Analysis and Design of Composite Beams with Composite Deck Slab.docx

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1 Analysis and Design of Composite Beams with Steel Decks
Analysis and Design
Of
Composite Beams with
Steel Deck
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
Ahmed & Alan
Supervised by
Assistant lecturer, A. N. LAZEM
(M.Sc., in Structural Engineering)
July /2008

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Analysis and Design
Of
Composite Beams with
Steel Deck

3
I certify that study entitled “Analysis and Design of Composite Beams with Steel
Decks”, was prepared by ( and )
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.

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Supervisor
Signature:
Name: A. N. LAZEM
Assistant lecturer
(M.Sc., in Structural
Engineering)
Date:
We certify that we have read this study “Analysis and Design of Composite Beams
with Steel Decks” and as examining committee examined the students in its content
and in what are 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:

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Signature:
Name:
Head of Civil Engineering Department
College of Engineering
Baghdad University
Date:
Abstract:
This objective of this study is to develop a better understanding for the basic
principles of the “Analysis and Design of Composite Beams with Steel Decks” so
they can be efficiently implemented on modern computers.
Demonstrate the effects of load and geometry (supports) changes on the behavior of
in-plane structure (composite beam). Studying the effects of load and geometry
(supports) changes on the internal stresses of in-plane structure.
Develop a stiffness matrices for in-plane structure (composite beam), that take into
account the interaction between concrete slab and steel girder.
In addition several important parameters have been incorporated in the stiffness
matrices development; the shear connecters design (numbers and spacing) between
concrete deck and steel girder, the effects of steel decks (height of ribs =hr) on
section moment of inertia, and using equivalent lane loads instead of actual moving
loads (series of axel loads) according to AASHTO design manual.
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

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The project is divided into five chapters as follows:
Chapter one: presents a general introduction to the subject of Composite Beams
with Steel Decks and stiffness matrix method.
Chapter two: presents the previous literatures published about this subject.
Chapter three: presents the theoretical bases for the analysis method and the
Design of Composite Beams.
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.
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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Chapter one
Introduction

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1.1. General Introduction
A typical bay floor framing of a high-rise building consists of steel floor beams
framing into steel girders along bay lines (Fig. 1.1). The floor beams generally are designed
for only gravity loads imposed by the floor as simply supported composite beams with the
slab essentially in compression over the full span of the beam. The beam normally is of a
standard rolled, wide-flange shape designed to interact compositely with the concrete floor
slab by means of shear studs placed in metal decking troughs. In such a metal deck composite
beam system, the ribs or corrugations generally run perpendicular to the supporting floor
beams
1.2. COMPOSITE CONSTRUCTION
1.21 Definition
When the dimensions of a concrete slab supported on steel beams are such that the slab can
effectively serve as the flange of a composite T-beam, and the concrete and steel are
adequately tied together so as to act as a unit, the beam can be proportioned on the
assumption of composite action. Two cases are recognized: fully encased steel beams, which
depend upon natural bond for interaction with the concrete, and those with mechanical an-
chorage to the slab (shear connectors), which do not have to be encased.
For composite beams with formed steel deck, studies have demonstrated that the total slab
thickness, including ribs, can be used in determining effective slab width.
1.2.2 Design Assumptions

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Unless temporary shores are used, beams encased in concrete and interconnected only by
means of natural bond must be proportioned to support all of the dead load, unassisted by the
concrete, plus the superimposed live load in composite action, without exceeding the
allowable bending stress for steel provided in Sect. 1.5.1.
Because the completely encased steel section is restrained from both local and lateral
buckling, an allowable stress of (0.66 Fy), rather than (0.60 Fy), can be applied when the
analysis is based on the properties of the transformed section. The alternate provision to be
used in designs where a fully encased beam is proportioned, on the basis of the steel beam
alone, to resist all loads at a stress not greater than (0.76 Fy), reflects a common engineering
practice where it is desired to eliminate the calculation of composite section properties.
It is accepted practice that when shear connectors are used to obtain composite action, this
action may be assumed, within certain limits, in proportioning the beam for the moments
created by the sum of live and dead loads, even for unshored construction. This liberalization
is based upon an ultimate strength concept, although the provisions for proportioning of the
member are based upon the elastic section modulus of the transformed cross section.
The flexural capacity of composite steel-concrete beams designed for complete composite
action is the same for either lightweight or normal weight concrete, given the same area of
concrete slab and concrete strength, but with the number of shear connectors appropriate to
the type of concrete. The same concrete design stress level can be used for both types of
concrete.
In order that the maximum bending stress in the steel beam, under service loading, will be
well below the level of initial yielding, regardless of the ratio of live-load moment to dead-
load moment, the section modulus of the composite cross section, referred to the bottom of
the beam, for unshored construction, is limited in calculations to (1.35 + 0.35 ML/MD) times
the section modulus of the bare beam.*
On the other hand, the requirement that flexural stress in the concrete slab, due to composite
action, be computed on the basis of the transformed section modulus, referred to top of
concrete, and limited to the generally accepted working stress limit, is necessary in order to
avoid excessively conservative slab-to-beam proportions.
Research at Lehigh University* * has shown that, for a given beam and concrete slab, the
increase in bending strength intermediate between no composite action and full composite
action is dependent upon the shear resistance developed between the steel and concrete, i.e.,
the number of shear connectors provided between these limits. Usually, it is not necessary,
and occasionally it may not be feasible, to provide full composite action. Therefore, the
Specification recognizes two conditions: full and partial composite action.
1.3. GENERAL CONSIDERATIONS
Composite construction is appropriate for any loading. It is most efficient with heavy
loading, relatively long spans, and beams spaced as far apart as permissible. The decision to
use composite beams is usually economic and will generally be based on a comparison of the
installed cost of the shear connectors and the savings in beam weight.
For unshored construction, concrete compressive stress will seldom be critical for the
beams listed in the Composite Beam Selection Tables if a full width slab and Fy = 36 ksi steel
are used. It is more likely to be critical when a narrow concrete flange or Fy = 50 ksi steel is
used, and is frequently critical if both Fy = 50 ksi steel and a narrow concrete flange are used.
Shored construction also increases the concrete stress.

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Because composite construction usually involves relatively long spans and wide
spacing of beams, the Specification rule that governs effective slab width is usually the
provision limiting the projection beyond the edge of each beam flange to eight times the slab
thickness (see AISC, Sect. 1.11.1). in addition to that, the best advantage of composite
construction that: any Steel or Concrete materials of various strengths may be used.
Fig.(1.1)
Chapter two

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Literature
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.

