Bearing Behaviour of Shallow Foundations Under Heavy Train

Master Thesis

Bearing Behaviour of Shallow

Foundations Under Heavy Train Load

Nelson García Iglesias

Luleå University of Technology

Department of Civil, Environmental and Natural Resources Engineering

2019

Acknowledgements

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Acknowledgements

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ACKNOWLEDGEMENTS

The present work is the end of the master studies in Mining engineering which I had the

opportunity to finish in the Luleå University of Technology Soil Mechanics department.

In first place, I would like to thank my supervisor and examiner, Professor Jan Laue who

has guided me with his knowledge and experience during the development of this thesis.

Finally, I would also like to thank Per Gunnvard, PhD student in the Soil Mechanics

department of Luleå University of Technology; who helped me with his expertise in the finite

element program Plaxis.

Luleå, June 2019

Nelson García Iglesias

Abstract

4

ABSTRACT

Foundations related to slopes are commonly found holding railway bridges. Traditionally

this type of foundations is assessed from a foundation’s static point of view and with very

conservative factors of safety. The bearing behaviour of this foundations is not well studied

yet, neither how cyclic loads affect it since these loads are normally neglected. The aim of this

thesis is to get a better understanding of the bearing behaviour of foundations related to

slopes under heavy train loads in static and dynamic conditions with the help of the finite

element program Plaxis.

The models have been built regarding a square foundation of 2,8 m side and 1,14 m of

embedment. This foundation will be placed in top and in the middle of a sand slope. 20º and

30º angle slopes are considered. The same models are made with the foundation’s

embedment doubled. To all the models, bearing capacity calculations are performed from an

analytical and numerical point of view. In addition, the models are cyclically loaded (along 100

cycles) in three ways: first for the foundations with the initial embedment, a load keeping a

ratio of 3,7 between maximum bearing capacity and applied load is used; for the cases with

double embedment a calculated load form a train passing through the bridge is placed; finally,

in the case of double embedment and foundation on top of a 20º slope the horizontal force

and momentum generated by the train braking is also added to the foundation.

From the ultimate bearing capacity calculations, it is seen that a local shear failure is

present in all cases. In situations with bigger slope angles and the foundation situated in the

middle of the slope a combination of local shear failure and slope failure is present. The main

parameter affecting the bearing capacity is the slope angle, bigger slope angles reduce

significantly the bearing capacity. Positions in the middle of the slope and bigger embedment

deals higher bearing capacities.

When taking a look to the cyclic models a punching like failure is developing in all cases,

combined with the start of a slope failure in cases with the foundation in the middle of a 30º

angle slope. In all cases the vertical displacement along the cycles presents an inflexion point

which separates an initial slower settlement speed from the bigger settlement speed after the

inflexion. This interesting point is related with the generation of an active area under the

foundation. The amplitude of the plastic deformation along the cycles follows the same patter

for all cases, it increases until the inflexion point for staying constant after it. The speed of

settlement is affected mainly by the slope angle.

Abstract

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TABLE OF CONTENTS

Acknowledgements ........................................................................................................................ 3

Abstract ........................................................................................................................................... 4

1 Introduction ............................................................................................................................ 7

2 Background and Motivation ................................................................................................... 8

2.1 Traditional Bearing Capacity Calculations ..................................................................... 8

2.2 Bearing Capacity Problem vs Slope Stability Problem. ................................................ 13

2.3 Strain Accumulation Under Cyclic Loading .................................................................. 13

2.4 Failure Behaviour of Horizontally Loaded Foundations in Slopes ............................... 14

2.5 Small Strain Hardening Soil Model............................................................................... 15

2.6 Phi-C Reduction Calculation ......................................................................................... 16

2.7 Aim and Objectives....................................................................................................... 17

3 Methodology ........................................................................................................................ 18

3.1 Ultimate Bearing Capacity: Analytical Methods .......................................................... 18

3.2 Plaxis Models Build ....................................................................................................... 20

3.3 Ultimate Bearing Capacity Plaxis.................................................................................. 22

3.4 Cyclic Loading Models .................................................................................................. 22

3.4.1 Calculation of Vertical Train Load ........................................................................ 22

3.4.2 Calculation of Horizontal Train Load .................................................................... 24

3.4.3 Constant Force Ratio Loading .............................................................................. 24

3.4.4 Real Force Loading ............................................................................................... 24

3.4.5 Plaxis Modelling for Cyclic Loading ...................................................................... 25

4 Results and Analysis ............................................................................................................. 26

4.1 Ultimate Bearing Capacity ............................................................................................ 26

4.1.1 Input ..................................................................................................................... 26

4.1.2 Calculations .......................................................................................................... 26

4.1.3 Output .................................................................................................................. 27

4.2 Cyclic Loading: Constant Vertical Force Ratio .............................................................. 49

4.2.1 Input ..................................................................................................................... 49

4.2.2 Output .................................................................................................................. 49

4.3 Cyclic Load: Vertical Real Load ..................................................................................... 63

4.3.1 Input ..................................................................................................................... 63

4.3.2 Output .................................................................................................................. 63

4.4 Cyclic Loading: Vertical and Horizontal Real Load ....................................................... 82

4.4.1 Input ..................................................................................................................... 82

Abstract

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4.4.2 Output .................................................................................................................. 82

4.5 Analysis of the Results .................................................................................................. 90

4.5.1 Cyclic Loading with Vertical Load ......................................................................... 92

4.5.2 Cyclic Loading with Horizontal and Vertical Load ................................................ 95

5 Discussion ............................................................................................................................. 97

5.1 Bearing Capacity ........................................................................................................... 97

5.2 Cyclic Vertical Loading .................................................................................................. 99

5.3 Cyclic Vertical and Horizontal Loading ....................................................................... 101

6 Conclusion and Future Work .............................................................................................. 103

7 References .......................................................................................................................... 104

Appendix I: Soil Model Selection ................................................................................................ 106

Appendix II: Load Control vs Prescribed Displacement ............................................................. 108

Appendix III: Boundary effects ................................................................................................... 113

Appendix IV: Ultimate Bearing Capacity, Shear Surfaces Development ................................... 116

Introduction

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1 INTRODUCTION

Bridges supported by foundations located close or in slopes is not an uncommon find in

railway lines. This generates a relation between the slope and the foundation in which both

are affected by the others characteristics, changing their isolated ability to support load

without failure.

Traditionally, these problems have been assessed with static calculations that combine

bearing capacity of foundations theories and slope stability. These calculations are also

supported by very conservative factors of safety. Many bridge foundations falling into this

category which have been calculated with these methods are in use today. With time the load

of the trains has been increased leading to a reduction in the safety of these foundations and

making them deal with heavy loads.

It can be said that the failure mechanism of this problem is still not fully studied in order

to understand what kind of failure and how it develops under the heavy load of the foundation

and the slope characteristics.

Other topic traditionally disregarded is the cyclic load that the trains generate when they

travel through the bridge as well as the horizontal forces generated due to braking. With

enough number of cycles and the added effect of the big loads it can change the bearing

behaviour of the foundation in relation to a static situation helping to trigger a failure in the

foundation – slope system.

The present work studies this topic from a theoretical point of view, analysing the static

bearing capacity and developed failures in several foundation – slope cases under heavy loads.

A cyclic analysis of the same cases will be performed as well to see the differences with the

static situation and how the cyclic load affects the bearing capacity and failure mechanisms.

Static and cyclic situations will be studied developing finite element models of the cases in the

program Plaxis.

Background and Motivation

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2 BACKGROUND AND MOTIVATION

2.1 TRADITIONAL BEARING CAPACITY CALCULATIONS One of the most popular bearing capacity theories is the one developed by Karl Terzaghi.

He considers the soil to be in elastic equilibrium before placing any load. Once the load is

applied and until a certain amount of it, the soil is still in elastic conditions. If the load is

increased more, the soil enters a state of plastic equilibrium (Terzaghi, 1943)

In the case when the soil’s plastic flow during loading is preceded by a very small strain,

general shear failure takes place. In this case the foundation has very few settlements until

failure. If the strain that takes place before the failure is big, local shear failure takes place.

Showing important settlements before failure (Terzaghi, 1943). Figure 2-2 represents the

general look of the load – settlement curve related with both failure cases. General shear

failure surfaces are developed from the edges of the footing all the way until the surface,

ground surface heave is normally present in both sides of the foundation (Figure 2-1, a). In the

case of local shear failure, a state of plastic equilibrium is not fully developed, therefor the

failure surfaces do not reach the surface, heave is slightly present (Figure 2-1, b) (Knappett and

Craig, 2012).

Figure 2-1: General shear failure of a shallow foundation (a) and local shear failure of a shallow foundation (b). (M. Das 2011)


9

Figure 2-2: General look of a load settlement curve corresponding to general shear failure (solid line) and local shear failure (dashed line). (Terzaghi, 1943)

The Equation 1 is based on the method created by Terzaghi to calculate the bearing capacity of

foundations (Lang, Huder and Amann, 2003).

𝜎𝑓 = (𝛾𝑡 + 𝑞)𝑁𝑞𝑆𝑞𝑑𝑞𝑔𝑞 +1

2𝛾𝐵𝑁𝛾𝑆𝛾𝑑𝛾𝑔𝛾 (1)

Where:

𝛾= Density of the soil.

t= Embedment

q= Overburden pressure.

B= with of the foundation

𝑁𝑞 , 𝑆𝑞 , 𝑑𝑞, 𝑔𝑞 , 𝑁𝛾, 𝑆𝛾 , 𝑑𝛾 , 𝑔𝛾 =Bearing capacity factors from the Tables.

The bearing capacity factors are obtained from Figure 2-3.

The bearing capacity factors are determined according to several aspects of the foundation or

soil that affect the bearing capacity like friction angle, with and embedment of the foundation,

shape, etc. Originally was only regarding foundations in flat terrains but parameters covering

the effects of slopes has been added.


10

a)

b)

c)

d)

Figure 2-3: Bearing capacity factors for Terzaghi theory. (Lang, Huder and Amann, 2003)


11

Meyerhof also developed a bearing capacity theory which he combined with slope

stability (Meyerhof, 1957) for foundations on top and in slopes. He provides methods of

calculating bearing capacities in pure cohesive and pure frictional soils, as well as a combined

way for soils in between of this range.

According to his study, the plastic flow areas present in the side of the slope are

significantly reduced in comparison to the same foundation in a flat terrain, this also reduces

the bearing capacity of the foundation. A representation of common plastic flow areas is

shown in Figure 2-4 and 2-5. Also, in a pure cohesive soil the decrease in bearing capacity in

slopes with common use slope angles (<30º) is small but when it comes to pure frictional, this

reduction is quite significant and seems to decrease parabolically with increasing slope angle

since the slope angle gets closer to the friction angle of the soil, which is the maximum stable

for the slope (Meyerhof, 1957).

Figure 2-4: Plastic zones of a foundation in a slope (Meyerhof 1957)

Figure 2-5: Plastic zones of a foundation on top of a slope. (Meyerhof 1957)


12

Following the procedure in (Meyerhof, 1957), the bearing capacity for a foundation on the

crest or in a slope is described with Equation 2:

𝑞 = 𝑐𝑁𝑐𝑞 + 𝛾𝐵𝑁𝛾𝑞

2 (2)

Where

c=cohesion of the soil

𝛾=unit weight of the soil (in this case is dry weight)

B=with of the foundation

𝑁𝑐𝑞 and 𝑁𝛾𝑞= Bearing capacity factors obtained from Tables (since the studied soil is

cohesionless, only 𝑁𝛾𝑞 will be needed and can be found in Figure 2-6, a and 2-6, b for

each case)

a)

b)

Figure 2-6: Meyerhof bearing capacity factors for cohesionless soils. a) bearing capacity factor for foundations on top of a slope. b) bearing capacity factor for foundation in the slope. (Meyerhof, 1957)


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2.2 BEARING CAPACITY PROBLEM VS SLOPE STABILITY PROBLEM. The classification of a foundation’s bearing capacity related to a slope into a pure bearing

capacity or slope stability problem cannot be done straightforward. Neither the calculation

approach to the problem. Traditionally bearing capacity problems in foundations are assessed

by determining a factor of safety which relates the maximum load that the foundation can deal

with and the load that is going to carry. On the other hand, the stability of a slope is designed

by studying a safety factor regarding the shear strength of it. The safety factors that normally

are used in both approaches are generally very conservative. Despite this, none of the

approaches can be considered alone when studying a foundation – slope relation.

The way in which the failure mechanism of this problem can tend to one or the other

failure mode (or even have both at the same time), depends on many factors like magnitude of

the load and type of it, geometry of the slope and foundation, and characteristics of the soil.

Figure 2-5 shows the main failure modes in this type of problems. Generally, for light loaded

foundations, the main failure mode tends to be base and toe failure of the slope. When the

load is increased bearing capacity failure of the foundation starts to appear, coexisting with

the slope failure for a load interval until the main failure is bearing capacity of the foundation

(Pantelidis and Griffiths, 2014).

2.3 STRAIN ACCUMULATION UNDER CYCLIC LOADING

Figure 2-7: Load cycles and settlements on a foundation. (Wichtmann, 2005)

When a foundation is loaded in a cyclic way, the settlement of the foundation increases

with the number of cycles due to the residual strain left in the soil from each cycle. This

phenomenon is represented in Figure 2-7. The amount of this residual settlement depends on

the load characteristics and the soil.

The total strain (εtot) seen in the soil during one cycle can be decomposed in two parts:

• Elastic strain (εelast), part of the total strain which once the load goes to its

minimum in the cycle disappears.

