analysis of corrugated web beam to column extended end plate connection using
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ANALYSIS OF CORRUGATED WEB BEAM TO COLUMN EXTENDED
END PLATE CONNECTION USING
LUSAS SOFTWARE
ANIZAHYATI BINTI ALISIBRAMULISI
UNIVERSITI TEKNOLOGI MALAYSIA
PSZ 19:16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
ANALYSIS OF CORRUGATED WEB BEAM TO COLUMN EXTENDED END PLATE CONNECTION USING LUSAS SOFTWARE
ANIZAHYATI BINTI ALISIBRAMULISI
4
NO. 6, JALAN BUNGA KEMUNTING 2/10, 40000, SHAH ALAM, SELANGOR
19 MEI 2006
CATATAN: * Potong yang tidak berkenaan. ** Jika tesis ini SULIT atau TERHAD, sila lampirka
berkuasa/organisasi berkenaan dengan menyataka dikelaskan sebagai SULIT atau TERHAD.
υ Tesis dimaksudkan sebagai tesis bagi Ijazah Dokt penyelidikan, atau disertasi bagi pengajian secara Laporan Projek Sarjana Muda (PSM).
P.M DR SARIFFUDDIN SAAD
ok
2005/2006
19 MEI 2006
n surat daripada pihak n sekali sebab dan tempoh tesis ini perlu
r Falsafah dan Sarjana secara erja kursus dan penyelidikan, atau
“I hereby declare that I have read this project report and in
my opinion this report is sufficient in terms of scope and
quality for the award of the degree of Master of Engineering (Civil – Structure)”
Signature: ....................................................
Name of Supervisor:
Date:
.................................................... ASSOC. PROF. DR SARIFFUDDIN SAAD
.................................................... 19 MAY 2006
ANALYSIS OF CORRUGATED WEB BEAM TO COLUMN EXTENDED
END PLATE CONNECTION USING
LUSAS SOFTWARE
ANIZAHYATI BINTI ALISIBRAMULISI
A project report submitted in partial fulfillment of the
requirement for the award of the degree of
Master of Engineering (Civil – Structure)
Faculty of Civil Engineering
Universiti Teknologi Malaysia
MEI 2006
I declare that this project report entitled ‘Analysis of Corrugated Web Beam to
Column Extended End Plate Connection Using LUSAS Software’ is the result of my
own research except as cited in the references. The report has not been accepted for
any degree and is not concurrently submitted in candidature of any other degree.
Signature: ………………………………………..
Name: ……………………………………….. ANIZAHYATI BINTI ALISIBRAMULISI
Date: ……………………………………….. 19 MEI 2006
iii
ACKNOWLEDGEMENT
In the name of ALLAH, The Most gracious, Most merciful, with His
permission, Alhamdulillah this proposal report has completed. Praises to Prophet
Muhammad, his companies and those on the path as what he preached upon, may
ALLAH The All Mighty keep us in his blessings and tender care.
I would like to convey my highest appreciation to those who had sincerely,
without hesitation helped to make this report a possible success. My highest level of
appreciation to Associate Professor Dr Sariffuddin Saad for his guidance, without his
corporation, I would not be able to complete this proposal report. A special thanks to
Mr Arizu Sulaiman (PhD candidate – UTM), Mr Anis Sagaff (PhD candidate –
UTM), and Mr Che Husni for giving me the required information and guidance for
the completion of this study.
I would like to express my heartfelt appreciation to my husband (Ahmad
Saifuddin bin Abdul) and my children (Amiratul Soffiya and Amiratul Syuhada), for
their patient, love, prayers, support and also for understanding the sacrifices required
in completing this study. My sincere and special thanks also go to my beloved
friends and classmates for being supportive and for their contributions and
understanding.
Lastly but not least, thank you to all that have contributed either directly or
indirectly in making this study a success.
iv
ABSTRACT
Bolted extended end plate connections are commonly used in rigid steel
frame. Inappropriate or inadequate connections of beam and column are hazardous
and can lead to collapses and fatalities. Although laboratory testing is more accurate
in analyzing the connection, but unfortunately it is time consuming and quite
expensive. Thus, this project is intended to develop a Finite Element Analysis (FEA)
approach as an alternative method in studying the behavior of such connections. The
software being used is LUSAS 13.5 and the model used was an extended end plate,
welded to the end of a corrugated web beam and then bolted to the column-flange.
This type of connection will cause the column to bend about its major axis, and
affect the end plate, bolts and corrugated web beam. Therefore, the analysis will be
much more difficult as compared to plain web beam. The moment-rotation (M-φ)
response of the joint was plotted in term of a M-φ curve, and then, it was
superimposed with the curve taken from an existing experimental result. It was found
that the two curves shared the same stiffness at the elastic stage of the loading and
they started to diverge as the connection became plastic. However, the LUSAS
moment of resistance is 50% more than that obtained in the experiment. Further
investigations are necessary to improve the finite element prediction.
v
ABSTRAK
Sambungan rasuk kepada tiang dengan menggunakan skrew dan plat hujung
adalah satu perkara biasa dalam sambungan kerangka besi. Ketidaksesuaian dan
kelemahan sambungan rasuk dan tiang adalah berbahaya, dan boleh mengakibatkan
keruntuhan kerangka dan kemalangan jiwa. Walaupun ujikaji makmal merupakan
kaedah yang tepat untuk menganalisa jenis sambungan tersebut, tetapi ia memakan
masa yang lama dan memerlukan kos yang lebih tinggi. Oleh itu, projek ini bertujuan
untuk membangunkan analis unsur terhingga sebagai salah satu alternatif dalam
mengkaji kelakuan sebenar sambungan tersebut. Perisian yang digunakan bagi
analisis unsur terhingga ini adalah LUSAS 13.5 dan komponen-komponen ynag
terlibat dalam sambungan tersebut adalah; plat hujung yang dikimpal kepada hujung
rasuk yang ‘corrugated’ dan kemudiannya diskrewkan pada bebibir tiang.
Sambungan jenis ini akan menyebabkan tiang melentur pada paksi major dan
memberi kesan kepada plat hujung, skrew dan rasuk yang ‘corrugated’ tersebut.
Analisis ini adalah lebih kompleks berbanding dengan rasuk biasa. Tindakbalas
momen-putaran(M-φ) sambungan tersebut diplotkan dalam bentuk lengkungan M-φ,
yang kemudiannya di’super-impose’ dengan lengkungan M-φ ujikaji. Hasilnya
didapati, 2 lengkungan tersebut berkongsi nilai kekuatan yang sama pada tahap
elastik beban dan kemudiannya berpecah apabila sambungan mula bersifat plastik.
Walaubagaimanapun, keputusan momen kapasiti LUSAS adalah 50% melebihi
momen kapasati ujikaji. Oleh itu, lebih banyak penyelidikan diperlukan di masa
hadapan untuk memperbaiki keputusan analisis unsur terhingga ini.
vi
TABLE OF CONTENTS
CHAPTER TITLE PAGE
1 INTRODUCTION 1
1.1 PROBLEM BACKGROUND 1
1.2 PROBLEM STATEMENT 1
1.3 OBJECTIVES OF THE STUDY 2
1.4 SCOPE OF THE STUDY 2
1.5 SIGNIFICANCE OF RESEARCH 3
2 LITERATURE REVIEW 4
2.1 INTRODUCTION 4
2.2 CORRUGATED WEB BEAM AND EXTENDED
END PLATE
12
2.3 CLASSIFICATION OF CONNECTIONS 15
2.4 MOMENT-ROTATION (M-φ)
CHARACTERISTICS
19
2.5 ANALYSIS OF CONNECTIONS 22
2.5.1 EXPERIMENTAL SET-UP 22
3 RESEARCH METHODOLOGY 30
3.1 LUSAS SOFTWARE 30
3.1.1 FINITE ELEMENT MODEL 30
3.1.2 ELEMENT TYPES 31
3.1.3 NON-LINEAR ANALYSIS 35
3.1.4 BOUNDARY CONDITIONS 36
3.1.5 SUMMARY OF LUSAS FINITE ELEMENT
SYSTEM
38
vii
4 RESULTS AND DISCUSSIONS 39
5 CONCLUSION AND RECOMMENDATION FOR
FUTURE WORK
46
5.1 CONCLUSIONS 46
5.2 RECOMMENDATION FOR FUTURE WORK 46
REFERENCES 48
APPENDIX A (LUSAS Element Types) 51
APPENDIX B (Variations in K) 61
viii
LIST OF FIGURES
FIGURE TITLE PAGE
2.1 Corrugated plate 12
2.2 Corrugated web beam 12
2.3 Extended End Plate Connection 14
2.4 Connection Loading 14
2.5 Components of beam to column connection 14
2.6 Strength, stiffness and deformation capacity of steel and
connections
15
2.7 Failure modes for bolted T-stub connections 15
2.8(a) Moment-rotation curves of beam to column connections 16
2.8(b) Moment-rotation curves of beam to column connections 16
2.8(c) Moment-rotation curves of beam to column connections 16
2.9 Nominally pinned connections 17
2.10 Flush end plate connection 18
2.11 Rigid Connections 18
2.12 Semi rigid (semi flexible) connections 19
2.13 Characteristics of beam-to-column connections 20
2.14 Moment-rotation diagram of beam-to-column connections 21
2.15 Experimental set-up 22
2.16 Details of beam-to-column connections 23
3.1 Element types 32
3.2 Enlarged FEA Bolt Arrangement 32
3.3(a) Line mesh 32
3.3(b) Mesh Discretisation 33
3.3(c) Mesh BRS2 and JNT4 33
3.3(d) Attribute forms 34
3.3(e) Attribute forms 35
3.4 Attribute forms 36
ix
3.5 Attribute forms 37
3.6 FEA Supports and Loading 37
4.1 Comparison of M-φ Curve between Experimental and FEA
results (Nonlinear Analysis)
41
4.2 Experimental graph for Moment-Rotation Curve (N5
specimen) in determining Moment capacity, MR of the
connection.