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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.
Chapter three

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Theory
AISC Specification COMPOSITE DESIGN for building construction
3.1. GENERAL NOTES
The AISC Specification contains provisions for designing composite steel-concrete beams as
follows:
1. For totally encased unshored steel beams not requiring mechanical anchorage (shear
connectors), see Sects. 1.11.1 and 1.11.2.1.
2. 2. For both shored and unshored beams with mechanically anchored slabs, design of
the steel beam is based on the assumption that composite action resists
the total design moment (Sect. 1.11.2.2). In shored construction_ flexural stress in
the concrete slab due to composite action is determined from the total moment. In
unshored construction, flexural stress in the concrete slab due to composite action is
determined from moment ML, produced by loads imposed after the concrete has
achieved 75% of its required strength .Shored construction may be used to reduce
dead load deflection and must be used if Str > (1.35 + 0.35 ML/MD) Ss .
3. 3. For partial composite action, see Sect. 1.11.2.2.
4. For negative moment zones, see Sect. 1.11.2.2.
5. For composite beams with formed steel deck (FSD), see Sect. 1.11.5.
3.2. GENERAL CONSIDERATIONS

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1. Composite construction is appropriate for any loading. It is most efficient with heavy
loading, relatively long spans, and beams spaced as far apart as permissible. The
decision to use composite beams is usually economic and will generally be based on a
comparison of the installed cost of the shear connectors and the savings in beam
weight.
2. For unshored ed construction, concrete compressive stress will seldom be critical for
the beams listed in the Composite Beam Selection Tables if a full width slab and Fy =
36 ksi steel are used. It is more likely to be critical when a narrow concrete flange or
Fy = 50 ksi steel is used, and is frequently critical if both Fy = 50 ksi steel and a
narrow concrete flange are used. Shored construction also increases the concrete
stress.
3. Because composite construction usually involves relatively long spans and wide
spacing of beams, the Specification rule that governs effective slab width is usually
the provision limiting the projection beyond the edge of each beam flange to eight
times the slab thickness (see Sect. 1.11.1).
4. Steel and concrete materials of various strengths may be used.
3.2.3. END REACTIONS
End reactions for composite beams may be higher than for non-composite beams of the same
size. They must be calculated by the engineer and shown on the contract documents.
3.2.4. DEFLECTION
A composite beam has much greater stiffness than a non-composite beam of equal depth, size,
loads, and span length. Deflection of composite beams will usually be about 1/3 to 1/2 less
than deflection of non-composite beams. In practice, shallower beams are used and
deflections, particularly of the steel section alone under construction loads, should be
calculated and listed on the contract documents as a guide for cambering, or estimating slab
quantities.
If it is desired to minimize the transient vibration due to pedestrian traffic or other moving
loads when composite beams support large open floor areas free of partitions or other
damping sources, a suitable dynamic analysis should be made. (See AISC engineering
Journal, Vol. 12 No. 3, 3rd Quarter, 1975.)
Long term creep deflections are usually not significant for composite beams. However since a
portion of the section is concrete, which can be susceptible to creep, creep deflection should
be investigated if considered undesirable by the design professional. If it is desired to
investigate long term creep deflection, Itr should be based on a modular ratio,n, double that
used for stress calculations.
When lightweight concrete is used in composite construction, deflection should be calculated
using Itr based on the actual modulus of elasticity of the concrete, Ec , even though stress
calculations are based on the Ec of normal weight concrete.
3.2.5. USE OF COVER PLATES
Bottom cover plates are an effective means of increasing the strength or reducing the depth of
composite beams when deflections are not critical, but they should be used with overall
economy in mind. High labor cost has made their use rare. For this reason they have not been
retained in the selection table. However their use is allowed by the Specification. Section
properties may be calculated or obtained from other sources.
3.2.6. USE OF FORMED STEEL DECK (FSD)

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Although use of FSD in composite construction has been permitted by the AISC specification
for many years, specific provisions for it's use were not included in the specification until
1978. The following limitation on parameters were established :o keep composite
construction with FSD within the currently (1978) available research data (see Specification
Sect. 1.11.5.1):
1. Deck rib height (hr): Maximum 3 in.
2. Average width of concrete rib or haunch (wr): Minimum 2 in.
3. Shear connectors: Welded studs only, maximum 3/4 in. diameter.
4. Stud length: Minimum = rib height + 11/2 in.
5. Effective width of concrete flange: To be determined using total slab thickness,
including ribs.
6. Slab thickness above deck: Minimum 2 in.
Property and selection tables for composite beams with FSD are not included. There are
numerous proprietary FSD's and most manufacturers can furnish tables of properties for
beams using their own particular FSD. The FSD used in the examples is arbitrary and is not
intended to agree dimensionally with any proprietary deck.
The FSD itself may be either composite or non-composite. Under the 1978 AISC
Specification both will produce the same composite beam properties. Section moduli for
composite beams using FSD will be relatively close to those using a plain slab of the total
thickness. Therefore the plain slab selection tables can be used for at least preliminary
estimates of Str when using FSD.
3.2.7. OTHER CONSIDERATIONS
The AISC Specification provisions for the design of composite beams are based on ultimate
load considerations, even though they are presented in terms of working stresses. Because of
this, for unshored construction, actual stresses in the tension flange of the steel beam under
working load are higher than calculated stresses. The effect of Formula (1.11-2) in Sect.
1.11.2.2 is to limit the tension flange stress to 89% of the minimum yield stress (0.89Fy ).
Section 1.11.2.2 also provides requirements for limiting the steel beam compression flange
stress under construction loading.
Adequate lateral support for the compression flange of the steel section will be provided by
the concrete slab after hardening. During construction, however, lateral support must be
provided, or working stresses must be reduced in accordance with Sect. 1.5.1.4 of the
Specification. Steel deck with adequate attachment to the compression flange, or properly
constructed concrete forms, will usually provide the necessary lateral support for the type of
construction shown in the sketches accompanying the composite beam property tables. For
construction using fully encased beams, particular attention should be given to lateral support
during construction.
The design of the concrete slab should conform to the ACI Building Code. Supplement No. 3
to the 1969 AISC Specification disallowed the use of Specification Sect. 1.5.6 for unshored
construction. The 1978 AISC Specification disallows Sect. 1.5.6 only for stresses occurring in
a negative moment area. Therefore, the 1/3 increase in working stress allowed by this portion
of the Specification, when wind and seismic stresses are being investigated in positive
moment areas, may be used.
3.3. COMPOSITE BEAM SELECTION TABLES
Composite beam selection tables have been prepared for common conditions encountered in
building design and are based on the following:
1. 3.0 ksi concrete (n = 9); weight = 145 pcf.

