Analysis and Design of Plate Girder Bridges_.docx

STRUCTURAL ENGINEERING

1. 1

1 Analysis and Design of Plate Girder


Analysis and Design
of
Plate Girder


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
Athra’a & Halla


Supervised by
Assistant lecturer, Adnan N. LAZEM
(M.Sc., in Structural Engineering)
July /2008

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2 Analysis and Design of Plate Girder
Analysis and Design
of
Plate Girder

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3 Analysis and Design of Plate Girder

ABSTRACT:
The objective of this study is to develop a better understanding for the basic principles of the
structural analysis and design of plate girder so they can be efficiently implemented into modern
computers.
Develop an in-plane structure stiffness matrix that takes into the girder variable elements
prosperities (vertical, horizontal, and bearing stiffeners) into consideration.
In addition several important parameters have been incorporated in the analysis and design
process; Buckling and stability of web plate, web critical shear buckling, maximum allowable
deflection due to live load, maximum allowable flexural strength according to AISC-89, web
elements connections design, flange elements connections design, and flange curtailment lengths
design.
To evaluate the results of presented method were compared with result given in reference
number one. The agreement between both results was quit well.

PROJECT LAYOUT
The project is divided into five chapters as follows:
Chapter one: presents a general introduction to the subject of Plate Girder.
Chapter two: presents the previous literatures published about this subject.
Chapter three: presents the theoretical bases for the Matrix analysis method and Plate Girder
design.
Chapter four: presents a brief description of a computer program developed in this study.
Chapter five: discuses the results of this Analysis/Design method. And recommend future steps.

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4 Analysis and Design of Plate Girder

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

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5 Analysis and Design of Plate Girder









Chapter one
Introduction

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6 Analysis and Design of Plate Girder

1.1. GENERAL INTRODUCTION TO PLATE GIRDER
A plate girder is a built up beam and is normally fabricated from plate sections and angle-
iron sections. The web consists of a solid plate and the flanges are composed of angle-iron
sections and plates. Both welding and riveting can be adopted for fabrication but now-a-
days welding is preferred over riveting because of easier and faster fabrication even at sites.
Plate girders resist transverse bending like beams and are provided where loads are heavy.
For heavier loads, the section modulus required is not available in any standard rolled
section. In such cases a beam section is fabricated by riveting or welding plates and angle-
sections to form a plate girder. Plate girders are used extensively in every form of steel
construction, because of their adaptability. With different depths, different form of flanges
of different sizes, plate girders can be designed to serve a great variety of purposes. These
are used in buildings, factories and bridges for carrying heavy loads over spans greater than
15 m.
1.1.2. TYPICAL SECTIONS.
Plate girders may be composed of one or more web plates and with simple or composite
flanges consisting of angles, channels, and plates. Various forms of sections are shown in
Fig. 1.1. The most common type of section is as shown in Fig. 1.1 (a) & (b). It is made up
of a single solid web plate and four angles with or without flange plates. For heavier loads,
additional flange plates may be provided. These are also called cover plates. The cover
plates are usually curtailed as the bending moment decreases near the supports.

Fig. (1.1) Typical Plate Girder Cross-sections
Sections of type (c) & (d) are used where the top surface is required at a uniform elevation
as in the case of floors. These types of sections are not economical as these do not make
the best use of the flange area provided. Such sections are also provided at places where a
large number of cover plates are required so as to avoid the use of excessively long rivets.
Box girder sections of type (e) & (f) are provided in buildings where there are head-room
restrictions. Due to these restrictions, full depth of the girders cannot be provided and thus
additional webs are required to take the shear force. The box girder also provides greater
lateral stability.
In welded plate girders, Fig. 3.1 (g) angles are not used. The flange plates are directly
welded to the web plate. Also, a single flange plate of the required thickness is normally

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7 Analysis and Design of Plate Girder
provided instead of a number of thin plates. At the point of curtailment, the thicker flange
plate is cut and thinner plate is provided which is butt welded to the thicker plate.
Girder types
Simple deck beam bridges are usually metal or reinforced concrete. Other beam and girder
types are constructed of metal. The end section of the two deck configuration shows the
cross-bracing commonly used between beams. The pony end section shows knee braces
which prevent deflection where the girders and deck meet.






One method of increasing a girder's load capacity while minimizing its web depth is to add
haunches at the supported ends. Usually the center section is a standard shape with parallel
flanges; curved or angled flanged ends are riveted or bolted using splice plates. Because of
the restrictions incurred in transporting large beams to the construction site, shorter, more
manageable lengths are often joined on-site using splice plates.

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8 Analysis and Design of Plate Girder








Chapter two
Literature

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9 Analysis and Design of Plate Girder
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 pre-computer 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 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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10 Analysis and Design of Plate Girder










Chapter three
Theory

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11 Analysis and Design of Plate Girder

3.1. ANALYSIS METHOD USING STIFFNESS MATRIX
Stiffness Matrix method is one of the most efficient means for solving a in-plane Elastic Structures
(frames and girders) type of problem based on following steps. It is easy to account for Boundary
Conditions, and self weight (Girder).
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) 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.
3.1.1 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.(3.1 )
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 ith
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.(3.2 )

Fig.(3.1) Beam Element, global and local forces-deformations designation.
An Equation relating internal-member deformation e at any node to the external nodal
displacements is:
{??????
??????}=[�
??????]{??????
??????}………………………..(a)

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12 Analysis and Design of Plate Girder
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 [A] matrix is exactly the transpose of the [B] 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 BKBT is inverted to obtain
{??????
??????}=([�][??????] [�]
??????
)
−1
{??????
??????}…………………………(f )
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.
The BKB
T
matrix above is often called Overall assembly 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 BKB
T
from the Local element EBKB
T
.
This is easily accomplished, since every entry in both the Global and Local BKBT with a unique
set of subscripts is placed into that subscript location in the BKBT; i.e., for i = 2, j = 5 all (2, 5)
subscripts in EBKB
T
are added into the (2, 5) coordinate location of the global BKBT.
3.1.2. DEVELOPING THE ELEMENT [B] MATRIX
Consider the in-plane structure, simple beam, shown in Fig.(3.2) 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.(3.2). 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.

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13 Analysis and Design of Plate Girder

Fig.(3.2) 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.(3.2d))
??????
1=�
1+0.�
2
Similarly, summing forces and noting that the soil reaction (spring) forces are Global and will be
considered separately, we have:
??????
2=
�
1
??????
+
�
2
??????

??????
3=0.�
1+�
2
And ??????
4=−
�
1
??????
−
�
2
??????

Placing into conventional matrix form, the Element Transformation Matrix [EB] in local coordinate
is:

Force-Displacement relationships (P-X indexing)
P1 P2
(a) Local force-displacement relationships (F-e indexing)

FEM

FEM

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14 Analysis and Design of Plate Girder
EB =
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.
3.1.3. DEVELOPING THE [k] MATRIX
Referring to Fig.(3.3) 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.(3.3) conjugate-beam method Moments and rotations of beam element.
Solving Eqs.(g) and (h) for F, obtaining:
�
1=
4�??????
??????
??????
1+
2�??????
??????
??????
2
�
2=
2�??????
??????
??????
1+
4�??????
??????
??????
2
Placing into matrix form, the Element Stiffness Matrix [ES] in local coordinate is:

Ek =
e1 e2
F1
4�??????
??????

2�??????
??????

F2
2�??????
??????

4�??????
??????

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15 Analysis and Design of Plate Girder
3.1.4. DEVELOPING THE ELEMENT [kB
T
] AND [BKB
T
] MATRICES
The EkB
T
matrix is formed by multiplying the [Ek] and the transpose of the [EB] matrix (in the
computer program this is done in place by proper use of subscripting) A
T
goes always with e and
X. The EBkB
T
will be also obtained in a similar.
Multiplying [Ek] and [EB
T
] matrices and rearrange them, yields:

EkB
T
=
1 2 3 4
1
4�??????
??????

6�??????
??????
2

2�??????
??????
−
6�??????
??????
2

2
2�??????
??????

