ultimate capacity of steel angles subjected to eccentric

Ultimate Capacity of Steel Angles Subjected to

Eccentric Compressive Load

by

Iftesham Bashar

MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURAL)

Department of Civil Engineering

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

2012

Ultimate Capacity of Steel Angles Subjected to

Eccentric Compressive Load

by

Iftesham Bashar

A thesis submitted to the Department of Civil Engineering of Bangladesh University of

Engineering and Technology, Dhaka, in partial fulfillment of the requirements for the

degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURAL)

2012

The thesis titled “Ultimate Capacity of Steel Angles Subjected to Eccentric

Compressive Load” submitted by Iftesham Bashar, Student No. 100704337F, and

Session: October 2007 has been accepted as satisfactory in partial fulfillment of the

requirement for the degree of M.Sc. Engg. (Civil and Structural) on 25th

January, 2012.

BOARD OF EXAMINERS

______________________________________

Dr. Khan Mahmud Amanat Chairman

Professor (Supervisor)

Department of Civil Engineering, BUET,

Dhaka-1000

______________________________________

Dr. Md. Mujibur Rahman Member

Professor and Head (Ex-officio)


Dhaka-1000

______________________________________

Dr. A. M. M. Taufiqul Anwar Member

Professor


Dhaka-1000

_____________________________________

Dr. Mahbuba Begum Member

Associate Professor


Dhaka-1000

______________________________________

Dr. Alamgir Habib Member

Professor (Retd.) (External)

Department of Civil Enginerring, BUET,

Dhaka-1000

Apartment # 2A, House # 124, Road # 9A

Dhanmondi, Dhaka.

DECLARATION

It is declared that, except where specific references are made to other investigators, the

work embodied in this thesis paper is the result of investigation carried out only by the

author under the supervision of Dr. Khan Mahmud Amanat, Professor, Department of

Civil Engineering, BUET. Neither the thesis nor any part of it has been submitted to or

is being submitted elsewhere for any other purposes.

Iftesham Bashar

(Author)

i

Dedicated

to

My Parents

ii

iii

ACKNOWLEDGEMENTS

I would like to express my heartfelt gratitude to the Almighty Allah for each and

every achievement of my life.

I would like to express my sincerest regards to my thesis supervisor, Dr. Khan

Mahmud Amanat for his guidance and encouragement during the course of this

research and throughout my master’s education. I am greatly indebted to him for all

his adept guidance, affectionate assistance, and enthusiastic encouragement

throughout the progress of this thesis. It would have been impossible to carry out this

study without his dynamic direction and critical judgment of the progress.

I wish to express my deepest gratitude to the Department of Civil Engineering,

BUET, the Head of the Department of Civil Engineering and all the members of the

BPGS committee to give me such a great opportunity of doing my M.Sc. and this

contemporary research work on eccentrically loaded single steel angles. I would also

like to specially thank the members of the defense board for their time and patience.

I greatly acknowledge all my friends and staffs of the university for their assistance

and encouragement. Finally, I would like to thank my parents and my family

members, for their undying love, encouragement and support at all stages of my life.

I would like to convey my special thanks to my mother for her perpetual patience

caring my babies while I was busy in my research works. The achievement of this

goal would have been impossible without their blessings.

Table of Contents

Page

No.

Declaration i

Dedication ii

Acknowledgements iii

Abstract iv

Chapter 1 Introduction

1.1 General 1

1.2 Background and Research Significance 1

1.3 Objectives of the Present Study 3

1.4 Methodology of the Study 3

1.5 Organization of the Thesis 4

Chapter 2 Literature Review

2.1 Introduction 5

2.2 Ultimate Compressive Load Capacity of Structural Steel Members 5

2.3 Development of Column Buckling Theory 8

2.4 Eccentrically Loaded Column Buckling Theory 11

2.4.1 General 11

2.4.2 Development of Eccentrically Loaded Column Buckling

Theory

12

2.4.2.1 Theoretical Investigations 12

2.4.2.2 Experimental Investigations 14

2.4.2.3 Numerical Investigations 17

2.5 Experiment of Elgaaly et al (1991) 18

2.5.1 Test Procedure 20

2.5.2 Test Results (Elgaaly et al 1991) 20

2.6 Code Provisions to Determine Ultimate Compressive Load Capacity

of Structural Steel Angles

23

2.6.1 ASCE Standard 10-97: Design of Latticed Steel

Transmission Structures

23

2.6.2 AISC 2005 Specification 25

2.6.3 BS (8100) Formula (1999) 26

2.6.4 AASHTO Formula 29

2.6.5 IS Code Formula 29

2.6.6 CRC Formula 31

2.7 Example Calculation 32

2.8 Remarks 41

v

Chapter 3 Methodology for Finite Element Analysis

3.1 Introduction 43

3.2 Finite Element Modeling of the Truss System 43

3.2.1 Modeling Methodology 43

3.2.2 Material Properties 52

3.2.3 Loading and Boundary Conditions 53

3.2.3.1 Restraints 53

3.2.3.2 Loads 53

3.2.4 Meshing 53

3.2.5

3.2.6

Solution Method

Mesh Sensitivity

55

56

3.3 Typical Analysis Results 56

3.3.1 Typical Load-Deflection Behavior 56

3.3.2 Deflected Shapes 57

3.4 Validation of Finite Element Model 60

Chapter 4 Results and Discussions

4.1 Introduction 62

4.2 Principal Features of Present Analysis 62

4.3 Presentation of Results 63

4.3.1 Axial Force vs Lateral Displacement Response 63

4.3.2 Deformation Characteristics of the Target Angle 66

4.3.2.1 General 66

4.3.2.2 Initial Stage 67

4.3.2.3 Final Stage 68

4.3.3 Comparative Results 69

4.3.3.1

4.3.3.2

Typical Comparative Load vs Deflection Graphs

Correlation between Test and Present Study

69

72

4.3.3.2 Comparison with Codes and Discussions 75

4.4

4.5

Interpretation and Explanation of Observations

Remarks

80

84

Chapter 5 Conclusion

5.1 General 85

5.2 Findings 85

5.3 Scpoe for Future Investigation 86

References

87

Appendices

Appendix-A : Results for single bolted angles 91

Results for double bolted angles

92

vi

Appendix-B : Ultimate Capacity of Angles Using Code Formulas

Appendix-C : Comparative Results of Single Bolted Angles

Comparative Results of Double Bolted Angles

93

103

105

vii

ABSTRACT

Steel angles have a wide range of structural applications requiring a comprehensive

