This thesis for the Master of Science
degree by
Farrokh Jalalvand
has been approved
by
Kevin L. Rens
Date
Jalalvand, Farrokh (M.S., Civil Engineering)
A Theoretical Derivation of Initial Stiffness of the Semirigid Structural Tee
Connection
Thesis directed by Associate Professor Judith J. Stalnaker
ABSTRACT
That a semirigid connection allows some rotation causes reduced beam end
moments compared to a rigid connection. This study develops a theoretical
equation for the initial elastic stiffness of a steel semirigid structural tee
connection. The initial stiffness is not only a function of flexural and shear
deformations in the connection, but is also a function of deformations of the
bolt, column flange, and column web.
A lateralforce resisting frame of a threestory building is analyzed as a rigid
frame for gravity and gravity plus wind, using a MATLAB computer program.
Then, the members and rigid connections are designed. Connection initial
stiffness is determined from the theoretical equation and is used for a
computer semirigid analysis. New members and connections are chosen
based on the reduced moments.
Column flange and column web deformations were found to be particularly
significant in contributing to connection rotational deformation.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
IV
DEDICATION
I would like to dedicate this thesis to the memory of my late Iranian father,
who, despite his being illiterate, zealously valued formal education and very
much wanted me to become a civil engineer and to my American wife Jean E.
Johnson who, through her support, patience, and encouragement, made it all
happen here in the USA.
ACKNOWLEDGEMENT
I would like to thank Dr. Judith J. Stalnaker for her directing this thesis and
also for her generous assistance and encouragement in the course of my
thesis research. I would also like to thank Professor John Mays for developing
the computer program used in this thesis and for his valuable assistance in
the use and interpretation of the program results. Additionally, 1 am indebted
to both Dr. Stalnaker and Dr. Mays for all the valuable knowledge I received
from them through their class lectures and by tutoring their structural analysis
students.
Finally, I would like to express my appreciation to Professor K. H. Gerstle of
the University of Colorado at Boulder for providing input, ideas, and his own
written and published papers on the subject of flexible connections for my
thesis research.
CONTENTS
Figures.................................................................xii
Tables..................................................................xiv
Chapter
1. Introduction....................................................1
1.1 Definition...................................................... 1
1.2 Names.............................................................2
1.3 Types.............................................................2
1.4 Design Method.....................................................2
1.5 Purpose of the Study..............................................2
1.6 Scope of the Study................................................3
2. Semirigid Connection.............................................4
2.1 Response Overview.................................................4
2.2 Definition........................................................5
2.3 Degree of Rigidity................................................5
2.4 Physical Look.....................................................6
2.5 Literature Review.................................................9
2.6 Design Practice .................................................14
vii
3. Response of a Semirigid Structural Tee......................15
3.1 Elastic M
3.2 Total Initial Stiffness Kl(tolal)............................17
3.3 Comparison with Test Data................................... 22
4. Design Example...............................................26
4.1 Computation of Loads.........................................28
4.2 Approximate Rigid Frame Analysis.............................29
4.3 Preliminary Design of Rigid Frame............................33
4.31. Design for Gravity Alone.....................................33
4.3.2 Design for Combined Gravity and Wind.........................39
4.4 Exact Rigid Frame Analysis.................................. 43
4.4.1 HandPrepared Input Data.....................................43
4.4.2 MATLAB Program Tasks.........................................46
4.4.3 Results of Exact Rigid Frame Analysis....................... 48
4.5 Exact Design of Rigid Frame..................................51
4.5.1 Beams........................................................51
4.5.2 Columns......................................................53
4.5.3 Rigid Connection.............................................60
4.6 Exact Semirigid Frame Analysis with K^totai)................61
VIII
4.6.1 Calculation of Connection Stiffness Kj(totai)....................63
4.6.2 Results of Semirigid Frame Analysis............................ 65
5. Modified Initial Stiffness Kmiolal)..............................70
5.1 Column Flange Flexure............................................71
5.2 Bolt Elongation..................................................72
5.3 Column Web Shear Deformation.....................................77
5.4 Column Web Compression Deformation...............................79
5.5 Total Deformation................................................83
5.6 Exact Semirigid Frame Analysis with KMi(totai)..................86
5.6.1 Calculation of Connection KMi(totai).............................86
5.6.2 Results of Exact Semirigid Analysis with KMi(totai).............90
5.7 Exact Design of Semirigid Frame with l
5.7.1 Beams......................................................... 93
5.7.2 Columns..........................................................94
5.8 Comparison of Rigid and Semirigid...............................95
6. Discussion, Conclusion, and Further Study........................96
6.1 Discussion.......................................................96
6.2 Conclusion.......................................................97
6.3 Further Study....................................................97
IX
Appendix
A. Derivation of Semirigid Structural Tee
Total Initial Stiffness K^otai)................................98
A. 1 General Assumptions..............................................99
A.2 Notations.......................................................100
A.3 Sign Convention.................................................102
A.4 Initial Stiffness for Top Tee Ki{lop)...........................102
A.4.1 Deflection ( Ab) Due to Flexure.................................105
A.4.2 Deflection (8B) Due to Shear....................................110
A.5 Initial Stiffness for Bottom Tee Kj{bot)........................115
A. 5.1 Assumptions for Bottom Tee......................................115
B. Frame Analysis..................................................118
B. 1 Computation of Loads............................................119
B.1.1 Gravity Loads...................................................119
B.1.2 Wind Loads......................................................122
B.2 Approximate Rigid Frame Analysis................................125
B.2.1 Gravity Acting Alone............................................125
B.2.2 Wind Acting Alone...............................................130
B.3 MATLAB Input/Output, Rigid Frame Analysis.......................138
B.3.1 Gravity Alone Input/Output Data.................................139
B.3.2 Gravity and Wind Input/Output data..............................145
x
B.4 Rigid Connection Design..........................................151
B.5 MATLAB Input/Output, Semirigid Analysis with Kj(totai).......157
B.5.1 Gravity Alone Input/Output data..................................158
B.5.2 Gravity and Wind Input/Output Data...............................165
B.6 MATLAB Input/Output, Semirigid Analysis with KMi(totai)......172
B.6.1 Gravity Alone Input/Output Data..................................173
B.6.2 Gravity and Wind Input/output Data...............................180
References...............................................................187
XI
FIGURES
Figure
1.1 Structural Tee under Study......................................1
2.1 Comparing Rigid and Semirigid Connections......................4
2.2 A Sample of Semirigid Connections..............................7
2.3 Typical M<> Curves...........................................11
3.1 Initial Stiffness as Slope of Tangent Line to
M<> Curve at Origin.........................................16
3.2 Top Structural Tee as a Frame
under Tension (T).............................................18
3.3 Comparing Initial Stiffness with
Rathbuns M
4.1 A Typical Interior Frame under Study...........................27
4.2 Unbraced Frame under Design Loads..............................28
4.3 Degrees of Freedom and Member
Numbering for Rigid Frame.....................................44
4.4 Member Local Coordinate System with
Sign Convention for Positive..................................45
4.5 Flowchart of MATLAB Tasks for
Rigid Frame Analysis..........................................47
4.6 Degrees of Freedom and Member
Numbering for Semirigid Frame................................62
XII
5.1 Tstub Modeling of Column Flange..............................71
5.2 Bolts Pretensioned...........................................72
5.3 Tee Connection under External Load (T)........................73
5.4 Effective Length of Bolt......................................75
5.5 Column Web under Shear...................................... 77
5.6 Column Web under Compression................................. 79
5.7 Relative Rotation of Beam End
with Respect to Column.......................................83
A.1 Modeling of Top Tee as a Frame...............................102
A.2 Top Structural Tee as a Frame
under Tension (T)........................................... 104
A.3 Flange of Top Structural Tee under Shear.....................111
A.4 Rotation of Beam End Relative to Column......................113
A. 5 Bottom Web Tee under
Bearing and Rotation........................................115
B. 1 Unbraced Frame under Wind Pressure...........................123
B.2 Unbraced Frame under Joint Wind Loads........................124
B.3 Simple Beams & Cantilever Columns for
Gravity Load Analysis.......................................126
B.4 Approximate Moment Diagram Due to Gravity....................129
B.5 Frame Story Midheight Cuts..................................130
B.6 Approximate Moment Diagram Due to Wind.......................137
XIII
TABLES
Table
4.1 Member Moments and Axial Forces Due to Gravity.................30
4.2 Member Moments and Axial Forces Due to Wind....................31
4.3 Combined Effect of Gravity and Wind Loads......................32
4.4 Preliminary Beam Sizes for Gravity Loading.....................34
4.5 Preliminary Column Sizes for Gravity Loading...................38
4.6 Final Preliminary Column Sizes for
Approximate Analysis..........................................42
4.7 Exact Rigid Analysis (Moments and
Axial Forces) Due to Gravity..................................49
4.8 Exact Rigid Analysis (Moments and
Axial Forces) Due to Gravity and Wind.........................50
4.9 Exact Design of Beams for Rigid Frame..........................52
4.10 Exact Design of Columns for Rigid Frame........................59
4.11 Rigid Structural Tee Connection Sizes..........................60
4.12 Initial Stiffness of Designed Tee Connections..................64
4.13 Exact Semirigid Analysis (Moments and
Axial Forces) Due to Gravity with K^otai).....................66
4.14 Exact Semirigid Analysis (Moments and
Axial Forces) Due to Gravity and Wind with Kj(totai)..........67
XIV
4.15 Comparing Connection Initial Stiffness with
Beam Rotational Stiffness............................................68
5.1 Comparison among Kj(totai), KMi(totai), and
Beam Rotational Stiffness............................................89
5.2 Exact Semirigid Analysis with l
(Moments and Axial Forces) for Gravity...............................91
5.3 Exact Semirigid Analysis with Ki^tai)
(Moments and Axial Forces) for Gravity and Wind.....................92
5.4 Exact Beam Design for Semirigid Frame with KMi(totai)...............93
5.5 Exact Column Design for Semirigid Frame with KMi(totai).............94
xv
1. Introduction
The American Institute of Steel Construction (AISC, 1989) recognizes three
types of connections in steel frame construction. They are type 1 rigid, type
2 simple (pin), and type 3 semirigid. The subject of this thesis will be type 3
connections. In particular, a structural tee will be investigated as a semirigid
connection. This chapter provides a definition, names, types, and design
methods for a structural tee, and the purpose and scope of the study.
1.1 Definition
A Structural tee connection is a steel momentresisting connection that is
used to connect beams to columns by such fasteners as rivets or bolts in
steel frame construction (Fig. 1.1). Welding is also possible in combination
with bolts or rivets (McGuire, 1968, p. 964). The noun tee in the term
structural tee comes from the fact that the connection resembles the letter
T in the English alphabet if it is viewed in cross section.
1
1.2 Names
A structural tee connection is also known by three other names. One is split
beam" connection as a beam is spljt in making it (Lothers, 1965, p. 195).
Another is teestub connection (McGuire, 1968, p. 964). The noun stub
comes from the fact that a short length is cut from a steel beam in making it.
The third is "splitbeam tee connection (Salmon and Johnson, 1980, p. 785).
1.3 Types
There are three types of structural tees, each of which is obtained from a
specific steel beam shape. The beam is split along its web at the desired
length. A WT is split from a Wshape, an MT from an Mshape, and an ST
from an Sshape (AISC, part 1, 1989).
1.4 Design Method
A structural tee connection is designed by two methods. In the simplest and
most common method, the tee is designed with web angles such that the tee
transmits moment, and the web angles carry shear. In another method, the
tees are designed without web angles such that the tees transmit both
moment and shear (Lothers, 1965, p. 195).
1.5 Purpose of the Study
A structural tee is often regarded as a rigid connection (Bakers, 1954, p. 116,
Lothers, 1960, p. 141, and Tall, 1974, p. 619) and designed as such (Lothers,
1965, p. 201, Tall, 1974, p. 623). However, a structural tee can be designed
as a semirigid connection (defined in chapter 2), if possible prying action is
checked. They are not economical as rigid connections since they cannot
develop the full moment capacity of the beam (Marcus, 1977, p. 343).
2
Actually, a structural tee is designed in the elastic range as a semirigid
connection for a given design moment (Johnston et al., 1980, p.192 and
Fanella et al., 1992, p. 294 and p. 318). But, neither source specifies how to
determine the design moment for such a semirigid connection.
The present study attempts an answer to that question: how to find the design
moment for a semirigid structural tee. To achieve this, a theoretical approach
is used to derive a linear momentrotation relationship for the connection.
First, the rotational deformation of the connection itself will be determined.
Next, the column deformations believed to be contributing to the deformation
of the connection will be derived and incorporated into the relationship. The
results will then be utilized in computer matrix analysis to find the design
moment of the connection.
1.6 Scope of the Study
This study contains the following.
 Theoretical derivation of an elastic moment rotation relationship for
structural tee connection as semirigid
 Application of the results in the analysis of unbraced portal frames with
semirigid structural tees under gravity and wind loads using computer
matrix analysis.
 Design of an unbraced portal frame with semirigid and rigid structural
tees and comparison of the results.
3
2. Semirigid Connection
This chapter provides a response overview, a definition, the degree of rigidity,
and physical look of a semirigid connection. It also briefly reviews literature
and design practice.
2.1 Response Overview
A beamcolumn semirigid connection subjected to inplane loadings deforms
primarily in rotation (Faella et al., 2000, p. 38). Thus, the loaddeformation
relationship for such a connection would be a relationship between the
applied moment and the rotation of the connection.
Although it is the semirigid connection that rotationally deforms under the
action of inplane loading, in experimental work the deformation within the
connection is measured in terms of the rotation (<)) of the beam end relative to
the column (Maugh, 1964, p. 397). This rotation () will be explained in this
section by comparing the behavior of a rigid connection with that of a semi
rigid connection when both are subjected to inplane loadings (Fig. 2.1).
0
0
9
(a) Rigid Connection
0
(b) Semirigid Connection
Fig. 2.1 Comparing Rigid and Semirigid Connections
4
Fig. 2.1(a) shows schematically a rigid connection before and after loading.
Before loading, the tangents to the undeformed elastic curves of the beam
and column had intersected at a 90degree angle at the joint (Ji). After
loading, the two elastic curves deform in flexure causing the joint (J0, the
beam end, and the column section at (Ji) to rotate the same amount (0), while
the original angle between the two tangents at (Ji) still remains 90 degrees.
Fig. 2.1(b) shows schematically a semirigid connection before and after
loading. Before loading, the situation for this connection would be identical to
that of the rigid connection before loading; nothing is deformed. After loading,
however, the two elastic curves deform in flexure, and the tangent to the
elastic curve at the beam end at the joint (J2) rotates an extra amount () in
addition to the rotation (0) of the joint and column. Another word, the end of
the beam rotates by an amount () in addition to the rotation (0) of the column
(Lothers, 1960, p.370). This is so because of the deformation within the
semirigid connection. Note from Fig. 2.1(b) that the original angle between
the two tangents to the elastic curves no longer remains 90 degrees since the
end of the beam has rotated by an amount (cj) relative to the column. In order
to sum the above response, a definition will be presented in the next section.
2.2 Definition
A semirigid connection is a momentresisting, beamtocolumn connection
that allows the end of the connected beam to rotate some amount relative to
the column. In this definition the relative amount of rotation depends on the
degree of rigidity of such a connection.
2.3 Degree of Rigidity
Degree of rigidity of a semirigid connection is actually a comparison that is
made between the moment such a connection can take and the moment
taken by a rigid connection. The degree of rigidity of a semirigid connection
is between the rigidity of a simple (pinned) connection and a rigid (fixed) one
and varies with the type of semirigid connection. The degree of rigidity of a
simple connection is from 020%, that of a rigid connection is above 90%, and
that of a semirigid connection is from 2090%, with the degree of rigidity
being a percentage of the ratio of the actual moment to the fixedend moment
(McCormac, 1992, p. 406).
5
2.4 Physical Look
A semirigid connection looks like any one of the commonly used beamto
column connections. The only requirement is that its momentrotation
response be considered in the frame analysis and design. Fig.2.2(a) and
Fig.2.2(b) show several types of steel frame connections that can be used as
semirigid.
Some of these beamtocolumn connections are idealized as rigid; others are
idealized as simple (pin), and the frame is designed accordingly. However,
neither of these cases can be materialized in practice; so, they should be
regarded as semirigid (Maxwell, et al., 1981 p. 2.72, Faella,et al., 2000, p.
38).
6
(a) Type A
The webangle connection
(c) Type C
Combination of types A;B
(b) Type B
Top and seatangle connection
(d) Type 0
The neckeddown connection
(f) Rigid connection
(e) Top view
Fig.2.2 A Sample of Semirigid Connections
Source: (Lothers, I960, Fig.803)
7
(a) Web angles with seat
angle and top angle
Fig.2.2 A Sample of Semirigid Connections (cont'd)
Source: (McCormac, 1992, Fig. 14.4)
8
2.5 Literature Review
Research into the response of steel beamtocolumn connections under loads
spans back over 80 years in both the United States and abroad. The research
has been both experimental and analytical. In experimental work, the focus
was mostly on the momentrotation relationship in the form of a plot of M<().
Thus, tested were common types of beamtocolumn connections under
applied moment; measured was the rotational deformation of each connection
type in the form of the rotation of the beam end relative to the column.
In analytical work, the emphasis was to derive a formula that can predict the
linear initial stiffness of a connection. Here, initial stiffness is the initial slope
of the momentrotation curve of the connection (Azizinamini, 1987, p. 73).
It is observed from experimental momentrotation curves that every
connection type tested allowed some rotation of the beam end relative to the
column. Hence, some researchers called them semirigid; others called them
flexible.
It is not clear who first introduced the term semirigid, and who introduced
the term "flexible in the literature. The term semirigid has been used more
often than the term flexible. It is even now in the title of two recent books on
the subject (Chen, 1993 and Faella et al. 2000).
The mere fact that there are two names for the same connection behavior
shows that structural researchers have not reached an agreement on what to
call a connection when not idealized as rigid or simple (pin). Professor Gerstle
of the University of Colorado at Boulder (Gerstle, 2000) argues that he would
call such connections flexible since semirigid means half rigid11, and since
there is a range for connection response that falls between the two extreme
cases of simple and rigid connections. It was noted earlier (Section 2.3) that
the rigidity of a semirigid connection is from 2090%. This statement does
not imply "halfrigid. However, in the memory of the pioneers on the subject
who first studied connection response and used the term semirigid in their
work, this study uses the term semirigid" in its work.
9
As pointed out in section 2.1, the result of an experimental test on a semirigid
connection is represented by a relationship between the moment (M) applied
to the connection and the rotation () of the beam end relative to the column.
This relationship is plotted as a nonlinear M<> curve. Figure 2.3 is redrawn
qualitatively from a more involved plot of M<> curves (Tall, 1974, p. 625) to
suit our purpose of observing the nonlinear nature of these curves with the
related connections. In this redrawn sketch, the beams (originally vertical) are
positioned horizontally, the connection names (lacking in the original) are
added, and an explanation for the axes of the M
original) is provided. Similar experimental M curves, not shown in this study,
are also available for welded connections (McGuire, 1968, p. 893).
10
Fig. 2.3 Typical M<> curves
(Redrawn from Tall, 1974)
11
Note from Fig.2.3 that the horizontal axis represents the idealized case of a
simple (pinned) connection for M = 0, and the vertical axis represents the
idealized case of rigid (fixed) connection for cj = 0. Curve (a) represents a
more rigid connection than curve (b), and each subsequent curve is more
rigid than the other.
The following experimental and theoretical or analytical citations are by no
means exhaustive. The interested reader is referred to Jones et al., 1983,
Chen, 1993, and Faella, et al., 2000 for more information and references on
the subject.
In the United States, in 1917 Wilson and Moore, who were reportedly the first
to have carried out experiments on connections (Jones et al., 1983, p. 2, and
Chen, 1993, p. 233) tested riveted brackets, top and seat angles, and double
web angles for their rigidity in rectangular frames (Wilson and Moore, 1917).
In Canada, in 1928 Young, C. R. and in 1934 Young, C. R. and Jackson, K.
B. reportedly (Rathbun, 1936, p. 525, Jones et al., 1983, p. 3, and Chen,
1993, p. 234) tested riveted and welded connections.
In Great Britain, in 1934 Batho and Rowan tested riveted and bolted double
web angles, top and seat angles with and without doubleweb angles, and
structural tees. They also provided a method of finding the end moments in
the beams from experimental results (Batho and Rowan, 1934, pp. 61137).
In the United States, in 1936 Rathbun conducted a series of tests on different
sizes of doubleweb angles, top and seat angles with and without double
web angles, and structural tees. In 19411942 Hechtman and Johnston,
under the sponsorship of AISC (American Institute of Steel Construction)
tested forty seven riveted connections of different sizes. Connection types
were top and seat angles, top and seat angles with web angles, tees and seat
angle, web clip and seat angles, and tees with and without web angles
(Hechtman and Johnston, 1947).
In the United States, in 1951 Lothers analytically derived an initial elastic
restraint equation for the semirigid connection factor, Z, for doubleweb
angles. He then applied (1/Z) as the initial tangent slope to some of Rathbuns
momentrotation curves (Rathbun, 1936) for similar connections and reported
consistent agreement (Lothers, 1951, p. 490). In 1953 Yu, under Lothers,
derived similar equations for semirigid top and seat angle connections (Yu,
Shan Yuan, 1953). In 1955 Yu, also under Lothers, derived similar equations
for semirigid top and seat angles with web angles (Yu, Wei Wen, 1955). In
12
1958, Huang derived similar equations for the split beam semirigid
connection (Huang, 1958).
In the United States, in 1956 Pray and Jensen tested welded top plate and
seated angle connections to verify their proposed analysis and design of a
twoway beamcolumn system (Pray and Jensen, 1956). In 1965, Douty and
McGuire tested highstrength bolted moment connections, such as Tstub
with web angles and end plates for their response and design in both the
elastic and plastic range (Douty and McGuire, 1965).
In the United States, in 1979 Herzl performed a series of tests on bolted top
and seat angles as well as bolted flange plates for their momentrotation
response and on frames with bolted top and seat angles for their actual
response (Herzl, 1979). In 1986 Stelmack, Marley, and Gerstle conducted
tests on a particular flexible top and seat angle connection to verify analytical
methods of steel frame behavior with such a connection (Stelmack, Marley,
and Gerstle, 1986). Also, in 1987 Azizinamini, Bradburn, and Radziminski
derived an expression for the initial stiffness of bolted semirigid top and
bottom angles with web angles, testing the connection (Azizinamini,
Bradburn, and Radziminski, 1987).
In the United States, in 1993 Chen published analytical derivations of the
initial stiffness of the momentrotation curve for top angles, seat angles, and
double web angles with rivet or bolt fasteners (Chen, 1993, Appendix B).
More recently, a whole book has been published on the subject of semirigid
connections (Faella, et al., 200).
In short, structural researchers have produced experimental data as well as
analytical elastic and inelastic equations to help predict the semirigid
connection response to moment. In both cases, connections were mostly
riveted angle connections. One notices that the majority of the research has
been experimental, which is time consuming and expensive. More analytical
research based on the standard principles of Structural Mechanics should be
pursued to cover the elastic response of connections using bolts as fasteners,
which are now common. Analytical methods have the advantage of being less
time consuming, less expensive, and accessible to each design office. After
all, the design practice is mostly based on linearly elastic (not plastic)
response of steel frames. Such research can help the structural designer to
design for a realistic frame behavior, a practice reportedly lacking in todays
design practice.
13
2.6 Design Practice
The customary design practice in steel frames is to assume that a connection
is either rigid (fixed) or simple (pinned). Based on such assumptions, the
members in the frame are analyzed and designed. There are some concerns
about such a practice. One is that it does not reflect the true behavior of the
frames; it does not consider the rotational deformation of the connections in
the frames. Such a deformation influences the deflection of an unbraced
frame and internal force distribution in its members (Gerstle, 1988, pp. 241
242).
The other concern about such a design practice is that it is not a balanced
design. Lui and Chen in their study of frames with "flexible joints conclude
the following (Lui and Chen, 1986):
It should be noted that, in reality, fully rigid and ideallypinned connections do
not exist. All connections exhibit behaviour somewhere in between these two
extreme cases. The fully rigid and ideallypinned joint idealizations are just
design simplification. Generally speaking, the rigidjoint assumption will lead
to an underestimate of frame drift and overestimate of frame strength,
whereas the pinnedjoint assumption will result in an overdesign of the girder
and underdesign of the columns.
14
3. Response of a Semirigid Structural Tee
This chapter presents a method for the derivation of the linearly elastic M<>
response of semirigid structural tee connections. It then compares the results
with available M<> curves.
3.1 Elastic M<)> Derivation
It was mentioned earlier (Section 2.5) that a semirigid connection M<> curve
was nonlinear. The slope of this curve at any point is the rotational stiffness
of the connection (Gerstle, 1988, p.241). For calculation purposes, various
formulas (polynomial, cubic Bspline, power, and exponential) have been
devised to represent such a curve (Chen, 1993, pp. 236 237). However, the
initial elastic stiffness can represent the linear response of a connection
(Gerstle, 1988, p. 244, and Chen, 1993, p. 235).
By definition (Section 2.2), a semirigid connection allows some rotation (<)>) at
the end of the beam relative to the column. This extra rotation (<>) increases
with the applied moment (M) at the end of the beam and varies inversely with
the stiffness of the semirigid connection (Lothers, 1960, p. 371). The
relationship between the applied moment (M) and the rotation (<>) can be
expressed as
or, alternatively as
M
M (3.1)
_K_
(KH (3.2)
15
K in both (3.1) and (3.2) is the semirigid connection initial stiffness. That is, K
is the slope of the momentrotation (M) curve of the connection at its origin.
See Fig.3. 1.
Fig.3.1 Initial Stiffness as Slope of
Tangent Line to M Curve at Origin
The relationship in (3.2) will be used in this study, and K will be called the
total initial stiffness (.Ki(lolal)) of the semirigid structural tee connection.
16
3.2 Total Initial Stiffness Ki(totai)
One semirigid structural tee connection consists of two tees. One tee would
be installed on the top flange of the beam and thus called top tee; the other
would be installed on the bottom flange of the beam and thus called bottom
tee. The task is to derive the following:
1. Initial stiffness for the top tee
2. Initial stiffness for the bottom tee
3. Initial stiffness for the whole connection
The method of analysis used was that a oneinch strip of the top tee was
taken out and treated as an indeterminate frame for the determination of its
initial stiffness. A oneinch strip of the bottom tee was taken out and treated
as a cantilever beam for the determination of its initial stiffness. The slope
deflection method and Castiglianos theorem on deflection were used for
analysis. The initial stiffness for the whole connection would then be the sum
of stiffness of top and bottom tees. In this section the steps of the derivations
will be presented. The derivation details along with assumptions and notation
used are provided in Appendix A. See Fig.3.2 to help visualization.
Step 1. Write slopedeflection equations for the three members of the frame
(members AB, BC, and BD).
Step 2. Eliminate the terms that are zero in the equations of Step 1. Also,
using the sign convention in Section A.3, set Aab = Abc = Ab
Step 3. Apply the moment equilibrium equation to joint B of the frame.
Step 4. Substitute for the moments in the equation of Step 3 using the slope
deflection equations of Step 1. (The rotation 0 of joint B is found to be zero.)
Step 5. Write one shear equilibrium equation for free body diagram (FBD) of
the frame cut near the supports A, B, and D.
Step 6. Write two moment equilibrium equations for the FBD of members AB
and BC of the frame by taking moments at B on each member.
17
A 12, b
(a) (b)
Fig 3.2. Top Structural Tee as a Frame under Tension (T)
18
Step
Step
Step
Step
Step
Step
Step
7. Solve for the shear forces in the equation of Step 6.
8. Substitute the shear forces of Step 7 into the equation of Step 5.
9. Solve the equation of Step 8 for the horizontal deflection (Ab).
10. Write Castiglianos theorem for shear deflection of AB and BC.
11. Solve the equation of Step 10 for the horizontal deflection (5b).
12. Add the deflections of Step 9 and Step 11.
13. Write <(> =
(stepll)
d
Step 14. Write M = [ d ] T.
Step 15. Substitute T of the equation of Step 12 into that of Step 14 to find
M(top) = [ Kj(top)] <]).
Step 16. Find the deflection [At,0t(web)] of the free end of the bottom tee from
the beam deflection formula for the cantilever case.
Wfog L bol(web)
* hot (web)
8Â£7
Step 17. Write <> =
^bol(weh)
^bo/( web)
Step 18. Take the moment of the uniform bearing reaction force of the beam
acting on the bottom tee web about the center of rotation of the connection.
Step 19. Substitute Wbrg of Step 16 into Step 18 and Abot(web) of Step 17 into
Step 18 to find
M(bot) [ Ki(bot) ] (j*
19
Step 20. Find the total initial stiffness of the connection by forming
Ki(total) = Kj(top) + Kj(bot)
These 20 steps were carried out to derive the initial stiffness for the top tee,
the initial stiffness for the bottom tee, the total initial stiffness for the
connection, and the connection moment resistance. Below are the results.
1. Top initial stiffness:
K
'(lop)
d2
L\ 3A
24 El 5GA
(A.28)
2. Bottom initial stiffness:
K
i(bot)
bot(web)
^bo/(web)
(A.33)
3. Connection total initial stiffness:
^i(lolal) ^i(lop) ^i(bol)
(A. 34)
4EI
W  3Z,
24 El 5GA
+
bo/(web)
r
hot (web)
(A. 35)
20
4. Finally, connection moment resistance:
M =
d bot(weh)
L31 + Lbol(web)
24 El 5 GA
W
(A. 36)
21
3.3 Comparison with Test Data
The connection initial stiffness, Equation (A.35), will be tested using data from
Rathbuns experimental M curve for specimen 16 obtained for a similar
connection (Rathbun, 1936, p. 528). To do this, the connection moment
resistance, Equation (A.36), will be plotted on this curve. If the straight line
representing Equation (A.36) becomes tangent the M<() curve at its origin,
Equation (A.35) is considered a good prediction of the initial stiffness of the
connection. Others have applied such a method in the past to compare their
initial stiffness equations for other types of connections with test results
(Lothers, 1951, pp. 489490, Lothers, 1960, pp.394395, Yu, 1953, pp. 1319,
Yu, 1955, pp.1316). Yuang, 1958 also applied the same method to test the
initial stiffness of the split beam connection (Yuang, 1958, pp.910).
Rathbuns M Curve
Rathbun's specimen 16 shows a vertical plate simulating a column, and two
beams, one on each side, are abutted to the plate with rivets. It should be
noted that since this specimen does not have a column, it is not considered
as a beamcolumn arrangement that is typical in actual building frames
(Chen, 1993, p. 252). However, since the present study did not find an
experimental M curve for a tee connection using beams and columns
whose dimensions could be found in the AISC manual, the connection total
initial stiffness, Equation (A.35), will be computed with the data for Rathbuns
specimen 16, reprinted in Fig.3.3.
Kl {total) =
AEI,
+ 
3 L
A
. 24Â£7 5 GA
+ 
hot (web)
r
~'bot(web)
(A.35)
22
From the data for the specimen,
d 22.000 in.
U 2.250 in.
A 18.060 in.
I 2.182 in.
lbot(web) 0.852 in.
E 29.0 x 106 psi.
G 11.2 x106 psi.
Lbot(web) 17.398 in.
Substitution of these data into Equation (A.35) yields
K
i(fotal)
__________________(22f___________________
(2.25)3 3(2.25)
24(29 x 106 )(2.182) 5(11.2 x 106 )(18.06)
4(29 xl06)(0.852)
(17.398)
rad
Thus, the connection moment resistance, Equation (A.36) becomes
M = (3.4xl0')(^).
