Beams on Elastic Foundation using Winkler Model.docx
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About This Presentation
STRUCTURAL ENGINEERING
Slide Content
1
1 Beam on Elastic Foundation (Winkler Model)
Beam on elastic Foundation
A graduation project
Submitted to the department of civil engineering at
The University of Baghdad
Baghdad - Iraq
In partial fulfillment of the requirement for the
degree of Bachelor of Science in civil engineering
By
Amjad Salem Rashid Rafife Sa’ad Hadi Sarah Mohamed Saleh
Supervised by
Assistant lecturer, Adnan Najem (M.Sc., in Structural Engineering)
July /2007
1 Beam on Elastic Foundation (Winkler Model)
Beam on elastic Foundation
A graduation project
Submitted to the department of civil engineering at
The University of Baghdad
Baghdad - Iraq
In partial fulfillment of the requirement for the
degree of Bachelor of Science in civil engineering
By
Amjad Salem Rashid Rafife Sa’ad Hadi Sarah Mohamed Saleh
Supervised by
Assistant lecturer, Adnan Najem (M.Sc., in Structural Engineering)
July /2007
2
2 Beam on Elastic Foundation (Winkler Model)
Beam on Elastic
Foundation
2 Beam on Elastic Foundation (Winkler Model)
Beam on Elastic
Foundation
3
3 Beam on Elastic Foundation (Winkler Model)
I certify that study entitled “BEAM ON ELASTIC FOUNDATION”, was prepared by under
my supervision at the civil engineering department in the University of Baghdad, in
partial fulfillment of requirements for the degree of Bachelor of Science in civil
engineering.
Supervisor:
Signature:
Name: Adnan Najem
Assistant lecturer (M.Sc., in Structural Engineering)
Date:
3 Beam on Elastic Foundation (Winkler Model)
I certify that study entitled “BEAM ON ELASTIC FOUNDATION”, was prepared by under
my supervision at the civil engineering department in the University of Baghdad, in
partial fulfillment of requirements for the degree of Bachelor of Science in civil
engineering.
Supervisor:
Signature:
Name: Adnan Najem
Assistant lecturer (M.Sc., in Structural Engineering)
Date:
4
4 Beam on Elastic Foundation (Winkler Model)
We certify that we have read this study “BEAM ON ELASTIC FOUNDATION” and as
examining committee examined the students in its content and in what is connected to
with it, and that in our opinion it meets the standard of a study for the degree of
Bachelor of Science in civil engineering.
Committee Member: Committee Member:
Signature: Signature:
Name: Name:
Date: Date:
Committee Chairman:
Signature:
Name:
Date:
Signature:
Name:
Head of Civil Engineering Department
College of Engineering
Baghdad University
Date:
4 Beam on Elastic Foundation (Winkler Model)
We certify that we have read this study “BEAM ON ELASTIC FOUNDATION” and as
examining committee examined the students in its content and in what is connected to
with it, and that in our opinion it meets the standard of a study for the degree of
Bachelor of Science in civil engineering.
Committee Member: Committee Member:
Signature: Signature:
Name: Name:
Date: Date:
Committee Chairman:
Signature:
Name:
Date:
Signature:
Name:
Head of Civil Engineering Department
College of Engineering
Baghdad University
Date:
5
5 Beam on Elastic Foundation (Winkler Model)
Thanks:
We would like to present our great thanks to the head of civil engineering department
and their teaching stuff for all their great help and assistance along our study journey.
5 Beam on Elastic Foundation (Winkler Model)
Thanks:
We would like to present our great thanks to the head of civil engineering department
and their teaching stuff for all their great help and assistance along our study journey.
6
6 Beam on Elastic Foundation (Winkler Model)
Abstract:
This objective of this study is to develop a better understanding for the basic principles
of structural analysis of beams resting on elastic half space foundation so they can be
efficiently implemented on modern computers.
Demonstrate the effect of elastic foundation on the behavior of in-plane structures
resting on it.
Develop a foundation’s stiffness matrices that take into the interaction between
adjacent points of the elastic foundation into consideration.
In addition several important parameters have been incorporated in the stiffness
matrices development; the horizontal contact pressures at the interface between
structure and foundations, the effects due to separation of contact surfaces due to
uplift forces, and discrepancy between contact surfaces (soil contact surface and neutral
axis of superstructure element).
Beams on elastic foundation are analysis is divided into parts. Firstly for superstructure;
the structural members (beams) were analyzed using linearly elastic methods such as
stiffness method. Secondly for substructure; elastic foundation continuum is modeled
according to the elastic continuum theory (elastic half space model).Then both parts
were assembled in matrix forms and analyzed by stiffness method.
Project layout
The project is divided into five chapters as follows:
Chapter one: presents a general introduction to the subject of stiffness method and
elastic foundation models.
Chapter two: presents the previous literatures published about this subject.
Chapter three: presents the theoretical bases for the analysis method and the elastic
foundation models derivations.
Chapter four: presents a brief description of a computer developed in this project.
Chapter five: discuses the results of this analysis method. And recommend future steps.
6 Beam on Elastic Foundation (Winkler Model)
Abstract:
This objective of this study is to develop a better understanding for the basic principles
of structural analysis of beams resting on elastic half space foundation so they can be
efficiently implemented on modern computers.
Demonstrate the effect of elastic foundation on the behavior of in-plane structures
resting on it.
Develop a foundation’s stiffness matrices that take into the interaction between
adjacent points of the elastic foundation into consideration.
In addition several important parameters have been incorporated in the stiffness
matrices development; the horizontal contact pressures at the interface between
structure and foundations, the effects due to separation of contact surfaces due to
uplift forces, and discrepancy between contact surfaces (soil contact surface and neutral
axis of superstructure element).
Beams on elastic foundation are analysis is divided into parts. Firstly for superstructure;
the structural members (beams) were analyzed using linearly elastic methods such as
stiffness method. Secondly for substructure; elastic foundation continuum is modeled
according to the elastic continuum theory (elastic half space model).Then both parts
were assembled in matrix forms and analyzed by stiffness method.
Project layout
The project is divided into five chapters as follows:
Chapter one: presents a general introduction to the subject of stiffness method and
elastic foundation models.
Chapter two: presents the previous literatures published about this subject.
Chapter three: presents the theoretical bases for the analysis method and the elastic
foundation models derivations.
Chapter four: presents a brief description of a computer developed in this project.
Chapter five: discuses the results of this analysis method. And recommend future steps.
7
7 Beam on Elastic Foundation (Winkler Model)
Contents:
Title……………………………………………………………………………………2
Supervisor words……………………………………………………………….3
Committee words………………………………………………………………4
Thanks……………………………………………………………………………….5
Abstract…………………………………………………………………………….6
Project Layout………………………….……………………………………….6
Contents…….……..………………………………………………………………7
Chapter one; introduction…………..….……………………………… …8
Chapter two; literature………………….………………………………. .12
Chapter three; theory………………………………………………………17
Chapter four; computer program…………………………………….25
Chapter five; conclusions and recommendations…………….36
References………………………………………………………………… ..……39
Appendix I…………………………………………………………… ……..……40
7 Beam on Elastic Foundation (Winkler Model)
Contents:
Title……………………………………………………………………………………2
Supervisor words……………………………………………………………….3
Committee words………………………………………………………………4
Thanks……………………………………………………………………………….5
Abstract…………………………………………………………………………….6
Project Layout………………………….……………………………………….6
Contents…….……..………………………………………………………………7
Chapter one; introduction…………..….……………………………… …8
Chapter two; literature………………….………………………………. .12
Chapter three; theory………………………………………………………17
Chapter four; computer program…………………………………….25
Chapter five; conclusions and recommendations…………….36
References………………………………………………………………… ..……39
Appendix I…………………………………………………………… ……..……40
8
8 Beam on Elastic Foundation (Winkler Model)
Chapter one
Introduction
8 Beam on Elastic Foundation (Winkler Model)
Chapter one
Introduction
9
9 Beam on Elastic Foundation (Winkler Model)
INTRODUCTION TO STIFFNESS METHOD
This method of analyzing structures is probably
(14)
used more widely than the flexibility
method, especially for large and complex structures (with multiple nodes). Such
structures require the use of electronic computers for carrying out the extensive
numerical calculations, and the stiffness method is much more suitable for computer
programming than the flexibility method!
The reason is that the stiffness method can be put into the form of a standardized
procedure which dose not requires any engineering decisions during the calculation
process. And also the unknown quantities in the stiffness method are prescribed more
clearly than the flexibility method.
When analyzing a structure by the stiffness method, normally we use the concepts of
kinematic indeterminacy, fixed-end reactions, and stiffnesses. These definitions will be
explained as follows:
KINEMATIC INDETERMINACY
In stiffness method the unknown quantities in the analysis are the joint displacements
of the structure, rather than the redundant reactions and stress resultants as is the case
of flexibility method. The Joints in any structure will be define as points where two or
more members intersect, the points of support, and the free ends of any projecting
members.
When the structure is subjected to loads, all or some of the joints will undergo
displacements in the form of translations and rotations. Of course, some of the joints
displacements will be zero because of the restraint conditions; for instance, at a fixed
support there will be no displacements of any kind.
The unknown joint displacements are called kinematic unknowns and their number is
called either the degree of kinematic indeterminacy or the number of degrees of
freedom (DOF) for joint displacements.
FIXED-END ACTIONS
In stiffness method we regulatory encounter fixed-end beam, because one of the first
steps in this method is to restrain all of the unknown joint displacements. The
imposition of such restrains causes a continuous beam or plane frame to become an
assemblage of fixed-end beams. Therefore, we need to have readily available a
collection of formulas for the reactions of fixed-end beams for multiple case. These
reactions which consist of both; forces and couples (moments), are known collectively
as Fixed-End actions. Values of fixed-end actions for multiple cases are shown in
Appendix I.
STIFFNESSES
9 Beam on Elastic Foundation (Winkler Model)
INTRODUCTION TO STIFFNESS METHOD
This method of analyzing structures is probably
(14)
used more widely than the flexibility
method, especially for large and complex structures (with multiple nodes). Such
structures require the use of electronic computers for carrying out the extensive
numerical calculations, and the stiffness method is much more suitable for computer
programming than the flexibility method!
The reason is that the stiffness method can be put into the form of a standardized
procedure which dose not requires any engineering decisions during the calculation
process. And also the unknown quantities in the stiffness method are prescribed more
clearly than the flexibility method.
When analyzing a structure by the stiffness method, normally we use the concepts of
kinematic indeterminacy, fixed-end reactions, and stiffnesses. These definitions will be
explained as follows:
KINEMATIC INDETERMINACY
In stiffness method the unknown quantities in the analysis are the joint displacements
of the structure, rather than the redundant reactions and stress resultants as is the case
of flexibility method. The Joints in any structure will be define as points where two or
more members intersect, the points of support, and the free ends of any projecting
members.
When the structure is subjected to loads, all or some of the joints will undergo
displacements in the form of translations and rotations. Of course, some of the joints
displacements will be zero because of the restraint conditions; for instance, at a fixed
support there will be no displacements of any kind.
The unknown joint displacements are called kinematic unknowns and their number is
called either the degree of kinematic indeterminacy or the number of degrees of
freedom (DOF) for joint displacements.
