Arthur Godwin Addiah

7/31/2019 Arthur Godwin Addiah

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INVESTIGATION OF THE FAILURE MECHANISM AND MOMENT CAPACITY

PREDICTION IN A TEN BOLT FLUSH END PLATE MOMENT CONNECTION

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

Godwin Addiah Arthur

August, 2010



ii

INVESTIGATION OF THE FAILURE MECHANISM AND MOMENT CAPACITY

PREDICTION IN A TEN BOLT FLUSH END PLATE MOMENT CONNECTION

Godwin Addiah Arthur

Thesis

Approved: Accepted:

_____________________________ ______________________________

Advisor Dean of the College

Dr. Craig Menzemer Dr. George K. Haritos

_____________________________ ______________________________

Faculty Reader Dean of the Graduate School

Dr. Anil Patnaik Dr. George R. Newkome

_____________________________ ______________________________

Faculty Reader Date

Dr. Kallol Sett

_____________________________

Department Chair

Dr.Wieslaw Binienda



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ABSTRACT

Moment end plate connections have found their usage in a number of applications

and extensive research has been done on smaller capacity connections of such type.

This study seeks to investigate the failure mechanism and the subsequent prediction

of the moment capacity of a ten bolt flush end plate connection. The model under

study is typical of a flush end plate connection present in an existing steel parking

deck. Yield line analysis methods have been employed in analyzing the various

failure mechanisms and have been used in developing prediction equations for the

design of the end-plate. The modified Kennedy method has also been used in the

prediction of the bolt forces for all cases of the analysis with failed bolts. A three

dimensional finite element model using ABAQUS Standard has been employed to

validate the results from the analytical methods taking into consideration, the

interactions between the connection components, boundary conditions and material

non-linearities. The moment capacity and the corresponding failure mechanism is

predicted under two cases of bolt loss in rows of the connection. Limit state of end

plate yielding governs the design for the ten bolt connection and that with one row

of bolt failure at the top. Bolt fracture governs the design when there are two rows

of failed bolts.



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ACKNOWLEDGEMENTS

My sincere thanks to all Faculty members of the Civil Engineering Department

at the University of Akron for their immense contribution and help especially my

advisor, Dr. Menzemer through whose guidance this work has been completed

succesfully.

To the other members of my thesis committee, Dr. Patnaik and Dr. Sett, for

their help, suggestions and contributions. To my friends, who through their

encouragement saw me through the completion of this research work. Finally, to

my family for the love, support and encouragement throughout the entire duration

of my study. To them I owe everything I have accomplished.



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TABLE OF CONTENTS

Page

LIST OF TABLES ………………………………………………………………….…ix

LIST OF FIGURES …………………………………………………………………... x

CHAPTER

I. INTRODUCTION ………………………………………………………………. ...1

1.1 Background ………………………………………………………………………1

1.2 Statement of Problem …………………………………………………………….3

1.3 Justification …………………………………………………………………… ...4

1.4 Objectives ……………………………………………………………………......5

1.5 Scope of Thesis …………………………………………………………………..5

1.6 Outline of Thesis ………………………………………………………………....6

II. LITERATURE REVIEW ………………………………………………..................7

2.1 Overview …………………………………………………………………………7

2.2 End -Plate Moment Connections ……………………………………………….10

2.2.1 Moment Capacity Prediction and Moment rotation curves………………...14

2.3 End Plate Design and performance……………………………………………...17



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2.4 Flush End Plate Moment Connections ………………………………………….19

2.5 Finite Element Analysis ………………………………………………………...22

2.6 Need for Further Research ……………………………………………………...26

III. DEVELOPMENT OF MOMENT PREDICTION EQUATIONS ………….........27

3.1 Overview ………………………………………………………………………..27

3.2 Yield Line Analysis of Steel End Plates ………………………………..............27

3.3 Earlier Moment Prediction Equations …………………………………………..30

3.3.1 Two Bolt Flush End Plate connection …………………………….........31

3.3.2 Four Bolt Flush End Plate Connection ………………………………....32

3.4 End Plate prediction Equations …………………………………………………35

3.4.1 Case one – ten bolt connection …………………………………………35

3.4.2 Case two – eight bolts …………………………………………………..37

3.4.3 Case three – six bolts …………………………………………………...39

3.5 Bolt Force Predictions ………………………………………………………….40

3.5.1 Prying Action ………………………………………………...................41

3.5.2 Determination of Plate Thickness Limits ………………………………43

3.5.3 Modified Kennedy Model ………………………………………………45

3.5.4 Moment Capacity Prediction based on Bolt Forces …………………….47

IV. RESULTS VERIFICATION AND FINITE ELEMENT ANALYSIS ………….51

4.1 Overview ………………………………………………………………………..51

4. 2 Geometry of Structural components …………………………………………...51



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4.3 Moment Capacity Predictions ………………………………………………….53

4.3.1 Limit State of End Plate Yielding ……………………………................53

4.3.2 Limit State of Bolt Fracture …………………………………………….55

4.3.3 Determination of Predicted Capacity …………………………………...57

4.4 Finite Element Analysis ………………………………………………………..58

4.4.1 Model of the Connection Components …………………………………58

4.4.2 Material Properties ……………………………………………………...59

4.4.3 Assembly ……………………………………………………………….61

4.4.4 Contact Interactions …………………………………………………….62

4.4.5 Boundary Conditions …………………………………………………...62

4.4.6 Loading and Analysis Steps …………………………………………….63

4.4.7 Element Type and Meshing …………………………………………….64

4.4.8 Visualization of Results and Failure patterns …………………………..65

4.5 Moment Capacity Prediction from Finite Element Analysis …………………..67

4.5.1 Comparison and Summary of Results ………………………………….69

V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS …………………71

5.1 Summary ………………………………………………………………………..71

5.2 Conclusions ……………………………………………………………………..72

5.3 Recommendations ……………………………………………………………...73

REFERENCES ……………………………………………………………………….74

APPENDICES ………………………………………………………………………..79



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APPENDIX A – Assumed Yield Line Mechanisms ………………………………....80

APPENDIX B – Finite Element Analysis Output Summary …………………………88



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LIST OF TABLES

Table Page

4.1 Section Properties for End Plate and Beam…………………………………...53

4.2 Summary of Predicted Moment Capacities with Failure mechanisms………..57

4.3 Comparison of the Moment Capacity values …………………………………70



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LIST OF FIGURES

Figure Page

1.1 Typical Parking Deck with failed tension bolts, (Courtesy: UA) ………….....4

2.1 End plate moment connection types…………………………………………...8 3.1 Virtual Displacement in a two bolt flush unstiffened end plate ……………...30

3.2 Controlling Yield Mechanism for Four bolts Flush End Plate ……………….33

3.3 Controlling Yield Line Mechanism for Four Bolt (AISC)……………………34

3.4 Equivalent form of the 10 bolt Flush onto a Four Bolt

Flush Configuration …………………………………………………………..36

3.5 Controlling Yield line Mechanism for the 10-bolt connection ……………….37

3.6 Controlling Yield line Mechanism for the 8-bolt connection ………………...38

3.7 Controlling Yield line Mechanism for the 6-bolt connection ………………...40

3.8 Stages of Plate Behavior based on the Kennedy Model ……………………...43

3.9 Modified Kennedy Model for cases of the flush end plate connection……… 46

3.10 Case One bolt forces with (a) No Prying Action (b) Prying Action…........... 48

3.11 Case Two bolt forces with (a) No Prying Action (b) Prying Action………...49

3.12 Case Three bolt forces with (a) No Prying Action (b) Prying Action…….....50



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4. 1 Connection Details for End Plate with Adjoining Beam…………………….. 52

4.2 Components of the Flush End Plate connection in ABAQUS CAE…………..59

4.3 (a) Stress Strain Relationship for Beam, Endplate and Column Material…….60

4.3 (b) Stress Strain Relationship for Bolts and Nut ……………………………...60

4.4 An Assembly of the 10-bolt Flush end plate connection ……………………..61

4.5 Connection Boundary Conditions …………………………………………….63

4.6 Mesh of Connection Assembly ……………………………………………….64

4.7 Linear element, 8-node brick, C3D8 (ABAQUS Analysis Manual) …………65

4.8 Patterns in different cases of connection failure ……………………………...67

4.9 Graph of applied moment against end plate separation for case 1……………68

4.10 Graph of applied moment against endplate separation for case 2…………….69



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CHAPTER I

INTRODUCTION

1.1 Background

The study and research on structural connections has been a subject of interest

in improving the overall performance of steel structures. Although some improvements

have been made to ensure economy and performance, a number of connections in

existing structures have been designed based on empirical methods and must be

examined in evaluating or determining the structural soundness of older infrastructure.

This has become necessary and evident due to the dilapidated state in which a good

deal of infrastructure has been found (ASCE Report Card, 2009). In this research, a

ten bolt flush end plate connection of an existing typical steel parking deck is

considered.

Upon a routine maintenance check, it had been established that some bolts of

the flush end plate connections had failed. Although the problem might be due to

causes other than over stressing, the structural adequacy of these connections is

investigated and the moment capacity of the 10-bolt flush end plate connection is

examined using prediction equations derived from existing methods and validated with

results from Finite Element Analysis (FEA). The predicted capacity is based on the

number of failed bolts, with the moment capacity of the connection determined at each



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stage as a function of the number of failed bolts. Two cases of rows of failed bolts

have been considered. The neutral axis is directly affected by the loss of bolts and this

directly affects the number of bolts in tension. The various failure mechanisms that

arise are also of interest.

