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 Loading

Figure 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