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2. An effective flange width of 16t + b f .
3. Plain (no FSD) concrete slab thicknesses, t, of 4, 41/2, 5, 51/2 and 6 in. placed
directly on the steel beam.
4. Selected steel beams ranging from 8 to 36 in. in depth.
Selection tables are provided for only a limited number of the possible beam selections. They
are intended to assist the designer in selecting his initial trial section. after which he must
calculate (or obtain from other sources) the necessary section properties to complete the
design.
The listed composite sections are arranged in descending order of transformed section
modulus, Str for the 4 in. slab thickness. For other slab thicknesses, slight inconsistencies may
occur in the descending order. Only concrete above the neutral axis is used in calculating
section properties. The data is applicable to all structural grades of steel. Interpolation
between the tabulated slab thickness is considered proper. Errors resulting from such
interpolation are usually negligible. A reduction in slab width, b, of nearly 60% reduces the
transformed section modulus by only about 6 to 8%. For this reason the tables may also be
used with interpolation if desired to select trial sections for other concrete strengths, slab
thicknesses, slab widths, or slabs on formed steel deck (FSD).
3.4. COMPOSITE CONSTRUCTION
3.4.1 Definition
When the dimensions of a concrete slab supported on steel beams are such that the slab can
effectively serve as the flange of a composite T-beam, and the concrete and steel are
adequately tied together so as to act as a unit, the beam can be proportioned on the
assumption of composite action. Two cases are recognized: fully encased steel beams, which
depend upon natural bond for interaction with the concrete, and those with mechanical an-
chorage to the slab (shear connectors), which do not have to be encased.
For composite beams with formed steel deck, studies have demonstrated that the total slab
thickness, including ribs, can be used in determining effective slab width.
3.4.2 Design Assumptions
Unless temporary shores are used, beams encased in concrete and interconnected only by
means of natural bond must be proportioned to support all of the dead load, unassisted by the
concrete, plus the superimposed live load in composite action, without exceeding the
allowable bending stress for steel provided in Sect. 1.5.1.
Because the completely encased steel section is restrained from both local and lateral
buckling, an allowable stress of (0.66 Fy), rather than (0.60 Fy), can be applied when the
analysis is based on the properties of the transformed section. The alternate provision to be
used in designs where a fully encased beam is proportioned, on the basis of the steel beam
alone, to resist all loads at a stress not greater than (0.76 Fy), reflects a common engineering
practice where it is desired to eliminate the calculation of composite section properties.
It is accepted practice that when shear connectors are used to obtain composite action, this
action may be assumed, within certain limits, in proportioning the beam for the moments
created by the sum of live and dead loads, even for unshored construction. This liberalization
is based upon an ultimate strength concept, although the provisions for proportioning of the
member are based upon the elastic section modulus of the transformed cross section.
The flexural capacity of composite steel-concrete beams designed for complete composite
action is the same for either lightweight or normal weight concrete, given the same area of
concrete slab and concrete strength, but with the number of shear connectors appropriate to

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the type of concrete. The same concrete design stress level can be used for both types of
concrete.
In order that the maximum bending stress in the steel beam, under service loading, will be
well below the level of initial yielding, regardless of the ratio of live-load moment to dead-
load moment, the section modulus of the composite cross section, referred to the bottom of
the beam, for unshored construction, is limited in calculations to (1.35 + 0.35 ML/MD) times
the section modulus of the bare beam.*
On the other hand, the requirement that flexural stress in the concrete slab, due to composite
action, be computed on the basis of the transformed section modulus, referred to top of
concrete, and limited to the generally accepted working stress limit, is necessary in order to
avoid excessively conservative slab-to-beam proportions.
Research at Lehigh University* * has shown that, for a given beam and concrete slab, the
increase in bending strength intermediate between no composite action and full composite
action is dependent upon the shear resistance developed between the steel and concrete, i.e.,
the number of shear connectors provided between these limits. Usually, it is not necessary,
and occasionally it may not be feasible, to provide full composite action. Therefore, the
Specification recognizes two conditions: full and partial composite action.
For the case where the total shear ( V'h ) developed between steel and concrete each side of
the point of maximum moment is less than Vh, Formula (1.11-1) can be used to derive an
effective section modulus, Seff, having a value less than the section modulus for fully effective
composite action, Str but more than that of the steel beam alone. In the 1969 Specification, the
obviously conservative straight line function of Formula (1.11-1) was adopted pending the
results of research. The completed research indicated that a parabolic function using (
V'h/Vh)1/2 provided a good fit to the test results.
3.4.3. Shear Connectors
Composite beams in which the longitudinal spacing of shear connectors, as shown in fig.(3.1),
has been varied according to the intensity of statical shear, and duplicate beams where the
required number of connectors were uniformly spaced, have exhibited the same ultimate
strength and the same amount of deflection at normal working loads. Only a slight
deformation in the concrete and the more heavily stressed shear connectors is needed to
redistribute the horizontal shear to other less heavily stressed connectors. The important
consideration is that the total number of connectors be sufficient to develop the shear, VjZ ,
either side of the point of maximum moment. The provisions of the Specification are based
upon this concept of composite action.
Fig(3.1)

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In computing the section modulus at points of maximum negative bending, reinforcement
parallel to the steel beam and lying within the effective width of slab may be included,
provided such reinforcement is properly anchored beyond the region of negative moment.
However, enough shear connectors are required to transfer, from the slab to the steel beam,
1/2 of the ultimate tensile strength of the reinforcement.
Studies have defined stud shear connection strength, Q, , in terms of normal weight and
lightweight aggregate concretes, as a function of both concrete modulus of elasticity and
concrete strength:
c
c
s
u E
f
A
Q '
5
.
0

Eq.(3.1)
Where:
Ac = cross-sectional area of stud, square inches.
f''c = concrete compressive strength, kips per square inch.
Ec = concrete modulus of elasticity, kips per square inch.
Tests(1) have shown that fully composite beams designed using the values in Tables 1.11.4,
AISC, and/or 1.11.4A, as appropriate, and concrete meeting the requirements of Part 3, Chap.
4, "Concrete Quality", of ACI Standard 318-71, made with ASTM C33 or C330 aggregates,
develop their full flexural capacity. For normal weight concrete, compressive strengths
greater than 4.0 kips per square inch do not increase the shear capacity of the connectors, as is
reflected in Table 1.11.4. For lightweight concrete, compressive strengths greater than 5.0
kips per square inch do not increase the shear capacity of the connectors. The reduction
coefficients in Table 1.11.4A are applicable to both stud and channel shear connectors and
provide comparable margins of safety.
When partial composite action is counted upon to provide flexural capacity, the restriction on
the minimum value of V'h is to prevent excessive slip, as well as substantial loss in beam
stiffness. Studies indicate that Formulas (3.1) adequately reflect the reduction in strength and
beam stiffness, respectively, when fewer connectors than required for full composite action
are used.
Where adequate flexural capacity is provided by the steel beam alone, that is, composite
action to any degree is not required for flexural strength, but where it is desired to provide
interconnection between the steel frame and the concrete slab for other reasons, such as to
increase frame stiffness or to take advantage of diaphragm action, the minimum requirement
that V'h be not less than Vh/4 does not apply.
The required shear connectors can generally be spaced uniformly between the points of
maximum and zero moment.* However, certain loading patterns can produce a condition
where closer connector spacing is required over a part of this distance.
For example, consider the case of a uniformly loaded simple beam also required to support
two equal concentrated loads, symmetrically disposed about midspan, of such magnitude that
the moment at the concentrated loads is one slightly less than the maximum moment at
midspan. The number of shear connectors (N2) required between each end of the beam and
the adjacent concentrated ' load would be only slightly less than the number (N1) required
between each end and midspan.
Equation (3.2) is provided to determine the number of connectors, N2, required between one
of the concentrated loads and the nearest point of zero moment. It is based upon the following
requirement:

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1
1
1
2















s
eff
s
eff
eff
s
eff
s
S
S
S
S
S
S
S
S
S
S
N
N
Eq.(3.2)
Where:
S = section modulus required at the concentrated load at which location moment equals M, inches
Seff = section modulus required at Mmax (equal to St, for fully composite case), inches3
Ss = section modulus of steel beam, inches
Nl = number of studs required from Mmax to zero moment N2 = number of studs required from M to
zero moment
M = moment at the concentrated load point
Mmax = maximum moment in the beam. Noting that S/Seff = M/Mmax, and defining ß as Seff /Ss, the
above equation is equivalent to Formula (1.11-7).
With the issuance of Supplement No. 3 to the 1969 AISC Specification, the requirement for
1-inch cover over the tops of studs was eliminated. Only the concrete surrounding the stud
below the head contributes to the strength of the stud in resistance to shear. When stud shear
connectors are installed on beams with formed steel deck, concrete cover at the sides of studs
adjacent to sides of steel ribs is not critical. Tests have shown that studs installed as close as is
permitted to accomplish welding of studs does not reduce the composite beam capacity.
Stud welds not located directly over the web of a beam tend to tear out of a thin flange before
attaining their full shear-resisting capacity. To guard against this contingency, the size of a
stud not located over the beam web is limited to 21/2 times the flange thickness.
3.5. Composite Beams and Girders with Formed Steel Deck
The 6 diameter minimum center-to-center spacing of studs in the longitudinal direction is
based upon observation of concrete shear failure surfaces in sectioned flat soffit concrete slab
composite beams which had been tested to full ultimate strength. The reduction in connection
capacity of more closely spaced shear studs within the ribs of formed steel decks is accounted
for by the parameter 0.85/x/N, in Formula (1.11-8, AISC).
When studs are used on beams with formed steel deck, they may be welded directly through
the deck or through prepunched or cut-in-place holes in the deck. The usual procedure is to
install studs by welding directly through the deck; however, when the deck thickness is
greater than 16 gage for single thickness, or 18 gage for each sheet of double thickness, or
when the total thickness of galvanized coating is greater than 1.25 ounces per square foot,
special precautions and procedures recommended by the stud manufacturer should be
followed. Fig.(3.2) is a graphic presentation of the terminology used in Sect. 1.11.5.

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Fig.(3.2-a)
Fig.(3.2-b)

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Fig.(3.3-a) Computation of Elastic Moment Capacity for (-ve )Negative Bending Sections
Fig.(3.3-b) Computation of Plastic Moment Capacity for (+ve )Posative Bending Sections
The design rules which have been added for composite construction with formed steel deck
are based upon a study36 at Lehigh University of all available test results. The limiting
parameters listed in Sect. 1.11.5.1 were established to keep composite construction with
formed steel deck within the available research data.
Seventeen full size composite beams with concrete slab on formed steel deck were tested at
Lehigh University and the results supplemented by the results of 58 tests performed
elsewhere. The range of stud and steel deck dimensions encompassed by the 75 tests were
limited to:
Elastic
Neutral
axis

22
1. Stud dimensions: 3/4-in. diam. x 3.00 to 7.00 in.
2. Rib width: 1.94 in. to 7.25 in.
3. Rib height: 0.88 in. to 3.00 in.
4. Ratio wr/hr : 1.30 to 3.33
5. Ratio Hs/hr: 1.50 to 3.41
6. Number of studs in any one rib:1, 2, or 3
Based upon all tests, the strength of stud connectors in flat soffit composite slab beams,
determined in previous test programs,41 when multiplied by values computed from Formula
(1.11-8), reasonably approximates the strength of stud connectors installed in the ribs of
concrete slabs on formed steel deck with the ribs oriented perpendicular to the steel beam.
Hence, Formula (1.11-8) provides a reasonable reduction factor to be applied to the allowable
design stresses in Tables 1.11.4 and 1.11.4A.
For the case where ribs run parallel to the beam, limited testing36 has shown that shear
connection is not significantly affected by the ribs. However, for narrow ribs, where the ratio
w,/hr is less than 1.5, a shear stud reduction factor, Formula (1.11-9), has been suggested in
view of lack of test data.
The Lehigh study36 also indicated that Formula (1.11-1) for effective section modulus and
Formula (1.11-6) for effective moment of inertia were valid for composite construction with
formed steel deck.
When metal deck includes units for carrying electrical wiring, crossover headers are
commonly installed over the cellular deck, perpendicular to the ribs, in effect creating
trenches which completely or partially replace sections of the concrete slab above the deck.
These trenches, running parallel to or transverse to a composite beam, may reduce the
effectiveness of the concrete flange. Without special provisions to replace the concrete
displaced by the trench, the trench should be considered as a complete structural discontinuity
in the concrete flange.
When trenches are parallel to the composite beam, the effective flange width should be
determined from the known position of the trench.
Trenches oriented transverse to composite beams should, if possible, be located in areas of
low bending moment and the full required number of studs should be placed between the
trench and the point of maximum positive moment. Where the trench cannot be located in an
area of low moment, the beam should be designed as non-composite.
3.6 Design of Partially or Fully Composite Beams, with Ribbed Metal
Deck, Using LRFD Specifications
A Hindu saying state (2), "One picture is worth a thousand words (numbers)." With this optic
in mind, the design aspects of steel-concrete composite beams using the recently adopted
LRFD specifications1 are analyzed critically and the significance of several parameters is
brought out clearly.
Charts are then constructed to facilitate the design of partially or fully composite beams using
rolled-steel, wideflange sections of A36 steel or A572 Gr. 50 steel. The slab may be a
composite metal deck slab with ribs perpendicular to the beam, a haunched slab or, simply, a
flat soffit concrete slab. The charts cover both adequate and inadequate slabs.
The charts provided are a valuable tool from the practical standpoint, and also familiarity with
them should contribute to the student's and young engineer's overall feel of the composite