6�??????
??????
2

4�??????
??????
−
6�??????
??????
2


Multiplying [EB] and [EkB
T
] matrices and rearrange them, yields:
EBkB
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
??????
=0.0
Since (F1+F2)/L is already included in the Global BkB
T
, we could rewrite above equation to:
??????
2=�??????�
2??????2
??????
??????
2=[�??????�
2??????2
??????
]??????
2
A check on the correct formation of the EBkB
T
and the global BkB
T
is that it is always symmetrical
and there cannot be a zero on the diagonal.

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16 Analysis and Design of Plate Girder
3.1.5. 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:
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 Actions 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.
3.1.6. 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).








3.2. DESIGN OF PLATE GIRDER COMPONENTS

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17 Analysis and Design of Plate Girder
A plate girder consists of a number of components, as shown in Fig. 3.4. These are
(a) Web plate
(b) Flange angles
(c) Cover plates
(d) Longitudinal stiffeners
(e) Transverse stiffeners
(f) Bearing stiffeners
(g) Web splice
(h) Flange splice
(i) End bearings
(j) Rivets or welds Connections
(k) Filler plates
3.2.1. WEIGHT AND ECONOMIC WEB DEPTH OF PLATE GIRDER:
The depth of plate girders normally varies between one-eighth of span for short girders and one-
twelfth of span for long girders except for girders having headroom restrictions.

Fig(3.4), Distribution of bending stress over I section.

As the cost of a plate girder depends upon its total weight, the depth should be kept such that the
weight of the plate girder is the minimum. As the depth of the web is increased, the area of flanges
decreases but the area of web and weight of striffeners, splices, etc. increases. However, in some

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18 Analysis and Design of Plate Girder
cases, the depth of a plate girder may infringe upon head-room or other clearance requirements and
thus be limited by considerations other than the minimum weight.
Let
M = Maximum bending moment
F = moment of inertia of flange
Iw = moment of inertia of web
dw = depth of web
d = distance between the centroid of flanges
tw. = thickness of web
Aw = gross area of web
Af = net area of flange
Fbt = permissible stress in bending.

Neglecting the moment of inertia of the flanges about their centroids (being very small), the
moment of inertia of flanges is given by

In most plate girders, d and df are nearly equal. Therefore.
...3.1
If only the flanges resist the bending` moment, then


or Area of flange ...3.2

As the flange plates are normally curtailed, the average flange area is taken as eighty percent of the
maximum flange area. For both flanges, the total area is


The moment of inertia (I) of the plate girder section is

or ..3.3
Area of web.
..3.4
or

The term is called: ‘Effective Flange Area' and is called the 'Web equivalent'.

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19 Analysis and Design of Plate Girder
In a riveted plate girder, the area of rivet holes is deducted from the gross area. If the rivet holes in
the web are of diameter ‘Φ' and at a pitch of 4 ‘Φ ', then the net area of the web will be seventy-
five percent of the gross area.
Therefore, effective flange area for a riveted plate girder is
...3.5

Taking into account the contribution of web, the area of flange required is

And
For both flanges, area required


The weight of stiffeners is usually taken as 30% and 60% of the weight of web for welded girders
and riveted girders respectively. If y, is the unit weight of steel then for unit length, the weight of
various components of the riveted girder is
(a) Flanges
(b) Web
(c) Stiffeners

The total weight of riveted plate girder per unit length is

or
or ...3.6
For minimum weight,


or ...3.7
and ...3.8

For welded plate girder, the total weight per unit length is given by

or

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20 Analysis and Design of Plate Girder
or ...3.9
For minimum weight, = 0
or
or ...3.10
and …3. I 1
Substituting the above value of tw in Equation 3.9, we get

or …3.12
if
where;
W = total load on girder in N
L = Span of girder in mm

Putting values in Equation 3.12, we get

Or

Adopt a slightly higher self-weight for designing i.e.
...3.13
Substituting value of tw from Equation 3.8 in Equation 3.6 for riveted girder we get,
or
If the flanges are not curtailed, then the economical depth of the plate girder is given by
...3.14

A small deviation from the economical depth does not increase the weight of the girder appreciably.
A reduction in the economical depth by 20% increases the weight by about 2%. Thus, the depth is
usually kept about 10% smaller than the economical depth.

3.2.2. DESIGN OF FLANGES
The flanges of a plate girder are designed for resisting the maximum bending moment. It is assumed
that the intensity of stress is uniform on the flange angles & plates. The stress distribution over the
web varies with maximum at the flanges and zero at the neutral axis. The assumed and the actual
stress distribution over the section of the plate girder is shown in Fig. 3.3. From Fig. 3.3. it is evident

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21 Analysis and Design of Plate Girder
that the average stress on the flange is smaller than the actual maximum stress. This difference
depends upon the ratio of do and d.
Where;
do = distance between the centroids of the flanges
d = overall depth of girder. .

For shallow girders, the ratio do/d is less than unity and thus the assumption of uniform stress in
flanges does not hold true. But for deep girders, as the depth increases, the ratio ddd approaches
unity and thus, the assumption becomes approximately true. Therefore, for all depths of girders, it
is necessary to check the maximum stress by computing the moment of inertia after designing the
flanges on uniform stress.
The area of the flanges should be proportioned such that the maximum bending stresses are within
the permissible limits. Solid web girders should preferably be proportioned on the basis of the
moment of inertia of the gross cross section with the neutral axis taken at the centroid of that
section, but it shall be permissible to use the net moment of inertia. In arriving at the maximum
flexural stresses, the stresses calculated on the basis of the gross moment of inertia shall be
increased in the ratio of gross area to effective area of the flange section. For this purpose the flange
sectional area in riveted or bolted construction shall be taken to be that of the flange plate, flange
angles and the portion of the web and side plates (if any) between the flange angles. In welded
construction and flange sectional area shall be taken to be that of the flange plates plus that of the
tongue plates (if any) up to a limit of eight times their thickness, which shall be not less than twice
the thickness of the web.
The effective sectional area of compression flanges shall be the gross area with deductions for
excessive width or outstand of plates as specified for compression members (Art.6.9) and for open
holes (including holes for pins and black bolts) occurring in a plane perpendicular to the direction
of stress at the section being considered. The effective sectional area of tension flanges shall be the
gross sectional area with deductions for holes as specified in Art 5.4. In riveted or bolted
construction, flange angles shall form as large a part of the area of the flange as practicable
(preferably not less than one-third) and the number of flange plates shall be kept to a minimum:
(a) In exposed situations where flange plates are used, at least one plate of the top flange shall
extend the full length of the girder, unless the top edge of the web is machined flush with
the flange angles. Where two or more flange plates are used on the one flange, tacking
rivets shall be provided.
(b) Each flange plate shall be extended beyond its theoretical cut-off point, and the extension
shall contain sufficient rivets or welds to develop in the plate the load calculated for the
bending moment on the girder section (taken to include the curtailed plate) at the theoretical
cut-off point.
(c) The outstand of flange plates, that is the projection beyond the outer line of connections to
flange angles, channel of joist flanges, or, in the case of welded constructions, their
projection beyond the face of the web or tongue plate, shall not exceed the values given in
Chp. 6.9 and 5.4.
(d) In case of box girders, the thickness of any plate, or the aggregate thickness of two or more
plates when these plates are tacked together to form the flange, shall satisfy the
requirements of maximum widths for box girders with width/depth ratio less than 0.2.

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22 Analysis and Design of Plate Girder
The maximum bending stress is given by M;
where,
= maximum tensile or compressive bending stress.
M = bending moment
y = distance of the farthest fiber from the centroid of the section 1= moment of Inertia about the centroid of
the section.
The moment of inertia of the section of plate girder shown in Fig. 3.4 is given by

or ...3.15

where,
IF, Iw = moment of inertia of both flanges), web about the centroid of section
If= moment of inertia of flange about its own centroid
Af = gross area of one flange
do = distance between the centroids of flanges
d = maximum depth of girder
dw = depth of web
tw = thickness of web

As IF, is very small compared to other terms, it may be neglected and Eqn. 3.15 reduces to

Gross area of web,

For most girders, d, dw and do are almost equal. Thus
...3.16
The moment of resistance from the basic flexure formula is

or ...3.17
From the above equation, the flange area can be calculated for the known depth of girder.