FIXED-END ACTIONS
In stiffness method we regulatory encounter fixed-end beam, because one of the first
steps in this method is to restrain all of the unknown joint displacements. The
imposition of such restrains causes a continuous beam or plane frame to become an
assemblage of fixed-end beams. Therefore, we need to have readily available a
collection of formulas for the reactions of fixed-end beams for multiple case. These
reactions which consist of both; forces and couples (moments), are known collectively
as Fixed-End actions. Values of fixed-end actions for multiple cases are shown in
Appendix I.
STIFFNESSES
10
10 Beam on Elastic Foundation (Winkler Model)
In the stiffness method we make use of actions caused by unit displacement. These
displacement may be either unit translation or unit rotation, and the resulting actions
are either forces of couples (moments). These actions caused by unit displacement are
known as stiffness influence coefficients, or stiffnesses. These coefficients called also
member stiffnesses which they are frequently used in this method. Here by two of the
most useful cases as shown in fig. (1.1).
Fig.(1.1) member stiffnesses
GENERAL EQUATION OF STIFFNESS METHOD
Now most of the preliminary ideas and definitions have been set fourth, and the
problem of analyzing a structure can be established. Interpreting of Equilibrium
Equations, and making use of the Principles of Superposition, for the case of a structure
having (n x n) Degrees of Kinematic Indeterminacy will lead to the following sets of
linear equations are obtained:
�
11�
1
�
21�
1
+ �
12�
2
+ �
22�
2
+ �
13�
3
+ �
23�
3
: : :
�
�1�
1+ �
�2�
2+ �
�3�
3
…….+ �
1�
…….+ �
2�
:
…….+ �
��
�
�
�
�
+ �
1
+ �
2
=??????
1
=??????
2
: : :
�
�+ �
�=??????
�
……………….Eq. (1.1)
This can be reduced to General Equation form:
[??????]|∆|=|�|…………..Eq. (1.2)
Hence, the principles of superposition are used in developing fixed-end actions (forces),
therefore, this method is limited to linearly elastic structures with small displacements.
The n equations can be solved for the n unknown joint displacement of the structure.
The important fact which need to be established: that Equilibrium Equations of the
Stiffness Method express the superposition of actions (forces) corresponding to
unknown displacements. While the compatibility equations of the Flexibility Method
express the superposition of displacements corresponding unknown actions (forces).
Also; it should be noticed that above equilibrium equations (1.1) are written in a form
which takes into account only the effects of applied loads on the structure, but the
10 Beam on Elastic Foundation (Winkler Model)
In the stiffness method we make use of actions caused by unit displacement. These
displacement may be either unit translation or unit rotation, and the resulting actions
are either forces of couples (moments). These actions caused by unit displacement are
known as stiffness influence coefficients, or stiffnesses. These coefficients called also
member stiffnesses which they are frequently used in this method. Here by two of the
most useful cases as shown in fig. (1.1).
Fig.(1.1) member stiffnesses
GENERAL EQUATION OF STIFFNESS METHOD
Now most of the preliminary ideas and definitions have been set fourth, and the
problem of analyzing a structure can be established. Interpreting of Equilibrium
Equations, and making use of the Principles of Superposition, for the case of a structure
having (n x n) Degrees of Kinematic Indeterminacy will lead to the following sets of
linear equations are obtained:
�
11�
1
�
21�
1
+ �
12�
2
+ �
22�
2
+ �
13�
3
+ �
23�
3
: : :
�
�1�
1+ �
�2�
2+ �
�3�
3
…….+ �
1�
…….+ �
2�
:
…….+ �
��
�
�
�
�
+ �
1
+ �
2
=??????
1
=??????
2
: : :
�
�+ �
�=??????
�
……………….Eq. (1.1)
This can be reduced to General Equation form:
[??????]|∆|=|�|…………..Eq. (1.2)
Hence, the principles of superposition are used in developing fixed-end actions (forces),
therefore, this method is limited to linearly elastic structures with small displacements.
The n equations can be solved for the n unknown joint displacement of the structure.
The important fact which need to be established: that Equilibrium Equations of the
Stiffness Method express the superposition of actions (forces) corresponding to
unknown displacements. While the compatibility equations of the Flexibility Method
express the superposition of displacements corresponding unknown actions (forces).
Also; it should be noticed that above equilibrium equations (1.1) are written in a form
which takes into account only the effects of applied loads on the structure, but the
11
11 Beam on Elastic Foundation (Winkler Model)
equation can be readily modified to include the effects of temperature changes,
prestrains, and support settlements. It is only necessary to include these effects in the
determination of the actions (forces) A1, A2,…, An. Furthermore, Eq. (1.2) apply to many
types of structures, including trusses and space frames, although in this project is
limited to in-plane structure (beams), and hence the stiffness method is applicable only
to linearly elastic structures.
STIFFNESS METHOD VERSUS FINITE ELEMENT METHOD (FEM)
Stiffness method can be used to analyze structures only, finite element analysis, which
originated as an extension of matrix (stiffness and flexibility), it is detected to analyze
surface structures (e. g. plates and shells). FEM has now developed to the extent that it
can be applied to structures and solids of practically any shape or form. From theoretical
viewpoint, the basic difference between the two is that, in stiffness method, the
member force-displacement relationships are based on the exact solutions of the
underlying differential equations, whereas in FEM, such relations are generally derived
by Work-Energy Principles from assumed displacement or stress functions.
Because of the approximate nature of its force-displacements relations, FEM analysis
yield approximate results. However, FEM is always more accurate than stiffness matrix
especially in nonlinear analysis.
11 Beam on Elastic Foundation (Winkler Model)
equation can be readily modified to include the effects of temperature changes,
prestrains, and support settlements. It is only necessary to include these effects in the
determination of the actions (forces) A1, A2,…, An. Furthermore, Eq. (1.2) apply to many
types of structures, including trusses and space frames, although in this project is
limited to in-plane structure (beams), and hence the stiffness method is applicable only
to linearly elastic structures.
STIFFNESS METHOD VERSUS FINITE ELEMENT METHOD (FEM)
Stiffness method can be used to analyze structures only, finite element analysis, which
originated as an extension of matrix (stiffness and flexibility), it is detected to analyze
surface structures (e. g. plates and shells). FEM has now developed to the extent that it
can be applied to structures and solids of practically any shape or form. From theoretical
viewpoint, the basic difference between the two is that, in stiffness method, the
member force-displacement relationships are based on the exact solutions of the
underlying differential equations, whereas in FEM, such relations are generally derived
by Work-Energy Principles from assumed displacement or stress functions.
Because of the approximate nature of its force-displacements relations, FEM analysis
yield approximate results. However, FEM is always more accurate than stiffness matrix
especially in nonlinear analysis.
12
12 Beam on Elastic Foundation (Winkler Model)
Chapter two
Literature
12 Beam on Elastic Foundation (Winkler Model)
Chapter two
Literature
13
13 Beam on Elastic Foundation (Winkler Model)
2.1 LINEAR ANALYSIS OF IN-PLANE STRUCTURES USING STIFFNESS MATRIX METHOD
The theoretical foundation for matrix (stiffness) method of structural analysis was laid
and developed by many scientists:
James, C. Maxwell, [1864] who introduced the method of Consistent Deformations
(flexibility method).
Georg, A. Maney, [1915] who developed the Slope-Deflection method (stiffness
method).
These classical methods are considered to be the precursors of the matrix (Flexibility
and Stiffness) method, respectively. In the precomputer era, the main disadvantage of
these earlier methods was that they required direct solution of Simultaneous Equations
(formidable task by hand calculations in cases more than a few unknowns).
The invention of computers in the late-1940s revolutionized structural analysis. As
computers could solve large systems of Simultaneous Equations, the analysis methods
yielding solutions in that form were no longer at a disadvantage, but in fact were
preferred, because Simultaneous Equations could be expressed in matrix form and
conveniently programmed for solution on computers.
Levy, S., [1947] is generally considered to have been the first to introduce the flexibility
method, by generalizing the classical method of consistent deformations.
Falkenheimer, H., Langefors, B., and Denke, P. H., [1950], many subsequent researches
extended the flexibility method and expressed in matrix form are:
Livesley, R. K., [1954], is generally considered to have been the first to introduce the
stiffness matrix in 1954, by generalizing the classical method of slop-deflections.
Argyris, J. H., and Kelsey, S., [1954], the two subsequent researches presented a
formulation for stiffness matrices based on Energy Principles.
Turner, M. T., Clough, R. W., and Martin, H. C., [1956], derived stiffness matrices for
truss members and frame members using the finite element approach, and introduced
the now popular Direct Stiffness Method for generating the structure stiffness matrix.
Livesley, R. K., [1956], presented the Nonlinear Formulation of the stiffness method for
stability analysis of frames.
Since the mid-1950s, the development of Stiffness Method has been continued at a
tremendous pace, with research efforts in the recent years directed mainly toward
formulating procedures for Dynamic and Nonlinear analysis of structures, and
13 Beam on Elastic Foundation (Winkler Model)
2.1 LINEAR ANALYSIS OF IN-PLANE STRUCTURES USING STIFFNESS MATRIX METHOD
The theoretical foundation for matrix (stiffness) method of structural analysis was laid
and developed by many scientists:
James, C. Maxwell, [1864] who introduced the method of Consistent Deformations
(flexibility method).
Georg, A. Maney, [1915] who developed the Slope-Deflection method (stiffness
method).
These classical methods are considered to be the precursors of the matrix (Flexibility
and Stiffness) method, respectively. In the precomputer era, the main disadvantage of
these earlier methods was that they required direct solution of Simultaneous Equations
(formidable task by hand calculations in cases more than a few unknowns).
The invention of computers in the late-1940s revolutionized structural analysis. As
computers could solve large systems of Simultaneous Equations, the analysis methods
yielding solutions in that form were no longer at a disadvantage, but in fact were
preferred, because Simultaneous Equations could be expressed in matrix form and
conveniently programmed for solution on computers.
Levy, S., [1947] is generally considered to have been the first to introduce the flexibility
method, by generalizing the classical method of consistent deformations.
Falkenheimer, H., Langefors, B., and Denke, P. H., [1950], many subsequent researches
extended the flexibility method and expressed in matrix form are:
Livesley, R. K., [1954], is generally considered to have been the first to introduce the
stiffness matrix in 1954, by generalizing the classical method of slop-deflections.
Argyris, J. H., and Kelsey, S., [1954], the two subsequent researches presented a
formulation for stiffness matrices based on Energy Principles.
Turner, M. T., Clough, R. W., and Martin, H. C., [1956], derived stiffness matrices for
truss members and frame members using the finite element approach, and introduced
the now popular Direct Stiffness Method for generating the structure stiffness matrix.
Livesley, R. K., [1956], presented the Nonlinear Formulation of the stiffness method for
stability analysis of frames.
Since the mid-1950s, the development of Stiffness Method has been continued at a
tremendous pace, with research efforts in the recent years directed mainly toward
formulating procedures for Dynamic and Nonlinear analysis of structures, and
14
14 Beam on Elastic Foundation (Winkler Model)
developing efficient Computational Techniques (load incremental procedures and
Modified Newton-Raphson for solving nonlinear Equations) for analyzing large
structures and large displacements. Among those researchers are: S. S. Archer, C.