design methodology. Steel angles, although used universally, such as in lattice

microwave towers and transmission towers have not received a comparable attention

for various reasons. The behavior of angles is different to some extent from that of

other commonly used steel shapes because they are unsymmetric sections and are

usually attached to other members by one leg only resulting the load to be applied

eccentrically. Eccentrically loaded single steel angles are one of the most difficult

structural members to analyze and design. They are prone to failure modes that are

not usually associated with other steel shapes. The ultimate compressive load

carrying capacity of single steel angles subjected to eccentrically applied axial load

is investigated in this project as part of a three dimensional truss. In this thesis, a

previously conducted experimental study is simulated. A finite element study was

conducted to properly understand the complex load carrying behavior of single

angles. Account is taken of member eccentricity, local deformation as well as

material and geometric non-linearity. Results are then compared with experimental

records and with those found by means of internationally adopted standard codes. It

is demonstrated that the finite element model closely predicted the experimental

ultimate loads and the behavior of steel angles. Hence, finite element analysis of

structures composed of single angles may be an easy alternative to physical testing

of these structures. From the present study it has been observed that only the ASCE

Standard 10-97 predicts angle capacity to a reasonable extent when compared to

both FE analyses and experimental results. Generally, AISC, BS and IS Codes

underestimate the capacity while AASHTO and CRC Formulas overpredict the

capacity. It is therefore recommended that in the design of three-dimensional trusses

and lattice towers, the provisions of ASCE Standard 10-97 should be followed.

iv

Chapter 1

INTRODUCTION

1.1 GENERAL

Angles are one of the most important sections used in steel structures. Sometimes an

entire structure is composed of steel angle sections such as lattice towers used in

telecommunication and power transmission sectors. The lattice tower is analyzed and

designed assuming that each member is a two-force member of truss (which is

subjected to tension and compression only). But in practical cases, in latticed towers,

trusses, etc. many members are connected by one leg to carry compressive loads.

This loads the member in axial compression with end moments due to the eccentric

connection. The resulting problem is rather complex to be analyzed because of the

eccentricity of load with respect to both principal axes and the uncertain nature of the

end restraints which would render the problem of finding an effective length factor

difficult. There are different codes worldwide for the analysis and design of single

steel angles having different design provisions. Until today, the electrical towers

have been designed without considering the effect of eccentricity on the ultimate

load carrying capacity of single steel angles, which is a prime limitation for

designing safe towers. Hence, there is a significant scope to investigate this matter.

This investigation is expected to provide the design engineer some definite

guidelines and recommendations for designing suitably load resistant tower

structures.

1.2 BACKGROUND AND RESEARCH SIGNIFICANCE

Single angles are used as primary structural components and members. Their

analysis and design should be done properly to assure that inadequate single angles

do not result. The loading of single angle struts is typically eccentric, producing

flexure about a non-principal axis. Furthermore, analysis may be complicated by the

possibility of torsional or lateral buckling of the angle. Often the most difficult aspect

Introduction

2

of evaluation of column capacity for single angles is the determination of the

effective slenderness ratio. The effective length factor can often be evaluated or

estimated about the x and y axes of the angle. However, these are not the principal

axes for the angle, so the determination of the governing slenderness ratio is not

easy. The most common situation has the ends of the angles attached with one leg to

a chord stem or gusset plate. This connection generally produces a relatively rigid

rotational restraint in the plane of the attached leg. The perpendicular leg usually has

a small restraint due to the flexibility of the stem or gusset about the chord's axis.

Due to the difference in effective lengths about the two geometric axes, the radius of

gyration does no longer represent the critical value. Therefore, the design and

analysis of steel angles become very complex. Traditionally, various international

design practices on eccentrically loaded single angle struts differ from each other

very widely. The specifications that deal with general steel construction are found

very conservative in estimating the design strength of eccentrically loaded single-

angle members. On the other hand, the specifications and manuals that deal with

lattice transmission towers that predominantly use steel angle members are found to

be much less conservative. As for example, the slenderness ratios of eccentrically

loaded single angle struts are modified in ASCE Manual No. 52 (American Society

of Civil Engineers, 1988) to make use of the formulas applicable to concentrically

loaded struts; Canadian tower design practice CSA-S37 (Canadian Standards

Association, 1986) and British practice (British Standards Institution, 1985) is to

ignore the eccentricity and limit the strength of eccentrically loaded single angle

struts to a certain percentage of the strength of corresponding concentric axially

loaded struts (reported by Adluri and Madugula (1992)). More detailed provisions

have probably not been put into code form for several reasons - perceived lack of

importance and analysis complexity. There were some efforts in the previous works

of the researchers to provide experimental data in an area of steel research that

clearly needs the information to revise the existing design procedures. Although

angle members are seemingly simple structural shapes used in several kinds of

applications, their design is quite complicated and has not been resolved completely

to the satisfaction of design engineers. Part of the reason for this is the lack of

sufficient experimental data for angles in comparison with some of the other standard

Introduction

3

structural shapes, e.g., wide flange sections. It becomes therefore obvious to make a

formulation for predicting ultimate compressive load carrying capacity of the single

steel angles. Proper analysis and design of single angle struts can only be

accomplished with some effort which is required to find appropriate governing

provisions as well as applying them correctly.

1.3 OBJECTIVES OF THE PRESENT STUDY

The objective of the study is to investigate latticed structures composed of single

angles by means of numerical finite element analysis. Three dimensional finite

element study will be carried out to simulate previously done experimental works by

the researchers. Comparison of compressive load capacity of the single angles

obtained by finite element analysis, different code formulas and previous

experimental results shall be made. Attempt will be made to provide a guideline for

rational design of single angle structures.

1.4 METHODOLOGY OF THE STUDY

For the purpose of carrying out the investigation, the compressive load carrying

behavior of a number of equal leg single steel angle members have been studied

using finite element method as part of a three dimensional truss as was tested by

Elgaaly et al (1991). Both for single bolted and double bolted configuration, the

effective length of the angle members under the analysis, have been modified to

simulate the actual conditions as close to the practical. The finite element analysis

incorporates shell element for modeling the entire system. Both material and

geometric non-linearity have been included during analysis. The finite element

problem is inherently nonlinear due to the plasticity of the model. Therefore, it

requires that, in addition to multiple iterations per load step for convergence, the

loads be applied in small increments, to characterize the actual load history. Arc-

length method has been applied to solve the concerned non-linear problem. Then,

compressive load capacities of the same angle members have been evaluated using

different code formulas. And, finally comparisons of results from previous test, code

formulas and finite element analysis have been presented.