According to the AISC Specification (2005), the connection behavior of a

semi-rigid connection requires a good knowledge of the moment capacity, rotational

capacity and stiffness of the connection. Though moment end plate connections are

normally classified as type one (rigid), study have shown that they exhibit semi-rigid

properties especially in this case where there might be considerable rotation due to

some failed bolts in tension. (McGuire, 1988)

A typical flush end plate bolted connection has a rectangular steel plate of

almost the same depth of beam section and welded to the end of the beam. The end

plate is bolted to the column flange with bolts both in the tension and compression side

of the beam. Analysis of bolted end-plate connections seem quite complex due to the

behavior of the individual components of the connection. The varying properties

which include material, geometry, and contact interactions between the various parts,

bolt size, beam depth, grade of steel, web thickness and the column side parameters

make the analysis complicated. The effect of prying forces resulting from the column

and end-plate interfaces after an extent of load application complicates the problem.

In order to investigate the failure mechanism of the connection, an experimental

set-up usually needed to accurately predict the behavior and performance of the structural

components, will be a cost intensive option since connection behavior results from varied

parameters of the various components.



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It would be difficult to study the various interactions between the end plate and

column flange. A three dimensional finite element analysis provides a convenient way to

study the behavior and interaction of the various parts of the 10-bolt flush end plate

connection using ABAQUS Standard commercial code. (ABAQUS Manual, 2006)

1.2 Statement of Problem

The loss of structural capacity of older structures has been of major concern

since the collapse of the I-35 Bridge in Minneapolis. The situation is not isolated to

bridges alone but to other structural systems which are either structurally deficient, due

to inadequate technology for analysis at the time of design or as a result of fatigue,

corrosion or a combination of other factors. In steel structures, connections play an

important role and account for the overall structural rigidity and stability of the

structure. Flush end plate connections have found use in a number of structures

including parking decks primarily due to the ease of erection and its added ability to

resist lateral loads in light frames.

Upon a routine inspection, it was found that some bolts at the flush end plate

connections of the beams had failed, thus affecting the overall structural performance

and moment capacity of the connection. Though the failed bolts have been replaced,

the extent to which the moment capacity of the flush-end plate connection was affected

needs to be evaluated to find out the impact on load carrying capacity and the probable

failure mechanisms which might result.



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Figure 1.1 - Typical Parking Deck with failed tension bolts, (Courtesy: UA)

1.3 Justification

It is important therefore to have design equations for the prediction of the

moment capacity of these flush end plate connections. Murray and Shoemaker (AISC

Design Guide 16) provides a methodology for the design of extended and flush end

plate moment connections, but the configurations are only up to four bolts in both the

stiffened and unstiffened case. Extensive work has been carried out on the design of

extended end plate moment connections but very few on large capacity connections.

In studying the 10-bolt flush end plate connection of the parking deck, an

extrapolation of the Design equations in AISC Design Guide 16 is employed and

compared to a new set of design equations to be developed from yield line analysis and

a modified form of the Kennedy method. The behavior of the connection would further

be validated using finite element analysis and compared with other design equations

proposed.



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1.4 Objectives

In studying the probable failure mechanisms resulting from the failed bolts and

predicting the moment capacity, the following will be examined:

• Identifying possible failure modes that could result due to a certain number of

failed bolts, analyzing the failure mechanisms with the yield line method in

evaluating end plate yielding and the prediction of the bolt forces using a

modified form of the Kennedy method. The controlling failure mechanism will

be of upmost interest and will serve as a basis for the developed equation.

• Accessing the effect of the various components of the connection on the overall

structural capacity including moment capacity and rotational stiffness, after a

certain number of failed bolts.

• Prediction of the moment capacity of the 10-bolt Flush End plate Connection

and observation of various levels of failure using Finite Element Analysis

(ABAQUS Standard)

• Design methodology for the prediction of the moment capacity for a 10 bolt

unstiffened flush end plate connection with a predetermined number of failed

bolts.

1.5 Scope of Thesis

The scope of this research involves predicting the moment capacity of a 10-bolt flush

end plate connection based on available models and equations, development of

prediction equations to be developed as part of this study using the yield line method



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and examining bolt forces using a modified form of the Kennedy method. The failure

mechanism in the 10 - bolt flush end plate connection is to be investigated using finite

element analysis with ABAQUS Standard, a commercial software code.

1.6 Outline of Thesis

The thesis consists of five chapters. The first chapter gives a description of the

research, the scope and the objectives. A review of existing work in the area of end

plate moment connections, the behavior of components, failure modes, design

methodology and finite element modeling with emphasis on flush end plate

connections is highlighted in the second chapter. The assessment of the moment

capacity of the connection based on existing models and the development of prediction

equations based on the yield line analysis of the end-plate and the prediction of bolt

forces using the modified Kennedy method is presented in the third chapter. The finite

element model of the connection, geometry, boundary conditions, material properties

and nonlinearity together with the modeling of the contact surfaces, mesh generation

and analysis is evaluated in chapter four. It also presents a comparison of the

prediction equations with verification from the analysis of the finite element results.

The summary, major conclusions and recommendations derived from the research are

presented in the last chapter.



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CHAPTER II

LITERATURE REVIEW

2.1 Overview

The role that structural connections play in the rigidity and performance of a

structure cannot be overemphasized, serving as points for the transfer of both service

loads and in some instances lateral loads. Welded and bolted connections have been

used in connecting structural members for decades after riveting outlived its

usefulness. Structural bolts and welds lend themselves to fabrication and erection of

structures. End plate connections have found usage in various parts of steel buildings

and frames but are mostly in beam – column intersections. They are often preferred for

this purpose. The behavior of the end-plate connection will depend on variable

parameters, including bolt diameters, number of bolt rows, bolt spacing, grade, end-

plate dimensions, stiffeners, column and beam sizes, bolt pretension force, yield

strength of steel and the slip coefficient of contact surfaces.

According to the AISC Load and Resistance Factor Design specification

(2005), moment connections based on the stiffness may either be fully restrained (FR)

or partially restrained (PR) based on the category of construction. Fully restrained

connections must possess sufficient strength and stiffness to transfer



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moment and still maintain an angle between connected members while a partially

restrained moment connection are designed to transfer moments but allow rotation

between connected members as the loads are resisted. Due to the complications

involved in the design of partially restrained moment connections, AISC states that the

response characteristics of a particular partially restrained connection must be

documented or established by analytical or experimental means.

End plate moment connections are made of several components involving a

steel plate welded to the end of a beam section and attached to a member with rows of

high strength bolts. Two types of end plate moment connections are commonly found

in practice namely the Flush end plate moment connection and the extended end plate

connection as shown in Figure 2.1

(a) Flush end plate Type (b) Extended End Plate Type

Figure 2.1 – End plate moment connection types



9

In a flush end plate connection, the plate does not extend beyond the beam flanges and

the bolt rows are therefore confined within the beam flanges. Stiffeners might be

present to increase the overall strength rigidity. The flush end plate connection has not

been extensively used due to its lesser capacity to resist high lateral loads. It is

therefore used mainly in frames subjected to light lateral loads.

A number of experimental tests have been done on both rigid and semi-rigid

connections. Advances in structural analysis have allowed for the analysis of semi-

rigid steel framed structures so that, the real properties of the connections are

considered instead of typical simple assumptions of either pinned or fixed connections.

In the simulation of end plate connections, tee-stub models have long been used. The

bolts are subjected to direct tensile loading and thus are frequently used to transfer load

to the columns from the supporting beams and girders.

Accurate prediction of the structural behavior of steel beam-to column

connections, by estimating the local deformations and induced stresses is necessary in

assessing the capacity of the connections and preventing their failure. There are

factors affecting the study of these moment end plate connections. Experimental

testing forms a major basis for research. It has its limitations due to increased cost and

the inability to model certain testing conditions. Though finite element modeling only

gives an approximation of the actual solution, extensive research has shown that it

gives result comparable to experimental testing and thus suitable for this analysis.

(Choi and Chang, 1996)



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2.2 End -Plate Moment Connections

End plate moment connections found application in the 1960’s after

development from tee stub moment connections in the 1950’s. A great deal of work

has been carried out on moment end-plate connections after a surge in usage dating

back to the 1960s. Extensive work has been carried out in the past and studies are still

ongoing especially on the performance of these connections in seismic regions,

(FEMA 2002).Experimental testing of various end plate connections over time and the

progress that has been made and projects ongoing currently is considered.

End plate connections have the advantage of being suitable for erection even in bad

weather since only field bolting is required. Shop welding of the plates eliminates

those problems that would have been encountered in the field and the erection process

is simplified. A major disadvantage is the presence of prying forces that may result due

to bolts loaded in tension.

Design procedures have been available for both flush and extended moment

end plates. Earlier methods only considered the use of statics and did not fully embrace

the concept of the existence of prying forces. The design procedure proposed resulted

in both thick end plates and large bolt diameters. Various techniques have been

employed in the analysis and the behavior of these connections. Yield line analysis

theory has been used in analyzing the various failure mechanisms that develop in the

end-plate and design equations have been proposed to better design end-plates. The

tee-stub analogy has found its use in predicting bolt forces. The major failure modes

occurring in end-plate connections are plate yielding and bolt fracture.



11

More recently, finite element methods have been employed to study connection

behavior and investigate various failure mechanisms. Regression analysis results have

been used in formulating design equations for these connections. The behavior of

connections can be established making a number of assumptions:

• The components of the connection can be studied individually and each

contribution to the overall response of the connection under a given set of

forces.