• Plastic strain (εplast), amount of the total strain that remains in the soil.

The relation between these strains during a cyclic loading can be seen in Figure 2-8. Also, it is

notable how the biggest plastic strain happens in the first loading. The following cycles have

slightly decreasing deformation with the cycles until a certain number of cycles.


14

Figure 2-8: Representation of a strain history from a cyclic loading. (Adapted from Wichtmann, 2005)

Not every case behaves in the same way. Three main patterns can be defined according to

the plastic strain along the cycles (Goldscheider and Gudehus, 1976). In the case of stepwise

failure, the plastic strain that remains from each cycle is constant; in a shakedown case, the

remaining plastic strain reduces along the cycles until some point that only elastic

deformations occur; in an abation case the remaining plastic strain diminishes with the cycles

but never disappears completely.

2.4 FAILURE BEHAVIOUR OF HORIZONTALLY LOADED FOUNDATIONS IN SLOPES When a foundation is affected by a horizontal load, rotation or sliding of it are common

occurrences. The amount of rotation depends on geometrical and soil parameters, also it is

strongly influenced by the load on the foundation and its horizontal-vertical ratio. The next

situations can occur for a surface foundation which suffers from important rotation. Figure 2-9

offers a clearer view of the phenomenon: (Taeseri, 2017)

• For big factors of safety (against vertical load), the failure surface is generated in the

same direction as the rotation.

• In case of safety factors close to 1 the failure is developed in the opposite direction as

the rotation.

This effect can also be noticed for embedded foundations, but this type is more resistant to

rotation than a foundation in the surface.


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Figure 2-9: Relation between factor of safety and failure mechanism in foundations with rotation due to horizontal loading. (Taeseri, 2017)

2.5 SMALL STRAIN HARDENING SOIL MODEL. Soils can be considered elastic in a very short strain range. Also, when the strain

amplitude during loading and unloading increases, the soil stiffness reduces in a nonlinear way

(Plaxis, 2018). This relation is demonstrated in Figure 2-10.

Figure 2-10: Stiffness-Strain relation. (Plaxis, 2018)

This phenomenon happens due to a higher loss of intermolecular and surface forces in the

soil particles when the strain increases. When the loading is released, the soil returns to its

maximum stiffness which is in the order of the initial one (Plaxis, 2018).


16

The behaviour of materials under loading-unloading cycles has been described by (Masing,

1926) and is illustrated in Figure 2-11:

• When unloading, the shear modulus is the same as the original tangent modulus in the

first loading.

• The loading-unloading curve’s shape is the same as the first loading curve but two

times bigger.

This idea has been implemented in this model with an increase of the plastic stiffness along

the cycles. At the start of each loading, the stiffness is in the range of the original one, but

when the strain increases, the stiffness decreases in a non-linear way. In the successive cycles,

the stiffness is higher than in the original loading. Also, during the initial loading, the stiffness

reduces more rapidly within the cycle than in the rest of the cycles (Figure 2-12).

Figure 2-11: Stiffness variation in first loading and successive ones (Plaxis, 2018)

Figure 2-12: Loading-Unloading materials behaviour. (Plaxis, 2018)

2.6 PHI-C REDUCTION CALCULATION Phi-c reduction is provided in Plaxis as a safety calculation method. The provided safety is

related with the shear strength of the soil. Along the calculation, the shear strength

parameters (friction angle, cohesion and tensile strength) are progressively reduced until

failure (Plaxis, 2018). Dilatancy angle is kept without reduction until the friction angle reaches

the same value as it, then they are reduced equally.

The safety factor is given by the parameter Msf. It is calculated by dividing the original soil

parameter by the reduced one. The Msf value at failure corresponds to the safety factor and

express the relation of available shear strength and shear strength at failure. Other relevant

result of this type of calculation is the developed failure mechanism.

Care must be taken when performing this calculation and checking that the obtained Msf

is constant while there is still deformation. This means that a failure has truly developed

(Plaxis, 2018)


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2.7 AIM AND OBJECTIVES Two main cases are going to be regarded. A square foundation placed on top of the crest

of a slope and a foundation in the slope. In the second case, the slope is divided in two parts

with a small flat part in between where the foundation is placed. This can be normally seen in

real foundations due to the need of a working platform for the construction of it. These two

cases represent railway bridge foundations heavily loaded with the weight of the bridge and a

cyclic load corresponding to a train passing on top of them.

The principal aim of this work is to have a better understanding of the bearing behaviour

and failure mechanisms developed in foundations under the described conditions. The

stablished objectives are:

• Model foundations on slopes in Plaxis 2D.

• Analyse the ultimate bearing capacity of the different cases, the developed failure

mechanisms and understand some of the main parameters that influence the bearing

capacity.

• Study the bearing behaviour and how failures develop of the chosen cases under

several loading cycles.

• Find any difference, if it exists, between the failure mechanism present in a cyclic

loading and when a static bearing capacity failure happen.

Methodology

18

3 METHODOLOGY

In order to fulfil the stablished objectives, the problem was divided into several steps

increasing the number of forces acting in the foundation, this will allow a better understanding

of the effect that each one has in the model and to pinpoint possible problems during the

modelling. For each of the regarded cases the same analysis procedure is going to be

followed. The original geometry of the problem can be seen in Figure 3-1, 3-2 and Table 1.

The soil in all case is going to be dry, with the water Table under the slope. This means

that no pore pressure is developed, making the problem independent of time and simplifying

the modelling since it can be done with static plastic calculations steps alone.

The analysis performed for each case includes:

• Vertical push analysis with static load until failure.

• Cyclic analysis simulating the force of a train passing on top of the foundation. This will

be made in two different ways:

o Maintaining a constant ultimate bearing capacity-real load ratio for each case.

This will give a different loading situation for each of the studied foundations,

but the traditional safety factor used in foundation calculation will be kept

constant.

o Maintaining real force constant and modifying the foundation.

• A horizontal force with a momentum will be added to one of the cases to simulate the

braking force of the train.

The ultimate bearing capacity of each case will be contrasted with well stablished

analytical bearing capacity methods to check if the results from the simulations are reliable.

The chosen soil for the model is Perth sand. This is a well characterized sand used in some

studies in the same topic either computer modelling or with centrifuge models (Taeseri, 2017).

3.1 ULTIMATE BEARING CAPACITY: ANALYTICAL METHODS In order to compute the bearing capacity, developed failure mechanism and affecting

parameters for each case, ultimate bearing capacity calculations are going to be performed for

each model using Plaxis. To check if the models are correct, the bearing capacity will also be

calculated with well stablished analytical methods:

• Bearing capacity calculated with Meyerhof’s theory. Using Equation 2 in Chapter

2.1

• Bearing capacity calculated with Terzaghi’s theory. Using Equation 1 in Chapter

2.1

• French code. These calculations are performed as follows:

Methodology

19

French code calculates the bearing capacity according to the serviceability limit stage

(ULS). The load corresponding to this limit is calculated with Equation 3:

𝑞 ≤1

𝛾𝑞

(𝑞𝑢 − 𝑞0)𝑖𝛿𝛽 + 𝑞𝑜 (3)

Where:

𝑞 = Load on the foundation.

𝛾𝑞 = Safety factor. Normally takes the value of 2 but since the ultimate bearing

capacity is the aim a value of 1 will be used.

𝑞𝑢 = Bearing capacity of the soil.

𝑞0 = Overburden pressure.

𝑖𝛿𝛽 = Reduction factor considering the inclination of the load and the soil geometry.

The bearing capacity of the soil is calculated according to Equation 4:

𝑞𝑢 − 𝑞0 = 𝑘𝑝𝑝𝑙𝑒∗ (4)

Where:

𝑘𝑝 = Bearing capacity factor.

𝑝𝑙𝑒∗ = Equivalent limit pressure.

The bearing capacity factor and the equivalent limit pressure are obtained with Equations 5

and 6:

𝑘𝑝 = 1 + 0,5 (0,6 + 0,4𝐵

𝐿) ∗

𝐷

𝐵 (5)

Where:

B and L = With and length of the foundation (Same value in case of square foundation).

D = Embedment of the foundation.

𝑝𝑙𝑒∗ = 𝑝𝑙

∗(𝑧𝑒) (6)

Where:

𝑧𝑒 = 𝐷 +2

3𝐵

𝑝𝑙∗(𝑧) = 𝑎 ∗ 𝑧𝑒 + 𝑏

Having a=123 and b=56 (which are empirically obtained parameters used for every

case).

Methodology

20

Finally, the reduction factor is computed with Equation 7:

𝑖𝛿𝛽 = 1 − 0,9 tan(𝛽) (2 − tan(𝛽)) [max ((1 −𝑑

8𝐵) ; 0)]

2

(7)

Where:

𝛽 = Inclination of the slope.

d = Distance from the foundation to the edge of the crest.

3.2 PLAXIS MODELS As two geometries are regarded, two basic models are developed. These models will be

the base on where further modifications will be made.

The foundation studied in the bearing capacity calculations and in the constant force ratio

is a square foundation with the characteristic shown in Table 1. For the real force calculations,

the geometry of the foundation will be modified to fit the load requirements. The geometry

from Table 1 is based on foundations studied in centrifuge tests in Taeseri, 2017.

Table 1: Original geometry of the foundation.

Side Length Thickness Embedment Ration Material

B= 2,8 m D=1,12 m 0,4 Aluminium

In the mentioned study made by Taeseri, Aluminium is chosen as the foundation’s

material in the physical models for having a similar density to concrete, which will be also

followed here. Also, the foundation will be made fairly rigid. The point of this study is to

analyse the interaction between the soil and the foundation and not the structural behaviour

of the foundation.

Figure 3-1 and Figure 3-2 show the original geometry of the complete model for both

studied cases. The original model is made with a slope of 20º angle. In order to study the effect

of the slope angle in the foundation, a 30º angle slope will also be modelled. For consistency,

the height of the slope is 8 m in every case.

The chosen size for the model is 40 x 20 m, being big enough to avoid boundary effects in

the models. More information about how the boundaries were set can be found in Appendix

III.

Methodology

21

Figure 3-1: Original geometry of the foundation on top of a 20º slope.

Figure 3-2: Original geometry of the foundation in the middle of a 20º slope

The chosen soil model is Small Strain Hardening soil, with it, the non-linear relation

between stiffness of the soil and strain as well as the hardening material law in it will allow to

simulate more realistically the behaviour of the soil, see Chapter 2.5. In Appendix I more

information regarding the selection of the soil model can be found as well. The properties of

the Perth sand for the small strain hardening soil can be found in Table 3.

All soil-foundation interfaces have been set to a friction coefficient of 0,5. (Taeseri, 2017).

No water flow conditions are needed because of the dry nature of the problem.

Methodology

22

3.3 ULTIMATE BEARING CAPACITY PLAXIS For calculating the ultimate bearing capacity of the foundation, the way Plaxis applies a

load during a plastic phase is going to be used. This load is not applied at once, it goes

gradually increasing with the iterations until it reaches the defined value or the soil collapses.

The amount of the applied load during the calculations is denoted by the program with the

Mstage parameter. This parameter ranges from 0 to 1, being 0 equal to no applied load and 1

the total of the designated load is applied. As an example, if a load of -100 kN/m/m is

designated, the Mstage will be 0,5 when -50 kN/m/m are applied in the calculations.

The way to find the ultimate bearing capacity is: first an arbitrary big load is placed in the

foundation, which makes the soil body collapse since is bigger than the bearing capacity. At

this point, the reached Mstage is checked and multiplied by the applied load. The result is the

maximum load that could be applied before collapse or, in other words the ultimate bearing

capacity. Finally, the model is calculated once again with the calculated load. Doing this the

load is checked and the failure mechanism can be observed in the results.

The calculated models are the ones with the original geometry shown in Figure 3-1 and 3-

2 with a slope angle of 20º and 30º. Also, the same models are calculated once again with the

embedment of the foundation doubled.

Several mesh sizes were tried until the adequate one was found. The selected one is a bit

finer than the minimum required, since the calculation time in these models is not a problem.

Also, this mesh provides good quality.

The stages used to calculate these models are:

• Initial phase where the initial stresses are computed. This is done only with gravity

loading since there is no phreatic in the model.

• During the second phase, the foundation is placed in the soil and the interfaces

activated. After this stage all displacement are reset.

• In the next phase the force is activated.

• Two more phases are added with halve and 2/3 of the ultimate bearing capacity.

3.4 CYCLIC LOADING MODELS When a train passes on top of a foundation, the loads generated by it can be divided into

three steps. Initially the train starts entering the influence area of the foundation increasing

the vertical load on it. Once the head of the train has passed the foundation and left the

influence area, the load stops increasing and stays stable until the end of the train enters the

influence area. At this point the load starts decreasing until the end of the trains lives the

influence area when the train load becomes zero again.

3.4.1 Calculation of Vertical Train Load

For calculating the loads related with a train passing on top of the foundation, several

assumptions and simplifications are made:

Methodology

23

• The bridge that the foundation supports has 30m span, being the smallest span where the

inertia of moving vehicles can be neglected (Unsworth, 2017). Also, the dead load of the

bridge is assumed to be 96 tons (being a reasonable load for a bridge of medium size

(Taeseri, 2017) which gives a force of 940 kN or a dead load of 336 kN/m (as a point load

in Plaxis 2D the units are kN/m).

• The wagons of the passing train have a weight of 30 tons (no difference made between

wagons and locomotive) with a length of 9,4 m. This leads to a line load of 31,3 kN/m. The

speed of the train is chosen to be 30 km/h (8,33 m/s). With very long trains, the interval

when the load is constant at its maximum is also long. This is important in cases when

pore pressure can develop, although this fact is not applicable for this case.