23
beam design problem. The design charts given here complement the composite beam design
tables provided in the LRFD manual.
3.6.1 General Introduction
A typical bay floor framing of a high-rise building consists of steel floor beams framing into
steel girders along bay lines (Fig. 3.3-a). The floor beams generally are designed for only
gravity loads imposed by the floor as simply supported composite beams with the slab
essentially in compression over the full span of the beam. The beam normally is of a standard
rolled, wide-flange shape designed to interact compositely with the concrete floor slab by
means of shear studs placed in metal decking troughs. In such a metal deck composite beam
system, the ribs or corrugations generally run perpendicular to the supporting floor beams
The Load and Resistance Factor Design Specification (LRFD) for Structural Steel Buildings
adopted by AISC in September 1986 uses the ultimate strength of composite beams as the
basis of their design. According to LRFD, composite beam designs are classified as fully
composite and partially composite designs.2,3,4,5,6,7,8,9,10(2) The present report on the
design of partially composite beam design is a generalization of the work presented earlier in
Ref. 11.
3.6.2. COMPOSITE METAL DECK SLABS
Composite metal deck slabs consist of light-gage, ribbed metal deck forms which interact
with structural concrete topping as a composite unit to resist floor loads (Fig. 3.1).
Special embossments, dimples or lugs cold-rolled into the decking increase bond and act as
shear connectors. Uplift is prevented either by the shape of the profile or by inclining the lugs
to the vertical in opposite directions, on the two sides of the rib. It is usual practice to design
the slab as a one-way, simply supported beam, for the ultimate limit state (with the metal
decking acting as reinforcement steel in the span direction), even though the slab and the
decking may be continuous over the floor beams. The slab is usually provided with square
mesh steel reinforcement at, or above, mid depth of the slab to minimize cracking due to
shrinkage and temperature effects and to help distribute concentrated loads.
The variables for a composite deck slab include span length, gage thickness, rib depth, slab
thickness, unit weight of concrete and concrete strength.12,13,14(2) Thickness of metal deck
plate elements usually varies from 22ga. (0.0336 in.) to 12ga. (0.1084 in.), depending on
configuration of the section. For noncellular decks 1½ in., 2 in. and 3 in. deep decks are
generally used for spans up to 8 ft, 10 ft and 15 ft, respectively. The thickness of concrete
above the metal deck typically varies from a minimum of 2½ in. to 4 in., which may be
controlled by fire-rating requirements of the slab instead of structural requirements. The
choice of lightweight concrete or normal weight concrete, depends on economics and fire-
rating considerations. For high-rise buildings, light-weight concrete weighing about 110 pcf is
often used.
The advantage of the metal deck slab system is the elimination of the formwork and shoring
and the consequent increase in speed of construction. The metal deck can be used as a
working and erection platform. Also, it acts as a diaphragm to help stabilize the steel skeleton
by integrating all members into a system.
The steel deck alone has to withstand the weight of the wet concrete, plus any construction
loads for placement of concrete, for the desired non-shored condition. The depth of the ribs
necessary is generally controlled by this loading. The composite steel deck (steel deck +
concrete) has to withstand the factored superimposed dead and live loads. Design of the steel
deck should conform to the latest edition of the specifications for the design of light gage,
cold formed steel structural members of the American Iron and Steel Institute (AISI). The
permissible superimposed floor loads are tabulated for different deck sections by their
manufacturers.

24
The tabulated capacity is obtained as the smallest load of the following conditions:
1. Bending stress at the bottom fiber of the deck;
2. Bending stress in the top fiber of the concrete;
3. Bending stress at the top fiber of deck under dead load only;
4. Value of the transverse shear for shear bond failure and;
5. Deflection at mid-span. Hence, a detailed design is not performed by the structural
engineer.
3.6.3. SHEAR CONNECTORS
The purpose of shear connectors in a composite beam is to tie the slab and steel beam together
and force them to act as a unit. For this, the connectors must resist the horizontal shear force
that develops between the slab and beam as the composite member is loaded, and they should
prevent vertical separation or uplift of the concrete slab from the steel beam.
Presently, stud connectors are the most commonly used shear connectors in the U.S. The stud
shear connector is a short length of round steel bar, welded to the steel beam at one end, with
an upset end or head at the other end. They range in diameter from ½ in. to 1 in. and lengths
from 2 to 8 in. The ratio of the overall length to the diameter of the stud is not less than 4. The
most commonly used sizes in building structures are ¾ in. or 7/8 in. dia. The head diameter is
½ in.
Larger than stud diameter and the head thickness is 3/8 in. or ½ in. The anchorage provided
by the head on the stud ensures the required uplift resistance. The studs are made with
ASTM-A108, AISI Grades C1010, C1015, C1017 or C1020 cold-drawn steel with a
minimum tensile strength of 60 ksi and a minimum elongation of 20% in 2-in. gage length, as
specified in the AWS Structural Welding Code D1.1-75. To prevent premature failure of
studs because of tearing of base metal, the size of a stud not located over the beam web is
limited to 2½ times the flange thickness.1 The strength of stud connectors increases with stud
length up to a length of about four diameters and remains approximately constan beyond this
limit.

25
3.7. 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.
3.7.1. 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:
𝑃𝑖 = 𝐵𝑖𝐹𝑖……………………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:
{𝑃𝑖} = [𝐵𝑖]{𝐹𝑖}……………………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:
{𝑒𝑖} = [𝐴𝑖]{𝑋𝑖}………………………..(a)

26
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:
{𝑒𝑖} = [𝐵]𝑇
{𝑋𝑖}……………………..(b)
The internal-member forces {F} are related to the internal-member displacements {e} as:
{𝐹𝑖} = [𝑘]{𝑒𝑖}…………………………(c)
These three equations are the fundamental equations in the Stiffness Matrix Method of
analysis:
Substituting (b) into (c),
{𝐹𝑖} = [𝑘]{𝑒𝑖} = [𝑘][𝐵]𝑇
𝑋…………………………(d)
Substituting (d) into (a),
{𝑃𝑖} = [𝐵]{𝐹𝑖} = [𝐵][𝑘][𝐵]𝑇
𝑋…………………………(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
{𝑋𝑖} = ([𝐵][𝑘] [𝐵]𝑇
)−1
{𝑃𝑖}…………………………(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 BKBT
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 BKBT
from the
Local element EBKBT
. 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 EBKBT
are added into the (2, 5) coordinate
location of the global BKBT.
3.7.2. 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.
Now at node (1), summing Moments (Fig.(3.2d))
𝑃1 = 𝐹1 + 0. 𝐹2
Similarly, summing forces and noting that the soil reaction (spring) forces are Global and will
be considered separately, we have:
𝑃2 =
𝐹1
𝐿
+
𝐹2
𝐿
𝑃3 = 0. 𝐹1 + 𝐹2
And 𝑃4 = −
𝐹1
𝐿
−
𝐹2
𝐿
Placing into conventional matrix form, the Element Transformation Matrix [EB] in local
coordinate is:
EB =
F1 F2
P1 1 0