Introduction

4

1.5 ORGANIZATION OF THE THESIS

The thesis is organized to best represent and discuss the problem and findings that

come out from the studies performed. Chapter 1 introduces the problem, in which an

overall idea is presented before entering into the main studies and discussion.

Chapter 2 is Literature Review, which represents the work performed so far in

connection with it collected from different references. It also describes the strategy of

advancement for the present problem to a success. Chapter 3 is all about the finite

element modeling exclusively used in this problem and it also shows some figures

associated with this study for proper presentation and understanding. Chapter 4 is the

corner stone of this thesis write up, which solely describes the computational

investigation made throughout the study in details with presentation by many tables

and figures followed by some discussions. Chapter 5, the concluding chapter,

summarizes the whole study as well as points out some further directions.

Chapter 2

LITERATURE REVIEW

2.1 INTRODUCTION

Steel angles are one of the most important and simplest type of structural

compression members. They are used in a variety of structures because of the ease

with which they can be fabricated and erected into structure or structural

components. This is facilitated to a large extent by the basic simplicity and

adaptability of the angle shape (figure 2.2). They are extensively used as primary leg

and diagonal members of latticed electrical transmission line towers (figure 2.1) and

antenna-supporting tower; as the chord members in plane trusses; as web and

bracing members in latticed towers, trusses; open-web steel joists and frames; as

lintels spanning openings over doors, windows etc. These structural components can

be either single or built-up angles; equal or unequal-leg angles; hot-rolled or cold-

formed. They are subjected to axial (either concentric or eccentric) or transverse

loads, or a combination of axial and transverse loads and moments, producing

stresses either below or above the proportional limit of the material. The analysis of

members composed of angle shapes is relatively complicated than, for instance, that

of wide flange shapes. Because, angle is an unsymmetric shape loaded eccentrically

through one side of the section (figure 2.3). Eccentrically loaded single-angle struts

are among the most difficult structural members to analyze and design.

2.2 ULTIMATE COMPRESSIVE LOAD CAPACITY OF

STRUCTURAL STEEL MEMBERS

Ultimate compressive load capacity of a structural member may be defined as the

load carrying capacity of that member at which it fails by buckling or yielding or a

combination of the two. Usually short members fail by yielding, whereas, long and

slender members fail by buckling characterized by large lateral deflections.

Literature Review 6

Figure 2.1 Four-legged electrical transmission tower (pylon) with single

steel angles

Literature Review 7

In between these extremes, columns of intermediate slenderness exhibit a combined

failure mode involving both yielding and large lateral deflections (figure 2.4).

Figure 2.2 Image of single equal leg angle members

Figure 2.3 Image of a typical bolted connection of single angle members

Literature Review 8

Figure 2.4 Failure characteristics of short and long compression members

2.3 DEVELOPMENT OF COLUMN BUCKLING THEORY

The behavior of single steel angles subjected to buckling is almost similar to that of

columns. The theory of elastic flexural buckling of concentrically loaded columns

was formulated by Leonhard Euler (1757). As described by Popov (2003), Euler

considered a perfectly straight column with no eccentric axial load (both end pinned).

The Euler formula describes the critical load for elastic buckling and is valid only for

long columns. The theory clearly fails to describe short columns, since it predicts

infinite resistance to compressive loads for slenderness ratio tending to zero (figure

2.5).

http://www.efunda.com/formulae/solid_mechanics/columns/columns.cfm

http://www.efunda.com/formulae/solid_mechanics/columns/theory.cfm

Literature Review 9

Figure 2.5 The short/intermediate/long classification of columns in terms of

stress-slenderness ratio curve

Engesser extended the elastic column buckling theory in 1889 (Engesser formula, as

described by Bleich (1952)). He assumed that inelastic buckling occurs with no

increase in load, and the relation between stress and strain should be defined by

tangent modulus Et.

The principal assumption which caused the tangent modulus theory to be erroneous

is that as the member changes from a straight to bent form, no strain reversal takes

place. So, Engesser in 1895 corrected his original theory (as described by Salmon

and Johnson in 1996) by accounting for the different tangent modulus of the tensile

increment. This is known as the reduced modulus or double modulus. The

assumptions are the same as before. That is, there is no increase in load as buckling

occurs.

Both the Tangent-Modulus Theory and Reduced-Modulus Theory were accepted

theories of inelastic buckling. But the engineers were faced with the confusion that

the reduced modulus theory was correct, but the experimental data was closer to the

Literature Review 10

tangent modulus theory. Shanley (1947) eventually resolved the problem by

conducting very careful experiments on small aluminum columns. He observed that

lateral deflection started very near the theoretical tangent modulus load and the load

capacity increased with increasing lateral deflections. The column axial load

capacity never reached the calculated reduced or double modulus load. Therefore, he

developed a column model to explain the observed phenomenon. He reasoned that

the tangent modulus theory is valid when buckling is accompanied by a simultaneous

increase in the applied load of sufficient magnitude to prevent strain reversal in the

member. The applied load given by the tangent modulus theory increases

asymptotically to that given by the double modulus theory.

In practice, however, most structures suffer plastic knockdown and the

experimentally obtained buckling loads are much less than the Euler (1757)

predictions. For structures in this category, a suitable formula is the Rankine-Gordon

formula (as mentioned by Tall in 1974) which is a semi-empirical formula, and takes

into account the crushing strength of the material, its Young's modulus and its

slenderness ratio, namely l/r. This criteria is based on experimental results.

Another equation is Tetmajer Equation (reported by Tall in 1974) which is a linear

formula and is valid in the range of inelastic buckling when the current slenderness

ratio (L/r) is less than the critical slenderness ratio.

For very short columns the yield stress (with appropriate design factor) can be used

for predicting the ultimate load capacity. But for columns that are not short, and

where the Euler formula gives stress above the yield stress, empirical methods of

design should be used. One popular equation in use since the early 1900s is the

Johnson formula (as described by Tall in 1974) which can be used for columns with

slenderness ratios below a transition slenderness ratio. The Johnson Parabola is one

of these curve fitting methods, and has been used commonly in structural

engineering. It is an inverted parabola, symmetric about the point (0, σy) tangent to

the Euler curve.