For <(> = 0.001 radians (small deformation theory),
M = (3.4xl0')(0.001)
= 34.0xl06 lbin.
23
The plot of the point whose ordered pair is (0.001, 34 x 106) on the Rathbuns
specimen 16 curve (Fig.3.3) falls beyond the scale on the moment axis.
However, if the scale on the moment axis is extended, the point plotted, and
the point connected to the origin (0, 0) with a straight line, the line will be
tangent to the curve at its origin.
The slope of this line is Kj(t0tai) = 3.4 x 1010 lbin. per radians. This slope is
about twice as large as the slope (initial stiffness) value of 1.6 x 1010 lbin. per
radians that has been reported for the Rathbuns specimen 16 (Yuang, 1958,
p. 10). Considering the reported slope value, one may obtain the connection
moment resistance of this specimen as (1.6 x 101)(0.001) = 16x10 lbin.,
which is about half as large as M = 34.0 x 10 lbin. predicted above.
Yuang, in his Masters thesis study at the Oklahoma State University,
reported the abovementioned value for comparison with the initial stiffness
value of 10.8 x 101 lbin. per radians his thesis had predicted for the split
beam connection (Yuang, 1958, p.10). His predicted value is larger than the
value he reported for Rathbun's specimen 16. Yuang obtained the predicted
initial stiffness value by deriving an elastic restraint equation (Z) based on
flexural deformation of Tstub connections whose reciprocal (1/Z) represents
the initial stiffness of the connection, concluding ... the elastic deformation of
rivets and other factors... needed to be accounted for (Yuang, 1958, p. 40).
Aspects such as these are investigated in Chapter 5.
24
Ptan lSxlxJ'3j*
4'llf
; 16:
4Q3^ro;
Specimen 16. Series r jv j't"
2? in 5 Bem 101 lb 4 lonj J ] MM
*oT j 4.30 r. GBcams f? 240 lb T 3" 'ong
it
ijr
"'.I vO n _
Jo
i * : r L*,
10 0*
niF~~r
d
3 JIIM^
Fiq.3.3 Comparing Initial Stiffness with Rathbuns M) curve
Source: (Rathbun, 1936)
25
4. Design Example
This chapter presents load computations, analysis, and design of a portal
frame for two cases of rigid and semirigid construction to illustrate the
application of the stiffness derivation developed in Chapter 3. The results of
these cases will then be compared.
For an assumed typical 3story, steel frame office building, a typical interior
onebay frame is selected for analysis and design. The building uses planar
portal frames spaced at 30 feet in both directions and is 41 feet high. This
frame acts as the main lateralforce resisting system for the building by
means of momentresisting structural tee connections. It is unbraced in the
short direction of the building, but braced diagonally in the long direction such
that the column ends are free to rotate about the weak axis and not free to
translate perpendicular to that axis. See Fig.4.1.
The building is assumed to be located in downtown Denver, Colorado. The
applicable Uniform Building Code (UBC) can then be used.
26
/~rv s~r~,
X*
30
Sec. 11
71^
Sec. 22
13
13
15
Fig.4.1 A Typical Interior Frame under Study
27
4.1 Computation of Loads
The selected interior frame (Fig.4.1, Sec.11) is subjected to gravity and wind
loads. Both loading conditions are developed in Appendix B. This section
shows the results, which are the design loads, acting on the frame (Fig.4.2).
1.8 k/ft
yf
13
13
15
Fig.4.2 Unbraced Frame under Design Loads
28
4.2 Approximate Rigid Frame Analysis
Since the frame is statically indeterminate, an approximate analysis is to be
carried out to determine its preliminary member sizes for a subsequent exact
analysis. Two load cases are considered: gravity and wind. Both approximate
analyses, along with their moment diagrams, are provided in Appendix B.
This section shows the results in Tables 4.1 and 4.2. The combined effect of
gravity and wind is as shown in Table 4.3 for maximum moments and axial
forces.
29
4
8
3
2
7
6
1 5
Table 4.1 Member Moments and Axial Forces Due to Gravity
Member Max. M (ftk) Min. M (ftk) Axial N (k)
12 37.3 (cw) 18.7 (cw) 119.30 (c)
23 43.7 (cw) 40.5 (cw) 74.63 (c)
34 72.9 (cw) 40.5 (cw) 38.75 (c)
56 37.3 (ccw) 18.7 (ccw) 119.30 (c)
67 43.7 (ccw) 40.5 (ccw) 74.63 (c)
78 72.9 (ccw) 40.5 (ccw) 38.75 (c)
26* 144.0 81.0 neglected
37* 144.0 81.0 neglected
48* 129.6 72.9 neglected
CW Clockwise
CCW Counterclockwise
C Compression
(*) Beam
30
4
8
3
2
7
6
1 5
Table 4.2 Member Moments and Axial Forces Due to Wind
Member Top & Bottom M (ftk) Axial N (k)
12 81.45 (ccw) 13.55 (T)
23 44.85 (ccw) 5.13 (T)
34 16.05 (ccw) 1.07 (T)
56 81.45 (ccw) 13.55 (c)
67 44.85 (ccw) 5.13 (c)
78 16.05 (ccw) 1.07 (c)
26* 126.3 (ccw) neglected
37* 60.90 (ccw) neglected
48* 16.05 (ccw) neglected
CCW Counterclockwise
T Tension
C Compression
(*) Beam
31
Table 4.3 Combined Effect of Gravity and Wind Loads
Beam Max. M (ftk)
48 129.60 + 0.0 = 129.60
37 144.00 +0.0 = 144.00
26 81.0 + 126.30 = 207.30
Column Max. end M (ftk) Min. end M (ftk) Axial N (k)
78 72.9+6.05=88.95 40.5+16.05=56.55 38.75+1.07=39.82
67 43.7+44.85=88.55 40.5+44.85=85.35 74.63+5.13=79.76
56 37.3+81.45=118.75 18.7+81.45=100.15 119.33+13.55=132.88
32
4.3 Preliminary Design of Rigid Frame
The allowable stress design method (AISC, 1989) is used to select
preliminary member sizes for beams and columns. The member shapes are
rolled W shapes.
When wind acts with gravity, the allowable stresses can be increased by 1/3
(AISC, 1989, A5.2). Alternatively, this provision can be observed by reducing
the combined gravity and wind loads to 75% of their total value. The latter is
used in this study.
Beams and columns are designed for two loading cases: gravity alone and
combined gravity and wind. Controllingdesign sections are then selected.
4.3.1 Design for Gravity Alone
Beams. For gravity loading the horizontal reactions at the base of columns
are small; thus, the axial forces in beams are small and assumed zero (Wang,
1983, p.507). Therefore, the beams in the frame are designed for bending
moment only. The beams are assumed to have their compression flange
braced at every 6 feet against lateral displacement (lateral buckling).
33
Beam 48 (roof)
Assume the section is compact.
Fb = 0.66 Fy = 0.66 (36) = 24 ksi
M = 129.6 ftk
Sx = M / Fb = l (129.6) (12)/ (24)] = 64.80 in3
Try W16x40(Sx= 64.7, Mr = 128 ftk). Say o.k. for preliminary selection.
Fy > 65 ksi > Fy = 36 ksi, flange is compact
For fa = 0 (axial force neglected),
d 16.01 640
== 52.5 < j= = 106.7, web is compact
tw 0.305 V36
Lc = 7.4 ft > 6 ft, bracing is adequate
Use W16 x 40 for roof beam.
Using the same beamselection procedure, the floor beams are selected. The
beam sizes are summarized in Table 4.4.
Table 4.4 Preliminary Beam Sizes for Gravity Loading
Beam Size Mp(ftk) lx (in4) Sx (in3) A (in2)
48 W 16x40 128 518 64.7 11.8
37 W 21 x44 162 843 81.6 13.0
26 W 21 x 44 162 843 81.6 13.0
34
Columns. The frame does not experience a sidesway because of symmetry
in gravity loading and structure.
Column 34 (third story)
Assume KyLy =1x13 =13ftto control and check later.
Assume Cm = 0.85
m = [(2.3 + 2.2)/ (2)] = 2.25
Peff = Po + Mx m
= 38.75 + 72.90 x 2.25 = 202.8 k
Try W 12x45 (Pa = 202k).
Subsequent approximation
m =[(2.1+2)/(2)] = 2.05
Peff = 38.75 + 72.9 x 2.05 = 188 k
Try W12 x 45 ( Pa = 202 k).
lx
Sx
rx/r
Lc
Lu
Fex(KxLx)2/(10)2 =
13.20
350.00
58.10
2.65
5.15
1.94
8.50
17.70
275.00
in
in4
in3
in
in
ft
ft
k
Check W 12 x 45 for possible buckling about Xaxis. Use the alignment chart
(AISC, 1989, Fig. 1, p.3.5) for this purpose.
G =
(AISC, 1989, p.3.5)
35
Top of column
350
G*=r'56
30
Bottom of column
Assume the secondstory column to have approximately the same size as the
one in the third story.
Gb =
350 350
13 + 13
843
30
1.92
Use Kx = 0.84
If Xaxis buckling governs,
> hhL or KXLX >KrLr 
rx ry ry
KXLX = 0.84 x 13 = 10.92
KyLy^~ 1 x 13x2.65 = 34.5
Thus, YY axis governs.
Check W 12 x 45 for adequacy regarding interaction equations (AISC, 1989,
Chapter H, p. 5.54).
38.75
13.2
= 2.94 ksi
KXLX 0.84x13x12
rx ~ 5.15
36
KrLy = 1x13x12 = 804. say 80.5.
r, 1.94
^ = 15^4 + 1^36 =15 3 ksj.
Z94 _Q 192 > 0.15
Fa 15.3
Use equations H1.1 and H1.2 (ASD of AISC, 1989).
Since no transverse loads act on the column, the maximum moment at the
ends of the unbraced length is used for ASD equation H1 (McCormac, 1992,
p.263).
f =M= 7290x12 = 15.06 ksi
Jbx S. 58.1
Since Lc = 8.5 ft < 13 ft < Lu = 17.7 ft,
Fbx = 0.66 Fy = 0.66 x 36 = 24 ksi
column ends
Fbx = 0.60 F = 0.60 x 36 = 22 ksi @ column middepth
= 0.6 0.4 (M1 / M2) = 0.6 0.4 (+ 40.5 / 72.9) = 0.4 (AISC, Chapter H)
F'
(275)(10)2
(0.84 xl3)2
= 230.6 k
0.4
1
fa
1
2.94
= 0.41 < 1
Fex. 230.6
Use 1.0 (McCormac, 1992, p. 263).
A +___________=2:94 +1x15.06 = a87?<1 Q k (AISC, 1989, EQ. H11)
1
fa
15.3
22
bx
ex' J
37
(AISC, H12)
i+lM = 0.76,
0.6 F, F 22 24
o.k.
Use W 12 x 45 for column 34 (third story).
Using the same columnselection procedure, the remaining story columns are
selected. The column sizes are summarized in Table 4.5.
Table 4.5 Preliminary Column Sizes for Gravity Loading
Column Size Pa(k) lx(in4) Sx(in3) A(in2>
34 W 12x45 202 350 58.1 13.2
23 W 12x45 202 350 58.1 13.2
12 W 12x50 206 394 64.7 14.7
Design conditions are the same for columns 78, 67, and 56. Thus, these
columns will have the same sizes as the opposite columns.
38
4.3.2 Design for Combined Gravity and Wind
Beam. The 1/3 increase in allowable stress due to the combined effect of
gravity and wind resulted in smaller beam sizes than required by gravity dead
and live loads alone, except for the second floor beam where both the gravity
and combined effect furnished the same size beam. Therefore, the gravity
loads govern the beam design, and the beam sizes selected in Table 4.4 are
considered the final preliminary beam sizes for approximate design.
Columns. The columns are designed for the maximum end moments and the
axial forces shown in Table 4.3. The frame under the combined loading of
gravity and wind is free to sideway in the plane of loading, but it is diagonally
braced in the perpendicular plane.
Column 78 (third story)
Assume KyLy =1x13 = 13ftto control and check later.
Assume Cm = 0.85
m = [(2.3 + 2.2) / (2)] = 2.25
Peff = Po + Mx m
= [39.82 + 88.95 x 2.25] (0.75) = 179.97 k
Try W 12x40 (Pa= 180 k).
Subsequent approximation
m = [ (2.1 + 2) / (2)] = 2.05
Peff = [39.82 + 88.95 x 2.05] (0.75) = 166.63 k
Try W 12x40 (Pa =180 k).
A 11.80 in2
lx 310.00 in4
Sx 51.90 in3
rx/ry 2.66 in
rx 5.13 in
ry 1.93 in
Lc 8.40 ft
Lu 16.00 ft
39
Fex (KXLX)2 / (10)2 = 273 k
Check W12 x 40 for possible buckling about Xaxis. Use the alignment chart
(AISC, 1989, Fig. 1 p. 3.5) for this purpose.
Top of column
310
g*=M=1'38
30
Bottom of column
Assume the column in the second story to have approximately the same size
as the one in the third story.
Gb 
2x310
13
843
30
1.70
Use Kx = 1.47 (sidesway uninhibited)
If Xaxis buckling governs,
KXLX
KVL
, or KXLX > KrLy
KXLX =1.47x13 = 19.11/
40
KYLY = 1 x 13 x 2.66 = 34.58ft
ry
Thus, YY axis governs.
Check W 12 x 40 for adequacy regarding interaction equations (AISC, 1989,
Chapter H, p.5.54).
39.42
11.8
= 3.34 ksi
KyLy _1x13x12
r, 1.93
Fa = 15.24 ksi (AISC, 1989, Table C36, p.3.16)
A
Fa
3.34
15.24
= 0.219 >0.15
Use ASD equations H1.1 and H1.2.
fbx ~
82.95x12
51.90
19.18 ksi
Since Lc = 8.40 ft < Lb = 13 ft < Lu = 16 ft,
Fbx = 0.66 Fy = 0.66 x 36 =24 Ksi @ column ends
Fbx = 0.60 Fy = 0.60 x 36 =22 ksi @ column middepth
Cmx = 0.85 (AISC, 1989, Chapter H, p.5.55)
F'
ex
(273Y10)2 . .
= 74.76 ksi
(1.47xl3)2
0.85
1
fa
F\
3.34
74.76
= 0.89 < 1
1
41
Use 1.0 (McCormac, 1992, p.263).
Cnlx fbx
1
fa_
F'
bx
cx J
3 34 1x19 18
~~~1~ =1.09 <1.33 o.k. (AISC, 1989, EQ.H1.1)
15.24 22
f f 1 14 1Q 18
^^ + =  +  = 0.951 <1.33 o.k. (AISC, 1989, EQ. H1.2)
0.6F, Fhx 22 24
The factor 1.33 is due to the consideration of wind loads. W 12 x 40 is
adequate for the combined effect of gravity and wind, but it is smaller than
required by gravity alone. Thus, gravity governs.
Use W 12 x 45 for column 78 (third story).
Using the same columnselection procedure, the remaining design process
for the combined effect of gravity and wind indicates that both gravity and the
combined effect produced similar column selections for the secondstory
column. However, the combined effect controlled the column design for the
firststory column. Table 4.6 summarizes the final preliminary column sizes for
the approximate analysis of the rigid frame under study.
Table 4.6 Final Preliminary Column Sizes for Approximate Analysis
Column Size Pa (k) lx (in4) s* (in3) A (in2)
78 W 12x45 202 350 58.1 13.2
67 W 12x45 202 350 58.1 13.2
56 W 12x58 276 475 78.0 17.0
It should be noted that the columns in each story are of the same size.
42
44 Exact Rigid Frame Analysis
The final preliminary member sizes for the approximate analysis summarized
in Tables 4.4 and 4.6 are used as input for an exact analysis of the same rigid
frame. The exact analysis uses a MATLAB computer program developed for
plane structures by Professor John Mays of the University of Colorado at
Denver as part of a course in matrix structural analysis.
The program matrix formulation for analysis uses the stiffness method. The
stiffness method treats an entire structure, such as the frame under study, as
an assembly of members connected together at their end points. These
points are called nodal points, whose independent displacements are primary
unknowns. The formulation is built on three requirements of compatibility,
material law, and equilibrium essential for the analysis of any indeterminate
structure.
The program task is to first compute these primary displacements. Then, the
secondary unknowns, memberend forces (axial, shear and moment), are
computed using the forcedeformation relationship for each member. To do
this task, certain handprepared input data are needed (Section 4.4.1). In
Section 4.4.2, the MATLAB program main logical tasks are outlined. Section
4.4.3 presents the results of the exact analysis.
4.4.1 HandPrepared Input Data
For the analysis of the frame under study with MATLAB, the following input
data preparations are in order. Use units of kips and feet.
1. Determine in the global coordinate system the number of independent
displacements at each nodal point (points whose displacements are not
restrained by a support). Call these displacements alphatype degrees of
freedom. Number alphatype degrees of freedom starting from 1 for
translation in global Xdirection, followed by 2 in global Ydirection and by 3
for rotation about global Zdirection. Use righthand rule for this purpose. For
this study, alphatype degrees of freedom are 18. See Fig.4.3 (a).
43
411
 1 0
12
/N 14
> 13
15
/N
17
16
23
22
24
(a) Alphaand Betadegrees
of Freedom
(b) Connectivity and member
numbers
Fig.4.3 Degrees of Freedom and Member
Numbering for Rigid Frame
44
2. Determine in the global coordinate system the number of independent
displacements restrained at each support. Call these displacements betatype
degrees of freedom. Number betatype degrees of freedom starting from
(alpha+1) in X, Y, and Z directions. For this study, starting from 19, betatype
degrees of freedom are 6. See Fig.4.3(a).
3. Form the sum of alphatype and betatype degrees of freedom and call the
sum n. That is, n = alpha + beta. The number n represents the number of
simultaneous linear equations to be solved by the MATLAB computer
program for the nodal displacements. For this study, n = 18 + 6 = 24.
4. Form a row matrix of member connectivity, which is called Im matrix in the
MATLAB computer program for this study. The member connectivity flags the
program which nodal point members are connected to. The program uses this
information to assemble the structural stiffness matrix. For this study, the
member connectivity is designated as i and j. See Fig.4.3 (b).
4. Number the members sequentially. See Fig. 4.3(b).
5. Determine L, E, and section properties I, A for the members.
6. Convert in the member local coordinate system and conforming to local
degrees of freedom any concentrated or distributed load between nodal
points on the frame to concentrated loads at those points at the end of each
member. See Fig.4.4 for member local coordinate system with the sign
convention for memberend degrees of freedom (righthand rule). Use the
standard formulas for fixedend force calculations for beams, whether by
hand or by a computer program written for such a conversion and place them
in a column matrix. In this study, the conversion is done by hand.
V7I 3
\~7I 6
Fig. 4.4 Member Local Coordinate System
with Sign Convention for Positive
45
7. Find the angle of inclination (theta) of the positive local xaxis of each
member from positive global Xaxis. This angle, measured in radians, is
positive, if its direction is counterclockwise.
4.4.2 MATLAB Program Tasks
Once the MATLAB computer program receives the input data, it carries out
the frame analysis. The main logical tasks in the analysis procedure are as
outlined in the flowchart of Fig.4.5.
46
Fig.4.5 Flowchart of MATLAB Tasks
for Rigid Frame Analysis
47
4.4.3 Results of Exact Rigid Frame Analysis
The exact analysis is performed for two load cases of gravity alone and
combined gravity and wind. The computer input / output data used and
produced by the MATLAB program for both loading cases are provided in
Section B.3 of Appendix B. In this section, the results of the analyses, which
are the moments and axial forces, are presented in Tables 4.7 and 4.8.
48
Table 4.7 Exact Rigid Analysis (Moments and Axial Forces)
Due to Gravity
Member Max. M (ft.k) Min. M (ft.k) Axial N (k)
12 58.5 (cw) 29.4 (cw) 87.0 (c)
23 65.7 (cw) 57.1 (cw) 57.0 (c)
34 105.7 (cw) 76.0 (cw) 27.0 (c)
56 58.5 (ccw) 29.4 (ccw) 87.0 (c)
67 65.7 (ccw) 57.1 (ccw) 57.0 (c)
78 105.7 (ccw) 76.0 (ccw) 27.0 (c)
26* 124.2 100.8 neglected
37* 133.0 92.0 neglected
48* 105.7 96.8 neglected
CW Clockwise
CCW Counterclockwise
C Compression
(*) Beam
49
Table 4.8 Exact Rigid Analysis (Moments and Axial Forces)
Due to Gravity and Wind
Member Max. M (ftk) Min. M (ftk) Axial N (k)
12 71.57 (ccw) 4.03 (ccw) 74.7 (c)
23 27.05 (cw) 6.11 (cw) 51.5 (c)
34 85.10 (cw) 64.7 (cw) 25.6 (c)
56 129.9 (ccw) 120.6 (ccw) 99.3 (c)
67 108.05 (ccw) 104.51 (ccw) 62.5 (c)
78 126.31 (ccw) 87.54 (ccw) 28.37 (c)
26* 225.24 22.97 neglected
37* 195.6 70.8 neglected
48* 126.3 85.1 neglected
CCW Counterclockwise
CW Clockwise
C Compression
n Beam
50
4.5 Exact Design of Rigid Frame
With reference to Tables 4.7 and 4.8, the controlling loading case is used to
design beams and columns of the rigid frame, followed by rigid connection
design.
4.5.1 Beams
Beam 48. The case of gravity alone is larger than that of gravity and wind, for
which 75% of its value is allowed by codes (or a 1/3 increase in allowable
stress).
Assume a compact section and a 6ft lateral bracing interval for the
compression flange of the beam.
Fb = 0.66 Fy = 0.66 x 36 = 24 ksi
M = 105.7//k> 75%(126.3) = 94.7ftk
s.=M =
1057xl2 = 52.85 in3
24
Try W 18x35 (Sx = 57.6, Mr =114 ftk).
Fy =>65 ksi >Fy = 36 ksi, For fa = 0 (axial force neglected), flange compact
d = 17'7 = 59 <6^=6= 106.7, K 03 JFy V36 web compact
Lc = 6.3 ft, bracing is adequate
Use W 18 x 35 for roof beam.
51
Beam 37. The case of combined gravity and wind is larger than that of
gravity alone.
M = 75%(195.6) = 146.7/rl > 133//.A:
Using the same beamselection procedure, the size of beam 37 is selected.
Use W 21 x 44 for thirdfloor beam.
Beam 26. The case of combined gravity and wind is larger than that of
gravity alone.
M = 75%(225.24) = 168.93ft.k > 124.20ft.k
Using the same beamselection procedure, the size of beam 26 is selected.
Use W 18 x 50 for secondfloor beam.
The results of exact design of beams are shown in Table 4.9.
Table 4.9 Exact Design of Beams for Rigid Frame
Beam Size Mr (ftk) lx (in4) Sx (in3) A (in2)
48 W 18x35 114 510 57.6 10.3
37 W 21 x 44 162 843 81.6 13
26 W 18x50 176 800 88.9 14.7
52
4.5.2 Columns
Column 78. The case of gravity alone is larger than that of combined gravity
and wind, which is reduced to 75% of its total value.
Assume sidesway for the frame even though gravity is going to control the
design, since as will be seen shortly, for the other columns in the frame the
case of gravity and wind is going to control the design.
Assume FCL = 1 x 13 ft to control and check later.
Cm = 0.85
m
m = 2.25 (AISC, 1989, Table B, p. 3.10)
PeJf =P0 +Mxm
= [27 + 105.7(2.25)] = 264M > [28.37 +126.31(2.25)](75%) = 234.4k
Try W 12x53 (Pa = 268k)
Subsequent approximation
m = 2.05
PeM = [27 + 105.7(2.05)] = 243.7A:
Try W 12x53
10.60 ft
22.00 ft
15.60 in2
425.00 in4
70.60 in3
5.23 in
2.11
.flgg2 / (10)2 = 284 k
X
Sx
rx
rx/r.
F
Check W 12 x 53 for possible buckling about Xaxis. Use the alignment chart
(AISC, 1989, Fig1, p.3.5) for this purpose.
53
G =
z
Z
^4
Top of column
425
gM=1'92
30
Bottom of column
Assume the secondstory column to have approximately the same size as the
one in the third story.
<4 =
425 425
JJ3___13.
843
30
2.33
^=1.6 (sidesway uninhibited)
If Xaxis buckling controls,
KXLX
KL
y y
,orKxLx >KyL
KXLX =1.6x13 = 20.8/r
KVL^ = lx 13x2.11 = 27.43
y r
y
Thus, YY axis governs.
Check W 12 x 53 for adequacy regarding interaction equations (AISC, 1989,
Chapter H, p. 5.54).
54
27
15.6
= 1.73 ksi
KL 1.6x13x12
J J == 47.72
rx 5.23
KyL 1x13x12
== 63
r, 2.48
Fa =17.14 ksi (ASIC, 1989, Table C.36, p. 3.16).
^ = = 0.10 < 0.15
Fa 17.14
Use equation H1.3 (ASD of AISC, 1989).
fbx~
f.
Mr 105.7x12
+
S,
fbx
K
70.6
= 17.97 ksi
1 73 17 97
 + ^ = 0.10 + 0.75 = 0.85 <1 o.k.
17.14 24
(ASIC,1989, EQ. H1.3)
Use W 12 x 53 for thirdstory columns.
Column 67. The case of combined gravity and wind is larger than that of
gravity alone. The frame is free to have a sidesway.
Assume KyLy = 1 x 13 ft to control and check later.
Cm = 0.85
m =2.25
PeIf = [62.5 +108.05(2.25)](75%) = 22921k > [57 + 65.7(2.25)] = 204.83*
Try W 12x53
Subsequent approximation
m = 2.05
Pelf = [62.5 +108.05(2.05)](75%) = 2Uk > [57 + 65.7(2.05)] = 191.7*
This axial load results in W 12 x 50. However, to have one less column splice,
try W 12 x 53 to run the height of the second and third stories. The top of the
firststory column will be spliced to the bottom of the secondstory column at
the appropriate height above the second floor. The design of column splice is
beyond the scope of this study; thus, it is not included.
55
Just as the thirdstory column, this column satisfies all the applicable code
requirements of AISC, 1989.
Use W 12 x 53 for secondstory columns.
Column 56. The case of combined gravity and wind is larger than that of
gravity alone. The frame is free to have a sidesway.
Assume KL =1x15=15ftto control and check later.
Cm =0.85
m
m =2.2
PeJf = [99.3 +129.9(2.2)](75%) = 288.81* > [87 + 58.5(2.2)] = 215.7*
TryW 12x58
Subsequent approximation
m =2
Peff = [99.3 +129.9(2)1(75%) = 269.3* > [87 + 58.5(2)] = 204*
TryW 12x58 (Pa = 276 k)
A 17.00 in2
lx 475.00 in4
sx 78.00 in3
rjr 2.10
rx 5.28 in
Fy 2.51 in
LVC 10.60 ft
K 24.40 ft
F'eAK,Lx) _2g9A
(10)2
Check W 12 x 58 for possible buckling about Xaxis. Use the alignment chart
(AISC, 1989, fig.1, p. 3.5) for this purpose.
56
G =
Top of column
425 475
G 13 + 15 _
" 800
30
Bottom of column
2.41
Gb 1.
(AISC, 1989, p.3.5)
Use Kx = 1.5 (sidesway uninhibited)
If Xaxis buckling governs,
KL KVL
y y
,or KxLx>KyLy^
K.L =1.5x15 = 22.5 ft
Â£ Z, = 1x15x2.10 = 31.5 ft
y y r
y
Thus, YY axis governs.
Check W 12 x 58 for adequacy regarding interaction equations (AISC, 1989,
Chapter H, p. 5.54).
99 3
= 5.84 ksi
17
KXLX 1.5x15x12 c
== 5
rx 5.28
K>L, Ixl5xl2?1 yl
r, 2.51
Fa = 16.25 ksi
57
L
F,
5 84
= _Â£^Z_ o.36 > 0.15
16.25
Use ASD equations H1.1 and H1.2.
fbx
129.9x12
S.
78
= 19.98 ksi
Since Lc = 10.6 ft < Lb = 15 ft < Lu = 24.4 ft,
Fbx = 0.60 Fy =0.60 x 36 = 22 ksi @ column middepth
Fbx = 0.66 Fy =0.66 x 36 = 24 ksi (5} column ends
C = 0.85
m
F' = 289(1Q)2 = 57 ksi
(AISC, 1989, Chapter H, p. 5.55)
(1.5x15)
0.85
1
A
F'
1
5.84
57
= 0.95 < 1
Usel.O (McCormac, 1992, p. 263).
Ctnxfbx
(1
Wbx
_5.84_ + 1x19.98 =1 267<1 33 o k (ASC, 1g89i EQ H1.1)
16.25 22
Jjl + L* = 5^4 + 19i98 =1 098<1 33 o.k. (AISC,1989, EQ. H1.2)
0.6^ Fbx 22 24
Use W 12 x 58 for firststory columns.
58
The results of exact design of columns are shown in Table 4.10
Table 4.10 Exact Design of Columns for Rigid Frame
Column Size Pa (k) lx (in4) Sx (in3) A (in2)
78 W 12x53 268 425 70.60 15.60
67 W 12x53 268 425 70.60 15.60
56 W 12x58 276 475 78.00 17.00
It should be noted that the columns in each story are of the same size.
59
4.5.3 Rigid Connection
To determine the initial stiffness of structural tee connections to be used as
input in the MATLAB computer program for semirigid frame analysis, the
connections must be designed first. Each connection is designed for the
maximum moment at the end of its respective beam. The maximum moment
results from the controlling loading case used to design the beam itself. The
connections are WTshape.
The results of the rigid connection design are summarized in Table 4.11. The
design details are provided in Appendix B.
Table 4.11 Rigid Structural Tee Connection Sizes
Connection Location Beam
WT 13.5x64.5 Roof level W 18x35
WT 13.5x64.5 Third floor W 21 x 44
WT 15x74 Second floor W 18x50
60
4.6 Exact Semirigid Frame Analysis with Kj(totai)
To analyze the same steel frame when semirigid structural tee connections
are used between the beams and columns, each connection is modeled as a
linear rotational spring, as shown in Fig.4.5. Each spring has two rotational
degrees of freedom at its ends and thus will have a stiffness matrix of 2 x 2.
Each spring stiffness matrix contains four values, each of which is the value
calculated manually (Section 4.6.1) from Equation (A.35) for the connection
represented by that spring.
The same MATLAB computer program, whose flowchart of tasks for rigid
frame analysis appears in Fig.4.5, will be used to analyze the frame having
semirigid connections, with one extra input. The input will be the initial
stiffness of each connection manually calculated and placed in the structural
data portion for the program to read. The program then forms a 2 x 2 stiffness
matrix for each spring representing each connection. It is noted that this part
of the MATLAB computer program was also developed by Professor John
Mays of the University of Colorado at Denver. The rest of the semirigid
analysis follows the way the flowchart in Fig.4.5 depicts, with one extra out
put. The extra output will be the moment in each spring, which is actually the
moment transmitted by each connection. Section 4.6.2 presents the results of
the analysis.
61
* 14
/K 18
> 17
19
21
12 15
(a) Alphaand Betadegrees
of Freedom
(b) Connectivity and member
numbers
Fig.4.6 Degrees of Freedom and Member
Numbering for Semirigid Frame
62
4.6.1 Calculation of Connection Stiffness Kj(totai)
The MATLAB results from exact analysis (Section 4.4.3) were used to do the
exact design of rigid frame (Section 4.5) for beams and columns. The beam
end moments were used to design rigid connections (Section 4.5.3) for the
frame. In this section the rigid connection dimensions are used to calculate
the initial stiffness of each connection to be used in MATLAB computer
program for semirigid frame analysis (Section 4.6.2).
