Analysis of Beams on Elastic Foundations

Analysis of In-plane Structure Resting on Elastic Half-Space Foundation
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Analysis of inplane structure resting
on elastic Half Space 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
Thourra Muhsin Khaleel Raghed Adnan Hameed
Supervised by
Assistant lecturer, Adnan Najem (M.Sc., in Structural Engineering)
July /2007

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Analysis of inplane structure
resting on elastic Half Space
Foundation

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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:

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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:

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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.

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ABSTRACT
The 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).
Analysis of the Beams on elastic foundation process 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.

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CONTENTS
Title……………………………………………………………………………………2
Supervisor words……………………………………………………………….3
Committee words………………………………………………………………4
Thanks……………………………………………………………………………….5
Abstract…………………………………………………………………………….6
Project Layout………………………….……………………………………….6
Contents…….……..………………………………………………………………7
Notation…………….……………………………………………………………….8
Chapter one; introduction…………..….…………………………………9
Chapter two; literature………………….………………………………..13
Chapter three; theory………………………………………………………17
Chapter four; computer program…………………………………….26
Chapter five; conclusions and recommendations…………….38
References………………………………………………………………………40

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NOTATION
E: Young's modulus of in-plane structure members (beam).
µ: Poisson's ratio of in-plane structure members (beam).
E0: Young's modulus of the elastic foundation.
µ0: Poisson's ratio of elastic foundation.
wmn: vertical displacement of foundation due to vertical force.
umn: horizontal displacement of foundation due to vertical force.
wMN: vertical displacement of foundation due to horizontal force.
uMN: horizontal displacement of foundation due to horizontal force.
Vn: vertical force at point n.
Hn: horizontal force at point n.
DOF: degree of freedom.
In: logarithm to the base e

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Chapter one
Introduction

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Introduction to elastic half space (foundation model)
This model is based on the elastic continuum theory. Its basic assumptions are
homogenous, isotropic, elastic and infinite depth. This model treat the foundation mass
as one unit consist of number of finite elements that connected by nodes. Through
these nodes the contact pressure will be transmitted to the foundation mass.
In order to derive the foundation stiffness terms the pressure-displacement relationship
should be established. The direct derivations of these displacements (normal and
tangential) beneath a contact pressure will lead to complex integrations, hence, the
problem simplified to the point load acting (normal and tangential) to the surface of
elastic half-space which are also known as (Boussinesq’s and Cerruti’s problems). The
general expression of the displacements produced by contact pressure over a
rectangular area can be obtained by integrating Boussinesq’s and Cerruti’s solutions
over the rectangular area (a × b), figure (1.2) and (1.1).
Fig.(1.1) Boussinesq problem

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Fig.(1.2) Cerruti problem
It was observed that for flexible beams the linear elastic analysis will yield tensile as well
as compressive contact pressures, as can be seen from Fig.(1.3). Tensile contact
pressures can also result from local uplifting forces due to wind load. However, for most
materials the contact surfaces cannot transmit tensile forces and will tend to separate,
thus causing non-linear behavior. An iterative approach has been adopted in this study
to deal with such a situation and convergence is usually reached after several cycles,
depending on the relative stiffness of the foundation.
It was also observed that the vertical contact pressures on the elastic foundation
produce horizontal displacements, which are resisted by the beam through friction and
shear bonding with foundation beneath it, resulting in developing horizontal contact
pressures even when only vertical loadings are present. These horizontal contact
pressures can affect the stresses in the beams to a significant extent for relatively stiff
foundations, and should be included in the analysis under such conditions.
An approximate approach was adopted in this analysis; which presume that a constant
pressure will be developed under each nodal points to spread over on the rectangle (a x
b) around each nodal point, such that the pressure around node i will be of the
magnitude Pi/ab. This approximation can be enhanced by reducing the dimension of
these rectangles (i.e. more finite elements).
The flexibility matrix of the foundation due to the step loads (vertical and horizontal
uniform loads) is first developed out for the appropriate nodal points, for two degree of
freedom for each node (i.e. 2DOF X 2DOF) which will be denoted by [2x2].Then the
discrepancy effect incorporated in the analysis, resulting in new Moment’s coefficients.
Therefore the total number of degree of freedom becomes [3x3].
Then flexibility matrix will be inverted (inverse process of matrices) to obtain the
stiffness matrix [3X3]. This Stiffness Matrix of the elastic foundation is then
appropriately added to the opposite assembly of overall stiffness matrix of in-plane
structure.

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This Total stiffness matrix will be solved to for the unknowns nodal displacements
(translations in X, Y axes and rotations in Z axis), subsequently the contact pressures
(vertical in Y direction and horizontal in X direction), and element internal forces (Axial,
Shear, and Bending Moments) for each element of the in-plane structure.
Fig.(1.3) vertical contact pressure of square plate on elastic foundation
(Represented by Isotropic Elastic Half-Space Model)
With concentrated load (P) at center.

