11
Compu Research Paper The influence of sup cap Faculty of Science and (Rece Abstract. In 1920 Prandtl p centric loaded strip footing this solution for a surroun weight. Terzaghi (1943) wro components for the cohesion tent the currently used com superposition is correct, bec not the same. A number of f which is luckily not too large tended the equation of Terza inclination of the load. For common practice is to reduce cides with that of the load, w tive area, is completely negl foundation area is an accur due to eccentric loading. A n the case. Keywords: Bearing capacity; Fo 1. Introduction In 1920, Ludwig Prandtl p soil under a load, p, causing neath. The strength of the ha cohesion, c. The failure mec Prandtl in Figure 1) has been by Jumikis (1956), Selig and Prandtl subdivided the slid 1 Corresponding author. [email protected] utations and Materials in Civil Engineering 2016, Volume 1, No. 3, 121-131 Golden Spiral Publishing Inc. http://gspiralpublishing.com/index.php/cmce perposition and eccentric loading o pacity of shallow foundations Stefan Van Baars 1 d Technology, University of Luxembourg, Luxembourg, Lux eived 12 September 2016; accepted 25 October 2016) published an analytical solution for the bearin on a weightless in-finite half-space. Reissner nding surcharge and Keverling Buisman (19 ote this as a superposition of three separate b n, surcharge and soil-weight. The first questio mponents are correct. The second question is to cause the failure mechanisms for these three finite element calculations show that there is e and leads to predictions on the safe side. Mey aghi with correction factors for the shape of the eccentric loading however, there are no correct e the contact area of the foundation such that i which means that, the area of the foundation o lected. Therefore the third question is, if this rate method to describe the reduction of the b number of finite element calculations show tha ootings; Foundations; Finite Element Modelling. published an analytical solution for the be kinematic failure of the weightless infinite alf-space is given by the angle of internal chanism proposed by Prandtl (see the o n validated by laboratory tests, for examp McKee (1961), and Muhs and Weiß (1972). ding soil part into three zones (Figure 2): on the bearing xembourg ng capacity of a (1924) extended 940) for the soil bearing capacity on is to what ex- what extent the components are indeed an error, yerhof (1953) ex- e footing and the tion factors. The its centroid coin- outside the effec- reduction of the bearing capacity at this is indeed earing capacity of a e half-space under- friction, , and the original drawing of ple those performed .

The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

Embed Size (px)

Citation preview

Page 1: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

Computation

Research Paper

The influence of superposition and eccentric loading on capaci

Faculty of Science and Technology (Received

Abstract. In 1920 Prandtl published an analytical solution for the bearing capacity of a centric loaded strip footing on a weightless inthis solution for a surrounding surcharge and Keverling Buismanweight. Terzaghi (1943) wrote this components for the cohesion, surcharge and soiltent the currently used components are correct. The second question is to what extent superposition is correct, because the failure mechanisms for these three not the same. A number of finite element calculwhich is luckily not too large and tended the equation of Terzaghiinclination of the load. For eccentric loading however, there are no correction factors. The common practice is to reduce the contact area of the foundation such that its centroid coicides with that of the load, which means that, the area of the foundation outside the effetive area, is completely neglected. Thefoundation area is an accurate method to describe the reduction of the bearing capacity due to eccentric loading. A number of finite element calculations show that this isthe case.

Keywords: Bearing capacity; Footings; Foundations; Finite Element Modelling.