27
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.
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.
3.7.3. DEVELOPING THE [k] MATRIX
Referring to Fig.(3.3) and using conjugate-beam (Moment Area Method)principle, the end
slopes e1, and e2 are:
𝑒1 =
𝐹1𝐿
3𝐸𝐼
−
𝐹2𝐿
6𝐸𝐼
………………………(g)
𝑒2 = −
𝐹1𝐿
6𝐸𝐼
+
𝐹2𝐿
3𝐸𝐼
…………………….(h)
Force-Displacement relationships (P-X indexing)
P1 P2
(a) Local force-displacement relationships (F-e indexing)
FEM FEM

28
Fig.(3.3) conjugate-beam method Moments and rotations of beam element.
Solving Eqs.(g) and (h) for F, obtaining:
𝐹1 =
4𝐸𝐼
𝐿
𝑒1 +
2𝐸𝐼
𝐿
𝑒2
𝐹2 =
2𝐸𝐼
𝐿
𝑒1 +
4𝐸𝐼
𝐿
𝑒2
Placing into matrix form, the Element Stiffness Matrix [ES] in local coordinate is:
Ek =
e1 e2
F1
4𝐸𝐼
𝐿
2𝐸𝐼
𝐿
F2
2𝐸𝐼
𝐿
4𝐸𝐼
𝐿
3.7.4. DEVELOPING THE ELEMENT [kBT
] AND [BKBT
] MATRICES
The EkBT
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) AT
goes always
with e and X. The EBkBT
will be also obtained in a similar.
Multiplying [Ek] and [EBT
] matrices and rearrange them, yields:
kBT
=
1 2 3 4
1
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
−
6𝐸𝐼
𝐿2
2
2𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
4𝐸𝐼
𝐿
−
6𝐸𝐼
𝐿2
Multiplying [EB] and [EkBT
] matrices and rearrange them, yields:
EBkBT
=
X1 X2 X3 X4
P1
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
−
6𝐸𝐼
𝐿2
P2
6𝐸𝐼
𝐿2
12𝐸𝐼
𝐿3
6𝐸𝐼
𝐿2
−
12𝐸𝐼
𝐿3

29
P3
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
4𝐸𝐼
𝐿2
−
6𝐸𝐼
𝐿2
P4 −
6𝐸𝐼
𝐿2
−
12𝐸𝐼
𝐿3
−
6𝐸𝐼
𝐿2
12𝐸𝐼
𝐿3
From Fig.(3.4), summing of the vertical forces on a node 1 will produce:
𝑃2 −
𝐹1 + 𝐹2
𝐿
= 0.0
Since (F1+F2)/L is already included in the Global BkBT
, we could rewrite above equation to:
𝑃2 = 𝐵𝑘𝐵2𝑋2
𝑇
𝑋2 = [𝐵𝑘𝐵2𝑋2
𝑇
]𝑋2
A check on the correct formation of the EBkBT
and the global BkBT
is that it is always
symmetrical and there cannot be a zero on the diagonal.
3.7.5. 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.
3.7.6. 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).

30
Chapter four
Computer program

31
4.1. INTRODUCTION
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 Composite Structures” using
Stiffness Matrix Method.
4.2. PROGRAM PROCEDURE
Based on theoretical equations presented in Chapter Three; the following step-by-step
procedure for the computer program will be develop:
At first; establish sign convention which to be used in this analysis. Therefore; 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:
Then prepare the analytical model of in-plane structure, as follows:
1. Draw a line diagram of the in-plane structure (beam), and identify each joint member by a
number.
2. Determine the origin of the global (X-Y) coordinate system (G.C.S.). It is usually located to
the lower left joint, with the X and Y axes oriented in the horizontal (positive to the right) and
vertical (positive upward) directions, respectively.
3. For each member, establish a local (x-y) coordinate system (L.C.S), with the left end
(beginning) of the member, and the x and y axes oriented in the horizontal (positive to the
right) and vertical (positive upward) directions, respectively.
4. Number the degrees of freedom and restrained coordinates of the beam elements and nodes.
5. Evaluate the Overall Stiffness Matrix [k], and Fixed-End forces Vector {Pf}. The number of
rows & columns of [S] must be equal to the number of DOF of the structure. For each
element of the in-plane structure, perform the following operations:
a) Compute the Element stiffness matrix [ke
] in (L.C.S) by apply the basic stiffness
equation, as follow:

32
a. {𝑓} = [𝑘𝑒]{𝑒}.
b) Transform the force vector {𝑓} form (L.C.S) to {𝑃} in (G.C.S.) using transformation
matrix [A], as follow:
a. {𝑃} = [𝐴]{𝑓}.
c) Transform the deformation vector {𝑒}form (L.C.S) to {𝑋} (G.C.S.) using
transformation matrix [B], as follow:
a. {𝑒} = [𝐵]{𝑋}.
d) It is evident that matrix [B] is the transpose of matrix [A], therefore ;
a. {𝑒} = [𝐴]𝑇{𝑋}.
e) Substituting step (d) in step (a), resulting in:
a. {𝑓} = [𝑘𝑒][𝐴]𝑇{𝑋}.
f) Substituting step (e) in step (b), resulting in:
a. {𝑃} = [𝐴][𝑘𝑒][𝐴]𝑇{𝑋}.
g) Inverting equation in step (f), resulting in:
a. {𝑋} = [[𝐴][𝑘𝑒][𝐴]𝑇
]−1{𝑃}.
h) Store the element stiffness matrix, in (G.C.S.), [𝑘𝑒
] = [[𝐴][𝑘𝑒][𝐴]𝑇
]−1
, for each
element.
i) Compute the Fixed-Ends (lateral loads) forces Vector {Pf}. Knowing that this step
working only if there are existing lateral loading on the element. Using their proper
positions in the Element Stiffness Matrix [ke
] in (G.C.S.).
6. Assemble Overall Stiffness Matrix [K] for the System of in-plane structure. By assembling
the element stiffness matrices for each element in the in-plane structure, using their proper
positions in the in-plane structure Stiffness Matrix [K], and it must be symmetric.
7. Compute the Joint load vector {Pj} for each joint of the in-plane structure.
8. Added the Fixed-Ends (lateral loads) forces Vector {Pf} to their corresponding Joint load
vector {P} using their proper positions in the in-plane structure Stiffness Matrix [K].
9. Determine the structure joint displacements {X}. Substitute {P}, {Pe}, and [K] into the
structure stiffness relations, {𝑃𝑗 + 𝑃𝑓} = [𝐾]{𝑋} .and solve the resulting system of
simultaneous equations for the unknown joint displacements {X}.
10. Compute Element end displacement {e} and end forces {f}, and support reactions. For each
Element of the beam, as following:
11. Obtain Element end displacements {e} form the joint displacements {X}, using the Element
code numbers.
12. Compute Element end forces {f}, using the following relationship:
{𝑓} = [𝑘𝑒]{𝑒} + {𝑃𝑓}.
13. Using the Element code numbers, store the pertinent elements of {f}, in their proper position
in the Support Reaction Vector {R}
14. Check the calculation of the member end-forces and support reactions by applying the
Equation of Equilibrium to the free body of the entire in-plane structure;
∑ Fy = 0
𝑛
𝑖=0 , ∑ Mz = 0
𝑛
𝑖=0