3.2.2.1. TENSION FLANGE:
For tension flange, the area of rivet of bolt holes is to be deducted. The effective sectional area
(Af) of the tension flange is equal to gross area minus the area of holes (AR)
Af = AF-AR
As there are rivet holes in the web plate also, the net effective area will be equal to gross area
minus area of rivet or bolt holes. If the holes are of diameter ¢ and are at a pitch of 4 ¢, then the net
area of the web is reduced to 75% of the gross area. Therefore, the effective flange area will be
= Af +3/4 * Aw/6

23. 23

23 Analysis and Design of Plate Girder
= Af + Aw/8
The maximum bending stresses are computed by increasing the stresses based on gross moment of
inertia by the ratio of gross area of flange to net area of flange.
Therefore, maximum tensile stress,

or
or ...3.18
or ...3.19

From the above equation, the net area of tension flange can be calculated.
For welded girders, the gross area of flanges & web is effective as there are no holes. Therefore.
Eqns. 3.18 & 3.19 can be rewritten for welded girders as follows

...3.20
...3.21

3.2.2.2. COMPRESSION FLANGE.
The effective sectional area of the compression flange is taken as the gross area with deductions
for excessive outstands/widths of plates and for open holes. The compression flange is
quite strong in the vertical plane due to the rigidity provided by the web plate. However, as
explained in Art. 4.5, the compression flange has a tendency to bend sideways or buckle laterally
under the compression forces. If the compression flange is restrained against lateral movement,
then buckling would not take place. Due to this tendency to buckle. The allowable compressive
stress is reduced depending upon the slenderness ratio of the compression flange. The permissible
stress depends upon the section of the girder and the section of the girder can he computed only if
the permissible stress is known. Thus, the design of compression flange is by trial and error.
Normally the tension flange is designed first and the area of compression flange is kept equal to
the area of tension flange. The maximum compressive stress is then checked and should be within
the permissible limits.
To safe-guard the flange plates against local buckling, their outstand beyond the line of rivets or
width between two lines of rivets should be within the permissible limits.
3.2.3. DESIGN OF WEB
The shear stress diagram for a plate girder or I-section is shown in Fig. 3.5. The distribution of
shear stress across the depth of the section shows that more than 90% of the shear is taken by the
flanges. Thus, the web of a plate girder is designed for shear force. It is assumed that the shear is
carried wholly by the web and that the intensity of shear stress is uniform over the depth of the
section. The shear stress in web, ti, is given by

24. 24

24 Analysis and Design of Plate Girder

where,
V = maximum shear force
tw = thickness of web plate
dw = depth of web plate

The shear stress τυ should be less than the permissible average shear stress. The depth of a plate
girder normally ranges from one-twelfth to one-eighth of span. However, it is not always possible
to provide the economical depth due to headroom or clearance requirements. The web of a plate
girder is thin and deep and, thus, has a tendency to buckling. The web plate is strengthened by
providing stiffeners.
3.2.3.1 WEB BUCKLING.
Thin and deep web plates of plate girders are liable to fail in buckling if the shear stress exceeds
the critical stress in shear.
The critical stress in shear for a plate is given by

...3.23
where,
τcr = critical stress in shear
k = constant, depending upon the aspect ratio c/d
c = the larger plate dimension
d = smaller plate dimension
t = thickness of plate
µ= Poisson's ratio

The values of k for various aspect ratios are given in Table 3.1. For unstiffened webs, the c/d ratio
is around 10 and thus the value of k is 5.3. Putting the value of ;.t = 0.3 & E=200000 N/mm
2
in
Eqn. 3.23, we get

or
Fig.(3.5), Web panels of Plate Girder.
…3.22
Fig.(3.6), Shear stress distribution.

25. 25

25 Analysis and Design of Plate Girder
If the yield stress is shear is 150 MPa, then the (d/t) ratio will be d



In order to avoid shear buckling, the d/t of web plate should be lesser than 80. If d/t is less than 80,
then the failure of the web plate will be by yielding and when d/t is greater then 80, then the web
plate will fail by buckling.
For plate girders, the distance `d' is the distance between the flange angles or between the flanges
if there are no flange angles (Fig. 3.6). As the d/t ratio of plate girders is quite often more than 80,
the critical stress or the allowable shear stress gets decreased accordingly. With the increase in d/t
ratio, the critical stress decreases rapidly. Thus, it becomes necessary to provide transverse
stiffeners so as to decrease the d/t ratio. By decreasing the spacing of transverse stiffeners, the value
of 'k' increases and accordingly, the critical stress increases. Thus, transverse stiffeners are provided
to prevent shear buckling of the web plate.
3.2.3.2. WEB STABILITY.
The web of a plate girder also carries bending stresses in addition to shear stress. The bending
stress varies along the depth of the section as shown in (Fig. 3.4). The compressive bending stress
tends to buckle the web plate in the lateral direction if it is not adequately restrained. The buckling
of the web is shown in (Fig. 3.7). The critical longitudinal buckling stress of a plate is also given
by Equation 3.23, but the values of coefficient `k' are different. It has been found that the buckling
due to longitudinal compression is of a local nature (Fig. 3.7) and, therefore, does not affect the
ultimate strength of the beam.


Fig.(3.7), Web buckling sue to longitudinal compression.
Table (3.1)

26. 26

26 Analysis and Design of Plate Girder
The maximum allowable value of d/t for web plates stiffened by transverse stiffeners only is 200.
When d/t exceeds 200, longitudinal stiffeners are required to prevent buckling. For d/t up to 250, a
horizontal stiffener is provided at a distance from the compression flange equal to 2/5 of the
distance from the compression flange to the neutral axis. A second horizontal stiffener is provided
at the neutral axis when the value of d/t exceeds 250. Thus, longitudinal or horizontal stiffeners are
required in the compression zone of the web plate to provide lateral restraint.
3.2.3.3. DEPTH OF WEB.
As per S.I.800-1984, the effective depth of plate girders is defined as follows (Fig. 3.8)

Fig.(3.8), Effective Depth of Plate Girders.
(a) d1, for the web of beams without horizontal stiffeners = the clear distance between the flanges,
neglecting fillets or the clear distance between the inner toes of the flange angles.
(b) d1, for the web of beams with horizontal stiffeners = the clear distance between the horizontal
stiffeners and the tension flange, neglecting fillets or the inner toes of the tension flange angles as
appropriate.
(c) d2 = twice the clear distance from the neutral axis of a beam to the compression flange, neglect-
ing fillets or the inner toes of the flange angles.
(d) Where tongue plates having a thickness of not less than twice the thickness of the web plate are
used, the depth shall be taken as the depth of the girder between the flanges less the sum of the
depths of the tongue plates or eight times the sum of the thickness of the tongue plates, whichever
is less,
The depth of the web plate is kept as per the economical depth or headroom restrictions, if any.
3.2.3.4. THICKNESS OF WEB.
1. Minimum thickness. The thickness of the web plate shall be not less than the following (a) For
unstiffened webs : the greater of and but not less than
where,
d, = depth of web as defined in 3.3(iii) and
τυa.cal= calculated average stress in the web due to shear force.

(b) For vertically stiffened web: the greater of 1/180 of the smallest clear panel dimension and
but not less than

27. 27

27 Analysis and Design of Plate Girder
(c) For webs stiffened both vertically and horizontally with a horizontal stiffener at a distance from
the compression flange equal to 2/5 of the distance from the compression flan to the neutral axis:
the greater of 1/I80 of the smaller dimension in each panel, and but not less than d2/250
(d) When there is also a horizontal stiffener at the neutral axis of the girder: the greater of 1/180 of
the smaller dimension in each panel, and but not less than
In (b), (c) and (d) above, d2 is twice the clear distance from the compression flange angles, or plate,
or tongue plate to the neutral axis.
In the case of welded crane gantry plate girders intended for carrying cranes with a lifting load of
15 tones or more, the thickness of web plate shall be not less than 8 mm.
The minimum thickness of web plates for different yield stress values are given in Table 3.2 for
information.
In no case shall the greater clear dimension of a web panel exceed 270 t, nor the lesser clear
dimension of the same panel exceed (180 t) where t is the thickness of the web plate. Therefore,
the minimum thickness that can be provided for a web plate is d2/400. It is economical to provide
minimum thickness of web plate and provide stiffeners if required. The minimum thickness is
limited to 6 mm for girders exposed to weather but accessible for painting & 8 mm if inaccessible
for painting. It is 8 mm for highway bridges and 10 mm for railway bridges.
2. Riveted construction. For girders in exposed situations and which do not have flange plates for
their entire length, the top edge of the web plate shall be flush with or above the angles, as specified
by the Engineer, and bottom edge of the web plate shall be flush with or set back from the angles,
as specified by the engineer.
3. Welded construction. The gap between the web plates and flange plates shall be kept to a
minimum, and for fillet welds it shall not exceed 1 mm at any point before welding.