• All the generated loads will be transferred to a point load in the middle of the foundation.

The load is assumed to affect the foundation from the half of the spam before and after the

foundation (15 m on each side of the foundation). In this way, the line load of the train is

calculated as a moving point load with increasing value until the train is covering the full

influence length. The same calculations are made when the train leaves the influence length of

the bridge. In Figure 3-3 the profile of the calculated load can be seen.

Figure 3-3: Load profile on the foundation generated by a train passing on top, according to the assumptions and simplifications made.

The maximum reached force is 939 kN, which transferred to a point load in the foundation

gives 335 kN/m. The calculated loads that are going to be applied are summarised in Table 2.

From now on these loads are going to be referred as real loads

Table 2: Real loads applied on the foundation.

Dead Load [kN/m] Train Load [kN/m] Maximum Load [kN/m]

336 335 671

Each load corresponds to the 50 % of the maximum one.

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25 30

Load

[kN

]

time [s]

Train Load

Methodology

24

3.4.2 Calculation of Horizontal Train Load

The horizontal forces in the bridge are generated by the movement of trains along the

bridge. These horizontal forces are transferred to the tracks by friction forces between wheels

and tracks, and from the tracks to the superstructure and substructure of the bridge. The

horizontal forces generated by the train moving at constant speed are small, but they reach

bigger values when the train brakes or accelerates (Fryba, 1996).

Traditionally this force does not play an important role when designing a foundation for a

small or medium bridge. In these cases, it is considered that this force act only in the

abutments and any horizontal load in the foundations is just covered by the conservative

safety factors used for its design. But even being small, this force generates a cyclic loading in

the foundation that can affect the bearing capacity in a long term.

The horizontal force generated by a train braking is taken as 25% of the axle load (Chen

and Duan, 2000). According to this and having a wagon with 30 tons weight and four axles,

each axle deals with a load of 73,5 kN. Giving a braking force of 18,375 kN. Which transferred

to a point load in Plaxis is 6,56 kN/m.

Since this load is applied on the bridge it will be transferred to the foundation by a

horizontal component to the load of 6,56 kN/m and a momentum. This momentum is

calculated according to the height of the pier that connects the foundation with the bridge,

this height is assumed to be 4m (Taeseri, 2017). For this situation the momentum is 26,25

kN·m/m. The momentum will be applied in a clockwise direction with a horizontal force

pointing right to simulate a train passing through the bridge from left to right. The opposite

force and momentum will be applied to simulate a train going from right to left.

3.4.3 Constant Force Ratio Loading

For this situation, all cases have a ratio between bearing capacity and applied load of 3,7

(Equation 8). This can also be seen as a traditional safety factor of 3,7 (A value between 3 and

4 is commonly used for design of foundations). The ratio between dead load and maximum

train load is also maintained, corresponding 50% of the total load to the dead load and 50% to

the maximum train load. The calculated loads that are going to be applied can be found in

Table 6.

𝑈𝑙𝑡𝑖𝑚𝑎𝑡𝑒 𝐵𝑒𝑎𝑟𝑖𝑛𝑔 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦

𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐿𝑜𝑎𝑑= 3,7 (8)

3.4.4 Real Force Loading

The real load is too big for the original foundation’s geometry. In real situations with this

kind of foundations it is common to have a limited building area so increasing the area of the

foundation is not an option. The common solution is to increase the depth of embedment.

Following this praxis, the embedment of the foundation will be doubled, being the new one

2,24 m.

The applied load for every case can be found in Table 2.

Methodology

25

3.4.5 Plaxis Modelling for Cyclic Loading

The used geometry of the models is the same as used for the bearing capacity calculations

(Figure 3-1 and 3-2 with 20º and 30º slope angle), except for the real force cases which will

have the embedment doubled.

The soil is modelled with Small Strain Hardening Soil model, the used parameters are

gathered in Table 3. Motivation of the use of this soil model can be found in Appendix I and in

the Chapter 2.5.

Points to obtain data for plotting are chosen in the top part of the foundation. Being their

location in the middle, left and right edge. This will allow to track the vertical displacement of

the foundation and if there is any rotation in it.

Force control with a point load and a momentum (In cases with horizontal load) is used as

the way to load the foundation. Also, a very stiff plate is added on the surface of the

foundation making possible to simulate the momentum in the foundation.

Every case is modelled following the same phases pattern:

• Initial Phase: Gravity stage to calculate initial stresses.

• Plastic Phase: Activation of the foundation and interfaces.

• Plastic Phase: Activation of the dead load (all deformations before the phase are reset

to 0.

• Plastic Phase: Increase of the load until maximum load. This phase with the previous

one are the first loading cycle.

• The previous two phases are repeated until a total of 100 cycles.

• In the cases where the braking force is applied, the horizontal load component and

momentum is applied in the same phase as the increasing of the load until maximum.

A model regarding a foundation in a flat surface will also be made following this

procedure. The real load of the train will be applied to it and the foundation geometry related

with this loading case will also be used.

Results and Analysis

26

4 RESULTS AND ANALYSIS

4.1 ULTIMATE BEARING CAPACITY

4.1.1 Input

The soil parameters of the Perth sand used in the Small Strain Hardening Soil for the computer

calculations are gathered in Table 3.

Table 3: Perth sand parameters for Small Strain Hardening Soil model

Parameters Perth Sand

Effective particle size, 𝐷10 (𝑚𝑚) 0.14

Average particle size, 𝐷50 (𝑚𝑚) 0.23

Uniformity coefficient, 𝐶𝑢 1.79

Coefficient of curvature, 𝐶𝑐 1.26

Specific density, 𝜌𝑠 (𝑘𝑔/𝑚3) 2700

Dry density, 𝜌𝑑 (𝑘𝑔/𝑚3) 1700

Relative density, 𝐷𝑟 (%) 80

Void ratio, 𝑒𝑚𝑖𝑛 … 𝑒𝑚𝑎𝑥 0.502…0.752

Friction angle, 𝜑′𝑚𝑎𝑥(°) 38

Dilatancy angle, 𝜓 (°) 8

Surface shear modulus, 𝐺0 (𝑘𝑃𝑎) 35000

Surface shear wave velocity 𝑉𝑠,0 (𝑚/𝑠) 150

Secant stiffness, 𝐸50𝑟𝑒𝑓(𝑘𝑃𝑎) 33000

Tangent stiffness, 𝐸𝑜𝑒𝑑𝑟𝑒𝑓 (𝑘𝑃𝑎) 27150

Unloading reloading stiffness, 𝐸𝑢𝑟𝑟𝑒𝑓 (𝑘𝑃𝑎) 99000

Shear strain, 𝛾0.72 (−) 2x10-4

Shear modulus at very small strains, 𝐺0𝑟𝑒𝑓 (𝑘𝑃𝑎) 190000

Reference stress level, 𝑃𝑟𝑒𝑓 (𝑘𝑃𝑎) 100

In the case of the hand calculations, dry density is used from Table 3, a friction angle of 30º

(which comes from the subtraction of the dilatancy, 8º; to the real friction angle 38º), and the

specific parameters needed in each case are obtained from the Figures presented in Chapter

2.1

4.1.2 Calculations

Analytical calculations are made with the formulas related with each of the three methods

presented in Chapters 2.1 and 3.1

The Plaxis models are calculated following the stages from Chapter 3.3.


27

4.1.3 Output

4.1.3.1 Analytical Calculations

Table 4 gathers the ultimate bearing capacity obtained from the three analytical methods

for each case. Terzaghi and French Code methods are only applied for the cases with the

foundation on top of the slope.

Table 4: Ultimate bearing capacity results from the three analytical calculations

Case Meyerhof Terzaghi French Code

Top, 20º slope 671,73 kN/m2 552,04 kN/m2 527,184 kN/m2

Top, 30º slope 480,47 kN/m2 268,46 kN/m2 151,38 kN/m2

Middle, 20º slope 848,99 kN/m2 - -

Middle, 30º slope 587,76 kN/m2 - -

4.1.3.2 Original Geometry Cases

A) FOUNDATION ON TOP OF A 20º ANGLE SLOPE, ORIGINAL GEOMETRY

The following set of figures shows the incremental displacements and incremental

deviatoric strain generated by the ultimate load (Mstage = 1). Vertical displacement- load

curve is also included.

Figure 4-1: Incremental displacements plot from ultimate bearing capacity calculations of a foundation on top of a 20º slope


28

Figure 4-2: Deviatoric strain from ultimate bearing capacity calculation of foundation on top of a 20º slope. Corresponding to Mstage = 1 in Figure 4-3

Figure 4-3: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation on top of a 20º slope


29

Figure 4-4: Phi-c reduction plot of the ultimate bearing capacity calculations from a foundation on top of a 20º slope

In this case the maximum allowed load is -554,4 kN/m/m with a vertical displacement of -

0,138 m according to the load-settlement curve in Figure 4-3. Figure 4-1 and 4-2 show a failure

mechanism that resembles mainly a local failure of the foundation (see Figure 2-5). In the

same two figures a deeper failure mechanism seems to be starting to develop which looks like

a toe or base failure of the slope. When looking at Figure IV -1 and IV -2 from appendix IV (a),

the origin of the failure mechanism starts slightly developing an active zone under the

foundation (Figure IV -1), it can be seen in Figure IV -2 that when the load is the 75% of the

ultimate load the active zone seems to lose importance and the main shear surface is the one

developing in direction to the slope surface, which can be seen fully developed in Figure 4-2

when the ultimate load is fully reached, although at this stage the active zone is still present.

Results from the phi-c reduction calculation in Figure 4-4 are not reliable, the calculation

doesn’t converge to a clear value presenting big oscillations. This occurs in all phi-c calculations

made for the bearing capacity calculations. For this reason, only in the present case, this plot

will be shown as an example.


30

B) FOUNDATION ON TOP OF A 30º ANGLE SLOPE, ORIGINAL GEOMETRY




Figure 4-5: Incremental displacements plot from ultimate bearing capacity calculations of a foundation on top of a 30º slope

Figure 4-6: Deviatoric strain from ultimate bearing capacity calculation of foundation on top of a 30º slope. Corresponding to Mstage = 1 in Figure 4-7


31

Figure 4-7: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation on top of a 30º slope

In Figure 4-5 and 4-6 is possible to see the failure mechanism of this case. At the look of

Figure 4-6 failure in the foundation can be seen, the active zone under the foundation, as well

as the transition and passive zone delimited by, a shear surface reaching almost the toe of the

slope (Figure 4-6), resembles a local shear failure. Regarding the history of the failure, in

appendix IV (b), Figures IV -3 and IV -4 show a late formation of the failure, event at 75% of the

ultimate load (Figure IV -4) the active zone is not fully formed.

As Figure 4-7 shows, the maximum allowed load corresponds to -223,4 kN/m/m with a vertical

displacement of -0,067 m.


32

C) FOUNDATION IN THE MIDDLE OF A 20º ANGLE SLOPE, ORIGINAL GEOMETRY


deviatoric strain generated by the ultimate load (Mstage = 1. Vertical displacement- load curve

is also included.

Figure 4-8: Incremental displacements plot from ultimate bearing capacity calculations of a foundation in the middle of a 20º slope

Figure 4-9: Deviatoric strain from ultimate bearing capacity calculation of foundation in the middle of a 20º slope


33

Figure 4-10: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation in the middle of a 20º slope

For this case two main failure surfaces are present (Figure 4-8). One resembles a local

failure of the foundation with an active zone and a shear surface running from the edge of the

active zone towards the toe of the slope which delimits the passive zon3 (Figure 4-9). The

other, seems like a base failure of the slope. Also, small shear surfaces along the part of the

slope above the foundation denote that this part is also activated and applies horizontal load

on the foundation’s side. When taking a look to Figures IV -5 and IV -6, in appendix IV (c), it can

be said that when the load is 50% of the ultimate load, the active zone under the formation is

starting to be present. It is interesting when the load reaches 75% of the ultimate load, how

the shear surfaces in the slope above the foundation are already present and also two shear

surfaces starting in the foundation running in the opposite direction to the slope are also

present, showing a clockwise tilting direction of the foundation. In a point between 75% and

100% of the ultimate load, the shear surfaces form Figure 4-9 start to be more critical and the

tilt of the foundation changes to counter clockwise direction.

The maximum vertical load allowed is -666 kN/m/m with a vertical displacement of -0,155 m.

This point is the end of the curve in Figure 4-10.


34

D) FOUNDATION IN THE MIDDLE OF A 30º ANGLE SLOPE, ORIGINAL GEOMETRY

The following set of Figures shows the incremental displacements and incremental



Figure 4-11: Incremental displacements plot from ultimate bearing capacity calculations of a foundation in the middle of a 30º slope

Figure 4-12: Deviatoric strain from ultimate bearing capacity calculation of foundation in the middle of a 30º slope


35

Figure 4-13: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation in the middle of a 30º slope

As seen in Figure 4-11 and 4-12, there is a foundation failure with a fully developed active

zone under the foundation and a shear surface running from this zone until the toe of the

slope. The start of a deeper failure surface in the slope is also visible (Figure 4-11 and 4-12).

The shear surface showing the biggest amount of strain is the one located in the slope above

the foundation (Figure 4-12). When looking at Figures IV -7 and IV -8 from appendix IV (d), it

can be said that the development of the shear surfaces is quite late, when the 50% of the

ultimate load the active zone is not fully developed (Figure IV -7), and when the 75% of the

load is applied the active zone is present as well as the shear surface from the active zone

towards the toe of the foundation and the shear surface in the part of the slope above the

foundation (Figure IV -8).