• The behavior of the connection on the overall stiffness and strength of

members joined.

The structural behavior of the connection can best be described with its rotational

stiffness and moment resistance depicted in a moment rotation diagram. This behavior

clearly differentiates the various types of construction as presented in the AISC

specification (2005). Connection stiffness is taken as the slope of the moment rotation

curve. Earlier design procedures, have proportioned connections so that joint

deformation is prevented or kept to a minimal level to simplify the analysis of the

connection. However, it has been established that most connections are actually semi-

rigid.

Murray (1988) presents developments based on research in the design of

moment end-plate connections. The various design methodologies and procedures that

had been proposed in the design of end plate connections were considered. The

procedures used in most of this analysis involved the determination of the end plate



12

using yield line analysis and the prediction of bolt forces using the modified Kennedy

method.

Douty and McGuire (1965) performed test on both tee-stub and end plate

moment connections. They investigated the forces induced in the bolts by the tension

flanges by testing both connection types. There was good correlation between

theoretical and experimental results and an increase in bolt tension in thinner plates

was observed.

Mann (1968) developed prediction equations for end plate strength based on

experimental testing. Nair et al (1974) further investigated bolt prying forces by

conducting tests on tee-stubs. The effect of prying forces in reducing the ultimate load

carrying capacity and the fatigue strength of bolts was clearly observed. A prediction

equation for bolt forces including prying action was developed.

Zoetemeijer (1974) used the yield line theory in his analysis of the yielding of

the end plate. He proposed a design method based on two different collapse

mechanisms. Results from series of tests were used to test his analytical model and

there was good agreement.

Packer and Morris (1977) also used yield line theory to predict end-plate

moment capacity. Their tests investigated failure of the column flange. Straight and

curved yield lines were considered in the yield line analysis of the plate, though the

work was not conclusive. Phillips and Packer (1981) continued the earlier work of

Packer and Morris to establish the role that the end-plate thickness had on moment-

rotation characteristics and on end-plate failure mechanisms.



13

After Mann and Morris (1979) had reviewed several research programs and

proposed design procedures for extended end plate moment connections, they

suggested two very basic requirements that needed to be fulfilled in plastic design:

• Connections must be strong enough to withstand hinge movements.

• Connections must provide adequate hinge rotation while sustaining these

moments.

The design procedure proposed included methods in determining bolt size and end

plate thickness.

In the prediction of bolt forces, Kennedy et al (1981) derived a method using

the tee-stub analogy. His method identified three stages of tee-stub flange plate

behavior. The first stage of loading occurred at lower loads and the behavior of the

plate was described as thick. Prying forces were absent at this stage. With an increase

in the load, the second stage of behavior starts with the formation of plastic hinges at

the base of the stem of the tee. The prying forces at this stage of loading are between

the two extremes, not being present and the maximum. A subsequent plastic hinge

forms through the bolt line in the third stage and the plate is said to be thin, and prying

forces are at a maximum. Kennedy developed equations which set the various stages of

plate behavior with geometric properties and yield stress values of the plate and the

applied flange force. A modified form of the Kennedy method has been found to be

ideal for the prediction of bolt forces after it had been tested to correlate well with

experimental results.



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Tarpy and Cardinal (1981) discuss the effect of end-plate thickness on the

deformed shape and magnitude of displacements of the column flange. Two types of

end-plates are discussed and are described as “thick” when the end plate thickness is

about four times the thickness of the column flanges and “thin” when about one and a

half times the thickness of the column flange.

2.2.1 Moment Capacity Prediction and Moment rotation curves

Sherbourne and Bahaari (1994) proposed a methodology which involved finite

element modeling in establishing the moment-rotation relationships for steel

connections. As part of the findings, it was stated that one (beam) or two dimensional

(plate) models of connections do not accurately model the behavior when one of the

plates is in contact, and either the column flange or end plate is relatively thin. Prying

forces in the projected portion of the end plate increase with decrease of end-plate

thickness. The bolts in the projected portion of the thin end-plate connection are under

significant biaxial bending.

Foley and Vinnakota (1994) simplified the development of moment-rotation

curves for end plate beam to column connections. They used a semi-analytical

approach to develop moment-rotation curves for unstiffened extended end plate

connections. They used a modified form of the power model proposed by Kishi and

Chen (1990) which used a three parameter (Mp,Ki, β ) model based on initial stiffness,

moment capacity and an experimentally based parameter.



15

β β

θ / 1

1

+

=

p

i

i

M

K

K M

Where M and θ are the resisting moment of the connection and the ro tation of the

connection respectively, Mp is the plastic moment capacity of the connection and K i is

the initial stiffness of the connection. Ki and Mp are determined analytically, while the

decay parameter β is evaluated based on experimental data. The plastic moment

capacity of the connection was determined based on the failure modes listed below:

• Failure of the bolts in the tension region.

• Failure of the end-plate in the tension region.

• Failure of the column flange in the tension region

• Web yielding of the column in the compression zone

• Web crippling of the column in the compression zone

Their proposed method was supposed to lead to easier creation of adequate moment-

rotation curves for design office use.

Maggi et al (2005) utilized parametric analyses to study the behavior of bolted

extended end plate connections using finite element analysis. T-stub failure models

were used for calculations of the flexural strength for the end plate. Failure modes

included formation of yield-lines in the plate and bolt tension failure was well defined.

Failure due to a combination of these mechanisms represents levels of interaction

between the end plate and bolts which is difficult to predict accurately.



16

Hongxia, et al.(2008) developed a tee stub model for end plate behavior at large

deformations. He proposed a yield line model based on the virtual work principle

considering material hardening for both the T-stub and bolts after yielding. His yield

line model was compared to end plate connections on the basis of equivalent yield-

lines. The model represented the behavior of endplate connections well. A single row

of bolts in an end plate connection was modeled as a tee stub. The effect of material

properties and varying end plate thickness was highlighted in his finite element

analysis. His model considered the yield strength and the hardening phase of the plate

and the bolt, as well as compatibility between them.

Kukreti, Murray and Abolmaali (1987) developed a methodology for

establishing moment-rotation relationships for bolted steel connections based on finite

element modeling. The methodology was demonstrated for a flush end plate

connection. A few specimens were experimentally tested to verify the finite element

solution and computer analysis. The data collected were regressed to develop a

prediction equation characterizing the general behavior of the flush end plate

connection.

Murray (1990) presented design procedures for three configurations namely

four bolt unstiffened, four bolt wide unstiffened and the eight bolt extended stiffened

end-plate moment connections. This was based on earlier methods of Krisnamurthy

(1978), Ghassemieh et al (1983) and Murray and Kukreti (1988).

Borgsmiller (1995) presented a simplified method for the design of four flush

and five bolt extended end plate moment connection using the simplified version of



17

the modified Kennedy method in predicting bolt forces and the yield line analysis

method in the design of the end plate. His studies established that prying forces in

connections become significant when ninety percent of the end plate strength is

achieved, establishing a threshold where prying forces can be ignored. Good

correlation with past results was obtained.

Summer and Murray (2001) investigated extended end plate connections with

tests on 6 multiple row extended end plate connections to investigate the validity of

current design procedures for gravity, wind and low seismic loading. The test also

investigated the effects of the standard and large inner pitch distances and the

connections utilized in both A325 and A490 bolts.

Kukreti & Biswas (1997) developed a computer program to analyze the

moment-rotation behavior of end plate connections subjected to seismic loading. The

program had the capability of applying pretension loads to bolts in the model. The

predicted results were validated with experiments conducted on three connection

geometries in which the end plate thickness is varied. The failure modes in the

analytical model compared well with that of the experimental with a maximum

difference of about 18%.

2.3 End Plate Design and performance

The underlying philosophy in the design of end plate connections lies in the

yielding of the endplate, fracture of the bolts and deformation in the column flange.

The performance of the endplate connection is affected by varying conditions notable



18

among them being seismic loading and in fire conditions. The performance of the

connections is reviewed with respect to these two conditions. The extended moment

end plate connection has been shown to provide the required ductile behavior during

earthquakes and thus performs better than flush end plate connections.

Broderick and Thomson (2001) studied the behavior and response of flush end

plate joints under earthquake loading. They performed an experimental investigation

under monotonic and cyclic loading; they observed that individual connections

displayed the same failure mode in both cyclic and monotonic loading. They

concluded that specimens under cyclic loading displayed large rotation ductility

capacities and their modes of failure were similar under monotonic loads. In many

instances, failure occurred due to thread stripping on nuts and bolts. They stated that

the European codes appeared to identify the failure modes for analysis correctly, but

under-predicted the moment capacity. They re-emphasized the low moment capacity

possessed by flush end-plate connections and that its use must be restricted to areas of

low to medium seismicity.

Gang Shi et al (2006) studied the behavior of end-plate moment connections

under earthquake loading. They performed eight full-scale tests in investigating end-

plate thickness, bolt diameter and end-plate stiffeners, with both flush and extended

end-plates. Their results showed that an extended end plate provided adequate strength,

joint rotational stiffness, ductility and energy dissipation capacity to perform better in

seismic frames than flush end-plate connections. The flush end plates did not provide

enough structural rigidity and showed large rotations between the column flange and

end plate under such conditions.



19

Connection behavior during fires might indicate the extent to which damage or

failure occurs in the structure, since connections contribute to the overall structural

stability of the frames. The performance of connections in fire is of upmost importance

in structures.