Table 3.2. Minimum Thickness of Web
Minimum Thickness of Web for Yield Stress (in MPa) of

28. 28

28 Analysis and Design of Plate Girder
3.2.3.5. EFFECTIVE SECTIONAL AREA .
(a) Web of plate girder. The effective cross-sectional area shall be taken as the full depth of the
web plate multiplied by the thickness.
Where webs are varied in thickness in the depth of the section by the use of tongue plates or the
like or where the proportion of the web included in the flange area is 25 percent or more of the
overall depth, the above approximation is not permissible and the maximum shear stress shall be
computed.
(b) Rolled beams and channels. The effective cross - sectional area for shear shall be taken as the
full depth of the beam or channel multiplied by its web thickness. For other sections the maximum
shear stress shall be computed from the whole area of the cross section, having regard to the actual
distribution of shear stress.
(c) Webs which have openings larger than those normally used for rivets or other fastenings require
special analysis to ensure that the permissible stress as specified in the Indian Standard are not
exceeded.
3.2.3.6. UNSTIFFENED WEBS.
In rolled beam sections and built-up rolled beam sections, the web is safe against buckling and thus
does not require stiffening. the average shear stress for unstiffened webs calculated on the cross-
section of the web should not exceed 0.4 f,. or
...3.24
where,
τυa =maximum permissible average shear stress
fy.= yield stress of steel. (The yield stress in shear is x the yield stress of steel in direct tension).

The maximum shear stress on any part of the plate girder should not exceed 0.45 fy or
...3.25
where, τυa =maximum permissible shear stress.
3.2.3.7. STIFFENED WEBS.
The webs of plate girders are normally stiffened by transverse and/or longitudinal stiffeners. For
stiffened webs, the average shear stress calculated on the cross-section of the web should not exceed
the values given in by the following formulae, provided that the average stress τυa shall not exceed
0.4 fy.
(i) For webs where the distance between the vertical stiffeners is less than 'd'
...3.26
(ii) For webs where the distance between the vertical stiffeners is more than 'd'

29. 29

29 Analysis and Design of Plate Girder

where,
τυa = maximum permissible average shear stress.
c = distance between vertical stiffeners

d = (i) For vertically stiffened webs without horizontal stiffeners - the clear distance between flange
angles or, where there are no flange angles, the clear distance between flanges, ignoring fillets.
Where tongue plates (See Fig. 3.8 (e)) having a thickness of not less than twice the thickness of the
web plate are used, the depth d shall be taken as the depth of the girder between the flanges less the
sum of the depths of the tongue plates or eight times the sum of the thickness of the tongue plates,
whichever is less.
For vertically stiffened webs with horizontal stiffeners-the clear distance between the tension
flanges (angles, flange plate or tongue plate) and the horizontal stiffener.
t = the thickness of the web.
3.2.4. CURTAILMENT OF FLANGE PLATES
The bending moment for a simply supported plate girder is maximum at the centre and decreases
towards the supports. The section of the plate girder is designed for the maximum moment. As the
bending moment decreases near the supports, some of the flange plates can be curtailed as the full
area of the birder is required only at the point of maximum moment. The points, as per the bending,
moment diagram, at which the flange plates can be curtailed are called the theoretical cut-off
points. The theoretical cut-off points can be determined by two methods:
3.2.4.1. ANALYTICAL METHOD.
Consider a plate girder carrying a uniformly distributed load and simply supported at its ends.
Let
w = uniformly distributed load per unit length
n = number of flange plates to be curtailed (numbered 1, 2, 3 n from the outermost
ln = distance between theoretical cut-off points for nth plate.
I = span of plate girder.
Maximum bending moment =
From Equation 3.17, moment of resistance Mr is given by

If the moment of resistance of the plate) girder section is equal to the maximum bending moment,
then
...(i)
At the point of curtailment of nth plate, the above equation becomes

30. 30

30 Analysis and Design of Plate Girder

where, A1, A2, .a = area of flange plates No. l, 2....n Dividing Equation (ii) by (i), we get

or …3.28
or
Thus l1, l2, ln define the theoretical cut-off points.
For other types of loadings, the analytical method is not as simple as for wl formly distributed
loading. It is simpler to use the graphical method.

3.2.4.2. GRAPHICAL METHOD.
in the graphical method, the maximum bending moment diagram is drawn and then the moment
of resistance diagram is superimposed upon it to pct the theoretical cut-off points.
From Equation 3.17, the moment of resistance M, is given by

The area of flange AF is composed of area of flange angles Aa plus area of flange plates A,, AZ
..An, where n is the number of flange plates to be curtailed. The flange plates are numbered l, 2....n
from the outermost plate. Thus, the above equation can be rewritten as

The above equation shows that the contribution of each flange component towards the moment of
resistance of the section is proportional to the area of that component.
Draw the bending moment diagram ABC for the plate girder of span `I' (Fig. 3.9). From A draw
any line AD and plot on it the areas of various components of the flange i.e.
As shown in Fig. 3.9. The area is plotted first and then the areas of plates are plotted starting
from the highest numbered or the innermost plate to the lowest numbered or the outermost plate.
Next draw a line from A perpendicular to AB and plot a point 'D' on it such that AD' represents the
moment of resistance of the full section. Join points, D and D' Draw lines EE',FF'….. NN' parallel
to line DD'.
The points E', F' N' represent the moment of resistance of the section after curtailment of plate 1, 2
.... N. From points E', F' ……. N' draw lines parallel to AB. The points of intersection of lines
from E' F' …….. N' and the bending

31. 31

31 Analysis and Design of Plate Girder
moment curve ACB are the theoretical cut-off points for plates 1, 2 ... N, i.e. the points of
interception (1, 2) of line through E' and the bending moment curve are the theoretical cut-off points
for plate I having an area A1, the theoretical required length of plate t is thus lt.

For symmetrical sections, the moment of resistance is proportional to the flange area. But for
unsymmetrical sections, it is not so as the centre of gravity of the section changes with the
curtailment of plates. Thus, the moment of resistance should be calculated at each curtailment. The
moment of resistance at each curtailment is then plotted on line AD' such that AD' represents the
moment of resistance at full section, AE' at the first curtailment, AF' at the second curtailment &
so on. From points D', E', F……. N', lines are then drawn parallel to AB which intersect the
bending moment rung to Get the theoretical cut - off points for each curtailment.
(a) In exposed situations where flange plates are used, at least one plate of the top flange shall
extend the full length of the girder, unless the top edge of the web is machined flush with the flange
angles. Where two or more flange plates are used on the one flange, tacking rivets shall be provided.
(b) Each flange plate shall he extended beyond its theoretical cut-off point. and the extension shall
contain sufficient rivets or welds to develop in the plate the load calculated for the bending moment
on the girder section (taken to include the curtailed plate) at the theoretical rut-off point.
3.2.5. DESIGN OF STIFFENERS
3.2.5.1. INTERMEDIATE VERTICAL STIFFENER.
Due to the use of deep and thin web plates, the web of a plat: girder has a tendency to buckle as
explained earlier in 3.3 (i). To prevent buckling due to shear, vertical stiffeners are provided along
the length of the web. Thus, the primary purpose of the intermediate
vertical stiffeners is to prevent the web plate from buckling. The provision of vertical stiffeners at
regular intervals break up the web plate into small panels and thus provide supports to these panels.
Due to this, the resistance of the plate to buckling is considerably increased.
The Indian Code I.S. 800-1984 recommends that when the thickness of the web is less than the
limits specified in 3.3 (iv) I (a), vertical stiffeners shall be provided throughout the length of the
girder. The vertical stiffeners are provided at a distance not greater than 1.5 d1 and not less than
0.33 d1, where dl is as defined earlier in 3.3 (iii). The spacing of the stiffeners is kept such that the
greater unsupported clear dimension of a web panel does not exceed 270 t,,, nor the lesser
unsupported clear dimension of the same panel exceed 180 t,,,, where t,,, is the thickness of the web
plate.
Intermediate vertical stiffeners should extend from flange to flange, but the fitting of the ends need
not provide a tight bearing on the flanges. These may be crimped or joggled and can also be fixed
straight with filler plates. (Fig. 3.10)
Fig.(3.9), Curtailment of flange plates- Graphical Method.