The maximum load obtained from the end point of Figure 4-13 is -423,5 kN/m/m with a

vertical displacement of -0,118 m.


36

E) FOUNDATION IN FLAT TERRAIN, ORIGINAL GEOMETRY




Figure 4-14: Incremental displacements plot from ultimate bearing capacity calculations of a foundation on flat terrain.

Figure 4-15: Deviatoric strain from ultimate bearing capacity calculation of foundation on flat terrain.


37

Figure 4-16: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation on flat terrain.

The failure mechanism seen in Figure 4-14 and 1-15, resembles a local shear failure

(Figure 2-1). As it is on a flat surface the shear has symmetry in both sides of the foundation.

One active zone is developed under the foundation with one passive area on each side of the

active zone. The development of this failure mechanism can be described using Figure IV -9

and IV -10 in appendix IV (e). When the load is 50% the ultimate load, a small active zone is

starting to be present under the foundation (Figure IV -9), in the moment when the 75% of the

ultimate load is reached a bigger active zone is present (Figure IV -10) showing that the

development of the passive zone is the last part of the failure to be present.

The maximum obtained load from the curve in Figure 4-16 is -1105 kN/m/m with a vertical

displacement of -0.175 m


38

4.1.3.3 Double Embedment Geometry

F) FOUNDATION ON TOP OF A 20º ANGLE SLOPE, DOUBLE EMBEDMENT GEOMETRY




Figure 4-17: Incremental displacements plot from ultimate bearing capacity calculations of a foundation on top of a 20º slope.

Figure 4-18: Deviatoric strain from ultimate bearing capacity calculation of foundation on top of a 20º slope.


39

Figure 4-19: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation on top of a 20º slope.

Figure 4-17 and 4-18 show three main failure mechanism toward the slope, one starting

from the active zone under the foundation that seems to go towards the toe of the

foundation, a deeper shear surface resembling the start of a base failure and one shallower.

Also, in the left side of the foundation one shear surface is present form the bottom left corner

of the foundation to the surface (also looks like it has the same direction as the shallower

shear surface on the right. At the look of Figure IV -11 and IV -12 (appendix IV, (f)), when the

50% of the ultimate load is applied, the active zone under the foundation is present (Figure IV -

11), the moment the load reaches the 75% of the ultimate load, an active zone under the

foundation and what seems a passive zone at each side of the active is visible, one of the shear

surfaces is already getting close to the slope surface (it seems to be the shallower shear

surface in the ultimate stage of the failure).

The ultimate reached load is -871,3 kN/m/m with a vertical displacement of -0,176 m, from

Figure 4-19.


40

G) FOUNDATION ON TOP OF A 30º ANGLE SLOPE, DOUBLE EMBEDMENT GEOMETRY




Figure 4-20: Incremental displacements plot from ultimate bearing capacity calculations of a foundation on top of a 30º slope.

Figure 4-21: Deviatoric strain from ultimate bearing capacity calculation of foundation on top of a 30º slope.


41

Figure 4-22: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation on top of a 30º slope.

Figure 4-21 and 4-20 show a similar mechanism than in the 20º case but shallower, the

main shear surface that goes until the slope surface seems to cut the active zone and has the

same direction as the failure surface on the left of the foundation. A deeper shear surface

starting from the active zone is also present, as well as the start of what it seems a base failure.

From Figures IV -13 and IV -14 in appendix IV (g), when the load is 50% of the ultimate load,

the active zone is already present, in this point the shear surface from the active zone to the

surface of the slope is starting to develop. This shear surface seems to move deeper and reach

the horizontal surface on the left of the foundation when the applied load reaches 70% of the

ultimate load, the start of what seems a base failure is also present, but it does not present big

strains.

From Figure 4-22, the ultimate reached load is -445,2 kN/m/m with a vertical displacement of -

0,111 m.


42

H) FOUNDATION IN THE MIDDLE OF A 20º ANGLE SLOPE, DOUBLE EMBEDMENT

GEOMETRY




Figure 4-23: Incremental displacements plot from ultimate bearing capacity calculations of a foundation in the middle of a 20º slope.

Figure 4-24: Deviatoric strain from ultimate bearing capacity calculation of foundation in the middle of a 20º slope.


43

Figure 4-25: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation in the middle of a 20º slope.

Figure 4-23 and 4-24 shows a failure mechanism that resembles a local shear failure, with

an active zone under the foundation and the passive zones on the sides. The right passive zone

seems to be bigger and more developed than the left one, it also reaches depths bigger than

the slope toe. The part of the slope above the foundation is affected by several shear surfaces

running along it. Finally, a failure surface that starts form the base of the foundation until the

surface of the slope is also present, this surface may be generated in such a shallow depth due

to the horizontal load generated by the slope above the foundation and the big depth of the

active part. Figure IV -15 and IV -16 from appendix IV (h), the failure mechanism seems to start

with the development of the active zone, already present when the applied load reaches 50 %

of the ultimate load (Figure IV -15). When the load reaches 75% of the ultimate load, the main

shear surface is located from the slope above the foundation passing throw its bottom left

corner and continuing in direction to the slope. This surface presents great strains at this point

what indicates a counter clockwise tilting of the foundation. The rest of the failure mechanism

seen in Figure 4-24 is developed after 75% of the ultimate load. The slope above the

foundation is more affected by shear surfaces which increases the horizontal force in the

foundation, this generate the local shear failure that is present in the figure as well as a change

in the direction of tilting.

From Figure 4-25, an ultimate load before failure of -1012 kN/m/m is reached with a vertical

displacement of -0,162 m.


44

I) FOUNDATION IN THE MIDDLE OF A 30º ANGLE SLOPE, DOUBLE EMBEDMENT

GEOMETRY




Figure 4-26: Incremental displacements plot from ultimate bearing capacity calculations of a foundation in the middle of a 30º slope.

Figure 4-27: Deviatoric strain from ultimate bearing capacity calculation of foundation in the middle of a 30º slope.


45

Figure 4-28: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation in the middle of a 30º slope.

Figures 4-26 and 4-27 show the failure mechanism of this case. On them a slope failure is

triggered by the local shear failure of the foundation. The slope above the foundation is heavily

affected by shear surfaces running along all its length. Also, the toe of the slope under the

foundation is affected by nearly horizontal shear surfaces, what indicates and horizontal force

pushing from the surface of the slope all along until the shear surface. At the look of Figure IV -

17 and IV -18 from the appendix IV (i), the development of the final failure mechanism starts

quite late on the loading history. When the applied load reaches 50% of the ultimate load, the

active zone under the foundation is present but not fully developed as well as the transitions

zones on both sides. When the applied load reaches 75% of the ultimate load, the part of the

slope above the foundation starts to be affected by several shear failures, this increases the

horizontal load on the foundation. At this point the active and transition zones under the

foundation are clearly present, the strain in the left side of the active zone is also considerable

in relation to the rest, resembling some counter clockwise tilting. In the last part of the

loading, the failure in the top part of the slope increases considerably, and with it the

horizontal load in the foundation. This horizontal load generates the nearly horizontal failure

surfaces reaching the toe of the foundation. The strain in the foundation failure (active and

transition zones) is also increased.

The reached load before failure is -600,2 kN/m/m with a vertical displacement of -0,098 m


46

J) FOUNDATION ON FLAT TERRAIN, DOUBLE EMBEDMENT GEOMETRY




Figure 4-29: Incremental displacements plot from ultimate bearing capacity calculations of a foundation on flat terrain

Figure 4-30: Deviatoric strain from ultimate bearing capacity calculation of foundation on flat terrain


47

Figure 4-31: Vertical displacement vs load plot of the centre point from ultimate bearing capacity calculation of a foundation on flat terrain.

Figure 4-29 and 4-30 show the failure mechanism for this case. It is a local shear failure

from a foundation. The mechanism is symmetrical due to the isotropy of the model. It has an

active zone under the foundation and one passive zone on each side. Since is local shear failure

the shear surfaces don’t reach the surface of the soil. When looking at Figures IV -19 and IV -20

from appendix IV (j), the development of the failure mechanism can be explained. Initially the

active zone becomes present, when the load reaches 50% of the ultimate load this zone is still

developing (Figure IV -19). Then the passive zones start to appear, and the active zone

becomes smaller adopting the characteristic triangular shape. This can be seen when the load

reaches 75% of the ultimate load (Figure IV -20). Having this failure mechanism, symmetrical

and so close to the described by the theory reassures that the model is made correctly.

For the present case, the reached load before failure is – 1379 kN/m/m with a vertical

displacement of – 0,173 m


48

REACHED LOADS AND VERTICAL DISPLACEMENTS

Table 5: Ultimate bearing capacity results from the Plaxis calculations for each case

Case Maximum Load (kN/m/m) Maximum Vertical Displacement (m)

Original Geometry

Double Embedment

Original Geometry

Double Embedment

Top, 20º -554,4 -871,3 -0,138 -0,176

Top, 30º -223,4 -445,2 -0,067 -0,111

Middle, 20º -666 -1012 -0,155 -0,162

Middle, 30º -423,5 -600,2 -0,118 -0,098

Flat -1105 -1379 -0,175 -0,173

Table 5 collects all the results from the bearing capacity calculation using Plaxis. This will allow

an easier comparison later on. Figure 4-32 collects the vertical displacement vs load curves

from all the cases allowing to compare them under the same scale.

Figure 4-32: Load-Settlement curve gathering all cases from original and double embedment geometry

-0,2

-0,18

-0,16

-0,14

-0,12

-0,1

-0,08

-0,06

-0,04

-0,02

0

-1600-1400-1200-1000-800-600-400-2000

Ver

tica

l Dis

pla

cem

ent

[m]

Load [kN/m/m]

Load-Settlement

flat, original top 20º, originaltop 30º, original middle 20º, originalmiddle 30º, original flat, double embedmenttop 20º, double embedment top 30º, double embedmentmiddle 20º, double embedment middle 30º, double embedment


49

4.2 CYCLIC LOADING: CONSTANT VERTICAL FORCE RATIO

4.2.1 Input

The used soil parameters can be found in Table 3, being the same ones used for the

bearing capacity calculations. The applied forces for each case are calculated according to the

method described in Chapter 3.3.3 for keeping a constant ratio between bearing capacity and

applied load of 3,7. The used loads can be found in Table 6.

Table 6: Calculated forces for each case according to the method described in Chapter 3.3.3

Case Dead Load [kN/m] Maximum Load [kN/m]

Top of slope (20º) -210,7 -421,1

Top of slope (30º) -84,53 -169,1

Middle of slope (20º) -252 -504

Middle of slope (30º) -160 -320,5

4.2.2 Output

FOUNDATION ON TOP OF A 20º SLOPE

The next set of Figures shows the obtained deformations in the mesh, and the total

principal strains after the first cycle and after 100 cycles.

Figure 4-33: Deformations from a foundation on the top of a 20º slope with cyclic loading at constant load rate after 100 cycles


50

Figure 4-34: Total principal strain from a foundation on top of a 20º slope cyclically loaded with constant rate load ratio after 100 cycles

Figure 4-35: Total principal strain from a foundation on top of a 20º slope cyclically loaded with constant rate load ratio after the first cycle


51

The three following Figures gather the information related with vertical displacements as

well as amplitude of deformations within each cycle.

Figure 4-36: Vertical displacement of a foundation on top of 20º slope loaded with constant load ratio after 100 cycles. Vertical displacements taken in three points: middle (blue), right edge (red), left edge (purple).

Figure 4-37: Vertical displacement from a foundation on top of a 20º slope cyclically loaded with constant load ratio. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Ver

tica

l dis

pla

cem

ent

[m]

No. Cycles

Vertical Displacement

Displacement Dead Load Displacement Max Load


52

Figure 4-38: Deformations from a foundation in the top of a 20º slope loaded with constant load ratio. Total, plastic and elastic deformation within each cycle.

The shear surface develops starting from the edges of the foundation (Figure 4-35). At the

end of the cycles the active zone is fully present (Figure 4-34), the shear surface at the end

looks like the start of a foundation failure. The vertical displacement shows an inflexion point,

before the vertical displacement in each cycle is smaller than after. Passed the inflexion the

deformation per cycle is also fairly constant (Figure 4-37 and 4-38). Clockwise tilting is also

seen after the inflexion point (Figure 4-33 and 4-36). The plastic deformation in each cycle

seems to slightly increase until the inflexion point and remain with constant amplitude for the

rest of the cycles (Figure 4-38).

• Initial displacement with dead load is applied for first time: -0,012 m

• Displacement when first maximum train load is reached: -0,027 m

• Displacement at maximum train load in the last cycle: -0,061 m

• Displacement after last cycle: -0,059 m

• Inflexion point around cycle 32

• Tilt of the vertical displacement curve after inflexion: 0,02437

-0,016

-0,014

-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0,002

0 20 40 60 80 100 120D

efo

rmat

ion

[m

]

No. Cycle

Deformations/Cycle

Total Deformation

Plastic Deformation

Elastic Deformation


53




Figure 4-39: Deformations from a foundation on the top of a 30º slope with cyclic loading at constant load rate after 100 cycles

Figure 4-40: Total principal strain from a foundation on top of a 30º slope cyclically loaded with constant rate load ratio after 100 cycles


54

Figure 4-41: Total principal strain from a foundation on top of a 30º slope cyclically loaded with constant rate load ratio after the first cycle

The three following Figures gather the information related with vertical displacements as well

as amplitude of deformations within each cycle.

Figure 4-42: Vertical displacement of a foundation on top of 30º slope loaded with constant load ratio after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).


