.4 DIB=0.4

. ; . ,· ,·, .

D/8=2.0 D/8=5.0

Figure 8: Typical contours and vectors of tota l displacement for square footings (not to sca le)



I ,

... . . ,

, . , /

DIB=0.2 DIB=0.6

j • •

I • .. . . . . .

. .. . . . .

DIB=l.O D/8=3.0

Figure 9: Typical vectors of tota l displacement for circu lar footings (not to scale).

Austral ian Geomechanics Vol 47 No 2 June 2012 35


N (sq11are-s111fi1ce-foo1i11g ] 5

_59

F. = c = -- = 1.1 4 <S N( strip-swface-footing] 5 .24

c

(8)

where:

NJsqnnre-swface-footing] = 5.95 is the bearing capacity factor of a square surface footing, from Table 1;

NJsirip-swface-footing] = 5.24 is the bearing capacity factor of a strip surface footing, from Table I .

For a circular footi ng, the shape factor is determined as: N(circ11/ar-s11rface-fooling]

6 05 F = c =-· -= 1.155

cs N (s1rip -s111fnce - footing] 5.24 c

(9)

\\lllel·e Nicircular- s111fi1ce-footing ] __ 6.05 · l b · · f: f · I f: c · c T bl I 1s t 1e earmg capacity actor o a c1rcu ar sur ace 1ootmg, 1rom a e .

For a rectangular footing, the shape factor is determined as: N(rec1<111gnlar-s111fi1ce- footing J

F = _c ________ _

cs N (s1rip-s111face-footing) c

(I 0)

where NJrectongular-.wiface-footing] is the bearing capacity factor of a rectangular surface footi ng (Table I).

Fcs = 5

·73

= 1.094 for BI l = 0.5; F,5

= 5

·613

= 1.071 for B I L = 0.33; Fcs = 5

·54

= 1.057 for BIL = 0.25; 5.24 ( 5.24 5.24

Fe,· = 5

·513

= 1.052 for BIL = 0.2 ; . 5.24

All of these values of shape factor Fcs are summarized in Table 2.

Table 2: Shape factor for various footi ng geometries

Footing types BIL Fcs

Circular foot ings 1 1.1 55

Square footings 1 1. 136

0.50 1.094

Rectangular 0.33 1.071

footings 0.25 1.057

0.20 1.052

The shape factor F,_ ... increases as the ratio of relative dimensions of footing B I l rises from 0 (for strip footing) to I

(for square and circular footing). For 0 ~ BIL~ 1 , the values of the shape factor Fcs (in Table 2) can be approximated by the following equation:

2

Fcs = 1+0.14( ~ y for 0 ~ B I l ~ 1 ( II )

Equation (11) provides estimates of the shape factor to within ± 1% of the finite element solution (ABAQUS) and therefore can be used with confidence to solve practical design problems. The shape factors calculated from ABAQUS are presented in Figure I 0, along with those estimated using equation ( 11 ), equation (2) of Meyerho f ( 195 1 ), and equation (7) of Salgado et al. (2004).

This actual shape factor F,, will be used when calculating the depth factor F,.., . Therefore equation ( 1 1) can be used to

determine shape factors F,, for both surface footi ngs and embedded footings.



1.25 -r;:::=:=============::;-------i

1.20

1.15

1.10

1.05

• Abaqus .ii. Salgado

Meyerhof 1951 Equation 11

- Salgado Equalion 7

1.00 +----~--~--~---~--! 0.0 0.2 0.4 0.6 0.8 1.0

Bi l

Figure 10: Comparison ofthe shape factor F,,.

4.2 DEPTH FACTOR FOR STRIP FOOTINGS

D 8

2

3

4

' "

Abaqus - Salgado Equation (6)

Skempton Equation(4)

'

s~-~-~----~-~--~-_,_.-~

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Figure 11: Comparison of depth factors F,., for strip footings

The depth factor Fed of a strip footing can be calculated using values in Table I in conjunction with equation (I). The

depth factors F;_.d are shown in Table 3. F;_.d increases gradually with the ratio of D I B. In this study, the depth factors

Fed are determined as equations of two variables of DI B and BIL and calculated by dividing the bearing capacity

factor Ne al various DI B values by N,, for surface footing. For strip footing, B I L =0, the depth factors F;_.d depend

on D I B only. The depth factors Fed results of Salgado et al. (2004) ( F;_.,, = I+ 0.27 J%) and of Skempton (195 1)

( F,." = I+ Jo.053 ~) , and of ABAQUS are shown in Figure 11 .