32. 32

32 Analysis and Design of Plate Girder
Intermediate vertical stiffeners may be single or in pairs placed one on each side of the web. Where
single stiffeners are used, they should preferably be placed alternatively on opposite sides of the
web.


The vertical stiffeners shall be designed so that I is not less than
...3.29
Where;
I = the moment of inertia of a pair of stiffeners about the centre of the web, or a single stiffener about the
face of the web,
t = the minimum required thickness of the web, and
c = the maximum permitted clear distance between vertical stiffener for thickness t.

If the thickness of the web is made greater, or the spacing of stiffeners made smaller than that
required by the standard, the moment of inertia of the stiffener need not be correspondingly
increased.
External forces on intermediate stiffeners-When vertical intermediate stiffeners are subjected to
bending moments and shears due to eccentricity of vertical loads, or the action of transverse forces,
the moment of inertia of the stiffeners calculated above shall be increased as follows
(a) Bending moment on stiffener due to eccentricity of vertical loading with respect to the vertical
axis of the web:
Increase of
(b) Lateral loading on stiffener
Increase of.
Where;
M = the applied bending moment, kNm;
D = overall depth of girder, in mm;
E = Young's modulus, 2 x l0' MPa;
r = thickness of web, min; and
V = the transverse force in kN to be taken by the stiffener and deemed to be applied at the compression flange
of the girder outstand of stiffeners. Unless the outer edge of each stiffener is continuously stiffened, the
outstand of all stiffeners from the web should bb not more than for sections and 12t for flats where
t is the thickness of the section or flat.

(a) (b)
Fig.(9.10), Vertical Intermediate stiffener

33. 33

33 Analysis and Design of Plate Girder
3.2.5.2. HORIZONTAL STIFFENERS.
In addition to shear, the web of a plate girder also carries bending stresses. The compressive
bending stress tends to buckle the plate in the lateral direction as explained earlier in 3.3 (ii). In
order to restrain the web plate from buckling, horizontal or longitudinal stiffeners are provided.
As per I.S.800-1984, horizontal stiffeners should be provided in addition to vertical stiffeners when
the thickness of the web is less than the limits specified in 3.3 (iv) I (b).
Where horizontal stiffeners are used in addition to vertical stiffeners, they shall be as follows
(a) One horizontal stiffener shall be placed on the web at a distance I form the compression flange
equal to 2/5 of the distance from the compression flange to the neutral axis when the thickness of
the web is less than limits specified in 3.3(iv) I (b). This stiffener shall be designed so that I is not
less than 4 c.t3 where I and t are as defined for vertical stiffeners and c is the actual distant: between
the vertical stiffeners;
(b) A second horizontal stiffener (single or double) shall be placed at the neutral axis of the girder
when the thickness of the web is less than the limit specified in 3.3 (iv) I (c). This stiffener shall he
designed so that I is not less than d2 . t3 where d2 also in mm, I and t are as defined above and d2
is as defined in 3.3 (iv.)
(c) Horizontal web stiffeners shall extend between vertical stiffeners but need not be continuous
over them ; and
(d) Horizontal stiffeners may be in pairs arranged on each side of the web, or single.
The outstand of all stiffeners from the web should not be more for sections and 12t for
flats, where t is the thickness of the section or the flat except for stiffeners with stiffened edges.
Connections of intermediate stiffeners to web
Intermediate vertical and horizontal stiffeners not subjected to external loads shall be connected to
the web by rivets or welds, so as to withstand a shearing force, between each component of the
stiffener and the web of not less than
...3.32
where,
t = the web thickness in mm.
h = the outstand of stiffener in mm.

For stiffeners subjected to external loads, the shear between the web and stiffeners due to these
loads shall he added to the above values.
3.2.5.3. BEARING STIFFENER
Bearing stiffeners are provided at supports and at the points of concentrated loads. The function of
the bearing stiffeners is to transmit the concentrated loads and thereby avoid local bending failure
of the flange and local crippling or buckling of the web. When a column applies load to a plate
girder, either from above or as a reaction support from below, bearing stiffeners should be provided
in pairs such that they line up approximately with the flanges of the column. Bearing stiffeners
normally consist of single angles with one on either side of the web or a pair of angles on each side
of the web, These stiffeners must bear tightly between the flanges as these are to transmit the

34. 34

34 Analysis and Design of Plate Girder
vertical concentrated load directly to the bearing stiffener through bearing. The ends of the bearing
stiffeners should be milled
for direct bearing. For any section, load bearing stiffeners should be provided at points of
concentrated load (including. points of support) where the concentrated load or reaction exceeds
the value of
...3.33
Where;
σac = the maximum permissible axial stress for columns as given in 6.4 for a slenderness ratio
t = web thickness;
B = the length of the stiff portion of the bearing plus the additional length given by dispersion at 45° to the
level of the neutral axis, plus the thickness of the seating angle, if any. The stiff portion of a bearing is that
length which cannot deform appreciably in bending and shall not be taken as greater than half the depth of
beam for simply supported beams and the full depth of the beams continuous over a bearing; and
d1 = clear depth of web between root fillets.

Load bearing stiffeners shall be symmetrical about the web, where possible in Plate Girders. In
addition to the requirements given above load bearing stiffeners shall be provided also at the
supports where either:
(a) the web is overstressed in shear, or
(b) the web is otherwise overstressed at support or at the web, connection. The design of load
bearing stiffeners should take into account the following
(a) Load bearing stiffeners shall be designed as columns assuming the section to consist of the pair
of stiffeners together with a length of web on each side to the centre line of the stiffeners and equal,
where available, to 20 times the web thickness (Fig. 3.11). The radius of gyration shall be taken
about the axis parallel to the web of the beam or girder, and the working stress shall be in
accordance with the appropriate allowable value for a compression member as-suming an effective
length equal to 0.7 of the length of the stiffeners;

(b) The outstanding legs of each pair of stiffeners shall be so proportioned that the bearing stress
on that part of their area clear of the root of the flange or flange angles or clear of the welds does
not exceed the permissible bearing stress of 0.75 fy.
The connected legs of the bearing stiffener angles have to be chamfered at the flanges so as to clear
the fillet of the flange angles. Thus, the net bearing area of the outstanding legs of the bearing
stiffener should be sufficient such that the permissible bearing stress is not exceeded.
(c) Stiffeners shall be symmetrical about the web, where possible and at points of support shall
project as nearly as practicable to the outer edges of the flanges;
Fig.(3.11), Bearing Stiffener.