Birnstiel, R. H. Gallagher, J. Padlog, J. S. przemieniecki, C. K. Wang, and E. L. Wilson
and many others.
LIVESLEY, R. K. [1964] described the application of the Newton- Raphson procedure to
nonlinear structures. His analysis is general and no equations are presented for framed
structures. However, he did illustrate the analysis of a guyed tower.
2.2 SOIL-STRUCTURE INTERACTION
This subject has attracted the attention of both structural and geotechnical engineers,
because it has a mutual effect on both superstructure and substructure elements. For
this reason, various procedures have been and are being periodically proposed to
develop, enhance, or try to simulate reality in soil and/or foundation analysis. This
section is devoted to survey the most significant research in this field along the history,
for both foundation models and foundation analysis methods, with a quick summary of
their projecting attributes. The sequence of time has been taken into account.
A considerable amount of literature work on the soil-structure interaction problem has
been published. Historically, there are three basic approaches to the problem of the
beams or plate on elastic foundation:
2.2.1 BEAMS ON ELASTIC FOUNDATION (WINKLER MODEL)
The Winkler hypothesis [1867], proposed by Winkler, E. in about 1867, treats the soil
mass supporting the foundation as a series of springs on which the structural member is
supported.
WINKLER, E. [1867] presented for the first time the conventional analysis of beams on
elastic subgrade based on the assumption that the ratio of contact pressure to the
deflection is the same at every point of the beam. Denoting the pressure at any point by
P, and the beam deflection at the same point by w, this assumption, often called
Winkler's hypothesis, may be written:
??????
??????=
??????
�
……………………..Eq.(2.1)
Where ks: is a constant called Modulus of Subgrade Reaction.
For about seventy years since of this hypothesis in the theory of bending of beams on
elastic subgrade, most of the investigators in this field worked on solutions of the basic
differential equation of the problem. Little attention was given to the question of
reliability of the basic hypothesis.
14 Beam on Elastic Foundation (Winkler Model)
developing efficient Computational Techniques (load incremental procedures and
Modified Newton-Raphson for solving nonlinear Equations) for analyzing large
structures and large displacements. Among those researchers are: S. S. Archer, C.
Birnstiel, R. H. Gallagher, J. Padlog, J. S. przemieniecki, C. K. Wang, and E. L. Wilson
and many others.
LIVESLEY, R. K. [1964] described the application of the Newton- Raphson procedure to
nonlinear structures. His analysis is general and no equations are presented for framed
structures. However, he did illustrate the analysis of a guyed tower.
2.2 SOIL-STRUCTURE INTERACTION
This subject has attracted the attention of both structural and geotechnical engineers,
because it has a mutual effect on both superstructure and substructure elements. For
this reason, various procedures have been and are being periodically proposed to
develop, enhance, or try to simulate reality in soil and/or foundation analysis. This
section is devoted to survey the most significant research in this field along the history,
for both foundation models and foundation analysis methods, with a quick summary of
their projecting attributes. The sequence of time has been taken into account.
A considerable amount of literature work on the soil-structure interaction problem has
been published. Historically, there are three basic approaches to the problem of the
beams or plate on elastic foundation:
2.2.1 BEAMS ON ELASTIC FOUNDATION (WINKLER MODEL)
The Winkler hypothesis [1867], proposed by Winkler, E. in about 1867, treats the soil
mass supporting the foundation as a series of springs on which the structural member is
supported.
WINKLER, E. [1867] presented for the first time the conventional analysis of beams on
elastic subgrade based on the assumption that the ratio of contact pressure to the
deflection is the same at every point of the beam. Denoting the pressure at any point by
P, and the beam deflection at the same point by w, this assumption, often called
Winkler's hypothesis, may be written:
??????
??????=
??????
�
……………………..Eq.(2.1)
Where ks: is a constant called Modulus of Subgrade Reaction.
For about seventy years since of this hypothesis in the theory of bending of beams on
elastic subgrade, most of the investigators in this field worked on solutions of the basic
differential equation of the problem. Little attention was given to the question of
reliability of the basic hypothesis.
15
15 Beam on Elastic Foundation (Winkler Model)
However, the investigations preformed during the past decades have shown that the
distribution of vertical pressure at the contact surface between beams or slabs and
elastic subgrade may be quite different from that obtained by the conventional analysis.
Winker's hypothesis seemed not to be justified at least for beams and slabs on subgrade
such as concrete, rock, or soils. Consequently, from a theoretical point of view, the
coefficient of subgrade reaction k was considered as an artificial concept. It appeared
that an analysis based on that concept was in this case a crude estimate
(45)
.
LEVINTON, Z. [1949] suggested a simplified method for analyzing beams on elastic
foundation. In this method, the contact pressure is represented by a number of
redundant reactions and by creating a set of simultaneous equations in terms of
pressure diagram coordinates and elasticity constants. For each of beam and
foundation, it is possible to find the bending moment and shear force at any point along
the beam and also to calculate the contact pressure.
TERZAGHI, K. [1955] established a number of equations to calculate the modulus of
subgrade reaction for cohesive and cohesion less soils, depending on plate load test
results. He stated that the theories of subgrade reaction are based on the following
assumptions:
1. The ratio ks between the contact pressure p and the corresponding displacement w is
independent of the pressure p.
1. The coefficient of subgrade reaction ks has the same value for every point on the
surface acted upon by the contact pressure. Terzaghi concluded that, provided that p is
smaller than one-half of the ultimate bearing pressure (as well as the fact that ks is
dependent on the dimension of the loaded area), the theories of subgrade reaction
could furnish reliable estimates of stresses and bending moments, although they were
not good in estimating displacements.
BOWLES, J. E. [1974] developed a computer program to carry out the analysis of beams
on elastic foundation using finite element method, in which Winkler model is adopted
to represent the elastic foundation. Several boundary conditions can be entered easily.
From comparison with Vesic method, it is shown that results provide a more realistic
distribution of longitudinal bending moment in the member.
BOWLES, J. E. [December, 1986] introduced a brief survey of computerized methods for
mat design with particular advantages and disadvantages of the three common discrete
element methods, the finite difference, the finite grid, and the finite element. The
modulus of subgrade reaction (ks) is considered in some detail both in obtaining
reasonable initial design estimates and simple methods to couple node effects. Also, a
mat foundation is analyzed by three methods with three ways of springs coupling.
Bowles deduced that there is no serious difference between the coupled and uncoupled
springs and that the finite grid method is a good alternative to the finite element
method.
15 Beam on Elastic Foundation (Winkler Model)
However, the investigations preformed during the past decades have shown that the
distribution of vertical pressure at the contact surface between beams or slabs and
elastic subgrade may be quite different from that obtained by the conventional analysis.
Winker's hypothesis seemed not to be justified at least for beams and slabs on subgrade
such as concrete, rock, or soils. Consequently, from a theoretical point of view, the
coefficient of subgrade reaction k was considered as an artificial concept. It appeared
that an analysis based on that concept was in this case a crude estimate
(45)
.
LEVINTON, Z. [1949] suggested a simplified method for analyzing beams on elastic
foundation. In this method, the contact pressure is represented by a number of
redundant reactions and by creating a set of simultaneous equations in terms of
pressure diagram coordinates and elasticity constants. For each of beam and
foundation, it is possible to find the bending moment and shear force at any point along
the beam and also to calculate the contact pressure.
TERZAGHI, K. [1955] established a number of equations to calculate the modulus of
subgrade reaction for cohesive and cohesion less soils, depending on plate load test
results. He stated that the theories of subgrade reaction are based on the following
assumptions:
1. The ratio ks between the contact pressure p and the corresponding displacement w is
independent of the pressure p.
1. The coefficient of subgrade reaction ks has the same value for every point on the
surface acted upon by the contact pressure. Terzaghi concluded that, provided that p is
smaller than one-half of the ultimate bearing pressure (as well as the fact that ks is
dependent on the dimension of the loaded area), the theories of subgrade reaction
could furnish reliable estimates of stresses and bending moments, although they were
not good in estimating displacements.
BOWLES, J. E. [1974] developed a computer program to carry out the analysis of beams
on elastic foundation using finite element method, in which Winkler model is adopted
to represent the elastic foundation. Several boundary conditions can be entered easily.
From comparison with Vesic method, it is shown that results provide a more realistic
distribution of longitudinal bending moment in the member.
BOWLES, J. E. [December, 1986] introduced a brief survey of computerized methods for
mat design with particular advantages and disadvantages of the three common discrete
element methods, the finite difference, the finite grid, and the finite element. The
modulus of subgrade reaction (ks) is considered in some detail both in obtaining
reasonable initial design estimates and simple methods to couple node effects. Also, a
mat foundation is analyzed by three methods with three ways of springs coupling.
Bowles deduced that there is no serious difference between the coupled and uncoupled
springs and that the finite grid method is a good alternative to the finite element
method.
16
16 Beam on Elastic Foundation (Winkler Model)
2.2.2 BEAMS ON ELASTIC FOUNDATION (DIFFERENTIAL EQUATION)
Differential Equation or Classical Solution of the soil-structure interaction problem, this
equation may be written:
�??????
��
4
��
4
+??????
??????.??????=�…………………………………Eq.(2.2)
Where ks: is a constant called Modulus of Subgrade Reaction.
HETENYI, M. [1946] presented a textbook for the theory and applications of elastically
supported beams in the fields of civil and mechanical engineering. The subject of this
textbook is the analysis of elastically supported beams using classical differential
equation. Different variation parameters such as beam end conditions, beam flexural
rigidity, elasticity properties of the foundation, and applied loading are studied and
resolved. Two basic types of elastic foundation were considered, Winkler model and
elastic solid which, in contrast to Winkler type, represents the case of complete
continuity in the supporting medium.
There are many of literature has been published on this problem, especially for
nonlinear analysis, but it’s out the scope of this study.
16 Beam on Elastic Foundation (Winkler Model)
2.2.2 BEAMS ON ELASTIC FOUNDATION (DIFFERENTIAL EQUATION)
Differential Equation or Classical Solution of the soil-structure interaction problem, this
equation may be written:
�??????
��
4
��
4
+??????
??????.??????=�…………………………………Eq.(2.2)
Where ks: is a constant called Modulus of Subgrade Reaction.
HETENYI, M. [1946] presented a textbook for the theory and applications of elastically
supported beams in the fields of civil and mechanical engineering. The subject of this
textbook is the analysis of elastically supported beams using classical differential
equation. Different variation parameters such as beam end conditions, beam flexural
rigidity, elasticity properties of the foundation, and applied loading are studied and
resolved. Two basic types of elastic foundation were considered, Winkler model and
elastic solid which, in contrast to Winkler type, represents the case of complete
continuity in the supporting medium.
There are many of literature has been published on this problem, especially for
nonlinear analysis, but it’s out the scope of this study.