1. Introduction

In 1920, Ludwig Prandtl published an analytical solution for the bearing capacity of a soil under a load, p, causing kinematic failure of the neath. The strength of the halfcohesion, c. The failure mechanism proposed by Prandtl (see the original drawing of Prandtl in Figure 1) has been validated by by Jumikis (1956), Selig and McKee (1961), and Muhs and Weiß (1972)

Prandtl subdivided the sliding soil part into three zones (Fig

1Corresponding author. [email protected]

Computations and Materials in Civil Engineering 2016, Volume 1, No. 3, 121-131

Golden Spiral Publishing Inc.

http://gspiralpublishing.com/index.php/cmce

influence of superposition and eccentric loading on capacity of shallow foundations

Stefan Van Baars 1

Faculty of Science and Technology, University of Luxembourg, Luxembourg, Luxembourg(Received 12 September 2016; accepted 25 October 2016)

In 1920 Prandtl published an analytical solution for the bearing capacity of a strip footing on a weightless in-finite half-space. Reissner

this solution for a surrounding surcharge and Keverling Buisman (1940)wrote this as a superposition of three separate bearing capacity

for the cohesion, surcharge and soil-weight. The first question is to what ethe currently used components are correct. The second question is to what extent

, because the failure mechanisms for these three not the same. A number of finite element calculations show that there is indeed an error, which is luckily not too large and leads to predictions on the safe side. Meyerhof

of Terzaghi with correction factors for the shape of the footing and the eccentric loading however, there are no correction factors. The

common practice is to reduce the contact area of the foundation such that its centroid coicides with that of the load, which means that, the area of the foundation outside the effe

, is completely neglected. Therefore the third question is, if this reduction of the foundation area is an accurate method to describe the reduction of the bearing capacity due to eccentric loading. A number of finite element calculations show that this is

Bearing capacity; Footings; Foundations; Finite Element Modelling.

In 1920, Ludwig Prandtl published an analytical solution for the bearing capacity of a , causing kinematic failure of the weightless infinite half

neath. The strength of the half-space is given by the angle of internal friction, . The failure mechanism proposed by Prandtl (see the original drawing of

1) has been validated by laboratory tests, for example those performed by Jumikis (1956), Selig and McKee (1961), and Muhs and Weiß (1972).

Prandtl subdivided the sliding soil part into three zones (Figure 2):

influence of superposition and eccentric loading on the bearing

Luxembourg

In 1920 Prandtl published an analytical solution for the bearing capacity of a (1924) extended

(1940) for the soil three separate bearing capacity

e first question is to what ex-the currently used components are correct. The second question is to what extent the

, because the failure mechanisms for these three components are ations show that there is indeed an error,

Meyerhof (1953) ex-with correction factors for the shape of the footing and the

eccentric loading however, there are no correction factors. The common practice is to reduce the contact area of the foundation such that its centroid coin-cides with that of the load, which means that, the area of the foundation outside the effec-

question is, if this reduction of the foundation area is an accurate method to describe the reduction of the bearing capacity due to eccentric loading. A number of finite element calculations show that this is indeed

In 1920, Ludwig Prandtl published an analytical solution for the bearing capacity of a weightless infinite half-space under-

space is given by the angle of internal friction, , and the . The failure mechanism proposed by Prandtl (see the original drawing of

, for example those performed .

Page 2: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars122

1. Zone 1: A triangular zone below the strip load.

ground surface, the directions of the principal stresses are horizontal and vertical;

the largest principal stress is in the vertical direction.

2. Zone 2: A wedge with the shape of a logarithmic spiral, in which the princi

stresses rotate through

equals the angle of internal friction;

Zone 1 and Zone 3.

3. Zone 3: A triangular zone adjacent to the strip load. Since there is no friction on the

surface of the ground, the directions of principal stress are horizontal and vertical;

the largest principal stress is in the horizontal direction.

Fig. 1. The Prandtl-wedge failure mec

The failure mechanism does not have to be symmetrical, with sliding in two directions,

as proposed by Prandtl in Figdirection only, as shown in Fig

Fig. 2. The Prandtl

The solution of Prandtl was extended by Hans J. Reissner

charge, q, and was based on the same failure mechanism. Albert S. Keverling Buisman (1940) and Karl Terzaghi (1943) extendweight, . It was Terzaghi (1943) who wrote the superposition of the three separate bearing capacity

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

Zone 1: A triangular zone below the strip load. Since there is no friction on the

ground surface, the directions of the principal stresses are horizontal and vertical;

the largest principal stress is in the vertical direction.

