Journal Grupos de Pilotes

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Mathematical models and solution algorithms for computational design

of RC piles under structural effects

Abdurrahman Sahin

Bogazici University, Kandilli Observatory and Earthquake Research Institute, Department of Earthquake Engineering, 34684 Cengelkoy, Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 17 May 2010

Received in revised form 26 November 2010

Accepted 26 January 2011

Available online 4 February 2011

Keywords:

Pile foundation

Design of piles

Optimum structural design

RC design

Analytical method

Integral transforms

a b s t r a c t

The paper presents mathematical models and solution algorithms for RC pile design,

through scanning soil stratums from top to downwards with an interactive scanner band.

The equilibrium of transferred loads from the superstructure, friction forces and tip bearing

forces are considered for the design, which leads to optimum pile length. The most impor-

tant contribution of this research for designers is supplying an efficient tool to obtain opti-

mum pile length and reinforced concrete design of pile foundation systems. A program

package has been developed in MATLAB depending on the proposed algorithm. Soil behav-

iors depending on external effects, active and passive zone distributions are considered. All

possible effects in all freedom degrees are taken into account in design process. Stress and

strain distributions due to axial loads, bending moments, shear forces and torsional

moments may be monitored. The optimum pile length, cross section dimension and rein-

forcement details may be found by using developed algorithm.

In this study, a dynamic layer object has been produced to find optimum pile length. In

solution process, this layer is compressed to a scanner band to find more optimal pilelength. This scanner band moves from up to down and stops whenever optimum length

is found. During this movement, all bearing capacity controls are carried out considering

soil effects. Reinforced concrete design is also achieved depending on external loads. The

flowcharts and computer codes developed for pile design is presented in the paper in

detail. Finally, numerical solutions of axially and multi-loaded piles as deep foundation

systems are presented. The design procedure of given examples are demonstrated through

the proposed algorithms. The optimum length, cross section dimension and reinforcement

details are presented.

2011 Elsevier Inc. All rights reserved.

1. Introduction

Buildings and bridges on loose to medium dense sands are generally built on piles to limit settlements because the sur-

face ground layers are often not stiff enough to support the structures. Piles are column structures embedded in the ground

to support loads coming from superstructures. Piles are generally subjected to compressive (downward) loads. Hence the

majority of studies reported in the literature were on the behavior of piles under compressive loads.

The purpose of a pile foundation is to transmit the loads of a superstructure to the underlying soil while preventing exces-

sive structural deformations. The capacity of the pile foundation is dependent on the material and geometry of each individ-

ual pile, the pile spacing (pile group effect), the strength and type of the surrounding soil, the method of pile installation, and

0307-904X/$ - see front matter 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2011.01.037

Tel.: +90 216 516 34 21/539 637 53 38.

E-mail addresses:[email protected],[email protected]

Applied Mathematical Modelling 35 (2011) 36113638

Contents lists available at ScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a te / a p m
http://dx.doi.org/10.1016/j.apm.2011.01.037mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.apm.2011.01.037http://www.sciencedirect.com/science/journal/0307904Xhttp://www.elsevier.com/locate/apmhttp://www.elsevier.com/locate/apmhttp://www.sciencedirect.com/science/journal/0307904Xhttp://dx.doi.org/10.1016/j.apm.2011.01.037mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.apm.2011.01.037


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the direction of applied loading (axial tension or compression, lateral shear and moment, or combinations). Except in unu-

sual conditions, the effects of axial and lateral loads may be treated independently.

Prior to the development of reliable computer programs, the design of a single pile was based primarily on the ultimate

load capacity of the pile as determined from a load test or from semi-empirical equations.

In literature, many analytical and numerical methods have been developed for analyzing piles[118]. A majority of the

methods are based on the elastic continuum theory. Among these methods, the infinite layer approach, the finite layer

method, the boundary element method, and the variational approach can be considered as analytical approaches with the

advantages of efficiency and accuracy. Some computer packages have been developed for solving and designing piles[19,20].

This paper presents an alternative approach for analyzing the axially and multi-loaded piles in multilayered and non-

homogeneous soils. A novel optimum design approach for analyzing the reinforced concrete piles is presented. The paper

adopts the analytical solution of a multilayered elastic medium subjected to concentrated point loads. Then the analytical

solution of the multilayered elastic medium is applied to analyze the axially or multi-loaded piles in multilayered elastic

soils. It is found that the analytical solution can be used with ease for the analysis and prediction of piles in layered elastic

soils.

2. Analytical system modeling

Rational designs must be based on solutions in which equilibrium and force displacement compatibility are simulta-

neously satisfied. Ongoing research has resulted in the development of mathematical models for the pile/soil system which

permit analysis of the entire range of loaddisplacement response for single piles subjected to axial and/or lateral loads.