Al-Jabri (2005) investigated the behavior of unprotected flush end plate

connections at elevated temperatures using ABAQUS, commercial software code. A

three dimensional model was used to establish the moment-rotation characteristics of

the connections under load and in elevated temperatures.

Hongxia,et al.(2008) performed tests at both ambient and elevated temperatures

to investigate the behavior of connections at the end of unprotected beams in fire

situations. At higher temperatures, failure always occurred in the bolts. They

emphasized the role of thicker end plates at elevated temperatures in enhancing peak

resistance, but reducing the rotational capacity of the connection.

2.4 Flush End Plate Moment Connections

In the United States, flush end plate connections found widespread use in pre-

engineered metal buildings until the early 1980’s. Application of the flush end plate

found extensive use in the United Kingdom. Flush end-plate connections have been

used in the structure under study that was built in 1979. Its usage has been limited and

currently available research shows extended end plate connections with stiffeners

perform better than flush end plates in frames and have the added advantage of

resisting lateral loads. Nevertheless, flush end plates have been used in a number of



20

structures where geometry does not allow the usage of extended end plates and where

lateral loads are minimal.

Phillips and Parker (1981) studied the effect of the thickness of the end plate on

flush end plate connections through experimental testing. There were two rows of bolts

in the tension region in order to study the influence of the second row of bolts on the

stiffness of the connection. They concluded that flush end-plates with two rows of

bolts in the tension region are suitable for semi-rigid construction, explaining that the

second row of tension bolts were effective to an extent than had earlier been under

estimated.

Zoetmeijer (1981) proposed a chart for approximating the ultimate load

capacity of a stiffened column flange or flush-end plate when the distance from the

bolt to the flange and web of the beam are known.

Srouji(1983), after a review of earlier work by Douty and McGuire(1965),

Zoetmeijer(1981) and Kennedy et al(1981) developed a methodology for the design of

four configurations of end plate connections including two-bolt flush, four bolt flush,

unstiffened four-bolt extended and stiffened four bolt extended. He used the yield line

method to determine end plate thickness, and a modification of the Kennedy method to

predict the bolt forces considering prying effects. Experimental testing of the various

configurations produced excellent agreement for both the end-plate strength and bolt

force magnitudes. Hedrick (1985) extended the work of Srouji and presented a unified

yield-line based design procedure for four types of flush end-plate configuration.



21

Li, et al.(1996) developed a method for the prediction of the moment capacities

of flush end plate composite connections. In as much as the tension side of the

connection had to work in conjunction with reinforced concrete, much of their findings

are applicable to this research. Possible failure modes based on careful study of various

experimental data included yield or fracture of bolts in tension, yield of the column

flange in bending, weld failure between the end-plate and steel beam, yield or buckling

of the lower beam flange in compression and shear, and yielding or buckling of the

column web in transverse compression. They investigated the effect of the slab

reinforcement on the flush end plate connection and derived equations for predicting

moment capacity of such connections. They proposed equations based on the number

of bolts in both the tension and compression zones. It was also noted that the effect of

the composite section on the connection could be treated as a row of bolts and will

give satisfactory results.

Bursi and Jaspart (1997) studied the behavior of the plastic and failure

mechanism of a tee stub connection which was later extended to investigate the

behavior of end plate connections. They used brick as well as contact elements and

based their conclusions on results from the finite element model.

Bose, Wang and Sarkar (1997) developed a sophisticated three dimensional

model to investigate the behavior of unstiffened flush end-plate steel bolted joints

using LUSAS commercial code. They compared the results with experimented data.

The results confirmed the accuracy of their finite element method prediction model.



22

Olsen (1997) proposed design formulae for bolted flush and extended end

plates based on plasticity theory, His model equation did not consider the effect of

prying forces for flush end plates. The design formulae was written in a generalized set

for both flush and extended end plate connections.

Murray and Shoemaker (2002) presented a guide for the design of both

extended and flush end plate moment connections. Flush end plate connections that

could be designed was limited to four types, four bolt unstiffened, two bolt

unstiffened, four bolt stiffened with stiffeners between the tension flanges and four bolt

stiffened with stiffeners inside the bolt rows. The provisions in the design guide are

limited to gravity and low lateral loads. A unified design based on the Borgsmiller

(1995) method was used.

2.5 Finite Element Analysis

Krishnamurthy and Graddy (1976), in their attempt to predict deformation in

end plates for an extended four-bolt extended connection, were constrained by

computer size, speed and the complexity of the mesh to better model connection

behavior. Krisnamurthy (1979) used finite element techniques to model three different

types of connection namely top-angle connections, tee stub connections and end plate-

connections. 2D and 3D models were created of the various types of connections.

Parametric studies were then performed on each to establish a correlation between the

two different models leading to the development of a set of prediction equations.



23

Kukreti et al (1987) in a comparison of two dimensional and three dimensional models

considering complexity and analysis accuracy, concluded that two dimensional models

provided adequate reliable modeling of moment end plate connections at the time.

Ahuja (1982) used finite element analysis to investigate the elastic properties of eight-

bolt stiffened connections. The investigation was continued by Ghassemieh(1983) to

include the non-linear behavior of the end-plate and bolts.

Abolmaali et al (1984) developed correlation coefficients from 2D and 3D

finite analysis for two bolt flush end-plate moment connection configurations. Finite

element 2D analysis served to generate regression equations for the design of the

connections adjusting them by correlation coefficients to closely match experimental

results.

Kukreti et al (1990) modeled an eight bolt connection and conducted

parametric studies to predict end plate displacement and inner bolt forces. Regression

analysis of the data was conducted after comparing with experimental data for

correlation.

Gebbeken, et al. (1994), Bahaari and Sherbourne (1994) used various finite

element analysis codes in studying the behavior of four bolt unstiffened end plate

moment connections. The non-linear material behaviors as well as the contact

interactions between the component parts were modeled using different elements.

Bahaari, et al. recommended that three dimensional models be used to generate

analytical formulations to predict the behavior and strength of the connection

components.



24

Choi and Chung (1996) studied several techniques involved in finite element

modeling of connections and pointed out flaws involved in earlier methods. They

developed a refined three-dimensional finite element model for end-plate connections

using an elastoplastic non conforming solid element with variable nodes. Effect of bolt

pretensioning and the shapes of bolt shank, head and nut were taken into consideration.

An algorithm for contact with a new gap element was used to simulate the interaction

between the end plate and column flange. Their model clearly established in detail,

moment-rotation relationships, contact phenomenon and effective stress distributions.

Bursi and Jaspart (1998) addressed some of the basic issues relating to finite

element modeling of end plate connections. According to their study, finite element

models depending on constitutive relationships, step size, number of integration points,

kinematic descriptions, element types and discretizations together with the non- linear

behavior of the components makes the modeling process cumbersome. They proposed

a suitable analysis methodology for end plate moment resisting connections and proper

three dimensional finite element models. The individual components of the connection

were critically examined.

Bahaari & Sherbourne (2000) studied the behavior of eight-bolt large capacity

endplate connections. They highlighted extensive work that had already been done on

end-plate connections, but those were limited to those with smaller capacity, thereby

making it necessary to study large capacity connections, as been done in this research.

They used ANSYS, a commercial finite element code in studying the structural

behavior of unstiffened eight bolt end plate. The endplate, beam and column flanges

were modeled with plate elements, and the bolt shank with six spar elements. They



25

proposed that an inelastic finite element model previously used for common extended

end plate connections may be employed for modeling the end plate with more bolts in

the tension region. As an alternative to the two row eight bolt connection, a hybrid

configuration was suggested in which four bolt rows are above, and two rows of two-

bolt are under the beam flange. The latter, offers slightly more initial stiffness,

especially for thick end plates, and the same ultimate strength. Olsen (2002) proposed

a new design method based on an easy to understand approach for developing the set

of equations derived from mechanics principles as presented in Eurocode 3.

Abolmaali, et al. (2005) developed a three parameter power model prediction

equation for the moment-rotation behavior of flush end plate connections with one row

of bolt below the tension and compression flanges. A finite element model of the

connection region along with the connected beam and column was developed for load

deformation analysis.

Shrih, et al. (2008) studied the behavior of flush end plate connections using a

highly detailed three dimensional finite element model in ANSYS. The connection

used in the analysis was typical of what is presently used in steel framed buildings.

Good correlation between experimental results and the finite element solution was

established with a deviation of about 12%. Serving as a methodology towards the

creation of finite element models, a sub-frame constructed with similar connection

properties was used to generate temperature – rotation diagrams that described the

behavior of the connection and predicts the failure mechanism of such a frame in

extremely high temperatures.



26

Ju, et al. (2003) used three dimensional elastoplastic finite element methods to

study the structural behavior of butt-type steel bolted joint. They observed that when

steel reaches non-linear behavior, the nominal bolt forces obtained from the finite

element analysis were almost linearly proportional to the bolt number arranged in the

connection. Bolt failure depended marginally on plate thickness that dominates the

magnitude of the bending effect.

2.6 Need for Further Research

Extensive research work has been conducted in the area of end plate moment

connections. However, only a small percentage has considered the investigation of the

failure mechanism in large capacity flush end plate connections and moment capacity

predictions. Available literature provides for the design of flush end plate connections

of smaller capacity. There is therefore a need to investigate the behavior of flush end

plate connections with reduced capacities due to the failure of some tension bolts.