Zone 2: A wedge with the shape of a logarithmic spiral, in which the princi

stresses rotate through 90 , from Zone 1 to Zone 3. The pitch of the sliding surface

equals the angle of internal friction; , creating a smooth transition between

zone adjacent to the strip load. Since there is no friction on the

surface of the ground, the directions of principal stress are horizontal and vertical;

the largest principal stress is in the horizontal direction.

wedge failure mechanism (Original drawing of Prandtl, 1920).

The failure mechanism does not have to be symmetrical, with sliding in two directions, as proposed by Prandtl in Figure 1, but can also be unsymmetrical, with sliding in one

Figure 2.

Fig. 2. The Prandtl-wedge with three zones.

The solution of Prandtl was extended by Hans J. Reissner (1924) for , and was based on the same failure mechanism. Albert S. Keverling Buisman

(1940) and Karl Terzaghi (1943) extended the Prandtl-Reissner formula for the soil . It was Terzaghi (1943) who wrote the equation for the bearing capacity

the three separate bearing capacity components for the cohesion, su

Since there is no friction on the

ground surface, the directions of the principal stresses are horizontal and vertical;

Zone 2: A wedge with the shape of a logarithmic spiral, in which the principal

, from Zone 1 to Zone 3. The pitch of the sliding surface

, creating a smooth transition between

zone adjacent to the strip load. Since there is no friction on the

surface of the ground, the directions of principal stress are horizontal and vertical;

hanism (Original drawing of Prandtl, 1920).

The failure mechanism does not have to be symmetrical, with sliding in two directions, 1, but can also be unsymmetrical, with sliding in one

a surrounding sur-, and was based on the same failure mechanism. Albert S. Keverling Buisman

Reissner formula for the soil bearing capacity as a for the cohesion, sur-

Page 3: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131 123

charge and soil-weight. George G. Meyerhof (1953) was the first to propose equations for inclined loads. He was also the first in 1963 to write the formula for the (vertical) bearing capacity pv with dimensionless bearing capacity factors (N), inclination factors (i) and shape factors (s), for the three independent bearing components; cohesion (c), surcharge (q) and soil-weight (), in a way it was adopted by Jørgen A. Brinch Hansen (1970) and is still used nowadays:

1

2.v c c c q q qp i s cN i s qN i s BN

(1)

where c is the cohesion (kPa), q is the surcharge (kPa), is the effective soil weight (kN/m3), and B is the width of the strip load (m).

The purpose of this study is threefold: to check the correctness of the currently used

bearing capacity factors for non-dilatant soil, to check the influence of the superposition on these factors, and to check the influence of eccentric loading on these factors.

2. The correctness of the bearing capacity factors for non-dilatant soil

2.1. Theory

The cohesion bearing capacity factor follows from the solution of Prandtl (1920):

tan 1 cotc pN K e with: 1 sin

1 sinpK

(2)

where is the friction angle of the soil () and Kp is the passive earth pressure coeffi-cient (-).

The surcharge bearing capacity factor follows from the solution of Reissner (1924):

2

tan3

1

q p p

rN K K e

r

with: 1 sin

1 sinpK

(3)

Keverling Buisman (1940), Terzaghi (1943), Caquot and Kérisel (1953), Meyerhof (1951; 1953; 1963; 1965), Brinch Hansen (1970), Vesic (1973, 1975), and Chen (1975) subse-quently proposed different equations for the soil-weight bearing capacity factor N. There-fore the following equations for the soil-weight bearing capacity factor can be found in the literature:

tan

tan

tan

tan

1 tan 1.4 (Meyerhof, 1963),

1.5 1 tan (Brinch Hansen, 1970),

2 1 tan (Vesic, 1973),

2 1 tan (Chen, 1975).

p

p

p

p

N K e

N K e

N K e

N K e

(4)

The equation from Brinch Hansen is, as he writes, “…based on calculations first from Lundgren and Mortensen (1953) and later from Odgaard and N. H. Christensen”. The equation of Vesic is almost identical to the solution of Caquot and Kérisel (1953) because it is, as he writes, based on “…the numerical results of an analysis made by them”.