Methods have been developed for the design of pile groups in which the soilstructure interaction characteristics of singlepiles have been incorporated.

For analysis of a pile subjected to all three dimensional effects are described inFig. 1. In this figure, a schematic of an

embedded pile is shown in detail. In the discussions which follow, the pile is assumed to be in contact with the surrounding

soil over its entire length. Consequently, the embedded length and the total length of the pile are the same.

Fig. 1. Pile subjected to centric loads at the head and interaction with the soil.

3612 A. Sahin/ Applied Mathematical Modelling 35 (2011) 36113638


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The pile is assumed to have a straight centroidal axis (the z-axis, positive upward) and is subjected to centric loads

(P1, P2, P3, P4, P5 and P6) at the head of the pile. P1 andP2 loads are shear forces, P3 load is axial force, P4 and P5 loads are

bending moments and P6 load is torsional moment, respectively. These loads are transferred to pile from the structural

system.

Displacements parallel to the axis of the pile are denoted wand are negative in the positive z-direction. The pile material

is assumed to be linearly elastic for all levels of applied loads. Ultimate conditions referred to subsequently indicate that a

limit has been reached in which any additional head load would cause excessive displacements.

2.1. Pile behaviors depending on the load effects

Lateral earth stress coefficient is the ratio of lateral stress to vertical stress for cohesionless soils. There are three

situations for defining the lateral earth stress coefficient. These are at-rest, active, and passive situations. At-rest situation,

stress is determined as the lateral stress in the ground before any movements occurs. The active stress state is obtained

when pile moves away from the soil under the effect of lateral stress, and results from shear failure due to reduction of

lateral stress. The passive stress state is obtained when pile is pushed into the soil far enough to cause shear failure

within the mass due to increase of lateral stress. In literature, many empirically based and analytically derived theories

have been proposed for estimating lateral earth stress. In this study, active and passive zones are determined depending

on pile behaviors due to loading conditions. The transition from active to passive parts with transition areas have not

been taken into account. The active and passive zone distributions have been determined suitable with the technical

literature.

Pile behaviors in the soil differ depending on the transferred loads from the superstructure. The soil part starting activitywith pile movement is active zone and the activity is expressed withkacoefficient. The soil part resisting the pile movement

is passive zone and the behavior is expressed withkpcoefficient. Bothkaandkpare the horizontal earth pressure coefficients

and depend on transferred loads from the superstructure.

2.1.1. Pile behavior depending on shear forces (P1 and P2)

Although the usual application of a pile foundation results primarily in axial loading, there exist numerous situations in

which components of load at the pile head produce significant lateral displacements as well as bending moments and shears.

Unlike axial loads, which only produce displacements parallel to the axis of the pile (a one-dimensional system), lateral loads

may produce displacements and rotations in any direction.

A schematic of a laterally loaded pile in 1 and 2 directions is shown inFig. 2. Thexyplane is assumed to be a principal

plane of the pile cross section. Due to the applied head shears P1and P2, each point on the pile undergoes a translationu in

thex and y direction. Displacements and forces are positive if their senses are in a positive coordinate direction. The sur-

rounding soil develops pressures, denoted p in Fig. 2, which resists the lateral displacements of the pile.When there is a movement or effect in the direction ofP1andP2forces, the zone in back of the loads supports the activity

and the zone in front of the loads resists the activity.

2.1.2. Pile behavior depending on axial load (P3)

A schematic of an axially loaded pile is shown inFig. 3. In the discussion which follow, the pile is assumed to be in contact

with the surrounding soil over its entire length. Consequently, the embedded length and the total length of the pile are the

same.

When there is a movement or effect in the direction ofP3force, all soil zones resist the activity and become passive. Soil

surrounds the pile and resists the activity because of the friction forces. The distribution of passive zone in the soil is shown

inFig. 3.

The head loadP3is transferred to the surrounding soil by shear stresses (skin friction) along the lateral pile/soil interface

and by end-bearing at the pile tip (bottom of the pile).

2.1.3. Pile behavior depending on bending loads (P4 and P5)

A schematic of a laterally loaded pile in P4andP5directions is shown inFig. 4. Thexyplane is assumed to be a principal

plane of the pile cross section. Due to the applied head moment (P4andP5), each point on the pile undergoes rotationhabout

thexandyaxes. Displacements and forces are positive if their senses are in a positive coordinate direction. The surrounding

soil develops pressures, denoted p in Fig. 4, which resist the lateral displacements of the pile.

When there is a movement or effect in the direction ofP4 and P5 forces, the zone in back of the loads supports the

activity and the zone in front of the loads resists the activity. The distribution of active and passive zones in the soil is shown

inFig. 4.

2.1.4. Pile behavior depending on torsional load (P6)

Three-dimensional analysis of a single pile requires a relationship between the resistance of the soil and the torsional dis-

placement of the pile. There has been only limited investigation [2123]of this torquetwist relation because its effect issmall compared to the axial and lateral effects.