Initial stiffness for roof beam connection:
Roof beam W 18 x 35 (d = 17.7)
Connection WT 13.5x64.5
tf 1.100 in
bf 10.010 in
t^, 0.610 in
d 13.815 in
g 4.000 in
L 8.000 in (length of tee connection)
L
2 2
(8)(1.10)3 =0.8873 in4
12
A
(8)(1.10) =8.8 in2
bot(web)
(8)(0.61)3 =0.151 in4
12
L,
bot(web)
13.815 1:1^. = 13.265 in
2
63
E 29.0x10 psi
G 11.2x10 psi
K
i (total)
d2
L\ 3Z,
2AEI+ 5GA
4 FI
^Â£jlbot(web)
^ hot (web)
(A.35)
(17.7)2
(2)
3(2)
+
(4)(29x106)(0.151)
13.265
(24)(29 x 106 )(0.8873) (5)(11.2 x 1 Ob )(8.8)
= 12467xlO6 + 1.32x 106
rad
in lb
rad
ftk
= 1039000 
rad
The initial stiffnesses of the remaining connections are obtained in similar
fashion. The results of the calculations are as shown in Table 4.12 for all
three connections.
Table 4.12 Initial Stiffness of Designed Tee Connections
Connection Location Initial Stiffness (ftk / rad)
W 13.5x64.5 Roof level 1039000
W 13.5x64.5 Third floor 1416000
W 15x74 Second floor 1235000
64
4.6.2 Results of Semirigid Frame Analysis
The initial stiffness values from Table 4.12 are used in the MATLAB computer
program for semirigid frame analysis. The load cases used for this analysis
are the same as those for rigid frame analysis: (1) gravity alone, and (2)
gravity and wind. The computer input / output data used and produced by the
MATLAB program are provided in Section B.4 of Appendix B. This section
presents the results of the analysis, which are the moments and axial forces
in Tables 4.13 and 4.14.
65
Table 4.13 Exact Semirigid Analysis (Moments and Axial Forces)
Due to Gravity with Kj(totai)
Member Max. M (ft.k) Min. M (ft.k) Axial N (k)
12 58.1 (cw) 29.2 (cw) 87 (c)
23 65.2 (cw) 56.7 (cw) 57 (c)
34 105.1 (cw) 75.6 (cw) 27 (c)
56 58.1 (ccw) 29.2 (ccw) 87 (c)
67 65.2 (ccw) 56.7 (ccw) 57 (c)
78 105.1 (ccw) 75.6 (ccw) 27 (c)
26* 123.3 101.7 neglected
37* 132.2 92.8 neglected
48* 105.1 97.4 neglected
CW Clockwise
CCW Counterclockwise
C Compression
(*) Beam
66
Table 4.14 Exact Semirigid Analysis (Moments and Axial Forces)
Due to Gravity and Wind with Kj(total)
Member Max. M (ft.k) Min. M (ft.k) Axial N (k)
12 72.2 (ccw) 4.1 (ccw) 74.8 (c)
23 26.7 (cw) 5.5 (ccw) 51.5 (c)
34 84.4 (cw) 64.3 (cw) 25.6 (c)
56 130.1 (ccw) 119.9 (ccw) 99.2 (c)
67 107.7 (ccw) 103.9 (ccw) 62.5 (c)
78 125.8 (ccw) 86.9 (ccw) 28.4 (c)
26* 222.8 22.6 neglected
37* 194.7 69.9 neglected
48* 125.8 84.4 neglected
CW Clockwise
CCW Counterclockwise
C Compression
(*) Beam
67
Observations: A comparison of member moments due to the exact rigid frame
analysis (Table 4.7) with those due to the exact semirigid analysis using
Ki(totai) (Table 4.13) for gravity alone shows a minimal reduction in the values
of end moments for beams and columns of semirigid frame (1 ft.k and 1 ft.k).
For beams, the reduction was distributed to the middle of the beams; for
columns, to the other ends. This is important in that it confirms that semirigid
connections redistribute moment in a member (Gerstle, 1988, p. 241). Also, a
similar comparison for the loading case of gravity and wind (Table 4.8 and
Table 4.14) shows a similar minimal reduction, except a reduction of 2.4 ft.k
at one end of beam 26 and of 9.3 ft.k between its ends. Again, some
redistribution of moments occurred.
This observation substantiates that the connections remained rigid, as they
were designed for. As such, Professor Mays of the University of Colorado at
Denver notes, it is not surprising to see that the connections are so stiff as
compared to the rotational stiffness of the connected beam (4EI / L) that
nothing would be gained by including their initial stiffness in the frame
analysis (Mays, 2001). Table 4.15 shows this comparison.
Table 4.15 Comparing Connection Initial Stiffness with Beam
Rotational Stiffness
Connection Location Ki(totai) (ft.k/rad) Beam (ft.k / rad)
WT 13.5x64.5 Roof Level 1039000 13909
WT 13.5x64.5 Third floor 1416000 22636
WT 15x74 Second floor 1235000 22636
68
Another observation is that Equation (A.35) is sensitive to variations in three
parameters. One is the moment of inertia of the tee flange cross section.
Equation (A.35) is directly proportional to this parameter. An increase in the
cross section moment of inertia increases the connection initial stiffness. The
second parameter is d, the overall depth of the beam that is attached to the
connection. This d is used as the moment arm of the moment resistance of
the connection. The initial stiffness equation is directly proportional to the
square of d. Even taking the moment arm as the distance from center to
center of the top and bottom flanges of the beam reduces the initial stiffness
of the corresponding connection.
The third parameter whose variation makes Equation (A.35) sensitive is L.
Li is taken as half the bolt gage in the tee flange. The initial stiffness defined
by this equation is inversely proportional to the cube of Li. A small (one or two
inches) increase in bolt gage (limited in length as it is by practice) reduces the
initial stiffness appreciably. A bolt gage as small as 4 inches in the design of
connections (Section 4.5.3) may, therefore, partly explain the lack of
appreciable flexural and shear deformations in the connection, as these
deformations were the basis of the derivation of Equation (A.35). However, an
increase in bolt gage may activate such a prying action that the connection
may have to be designed against it.
In the light of these observations, the connection deformations due to flexure
and shear alone do not seem to be enough to render it semirigid. Other
sources of deformation must contribute to the deformation of this connection if
it must function as semirigid. Chapter 5 investigates these sources.
69
5. Modified initial Stiffness KMi(totai)
It was mentioned in the last paragraph of Section 4.6.2 that Other sources of
deformation must contribute to the deformation of this connection if it must
function as semirigid. What are those sources? Several come to mind, all in
contact with the web and flanges of the connection. Those in contact with the
connection web are beam ends and bolts fastening the web to the beam
flanges (not investigated in the present study). Those in contact with the
connection flanges are bolts fastening the flanges to the supporting column
and the areas of the column around the connection, all treated together as
column deformations and investigated in this section.
When a frame deforms every element of it deforms. A column is an element
of the frame. Therefore, the column deforms. Ignoring column deformations
assumes that connection has a rigid support (Faella et al., 2000, p.71). It
might be the case that a connection has such a low stiffness that the
assumption of the supporting column being rigid may be reasonable. But, for
connection with relatively high stiffness, column deformation investigation can
prove important as far as the deformational behavior of the connection.
A number of column deformations have been identified as contributing to the
deformation of extended endplate connections (Yee and Melchers, 1986, p.
p. 621). They are: column flange flexure, bolt elongation, column web shear
(including stiffener where applicable), and column web compression. As part
of their study, Yee and Melchers used these column deformations to derive
an equation to predict the initial stiffness of the connection when material is
linearly elastic (Yee and Melchers, 1986, p. 620).
In the present study the above column deformations are considered to
contribute to the deformation of structural tee connections as well. An
equation is derived independently of the Yee and Melchers derivation for
each column deformation, and the results are used to derive a modified initial
stiffness. The principles of Mechanics of Materials and of Structural
Mechanics are used for this purpose.
70
5.1 Column Flange Flexure
A portion of the column in contact with the connection at the beam tension
flange level is modeled as a Tstub with its flange flexural deformation
representing the column flange flexural deformation (Yee and Melchers,
1983, p. 622). Half of the column web is assumed to act as the web of theT
stub model. See Fig.5.1. It is assumed in this study that the column flange is
not stiffened. If the analysis shows that the column flange needs transverse
stiffener, the Tstub model will be rotated 90 degrees such that the flange
stiffener becomes the Tstub web.
It is assumed that the Tstub model flange is fixed at bolt lines. It can be
shown that, due to the pull from the bolts, the state of deformation (flexure
and shear) of the Tstub model flange is similar to Equation (A.23), except it
uses the column dimensions, and its bolt gage is in a horizontal position.
Thus,
The symbol Ac/ represents the column flange deformation, and m stands for
Tstub model. Other symbols are as defined in Section A.2 of Appendix A.
, where the m stands for model.
(5.1)
Column
Connection
Fig.5.1 Tstub Modeling of Column Flange
71
5.2 Bolt Elongation
The elastic forcedeformation (A = PL / AE), found in Mechanics of Materials
textbooks, enables one to find the elastic elongation of a bolt for a known
axial load. In its present form, this relationship is not applicable to the case of
multiple pretensioned bolts in a beamcolumn structural tee connection. The
reason is that the tension force on each of these bolts is indeterminate. As a
result, a modified version of the above equation is developed in this section
based on the method of consistent deformation for the solution of such a
statically indeterminate problem.
In 1980, Popov developed an elastic forcedeformation relationship based on
the same method to find the force in a single pretensioned bolt gripping two
washers of total thickness L, where L represented the gripping length of the
bolt (Popov, 1980, p. 413). A similar but more involved relationship is
presented in the present study to account for the presence of multiple bolts
and two connected flanges (connection and column) of different thicknesses.
It is assumed that the bolts in the tee connection are pretensioned to a
tension force prescribed by ASIC, 1989, p. 274 so that the flanges of the
connection and column are gripped (clamped) together firmly.
Fig.5.2 shows the connection bolts pretensioned and no external load acting.
(a) Col. & Connection (b) Free Body Diagram
Fig.5.2 Bolts Pretensioned
The symbols in Fig.5.2 are as follows.
72
n number of bolts in tee connection flange
Ti pretension force in each bolt
Cj clamping force (assumed uniform) on contact area around each bolt
Writing force equilibrium equation for the free body diagram in Fig.5.2(b),
SFy =0.
Ti = Ci (5.2)
After the external load (T) is applied to the connection, a free body diagram is
obtained as shown in Fig.5.3.
Fig.5.3 Tee Connection under External Load (T)
The new symbols in Fig.5.3 are as follows.
Y portion of applied load (T) reducing clamping force on contact areas
X portion of applied load (T) increasing tension in each bolt
Writing force equilibrium equation for the free body diagram in Fig.5.3,
S FY = 0.
T + n (Ci Y) n (Tj + X) = 0. (5.3)
Substitution of Equation (5.2) into equation (5.3) yields
X + Y =  (5.4)
n
73
Equation (5.4) has two unknowns X and Y; the problem is statically
indeterminate. If the flanges stay in contact, compatibility condition is
Abolts = Afar,ges (5.5)
AhE
(5.6)
and
* flanges
(Y)(LfQ
Aco,pE
(5.7)
Substitution of the right hand sides of Equations (5.6) and (5.7) into Equation
(5.5), yields
(X)(LeJf) (Y)(Lfh)
AE AompE
In Equation (5.8),
Ab bolt area
Acomp compression area around each bolt under clamping force. ACOmp is
assumed to be a circular area of about 3 bolt diameters (Kuzmanvic, 1983, p.
326):
n
4
Acomp = ^(3 A)2 =9
= 9Ah
(5.9)
Dp bolt diameter
E modulus of elasticity
Uff bolt effective length (Ballio et al., 1983, p. 289) equal to sum of
thicknesses of connected plates (flanges, in this study) and onehalf the sum
of thicknesses of bolt head and bolt nut
Lfh sum of thicknesses of tee and column flanges
tf tee flange thickness
tcf column flange thickness
See Fig.5.4 for effective length of a bolt.
74
Leff
Fig.5.4 Effective Length of Bolt
From Fig.5.4,
Lcff = (tf +tcf)+2^bh +tbn) (510)
=('/+'*) (511)
Solving Equations (5.4) and (5.8) as a system of two equations with two
unknowns and using Equation (5.9), one obtains
\
Y
V. Lc ff
(5.12)
75
and
( \
xL n T
ULJ x9
l LJ\% 1 )
(5.13)
Substitution of Equation (5.13) into Equation (5.6) yields bolt elongation as
r
\
A
boh
1 T (L \
n Lf Q 1 + eJf x * Kae)
l Z/lE 1 J
(5.14)
Similarly, one can obtain an equation for A, which is not used in the
present study.
76
5.3 Column Web Shear Deformation
Yee and Melchers consider a portion of the column as a short column whose
height is the vertical distance between the centerlines of the beam flanges.
This short column, assumed fixed at the beam bottom flange level, is then
subjected to inplane shear force from the beam top flange (Yee and
Melchers, 1986, p. 626). See Fig.5.5.
tbf
jf
(d tbf)
> T
M
77777 <; C
Fig.5.5 Column Web under Shear
A shear deformation equation has been derived by Maugh based on the
principle of virtual work applied to beams (Maugh, 1946, p.37). This equation,
which is in integral form, is used in the present study to develop an inplane
column web shear deformation equation. Maughs equation is as follows.
KV)(y)
GACW J
dx
(5.15)
where
V maximum shear in column web equal to beam top flange tension force
v maximum shear in column web due to a unit load at top of column web
G shear modulus
Am column web area equal to (tew x dc), where tew and dc are column web
thickness and column overall depth, respectively.
77
Evaluation of the integral in Equation (5.15) between its limits (entire length of
column web) leads to the column web shear deformation equation offered in
Yee and Melchers study, where they suggest k=1 for Ibeams (Yee and
Melchers, 1986, p. 627). This equation is as follows.
A. =
Cdhf)
s
frTXri'h k(TXdtb/)
GACW J GACW
(5.16)
78
5.4 Column Web Compression Deformation
Lee and Melchers consider a square region of the column as a plate having
the size of drx dr, where dr is the column depth between root fillets. They
assume the plate to be fixed at its top and bottom in the Ydirection and
subjected to a uniform compression force from the beam compression flange
in the Xdirection (Yee and Melchers, 1986, p. 627). To aid visualization, a
brief sketch is presented in Fig.5.6.
dr
C=T
Fig.5.6 Column Web under Compression
The following derivation is offered in the present study leading to Yee and
Melchers Equation (44) for column web compression deformation (Yee and
Melchers, 1986, p. 628).
79
From generalized Hookes law for multiaxial loading (Beer and Johnson,
1992, p. 81),
Â£
X
E
vc
vc
, and s =
E y E
E E
where
(5.17)
ex normal strain in Xdirection
Ey normal strain in Ydirection
cx normal stress in Xdirection
v poissons ratio
ay normal stress in Ydirection
crz normal stress in Zdirection
E modulus of elasticity
For the case of plane stress, c
rr VC C
^ .^d sy=Â£
E E
z = 0. Thus, Equations (5.17) become
E
(5.18)
From the first relation in (5.18),
rr a*~EÂ£*
Gy =
V
Substitution of Equation (5.19) into the second relation in (5.18) yields
(5.19)
Â£yEv + Esx =cx( 1v2)
(5.20)
80
Since the plate is assumed fixed at its top and bottom in Ydirection, ey=0.
Thus, Equation (5.20) becomes
CT^lV2)
E
or
Eex = cx (1 v2)
The normal stress
T
" frWO
Substitution of Equation (5.23) into Equation (5.22) yields
Ee _ZXlzz!)
J (rfrXO
Rearrangement of Equation (5.24) yields
g, T
iv2 (d,yt)E
The normal strain ex on the plate is
where
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
81
deflection of column web in Xdirection
Substitution of Equation (5.26) into Equation (5.25) yields
sc*x = T
1v2 {tcw)E
(5.27)
Multiplying both sides of Equation (5.27) by (1v2) leads to the column web
compression deformation Equation (44) of Yee and Melchers, which is
niv2)
E(tcw)
(5.28)
82
5.5 Total Deformation
The total deformation in the tension zone of the column due to the tension
force from the top flange of the connected beam is obtained by adding
Equations (5.1), (5.14), and (5.16). This total deformation is as follows.
^ total (ten) = &cf +^bolt+^s (529)
When the column web compression deformation, Equation (5.28), is added to
Equation (5.29), the total column deformations contributing to the deformation
of the structural tee connection, Equation (A.23), is obtained as
Ac(totaI) = Atotal (ten) + &cwx = (Acf + ^bolt + Af ) + ^cwx (5.30)
The sum of Equations (5.29) and (A.23), both at the beam top flange level, as
well as Equation (5.28) cause the relative rotation (<])) of the end of the beam
with respect to the column. This sum is as follows.
Asum total (ten) AB (total)) ^cwx (5.31)
A simplified deformation diagram is in order to show the relationship between
Equation (5.31) and the relative rotation () of the end of the beam with
respect to the column. See Fig.5.7.
A
total (ten)
+ A
B (total)
Fig.5.7 Relative Rotation of Beam End
with Respect to Column
83
From Fig.5.7, one obtains
=
(A total (ten) ~^~^B(total) ~^~dcm)
(5.32)
The moment resistance of the connection still is as Equation (A.25), which is
repeated here for convenience.
M=
Finally, the modified initial stiffness of the connection is defined as
K
M,
top
Mi (total)
+ K;,
i(bot)
(5.34)
where
Kj(bot) represents the initial stiffness contribution from the bottom tee web,
whicn was introduced in Equation (A.33).
Substitution of Equations (5.32) and (5.33) into Equation (5.34) yields
K
Mi (total)
(T)(d)
(^ total (ten) + AB (total) +
^ + K;
i(bot)
CT)(d2)
(Atotal (ten) ^ A B (total) ~^~dCWJ[)
+ K:
i(bol)
(5.35)
84
Substitution for the deformations a , a and s within the
parentheses on the righthand side of Equation (5.35) yields the modified
initial stiffness for structural tee connection as follows, with T dropping off.
^Mi(total)
Aj + A2 + A3 + A4 + As
+ K
i(bot)
where
(5.36)
4 =
r A3 + 3A A
24 El 5 GA
, from Equation (A.23)
(5.37)
A2
' Lj
3
24 EIm 5GA
m m J
, from Equation (5.1)
(5.38)
4=~
r ^
1
1 +
9
Lfl 1
(L \
\4EJ
, from Equation (5.14)
Aa =
\ "/ib 1 y
GA,
, from Equation (5.16)
(5.39)
(5.40)
A = ^^ from Equation (5.28)
E{tcw)
(5.41)
K
i(bOt)
4 El
bot(web)
~'bot(web)
, from Equation (A.33)
(5.42)
Finally, the moment resistance of the semirigid structural tee connection is
M = (KM(lolalM
(5.43)
85

PAGE 1
A THEORETICAL DERIVATION OF INITIAL STIFFNESS OF THE SEMIRIGID STRUCTURAL TEE CONNECTION by Farrokh Jalalvand B.S., University of Oklahoma, 1980 M.S., Oklahoma State University, 1984 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering 2001
PAGE 2
by Farrokh Jalalvand All rights reserved
PAGE 3
This thesis for the Master of Science degree by Farrokh Jalalvand has been approved by j'JUdit:stalnaker Kevin L. Rens
PAGE 4
Jalalvand, Farrokh (M.S., Civil Engineering) A Theoretical Derivation of Initial Stiffness of the Semirigid Structural Tee Connection Thesis directed by Associate Professor Judith J. Stalnaker ABSTRACT That a semirigid connection allows some rotation causes reduced beam end moments compared to a rigid connection. This study develops a theoretical equation for the initial elastic stiffness of a steel semirigid structural tee connection. The initial stiffness is not only a function of flexural and shear deformations in the connection, but is also a function of deformations of the bolt, column flange, and column web. A lateralforce resisting frame of a threestory building is analyzed as a rigid frame for gravity and gravity plus wind, using a MATLAB computer program. Then, the members and rigid connections are designed. Connection initial stiffness is determined from the theoretical equation and is used for a computer semirigid analysis. New members and connections are chosen based on the reduced moments. Column flange and column web deformations were found to be particularly significant in contributing to .connection rotational deformation. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. iv
PAGE 5
DEDICATION I would like to dedicate this thesis to the memory of my late Iranian father, who, despite his being illiterate, zealously valued formal education and very much wanted me to become a civil engineer and to my American wife Jean E. Johnson who, through her support, patience, and encouragement, made it all happen here in the USA.
PAGE 6
ACKNOWLEDGEMENT I would like to thank Dr. Judith J. Stalnaker for her directing this thesis and also for her generous assistance and encouragement in the course of my thesis research. I would also like to thank Professor John Mays for developing the computer program used in this thesis and for his valuable assistance in the use and interpretation of the program results. Additionally, I am indebted to both Dr. Stalnaker and Dr. Mays for all the valuable knowledge I received from them through their class lectures and by tutoring their structural analysis students. Finally, I would like to express my appreciation to Professor K. H. Gerstle of the University of Colorado at Boulder for providing input, ideas, and his own written and published papers on the subject of flexible connections for my thesis research.
PAGE 7
CONTENTS Figures ........................................................................................................... xii Tables ............................................................................................................ xiv Chapter 1. Introduction ............... ......................................................................... 1 1.1 Definition ........................................................ : ................................... 1 1.2 Names ............................................................................................... 2 1.3 Types ................................................................................................. 2 1.4 Design Method .................................................................................. 2 1.5 Purpose of the Study ......................................................................... 2 1.6 scope of the Study ............................................................................ 3 2. Semirigid Connection ....................................................................... 4 2.1 Response Overview ........................................................................... 4 2.2 Definition ............................................................................................ 5 2.3 Degree of Rigidity ......................................................................... .... 5 2.4 Physical Look .................................................................................... 6 2.5 Literature Review ............................................................................... 9 2.6 Design Practice .............................................................................. 14 vii
PAGE 8
3. Response of a Semirigid Structural Tee ......................................... 15 3.1 Elastic M$ Derivation ..................................................................... 15 3.2 Total Initial Stiffness K1(totaJ) 17 3.3 Comparison with Test Data ............................................................. 22 4. Design Example .............................................................................. 26 4.1 Computation of Loads ...................................................................... 28 4.2 Approximate Rigid Frame Analysis .................................................. 29 4.3 Preliminary Design of Rigid Frame .................................................. 33 4.31. Design for Gravity Alone .................................................................. 33 4.3.2 Design for Combined Gravity and Wind ........................................... 39 4.4 Exact Rigid Frame Analysis ............................................................. 43 4.4.1 HandPrepared Input Data ............................................................... 43 4.4.2 MATLAB Program Tasks ................................................................. 46 4.4.3 Results of Exact Rigid Frame Analysis ........ ................................... 48 4.5 Exact Design of Rigid Frame ........................................................... 51 4.5.1 Beams ....... ................................................................................... 4.5.2 Columns .......................................................................................... 53 4.5.3 Rigid Connection ............................................................................. 60 4.6 Exact Semirigid Frame Analysis with Ki(total) ................................... 61 viii
PAGE 9
4.6.1 Calculation of Connection Stiffness Ki(total) ....................................... 63 4.6.2 Results of Semirigid Frame Analysis .............................................. 65 5. Modified Initial Stiffness KMi(totaJ) ...................................................... 70 5.1 Column Flange Flexure ................................................................... 71 5.2 Bolt Elongation ....... ; ........................................................................ 72 5.3 Column Web Shear Deformation ..................................................... 77 5.4 Column Web Compression Deformation ......................................... 79 5.5 Total Deformation ............................................................................ 83 5.6 Exact Semirigid Frame Analysis with KMi(total) ................................. 86 5.6.1 Calculation of Connection KMi(total) ............................................... 86 5.6.2 Results of Exact Semirigid Analysis with KMi(total) ............................ 90 5.7 Exact Design of Semirigid Frame with KMi(total) ............................... 93 5.7.1 Beams .......................................................... ', .................................. 93 5.7.2 Columns .......................................................................................... 94 5.8 Comparison of Rigid and Semirigid ................................................ 95 6. Discussion, Conclusion, and Further Study ..................................... 96 6.1 Discussion ....................................................................................... 96 6.2 Conclusion ....................................................................................... 97 6.3 Further Study ................................................................................... 97 ix
PAGE 10
Appendix A. Derivation of Semirigid Structural Tee Total Initial Stiffness Ki(total) ............................................................... 98 A.1 General Assumptions ...................................................................... 99 A.2 Notations ....................................................................................... 1 00 A.3 Sign Convention ............................................................................ 102 A.4 Initial Stiffness for Top Tee K;(top) ................................................. 102 A.4.1 Deflection ( Ae) Due to Flexure ...................................................... 105 A.4.2 Deflection (38 ) Due to Shear ......................................................... 110 A.5 Initial Stiffness for Bottom Tee K;(hot> ............................................ 115 A.5.1 Assumptions for Bottom Tee ......................................................... 115 B. Frame Analysis ............................................................................. 118 B.1 Computation of Loads ................................................................... 119 B.1.1 Gravity Loads ................................................................................ 119 B.1.2 Wind Loads ................................................................................... 122 B.2 Approximate Rigid Frame Analysis ............................................... 125 8.2.1 Gravity Acting Alone ...................................................................... 125 B.2.2 Wind Acting Alone ......................................................................... 130 B.3 MATLAB Input/Output, Rigid Frame Analysis ............................... 138 B.3.1 Gravity Alone Input/Output Data ................................................... 139 B.3.2 Gravity and Wind Input/Output data ................................. ............ 145 X
PAGE 11
8.4 Rigid Connection Design ............................................................... 151 8.5 MATLA8 Input/Output, Semirigid Analysis with Ki(total) ..... 157 8.5.1 Gravity Alone Input/Output data .................................................... 158 8.5.2 Gravity and Wind Input/Output Data .............................................. 165 8.6 MATLA8 Input/Output, Semirigid Analysis with KMi(total) ..... 172 8.6.1 Gravity Alone Input/Output Data ................................................... 173 8.6.2 Gravity and Wind Input/output Data .............................................. 180 References .................................................................................................. 187 xi
PAGE 12
FIGURES Figure 1.1 Structural Tee under Study ................................................................ 1 2.1 Comparing Rigid and Semirigid Connections .................................. .4 2.2 A Sample of Semirigid Connections ................................................. 7 2.3 Typical Mcp Curves .......................................................................... 11 3. 1.. Initial Stiffness as Slope of Tangent Line to Mel> Curve at Origin ......................................................................... 16 3.2 Top Structural Tee as a Frame under Tension (T) ............................................................................ 18 3.3 Comparing Initial Stiffness with .Rathbun's Mcp Curve ....................................................................... 25 4.1 A Typical Interior Frame under Study .............................................. 27 4.2 Unbraced Frame under Design Loads ............................................. 28 4.3 Degrees of Freedom and Member Numbering for Rigid Frame ............................................................. 44 4.4 Member Local Coordinate System with Sign Convention for Positive ............................................................ 45 4.5 Flowchart of MATLAB Tasks for Rigid Frame Analysis ....................................................................... 47 4.6 Degrees of Freedom and Member Numbering for Semirigid Frame ..................................................... 62 xii
PAGE 13