41
4.3 Finite Element Analysis graph for Moment-Rotation Curve
(N5 specimen) in determining Moment capacity, MR of the
connection.
43
4.4(a) Position of nodes selected in determining the displacement 43
4.4(b) Position of nodes selected in determining the displacement 43
4.5 N5 specimen after failure 45
4.6 FEA deformed mesh for N5 45
1
CHAPTER 1
INTRODUCTION
1.1 PROBLEM BACKGROUND
To date, the experimental approach to study the behaviour of connection in
steel structures will certainly remain the most popular for still some years but
because of the highly cost involved, researchers are increasingly looking for
less costly but acceptable alternatives. The most obvious alternative is
modeling by the finite element method. Due to the highly complex nature of
connections and the large number of parameters involved, numerous tests are
required before an adequate set of empirical formulae is developed for the
design of a specific type of connection. It appears to be more rational and
more economical to develop numerical models to play with the various
parameters and to check the accuracy of the numerical models against the
results of an appropriate number of experimental tests. Not only are
experimental tests needed to validate the models but they are also required for
calibration purposes.
1.2 PROBLEM STATEMENT
Accurate analysis of the connection is difficult due to the number of
connection components and their inherit non-linear behaviour. The bolts,
welds, beam and
2
column sections, connection geometry and the end plate itself can all have a
significant effect on connection performance. Any one of these can cause
connection failure and some interact. The most accurate method of analysis is
of course to fabricate full scale connections and test these to destruction.
Unfortunately this is time consuming, expensive to undertake and has the
disadvantage of only recording strain readings at pre-defined gauge locations
on the test connection. A three dimensional materially static non-linear finite
element analysis approach has therefore been developed as an alternative
method of connection appraisal. For this research, extended end plate and
corrugated web beam will be used, since not much research is done on such
connections.
1.3 OBJECTIVES OF THE STUDY
The main objective of this research is to study the moment-rotation behaviour
of corrugated web beam to column connections. A static non-linear finite
element analysis will be used to model and analyze the bolted connection.
Extended end plate and non linear elastic-plastic behaviour will be considered
in the analysis. The moment-rotation curve plotted from the result will be
compared with the relevant data available from experimental testing.
1.4 SCOPE OF THE STUDY
There are various types and shape of connection in structural steelwork. This
study focused mainly on extended end plate bolted connection and corrugated
web beam, particularly, trapezoidal web beam. The plate has 8 holes and M20
bolts will be used. The column size is 305x305x118 UC (S275) and its length
is 3 m and the beam size is 400x140x39.7/12/4 � 1.5m, Flange � S355, Web
- S275. A static point load was applied incrementally at the end of the
cantilever beam. LUSAS software [1] will be used to model the connections.
The result from the finite element analysis, mainly moment-rotation curve,
will be compared with the existing experimental result.
3
1.5 SIGNIFICANCE OF RESEARCH
Research significance to be obtained from this study will be the results and
analysis of the behavior of beam to column connection, when extended end
plate and corrugated web beam is used. It is necessary to compare the
moment �rotation curve of the result from the finite element analysis and
experimental testing. The aim was to determine the accuracy of the analytical
method and to verify the strength of the corrugated web beam as compared to
a plane web. Corrugated web beam is still new in the industry, so if much
research is done on it, more application of it can vary our steel industry
products.
4
CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION
Despite numerous years of extensive research, particular in the 1970�s, no
fully agreed design method exists. Many areas of connection behaviour still
require investigation. More recently Bose, Sarkar and Bahrami [2] used FEA
to produce moment rotation curves, Bose, Youngson and Wang [3] reported
on 18 full scale tests to compare moment resistance, rotational stiffness and
capacity. The latest design method utilizes plastic bolt force distribution to
create an increased moment connection capacity and reduced column
stiffening. In 1995 when the SCI and the BCSA produced the Green Book
guide, based on the EC3 design model, the editorial committee felt a number
of areas, particularly bolt force distribution and compression flange overstress
required further investigation.
Krishnamurthy (1979) [4] conducted early finite element analysis of moment
endplate connections. This study included thirteen finite element models of
�benchmark connections, with dimensions spanning values commonly used
in the industry�. The study was limited greatly by the technology at that time.
A 2-dimensional/3-dimensional finite element analysis was conducted to
determine adequate correlation between results. If such correlation could be
found for the thirteen connections considered, then two dimensional analyses
could be used
5
with factors that correlated to the three-dimensional model of the same
connection. According to this paper, it would have been impossible to
feasibly conduct an exhaustive three-dimensional analysis of end-plate
connections because of the time involved in the programming, as well as the
computational time required. Additionally, the task of creating a three-
dimensional model with every detail and adequate proportions was deemed
impossible at that time. Therefore some simplifying assumptions were made,
and a three-dimensional model was created based on a constant strain triangle
and eight-node sub parametric brick elements. Bolt heads were omitted, and
the bolts were modeled as rectangular shanks, having the same cross-
sectional area as round bolts. No �contact elements� were used. The bolts
were assumed to be in tension, and an effective square area at the
compression flange was assumed to be compressed against the column
flange. This study produced stress distribution plots at different loading
magnitudes on the plate. Some correlation between the two-dimensional and
three-dimensional models was observed. The three-dimensional model had
less stiffness than the two-dimensional model because of the �prevention of
the transverse variation of deformations and stresses�. Seven correlation
factors, relating the two different models, were tabulated for each of the
thirteen benchmark connections. Krishnamurthy (1979) [4] concluded that
prying forces do not exist in moment end-plate connections based on this
study.
Bursi and Leonelli (1994) [5] presented some additional results that had not
been discussed by Bursi and Jaspart (1997b) [6]. Twenty-node brick elements
were used to model the beam and plate material. Contact elements were used
to represent the end-plate/column-flange interaction problem. Once again,
beam elements were used to model the bolts, but here the bolts were
pretensioned to a snug tight condition. The column flange was considered
rigid. End-plate rotation and bolt loads were examined using the finite
element model. Fairly good correlation with experimental results wass
obtained. A direct application of the model suggested by Richard and Abbott
(1975) [7] is used to describe the analytical results obtained from the finite
element model. Using the finite
6
element method to obtain the elastic stiffness Ke,th, the inelastic stiffness
Kp,th, the plastic failure moment Mp,th, and the ultimate applied moment Mu,th
for the connection, the moment rotation plot or M-θ relationship can be
described by;
Where n is the shape factor.
Gebbeken et al. (1994) [8] investigated the different finite element modeling
techniques to uncover the important criteria for describing moment end-plate
connection behavior. Also, the authors discussed the results of a parametric
study to determine which elements of the connection provided significant
amounts of connection flexibility. The four-bolt unstiffened extended end-
plate connection was considered. First, a two-dimensional model was used.
The material stress/strain relationship was represented as a bilinear function.
Friction between the column flange and the end-plate was neglected. The
results from this analysis were poor since strength predictions were found to
be very unconservative when compared to the experimental results. The
three-dimensional model used by the authors provides some limited success
in predicting the moment-rotation characteristics of the connection. The
description of the finite element model was vague, yet it was mentioned that
brick elements were used. Also, the figures in the paper made it to appear that
a tee stub and not an actual end-plate was considered. In some cases the
results were accurate, but in others the strength were off by 50% or more,
possibly suggesting inadequate modeling assumptions. Rothert et al. (1992)
[9] presents similar results and findings based on the same research.
Sherbourne and Bahaari (1997) [10] developed a methodology based on three
dimensional finite element designs, to analytically evaluate the moment
rotation
7
relationships for moment end-plate connections. ANSYS 4.4 was the
software package used. The purpose for this research was to provide
designers with a method of determining stiffness for these connections. It was
apparent at the time that the ability of designers to produce a moment-rotation
curve for moment end-plate connections was limited. Because of
advancements in computer technology, Sherbourne and Bahaari�s models
included plate elements for the flange, webs, and stiffeners of the column and
beam, as well as taking into account the bolt shank, nut, head of the bolt, and
contact regions. However, bolt pre-stressing was not included. It was
determined that the behavior of a moment rotation curve for an end-plate
connection throughout an entire loading history, up to and including failure,
can be feasibly and accurately modeled by performing a three-dimensional
finite element analysis. This is particularly useful when one of the plates in
contact, either the column flange or the end plate, is thin. The analysis of such
a plate is inaccurate when using two-dimensional models. An additional
advantage to the use of the three-dimensional model is the separation of the
column, bolt, plate, and beam stiffness contributions to the overall behavior
of the connection.
Bahaari and Sherbourne (1997) [10] presented part two of their finite element
study on moment end-plate connections. Based on the parametric study found
in
Sherbourne and Bahaari (1997) [10], this paper uses the Richard-Abbott
power function (similar to that suggested by Bursi and Leonelli (1994) [5]) to
describe the moment-rotation behavior of four-bolt unstiffened extended
moment end-plate connections of known geometrical configuration. The
proposed moment-rotation relationship is;
where the elastic stiffness Ki, the inelastic stiffness Kp, and the plastic failure
moment Mp were all obtained from the results of a finite element analysis. Mo
8
and n are the connection-dependent reference moment and shape factor,
respectively. If Ki and Kp are equal, the function becomes linear. Likewise, if
Kp is zero, the curve becomes an elastic plastic model of the connection�s
behavior. For large values of n, the model approaches a bilinear model of
behavior. A curve fitting technique is used to determine the best set of values
for the variables of numerous connection configurations. Using these results,
an empirical equation was developed to describe the moment-rotation
characteristics based on the end-plate configuration, bolt size, beam
dimensions, and column dimensions. The results of this paper are eminent for
the application of four-bolt unstiffened extended end-plates to semi-rigid
connection philosophy. Although the moment-rotation plots given in the
application examples included in the paper have decent correlation, the
connection strength predicted by the method is off by as much as 75% in
some cases.