A. Sahin/ Applied Mathematical Modelling 35 (2011) 36113638 3613


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When there is a movement or effect in the direction ofP6 force, all soil components resists against the activity with the

pile movement. The distribution of passive zone in the soil is shown in Fig. 5.

In case of simultaneous loading of several basic loads, superposition is a powerful idea of obtaining solutions. According

to the superposition method, the total response at a given place and time caused by two or more stimuli is the sum of the

responses which would have been caused by each stimulus individually.

2.2. Numerical statements of soil parameters

According to Terzaghi Method the numerical soil parameters are calculated as follows[24,25]:

Active and passive lateral earth pressure is,

ka tg2 p

4U

2

2c tg

p4

U

2

; 1

kp tg2 p

4U

2

2c tg

p4

U

2

: 2

Bearing value of soil is,

qu k1 c Nc c1 Df Nq k2 c2 B Nc; 3

Fig. 2. A schematic of a laterally loaded pile in P1 and P2 directions and soil behaviors depending on these effects.



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where

Nq a2

2cos2 p4

U2

1; a e3p4 U2tgU; 4Nc cotU

a2

2cos2 p4

U2

1" #

; 5

Nc 1

2tgU

kpc

cos2 U 1

; kpc 3tg2

p4

U 33

2

: 6

For circular cross section,

L B D; k1 1:3; k2 0:3:

2.3. Determining active and passive lateral earth pressure coefficients (ka, kp) for each layers

Lateral earth pressure coefficients of stratums depend on not only soil parameters but also the behavior of the piles. All

soil parameters for each stratum are shown inFig. 6. The parameter symbols are written for programming languages. The

real symbols and program symbols of soil parameters and expressions are presented in following statements.

c {c} = Cohesion intercept of the soil at the point at which the strength is desired.

U {fi} = Angle of internal friction of the soil at the point at which the strength is desired.

c {gama} = Unit weight of soil.

Active and passive lateral earth pressure coefficients are calculated by using Terzaghi Method. Active lateral earth pres-

sure coefficient (ka) is defined as follows [24,25]:

ka tg

2 p

4

U

2

2c tg

p

4

U

2

: 7

Fig. 3. A schematic of axially loaded pile inP3 direction and soil behavior depending on this effect.



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Passive lateral earth pressure coefficient (kp) is defined as follows:

kp tg2 p

4U

2

2c tg

p4

U

2

: 8

3. Algorithm for assigning lateral earth pressure coefficients for each stratum

The lateral earth pressure coefficients of soil stratums vary depending on the soil activity. If soil components start activity

due to transferred loads from the superstructure, the lateral earth pressure coefficients are expressed with ka. If soil com-

ponents are passive and resist the pile activity due to transferred loads from the superstructure, the lateral earth pressure

coefficients are expressed with kp. In analysis package, these coefficients must be calculated for each stratum. The flow chart

of assigning lateral earth pressure coefficients for each stratum is shown in Fig. 7.

4. Reinforced concrete design of circular piles under different structural effects

Although the usual application of a pile foundation results primarily in axial loading, there exist numerous situations in

which components of load at the pile head produce significant lateral displacements as well as bending moments and shears.

Unlike axial loads, which only produce displacements parallel to the axis of the pile (a one-dimensional system), lateral loadsmay produce displacements in any direction. Unless the pile cross section is circular, the laterally loaded pile/soil system

Fig. 4. A schematic of a laterally loaded pile in P4 and P5 directions and soil behaviors depending on these effects.



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represents a three-dimensional problem. Most of the research on the behavior of laterally loaded piles has been performed

on piles of circular cross section in order to reduce the three-dimensional problem to two dimensions. Little work has been

done to investigate the behavior of noncircular cross section piles under generalized loading. In many applications, battering

of the piles in the foundation produces combined axial and lateral loads. However, the majority of the research on lateral

Fig. 5. A schematic of axially loaded pile inP6 direction and soil behavior depending on this effect.

Fig. 6. Soil parameters for each stratum.



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load behavior has been restricted to vertical piles subjected to loads which produce displacements perpendicular to the axisof the pile. In the discussion which follow, it is assumed that the pile has a straight centroidal vertical axis. If the pile is

Fig. 7. Developed flow chart for determining lateral earth pressure coefficients for each stratum.



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N Nb Nc rb ab rc ac rb ab 1 rcrb

ac

ab

: 14

If we assume that m rcrb

N rb ab1 l m rb1 l mab;

N ri ab;15

where

ri rb1 l m: 16

rinominal stress and represents associate behavior of concrete and steel. The nominal stress depends on material type andpercentage ratio as seen in Eq.(16).