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Chapter two
Literature

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2.1 SOIL-STRUCTURE INTERACTION
Many problems related to soil-structure interaction can be modeled by means of beams
on elastic foundation. Examples of these are railway tracks, continuously supported
pipelines and strip and ring foundations. In frameworks with some of their members
supported on or driven into soil, the structural behavior of frameworks will be
influenced significantly by the restraint caused by the foundation, and the amount of
influence will be dependent on the flexural rigidity of the embedded members and the
soil stiffness. Most of the available published works on the analysis of beam-column on
elastic foundation have been so far assuming that the degrees of freedom at the nodes
to be lateral displacement and flexural rotation, (i.e., infinite axial stiffness is assumed
and thus, axial displacement is neglected).
2.1.1 BEAMS ON ELASTIC FOUNDATION (ELASTIC HALF-SPACE MODEL)
Generally credit to Biot [1937] with elaborating by Ohde [see appendix of Vesic and
Johnson ], treats the foundation as Elastic Half-Space medium. A few amount of
literature work on the elastic half space foundation are available. Analysis of in-plane
structures resting on elastic half space foundation also are quit few.
BOUSSINESQ, J. [1878] analyzed the problem of semi-infinite homogeneous isotropic
linear elastic solid subjected to a concentrated force, which acts normal to the plane
boundary. This problem was solved not for application to geotechnical pursuits, but
simply to answer basic questions about elasticity and the behavior of elastic bodies.
Boussinesq theory has overcome the deficit of Winkler's hypothesis in the discreteness
phenomenon, where it accounts for the continuous behavior of deforming soil media.
BIOT, M. A., [1937] considered the problem of bending, under a concentrated load, of
infinite flexible beams on a homogenous Elastic-Isotopic subgrade. He derived the
expressions for shear, bending moment at any point x of the beam. It is shown that the
Winkler's hypothesis is practically satisfied for infinite beams (AL > 5.0).
VESIC, A. B., [1961] extended Biot's solution of an infinite beam on semi-infinite elastic
solid and presented approximate analytical expressions for the integrals appearing in
the solution evaluated (47). These integrals were found to be presented by curves of
damped-wave type, very similar to the corresponding curves obtained by the
CHEUNG, Y. K., and ZIENKIEWICZ, 0. C., [1964-1965] solved the problem of slabs and
tanks resting either on a semi-infinite elastic continuum or on individual springs
(Winkler) using the finite element method. In the first model, they have depended the
Boussinesq equation in deriving the soil stiffness matrix that have been combined with
plate bending finite element to form the overall stiffness matrix of the whole system.
Also, they have made comparisons between the contact pressure distributions beneath

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foundation using the two models. Re-entrant corners, rigid walls on the slab edges,
concentrated moments due to bending of column, etc., involve little computational
difficulty in the method presented.
CHEUNG, Y. K., and NAG, D. K. [1968] extended the work of Cheung and Zienkiewicz
(1965) by allowing for horizontal contact pressures beneath the foundation. The effects
due to separation of contact surfaces due to uplift forces have also been investigated. In
addition, they have enabled the prediction of the bending and torsion moments in the
plate sections by adopting three degrees of freedom per node that the previous workers
did not.
DAVIS, R. 0., and Selvadurai, A. S., [1996] present a complete survey of fundamental
elasticity solutions of Geomechanics problems, most of these problems were solved in
the latter part of the nineteenth century, and they were usually solved for both
geotechnical application and to answer the basic questions about elasticity and behavior
of elastic bodies.
2.1 LINEAR ANALYSIS OF IN-PLANE STRUCTURES USING STIFFNESS MATRIX METHOD
A considerable amount of literature work on the Stiffness Matrix method has been
published. Historically, the Matrix (stiffness) method of structural analysis was laid and
developed by:
 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 were:.

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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 S. 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
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.