5.1 Tstub Modeling of Column Flange .................................................. 71 5.2 Bolts Pretensioned ......................................................................... 72 5.3 Tee Connection under External Load (T) ......................................... 73 5.4 Effective Length of Bolt. ................................................................... 75 5.5 Column Web under Shear ....................................................... ; ....... 77 5.6 Column Web under Compression .................................................... 79 5. 7 Relative Rotation of Beam End with Respect to Column ................................................................... 83 A.1 Modeling of Top Tee as a Frame .................................................. 102 A.2 Top Structural Tee as a Frame under Tension (T) .......................................................................... 104 A.3 Flange of Top Structural Tee under Shear .................................... 111 A.4 Rotation of Beam End Relative to Column .................................... 113 A.5 Bottom Web Tee under Bearing and Rotation ..................................................................... 115 8.1 Unbraced Frame under Wind Pressure ......................................... 123 8.2 Unbraced Frame under Joint Wind Loads ..................................... 124 8.3 Simple Beams & Cantilever Columns for Gravity Load Analysis .................................................................... 126 8.4 Approximate Moment Diagram Due to Gravity .............................. 129 8.5 Frame Story Midheight Cuts ........................................................ 130 8.6 Approximate Moment Diagram Due to Wind ................................. 137 xiii
PAGE 14
TABLES Table 4.1 Member Moments and Axial Forces Due to Gravity ........................ 30 4.2 Member Moments and Axial Forces Due to Wind ........................... 31 4.3 Combined Effect of Gravity and Wind Loads ................................... 32 4.4 Preliminary Beam Sizes for Gravity Loading ................................... 34 4.5 Preliminary Column Sizes for Gravity Loading ................................ 38 4.6 Final Preliminary Column Sizes for Approximate Analysis 4.7 Exact Rigid Analysis (Moments and Axial Forces) Due to Gravity ............................................................ 49 4.8 Exact Rigid Analysis (Moments and Axial Forces) Due to Gravity and Wind ............................................ 50 4.9 Exact Design of Beams for Rigid Frame .......................................... 52 4.10 Exact Design of Columns for Rigid ...................................... 59 4.11 Rigid Structural Tee Connection Sizes ............................................ 60 4.12 Initial Stiffness of Designed Tee Connections ................................. 64 4.13 Exact Semirigid Analysis (Moments and Axial Forces) Due to Gravity with Ki(total) .......................................... 66 4.14 Exact Semirigid Analysis (Moments and Axial Forces) Due to Gravity and Wind with Ki(total) .......................... 67 xiv
PAGE 15
4.15 Comparing Connection Initial Stiffness with Beam Rotational Stiffness ............................................................... 68 5.1 Comparison among Ki(total), KMi(total), and Beam Rotational Stiffness ............................................................... 89 5.2 Exact Semirigid Analysis with KMi(total) (Moments and Axial Forces) for Gravity ........................................... 91 5.3 Exact Semirigid Analysis with KMi(total) (Moments and Axial Forces) for Gravity and Wind ........................... 92 5.4 Exact Beam Design for Semirigid Frame with KMi(total) ........ 93 5.5 Exact Column Design for Semirigid Frame with KMi(total) ... 94 XV
PAGE 16
1. Introduction The American Institute of Steel Construction (AISC, 1989) recognizes three types of connections in steel frame construction. They are type 1rigid, type 2simple (pin), and type 3semirigid. The subject of this thesis will be 'type 3 connections. In particular, a structural tee will be investigated as a semirigid connection. This chapter provides a definition, names, types, and design methods for a structural tee, and the purpose and scope of the study. 1.1 Definition A Structural tee connection is a steel momentresisting connection that is used to connect beams to columns by such fasteners as rivets or bolts in steel frame construction (Fig.1.1 ). Welding is also possible in combination with bolts or rivets (McGuire, 1968, p. 964). The rioun "tee" in the term "structural tee" comes from the fact that the connection resembles the letter "T" in the English alphabet if it is viewed in cross section. Fig. 1.1 Structural Tee under Study 1
PAGE 17
1.2 Names A structural tee connection is also known by three other names. One is split beam" connection as a beam is split in making it (Lathers, 1965, p. 195). Another is "teestub" connection (McGuire, 1968, p. 964). The noun "stub" comes from the fact that a short length is cut from a steel beam in making it. The third is "splitbeam tee" connection (Salmon and Johnson, 1980, p. 785). 1.3 Types There are three types of structural tees, each of which is obtained from a specific steel beam shape. The beam is split along its web at the desired length. A wr is split from a Wshape, an MT from an Mshape, and an ST from an Sshape (AISC, part 1, 1989). 1.4 Design Method A structural tee connection is designed by two methods. In the simplest and most common method, the tee is designed with web angles such that the tee transmits moment, and the web angles carry shear. In another method, the tees are designed without web angles such that the tees transmit both moment and shear (Lathers, 1965, p. 195). 1.5 Purpose of the Study A structural tee is often regarded as a rigid connection (Bakers, 1954, p. 116, Lathers, 1960, p. 141, and Tall, 1974, p. 619) and designed as such (Lathers, 1965, p. 201, Tall, 1974, p. 623). However, a structural tee can be designed as a semirigid connection (defined in chapter 2), if possible prying action is checked. They are not economical as rigid connections since they cannot develop the full moment capacity of the beam (Marcus, 1977, p. 343). 2
PAGE 18
Actually, a structural tee is designed in the elastic range as a semirigid connection for a given design moment (Johnston et al., 1980, p.192 and Fanella et al., 1992, p. 294 and p. 318). But, neither source specifies how to determine the design moment for such a semirigid connection. The present study attempts an answer to that question: how to find the design moment for a semirigid structural tee. To achieve this, a theoretical approach is used to derive a linear momentrotation relationship for the connection. First, the rotational deformation of the connection itself will be determined. Next, the column deformations believed to be contributing to the deformation of the connection will be derived and incorporated into the relationship. The results will then be utilized in computer matrix analysis to find the design moment of the connection. 1.6 Scope of the Study This study contains the following. Theoretical derivation of an elastic momentrotation relationship for structural tee connection as semirigid Application of the results in the analysis of unbraced portal frames with semirigid structural tees under gravity and wind loads using computer matrix analysis. Design of an unbraced portal frame with semirigid and rigid structural tees and comparison of the results. 3
PAGE 19
2. Semirigid Connection This chapter provides a response overview, a definition, the degree of rigidity, and physical look of a semirigid connection. It also briefly reviews literature and design practice. 2.1 Response Overview A beamcolumn semirigid connection subjected to inplane loadings deforms primarily in rotation (Faella et al., 2000, p. 38). Thus, the loaddeformation relationship for such a connection would be a relationship between the applied moment and the rotation of the connection. Although it is the semirigid connection that rotationally deforms under the action of inplane loading; in experimental work the deformation within the connection is measured in terms of the of the beam end relative to the column (Maugh, 1964, p. 397). This rotation be explained in this section by comparing the behavior of a rigid connection with that of a semi rigid connection when both are subjected to inplane loadings (Fig. 2.1) . 8 (a) Rigid Connection (b) Semirigid Connection Fig. 2.1 Comparing Rigid and Semirigid Connections 4
PAGE 20
Fig. 2.1 (a) shows schematically a rigid connection before and after loading. Before loading, the tangents to the undeformed elastic curves of the beam and column had intersected at a 90degree angle at the joint (J1). After loading, the two elastic curves deform in flexure causing the joint (J1), the beam end, and the column section at (J1) to rotate the same amount (8), while the original angle between the two tangents at (J1 ) still remains 90 degrees. Fig. 2.1(b) shows schematically a semirigid connection before and after loading. Before loading, the situation for this connection would be identical to that of the rigid connection before loading; nothing is deformed. After loading, however, the two elastic curves deform in flexure, and the tangent to the elastic curve at the beam end at the joint (J2) rotates an extra in addition to the rotation (8) of the joint and column. Another word, the end of the beam rotates by an in addition to the rotation (8) of the column (Lathers, 1960, p.370). This is so because of the "deformation within" the semirigid connection. Note from Fig. 2.1 (b) that the original angle between the two tangents to the elastic curves no longer remains 90 degrees since the end of the beam has rotated by an relative to the column. In order to sum the above response, a definition will be presented in the next section. 2.2 Definition A semirigid connection is a momentresisting, beamtocolumn connection that allows the end of the connected beam to rotate some amount relative to the column. In this definition the relative amount of rotation depends on the degree of rigidity of such a connection. 2.3 Degree of Rigidity Degree of rigidity of a semirigid connection is actually a comparison that is made between the moment such a connection can take and the moment taken by a rigid connection. The degree of rigidity of a semirigid connection is between the rigidity of a simple (pinned) connection and a rigid (fixed) one and varies with the type of semirigid connection. The degree of rigidity of a simple connection is from 020%, that of a rigid connection is above 90%, and that of a semirigid connection is from 2090%, with the degree of rigidity being a percentage of the ratio of the actual moment to the fixedend moment (McCormac, 1992, p. 406). 5
PAGE 21
2.4 Physical Look A semirigid connection looks like any one of the commonly used beamto column connections. The only requirement is that its momentrotation response be considered in the frame analysis and design. Fig.2.2(a) and Fig.2.2(b) show several types of steel frame connections that can be used as semirigid. Some of these beamtocolumn connections are idealized as rigid; others are idealized as simple (pin), and the frame is designed accordingly. However, neither of these cases can be materialized in practice; so, they should be regarded as semirigid (Maxwell, et al., 1981 p. 2.72, Faella,et al., 2000, p. 38). 6
PAGE 22
lal Type A The webangle connection h. _, ' 1 "' r(cl Type C Combination of typn A;B lbl Type 8 Top and seatangle connectron I(dl Type 0 The neckeddown connection (e) Top view Fig.2.2 A Sample of Semirigid Connections Source: (Lathers, 1960, Fig.803) 7
PAGE 23
(a) Web angles with seat angle and top angle (b) Structural tee connection (c) Welded connection Fig.2.2 A Sample of Semirigid Connections (cont'd) Source: (McCormac, 1992, Fig.14.4) 8
PAGE 24
2.5 Literature Review Research into the response of steel beamtocolumn connections under loads spans back over 80 years in both the United States and abroad. The research has been both experimental and analytical. In experimental work, the focus was mostly on the momentrotation relationship in the form of a plot of Thus, tested were common types of beamtocolumn connections under applied moment; measured was the rotational deformation of each connection type in the form of the rotation of the beam end relative to the column. In analytical work, the emphasis was to derive a formula that can predict the linear initial stiffness of a connection. Here, initial stiffness is the initial slope of the momentrotation curve of the connection (Azizinamini, 1987, p. 73). It is observed from experimental momentrotation curves that every connection type tested allowed some rotation of the beam end relative to the column. Hence, some researchers called them semirigid; others called them flexible. It is not clear who first introduced the term "semirigid", and who introduced the term "flexible" in the literature. The term "semirigid" has been used more often than the term "flexible." It is even now in the title of two recent books on the subject .(Chen, 1993 and Faella et al. 2000). The mere fact that there are two names for the same connection behavior shows that structural researchers have not reached an agreement on what to call a connection when not idealized as rigid or simple (pin). Professor Gerstle of the University of Colorado at Boulder (Gerstle, 2000) argues that he would call such connections flexible since "semirigid means half rigid", and since there is a range for connection response that falls between the two extreme cases of simple and rigid connections. It was noted earlier (Section 2.3) that the rigidity of "a semirigid connection is from 2090%." This statement does not imply "halfrigid." However, in the memory of the pioneers on the subject who first studied connection response and used the term "semirigid" in their work, this study uses the term "semirigid" in its work. 9
PAGE 25
As pointed out in section 2.1, the result of an experimental test on a semirigid connection is represented by a relationship between the moment (M) applied to the connection and the rotation (cp) of the beam end relative to the column. This relationship is plotted as a nonlinear Mcp curve. Figure 2.3 is redrawn qualitatively from a more involved plot of Mcp curves (Tall, 1974, p. 625) to suit our purpose of observing the nonlinear nature of these curves with the related connections. In this redrawn sketch, the beams (originally vertical) are positioned horizontally, the connection names (lacking in the original) are added, and an explanation for the axes of the Mcp curves (lacking in the original) is provided. Similar experimental Mcp curves, not shown in this study, are also available for welded connections (McGuire, 1968, p. 893). 10
PAGE 26
a Tstub M b Top, seat, & Web angles a c Top & seat angles Fig. 2.3 Typical curves (Redrawn from Tall, 1974) 11 d Web angle(s)
PAGE 27
Note from Fig.2.3 that the horizontal axis represents the idealized case of a simple (pinned) connection for M = 0, and the vertical axis represents the idealized case of rigid (fixed) connection for = 0. Curve (a) represents a more rigid connection than curve (b), and each subsequent curve is more rigid than the other. The following experimental and theoretical or analytical citations are by no means exhaustive. The interested reader is referred to Jones et al., 1983, Chen, 1993, and Faella, et al., 2000 for more information and references on the subject. In the United States, in 1917 Wilson and Moore, who were reportedly the first to have carried out experiments on connections (Jones et al., 1983, p. 2, and Chen, 1993, p. 233) tested riveted brackets, top and seat angles, and double web angles for their rigidity in rectangular frames (Wilson and Moore, 1917). In Canada, in 1928 Young, C. R. and in 1934 Young, C. R. and Jackson, K. B. reportedly (Rathbun, 1936, p. 525, Jones et al., 1983, p. 3, and Chen, 1993, p. 234) tested riveted and welded connections. In Great Britain, in 1934 Batho and Rowan tested riveted and bolted double web angles, top and seat angles with and without doubleweb angles, and structural tees. They also provided a method of finding the end moments in the beams from experimental results (Batho and Rowan, 1934, pp. 61137). In the United States, in 1936 Rathbun conducted a series of tests on different sizes of doubleweb angles, top and seat angles with and without double web angles, and structural tees. In 19411942 Hechtman and Johnston, under the sponsorship of AISC (American Institute of Steel Construction) tested fortyseven riveted connections of different sizes. Connection types were top seat angles, top and seat angles with web angles, tees and seat angle, web clip and seat angles, and tees with and without web angles (Hechtman and Johnston, 1947). In the United States, in 1951 Lathers analytically derived an initial elastic restraint equation for the semirigid connection factor, Z, for doubleweb angles. He then applied (1/Z) as the initial tangent slope to some of Rathbun's momentrotation curves (Rathbun, 1936) for similar connections and reported "consistent agreement" (Lathers, 1951, p. 490). In 1953 Yu, under Lathers, derived similar equations for semirigid top and seat angle connections (Yu, Shan Yuan, 1953). In 1955 Yu, also under Lathers, derived similar equations for semirigid top and seat angles with web angles (Yu, Wei Wen, 1955). In 12
PAGE 28
1958, Huang derived similar equations for the split beam semirigid connection (Huang, 1958). In the United States, in 1956 Pray and Jensen tested welded top plate and seated angle connections to verify their proposed analysis and design of a twoway beamcolumn system (Pray and Jensen, 1956). In 1965, Douty and McGuire tested highstrength bolted moment connections, such as Tstub with web angles and end plates for their response and design in both the elastic and plastic range (Douty and McGuire, 1965). In the United States, in 1979 Herzl performed a series of tests on bolted top and seat angles as well as bolted flange plates for their momentrotation response and on frames with bolted top and seat angles for their actual response (Herzl, 1979). In 1986 Stelmack, Marley, and Gerstle conducted tests on a particular flexible top and seat angle connection to verify analytical methods of steel frame behavior with such a connection (Stelmack, Marley, and Gerstle, 1986). Also, in 1987 Azizinamini, Bradburn, and Radziminski derived an expression for the initial stiffness of bolted semirigid top and bottom angles with web angles, testing the connection (Azizinamini, Bradburn, and Radziminski, 1987). In the United States, in 1993 Chen published analytical derivations of the initial stiffness of the momentrotation curve for top angles, seat angles, and double web angles with rivet or bolt fasteners (Chen, 1993, Appendix B). More recently, a whole book has been published on the subject of semirigid connections (Faella, et al., 200). In short, structural researchers have produced experimental data as well as analytical elastic and inelastic equations to help predict the semirigid connection response to moment. In both cases, connections were mostly riveted angle connections. One notices that the majority of the research has been experimental, which is time consuming and expensive. More analytical research based on the standard principles of Structural Mechanics should be pursued to cover the elastic response of connections using bolts as fasteners, which are now common. Analytical methods have the advantage of being less time consuming, less expensive, and accessible to each design office. After all, the design practice is mostly based on linearly elastic (not plastic) response of steel frames. Such research can help the structural designer to design for a realistic frame behavior, a practice reportedly lacking in today's design practice. 13
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2.6 Design Practice The customary design practice in steel frames is to assume that a connection is either rigid (fixed) or simple (pinned). Based on such assumptions, the members in the frame are analyzed and designed. There are some concerns about such a practice. One is that it does not reflect the true behavior of the frames; it does not consider the rotational deformation of the connections in the frames. Such a deformation influences the deflection of an unbraced frame and internal force distribution in its members (Gerstle, 1988, pp. 241242). The other concern about such a design practice is that it is not a balanced design. Lui and Chen in their study of frames with "flexible joints" conclude the following (Lui and Chen, 1986): It should be noted that, in reality, fully rigid and ideallypinned connections do not exist. All connections exhibit behaviour somewhere in between these two extreme cases. The fullyrigid and ideallypinned joint idealizations are just design simplification. Generally speaking, the rigidjoint assumption will lead to an underestimate of frame drift and overestimate of frame strength, whereas the pinnedjoint assumption will result in an overdesign of the girder and underdesign of the columns. 14
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3. Response of a Semirigid Structural Tee This chapter presents a method for the derivation of the linearly elastic response of semirigid structural tee connections. It then compares the results with available curves. 3.1 Elastic Derivation It was mentioned earlier (Section 2.5) that a semirigid connection curve was nonlinear. The slope of this curve at any point is the rotational stiffness of the connection (Gerstle, 1988, p.241 ). For calculation purposes, various formulas (polynomial, cubic 8spline, power, and exponential) have been devised to represent such a curve (Chen, 1993, pp. 236237). However, the initial elastic stiffness can represent the linear response of a connection (Gerstle, 1988, p. 244, and Chen, 1993, p. 235). By definition (Section 2.2), a semirigid connection allows some at the end of the beam relative to the column. This extra increases with the applied moment (M) at the end of the beam and varies inversely with the stiffness of the semirigid connection (Lathers, 1960, p. 371). The relationship between the applied moment (M) and the rotation ( can be expressed as (3.1) or, alternatively as M = (K) (3.2) 15
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K in both (3.1) and (3.2) is the semirigid connection initial stiffness. That is, K is the slope of the momentrotation curve of the connection at its origin. See Fig.3. 1. M "'Experimental Response Fig.3.1 Initial Stiffness as Slope of Tangent Line Curve at Origin The relationship in (3.2) will be used in this study, and K will be called the total initial stiffness (Ki(totar)) of the semirigid structural tee connection. 16
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3.2 Total Initial Stiffness Ki(total) One semirigid structural tee connection consists of two tees. One tee would be installed on the top flange of the beam and thus called top tee; the other would be installed on the bottom flange of the beam and thus called bottom tee. The task is to derive the following: 1. Initial stiffness for the top tee 2. Initial stiffness for the bottom tee 3. Initial stiffness for the whole connection The method of analysis used was that a oneinch strip of the top tee was taken out and treated as an indeterminate frame for the determination of its initial stiffness. A oneinch strip of the bottom tee was taken out and treated as a cantilever beam for the determination of its initial stiffness. The slope deflection method and Castigliano's theorem on deflection were used for analysis. The initial stiffness for the whole connection would then be the sum of stiffness of top and bottom tees. In this section the steps of the derivations will be presented. The derivation details along with assumptions and notation used are provided in Appendix A. See Fig.3.2 to help visualization. Step 1. Write slopedeflection equations for the three members of the frame (members AB, BC, and BD). Step 2. Eliminate the terms that are zero in the equations of Step 1. Also, using the sign convention in Section A.3, set dAB = dec = de. Step 3. Apply the moment equilibrium equation to joint B of the frame. Step 4. Substitute for the moments in the equation of Step 3 using the slope deflection equations of Step 1. (The rotation 8 of joint B is found to be zero.) Step 5. Write one shear equilibrium equation for free body diagram (FBD) of the frame cut near the supports A, B, and D. Step 6. Write two moment equilibrium equations for the FBD of members AB and BC of the frame by taking moments at B on each member. 17
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A L2, b Bolt Row (Fixed Support) (, r1 "' 1 T T L1 I I D ... ... L1, I Bolt Row (Fixed Support) c (a) (b) Fig 3.2. Top Structural Tee as a Frame under Tension (T) 18
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Step 7. Solve for the shear forces in the equation of Step 6. Step 8. Substitute the shear forces of Step 7 into the equation of Step 5. Step 9. Solve the equation of Step 8 for the horizontal deflection (L1s). Step 10. Write Castigliano's theorem for shear deflection of AB and BC. Step 11. Solve the equation of Step 10 for the horizontal deflection (8s). Step 12. Add the deflections of Step 9 and Step 11. Step 13. Step 14. Write M = [ d] T. Step 15. Substitute T of the equation of Step 12 into that of Step 14 to find M(top) = [ Ki(top)] Step 16. Find the deflection [L1bot(web)] of the free end of the bottom tee from the beam deflection formula for the cantilever case. WbrgL4 boi(webJ L1 _.:::.._ __ boi(web) SEJ Step 17. [L1bot(wcb)]. Lboi(web) Step 18. Take the moment of the uniform bearing reaction force of the beam acting on the bottom tee web about the center of rotation of the connection. Step 19. Substitute Wbrg of Step 16 into Step 18 and L1bot(web) of Step 17 into Step 18 to find M(bot) = [ Ki(bot) ] 19
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Step 20. Find the total initial stiffness of the connection by forming Ki(total) = Ki(top) + Ki(bot) These 20 steps were carried out to derive the initial stiffness for the top tee, the initial stiffness for the bottom tee, the total initial stiffness for the connection, and the connection moment resistance. Below are the results. 1. Top initial stiffness: Ki(top) = 3 L1 3L1 +(A.28) 24EI 5GA 2. Bottom initial stiffness: K [4E/bot(web)] i(bot) Lbot(web) (A.33) 3. Connection total initial stiffness: Ki(toral) = Ki(top) + Ki(hot) (A.34) = d 2 4/bot(weh) 3 + Ll 3LI Lbot(weh) +(A.35) 24/ 5GA 20
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4. Finally, connection moment resistance: M= d 2 + 4EJbot(weh) ( ) fjJ L\ 3ELt Lbot(web) +(A.36) 24EI SGA 21
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3.3 Comparison with Test Data The connection initial stiffness, Equation (A.35), will be tested using data from Rathbun's experimental Mlj> curve for specimen 16 obtained for a similar connection (Rathbun, 1936, p. 528). To do this, the connection moment resistance, Equation (A.36), will be plotted on this curve. If the straight line representing Equation (A.36) becomes tangent the Mlj> curve at its origin, Equation (A.35) is considered a good prediction of the initial stiffness of the connection. Others have applied such a method in the past to compare their initial stiffness equations for other types of connections with test results (Lathers, 1951, pp. 489490, Lathers, 1960, pp.394395, Yu, 1953, pp. 1319, Yu, 1955, pp.1316). Yuang, 1958 also applied the same method to test the initial stiffness of the split beam connection (Yuang, 1958, pp.910). Rathbun's Mlj> Curve Rathbun's specimen 16 shows a vertical plate simulating a column, and two beams, one on each side, are abutted to the plate with rivets. It should be noted that since this specimen does not have a column, it is not considered as a beamcolumn arrangement that is typical in actual building frames (Chen, 1993, p. 252). However, since the present study did not find an experimental Mlj> curve for a tee connection using beams and columns whose dimensions could be found in the AISC manual, the connection total initial stiffness, Equation (A.35), will be computed with the data for Rathbun's specimen 16, reprinted in Fig.3.3. K;(total) = d2 4EJhot(weh) 3 +!.'Ll 3L! Lbot(weh) +(A.35) 24EI SGA 22
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From the data for the specimen, d L1 A I lbot(web) E G Lbat(web) 22.000 in. 2.250 in. 18.060 in. 2.182in. 0.852 in. 29.0 x 106 psi. 11.2 x 1 06 psi. 17.398 in. Substitution of these data into Equation (A.35) yields Ki(total) = (22)2 4(29x106)(0.852) (2.25)3 3(2.25) (17.398) ':'+',.:. __ 24(29 X 106 )(2.182) 5(11.2 X 106 )(18.06) =3.4x101o !bin rad Thus, the connection moment resistance, Equation (A.36) becomes M = (3.4x1010)(). For = 0.001 radians (small deformation theory), M = (3.4x1010)(0.001) = 34.0 x 106 lbin. 23
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The plot of the point whose ordered pair is (0.001, 34 x 1 06 ) on the Rathbun's specimen 16 curve (Fig.3.3) falls beyond the scale on the moment axis. However, if the scale on the moment axis is extended, the point plotted, and the point connected to the origin (0, 0) with a straight line, the line will be tangent to the curve at its origin. The slope of this line is Ki(total) = 3.4 x 1010 lbin. per radians. This slope is about twice as large as the slope (initial stiffness) value of 1.6 x 1010 lbin. per radians that has been reported for the Rathbun's specimen 16 (Yuang, 1958, p. 1 0). Considering the reported slope value, one may obtain the connection moment resistance of this specimen as (1.6 x 1010)(0.001) = 16 x 1061bin., which is about half as large as M = 34.0 x 106 lbin. predicted above. Yuang, in his Master's thesis study at the Oklahoma State University, reported the abovementioned value for comparison with the initial stiffness value of 10.8 x 1010 I bin. per radians his thesis had predicted for the split beam connection (Yuang, 1958, p.1 0). His predicted value is larger than the value he reported for Rathbun's specimen 16. Yuang obtained the predicted initial stiffness value by deriving an elastic restraint equation (Z) based on flexural deformation ofTstub connections whose reciprocal (1/Z) represents the initial stiffness of the connection, concluding ... the elastic deformation of rivets and other factors ... needed to be accounted for (Yuang, 1958, p. 40). Aspects such as these are investigated in Chapter 5. 24
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16. Seroes ; i 4 JOon. GBcom < I x3" Jf .... i"'_..__ 0.001 0.002 Aolltioo in RdllnS 0.003 0.004 0.005 0.006 0.007 Fig.3.3 Comparing Initial Stiffness with Rathbun's curve Source: (Rathbun, 1936) 25