Bursi and Jaspart (1997a) [11] presented part one of a two-part investigation
of
finite element modeling of bolted connections. Unlike its companion paper
(Bursi and Jaspart, 1997b) [6], this paper did not consider moment end-plate
connections themselves. It did, however, present the results of which showed
that finite element programs could be used to accurately predict the behavior
of moment endplate connections. Hence, it is included here. Tee stub
connections were first modeled to determine the accuracy and/or calibration
required when using finite elements to model connection behavior. Using the
LAGAMINE software package, the models were constructed using both
hexahedron (more commonly called brick) and contact elements. The contact
elements utilize what is called a penalty technique. Here, a value was chosen
as a penalty parameter and this is similar to placing a spring between two
bodies. Contact is simulated only for displacements within this given penalty
value. Friction caused by the sliding and sticking between bodies is modeled
with an isotropic Coulomb friction law. Nonlinear finite element analysis that
considers large displacements, large rotations, and large deformations is used.
Loads were applied using displacement as the controlling parameter. When
considering the bolts, the
9
additional flexibility provided by the nut and threaded region of the bolt were
taken into account by using an effective length of the bolt. Due to the
symmetry of the tee stub connection, only a quarter of the connection was
modeled. Preloading forces in the bolts were taken into account by using
applied initial stresses. The material properties are modeled using piece-wise
linear constitutive laws for the material from experimentally tested
connections. For several of these experimentally tested connections, a finite
element analyses were performed. The finite element results compared quite
nicely to experimental results. There was a slight difference in deflection
values at the onset of yielding, which was primarily due to the presence of
residual stresses in the actual tee stubs which was neglected in the finite
element models of these members.
Bursi and Jaspart (1997b) [6] presented the second part of the two-part
investigation by the authors. They used ABAQUS finite element code to
analyze four-bolt unstiffened extended moment end-plate connections under
static loading. The purpose of the study was to examine the stiffness and
strength behavior of these connections. The finite element results were
compared with those from an experimental study. End-plate rotation and bolt
forces were both considered. The authors� intent was to show the feasibility
of using the finite element method via commercial codes to determine
moment�rotation characteristics of semi-rigid connections. Although
dynamic characteristics of these connections was not considered, the authors
did consider thin endplates mainly for their ability to behave in a ductile
manner when plate yielding occurs. The finite element model considered by
the authors was quite complex. The bolt and bolt head were modeled using
beam elements. Both preloaded and non-preloaded bolts are considered, but
only bolts in the tension region were included. The end-plate and beam
elements were generated using eight-node brick elements that allow
plasticity. Contact elements were used to describe the interaction between the
end-plate and the rigid column flange. Around the bolt holes, nodes were
constrained in the direction perpendicular to the face of the endplate. This
assumption was used, as tests and other finite element studies
10
have shown that end-plates tended to pull away from the column flange even
at the bolt locations. Other than friction forces taken care of by the contact
elements, there were no lateral constraints mentioned in the paper. However,
results were obtained even for the zero friction case, which should result in
divergence due to a singular stiffness matrix. Thus it was assumed that some
other boundary conditions were provided, but this was not discussed. By
comparison with experimental results, the results indicated that the model
predicts the end-plate moment-rotation characteristics quite accurately.
However, the bolt forces were not recorded experimentally and no
comparison is made. The bolt axial force versus beam flange force seems
reasonable in the plots provided. Bursi and Jaspart (1998) [12] presented
basically the same results as the paper discussed in this section and is not
considered separately.
Ribeiro et al. (1998) [13] discussed results of an experimental study of beam-
to-column moment end-plate connections. This study included testing of
twelve cruciform built-up sections to validate design criteria used for rolled
shapes for the design of built-up sections. Specimens were designed,
specifically to check the method proposed by Krishnamurthy (1979) [4]. The
following observations among others were made:
(a) Applied moments were about 20% greater than the plastic moment
capacities predicted.
(b) The greater the bolt diameter, the greater was the influence of end-plate
thickness.
(c) Krishnamurthy�s method was found to be non-conservative.
(d) Bolt rupture occurred in the tests in which the Krishnamurthy method
Predicted otherwise.
(e) Results involving the collapse modes of the specimen led to a
hypothesis concerning prying forces which was not accepted by
Krishnamurthy.
11
In 1998, Troup et al. (1998) [14] presented a paper describing finite element
modeling of bolted steel connections. The ANSYS software was used for this
study, which included an extended moment end plate model as well as a tee-
stub model. The model utilized a bilinear stress-strain relationship for the
bolts. Also, special contact elements were used between the end-plate and the
column flange for the extended end-plate model, and between the tees for the
tee model. By using the contact elements between the contact surfaces of the
models, the geometric non-linearities that are present between the surfaces as
separation occurs due to increased load can be realistically modeled. Both
models were calibrated with experimental test data to show excellent
correlation between analytical and experimental stiffness. Bolt forces were
also analyzed. It was found that for the simple four-bolt arrangement about
the tension flange, the tee design prediction was accurate. However, for more
complex bolt patterns, the distribution of prying forces was not as clear.
Troup, et al. (1998) [14] concluded the following:
(a) Tee-stub analogy was a useful benchmark to provide an indication of
the performance of analysis techniques.
(b) Shell elements are more accurate for modeling beam and column sections.
Thick endplate design could provide additional rotational stiffness and
moment capacity but may result in bolt fracture.
(c) Thin end plates could provide enough deformation capacity to allow semi-
rigid connection design, but may result in excessive deflection.
(d) The moment capacity prediction of Eurocode 3 had been shown to be
reasonable, but conservative, for simple end-plate bolt configurations.
However, the code is inaccurate when analyzing more complicated bolt
arrangements. If these inaccuracies did not lead to bolt failure, then they
might be acceptable.
Maggi et al. (2004) [15] focused on the behavioral variations of bolted
extended end plate connections due to changes in plate thickness and bolt
diameter. It also discussed the application of FE model as tools to perform
parametric analysis in
12
order to assess the accuracy of commonly used design procedures and to
provide data for development of new analytical models.
2.2 CORRUGATED WEB BEAM AND EXTENDED END PLATE
In this research, the focus is to obtain the M-φ characteristics of an extended
end plate connection involving a corrugated web beam.
A corrugated web beam is a built-up girder with a thin-walled,
corrugated web and plate flanges. In this case, the profiling of the web
attributes to its high load-bearing capacity at low design weight, which
represents a particular economical solution for wider spans. See Figure 2.1
and 2.2.
Figure 2.1: Corrugated plate
Figure 2.2: Corrugated web beam
13
Nowadays, corrugated webs are used to allow the use of thin plates without
stiffener for use in building and bridges. Thus, resulted in the reduction of the
beam weight and cost. The early investigation of such beam is carried out by
Elgaaly [16] and has been further developed to the practical stage. Most of
these analytical and experimental studies concentrated on the trapezoidal
vertically corrugated webs.
Elgaaly et al [16] investigated the failure mechanism of these beams
under shear, bending and compressive patch loads. It was found that the
failure of beams under shear loading is due to the buckling on the web, where
local buckling and global buckling occurred for coarse and dense corrugation
respectively. Similarly under bending, the compression flange vertically
buckled into the crippled web when the yield stress was reached. It was also
found that the ultimate moment capacity could be calculated considering the
flange and neglecting the web as its contribution to the beam�s moment
carrying capacity was considered to be insignificant. Nevertheless, under
compressive patch loads, two distinct modes of failure were observed. These
involved the formation of collapse mechanism on the flange followed by the
web crippling or yielded web crippled followed by vertical bending of the
flange into the crippled web. The failure of these beams was found to be
dependent on the loading position and also on the corrugation parameters
where it can be a combination of the aforementioned modes.
Zhang et al. [17] and Li et al [18] studied the influence of the
corrugation parameters and developed a set of optimized parameters for the
wholly corrugated web beams based on the basic optimization on the plane
web beams. It was also found that the corrugated web beam had 1.5 � 2 times
higher buckling resistance than the plane web beam.
An extended end plate connection consists of a plate welded in the
fabrication shop to the end of the steel beam as shown in Figure 2.3. The end
14
plate is pre-drilled and then bolted at site through corresponding holes in the
column flange. The plate extends above the tension flange in order to increase
the lever arm of the bolt group and subsequently the load carrying capacity.
The connection is usually loaded by a combination of vertical shear force,
axial force in the beam member and a moment as shown in Figure 2.4. The
overall components of the beam to column connection are shown in Figure
2.5.
Figure 2.3: Extended End Plate Connection
Figure 2.4: Connection Loading
Figure 2.5: Components of beam to column connection
The beam to column connection should have comparable properties to the
structural steel. Relevant properties of steel are its strength, its stiffness and
its ductility or deformation capacity. These properties can be demonstrated in
a tensile test (see Figure 2.6). A well designed steel structure should possess
the same good properties.
15
Figure 2.6: Strength, stiffness and deformation capacity of steel and connections
The failure of such connection is shown in Figure 2.7 below.
Mode III: Bolt failure
Mode I: Complete flange yielding
Mode II: Bolt failure with flange
yielding
Figure 2.7: Failure modes for bolted T-stub connections
2.3 CLASSIFICATION OF CONNECTIONS
The structural properties of connection can also be presented in a M-φ
diagram. In Figure 2.8(a), 2.8(b) & 2.8(c) below, shows a set of M-φ curves
for connections with different types of behavior.
16
Figure 2.8(a): Moment-rotation curves of beam to column connections
Figure 2.8(b): Moment-rotation curves of beam to column connections
Figure 2.8(c): Moment-rotation curves of beam to column connections
17
For the use in plastic design, the connections can be classified in the
following categories, namely; nominally pinned connections, full strength
connections and partial strength connections.