As can be seen from Figure I I, at the same embedment depths D I B , the depth factor of this study is very close to that of Salgado et al. ( 1964) equation (higher than -13%) but slightly higher than result of Skempton ( 1951 ). These differences are larger as footing depths are deeper.

4.3 DEPTH FACTOR FOR SQUARE FOOTINGS

For embedded square footings the depth factors Fed are calculated by dividing the bearing capacity factors Nc of

embedded square footing at the various D I B by Nc of square surface footing. The depth factors Fed from this study

are presented in Table 3 and shown in Figure 12.

For embedded square, circular and rectangular footings, the depth factors F;_.d from this study are different from that of

Skempton ( 195 1) and Salgado et al. (2004). The depth factors F;_." of Skempton ( 1951 ) and Salgado et al. (2004) are the

same for any shape footing, while depth factors Fed in this study change from footi ng shape to footing shape.



2.4

2.2

2.0

1.B

Fed 1.6

1.4

1.2

1.0

0

Square Circular B/L=0.5 B/L=0.33 B/L=0.25 B/L=0.2 ~

Strip /'.'.: '--------~ .......-:: /

/.

2

D I B

-- ....- --....... --

3 4

Figure 12: Depth factors for embedded footings from ABAQUS

4.4 DEPTH FACTOR FOR CIRCULAR FOOTINGS

5

The bearing capacity factor Ne of this study was found to be almost identical for square and circular footings for all

cases. In the case of D I B ~ 3.0 , the bearing capacity factor for square footings was aroun&lfo above that of a circular footing. But in the cases of D I B > 3.0 the bearing capacity factor for circular footings was around 0.5% above that of a square footing. This indicates some level of mesh dependency.

For embedded circular footings the depth factors F;_." are also calculated by dividing the bearing capacity factors Ne of

embedded circular footi ngs a t the various D I B by Ne of the surface circular footing. The depth factors Fed for

embedded circular footing are presented in Figure 12 and in Table 3.

Table 3: Depth factors Fed for embedded footings from ABAQUS

Strip foot ing Square footing Circular footing Rectangular footing

DIB BIL=O B/L=l BIL= l BIL=0.5 BIL=0.33 BIL=0.25 BIL=0.2 0.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0 .010 1.002 1.044 1.036 1.0 16 1.010 l .006 1.0 11

0 .050 1.046 1.1 19 1. 133 1.080 1.066 1.069 1.059

0 .1 00 1.082 1.187 1.2 11 1.1 5 1 1.096 1.105 1. 11 3

0.200 l.11 8 1.267 1.297 1.2 12 1.1 8 1 1.177 1.184

0.400 1.197 1.365 1.378 1.300 1.267 1.257 1.26 1

0.600 1.246 1.437 1.45 1 1.367 1.32 1 1.307 1.308

0.800 1.292 1.499 1.5 18 1.428 1.373 1.35 1 1.346

1.000 1.3 11 1.558 1.583 1.480 1.424 1.395 1.382

2.000 1.435 1.783 1.8 19 1.691 1.627 1.574 1.546

3.000 1.5 14 2.035 1.998 1.849 1.779 1.723 1.689

4.000 1.558 2. 197 2. 146 1.986 1.906 1.852 1.807

5.000 1.622 2.327 2.265 2. 102 2.022 1.966 1.919

38 Aus tralian Geomechanics Vol 47 No 2 June 201 2

TWO· AND THREE-DIMENSIONAL UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS NGUYEN & MERIFIELD

4.5 DEPTH FACTOR FOR RECTANGULAR FOOTINGS

In general, the results of this study are similar to those obtained from the limit analysis of Salgado et al. ( 1964) tending to be closer to the upper bound than the lower bound results. Bearing capacities of embedded rectangular footing increase as ratios D I B and B I l increases.

In this study, for embedded rectangular footings, depth factors Fed are also calculated by dividing the bearing capacity

factors Ne of embedded rectangular footings at the various D I B by Ne of a surface rectangular footing. The depth

factors Fed for embedded rectangular footi ngs calculated from results of this study are presented in Table 3 and shown

in Figure 12. The depth factors Fed contained both parameters DI B and BI l. For practical uses, the depth factors

Fed Table 3 and Figure 1 can be approximated by following equation of two parameters D I B and B I L.

Fed = l +J0.48- 0.4e- Bfl ~ (12)

Equation (8) provides estimates of the depth factors to within ±3.5% of the finite element solutions (ABAQUS) and therefore can be used with confidence to solve practical design problems. Figure 13 presents graphica lly the depth factors from ABAQUS and from the est imated equation (8).