55

Figure 4-43: Vertical displacement from a foundation on top of a 30º slope cyclically loaded with constant load ratio. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

Figure 4-44: Deformations from a foundation in the top of a 30º slope loaded with constant load ratio. Total, plastic and elastic deformation within each cycle

-0,016

-0,014

-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles



-0,007

-0,006

-0,005

-0,004

-0,003

-0,002

-0,001

0

0,001

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles

Deformation/Cycle

Total Deformation

Plastic Deformation

Elastic Deformation


56

The shear surface also develops from the edge of the foundation until the active zone is

developed (Figure 4-40 and 4-41). In this case the shear surfaces after 100 cycles seems to be

tilted towards the slope. The inflexion point is also present but not so obvious as in other cases

(Figure 4-43). A clockwise tilting is happening from the start of the loading, this can be seen in

Figure 4-42. The plastic deformation within the cycles Is small for this case also increasing after

the inflexion (Figure 4-44), the elastic deformation is almost constant for every cycle with a

very slight increase. The late development of the inflexion point can be an explanation of the

small plastic deformations obtained in this model.







FOUNDATION IN THE MIDDLE OF A 20º SLOPE



Figure 4-45:Deformations from a foundation in the middle of a 20º slope with cyclic loading at constant load rate after 100 cycles


57

Figure 4-46: Total principal strain from a foundation in the middle of a 20º slope cyclically loaded with constant rate load ratio after 100 cycles

Figure 4-47: Total principal strain from a foundation in the middle of a 20º slope cyclically loaded with constant rate load ratio after the first cycle




58

Figure 4-48: Vertical displacement of a foundation in the middle of 20º slope loaded with constant load ratio after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).

Figure 4-49:Vertical displacement from a foundation in the middle of a 20º slope cyclically loaded with constant load ratio. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles




59

Figure 4-50: Deformations from a foundation in the middle of a 20º slope loaded with constant load ratio. Total, plastic and elastic deformation within each cycle

As seen in the previous cases the shear surface increases from the edges of the

foundation (Figure 4-46 and 4-47). At the end of the cycles the active zone is fully developed.

For this case the shear surfaces are tilted towards the slope (Figure 4-46). Also, the lower part

of the slope above the foundation is slightly affected by this shear surfaces. A clear inflexion

point is present. A bit before it the plastic deformations within each cycle starts increasing

(Figure 4-50 and 4-49) until they remain constant when the inflexion is passed. The clockwise

tilting of the foundation is also increasing until this inflexion, after it the foundation settles

evenly (Figure 4-48). The elastic deformation remains nearly constant with a slight increase

(Figure 4-50).







-0,016

-0,014

-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0,002

0 20 40 60 80 100 120D

efo

rmat

ion

[m

]

No. Cycles

Deformations/Cycle

Total Deformation Plastic Deformation Elastic deformation


60




Figure 4-51: Deformations from a foundation in the middle of a 30º slope with cyclic loading at constant load rate after 100 cycles

Figure 4-52: Total principal strain from a foundation in the middle of a 30º slope cyclically loaded with constant rate load ratio after 100 cycles


61

Figure 4-53: Total principal strain from a foundation in the middle of a 30º slope cyclically loaded with constant rate load ratio after the first cycle



Figure 4-54: Vertical displacement of a foundation in the middle of 30º slope loaded with constant load ratio after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).


62

Figure 4-55:Vertical displacement from a foundation in the middle of a 30º slope cyclically loaded with constant load ratio. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

Figure 4-56: Deformations from a foundation in the middle of a 30º slope loaded with constant load ratio. Total, plastic and elastic deformation within each cycle

At the end of the 100 cycles the active part of a shear failure in a foundation is fully

developed. In this case also a shear surface reaching the toe of the foundation is present and

one affecting great part of the slope above the foundation (Figure 4-52). Already in the first

cycle the shear surface affecting the slope above the foundation is present (Figure 4-53). A

inflexion point in the vertical displacement is again present (Figure 4-55), before the inflexion

the clockwise tilt of the foundation increases with the cycles remaining sTable after this point

(Figure 4-54). Also the plastic deformation can be related with the inflexion point, increasing in

each cycle before the inflexion and remaining with constant amplitude in the cycles after it

-0,045

-0,04

-0,035

-0,03

-0,025

-0,02

-0,015

-0,01

-0,005

0

0 20 40 60 80 100 120D

isp

lace

men

t [m

]

No. Cycle



-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0,002

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycle

Deformations/Cycle



63

(Figure 4-56). The elastic component of the deformation seems to be fairly constant with a

slight increase with the cycles.







4.3 CYCLIC LOAD: VERTICAL REAL LOAD

4.3.1 Input

The used soil parameters can be found in Table 3. Also, Table 7 shows the calculated

forces applied to each case. Since the load is too big for using with the original geometry, the

side of the foundation will be maintained at 2,8 m but the embedment will be doubled, 2,24m.

Table 7: Applied load for real load cases. Dead load corresponds to the dead load of the bridge that acts permanently, and maximum load is the maximum total load reached when the train passes

Case Dead Load [kN/m] Maximum Load [kN/m]

(Same to all) -336 -671,1

4.3.2 Output





64

Figure 4-57: Deformations from a foundation on top of a 20º slope with cyclic loading at real load after 100 cycles

Figure 4-58: Total principal strain from a foundation on top of a 20º slope cyclically loaded with real load after 100 cycles


65

Figure 4-59: Total principal strain from a foundation on top of a 20º slope cyclically loaded with real load after the first cycle



Figure 4-60: Vertical displacement of a foundation on top of 20º slope loaded with real load after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).


66

Figure 4-61:Vertical displacement from a foundation on top of a 20º slope cyclically loaded with real load. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

Figure 4-62: Deformations from a foundation on top of a 20º slope loaded with real load. Total, plastic and elastic deformation within each cycle

The active part of the shear failure is fully developed after the 100 cycles (Figure 4-58)

starting at the end of the first cycle from the edges of the foundation (Figure 4-59). In the

vertical displacements an inflexion point is present quite early in the cycles (Figure 4-61) after

the inflexion, the foundation tilts counter clockwise, some cycles after tilts clockwise (Figure 4-

60) but in general remains fairly flat. This can also be related with the plastic deformations

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles


Deformation Dead Load Deformation Max Load

-0,02

-0,015

-0,01

-0,005

0

0,005

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Deformations/Cycle

Total Deformation

Plastic Deformation

Elastic Deformation


67

within each cycle shown in Figure 4-62. The plastic deformation amplitude increases after the

inflexion point, remaining stable for some cycles and slightly decreasing, the start of this

decrease can be matched with the cycles when the foundation starts tilting clockwise. The

elastic deformation is nearly constant.










Figure 4-63: Deformations from a foundation on top of a 30º slope with cyclic loading at real load after 100 cycles


68

Figure 4-64: Total principal strain from a foundation on top of a 30º slope cyclically loaded with real load after 100 cycles

Figure 4-65: Total principal strain from a foundation on top of a 30º slope cyclically loaded with real load after the first cycle




69

Figure 4-66: Vertical displacement of a foundation on top of 30º slope loaded with real load after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).

Figure 4-67:Vertical displacement from a foundation on top of a 30º slope cyclically loaded with real load. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

-0,12

-0,1

-0,08

-0,06

-0,04

-0,02

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles


Displacement Dead Load Displacement Dead Load


70

Figure 4-68: Deformations from a foundation on top of a 30º slope loaded with real load. Total, plastic and elastic deformation within each cycle

For this case the active area of the shear failure is developed quite early in the cycles, in

Figure 4-65 is possible to see that after the first cycle it is almost formed. At the end of the 100

cycles it is fully developed, and it reaches great depth, seems to be tilted towards the slope.

Also, a shear surface towards the toe of the slope is starting to be present as well as one

affecting the soil on the left of the foundation. These two last surfaces can be the start of a toe

failure of the slope (Figure 4-64). The inflexion point in the vertical displacement curve (Figure

4-67) develops very early in the cycles. The foundation starts tilting form the first cycle,

increasing the rotation after the inflexion and remaining stable for the last cycles with even

settlement (Figure 4-66). This can be matched with the plastic deformation within each cycle

from Table 4-68, which increases the amplitude after the inflexion for later on decreasing it

and finally remain stable. The cycles with constant plastic amplitude are the same where the

foundation stops tilting.







-0,03

-0,025

-0,02

-0,015

-0,01

-0,005

0

0,005

0 20 40 60 80 100 120D

efo

rmat

ion

[m

]

No. Cycles

Deformations/Cycle

Total deformation

Plastic Deformation

Elastic Deformation


71




Figure 4-69: Deformations from a foundation in the middle of a 20º slope with cyclic loading at real load after 100 cycles

Figure 4-70: Total principal strain from a foundation in the middle of a 20º slope cyclically loaded with real load after 100 cycles


72

Figure 4-71: Total principal strain from a foundation in the middle of a 20º slope cyclically loaded with real load after the first cycle



Figure 4-72: Vertical displacement of a foundation in the middle of 20º slope loaded with real load after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).


73

Figure 4-73:Vertical displacement from a foundation in the middle of a 20º slope cyclically loaded with real load. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

Figure 4-74: Deformations from a foundation in the middle of a 20º slope loaded with real load. Total, plastic and elastic deformation within each cycle

The present shear surfaces after 100 cycles are a fully developed active zone under the

foundation as well as some shear surfaces in the part of the slope above the foundation, as it

can be seen in Figure 4-70. As in the other cases the shear surfaces start from the edges of the

foundation (Figure 4-71). The vertical displacement curve from Figure 4-73 resembles an

inflexion point (Figure 4-73). From this inflexion point on the plastic deformation seems to

increase in amplitude in the following cycles for later on slightly decrease and remain stable

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120D

isp

lace

men

t [m

]

No. Cycles



-0,016

-0,014

-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0,002

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Deformations/Cycle

Plastic Deformation Elastic deformation Total Deformation


74

(Figure 4-74). The tilting of the foundation is present since the first cycle but increase more

rapidly after the inflexion point it also seems to stop once the plastic deformation passes its

maximum and stabilizes (Figure 4-72).










Figure 4-75: Deformations from a foundation in the middle of a 30º slope with cyclic loading at real load after 100 cycles


75

Figure 4-76: Total principal strain from a foundation in the middle of a 30º slope cyclically loaded with real load after 100 cycles

Figure 4-77:Total principal strain from a foundation in the middle of a 30º slope cyclically loaded with real load after the first cycle




76

Figure 4-78: Vertical displacement of a foundation in the middle of 30º slope loaded with real load after 100 cycles. Vertical displacements taken in three points: middle (blue), right edge (red), left edge (purple).

Figure 4-79:Vertical displacement from a foundation in the middle of a 30º slope cyclically loaded with real load. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

-0,09

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles


Displacemenet Dead Load Displacement Max Load


77

Figure 4-80: Deformations from a foundation in the middle of a 30º slope loaded with real load. Total, plastic and elastic deformation within each cycle

From Figure 4-76, a fully active failure zone under the foundation is developed after 100

cycles. Also, there is a shear surface reaching the toe of the slope and one running along the

part of the slope above the foundation, strains in these two areas can be seen since the first

cycle (Figure 4-77). An inflexion point is present in the curve from Figure 4-79, developed early

during the cyclic loading. Clockwise tilting is present from the first cycle, seeming to be in the

middle part of the cycles but increasing again during the last part of the cyclic loading (Figure

4-78 and 4-75). Plastic deformation increases within each cycle after the inflexion point

remaining fairly constant after cycle 20. According to the elastic deformation it seems to be

constant (Figure 4-80).







-0,02

-0,015

-0,01

-0,005

0

0,005

0 20 40 60 80 100 120D

efo

rmat

ion

[m

]

No. Cycle

Deformations/Cycle



78

FOUNDATION IN FLAT TERRAIN



Figure 4-81: Deformations from a foundation on flat terrain with cyclic loading at real load after 100 cycles

Figure 4-82: Total principal strain from a foundation on flat terrain cyclically loaded with real load after 100 cycles


79

Figure 4-83:Total principal strain from a foundation on flat terrain cyclically loaded with real load after the first cycle



Figure 4-84: Vertical displacement of a foundation on flat terrain loaded with real load after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red), right edge (purple).


80

Figure 4-85:Vertical displacement from a foundation on flat terrain cyclically loaded with real load. Displacement of the middle point at no train load (blue) and with the maximum train load (orange)

Figure 4-86: Deformations from a foundation on flat terrain loaded with real load. Total, plastic and elastic deformation within each cycle

A fully developed active zone can be seen under the foundation in Figure 4-82, extending

from the edges of the foundation from the first cycle (Figure 4-83). The inflexion point in the

vertical displacement curve is also present but several cycles after it the vertical displacement

along the cycles seems to slow down (Figure 4-85). From Figure 4-81 and 4-84 no tilting is

present. Very small plastic deformation happens until the inflexion point is reached, after it the

amplitude of the plastic deformation within each cycle increases until cycle 60 approximately

when it reduces the amplitude and remain fairly constant for the rest of the cycles (Figure 4-

86). This reduction in the plastic deformation amplitude seems to match with the point when

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120D

isp

lace

men

t [m

]

No. Cycles

Vertical Displacements


-0,016

-0,014

-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0,002

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Deformation/Cycle



81

the vertical displacement starts increasing more slowly. The elastic deformation is nearly

constant with a slight increase.