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4. Design Example This chapter presents load computations, analysis, and design of a portal frame for two cases of rigid and semirigid construction to illustrate the application of the stiffness derivation developed in Chapter 3. The results of these cases will then be compared. For an assumed typical 3story, steel frame office building, a typical interior onebay frame is selected for analysis and design. The building uses planar portal frames spaced at 30 feet in both directions and is 41 feet high. This frame acts as the main lateralforce resisting system for the building by means of momentresisting structural tee connections. It is unbraced in the short direction of the building, but braced diagonally in the long direction such that the column ends are free to rotate about the weak axis and not free to translate to that axis. See Fig.4.1. The building is assumed to be located in downtown Denver, Colorado. The applicable Uniform Building Code (UBC) can then be used. 26
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1 30' Sec. 11 2 2 Framing Plan 1 3@ 30'=90' Sec. 22 Fig.4.1 A Typical Interior Frame under Study 27 13' 13' 15'
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4.1 Computation of Loads The selected interior frame (Fig.4.1, Sec.11) is subjected to gravity and wind loads. Both loading conditions are developed in Appendix B. This section shows the results, which are the design loads, acting on the frame (Fig.4.2). 2 k /ft 13' 2 k /ft 13' 15' 30' Fig.4.2 Unbraced Frame under Design Loads 28
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4.2 Approximate Rigid Frame Analysis Since the frame is statically indeterminate, an approximate analysis is to be carried out to determine its preliminary member sizes for a subsequent exact analysis. Two load cases are considered: gravity and wind. Both approximate analyses, along with their moment diagrams, are provided in Appendix B. This section shows the results in Tables 4.1 and 4.2. The combined effect of gravity and wind is as shown in Table 4.3 for maximum moments and axial forces. 29
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4., 8 31i 7 2 1i 6 1 5 Table 4.1 Member Moments and Axial Forces Due to Gravity Member Max. M (ftk) Min. M (ftk) Axial N (k) 12 37.3 (cw) 18.7 (cw) 119.30 (c) 23 43.7 (cw) 40.5 (cw) 74.63 (c) 34 72.9 (cw) 40.5 (cw) 38.75 (c) 56 37.3 (ccw) 18.7 (ccw) 119.30 (c) 67 43.7 (ccw) 40.5 (ccw) 74.63 (c) 78 72.9 (ccw) 40.5 (ccw) 38.75 (c) 26" 144.0 81.0 neglected 37" 144.0 81.0 neglected 48 129.6 72.9 neglected CW Clockwise CCW Counterclockwise C Compression (*) Beam 30
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4., 8 3 t1 7 2 t1 6 1 5 Table 4.2 Member Moments and Axial Forces Due to Wind Member Top & Bottom M (ftk) Axial N (k) 12 81.45 (ccw) 13.55 (T) 23 44.85 (ccw) 5.13 (T) 34 16.05 (ccw) 1.07 (T) 56 81.45 (ccw) 13.55 (c) 67 44.85 (ccw) 5.13 (c) 78 16.05 (ccw) 1.07 (c) 26* 126.3 (ccw) neglected 37 60.90 (ccw) neglected 48* 16.05 (ccw) neglected CCW Counterclockwise T Tension C Compression (*) Beam 31
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Table 4.3 Combined Effect of Gravity and Wind Loads Beam Max. M (ftk) 48 129.60 + 0.0 = 129.60 37 144.00 +0.0 = 144.00 26 81.0 + 126.30 = 207.30 Column Max. end M (ftk) Min. end M (ftk) Axial N (k) 78 72.9+6.05=88.95 40.5+16.05=56.55 38.75+1.07=39.82 67 43.7+44.85=88.55 40.5+44.85=85.35 74.63+5.13=79.76 56 37 .3+81.45=118. 75 18.7+81.45=1 00.15 119.33+13.55=132.88 32
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4.3 Preliminary Design of Rigid Frame The allowable stress design method (AISC, 1989) is used to select preliminary member sizes for beams and columns. The member shapes are rolled W shapes. When wind acts with gravity, the allowable stresses can be increased by 1/3 (AISC, 1989, A5.2). Alternatively, this provision can be observed by reducing the combined gravity and wind loads to 75% of their total value. The latter is used in this study. Beams and columns are designed for two loading cases: gravity alone and combined gravity and wind. Controllingdesign sections are then selected. 4.3.1 Design for Gravity Alone Beams. For gravity loading the horizontal reactions at the base of columns are small; thus, the axial forces in beams are small and assumed zero (Wang, 1983, p.507). Therefore, the beams in the frame are designed for bending moment only. The beams are assumed to have their compression flange braced at every 6 feet against lateral displacement (lateral buckling). 33
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Beam 48 (roof) Assume the section is compact. Fb = 0.66 Fy = 0.66 (36) = 24 ksi M = 129.6 ftk Sx = M I Fb = [ (129.6) (12)/ (24)] = 64.80 in3 Try W 16 x 40 ( Sx = 64.7, MR = 128 ftk). Say o.k. for preliminary selection. F'y > 65 ksi > Fy = 36 ksi, flange is compact For fa= 0 (axial force neglected), .!!_ = 16 01 = 52.5 < 640 = 106.7, web is compact t w 0.305 .J36 Lc = 7.4 ft >6ft, bracing is adequate Use W 16 x 40 for roof beam. Using the same beamselection procedure, the floor beams are selected. The beam sizes are summarized in Table 4.4. Table 4.4 Preliminary Beam Sizes for Gravity Loading Beam Size MR(ftk) lx {in4 ) Sx (in3 ) A (in2 ) 48 W16x40 128 518 64.7 11.8 37 W21 x44 162 843 81.6 13.0 26 W21 x44 162 843 81.6 13.0 34
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Columns. The frame does not experience a sidesway because of symmetry in gravity loading and structure. Column 34 (third story) Assume Kyly = 1 x 13 = 13 ft to control and check later. Assume Cm = 0.85 m = [(2.3 + 2.2)/ (2)] = 2.25 Pett =Po+ Mx m = 38.75 + 72 .90 X 2.25 = 202.8 k Try W 12 x 45 ( Pa = 202 k). Subsequent approximation m = [(2.1 + 2)/ (2)] = 2.05 Pett = 38.75 + 72.9 X 2.05 = 188 k Try W 12 x 45 ( Pa = 202 k). A 13.20 in2 lx 350.00 in4 Sx 58.10 in3 rx/ ry 2.65 rx 5.15 in ry 1.94 in Lc 8.50 ft Lu 17.70 ft F'ex (Kxlx)2 I (1 0)2 = 275.00 k Check W 12 x 45 for possible buckling about Xaxis. Use the alignment chart (AISC, 1989, Fig. 1, p.3.5) for this purpose. ( AISC, 1989, p.3.5) 35
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Top of column 350 GA = 5?8 = 1.56 30 Bottom of column Assume the secondstory column to have approximately the same size as the one in the third story. 350 350 +G = 13 13 = 1 92 B 843 30 Use Kx = 0.84 If Xaxis buckling governs, r KyLy 2... = 1 X 13 X 2.65 = 34.5 rY Thus, YY axis governs. Check W 12 x 45 for adequacy regarding interaction equations (AISC, 1989, Chapter H, p. 5.54). f. = P = 3875 = 2.94 ksi a A 13.2 KxLx = 0.84x13x12 = 25.4 r:r 5.15 36
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KrLr = 1 x 13 x 12 = 80.4' say 80.5. rY 1.94 Fa = 15.24 + 15.36 = 153 ksi 2 fa = 2 94 = 0.192 > 0.15 Fa 15.3 Use equations H1.1 and H1.2 (ASD of AISC, 1989). Since no transverse loads act on the column, the maximum moment at the ends of the unbraced length is used for ASD equation H1 (McCormac, 1992, p.263). = = 72.90xl2 =} 5 06 ksi 58.1 Since Lc = 8.5 ft < 13ft< Lu = 17.7 ft, Fbx = 0.66 FY = 0.66 x 36 = 24 ksi@ column ends Fbx = 0.60 F = 0.60 x 36 = 22 ksi @ column middepth Cmx = 0.60.4 (M1 / M2 ) = 0.60.4 (+ 40.5/72.9) = 0.4 (AISC, Chapter H) F' (275 )(10)2 = 230.6 k u (0.84x 13)2 c= = 0.4 = 0.41 < 1 }fa }2.94 Fez' 230.6 Use 1.0 (McCormac, 1992, p. 263). fa+ =2.94+1xl5.06=0.877<1 O.k. (AISC,1989,EQ.H11) Fa (}fa JF. 15.3 22 F u' 37
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___b_+ fb = 2.94 + 15.06 = 0 _761 <1 O.k. (AISC, H12) 0.6FY Fbx 22 24 Use W 12 x 45 for column 34 (third story). Usjng the same columnselection procedure, the remaining story columns are selected. The column sizes are summarized in Table 4.5. Table 4.5 Preliminary Column Sizes for Gravity Loading Column Size pa (k) lx(in4) Sx(in3 ) A(in2> 34 W 12x45 202 350 58.1 13.2 23 W 12 x45 202 350 58.1 13.2 12 W 12 X 50 206 394 64.7 14.7 Design conditions are the same for columns 78, 67, and 56. Thus, these columns will have the same sizes as the opposite columns. 38
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4.3.2 Design for Combined Gravity and Wind Beam. The 1 I 3 increase in allowable stress due to the combined effect of gravity and wind resulted in smaller beam sizes than required by gravity dead and live loads alone, except for the second floor beam where both the gravity and combined effect furnished the same size beam. Therefore, the gravity loads govern the beam design, and the beam sizes selected in Table 4.4 are considered the final preliminary beam sizes for approximate design. Columns. The columns are designed for the maximum end moments and the axial forces shown in Table 4.3. The frame under the combined loading of gravity and wind is free to sideway in the plane of loading, but it is diagonally braced in the perpendicular plane. Column 78 (third story) Assume Kyly = 1 x 13 = 13 ft to control and check later. Assume Cm = 0.85 m = [(2.3 + 2.2) I (2)] = 2.25 Pett =Po+ Mx m = (39.82 + 88.95 X 2.25] (0.75) = 179.97 k Try W 12 X 40 (Pa = 180 k). Subsequent approximation m = [ (2.1 + 2) I (2)] = 2.05 Pett = [39.82 + 88.95 X 2.05] (0.75) = 166.63 k Try W 12 x 40 (Pa =180 k). A 11.80 in2 lx 310.00 in4 Sx 51.90 in3 rxlry 2.66 in rx 5.13 in ry 1.93 in Lc 8.40 ft Lu 16.00 ft 39
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Check W 12 x 40 for possible buckling about Xaxis. Use the alignment chart (AISC, 1989, Fig. 1 p. 3.5) for this purpose. G = Lc Lg Top of column 310 GA = 51(8 = 1.38 30 Bottom of column Assume the column in the second story to have approximately the same size as the one in the third story. Ge= 2x310 13 843 = 1.70 30 Use Kx = 1.47 (sidesway uninhibited) If Xaxis buckling governs, 40
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KyLy rz = 1 X 13 X 2.66 = 34.58.ft ry Thus, YY axis governs. Check W 12 x 40 for adequacy regarding interaction equations (AISC, 1989, Chapter H, p.5.54). f. = P = 39.42 = 3.34 ksi a A 11.8 KrLr =1x13x12=81 rY 1.93 Fa= 15.24 ksi (AISC, 1989, Table C36, p.3.16) fa = 3 34 = 0.219 > 0.15 Fa 15.24 Use ASD equations H1.1 and H1.2. = M z = 82.95 x 12 19 _18 ksi J bz Sz 51.90 Since Lc = 8.40 ft < Lb =13ft< Lu =16ft, Fbx = 0.66 Fy = 0.66 x 36 = 24 Ksi @ column ends Fbx = 0.60 Fy = 0.60 x 36 = 22 ksi @ column middepth Cmx = 0.85 (AISC, 1989, Chapter H, p.5.55) F' = ( 273 )(10)2 = 74.76 ksi e.r (1.47 X 13)2 = 0.89 < 1 174.76 41
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Use 1.0 (McCormac, 1992, p.263). f. + ( c_t) = 3 34 + 1 x 19 18 = 1.09 <1.33 a.k. (AISC, 1989, EQ.H1.1 l Fa 1 fa 15.24 22 F' hz t!Z fbx = 3 34 + 19.18 =0.951 <1.33 o.k. (AISC, 1989, EQ. H1.2) 0.6Fy Fhx 22 24 The factor 1.33 is due to the consideration of wind loads. W 12 x 40 is adequate for the combined effect of gravity and wind, but it is smaller than required by gravity alone. Thus, gravity governs. Use W 12 x 45 for column 78 (third story). Using the same columnselection procedure, the remaining design process for the combined effect of gravity and wind indicates that both gravity and the combined effect produced similar column selections for the secondstory column. However, the combined effect controlled the column design for the firststory column. Table 4.6 summarizes the final preliminary column sizes for the approximate analysis of the rigid frame under study. Table 4.6 Final Preliminary Column Sizes for Approximate Analysis Column Size Pa (k) lx {in4 ) Sx (in3 ) A (in2 ) 78 W 12 x45 202 350 58.1 13.2 67 W 12 X 45 202 350 58.1 13.2 56 W 12 X 58 276 475 78.0 17.0 It should be noted that the columns in each story are of the same size. 42
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4.4 Exact Rigid Frame Analysis The final preliminary member sizes for the approximate analysis summarized in Tables 4.4 and 4.6 are used as input for an exact analysis of the same rigid frame. The exact analysis uses a MATLAB computer program developed for plane structures by Professor John Mays of the University of Colorado at Denver as part of a course in matrix structural analysis. The program matrix formulation for analysis uses the stiffness method. The stiffness method treats an entire structure, such as the frame under study, as an assembly of members connected together at their end points. These points are called nodal points, whose independent displacements are primary unknowns. The formulation is built on three requirements of compatibility, material law, and equilibrium essential for the analysis of any indeterminate structure. The program task is to first compute these primary displacements. Then, the secondary unknowns, memberend forces (axial, shear and moment), are computed using the forcedeformation relationship for each member. To do this task, certain handprepared input data are needed (Section 4.4.1). In Section 4.4.2, the MATLAB program main logical tasks are outlined. Section 4.4.3 presents the results of the exact analysis. 4.4.1 HandPrepared Input Data For the analysis of the frame under study with MATLAB, the following input data preparations are in order. Use units of kips and feet. 1. Determine in the global coordinate system the number of independent displacements at each nodal point (points whose displacements are not restrained by a support). Call these displacements alphatype degrees of freedom. Number alphatype degrees of freedom starting from 1 for translation in global Xdirection, followed by 2 in global Ydirection and by 3 for rotation about global Zdirection. Use righthand rule for this purpose. For this study, alphatype degrees offreedom are 18. See Fig.4.3 (a). 43
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T 11 ( 7 1 0 4 i 9 12 7 3 i i 7 51 14 7 4 c T 7 13 5 6( 15 7 j j 2 i i 8 2 17 3(1 7 1 ( 17 16 6 18 7 201 123 1 1' i 9 719 ( .7 22 21(/7 /7 24 /7 /7 (a) Alphaand Betadegrees (b) Connectivity and member of Freedom numbers Fig.4.3 Degrees of Freedom and Member Numbering for Rigid Frame 44
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2. Determine in the global coordinate system the number of independent displacements restrained at each support. Call these displacements betatype degrees of freedom. Number betatype degrees of freedom starting from (alpha+1) in X, Y, and Z directions. For this study, starting from 19, betatype degrees of freedom are 6. See Fig.4.3(a). 3. Form the sum of alphatype and betatype degrees of freedom and call the sum n. That is, n = alpha + beta. The number n represents the number of simultaneous linear equations to be solved by the MATLAB computer program for the nodal displacements. For this study, n = 18 + 6 = 24. 4. Form a row matrix of member connectivity, which is called "lm matrix" in the MATLAB computer program for this study. The member connectivity flags the program which nodal point members are connected to. The program uses this information to assemble the structural stiffness matrix. For this study, the member connectivity is designated as i and j. See Fig.4.3 (b). 4. Number the members sequentially. See Fig. 4.3(b). 5. Determine L, E, and section properties I, A for the members. 6. Convert in the member local coordinate system and conforming to local degrees of freedom any concentrated or distributed load between nodal points on the frame to concentrated loads at those points at the end of each member. See Fig.4.4 for member local coordinate system with the sign convention for memberend degrees of freedom (righthand rule). Use the standard formulas for fixedend force calculations for beams, whether by hand or by a computer program written for such a conversion and place them in a column matrix. In this study, the conversion is done by hand. ly 4 t_2 ____ t 47 X ".71 3 ".71 6 Fig. 4.4 Member Local Coordinate System with Sign Convention for Positive 45
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7. Find the angle of inclination (theta) of the positive local xaxis of each member from positive global Xaxis. This angle, measured in radians, is positive, if its direction is counterclockwise. 4.4.2 MATLAB Program Tasks Once the MATLAB computer program receives the input data, it carries out the frame analysis. The main logical tasks in the analysis procedure are as outlined in the flowchart of Fig.4.5. 46
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I I I I Read Structural Data I '\ v I Form Member Stiffness Matrix I '\ 1/ Assemble Structural Stiffness Matrix I '\ / Compute Nodal Displacements '\ / Compute Member End Forces I '\ / Write Nodal '\ / I Write Member End Forces J / I End I Fig.4.5 Flowchart of MATLAB Tasks for Rigid Frame Analysis 47 I I I
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4.4.3 Results of Exact Rigid Frame Analysis The exact analysis is performed for two load cases of gravity alone and combined gravity and wind. The computer input I output data used and produced by the MATLAB program for both loading cases are provided in Section 8.3 of Appendix B. In this section, the results of the analyses, which are the moments and axial forces, are presented in Tables 4.7 and 4.8. 48
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Table 4.7 Exact Rigid Analysis (Moments and Axial Forces) Due to Gravity Member Max. M (ft.k) Min. M (ft.k) Axial N (k) 12 58.5 (cw) 29.4 (cw) 87.0 (c) 23 65.7 (cw) 57.1 (cw) 57.0 (c) 34 105.7 (cw) 76.0 (cw) 27.0 (c) 56 58.5 (ccw) 29.4 (ccw) 87.0 (c) 67 65.7 (ccw) 57.1 (ccw) 57.0 (c) 78 105.7 (ccw) 76.0 (ccw) 27.0 (c) 26 124.2 100.8 neglected 37* 133.0 92.0 neglected 48 105.7 96.8 neglected CW Clockwise CCW Counterclockwise C Compression (*) Beam 49
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Table 4.8 Exact Rigid Analysis (Moments and Axial Forces) Due to Gravity and Wind Member Max. M (ftk) Min. M (ftk) Axial N (k) 12 71.57 (ccw) 4.03 (ccw) 74.7 (c) 23 27.05 (cw) 6.11 (cw) 51.5 (c) 34 85.10 (cw) 64.7 (cw) 25.6 (c) 56 129.9 (ccw) 120.6 (ccw) 99.3 (c) 67 108.05 (ccw) 104.51 (ccw) 62.5 (c) 78 126.31 (ccw) 87.54 (ccw) 28.37 (c) 26 225.24 22.97 neglected 37 195.6 70.8 neglected 48 126.3 85.1 neglected CCW Counterclockwise CW Clockwise C Compression (*) Beam 50
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4.5 Exact Design of Rigid Frame With reference to Tables 4.7 and 4.8, the controlling loading case is used to design beams and columns of the rigid frame, followed by rigid connection design. 4.5.1 Beams Beam 48. The case of gravity alone is larger than that of gravity and wind, for which 75% of its value is allowed by codes (or a 1/3 increase in allowable stress). Assume a compact section and a 6ft lateral bracing interval for the compression flange of the beam. Fb = 0.66 Fy = 0.66 x 36 = 24 ksi M = 105.7 ftk > 75%(126.3) = 94.7 ftk S = M = 105.7x12 =52 _85 in3 ;r F. 24 b Try W 18 X 35 (Sx = 57.6, MR = 114 ftk). FY. = > 65 ksi > FY = 36 ksi, For fa= 0 (axial force neglected), !!____ = 17.7 =59< 640 = 640 = 1 06.7, t,. 0.3 JF: .J36 Lc = 6.3 ft, Use W 18 x 35 for roof beam. flange compact web compact bracing is adequate 51
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Beam 37. The case of combined gravity and wind is larger than that of gravity alone. M = 75%(195.6) = 146.7 ft.k > l33ft.k Using the same beamselection procedure, the size of beam 37 is selected. Use W 21x 44 for thirdfloor beam. Beam 26. The case of combined gravity and wind is larger than that of gravity alone. M = 75%(225.24) = 168.93ft.k > l24.20ft.k Using the same beamselection procedure, the size of beam 26 is selected. Use W 18 x 50 for secondfloor beam. The results of exact design of beams are shown in Table 4.9. Table 4.9 Exact Design of Beams for Rigid Frame Beam Size MR (ftk) lx (in4 ) Sx (in3 ) A (in2 ) 48 W 18 X 35 114 510 57.6 10.3 37 W21 x44 162 843 81.6 13 26 W 18 X 50 176 800 88.9 14.7 52
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4.5.2 Columns Column 78. The case of gravity alone is larger than that of combined gravity and wind, which is reduced to 75% of its total value. Assume sidesway for the frame even though gravity is going to control the design, since as will be seen shortly, for the other columns in the frame the case of gravity and wind is going to control the design. Assume = 1 x 13 ft to control and check later. em = 0.85 m = 2.25 (AISC, 1989, Table 8, p. 3.1 0) pelf= Po +Mxm = [27 + 105.7(2.25)] = 264.8k > [28.37 + 126.31(2.25)](75%) = 234.4k TryW 12 x 53 (P8 = 268 k} Subsequent approximation m = 2.05 Pelf = [27 + 105.7(2.05)] = 243.7k TryW 12 X 53 Lc 10.60 ft Lu 22.00 ft A 15.60 in2 lx 425.00 in4 Sx 70.60 in3 rx 5.23 in rx I rJ. 2.11 (10} 2 = 284 k Check W 12 x 53 for possible buckling about Xaxis. Use the alignment chart (AISC, 1989, Fig1, p.3.5) for this purpose. 53
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I_&_ G= Lc Lg Top of column 425 13 G A = 51 0 = 1.92 30 Bottom of column Assume the secondstory column to have approximately the same size as the one in the third story. 425 425 +13 13 GB = 843 = 2.33 30 1.6 (sidesway uninhibited) If Xaxis buckling controls, r KYL2= 1 X 13 X 2.11 = 27.43 rY Thus, YY axis governs. Check W 12 x 53 for adequacy regarding interaction equations (AISC, 1989, Chapter H, p. 5.54). 54
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J: = P = _E._ = 1. 73 ksi a A 15.6 KxLx = 1.6xl3x12 = 47 72 rx 5.23 KYLY = 1xl3x12 = 63 rY 2.48 Fa = 17.14 ksi (ASIC, 1989, Table C.36, p. 3.16). fa = 1.73 = 0.10 < 0.15 Fa 17.14 Use equation H1.3 (ASD of AISC, 1989). fhx= Mx = 105.7x12 =17.97 ksi Sx 70.6 fa + fbx = 1.73 + 17 97 = 0.10 + 0.75 = 0.85 < 1 o.k. (ASIC, 1989, EQ. H1.3) Fa Fhx 1 7.14 24 Use W 12 x 53 for thirdstory columns. Column 67. The case of combined gravity and wind is larger than that of gravity alone. The frame is free to have a sidesway. Assume Kyly = 1 x 13 ft to control and check later. Cm = 0.85 m = 2.25 Pelf = [62.5 + 108.05(2.25)](75%) = 229.21k >[57+ 65.7(2.25)] = 204.83k Try W 12 X 53 Subsequent approximation m = 2.05 Pelf = [62.5 + 108.05(2.05)](75%) = 213k >[57+ 65.7(2.05)] = 191.7 k This axial load results in W 12 x 50. However, to have one less column splice, try W 12 x 53 to run the height of the second and third stories. The top of the firststory column will be spliced to the bottom of the secondstory column at the appropriate height above the second floor. The design of column splice is beyond the scope of this study; thus, it is not included. 55
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Just as the thirdstory column, this column satisfies all the applicable code requirements of AISC, 1989. Use W 12 x 53 for secondstory columns. Column 56. The case of combined gravity and wind is larger than that of gravity alone. The frame is free to have a sidesway. Assu_me K.,LY = 1 x 15 = 15 ft to control and check later. em 0.85' m =2.2 Pelf = [99.3 + 129.9(2.2)](75%) = 288.81k > [87 + 58.5(2.2)] = 215.7k TryW 12 X 58 Subsequent approximation m =2 Pelf = [99.3 + 129.9(2)](75%) = 269.3k > [87 + 58.5(2)] = 204k Try W 12 X 58 (P a = 276 k) A 17.00 in2 lx 475.00 in4 Sx 78.00 in3 rx/ry 2.10 rx 5.28 in ry 2.51 in Lc 10.60 ft Lu 24.40 ft Check W 12 x 58 for possible buckling about Xaxis. Use the alignment chart (AISC, 1989, fig.1, p. 3.5) for this purpose. 56
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(AISC, 1989, p.3.5) Top of column 425 475 +13 15 Ga = 800 = 2.41 30 Bottom of column Gs = 1. Use Kx = 1.5 (sidesway uninhibited) If Xaxis buckling governs, KxL>= KYLY K L K L rx >,or l: x> y yr_T rY rY r KYLY.2...=1x15x2.10=31.5 ft rY Thus, YY axis governs. Check W 12 x 58 for adequacy regarding interaction equations (AISC, 1989, Chapter H, p. 5.54). + = P = 99 3 = 5.84 ksi Ja A 17 K:xL:x = 1.5x 15 x12 = 51.14 r" 5.28 KYLY = lx15x1271.7 l rY 2.51 Fa = 16.25 ksi 57
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fa = 5 84 = 0.36 > 0.15 Fa 16.25 Use ASD equations H1.1 and H1.2. I' =129.9x12=19.98 ksi s 78 Since Lc = 10.6 ft < Lb =15ft< Lu = 24.4 ft, Fbx = 0.60 FY =0.60 x 36 = 22 ksi@ column middepth Fbx = 0.66 FY =0.66 x 36 = 24 ksi@ column ends Cm = 0.85 (AISC, 1989, Chapter H, p. 5.55) F' = 289(10)2 = 57 ksi e.r (1.5 X 15)2 cna = 0.85 = 0.95 < 1 1 fa 1 5.84 F' 51 e.r Use1.0 (McCormac, 1992, p. 263). Ia + = 5.84 + 1x19.98 =1.267 <1.33 O.k. (AISC, 1989, EQ. H1.1) Fa (1fa )F. 16.25 22 F' e.r fbx = 5.84 19.98 = 1.098 < 1.33 o.k. (AISC,1989, EQ. H1.2) 0.6FY 22 24 Use W 12 x 58 for firststory columns. 58
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The results of exact design of columns are shown in Table 4.10 Table 4.10 Exact Design of Columns for Rigid Frame Column Size Pa (k) lx (in4 ) Sx (in3 ) A (in2 ) 78 W 12 X 53 268 425 70.60 15.60 67 W 12 X 53 268 425 70.60 15.60 56 W 12 X 58 276 475 78.00 17.00 It should be noted that the columns in each story are of the same size. 59
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4.5.3 Rigid Connection To determine the initial stiffness of structural tee connections to be used as input in the MATLAB computer program for semirigid frame analysis, the connections must be designed first. Each connection is designed for the maximum moment at the end of its respective beam. The maximum moment results from the controlling loading case used to design the beam itself. The connections are WTshape. The results of the rigid connection design are summarized in Table 4.11. The design details are provided in Appendix B. Table 4.11 Rigid Structural Tee Connection Sizes Connection Location Beam WT 13.5 X 64.5 Roof level W 18 X 35 ... WT 13.5 X 64.5 Third floor W21 x44 WT15x74 Second floor W 18 X 50 60
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4.6 Exact Semirigid Frame Analysis with Ki(total) To analyze the same steel frame when semirigid structural tee connections are used between the beams and columns, each connection is modeled as a linear rotational spring, as shown in Fig.4.5. Each spring has two rotational degrees of freedom at its ends and thus will have a stiffness matrix of 2 x 2. Each spring stiffness matrix contains four values, each of which is the value calculated manually (Section 4.6.1) from Equation (A.35) for the connection represented by that spring. The same MATLAB computer program, whose flowchart of tasks for rigid frame analysis appears in Fig.4.5, will be used to analyze the frame having semirigid connections, with one extra input. The input will be the initial stiffness of each connection manually calculated and placed in the structural data portion for the program .to read. The program then forms a 2 x 2 stiffness matrix for each spring representing each connection. It is noted that this part of the MATLAB computer program was also developed by Professor John Mays of the University of Colorado at Denver. The rest of the semirigid analysis follows the way the flowchart in Fig.4.5 depicts, with one extra out put. The extra output will be the moment in each spring, which is actually the moment transmitted by each connection. Section 4.6.2 presents the results of the analysis. 61
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62
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4.6.1 Calculation of Connection Stiffness Ki(total) The MATLAB results from exact analysis (Section 4.4.3) were used to do the exact design of rigid frame (Section 4.5) for beams and columns. The beam end moments were used to design rigid connections (Section 4.5.3) for the frame. In this section the rigid connection dimensions are used to calculate the initial stiffness Ki(total) of each connection to be used in MATLAB computer program for semirig1d frame analysis (Section 4.6.2). Initial stiffness for roof beam connection: Roof beam Connection A lbot(web) Lbot(web) W 18 X 35 ( d = 17. 7) WT 13.5 X 64.5 1.100 in br 10.010 in \v 0.610 in d 13.815 in g 4.000 in L 8.000 in (length of tee connection) g = 4 =2 in 2 2 _!_ (8)(1.10)3 = 0.8873 in4 12 (8)(1.1 0) = 8.8 in2 __!_ (8)(0.61) 3 =0.151 in4 12 13.815l.IO = 13.265 in 2 63
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E 29.0 x 106 psi G 11 .2 X 1 06 psi d2 4E/hot(web) Ki(totar> = L3 3 L + L _1_ + __ hot( web) (A.35) 24EI 5GA (17.7)2 (4)(29 X }06)(0.151) = (2)3 3(2) 13.265 ''+''':::(24)(29 X I 06)(0.8873) (5)(11.2 X 106 )(8.8) = 12467 x 106 in lb + 1.32 x 106 in lb rad rad =1039000 ftk rad The initial stiffnesses of the remaining connections are obtained in similar fashion. The results of the calculations are as shown in Table 4.12 for all three connections. Table 4.12 Initial Stiffness of Designed Tee Connections Connection Location Initial Stiffness (ftk I rad) W 13.5 x64.5 Roof level 1039000 W 13.5 X 64.5 Third floor 1416000 W 15 x74 Second floor 1235000 64
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4.6.2 Results of Semirigid Frame Analysis The initial stiffness values from Table 4.12 are used in the MATLAB computer program for semirigid frame analysis. The load cases used for this analysis are the same as those for rigid frame analysis: (1) gravity alone, and (2) gravity and wind. The computer input I output data used and produced by the MATLAB program are provided in Section 8.4 of Appendix B. This section presents the results of the analysis, which are the moments and axial forces in Tables 4.13 and 4.14. 65
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Table 4.13 Exact Semirigid Analysis (Moments and Axial Forces) Due to Gravity with Ki(totat) Member Max. M (ft.k) Min. M (ft.k) Axial N (k) 12 58.1 (cw) 29.2 (cw) 87 (c) 23 65.2 (cw) 56.7 (cw) 57 (c) 34 105.1 (cw) 75.6 (cw) 27 (c) 56 58.1 (ccw) 29.2 (ccw) 87 (c) 67 65.2 (ccw) 56.7 (ccw) 57 (c) 78 105.1 (ccw) 75.6 (ccw) 27 (c) 26 123.3 101.7 neglected 37* 132.2 92.8 neglected 48 105.1 97.4 neglected CW Clockwise CCW Counterclockwise C Compression (*) Beam 66
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Table 4.14 Exact Semirigid Analysis (Moments and Axial Forces) Due to Gravity and Wind with Ki(total) Member Max. M (ft.k) Min. M (ft.k) Axial N (k) 12 72.2 (ccw) 4.1 (ccw) 74.8 (c) 23 26.7 (cw) 5.5 (ccw) 51.5 (c) 34 84.4 (cw) 64.3 (cw) 25.6 (c) 56 130.1 (ccw) 119.9 (ccw) 99.2 (c) 67 107.7 (ccw) 103.9 (ccw) 62.5 (c) 78 125.8 (ccw) 86.9 (ccw) 28.4 (c) 26 222.8 22.6 neglected 37 194.7 69.9 neglected 48 125.8 84.4 neglected C W Clockwise CCW Counterclockwise C Compression (*) Beam 67