33
4.3. FLOW CHART OF COMPUTER PROGRAM
Solve for unknowns displacement {d}=[K]-1
{d}
Assemble Overall Stiffness
Matrix of system [K]
Build Element Stiffness
Matrix in G.C.S.
[ke
] (I, ND, ND)
START
Re-correct
cross-section to
satisfy required
conditions
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)

34
4.4. COMPUTER PROGRAM APPLICATION:
4.4.1. First Case study:
Design a simply supported Composite Bridge with Steel Deck, as shown in fig.(4.1), with a
span of (70’.0”) and a clear width of (28’.0”), to carry the following conditions: a live load of
HS20, wearing surface (30 psf), Concrete strength (5000 psi), A36 steel girder, and
reinforcement bars of (grade 50). The computed results are compared with those obtained by
G. Winter and A. Nilson (3)
using a (S.M.M.). The agreement was very good. Each case was
investigated for three load cases in order to demonstrate the effects of each factor.
END
Evaluate Internal Forces, in L.C.S., of in-plane structure
elements: F (NE, 6). Then calculate Reaction forces
Is all checks
is OK?
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)

35
Fig.(4.1) In-plane structure layout (continued)
P
70’.0”
w
Abutment wall
(6 x 32) ft
Abutment wall
(6 x 32) ft
(a) Bridge Profile
(b) Span Arrangement
70’-0”
(c) Longitudinal Girder Section
Under Concentrated Load stiffener
Bearing stiffener Bearing stiffener
Bottom Cover Plate
Reinforcement bars Concrete Slab with Steel Decks

36
Fig.(4.1) In-plane structure layout
Fig.(4.1-d) Transverse Bridge Section
32’-0”
10’-0”
10’-0” 4’-0”
4’-0”
2’-0”
3’-6”
4 Spaces @ 6’-9” = 27’-0” 2’-6”
2’-6”
Girder Cross-Section Girder Cross-Section with
Bearing & under P, Stiffener
(e) Transverse Girder Sections
Girder Cross-Section with
Intermediate Stiffener
(d) Transverse Bridge Section
32’-0”
10’-0”
10’-0” 4’-0”
4’-0”
2’-0”
3’-6”
4 Spaces @ 6’-9” = 27’-0” 2’-6”
2’-6”

37
Fig.(4.2) Vertical Displacements Diagram
Fig.(4.3) Rotational Displacements Diagram
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 42 84 126 168 210 252 294 336 378 420
Displacements
(in)
Joist Girder length (in)
load case one
load case two
load case three
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0 42 84 126 168 210 252 294 336 378 420
Rotation
(Radians)
Gidrder length (in)
load case one
load case two
load case three

38
Fig.(4.4) Shear Force Diagram
Fig.(4.5) Bending Moment Diagram
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 42 84 126 168 210 252 294 336 378 420
Shear
Force
(kip)
Gidrder length (in)
load case one
load case two
load case three
0
5000
10000
15000
20000
25000
0 42 84 126 168 210 252 294 336 378 420
Bending
Moment
(kip.in)
Gidrder length (in)
load case one
load case two
load case three

39
Fig.(4.6) Flexural Stresses Of Concrete Deck
Fig.(4.7) Flexural Stresses Of Steel Girder
0
5
10
15
20
25
30
35
40
45
0 42 84 126 168 210 252 294 336 378 420
Compressive
Stress
(ksi)
Gidrder length (in)
Allowable Compressive Stress
load case one
load case two
load case three
0
5
10
15
20
25
0 42 84 126 168 210 252 294 336 378 420
Tensile
Stress
(ksi)
Gidrder length (in)
Allowable Tensile Stress
load case one
load case two
load case three

40
Fig.(4.8) Flexural Stresses Of Reinforcement Steel Bars
Fig.(4.9) Web Shear Stresses
0
5
10
15
20
25
30
35
0 42 84 126 168 210 252 294 336 378 420
Compressive
Stress
(ksi)
Gidrder length (in)
load case one
load case two
load case three
0
2
4
6
8
10
12
14
16
0 42 84 126 168 210 252 294 336 378 420
Shear
stress
(ksi)
Gidrder length (in)
Allowable Shearing Stress
load case one
load case two
load case three

41
4.4.2. Second Case study:
Design a continuously supported Composite Bridge with Steel Deck, as shown in fig.(4.10),
with a span of (70’.0”) and a clear width of (28’.0”), to carry the following conditions: a live
load of HS20, wearing surface (30 psf), Concrete strength (5000 psi), A36 steel girder, and
reinforcement bars of (grade 50). The computed results are compared with those obtained by
G. Winter and A. Nilson (3)
using a (S.M.M.). The agreement was very good. Each case was
investigated for three load cases in order to demonstrate the effects of each factor.
Fig.(4.10) In-plane structure layout (continued)
P P
35.0 ft 35.0 ft 35.0 ft 35.0 ft
w
Abutment wall
(6 x 32) ft
Abutment wall
(6 x 32) ft
70 ft 70 ft
140 ft
(a) Bridge Profile
(b) Span Arrangement
(c) Longitudinal Cross-Section of Continuous Girder
Top & Bot. Cover Plates
Reinforcement Steel Bars
Concrete Slab with Steel Deck
Supports & Under-Load
& Vertical Stiffeners