Nominally pinned connections
This type of connection is designed to transfer shear and normal force only.
The rotation capacity of the hinge should be sufficient to enable all the plastic
hinges necessary for the collapse mechanism to develop.
Full strength connections (Connections A and B in Figure 2.8(c))
The moment capacity is greater than that of the member. A plastic hinge will
not be formed in the connection but in the member adjacent to the
connection. In theory, no rotation capacity is required for the connection.
Partial strength connections (Connections C, D and E in Figure 2.8(c))
The moment capacity is less than that of the member. A plastic hinge will be
formed in the connection, so sufficient rotation capacity is required.
In elastic design, traditionally two categories of connections were considered:
Nominally pinned connections (Figure 2.9)
The connections are assumed to transfer only the end reaction of the beam
(vertical shear force and eventually normal force) to the column.
Figure 2.9: Nominally pinned connections
They should be capable of accepting the resulting rotation without
developing significant moments, which might adversely affect the stability of
the column. It is a common practice to design structures on a simply
supported basis
18
and then to provide connections which are in effect semi-rigid. A typical
example is the flush end plate as shown in Figure 2.10. This may be unsafe
due to insufficient rotation capacity of the connection.
Figure 2.10: Flush end plate connection
Rigid connection (Figure 2.11)
Rigid connections are used to transfer moments as well as end reactions.
Design assumes joint deformation to be sufficiently small that may influence
the moment distribution and the structure�s deformation may be neglected.
Figure 2.11: Rigid Connections
To fill the gap between pinned and rigid connections, a third category is
defined and accepted in most modern codes.
Semi rigid (semi flexible) connections (Figure 2.12)
These connections are designed to provide a predictable degree of interaction
between members based on actual or standardized design M-φ Characteristics
of the joints.
19
Figure 2.12: Semi rigid (semi flexible) connections
2.4 MOMENT-ROTATION (M-φ) CHARACTERISTICS
Because the flexural rigidity of each connection plays an important role in the
behavior of the entire structural steel frame, most of the research on various
connection types are focused on the investigation of moment-rotation
relationships. For this purpose, many experimental tests have been conducted
to obtain moment-rotation curves. Considering the moment-rotation curves
obtained from experimental tests are available, a simplified analytical model
is proposed in this project to predict the behavior of the connection by the
application of Finite Element Analysis (FEA).
The main structural elements of steel framed multi-storey structures are the
columns, the beams and their connections. Conventionally the beam-to-
column connections are considered to be either pinned or rigid. In the case of
pinned or 'simple' connections, the frames have to be stabilized by
appropriate bracing systems. Such frames are named braced frames by
Eurocode 3.
The term 'rigid' in this context implies that the connection is capable of
resisting moments with a high stiffness, i.e., the connection flexibility has a
negligible influence on the distribution of movements in the frame
connections. When the connections are rigid, the overall stability may be
provided by the frame itself without the inclusion of specific bracing systems.
Although the idealisation of connection stiffness as pinned or rigid has been
applied exclusively in the past it is generally recognized that the real
behaviour
20
of the connections is never as ideal as assumed in the analysis (Figure 2.13).
The two cases, pinned and fully rigid, actually represent extremes of
connection behaviour. In reality, the connections behave somewhere between
those limits, that is they behave as semi-rigid.
Figure 2.13: Characteristics of beam-to-column connections
A further classification of moment resisting connections relates to their
strength. A 'full-strength' connection is a connection that can at least develop
the bending strength of the elements it connects. A 'partial-strength'
connection has a lower design strength than that of the elements it connects.
The rotation capacity of a moment-resisting connection can also be important.
For example a beam with partial-strength end connections can be designed
plastically if the connection rotation capacity is sufficient to ensure the
development of an effective hinge at midspan.
For practical design situations the actual non-linear connection behaviour has
to be approximated. The connection behaviour is characterised by its moment
resistance MRd, its rotational capacity φcd and its rigidity s = M/φ.
21
Figure 2.14 shows the moment/rotation diagram of a beam to column
connection. For design purposes, the real connection behavior can be
represented by a bi-linear diagram in which the following properties can be
distinguished.
Figure 2.14: Moment-rotation diagram of beam-to-column connections
(a) The design resistance of the connection
(b) The stiffness of the connection when subjected to small moments
(c)The stiffness of the connection when subject to ultimate moments
(d) The rotation capacity
22
2.5 ANALYSIS OF CONNECTIONS
2.5.1 EXPERIMENTAL SET-UP
Figure 2.15: Experimental set-up
23
The arrangement of the experimental set-up is as shown in the Figure 2.15
above. Load was applied near end of the cantilever beam and added
progressively. Clinometer is attached at the beam and column as shown. It
gives rotation reading for each beam and column. The differences between
these two rotation values, will give the value of rotation, φ needed for plotting
the M-φ curve. The details of the connection are shown in Figure 2.16 below.
Figure 2.16: Details of beam-to-column connections
24
2.5.2 FINITE ELEMENT METHOD (FEM)
In the finite element method, the response of a complex shape to any external
loading, can be calculated by dividing the complex shape into lots of simpler
shapes. These are the finite elements that give the method its name. The
shape of each finite element is defined by the coordinates of its nodes.
Adjoining elements with common nodes will interact.
(a) Definition
(i) FEM is a numerical procedure of finding solution to a
complicated problem establishing the response of
interconnected elements of finite dimensions with continuity
and equilibrium considerations (Desai 1985)
(ii) FEM is a computer-aided mathematical technique for
obtaining approximate numerical solutions to the abstract
equations of calculus that predict the response of physical
systems subjected to external influences (Burnet 1998)
(iii) FEM is a numerical method for solving problems of
engineering and mathematical physics which include structural
analysis, heat transfer, fluid flow, mass transport and
electromagnetic potential (Logan 1981)
(b) The basic concept
In the finite element method, the structure under consideration is
divided into smaller zones, known as elements. The elements are
assumed to be connected to each other at certain points (usually at the
corners) called nodes. It is at the nodes that we compute the
displacements. Thus the body with infinite number of degrees of
freedom is approximated by a body having degrees of freedom equal
to two or three times the number of nodes. It is obvious, though there
are rigorous mathematical proofs available, that as the number of
nodes is increased, a better (closer to the exact) solution is obtained.
The displacements at any point within an element are related to the
25
displacements at the nodes by making certain assumptions.
Displacements are fundamental variables. From the displacements, the
strains can be obtained, and then using the stress-strains relationships,
the stresses can be calculated.
(c) Solution process
The process for solving a problem using the finite element method
involves six major steps:
Step 1. Establish governing equations and boundary conditions.
In order to generate a valid approximate solution to a problem, the
differential equation that governs the behavior and the corresponding
boundary conditions for the problem must be determined. Once this is
done the appropriate finite element formulation can be used to
generate
the solution.
Step 2. Divide solution domain into elements.
In this step, the entire solution domain is subdivided into �small�
elements. Care is taken to make sure that enough elements are
included to capture the behavior of the solution over the entire
domain. Areas of particular interest and care are locations where
critical values are expected, locations with large stress gradients,
locations where the geometry changes suddenly, locations where
boundary conditions and loads are applied. Typically, the larger the
number of elements the better the approximation of the solution to the
differential equation.
Step 3. Determine element equations.
Once the elements are formed, the algebraic equations to be solved
are developed for each individual element. The form of the algebraic
equations for every element will be the same. Differences
26
from one element to the next will be due to changes in element size
and properties. This is the power of the finite element method, the
equations can be written once for a general element then they only
need to be modified to reflect particular elements geometry and
properties.
Step 4. Assembly of global equations.
Once all the element equations are generated, they are put together to
form a system of equations for the entire solution domain.
Step 5. Solution of global equations.
This system of equations is solved for the value of the dependent
variable in the original differential equation at discreet points
throughout the solution domain. Depending on the problem types
there may be hundreds, thousands, tens of thousands, or even
hundreds of thousands of points at which the solution to the
differential equation is approximated.
Step 6. Solution verification.
The accuracy of the solution must be verified before the results can be
considered valid. One way to do this is to refine the mesh (increase
the number of elements) and rerun the analysis. If the value of the
dependent variable at the discreet points in the mesh does not change
significantly as the mesh is refined, the solution is deemed to be
accurate.
(d) Joint Element
Joint elements may be introduced into the structural idealisation in
order to model releases, springs or restraints between any two nodes
in arbitrary directions. Joint elements are available for use in two and
three dimensions and comprise a range of nonlinear material and
boundary condition models. These nonlinear models enable the
realistic modelling of hardening elastoplastic compressive and tensile
joint behaviour as well as contact and friction types of nonlinear
boundary condition.
27
directioncoordinatelocalx�
freedomofreed
forcenodallocalf
x
x
deg�
�
�2
�2
node
k - spring constant
node
Uniaxial Bar k = AE/L
(e) Decision in finite element modelling
Finite element analysis in the forms of computer software packages
has now been made available. These products are presented and
displayed very impressively and allow interactive modelling and
checking, with colour graphics, windowing, etc. With the
development of the sophisticated software plus a decrease in the price
of hardware, it has now become a competitive business amongst the
software suppliers.
Some of the factors that govern the choice of this software are
namely:
(a) Availability
(b) Degree of sophistication
(c) Limitations
(d) Ease of use
(e) Accuracy
(f) Special features
(g) Costs
Currently, the complete FEA software packages available are;
LUSAS, ANSYS, COSMOS-M, PAFEC, IMAGES-3D, GT-
STRUDL, SAP80/90, FESDEC, SUPERSAP, GIFTS, ESDUFINE,
etc.
28
In this research, LUSAS 13.5 [1] software is being used for the
numerical analysis. This is mainly due to its availability, ease of use,
accuracy and cost.