In practice a structure cannot be perfectly straight, and the analysis for such a

structure would become more realistic if account could be taken of the slight


deviations from straightness of the centroidal axis of the strut. From this

consideration, Perry-Robertson Approach (as reported by Trahair and Bradford in

1988) has been initiated.

In addition to above theories, further observations regarding inelastic buckling were

made by Lin (1950), Duberg and Wilder (1950). Most of their analysis was based on

a lot of experiments on inelastic behavior of column member.

The maximum load lying between the tangent modulus load and the double modulus

load for any time-independent elastic-plastic material and cross-section was

accurately determined by Lin (1950).

Duberg and Wilder (1950) have further concluded that for materials whose stress -

strain curves change gradually in the inelastic range, the maximum column load can

be appreciably above the tangent modulus load. If, however, the material in the

inelastic range tends rapidly to exhibit plastic behavior the maximum load is only

slightly higher than the tangent modulus load. These researches also included elastic-

plastic nature of materials.

2.4 ECCENTRICALLY LOADED COLUMN BUCKLING

THEORY

2.4.1 General

In reality, columns are often subjected to eccentric loading which causes

development of bending stress as well as generates possible buckling behavior.

There is always some eccentricity in the applied loading of a member due to initial

imperfections in the section or in its boundary conditions. When a compressive load

is increased, eccentricity sets up bending in the member causing it to deflect. In turn,

the deflection increases the eccentricity, which increases the bending. This may

progress to where the bending increases at a greater rate than the compressive

loading and the member becomes unstable. This phenomenon can occur in columns,

flanges, plates and shells subject to compression. In figure 2.6, illustration of the

stress that a column experiences under increasing eccentric loading has been shown.


Figure 2.6 Form of the stress prism changes from an even distribution to a very

uneven distribution due to increasing eccentricity of loading

It is clear that with increasing eccentricity of loading, the stress becomes bending in

nature. This bending stress introduces bending moment, which a column section

must resist in addition to compressive stress under equilibrium condition.

2.4.2 Development of Eccentrically Loaded Column Buckling

Theory

2.4.2.1 Theoretical Investigations

In a brief account of the development of the theory of eccentrically loaded columns,

Ostenfeld (1898), over a century ago, made an attempt to derive design formulas for

centrally and eccentrically loaded columns. His method was based upon the concept

that the critical column load is defined as the loading which first produces external

fiber stresses equal to the yield strength.

Ros (1928) established a simplified stability theory of eccentrically loaded columns

and proved the theoretical results by a number of tests. They assumed that the

deflected center line of the column can be represented by the half wave of a sine

curve but based the computation of the critical load upon the actual stress-strain


diagram.

Westergaard and Osgood (1928) presented a paper in which the behavior of

eccentrically loaded columns and initially curved columns were discussed

analytically. The method assumes the deflected center line of eccentrically loaded

compression members to be part of a cosine curve without impairing the practical

accuracy of the results.

Chwalla (1928) in a series of papers between 1928 and 1937 investigated in a very

elaborate manner the stability of eccentrically loaded columns and presented the

results of his studies for various shapes of column cross section in tables and

diagrams. He based all his computations on one and the same stress-strain diagram

adopted as typical for structural steel. The significance of his laborious work is that

the numerous tables and diagrams brought insight into the behavior of eccentrically

loaded columns as influenced by shape of the column cross section, slenderness

ratio, and eccentricity and that his exact results can serve as a measure for the

accuracy of approximate methods.

In his two subsequent papers, Jezek (1935, 1936) presented an approximate and

simple to use method for the flexural buckling of eccentrically loaded columns

giving satisfactory results. In this method, the shape of the deflected column is

assumed to be one-half a sine wave.

A different approach to the complex problem of eccentrically loaded columns,

starting from the secant formula, was made by Young (1936). He considers, as

Ostenfeld and others did, the failure load as the load which produces the beginning

of yielding in the highest stressed fiber. For structural steel having 36 kips/in.2 yield

strength, he develops column curves for various values of the eccentrically, and he

treated initially curved columns by the same method.

The first to consider the determination of the buckling load of eccentrically loaded

columns as a stability problem was Karman (1940) who gave, in connection with his


investigations on centrally loaded columns, a complete and exact analysis of this

rather involved problem. He called attention to the sensitiveness of short and

medium-length columns to even very slight eccentrically of the imposed load, which

reduce the carrying capacity of straight columns considerably.

Bleich (1952) also developed simple algebraic formulas for the approximate method

of design of eccentrically loaded columns which fail by flexural buckling about the

weak axis.

Trahair (2007) in a series of papers (from 2001 to 2007) investigated the behavior of

single angle sections thoroughly. These papers summarize a design method for angle

sections which is rational, consistent and economical. Though his research works are

confined to assume angles as flexural members, they are quite informative and give a

clear and concise idea about complex load behaviour characteristics of angles.

2.4.2.2 Experimental Investigations

Stang and Strickenberg (1922) conducted the first compression tests on angle

members in the United States. One hundred and seventy tests were performed on

hot-rolled single angles, and a wide variety of end connections and slenderness ratios

were considered. End fixity factors were determined for different end conditions. For

large slenderness ratios, the results were in good agreement with Euler's buckling

formula and the reduced modulus theory agreed well with the average test values for

single angles with smaller slenderness ratios. It was also found that load eccentricity

and end restraint are very important in evaluating single-angle strength, with load

eccentricity having more effect than end restraint for slenderness ratios below about

85. Further, unequal leg angles were stronger when connected by their short leg.

Wakabayashi and Nonaka (1965) studied 10 equal-leg hot-rolled 79090 mm

angles under concentric loading for slenderness ratios ranging between 40 and 150.

Hemispherical ends were employed and the results were used to develop a design

method. Of the 10 test specimens, seven failed in flexural buckling.


Yokoo et al. (1968) performed a study that included the testing of hot-rolled single-

angle members loaded concentrically in compression using a ball-joint connection.

Torsional deformations were predominant in concentrically loaded specimens. They

also showed that boundary conditions for twisting do not significantly affect the

failure load.

Equal-leg high-strength steel angles 67575 mm and 66565 mm in size were

tested by Ishida (1968).The tests showed that the load-carrying capacity of mild steel

angles was generally higher compared to hot-rolled high-strength steel angles, which

contain large residual stresses.