17
17 Beam on Elastic Foundation (Winkler Model)
Chapter three
Theory
17 Beam on Elastic Foundation (Winkler Model)
Chapter three
Theory
18
18 Beam on Elastic Foundation (Winkler Model)
STIFFNESS MATRIX METHOD OF BEAM ON ELASTIC FOUNDATION
Stiffness Matrix method is one of the most efficient means for solving a Beam on Elastic
Foundation type of problem based on the following Eq. (2.2). It is easy to account for Boundary
Conditions, beam weight, and nonlinear soil effects caused by footing separation.
It is more versatile (multi-purposes) than the Finite Difference method, which requires a
different equation formulation for ends and the boundary conditions, and great difficulty is had
if the Beam elements are of different lengths.
Only the basic elements of the Stiffness Matrix Method will be introduce here, and the
researcher is referred to KassimAli (1999)
(15)
or Bowles (1974)
(11)
if more background is
required. This method was interpolated to computer program which is given in appendix A. The
program algorithm is explained in details in chapter four and it conveniently coded for the user.
Also the same program was used to obtain the results of the numerical examples given in
chapter four of this study.
??????
??????=�
??????+�
????????????
�
……………………Eq.(3.1)
GENERAL EQUATION AND THEIR SOLUTION
For the Beam Element, shown in Fig.(3.1), at any node (i) (junction of two or more members) on
the in-plane structure the equilibrium equation is:
??????
�=�
��
�……………………Eq.(3.2)
Which states that the external node force P is equated to the internal member forces F using
bridging constants A. It should be is understand that (Pi, Fi) are used for either Forces (Shear) or
Bending Moments. This equation is shorthand notation for several values of Ai, Fi summed to
equal the i
th
nodal force.
For the full set of nodes on any in-plane structure and using matrix notation where P, F are
Columns Vectors and A is a Rectangular Matrix, this becomes:
{??????
�}=[�
�]{�
�}……………………Eq.(3.3)
18 Beam on Elastic Foundation (Winkler Model)
STIFFNESS MATRIX METHOD OF BEAM ON ELASTIC FOUNDATION
Stiffness Matrix method is one of the most efficient means for solving a Beam on Elastic
Foundation type of problem based on the following Eq. (2.2). It is easy to account for Boundary
Conditions, beam weight, and nonlinear soil effects caused by footing separation.
It is more versatile (multi-purposes) than the Finite Difference method, which requires a
different equation formulation for ends and the boundary conditions, and great difficulty is had
if the Beam elements are of different lengths.
Only the basic elements of the Stiffness Matrix Method will be introduce here, and the
researcher is referred to KassimAli (1999)
(15)
or Bowles (1974)
(11)
if more background is
required. This method was interpolated to computer program which is given in appendix A. The
program algorithm is explained in details in chapter four and it conveniently coded for the user.
Also the same program was used to obtain the results of the numerical examples given in
chapter four of this study.
??????
??????=�
??????+�
????????????
�
……………………Eq.(3.1)
GENERAL EQUATION AND THEIR SOLUTION
For the Beam Element, shown in Fig.(3.1), at any node (i) (junction of two or more members) on
the in-plane structure the equilibrium equation is:
??????
�=�
��
�……………………Eq.(3.2)
Which states that the external node force P is equated to the internal member forces F using
bridging constants A. It should be is understand that (Pi, Fi) are used for either Forces (Shear) or
Bending Moments. This equation is shorthand notation for several values of Ai, Fi summed to
equal the i
th
nodal force.
For the full set of nodes on any in-plane structure and using matrix notation where P, F are
Columns Vectors and A is a Rectangular Matrix, this becomes:
{??????
�}=[�
�]{�
�}……………………Eq.(3.3)
19
19 Beam on Elastic Foundation (Winkler Model)
Fig.(3.1) Beam Element, external and internal forces and their deformations.
An Equation relating internal-member deformation e at any node to the external nodal
displacements is:
{�
�}=[�
�]{??????
�}
Where both e and X may be rotations (in radians) or translations. From the Reciprocal Theorem
in structural mechanics it can be shown that the [B] matrix is exactly the transpose of the [A]
matrix, thus:
{�
�}=[�]
??????
{??????
�}……………………..(b)
The internal-member forces {F} are related to the internal-member displacements {e} as:
{�
�}=[�]{�
�}…………………………(c)
These three equations are the fundamental equations in the Stiffness Matrix Method of
analysis:
Substituting (b) into (c),
{�
�}=[�]{�
�}=[�][�]
??????
??????…………………………(d)
Substituting (d) into (a),
{??????
�}=[�]{�
�}=[�][�][�]
??????
??????…………………………(e)
Note the order of terms used in developing Eqs. (d) and (e}. Now the only unknowns in this
system of equations are the X’s: so the ASA
T
is inverted to obtain
{??????
�}=([�][�] [�]
??????
)
−1
{??????
�}…………………………(e)
And with the X’s values we can back-substitute into Eq. (d) to obtain the internal-member forces
which are necessary for design. This method gives two important pieces of information: (1)
design data and (2) deformation data.
19 Beam on Elastic Foundation (Winkler Model)
Fig.(3.1) Beam Element, external and internal forces and their deformations.
An Equation relating internal-member deformation e at any node to the external nodal
displacements is:
{�
�}=[�
�]{??????
�}
Where both e and X may be rotations (in radians) or translations. From the Reciprocal Theorem
in structural mechanics it can be shown that the [B] matrix is exactly the transpose of the [A]
matrix, thus:
{�
�}=[�]
??????
{??????
�}……………………..(b)
The internal-member forces {F} are related to the internal-member displacements {e} as:
{�
�}=[�]{�
�}…………………………(c)
These three equations are the fundamental equations in the Stiffness Matrix Method of
analysis:
Substituting (b) into (c),
{�
�}=[�]{�
�}=[�][�]
??????
??????…………………………(d)
Substituting (d) into (a),
{??????
�}=[�]{�
�}=[�][�][�]
??????
??????…………………………(e)
Note the order of terms used in developing Eqs. (d) and (e}. Now the only unknowns in this
system of equations are the X’s: so the ASA
T
is inverted to obtain
{??????
�}=([�][�] [�]
??????
)
−1
{??????
�}…………………………(e)
And with the X’s values we can back-substitute into Eq. (d) to obtain the internal-member forces
which are necessary for design. This method gives two important pieces of information: (1)
design data and (2) deformation data.
20
20 Beam on Elastic Foundation (Winkler Model)
The ASA
T
matrix above is often called Overall Global Matrix, since it represents the system of
equations for each P or X nodal entry. It is convenient to build it from one finite element of the
structure at a time and use superposition to build the global ASA
T
from the Local element EASA
T
.
This is easily accomplished, since every entry in both the Global and Local ASA
T
with a unique set
of subscripts is placed into that subscript location in the ASA
T
; i.e., for i = 2, j = 5 all (2, 5)
subscripts in EASA
T
are added into the (2, 5) coordinate location of the global ASA
T
.
DEVELOPING THE ELEMENT A MATRIX
Consider the in-plane structure, simple beam, shown in Fig.(3.4) coded with four values of P-X
(note that two of these P-X values will be common to the next element) and the forces on the
element Fig.(3.4). The forces on the element include two internal Bending Moments and the
shear effect of the Bending Moments. The sign convention used is consistent with the
developed computer program BEF.
20 Beam on Elastic Foundation (Winkler Model)
The ASA
T
matrix above is often called Overall Global Matrix, since it represents the system of
equations for each P or X nodal entry. It is convenient to build it from one finite element of the
structure at a time and use superposition to build the global ASA
T
from the Local element EASA
T
.
This is easily accomplished, since every entry in both the Global and Local ASA
T
with a unique set
of subscripts is placed into that subscript location in the ASA
T
; i.e., for i = 2, j = 5 all (2, 5)
subscripts in EASA
T
are added into the (2, 5) coordinate location of the global ASA
T
.
DEVELOPING THE ELEMENT A MATRIX
Consider the in-plane structure, simple beam, shown in Fig.(3.4) coded with four values of P-X
(note that two of these P-X values will be common to the next element) and the forces on the
element Fig.(3.4). The forces on the element include two internal Bending Moments and the
shear effect of the Bending Moments. The sign convention used is consistent with the
developed computer program BEF.
21
21 Beam on Elastic Foundation (Winkler Model)
Fig.(3.4)
a) in-plane structure divided into finite element
b) Global coordinate system coding in (P-X) form.
c) Local coordinate system coding in (F-e) form.
d) Summing of external and internal nodal forces.
Now at node (1), summing Moments (Fig.(3.4d))
??????
1=�
1+0.�
2
Similarly, summing forces and noting that the soil reaction (spring) forces are Global and will be
considered separately, we have:
21 Beam on Elastic Foundation (Winkler Model)
Fig.(3.4)
a) in-plane structure divided into finite element
b) Global coordinate system coding in (P-X) form.
c) Local coordinate system coding in (F-e) form.
d) Summing of external and internal nodal forces.
Now at node (1), summing Moments (Fig.(3.4d))
??????
1=�
1+0.�
2
Similarly, summing forces and noting that the soil reaction (spring) forces are Global and will be
considered separately, we have:
22
22 Beam on Elastic Foundation (Winkler Model)
??????
2=
�
1
�
+
�
2
�
??????
3=0.�
1+�
2
And ??????
4=−
�
1
??????
−
�
2
??????
Placing into conventional matrix form, the Element Transformation Matrix [EA] in local
coordinate is:
EA =
F1 F2
P1 1 0
P2 1/L 1/L
P3 0 1
P4 -1/L -1/L
In same manner the EA matrix for element (2) would contain P3 to P6.
DEVELOPING THE [S] MATRIX
Referring to Fig.(3.5) and using conjugate-beam (Moment Area Method)principle, the end slopes
e1, and e2 are:
�
1=
�
1??????
3�??????
−
�
2??????
6�??????
………………………(g)
�
2=−
�
1??????
6�??????
+
�
2??????
3�??????
…………………….(h)
Fig.(3.5) conjugate-beam method Moments and rotations of beam element.
Solving Eqs.(g) and (h) for F, obtaining:
22 Beam on Elastic Foundation (Winkler Model)
??????
2=
�
1
�
+
�
2
�
??????
3=0.�
1+�
2
And ??????
4=−
�
1
??????
−
�
2
??????
Placing into conventional matrix form, the Element Transformation Matrix [EA] in local
coordinate is:
EA =
F1 F2
P1 1 0
P2 1/L 1/L
P3 0 1
P4 -1/L -1/L
In same manner the EA matrix for element (2) would contain P3 to P6.
DEVELOPING THE [S] MATRIX
Referring to Fig.(3.5) and using conjugate-beam (Moment Area Method)principle, the end slopes
e1, and e2 are:
�
1=
�
1??????
3�??????
−
�
2??????
6�??????
………………………(g)
�
2=−
�
1??????
6�??????
+
�
2??????
3�??????
…………………….(h)
Fig.(3.5) conjugate-beam method Moments and rotations of beam element.
Solving Eqs.(g) and (h) for F, obtaining:
23
23 Beam on Elastic Foundation (Winkler Model)
�
1=
4�??????
�
�
1+
2�??????
�
�
2
�
2=
2�??????
�
�
1+
4�??????
�
�
2
Placing into matrix form, the Element Stiffness Matrix [ES] in local coordinate is:
ES =
e1 e2
F1
4�??????