Optimum design may be achieved by using axial loads, material properties, restrictions of codes and technical specifica-

tions. All data used in the design process are shown in Table 1. Their symbols used in program modules, data types and data

structures are also presented in the grid system.

By consideringFigs. 2 and 8,Table 1and Eqs.(9)(16), functions and procedures are prepared for design and are installed

into pile foundations design functions in the program.

Pile foundation system is designed considering transferred loads from the superstructure, pile own weight and soil ef-

fects. Section control is carried out considering dynamic loads like earthquake loads, wind loads, etc. Optimum design algo-

rithm is prepared depending on the grid system in Table 1and prepared functions. The flow chart for optimum design of

axially loaded pile is shown inFig. 9.

InFig. 9, the function which is presented with label 2 is increasing module. When a single pile is not able to bear the

transferred loads, number of piles is increased and pile cross section diameters are recalculated considering new pile num-

bers. The flow chart of the program module increasing pile numbers and redesigning is shown inFig. 10.

As it can be seen from the algorithm inFig. 10, the design module named as axial design is recalled here. All modules call

others in the program structure. These procedures continue until optimum design is achieved.

4.2. RC design considering simple bending effect

Firstly, numerical model of pile cross section under the effect of simple bending is modeled. The cross section is designed

considering P4 or P5 effect. The compression and tension zones and stress distributions of concrete and steel are shown in

Fig. 11. In this figure, the colorful area is compression zone.

The stress formulation is as follows:

r e E: 17

Table 1

All data, symbols, data types and data structures used in the program for optimum design of axially loaded piles.

Data used in software algorithm Unit Numerical

Symbol

MATLAB

Symbol

Data

Type

Concrete max. stress kg/cm2 rbu bgeru float U UConcrete min. stress kg/cm2 rba bgera float U UPercentage top limit % lu mu float U UPercentage bottom limit % la ma float U UPercentage % l pur float U USteel max. stress kg/cm2 rcu cgeru float U U

Steel min. stress kg/cm2

rca cgera float U UConcrete area cm2 ab ab float U

Steel area cm2 ac ac float U

Concrete cross section diameter cm Ub cpb float U

Rust part cm pp pp float U U

Nominal stress kg/cm2 ri bgeri float UBar diameter cm Ud cpc int U

Area of a bar cm2 ac1 ac1 float U U

Number of bars tam nc nc int U

Axial effect kg F3 (N) nk float U

m coefficient Non-

dimensional

m m float U

Min. reinforced concrete cross section diameter cm Uba cpba int U U

Max. reinforced concrete cross section diameter cm Ubu cpbu int U U

Diameter increase quantity of RC cross section

diameter

cm dUb dcpb int U U

Diameter increase quantity of steel bars cm dUc dcpc int U U



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The strain values in compression and tension zone are,

euruEu

; earaEa

: 18

The ratio of these values is,

euea

ruEu

Eara

: 19

If we assume that,

EaEu

n and raru

m: 20

The following statement is obtained

euea

n

m: 21

The following equation is attained by taking advantage of the similarity of top and bottom strains inFig. 11

euea

xz

h xz : 22

Fig. 9. The flow chart for optimum design of axially loaded pile.



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If it is assumed that,

xz kxh: 23

Fig. 10. The flow chart of the program module increasing pile numbers and redesigning for axial force.

Fig. 11. Stress distribution of the cross section under the effect of simple bending.



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kxh

h kxh n

m; 24

kx n

n m: 25

Hereby, the coefficients which are necessary for design are attained.

For designing of circular reinforced concrete pile cross sections, firstly a differential line is considered. We assume that thecoordinates of this line are x,zand the angular distance from the origin isu.

The coordinates are,

x r cosu; z r sinu: 26

Length of the differential line is,

bx 2x 2rcosu: 27

Thickness of the differential line is,

dz rcosudu: 28

The area of this differential line is,

dA bx dz 2xdz 2r2 cos2 udu: 29

The compressive stress effecting differential area is defined as rux. The following statement is obtained using similarity ofstress graph inFig. 11

ruxru

z rxz

xz! rux ru

z rxzxz

: 30

By considering,

z r sinu: 31


rux rur sinu rxz

xz: 32

The compression force effect differential line is attained by multiplying line area and stress which affects line

dDbe rux dA: 33

From Eqs.(29) and (32), the following statements are obtained

dDbe rur sinu rxz

xz2r2 cos2 udu;

dDbe 2r2ruxz

r sinu rxz cos2 udu;

dDbe 2r2ruxz

r sinu rxz1 sin2udu;

dDbe 2r2ruxz

r sinu r sin3 u r r sin2 u xzxz sin2 u:

34

If this differential value is integrated along compression area, total compressive force affecting compressive zone may be

calculated

Dbe

2r2ruxz

Z pubub r sinu r sin

3

u r r sin

2

u xzxz sin

2

u

du: 35



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This integral is solved as follows:

Dbe 2r2ruxz

Z pubub

rsinudu

Z pubub

rsin3udu

Z pubub

r du

Z pubub

rsin2udu

Z pubub

xzdu

Z pubub

xzsin2udu

!;

Z pub

ub

sinudu 2cosub;

Z pubub

sin3udu

2

3cos3 ub 2cosub;

Z pubub

du p 2ub;

Z pubub

sin2udu

1

2 p 2ub sin2ub

Dbe 2r2ruxz

r 2cosub r 23

cos ub 2cosub

rp 2ub

r1

2

p 2ub sin2ub xzp 2ub xz1

2

p 2ub sin2ub

" #:

36

After applying some corrections, the following equation is attained

Dbe 2r2ru

1 sinub

2

3cos3 ub

1

2sin ubp 2ub sin

2ubcosub

: 37

If it is assumed that,

ks1 2

3cos3 ub

1

2sin ubp 2ub sin

2 ub cosub

; 38

ks2 2ru

1 sinub: 39

The equation may be written as follows:

Dbe r2 ks1 ks2: 40

Bending moment affecting cross section is attained by multiplying the force effecting compressive zone and level arm.

The length of level arm is

ze h 1

3xz h 0:333xz: 41

The maximum bending moment which cross section can bear is

me Dbe ze r2 ks1 ks2 ze: 42

Radius of cross section is found for using in design of cross section as follows:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

meks1 ks2 ze

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiM

ks1 ks2 ze

s : 43

4.2.1. Determining compressive areas

The origin angle value of compressive zone ub is shown inFig. 11. We can determine this value using trigonometry

sinub rxz

r ! ub sin

1 rxz

r

: 44

The area of compressive zone is attained by integrating the differential line from the beginning to the end of the compressive

zone

ab

Z usub

dA

Z pubub

2r2 cos2 udu;

ab r2 p 2ub sin2ub :

45



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4.2.2. Creating pile design program modules considering simple bending effects

By consideringFig. 11and Eqs.(17)(45), functions and procedures are prepared for design and installed into pile foun-

dations design functions in the program.

Fig. 12. Pile design algorithm flow chart considering simple bending effect.



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The design algorithm flow chart of pile considering simple bending effect is as shown inFig. 12. In this flow chart, the

program module shown with label 3 is pile increase module for bending moment. When a single pile cannot bear the trans-

ferred loads, number of piles is increased and pile cross section diameters are recalculated considering new pile numbers.

The flow chart of the program module increasing pile numbers and redesigning is shown inFig. 13.

Fig. 13. The flow chart of the program module which is increasing pile numbers and redesigning for bending moment.



17/28

As it can be seen from the algorithm inFig. 13, the design module named as bending_design is recalled here. All modules

call others in the program structure. These procedures continue until optimum design is achieved.

4.3. Compound bending effect

When bending moment and axial force affects cross section at same time, compound bending effects must be considered.

Firstly, equivalent axial load distance for bending moment is calculated. This distance is called as eccentricity distance. The

multiplication of this eccentricity distance and axial load is equal to bending moment. Hereby, we can have a loading system

representing axial load and bending moment by moving axial load to the eccentricity distance.

The solution algorithm differs when eccentricity distance length is longer or smaller than the radius of the cross section.

4.3.1. Situation: eccentricity is smaller than the radius

InFig. 14, the compound bending effect considering axial force with small eccentricity is shown. The stress distributions

for bending and axial effects are also seen.

The eccentricity value is

e M

N; e < r: 46

The top and bottom stress values are,

ruN

Fi

N e

xi;

raN

Fi

N e

xi:

47

Nominal area of cross section is,

Fi pr2 n Fe; 48

where

n Ec

Eb 15C16C20: 49

Resistance moment of cross section is,

xi pr3

4

n

2Fe

r2e

r : 50

Total steel area is,

Fe lFb lpr2: 51

Concrete area is,

Fb pr2: 52

Hereby, we can attain nominal area as follows:

Fi pr2 n Fe Fb n l Fb Fb1 n l: 53

Fig. 14. Compound bending effect when eccentricity is smaller than the radius.



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Resistance moment is,

xi pr3

4

n

2Fe

r2er

pr2r

4

n

2lFb

r2er

Fbr

4

n

2lFb

r2er ; 54

xi Fbr

4

r

n l r2e

2r

2

0

BBB@

1

CCCA Fbr2 2nlr2e

4r :

The top and bottom stresses are,

r1;2 N

Fb1 nl

N e

Fbr22nlr2e

4r

: 55If it is assumed thatu re

r andre= u r, the following statement may be attained.

r1;2 N

Fb

1

1 nl

4 e r

r2 2nlr2e

N

Fb

1

1 nl

4 e r

r2 2nlu2r2

56

N

Fb

1

1 nl

4 e

r 2nlu2r N

Fb

1

1 nl

4

1 2nlu2e

r ru

N

pr21

1 nl

4

1 2nlu2e

r

; 57

ra N

pr21

1 nl

4

1 2nlu2e

r

: 58

4.3.2. Situation: eccentricity is longer than the radius

InFig. 15, the compound bending effect considering axial force with long eccentricity is shown. The stress distributions

for bending and axial effects are also seen.