35. 35

35 Analysis and Design of Plate Girder
(d) Load bearing stiffeners shall be provided with sufficient rivets or welds to transmit to the web
the whole of the concentrated load;
(e) The ends of load bearing stiffeners shall be fitted to provide a tight and uniform bearing upon
the loaded flange unless welds or rivets designed to transmit the full reaction or load are provided
between the flange and stiffener. At points of support this requirement shall apply at both flanges.
(f) Bearing stiffeners shall not be joggled and shall be solidly packed throughout; and
(g) For plate girders, where load bearing stiffeners at supports are the sole means of providing
restraint against torsion the moment of inertia, I, of the stiffener about the centre line of the web
plate, shall he not less than
..3.34
Where,;
D = overall depth of the girder,
T= maximum thickness of compression flange,
R = reaction of the beam at the support, and
W = total load on the girder between supports.
Where a load is applied directly to the top flange, it may be considered as dispersed uniformly at an angle of
30° to the horizontal.
3.2.6. DESIGN OF SPLICES
A splice becomes necessary when the desired length of material is not- available or when it is
impracticable to transport the whole member in one piece. As the span of plate girders is large, the
webs and flanges have to be spliced. Splicing is also done sometimes to facilitate fabrication
worked.
3.2.6.1. WEB SPLICES.
Splice plates connecting the web are called web splices. As the web plate carries both bonding and
shear stresses, the splice is designed for bending and shear. The splice should be able to transmit
all the stresses which actually occur in the web plate. There are three types of web splices as shown
in Fig. 3.13. Splice plates should be provided on both sides of the web. In welded plate girders web
splices should be made with complete penetration butt welds.
(a) Type 1. This type of web splice (Fig. 3.12(a)) is the most efficient as it transmits the stresses
directly. It consists of two splice plates ‘A' and `B' fitted on both sides of the web. The minimum
thickness of the plates A and B kept equal to half The width of the splice plate 'A' should be
sufficient to accommodate all the rows or rivets. The splice plates 'A' extend from flange angle to
flange angle and the stresses in the web underneath the flange angles are transmitted by splice plates
'B'.


Where;
(a) (b) (c)
Fig.(3.12), Web Splices.

36. 36

36 Analysis and Design of Plate Girder
p = vertical pitch of rivets
n = number of rivets in a horizontal line on one side of the splice.
t = thickness of web
t = shear stress in web
G = bending stress in web at the rivet line under consideration
R = rivet value.

Then for splice plate 'A'.
Horizontal force on each rivet,
Vertical force on each rivet.
Resultant force
…3.35
The shear stress is uniform over the web, whereas the bending stress varies linearly with maximum
value at the flanges. The pitch should be fixed such that no rivet is overstressed. As the top most
rivets are the maximum stressed, the bending stress at this level should be taken for design.
Therefore, if the outermost rivets are not overstressed the inner rows of rivets will also be safe.
Thus the value or 'a' in Equation 3.35 will be taken as the bending stress at the top most rivet line.
The splice plates `B' transmit the stresses in the part of web covered by flange angles. In addition
to theses, the splice plates `B' also take up the horizontal shear between the flange angles and the
web.
Where;
m = number of rivets required on splice plate B.
b = width of web plate under flange angles.

Horizontal shear force in flange angles per unit length;

Horizontal shear force in flange angles on length m x p

And bending force in splice plate

If R is the rivet value, then the total force carried by rivets = m R

37. 37

37 Analysis and Design of Plate Girder

From the above equation, the number of rivets can be determined. The splice plates 'B' can be
omitted if the flange area is more than that required.
(b) Type 2. For the web splice of type 2 as shown in Fig. 3.12(b) separate plates are used to transmit
shear and moment of the web. The splice plate A is designed to transmit only shear and the splice
plates B are designed for moment. Splice plates A are, therefore, called shear plates and plates B
are called moment plates.
The shear on shear plate A is assumed as uniform and the spacing of rivets, in a riveted girder, is
kept uniform. The rivets connecting these plates to the web are designed to transmit shear.
The moment plates B resist the bending moment carried by the web. As the web is rigidly fixed to
the flanges, the curvature on bending will be the same for both. Then
...3.37
Where,
Mw = moment taken by web
1w = moment of inertia of web
I = moment of inertia of plate girder
M = total bending moment on the plate girder.

Bending force on plate B

Where;
h = vertical distance between the outer edges of plates B
b = depth of plates B
t = thickness of plates $

Average bending stress on the section at the level of consideration.

Bending force = (average bending stress) x (area of splice plates on both sides of web) or

The splice plates are provided on both sides of the web. From the above equation, the thickness
required for moment plates can be determined.
...3.38
The rivets are designed to transmit the bending force. If `m' is the number of rivets required and `R'
is the rivet value, then

38. 38

38 Analysis and Design of Plate Girder
...3.39
(c) Type 3. In type 3, (Fig. 3.12(c)), a single splice plate is provided on each side of the web. These
plates and their connections to the web are designed to take full shear and moment carried by the
web at that point. The splice plates are provided only between the flange angles. Additional plates
are not provided for the web underneath the angles even if there is no excess flange area.
The bending stress at extreme fiber of the splice plate is given by

Where;
M = bending moment on the girder
I = moment of inertia of the girder
Y = distance of extreme fiber of splice plate.

If ‘t' is the thickness of each splice plate and d is its depth, then section modulus (Z) for each plate
is

…3.40
The number of rivets are calculated from equation (3.40.a) and then checked for bending moment
and shear force.

3.2.6.2. FLANGE SPLICES.
Due to long spans of girders, the flange angles and cover plates are also spliced. All element of the
flange should not be spliced at one section but should be suitably staggered.
Flange joints preferably should not be located at points of maximum stress. Where splice plates are
used, there area shall be not less than 5 percent in excess of the area of the flange element spliced;
their centre of gravity shall coincide, as nearly as possible, with that of the element spliced. There
shall be enough rivets or welds on each side of the splice to develop the load in the element spliced
plus 5 percent but in no case should the strength developed be less than 50 percent of the effective
strength of the material spliced. In welded construction, flange plates shall be joined by complete
penetration butt welds, wherever possible. These butt welds shall develop the full strength of the
plates.
Flange angles can be spliced by a single angle or a single angle on one side and a flat on the other
side, as shown in (Fig. 3.13). The splicing angles should be shaped at the heel to clear the fillet of
the flange angle and also the splicing angle or Elates should not project beyond the flanges.
A single angle splice is a direct splice and is in contact with the flange angle. In this case, the shear
between then flange angle and the web is not affected and thus additional rivets are not required
for connection of splice. However, the length of the splice angle should be sufficient to
..3.40.a

39. 39

39 Analysis and Design of Plate Girder
accommodate all the rivets required to develop the strength of the splice angle. The pitch of the
rivets can also be decreased to reduce the length of the splice angle.

When the splice consists of an angle and a plate placed on each side of the web, the rivets arc
subjected to an additional shear force acting between the web and the flange angle. In this case the
stress is transferred from the splice angle to the splice plate through the web.
Where;
M = moment at the section
V = shear force at the section
n = number of rivets
p = pitch of rivets
R = rivet value
I = moment of inertia
b = width of splice plate
r = thickness of splice plate.

Horizontal shear force in the flange angles per unit length

and Horizontal shear force on length n x p

Bending force on the splice plate

Shear force between the web plate and the flange angle = horizontal shear in one plane + bending
force
or
or ...3.41

3.2.7. CONNECTIONS
The flanges and the web of a plate girder are connected by riveting or welding. Welded connections
have now-a-days replaced riveted connections. Design of both riveted and welded connections is
described.



3.2.7.1. BOLTS CONNECTING FLANGE ANGLES TO WEB.
(a) (b)
Fig.(3.13), Flange Splices.

40. 40

40 Analysis and Design of Plate Girder
The flanges of plate girders should be connected to the web by sufficient :vets or bolts so as to
transmit all the forces acting at that level.The force acting on these bolts is the horizontal shear
resulting from bending moment and force due to vertical loads,as the bolts connecting the flange
angles to web also transfer the vertical load acting on the top flange to the web. (Fig. 3.14).

Horizontal shear stress (τ) at a level is

Where;
V = vertical shear force
A = area of section I
y = distance of cetnre of gravity of section from neutral axis. = moment of inertia
b = width of section.

Per 'unit length, horizontal shear at the level of bolts is

If the pitch of bolts is 'p' and the bolt value is 'R', then horizontal shear force per pitch length is

or
or
or ...(3.42)
If vertical load on top flange is `w' per unit length, then vertical force per pitch length is =p.w
The force on bolts will be the vector sum of force due to the horizontal shear and force due to
vertical load.
Therefore resultant force per pitch length

As resultant force per pitch length is equal to the bolt value, we get

Fig.(3.14),Forces on Top Flange Rivets.