�
2�??????
�
F2
2�??????
�
4�??????
�
DEVELOPING THE ELEMENT SA
T
AND ASA
T
MATRICES
The ESAT matrix1 is formed by multiplying the [ES] and the transpose of the [EA] matrix (in the
computer program BEF this is done in place by proper use of subscripting) A
T
goes always with e
and X. The EASA
T
will be also obtained in a similar.
The node soil "spring" will have units of FL
-1
obtained from the modulus of subgrade reaction
and based on contributory node area. When ks = constant they can be computed as
�
1=
??????
1
2
�??????
?????? and �
2=
??????
1+??????
2
2
�??????
??????
J. Bowles
(100)
, shows that best results are obtained by doubling the end springs. This was done
to make a best fit of the measured (experimental results) data of Vesic and Johnson (1963)
with
computed results (by computer).
This is incorporated into the computer program. There is some logic in this in that if higher edge
pressures are obtained for footings, then this translates into "stiffer" end soil springs. For above
use K1=L1.B.KS and similarly for K5 of Fig.(3.4).
Multiplying [ES] and [EA
T
] matrices and rearrange them, yields:
ESA
T
=
1 2 3 4
1
4�??????
�
6�??????
�
2
2�??????
�
−
6�??????
�
2
2
2�??????
�
6�??????
�
2
4�??????
�
−
6�??????
�
2
Multiplying [EA] and [ESA
T
] matrices and rearrange them, yields:
23 Beam on Elastic Foundation (Winkler Model)
�
1=
4�??????
�
�
1+
2�??????
�
�
2
�
2=
2�??????
�
�
1+
4�??????
�
�
2
Placing into matrix form, the Element Stiffness Matrix [ES] in local coordinate is:
ES =
e1 e2
F1
4�??????
�
2�??????
�
F2
2�??????
�
4�??????
�
DEVELOPING THE ELEMENT SA
T
AND ASA
T
MATRICES
The ESAT matrix1 is formed by multiplying the [ES] and the transpose of the [EA] matrix (in the
computer program BEF this is done in place by proper use of subscripting) A
T
goes always with e
and X. The EASA
T
will be also obtained in a similar.
The node soil "spring" will have units of FL
-1
obtained from the modulus of subgrade reaction
and based on contributory node area. When ks = constant they can be computed as
�
1=
??????
1
2
�??????
?????? and �
2=
??????
1+??????
2
2
�??????
??????
J. Bowles
(100)
, shows that best results are obtained by doubling the end springs. This was done
to make a best fit of the measured (experimental results) data of Vesic and Johnson (1963)
with
computed results (by computer).
This is incorporated into the computer program. There is some logic in this in that if higher edge
pressures are obtained for footings, then this translates into "stiffer" end soil springs. For above
use K1=L1.B.KS and similarly for K5 of Fig.(3.4).
Multiplying [ES] and [EA
T
] matrices and rearrange them, yields:
ESA
T
=
1 2 3 4
1
4�??????
�
6�??????
�
2
2�??????
�
−
6�??????
�
2
2
2�??????
�
6�??????
�
2
4�??????
�
−
6�??????
�
2
Multiplying [EA] and [ESA
T
] matrices and rearrange them, yields:
24
24 Beam on Elastic Foundation (Winkler Model)
EASA
T
=
X1 X2 X3 X4
P1
4�??????
�
6�??????
�
2
2�??????
�
−
6�??????
�
2
P2
6�??????
�
2
12�??????
�
3
+??????
�
6�??????
�
2
−
12�??????
�
3
P3
4�??????
�
6�??????
�
2
4�??????
�
2
−
6�??????
�
2
P4 −
6�??????
�
2
−
12�??????
�
3
−
6�??????
�
2
12�??????
�
3
+??????
�
From Fig.(3.4), summing of the vertical forces on a node 1 will produce:
??????
2−
�
1+�
2
�
−�
1??????
2=0.0
Since (F1+F2)/L is already included in the Global ASA
T
, we could rewrite above equation to:
??????
2=���
2??????2
??????
??????
2+�
1??????
2=(���
2??????2
??????
+�
1)??????
2
It simply means the node spring will be directly added to the appropriate diagonal terms,
subscripted with (i, i).
This is the most efficient method of including the soil stiffness (represented as elastic springs)
since they can be built during element input into a "spring" array.
Later the global ASA
T
is built (and saved for nonlinear cases) and the springs then added to the
appropriate diagonal terms (or column 1 of the banded matrix usually used).
A check on the correct formation of the EASA
T
and the global ASA
T
is that it is always
symmetrical and there cannot be a zero on the diagonal. Note that the soil spring is an additive
term to only the appropriate diagonal term in the global AS A1 matrix. This allows easy removal
of a spring for tension effect while still being able to obtain a solution, since there is still the
shear effect at the point (not having a zero on the diagonal). This is the procedure used in
program B-5 using subroutine MODIF. This procedure has an additional advantage that the ASA7
does not have to be rebuilt for nonlinear soil effects if a copy is saved to call on subsequent
cycles for nodal spring adjustments.
DEVELOPING THE P MATRICES
The P matrix (a column vector) consists in zeroing the array and then inputting those node loads
that are nonzero. The usual design problem may involve several different loading cases or
conditions, as shown in Appendix II, so the array is of the form P(I, J) where (i) identifies the load
entry with respect to the node and P-X coding and (j) the load case.
It is necessary to know the sign convention of the (P-X) coding used in forming the [EA] matrix or
output may be in substantial error. Therefore; the sign convention will be as follow: the joint
translations are considered positive when they act in positive direction of Y-axis, and joint
rotations are considered positive when they rotate in counterclockwise direction.
For columns that are intermediate between two nodes, we may do one of two things:
24 Beam on Elastic Foundation (Winkler Model)
EASA
T
=
X1 X2 X3 X4
P1
4�??????
�
6�??????
�
2
2�??????
�
−
6�??????
�
2
P2
6�??????
�
2
12�??????
�
3
+??????
�
6�??????
�
2
−
12�??????
�
3
P3
4�??????
�
6�??????
�
2
4�??????
�
2
−
6�??????
�
2
P4 −
6�??????
�
2
−
12�??????
�
3
−
6�??????
�
2
12�??????
�
3
+??????
�
From Fig.(3.4), summing of the vertical forces on a node 1 will produce:
??????
2−
�
1+�
2
�
−�
1??????
2=0.0
Since (F1+F2)/L is already included in the Global ASA
T
, we could rewrite above equation to:
??????
2=���
2??????2
??????
??????
2+�
1??????
2=(���
2??????2
??????
+�
1)??????
2
It simply means the node spring will be directly added to the appropriate diagonal terms,
subscripted with (i, i).
This is the most efficient method of including the soil stiffness (represented as elastic springs)
since they can be built during element input into a "spring" array.
Later the global ASA
T
is built (and saved for nonlinear cases) and the springs then added to the
appropriate diagonal terms (or column 1 of the banded matrix usually used).
A check on the correct formation of the EASA
T
and the global ASA
T
is that it is always
symmetrical and there cannot be a zero on the diagonal. Note that the soil spring is an additive
term to only the appropriate diagonal term in the global AS A1 matrix. This allows easy removal
of a spring for tension effect while still being able to obtain a solution, since there is still the
shear effect at the point (not having a zero on the diagonal). This is the procedure used in
program B-5 using subroutine MODIF. This procedure has an additional advantage that the ASA7
does not have to be rebuilt for nonlinear soil effects if a copy is saved to call on subsequent
cycles for nodal spring adjustments.
DEVELOPING THE P MATRICES
The P matrix (a column vector) consists in zeroing the array and then inputting those node loads
that are nonzero. The usual design problem may involve several different loading cases or
conditions, as shown in Appendix II, so the array is of the form P(I, J) where (i) identifies the load
entry with respect to the node and P-X coding and (j) the load case.
It is necessary to know the sign convention of the (P-X) coding used in forming the [EA] matrix or
output may be in substantial error. Therefore; the sign convention will be as follow: the joint
translations are considered positive when they act in positive direction of Y-axis, and joint
rotations are considered positive when they rotate in counterclockwise direction.
For columns that are intermediate between two nodes, we may do one of two things:
25
25 Beam on Elastic Foundation (Winkler Model)
1. Transfer the column loads to adjacent nodes prier to make problem sketch using
superposition concept.
2. Transfer the column loads to adjacent nodes as if the element has Fixed-Ends so the values
include Fixed-End moments and shears (vertical loads).This procedure is strictly correct but the
massive amount of computations is seldom worth the small improvement in computational
precision.
BOUNDARY CONDITIONS
The particular advantage of the Stiffness Matrix method is to allow boundary conditions of
known displacement (translations or rotations). It is common in foundation analysis to have
displacements which are known to be zero (beam on rock, beam embedded in an anchor of
some type, etc.). There are two major cases of boundary conditions:
a. When the displacements are restrained (zero) in any particular node then the
corresponding rows and columns in the overall stiffness matrix will be eliminated
(substitute by zeros).
b. When the (i) displacements are known (δ) in any particular node then the opposite
position in load vector [p] will have this known value (δ), and corresponding rows and
columns in the overall stiffness matrix will be eliminated (substitute by zeros) except the
location of (i,i) which will have unit value of (1.0).
SPRING COUPLING
From a Boussinesq analysis it is evident that the base contact pressure contributes to
settlements at other points, i.e., causing the center of a flexible uniformly loaded base to settle
more than at the edges. Using a constant ks on a rectangular uniformly loaded base w^ill
produce a constant settlement (every node will have the same AH within computer round-off) if
we compute node springs based on contributing node area. This is obviously incorrect and many
persons do not like to use ks because of this problem. In other words the settlement is
"coupled" but the soil springs from ks have not been coupled.
It is still desirable, however, to use ks (some persons call this a "Winkler" foundation) in a spring
concept since only the diagonal translation terms are affected. When we have true coupling,
fractions of the springs X, are in the off-diagonal terms making it difficult to perform any kind of
nonlinear analysis (soil-base separation or excessive displacements). We can approximately
include coupling effects in several ways:
1. Double end springs this effectively increases ks in the end zones. This is not applicable to sides
of very long narrow members.
2. Zone ks with larger values at the ends which transitions to a minimum at the center. For
beam-on-elastic foundation problems where concentrated loads and moments are more
common than a uniform load, doubling the end springs is probably sufficient coupling.
25 Beam on Elastic Foundation (Winkler Model)
1. Transfer the column loads to adjacent nodes prier to make problem sketch using
superposition concept.
2. Transfer the column loads to adjacent nodes as if the element has Fixed-Ends so the values
include Fixed-End moments and shears (vertical loads).This procedure is strictly correct but the
massive amount of computations is seldom worth the small improvement in computational
precision.
BOUNDARY CONDITIONS
The particular advantage of the Stiffness Matrix method is to allow boundary conditions of
known displacement (translations or rotations). It is common in foundation analysis to have
displacements which are known to be zero (beam on rock, beam embedded in an anchor of
some type, etc.). There are two major cases of boundary conditions:
a. When the displacements are restrained (zero) in any particular node then the
corresponding rows and columns in the overall stiffness matrix will be eliminated
(substitute by zeros).
b. When the (i) displacements are known (δ) in any particular node then the opposite
position in load vector [p] will have this known value (δ), and corresponding rows and
columns in the overall stiffness matrix will be eliminated (substitute by zeros) except the
location of (i,i) which will have unit value of (1.0).