Corresponding failure mechanisms resulting from these failures is also of major

concern. Research findings illustrate the use of the yield line analysis in predicting

moment capacity of the end plates and the modified Kennedy method for the

prediction of the bolt forces. Three dimensional finite element analyses can accurately

predict the behavior of flush end plate connections, and has been utilized in the

analysis. Development of design equations from both the yield-line analysis method

and the modified Kennedy method are presented. Details of the study have been

presented in subsequent chapter.



27

CHAPTER III

DEVELOPMENT OF MOMENT PREDICTION EQUATIONS

3.1 Overview

The development of a moment prediction equation for the 10-bolt flush end plate

connection involves the investigation of the various failure mechanisms in the end-

plate and in the prediction of the maximum load that causes bolt fracture. Therefore,

two limit states have therefore been considered in the analysis namely end plate

yielding and bolt fracture. Yield line theory has been used in the investigation of the

steel end plates. The modified Kennedy method is employed to predict bolt forces.

Both methods have been used in previous studies and it has been shown to accurately

predict the moment capacity of end plate connections. (Sumner, 2001)

3.2 Yield Line Analysis of Steel End Plates

Local collapse mechanisms have been investigated using yield line theory. Though

originally designed for reinforced concrete, it has found its usefulness in the design

and analysis of steel plates. Assumed yield line modes are normally based on

experimental observations but since experimental testing has not been done in this

study, extensions of assumed yield lines based on earlier work by Srouji (1983) and the

AISC Design Guide 16 (AISC, 2002) have been employed.



28

Two major components of the method include:

(a) An assumed yield line mechanism model for all possible failure patterns based on

the geometry of the end plate.

(b) Generation of work energy equations for both external and internal work and the

subsequent determination of the moment capacity.

Srouji, et al. (1983) set out guidelines for the location of yield lines in steel plates by

making the following assumptions:

(a) Axes of rotation generally lie along lines of support

(b) Yield lines pass through the intersection of the axes of rotation of adjacent plate

elements

(c) Along a yield line , the bending moment is assumed to be constant and equal to the

plastic moment of the plate

Two methods of analysis are generally employed in yield line analysis namely the

equilibrium method and the virtual work method. The latter has been shown to be

more suitable for the analysis of steel plates and is therefore employed in the analysis.

This involves a small arbitrary displacement in the direction of the applied load and

the external work generated. The internal work done by the plate is estimated by the

rotation of the plate along the yield lines. By equating the internal and external work,

the unknown moment capacity of the plate can be predicted. It is an upper bound

theory and therefore several failure patterns need to be evaluated to get the least or

smallest upper bound. The least upper bound solution is the pattern that produces the

lowest failure moment.



29

The internal work for a yield line analysis is the summation of the internal work

stored in each of the yield lines forming the mechanism. The internal work per unit

length is equal to the normal moment on the yield line multiplied by the normal

rotation of the yield line. Therefore the work of the nth yield line is as given by the

expression:

nn pn

Ln

pi Lmdsmw θ θ == ∫

Total work for the entire mechanism is therefore given by

nn

N

n

p

N

n

ii LmwW θ ∑∑ ====

11

Where mp is the plastic moment capacity of the plate, θn is the normal rotation of yield-

line n, and Ln is the length of the yield line projected to the axis of rotation.

Complicated patterns may be simplified by separating the internal work into its

components, that is:

( ) yny px xnx px

N

n

i Lm LmW θ θ +=∑=1

Where θnx and θny are the x- and y- components of the relative rotation of the rigid

plate segments along the yield line, Lx and Ly are the x and y components of the yield

line length, and mp is the plastic moment strength of the plate per unit length.



30

h

pt

b

s

g

tw

Mpl

h

p

pf

δ =1

θ

Fig 3.1 - Virtual Displacement in a two- bolt flush unstiffened end plate

Considering the two bolt flush end plate configuration in Figure 3.1, the external work

due to the unit virtual displacement is given

==

h M M W pl ple

1θ For small angles, θ

Mpl is the yield moment for the connection and θ is the virtual rotation of the

connection

3.3 Earlier Moment Prediction Equations

Extensive work has been done for the prediction of moment capacity in a two bolt and

four bolt flush end plate configuration. The work of Srouji (1983) and Murray (2002)



31

as highlighted in the AISC Design guide provide equations for the analysis of such

connections.

3.3.1 Two Bolt Flush End Plate connection

Srouji (1983) used yield line analysis to develop moment prediction equations for a

two bolt flush unstiffened end plate. Analysis of the possible failure modes led to the

choice of the controlling mechanism as shown in Figure 3.2, and is used in the

derivation of the internal work as:

( ) ( )

++

+−= s p

gs p

b ph

h

mW f

f

p

t

p

i

211

2

4

In the expression, s is an unknown expression that is found by differentiating the

internal work equation leading to the value of s as:

gbs p2

1=

Equating the external Energy to the Internal energy yields the following expression for

the predicted moment of the end plate.

( ) ( )

++

+−= s p

gs p

b ph

h

m

h

M f

f

p

t

p pl 211

2

4

( ) ( )

++

+−= s pgs p

b

phm M f

f

p

t p pl

211

24

The plastic moment per unit length of plate (mp) for yield line analysis is given by



32

yp p p F t m ⋅⋅= 2

4

1

Where Fyp is the yield stress of the end plate and tp is the end plate thickness.

The design procedure presented in the AISC Design Guide 16 on Flush and extended

end plate moment connection employs the above moment prediction equation for the

design of a two bolt flush end plate configuration. The predicted moment capacity of

the flush end plate is given by:

( ) ( )

++

+−= s p

gs p

b pht F M f

f

p

t p yp pl

211

2

2

3.3.2 Four Bolt Flush End Plate Connection

Srouji (1983) developed equations for the four bolt flush end plate after yield line

analysis of several possible failure mechanisms. The controlling yield line mechanism

is shown in Figure 3.2. The total internal energy is

( )

−+++

−+

−=

g

phs p p

u

ph

p

phb

h

mW t

b f

t

f

t p p

i 22

42

In the expression, u is an unknown expression that is found by differentiating the

internal work equation leading to the value of u as:

−

−=

t

t p ph

phgbu 2

2

1



33

h

u

g

pf

bp

pt

pb

pt2

Figure 3.2 –Controlling Yield Mechanism for Four bolts Flush End Plate (Srouji 1983)

The resulting moment capacity predicted is given by;

( )

−+++

−+−=g

phs p p

u

ph

p

phbt F M t

b f

t

f

t p

p yp pl 22

22

The controlling mechanism for the four bolts unstiffened flush end plate in the

AISC Design Guide Manual for Flush End Plate and Extended End Plate connection is

different from that presented by Srouji (1983). This mechanism was reached after

further experimental testing of the end plate connection.



34

hg

s

tw

tp

h1

h2

pt

bp

pb

pt2

pf

Figure 3.3 – Controlling Yield Line Mechanism for 4-Bolt (AISC Design Guide 16)

The total internal energy is

( ) ( )[ ]

+++++

+= 225.075.0

2

2

4

21

21 g

psh p phgs

h

p

hb

h

m

W bb f f

p p

i

In the expression, s is an unknown expression that is found by differentiating the

internal work equation leading to the value of s as:

gbs p2

1=

The value of pf is set as equal to s, if the value of pf is greater than s. The predicted

Moment Capacity is given by

( ) ( )[ ]

+++++

+=

225.075.0

2

221

212 g psh p ph

gs

h

p

hbt F M bb f

f

p

p yp pl



35

3.4 End Plate prediction Equations

The moment carrying capacity of the connection depends on the resistance of

the components in the tension region of the connection. The neutral axis of the bolt

group directly affects the behavior of the connection because it provides a mechanism

for determining the number of bolts in the group that will be in tension. Different cases

of the flush end plate connection based on the number of rows of bolts are analyzed.

3.4.1 Case One –Ten bolt connection. ( All rows of tension bolts present)

With the 10 bolt flush end plate connection under consideration, it is assumed

that the neutral axis lie directly in the third row of bolts. Two rows of bolts will be

found in the tension zone of the connection, and therefore will be synonymous to a

four bolt flush end plate configuration. The prediction equations for a four bolt flush

end plate configuration in the AISC Design Guide 16 can be employed. The equations

dealt with in this report do not include the safety factors employed in the process of

design. The study involves a measure of the actual strength of the connections.

A number of assumptions have been made in applying the equivalent form of

the proposed model below:

• The minimum distance, s is less than pb (internal bolt spacing). Where s is

greater than pb, the value of pb must be used.

• The bolts in compression only provide resistance against shear.



36

h

bp

Neutral Axis

Compression Zone

Tension Zone

h

bp

pb

Figure 3.4 – Equivalent form of the 10 bolt Flush onto a Four Bolt Flush Configuration

A yield analysis of the 10-bolt connection has been done as part of the study to

investigate the failure patterns. Different yield line patterns as presented in Appendix

A have been evaluated to determine the controlling yield line mechanism. The

controlling yield line pattern is as shown in Figure 3.5.

The total internal work of this mechanism is given by:

( )[ ]

++

+= b f

b f

p p

i p phg p

h

p

hb

h

mW 2

2

2

41

21



37

g

h h1

h2

h3

pt

bp

pb

pb

pf

pt2

Figure 3.5 – Controlling Yield line Mechanism for the 10-bolt connection

The predicted moment capacity is given by:

( )[ ]

++

+= b f

b f

p

p yp pl p ph

g p

h

p

hbt F M 2

2

21

212

3.4.2 Case Two – 8 bolts (One row of failed bolts)

An assessment of the connection strength of older structures makes it necessary

to consider Case 2. The upper bolts of a connection might have failed due to a number

of factors. A careful evaluation of the structure when the bolt loss has occurred is

important in investigating the overall effect on the moment capacity of the connection.