82

4.4 CYCLIC LOADING: VERTICAL AND HORIZONTAL REAL LOAD

4.4.1 Input

For these calculations, soil parameters from Table 3 where used. As cyclic vertical load,

the values can be found in Table 7. The cyclic horizontal load and momentum will be applied

simulating a train passing from left to right in one case and from right to left in the other. The

values can be seen in Table 8. The geometry for a foundation on top of a 20º slope cyclically

loaded with vertical real load will be used for these cases.

Table 8: Cyclic horizontal load and momentum for simulating braking force.

Case Horizontal Load [kN/m] Momentum [kN·m/m]

Left to right 6,56 -26,25

Right to left -6,56 26,25

4.4.2 Output

BRAKING FORCE LEFT TO RIGHT



Figure 4-87: Deformations from a foundation on top of a 20º slope with cyclic loading (vertical and horizontal left to right) at real load after 100 cycles


83

Figure 4-88: Total principal strain from a foundation on top of a 20º slope cyclically loaded (vertical and horizontal left to right) with real load after 100 cycles

Figure 4-89:Total principal strain from a foundation on top of a 20º slope cyclically loaded (vertical and horizontal left to right) with real load after the first cycle




84

Figure 4-90: Vertical displacement of a foundation on top of a 20º slope loaded with real load (vertical and horizontal left to right) after 100 cycles. Vertical displacements taken in three points: middle (blue), left edge (red),

right edge (purple).

Figure 4-91:Vertical displacement from a foundation on top of a 20º slope cyclically loaded (vertical and horizontal left to right) with real load. Displacement of the middle point at no train load (blue) and with the maximum train

load (orange)

-0,1

-0,09

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles




85

Figure 4-92: Deformations from a foundation on top of a 20º slope loaded with real load (vertical and horizontal left to right). Total, plastic and elastic deformation within each cycle

After 100 cycles a fully developed active zone under the foundation is present, bigger

strain can be seen in the right shear surface of the active zone (Figure 4-88). It started from the

edges of the foundation after the first cycle (Figure 4-89). Also, a shear surface is located on

the left of the foundation starting in the bottom left edge and reaching the surface of the flat

terrain. Increasing tilting of the foundation can be seen in Figure 4-90, starting from the first

loading cycle but increasing more rapidly along the cycles after the inflexion point in the

vertical displacement curve (Figure 4-91). Plastic deformation within each cycle seems to

increase around the inflexion point until its maximum around cycle 20. After this is remains

relatively constant with a slight decrease (Figure 4-92). Elastic deformation seems constant for

all cycles.







-0,02

-0,015

-0,01

-0,005

0

0,005

0 20 40 60 80 100 120D

efo

rmat

ion

[m

]

No. Cycle

Deformations/Cycle



86

BRAKING FORCE RIGHT TO LEFT



Figure 4-93: Deformations from a foundation on top of a 20º slope with cyclic loading (vertical and horizontal right to left) at real load after 100 cycles

Figure 4-94: Total principal strain from a foundation on top of a 20º slope cyclically loaded (vertical and horizontal right to left) with real load after 100 cycles


87

Figure 4-95:Total principal strain from a foundation on top of a 20º slope cyclically loaded (vertical and horizontal right to left) with real load after the first cycle



Figure 4-96: Vertical displacement of a foundation on top of a 20º slope loaded with real load (vertical and horizontal right to left) after 100 cycles. Vertical displacements taken in three points: middle (blue), right edge (red),

left edge (purple).


88

Figure 4-97:Vertical displacement from a foundation on top of a 20º slope cyclically loaded (vertical and horizontal right to left) with real load. Displacement of the middle point at no train load (blue) and with the maximum train

load (orange)

Figure 4-98: Deformations from a foundation on top of a 20º slope loaded with real load (vertical and horizontal right to left). Total, plastic and elastic deformation within each cycle

At the look of Figure 4-94, a fully developed active zone is present under the foundation.

The left shear surface delimiting this zone presents bigger strains than the right one. Also, this

can be seen in Figure 4-95 which shows the strain after the first cycle. This makes perfect sense

when looking at Figure 4-96 and 4-93, which show a counter clockwise tilting of the

foundation. This tilting is present since the first cycles but increases more rapidly in the last

cycles (Figure 4-96). An inflexion point is present in the vertical displacement curve (Figure 4-

-0,09

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Dis

pla

cem

ent

[m]

No. Cycles



-0,02

-0,015

-0,01

-0,005

0

0,005

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Deformation/Cycle

Toatal Deformation Plastic Deformation Elastic deformation


89

97), during the cycles after the inflexion the amplitude of the plastic deformations within each

cycle starts increasing until it reaches the maximum around cycle 30. Elastic deformation

seems fairly constant with very slightly increase (Figure 4-98).








90

4.5 ANALYSIS OF THE RESULTS Several plots and tables are developed with relevant obtained data.

Table 9 shows the reached safety factors obtained from the phi-c reductions made in each

case. Three calculations of this type are performed for each model: the first when the dead

load of the bridge is placed on the foundation for the first time, the second when the

maximum train load is reached for the first time and the last after the cycle 100 is over (this

last calculations are not shown in this table because due to the limitations of the model the

results from them are not realistic, a more detailed explanation about this can be found in

Chapter 5.2). Since this safety factors are performed with phi-c reduction calculations they

regard the shear resistance of the soil.

In Table 10, the reached vertical displacements obtained from the Plaxis models can be

found. The displacements are taking at four interesting points of the model’s load history: The

first time the dead load of the bridge is fully applied, the first time the maximum train load is

reached, the last time the maximum train load is reached and after 100 loading cycles only

with the dead load of the bridge.

Table 9: Safety factors reached with phi-c reduction calculations.

Case 1st Dead Load 1st Max Load

Constant Load Ratio

Top of slope (20º) 2,0 1,7


Middle of slope (20º) 1,9 1,6


Vertical Real Load





Flat Surface 3,7 2,6

Vertical and Horizontal Real Load

Top of slope (20º)-Left to right 2,0 1,7

Top of slope (20º)-Right to left 2,0 1,7


91

Table 10: Reached vertical displacements from each case

Case 1st Dead Load [m]

1st Max Load [m]

100 Cycles Max Load [m]

After 100 Cycles [m]

Inflexion Point Cycle Nº

Tilting after inflexion

Constant Load Ratio

Top of slope

(20º)

-0,012 -0,027 -0,061 -0,059 32 0,02437

Top of slope (30º)

-0,004 -0,011 -0,015 -0,014 84 0,03720

Middle of slope (20º)

-0,011 -0,026 -0,062 -0,060 40 0,02895


-0,006 -0,018 -0,039 -0,037 55 0,01849

Vertical Real Load

Top of slope

(20º)

-0,015 -0,033 -0,075 -0,072 17 0,02580

Top of slope (30º)

-0,018 -0,043 -0,108 -0,105 11 0,03735


-0,01 -0,025 -0,055 -0,052 32 0,01994


-0,011 -0,028 -0,083 -0,081 9 0,0339

Flat Surface -0,011 -0,026 -0,056 -0,054 53 0,01740

Vertical and Horizontal Real Load

Top of slope (20º)-Left to right

-0,015 -0,033 -0,091 -0,089 12 0,03673

Top of slope

(20º)-Right to left

-0,015 -0,033 -0,082 -0,079 15 0,02661

In order to see and compare the vertical displacements and deformations per cycle from

each case with the others, they will be placed in a Figure with the same scale. All the cyclic

loading cases with vertical load are gathered in Figure 5-1. With the deformation per cycle

data, two sets of plots are done: one for the cases with constant load ratio and other for the

cases with vertical real load. Each set will have three Figures regarding total deformation per

cycle, plastic deformation per cycle and elastic deformation per cycle. The elastic deformation

plot is quite clear but the other two have too much noise due to the numerical calculations. A

polynomial interpolation has been done to clean the noise and obtain the general trend of the

results. Other set of Figures including all cases with cyclic real load (vertical, and vertical and

horizontal) related with a foundation on top of a 20º slope and the flat case is also made, the

same procedure as commented before is also followed for this set.


92

4.5.1 Cyclic Loading with Vertical Load

The following Figures shows the vertical displacement curves for the cyclic loading cases

only with horizontal loads and the deformations within each cycle, one set of Figures for the

cases with constant load ratio and other for the real force cases

Figure 4-99: Vertical displacement curves from all cases with cyclic vertical load

-0,12

-0,1

-0,08

-0,06

-0,04

-0,02

0

0 20 40 60 80 100 120

Ver

tica

l Dis

pla

cem

ent

[m]

No. Cycles


top 20º, Real Load top 20º, Constant ratiotop 30º, Real Load top 30º, constant ratioMiddle 20º, Real Load middle 20º, constant ratiomiddle 30º, Real Load middle 30º, Constant Ratioflat, Real Load


93

CONSTANT FORCE RATIO DEFORMATIONS

Figure 4-100: Amplitude of the total deformation within each cycle for constant load ratio cases

Figure 4-101: Amplitude of the plastic deformation within each cycle for constant load ratio cases

Figure 4-102: Amplitude of the elastic deformation within each cycle for constant load ratio cases

-0,004

-0,003

-0,002

-0,001

0

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Total Deformation/Cycle. Constant force ratio

top 20º, constant ratio top 30º, constant ratio

middle 20º, constant ratio middle 30º, constant ratio

-0,0008

-0,0006

-0,0004

-0,0002

0

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Plastic Deformation /Cycle. Constant force ratio

top 20º, constant ratio top 30º, constant ratio

middle 20º, constant ratio middle 30º, constant ratio

-0,003

-0,0025

-0,002

-0,0015

-0,001

-0,0005

0

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Elastic Deformation/Cycle. Constant force ratio

top 20º, Constant Ratio top 30º, Constant Ratio

middle 20º, Constant Ratio middle 30º, Constant Ratio


94

VERTICAL REAL LOAD DEFORMATIONS

Figure 4-103: Amplitude of the total deformation within each cycle for vertical real load cases

Figure 4-104: Amplitude of the plastic deformation within each cycle for vertical real load cases

Figure 4-105: Amplitude of the elastic deformation within each cycle for vertical real load cases

-0,005

-0,004

-0,003

-0,002

-0,001

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Total Deformation/Cycle. Real Load

top 20º, real loadtop 30º, real load middle 20º, real loadmiddle 30º, real load flat, real load

-0,001

-0,0005

0

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Plastic Deformation /Cycle. Real Load

top 20º, real load

top 30º, real load middle 20º, real load

middle 20º, real load flat, real load

-0,0035

-0,003

-0,0025

-0,002

-0,0015

-0,001

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Elastic Deformation/Cycle. Real Load

top 20º, Real Load top 30º, Real Load

middle 20º, Real Load middle 30º, Real Load

flat, Real Load


95

4.5.2 Cyclic Loading with Horizontal and Vertical Load

Next four Figures contain the data related with the vertical displacement of the cyclic

cases with vertical and horizontal real load as well as the data for the deformations within

each cycle, the flat case and the foundation on top of a 20º slope with real load are also

present to allow comparisons.

Figure 4-106: Vertical displacement curves for all foundations on top of a 20º slope with real loading

-0,1

-0,09

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 20 40 60 80 100 120

Ver

tica

l Dis

pla

cem

ent

[m]

No. Cycle

Vertical displacement

top 20º, vertical force top flat, vertical load

top 20º, vertical and horizontal left to right top 20º, vertical and horizontal right to left


96

FOUNDATION ON TOP OF A 20º SLOPE REAL LOAD DEFORMATIONS

Figure 4-107: Amplitude of the total deformation within each cycle for real load foundations on top of a 20º slope

Figure 4-108: Amplitude of the plastic deformation within each cycle for real load foundations on top of a 20º slope

Figure 4-109: Amplitude of the elastic deformation within each cycle for a real load foundation on top of a 20º slope

-0,004

-0,003

-0,002

-0,001

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Total Deformation/Cycle

top 20º vertical flat, vertical


-0,0008

-0,0006

-0,0004

-0,0002

0

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Plastic Deformation/Cycle

top 20º, vertical flat, vertical


-0,003

-0,0025

-0,002

-0,0015

-0,001

0 20 40 60 80 100 120

Def

orm

atio

n [

m]

No. Cycle

Elastic Deformation/Cycle

top 20º, vertical top flat, vertical


Discussion

97

5 DISCUSSION

5.1 BEARING CAPACITY When comparing the results from the numerical simulations and the analytical methods

(Table 4 and Table 5), a good match can be seen. This increases the reliability of the results. As

the mentioned Tables show, the results from the models are very similar to the ones obtained

using Terzaghi’s method. Results from the French code are also in the range of the two

mentioned. These two methods only regard the case of foundations on top of a slope. Only

Meyerhof theory regards the case of foundations within the slope. For the case of a

foundation on top of the slope, Meyerhof results are in the range of 200 kN/m/m bigger than

the rest of the results. This is not a surprise since the theory is based on deep foundations.

When it comes to the foundations in the middle of the slope, Meyerhof results are also around

200 kN/m/m bigger than the ones obtained from Plaxis, this stable difference in the results

leads to think that the results from Plaxis related with the foundation in the middle of the

slope are also reasonable.

According to the results from Table 5, foundations placed in the middle of the slope have

bigger bearing capacity than the same ones on top of the slope. Also, bigger slope angles lead

to bigger reductions in bearing capacity. In the regarded cases, this effect is very remarkable

possibly by the proximity of the slope angle to the friction angle of the soil (since there is no

cohesion in the soil, the maximum allowed slope angle is equal to the friction angle). In

general, the cases with smaller or less shear surfaces tend to have less bearing capacity.