41. 41

41 Analysis and Design of Plate Girder
or
or ...(3.43)
From equation 3.16,

Substituting the values of l and Ay in Equation 3.43, we get
...(3.44)
If w = 0, then the equation reduces to
...(3.45)
Thus for the upper flange, equation 3.44 is used and for the lower flange, Equation 3.45 is used.
As the flange comprise of two angles, the bolts will be in double shear and in bearing. These bolts
should be staggered with respect to the bolts connecting the flange angles and flange plates so that
the deduction for bolts holes is not concentrated at one section.


3.2.7.2 BOLTS CONNECTING FLANGE ANGLES TO COVER PLATES.
The bolts connecting the cover plates to the flange angles are designed for the horizontal shear
between the angles and the cover plate. As the vertical load is transferred by direct bearing, there
will be no vertical force in the bolts.
If AP is the area of flange plates, the pitch of bolts (Equation 3.42).
...(3.46)
where, y = distance of centre of gravity of cover plates from the neutral axis.
However, a uniform weld size is provided throughout.
If the ends of the web are machined and are in close contact with the flange plates before welding,
then the vertical loads on the flange are transmitted to the web by direct bearing. The dispersion of
load through flange and web may be taken as 300 to the horizontal. The web should be save in
bearing.

42. 42

42 Analysis and Design of Plate Girder









Chapter four
COMPUTER PROGRAM










4.1 INTRODUCTION

43. 43

43 Analysis and Design of Plate Girder
This chapter presents a brief description of the computer program developed in this study which
governs the problem of analysis and design of in-plane structure (PLATE GIRDER) using Stiffness
Matrix as Method (S.M.M.). The program was written using MATLAB (version 7.2), it is consist
of three major parts; first the analysis of the in-plane structure. Second part is the design of plate
girder elements. Third part is the checking of plate girder elements according to the AISC-89
Design Manual using ASD designing method.

4.2 DEVELOPMENTS OF COMPUTER PROGRAM
4.2.1. First part; Analysis of in-plane structure using Stiffness Matrix Method.
At first, the computer program will develop the [EB] and [Ek] for each in-plane (beam) element
from input data describing the member geometry (coordinates) and cross-section properties
(modulus of elasticity, moment of inertia, area, angle of rotation…etc). Then, the program will
develop element stiffness matrix [EBkB
T
] for each element in global coordinate system, throughout
series of matrix operations (inverse, multiplication, and addition). Later on, the program will
assemble the overall stiffness matrix [BkB
T
] which is also represented by [K]. Finally, a direct
solution of the general stiffness equation {P}=[K]{d}, where {P} matrix containing the known
externally applied loads, will yield the global displacements {X} (translations and rotations). The
computer program then rebuilds the [EB] and [Ek] to obtain the [EkB
T
] and computes the internal
element forces (axial, shear and moments) and node reactions.
The sign convention used in this program 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.
It should be noticed that all above steps should be carried out with proper indices that identifies the
(P-X) coding so that the entries are correctly inserted into their right position of matrix.
Let the number of nodes NN and since DOF is three for each node. Each element stiffness element
[EBkB
T
] has (6x6) size but the overall assembled stiffness matrix [BkB
T
] or [K] has (NPxNP) size
because of the assembling process, where NP = NN * 3, therefore;
{??????
????????????}=[�??????�
???????????? × ????????????
??????
]{??????
????????????}
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.(3.4).
The global [BkB
T
] is banded with all zeros except for a diagonal strip of nonzero entries that is
twelve values wide. These twelve nonzero entries, six 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 6) entries instead of (NP x
NP).
It may be convenient to store the [EkB
T
] on a separate array when the [BkB
T
] is being built and
recall it to compute the internal element forces of the {F} matrix.


4.2.2. Second part; Design of plate Girder elements:

44. 44

44 Analysis and Design of Plate Girder
1. Economic Girder depth
2. Flanges :
a. Tension flange
b. Compression flange
3. Flange elements:
a. Flange angles
b. Flange plate (cover plate)
4. Web elements:
a. Stiffened and unstiffened type
b. Web depth
c. Web Thickness
d. Curtailment lengths
5. Stiffeners:
a. Intermediate stiffener
b. Horizontal stiffener
i. 1
st
hor. Stiff. At 20% form top of flange plate
ii. 2
nd
hor. Stiff. At 50% form top of flange plate
c. Bearing stiffener
6. Connections
a. Flange elements
b. Web elements
4.2.3. Third part: Checking of Plate Girder elements:
1. Checking of assumed self weight of plate girder.
2. Checking of selected flange cross-sectional area.
3. Checking of web plate failure mode.
4. Checking of web plate buckling.
5. Checking of web plate stability.
6. Checking of web plate buckling shear stress (Euler equation).
7. Checking of flexural stresses at curtailed cross-section of plate girder.
8. Checking the required kind of stiffeners (vertical, horizontal (20% height), and horizontal
(50% height)).
9. Checking of checking of the vertical stiffener stability (two conditions).
10. Checking of selected Ver. Stiff. cross-sectional area.
11. Checking of selected 1
st
Hor. Stiff. cross-sectional area.
12. Checking of selected 2
nd
Hor. Stiff. cross-sectional area.
13. Checking of selected Bearing Stiff. (under Concentrated loads) cross-sectional area.
14. Checking of selected Bearing Stiff. (at supports) cross-sectional area.
15. Checking of bolts spacing (shearing and bearing capacities) at top web plate connections.
16. Checking of bolts spacing (shearing and bearing capacities) at bottom web plate
connections.
17. Checking of bolts spacing (shearing and bearing capacities) at top flange plate connections.
18. Checking of bolts spacing (shearing and bearing capacities) at bottom flange plate
connections.
Developed computer program, is listed in Appendix I, can be also used to solve a number of
structural problems.

4.3 FLOW CHART OF COMPUTER PROGRAM

45. 45

45 Analysis and Design of Plate Girder












START



Build Element Stiffness
Matrix in G.C.S.
[k
e
] (I, ND, ND)
Assemble Overall Stiffness
Matrix of system [K]
Solve for unknowns displacement {d}=[K]
-1
{d}
Is all
checks is
OK?

Evaluate Internal Forces, in L.C.S., of in-plane structure
elements: F (NE, 6). Then calculate Reaction forces

END

Re-correct
cross-section
to satisfy
required
conditions
No
Yes
OUTPUT UNIT
Printout global 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)
INPUT UNIT FOR LOADING CONDITIONS
For each node (1  NN) of the In-plane Structure read the following:
-Lateral loads (Fixed-End Forces).
-Joint loads.
-Predefined displacements.
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…),
Cross-section properties (Ag, Ix, rx, d, b, t…), and Boundary conditions (DOF)

46. 46

46 Analysis and Design of Plate Girder
4.4 COMPUTER PROGRAM APPLICATION:
First Case study: Analysis and design a simply supported plate girder (riveted type), as shown in
fig.(4.1), with a span of (20.0 m) and carrying a uniformly distributed load of (WLL= 12.5 kN/m,
WDL=1.5 kN/m). In addition it carries two concentrated loads each of (PL = 325.0 kN, PIM = 84.5
kN) each at middle third points. The computed results are compared with those obtained by I. C.
Syal
(1)
using a (S.M.M.). The agreement was very good. Each case was investigated for three
load cases in order to demonstrate several factors effects.