In the research, the moment capacity of the connection is predicted when one row of



38

tension bolts has failed. Different yield patterns have been investigated to determine

the controlling collapse mechanism. The controlling yield line mechanism is as shown

in Figure 3.6

g

h h1

h2

h3

pt

bp

pb

pb

pt2pf

s

Figure 3.6 – Controlling Yield line Mechanism for the 8-bolt connection

The total Internal Work of this mechanism is given by:

( )[ ]

+++

+

+= b f

b f

p p

i ps phgs

h

p p

hb

h

mW 2

2

22

42

2


( )[ ]

+++

+

+= b f

b f

p

p yp pl ps phgs

h

p p

hbt F M 2

2

222

22



40

h

h3g

pf

bp

s

pb

Figure 3.7 - Controlling Yield line Mechanism for the 6-bolt connection


( )[ ]

++

+= s phgs

h

p

hbt F M f

f

p

p yp pl 33322

2

When the value of pf is greater than s, its value should be set to s to give the least

moment capacity of the connection.

3.5 Bolt Force Predictions

The ultimate limit state of the connection depends on either the yielding of the end

plate or the failure of the tension bolts. Moment capacity of the connection has earlier

been predicted based on end plate yielding. The prediction of forces in the bolts as a



41

result of loading on the connection is necessary in order to establish the overall

capacity of the connection.

When a connection is subjected to flexure, its bolts in tension are acted on by the

tensile loads through the bending of the plates. Connection failure occurs when the

bolts in tension reach their tensile capacity. A major step involved in the prediction of

these forces is the determination of all the prying forces.

3.5.1 Prying Action

The effect of prying action in end plates tends to increase the effective tensile load that

is transferred to the bolts and therefore its effect needs to be evaluated. A modified

form of the Kennedy model (1981) has been employed in different studies which

involve end plate moment connections such as that by Srouji (1983) and Borgsmiller

(1995).

Three stages of plate behavior occur during the process of the application of the

load. The initial behavior of the plate stage (Figure 3.8a) is referred to as “thick” when

the load as applied. At this stage, no prying forces are acting and thus the bolts

experience only direct tensile forces as a result of the applied load. As the load is

increased, an initial plastic hinge is formed in the plate at the base of the tee stem; the

behavior of the plate is same to be “intermediate”. At this stage, prying forces are

present but not a maximum (Figure 3.8b). Another plastic hinge develops at the bolt

line in addition to that initially occurring at the stem of the tee after the load is further

increased. The end plate behavior at this stage is referred to as “thin” (Figure 3.8c)



43

2F

B+Qmax B+QmaxQmax Qmax

(c) Thin Plate Behavior (Prying forces maximum)

Figure 3.8 – Stages of Plate Behavior based on the Kennedy Model

A simplified method proposed by Borgsmiller (1995) has been used in this study

which considers only two stages of the behavior of the plate being the thick plate limit

where prying forces are absent and the thin plate limit where maximum prying forces

occur.

3.5.2 Determination of Plate Thickness Limits

Srouji (1983) proposed several thicknesses limits equations upon which the behavior

of the plate might be described as thick, intermediate or thin. After the resulting flange

force, F, has been determined with a flange stress, σf , the thick plate limit tthick is found

by a set of two equations given by:



44

py

f f f

F

t pt

σ 11.21 =

Using the iterative equation below, the thick plate limit is found.

2

1

2

23

2

−

=

t

t F

pt t

f f

py

f f f

thick

σ

σ

If the actual plate thickness, tp is greater than tthick , then thick plate behavior has been

established and thus prying forces are absent, Q = 0.

For the thin plate limit, thin , another set of two equations are involved given by :

( )'80.085.0

162 3

2wbF

F d pt b

t f py

ybb f f f f

+

−

=

π σ

Where w’= width of end plate per bolt at bolt line minus bolt hole diameter. Fyb is the

yield stress of the bolt. Then using the interactive equation

2

1

2

2

2

2

2

2

3

'23'

23

162

−+

−

−

=

t w

t bF w

t

t F b

F d pt b

t

f f f

py

f f

yp f

ybb f f f f

thin

σ σ

π σ

If tp is greater than tthin, then intermediate plate behavior occurs and thus prying forces

exist and should therefore be accounted for. In the same way, when tp is less than tthin,

maximum prying forces occur and is therefore considered in the estimation of the

capacity.



45

Borgsmiller (1995) simplified the requirement for the determination of the limits for

plate behavior through further experimental testing. Only the thin plate and thick plate

behavior is taking into consideration in his approach. The effect of the prying forces

becomes evident when 90% of the full strength of the end plate has been developed as

a result of the application of load. Therefore no prying forces need to be considered for

the prediction of the forces in the bolts when

plnp M M 90.0≤

3.5.3 Modified Kennedy Model

A modified form of the Kennedy model has been used in the prediction of bolt forces

by taking half of the original Kennedy model to represent a flush end plate connection.

A simplified form of the Kennedy model had earlier been used by Srouji(1983) in the

prediction of the bolt forces. The various models have been employed to help predict

the models for each case of bolt loss.

P P a

F

Mb

B1

f b

B2 Q

Mb

M M2 3M1

(a) Case 1 – 10 bolts present



46

P a

F

Mb

f

B2 Q

M3M1

Pb

Mb

M4

B3

(b) Case 2 -8 bolts present ( one row of failed bolts)

P a

F

Mb

f

B Q

M4M1

(c) Case 3- 6 bolts present ( 2 rows of bolts missing)

Figure 3.9 - Modified Kennedy Model for cases of the flush end plate connection.



47

3.5.4 Moment Capacity Prediction based on Bolt Forces

The maximum prying forces, Qmax,I is given in the AISC design manual of flush and

extended moment end plate connections as :

2'

2

2

max,'

34

'

−=

p

i

py

i

p

it w

F F

a

t wQ

where

+−=

16

1

2' b

pd

bw , 085.0682.3

3

−

=

b

p

id

t a

i f

t b p

py p

i p

F d

w

b

F t F

,

3

2

'

4

8'80.0285.0

π

+

+=

If the radical in either expression for Qmax,i is negative, combined flexural and shear

yielding of the end plate is the controlling limit state and the end-plate is not adequate

for the specified moment. The last term in the numerator of the F i term represents the

contribution of the bolt shank bending.

In determining the maximum moment leading to maximum forces in the bolts,

the static moment of the bolts about the centerline of the compression flange is taken.

Moment capacity equations have therefore being derived for the three different cases

considered with or without prying forces.

Case One – Ten bolts

Taking moments about the compression flange (Figure 3.10) gives the expression

i. Without Prying Action



48

( ) 21 22 hPhP M t t np +=

( )212 hhP M t np +=

Mnp

g

2P

2Pt

t

Mnp

2(P - Qmax )t

2(P - Qmax)t

(a) (b)

h h1

h2

h3

pt

bp

pb

pb

pf

pt2

Figure 3.10 - Case One bolt forces with (a) No Prying Action (b) Prying Action

ii. With Prying Action

The bolt tension is increased by the maximum prying, Qmax. The expression for the

moment is given by the maximum of either one of the two equations.

( ) ( ) 2max1max 22 hQPhQP M t t np −+−=

( )( )21max2 hhQP M t np +−=

This expression is written in terms of the pretension Force, Tb as

( )212 hhT M bnp +=



49

Case Two – Eight bolts

g

Mnp

2Pt

Mnp

2(P - Qmax)t

2(P - Qmax)t

2Pt

(a) (b)

h h1

h2

h3

pt

bp

pb

pb

pt2pf

s

Figure 3.11 - Case two bolt forces with (a) No Prying Action (b) Prying Action

Taking moments about the compression flange (Figure 3.11) gives the expression:


( )322 hhP M t np +=


The expression for the moment is given by the maximum of either one of the two

equations.

( )( )32max2 hhQP M t np +−=


( )322 hhT M bnp +=



50

Case Three – Six bolts

h

h3

pb

g

pf

Mnp Mnp

2Pt

s

(a) (b)bp

2(P - Qmax)

Figure 3.12 - Case three bolt forces with (a) No Prying Action (b) Prying Action

Taking moments about the centre line of the compression flange gives the expression


( )32 hP M t np =


The expression for the moment is given by the maximum of either one of the two

equations.

( )max32 QPh M t np −=


( )32 hT M bnp =

Tb = specified bolt pretension



51

CHAPTER IV

RESULTS VERIFICATION AND FINITE ELEMENT ANALYSIS

4.1 Overview

The behavior of the flush end plates with 10-bolts and the two cases of the connection

with failed bolts have been evaluated and consequently verified with results from an

investigation using a finite element analysis. Though there has not been experimental

testing of the connection, earlier studies (Choi and Chang, 1996) have shown that finite

element analysis can accurately predict the behavior of such connections. A finite

element model has been developed for the ten bolt flush end plate connection, and the

two cases of failed bolts which occur in the tension regions of the connection. The

detail of the proposed structure used in the study is a ten bolt flush end plate

connection of an existing steel parking deck.