As mentioned in Chapter 2.2, the developed failure can be a foundation failure or slope

failure, or a combination of both. Ultimately, all the obtained failures resemble that the main

failure in the majority of the cases is related with the foundation. Since the aim of this set of

models is to find the bearing capacity, they can be considered as heavily loaded. As

commented in Pantelidis and Griffiths, 2014; heavily loaded foundations tend to show failures

related with foundations more than the slope.

When looking at the deviatoric strain plots from Chapter 4.1.3 al the failure mechanism

has at least one structure in common: a triangular active zone under the foundation with

transition zones on the sides is present resembling a local shear failure. This is also supported

by the shape of the curves in Figure 4-32, showing a great amount of plastic deformation

before failure (comparing with Figure 2-2). This basic structure is of course very affected by the

different characteristics of each case.

Figures 4-15 and 4-30 corresponding to the cases with foundations on flat terrain show a

clear local shear failure (compare with Figure 2-1). In these cases, the failure is symmetrical in

both sides of the foundation, changing the depth of the shear surfaces due to the change in

embedment. The differences from the rest of the models with this two are product only of

different slope angle and different position of the foundation in relation with the slope.

The cases with foundations on top of the slope with original embedment (Figure 4-2 and

4-6) present the same failure mechanism. The main parts of the local shear failure are present,

and they differ from the flat case on its lack of symmetry, being the failure developed on the

side of the slope. Both present also a shear surface running from the active zone towards the

surface of the slope but not reaching it. The case with 20º slope shows a longer shear surfaces

Discussion

98

and they affect the upper half part of the slope, when the slope is 30º the main shear surface

reaches nearly the toe of the slope.

Cases with original geometry and foundations in the middle of the slope have also the

same failure mechanism. The failure of this cases beside having the local shear failure

structure, seems that they also have an important component of slope failure. The active,

transition and passive zone is present in all of them. The part of the slope above of the

foundation presents failure signs in both cases (more important in the case with 30º slope).

This generates horizontal force in the foundation making the shear surface running from the

active zone towards the slope more active. In the case of 30º slope angle this failure reaches

the toe of the slope and is connected with the failure un the foundation above the foundation

(Figure 4-12). A deeper shear surface is present in both cases, it starts from the bottom left

corner of the foundation and resembles a base failure of the slope, it is more relevant in the

case with 20º angle slope (Figure 4-9).

For the cases with foundations on top of the slope and double embedment, the failure

mechanism seems very similar to the foundation in the middle of a slope with original

geometry but deeper. The local shear failure is again present with a bigger active zone than in

the cases with original geometry. Three well defined shear surfaces are also present but in

general does not reach the surface of the slope. The deepest starts from the bottom left

corner of the foundation and resembles the start of a base failure of the slope, more important

in the 20º case (Figure 4-18). Other runs from the active zone and seems to go in direction to

the toe of the slope but in both cases, it does not get close to the slope surface. The third one

starts from the middle of the active zone, nearly reaching the toe of the foundation in the case

of the 30º slope angle (Figure 4-21), the start of this surface seems to point towards the small

shear surface on the left of the foundation reaching the flat surface.

Finally, the cases with foundations in the middle of a slope and double of the original

embedment, are the ones with more complex failure mechanism. The active zone under the

foundation, and the transition zones are present in both cases what shows the local failure

component of the foundation. In the case of 20º slope angle (Figure 4-24) the transition and

passive zone in right are well developed and deep, no shear failure touches the slope surface.

In both cases the part of the slope above the foundation is heavily affected (important shear

surfaces along the full length of it in the case of 30º slope). This generates important horizontal

forces in the foundation, producing even a horizontal flat surface reaching the toe of the slope

in the 30º slope case (Figure 4-27). In both cases, shear surfaces with important strains are

present starting from the right bottom corner of the foundation and running in the opposite

direction than to the slope, this can be due to the horizontal force in the foundation that

makes it tilt clockwise. This surface is more relevant in the case with 20º slope angle.

The way failure mechanisms are developed is also quite similar for all the cases. First the

active zone is developed, in cases of 20º slope is normally already present when the load is

50% the ultimate load (Figure IV-11), cases with 30º slope tend to develop it after this load

(Figure IV-3). Also, before 50% of the ultimate load some strain is happening in the part of the

slope above the foundation for the cases with foundations in the middle of the slope (Figure

IV-7). In between 50% and 75% of the ultimate load, the part of the failure corresponding the

local shear failure of the foundation is generally developed (Figure IV-12). At this point nearly

all cases on slope, develop the shear failure starting from the active zone and pointing towards

the slope. In cases with foundation in the middle of the slope, the strain in the part above the

Discussion

99

foundation increases (This is more notable in cases with 30º slope). Finally, between 75% and

100% of the ultimate load the rest of the failure mechanism is developed.

The behaviour of the failure in foundations horizontally loaded according to the amount

of vertical load on them described by Taeseri 2017 (Figure 2-9), can be seen in some of the

models with bigger horizontal loads (foundation on the middle of a slope with double

embedment). When the foundations are not fully loaded, the principal shear failure tends to

develop in the same direction as the applied horizontal load (for these cases from left to right)

and generate a counter clockwise tilting. Figures resembling this are IV-16 and IV-18. When the

full load is applied, the mentioned shear surfaces seems to loose importance and the strain is

concentrated in surfaces with the opposite direction to the horizontal load applied, changing

the tilting direction (Figure 4-24 and 4-27).

5.2 CYCLIC VERTICAL LOADING Continuing with the topic of slope failure vs foundation failure, the constant load ratio

cases have a ratio of 3,7 between bearing capacity and applied load. This can be seen as a

traditional safety factor against vertical load for a foundation. When comparing the safety

factors obtained from the phi-c reduction calculations of this cases after applying the

maximum load (dead load and maximum train load) the results are around 1,7 for the cases

with 20º slopes and 1,3 for the cases with 30º slope. This safety factors are related with the

shear strength of the model, meaning that is more similar to the normal approach of analysing

slopes. It can be seen that the factors of safety about the half of the one against vertical load.

This proves two things: first the strong influence of the slope failure in the problem, and

second the necessity of checking both failure mechanism when assessing this kind of

problems.

On the other hand, this safety factors shows one limitation of the model. The safety

factors at the start and at the end of the cycles ares the same for every case. In reality, this is

not possible once deformation takes place. This is related with the activated dilatancy that the

model does not take into account for the safety calculation. To check it in the case of the

foundation on top of a 20º slope with double embedment geometry, the ultimate bearing

capacity load was placed in the foundation after the cycles. The result is that the foundation

can no longer hold this load, being reduced the bearing capacity by around a 10%.

Figure 4-99 and 4-106 show the vertical displacements along the cycles of the middle

point of the foundation in each case. It is possible to see that regardless of the case the general

shape of the displacement is very similar between all, showing an inflexion point that divides

the curve in two zones, before with a slower vertical displacement and after with faster

vertical displacement. Many of the obtained results can be related with this inflexion point.

The approximate cycle when the inflexion takes place for each case can be seen in Table 10.

Discussion

100

Beside when the inflexion appears the mechanism seems to be the same in all cases.

When comparing the plots of total principal strain from all cyclic cases (Chapter 4.2, 4.3 and

4.4) it can be seen for all cases that at the end of the 100 cycles a fully developed active zone

under the foundation is present. In the same way, after the first loading cycle a small start of

the shear surfaces that delimitate the active zone is present in the bottom edges of the

foundation. If this same plot is checked few cycles before the inflexion (Figure 5-1 a and c) the

two shear surfaces are already developed but they have not formed the triangular active zone.

Once this triangle is fully formed the inflexion takes place (Figure 5-1 b and d).

According to the sinking speed after the inflexion point, it seems to be highly related with

slope angle. Table 10 shows that the slowest sinking speed is the foundation in the flat terrain,

the foundations in slopes of 20º have slower sinking speed than the cases in slopes of 30º.

Beside the foundation in flat terrain (Figure 4-62), all the cases present tilting of the

foundation. Except a couple of exceptions where there is a great amount of tilting in the first

cycle (Figure 4-23 and 4-35), the tilt of the foundation starts some cycles before the inflexion

and increases the amplitude per cycle until some cycles after the inflexion. At this point it

becomes stable. Also, a bigger amount of tilting is present in the cases with 30º slope.

Regarding the generated shear surfaces in the models, all of them show a fully developed

active zone under the foundation at the end of 100 cycles (as an example Figure 4-15 and 4-57

can be seen). The active appear tilted towards the slope in the cases of the foundation in the

middle of the slope, with a bigger tilt when the slope is 30º. Also, these cases show shear

surfaces running along the slope above the foundation, being maybe an explanation about the

a)

b)

c)

d)

Figure 5-1: Total principal strains few cycles before the inflexion point (a, c) and few cycles after (b, d) for

foundation on top of a slope with real load (a, b) and in the middle of a slope with constant load ratio (c, d).

Discussion

101

tilt of the active zone due to the horizontal force that the soil between the shear surface and

the surface of the slope generate in the left side of the foundation (explaining also why these

cases show the biggest rotations in the foundation). Also some cases with 30º slope show

shear surfaces resembling toe slope failures (see Figure 4-33 and 4-45) reassuring the

important role of the slope angle in the bearing behaviour of this problems.

When comparing the shear surfaces of the cyclic models and the bearing capacity ones is

clear that they do not follow the same path. Shear surfaces that look like slope failures are

present in some of the cyclic loading, but they are more common in the models loaded until

failure. The foundation failure developed in the bearing capacity cases is a local shear failure

compared to the cyclic cases which resembles a punching failure.

For concluding a look to the deformations per cycle will be made. From Figure 4-100 to

Figure 4-105 the curves with total, plastic and elastic deformations in the cases with cyclic

vertical load are gathered. Generally, the amplitude of the elastic deformation in each cycle is

rather constant with a very small increase along the cycles. The plastic deformation increases

the amplitude in each cycle until the inflexion point. After, it decreases the amplitude and

remains fairly constant (some fluctuations can be seen in certain cases, but it can be more to

the interpolation from the original results than a real fluctuation). Again, it is seen that the

value of the amplitude is more related to the slope angle than to other parameters. Slopes

with more inclination give more plastic deformation amplitude in the cases of constant load

ratio and smaller one in the cases with real load. The shape of the curves in Figure 5-6 can also

be related with the position of the slope, the increase and decrease in amplitude around the

inflexion point is bigger in the cases with the foundation on top of the slope.

In general, the cases can be classified as stepwise failure according to Goldscheider and

Gudehus, 1976. Even if there are some small variations in the amplitude of the plastic

deformation, it never disappears completely and it does not decreases constantly with the

cycles. Also, as pointed by Wichtmann, 2005; the first cycle is where the biggest plastic

deformation takes place. Although, for surely know what kind of behaviour related with the

plastic deformation more cycles have to be done.

5.3 CYCLIC VERTICAL AND HORIZONTAL LOADING Many of the observations made for the cyclic vertical loading cases are also applicable to

these cases. Figure 5-8 shows the relation between the vertical displacement curves, the flat

case and the foundation on top of a 20º slope with real loading are also added for comparison.

The case in flat terrain is the one with less vertical deformation and less deformation speed. Is

interesting how the rest cases follow the same curve at the start, until the case with horizontal

force from left to right reaches its inflexion point (this case hast the fastest vertical

displacement between the ones in this Figure). The other two cases continue throw the same

path some more cycles until the one with horizontal force from right to left increases the

sinking speed. When comparing the reached vertical displacements in Table 10, all the cases

on slope have the same displacements during the first cycle.

The foundations tilt according to the applied momentum and horizontal force, showing an

always increasing clockwise rotation for the case with horizontal force left to right (Figure 4-

71) and a counter clockwise for the other case (Figure 4-77), in this last case the rotation only

seems to increase with the cycles along the last cycles.

Discussion

102

The developed shear surfaces are, as in every case, the ones related with the active zone

under the foundation. These cases also show some particular shear surfaces from the bottom

edge of the foundation to the surface of the soil, in the right of the foundation for the case

with horizontal force from right to left (Figure 4-75) and on the left for the other case (Figure

4-69). These surfaces are developed by the rotation of the foundation.

Elastic deformation has the same behaviour as the cases with vertical load and does not

seem very affected by the horizontal loads and momentums (Figure 4-109). The biggest plastic

deformation amplitude in the cycle is seen in the case with horizontal force from left to right.

Also, when this deformation decreases after the inflexion, it has the slowest decrease from all

the cases. Beside the flat terrain the three other curves have the same shape.

Conclusion and Future Work

103

6 CONCLUSION AND FUTURE WORK

The purpose of the present thesis has been to study the bearing capacity of railway bridge

foundations related to slopes under heavy loads. As a conclusion it can be said:

• The bearing capacity and failure mechanism depends strongly in the position of the

foundation, angle of the slope and embedment. The angle of the slope seems to be

the one affecting more the bearing capacity, reducing it considerable with bigger slope

angles. Bigger embedment gives bigger bearing capacities and foundations in the

middle of the slope generally have also bigger bearing capacities, but they have the

inconvenient of the horizontal load generated by the part of the slope above them.

• Local shear failure is the predominant failure mode when the models where statically

loaded until failure. Some cases, generally related with big slope angles or big

horizontal loads generated by the slope, present a combination between slope failure

and foundation failure.

• Traditional bearing capacity safety factors are not accurate for this type of problems.

The influence of the slope has also to be taken into account with shear strength

calculations.

• The present shear failures when the foundations are cyclically loaded resembles a

punching failure. As in the static case, models with bigger slopes and horizontal forces

also have a slope failure component in the mechanism.

• An inflexion point from where the vertical displacement increases the speed during

the cyclic loading is present in all cases. This point is related with the formation of an

active zone under the foundation and mainly is affected by the slope angle and the

embedment.