Kennedy and Murty (1972) presented a rational buckling analysis that was designed

to overcome limitations in the American Institute of Steel Construction (AISC)

Specifications and the Canadian Standards Association (CSA) design code. As part

of the testing program designed to verify the analytical buckling analysis, 72 single-

angle struts were tested with ends both fixed and hinged. All angles were designed to

fail inelastically, and actual dimensions and yield stresses were measured as part of

the testing program.

Wood (1975) reported that a series of tests were performed on 153 crossed diagonal

angles in lattice transmission towers. All of the tests were conducted using tower

substructures in order to duplicate actual end conditions. One-, two-, and three-bolt

connections were evaluated for members of different slenderness ratios. All of the

test specimens failed in the elastic buckling range and their strengths were compared

with the Euler buckling load.

Mueller and Erzurumlu (1983) investigated the overall performance of single-angle

columns. The parameters studied were yield stress, load eccentricity, and end

restraint. To conduct parametric studies, an analytical study was also performed.

Mueller and Wagner (1983) performed further testing to gain more knowledge

regarding the post buckling performance of angle members. Several parameters were


studied, including end restraint, slenderness ratio, load eccentricity, and intermediate

supports.

Kitipornchai and Lee (1984) performed a series of tests on hot-rolled single-angle,

double-angle, and tee struts failing in the inelastic range. A total of 54 specimens

were tested under concentric loading with special pinned-end supports designed to

restrain twisting about the longitudinal axis. The results were compared to theoretical

predictions as well as various design codes.

Al-Sayed and Bjorhovde (1989) reported the results of an experimental investigation

on 12 steel-angle columns failing in flexural or torsional-flexural buckling modes.

The tests were conducted under concentric loads using spherical ends that permitted

both bending and twisting deformations. The results include residual stresses, stub-

column strengths, and other observations. Initial out-of-straightness was not

measured.

The ultimate compressive load capacity of single steel angles subjected to

eccentrically applied axial load has been investigated by Leander Bathon (1993).

Seventy-five full-scale tests (thirty one single angle equal-leg and forty four single

angle unequal-leg specimens) were done with data collected in the elastic, inelastic

and post buckling regions of member’s performance. Test specimens were made of

A36 steel angle sections connected to the end plate by bolts. The end plates were

supported using a ball joint in an attempt to model end conditions that were

unrestrained against rotation. The results of the tests were compared with predicted

load capacities obtained from ASCE Manual 52 for the Design of Steel Transmission

Towers (1988), which is the design guide for lattice steel transmission towers.

Experiments were carried out by Adluri and Madugula in 1996. They comprised 26

tests and had six different sizes and 23 different slenderness ratios. They produced

test data with slenderness parameter λ between 0.91 and 2.44.The ratio of the

nominal leg width to leg thickness ranged from 6 to 16.All the test specimens

including those prone to local buckling failed in flexural buckling before exhibiting


some local failure. Finally they were able to develop several column curves to verify

test results. It was observed that the generated column curves were very close to test

results.

2.4.2.3 Numerical Investigations

Adluri (1994) used Finite Element Method to simulate the behavior of steel angles

under flexural buckling. For the purpose of analysis, the steel angle is discretized

into several strips of elements along the length. Each strip is subdivided into

individual finite elements. Eight-node finite elements with six degrees of freedom at

each node were used. The element stiffness was computed using a reduced

integration scheme with four Gause points per element. The angle geometry was

defined with a sinusoidal initial out-of-straightness along the major axis. The

maximum midheight out-of-straightness was prescribed as length/1,500. The results

show good agreement between theory and experiments. Most of the predicted

strengths are close to but below test results by up to 10%.

Studies on the use of the finite-element method (FEM) for steel angles were reported

by Lu et al. (1983) and Chuenmei (1984). Applicability of the FEM for a failure

analysis of schifflerized steel-angle members was shown by Adluri et al. (1991).

However, the application of the method is not very suitable for the development of

column curves, because of the large amount of computational effort required even

with the computing power available today.

An exploratory study was conducted by Haaijer, Carskaddan and Grubb (1981) to

investigate the feasibility of using a finite element analysis in lieu of a physical test

of an eccentrically loaded single angle column connected by one leg. Only elastic

behavior was considered so that the results are applicable to relatively slender

members.

A combination of finite element and finite segment approaches has been used by Hu

and Lu (1981) to determine the complete load-deflection relationships of single-

angle struts subjected to eccentric compressive loads, with or without end restraints.


A rational design procedure for eccentrically loaded single angles is also being

developed.

2.5 EXPERIMENT OF ELGAALY et al (1991)

In 1991, Elgaaly et al conducted tests on 50 non-slender single steel angles as part of

a three dimensional latticed truss. Both the specimens with eccentric single bolted

and double bolted end connections were investigated. Of the specimens, 25 were

double bolted and the rest 22 were single bolted at their ends. Table 2.1 lists the

angle sections by groups depending on difference in cross-sectional dimensions,

slenderness ratios (l/r) and end conditions.

Table 2.1: Test specimens (Elgaaly et al. 1991)

The selection of specific member sizes for testing was based on both the capacity of

the truss and the need to cover slenderness ratio range from 0 to 120. The three

dimensional truss used to test each specimen is shown in figure 2.7. The truss was

designed so that the target angle would fail first without introducing significant

deformations in the remainder of the truss. Following each test, only the target angle

was replaced, allowing multiple tests to be conducted in the same setting. Load was

applied via two 100 kip capacity hydraulic jacks, which allowed the load on each

side of the truss to be kept balanced. Then the test results were given and six failure

modes were identified. These failure modes depend on the member slenderness ratio,

the angle leg width/ thickness ratio, and the end connection detail. These failure

modes can be generally classified as global with no appreciable local failures, or


Fig

ure

2.7

Tes

t se

tup o

f E

lgaa

ly e

t al

(1991)


local failures that triggered global failures in some cases. Finally, the design rules

given by the AISC Buildings Design Specification and the ASCE Manual 52 for the

design of Steel Transmission Towers were evaluated. The AISC Specification

requirements were found to be conservative. Meanwhile, the ASCE Manual 52

requirements were found to be unsafe, particularly in the cases where failure is due to

the local buckling of the angle leg. Table 2.1 lists the characteristics and numbers of

the test specimens chosen.

2.5.1 Test Procedure

After the initial readings were taken, the load was applied manually in small

increments via the two hydraulic jacks (figure 2.7). At the end of each increment, the

load was manually recorded, and the strain gage and transducer readings were taken

by the computer. When failure of the specimen was imminent, the size of the load

increments was reduced to approximately half its initial value. Each test was stopped

after excessive deformations to the test specimen indicating it had failed.