Page 4: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars124

2.2. Results

In order to check the correctness of the currently used bearing capacity factors for nondilatant soil, both displacement and stress controlled finite element model calculations have been made with Plaxis 2D, with a Mohrwith 15-node elements.

Figure 3 shows these results, together with the analytical solutions of the bearing cpacity coefficients of Prandtl and Reissner.

Fig. 3. Bearing capacity factors: Analytical versus FEM.

2.3. Discussion

Figure 3 shows a clear difference in bearing capacityical solutions, especially for higher friction angles. The reason for this is that tof Equations 2, 3 and 4 are all based on extreme dilatant (associated) soil, for dilatancy angle . Loukidis et al

lations, that non-dilatant (i.e. nonpacity factors) than extreme dilatant soil, pattern.

The difference between the (extreme dilatant or associated) analytical solutions and

the (non-dilatant or non-associated) numerical results has been explained by Knudsen and Mortensen (2016): The (larger pitch) of the PrandtlTherefore, the analytical formulas are only kinematically admissible for associated flow. The problem of associated soil is that such a high dilatancy angle is by far unrealistic for natural soils. This means as well that calculating the bearing capacity factor based on the analytical solutions is for higher friction angles also unrealistic, so it is better to use

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

In order to check the correctness of the currently used bearing capacity factors for nondilatant soil, both displacement and stress controlled finite element model calculations have been made with Plaxis 2D, with a Mohr-Coulomb soil model and a fine wide mesh

Figure 3 shows these results, together with the analytical solutions of the bearing cpacity coefficients of Prandtl and Reissner.

Fig. 3. Bearing capacity factors: Analytical versus FEM.

shows a clear difference in bearing capacity between the numerical and analy, especially for higher friction angles. The reason for this is that t

s 2, 3 and 4 are all based on extreme dilatant (associated) soil, for Loukidis et al. (2008) already noticed, from their numerical calc

dilatant (i.e. non-associated) soil is 15% - 30% weakerthan extreme dilatant soil, and it has also a rougher (less steady) failure

The difference between the (extreme dilatant or associated) analytical solutions and associated) numerical results has been explained by Knudsen

higher the friction angle, the wider the logarithmic spiral (larger pitch) of the Prandtl-wedge and the more the stresses reduce during failure. Therefore, the analytical formulas are only kinematically admissible for associated flow.

iated soil is that such a high dilatancy angle is by far unrealistic for natural soils. This means as well that calculating the bearing capacity factor based on the analytical solutions is for higher friction angles also unrealistic, so it is better to use

In order to check the correctness of the currently used bearing capacity factors for non-dilatant soil, both displacement and stress controlled finite element model calculations

and a fine wide mesh

Figure 3 shows these results, together with the analytical solutions of the bearing ca-

between the numerical and analyt-, especially for higher friction angles. The reason for this is that the solutions

s 2, 3 and 4 are all based on extreme dilatant (associated) soil, for which the (2008) already noticed, from their numerical calcu-

30% weaker (lower bearing ca-and it has also a rougher (less steady) failure

The difference between the (extreme dilatant or associated) analytical solutions and associated) numerical results has been explained by Knudsen

higher the friction angle, the wider the logarithmic spiral wedge and the more the stresses reduce during failure.

Therefore, the analytical formulas are only kinematically admissible for associated flow. iated soil is that such a high dilatancy angle is by far unrealistic for

natural soils. This means as well that calculating the bearing capacity factor based on the analytical solutions is for higher friction angles also unrealistic, so it is better to use nu-

Page 5: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars

merically derived bearing capacity factorssented by Van Baars (2015, 2016):

2 tan

tan

tan

cos e

1 cot

2 1 tan (Rough plate: Vesic)

4 tan 1 (Smooth plate)

q p

c q

p

N K

N N

N K e

N e

3. The influence of superposition

3.1. Theory

Terzaghi (1943) assumed that the failure mechanism, so that superposition

lowed 1

2c qp cN qN BN

3.2. Resuls

In order to check the influence of tions have been made with the finite element model Plaxisused with 15-node elements. The soil has been modelledand. The plate below the load is infinite stiff and rough.made for the three bearing components independentlweight) (Figure 4).