Under Concentrated Load stiffener
Bearing stiffener
Intermediate stiffener @ spacing c/c

(1
st
& 2
nd
) Horizontal stiffener
Top & Bottom Cover plate @ curtailment length

P P
6.67 m 6.66 m 6.67 m
Concrete wall
(0.40 x 2.64) m
Concrete wall
(0.46 x 2.64) m
w
Horizontal (1
st
& 2
nd
) stiffener Bearing & under P, Stiffener Intermediate stiffener

47. 47

47 Analysis and Design of Plate Girder
Fig.(4.1) In-plane structure layout

48. 48

48 Analysis and Design of Plate Girder

49. 49

49 Analysis and Design of Plate Girder

Fig.(4.2) deflection diagram

Fig.(4.3) contact pressure diagram
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
0 2000400060008000100001200014000160001800020000
displacements (mm)
Gidrder length (mm)
load case one
load case two
load case three
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 2000400060008000100001200014000160001800020000
Rotation (Radians)
Gidrder length (mm)
load case one
load case two
load case three

50. 50

50 Analysis and Design of Plate Girder

Fig.(4.4) shear force diagram

Fig.(4.5) bending moment diagram
-800000
-600000
-400000
-200000
0
200000
400000
600000
800000
0 2000400060008000100001200014000160001800020000
Shear Force (N)
Gidrder length (mm)
load case one
load case two
load case three
-4E+09
-3.5E+09
-3E+09
-2.5E+09
-2E+09
-1.5E+09
-1E+09
-5E+08
0
0 2500 5000 7500 1000012500150001750020000
Bending Moment(N.mm)
Gidrder length (mm)
load case one
load case two
load case three

51. 51

51 Analysis and Design of Plate Girder

Fig.(4.6) Flexural Stresses

Fig.(4.7) Web Shear Stresses
0
20
40
60
80
100
120
140
160
180
0 2500 5000 7500 10000 12500 15000 17500 20000
Bending Stress (Mpa)
Gidrder length (mm)
Allowable Bending
Stress
load case one
load case two
load case three
-20
0
20
40
60
80
100
120
0 2500 5000 7500 10000 12500 15000 17500 20000
Shear stress (Mpa)
Gidrder length (mm)
Allowable Shearing Stress
load case one
load case two
load case three

52. 52

52 Analysis and Design of Plate Girder
Second Case study:
Analysis and design a Continuous plate girder (riveted type), as shown in fig.(4.2), each span is
(20.0 m) and carrying a uniformly distributed load of (WLL= 12.5 kN/m, WDL=1.5 kN/m). In
addition it carries four Concentrated loads each of (PL = 325.0 kN, PIM = 84.5 kN) each at middle
third points. The computed results are compared with those obtained by I. C. Syal
(1)
using a
(S.M.M.). The agreement was very good. Each case was investigated for three load cases in order
to demonstrate several factors effects.

Fig.(4.8) In-plane structure layout

Under Concentrated Load stiffener
Bearing stiffener
Intermediate stiffener @ spacing c/c

(1
st
& 2
nd
) Horizontal stiffener
Top & Bottom Cover plate @ curtailment length

Concrete wall
(0.40 x 2.64) m
Concrete wall
(0.46 x 2.64) m
P P P P
6.67 m 6.66 m 6.67 m 6.67 m 6.66 m 6.67 m
w
Horizontal (1
st
& 2
nd
) Stiffener Bearing & under P, Stiffener Intermediate Stiffener

53. 53

53 Analysis and Design of Plate Girder

Fig.(4.9) deflection diagram

Fig.(4.10) contact pressure diagram
-12
-10
-8
-6
-4
-2
0
0 2000400060008000100001200014000160001800020000
displacements (mm)
Gidrder length (mm)
LOAD CASE ONE
LOAD CASE TWO
LOAD CASE THREE
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0 2000400060008000100001200014000160001800020000
Rotation (Radians)
Gidrder length (mm)
LOAD CASE ONE
LOAD CASE TWO
LOAD CASE THREE

54. 54

54 Analysis and Design of Plate Girder

Fig.(4.11) shear force diagram

Fig.(4.12) bending moment diagram
-1000000
-800000
-600000
-400000
-200000
0
200000
400000
600000
800000
1000000
0 2000400060008000100001200014000160001800020000
Shear Force (N)
Gidrder length (mm)
LOAD CASE ONE
LOAD CASE TWO
LOAD CASE THREE
-1.5E+09
-1E+09
-5E+08
0
500000000
1E+09
1.5E+09
2E+09
0 2000400060008000100001200014000160001800020000
Bending Moment (N.mm)
Gidrder length (mm)
LOAD CASE ONE
LOAD CASE TWO
LOAD CASE THREE

55. 55

55 Analysis and Design of Plate Girder

Fig.(4.13) Flexural Stresses

Fig.(4.14) Web Shear Stresses
-200
-150
-100
-50
0
50
100
150
200
0 2000400060008000100001200014000160001800020000
Bending Stress (Mpa)
Gidrder length (mm)
Allowable Bending
Stress
load case one
load case two
load case three
0
20
40
60
80
100
120
140
0 2000400060008000100001200014000160001800020000
Shear stress (Mpa)
Gidrder length (mm)
Allowable Shearing
Stress
load case one
load case two
load case three

56. 56

56 Analysis and Design of Plate Girder









Chapter five
CONCLUSIONS AND RECOMMENDATIONS

57. 57

57 Analysis and Design of Plate Girder
CONCLUSIONS
Depending on the results obtained from the present study, several conclusions may be established.
These may be summarized as follows:
Results indicate that in-plane structures (Plate Girder) can be can be dealt with successfully by the
Stiffness Matrix Method.
Developed Program in this study is quite efficient and reliable for this type of analysis, and the
process of analyses can be carried out rapidly on electronic computer. Design criteria given by
AISC-89 (allowable stress design) has been successfully implement inside presented program.
The presented results indicate that:
1. Increasing Girder Depth will increase both Bending and Shearing Strength Capacity of
girder.
2. Increasing Girder Depth will reduce the maximum deflection of girder.
3. Using Intermediate Vertical Stiffener will increase both Bending and Shearing Strength
Capacity of Girder and reduce buckling effects, due to Vertical loading, because of
additional stiffening.
4. Using (1
st
and 2
nd
) Horizontal Stiffener will increase both Bending Strength capacity of
girder and reduce buckling effects, due to horizontal shear, because of additional stiffening.
5. Simply supported girder is critical in bending stress than shearing stress.
6. Continuously supported girder is critical in shear stress than bending stress.
7. Using Bearing Stiffener will prevent both local buckling and web shearing failure of
girder at supports and uniformly transfer the reaction forces to supports.
8. Using Vertical Stiffener Under load (P) will prevent both local buckling and web
shearing failure of girder at concentrated load points and uniformly transfer the
concentrated loads to girder.
Presented results indicate also:
In order to overcome web critical shear stresses an additional vertical stiffener could be used at that
section or simply increase the girder depth, but it is not recommended.
In order to overcome beam critical bending stresses an additional horizontal stiffener could be used
at neutral axes of girder or simply increase the girder depth, but it is not recommended.
Although of their lack of shearing capacity; continuously supported girders are clearly much more
economical than simply supported girders, as shown in bellow table.

Load case No.
Simply supported girder Depth
(mm)
Continuously supported girder
Depth (mm)
(1) 800.0 420.0
(2) 1,540.0 720.0
(3) 1,720.0 860.0

Note: according to AASHTO design specifications the Impact load were included in this analysis,
taken as 30% of applied live loads, this case was presented in the third load case of both case
studies, which represent the critical loading case here.

58. 58

58 Analysis and Design of Plate Girder
RECOMMENDATIONS
Many important recommendations could be suggested, for the given analysis method for in-plane
structures, to include the following factors:
1. Axial deformation effects on bending moments (coupling).
2. Shear deformation since plate girder is much close to deep beam that long beam.
3. Torsional rigidity could be included to cover applied eccentric loads.
4. Composite girder (steel beam with concrete slab) could be included in same analysis
program.

REFERENCES
Syal, I. C., and Satinder S., "Design of steel structures.", Standard Publishers Distributers, Delhi, 2000.
Dayaramtnam. P., "Design of steel structures.", Chand S. Company ltd. for publishing , NewDelhi, 2003.
Livesley, R. K., and Chandler D. B., "Stability Functions for Structural Frameworks." Manchester University
Press, Manchester, 1956.
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.
Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis." Pergamon Press, London, 1964,
PP. 115-145.
Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London, 1964. PP. 241-252.
Bowles, J. E., "Analytical and Computer Methods in Foundation Engineering." McGraw-Hill Book Co., New
York, 1974, pp. 190-210.
Bowles, J. E., "Foundation analysis and design" McGraw-Hill Book Co., New York, 1986, fourth edition,
pp. 380-230.
Bowles, J. E., "Mat Design." ACI Journal, Vol. 83, No.6, Nov.-Dec. 1986, pp. 1010- 1017.
Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition, McGraw-Hill Book Company,
New York, 1961, pp. 1-17.
KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of Structural Engineering,
ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.
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.