2.5.2 Test Results (Elgaaly et al 1991)

Figures 2.8 to 2.11 are typical member force vs displacement in horizontal and in

vertical plane graphs obtained from the results of the test of Elgaaly. And the tables

2.2 and 2.3 list the cross sectional dimensions, width-thickness ratio, yield stress,

failure modes as well as failure loads for different angle specimens with buckling

load factors (n).

Figure 2.8 Member Force versus Strain and Displacement Specimen 34 (test of

Elgaaly et al 1991)



Elgaaly et al 1991)


Elgaaly et al 1991)


Elgaaly et al 1991)


Table 2.2: Test Results for Double-Bolted Specimens (Elgaaly et al 1991)


Table 2.3: Test Results for Single-Bolted Specimens (Elgaaly et al 1991)

2.6 CODE PROVISIONS TO DETERMINE ULTIMATE

COMPRESSIVE LOAD CAPACITY OF STRUCTURAL

STEEL ANGLES

2.6.1 ASCE Standard 10-97: Design of Latticed Steel

Transmission Structures

According to ASCE Standard 10-97, for angle compression members with

normal framing eccentricities at both ends of the unsupported panel,

r

L

r

KL5.060 for 1200

r

L (2.1)


y

cF

EC

2

The ratio w/t should not exceed 25, where, w = flat width and t = thickness of leg.

The design compressive stress (Fa) on the gross cross-sectional area shall be one of

the following equations (2.3) and (2.4):

y

c

a FC

rKLF

2

2

11

for, cCr

KL

22

rKL

EFa

for, cCr

KL

provided the largest value of w/t does not exceed the limiting value given by:

yFt

w 80

lim

If w/t exceeds the limiting value, the design compressive stress (Fa) shall be

according to equations no. (2.3) or (2.4) except with the replacement of Fcr for Fy

which is given by:

ycr Ftw

twF

lim

677.0677.1 for

yFt

w

t

w 144

lim

220332.0

tw

EFcr

for yFt

w 144

Where,

Fy = minimum guaranteed yield stress (MPa or ksi)

E = modulus of elasticity (MPa or ksi)

L = unbraced length (mm or inch)

r = radius of gyration (mm or inch)

K = effective length coefficient

Ψ = 1 for Fy in ksi and 2.62 for Fy in MPa

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)


2.6.2 AISC 2005 Specification

According to AISC Specifications, for single equal-leg or unequal-leg angle

compression members connected through the longer leg that are individual members

or are web members of planer trusses with adjacent web members attached to the

same side of the gusset plate or chord:

(i) For, 800 xr

L,

xr

L

r

KL5.072 (2.8) (2.8)

(ii) For, 80xr

L, 20025.132

xr

L

r

KL (2.9)

The nominal compressive strength, nP shall be determined based on the limit state

of flexural buckling,

gcrn AFP (2.10)

The flexural buckling stress, crF is determined as follows:

(a) When yF

E

r

KL71.4 (or ye FF 44.0 )

y

F

F

cr FF e

y

658.0 (2.11)

(b) When yF

E

r

KL71.4 (or ye FF 44.0 )

ecr FF 877.0 (2.12)

Where,

eF elastic critical buckling stress determined according to equation (2.13)


2

2

r

KL

EFe

(2.13)

2.6.3 BS (8100) Formula (1999)

According to British Standard (8100), for lattice towers, a single lattice is commonly

used where the loads are light and the length (Ld) is relatively short. For the case, the

slenderness ratio, λ for single steel angles should be taken as,

vv

d

rL

(2.14)

In order to calculate the design buckling resistance of the member, the effective

slenderness Λeff should be determined from:

Λeff = KΛ (2.15)

Where,

Λ = the relative slenderness of the member about the appropriate axis for which the

strength is required and is given by:

1

Λ (2.16)

Where,

λ = the slenderness ratio obtained from equation no. (2.14)

and

5.0

1

y

E

(2.17)

when,

E = 205000 MPa

K = effective slenderness factor depending on the structural configuration and is

given as follows:


33.1t

B

2

2

1.5

t

B

Er

For single and double bolted angle legs, which are normal to the frame and has

discontinuous both ends,

ΛK

58.07.0 (2.18)

For hot rolled angle sections, the reference stress, r is given by:

yr , if t

B (2.19)

t

Byr

2 , if 33.1

t

B (2.20)

, , if (2.21)

Where,

σy=The specific minimum yield stress of material of member (ksi or MPa)

B = the leg length of the angle (inch or mm)

t = the thickness of angle leg (inch or mm)

y

E

567.0 (2.22)

E= the modulus of elasticity (MPa or ksi)

The design of buckling resistance, N of a compression member should be taken as:

m

rAjN

(2.23)

Where,

j = 0.8 for single angle members connected by one bolt at each end,

0.9 for single angle members connected by one bolt at one end and continuous at

the other end and 1.0 in all other cases;

A= the cross-sectional area of the member (in2 or mm

2);

= the reduction factor for the relevant buckling mode and should be determined

from:


22.015.0 effeff ΛΛa

5.022

1

effΛ

(2.24)

Here,

(2.25)

a = an imperfection factor corresponding to the appropriate buckling curve

which should be obtained from the following table:

Buckling curve Imperfection factor, a

A 0.21

B 0.34

C 0.49

D 0.76

γm= the partial factor on strength as given in BS 8100-1 and -4,appropriate to the

quality class of the structure:

For angle section towers which have successfully been subjected to full-scale tests

under the equivalent factored loading or where similar configurations have been type

tested:

γm is 1.0 for Class A structures;

γm is 1.1 for Class B structures;

γm varies from about 1.2 to 1.35 for Class C structures depending on the performance

requirements.