• Deformations within each cycle tend to increase until the inflexion point and then

remain more or less stable. Elastic deformation is constant along all cycles with a slight

increase in amplitude. Plastic deformation increases its amplitude until the inflexion

point, after it the amplitude reduces slightly and the remains constant.

Since this is the first approach to the problem, further investigation has to be done.

Physical experiments can be performed as well as data collection from real sites. Also, the

cases can be tested with more slope angles in order to see the transition from 20º slope to

30º. Sea page can be added to the system to check how the failures and bearing capacities

vary with water in the system, this is important in cyclic loading due to the pore pressure

development. In this thesis the foundations where loaded with 100 cycles, to further

investigate the failure mechanism, would be interesting to see situation at a bigger number of

cycles. For a complete understanding, several geometries can be tested.

References

104

7 REFERENCES

• Chen, W.F. & Duan, L. (2000). Bridge Engineering Handbook. Boca Raton: CRC Press.

• Das, B.M. (2011). Principles of Foundation Engineering (7 ed.). Cengage Learning.

• DTU 13.11 & 13.12. French code of shallow foundations.

• Fryba, L. (1996). Dynamics of Railway Bridges. Academy of Sciences of the Czech

Republic, Prague.

• Goldscheider, M. & Gudehus, G. (1976). Einige bodenmechanische Probleme bei

Küstenund Offshore-Bauwerken. In Vortr•age zur Baugrundtagung 1976.

• Knappett, J.A. & Craig, F. (2012). Craig’s Soil Mechanics (8 ed.). New York: Spon Press.

• Lang, H.J. & Huder, J. & Amann, P. (2003). Bodenmechanik und Grundbau: Das

Verhalten von Böden und Fels und die wichtigsten grundbaulichen Konzepte. Springer

International Publishing.

• Masing, G. (1926). Eigenspannungen und Verfestigung Beim Messing. In In Proc. 2nd

Int. Congr. Appl. Mech. Zurich

• Meyerhof, G.G. (1957) The Ultimate Bearing Capacity of Foundations on Slopes. 4th

International Conference on Soil Mechanics and Foundation Engineering, 3, 384-386

• Pantelidis, L. & Griffiths, D.V. (2014). (2015). Footing on the Crest of Slope: Slope

Stability or Bearing Capacity?. In Engineering Geology for Society and Territory-Volume

2 (pp. 1231-1234). Springer International Publishing.

• PLAXIS. (2018). Material Models Manual. Retrieved 15 January 2019 from

https://www.plaxis.com/support/manuals/plaxis-2d-manuals/

• PLAXIS. (2018). Reference Manual. Retrieved 15 January 2019 from


• Taeseri, D. (2017). The Non-Linear Behaviour of Foundations in Slopes During Seismic

Events. Doctoral Thesis. ETH Zurich.

http://link.springer.com/chapter/10.1007/978-3-319-09057-3_215

http://link.springer.com/chapter/10.1007/978-3-319-09057-3_215



References

105

• Terzaghi, K. (1943). Theoretical Soil Mechanics, New York: Wiley.

• Unsworth, J.F. (2017). Design and Construction of Modern Steel Railway Bridges (2

ed.). Boca Raton: Taylor & Francis, CRC Press.

• Wichtmann, T. (2005). Explicit Accumulation Model for Non-Cohesive Soils Under Cyclic

Loading. Ruhr-University, Bochum.

Appendix I: Soil Model Selection

106

APPENDIX I: SOIL MODEL SELECTION

Two soil models were regarded before starting to make the calculations: Mohr-Coulomb

soil model and Small Strain Hardening soil.

Information about the Small Strain Hardening Soil model can be found in Chapter 2.5.

Mohr-Coulomb soil model is a perfectly plastic linear elastic model. The linear elastic

behaviour in this model is based in Hooke’s law and the plasticity rule in the Mohr-Coulomb

failure criterion (Plaxis, 2018). This criterion has a fixed yield surface independent on load or

strain history.

Two cases where tested with both soil models in order to determine which is more suitable:

bearing capacity of a foundation on top of a 20º slope with original geometry and bearing

capacity of a foundation in the middle of a 20º slope with original geometry.

Figure I-1: Incremental displacements of a foundation on top of a 20º slope, Mohr-Coulomb soil model.

Appendix I: Soil Model Selection

107

Figure I-2: Incremental displacements for a foundation in the middle of a 20º slope. Mohr-Coulomb soil model.

When comparing Figure I-1 and I-2 with Figure 4-1 and 4-8 (which corresponds to the

same cases calculated with Small Strain Hardening Soil), is quite clear that the results are not

the same. Taking a look to the reached maximum bearing capacity loads, in the case of the

foundation on top of the slope calculated with Mohr-Coulomb gives -315,95 kN/m/m; way

smaller than the reached value calculated with Small Strain and also not very close to the

results from the analytical methods. In the case of a foundation in the middle of the slope, the

reached loads are quite similar in both cases, being the maximum allowed load -622,71

kN/m/m from the Mohr-Coulomb calculations. Related to the failure mechanisms obtained

from the Mohr-Coulomb models, it is important to say that are completely different to the

ones from Small Strain Hardening Soil. They are way smaller and doesn’t resemble to any

expected failure mechanism.

Since the Small Strain hardening soil gives closer results to the analytical solutions, the

failure mechanisms seem more realistic and it properly models the non-linear changes in

stiffness of the soil, it is the models that is going to be used for all the studied models.

Appendix II: Load Control vs Prescribed Displacement

108

APPENDIX II: LOAD CONTROL VS PRESCRIBED DISPLACEMENT

Two ways of calculating the ultimate bearing capacity were tried:

• Force control: An arbitrary vertical big load is placed in the foundation and the model

is calculated. Once finished, the Mstage value of the load when the soil body collapse

is multiplied to the initial load giving the value of the maximum allowable load.

Chapter 3.3.

• Prescribed displacement: Follows the same procedure as the force control but

assigning a big vertical displacement to the foundation instead. The soil body collapses

before reaching the assigned vertical displacement and the load needed to reach this

displacement is checked in the program.

When comparing the results of these two calculation methods (figures in Chapter 4.1.3 for

load control and the figures in this appendix for prescribed displacement), some general

differences can be seen in all models. The loads and displacements obtained with prescribed

displacement are, in all cases, bigger than the ones from the calculations made with force

control. Also, the values of the loads calculated with prescribed displacement differ more from

the analytical results. This can be due to the way that prescribed displacement works, it

doesn’t allow rotation of the line where the displacement is applied (restricting also the

rotation of the foundation). This allows the deformation to continue once a great amount of

plasticity is reached (wobbly part in the vertical displacement vs load curves), increasing the

size of failure surfaces.

For these reasons, the force control seems to give more realistic results and was the

method used to calculate the rest of the models. Also, the cyclic loading cannot be done with

prescribed displacement since only the load is known.

Figure II-1: Incremental deviatoric strain from a foundation on top of a 20º slope. Prescribed displacement.


109

Figure II-2: Vertical displacement-load curve from a foundation on top o f a 20º slope. Prescribed displacement. Reached ultimate bearing capacity = -759,8 kN/m/m

Figure II-3: Incremental deviatoric strain from a foundation on top of a 30º slope. Prescribed displacement.


110

Figure II-4: Vertical displacement-load curve from a foundation on top of a 30º slope. Prescribed displacement. Reached maximum bearing capacity = -310 kN/m/m.

Figure II-5: Incremental deviatoric strain from a foundation in the middle of a 30º slope. Prescribed displacement.


111

Figure II-6: Vertical displacement-load curve from a foundation in the middle of a 30º slope. Prescribed displacement. Reached maximum bearing capacity = -451,7 kN/m/m.

An interesting observation was made when taking the vertical displacement from Figure

II-2 corresponding to the point right after the curve starts wobbling (-0,17 m) and ran the

model again with this value as prescribed displacement. The prescribed displacement results

are shown in figure II-7. The obtained failure, besides having a bigger surface, has a very

similar shape to the same case calculated with load control (Figure 4-1). The obtained load was

-580 kN/m/m, what is slightly bigger than the obtained for this model with force control. This

observation reassures that the models calculated with prescribed displacement does not fail

even with big amounts of plasticity, giving unrealistic ultimate bearing capacity loads.


112

Figure II-7: Incremental displacemets from a foundation on top of a 20º slope. Prescribed displacement = -0,17 m

Appendix III: Boundary effects

113

APPENDIX III: BOUNDARY EFFECTS

When using finite elements program like Plaxis, several considerations must be taking in

order to accurately model reality. One of the most important is properly set the boundaries of

the model and the structures in relation to them. To illustrate this, the model shown in Figure

III-1 represents a foundation in flat terrain cyclically loaded with real load. Same problem

without boundary conditions can be found in Chapter 4.3.2.

In this particular case, the foundation is placed on the left side of the model instead of in

the centre, what means that is closer from the left boundary than from the right.

Figure III-1: Deformed mesh of foundation in flat surface with boundary effects

Since the soil in the model is homogeneous and there is no other structures or loads

affecting the foundation, a vertical settlement with no tilting in the foundation is expected.

When comparing the principal strain plots from the same case with the foundation on the left

of the model (Figure IV-2, a) and in the centre (Figure IV-2, b) some differences can be seen.

With the foundation in the centre, the shear surfaces are symmetrical in both sides of the

foundation, thing that is not possible to say in the case with the foundation on the left.

An even clearer difference appears when comparing the vertical displacement plots of

both cases (Figure IV-3). In the case with no boundary effect (b), no tilting is present in the

foundation, what matches the expected result. When looking the case with boundary effect

(a), a counter clockwise tilt is present (notice right edge has less vertical displacement than the

left). This gives a result that cannot be trusted since it does not follow the real behaviour of the

foundation.


114

a)

b)

Figure III-2: Principal strain plot of foundation on flat terrain, cyclically loaded. a) Case with boundary effects. b) Case without boundary effects.

To avoid boundary effects in the calculated models, several considerations where taken:

• In the cases with foundations in flat terrain, the foundations where placed in the

centre of the model.

• The models are big enough to avoid having deformations and stresses affected by

the boundaries. This is checked comparing the stresses in the bottom boundary

from the first phase when there is no structures and loads, with the stresses in

the same place at the end of the calculation. The value must be close to ensure no

boundary effect.

• Vertical boundaries are set to allow vertical movements of the nodes within them

but not horizontal ones. The bottom boundary is fully fixed, and the surface is

fully free. In this way the effects of soil around the model are included.


115

a)

b)

Figure III-3: Vertical displacement of middle (blue), Left corner (red) and right corner (purple) of a foundation on flat terrain with cyclic loading. a) Case with boundary effects. b) Case without boundary effects.

Appendix IV: Ultimate Bearing Capacity, Shear Surfaces Development

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APPENDIX IV: ULTIMATE BEARING CAPACITY, SHEAR SURFACES

DEVELOPMENT

In the present appendix, Figures complementing the ones located in Chapter 4.1.3 will be

gathered. For each ultimate bearing capacity calculation case, two more Figures will be shown

at different Mstages during loading: at Mstage = 0,5 (halve of the ultimate load) and Mstage =

0,75 (three fourths of the ultimate bearing capacity).

This is made for the sake of finding how the shear surfaces develop along the loading

process and have a better understanding of the failure mechanism.

The comments about these figures can be found in Chapter 4.1.3 in the analysis of the

cases which they correspond.

A) FOUNDATION ON TOP OF A 20º SLOPE, ORIGINAL GEOMETRY

Figure IV-1: Incremental deviatoric strain for a foundation on top of a 20º slope with original geometry. Mstage = 0,5.


117


B) FOUNDATION ON TOP OF A 30º SLOPE, ORIGINAL GEOMETRY



118


C) FOUNDATION IN THE MIDDLE OF A 20º SLOPE, ORIGINAL GEOMETRY

Figure IV-5: Incremental deviatoric strain for a foundation in the middle of a 20º slope with original geometry. Mstage = 0,5.


119


D) FOUNDATION IN THE MIDDLE OF A 30º SLOPE, ORIGINAL GEOMETRY



120


E) FOUNDATION IN FLAT TERRAIN, ORIGINAL GEOMETRY

Figure IV -9: Incremental deviatoric strain for a foundation on flat terrain with original geometry. Mstage = 0,5.


121

Figure IV -10: Incremental deviatoric strain for a foundation on flat terrain with original geometry. Mstage = 0,75.

F) FOUNDATION ON TOP OF A 20º SLOPE, DOUBLE EMBEDMENT GEOMETRY

Figure IV -11: Incremental deviatoric strain for a foundation on top of a 20º slope with double embedment geometry. Mstage = 0,5.


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G) FOUNDATION ON TOP OF A 30º SLOPE, DOUBLE EMBEDMENT GEOMETRY



123


H) FOUNDATION IN THE MIDDLE OF A 20º SLOPE, DOUBLE EMBEDMENT GEOMETRY

Figure IV -15: Incremental deviatoric strain for a foundation in the middle of a 20º slope with double embedment geometry. Mstage = 0,5.


124


I) FOUNDATION IN THE MIDDLE OF A 30º SLOPE, DOUBLE EMBEDMENT GEOMETRY



125


J) FOUNDATION ON FLAT TERRAIN, DOUBLE EMBEDMENT GEOMETRY

Figure IV -19: Incremental deviatoric strain for a foundation on flat terrain with double embedment geometry. Mstage = 0,5.


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Figure IV -20: Incremental deviatoric strain for a foundation on flat terrain with double embedment geometry. Mstage = 0,75.

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Bearing Behaviour of Shallow Foundations Under Heavy Train