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Observations: A comparison of member moments due to the exact rigid frame analysis (Table 4.7) with those due to the exact semirigid analysis using Ki(total) (Table 4.13) for gravity alone shows a minimal reduction in the values of end moments for beams and columns of semirigid frame (1 ft.k and 1 ft.k). For beams, the reduction was distributed to the middle of the beams; for columns, to the other ends. This is important in that it confirms that semirigid connections redistribute moment in a member (Gerstle, 1988, p. 241 ). Also, a similar comparison for the loading case of gravity and wind (Table 4.8 and Table 4.14) shows a similar minimal reduction, except a reduction of 2.4 ft.k at one end of beam 26 and of 9.3 ft.k between its ends. Again, some redistribution of moments occurred. This observation substantiates that the connections remained rigid, as they were designed for. As such, Professor Mays of the University of Colorado at Denver notes, it is not surprising to see that the connections are so stiff as compared to the rotational stiffness of the connected beam (4EII L) that nothing would be gained by including their initial stiffness in the frame analysis (Mays, 2001 ). Table 4.15 shows this comparison. Table 4.15 Comparing Connection Initial Stiffness with Beam Rotational Stiffness Connection Location Ki(total> (ft.k I rad) Beam (ft.k I rad) wr 13.5 x 64.5 Roof Level 1039000 13909 wr 13.5 x 64.5 Third floor 1416000 22636 Wf15x74' Second floor 1235000 22636 68
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Another observation is that Equation (A.35) is sensitive to variations in three parameters. One is the moment of inertia of the tee flange cross section. Equation (A.35) is directly proportional to this parameter. An increase in the cross section moment of inertia increases the connection initial stiffness. The second parameter is d, the overall depth of the beam that is attached to the connection. This d is used as the moment arm of the moment resistance of the connection. The initial stiffness equation is directly proportional to the square of d. Even taking the moment arm as the distance from center to center of the top and bottom flanges of the beam reduces the initial stiffness of the corresponding connection. The third parameter whose variation makes Equation (A.35) sensitive is L1 L1 is taken as half the bolt gage in the tee flange. The initial stiffness defined by this equation is inversely proportional to the cube of L1 A small (one or two inches) increase in bolt gage (limited in length as it is by practice) reduces the initial stiffness appreciably. A bolt gage as small as 4 inches in the design of connections (Section 4.5.3) may, therefore, partly explain the lack of appreciable flexural and shear deformations in the connection, as these deformations were the basis of the derivation of Equation (A.35). However, an increase in bolt gage may activate such a prying action that the connection may have to be designed against it. In the light of these observations, the connection deformations due to flexure and shear alone do not seem to be enough to render it semirigid. Other sources of deformation must contribute to the deformation of this connection if it must function as semirigid. Chapter 5 investigates these sources. 69
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5. Modified Initial Stiffness KMr(total) It was mentioned in the last paragraph of Section 4.6.2 that "Other sources of deformation must contribute to the deformation of this connection if it must function as semirigid." What are those sources? Several come to mind, all in contact with the web and flanges of the connection. Those in contact with the connection web are beam ends and bolts fastening the web to the beam flanges (not investigated in the present study). Those in contact with the connection flanges are bolts fastening the flanges to the supporting column and the areas of the column around the connection, all treated together as column deformations and investigated in this section. When a frame deforms every element of it deforms. A column is an element of the frame. Therefore, the column deforms. Ignoring column deformations assumes that connection has a rigid support (Faella et al., 2000, p.71). It might be the case that a connection has such a low stiffness that the assumption of the supporting column being rigid may be reasonable. But, for connection with relatively high stiffness, column deformation investigation can prove important as far as the deformational behavior of the connection. A number of column deformations have been identified as contributing to the deformation of extended endplate connections (Yee and Melchers, 1986, p. p. 621). They are: column flange flexure, bolt elongation, column web shear (including stiffener where applicable), and column web compression. As part of their study, Yee and Melchers used these column deformations to derive an equation to predict the initial stiffness of the connection when material is linearly elastic (Yee and Melchers, 1986, p. 620). In the present study the above column deformations are considered to contribute to the defoimation of structural tee connections as well. An equation is derived independently of the Vee and Melchers' derivation for each column deformation, and the results are used to derive a modified initial stiffness. The principles of Mechanics of Materials and of Structural Mechanics are used for this purpose. 70
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5.1 Column Flange Flexure A portion of the column in contact with the connection at the beam tension flange level is modeled as a T stub with its flange flexural deformation representing the column flange flexural deformation (Vee and Melchers, 1983, p. 622). Half of the column web is assumed to act as the web of the Tstub model. See Fig.5.1. It is assumed in this study that the column flange is not stiffened. If the analysis shows that the column flange needs transverse stiffener, the Tstub model will be rotated 90 degrees such that the flange stiffener becomes the Tstub web. It is assumed that the T stub model flange is fixed at bolt lines. It can be shown that, due to the pull from the bolts, the state of deformation (flexure and shear) of the Tstub model flange is similar to Equation (A.23), except it uses the column dimensions, and its bolt gage is in a horizontal position. Thus, ( L3 3L J !J..cf = T m + __ m_ where the m stands for model. 24Elm 5GAm (5.1) The symbol !J..cf represents the column flange deformation, and m stands for T stub model. Other symbols are as defined in Section A.2 of Appendix A. Column Connection Fig.5.1 Tstub Modeling of Column Flange 71
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5.2 Bolt Elongation The elastic forcedeformation ( = PL I AE), found in Mechanics of Materials textbooks, enables one to find the elastic elongation of a bolt for a known axial load. In its present form, this relationship is not applicable to the case of multiple pretensioned bolts in a beamcolumn structural tee connection. The reason is that the tension force on each of these bolts is indeterminate. As a result, a modified version of the above equation is developed in this section based on the method of consistent deformation for the solution of such a statically indeterminate problem. In 1980, Popov developed an elastic forcedeformation relationship based on the same method to find the force in a single pretensioned bolt gripping two washers of total thickness L, where L represented the gripping length of the bolt (Popov, 1980, p. 413). A similar but more involved relationship is presented in the present study to account for the presence of multiple bolts and two connected flanges (connection and column) of different thicknesses. It is assumed that the bolts in the tee connection are pretensioned to a tension force prescribed by ASIC, 1989, p. 27 4 so that the flanges of the connection and column are gripped (clamped) together firmly. Fig.5.2 shows the connection bolts pretensioned and no external load acting. (a) Col. & Connection (b) Free Body Diagram Fig.5.2 Bolts Pretensioned The symbols in Fig.5.2 are as follows. 72
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n number of bolts in tee connection flange Ti pretension force in each bolt Ci clamping force (assumed uniform) on contact area around each bolt Writing force equilibrium equation for the free body diagram in Fig.5.2(b), L Fy = 0. Ti = ci (5.2) After the external load (T) is applied to the connection, a free body diagram is obtained as shown in Fig.5.3. T n (I;+ X) Fig.5.3 Tee Connection under External Load (T) The new symbols in Fig.5.3 are as follows. Y portion of applied load (T) reducing clamping force on contact areas X portion of applied load (T) increasing tension in each bolt Writing force equilibrium equation for the free body diagram in Fig.5.3, L Fv = 0. T + n (CiY) n (Ti +X) = 0. (5.3) Substitution of Equation (5.2) into equation (5.3) yields X+Y=T (5.4) n 73
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Equation (5.4) has two unknowns X andY; the problem is statically indeterminate. If the flanges stay in contact, compatibility condition is 8bolts = 8flanges but 8 _(X_)_( L"'e.ff_) holtA E b and (Y)(Lfig) 8 flanges = A E comp (5.5) (5.6) (5.7) Substitution of the right hand sides of Equations (5.6) and (5.7) into Equation (5.5), yields (5.8) In Equation (5.8), Ab bolt area Acomp compression area around each bolt under clamping force. Acomp is assumed to be a circular area of about 3 bolt diameters (Kuzmanvic, 1983, p. 326): A II 2 [II z] comp4(3Dh) = 9 4(Db) = 9Ab (5.9) Db bolt diameter E modulus of elasticity Lett bolt effective length (Ballio et al., 1983, p. 289) equal to sum of thicknesses of connected plates (flanges, in this study) and onehalf the sum of thicknesses of bolt head and bolt nut L11g sum of thicknesses of tee and column flanges tt tee flange thickness tcf column flange thickness See Fig.5.4 for effective length of a bolt. 74
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Fig.5.4 Effective Length of Bolt (5.10) (5.11) Solving Equations (5.4) and (5.8) as a.system of two equations with two unknowns and using Equation (5.9), one obtains (5.12) 75
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and (5.13) Substitution of Equation (5.13) into Equation (5.6) yields bolt elongation as 1 /!.bolt =n (5.14) Similarly, one can obtain an equation for which is not used in the present study. 76
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5.3 Column Web Shear Deformation Yee and Melchers consider a portion of the column as a short column whose height is the vertical distance between the centerlines of the beam flanges. This short column, assumed fixed at the beam bottom flange level, is then subjected to inplane shear force from the beam top flange (Yee and Melchers, 1986, p. 626). See Fig.5.5. 1.! de 1 "' (d4J) Col. Web Me Beam <> d , , Fig.5.5 Column Web under Shear A shear deformation equation has been derived by Maugh based on the principle of virtual work applied to beams (Maugh, 1946, p.37). This equation, which is in integral form, is used in the present study to develop an inplane column web shear deformation equation. Maugh's equation is as follows. A = s GA cw (5.15) where V maximum shear in column web equal to beam top flange tension force v maximum shear in column web due to a unit load at top of column web G shear modulus Acw column web area equal to (few x de ), where tcw and de are column web thickness and column overall depth, respectively. 77
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Evaluation of the integral in Equation (5.15) between its limits (entire length of column web) leads to the column web shear deformation equation offered in Yee and Melchers' study, where they suggest k=1 for 1beams (Yee and Melchers, 1986, p. 627). This equation is as follows. 11 = (k(T)(l) ... = k(T)(dtbf) .. 0 GAC,. r GAC,. (5.16) 78
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5.4 Column Web Compression Deformation Lee and Melchers consider a square region of the column as a plate having the size of dr x dr, where dr is the column depth between root fillets. They assume the plate to be fixed at its top and bottom in theYdirection and subjected to a uniform compression force from the beam compression flange in the Xdirection (Vee and Melchers, 1986, p. 627). To aid visualization, a brief sketch is presented in Fig.5.6. C=T Fig.5.6 Column Web under Compression The following derivation is offered in the present study leading to Vee and Melchers' Equation (44) for column web compression deformation (Yee and Melchers, 1986, p. 628). 79
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From generalized Hooke's law for multiaxial loading (Beer and Johnson, 1992, p. 81), (5.17) where GX normal strain in Xdirection sY normal strain in Ydirection ax normal stress in Xdirection v poisson's ratio crY normal stress in Ydirection az normal stress in Zdirection E modulus of elasticity For the case of plane stress, crz = 0. Thus, Equations (5.17) become ax vay cry vax s =and s =x E E' y E E (5.18) From the first relation in (5.18), (5.19) Substitution of Equation (5.19) into the second relation in (5.18) yields (5.20) 80
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Since the plate is assumed fixed at its top and bottom in Y direction, s Y = 0. Thus, Equation (5.20) becomes or The normal stress u x on the plate is T (j =X (d,)(tcw) Substitution of Equation (5.23) into Equation (5.22) yields Rearrangement of Equation (5.24) yields The normal strain sx on the plate is (}cwx s = X d r where 81 (5.21) (5.22) (5.23) (5.24) (5.25) (5.26)
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deflection of column web in Xdirection Substitution of Equation (5.26) into Equation (5.25) yields (5.27) Multiplying both sides of Equation (5.27) by ( 1v2 ) leads to the column web compression deformation Equation (44) ofYee and Melchers, which is (5.28) 82
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5.5 Total Deformation The total deformation in the tension zone of the column due to the tension force from the top flange of the connected beam is obtained by adding Equations (5.1), (5.14), and (5.16). This total deformation is as follows. (5.29) When the column web compression deformation, Equation (5.28), is added to Equation (5.29), the total column deformations contributing to the deformation of the structural tee connection, Equation (A.23), is obtained as (5.30) The sum of Equations (5.29) and (A.23), both at the beam top flange level, as well as Equation (5.28) cause the relative rotation of the end of the beam with respect to the column. This sum is as follows. (5.31) A simplified deformation diagram is in order to show the relationship between Equation (5.31) and the relative of the end of the beam with respect to the column. See Fig.5.7. D,.totol(ten) + D,. B(total) 7 T d (beam depth) Fig.5.7 Relative Rotation of Beam End with Respect to Column 83
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From Fig.5.7, one obtains f/J = (/1total(ten) +f1B(total) + b"cw.r) d (5.32) The moment resistance of the connection still is as Equation (A.25), which is repeated here for convenience. Mtop = (T)(d) (5.33) Finally, the modified initial stiffness of the connection is defined as Mtop K Mi(tota/) = f/J+ Ki(bot) (5.34) where Ki(botl represents the initial stiffness contribution from the bottom tee web, which was introduced in Equation (A.33). Substitution of Equations (5.32) and (5.33) into Equation (5.34) yields KMi(total) = [(/1 /1 b" )] + Ki(bot) tota/(len) + ;(total) + cw.r (T)(d) (5.35) 84
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Substitution for the deformations ll >, ll and g within the total(ten B(total) cw.r parentheses on the righthand side of Equation (5.35) yields the modified initial stiffness for structural tee connection as follows, with T dropping off. KMi(tota/) =[A +A +A +A +Ki(bot)] I 2 3 4 S where A = (_!l.__ + 3 L1 J from Equation (A23) I 24EJ 5GA A2 = ( Lm 3 + 3Lm J from Equation (5.1) 24El111 5GA"' 1 ( Lelf J from Equation (5.14) 1 + Left X 2_ A6E Lflg 1 k(d t ) A4 = "1 from Equation (5.16) GACW As = (1 v2 ) from Equation (5.28) E(tcw) K [4Elbot(web)] from Equation (A.33) i(bOt) I Lbot(web) (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) Finally, the moment 'resistance of the semirigid structural tee connection is M = {KMi(tota/)Xt/J) (5.43) 85
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5.6 Exact Semirigid Frame Analysis with KMi(total) The modified initial stiffness, Equation (5.36), is used in this section to re analyze the same frame analyzed before in Section 4.6. 5.6.1 Calculation of Connection KMi(total) The value of modified initial stiffness is manually calculated for each of the connections. The values are then used as input into the MATLAB program for a semirigid frame analysis (Section 5.6.2). Roof beam connection: Beam Connection Column W18 X 35 (d=17.7) WT13.5 X 64.5 W12 X 53 (d=12.06) From Equation (5.37), A = (2 )3 + 3 ( 2 ) = 0.02513 x 106 in 1 24(29 X 106 )(0.8873) 5(11.2 X} 06 )(8.8) /b I= _!_(8)(0.575) = 0.12674 in4 12 A= (8)(0.575) = 4.6 in2 Then, A = (2 )3 + 3 ( 2 ) =0.11400x106 in 2 24(29x106)(0.12674) 5(11.2x106)(4.6) lb For A3 n Bolt diameter Hex. Bolt head Hex. Bolt nut Column flange thickness, tct 4 bolts in each connection flange 0.750 in 0.500 in (AISC, 1989, p.142) 0.625 in (AISC, 1989, p.143) 0.575 in 86
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Tee flange thickness, = 1.100 in Lett = (tct + + 0.5 (bolt head +bolt nut) = (0.575 + 1.1) + 0.5 (0.5 + 0.625) = 2.2375 in = (tcf + = (0.575 +1.1) = 1.675 in Ab = n (0.75)2 4 = 0.442 in2 Then, A = _!.[ 1 ]( 2.2375 ) = 0.00335 x 106 in 3 4 1+ 2.2375 x2 0.442x29x106 lb 1.675 1 For A4 d tbf tcw de Acw = (tcw X de ) Then, = (0.345 X 12.060) = 4.16070 in 17.700 in 0.425 in 0.345 in 12.060 in A = 1(17.70.425) = 0.37071 x 106 in 4 11.2 X 10 6 X 4.16070 /b 0.30 (Poisson's ratio) 0.345 in 87
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Then, A = 1(0.30)2 = 0.090955x106 in S 29 X 106 (0.345) /b 6 inlb ( 64 ) Ki(bot) = 1.32 X 10 p. rad The sum of A1 through A5 is as follows. A1+A2+A3+fov.+A5 = (0.02513+0.1140+0.003351+0.37071+0.090955) x 10e = 0.604146 x 106 in lb Finally, Equation (5.36) yields K =[ (17 7 )2 +1 32x106 ] Mt(total) 0.604146x106 = 519.89 x 106 inlb rad = 43300 ftk rad The values of the modified initial stiffness of the connection at third floor and second floor are calculated in similar fashion. The results of modified initial stiffness for all three connections are summarized as shown in Table 5.1 along with Ki(total) and rotational stiffness of the corresponding beams for a comparison. As seen from this table, the column deformations influenced the rotational deformation of the connection with the result that the initial stiffness of the connections, as modified in this example, is reduced considerably. Also seen from this table is that the modified initial stiffness values are now closer to the values of the rotational stiffness of the corresponding beams. 88
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Table 5.1 Comparison among Ki(total), KMi(total). and Beam Rotational Stiffness Connection Location Ki(total) (ft.klrad) wr 13.5 x 64.5 Roof level 1039000 wr 13.5 x 64.5 Third floor 1416000 WT 15 X 74 Second floor 1235000 89 KMi(total) Beam (ft.klrad) (ft.klrad) 43300 13909 53500 22636 48900 22636
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5.6.2 Results of Exact Semirigid Analysis with KMi(total) The initial stiffness values for KMi(total) in Table 5.1 are used in the MATLA8 computer program for semirigid frame analysis. The analysis is performed for two loading cases: (1) gravity, and (2) gravity plus wind. The computer input I output used and produced for both cases are provided in Sections 8.5.1 and 8.5.2 of Appendix B. This section presents in Tables 5.2 and 5.3 the results obtained. The results are member moments and axial forces. 90
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Table 5.2 ExactSemirigid Analysis with KMi(total) (Moments & Axial Forces) for Gravity Member Max. M (ft.k) Min. M (ft.k) 12 46.6 (cw) 23.5 (cw) 23 60.4 (cw) 48.9 (cw) 34 97.0 (cw) 66.0 (cw) 56 46.6 (ccw) 23.5 (ccw) 67 60.4 (ccw) 48.9 (ccw) 78 97.0 (ccw) 66.0 (ccw) 26 118.0 107.0 37 114.9 110.1 48 105.5 97.0 CW Clockwise CCW Counterclockwise C Compression (*) Beam 91 Axial N (k) 87.0 (c) 57.0 (c) 27.0 (c) 87.0 (c) 57.0 (c) 27.0 (c) neglected neglected neglected
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Table 5.3 Exact Semirigid Analysis with KMi(totar) (Moments & Axial Forces) for Gravity and Wind Member Max. M (ft.k) Min. M (ft.k) Axial N (k) 12 87.3 (ccw) 6.2 (ccw) 75.4 (c) 23 25.4 (cw) 5.8 (ccw) 51.2 (c) 34 72.8 (cw) 58.0 (cw) 25.4 (c) 56 133.7 (ccw) 99.0 (ccw) 98.6 (c) 67 103.6 (ccw) 95.5 (ccw) 62.8 (c) 78 121.8 (ccw) 73.8 (ccw) 28.6 (c) 26 194.6 19.3 neglected 37* 177.4 52.5 neglected 48 121.8 72.8 neglected CW Clockwise CCW Counterclockwise C Compression (*) Beam 92
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5.7 Exact Design of Semirigid Frame with KMi(totaJ) The controlling loading cases in Tables 5.2 and 5.3 are used to design each member of the semirigid frame. The design of semirigid connections for this frame and a check on prying action in the connections follow the procedures used to design rigid connections (Section 4.5.3) and will not be demonstrated. 5.7.1 Beams All three beams in the semirigid frame are designed for the controlling loading case. The procedures of design are the same as those used to design the rigid frame beams (Section 4.5.1). The results are as shown in Table 5.4. Table 5.4 Exact Beam Design for Semirigid Frame with KMi(total) Beam Size MR (ft.k) Sx (in3 ) Controlling Load 48 W 18 X 35 114.0 57.6 Gravity 37 W 18 x40 135.0 68.4 Gravity + Wind 26 W21 x44 162.0 81.6 Gravity + Wind Observation. A comparison of member moments due to exact rigid frame analysis (Table 4.7) with those due to the exact semirigid analysis using KMi(total) (Table 5.2) for gravity alone shows a major reduction in beamend moments (reduced by a maximum value of 18.1 ft.k). The reductions were distributed to the middle of the beams; thus, a redistribution of moments occurred by semirigid connections. All column end moments were reduced (reduced by a maximum value of 11.9 ft.k). A similar comparison for the loading case of gravity and wind (Tables 4.8 and 5.3) also shows reduction in beam and column moments. At beam ends, the moments are reduced by a maximum value of 30.6 ft.k and by a minimum value of 4.5 ft.k; redistribution of these moments in between the beam ends are from 17.8 ft.k to 4.3 ft.k. At column ends, column 12 experiences an increase in moments of 15.7 ft.k 93
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and 2.17 ft.k, and column 56 a decrease of 21.9 ft.k with an increase of 3.7 ft.k. The remaining columns experience moment reduction (max. 12.3 ft.k). 5.7.2 Columns All six columns in the semirigid frame are designed for the controlling loading case. The procedures of design are the same as those used to design the rigid fame columns (Section 4.5.2). The results are as shown in Table 5.5. Table 5.5 Exact Column Design for Semirigid Frame with KMi(total) Column Size Pa (k) Sx (in3 ) Controlling Load 78 W12x53 268.0 70.6 Gravity 67 W 12 X 53 268.0 70.6 Gravity + Wind 56 W 12 X 58 276.0 78.0 Gravity + Wind It should be noted that the columns in each story are of the same size. 94
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5.8 Comparison of Rigid and Semirigid Although the exact semirigid analysis with KMi(total) reduced and redistributed moments in comparison with the exact rigid analysis of the same frame, the design of members based on the former produced lighter members in only two of the beams (combined gravity and wind controlled the design), but did not change the design of others. Savings from .these lighter sections are as follows. Third floor beam: W 18 x 40 (Semirigid design) v.s. W 21 x 44 (Rigid design) Saving in weight (30 ft)(44)(30 ft)(40) = 120 lb Saving in Sx [ (81.668.4) I (81.6) 1 = 16% Second floor beam: W 21 x 44 (Semirigid design) v.s. W 18 x 50 (Rigid design) Saving in weight (30 ft)(50) (30 ft)(44) = 180 lb Saving in Sx [ (88.981.6) I (88.9) 1 = 8% In case of steel frames with many members, the savings can be considerable. 95
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6. Discussion, Conclusion, and Further Study 6.1 Discussion A theoretical approach is used in this thesis to analytically derive an elastic momentrotation relationship to predict the initial stiffness of a semirigid structural tee connection in beamcolumn steel portal frames. First, an elastic initial stiffness equation is derived based on the flexural and shear deformations of the connection itself. Second, a series of elastic relationships is derived for column deformations contributing to the deformation of such a connection. The results are then combined to present a single elastic equation to predict the initial stiffness of the connection used as semirigid. The objective of these derivations is to present an initial stiffness equation for use in the computer matrix analysis of semirigid steel portal frames using a structural tee for connection, aiming at capturing connection behavior in the analysis of such frames and thus reducing member weight. It is emphasized that the equation is only theoretical; it has not been tested experimentally. There are three steps in using this initial stiffness equation in a semirigid frame analysis. Step 1. The frame is analyzed and designed as rigid, and rigid structural tee connections are designed for it accordingly. Step 2. The dimensions of the connections designed in Step 1 are then used to compute the initial stiffness value for each connection to be used in sub sequent analysis using a computer program with semirigid analysis capabilities. Step 3. The results of the semirigid analysis in Step 2 are used to redesign members and semirigid connections. Prying action on the connections are then checked. 96
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6.2 Conclusion It was shown in the present thesis that a semirigid structural tee connection can reduce moments in beams and columns of steel portal frames. For some beams the reduction in moments resulted in lighter sections. In addition, it was shown that the connection deformation based on flexure and shear alone could not render the connection semirigid; various column deformations contributed to the connection rotational deformation with the result that moment reduction occurred leading to savings in weight for two beams. 6.3 Further Study Areas needing further study are suggested as follows. 1. A laboratory test be conducted on a bolted structural tee connection in a beamcolumn assembly to measure its momentrotation response under the application of static loading. The momentrotation curve from the test can be used to compare the initial stiffness equations Ki(total) and KMi(totaJ) with the test results. 2. A parametric study be carried out to determine a practical range of variations in three variables of moment of inertia (a function of the connection flange thickness), overall depth of a beam, and bolt gage (Section 4.6.2) that could help a structural tee function as semirigid. 3. A study be carried out to determine what value of a structural tee initial stiffness warrants its inclusion in semirigid frame analysis. The value(s) can be compared with the rotational stiffness of the connected beam (4EI/ L) to set a limit that, for example, if the initial stiffness is three times (or less) as large as the beam rotational stiffness, the inclusion of the initial stiffness can result in moment reduction in the connected beams and/or columns. 4. A study be carried out using a computer program for semirigid frame analysis to include sidesway of the frame using structural tee connections. This was originally suggested to the writer by professor Gerstle of the University of Colorado at Boulder(Gerstle, 2000). 97
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APPENDIX A A. Derivation of Semirigid Structural Tee Total Initial Stiffness K;uorat} A. 1 General Assumptions A.2 Notations A.3 Sign Convention A.4 Initial Stiffness for the Top Tee A.5 Initial Stiffness for the Bottom Tee A.6 Total Initial Stiffness for the Connection 98
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A. Derivation of Semirigid Structural Tee Total Initial Stiffness K;(1o1ar) This appendix provides general assumptions, notations, sign convention, and derivation details of semirigid structural tee total initial stiffness Ki(lolal) A.1 General Assumptions The following general assumptions apply to the entire derivation. 1. Material behaves linearly elastic, implying the use of superposition. 2. Axial deforrf!ation is negligible. 3. Small deformation theory is applied. 4. Bolts are highstrength and prtensioned; so, bolt line acts as fixed support. 5. Bolts will not yield. 6. Beam end will not deform locally. 99
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A.2 Notations d E FEMAB FEMBA FEM/Jc. FEMcB FEMBD FEMDB HA HB He I 12 ]bot( web) Ki(bot) Ki(top) Ki(total) Lt L2 Lbot(web) M Mbot(web) Mlop MAB MBA MBC McB Beam depth Modulus of elasticity Fixedend moment at A end of member AB Fixedend moment at B end of member AB Fixedend moment at B end of member BC Fixedend moment at C end of member BC Fixedend moment at B end of member BD Fixedend moment at D end of member BD Shear force at A end of member AB Shear force at B end of member AB Shear force at C end of member BC Moment of inertia of members AB and BC Moment of inertia of member BD Moment of inertia of bottom tee web Bottom tee web initial stiffness Top tee initial stiffness Connection initial stiffness Length of members AB and BC Length of member BD Length of bottom tee web Semirigid connection moment resistance Moment resistance of bottom tee web Moment resistance of top tee Moment at A end of member AB Moment at B end of member AB Moment at B end of member BC Moment at C end of member BC 100
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flop( web) f bot(web) w"rg Moment at B end of member BD Moment at D end of member BD Neutral axis Tensile force at top flange of the beam Thickness of structural tee flange Thickness of top tee web Thickness of bottom tee web Beamend bearing reaction Greek Symbols eA eB E>c E)D /!,.B OB /!,.bot( web) /!,.B(total) Rotation of joint A Rotation of joint B Rotation of joint C Rotation of joint D Rotation of beam end relative to column Horizontal deflection of joint B due to bending stress Horizontal deflection of joint B due to shear stress Deflection of bottom tee web Total horizontal deflection of joint B due to flexure and shear 101
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A.3 Sign Convention The sign convention for the slopedeflection method used in this study is as follows. If the moment at the end of the member is clockwise, the moment will be taken as positive; if the moment at the joint is counterclockwise, it will be taken as positive; if the chord rotation is clockwise, it will be taken as positive. A.4 Initial Stiffness for Top Tee K;cwp) The top tee of the semirigid structural tee connection is modeled as a plane frame having a depth of one inch into the paper. This oneinch frame will be obtained by passing two parallel planes one inch apart and parallel to the plane of the web of the beam the connection supports. See Fig.A.1. 1 Flange<: z Fig.A.1 Modeling of Top Tee as a Frame 102