4. 2 Geometry of Structural components

The structure is a steel frame structure with pre-tensioned concrete decking. The beams

between both sides of column along the span are castellated. The castellated beam,

which is a built up section using a W21 x 62 as the top chord and a W24 x 76 as the

bottom chord has been modeled as a combined section with properties as shown in



52

Figure 4.1. The beam is connected to a W10 x 39 column. The steel used for all

sections is Grade 60. The end-plate which has been welded to the end of the beam has

a thickness of 0.5 inches. The beam with the welded end plate is bolted to the column

flange with A325 high strength bolts of ¾” diameter. Details of the geometry of the

beam and end plate is shown in Figure 4.1 with dimensions presented in Table 4.1

h h1

h2

h3

pt

bp

pb

pb

g

pt2

Figure 4.1- Connection Details for End Plate with Adjoining Beam



53

Table 4.1 – Section Properties for End Plate and Beam

Component Dimensions

Beam

h = 22 in.bp = 8.24 in

tf = 0.615 in.

tw = 0.4 in.

End Plate

bp = 8.24 in.

g = 4.12 in.

pf = 2.585 in.

pb = 3.9 in.tp = 0.5 in

4.3 Moment Capacity Predictions

Moment capacity prediction equations developed in the earlier chapter have been used

to predict the moment capacity of the connection described above. This involved

prediction based on the yielding of the end plate and the rupture of the bolts.

4.3.1 Limit State of End Plate Yielding

Yield strength of end plate, Fy = 60ksi

ingbs p 91.212.424.82

1

2

1 =×==



54

AISC Design Guide 16 – nominal connection strength

( ) ( )[ ]

+++++

+=

225.075.0

2

221

212 g psh p ph

gs

h

p

hbt F M bb f

f

p

p yp pl

( )[ ]

+×++×+

+×=

2

12.49.325.091.29.1451.58.18

12.4

2

91.2

9.14

585.2

8.18

2

24.85.060 2

pl M

Mpl = 1972 kip-in

Using Derived Prediction Equations

Case One – nominal connection strength

( )[ ]

++

+= b f

b f

p

p yp pl p phg p

h

p

hbt F M 2

2

21

212

( )[ ]

×++

+×= 9.32585.28.18

12.4

2

9.3

9.14

585.2

8.18

2

24.85.060 2

pl M

Mpl = 2106 kip-in

Case Two – nominal connection strength

( )[ ]

+++

+

+= b f

b f

p

p yp pl ps phgs

h

p p

hbt F M 2

2

222

22

( )[ ]

×+++

×+

+×= 9.3291.2585.29.14

12.4

2

91.22

22

9.3585.2

9.14

2

24.85.060 2

pl M

Mpl = 1820 kip-in



55

Case Three – nominal connection strength

( )[ ]

++

+= s ph

gs

h

p

hbt F M f

f

p

p yp pl 3332 2

2

( )[ ]

++

+×= 91.2585.211

12.4

2

91.2

11

585.2

11

2

24.85.060 2

pl M

Mpl = 937 kip-in

4.3.2 Limit State of Bolt Fracture

Bolt Data

Diameter, db = 3/4in

Tensile strength for A325 Bolts, Ft = 90ksi

Bolt Force, t

b

t F d

P4

2

π = = 39.765kips

Yield strength of End Plate, Fpy = 60ksi

Without Prying Action

Case One - nominal connection strength

( ) ( )1 22 2 39.765 18.8 14.9 2,680np t M P h h= + = × × + = Kip-in

Case Two –nominal connection strength

( ) ( )2 32 2 39.765 14.9 11 2,060np t M P h h= + = × × + = Kip-in



56

Case Three – nominal connection strength

( ) ( )32 2 39.765 11 875np t M P h= = × × = Kip-in

With Prying Action

2'

2

2

max,'

34

'

−=

p

i

py

i

p

it w

F F

a

t wQ

Where

8125.016

1

4

3

2

24.8

16

1

2' =

+−=

+−= b

pd

bw

0059.1085.075.0

5.0682.3085.0682.3

33

=−

=−

=

b

p

id

t a

465.7

4

8'80.0

285.0

3

2

' =

+

+

= f

t b p

py p

i

p

F d w

bF t

F

π

Ksi

567.25.08125.0

465.7360

0059.14

5.08125.02

22

max, =

×−

×

×=iQ Kips

Case One – nominal connection strength

( )( ) ( ) ( )max 1 22 2 39.765 2.567 18.8 14.9 2,510np t M P Q h h= × − + = × − × + = Kip-in

Case Two – nominal connection strength

( )( ) ( ) ( )max 2 32 2 39.765 2.567 14.9 11 1930np t M P Q h h= − + = × − × + = Kip-in



57

Case Three – Nominal connection strength

( )( ) ( )max 32 2 39.765 11 820np t M P Q h= − = × × = Kip-in

4.3.3 Determination of the Predicted Capacity

The failure mode with the minimum predicted value is chosen as the controlling failure

mechanism and its moment is taken as the predicted moment capacity of the

connection. The controlling failure mechanism can thus be determined based on these

moment capacity values. Experimentally, the predicted yield moment capacity is

determined from a moment – end separation response curve using a bilinear

approximation if there are no clearly defined yield points. This method can be used in

determining the yield moment from a Finite element analysis (FEA). The predicted

moment capacity values with their corresponding controlling yield mechanisms have

been summarized in Table 4.2

Table 4.2 – Summary of Predicted Moment Capacities with Failure mechanisms

Cases Mpl-End Plate

Yielding.(kip-in)

Mnp- Bolt Fracture (kip-in) Controlling

MechanismWithout

Prying

With

Prying

1 (10 bolts) 2106 2680 2510 End-plate

yielding

2(8 bolts) 1820 2060 1930 End-plate

yielding

3(6 bolts) 937 875 820 Bolt Fracture



58

4.4 Finite Element Analysis

ABAQUS Standard commercial software (a finite element code) has been used in

creating a three dimensional finite element model of the 10 bolt flush end plate

connection. In order to use the model to predict the moment capacity of the

connection, components of the connection including beam, column, end plate and bolts

have been modeled to represent their performance in service. Simplified modeling

techniques have been adopted to reduce analysis time and improve the accuracy of the

finite element solution. Symmetry about a vertical plane through the beam web and

flange and that of the column was employed and thus only half of the full model was

analyzed.

4.4.1 Model of the Connection Components

The components of the connection were modeled in ABAQUS CAE, using its graphic

building tools. For a simplified model, a typical connection in the structure was

modeled to include the column side, beam, end plate, bolts and nuts. A section of the

column which spans from floor to floor in the actual structure was modeled and the

necessary boundary conditions applied. The beam of the structure which is castellated

and directly supports the floor deck was modeled as a cantilever without the

perforations with movement restricted in the horizontal direction to represent a

continuous condition. The end plate was modeled to be in contact with the beam to

avoid complications which would arise in modeling the welds.



59

(b) Beam with End Plate Model

(a) Column Model (c) Bolt and Nut Model

Figure 4.2- Components of the Flush End Plate connection in ABAQUS CAE

4.4.2 Material Properties

Structural Steel was used as the material for the components of the model, but with

different stress -strain relationships. The modulus of elasticity for steel, E, was taken as

29,000ksi with a Poisson’s ratio, v, of 0.3. A bilinear stress strain curve was used in

modeling the material in the beam, end plate and column flanges, all having yield

strengths, Fy of 60ksi. The plastic behavior was considered to be linear after yield

(Figure 4.3a). A trilinear stress strain curve was used in modeling the bolts and nuts

with yield strengths, Fy of 90ksi and ultimate strengths of 100ksi (Figure 4.3b)



60

Stress (σ)

Strain(ε)

Fy

εy 11εy

E

Figure 4.3 (a) Stress Strain Relationship for Beam, End plate and Column Material

Stress (σ)

Strain(ε)

Fyb

εy 8εy

E

3.5εy

Fub

Figure 4.3 (b) Stress Strain Relationship for Bolt and Nut (Sumner, 2001)



62

4.4.4 Contact Interactions

The assembly module simply positions each component relative to each other under a

global coordinate system, but does not establish contact between faces even if the

surfaces are together. The contact between the various interfaces of the connection was

defined and established through the creation of both master and slave surfaces. Normal

and tangential behavior has been defined to represent the contact between adjoining

surfaces with or without friction. Contact was established between column and beam

interfaces as well as bolt and nut contact on both the beam and column sides arising as

a result of pretension. This was achieved in the first step of the analysis and the results

showed that there were no strains or deformations in the model before loading.

Penetration error between the interfaces was kept to a minimum and was within the

acceptable tolerance.

4.4.5 Boundary Conditions

The prescribed boundary conditions define states that various part of the assembly

exhibit both at the initial stage and within other stages in the analysis. It also describes

their behavior in terms of displacement and rotations during analysis steps. Two major

forms of boundary conditions including displacement and symmetry boundary

conditions were used. For an easy and economic analysis, only half of the connection

was modeled and a symmetric boundary condition applied to simulate the behavior of

the actual connection. The column ends were restrained in all the directions. The same

was applied to the bolts and the nuts in their initial state with movement in the



63

horizontal direction only when the load was applied. The free end of the beam was

restrained from movement in the horizontal direction. These conditions closely

represent the actual state in which the components undergo in an actual experimental

setup (Srouji, 1983).

Fig 4.5 – Connection Boundary Conditions

4.4.6 Loading and Analysis Steps

Different analysis steps were employed to establish contact and to apply the loads.

General static analysis was conducted without linear perturbation. Loading on the

connection was induced by applying two vertical loads (a couple) to the beam tip at the

free end. These loads were increased until the model had shown signs of instability.

Bolt Pretension was achieved by the application of a tensile load to simulate the

tightened condition of the bolt and nut with the code specified minimum bolt tension.