22

2 135000000

..

r

L

r

LSF

EF

ee

a

ce CrL

2.6.4 AASHTO Formula

AASHTO formula is a simple adoption of the AISC formula in which AASHTO

uses a different factor of safety (F.S) and a constant value of effective length factor

(K) for bolted or riveted connection. The AASHTO formulae for allowable stress

(Fa) are:

E

Fr

L

SF

FF

y

e

y

a 2

2

41

.. for ce CrL (2.26)

for (2.27)

Here y

cF

EC

22 , F.S = 2.12, K= 0.75 (2.28)

2.6.5 IS Code Formula

According to Indian Standard 800:2007, common hot rolled steel members used for

carrying axial compression, usually fail by flexural buckling. The buckling strength

of these members is affected by residual stresses, initial bow and accidental

eccentries of load. To account for all these factors, the strength of members

subjected to axial compression is defined by buckling class a, b, c, or d as given

below:

Buckling

Class

a b c d

Imperfection

Factor, α

0.21 0.34 0.49 0.76


The design compressive strength of a member is given by:

cded fAP (2.29)

Where,

Ae = effective sectional area (here, the gross sectional area shall be taken as the

effective sectional area for all compression members fabricated by welding,

bolting or riveting so long as the section is semi-compact or better)

fcd = design compressive stress obtained as per the following equation:

mo

y

mo

y

cd

fff

(2.30)

Where,

χ = stress reduction factor for different buckling class, slenderness ratio and

yield stress

= 5.022

1

(2.31)

fy = yield stress of the material

γmo = partial safety factor for material strength

= 1.1

22.015.0 (2.32)

λ = non-dimensional effective slenderness ratio (angles are usually loaded

eccentrically by connecting one of its legs either to a gusset or to an adjacent

member. Such angles will buckle in flexural-torsional mode in which there will be

significant twisting of the member. Such twisting may be facilitated by the flexibility

of the gusset plate and the other members connected to it. Thus, to account for the

reduction in strength due to flexural-torsional mode, the code gives an equivalent

slenderness rati0 ( λeq ) instead of λ as a function of the overall slenderness ratio and

the width-thickness ratio as given below:

2

3

2

21 kkk vveq (2.33)

Where,

k1, k2, k3 = constants depending upon the end condition as given below:


No of

bolts at

each end

connection

Gusset/connecting

member fixity 1k 2k 3k

2 Fixed 0.20 0.35 20

Hinged 0.70 0.60 5

1 Fixed 0.75 0.35 20

Hinged 1.25 0.50 60

250

2E

r

l

vv

vv

and

250

22

21

E

t

bb

(2.34)

Here,

λvv = slenderness ratio for flexural buckling

λφ = slenderness ratio for torsional buckling

l = centre to centre length of the supporting member

rvv = radius of gyration about the minor axis

b1, b2 = width of the two legs of the angle (mm)

t = thickness of the leg (mm)

ε = yield stress ratio

=

5.0

250

yf ; (2.35)

fy = yield stress of material (MPa)

2.6.6 CRC Formula

The basic column-strength curve adopted by the Column Research Council

(CRC) is based on parabolic equation proposed by Bleich as:

])(4

1[ 2

2 r

L

E

ey

ycr

(2.36)

In above equation 2.36,

cr = critical stress for column


5.13925.132 xr

L

r

KL

51.11671.4 yF

E

2.7 EXAMPLE CALCULATION

In the present study, single steel angles (both single and double bolted specimens)

which are examined by Elgaaly et al (1991) have been investigated using finite

element analysis and other code specifications. The purpose of this article is to

address the axial capacity evaluation of eccentrically loaded single angles. This

section presents sample calculations of evaluating the ultimate compressive load

capacity of the angle sections using various methods:

Problem Statement 1: Double bolted specimen (Test no. 20 as per the experiment

of Elgaaly et al (1991))

Angle size: width, d 992.1 inch, thickness, t 200.0 inch

86r

l,

2mm

kN200E , ksi4.47yF

AISC 2005 Specification:

For, 80xr

L,

Now,

As, the calculated yF

E

r

KL71.4 ,

So, 43.1012

2

r

KL

EFe

MPa

Now, 80.14344.0 yF MPa

So, ye FF 44.0 for which,

95.88877.0 ecr FF MPa

Therefore, kN43.4326.48895.88 gcrn AFP

Or, 99.8nP kips


ASCE Standard 10-97:

2596.8

200.0

200.0992.1

t

w

62.114.47

18080

lim

yFt

w 1 (for Fy in ksi)

Check: lim

t

w

t

w

907.1092

y

cF

EC

Now, for members with normal framing eccentricity at both ends of the

unsupported panel,

1035.060 r

L

r

KL

cCr

KL , for which,

y

c

a FC

r

KL

F

2

5.01

2mm

kN29.183

So, gan AFP

26.48829.183

Or, 49.89nP kN 12.20 kips

BS Standard Code:

m

rAjN

Where,

j=1.0

26.488200.0200.0984.12222 ttdA mm

2


222

1

effΛ

5684.0

Here,

4495.12427.12.02427.134.015.02.015.0 22 effeff ΛΛa

and 2427.1093.1137.1 KΛAeff

Now,

ΛK

58.07.0 ( both end discontinuous)

1

58.07.0

158.07.0

=1.23

Now,

96.9200.0

992.1

t

B

2.14567.0 y

E

Now,

for t

B,

823.326895.64.47 yr MPa

0.1m (let, for class-(1) structures)

Therefore,

0.1

823.326257.4885684.00.1 N

86

895.64.47

205000

58.07.0

5.0


39.20 kips

AASHTO Formula:

Allowable stress,

E

Fr

L

SF

FF

ye

y

a 2

2

41

. (for c

e Cr

L )

2050004

895.64.478675.01

0.1

895.64.472

2

89.24 kips

Though, AASHTO Code, factor of safety, F.S=2.12, but in the present study, it has

been assumed 1.0.

IS Code Formula:

875.0250

5.0

yf

106.1

250

2

E

r

l

vv

vv

1281.0

250

22

21

E

t

bb

9779.02

3

2

21 kkk vveq

1687.19779.02.09779.049.015.02.015.0 22 eqeq

553.0

15.022

eq

So, 697.180895.64.47553.0 mo

y

cd

ff

MPa


23.88697.18026.488 dP kN 84.19 kips

CRC Formula:

668.226])(4

1[ 2

2

r

L

E

ey

ycr

MPa

88.242571.488668.226 g

AP crn kips

Figure 2.12 Buckling load capacity of single steel angle member for

Elgaaly (test no.20, angle size:50.6×50.6×5.08) evaluated in

various codes

Angle capacity found by different codes as shown are graphically summarized in

figure 2.12. It has been observed that only the ASCE Standard 10-97 and IS Code

gives reasonably close values for the buckling load capacity of single steel angles as

part of a truss structure, whereas the AISC and BS (British Standard) underpredict at

a large extent from the test result obtained from Elgaaly et al (1991). And the other

codes AASHTO, CRC formulas give approximately same results, but overpredicts

the test results to some extent.