The eccentricity value is,

e

M

N; e > r: 59

The distance of neutral axis from the top is,

x n

n mr re; 60

where

m rcrb

; n EcEb

: 61

Fig. 15. Compound bending effect when eccentricity is longer than the radius.



19/28

Compressive force affecting cross section is,

Db rbr2

1 cosa

sin3 a

3 sina a cosa

!; 62

where

Dc: steel compressive force;

Zc: steel tensile force;

Compound steel force is,

Kc Dc Zc; 63

Kc Fcnrbcosa

1 cosa: 64

Ifx rthan? a> 90? Kc= Compressive load.Ifx =rthan? a= 90? Kc= 0.

Internal force moments are calculated as follows:

Mb rbr3

1 cosa

a

4 1

4sin a cosa 1

6cos a sin3

a

; 65

Mc rcr

2 n Fc r2 1 cosa

rb: 66

From the equilibrium equation, the following statements are obtained

N Db Kc; 67

M Mb Mc; 68

N rb r2

1 cosa

1

3sin

3a sina a cosa n p l cosa

; 69

kn 1

3

sin3 a sina a cosa n p l cosa ; 70

N rb r2

1 cosakn; 71

M rb r3

1 cosa

a

4

1

4sin a cosa

1

6cos a sin3 a

rcr

2 n p l2

; 72

km a

4

1

4sin a cosa

1

6cos a sin

3a

rcr

2 n p l2

; 73

M rb r3

1 cosakm: 74

Thea angle may be calculated as follows:

r0 M

r3 N

Fb; m

rcrb ; n

EcEb

; 75

e M

N; rc u r; 76

cosa mn

rcr

1 mn

m r n rc

n r

n

n m; 77

cosa m r n ur

rn m

m n u

n m ; 78

a cos1 m n u

n m

: 79

4.3.3. Design process

Input data:rb=rbem, r, rc,ac.Output data:N, M



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u rcr ; cosa

m nu

n m ! We can attain a; sina here;

Fb pr2 ! l

acab

percentage;

N rb r2

1 cosakn; M

rb r3

1 cosakm:

4.3.4. Design process

Input data: M, N, r.

Output data:rb

kn 1

3sin

3 a sina a cosa n p l cosa;

N rb r2

1 cosakn; M

rb r3

1 cosakm;

rbN1 cosa

r2kn;

MN1 cosa

r2 kn

r3

1 cosakm ! M

N r kmkn

:

4.4. Steel computation when neutral axis is on the origin

InFig. 16, the stress and strain distributions are shown when neutral axis is on the origin. The steel line is also shown for

the computation.

The parameters for differential line is,

x r cosu; z r sinu;

dz rcosudu;

bx 2x 2rcosu:

The area of differential line is,

dA bx dz 2r2 cos2 udu:

The stress which is effecting differential line is attained by taking advantage of the similarity of stress distribution graph is

Fig. 16

z

xzrbxrb

! rbx z

xzrb

r sinu

r rb rb sinu;

rbx rb sinu:

80

Fig. 16. The stress and strain distributions when neutral axis is on the origin.



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Compressive force which is affecting differential line is,

dDb rbx dA;

dDb rbx dA rb sinu2r2 cos2 udu;

dDb 2rbr2 sinu cos2 udu;

81

Differential bending moment is,

dme dDb z;dme 2rbr

2 sinu cos2 udu

r sinu 2rbr3 sin

2 u cos2 udu82

Bending moment affecting cross section is,

dme 2rbr3

Z p0

sin2 u cos2 udu 2rbr

3

Z p0

sinu cosusinu cosudu; 83

me 2rbr3

Z p0

sin2u

2

sin2u

2 du

1

2rbr

3

Z p0

sin2

2udu:

If sin2

2u 1cos4u2

n ostatement is written here, the following statement is obtained

me 2rbr3

Z p0

1 cos4u

2 du

1

4rbr

3p: 84

The compound force of steel is,

Fc ac rcz: 85

InFig. 16, we can take advantage of the similarities of triangles as follows:

zch xz

rczrc

! rcz zch xz

rc;

zc r sina;

rczr sina

h xzrc:

86

As s result, the compound force of steel is found as follows:

Fc ac rcz acr sina

h xzrc: 87

4.5. Steel computation when neutral axis is above the origin

InFig. 17, the stress and strain distributions are shown when neutral axis is above the origin. The steel line and differen-

tial point are also shown for the computation.