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Chapter three
Theory

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3.1 INTRODUCTION
A theoretical analysis is presented for estimating the in-plane displacement of elastic
framed structures (beams) where some/or all members are resting on elastic foundation
and subjected to static loads. The analysis method (stiffness method) was initially
developed for elastic structures and is extended to include soil-structure interaction.
The analysis adopts simple beam theory and models the structure members as beam
elements.
The influences of axial force on bending moment are neglected in stiffness matrix
assemblage. Also changes in member chord length due to axial deformation and bowing
effect, shear deformations are all neglected.
A computational technique utilizes an iterative procedure to satisfy joint equilibrium
and separation between soil and structure due to uplifts forces is taken into account.
3.2 BASIC ASSUMPTIONS:
In this chapter, major analysis assumptions will be outlined as follows:
Structural analysis, of beams resting on elastic half space foundation, is the prediction of
the performance of a given structure prescribed loads and/or other external effects.
Both matrix methods (stiffness and flexibility) of structural analysis are based on the
same fundamental principles. However, flexibility method (Δ=F.P) is developed to
analyze particular case such as soil flexibility matrix, where the elastic continuum theory
solutions in terms of stiffness matrix form (P=K.Δ) is more difficult to incorporate in this
position.
In-plane structures are composed of straight members whose lengths are significantly
larger than their cross-sectional dimensions.
An analytical model is a simplified (idealized) representation of a real structure for the
purpose of analysis. In-plane structures are modeled as assemblages of straight
members connected at their ends to joints, and these analytical models are represented
by line diagrams.
The analysis of structures involves three fundamental relationships: Equilibrium
Equations, Compatibility Equation (relate deformations and also called continuity
relations), and Constitutive Relations (stress-strain relations)
Linear structural analysis is based on two fundamental assumptions: the stress-strain
relationship for the structural material is linearly elastic, and the structures
deformations are so small that the equilibrium equations can be based on the
undeformed geometry of the structure. And the curvature of the structure flexural
members can be reduced to the following equation:
𝑑2𝑦
𝑑𝑥2
=
𝑀
𝐸𝐼
…………………………………………..Eq. (3.1)

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3.3 REVIEW OF THE FUNDAMENTAL PROBLEM
With reference to In-plane framed structure, element tangent stiffness matrix is entirely
consistent with the conventional Simple Beam theory. The local behavior of individual
element is first analyzed with respect to local reference system (L.C.S.) attached to
member itself. A transformation is then applied to pass from local to an arbitrary global
reference system (G.C.S.) as follow:
Tangent stiffness in local coordinates system:
{∆𝑆} = [𝑡]{∆𝑢}…………………………………..Eq. (3.2)
Tangent stiffness in global coordinates system:
{∆𝐹} = [𝑇]{∆𝑣}…………………………………..Eq. (3.3)
[𝑇] = [𝐵][𝑡][𝐵]𝑇
…………………………………..Eq. (3.4)
System (structure) equilibrium equations:
{∆𝑃} = [𝜏]{∆𝑥}…………………………………….Eq. (3.5)
3.4 FOUNDATION STIFFNESS MATRIX
Stiffness matrix elastic foundation will be developed for two kinds of representations according
to the used approximations elastic foundation:
First approach; Isotropic elastic half-plane model
The vertical and horizontal deflections of any point m due to a unit vertical point load at
(n) on the surface of an isotropic elastic half-plane are given by the Flamant equation
(Timoshenko & Goodier, 1951).
For the plane stress case:
𝑤𝑚𝑛 =
2
𝜋𝐸0
ln
𝑑
𝑟
……………………………Eq.(3.6)
𝑢𝑚𝑛 = ∓
1−𝜇0
2𝐸0
………………………….…Eq.(3.7)
For the plane strain case:
𝑤𝑚𝑛 =
2(1−𝜇0
2)
𝜋𝐸0
ln
𝑑
𝑟
……………………………Eq.(3.8)

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𝑢𝑚𝑛 = ∓
(1−2𝜇0)(1−𝜇0
2)
(1−𝜇0)𝐸0
……………………..Eq.(3.9)
For a strip under uniform load of magnitude (Vn/a), Fig.(3.2), with node (n) at its centre,
the deflections at any point m can be obtained by integrating equation (3.6) over the
loaded strip length (-a, +a):
𝑤𝑚𝑛 =
𝑉𝑛
𝜋𝐸0
𝐹
𝑚𝑛……………………………Eq.(3.10)
Where the coefficient (Fmn) varies only with (r/a), and is given in Table (3.1).
For a strip under uniform load of magnitude (Hn /a), with node (n) at its centre, a similar
set of formulae with identical coefficients can also be derived:
𝑤𝑀𝑁 = ∓
1−𝜇0
2𝐸0
𝐻𝑛……………………………Eq.(3.11)
𝑢𝑀𝑁 =
𝐻𝑛
𝜋𝐸0
𝐹𝑀𝑁………………………….……Eq.(3.12)
Table (3.1)
r/a Fmn r/a Fmn
0 0.000 5 -8.802
1 -3.296 6 -8.931
2 -4.751 7 -9.052
3 -5.574 8 -9.167
4 -6.154 9 -9.275
Fig.(3.2) vertical and horizontal relative displacements due to uniformly loaded strip on
isotropic half-plane.