SPRING COUPLING
From a Boussinesq analysis it is evident that the base contact pressure contributes to
settlements at other points, i.e., causing the center of a flexible uniformly loaded base to settle
more than at the edges. Using a constant ks on a rectangular uniformly loaded base w^ill
produce a constant settlement (every node will have the same AH within computer round-off) if
we compute node springs based on contributing node area. This is obviously incorrect and many
persons do not like to use ks because of this problem. In other words the settlement is
"coupled" but the soil springs from ks have not been coupled.
It is still desirable, however, to use ks (some persons call this a "Winkler" foundation) in a spring
concept since only the diagonal translation terms are affected. When we have true coupling,
fractions of the springs X, are in the off-diagonal terms making it difficult to perform any kind of
nonlinear analysis (soil-base separation or excessive displacements). We can approximately
include coupling effects in several ways:
1. Double end springs this effectively increases ks in the end zones. This is not applicable to sides
of very long narrow members.
2. Zone ks with larger values at the ends which transitions to a minimum at the center. For
beam-on-elastic foundation problems where concentrated loads and moments are more
common than a uniform load, doubling the end springs is probably sufficient coupling.
26
26 Beam on Elastic Foundation (Winkler Model)
Chapter four
Computer program
26 Beam on Elastic Foundation (Winkler Model)
Chapter four
Computer program
27
27 Beam on Elastic Foundation (Winkler Model)
4.1 INTRODUCTION
This chapter presents a detailed description of the computer program developed in this
study which governs the problem of Beam on Elastic Foundation using Stiffness Matrix
as analysis method and Winkler model for foundation representation.
4.2 DEVELOPMENTS OF COMPUTER PROGRAM FOR BEAM-ON-ELASTIC FOUNDATION
A computer program will develop the [EA] and [ES] for each beam element from input data
describing the member geometry and properties then a computations or reading of ks can be
made. The program performs matrix operations (multiplication, adding, and subtraction) to
form the [ESA
T
] and [EASA
T
] and with proper instructions identifies the (P-X) coding so that the
[EASA
T
] entries are correctly inserted into the element stiffness matrix in G.C.S. [ASA
T
] (called
also Global).
When this has been done for all the beam elements, let the number of nodes NN, and since DOF
is two for each node in beam element. Then in L.C.S. each stiffness element [A] has (NF X NF)
size and in G.C.S. the element stiffness [ASA
T
] will have (NP X NP) size, where NP = NN X 2, which
is have been developed as follows:
{??????
????????????}=[�
???????????? × ??????��
??????� × ??????��
??????� × ????????????
??????
]{??????
????????????}
And canceling interior terms (F) as shown gives
{??????
????????????}=[���
??????
???????????? × ????????????]{??????
????????????}
This indicates that the System of Equations is just sufficient, which yields a square coefficient
matrix [NPXNP], the only type which can be inverted.
It also gives a quick estimate of computer needs, as the matrix is always the size of (NP x NP) the
number of {P}. With proper coding, as shown in Fig.(3.12).
The global [ASA
T
] is banded with all zeros except for a diagonal strip of nonzero entries that is
eight values wide. Of these eight nonzero entries, four are identical (the band is symmetrical).
There are matrix reduction routines to solve these type half-band width problems. As a
consequence the actual matrix required (with a band reduction method) is only (NP x 4) entries
instead of (NP x NP).
The [ASA
T
] is inverted (a sub program reduces a band matrix) and multiplied by the {P} matrix
containing the known externally applied loads. This gives the nodal displacements {X} of rotation
and translation. The computer program then rebuilds the [EA] and [ES] to obtain the [ESA
T
] and
computes the internal element forces (shear and moments). Then node reactions and soil
pressures are computed
�
�=�
�??????
� and �
�=??????
????????????
�
It may be convenient to store the [ESA
T
] on a separate array when the [ASA
T
] is being built and
recall it to compute the internal element forces of the {F} matrix.
27 Beam on Elastic Foundation (Winkler Model)
4.1 INTRODUCTION
This chapter presents a detailed description of the computer program developed in this
study which governs the problem of Beam on Elastic Foundation using Stiffness Matrix
as analysis method and Winkler model for foundation representation.
4.2 DEVELOPMENTS OF COMPUTER PROGRAM FOR BEAM-ON-ELASTIC FOUNDATION
A computer program will develop the [EA] and [ES] for each beam element from input data
describing the member geometry and properties then a computations or reading of ks can be
made. The program performs matrix operations (multiplication, adding, and subtraction) to
form the [ESA
T
] and [EASA
T
] and with proper instructions identifies the (P-X) coding so that the
[EASA
T
] entries are correctly inserted into the element stiffness matrix in G.C.S. [ASA
T
] (called
also Global).
When this has been done for all the beam elements, let the number of nodes NN, and since DOF
is two for each node in beam element. Then in L.C.S. each stiffness element [A] has (NF X NF)
size and in G.C.S. the element stiffness [ASA
T
] will have (NP X NP) size, where NP = NN X 2, which
is have been developed as follows:
{??????
????????????}=[�
???????????? × ??????��
??????� × ??????��
??????� × ????????????
??????
]{??????
????????????}
And canceling interior terms (F) as shown gives
{??????
????????????}=[���
??????
???????????? × ????????????]{??????
????????????}
This indicates that the System of Equations is just sufficient, which yields a square coefficient
matrix [NPXNP], the only type which can be inverted.
It also gives a quick estimate of computer needs, as the matrix is always the size of (NP x NP) the
number of {P}. With proper coding, as shown in Fig.(3.12).
The global [ASA
T
] is banded with all zeros except for a diagonal strip of nonzero entries that is
eight values wide. Of these eight nonzero entries, four are identical (the band is symmetrical).
There are matrix reduction routines to solve these type half-band width problems. As a
consequence the actual matrix required (with a band reduction method) is only (NP x 4) entries
instead of (NP x NP).
The [ASA
T
] is inverted (a sub program reduces a band matrix) and multiplied by the {P} matrix
containing the known externally applied loads. This gives the nodal displacements {X} of rotation
and translation. The computer program then rebuilds the [EA] and [ES] to obtain the [ESA
T
] and
computes the internal element forces (shear and moments). Then node reactions and soil
pressures are computed
�
�=�
�??????
� and �
�=??????
????????????
�
It may be convenient to store the [ESA
T
] on a separate array when the [ASA
T
] is being built and
recall it to compute the internal element forces of the {F} matrix.
28
28 Beam on Elastic Foundation (Winkler Model)
If the footing tends to separate from the soil or the deflections are larger than Xmax it is desirable
to have some means to include the footing weight, zero the soil springs where nodes separate,
and apply a constant force to nodes where soil deflections exceed Xmax.
??????
�=−�
�(??????
�??????�)
Note the sign is negative to indicate the soil reaction opposes the direction of translation. Actual
sign of the computed P matrix entry is based on the sign convention used in developing the
general case as in Fig.(3.12).
The same developed computer program, listed in Appendix I, can also be used to solve a
number of structural problems by setting 0.0 for ks values.
4.3 PROGRAM PROCEDURE
Based on theoretical concepts presented in previous chapter, the following step-by-step
procedure for the analysis of In-plane structures (beams) resting on elastic foundation
(elastic spring model or Winkler model) by the Stiffness Method with modifications.
The sign convention used in this analysis is as follow: the joint translations are
considered positive when they act in positive direction of Y-axis, and joint rotations are
considered positive when they rotate in counterclockwise direction:
1. Prepare the analytical model of in-plane structure, as follows:
a. Draw a line diagram of the in-plane structure (beam), and identify each
joint member by a number.
b. Determine the origin of the global (XY) coordinate system (G.C.S.). It is
usually located to the farthest left joint, with the X and Y axes oriented in
the horizontal (positive to the right) and vertical (positive upward)
directions, respectively.
c. For each member, establish a local (xy) coordinate system (L.C.S.), with the
left end (beginning) of the member, and the x and y axes oriented in the
horizontal (positive to the right) and vertical (positive upward) directions,
respectively.
d. Number the degrees of freedom and restrained coordinates of the beam
elements and nodes.
2. Evaluate the Overall Stiffness Matrix [S], and Fixed-End forces Vector {Pf}. The
number of rows & columns of [S] must be equal to the number of DOF of the
structure. For each element of the in-plane structure, perform the following
operations:
a. Compute the Element stiffness matrix [S
e
] in (L.C.S) by apply the basic
stiffness equation, as follow: {�}=[�
�
]{�}.
b. Transform the force vector {�} form (L.C.S) to {??????} in (G.C.S.) using
transformation matrix [A], as follow: {??????}=[�]{�}.
c. Transform the deformation vector {�}form (L.C.S) to {??????} (G.C.S.) using
transformation matrix [B], as follow: {�}=[�]{??????}.
28 Beam on Elastic Foundation (Winkler Model)
If the footing tends to separate from the soil or the deflections are larger than Xmax it is desirable
to have some means to include the footing weight, zero the soil springs where nodes separate,
and apply a constant force to nodes where soil deflections exceed Xmax.
??????
�=−�
�(??????
�??????�)
Note the sign is negative to indicate the soil reaction opposes the direction of translation. Actual
sign of the computed P matrix entry is based on the sign convention used in developing the
general case as in Fig.(3.12).
The same developed computer program, listed in Appendix I, can also be used to solve a
number of structural problems by setting 0.0 for ks values.
4.3 PROGRAM PROCEDURE
Based on theoretical concepts presented in previous chapter, the following step-by-step
procedure for the analysis of In-plane structures (beams) resting on elastic foundation
(elastic spring model or Winkler model) by the Stiffness Method with modifications.
The sign convention used in this analysis is as follow: the joint translations are
considered positive when they act in positive direction of Y-axis, and joint rotations are
considered positive when they rotate in counterclockwise direction:
1. Prepare the analytical model of in-plane structure, as follows:
a. Draw a line diagram of the in-plane structure (beam), and identify each
joint member by a number.
b. Determine the origin of the global (XY) coordinate system (G.C.S.). It is
usually located to the farthest left joint, with the X and Y axes oriented in
the horizontal (positive to the right) and vertical (positive upward)
directions, respectively.
c. For each member, establish a local (xy) coordinate system (L.C.S.), with the
left end (beginning) of the member, and the x and y axes oriented in the
horizontal (positive to the right) and vertical (positive upward) directions,
respectively.
d. Number the degrees of freedom and restrained coordinates of the beam
elements and nodes.
2. Evaluate the Overall Stiffness Matrix [S], and Fixed-End forces Vector {Pf}. The
number of rows & columns of [S] must be equal to the number of DOF of the
structure. For each element of the in-plane structure, perform the following
operations:
a. Compute the Element stiffness matrix [S
e
] in (L.C.S) by apply the basic
stiffness equation, as follow: {�}=[�
�
]{�}.
b. Transform the force vector {�} form (L.C.S) to {??????} in (G.C.S.) using
transformation matrix [A], as follow: {??????}=[�]{�}.
c. Transform the deformation vector {�}form (L.C.S) to {??????} (G.C.S.) using
transformation matrix [B], as follow: {�}=[�]{??????}.