(f) LUSAS Version 13.5
A complete finite element analysis of LUSAS involves three stages:
i) Pre-Processing
ii) Finite Element Solver
iii) Results-Processing
i) Pre-Processing
Pre-processing involves creating a geometric representation of the
structure, then assigning properties, then outputting the information as
a formatted data file (.dat) suitable for processing by LUSAS.
Creating a Model
A created model in LUSAS is a graphical representation consisting of
Geometry (Points, Lines, Combined Lines, Surfaces and Volumes)
and Attributes (Mesh, Geometric, Materials, Support, Loading). Each
part of the model is created in two steps: First, �Define� the feature or
attribute, and second, �Assign� the attribute or attributes.
ii) Finite Element Solver
Once a model has been created, on the solve button is clicked to begin
the solution stage. LUSAS creates a data file from the model, solves
the stiffness matrix, and produces a result file (.mys). The results file
will contain some or all of the following data: Stresses, Strains,
Displacements, Velocities, Accelerations, Residuals, Reactions, Yield
flags, Potentials, Fluxes, Gradients, Named variables, Combination
datasets, Envelope definitions, Fatigue datasets and Strain energy.
29
iii) Results-Processing
Results-processing involves using a selection of tools for viewing and
analyzing the result file produced by the Solver. Many different ways
of viewing the results are available: Contour plots
(averaged/smoothed), Contour plots (unaveraged/unsmoothed),
Undeformed/Deformed Mesh Plots, Wood-Armer Reinforcement
Calculations, Animated Display of Modes/Load Increments, Yield
Flag Plots, Graph Plotting, Vector Plots.
30
CHAPTER 3
RESEARCH METHODOLOGY
3.1 LUSAS SOFTWARE
LUSAS is an associative feature-based Modeller. The model geometry is
entered in terms of features which are sub-divided (discretised) into finite
elements in order to perform the analysis. Increasing the discretisation of the
features will usually result in an increase in the accuracy of the solution, but
with a corresponding increase in solution time and disk space required. The
features in LUSAS form a hierarchy that is Volumes are comprised of
Surfaces, which in turn are made up of Lines or Combined Lines, which are
defined by Points.
3.1.1 FINITE ELEMENT MODEL
LUSAS FEA software was used for the finite element analysis. The
FEA models were created using command files rather than the CAD
interface tools even though this method was longer and initially
tedious. The command file could simply be copied and edited. The
command file also was more logical in order than command files
produced by the software after a model has been created. The
command file was also well described by 6 comments within the file
to provide a complete history of the model creation. FEA models can
often be a black box that provides answers without the user being
fully aware of what the model exactly entails. The extra work in
creating the command files has been well worth the effort and allowed
31
the subsequent models to be created quickly. The technique of FEA
lies in the development of a suitable mesh arrangement. The mesh
discretisation must balance the need for a fine mesh to give an
accurate stress distribution and reasonable analysis time. The optimal
solution is to use a fine mesh in areas of high stress gradients and a
coarser mesh in the remaining areas.
3.1.2 ELEMENT TYPES
Four element types were used as shown in Figure 3.1 (as used by Jim
Butterworth [19]). HX8M elements are three dimensional solid
hexahedral elements comprising 8 nodes each with 3 degrees of
freedom. Although the HX8M elements are linear with respect to
geometry, they employ an assumed internal strain field which gives
them the ability to perform as well as 20 noded quadratic iso-
parametric elements. These elements are used to model the beam
flanges, end plate and connecting column flange. QTS4 elements are
three dimensional flat facet thick shell elements comprising either 3 or
4 nodes each with 5 degrees of freedom and are used to model the
beam (web and flange) and column (web and column back flange).
JNT4 elements are non-linear contact gap joint elements and are used
to model the interface between the end plate and the column flange.
The bolts will be modeled by using BRS2 elements for the bolt shank
and HX8M elements for the head and nut as shown in Figure 3.2.
BRS2 are three dimensional bar elements comprising 2 nodes each
with 3 degrees of freedom. Each BRS2 element is connected to the
appropriate HX8M bolt head and nut to comprise the complete bolt
assembly. All bolts used were M20 grade 8.8 and were assigned an
area of 245mm2 which is equal to the tensile stress area. The bolt
holes were modeled as a square cut-out in the end plate and column
flange. Figure 3.3(a), 3.3(b) and 3.3(c) shows the FEA model with the
arrangement of mesh discretisation, whereas the following Figures
3.3(d) and 3.3(e) show the relevant attributes forms.
32
Figure 3.1: Element types
Figure 3.2: Enlarged FEA Bolt Arrangement
Appendix A shows the properties of the element types used in this
study.
Figure 3.3 (a): Line Mesh
33
Figure 3.3 (b): Mesh Discretisation
Figure 3.3(c): Mesh BRS2 and JNT4
34
Figure 3.3(d): Attribute forms
35
Figure 3.3(e): Attribute forms
3.1.3 NON-LINEAR ANALYSIS
Material non-linearity occurs when the stress-strain relationship ceases to be
linear and the steel yields and becomes plastic. The three sets of material data
will be as follows: For the elastic dataset all elements are defined as elastic
isotropic with a Young�s Modulus of Elasticity of 2.09 x 105 N/mm2 and
Poisson�s lateral to longitudinal strain ratio of 0.3. The actual materials test
certificates were obtained for all steel and enabled stress/strain curves to be
based on actual values rather than theoretical Tensile tests records on a
selection of bolts were available to enable the material properties used to be
as accurate as possible. Von Mises yield criteria was used for all material.
Figure 3.4 shows the relevant plastic attributes used in the model.
36
Figure 3.4: Attribute form
3.1.4 BOUNDARY CONDITIONS
Displacements of the nodes in the X, Y and Z directions were
restrained at the bottom of the column. Whereas at the top of the
column the displacement of all nodes in the X and Z direction were
restrained. The FEA model would have problems converging when
the beam end plate had no supports restraining movement in the Y
direction due to the lack of bending resistance in the bolt BRS2
elements. Therefore supports were added to the underside of the end
plate. This removed the shear force from the bolts but not of course
from the remaining connection elements. Shear in moment
connections is usually of minor importance but it is felt that the
supports are a compromise. The column flange to end plate interface
was modeled by using JNT4 joint elements with a contact spring
stiffness K of 0.1 kN/mm whereas, the bolt to end plate interface used
JNT4 with contact spring stiffness K of 1 kN/mm. An initial point
load of -20 kN was placed at 1300 mm from the column face. The
load was then factored in the control file to achieve the required range
of connection bending moments. Figure 3.5 shows the attribute form
for loads, whereas Figure 3.6 shows FEA supports and loading.
37
Figure 3.6: FEA Supports and LoadingFigure 3.5: Attribute form
38
3.1.5 SUMMARY OF LUSAS FINITE ELEMENT SYSTEM
The following chart shows the analysis processes involved in LUSAS;
Initialize Model
Feature Geometry
Preparing Model Attributes, Define mesh, geometry, material properties, supports and loading
Assign to Features
Analysis Create data file
Save Model Run Analysis
Successful
Pre-Processing (MYSTRO)
Check Data Input
NO
Post Processing
39
CHAPTER 4
RESULTS AND DISCUSSIONS
This chapter contains the results of the non-linear as well as linear analyses
involving connections containing corrugated web as well as plain web (for the sake
of comparison).
LUSAS does not provide moment as well as rotation values. So, steps were
taken to calculate these values using Excell Spreadsheet and using the loads and node
displacements data extracted from LUSAS.
Moment-Rotation Curve Calculation
For displacement analysis, the selected nodes are;
CORRUGATED WEB BEAM PLAIN WEB BEAM
Nonlinear analysis Nonlinear analysis
29054 (0,300,-183.2) � dz 37262 (0,300,-183.2) � dz
35037 (0,200,100) � dy 42890 (0,200,100) � dy
Linear analysis Linear analysis
43153 � dy 56670 � dy
Node 29054 and node 37262 (See Figure 4.4 (a)) is located along the column centre
line 100 mm above the intersection point between the beam and column centroidal
axes. Whereas, Node 35037 and node 42890 (See Figure 4.4 (b)) is located along
beam centre line 100 mm from the column face. All the 4 nodes are used for
40
nonlinear analysis. For linear analysis, nodes 43153 and node 56670 selected, were
located at the cantilever end (to plot load-deflection curve).
The LUSAS software doesn�t have the capability to produce moment-rotation curve
numerically, thus it has to be done manually.
For the applied moment, it is calculated by using the following formulae;
M = Total load factor * 20 * 1.3
20 kN is the initial point load and it is located 1300 mm @ 1.3m from the column
face.
The joint connection, rotation φj is the difference of beam rotation φb and column
rotation φc. It can be shown by the formulae below;
φj = φb - φc
The unit of φ is in radian and the displacement is in milimetres.
φb = Tan-1 (∆y / 100 mm)
φc = Tan-1 (∆z / 100 mm)
∆y is the vertical displacement of the node selected from the centre of rotation,
whereas ∆z is the horizontal displacement from the centre of rotation. 100 mm is the
distance of the node (inclinometer position) from the centre of rotation.
Table 4.1 shows the results of moment-rotation for the extended end plate
connection containing corrugated web beam with increasing loads. Table 4.2 shows
the moment-rotation for the connection using plain web beam. These results are
plotted as shown in Figure 4.1. The M-φ graph obtained from the experimental
testing is also plotted for the purpose of comparison.
41
Graph 1: Experimental results for M-Фcurve F
Moment-Rotation Curve
-50.000
0.000
50.000
100.000
150.000
200.000
250.000
300.000
-10 0 10 20 30 40 50
Rotation (mRad)
Mom
ent (
kNm
)
Experimental resultsinite Element Analysis (FEA) results corrugated web beam
Finite Element Analysis (FEA) plain web beam
Figure 4.1: Comparison of M-φ Curve between Experimental and FEA results
(Nonlinear Analysis)
Moment-Rotation Curve for Test EEP-1
-50.00
0.00
50.00
100.00
150.00
200.00
-5.000 0.000 5.000 10.000 15.000 20.000 25.000 30.000
Rotation, miliradians(mRad)
Mom
ent,
(kNm
)
Reading 1Reading 2
Figure 4.2: Experimental graph for Moment-Rotation Curve (N5 specimen)
in determining Moment capacity, MR of the connection.