42
Fig.(4.11) Vertical Displacements Diagram
Fig.(4.12) Rotational Displacements Diagram
-2.5
-2
-1.5
-1
-0.5
0
0 168 336 504 672 840 1008 1176 1344 1512 1680
Displacements
(in)
Joist Girder length (in)
Vertical Displacements
load case one
load case two
load case three
maximum deflection
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0 168 336 504 672 840 1008 1176 1344 1512 1680
Rotation
(Radians)
Gidrder length (in)
load case one
load case two
load case three

43
Fig.(4.13) Shear Force Diagram
Fig.(4.14) Bending Moment Diagram
-150
-100
-50
0
50
100
150
0 168 336 504 672 840 1008 1176 1344 1512 1680
Shear
Force
(kip)
Gidrder length (in)
load case one
load case two
load case three
-25000
-20000
-15000
-10000
-5000
0
5000
10000
15000
0 168 336 504 672 840 1008 1176 1344 1512 1680
Bending
Moment
(kip.in)
Gidrder length (in)
load case one
load case two
load case three

44
Fig.(4.15) Flexural Stresses Of Concrete Deck
Fig.(4.16) Flexural Stresses Of Steel Girder
-25
-20
-15
-10
-5
0
5
10
15
20
25
0 168 336 504 672 840 1008 1176 1344 1512 1680
Compressive
Stress
(ksi)
Gidrder length (in)
Allowable Stress
load case one
load case two
load case three
-30
-20
-10
0
10
20
30
0 168 336 504 672 840 1008 1176 1344 1512 1680
Stress
(ksi)
Gidrder length (in)
load case one
load case two
load case three

45
Fig.(4.17) Flexural Stresses Of Reinforcement Steel Bars
Fig.(4.18) Web Shear Stresses
-45
-35
-25
-15
-5
5
15
25
35
0 168 336 504 672 840 1008 1176 1344 1512 1680
Stress
(ksi)
Gidrder length (in)
load case one
load case two
load case three
0
2
4
6
8
10
12
14
16
0 168 336 504 672 840 1008 1176 1344 1512 1680
Stress
(ksi)
Gidrder length (in)
Allowable Shearing Stress
load case one
load case two
load case three

46
Chapter five
Conclusions and Recommendations

47
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. The main difference between simply and continuously supported composite girders
is: the later type is much more efficient (overall-response) in handling the excess
loading, developed from live and impact loading, where it can be seen form table(5-1)
that the increases in internal stresses and global displacements are significantly less in
continuous type than the discontinuous type. This might be justified; that in
continuous type there is a moment distribution facility which not exists in
discontinuous type.
2. Continuous type is weaker in Shearing Capacity than discontinuous type.
3. Discontinuous type is weaker in Bending Capacity than continuous type.
4. Shear connecters (studs) numbers is less in continuous than discontinuous type,
which is it not true because higher shear (in continuous type) require higher numbers
of shear connecters, this because the design of strength (AISC) govern than the
design of fatigue requirements and make the two types are equal in shear connecters.
5. Increasing higher steel decks (height of ribs) will increase both Bending and Shearing
Strength Capacity of girder.
6. Increasing Girder Depth will reduce the maximum deflection of girder. But it is not
recommended.
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 Cover Plate 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.
Additional requirement for the composite beams are:
 Bearing Stiffener at supports will prevent both local buckling and web shearing
failure of girder at supports and uniformly transfer the reaction forces to supports.
 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.
 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.

48
Table 5-1
Disp., Forces, & Stresses
Simply Supported Girder
Type
Continuously Supported
Girder Type
Ver. Displacements (in) 2.5 1.1
Shear Force (kip) 90 110
Bending Moment (kip-in) 22,000 20,000
Deck Compressive Stress (ksi) 23 13
Reinf. bars Tensile Stress (ksi) 20 10
Flexural Stress (ksi) 22 13
Shearing Stress (ksi) 4.2 5.5
No. of shear connecter rows, three
studs
84 76
Shear Connecters Spacing c/c (in) 10 11
Table 5-2
Simply Supported Girder Type
Disp., Forces, &
Stresses
1st
Load case
(DL)
2nd
Load case
(DL + LL)
3rd
Load case
(DL + LL + IM)
Ver. Displacements 1 1.65 1.8
S.F. 1 1.7 1.72
B.M. 1 1.7 1.85
Flexural Stress 1 1.6 1.79
Shearing Stress 1 1.67 1.68
Continuously Supported Girder Type
Ver. Displacements 1 1.66 1.7
S.F. 1 1.78 1.8
B.M. 1 1.78 1.79
Flexural Stress 1 1.69 1.7
Shearing Stress 1 1.73 1.75
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.

49
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
1. Winter G., and Nilson A., “Design of Concrete Structures”, Reinforced-concrete
Bridges, chapter twelve, 1974, PP 541-580.
2. S. VINNAKOTA, C. M. FOLEY and M. R. VINNAKOTA, “Design of Partially or
Fully Composite Beams with Ribbed Metal Deck, Using LRFD Specifications”
3. Winter, G. and Nilson, A., “Design of Concrete Structures”. McGraw-Hill Book Co.,
New York, 1977, 2nd edition, pp. 541-580.
4. AMERICAN NATIONAL STANDARD SJI-JG–1.1, SECTION 1001. Adopted by the
Steel Joist Institute November 4, 1985 (Revised to November 10, 2003 - Effective March
01, 2005).
5. Manual Of Steel Construction (AISC-1989, Allowable Stress Design), ninth edition.
6. Asalam Kassimali, “Matrix Analysis of Structures”, Brooks/ Cole Publishing Company,
1999.
7. Syal, I. C., and Satinder S., "Design of steel structures.", Standard Publishers
Distributers, Delhi, 2000.
8. Dayaramtnam. P., "Design of steel structures.", Chand S. Company ltd. for publishing ,
New-Delhi, 2003.
9. Livesley, R. K., and Chandler D. B., "Stability Functions for Structural Frameworks."
Manchester University Press, Manchester, 1956.
10. 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.
11. Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis." Pergamon
Press, London, 1964, PP. 115-145.
12. Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London,
1964. PP. 241-252.
13. Bowles, J. E., "Analytical and Computer Methods in Foundation Engineering."
McGraw-Hill Book Co., New York, 1974, pp. 190-210.
14. Bowles, J. E., "Foundation analysis and design" McGraw-Hill Book Co., New York,
1986, Fourth Edition, pp. 380-230.
15. Bowles, J. E., "Mat Design." ACI Journal, Vol. 83, No. 6, Nov.-Dec. 1986, pp. 1010-
1017.
16. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition, McGraw-
Hill Book Company, New York, 1961, pp. 1-17.
17. Timoshenko, S.P. and Gere, J.M., "Mechanics of Materials." 2nd Edition, Von Nostrand
Reinhold Book Company, England, 1978.
18. KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of
Structural Engineering, ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.

50
19. LAZEM, 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.

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