29
29 Beam on Elastic Foundation (Winkler Model)
d. It is evident that matrix [B] is the transpose of matrix [A]
(33)
, therefore
{�}=[�]
??????
{??????}.
e. Substituting step (d) in step (a), resulting in: {�}=[�
�
][�]
??????
{??????}.
f. Substituting step (e) in step (b), resulting in: {??????}=[�][�
�
][�]
??????
{??????}.
g. Inverting equation in step (f), resulting in: {??????}=[[�][�
�
][�]
??????
]
−1
{??????}.
h. Store the element stiffness matrix, in (G.C.S.), [�
�
�
]=[[�][�
�
][�]
??????
]
−1
, for
each element.
i. Compute the lateral loads forces Vector {P
e
}. Knowing that this step
working only if there are existing lateral loading on the element. Using their
proper positions in the Element Stiffness Matrix [S
e
] in (G.C.S.).
j. Assemble Overall Stiffness Matrix [S] for the System of in-plane structure.
By assembling the element stiffness matrices for each element in the in-
plane structure, using their proper positions in the in-plane structure
Stiffness Matrix [S], and it must be symmetric.
3. Compute the Joint load vector {Pj} for each joint of the in-plane structure.
4. Added the lateral loads forces Vector {Pe} to their corresponding Joint load
vector {P} using their proper positions in the in-plane structure Stiffness Matrix
[S].
5. Determine the structure joint displacements {X}. Substitute {P}, {P
e
}, and [S] into
the structure stiffness relations, {??????
�+??????
�}=[�]{??????} .and solve the resulting
system of simultaneous equations for the unknown joint displacements {X}.
6. Compute Element end displacement {e} and end forces {f}, and support
reactions. For each Element of the beam, as following:
a. Obtain Element end displacements {e} form the joint displacements {X},
using the Element code numbers.
b. Compute Element end forces {f}, using the following relationship: {�}=
[�
�
]{�}+{??????
�}.
c. Using the Element code numbers, store the pertinent elements of {f}, in
their proper position in the support reaction Vector {R}
7. Check the calculation of the member end-forces and support reactions by
applying the Equation of Equilibrium, ∑Fy=0
�
�=0 and ∑Mz=0
�
�=0 to the free
body of the entire plane structure.
29 Beam on Elastic Foundation (Winkler Model)
d. It is evident that matrix [B] is the transpose of matrix [A]
(33)
, therefore
{�}=[�]
??????
{??????}.
e. Substituting step (d) in step (a), resulting in: {�}=[�
�
][�]
??????
{??????}.
f. Substituting step (e) in step (b), resulting in: {??????}=[�][�
�
][�]
??????
{??????}.
g. Inverting equation in step (f), resulting in: {??????}=[[�][�
�
][�]
??????
]
−1
{??????}.
h. Store the element stiffness matrix, in (G.C.S.), [�
�
�
]=[[�][�
�
][�]
??????
]
−1
, for
each element.
i. Compute the lateral loads forces Vector {P
e
}. Knowing that this step
working only if there are existing lateral loading on the element. Using their
proper positions in the Element Stiffness Matrix [S
e
] in (G.C.S.).
j. Assemble Overall Stiffness Matrix [S] for the System of in-plane structure.
By assembling the element stiffness matrices for each element in the in-
plane structure, using their proper positions in the in-plane structure
Stiffness Matrix [S], and it must be symmetric.
3. Compute the Joint load vector {Pj} for each joint of the in-plane structure.
4. Added the lateral loads forces Vector {Pe} to their corresponding Joint load
vector {P} using their proper positions in the in-plane structure Stiffness Matrix
[S].
5. Determine the structure joint displacements {X}. Substitute {P}, {P
e
}, and [S] into
the structure stiffness relations, {??????
�+??????
�}=[�]{??????} .and solve the resulting
system of simultaneous equations for the unknown joint displacements {X}.
6. Compute Element end displacement {e} and end forces {f}, and support
reactions. For each Element of the beam, as following:
a. Obtain Element end displacements {e} form the joint displacements {X},
using the Element code numbers.
b. Compute Element end forces {f}, using the following relationship: {�}=
[�
�
]{�}+{??????
�}.
c. Using the Element code numbers, store the pertinent elements of {f}, in
their proper position in the support reaction Vector {R}
7. Check the calculation of the member end-forces and support reactions by
applying the Equation of Equilibrium, ∑Fy=0
�
�=0 and ∑Mz=0
�
�=0 to the free
body of the entire plane structure.
30
30 Beam on Elastic Foundation (Winkler Model)
4.4 FLOW CHART OF COMPUTER PROGRAM
A
Solve for unknowns displacement [d]
Evaluate Total Stiffness Matrix
T (ND, ND)
Build Elastic Foundation Stiffness Matrix (elastic spring)
in G.C.S. KSPNG (ND, ND)
SOILK (ND, ND)
Assemble Overall Stiffness
Matrix of system
H (ND, ND)
Build Element Stiffness
Matrix in G.C.S.
BEAMK (I, ND, ND)
START
ELASTIC FOUNDATION INPUT UNIT
For each node (1 NN) of the Elastic Winkler Foundation
Read the following:
-Geometry of In-plane structure (x, y).
-Elastic properties ( ks, Es, µ).
-Boundary conditions (DOF). NT = NN X 2
IN-PLANE STRUCTURE INPUT UNIT
For each element (1 NE) of the In-plane Structure read the following:
Geometry of In-plane structure (x, y), Elastic properties (E, G, µ), Boundary
conditions (DOF), and Loading pattern ND = NN X 3
Eliminate
corresponding
rows and columns
from the
foundation
stiffness matrix
(1ND)
No
Yes
Is there are negative
displacements exist?
30 Beam on Elastic Foundation (Winkler Model)
4.4 FLOW CHART OF COMPUTER PROGRAM
A
Solve for unknowns displacement [d]
Evaluate Total Stiffness Matrix
T (ND, ND)
Build Elastic Foundation Stiffness Matrix (elastic spring)
in G.C.S. KSPNG (ND, ND)
SOILK (ND, ND)
Assemble Overall Stiffness
Matrix of system
H (ND, ND)
Build Element Stiffness
Matrix in G.C.S.
BEAMK (I, ND, ND)
START
ELASTIC FOUNDATION INPUT UNIT
For each node (1 NN) of the Elastic Winkler Foundation
Read the following:
-Geometry of In-plane structure (x, y).
-Elastic properties ( ks, Es, µ).
-Boundary conditions (DOF). NT = NN X 2
IN-PLANE STRUCTURE INPUT UNIT
For each element (1 NE) of the In-plane Structure read the following:
Geometry of In-plane structure (x, y), Elastic properties (E, G, µ), Boundary
conditions (DOF), and Loading pattern ND = NN X 3
Eliminate
corresponding
rows and columns
from the
foundation
stiffness matrix
(1ND)
No
Yes
Is there are negative
displacements exist?
31
31 Beam on Elastic Foundation (Winkler Model)
Computer flow chart continued
4.5 COMPUTER PROGRAM APPLICATION:
First Example
A Tank structure resting on elastic foundation (Winkler model) has been simplified to the
general footing details, as shown in fig.(4.1), assuming that the loads are factored and they are
obtained from vertical walls. The results are compared with those obtained by J. E. Bowels
(11)
using a FEM. The agreement is very good.
Fig.(4.1) structure layout
kS = 22,000 Kn/m
3
, γc=25 kN/m
3
Ec = 21,700 Mpa, B = 2.64 m
2025.0 kN
5.0 m c/c
Concrete wall
(0.46 x 2.64) m
1350.0 kN
(0.60 x 2.64) m
Concrete wall
(0.40 x 2.64) m
81.0 kN.m
108.0 kN.m
1.18 m
END
Evaluate Internal Forces, in L.C.S., of in-plane structure
elements: F (NE, 6)
A
OUTPUT UNIT
Print in-plane structural displacement, in G.C.S.,
(Vertical, horizontal and rotations) (1NN)
OUTPUT UNIT
Print Internal Forces, in L.C.S., (Axial Force, Sear Force,
and Bending Moment) for left and right side of each
element: (1NE)
31 Beam on Elastic Foundation (Winkler Model)
Computer flow chart continued
4.5 COMPUTER PROGRAM APPLICATION:
First Example
A Tank structure resting on elastic foundation (Winkler model) has been simplified to the
general footing details, as shown in fig.(4.1), assuming that the loads are factored and they are
obtained from vertical walls. The results are compared with those obtained by J. E. Bowels
(11)
using a FEM. The agreement is very good.
Fig.(4.1) structure layout
kS = 22,000 Kn/m
3
, γc=25 kN/m
3
Ec = 21,700 Mpa, B = 2.64 m
2025.0 kN
5.0 m c/c
Concrete wall
(0.46 x 2.64) m
1350.0 kN
(0.60 x 2.64) m
Concrete wall
(0.40 x 2.64) m
81.0 kN.m
108.0 kN.m
1.18 m
END
Evaluate Internal Forces, in L.C.S., of in-plane structure
elements: F (NE, 6)
A
OUTPUT UNIT
Print in-plane structural displacement, in G.C.S.,
(Vertical, horizontal and rotations) (1NN)
OUTPUT UNIT
Print Internal Forces, in L.C.S., (Axial Force, Sear Force,
and Bending Moment) for left and right side of each
element: (1NE)
32
32 Beam on Elastic Foundation (Winkler Model)
Fig.(4.2) deflection diagram
Fig.(4.3) contact pressure diagram
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
1 2 3 4 5 6 7 8 9 10 111213
deflection (m)
Node Number
0
50
100
150
200
250
300
1 2 3 4 5 6 7 8 9 10 11 12 13
vertical contact pressure (kpa)
Node Number
32 Beam on Elastic Foundation (Winkler Model)
Fig.(4.2) deflection diagram
Fig.(4.3) contact pressure diagram
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
1 2 3 4 5 6 7 8 9 10 111213
deflection (m)
Node Number
0
50
100
150
200
250
300
1 2 3 4 5 6 7 8 9 10 11 12 13
vertical contact pressure (kpa)
Node Number
33
33 Beam on Elastic Foundation (Winkler Model)
Fig.(4.4) shear force diagram
Fig.(4.5) bending moment diagram
-1500
-1000
-500
0
500
1000
1500
1 2 3 4 5 6 7 8 9 10 11 12
Shear Force (KN)
Node Number
-1400
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
1 2 3 4 5 6 7 8 9 10111213
Bending Moment (KN.M)
Node Number
33 Beam on Elastic Foundation (Winkler Model)
Fig.(4.4) shear force diagram
Fig.(4.5) bending moment diagram
-1500
-1000
-500
0
500
1000
1500
1 2 3 4 5 6 7 8 9 10 11 12
Shear Force (KN)
Node Number
-1400
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
1 2 3 4 5 6 7 8 9 10111213
Bending Moment (KN.M)
Node Number
34
34 Beam on Elastic Foundation (Winkler Model)
Second Example
A combined footing shown in fig.(4.6) is represented as inplane structure resting on elastic
foundation (using Winkler model). The results are compared with those obtained by J. E. Bowels
(12)
using a FEM. The agreement is very good.