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Second approach; isotropic half-space model
Deflection formulae similar to equation (3) have been given by Boussinesq and Cerruti
(3) for vertical and horizontal points loads:
𝑤𝑚𝑛 =
1−𝜇0
2
𝜋𝐸0𝑟𝑛
…………………………………………..…Eq.(3.13a)
𝑢𝑚𝑛 = −
(1−𝜇0−2𝜇0
2)
2𝜋𝐸0
𝑥𝑛
𝑟𝑛
2………………………………Eq.(3.13b)
𝑤𝑀𝑁 =
(1−𝜇0−2𝜇0
2)
2𝜋𝐸0
𝑥𝑛
𝑟𝑛
2……………………………….Eq.(3.13c)
𝑢𝑀𝑁 =
(1−𝜇0
2)
𝜋𝐸0
1
𝑟𝑛
+
𝜇0(1−𝜇0)
𝜋𝐸0
𝑥𝑛
2
𝑟𝑛
3………………….Eq.(3.13d)
The above equations are integrated for a uniformly loaded rectangular area (axb),
obtaining:
𝑤𝑚𝑛 =
𝑉𝑛(1−𝜇0
2)
𝑎𝜋𝐸0
[𝐵 sinh−1 1
𝐵
+ sinh−1
𝐵 − 𝐶 sinh−1 1
𝐶
− sinh−1
𝐶] 𝑚 ≠ 𝑛…Eq.(3.14a)
𝑤𝑛𝑛 =
2𝑉𝑛(1−𝜇0
2)
𝑎𝜋𝐸0
[𝐵 sinh−1 1
𝐵
+ sinh−1
𝐵] 𝑚 = 𝑛……………………………………..……Eq.(3.14b)
𝑢𝑚𝑛 = −
𝑉𝑛(1−2𝜇0)(1−𝜇0)
2𝑎𝜋𝐸0
[𝐵 tan−1 1
𝐵
+ ln
1
√(1+B2)
− 𝐶 tan−1 1
𝐶
− ln
1
√(1+C2)
] 𝑚 ≠ 𝑛……..Eq.(3.14c)
𝑢𝑛𝑛 = 0.0 𝑚 = 𝑛……Eq.(3.14d)
𝑤𝑀𝑁 = 𝑢𝑚𝑛 𝑚 ≠ 𝑛……….Eq.(3.14e)
𝑤𝑁𝑁 = 𝑢𝑛𝑛 𝑚 ≠ 𝑛…….…Eq.(3.14f)
𝑢𝑀𝑁 =
𝐻𝑛(1−𝜇0
2)
𝑎𝜋𝐸0
[𝐵 sinh−1 1
𝐵
+ sinh−1
𝐵 − 𝐶 sinh−1 1
𝐶
− sinh−1
𝐶] +
𝐻𝑛𝜇0(1+𝜇0)
𝑎𝜋𝐸0
[𝐵 sinh−1 1
𝐵
+ sinh−1
𝐶] 𝑚 ≠
𝑛 .…Eq.(3.14g)
𝑤𝑁𝑁 =
2𝑉𝑛(1−𝜇0
2)
𝑎𝜋𝐸0
[𝐵 sinh−1 1
𝐵
+ sinh−1
𝐵] +
𝐻𝑛𝜇0(1+𝜇0)
𝑎𝜋𝐸0
[sinh−1
𝐵] 𝑚 = 𝑛………Eq.(3.14h)
Where:
𝐵 = {2(𝑚 − 𝑛) + 1} 𝑎 𝑏
⁄
𝐶 = {2(𝑚 − 𝑛) − 1} 𝑎 𝑏
⁄

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Fig.(3.3) vertical and horizontal relative displacements due to uniformly loaded
rectangular area on isotropic half-space.
Therefore, for any set of grid points, the deflections of both cases can be written as:
{𝑑} = [𝑓
𝑠]{𝑃}………………………Eq.(3.15)
Where
{𝑑} =
{
𝑑1
𝑑2
:
:
𝑑𝑛}
{𝑃} =
{
𝑃1
𝑃2
:
:
𝑃𝑛}
For single node above displacement and load vectors become:
{𝑑𝑖} = {
𝑢𝑖
𝑤𝑖
}
{𝑃𝑖} = {
𝐻𝑖
𝑉𝑖
}
If the horizontal contact pressure is ignored, then
{𝑑𝑖} = {𝑤𝑖}