Fig. 4. Failure of footing for the independent components.

Above: Incremental displacements at failure Below: Relative shear stress at failure

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

merically derived bearing capacity factors for non-dilatant soil, such as sented by Van Baars (2015, 2016):

2 tan

tan

tan

cos e

1 cot

2 1 tan (Rough plate: Vesic)

4 tan 1 (Smooth plate)

q pN K

N K e

N e

superposition on the bearing capacity factors

Terzaghi (1943) assumed that the three bearing capacity components failure mechanism, so that superposition of the three bearing capacity components

p cN qN BN .

the influence of this superposition, displacement tions have been made with the finite element model Plaxis 2D. A large mesh has been

node elements. The soil has been modelled with a Mohr-Coulomb soil model The plate below the load is infinite stiff and rough. First three calculations

made for the three bearing components independently (cohesion, surcharge and soil

Failure of footing for the independent components.

Above: Incremental displacements at failure Below: Relative shear stress at failure

125

, such as for example pre-

(5)

components have the same ng capacity components is al-

controlled calcula-. A large mesh has been

Coulomb soil model First three calculations were

y (cohesion, surcharge and soil-

Failure of footing for the independent components.

Above: Incremental displacements at failure Below: Relative shear stress at failure.

Page 6: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars126

The friction angle ( ) is taken

m. The bearing capacities of the three independent components were found to be:

24.5; 14.6; 12.6c qN N N

The upper part of Figure 4the different failure mechanisms. The lower pictures show the relative shear stress at failure (size of the Mohr-circle divided by failure envelope), indicating the different plastic zones.

In order to indicate the error due to the used superpositionbeen made for a footing in which the three bearing components are combined with

equal weight, by using a cohesion, surcharge and soil weight of

(Figure 5).

Fig. 5. Failure of footing with combined components.

Above: Incremental displacements at failure Below: Relative shear stress at failure.

3.3. Discussion

The different failure patterns in Figure 4 proof that the use of superposition is not co

rect, because the failure mechanisms belonging to the soil

and the surcharge ( qN ), are all three different.

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

) is taken 30 for all cases, and the width of the strip load (

. The bearing capacities of the three independent components were found to be:

(Figure 3).

4 shows the incremental displacements at failure, indicatinthe different failure mechanisms. The lower pictures show the relative shear stress at

circle divided by a maximum Mohr-circle touching the Coulomb failure envelope), indicating the different plastic zones.

the error due to the used superposition, anothermade for a footing in which the three bearing components are combined with

equal weight, by using a cohesion, surcharge and soil weight of c q B

Fig. 5. Failure of footing with combined components.

Above: Incremental displacements at failure Below: Relative shear stress at failure.

The different failure patterns in Figure 4 proof that the use of superposition is not co

the failure mechanisms belonging to the soil-weight ( N ) the cohesion (

), are all three different.

all cases, and the width of the strip load (B ) is 2

. The bearing capacities of the three independent components were found to be:

the incremental displacements at failure, indicating the different failure mechanisms. The lower pictures show the relative shear stress at

circle touching the Coulomb

nother calculation has made for a footing in which the three bearing components are combined with an

1

210 kPac q B

Above: Incremental displacements at failure Below: Relative shear stress at failure.

The different failure patterns in Figure 4 proof that the use of superposition is not cor-

N ) the cohesion ( cN )

Page 7: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars

The last calculation, in which the three bearing components

combined bearing capacity of

15.2% underestimation with the superposition ruleleads also to a prediction on the safmechanism for the combined loading is different from the for the three individual bearing capacity components.