DIB

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

6.0 1.0

" - =r .... ~x .~ .... M-,x ,. ~ ~ ·~· ~ \ ~'' ' \ • ·', • ', Circle, Square

\ ~'~· ', / \X . •' .Aaf-\ \' . \ -.', ' · ' ' \ \ ', ' f .. ~ \

\ \ \ . . \ \ . \

' '"*• ' ' ' ' ~ •.• ,. ·~

/ \ \\ '. +-,•

Strip >\ ·• •• \+ BIL= 0.2 0.25 0.33 0.5

1.2 1.4 1.6 1.8 2.0

' '

2.2

' -¥ ...

X - - - Strip Footing + - · - · Rectangular B/L=0.5

A Square Footing • - - - - - Rectangular B/L=0.33

+ Circular Footing • - • • - Rectangular B/L=0.25

- - - - • Circular, Square • ----Rectangular B/L=0.2

Note that: Marked points are ABAQUS results, the curves are estimated using Equation 12

2.4

Figure 13: Depth factors for embedded footings from ABAQUS (marked points) and equation (8) (curves)



5 CONCLUSION

The ultimate undrained bearing capacity of strip, square, circular and rectangular embedded footings has been investigated using the finite element method. The interactions between soil and the bottom of the footings are treated as being fully rough. The results obtained have been presented in terms of a normalised bearing capacity factors Ne in

both graphical and tabular form and the shape factors F; .. ,, and the depth factors ~'(/ to faci litate their use in solving practical design problems.

The following conclusions can be made based on the finite element results:

• In general , undrained bearing capacity of embedded foot ing compares well to the previously reported numerical limit analysis solutions. The finite element results are higher than that of Skempton ( 195 1 ).

• For strip footings, the bearing capacity factors are higher than that of both the upper and lower bound solutions of Salgado el al. (2004). This is somewhat unexpected and may suggest a certain level of mesh dependency in the FE results. For square and circular footing cases, the results are very close to the upper bound li mit analysis, while for rectangular footings, the bearing capacity factors lie between upper and lower bound solutions.

• For cases where D I BS 0.2 for strip footings, DI B $ 0.4 for rectangular footing and D I B S 0. 1 for square and rectangular footings the fai lure mechanism is similar to that of a surface foot ing. However when the DI B is larger, the fa ilure mechanism is locali sed near the footing.

• Bearing capacity continues to increase for D I B ~ 4, and this is different from the conclusion of Skempton ( 195 1) where the bearing capacity factor reaches a constant value in the case of D I B ~ 4. The inclusion of soil weight is likely to limit the growth in bearing capacity indefinitely.

• The present work also gives the shape and depth factors in the form of an equation and tables to aid in their use.

6 REFERENCES Drucker, D. C., Greenberg, H. J. & Prager, W. ( 195 1) The safety factor of an elastieplastic body in plane strain. J.

Appl. Mech. Trans. ASME 73, 37 1-378. Drucker, D. C. and W. Prager ( 1952). "Soil mechanics and plastic analysis or li mit design." Quarterly of Applied

Mathematics I 0(2): 157-165. Edwards, D. H., Zdravkovic, L. & Potts, D. M. (2005), "Depth factors for undrained bearing capacity". Geotechnique,

55, No. I 0, 755-758. Hansen, B. ( 1969), "Bearing Capac ity of Shallow Strip Footings in Clay". Proceedings of the 7th International

Conference on So il Mechanics and Foundation Engineering, Vol. 3, pp. I 07- 11 3. Hi ll , R., ( 195 1) The mathematical Theo1y of Plasticity, Oxford University Express. Hu, Y. and Randolph, M. F. (1998), "Deep penetration of shallow foundations on noomogeneous soil", Soil and

Foundation, Vol. 38, No. I, pp. 241 - 246. Meyerhof, G.G. ( 1951 ), 'The ultimate bearing capacity of fo undations", Geotechnique, 2: 30 1- 332. Salgado, R., Lyamin, A. V., Sloan, S. W. & Yu, H. S. (2004) "Two- and three-dimensional bearing capacity of

foundations in clay". Geotechnique, 54(5): 297-306. Skempton, A. W. ( 195 1 ). "The bearing capacity of clays". Proc. Building Research Cong. London, I, 180- 189. Terzaghi , K. ( 1943). "Theoretical soil mechanics". John Wiley, New York.


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