Fig.(4.6) structure layout
Fig.(4.7) deflection diagram
0
0.005
0.01
0.015
0.02
0.025
1 2 3 4 5 6 7 8 9 10 11
Deflection (m)
Node Nomber
kS = 7,540.32 Kn/m
3
, γc=23.6 kN/m
3
Ec = 22,408.75 Mpa, B = 3.048 m
1378.7 kN
4.877 m
c/c
Concrete wall
(0.40 x 3.048)m
1378.7 kN
(0.508 x 3.05) m
Concrete wall
(0.40 x 3.048)m
0.61 m 0.61 m
34 Beam on Elastic Foundation (Winkler Model)
Second Example
A combined footing shown in fig.(4.6) is represented as inplane structure resting on elastic
foundation (using Winkler model). The results are compared with those obtained by J. E. Bowels
(12)
using a FEM. The agreement is very good.
Fig.(4.6) structure layout
Fig.(4.7) deflection diagram
0
0.005
0.01
0.015
0.02
0.025
1 2 3 4 5 6 7 8 9 10 11
Deflection (m)
Node Nomber
kS = 7,540.32 Kn/m
3
, γc=23.6 kN/m
3
Ec = 22,408.75 Mpa, B = 3.048 m
1378.7 kN
4.877 m
c/c
Concrete wall
(0.40 x 3.048)m
1378.7 kN
(0.508 x 3.05) m
Concrete wall
(0.40 x 3.048)m
0.61 m 0.61 m
35
35 Beam on Elastic Foundation (Winkler Model)
Fig.(4.8) contact pressure diagram
Fig.(4.9) shear force diagram
0
20
40
60
80
100
120
140
160
180
1 2 3 4 5 6 7 8 9 10 11
Vetical Contact Pressure (Kpa)
Node Number
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
1 2 3 4 5 6 7 8 9 10
Shear Force (KN)
Node Number
35 Beam on Elastic Foundation (Winkler Model)
Fig.(4.8) contact pressure diagram
Fig.(4.9) shear force diagram
0
20
40
60
80
100
120
140
160
180
1 2 3 4 5 6 7 8 9 10 11
Vetical Contact Pressure (Kpa)
Node Number
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
1 2 3 4 5 6 7 8 9 10
Shear Force (KN)
Node Number
36
36 Beam on Elastic Foundation (Winkler Model)
Fig.(4.10) bending moment diagram
-1200
-1000
-800
-600
-400
-200
0
200
400
1 2 3 4 5 6 7 8 9 10 11
Bending Moment (KN.M)
Node Number
36 Beam on Elastic Foundation (Winkler Model)
Fig.(4.10) bending moment diagram
-1200
-1000
-800
-600
-400
-200
0
200
400
1 2 3 4 5 6 7 8 9 10 11
Bending Moment (KN.M)
Node Number
37
37 Beam on Elastic Foundation (Winkler Model)
Chapter five
Conclusions and Recommendations
37 Beam on Elastic Foundation (Winkler Model)
Chapter five
Conclusions and Recommendations
38
38 Beam on Elastic Foundation (Winkler Model)
CONCLUSIONS
Depending on the results obtained from the present study, several conclusions may be
established. These may be summarized as follows:
1. The results, indicate that in-plane structures (beam) resting on elastic foundation
can be can be dealt with successfully by the Stiffness Matrix Method together
with Winkler foundation model.
2. Developed Program in this study is quite efficient and reliable for this type of
analysis, and the process analyses can be carried out rapidly on electronic
computer.
3. Linear behavior of in-plane structures resting on elastic foundation can be
accurately predicted using Winkler concept as foundation model.
4. The results show that the increasing foundation rigidity (stiffness) will increase
vertical contact pressure.
5. The presented results indicate that suggested modulus of subgrade formula
given by J.E. Bowels
(12)
using spring technique is quite accurate comparing with
experimental results.
38 Beam on Elastic Foundation (Winkler Model)
CONCLUSIONS
Depending on the results obtained from the present study, several conclusions may be
established. These may be summarized as follows:
1. The results, indicate that in-plane structures (beam) resting on elastic foundation
can be can be dealt with successfully by the Stiffness Matrix Method together
with Winkler foundation model.
2. Developed Program in this study is quite efficient and reliable for this type of
analysis, and the process analyses can be carried out rapidly on electronic
computer.
3. Linear behavior of in-plane structures resting on elastic foundation can be
accurately predicted using Winkler concept as foundation model.
4. The results show that the increasing foundation rigidity (stiffness) will increase
vertical contact pressure.
5. The presented results indicate that suggested modulus of subgrade formula
given by J.E. Bowels
(12)
using spring technique is quite accurate comparing with
experimental results.
39
39 Beam on Elastic Foundation (Winkler Model)
RECOMMENDATIONS
Many important recommendations could be suggested:
1. Given analysis method presented in this study for inplane structures can be
extended to analyze three dimension (space) structures.
2. Given type of analysis presented in this study for inplane structures under static
loading can be extended to include dynamic loading cases.
3. More complicated examples should be investigated in order to examine the
program capability in nonlinear stage.
4. Driven and pored piles could be analyzed using same program with special
modifications for ks, to include depth effects.
5. Side wall friction and embodiments effects could be included in this analysis
using researchers published papers in this field, and include them in appropriate
method for the stiffness coefficients of inplane structure.
39 Beam on Elastic Foundation (Winkler Model)
RECOMMENDATIONS
Many important recommendations could be suggested:
1. Given analysis method presented in this study for inplane structures can be
extended to analyze three dimension (space) structures.
2. Given type of analysis presented in this study for inplane structures under static
loading can be extended to include dynamic loading cases.
3. More complicated examples should be investigated in order to examine the
program capability in nonlinear stage.
4. Driven and pored piles could be analyzed using same program with special
modifications for ks, to include depth effects.
5. Side wall friction and embodiments effects could be included in this analysis
using researchers published papers in this field, and include them in appropriate
method for the stiffness coefficients of inplane structure.
40
40 Beam on Elastic Foundation (Winkler Model)
REFERENCES
1. Livesley, R. K., and Chandler D. B., "Stability Functions for Structural Frameworks."
Manchester University Press, Manchester, 1956.
2. Livesley, R.K., "The Application of an Electronic Digital Computer to Some Problem
of Structural Analysis." The Structural Engineer, Vol. 34, no.1, London, 1956, PP. 1-
12.
3. Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis." Pergamon
Press, London, 1964, PP. 115-145.
4. Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London,
1964. PP. 241-252.
5. Winkler, E., "Die T.ehre Von Elasticitaet und Festigkeit." (H. Dominic us), Prague,
1867,pp.182-184
6. Hetenyi, M., "Beams on Elastic Foundations." The University of Michigan Press, Ann
Arbor, 1946, pp. 100-120.
7. Boussinesq, J. "Equilibre d'Elasticite d'Un Solide Isotrope Sans Pesanteur
Supporttant Differents Poids." C. Rendus Acad. Sci Paris.Vol. 86 , pp. 1260-1263,
1878, (in French).
8. Levinton, Z., "Elastic Foundation Analyzed By the Method of Redundant Reactions."
Transaction, ASCE, Vol. 114, 1949, pp. 40-78.
9. Terzaghi, K., "Evaluation of Coefficient of Subgrade Reaction." Geotechnique, Vol.5,
No.4, 1955, pp. 197-326.
10. Vesic, A. B., "Beams On Elastic Solid Subgrade and the Winkler Hypothesis." proc.,
5th International Conference on Soil Mechanics and Foundation Engineering, Vol. 1,
1961,pp.845-850.
11. Bowles, J. E., "Analytical and Computer Methods in Foundation Engineering."
McGraw-Hill Book Co., New York, 1974, pp. 190-210.
12. Bowles, J. E., "Foundation analysis and design" McGraw-Hill Book Co., New York,
1986, fourth edition, pp. 380-230.
13. Bowles, J. E., "Mat Design." ACI Journal, Vol. 83, No.6, Nov.-Dec. 1986, pp. 1010-
1017.
14. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition, McGraw-
Hill Book Company, New York, 1961, pp. 1-17.
15. KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of
Structural Engineering, ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.
16. Lazim, A. N., "Large Displacement Elastic Stability of Elastic Framed Structures
Resting On Elastic Foundation" M.Sc. Thesis, University of Technology, Baghdad,
2003, pp. 42-123.
40 Beam on Elastic Foundation (Winkler Model)
REFERENCES
1. Livesley, R. K., and Chandler D. B., "Stability Functions for Structural Frameworks."
Manchester University Press, Manchester, 1956.
2. Livesley, R.K., "The Application of an Electronic Digital Computer to Some Problem
of Structural Analysis." The Structural Engineer, Vol. 34, no.1, London, 1956, PP. 1-
12.
3. Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis." Pergamon
Press, London, 1964, PP. 115-145.
4. Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London,
1964. PP. 241-252.
5. Winkler, E., "Die T.ehre Von Elasticitaet und Festigkeit." (H. Dominic us), Prague,
1867,pp.182-184
6. Hetenyi, M., "Beams on Elastic Foundations." The University of Michigan Press, Ann
Arbor, 1946, pp. 100-120.
7. Boussinesq, J. "Equilibre d'Elasticite d'Un Solide Isotrope Sans Pesanteur
Supporttant Differents Poids." C. Rendus Acad. Sci Paris.Vol. 86 , pp. 1260-1263,
1878, (in French).
8. Levinton, Z., "Elastic Foundation Analyzed By the Method of Redundant Reactions."
Transaction, ASCE, Vol. 114, 1949, pp. 40-78.
9. Terzaghi, K., "Evaluation of Coefficient of Subgrade Reaction." Geotechnique, Vol.5,
No.4, 1955, pp. 197-326.
10. Vesic, A. B., "Beams On Elastic Solid Subgrade and the Winkler Hypothesis." proc.,
5th International Conference on Soil Mechanics and Foundation Engineering, Vol. 1,
1961,pp.845-850.
11. Bowles, J. E., "Analytical and Computer Methods in Foundation Engineering."
McGraw-Hill Book Co., New York, 1974, pp. 190-210.
12. Bowles, J. E., "Foundation analysis and design" McGraw-Hill Book Co., New York,
1986, fourth edition, pp. 380-230.
13. Bowles, J. E., "Mat Design." ACI Journal, Vol. 83, No.6, Nov.-Dec. 1986, pp. 1010-
1017.
14. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition, McGraw-
Hill Book Company, New York, 1961, pp. 1-17.
15. KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of
Structural Engineering, ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.
16. Lazim, A. N., "Large Displacement Elastic Stability of Elastic Framed Structures
Resting On Elastic Foundation" M.Sc. Thesis, University of Technology, Baghdad,
2003, pp. 42-123.
41
41 Beam on Elastic Foundation (Winkler Model)
Appendix I
41 Beam on Elastic Foundation (Winkler Model)
Appendix I
42
42 Beam on Elastic Foundation (Winkler Model)
42 Beam on Elastic Foundation (Winkler Model)
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