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{𝑃𝑖} = {𝑉1}
Inverting Equation (3.) leading to:
{𝑃} = [𝐾𝑆]{𝑑}………………………..Eq.(3.16)
Where:
[𝐾𝑠] = [𝑓
𝑠]−1
3.5 THE COMPLETE STIFFNESS FORMULATION
The foundation matrix has now to be combined with that of a beam or plate which is
divided into finite elements. Such matrices are given in standard texts (e.g. Livesley,
1964) in the form of equation (3.12) connecting the nodal forces {F} and displacements
{v}.
{𝐹} = [𝐾]{𝑣}………………………..Eq.(3.17)
The components of each nodal force {F} and displacements {v} will depend on the
nature of the problem. It should be noticed that equations (3.17) and (3.18) are not
compatible since the Moment rotation terms are present only in the latter. Therefore,
the foundation matrix should be augmented by appropriate rows and columns of zeros
or alternatively as described in the section on beams on elastic foundations with
horizontal contact pressure. However, the Moment-rotation terms can be incorporated
by adopting a more refined approach in which the effects due to a point couple on the
boundary are considered. Equation (3.16) can be rewritten as:
{𝑃} = [𝐾𝑠
′]{𝑣}………………………..Eq.(3.18)
Now if Qi represents an external applied load to a node, then the effective external
force acting on that node is:
{𝐹𝑖} = {𝑄𝑖} − {𝑃𝑖}
Substituted in Eq.(3.12), lead to:
{𝑄𝑖} = [𝐾𝐵]{𝑣𝑖} + [𝐾𝑆
′
]{𝑣𝑖}………………………..Eq.(3.19)
{𝑄𝑖} = [𝐾𝐵 + 𝐾𝑆
′
]{𝑣𝑖}……………………….………..Eq.(3.20)
Or:
{𝑣𝑖} = [𝐾𝐵 + 𝐾𝑆
′
]−1{𝑄𝑖}…………….………………..Eq.(3.21)
The contact forces can be obtained from equation (3.18).
3.6 SEPARATION OF CONTACT SURFACES WITH TENSILE CONTACT PRESSURES
In the, following analysis, only the vertical contact pressures have been considered,
although the horizontal contact pressures can also be taken into account without
difficulty. An iterative approach is adopted and convergence is usually reached after
three to four cycles.
The procedure can be outlined as follows:
(i) Perform the analysis as given in previous section. A linear elastic solution is
obtained.

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(ii) If all the contact pressures are compressive the problem is terminated. If
otherwise, proceed to next step.
(iii) Find out the^ nodes which are associated with tensile or zero contact
pressures and make the corresponding rows and columns in the original
flexibility matrix zero.
(iv) Invert the new flexibility matrix and repeat step (i).
After each iteration process the nodes components, which have been eliminated, will
always yield zero contact pressures.
3.7 BEAMS ON ELASTIC FOUNDATIONS WITH HORIZONTAL CONTACT PRESSURE
From Figure (3.2), it can be seen that the Beam stiffness matrix is given for forces and
displacements with respect to its neutral axes (x, u) and that the foundation matrices
are given with respect to the interface (x’, u’). Therefore modifications must be
introduced to overcome this discrepancy.
Even though, if only vertical contact pressure is considered, this modification still
workable; since vertical contact pressures on the elastic foundation will produce
horizontal displacements, which are resisted by the beam through friction and shear
bonding (at the interface level) with foundation beneath it, resulting in developing
couple forces (at the neutral axis level) that will effect the internal Bending Moment of
Beam and changes it.
Fig.(3.2) discrepancy between In-plane structure (Beam) and elastic foundation displacements.
At node (i) (using primed symbols for the foundation system) the following relationships
can be established.
𝑤𝑖
′
= 𝑤𝑖…………………………………….……..Eq.(3.22a)
𝑢𝑖
′
= 𝑢𝑖 − 𝜃𝑖(ℎ 2
⁄ )……………………………Eq.(3.22b)
𝑉𝑖
′
= 𝑉𝑖……………………………………….…....Eq.(3.22c)
𝐻𝑖
′
= 𝐻𝑖…………………………………………....Eq.(3.22d)
𝑀𝑖
′
= −𝐻𝑖
′
(ℎ 2
⁄ )………………………….……Eq.(3.22e)