The parameters for differential point in the steel line are as follows:

z r sinu;

dz rcosudu;

x rcosu:

88

Fig. 17. The stress and strain distributions when neutral axis is above the origin.



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InFig. 17, we can take advantage of the similarities of stress triangle as follows:

rzte z

h zte

rczrc

! rczrzte z

h zte rc;

dac 2ac1:

89

We carry out angular converting here

360 ! 2pr

360 us ub ! x

Fig. 18. All forces effecting pile in the soil and interactive layer object developed for solution.



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Contour length of the steel bar is as follows:

xcvr360 us ub

360 2pr: 90

Fig. 19. Optimum pile length design algorithm considering all effects and soil conditions.



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Lets take a differential section for computing steel tension force. The section has duangle anddcvr=r ducontour length.The area of this differential section is dac= 2ac1 r du.

The tension force affecting this differential section area is calculated as follows:

dZ dac rczrzte z

h zte rc 2ac1 r du;

dZ r

h zte

zte

h zte

r sinu

h zte ra 2ac1 r du;

dZ 2ac1 r rc

h zte r du zte du r sinu ;

Z2ac1 r rc

h zte r

Z du zte

Z du r

Z sinudu

:

91

5. Numerical solution procedure of piles as deep foundation systems

Embedded pile in the soil is affected by various loads. All forces effecting pile in the soil is shown in Fig. 18. In this figure,

all stratums and effect distributions are presented. For solution, an interactive layer object is developed. The object is green

colored. This interactive object is moved from up to down and used to calculate optimum pile length. In each stratum, the

friction forces by shear stresses (skin friction) along the lateral pile/soil interface affect the structure. Total soil pressure

forces affect pile sides from two zones. End-bearing force affects at the object tip (bottom of the object). When the sum

of all reaction forces is bigger than transferred loads from the super structure, the movement of the intelligent object stops.This control is carried out in each stratum. When the scanner object is moving in the stratums, reinforced concrete design is

carried out. In each steps, total reaction force is compared with pile bearing force. If total pile reaction force is bigger, the

Fig. 20. Soil parameters, loads transferred from the superstructure and scanner band for optimum pile design of axially loaded system.

Table 2

Soil parameters of all stratums for numerical example.

St ratum numbe r Unit wei ghtc (kg/m3) Cohesionc(kg/m2) Internal fricti on angle U () Friction coefficientl

1 1600 2000 5 0.2

2 1900 4000 15 0.25

3 1600 3000 17 0.3

4 1800 2000 20 0.45



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cross section is increased and the pile system is reanalyzed. Hereby, the optimum cross section, reinforcement and pile

length may be obtained. The algorithm of this solution method is shown in Fig. 19.The simplified programming codes of

this algorithm written in MATLAB[26]are shown inAppendix A.

6. Numerical Application

A design approach has been developed considering the algorithms expressed before and the approach has been pro-

grammed by using MATLAB. The program package solves piles using a scanner band. This band starts movement from upto down and stops whenever the bearing capacity of the system is larger than the external loads. The forces affecting the

pile in the scanner band are shown in Fig. 18. As shown in this figure, the scanner band is as big as a stratum. However,

Fig. 21. Optimum length of axially loaded pile calculated by scanner band.

Fig. 22. Calculated cross section and reinforcement for axially loaded pile.

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the pile doesnt have to reach to the end of the stratum. Therefore, a very thin scanner band should be taken into account to

find optimum pile length. The thickness of this band is chosen by program user (for example 0.5m). The stratums are divided

into small parts and are scanned to find optimum pile length. While soil stratums are being scanned, the bearing capacity

control is carried out at each step. While optimum pile length is being determined, reinforcement design is carried out,

too. If the calculated cross section diameter is bigger than the allowed limit diameter, the pile number is increased and

the design is repeated according to pile group.

In this part, an example for optimum design of an axially loaded pile is given. The scanner band movement direction, soil

parameters and load conditions for pile design are shown in Fig. 20. In this example, the transferred load from the

superstructure is axial load and equal to 1,000,000 kg. The surcharge load over the soil is assumed as 4000 kg/m. The

max. compressive stress of concrete is assumed as 2,000,000 kg/m2 and the max. tension stress of steel is assumed as

42,000,000 kg/m2. The percentage ratio is assumed as 0.01. The scanner band thickness is chosen as 0.5 m. This value is

optional and may be changed by user considering application restrictions. The stratum thicknesses are 5 m, 4 m, 8.5 m

and 5 m, respectively. The soil parameters of the stratums are shown inFig. 20andTable 2. When analysis starts, the scanner

band stops at 13.0 m far from the top as shown inFig. 21. The bearing capacity at that point is calculated as 1043923.1 kg and

this value is bigger than 1,000,000 kg. The optimum pile length is 13.0 m for these loading and soil conditions. In analysis

process, reinforced concrete design is also carried out. The minimum cross section diamater option for the piles is considered

as 0.8 m and the maximum cross section diameter option for the piles is considered as 1.8 m in this study. These limitations

depend on the possibilities of engineering company. The cross section of this single pile is shown in Fig. 22. Firstly, the

calculated cross section diameter is found as 0.7255 m and it is lower than the minimum possible diameter value, therefore

it is selected as 0.80 m. The reinforcement may be easily calculated depending on the given percentage by the user.