42
Table 4.1: Corrugated web beam data (nonlinear analysis)
dy t l factor dz
teta
beam
teta
column
rotation
mrad
moment
kNm
0 0 0 0 0 0 0
-0.12868 1 0.052729 0.001287 0.0005273 0.75954979 26
-0.3861 3 0.158208 0.003861 0.0015821 2.27893643 78
-0.78267 5.828427 0.309772 0.007827 0.0030977 4.72911635 151.5391052
-1.84377 8.656854 0.463158 0.01844 0.0046316 13.8081448 225.0782105
-2.03647 8.944215 0.49033 0.020368 0.0049033 15.4642112 232.5496021
-2.20023 9.142224 0.517218 0.022006 0.0051722 16.8336702 237.6978271
-2.41193 9.346859 0.555156 0.024124 0.0055516 18.5723162 243.0183356
-2.63961 9.516247 0.59956 0.026402 0.0059957 20.4065524 247.4224099
-2.86489 9.651213 0.647102 0.028657 0.0064711 22.1856095 250.9315271
-3.10805 9.773046 0.701902 0.03109 0.0070191 24.0713575 254.0992023
-3.38192 9.891128 0.773224 0.033832 0.0077324 26.0996631 257.1693346
-3.76746 10.032 0.871815 0.037692 0.0087184 28.9740984 260.8320711
-4.31824 10.2067 1.012284 0.043209 0.0101232 33.0860943 265.374231
-5.11394 10.43211 1.220651 0.051184 0.0122071 38.9768944 271.2347509
-6.28785 10.72996 1.536855 0.062962 0.0153698 47.59176 278.9788722
Table 4.2: Plain web beam data (nonlinear analysis)
dy t l factor dz
teta
beam
teta
column
rotation
mrad
moment
kNm
0 0 0 0 0 0 0
-0.30643 1 0.127602 0.003064 0.001276 1.7882848 26
-0.94769 3 0.38794 0.009477 0.003879 5.5977875 78
-1.21765 3.5 0.4591866 0.012177 0.004592 7.5851813 91
-1.7556 4 0.7571551 0.017558 0.007572 9.986142 104
-2.35666 4.183545 1.0377382 0.023571 0.010378 13.193229 108.7721613
-2.80337 4.242733 1.150645 0.028041 0.011507 16.534047 110.3110558
-3.21119 4.290365 1.2497375 0.032123 0.012498 19.624951 111.5494839
-3.58685 4.332549 1.3432876 0.035884 0.013434 22.45021 112.6462763
-4.65843 4.447822 1.628811 0.046618 0.01629 30.328473 115.643377
-4.89024 4.472146 1.693724 0.048941 0.016939 32.002575 116.2757908
43
Moment-Rotation Curve
0
50
100
150
200
250
300
0 5 10 15 20 25 30 35 40 45 50
Rotation (mRad)
mom
ent (
kNm
)
Figure 4.3: Finite Element Analysis graph for Moment-Rotation Curve (N5
specimen) in determining Moment capacity, MR of the connection.
Figure 4.4(b): Position of nodes
selected in determining the displacement
Figure 4.4(a): Position of nodes
selected in determining the displacement
44
These M-φ curves plotted are then used to determine the moment capacity, MR of the
connection. It is obtained by first, drawing the tangent line to the graph, second, the
angle of the curve is divided into two. From that, another line is drawn parallel to
that line. The intersection of vertical and horizontal tangent line is the moment
capacity of the connection. Figures 4.2 and 4.3 shows the way to calculate the
moment capacities, MR of test and LUSAS data of the extended end plate connection
containing corrugated web beam. The experimental and LUSAS value of MR are 100
kNm and 150 kNm respectively. A difference of 50%.
From Figure 4.1, it can be observed that all three graphs behaves in a similar
manner with increasing load. Initially, at small loading, the three graphs are straight
(indicating that the connections are elastic). At a rotation of about 3 mrad, the three
graphs curve showing that they become plastic. As indicated earlier, the difference
between moment of resistance of LUSAS and test is 50%. The use of plain web
beam, however, produces a moment of resistance MR of 75 kNm which is 25% lower
than the experimental value.
But, from the LUSAS moment of resistance result shown, the Moment-
Rotation curve is 1.5 times the value of the experimental Moment-Rotation curve as
well as its moment capacity (150 kNm vs 100 kNm). Thus, the model may not be
sufficient to validate the experimental result or being the alternative method in
replacing the actual case. It is believed, that the input data may not accurate in
analyzing the model. Proper material testing should be carried out to get the actual
data. The stiffness value for the contact elements, which plays an important role in
the semi rigid connection behavior, was obtained by trial and error to get to the best
moment-rotation curve as closely as possible to the experimental curve. And it was
found that to get the closest graph of moment-rotation curve as the experimental, the
value of K = 1.0 kN/mm for the contact interface between the bolt and the end plate,
whereas, between the column face and the end plate, a spring stiffness value of 0.1
kN/mm should be used. It took 1.5hrs � 2 hrs to run each model.
45
Also, the three M-φ curves from Figure 4.1 shows that the use of corrugated web
beam produces a stiffer connection compared to that of using plain web beam. All
the three curves indicate that they can be classified as semi rigid connections.
Figure 4.5 and Figure 4.6 shows the test and LUSAS deformed shape of the
connection respectively.
The effect of the various combinations of K values for the contact interface
between the bolt and the end plate and between the column face and the end plate on
the shape of the M-φ curves can be seen in Appendix B (Variations in K).
Figure 4.5: N5 specimen after failure Figure 4.6: FEA deformed mesh for
N5
46
CHAPTER 5
CONCLUSION AND RECOMMENDATION FOR FUTURE
WORK
5.1 CONCLUSIONS
The M-φ curve obtained from the finite element results are in accordance
with the experimental results. But the moment of resistance MR of LUSAS is
1.5 times the value of the experimental moment of resistance MR. Thus it
shows that the model may not sufficiently accurate to obtain a good MR result
for the extended end plate connection.
5.2 RECOMMENDATION FOR FUTURE WORK
The following recommendation can be useful for future investigations:
a) Proper material testing should be carried out to determine the actual
Young�s Modulus, uniaxial yield stress, hardening gradient slope, plastic
strain, etc. Thus, more accurate information can be input into the
software.
b) To get better results, whenever possible, the finite element mesh should
be relatively uniform. Special caution should be exercised in transition
from coarse to finer mesh. The aspect ratio between the element�s
longest and shortest dimensions should not be excessive. The optimum
aspect ratio is close to unity. Illegal element shapes must be avoided. For
triangular elements angles less than 30% are not desirable.
47
c) Since large stiffness variations between elements can lead to an ill-
conditioned stiffness matrix of the total structure, rendering meaningless
result, such conditions must be avoided by all means.
d) The beam can be modeled with other type of corrugation, like; horizontal
one arc corrugation, horizontal two arcs corrugation and vertical arcs
corrugation, instead of trapezoidal corrugation. The corrugated beam can
then be compared with the plain web beam.
e) Different types of meshing can be compared for the same type of model
to see the differences.
f) Different type of end plate thickness can be modeled to see the
connection behaviour.
48
REFERENCES
1. FEA Ltd, �LUSAS: Modeller User Manual, Version 13�, United Kingdom.
2. Bose B, Sarkar S, and Bahrami M, Finite Element Analysis of unstiffened
extended end plate connections, Structural Engineering Review, 3, 211-224,
1991.
3. Bose B, Youngson G K, and Wang Z M, An appraisal of the design rules in
Eurocode 3 for bolted end plate joints by comparison with experimental
results, Proceedings from the Institute of Civil Engineers Structures and
Buildings, 1996.
4. Krishnamurty, N. (1976), �Correlation between 2 & 3-D Finite Element
Analysis of Steel Bolted End-Plate Connections�, Computers and Structures,
6, 381-389.
5. Bursi, O.S. and Lionelli, L. (1994), �A Finite Element Model for the
Rotational Behavior of End Plate Steel Connections�, Proceedings of the
SSRC Annual Technical Session, Structural Stability Research Council,
Bethlehem P.A, 163-175.
6. Bursi, O.S., and Jaspart, J.P. (1997b), �Calibration of a Finite Element Model
for Isolated Bolted End-Plate Steel Connections�, Journal of Constructional
Steel Rersearch, 42, 225-262.
7. Richard, R.M., and Abbott, B.J., (1975), �Versatile Elastic-Plastic Stress-
Strain Formula�, Journal of the Engineering Mechanics Division, ASCE, 101,
511-515.
49
8. Gebbeken, N. Rothert, H. and Binder, B. (1994), �On the Numerical
Analysisi of Endplate Connections�, Journal of Constructional Steel
Research, 30, 177-196.
9. Rothert, H., Gebbeken, N. and Binder, B. (1992), �Nonlinear Three-
Dimensional Finite Element Contact Analysis of Bolted Connections in Steel
Frames�, International Journal for Numerical methods in Engineering, 34,
303-318.
10. Sherbourne, A.N and Bahaari, M.R. 91997), �Finite element Prediction of
End Plate Bolted Connection Behavior. I: Parametric study�, Journal of
Structural Engineering, ASCE, 123, 157-164.
11. Bursi, O.S. and Jaspart, J.P. (1997a), �Benchmarks for Finite Element
Modeling of Bolted Steel Connections�, Journal of Constructional Steel
Research, 42, 17-42.
12. Bursi, O.S, and Jaspart, J.P. (1998), �basic Issues in the Finite Element
Simulation of Extended End Plate Connections�, Computers and Structures,
69, 361-382.