Page 25
Expanding equation (3.17) and taking into account equations (3.22a) to (3.22d),
obtaining:
{
𝐻1
𝑉1
⋮
𝐻𝑛
𝑉
𝑛 }
=
[
𝑙11
𝑓11
𝑙12
𝑓12
𝑙13 …
𝑓13 …
⋮ ⋮ ⋮ ⋱
𝑙𝑛1
𝑓𝑛1
𝑙𝑛2
𝑓𝑛2
𝑙𝑛3 …
𝑓𝑛3 …
𝑙1𝑛
𝑓1𝑛
⋮
𝑙𝑛𝑛
𝑓
𝑛𝑛] {
𝑢1 − 𝜃1(ℎ 2
⁄ )
𝑤1
⋮
𝑢𝑛 − 𝜃𝑛(ℎ 2
⁄ )
𝑤𝑛 }
………………………….……Eq.(3.23)
Finally, incorporated Eq.(3.22e) into Eq.(3.23), resulting:
{
𝐻1
𝑉1
𝑀1
⋮
𝐻𝑛
𝑉
𝑛
𝑀𝑛 }
=
[
𝑙11
𝑓11
−
ℎ
2
. 𝑙11
𝑙12 −
ℎ
2
. 𝑙11
𝑓12
−
ℎ
2
. 𝑙12
−
ℎ
2
. 𝑓11
+
ℎ2
2
. 𝑙11
…
…
…
⋮ ⋮ ⋱
𝑙𝑛1
𝑓𝑛1
−ℎ
2
. 𝑙𝑛1
𝑙𝑛2 −
ℎ
2
. 𝑙𝑛1
𝑓𝑛2
−
ℎ
2
. 𝑙𝑛2
−
ℎ
2
. 𝑓𝑛1
+
ℎ2
2
. 𝑙𝑛1
…
…
…
] {
𝑢1
𝑤1
𝜃1
⋮
𝑢𝑛
𝑤𝑛
𝜃𝑛 }
………………………….……Eq.(3.24)
3.8 EFFECTS OF NEIGHBORING LOADS
The investigation of the effect of neighboring loads is of great importance when new
buildings are being constructed by the side of existing structures. The presences of the
neighboring loads will cause-displacements {vn} at the nodal points of the existing beam
or plate. It is evident that the same displacements can be caused by a set of fictitious
forces acting directly at the nodal points by using equation (3.17).
Thus
{𝑃𝑓𝑖𝑐𝑡.} = [𝐾𝑠
𝑛]{𝑣𝑛}………………………….……Eq.(3.25)
Where:
[𝐾𝑠
𝑛]: the foundation stiffness corresponding to {𝑣𝑛}.
The set of fictitious forces can now be treated as external forces applied to the, beam
and solved as outlined in the section on the complete stiffness formulation.

Page 26
Chapter four
Computer program

Page 27
4.1 INTRODUCTION
A computer program (software) is developed in this study to analyze in-plane structures
(beams) resting on elastic foundation (modeled as elastic half space medium) using
Stiffness Matrix.
The computer program is coded with FORTRAN 90 programming language. Most of the
new features, characteristic tools, and functions of this language, such as dynamic
arrays, have been employed in this program. A description of the program procedure
and flow charts are presented below. In addition, a complete program text, input data,
and outputs are presented in appendix A.
The computer program was originally introduced by {Lazim, A. N. } [based on theoretical
works presented by {Oran, C. & KassimAli } and {Y. K. Cheung & D. K. Nag}, it deals with a
large displacement elastic stability problems of In-plane structures subjected to static
and/or dynamic loading.
In the present study the computer program was modified to carryout the linear elastic
analysis of in-plane structures resting on elastic half-space foundation. The flow chart of
the program is listed below. In addition the Winkler-type foundation subroutine is also
included in the program for the purpose of comparison.
4.2 THE PROCEDURE OF COMPUTER PROGRAM
Based on the theory presented in chapter three, the following procedure of analysis for
the problem of In-plane structures (beams) resting on elastic foundation (elastic half
space model):
1. Make a sketch for the given structural system.
2. Code the structure nodes in Global Coordinate System (G.C.S.) and define the
degrees of freedom at each node. Note in practice the structure elements may
be connected to some other structural members that induce elastic restraints
which must be taken into account.
3. Read all geometrical quantities, material constants and loading pattern for each
member of the structure.
4. Calculate the properties of the elements such as moment of inertia, area, soil
modulus of elasticity...etc.
5. Calculate the applied load vector for each node {P}.
6. Calculate the Element Stiffness matrix [t] in (L.C.S) for each element.
7. Calculate the foundation flexibility matrices using Eqs.(3.14) for the substructure
members resting on an elastic continuum.

Page 28
8. Inverse the flexibility matrices to produce the soil stiffness matrices and modify
it by applying Eqs.(3.22) to overcome the discrepancy between the substructure
and the foundation as shown in Fig.(3.2).
9. Apply Eq.(3.25) to foundation stiffness matrices to calculate the neighboring
loading effects.
10. Assemble the system structure matrix [τ] using Eq.(3.5).
11. Apply the restrictions of the boundary conditions and calculate the reduced
system matrix.
12. Solve the set of equations using Gauss-Jordan elimination method to evaluate
the global displacements vector [V].
13. Check if there are negative displacements at any node (i).
14. If it is exist, then modify foundation flexibility matrices using by eliminate the
corresponding rows and columns of that node (i). Then inverse the flexibility
matrices to produce the modified soil stiffness matrices.
15. Repeat this process until there will no negative displacements at any node (i).
16. Modify the total stiffness matrix.
17. Solve the System of linear equations using Gauss-Jordan elimination method to
evaluate the global displacements vector [V].
18. Transform the displacements vector form global coordinate to local coordinate
system using transformation matrix [B].
19. Evaluate the contact pressure (vertical and horizontal) between the in-plane
structure and the elastic foundation.
20. Calculate the member forces (axial forces, shear forces, and bending moments)
for each element.
Note: The sign convention used in this analysis is as follows:
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:

Page 29
4.3 PROGRAM FLOW CHART
Solve for unknowns displacement [d]
Evaluate Total Stiffness Matrix
T (ND, ND)
Build Elastic Foundation Stiffness Matrix in G.C.S.
SOILK (ND, ND)
Inverse subprogram
Build Elastic Foundation Flexibility Matrix in G.C.S.
SOILF (NT, NT)
Assemble Overall Stiffness
Matrix of system
H (ND, ND)
Build Element Stiffness
Matrix in G.C.S.
BEAMK (I, ND, ND)
START
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
ELASTIC FOUNDATION INPUT UNIT
For each element (1  NE) of the Elastic Foundation
read the following:
-Geometry of In-plane structure (x, y).
-Elastic properties (E, ks, µ).
-Boundary conditions (DOF). NT = NN X 2
Eliminate
corresponding
rows and
columns from
the foundation
flexibility
matrix
(1NT)
No
Yes
Is there are negative
displacements exist?

Page 30
Program Flow chart continued
4.4 COMPUTER PROGRAM APPLICATION:
First example:
Variable thickness beam on elastic half-space with vertical contact-pressures only (Fig. 4.1).
The results are compared with those obtained 'by Zemochkin & Sinitzyp (1962)(3)
using a mixed
force and displacement method. The agreement is good.
END
Evaluate Internal Forces, in L.C.S., of in-plane structure
elements: F (NE, 6)
A
A
OUTPUT UNIT
Print in-plane structural displacement, in G.C.S.,
(Vertical, horizontal and rotations) (1NN)
OUTPUT UNIT
Print Internal Forces, in L.C.S., (Axial Force, Sear Force,
and Bending Moment) for left and right side of each
element: (1NE)

Page 31
Fig.(4.1) beam layout
Fig.(4.2) applied loading
Fig.(4.3) displacement diagram
40
100
20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
applied
load
(TON)
node number
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Vertical
Dispalcement
(mm)
Node Number
E = 2,100,000 T/m2
E0 = 300,000 T/m2, ν = 0.3
40 T
100 T
20 T
0.75m 1.5 m 0.5m 1.25m 0.5m 1.0 m
0.5m

Page 32
Fig.(4.4) contact pressure diagram
Fig.(4.5) shear force diagram
0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Vertical
Contact
Pressure
(TON/M2)
Node Number
-60
-40
-20
0
20
40
60
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Shear
Force
(TON)
Node Number

Page 33
Fig.(4.6) bending moment diagram
Second example:
Beam on elastic half space with vertical contact pressures only (Fig. 4.7).
The results are compared with those obtained by Zienkiewicz, 0. C, & Cheung, Y. K. 1964)(4)
using finite element method. The agreement is good.
-5
0
5
10
15
20
25
30
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Bending
Moment
(TON.M)
NodeNumber
E = 30.0E6 Psi, γc=150 lb/cu.ft
E0 = 30.0E6 Psi, ν = 0.15
4000.0 lb
2.0 ft
5.0 ft 5.0 ft
1.0 ft

Page 34
Fig.(4.8) deflection diagram
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Deflection
(in*1000)
Node Nomber
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Vetical
Contact
Pressure
(lb/sq.ft*1000)
Node Number

Page 35
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Shear
Force
(lb)
Node Number
-0.5
0
0.5
1
1.5
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Bending
Moment
(lb.ft)
Node Number

Page 36
Third example
Beam with both vertical and horizontal contact pressure on elastic half-space foundation.
The results are compared with those obtained by Vesic, A. S., and Johnson, W. H., (1963)(11)
using a mixed force and displacement method. The agreement is good.
Fig.(4.13) deflection diagram
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Deflection
(in)
Node Nomber
E = 30.0E6 Psi
E0 = 1173.0 Psi, ν = 0.15
8250.0 lb
1.0 Inch
36.0 Inch 36.0 Inch
8.0 Inch

Page 37
-15
-10
-5
0
5
10
15
20
25
30
35
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Vetical
Contact
Pressure
(lb/sq.in)
Node Number
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Shear
Force
(Kip)
Node Number

Page 38
Chapter five
-5
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Bending
Moment
(Kip.in)
Node Number

Page 39
Conclusions and Recommendations
5.1 CONCLUSIONS
Depending on the results obtained from the present study, several conclusions may be
established. These may be summarized as follows:
1. Depending on the comparisons 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 elastic half-space 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 elastic half space as foundation model as shown by
the comparisons results
4. Introducing horizontal contact pressure has actually reduced Bending Moment at
middle of beam when compared with the analysis due to vertical contact
pressure only, however when the foundation is relatively flexible the
modification in bending moment will be much more significant.