4. The influence of eccentric loading on the bearing capacit

4.1. Theory

Until now, nobody proposed correction factors for eccentric loading. The common pra

tice is to use the equation of Terzaghi (

area of the foundation such that its centroid coincides witthat, the area of the foundation outside the effective area, (Figure 6).

The question is, if this reduction of the foundation area is an accurate method to dscribe the reduction of the bearing eral people have performed finite element calculations of tric loaded strip footings, but the results are not completebased their results only on the limit analysis

dependently Khitas et al. (2016) only found results for frictionless soils

found that the error of the simplified method, with the reduction of the foundalimited to 5%, in case the eccentricity is limited to:

Fig. 6. Effective or reduced foundation, due to eccentric loading

4.2. Results

In order to check this conclusion, the 7) has been extended. The plate below the loading has been extended with 0.5 m (Case II) and 1.0 m (Case III), in order to create eccentric loading, but without changing the effetive area, B’. Also in this case, the calculations have been made for the three bearponents independently, but also for a loaded footing in which the three bearing comp

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

, in which the three bearing components are combined,

combined bearing capacity of 61.0c qN N N , showing a (61.0 - 24.5

underestimation with the superposition rule, which is luckily not too large and prediction on the safe side. This error is caused by the fact that the failure

mechanism for the combined loading is different from the optimized failure mechanisms for the three individual bearing capacity components.

The influence of eccentric loading on the bearing capacity factors

Until now, nobody proposed correction factors for eccentric loading. The common pra

tice is to use the equation of Terzaghi (1

2c qp cN qN B N ) and to reduce the contact

area of the foundation such that its centroid coincides with that of the load, which means that, the area of the foundation outside the effective area, B’, is completely neglected

The question is, if this reduction of the foundation area is an accurate method to dscribe the reduction of the bearing capacity due to vertical eccentric loading, or not. Seeral people have performed finite element calculations of both vertical and inclined ecce

, but the results are not complete. For example on the limit analysis and Knudsen and Mortensen (2016) and i

(2016) only found results for frictionless soils

found that the error of the simplified method, with the reduction of the foundalimited to 5%, in case the eccentricity is limited to: e/B < 0.30.

Effective or reduced foundation, due to eccentric loading

In order to check this conclusion, the centric example of Paragraph 3 (Case I in Fig. The plate below the loading has been extended with 0.5 m (Case II)

and 1.0 m (Case III), in order to create eccentric loading, but without changing the effeAlso in this case, the calculations have been made for the three bear

ponents independently, but also for a loaded footing in which the three bearing comp

127

are combined, resulted in a

24.5 - 14.6 - 12.6) / 61.0 =

which is luckily not too large and is caused by the fact that the failure

failure mechanisms

y factors

Until now, nobody proposed correction factors for eccentric loading. The common prac-

) and to reduce the contact

h that of the load, which means , is completely neglected

The question is, if this reduction of the foundation area is an accurate method to de-capacity due to vertical eccentric loading, or not. Sev-

vertical and inclined eccen-For example Hjiaj et al. (2004)

Knudsen and Mortensen (2016) and in-

(2016) only found results for frictionless soils 0 . They

found that the error of the simplified method, with the reduction of the foundation area, is

Effective or reduced foundation, due to eccentric loading.

example of Paragraph 3 (Case I in Figure . The plate below the loading has been extended with 0.5 m (Case II)

and 1.0 m (Case III), in order to create eccentric loading, but without changing the effec-Also in this case, the calculations have been made for the three bearing com-

ponents independently, but also for a loaded footing in which the three bearing compo-

Page 8: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars128

nents are combined such that the three bearing components; cohesion, surcharge and soil

weight, are: 1

210 kPac q B

Fig. 7.

Figure 8 shows the numerically calculated bearing capacities

components, for the three load cases, both centric and eccentric

Fig. 8. Individual components: Normalized stress versus relative

Figure 9 shows the numerically calculated bearing capacity, for the combined bearing

components, for the three load cases, both centric and eccentricthe failure mechanism and the plastic zone for thescales and sizes of the plate).