(AISC, 2005). Three different cases was simulated including the ten bolt case, one

row of failed bolts and finally, two rows of failed bolts.



64

4.4.7 Element Type and Meshing

Proper meshing in a finite element model leads to an economic analysis with higher

accuracy, both of which cannot be overemphasized. In order not to unnecessarily

increase the degrees of freedom of the various part instances, the mesh was created

with varying densities, being denser only around regions of interest and bolt holes

where stress concentrations and surfaces of contact between individual components

make up the model.

Figure 4.6 - Mesh of Connection Assembly

An 8 node brick (C3D8) solid continuum element was chosen from the ABAQUS

element library and used on all components of the connection. This element has nodes

only at their corners as shown in Figure 4.6, uses linear interpolation in each direction



65

and is often called a linear element or a first order-element. There is a constant

volumetric strain in the element which prevents shear locking when the material

response is incompressible. Full integration elements are defined to control hour glass.

Figure 4.7 – Linear element, 8-node brick-C3D8 (ABAQUS Analysis Manual)

4.4.8 Visualization of Results and Failure patterns

Visualization results for the three cases analyzed using ABAQUS is shown in Figure

4.7 for the three cases of the analysis with (a) being the undeformed shape and (b) the

deformed shape. It clearly indicates the likely failure patterns that would form in the

connection. Thick plate behavior could be inferred from the Von mises stress

distribution by examining the deformed shape in each analysis case. Areas in tension

showed high stresses in the tension zones and occurred along paths where yield lines

were likely to occur as outlined in the development of the yield line mechanism

models for the prediction of the moment capacity.



66

Case 1 -10 bolts

(a) Undeformed (b) Deformed

Case 2 – 1 row of bolt missing




67

Case 3 – 2 rows of bolts missing


Figure 4.8 – Failure Patterns in different cases of connection failure

4.5 Moment Capacity Prediction from Finite Element Analysis

Nodal displacements from the end plate and column flange were used to calculate end-

plate separation. The values were recorded at each analysis step and used in plotting a

moment – end plate separation curve for the various cases, with Case one (Figure 4.8)

and Case two (Figure 4.9) as examples in this report. The end plate yield moment was

then determined from these plots using bilinear approximation as shown in Figure 4.8.

There was a loss in stiffness in the connection when the applied moment was

increased. The ten bolt flush end plate connection had the highest stiffness in its elastic

region followed by the eight bolts and then the six bolts.



68

Figure 4.9 – Graph of applied moment against end plate separation for Case One



69

Figure 4.10 – Graph of applied moment against end plate separation for Case Two

4.5.1 Comparison and Summary of Results

A comparison of the predicted moment capacity values and that of the finite element

analysis showed very good correlation as outlined in Table 4.3. Predicted values were

within 6.. AISC design equations predicted the moment capacity of the ten bolt case

quite accurately based on the approximation that it represented a four bolt

configuration above the neutral axis.



70

Table 4.3 – Comparison of the Moment Capacity values.

CasesMpl - Developed

Equations

MFE-Finite Element

Solution)MPL /MFE

1(10 bolts) 2106 2200 1.04

2(8 bolts) 1820 1910 1.05

3(6 bolts) 937 995 1.06

The accuracy of the solution may have been affected by other factors such as

column side behavior and shear effects on the bolts resulting from the applied load.

Deformed shapes for the various situations emphasize the role that column side

behavior might have on the overall moment capacity of end plate connections.

An inference can be made for the parking deck considered in this study based

on the analysis results. The controlling failure mechanism for a 10 bolt flush end plate

connection has been established as the yielding of the end plate and not that of bolt

fracture. Therefore, if an end plate in the connection of the structure has not yielded

but a number of bolts have failed, then other factors such as corrosion or fatigue can be

inferred and needs further investigation.



71

CHAPTER V

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

5.1 Summary

The primary objective of this research was to predict the moment capacity of a

10-bolt flush end plate connection and assess the capacity of the connection with failed

bolts, taking into consideration the controlling failure mechanisms. This has been

achieved by developing moment prediction equations using yield line analysis to

predict failure in the end plates and the bolt force derived using a modified form of the

Kennedy method. A comprehensive literature review regarding end plate moment

connections was conducted to gather the knowledge in the field and also highlight the

needs for further research.

Analytical models were developed by considering yield line formation in the

end plate. Yield line analysis was employed to determine the moment capacity for the

various cases within the scope of the research. The modified Kennedy method with

some modifications as applied by Borgsmiller (2005), was utilized to predict the bolt

forces with or without prying action.

A three dimensional finite element model of the end plate connection was

developed for each research case using ABAQUS Standard commercial code to

investigate mechanism of failure in the plate. The yield moment capacity of the end



72

plate was then predicted based on deductions from moment – end separation curves

which has been generated from nodal displacements of end plate and column flange.

Comparison of the yield moment predicted from the set of equations developed and

that from the finite element showed good correlation within 6%.

5.2 Conclusions

After analysis of the 10 bolt flush end plate connection with the varying cases of failed

bolts, the following conclusions can be reached:

• Yield line analysis forms an effective method for the prediction of moment

capacity in steel plates. An elaborated procedure has been followed in this

report to predict the moment capacity of the 10 bolt-flush end plate connections

and also for the various cases of failed bolts.

• The AISC Design Guide on the Design of Flush and Extended End plates can

be used to predict the moment capacity of such a connection, provided the

neutral axis of the bolt arrangement lies in a position with exactly four bolts

above it.

• For a 10 bolt flush end plate connection, the controlling failure mechanism

depends on the thickness of the end-plate. The dominant failure mechanism is

yielding of the end plate.

• With one row of failed bolts, the controlling failure mechanism was due to the

yielding of the end plate.



73

• The controlling failure mechanism in a connection with two rows of failed bolt

was not due to the yielding of the end plate but due to the fracture of the bolts.

• Finite Element Analysis can be used to accurately predict the moment capacity

of a ten bolt flush end plate connection.

5.3 Recommendations

Moment capacity prediction was achieved for the 10-bolt flush end plate connection

using bolt analytical and finite element methods. The following recommendations can

be considered for further research:

• Experimental testing of the models would provide a more generalized method

for comparison and deductions.

• The column side behavior has been considered to be rigid in the analysis of the

endplate analytically, but the results from the finite element analysis showed

that the moment capacity of the flush endplate is affected by column flange

behavior and should be considered in future investigations.

• The loading involved in analysis of the research was mainly monotonic but

fatigue analysis of the connection can be studied with cyclic loading.

• In an unbalanced condition, or for connection with failed bolts, the effect of

compression on the column flange along the compression side of the end plate

needs further investigation on end plate failure and moment capacity

prediction.



74

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of the moment-rotation behavior of stiffened extended end-plate connections,

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Zoetemeijer, P.(1974),”A design method for the tension side of statically loaded bolted

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79

APPENDICES



80

APPENDIX A - Assumed Yield Line Mechanisms

Case One – Ten bolts

h h1

h2

h3

pt

bp

pb

pb

g

pf

pt2



81

g

h h1

h2

h3

pt

bp

pb

pb

pf

pt2

g

h h1

h2

h3

pt

bp

pb

pb

pf

pt2



82

g

h h1

h2

h3

pt

bp

pb

pb

pf

pt2

g

h h1

h2

h3

pt

bp

pb

pb

pfpt2



83

g

h h1

h2

h3

pt

bp

pb

pb

pf

pt2

g

h h1

h2

h3

pt

bp

pb

pb

pf

pt2



85

Case Two – Eight bolts (one row of failed bolts)

g

h h1

h2

h3

pt

bp

pb

pb

pt2pf

s

g

h h1

h2

h3

pt

bp

pb

pb

pfpt2

s



86

g

h h1

h2

h3

pt

bp

pb

pb

pt2pf

s

g

h h1

h2

h3

pt

bp

pb

pb

pt2pf

s



87

Case Three – Six bolts (Two rows of failed bolts)

h

h3g

pf

bp

s

pb



88

APPENDIX B – FINITE ELEMENT ANALYSIS OUTPUT SUMMARY

Case One

Time step Moment (kip-in) Beam column End Separation (in)

0 0 0 0 0.0000

0.1 360 0.00175574 -3.05E-05 0.0018

0.175 630 0.00307221 -5.33E-05 0.0031

0.25 900 0.00438841 -7.60E-05 0.00450.3625 1305 0.00636544 -1.10E-04 0.0065

0.53125 1912.5 0.00945429 -0.00016114 0.0096

0.78437 2823.732 0.0152431 -0.00024435 0.0155

1 3600 0.0219558 -0.00033924 0.0223

Case Two


0 0 0 0 0.0000

0.1 360 0.00365217 -1.03E-04 0.0038

0.2 720 0.00730344 -2.05E-04 0.0075

0.35 1260 0.0127787 -3.58E-04 0.0131

0.575 2070 0.0210336 -5.90E-04 0.0216

0.9125 3285 0.0342042 -0.00099443 0.0352

1 3600 0.0379633 -0.00111319 0.0391



Case Three


0 0 0 0 0.0000

0.1 360 0.00477048 -2.29E-04 0.0050

0.175 630 0.008348 -4.00E-04 0.0087

0.25 900 0.0119252 -5.72E-04 0.0125

0.35 1260 0.0172901 -8.28E-04 0.0181

0.5 1800 0.0253376 -0.00121864 0.0266

0.726 2614.02 0.0379369 -0.00187691 0.0398

1 3600 0.0521809 -0.00254078 0.0547

Arthur Godwin Addiah

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