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Fig.A.2 illustrates the frame modeled in Fig.A.1. The loaded frame is shown deflected away (explained below) from the supporting column in Fig.A.2(a), The same deflected frame is depicted as a line diagram in Fig. A.2(b). Note from Fig. A.2(b) that the frame has two fixed supports at A and 8, but it is free to translate horizontally under the action of a tension force. Now, consider a semirigid structural tee connection subjected to the moment (M) from the end of the beam. The moment can be considered as the moment of a couple consisting of two forces (T) and (C). The force (T) will be tensile from the top flange of the beam, and the force (C) will be compressive from the bottom flange of the beam, when the beam is loaded from the top. The forces (T) and (C) can be obtained as T= C= M/d. Note that the forces (T) and (C) are not known until after the frame analysis with semirigid structural tee connection is completed. As the force (T) is transmitted to the column, it subjects the beam top flange bolts to single shear, the top tee web to tension, the top tee flange to both flexural and shear stresses, and the top flange bolts to tension. As for the behavior of the top tee flange, being under flexure and shear, it is predicted that it tends to deflect away horizontally from the face of the column. Such a deflection is assumed to be maximum at midflange point B. See Fig.A.2(a). Therefore, there is a deflection (I:!. 8 ) due to flexure, and there is a deflection ( o 8 ) due to shear. Deflection (I:!. 8 ) will be treated first; deflection ( o 8 ) next. 103
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A L2, 12 Bolt line (Fixed Support) v v r"' 1 T T D L 1, I D ... ... As L 1, I As '\::__ Bolt line (Fixed Support) c (a) (b) Fig.A.2 Top Structural Tee as a Frame under Tension (T) 104
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A.4.1 Deflection Due To Flexure The frame in Fig.A.2(b) is statically indeterminate. The slopedeflection method will be used to solve the frame for the deflection Step 1. The slopedeflection equations for all three members of the frame in Fig.A.2(b) are as follows. 2El( MAB = FEMAB +2E>A +E>B 3L, L1 (A.1) (A.2) 2El( MBc = FEMBc +2E>B +E>c 3L1 L 1 (A.3) (A.4) (A.5) 2El2 ( MDB = FEMDB +2E>D +E>B 3L2 L2 (A.6) Step 2. The terms that are zero in Equations (A.1) through (A.6) will drop. Also, using the sign convention in Section A.3, Hence, (A.7) 105
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M = 2 E 1(2e BA L B L I I M = 2 E/(2e 3 BC L B L I I M = 2EI (e 3 CB L B L I I MBv = 2EI2 (2eB) L2 Step 3. Moment equilibrium equation for joint 8 is MBA ".._71 MBC MBD Step 4. Moments in (A.12) will be substituted for. Hence, 106 (A.8) (A.9) (A.1 0) (A.11) (A.12)
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This equation yields Therefore, there is no rotation at joint B in the frame, which is expected from the symmetry of the frame and loading. Step 5. Shear equilibrium equation for FBD of the frame is MAe HA A o7 T y B cl":../ He Mea (A.13) 107
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Step 6. Two moment equilibrium equations for FBD of members AB and BC are as follows. For member AB B MeA LMB. =0 (A.14) For member BC Mac B He l1 c He Mea LMB =0 Msc Mcs Hell =0 (A.15) 108
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Step 7. Shear forces from (A.14) and (A.15) are and Step 8. Shear forces HA and He are substituted into shear Equation (A.13). MAB +MBA_ Moe +Men =T LI LI or (A.16) Step 9. Moments in Equation (A.16) are substituted for, and the equation is solved for ( 1:10 ) This equation yields (A.17) 109
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A.4.2 Deflection (8B ) Due To Shear If the ratio of depth to length of a beam is less than 0.1, the effect of shear in strain energy is neglected (Beer and Johnson, 1992, p.579). The ratio of the thickness of member AB (and also of BC) to its length (L1 ) of the structural tee is greater than 0.1. This statement can be verified by taking such a ratio for most of structural tees in (AISC, 1989); set Equation (A.21) to zero if a ratio is found less than 0.1. Thus, the shear stress effect on strain energy and on deflection is considered for ratios greater than 0.1. Shear stress variation on transverse rectangular cross section is parabolic and given as follows (Beer and Johnson, 1992, p.287). The crosssections of the members in the frame are rectangular. Therefore, where shear stress on the cross section v shear force on the cross section A the area of the cross section = 1 x tt (Fig.A.3) y distance from the neutral axis of the cross section to the fiber in question c half the thickness of the cross section Step 10. The Castigliano's theorem on deflection (Beer and Johnson, 1992, p.611) will be applied to members AB and BC of the frame. See Fig.A.3. au I! =where '" av 2 u = f!._m,, where 2G u elastic strain energy stored in a member v shear force on the cross section = T/2 G shear modulus of elasticity 110 (A.18)
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fJv differential volume A N. A. B D c (c) Top Tee 1inch Strip Fig.A.3 Flange of Top Structural Tee under Shear Total strain energy in the frame = strain energy in AB + strain energy in BC That is, ulital = UAB +UBC [ 'Z'2 ] [ 'Z'2 ] 'Z'2 = Jdv + Jdv = 2 J=dv v2G v 2G v 2G Partial derivative of U,o,al with respect to shear force 01 = T /2) is au total = 2 J!_ ar dv aT vGBT' (A.19) where (A.20) Ill
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Step 11. Substitute (A20) into (A.19) and substitute the result into (A.18). Thus, 2 L, 3 y 1"dy J dx 4tf ( )' From this equation, 8 =A = 3TLt B .. 5GA (A.21) Note that the same result for deflection 0 B due to shear could be found using the principle of virtual work method. Actually, this study applied the method and found the same result as in Equation (A.21). Step 12. Total horizontal deflection of join 8 is obtained by superposition of Equations (A.17) and (A.21). Thus, AB(Iotal) = AB + OB (A.22) Hence, !1 = TL3t + 3TL1 B(lotal) 24 EJ SGA (A.23) 112
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Step 13. The rotation ( ) of the end of the beam relative to the column will be as shown in Fig.A.4. As mentioned earlier, the semirigid connection supplies this rotation to the end of the beam. In other words, the semirigid connection must rotate first under the tension force (T) in order for the beam end to have a rotation. Thus, the is also the rotation of the semirigid conneCtion. It is assumed that this rotation will take place about the center of rotation of the connection at the fillet face of the bottom tee (Fig.A.4). T ::,: f:ion @ fillet face Fig.A.4 Rotation of Beam End Relative to Column 113
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Thus, the .rotation of the connection about its center of rotation can be written from the geometry of deformation in Fig. A.4 as = 11 B(rora/) d A.24) Step 14. The resisting moment of the top tee is obtained by taking moment of T about the point of application of the compressive force (C). Therefore, M,op = (T Xd) Step 15. Solve (A.23) forT and substitute forT into (A.25) to obtain (d) L1 3L1 +24EI 5GA (A.25) (A.26) Substitute for 11 from (A.24) into (A.26) yields the relationship for the B(lolal) 'I' top tee as d2 ::L3 3L I +24EI 5GA (A.27) The coefficient in Equation (A.27) is the initial stiffness contribution from the top tee. Thus, d2 K,rop) = =L3c,3L, +24EI 5GA 114 (A.28)
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A.5 Initial Stiffness for Bottom Tee Ki(bot) The bottom tee flange will be under compressive force (C) and acts as a bearing pad (Douty and McGuire, 1965, p.1 07). Thus, the web of the bottom tee will be subject to bending. A.5.1 Assumptions for Bottom Tee 1. The web of the bottom tee is under uniform bearing reaction from the beam end and remains in contact with the bottom flange of the beam. This implies that the bottom tee web will act as a cantilever beam with a constant slope equal to the rotation ( ) of the end of the beam relative to the column. 2. The critical section for the bending of the bottom tee web is at the fillet face of the web and the center of rotation for the web will be at that face (Fig.A.5). The depthtolength ratio for the bottom web is less than 0.1.This statement can be verified by taking such a ratio for most structural tees in (AISC, 1 989); use Equation (A.21) with the load, L, and A of the web if a ratio is found larger than 0.1. Thus, the shear stress effect on the deflection of a member is ignored for ratios less than 0.1 (Beer and Johnson, 1992, p.579). The slope of the bottom tee web and the beam bearing reaction are shown in Fig.A.5. Center of rotation @ fillet face Fig A.5 Bottom Web Tee under Bearing & Rotation ( ) 115
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Step 16. The deflection of the bottom web tee is found from beam deflection formula. WbrgL4 bot( web) IJ. =bot(web) SEJ Step 17. The slope of the bottom web is t/J = IJ. bot (web) Lbot(web) (A.29) (A.30) Step 18. The moment of the beam bearing reaction about the center of rotation is { X { Lbot(web) J (wbrg )L2bot(web) Mhot(web) Lbot(web)\ 2 2 (A.31) Step 19. The beam bearing reaction (wbrg) from (A.29) is substituted into (A.31), and then !J.bot(web) from (A.30) will be substituted into (A.31) to yield the relationship for the bottom tee as (4EJbot(web) J Mbot(web) L t/J bot( web) (A.32) The coefficient in Equation (A.32) represents the initial stiffness from the bottom tee. Thus, [4EJbot(web)] K;(bot>L bot( web) (A.33) 116
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Step 20. The connection initial stiffness will be the sum of the initial stiffnesses from the individual components (Chen, 1993, p.245). Thus, the sum of equations (A.28) and (A.33) yields Ki(total) = Ki(top) + Ki(hot) (A.34) Thus, Ki(lota/) = d2 4E/hot(weh) 3 L I 3LI Lhot(weh) +(A.35) 24EI 5GA Finally, substitution of K. from Equation (A.35) into Equation (3.3) yields t(lotal) the resisting moment of the semirigid structural tee connection. Thus, M = 3 d 2 + 4E/hot(weh) {tp) L I 3LI Lhot(weh) +(A.36) 24EI 5GA 117
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APPENDIXB B. Frame Analysis 8.1 Computation of Loads 8.2 Approximate Rigid Frame Analysis 8.3 MATLA8 Input/Output, Rigid Frame Analysis 8.3.1 Gravity Alone Input/Output Data 8.3.2 Gravity and Wind Input/Output Data 8.3.3 Rigid Connection Design 8.4 MATLAB Input/Output, Semirigid Analysis with K;ctotal) 8.4.1 Gravity Alone Input/Output Data 8.4.2 Gravity and Wind Input/output Data 8.5 MATLA8 Input/Output, Semirigid Analysis with KMi(totaJ) 8.5.1 Gravity Alone Input/Output Data 8.5.2 Gravity and Wind Input/Output Data 118
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8.1 Computation of Loads Two load cases of gravity and wind are developed in this section. 8.1.1 Gravity Loads Local building code provisions on gravity loads on buildings for office use needs to be consulted for the specifics that apply. For the purpose of this study, the following gravity loads common to a typical office building are used (Ambrose and Parker, 1997, pp 320361). Roof live load Roofdeadload Floor finish Ceiling, light, ducks Interior walls Exterior walls 20 psf 40 psf 5 psf 15 psf 15 psf 25 psf Building Code for the City and County of Denver, Colorado, 1990, Table 23.A, prescribes Live load (office use) 50 psf No reduction in live load is taken for roof. Design load for roof beam= 30ft x 0.06 ksf = 1.8 kif However, reduction in live load is taken for floor beams (UBC, 1997). Tributary floor area to a beam = 30 ft x 30 ft = 900 ff > 150 ff R = 0.08% (900150)= 60% R = 23.1 (1 +D/L) = 39.3% R=40% Therefore, use R=39.3%. Reduced live load for floor beams = (50 psf0.393x50 psf) = 30.4 psf Design load for floor beams = 30ft (0.034 ksf + 0.035 ksf) = 2.0 kif 119
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Although roof and floor loads are delivered to columns by beams, the axial loads on columns are calculated based on the tributary area supported by a column (Ambrose and Parker, 1997, p. 358). Live load reduction is taken for floor load affecting a column, but such a reduction is not taken for roof load. The column axial loads are calculated at the base of each story column in the frame (Crawley, Dillon and Carter, 1984, p. 146). Thirdstory exterior column: Roof tributary area to column 15 ft x 30 ft = 450 tf DL 450 tf x 40 psf /1000 = 18 kips LL 450 ff x 20 psf /1000 = 9 kips Ext. wall 13ft x 30ft x 25 psf /1000 = 9.75 kips Total DL 18 + 9 + 9.75 = 29.75 kips Column selfwt.(assumed) 2 kips Total axial force at column base, Po 18 + 9 + 9.75 + 2 = 38.75 kips Secondstory exterior column: Floor tributary area to column DL LL Ext. wall Column selfwt.(assumed) Total DL LL reduction 15 ft X 30 ft = 450 ff 450 ff x 35 psf /1000 = 15.75 kips 450 ff x 50 pst /1000 = 22.50 kips 13ft x 30ft x 25 psf /1000 = 9.75 kips 3 kips 29.75 + 15.75 + 9.75 + 3 = 58.25 kips D 40 psf(roof) + 35 psf(third floor)= 75 psf L 20 psf(roof) +50 psf(third floor)= 70 psf Total LL 9 kips(roof) + 22.5 kips(third floor) = 31.5 kips R 23.1 (1+75nO) = 48% Reduced LL (148%) x 31.5 = 16.38 kips Total axial force at column base, Po 58.25 + 16.38 = 74.63 kips Firststory exterior column: Floor tributary area to column DL LL Ext. wall Column selfwt.(assumed) Total DL 15 ft X 30 ft = 450 ff 450 ff x 35 psf /1000 = 15.75 kips 450 ft2 x 50 psf /1000 = 22.50 kips 15 ft x 30 ft x 25 psf /1 000 = 11.25 kips 4 kips 58.25 + 15.75 + 11.25 + 4 = 89.25 kips LL reduction D L 40 psf(roof) + 35 psf(3rd floor) + 35 psf(2nd floor) = 110 psf 20 psf(roof) + 50 psf(3rd floor) + 50 psf (2nd floor) = 120 psf 120
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Total LL 9.0 kips (roof) + 22.5 kips(3rd floor) + 22.5 kips(2nd floor) = 54 kips R 23.1 (1 + 11 0/120) = 44.3% Reduced LL (144.3%) x 54= 30.08 kips Total axial force at column base, Po 89.25 + 30.08 = 119.33 kips 121
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B.1.2 Wind Loads The following data are used for wind load computation (Building Code for the City and County of Denver, Colorado, 1990, Section 2311 (b)). Basic wind speed qs (stagnation pressure) Exposure 85mph 19 psf B The following equation(s), parameters, and data are also used for wind load computation (International Conference of Building Officials. UBC, 1997). P = Ce Cq qs lw p design wind pressure combined height, exposure, and gust factor coefficient pressure coefficient wind stagnation pressure Importance factor Using method 2, projected area method (UBC, 1997, sec. 1621.3), Cq 1.4 on vertical projected area (Table 16.H) 0.7 [upward] on horizontal projected area (Table 16.H) lw 1.0 (Table 16.K) Also from Table 16.6 0.62 0.67 0.72 0.76 0.84 0.95 (0 15ft) (for 20ft) (for 25ft) (for 30ft) (for 40ft) (for 60ft) 122
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Horizontal windward pressures are computed for stepped pressure diagram. p 0.62 X 1.4 X 19 X 1 = 16.50 psf 0.67 X 1.4 X 19 X 1 = 17.80 psf 0.72 X 1.4 X 19 X 1 = 19.15 psf 0.76 X 1.4 X 19 X 1 = 20.22 psf 0.84 X 1.4 X 19 X 1 = 22.34 psf 0.95 X 1.4 X 19 X 1 = 25.27 psf Vertical pressure (uplift) is (0 15ft) (1520ft) (2025ft) (2530ft) (3040ft) (4060ft) P 0.95 x 0.7 x 19 x 1 = 12.64 psf (roof) To simplify the calculations involved, the horizontal windward pressures are shown in Fig.B.1 acting on the frame for heights of 15, 30, and 60 feet above the ground level. The uplift pressure is also shown in this figure. 12.64 psf 13' 13' 20.22 psf 17.80 psf 15' Fig.B.1 Unbraced Frame under Wind Pressures (psf) 123
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The horizontal windward and uplift loads per foot are obtained by multiplying the pressures in Fig.B.1 by their respective tributary width. These calculations follow. Horizontal windward loads: 17.80 X 30ft= 534.0 plf 20.22 X 30ft = 606.6 plf 25.27 X 30 ft = 758.1 plf Uplift load: 12.64 X 30 ft = 379.2 plf = 0.38 kif (0 15ft) (1530ft) (3060ft) This uplift load is smaller than the dead load acting on the roof beams: Dead load on roof beams = 40 psf x 30ft= 1200 plf /1000 = 1.20 kif. Thus, no design considerations are needed for uplift load. The horizontal windward loads at joints are obtained by multiplying the windward loads per foot by their respective tributary heights along the height of the frame. These calculations follow. F1 = {534.0 plf X 15/2 + 606.6 X 13/2) /1000 = 7.95 k F2 = (606.6 plf X 13/2 + 758.1 X 13/2) /1000 = 8.87 k F3 = (758.1 plf X 13/2) /1000 = 4.93 k These joint loads are shown acting on the frame in Fig.B.2. 4.93 k *., 13' 8.87 k *1 13' 7.95 k ?1t 15' 30' Fig.B.2 Unbraced Frame under Joint Wind Loads 124
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8.2 Approximate Rigid Frame Analysis An approximate analysis is performed on the unbraced frame subjected to the gravity and wind loads computed in Section 8.1 (Appendix 8). The frame and its loading are shown in Fig.4.2 (Chapter 4). First, the frame is analyzed for gravity loads alone; next, for wind loads. Then the results of these two load cases will be superimposed to obtain the combined effect on the frame. 8.2.1 Gravity Acting Alone In order to approximately analyze a rigid frame for gravity loads, assumptions are made regarding the frame response to those loads (Wolfgang, 1977, P.145, McCormac and Nelson, 1997, P. 463). The assumptions are required to reduce an indeterminate frame to a determinate one so that the equations of equilibrium alone can be used to determine its member forces. These assumptions are as follows: 1. Each beam has. two inflection points usually at onetenth of the beam span from its ends. 2. The axial force in each beam is small and thus neglected. Using these assumptions, each beam is considered as simply supported at its inflection points, and each column as a cantilever (Fig.8.3) 125
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1.8 klft 2 k /ft 3' 1' 24' 1' 3' 2 k I ft 3' 3' 3' Fig.B.3 Simple Beams & Cantilever Columns For Gravity Load Analysis 126
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The moments and reactions of the beams are as follows. Midspan moments: M48 = W(L)2/8 = 1.8(24)2/8 = 129.60 ftk M37 = M2s = 2 (24)2/8 = 144.00 ftk Reactions: R4 = R8 = W(L) /2 = 1.8 x 24/2 = 21.60 k R3 = R 7 = R2 = R6 = 2 x 24/2 = 24.00 k The moments at the ends of the beams connected to the columns are calculated as: M4 = 21.6 X 3 + 1.8 X 3 X 1.5 = 72.90 ftk (ccw) M3 =24x3+2x3x1.5 =81.00ftk (ccw) M 2 = 24 X 3 + 2 X 3 X 1.5 = 81.00 ftk (ccw) The moments at the right ends of the beams connected to the right columns are the same magnitude, but in opposite direction, because of symmetry in loading and structure. The columns must receive these beamend moments. The columns above and below a joint share the moments in proportion to their stiffness values (Tall, 1974, pp. 750751, AISC, 1989, p. 5.136, Wolfgang, 1990, p. 486). However, if the columns are the same length, it is conservatively assumed that half the moment at a joint is resisted by the column above, and the other half by the column below (Tall, 1974, p. 750751). If the height of the story below a joint is larger than the one above, assume equalsize columns and distribute the moment at the joint to the columns in .proportion to the reciprocal of their lengths; this results in a higher moment in the column above, which actually receives less than that (Tall, 1974, pp. 750751). Such moment distributions are as follows At joint 3 M34 = M3 x 0.5 = 81 x 0.5 = 40.50 ftk (cw) M3 2 = M3 X 0.5 = 81 X 0.5 = 40.50 ftk (cw) 127
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From moment equilibrium of joint 3, M34 + M32 M3 = 0 40.50 + 40.50 81 = 0 O.k. From moment equilibrium of joint 4, M43 = M4 = 72.90 ftk (cw) At joint 2 M23 = M2 [(1113) I (1113 + 1115)] = 81 x 0.54 = 43.74 ftk (cw) M21 = M 2 [(1115) I (1115 + 1113)] = 81 x 0.46 = 37.26 ftk (cw) From moment equilibrium of joint 2, M23 + M21 M2 = 0 43.74 + 37.2681 = 0 O.k. To find the moment at the lower end of the firststory column, one more assumption is required. This extra assumption concerns the inflection point of the firststory column. It is assumed that the location of inflection point in the firststory column is at onethird of its fixed base (Benjamin, 1959, p. 128). Thus, the moment is determined with reference to the following free body diagram. 37.26ftk 2 f10' 7y fv = 3.73 k 5' 1 7 "'../ v = 37.26110 = 3.73 k M12 = 3.73 X 5 = 18.65 ftk 128
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The approximate bendingmoment diagram for the frame under gravity loads is shown in Fig.B.4. 129.6 Fig.B.4 Approximate Moment Diagram Due to Gravity (ftk) 129
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8.2.2 Wind Acting Alone The portal method for lateral load analysis (Wang, 1983, pp. 510514) is used to approximately analyze the rigid frame under joint wind loads (Fig.B.2). The axial forces in the columns of the frame are obtained by passing a plane horizontally at midheight of each story to cut the columns in that story (Fig. 8.5). Then, the moments are taken at each cut at each right column of the wind forces above that cut. In the following calculations the letter T stands for Tension force, and the letter C for Compression. Wind is assumed to be from the left. 4.93k_1 ___ I_ 3 rtlstory cut !N3 1' 4.93 k 71, 8.87 k +1 4.93k 8.87 k 7.95 k 151story cut J,N1 1' Fig.8.5 Frame Story MidHeight Cuts 130
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At thirdstory cut N3 X 304.93 (13/2) = 0 N3 = 1.07 k (T) At secondstory cut N2 X 304.93 (13 + 6.5)8.87 (13/2) = 0 N2 = 5.13 k (T) At firststory cut N1 X 304.93 (13 + 13 + 7.5)8.87 (13 + 7.5)7.95 (7.5) = 0 N1 = 13.55 k (T) Similarly, the axial forces in the right columns are obtained by taking the moments at each similar cut at each left column. 131
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The shear forces at each beam's inflection point are obtained by writing the equation of equilibrium of forces in Ydirection y = 0). At joint 4 41 tV4=1.07k 15' N3= 1.07 k At joint 3 At joint 2 N2 = 5.13 k t 2L_ 1' v2 = 8.42 k J, N1 = 13.55 k 132
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The moments at the ends of the beams are obtained by writing the equation of equilibrium of moments (LM = 0) at the end of each beam. 4th story beam ftk 16.05 7 15" t v4 = 1.07 k 3rdstory beam ftk 60.9 J, 15' t v3 = 4.06 k 2"d story beam ftk J, 15' t v2 = 8.42 k 133
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The counterclockwise moments at the end of columns are obtained by transferring the clockwise moments at the end of each beam to the joint that connects the beam to its column and by writing the equation of equilibrium of moments at that joint. These calculations follow on next page. 134
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I 16.0511k 16.05ftk 4 16.05 ftk 3 16.05 ftk 16.05ftk 1 60.9 ftk 44.85 ftk 44.85 ftk 44.85 ftk 44.85ftk IP 126.3ftk 81.45ftk 2 81.45 ftk 1 81.45 ftk 135
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The shear forces in each column are obtained by dividing the moment at the end of that column by half its height. These calculations follow. 16.05ftk 4 7 6.5' v = 16.05/6.5 = 2.47 k 44.85ftk 3 7 6.5' v = 44.85 /6.5 = 6.9 k 81.45ftk 2 7 7.5' V=81.45/7.5=10.86k 136
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The approximate bendingmoment diagram for the frame under wind loads from the left is shown in Fig.B.6. Fig.B.6 Approximate Moment Diagram Due to Wind (ftk) 137
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8.3 MATLAB Input/Output, Rigid Frame Analysis The handprepared data mentioned in Section 4.4.1 of Chapter 4 are entered as input in the MATLA8 computer program for an exact analysis of the rigid frame. The data were prepared for two loading cases of gravity alone and the combined effect of gravity and wind. Section 8.3.1 presents input/output data for gravity alone, while Section 8.3.2 presents input/output data for the combined effect of gravity and wind. 138
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8.3.1 Gravity Alone Input/Output Data 139
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% The frame has element loads only; It uses preliminary member sizes from % tables 4.4 & 4.6 % file name: RgdFrmGravity2.m % Units: Kips, feet %Given info alpha=18; beta=6; num_loads=1; E=29000*144; % klft112 11=475/(12114); % ft114 A1=17/144; % ft112 12=350/(12114); % ft"4 A2=13.2/144; % ft112 13=350/(12114); % ft114 A3=13.2/144; % ft112 14=518/(12114); % ft114 A4=11.8/144; % ft"2 15=843/(12114); % ft"4 A5=13/144; % ft112 16=843/(12114); % ft114 A6=13/144; % ft"2 17=350/(12114); % ft"4 A7=13.2/144; % ft"2 18=350/(12114); % ft"4 A8=13,21144; % ft"2 19=475/(12114); % ft"4 A9=17/144; % ft112 FaNodalloads=zeros(alpha,num_loads); Ub=zeros(beta,num_loads); o/o preliminary calculations n=alpha+beta; K=zeros(n,n); Fequiv=zeros(n,num_loads); % zero out Fequivalent vector for element loads % one element at a time % element 1 i end at bottom theta=pi/2; theta_deg=theta*180/pi; % local coords % transformation matrix o/o global coords L1=15; kelp1=plbm(A1,11,E,L 1); T=rotate(theta_deg); kel1 =T'"kelpf"T; efrm1 =kelp1 *T; % recovers internal forces in local coords % when using U in global coords lm1=[19 20 21 1 2 3]; K=addstiff(K,kel1,1m1 ); %element2 theta=pi/2; theta_deg=theta*1 80/pi; L2=13; kelp2=plbm(A2,12,E,L2): T=rotate(theta_deg); kei2=T'*kelp2*T: efrm2=kelp2*T; lm2=[1 2 3 4 5 6]; K=addstiff(K.kel2,1m2); 140
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%element 3 theta=pi/2; theta_deg=theta*180/pi; L3=13; kelp3=plbm(A3,13,E,L3); T=rotate(theta_deg); kel3=r*kelp3*T; lm3=[4 5 6 7 8 9]; K=addstiff(K,kel3,1m3); % element4 %theta angle is zero L4=30; kelp4=plbm(A4,14,E,L4); kel4=kelp4; efrm4=kel4; lm4=[7 8 9 10 11 12]; K=addstiff(K,kel4,1m4); fef4L=(0;27;135;0;27;135]; %local coords fef4G=fef4L; % global coords Fequiv=addrowslm(Fequiv,fef4G,Im4); % update Fequiv. note neg. sign %element 5 % theta angle is zero L5=30; kelp5=plbm(A5,15,E,L5); kel5=kelp5; efrm5=kel5; lm5=[4 56 13 14 15]; K=addstiff(K,kel5,1m5); fef5L=[0;30;150;0;30;150]; fef5G=fef5L; Fequiv=addrowslm(Fequiv,fef5G,Im5); %elementS % theta angle is zero L6=30; kelp6=plbm(A6,16,E,L6); kel6=kelp6; efrm6=kel6; lm6=[1 2 3 16 17 18]; K=addstiff(K,kel6,1m6); fef6L=[0;30; 150;0;30;150]; fef6G=fef6L; Fequiv=addrowslm(Fequiv,fef6G,Im6); %element 7 theta=pi/2; theta_deg=theta*180/pi; L7=13; kelp7=plbm(A7,17,E,L7); T=rotate(theta_deg}; kel7=r*kelp7*T; efrm7=kelp7*T; lm7=[13 14 15 10 11 12]; K=addstiff(K,kel7,1m7); 141
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%element 8 theta=pi/2; theta_deg=theta*180/pi ; l8=13; kelp8=plbm(A8, 18, E,L8); T=rotate(theta_deg); kela=r*kelp8*T; efrm8=kelp8*T; lm8=[1617 1813 14 15); K=addstiff(K,kel8,1m8); % element9 theta=pi/2; theta_deg=theta*180/pi ; L9=15; kelp9=plbm(A9,19,E L9) ; T=rotate(theta_deg); kel9=r*kelp9*T; efrm9=kelp9*T; lm9=[22 23 241617 18); K=addstiff(K,ke19,1m9) ; % all elements completed % structure stiffness matrix now formed %now form Fa and Fbcorr Fa=FaNodalLoads+rmvsm(Fequiv, 1,1 ,alpha,num_loads): %add nodal loads to element loads Fbcorr=rrnvsm(Fequiv alpha+1, 1 ,beta,num_loads); % Fbcorr to correct reactions % solve for F and U [U F]=absolve(K Fa,Ub) ; % now correct reactions F=addsm(F,Fbcorr,alpha+1 ,1); %correct reactions (NOTE : the last 1 is anchor point.) % calculate beam end forces lf1=forcem(efrm1 ,lm1 ,U) 1ft= 87.0000 5.8639 29.4406 87.0000 5 8639 58.5177 142
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lf2=forcem(efrm2, 1m2, U) lf2 = 57.0000 9.4447 65.6802 57.0000 9.4447 57.1012 lf3=forcem(efrm3,1m3,U) lf3 = 27.0000 13.9777 76.0507 27.0000 13.9777 105.6587 lf4=forcem(efrm4,1m4,U); lf4=1f4+fef4L % correct internal forces lf4 = 13.9777 27.0000 105.6587 13.9777 27.0000 105.6587 lf5=forcem(efrm5,1m5,U); lf5=1f5+fef5L % correct internal forces lf5 = 4.5329 30.0000 133.1519 4.5329 30.0000 133.1519 lf6=forcem(efrrn6,1m6, U); lf6=1f6+fef6L % correct intemal forces Its= 3.5808 143
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30.0000 124.1979 3.5808 30.0000 124.1979 lf7=forcem(efrm7,1m7, U) lf7 = 27.0000 13.9777 76.0507 27.0000 13.9777 105.6587 lf8=forcem(efrm8,1m8,U) lf8 = 57.0000 9.4447 65.6802 57.0000 9.4447 57.1012 lf9=forcem(efrm9,1m9,U) lf9 = 87.0000 5.8639 29.4406 87.0000 5.8639 58.5177 >> 144
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8.3.2 Gravity and Wind Input/Output Data 145
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% CE Thesis,Fall 2001 (Rigid plane framegravity & wind acting) Farrokh Jalalvand % The frame has nodal loads & element loads: It uses preliminary member sizes from % tables 4.4 & 4.6 %file name: RigidFrame2.m % Units: Kips, feet %Given info alpha=18; beta=6; num_loads=1; E=29000*144; % klft112 11=475/(12114); % ft114 A1=17/144; % ft112 12=350/(12114); % ft114 A2=13.21144; % ft112 13=350/(12114); % ft114 A3=13.21144; % ft112 14=518/(12114); % ft114 A4=11.8/144; % ft112 15=843/(12114); % ft114 A5=13/144; % ft112 16=843/{12"4); % ft114 A6=13.0/144; % ft112 17=350/(12"4); % ft114 A7=13.21144; % ft112 18=350/(12114); % ft114 A8=13.21144; % ft"2 19=475/(12"4); % ft114 A9=171144; % flll2 FaNodalloads=zeros(alpha,num_loads}; FaNodalloads(1, 1 )=7. 95; FaNodalloads{4, 1 )=8.87; F aNoi:talloads(7, 1 )=4. 93; Ub=zeros(beta,num_loads); % preliminary calculations n=alpha+beta; K=zeros(n,n); Fequiv=zeros{n,num_loads); % zero out Fequivalent vector for element loads % one element at a time % element 1 i end at bottom theta=pi/2; theta_deg=theta*180/pi; L1=15; kelp1=plbm{A1 ,11 ,E,L 1); T=rotate(theta_deg); kel1 =r*kelp1 *T; % local coords % transformation matrix % global coords efrrn 1 =kelp 1 *T; % recovers internal forces in local coords % when using U in global coords lm1=[19 20 21 1 2 3]; K=addstiff(K,kel1 ,lm1 ); 146