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

nents are combined such that the three bearing components; cohesion, surcharge and soil

10 kPa .

. Eccentric loading: extension of the footing.

he numerically calculated bearing capacities, for the individual bearing for the three load cases, both centric and eccentric loading.

Individual components: Normalized stress versus relative displacement.

Figure 9 shows the numerically calculated bearing capacity, for the combined bearing for the three load cases, both centric and eccentric loading

the failure mechanism and the plastic zone for these three load cases

nents are combined such that the three bearing components; cohesion, surcharge and soil

or the individual bearing .

displacement.

Figure 9 shows the numerically calculated bearing capacity, for the combined bearing loading. Figure 10 shows

(note the different

Page 9: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars

Fig. 9. Combined components: Normalized stress versus relative displacement

Fig. 10. Influence of eccentric loaded footing with combined components.

Above: Total displacements at failure. Bel

4.3. Discussion

The numerically calculated bearing capacities, for the individual bearing components for eccentric loading, are found to be rather similar as for the centric loading (Figure 8). Also for the case with the combined components, the bearing capacities for eccentric loaing are almost the same as for centric loading (Figure 9). This means that the current method, which reduces the contact area of the foundation

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131

Combined components: Normalized stress versus relative displacement

Fig. 10. Influence of eccentric loaded footing with combined components.

Above: Total displacements at failure. Below: Relative shear stress at failure.

The numerically calculated bearing capacities, for the individual bearing components for eccentric loading, are found to be rather similar as for the centric loading (Figure 8).

ombined components, the bearing capacities for eccentric loaing are almost the same as for centric loading (Figure 9). This means that the current

reduces the contact area of the foundation for dealing with eccentric loads

129

Combined components: Normalized stress versus relative displacement.

Fig. 10. Influence of eccentric loaded footing with combined components.

ow: Relative shear stress at failure.

The numerically calculated bearing capacities, for the individual bearing components for eccentric loading, are found to be rather similar as for the centric loading (Figure 8).

ombined components, the bearing capacities for eccentric load-ing are almost the same as for centric loading (Figure 9). This means that the current

for dealing with eccentric loads,

Page 10: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131 130

is an accurate method. The reason for this is probably that the plastic zone is much larger than the failure mechanism (Figure 10), which means that a change in the failure mecha-nism (the additional rotation due to the eccentric loading as result of the extension of the plate), does not have to lead to another plastic zone. An extension of the plate has there-fore only a small consequence on the bearing capacity (maximum normalized vertical stress).

5. Conclusions

The currently used equations for the bearing capacity factors are, especially for higher friction angles, too high. Therefore more accurate equations have been given for non-dilatant soil.

Terzaghi wrote the equation for the bearing capacity of a shallow foundation as a su-perposition of three separate bearing capacity components, for the cohesion, surcharge and soil-weight. This is not correct because the three independent components have each their own failure mechanism. The Finite Element calculations with Plaxis 2D presented in this paper show that the used superposition results in an error, which is luckily not too large and leads also to predictions on the safe side.

Other Finite Element calculations show that the reduction of the foundation area is an accurate method to describe the reduction of the bearing capacity due to eccentric loading. The reason for this is probably that the plastic zone is much larger than the failure mech-anism, which means that an extension on one side of the foundation plate, without any loading on top, is causing a rotation of the plate, but not an increase in bearing capacity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

Brinch Hansen, J. A. (1970). Revised and extended formula for bearing capacity, Bulletin No 28, Danish Geotechnical Institute Copenhagen, 5-11.

Caquot, A., & Kerisel, J. (1953). Sur le terme de surface dans le calcul des fondations en milieu pulverulent. Proceedings of the third International Conference on Soil Mechanics and Founda-tion Engineering, Zurich, Switzerland.

Chen, W. F. (1975). Limit analysis and soil plasticity, Elsevier, Amsterdam, Netherlands.