13. Ribeiro, L.Calado, C.A. Castiglioni, C. Bernuzzi, Stability and Ductility of
Steel Structures Edited by T.Usami and Y.Itoh, Elsevier Science Ltd. 1998,
pp279-292, �Behavior of Steel Beam-To-Column Joints Under Cyclic
Reversal Loading: An Experimental Study�
14. Troup, S. Xiao, R.Y. and Moy, S.s.J. (1998), �Numerical Modeling of Bolted
Steel Connections� Journal of Constructional Steel Research, 46, Paper No.
362.
15. Y.I.Maggi, R.M.Goncalves, R.T.Leon, L.F.L.Ribeiro, �Parametric analysis of
steel bolted end plate connections using finite element modeling�, Journal of
Constructional Steel Research 61, Elsevier, 2005, pg 689-708
50
16. Elgaaly M, Hamilton RW, Seshadri A., �Shear Strength of beams with
corrugated webs�, Journal of Structural Engineering ASCE 1996; 122(4):390-
8.
17. Zhang W, Li Y, Zhou Q, Qi X, Widera GEO. �Optimization of the structure
of an H-beam, with either a flat or a corrugated web. Part 3: Development
and research on H-beams with wholly corrugated webs�, Journal of Materials
Processing Technology 2000;101(1):119-23.
18. Li Y, Zhang W, Zhou Q, Qi X, Widera GEO, �Buckling strength analysis of
the web of a WCW H-beam: Part 2. Development and research on H-beams
with wholly corrugated webs (WCW)�, Journal of Materials Processing
Technology 2000;101(1):115-8.
19. Jim Butterworth, �Finite Element Analysis of Structural Steelwork Beam to
Column Bolted Connections�, Constructional Research Unit, School of
Science & Technology, University of Teeside, UK
51
APPENDIX A (LUSAS Element Types)
Element Name JNT4
Element Group Joints
Element Subgroup 3D Joints
Element Description A 3D joint element which connects two nodes by three springs
in the local x, y and z-directions. Use JL43 for semiloof shell corner nodes.
Number Of Nodes 4. The 3rd and 4th nodes are used to define the local x-axis
and local xy-plane.
Freedoms U, V, W: at nodes 1 and 2 (active nodes).
Node Coordinates X, Y, Z: at each node.
Geometric Properties
Not applicable.
Material Properties
Linear Not applicable.
Matrix Stiffness: MATRIX PROPERTIES STIFFNESS 6 K1,..., K21 element
stiffness matrix (Not supported in LUSAS Modeller)
Mass: MATRIX PROPERTIES MASS 6 M1,..., M21 element mass matrix (Not
supported in LUSAS Modeller)
Damping: MATRIX PROPERTIES DAMPING 6 C1,..., C21 element damping
matrix (Not supported in LUSAS Modeller)
Joint Standard: JOINT PROPERTIES 3 (Joint: 3/Stiffness)
Dynamic general: JOINT PROPERTIES GENERAL 3 (Joint: 3/General)
Elasto-plastic: JOINT PROPERTIES NONLINEAR 31 3 (Joint: 3/Elasto-Plastic)
Elasto-plastic: JOINT PROPERTIES NONLINEAR 32 3 (Joint: 3/Asymmetric)
Nonlinear contact: JOINT PROPERTIES NONLINEAR 33 3 (Joint: 3/Initial
Gap)
52
Nonlinear friction: JOINT PROPERTIES NONLINEAR 34 3 (Joint: 3/Frictional)
Concrete Not applicable.
Elasto-Plastic Not applicable.
Rubber Not applicable.
Composite Not applicable.
Field Not applicable.
Stress Potential Not applicable.
Creep Not applicable.
Damage Not applicable.
Viscoelastic Not applicable.
Loading
Prescribed Value PDSP, TPDSP Prescribed variable. U, V, W: at active nodes.
Concentrated Loads CL Concentrated loads. Px, Py, Pz: at active nodes.
Element Loads Not applicable.
Distributed Loads Not applicable.
Body Forces CBF Constant body forces for element. Xcbf, Ycbf, Zcbf, Wx, Wy,
Wz, ax, ay, az
BFP, BFPE Not applicable.
Velocities VELO Velocities. Vx, Vy, Vz: at nodes.
Accelerations ACCE Accelerations. Ax, Ay, Az: at nodes.
Initial Stress/Strains SSI, SSIE Initial stresses/strains at nodes/for element. Fx,
Fy, Fz: spring forces in local directions. ex, ey, yz: spring strains in local directions.
SSIG Not applicable.
Residual Stresses Not applicable.
Temperatures TEMP, TMPE Temperatures at nodes/for element. T1, T2, T3, T1o,
T2o, T3o: actual and initial spring temperatures.
Field Loads Not applicable.
Temp DependentLoads Not applicable.
53
Element Name BRS2
Element Group Bars
Element Subgroup Structural Bars
Element Description Straight and curved isoparametric bar elements in 3D which
can accommodate varying cross sectional area.
Number Of Nodes 2 or 3.
Freedoms U, V, W at each node.
Node Coordinates X, Y, Z at each node.
Geometric Properties
A1 ... An Cross sectional area at each node.
Material Properties
Linear Isotropic MATERIAL PROPERTIES (Elastic: Isotropic)
Matrix Not applicable.
Joint Not applicable.
Concrete Not applicable.
Elasto-Plastic Stress resultant Not applicable.
Tresca: MATERIAL PROPERTIES NONLINEAR 61 (Elastic: Isotropic, Plastic:
Tresca, Hardening: Isotropic Hardening Gradient, Isotropic Plastic Strain or Isotropic
Total Strain)
Drucker-Prager: MATERIAL PROPERTIES NONLINEAR 64 (Elastic:
Isotropic, Plastic: Drucker-Prager, Hardening: Granular)
Mohr-Coulomb: MATERIAL PROPERTIES NONLINEAR 65 (Elastic:
Isotropic, Plastic: Mohr-Coulomb, Hardening: Granular with Dilation)
Von Mises (B/Euler): MATERIAL PROPERTIES NONLINEAR 75 (Elastic:
Isotropic, Plastic: Von Mises, Hardening: Isotropic & Kinematic)
Volumetric Crushing: Not applicable.
Rubber Not applicable.
Composite Not applicable.
Field Not applicable.
54
Stress Potential STRESS POTENTIAL VON_MISES
(Isotropic: von Mises, Modified von Mises)
Creep CREEP PROPERTIES (Creep)
Damage DAMAGE PROPERTIES SIMO, OLIVER (Damage)
Viscoelastic VISCO ELASTIC PROPERTIES
Loading
Prescribed Value PDSP, TPDSP Prescribed variable. U, V, W at each node.
Concentrated Loads CL Concentrated loads. Px, Py, Pz at each node.
Element Loads Not applicable.
Distributed Loads Not applicable.
Body Forces CBF Constant body forces for element. Xcbf, Ycbf, Zcbf, Wx, Wy,
Wz, ax, ay, az
BFP, BFPE Body force potentials at nodes/for element. 0, 0, 0, 0, Xcbf, Ycbf,
Zcbf
Velocities VELO Velocities. Vx, Vy, Vz at nodes.
Accelerations ACCE Acceleration Ax, Ay, Az at nodes.
Initial Stress/Strains SSI, SSIE Initial stresses/strains at nodes/for element.
(1) Resultants (linear material models): Fx , ex
(2) Components (nonlinear material models): 0, 0, sx , ex
SSIG Initial stresses/strains at Gauss points.
(1) Resultants (linear material models): Fx , ex
(2) Components (nonlinear model): 0, 0, sx , ex
Residual Stresses SSR, SSRE Not applicable.
SSRG Residual stresses at Gauss points.
Components (nonlinear material models): 0, 0, sx
Temperatures TEMP, TMPE Temperatures at nodes/for element. T, 0, 0, 0, To, 0, 0,
0 in local directions.
Field Loads Not applicable.
Temp DependentLoads Not applicable.
55
Element Name HX8M
Element Group 3D Continuum
Element Subgroup Solid Continuum
Element Description A 3D isoparametric solid element with an incompatible strain
field. This mixed assumed strain element demonstrates a much superior performance
to that of the HX8 element.
Number Of Nodes 8. The element is numbered according to a right-hand screw
rule in the local z-direction.
Freedoms U, V, W: at each node.
Node Coordinates X, Y, Z: at each node.
Geometric Properties
Not applicable.
Material Properties
Linear Isotropic: MATERIAL PROPERTIES (Elastic: Isotropic)
Orthotropic: MATERIAL PROPERTIES ORTHOTROPIC SOLID (Elastic:
Orthotropic Solid)
Anisotropic: MATERIAL PROPERTIES ANISOTROPIC SOLID (Elastic:
Anisotropic Solid)
Rigidities. Not applicable.
Matrix Not applicable.
Joint Not applicable.
Concrete MATERIAL PROPERTIES NONLINEAR 82 (Elastic:
Isotropic, Plastic: Cracking concrete)MATERIAL PROPERTIES NONLINEAR 84
(Elastic: Isotropic, Plastic: Cracking concrete with crushing)
Elasto-Plastic Stress resultant: Not applicable.