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% element2 theta=pi/2; theta_ deg=theta*180/pi ; L2=13; kelp2=plbm(A2, 12,E, L2); T=rotate(theta_deg) ; kel2=r"kelp2"T; efrm2=kelp2"T; lm2=[1 2 3 4 5 6]; K=addstiff(K, kel2, 1m2); %element 3 theta=pi/2; theta_deg=theta*180/pi; L3=13; kelp3=plbm(A3,13,E,L3); T=rotate(theta_deg) ; kel3=rkelp3*T; efrm3=kelp3*T; lm3=[4 5 6 7 8 9); K=addstiff(K, kel3, 1m3); % element4 % theta angle is zero L4=30; kelp4=plbm(A4,14 E,L4) ; kel4=kelp4; efrm4=kel4; lm4=[7 8 9 10 11 12); K=addstiff(K,kel4,1m4): fef4L=[0;27;135;0;27;] ; %local coords fef4G=fef4L; % global coords Fequiv=addrowslm(Fequiv,fef4G,Im4); %update Fequiv. note neg sign %element 5 % theta angle is zero L5=30; kelp5=plbm(A5,15,E L5); kel5=kelp5; efrm5=kel5 ; lm5=[4 56 13 14 15]; K=addstiff(K kel5,1m5); fef5L=[0;30 ; 150 ; 0;30;150]; fef5G=fef5L; Fequiv=addrowslm(Fequiv,fef5G,Im5); %elementS % theta angle is zero L6=30; kelp6=plbm(A6,16,E,L6); kel6=kelp6; efrm6=kel6; lm6=[1 2 3 16 17 18); K=addstiff(K,kel6,1m6) ; fef6L=[0;30 ; 150 ; 0;30;150]; fef6G=fef6L; Fequiv=addrowslm(Fequiv, fef6G,Im6); 147
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%element 7 theta=pi/2; theta_deg=theta*180/pi; L7=13; kelp7=plbm(A7,17,E,L7); T=rotate(theta_deg); lm7=[13141510 1112]; K=addstiff(K, kel7, lm7); %element 8 theta=pi/2; theta_deg=theta*180/pi; L8=13; kelp8=plbm(A8,18,E,L8); T=rotate(theta_deg); lm8=[16 17 18 13 1415]; K=addstiff(K,kel8,1m8); % element9 theta=pi/2; theta_deg=theta*180/pi; L9=15; kelp9=plbm(A9,19,E,L9); T=rotate(theta_deg); lm9=(22 23 24 16 17 18]; K=addstiff(K,kel9,1m9); % all elements completed % stn.icture stiffness matrix now formed % now form Fa and.Fbcorr Fa=FaNodalloads+rmvsm(Fequiv, 1,1 ,alpha,num_loads); % add nodal loads to element loads Fbcorr=rmvsm(Fequiv,alpha+1, 1 ,beta,num_loads); % get Fbcorr to correct reactions % solve for F and U (U F]=absolve(K,Fa,Ub); % now correct reactions F=addsm(F,Fbcorr,alpha+1, 1); %correct reactions (NOTE: the last 1 is anchor point.) % calculate beam end forces lf1 =forcem(efrm1 ,lm1 ,U) lf1 = 74.7232 5.0435 71.5718 74.7232 5.0435 4.0801 148
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lf2=forcem(efnn2,1m2,U) lf2 = 51.4658 2.5505 27.0457 51.4658 2.5505 6.1101 lf3=forcem(efnn3, 1m3, U) lf3 = 25.6258 11.5195 64.6743 25.6258 11.5195 85.0793 lf4=forcem(efnn4,1m4 U) ; lf4=1f4+fef4L % correct internal forces lf4 = 16.4495 25.6258 85.0793 16.4495 28.3742 126.3057 lf5=forcem(efnn5,1m5,U); lf5=1f5+fef5L % correct internal forces lf5= 0 0991 25.8400 70.7844 0.0991 34.1600 195.5844 lf6=forcem(efrrn6,1m6,U) ; lf6=1f6+fef6L % correct internal forces 149
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0.3561 23.2574 22 9656 0.3561 36.7426 225.2443 lf7=forcem(efrm7,1m7,U) lf7 = 28.3742 16.4495 87.5379 28.3742 16 4495 126.3057 lf8=forcem(efrm8,1m8,U) lf8 = 62.5342 16.3505 104.5093 62.5342 16.3505 108..0465 lf9=forcem(efrm9,1m9,U) lf9 = 99 2768 16.7065 129.8631 99.2768 16.7065 120.7350 >> 150
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8.4 Rigid Connection Design The moment at the end of each beam is used to design rigid structural tee connections for moment. The controlling loading case is used for this purpose. The notation, design procedure, and formulas of ASD method used in (AISC, 1989, pp. 4.894.92) are employed to design the connections. Notation 2P t b T Fy p a a p a a Allowable load on a structural tee, kips per inch, using maximum allowable bending stress of 27 ksi (0.75Fy) Thickness of tee flange, in. Distance from fastener line to the face of tee stem, in. Applied tension per bolt (exclusive of initial tightening and prying force), kips prying force per bolt at design load, kips Allowable tension per bolt, kips Flange thickness required to develop 8 in bolts with no prying action, in. = lBb' pFY Yield strength of the flange material, ksi Length of flange, parallel to stem, tributary to each bolt, in. Distance from bolt centerline to edge of tee flange but not more than 1.25b, in. bolt dia., in. Width of bolt hole parallel to tee stem, in. bd/2, in. a+ d/2, in. b'!a' (0 1.0). Ratio of moment at bolt line to moment at stem line = Mz/3M1 Value of a for which required thickness (treq'd) is a minimum or allowable applied tension per bolt (Tau) is a maximum. 151
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Ratio of net area (at bolt line) and the gross area (at the face of the stem) = 1d'/p Rooflevel beam 0fV18x35): This beam was designed for the controlling loading case of gravity alone, which resulted in Mmax = 105.7 ft.k at the end of the beam. The column connected to this beam is W12x53 (bt = 9.999"). 1. Tee flange Use g = 4 in., 3/4in. diameter (A325x) bolts, and p = 4 in. for tee flange. Force= M = 105 7x12 = 71.7 k, where d is the depth of the beam. d 17.7 B = 19.4 klbolt, allowable tension force (AISC, Table I.A, p.4.3). No. of bolts= Force= 71.7 = 3.7 bolts. B 19.4. Try 4 bolts minimum. T = Force = 71.7 = 17.9 k/bolt < 8 = 19.4 k/bolt o.k. 4 4 Tributary Width to one bolt = p = 4 in. P = T = 17 9 = 4.475 klbolt/in. p 4 b = g tw = 4 0 5 = 1.75 in., where 1 = web thickness of the flange of the 2 2 1V top tee connection with an assumed value of 0.5 in. 2 b 2 2(4.475) = 181 1.75 t = 0.933 in. Try WT1 0.5 x 46.5 as a preliminary selection: tf 0.930 in. tw 0.580 in. bt 8.420 in. d 10.810in. 152
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Use the preliminary selection to estimate required flange thickness. b = gtw = 4 0 580 = 1.71 > 1..!_in. wrench clearance o.k. 2 2 4 a= b1 g = 10.0104 = 2 .21in. 2 2 1.25b = 1.25(1.71) = 2.14 t=0.93in. N.G. 4 X 36(1 + 0. 797 X 0.235) Select a thickness, t, greater than or equal to treq'd =1.06 in. 153
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Try WT 13.5 x 64.5 tt 1.10 in. bt 10.01 in. tw 0.61 in. d 13.815 in. Check prying action. b= gtw = 4 0 610 =1.695in. > 1_!_ in., wrench clearance 2 2 4 a= bf g = 10.0104 =3.005 in. 2 2 1.25b = 1.25(1.695) = 2.12
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17.9 a =119.4 = 0.10 0.797 (1.10)2 1.19 t 2 1.1 Q = BrJap(f;) = (19.4 )(0. 797 )(0.1)(0.529 )( 1.19) = 0. 7 k 2. Tee web Use a length of 2p = 2(4) = 8 in. for the connection. Use% in. dia. A325x bolts, and p=4 in for bolts in the web. The connection between tee web and the top flange of the beam is considered to be bearingtype connection. Bolts are in single shear. Fv = 30. ksi (AISC, Table J32, p.5.73) Rv = AbFv = 7r (0.75)2 (30) = 13.3 klbolt 4 No. of bolts= 71.7 = 5.39 bolts 13.3 Try 6 bolts minimum, 3 bolts in each two rows in the web. Use a bolt gage of 4 in. in the web. Check WT 13.5 x 64.5, 8 in. long for the following stresses: Tension T = 0.6FyA9 = 0.6(36)(8 X 0.610) = 105 k > 71.7 k o.k. T = 0.6FuAn = 0.5(58)[82(0.75+1/8)](0.610)= 110.6 k > 71.7 k O.k. Bearing For an edge distance (parallel to tension force) = 2.30 in. > 1.5 bolt dia. = 1.5 x 0. 75 = 1.125 in. and centertocenter of holes = 4 in > 3 bolt dia. = 3 x 0.75 = 2.25 in, Fp = 1.2 Fu 1.2 X 58 = 69.6 ksi. fp = 71.7 = 26.1 ksi < 69.6 ksi o.k. 6x0.75x0.610 Use WT 13.5 x 64.5, 8 in. long for the roof beam. 155
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Using the same design procedure in AISC, 1989, rigid tee connections are selected for the third and second floor beams. They are as follows. Use WT 13.5 x 64.5, 8 in long for thirdfloor beam. Use four 1.0 in. dia. A325x highstrength bolts, 2 in each of the two rows, with g = 4 in. and p = 4 in. in the tee flange. Use eight% in. dia. A325x highstrength bolts, 4 in each of the two rows, with p = 4 in. in the tee web. The controlling loading case for design of this connection is the combined effect of gravity and wind. Prying action for this connection is Q = 1.8 k Use WT 15 x 74, 8 in long for secondfloor beam. Use four 1.0 in. dia. A325x highstrength bolts, 2 in each of the two rows, with g = 4 in. and p = 4 in. in the tee flange. Use six 1.0 in. dia. A325x highstrength bolts, 3 in each of the two rows, with p = 4 in. in the tee web. The controlling loading case for design of this connection is the combined effect of gravity and wind. Prying action for this connection is Q = 3.14 k Design of shear connection for the beams in the frame is beyond the scope of this study; thus, they are not included. 156
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8.5 MATLAB Input/Output, Semirigid Analysis with Ki(total) The handcalculated values of Ki(total) are used in the MATLA8 computer program for an exact analysis of semirigid frame. Here, the A and I from the approximate analysis are used for section properties; they could be used from the exact analysis. Two loading cases are considered: (1) gravity alone, and (2) gravity and wind. Section 8.5.1 presents input/output data for gravity alone, while Section 8.5.2 presents input/output data for gravity and wind. 157
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8.5.1 Gravity Alone Input/Output Data 158
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% CE Thesis, Fall 2001 (Semirigid plane frameGravity acting alone) Farrokh Jalalvand o/o The frame has element loads only; it uses A & I from approx. analysis & Ki(total). % file name: semirgdgrvty1.m o/o Units: Kips, feet o/o Given info alpha=24; beta=6; num_loads=1; E=29000*144; % klft"2 11=475/(12"4); % ft"4 A1=17/144; % ft"2 12=3501( 12"4); o/o ft"4 A2=13.21144; % ft"2 13=350/(12"4); % ft"4 A3=13.21144; % ft"2 14=518/(12"4); % ft"4 A4=11.81144; % ft"2 15=8431(12"4); % ft"4 A5=13/144; o/o ft"2 16=8431(12"4); % ft"4 A6=13/144; % ft"2 17=3501(12"4); o/o ft"4 A7=13.21144; o/o ft"2 18=3501(12"4); o/o ft"4 A8=13.21144; o/o ft"2 19=475/(12"4); % ft"4 A9=171144; o/o ft"2 K10=1235000; % kftfrad K11=1416000; o/o kftfrad K12=1039000; % kftfrad K13=K10; % kftfrad K14=K11; % kftfrad K15=K12; % kftfrad FaNodalloads=zeros(alpha,num_loads); Ub=zeros(beta,num_loads); o/o preliminary calculations n=alpha+beta; K=zeros(n,n); Fequiv=zeros(n,num_loads); % zero out Fequivalent vector for element loads o/o one element at a time o/o element 1 i end at bottom theta=pi/2; theta_deg=theta*180/pi; L1=15; % local coords % transformation matrix % global coords kelp1=plbm(A 1,11 ,E,L 1 ); T=rotate(theta_deg); kel1 =T*kelp1 *T; efrrn1=kelp1*T; % recovers internal forces in local coords o/o when using U in global coords lm1=[25 26 27 1 2 3]; K=addstiff(K,kel1 ,lm1); % element2 theta=pi/2; theta_deg=theta*1801pi; 159
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L2=13; kelp2=plbm(A2,12,E,L2); T=rotate(theta_ deg); kei2=T*kelp2*T; efrm2=kelp2*T; lm2=[1 2 3 5 6 7]; K=addstiff(K, kel2,1m2); % element3 theta=pi/2; theta_deg=theta*180/pi; L3=13; kelp3=plbm(A3, 13, E,L3); T=rotate(theta_deg); kei3=T*kelp3*T; efrm3=kelp3*T; lm3=(5 6 7 9 10 11); K=addstiff(K,kel3,1m3); % element4 % theta angle is zero L4=30; kelp4=plbm(A4,14,E,L4); kel4=kelp4; efrm4=kel4; lm4=[910 12 1314 16]; K=addstiff(K,kel4,lm4); fef4L=[0;27;135;0;27;135]; %local coords fef4G=fef4L; % global coords Fequiv=addrowslm(Fequiv,fef4G,Im4); %update Fequiv. note neg. sign %element 5 % theta angle is zero L5=30; kelp5=plbm(A5, 15, E, L5); kel5=kelp5; efrm5=kel5; lm5=[5 6 8 17 18 20); K=addstiff(K,kel5,1m5); fef5L=[0;30;150;0;30;150]; fef5G=fef5L; Fequiv=addrowslm(Fequiv ,fef5G Im5); %elements % theta angle is zero L6=30; kelp6=plbm(A6,16,E,L6); kel6=kelp6; efrm6=kel6; lm6=[1 2 4 21 22 24); K=addstiff(K,kel6,1m6); fef6L=[0;30;150;0;30;150]; fef6G=fef6L; Fequiv=addrowslm(Fequiv ,fef6G,Im6); %element 7 theta=pi/2; theta_deg=theta"180/pi; 160
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L7=13; kelp7=plbm(A7,17,E,L7); T=rotate(theta_deg); kei7=T*kelp7*T; efrm7=kelp7*T; lm7=[17 18 19 13 14 15] ; K=addstiff(K,ke17,1m7); %elementS theta=pi12; theta_deg=theta*180/pi ; L8=13; kelp8=plbm(A8,18,E L8); T=rotate(theta_deg); kei8=T'*kelp8*T; efrm8=kelp8*T; lm8=[21 22 23 17 18 19]; K=addstiff(K,kel8,1m8); %element 9 theta=pi/2; theta_deg=theta*180/pi; L9=15; kelp9=plbm(A9,19 E,L9); T=rotate(theta_deg); kei9=T*kelp9*T; efrm9=kelp9*T; lm9=[28 29 30 21 22 23]; K=addstiff(K,kel9,1m9); %element 10 ke11 O=SpringK(K1 0) ; efnn10=SpringFTM(K10); lm10=[3 4) ; K=addstiff(K,ke11 O,lm1 0); %element 11 kei11=SpringK(K11); efrm11=SpringFTM(K11); lm11=[7 8); K=addstiff(K,ke111,1m11 ); %element 12 kei12=SpringK(K12) ; efrm12=SpringFTM(K 12); lm12=[11 12]; K=addstiff(K,kel12,1m12); %element 13 kei13=SpringK(K13); efrm13=SpringFTM(K13); lm13={24 23]; K=addstiff(K,ke113,1m13); %element 14 ke114=SpringK(K14) ; efrm14=SpringFTM(K14); lm14=[20 19); K=addstiff(K,kel14,1m14); 161
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% elemnt 15 kei15=SpringK(K15); efrm15=SpringFTM(K15); lm15=[16 15]; K=addstiff(K,kel15,1m15); % all elements completed % structure stiffness matrix now formed % now form Fa and Fbcorr Fa=FaNodalloads+rmvsm(Fequiv, 1,1,alpha,num_loads); % add nodal loads to element loads Fbcorr=rmvsm(Fequiv,alpha+1,1 ,beta,num_loads); % get Fbcorr to correct reactions % solve for F and U [U F]=absolve(K,Fa,Ub); % now correct reactions F=addsm(F,Fbccirr,alpha+1,1 ); % correct reactions (NOTE: the last 1" is anchor point.) % calculate beam end forces lf1=forcem(efrm1,1m1 ,U) lf1 = 87.0000 5.8200 29.2203 87.0000 5.8200 58.0800 lf2=forcem(efrm2,1m2, U) lf2= 57.0000 9.3720 65.1811 57.0000 9.3720 56.6554 lf3=forcem(efrm3,1m3,U) lf3 = 27.0000 13.8979 75.5769 27.0000 13.8979 105.0964 lf4=forcem(efrm4,1m4,U); lf4=lf4+fef4L % correct internal forces 162
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lf4 = 13.8979 27.0000 105.0964 13.8979 27.0000 105.0964 lf5=lf5+fef5L % correct internal forces lf5 = 4.5259 30.0000 132.2323 4.5259 30.0000 132.2323 lf6=forcem(efrm6,1m6,U); lf6=1f6+fef6L % correct internal forces lf6 = 3.5520 30.0000 123.2611 3.5520 30.0000 123.2611 lf7=forcem(efrm7,1m7,U) lf7= 27.0000 13.8979 75.5769 27.0000 13.8979 105.0964 lfB=forcem(efrmB, 1mB, U) lfB = 57.0000 9.3720 65.1811 57.0000 9.3720 56.6554 163
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lf9=forcem(efrm9,1m9,U) lf9 = 87.0000 5.8200 29.2203 87.0000 5.8200 58.0800 lf1 O=forcem(efrm1 O,lm1 O,U) lf10 = 123.2611 lf11 =forcem(efrm11 ,lm11 ,U) lf11 = 132.2323 lf12=forcem(efrm12,1m12,U) lf12 = 105.0964 lf13=forcem(efrm13,1m13,U) lf13::;: 123.2611 lf14=forcem(efrm14,1m14,U) lf14= 132.2323 lf15=forcem(efrm15,1m15, U) lf15 = 105.0964 >> 164
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8.5.2 Gravity and Wind Input/Output Data 165
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% CE Thesis,Fall2001 (Semirigid plane frame under gravity & wind) Farrokh Jalalvand % The frame has nodal loads & element loads; it uses A & I from approx. analysis & Ki(total). % file name: semirgdcomnd1.m % Units: Kips, feet %Given info alpha=24; beta=6; num_loads=1; E=29000*144; % klft"2 11=475/(12"4); % ft"4 A1=17/144; % ft"2 12=350/(12"4); % ft"4 A2=13.21144; % ft"2' 13=350/(12"4); % ft114 A3=13.21144; % ft112 14=518/(12114); % ft"4 A4=11.8/144; % ft"2 15=843/(12114); % ft114 A5=13/144; % 16=843/(12"4); % ft"4 A6=13.0/144; % ft"2 17=350/(12"4); % ft"4 A7=13.21144; % ft"2 18=350/(12"4); % ft"4 A8=13.21144; % ft"2 19=475/(12"4); % ft"4 A9=17/144; % ft"2 K10=1235000; % kftlrad K11=1416000; % kftlrad K12=1039000; % kftlrad K13=K1 0; % kftlrad K14=K11; % kft/rad K15=K12; %kftlrad FaNodalloads=zeros(alpha,num_loads); FaNodalloads(1, 1 )=7 .95; FaNodalloads(5, 1 )=8.87; FaNodalloads(9, 1 )=4.93; Ub=zeros(beta,num_loads); % preliminary calculations n=alpha+beta; K=zeros(n,n); Fequiv=zeros(n,num_loads); %zero out Fequivalent vector for element loads % one element at a time % element 1 i end at bottom theta=pU2; theta_deg=theta*180/pi; L1=15; kelp1=plbm(A1,11,E,L1); %local coords T=rotate(theta_deg); % transformation matrix kei1=T*kelp1*T; %global coords efrrn1=kelp1*T; %recovers internal forces in local coords % when using U in global coords lm1=[25 26 27 1 2 3]; K=addstiff(K,kel1 ,lm1 ); 166
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% element2 theta=pi/2; theta_deg=theta*180/pi; L2=13; kelp2=plbm(A2,12,E,L2); T=rotate(theta_deg); kei2=T'*kelp2*T; efrm2=kelp2*T; lm2=[1 2 3 5 6 7]; K=addstiff(K,kel2,1m2); %element 3 theta=pi/2; theta_deg=theta*180/pi; L3=13; kelp3=plbm(A3,13,E,L3); T=rotate(theta_deg); kei3=T'*kelp3*T; efrm3=kelp3*T; lm3=[5 6 7 9 10 11]; K=addstiff(K,kel3,1m3); % element4 %theta angle is zero L4=30; kelp4=plbm(A4,14,E,L4); kel4=kelp4; efrm4=kel4; lm4=[9 10 12 13 14 16); K=addstiff(K,kel4,1m4); fef4L=[0;27;135;0;27;135]; %local coords fef4G=fef4L; % global coords Fequiv=addrowslm(Fequiv,fef4G,Im4); % update Fequiv. note neg. sign %elementS % theta angle is zero L5=30; kelp5=plbm(A5,15,E,L5); kel5=kelp5; efrm5=kel5; lm5=[5 6 8 17 18 20]; K=addstiff(K,kel5,1m5); fef5L=[0;30; 150;0;30;150]; fefSG=fef5L; Fequiv=addrowslm(Fequiv ,fef5G,Im5); %elementS % theta angle is zero L6=30; kelp6=plbm(A6,16,E,L6); kel6=kelp6; efrm6=kel6; lm6=[1 2 4 21 22 24]; K=addstiff(K,kel6,1m6); fef6L=[0;30; 150;0;30;150]; fef6G=fef6L; FeQuiv=addrowslm(Fequiv,fef6G,Im6); 167
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%element? theta=pU2; theta_deg=theta"180/pi; L7=13; kelp7=plbm(A7,17,E,L7); T=rotate(theta_deg); kel7=rkelp7"T; efrm7=kelp7T: lm7=[17 18 19 13 14 15]: K=addstiff(K,kel7,1m7); %elementS theta=pi/2; theta_deg=theta/pi; L8=13; kelp8=plbm(A8,18, E,LS); T=rotate(theta_ deg); kel8=rkelp8"T; efrm8=kelp8"T; lm8=[21 22 23 17 18 19]; K=addstiff(K,kel8,lm8); %element 9 theta=pU2; theta_deg=theta/pi; L9=15; kelp9=plbm(A9,19,E,L9); T=rotate(theta_deg); kel9=rkelp9"T; efrm9=kelp9"T; lm9=(28 29 30 21 22 23]; K=addstiff(K,kel9,1m9); %element 10 kel1 O=SpringK(K1 0); efrm1 O=SpringFTM(K1 0); lm10=[3 4]: K=addstiff(K,kel1 O,lm1 0); %element 11 kei11=SpringK(K11); efrm11=SpringFTM(K11 ); lm11=[7 8]; K=addstiff(K,kel11,1m11 ); %element 12 kel12=SpringK(K 12); efrm12=SpringFTM(K12); lm12=[ 11 12]; K=addstiff(K,kel12,1m12); %element 13 kei13=SpringK(K13); efrm13=SpringFTM(K13); lm13={24 23]; K=addstiff(K,kel13,1m13); %element 14 kei14=SpringK(K14); efrm14=SpringFTM(K14); 168
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lm14=[20 19]: K=addstiff(K,kel14,lm14 ): %element 15 kel15=SpringK(K15): efrm15=SpringFTM(K15); lm15=[16 15]: K=addstiff(K,kel15,1m15): % all elements completed % structure stiffness matrix now formed % now form Fa and Fbcorr Fa=FaNodalloads+rmvsm(Fequiv, 1,1 ,alpha,num_loads): % add nodal loads to element loads Fbcorr=rmvsm(Fequiv,alpha+1, 1 ,beta,num_loads): % get Fbcorr to correct reactions % solve for F and U [U F]=absolve(K,Fa,Ub): % now correct reactions F=addsm(F,Fbcorr,alpha+1, 1 ): % correct reactions (NOTE: the last 1 is anchor point.) % calculate beam end forces lf1=forcem(efrm1 ,lm1 ,U) lf1 = 74.7511 5.0873 72.2106 74.7511 5.0873 4.0986 lf2=forcem(efrm2,1m2,U) lf2 = 51.4581 2.4778 26.6625 51.4581 2.4778 5.5488 lf3=forcem(efrm3,1m3,U) lf3 = 25.6188 11.4398 64.3054 25.6188 11.4398 84.4117 lf4=forcem(efrm4,1m4, U): lf4=1f4+fef4L % correct internal forces 169
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lf4 = 16.3698 25.6188 84.4117 16.3698 28.3812 125.8483 lf5=forcem(efrm5 1m5,U); lf5=1f5+fef5L % correct internal forces lf5 = 0.0920 25.8393 69.8541 0.0920 34.1607 194.6751 lrS=forcem(efrm6,lm6,U); lf6=1f6+fef6L % correct internal forces lf6 = 0.3849 23.2930 22.5639 0. 3849 36.7070 223.7739 lf7=forcern(efrm7 ,lrn7 ,U) lf7 = 28.3812 16.3698 86.9587 28.3812 16.3698 125 8483 lf8=forcem(efrm8,1m8,U) lf8 = 62.5419 16.2778 103.8949 62.5419 16.2n8 107.7164 170
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lf9=forcem(efrm9,1m9,U) lr9 = 99.2489 16.6627 130.0618 99.2489 16.6627 119.8790 lf1 O=forcem(efrm1 O,lm1 O,U) lf10 = 22 5639 lf11 =forcem(efrm11 ,lm11 ,U) lf11 = 69 8541 lf12=forcem(efrm12,1m12,U) lf12 = 84.4117 lf13=forcem(efrm13,1m13,U) lf13 = 223.7739 lf14=forcem(efrm14,1m14,U) lf14 = 194.6751 lf15=forcem(efrm15,1m15,U) lf15 = 125.8483 >> 171
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8.6 MATLAB Input/Output, Semirigid Analysis with KMi(total) The handcalculated values of KMi(total) are used in the MATLA8 computer program for an exact analysis of semirigid frame. Here, A and I from the exact analysis are used for section properties; they could be used from the approximate analysis. Two loading cases are used: (1) gravity alone, and (2) gravity and wind. Section 8.6.1 presents the input/output data for gravity alone, while Section 8.6.2 presents input/output data for the combined effect of gravity and wind. 172
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8.6.1 Gravity Alone Input/Output Data 173
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% CE Thesis,Fall 2001 (Semirigid plane frameGravity acting alone) Farrokh Jalalvand %The frame has element loads only; it uses A & I from exact analysis & KMi(total). % file name: semirgdgrvty1.m % Units: Kips, feet %Given info alpha=24; beta=6; num_loads=1; E=29000*144; % klft"2 11=475/(12114); % ftl\4 A1=17/144; % ft"2 12=425/(12"4); % ft"4 A2=15.6/144; % ft"2 13=425/(12"4); % ft"4 A3=15.6/144; % ft"2 14=510/(12114); % ft"4 A4=10.3/144; % ft"2 15=843/(12114); % ft114 A5=13/144; % ft112 16=800/(12"4); % ft114 A6=14.7/144; % ft112 17=425/(12"4); % ft"4 A7=15.6/144; % ft"2 18=425/(12"4); % ft"4 A8=15.61144; % ft"2 19=475/(12"4); % ft114 A9=17/144; % ft"2 K10=48900; % kftlrad K11 =53500; % kftlrad K12=4.3300; % kftlrad K13=K10; % kftlrad K14=K11; % kft/rad K15=K12; % kftlrad FaNodaiLoads=zeros(alpha,num_loads): Ub=zeros(beta,num_loads); % preliminary calculations n=alpha+beta: K=zeros(n,n); Fequiv=zeros(n,num_loads); % zero out Fequivalent vector for element loads % one element at a time % element 1 i end at bottom theta=pi/2; theta_deg=theta*180/pi; L1=15; kelp1=plbm(A1 ,11 ,E,L 1); T=rotate(theta_deg); kel1=r*kelp1*T; % local coords % transformation matrix % global coords efrrn1 =kelp1 *T; % recovers internal forces in local coords % when using U in global coords lm1=[25 26 27 1 2 3]; K=addstiff(K,kel1 ,lm1 ); % element2 theta=pi/2; theta_deg=theta/pi; 174
PAGE 190
L2=13; kelp2=plbm(A2.12,E,L2); T=rotate(theta_deg); kel2=r*kelp2"T; efrm2=kelp2*T; lm2=(1 2 3 5 6 7]; K=addstiff(K,kel2,1m2); %element 3 theta=pi/2; theta_deg=theta*180/pi; L3=13; kelp3=plbm(A3,13,E,L3); T=rotate(theta_deg); kel3=r*kelp3"T; efrm3=kelp3"T; lm3=[5 6 7 9 10 11]; K=addstiff(K,kel3,1m3); % element4 % theta angle is zero L4=30; kelp4=plbm(A4,14,E,L4); kel4=kelp4; efrm4=kel4; lm4=(9 10 12 13 14 16]; fef4L=[0;27;135;0;27;135]; % local coords fef4G=fef4L; % global coords Fequiv=addrowslm(Fequiv,fef4G,Im4); % update Fequiv. note neg. sign %elementS % theta angle is zero L5=30; kelp5=plbm(A5,15,E,L5); kel5=kelp5; efrm5=kel5; lm5=[5 6 8 1718 20]; K=addstiff(K,kel5,1m5); fef5L=[0;30; 150;0;30;150]; fef5G=fef5L; Fequiv=addrowslm(Fequiv,fef5G,Im5); %elementS % theta angle is zero L6=30; kelp6=plbm(A6,16,E,L6); kel6=kelp6; efrm6=kel6; lm6=[1 2 4 21 22 24]; K=addstiff(K,kel6,1m6); fef6L=(0;30;150;0;30;150]; fef6G=fef6L; Fequiv=addrowslm(Fequiv,fef6G,Im6); %element 7 theta=pi/2; theta_deg=theta*180/pi; L7=13; 175
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kelp7=plbm(A7,17 ,E,L7); T=rotate(theta_deg); kel7=r*kelp7*T; efnn7=kelp7*T; lm7=[17 18 19 13 14 15]; K=addstiff(K,kel7 ,lm7); %element 8 theta=pi/2; theta_ deg=theta*180/pi; L8=13; kelp8=plbm(A8, 18,E,L8); T=rotate(theta_deg); kel8=r*kelp8*T; efnn8=kelp8*T; lm8=[21 22 23 17 18 19]; K=addstiff(K,kel8,1m8); %elements theta=pi/2; theta_deg=theta*180/pi; L9=15; kelp9=plbm(A9,19,E,L9); T=rotate(theta_deg); kel9=r*kelp9*T; efrm9=kelp9*T; lm9=[28 29 30 21 22 23]; K=addstiff(K,kel9,1m9); %element 10 kei10=SpringK(K1 0); efnn1 O=SpringFTM(K 1 0); lm10=[3 4]; K=addstiff(K,kel1 0, lm1 0); %element 11 kel11 =SpringK(K 11 ); efrm11=SpringFTM(K11); lm11=[7 8]; K=addstiff(K,kel11 ,1m 11 ); %element 12 kei12=SpringK(K12); efrm 12=SpringFTM(K12); lm12=[1112]; K=addstiff(K,kel12,1m12); %element 13 kei13=SpringK(K13); efnn13=SpringFTM(K13}; lm13=(24 23]; K=addstiff(K,kel13,1m13); %element 14 kei14=SpringK(K14); efnn14=SpringFTM(K14); lm14=(20 19]; K=addstiff(K,kel14,1m14); 176
PAGE 192
% elemnt 15 kei15=SpringK(K15); efrm15=SpringFTM(K15); lm15=[16 15]; K=addstiff(K,kel15,1m15); % all elements completed % structure stiffness matrix now formed % now form Fa and Fbcorr Fa=FaNodalloads+rmvsm(Fequiv,1,1,alpha,num_loads); % add nodal loads to element loads Fbcorr=rmvsm(Fequiv,alpha+1,1,beta,num_loads); %get Fbcorr to correct reactions % solve for F and U [U F]=absolve(K,Fa,Ub); % now correct reactions F=addsm(F,Fbcorr,alpha+1,1); %correct reactions (NOTE: the last 1 is anchor point.) % calculate beam end forces lf1=forcem(efrm1,1m1,U) lf1 = 87.0000 4.6745 23.4844 87.0000 4.6745 46.6335 lf2=forcem(efrm2,1m2, U) lf2 = 57.0000 8.4085 60.3695 57.0000 8.4085 48.9415 lf3=forcem(efrm3,1m3,U) lf3 = 27.0000 12.5430 66.0192 27.0000 12.5430 97.0393 lf4=forcem(efrm4,1m4,U); lf4=1f4+fef4l % correct internal forces 177
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lf4 = 12.5430 27.0000 97.0393 12.5430 27.0000 97.0393 lf5=forcern(efrm5,lrn5,U); lf5=1f5+fef5L % correct internal forces lf5 = 4 1344 30.0000 114.9606 4.1344 30.0000 114.9606 lf6=forcern(efrm6,1rn6,U); lf6=1f6+fef6L % correct internal forces lf6= 3.7340 30.0000 107.0030 3.7340 30.0000 107.0030 lf7=forcern(efrm7,1rn7,U) lf7= 27 0000 12.5430 66 0192 27.0000 12.5430 97.0393 lf8=forcern(efrm8,1rn8, U) lf8 = 57.0000 8.4085 60 3695 57.0000 8.4085 48.9415 178
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lf9=forcem(efrm9, lm9,U) lf9 = 87.0000 4.6745 23.4844 87.0000 4.6745 46.6335 lf10=forcem(efrm10,Im10,U) lf10 = 107.0030 lf11=forcem(efrm11 ,lm11,U) lf11 = 114.9606 tf12=forcem(efrm12,lm12,U) lf12 = 97.0393 lf13=forcem(efrm13,1m13,U) lf13 = 107 0030 lf14=forcem(efrm14,tm14,U) lf14 = 114.9606 lf15=forcem(efrm15,lm15,U) lf15 = 97.0393 >> 179
PAGE 195
8.6.2 Gravity and Wind Input/Output Data 180
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% CE Thesis,Fall2001 (Semirigid plane frame under gravity & wind) Farrokh Jalalvand % The frame has nodal loads & element loads; it uses A & I from exact analysis & KMi(total). % file name: semirgdcomnd1.m % Units: Kips, feet %Given info alpha=24; beta=6; num_loads=1; E=29000*144; % klft"2 11=475/(12"4); % ft"4 A1=17/144; % ft"2 12=425/(12"4); % ft"4 A2=15.6/144; % fl"2 13=425/(12"4); % ft"4 A3=15.6/144; % ft"2 14=510/(12114); % ft114 A4=10.3/144; % ft"2 15=843/(12114); % ft"4 A5=13/144; % ft112 16=800/(12114); % ft"4 A6=14.7/144; % ft112 17=425/(12"4); % ft114 A7=15.6/144; % ft112 18=425/(12"4); % ft114 A8=15.6/144; % ft112 19=475/(12"4); % ft"4 A9=17/144; % ft"2 K1 0=48900; % kftlrad K11=53500; % kftlrad K12=43300; % kftlrad K13=K10; % kftlrad K14=K11; % kftlrad K15=K12; % kftlrad FaNodalloads=zeros(alpha,num_loads); FaNodalloads(1,1)=7.95; FaNodalloads(5,1 )=8.87; FaNodalloads(9,1 )=4.93; Ub=zeros(beta,num_loads); % preliminary calculations n=alpha+beta; K=zeros(n,n); Fequiv=zeros(n,num_loads); % zero out Fequivalent vector for element loads % one element at a time % element 1 i end at bottom theta=pi/2; theta_deg=theta*180/pi; % local coords % transformation matrix % global coords L1=15; kelp1=plbm(A1,11,E,L 1); T=rotate(theta_deg); kel1 =T*kelp1"T; efrm1=kelp1"T; % recovers internal forces in local coords % when using U in global coords lm1=[25 26 27 1 2 3]; K=addstiff(K,kel1,1m1 ); 181
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% element2 theta=pi/2; theta_deg=theta*180/pi; L2=13; kelp2=plbm(A2,12,E,L2); T=rotate(theta_deg); lm2=[1 2 3 5 6 7]; K=addstiff(K,kel2,1m2); %element 3 theta=pi/2; theta_deg=theta*180/pi; L3=13; kelp3=plbm(A3,13,E,L3); T=rotate(theta_deg); lm3=[5 6 7 9 10 11]; K=addstiff(K,kel3,1m3); % element4 % theta angle is zero L4=30; kelp4=plbm(A4, 14,E,L4 ); kel4=kelp4; efrm4::ikel4; lm4=[910 12 13 1416]; K=addstiff(K,kel4,1m4); fef4L=[0;27;135;0;27;135]; %local coords fef4G=fef4L; % global coords Fequiv=addrowslm(Fequiv,fef4G,Im4); %update Fequiv. note neg. sign %elementS % theta angle is zero L5=30; kelp5=plbm(A5,15,E,L5); kel5=kelp5; efrm5=kel5; lm5=[5 6 8 17 18 20]; K=addstiff(K,kel5,1m5); fef5L=[0;30;150;0;30;150]; fef5G=fef5L; Fequiv=addrowslm(Fequiv ,fef5G,Im5); %elements % theta angle is zero L6=30; kelp6=plbm(A6,16,E,L6); kel6=kelp6; efrm6=kel6; lm6=[1 2 4 21 22 24]; K=addstiff(K,kel6,1m6); fef6L=[0;30;150;0;30;150]; fef6G=fef6L; Fequiv=addrowslm(Fequiv,fef6G,Im6); 182
PAGE 198
%element7 theta=pi/2 ; theta_deg=theta*180/pi; L7=13; kelp7=plbm{A7,17,E,L7); T=rotate(theta_deg); kel7=r*kelp7*T; efrm7=kelp 7*T; lm7=[17 18 19 13 14 15]; K=addstiff(K kel7 lm7) ; %element 8 theta=pU2; theta_deg=theta*180/pi; L8=1.3; kelp8=plbm(A8,18,E,L8); T=rotate(theta_deg); keiB=r*kelpB*T; efrm8=kelp8*T; lm8=[21 22 23 17 18 19]; K=addstiff(K,kel8,1m8); %element 9 theta=pU2; theta_deg=theta*180/pi; L9=15; kelp9=plbm(A9,19,E,L9); T=rotate(theta_deg); kel9=r*kelpS*T; efrm9=kelp9*T; lm9=[28 29 30 21 22 23]; K=addstiff(K,ke19,1m9); %element 10 kel1 O=SpringK{K1 0); efrm1 O=SpringFTM(K1 0); lm10=[3 4]; K=addstiff(K,kel1 O,lm10); %element 11 kei11=SpringK(K11); efrm11=SpringFTM(K11 ); lm11=[7 8); K=addstiff(K,kel11 ,lm11 ); %element 12 kei12=SpringK{K12); efrm12=SpringFTM(K12); lm12=[ 11 12]; K=addstiff{K,kel12,1m12); %element 13 kei13=SpringK{K 13); efrm13=SpringFTM(K13); lm13=[24 23]; K=addstiff{K,kel13,1m13); %element 14 ke114=SpringK(K14); efrm14=SpringFTM(K14); 183
PAGE 199
lm14=[20 19]; K=addstiff(K,ke114,1m14); %element 15 kei15=SpringK(K15); efrm15=SpringFTM(K15); lm15=[16 15]; K=addstiff(K,kel15,1m15); % all elements completed % structure stiffness matrix now formed %now form Fa and Fbcorr Fa=FaNodaiLoads+rrnvsm(Fequiv, 1,1 ,alpha,num_loads); % add nodal loads to element loads Fbcorr=rmvsm(Fequiv,alpha+1, 1 ,beta,num_loads); %get Fbcorr to correct reactions % solve for F and U [U F]=absolve(K,Fa,Ub); % now correct reactions F=addsm(F,Fbcorr,alpha+1, 1); %correct reactions (NOTE: the last 1 is anchor point.) % calculate beam end forces lf1=forcem(efrrn1 ,lm1 ,U) lf1 = 75.3746 6.2272 87.2557 75.3746 6.2272 6.1520 lf2=forcem(efrm2,1m2,U) lf2 = 51.2173 1.5109 25.4426 51.2173 1.5109 5.8013 lf3=forcem(efrrn3, 1m3, U} lf3= 25.3796 10.0847 58.3418 25.3796 10.0847 72.7591 lf4=forcem(efrm4,1m4,U); lf4=1f4+fef4L % correct internal forces 184
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lf4 = 15.0147 25.3796 72 7591 15.0147 28.6204 121.3704 lf5=forcem(efrm5,1m5 U); lf5=1f5+fef5L % correct internal forces lf5 = 0.2962 25.8377 52.5405 0 2962 34.1623 177.4091 lf6=forcem(efrm6 1m6,U); lf6=1f6+fef6L % correct internal forces lf6= 0.2120 24.1572 19.2906 0 2120 35.8428 194.5738 lf7=forcem(efrm7 ,1m7, U) lf7 = 28 6204 15 0147 73.8205 28.6204 15.0147 121.3704 lf8=forcem(efrm8,1m8,U) lf8 = 62.7827 15.3109 95.4526 62.7827 15.3109 103 5887 185
PAGE 201
lf9=forcem(efrm9 ,1m9, U) lf9 = 98.6254 15.5228 133.7212 98.6254 15.5228 99.1211 lf10=forcem(efrm10,Im10,U) lf10 = 19. 2906 lf11=forcem(efrm11 ,lm11 ,U) lf11 = 52.5405 lf12=forcem(efrm12,1m12,U) lf12 = 72.7591 lf13=forcem(efrm13,1m13,U) lf13 = 194.5738 lf14=forcem(efrm14,1m14,U) lf14 = 177.4091 lf15=forcem(efrm15,1m15,U) lf15 = 121.3704 >> 186
PAGE 202
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A THEORETICAL DERIVATION OF INITIAL STIFFNESS OF
THE SEMIRIGID STRUCTURAL TEE CONNECTION
by
Farrokh Jalalvand
B.S., University of Oklahoma, 1980
M.S., Oklahoma State University, 1984
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2001
IT
! AL'
pL r
2001 by Farrokh Jalalvand
All rights reserved