Hjiaj, M., Lyamin, A. V., & Sloan, S. W. (2004). Bearing capacity of a cohesive-frictional soil under non-eccentric inclined loading. Computers and Geotechnics, 31, 491–516.

Jumikis, A. R. (1956). Rupture surfaces in sand under oblique loads, Journal of Soil Mechanics and Foundation Design, 82(3), 1155-1159

Keverling Buisman, A. S. (1940). Grondmechanica, Waltman, Delft, the Netherlands, 243

Khitas, N. E. H., Mellas, M., Benmeddour, D. & Mabrouki, A. (2016). Bearing capacity of strip foot-ings resting on purely cohesive soil subjected to eccentric and inclined loading, 4th Internation-al Conference on New Development in Soil Mech. and Geotechn. Eng., Nicosia, Cyprus.

Knudsen, B. S., & Mortensen, N. (2016). Bearing capacity comparison of results from FEM and DS/EN 1997-1 DK NA 2013. Northern Geotechnical Meeting. Reykjavik(Island).

Page 11: The influence of superposition and eccentric loading on ... Journal... · The influence of superposition and eccentric loading on ... (Vesic, 1973), 2 1 tan (Chen ... equation of

S. V. Baars/ Comp. Mat. Civil. Eng. 1(3) (2016) 121-131 131

Loukidis, D., Chakraborty, T., & Salgado, R. (2008). Bearing capacity of strip footings on purely frictional soil under eccentric and inclined loads, Can. Geotech. J. 45, 768–787

Lundgren, H., & Mortensen, K. (1953). Determination by the theory of Plasticity of the bearing ca-pacity of Continuous Footings on Sand, Proceedings of the third International Conference on Conf. Soil Mech, Zürich, Switzerland.

Meyerhof, G. G. (1951). The ultimate bearing capacity of foundations, Géotechnique, 2, 301-332

Meyerhof, G. G. (1953). The bearing capacity of foundations under eccentric and inclined loads. Proceedings of the third International Conference on Soil Mechanics Found. Eng, Zürich, Switzerland.

Meyerhof, G. G. (1963). Some recent research on the bearing capacity of foundations, Canadian Geotechnics Journal. 1(1), 16-26.

Meyerhof, G. G. (1965). Shallow foundations, Journal of the Soil Mechanics and Foundations Division ASCE, 91(2), 21-32.

Muhs, H., & Weiß, K. (1972). Versuche über die Standsicheheit flach gegründeter Einzelfundamente in nichtbindigem Boden, Mitteilungen der Deutschen Forschungsgesellschaft für Bodenmechanik (Degebo) an der Technischen Universität Berlin, Heft 28, 122

Prandtl, L. (1920). Über die Härte plastischer Körper. Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl., 74–85.

Reissner, H. (1924). Zum Erddruckproblem. Proceedings of the 1st Intest international Congress for Applied Mechanics, Delft, Netherlands.

Selig, E. T. and McKee, K. E. (1961). Static and Dynamic behavior of small footings, Journal of Soil Mechanics and Foundation Design, ASCE, 87(6), 29-50.

Terzaghi, K. (1943). Theoretical soil mechanics, John Wiley and Sons, New York, USA.

Van Baars, S. (2015). The Bearing Capacity of Footings on Cohesionless Soils, The Electronic Journal of Geotechnical Engineering, 20, 12945-12955.

Van Baars, S. (2016). Failure mechanisms and corresponding shape factors of shallow foundations, Proceedings of the 4th International Conference on New Development in Soil Mech. and Geotechn. Eng, Nicosia, Cypres.

Vesic, A. S. (1973). Analysis of ultimate loads of shallow foundations. Journal of Soil Mech. Found. Div, 99(1), 45–76.

Vesic, A. S. (1975). Bearing capacity of shallow foundations, H.F. Winterkorn, H.Y. Fang (Eds.), Foundation Engineering Handbook, Van Nostrand Reinhold, New York, USA, 121–147