Page 40
5. The results show that the increasing foundation rigidity (stiffness) will increase
vertical contact pressure.
6. The results show that the variation of soil elastic properties (Poisson’s Ratio and
Hook’s Modulus) for different soil types can significantly change the structure
internal forces and contact pressure distribution.
5.2 RECOMMENDATIONS
Many important recommendations could be suggested:
1. This analysis method (structure stiffness matrices) could be extended to analysis
three dimensions structure (space structure), in same way the elastic foundation
stiffness (elastic half space stiffness matrices) should be extended to become
more general case as in space structures
2. The static type of analysis could be extended to be more general case as in
dynamic analysis; therefore, more complex problems could be analyzed such as
earth quick and dynamic response.
3. The Nonlinearity could be incorporated in this analysis to include geometry
nonlinearity and materials nonlinearity for both soil and structure.
4. The elastic half space model assume the soil modulus of elasticity is constant (Es )
which in reality not exactly truth, therefore, researcher could be encouraged
toward more advanced solutions of (Es ) and trying to insert that in this study.

Page 41
5. The developed soil stiffness matrix is quit important and easy going if one might
want to study the influence of neighboring loads on existing structure behavior.
REFERENCES
1. Love, A. E. IT. (1966). A treatise on the mathematical theory of elasticity.- London
Cambridge University Press.
2. Timoshenko, S. & Goodier, J. N. (1951). Theory of elasticity. 2nd ed. London:
McGraw-Hill.
3. Zemochkin, B. N., & Sinitzyp, A. P. (1962). Practical method of calculating beams and
plates on elastic foundations (in Russian). 2nd ed. Gosstroiizdat.
4. Zienkiewicz, 0. C, & Cheung, Y. K. 1964). The finite element method for the analysis
of elastic isotropic and orthotropic slabs. Proc. Instit. Civ. Engrs. 28, August, 471—
488.
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.

Page 42
3. Livesley, R. K. (1964). Matrix methods of structural analysis. Oxford Pergamon Press.
4. Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis." Pergamon
Press, London, 1964, PP. 115-145.
5. Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London,
1964. PP. 241-252.
6. Biot, M. A., "Bending of an Infinite Beam on an Elastic Foundation." Journal of
Applied Mechanics, ASME, Vol. 59, 1937, pp. A1-A7.
7. Vesic, A. S., and Johnson, W. H., "Model Studies of Beams Resting on Silt Subgrade."
Journal of the Soil Mechanics and Foundation Division, ASCE, Vol.89, No. SM I.
February, 1963, pp. 1-31.
8. 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).
9. Levinton, Z., "Elastic Foundation Analyzed By the Method of Redundant Reactions."
Transaction, ASCE, Vol. 114, 1949, pp. 40-78.
10. Terzaghi, K., "Evaluation of Coefficient of Subgrade Reaction." Geotechnique, Vol.5,
No.4, 1955, pp. 197-326.
11. 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.
12. Vesic, A. B., "Bending of Beams Resting on Isotropic Elastic Solid." Journal of
Engineering Mechanics Division, ASCE, Vol. 87, EM2, April, 1961, pp. 35-53.
13. Zienkiewicz, 0. C, & Cheung, Y. K., The finite element method for the analysis of
elastic isotropic and orthotropic slabs. Proc. Instit. Civ. Engrs. 28, August, 1964,
471—488.
14. Cheung, Y. K.., and Zienkiewicz, 0. C., "Plates and Tanks on Elastic Foundations - An
Application of the Finite Element Method." International Journal of Solids and
Structures, Vol.1 No.4, 1965, pp. 451-461.
15. Morris, D., "Interaction of Continuous Frames and Soil Media." Journal of the
Structural Division, ASCE, Vol. 92, No. ST5, October, 1966, pp. 13-44.
16. Cheung, Y. K., and Nag, D. K "Plates and Beams on Elastic Foundations –Linear and
Non-Linear Behavior." Geotechnique, Vol. 18 No.4, 1968 pp. 250 -260.
17. Davis, R. 0., and Selvadurai, A. P. S., "Elasticity and Geomechanics." Cambridge
University Press, 1996, pp. 76-111.
18. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition, McGraw-
Hill Book Company, New York, 1961, pp. 1-17.
19. KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of
Structural Engineering, ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.
20. 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.

Page 43

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