Tresca: MATERIAL PROPERTIES NONLINEAR 61 (Elastic: Isotropic, Plastic:
Tresca, Hardening: Isotropic Hardening Gradient, Isotropic Plastic Strain or Isotropic
Total Strain)
56
Drucker-Prager: MATERIAL PROPERTIES NONLINEAR 64 (Elastic:
Isotropic, Plastic: Drucker-Prager, Hardening: Granular)
Mohr-Coulomb: MATERIAL PROPERTIES NONLINEAR 65 (Elastic:
Isotropic, Plastic: Mohr-Coulomb, Hardening: Granular with Dilation)
Von Mises (B/Euler): MATERIAL PROPERTIES NONLINEAR 75 (Elastic:
Isotropic, Plastic: Von Mises, Hardening: Isotropic & Kinematic)
Volumetric Crushing: MATERIAL PROPERTIES NONLINEAR 81 (Volumetric
Crushing or Crushable Foam)
Rubber Ogden: MATERIAL PROPERTIES RUBBER OGDEN (Rubber: Ogden)
Mooney-Rivlin: MATERIAL PROPERTIES RUBBER MOONEY_RIVLIN
(Rubber: Mooney-Rivlin)
Neo-Hookean: MATERIAL PROPERTIES RUBBER NEO_HOOKEAN (Rubber:
Neo-Hookean)
Hencky: MATERIAL PROPERTIES RUBBER HENCKY (Rubber: Hencky)
Generic Polymer Isotropic MATERIAL PROPERTIES NONLINEAR 87
(Generic Polymer Model)
Composite Not applicable.
Field Not applicable.
Stress Potential STRESS POTENTIAL VON_MISES, HILL,
HOFFMAN
(Isotropic: von Mises, Modified von Mises
Orthotropic: Hill, Hoffman)
Creep CREEP PROPERTIES (Creep)
Damage DAMAGE PROPERTIES SIMO, OLIVER (Damage)
Viscoelastic VISCO ELASTIC PROPERTIES
Loading
Prescribed Value PDSP, TPDSP Prescribed variable. U, V, W: at each node.
Concentrated Loads CL Concentrated loads. Px, Py, Pz: at each node.
Element Loads Not applicable.
Distributed Loads UDL Not applicable.
FLD Face Loads. Px, Py, Pz: local face pressures at nodes.
57
Body Forces CBF Constant body forces for element. Xcbf, Ycbf, Zcbf, Wx, Wy,
Wz, ax, ay, az
BFP, BFPE Body force potentials at nodes/for element. 0, 0, 0, 0, Xcbf, Ycbf,
Zcbf
Velocities VELO Velocities. Vx, Vy, Vz: at nodes.
Accelerations ACCE Acceleration Ax, Ay, Az: at nodes.
Initial Stress/Strains SSI, SSIE Initial stresses/strains at nodes/for element. sx,
sy, sz, sxy, syz, sxz: global stresses. ex, ey, ez, gxy, gyz, gxz: global strains.
SSIG Initial stresses/strains at Gauss points sx, sy, sz, sxy, syz, sxz: global stresses.
ex, ey, ez, gxy, gyz, gxz: global strains.
Residual Stresses SSR, SSRE Residual stresses at nodes/for element. sx, sy,
sz, sxy, syz, sxz: global stresses.
SSRG Residual stresses at Gauss points. sx, sy, sz, sxy, syz, sxz global stresses.
Temperatures TEMP, TMPE Temperatures at nodes/for element. T, 0, 0, 0, To, 0, 0,
0
Field Loads Not applicable.
Temp DependentLoads Not applicable.
Element Name QTS4
Element Group Shells
Element Subgroup Thick Shells
Element Description A family of shell elements for the analysis of arbitrarily thick
and thin curved shell geometries, including multiple branched junctions. The
quadratic elements can accommodate generally curved geometry while all
elements account for varying thickness. Anisotropic and composite material
properties can be defined. These degenerate continuum elements are also
capable of modelling warped configurations. The element formulation takes
account of membrane, shear and flexural deformations. The quadrilateral
elements use an assumed strain field to define transverse shear which ensures
that the element does not lock when it is thin (see Notes).
58
Number Of Nodes 3, 4, 6 or 8 numbered anticlockwise.
Freedoms Default: 5 degrees of freedom are associated with each node U, V, W,
qa, qb. To avoid singularities, the rotations qa and qb relate to axes defined
by the orientation of the normal at a node, see Thick Shell Nodal Rotation.
These rotations may be transformed to relate to the global axes in some
instances (see Notes). Degrees of freedom relating to global axes: U, V, W,
qx, qy, qz may be enforced using the Nodal Freedom data input, or for all
shell nodes by using option 278 (see Notes).
Node Coordinates X, Y, Z: at each node.
Nodal Freedoms 5 or 6.
Geometric Properties
ez, t1... tn Eccentricity and thickness at each node.
Material Properties
Linear Isotropic: MATERIAL PROPERTIES (Elastic: Isotropic)
Orthotropic: MATERIAL PROPERTIES ORTHOTROPIC THICK (Elastic:
Orthotropic Thick)
Anisotropic: MATERIAL PROPERTIES ANISOTROPIC 5 (Elastic: Anisotropic
Thick Plate)
Rigidities. Not applicable.
Matrix Not applicable.
Joint Not applicable.
Concrete MATERIAL PROPERTIES NONLINEAR 82 (Elastic:
Isotropic, Plastic: Cracking concrete)MATERIAL PROPERTIES
NONLINEAR 84 (Elastic: Isotropic, Plastic: Cracking concrete with
crushing)
Elasto-Plastic Stress resultant: Not applicable.
Tresca: MATERIAL PROPERTIES NONLINEAR 61 (Elastic: Isotropic, Plastic:
Tresca, Hardening: Isotropic Hardening Gradient, Isotropic Plastic Strain or
Isotropic Total Strain)
59
Drucker-Prager: MATERIAL PROPERTIES NONLINEAR 64 (Elastic:
Isotropic, Plastic: Drucker-Prager, Hardening: Granular)
Mohr-Coulomb: MATERIAL PROPERTIES NONLINEAR 65 (Elastic:
Isotropic, Plastic: Mohr-Coulomb, Hardening: Granular with Dilation)
Volumetric Crushing: Not applicable.
Rubber Not applicable.
Composite Composite shell: COMPOSITE PROPERTIES
Field Not applicable.
Stress Potential STRESS POTENTIAL VON_MISES, HILL,
HOFFMAN
(Isotropic: von Mises, Modified von Mises
Orthotropic: Hill, Hoffman)
Creep CREEP PROPERTIES (Creep)
Damage DAMAGE PROPERTIES SIMO, OLIVER (Damage)
Viscoelastic Not applicable.
Loading
Prescribed Value PDSP, TPDSP Prescribed variable. 5 degrees of freedom: U,
V, W, qa, qb or 6 degrees of freedom: U, V, W, qx, qy, qz
Concentrated Loads CL Concentrated loads. 5 degrees of freedom: Px, Py, Pz,
Ma, Mb, where Ma and Mb relate to axes defined by qa and qb respectively.
6 degrees of freedom: Px, Py, Pz, Mx, My, Mz.
Element Loads Not applicable.
Distributed Loads UDL Uniformly distributed loads. Wx, Wy, Wz: mid-surface
local pressures for element.
FLD Not applicable.
Body Forces CBF Constant body forces for element. Xcbf, Ycbf, Zcbf, Wx, Wy,
Wz, ax, ay, az
BFP, BFPE Body force potentials at nodes/for element. j1, j2, j3, 0, Xcbf, Ycbf,
Zcbf, where j1, j2, j3 are the face loads in the local coordinate system.
Velocities VELO Velocities. Vx, Vy, Vz: at nodes.
Accelerations ACCE Accelerations. Ax, Ay, Az: at nodes.
60
Initial Stress/Strains SSI, SSIE Not applicable.
SSIG Initial stresses/strains at Gauss points. Stress/strain components relating to
local axes at Gauss points: sx, sy, sxy, syz, sxz, ex, ey, gxy, gyz, gxz. All of
these 10 terms are repeated for each fibre integration point through the
thickness (see Notes).
Residual Stresses SSR, SSRE Not applicable.
SSRG Residual stresses at Gauss points. Stress components relating to local axes at
Gauss points: sx, sy, sxy, syz, sxz all of these 5 terms are repeated for each
fibre integration point through the thickness (see Notes).
Temperatures TEMP, TMPE Temperatures at nodes/for element. T, 0, 0, dT/dz, To,
0, 0, dTo/dz
Field Loads Not applicable.
Temp Dependent Loads Not applicable.
61
APPENDIX B (Variations in K)
Variations in K (Spring stiffness for both column-endplate and bolt-endplate)
Unit of K is in kN/mm, Hardening gradient; Slope = 1, Plastic strain = 100
K=1 (C-E), K=10 (B-E) K=0.5 (C-E), K=5 (B-E)
Moment-Rotation Curve
0
100
200
300
400
500
600
-50 0 50 100 150 200 250 300 350 400 450 500
Rotation (mRad)
Mo
men
t (k
Nm
)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 70 80 90
Rotation (mRad)
mo
men
t (k
Nm
)
62
K=0.1 (C-E), K=1 (B-E) K=0.01 (C-E), K=0.1 (B-E)
Moment-Rotation Curve
0
50
100
150
200
250
300
0 5 10 15 20 25 30 35 40 45 50
Rotation (mRad)
mo
men
t (k
Nm
)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
0 20 40 60 80 100 120 140
Rotation (mRad)
mo
men
t (k
Nm
)
K=0.001 (C-E), K=0.01 (B-E) K=0.5 (C-E), K=1.0 (B-E)
Moment-Rotation Curve
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 400 450
Rotation (mRad)
mo
men
t (k
Nm
)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 70 80 90 100
Rotation (mRad)
mo
men
t (k
Nm
)
63
K=0.01 (C-E), K=1.0 (B-E) K=0.25 (C-E), K=7.5 (B-E)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
0 20 40 60 80 100 120 140 160
Rotation (mRad)
mo
men
t (k
Nm
)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 70 80 90
Rotation (mRad)
mo
men
t (k
Nm
)
K=0.05 (C-E), K=5.0 (B-E) K=0.15 (C-E), K=1.5 (B-E)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
400
0 20 40 60 80 100 120 140 160
Rotation (mRad)
mo
men
t (kN
m)
Moment-Rotation Curve
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 7
Rotation (mRad)
mo
men
t (kN
m)
0
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