If the axial load is increased, the system must be reanalyzed. The axial load is considered as 6,400,000 kg and the sur-

charge load over the soil is assumed as 4000 kg/m. The scanner band thickness is chosen as 0.5 m again. When analysis starts,

the scanner band stops at 17.5 m far from the top as shown inFig. 23. The minimum cross section diameter option for the

piles is onsidered as 0.80 m and the maximum cross section diameter option for the piles is considered as 1.8 m in this study.

The necessary cross section diameter is calculated as 1.84 m and this value is bigger than the maximum diameter option.

Therefore, the pile number is increased according to given loading conditions. The optimum pile length is 17.5 m for these

loading and soil conditions. The bearing capacity of pile group at that point is calculated as 6478740.6 kg and this value is

bigger than 6,400,000 kg. In analysis process, reinforced concrete design is also carried out. The cross sections of pile group is

shown inFig. 24. The computed cross section diameter is 1.3 m for each pile and reinforcement may be calculated depending

Fig. 23. Soil parameters, loads transferred from the superstructure and scanner band for optimum pile design of multi- loaded system.



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on the given percentage by the user. As emphasized before, if the cross section size is increased, the pile length may decrease.

This depends on the engineering desing and possibilies of the company.

7. Conclusions

In this study, a novel solution algorithm is developed for optimum design of reinforced concrete piles. One of the most

important contributions of this research is an efficient tool to obtain optimum pile length and reinforced concrete design

of pile foundation systems. The soil medium is modeled in three dimensional deterministic approach and all possible effectsare taken into consideration for design. Soil behaviors considering transferred loads from the superstructure are determined.

Active and passive zone distributions depending on the external effects in the soil are investigated. The lateral earth pres-

sures are computed considering these active and passive zone distributions. The reinforced concrete design algorithms have

been developed for all possible loads affecting the pile system. Design algorithms for axial loads, shear forces, bending mo-

ments and torsional moments have been developed. The flow charts for developed solution procedures are presented. The

algorithms are implemented in MATLAB program. All possible effects, all stratum soil parameters and behaviors are taken

into account for solution. A dynamic layer object has been developed to find optimum pile length. In solution process, this

layer is compressed to a scanner band to find more optimal pile length. This scanner band moves from up to down and stops

movement whenever optimum length is found. During this movement, all bearing capacity controls are carried out consid-

ering soil effects. Reinforced concrete design is also achieved depending on external loads. Finally, numerical solutions of

axially and multi-loaded piles as deep foundation systems are presented. The design procedure of given examples are dem-

onstrated through the proposed algorithm. The optimum length, cross section dimension and reinforcement details are

presented.The procedure suggested in this paper for computing the optimum design of piles by-passes the expensive and difficult

solution of classical approach. It can be used for practical engineering studies and supply economical gain because it gives

optimal pile length and prevents unneeded material use.

Appendix A

Main programming codes of optimum design algorithm implemented in MATLAB.

function [cpb, pile_number, lk, pk, ac, nc] = piledesign (p_effect, qs, k1, k2, bgeru, cgeru,

emb, emc, pur, pi, cpc, c, fi, zba, sk, dz, nk, pp, cpba, cpbu)

lk = dz (1);

pks = 0;

pk = 0;

dz0 = qs/zba (1);

dpy0_1 = fzet (k1(1), zba (1), dz0);

dpy0_2 = fzet (k2(1), zba (1), dz0);

[cpb1, nka] = axial_design (-p_effect (3), qs, k1, k2, bgeru, cgeru, emb, emc, pur, pi, cpc, c, fi,

zba, sk, dz, nk, pp, cpba, cpbu)

[cpb2, mka] = bending_design (p_effect (4), emc, emb, cgeru, bgeru, cpba, cpbu, pp)

[ac] = fac (pur, cpb, pi)

[nc] = fnc (pur, cpb, pi, cpc);

i = 1;

p_etk = abs (p_effect (3))/pile_number;

while (p_etk > pk & i < nk)

(continued on next page)

Fig. 24. Optimum length of multi-loaded pile calculated by scanner band.



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Journal Grupos de Pilotes

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