Problem Statement 2: Single bolted specimen (Test no. 53 as per the experiment of

Elgaaly et al)

Angle size: width, d=1.749 inch, thickness, t=0.133 inch

92r

l,

2mm

kN200E , ksi2.51yF


For, 80xr

L,

0.14725.132 xr

L

r

KL

Now,

11.11271.4 yF

E


E

r

KL71.4 ,

So, 35.912

2

r

KL

EFe

MPa

Now, yF44.0 33.155 MPa


11.8035.91877.0 crF MPa

So, 13.2374.28811.80 gcrn AFP kN

Or, 20.5nP kips


2515.12

133.0

133.0749.1

t

w

18.112.51

18080

lim

yFt

w 1 (for Fy in ksi)


13.202.51

1144144

yF

Check:

yFt

w

t

w 144

lim

, for which

19.48677.0677.1

lim

ycr F

t

w

t

w

F ksi

00.1092

cr

cF

EC

Now, for members with normal framing eccentricity at both ends of the unsupported

panel,

1065.060 r

L

r

KL

cCr

KL , for which,

cr

c

a FC

r

KL

F

2

5.01

= 2mm

kN15.175

So, 57.507381.28815.175 gan AFP kN

Or, 37.11nP kips

BS Standard Code:

m

rAjN

Where,

j=0.8

74.288133.0133.0749.12222 ttdA mm

2


222

1

effΛ

3688.0

Here,

733.143.12.043.134.015.02.015.0 22 effeff ΛΛa and

43.1215.1177.1 KΛΛeff

ΛK


=1.177

Now,

15.13133.0

749.1

t

B

66.13895.62.51

205000567.0567.0

y

E

Now, for t

B,

024.353895.62.51 yr MPa


Therefore,

0.1

024.35374.2883688.08.0 N

86.30073 N

76.6 kips

AASHTO Formula:

Allowable stress,

E

Fr

L

SF

FF

ye

y

a 2

2

41

. (for c

e Cr

L )

2000004

895.62.519275.01

0.1

895.62.512

2


24.14 kips

Though, in AASHTO Code, factor of safety, F.S=2.12, but in the present study, it

has been assumed 1.0.

IS Code Formula:

842.0895.62.51

2502505.05.0

yf

231.1

250

2

E

r

l

vv

vv

176.0

250

22

21

E

t

bb

378.12

3

2

21 kkk vveq

738.12.015.0 2 eqeq

357.0

15.022

eq

So, 172.1261

2.51357.0

xff

mo

y

cd

MPa

43.36172.12674.288 dP kN 19.8 kips

CRC Formula:

292.219])(4

1[ 2

2

r

L

E

ey

ycr

MPa

32.6374.288292.219 g

AP crn kN 24.14 kips


Figure 2.13 Buckling load capacity of single steel angle member for

Elgaaly (test no.53, angle size:50.6×50.6×5.08) evaluated in

various codes

Angle capacity found by different codes as shown are graphically summarized in

figure 2.13. It has been observed from figure 2.13 that as before like in the case of

double bolted specimens only the ASCE Standard 10-97 reasonably predicts the

buckling load capacity of single steel angles as part of a truss structure, whereas the

AISC, BS (British Standard) and IS Code underpredicts to a large extent from the

test result obtained from Elgaaly et al (1991). And the other codes AASHTO, CRC

formulas give approximately same results, but overpredicts the test results to some

extent. But the main distinguishing point is the difference in accuracy of the results

obtained from various methods due to the difference in the number of bolts in end

connections of target angles.

2.8 REMARKS

Single steel angles are one of the most important structural components. There have

been some researches carried out by past scientists throughout the world. Some of

the works aimed at the principal governing factors which influence the load carrying

nature of angles. In our country, little work has been carried out as per author’s


knowledge. Moreover, it has been shown that capacities of angles predicted by

different codes vary significantly. So, the author feels that there should be more

research oriented studies regarding the issue for providing more data to have proper

guideline for better estimation of compressive loads of angles eccentrically loaded in

structures. This will also help to bring any modification in the designing parameters

of angles in the codes. Hence, the proposed study is expected to provide the design

engineers with some definite guidelines on these areas.

Chapter 3

METHODOLOGY FOR FINITE ELEMENT

ANALYSIS

3.1 INTRODUCTION

The finite element method (FEM) is the most popular simulation method to predict

the physical behavior of systems and structures. Although the method was originally

developed to find a solution for problems of structural mechanics it can nowadays be

applied to a large number of engineering disciplines in which the physical

description results in a mathematical formulation with some typical differential

equations which can be solved numerically. Much research work has been done in

the field of numerical modeling during the recent years which enables engineers

today to perform simulations close to reality. Nonlinear phenomena in structural

mechanics such as nonlinear material behavior, large deformations or contact

problems have become standard modeling tasks. If experimental or analytical results

are available it is easily possible to verify any finite element result. In this chapter,

the actual work regarding the finite element modeling of a single steel angle

connected to horizontal and vertical angles as a component of a truss has been

described in detail. The representation of various physical elements with the FEM

(Finite Element Modeling) elements, properties assigned to them, boundary

conditions, material behavior and analysis types have also been discussed. The

various obstacles faced during modeling, material behavior used and details of finite

element meshing were also discussed in detail.

3.2 FINITE ELEMENT MODELING OF THE TRUSS SYSTEM

3.2.1 Modeling Methodology

FEM is a powerful technique originally developed for numerical solution of complex

problems in structural mechanics, and it remains the method of choice for complex

Methodology for Finite Element Analysis 44

systems. A large number of finite element analysis computer packages are available.

Of these packages, ANSYS 11.0 has been chosen for its versatility and relative ease

of use. ANSYS is capable of modeling and analyzing a vast range of two-

dimensional and three-dimensional practical problems. An example of the

configuration of the three dimensional finite element model of the truss structure is

shown in the figure 3.1. The model consists a target angle. The truss is designed so

that the target angle fails first without introducing significant deformations in the

remainder of the truss. Following each test, only the target angle is replaced,

allowing multiple tests to be conducted in the same setting as was conducted by

Elgaaly et al (1991).

Figure 3.1 General 3-D sketch of the problem

Both the provisions for single bolted and double bolted connections have been made

at the two ends of the target angle by modifying the effective length of that angle

member. Angle specimens were discretized into a mesh of elements using general-

purpose 4-node Shell 181 elements as specified in ANSYS. The target angle is

discretized into different mesh size considering the cross-sectional dimensions of the


target angle rather than the dimensions of other angle members. Discussion about the

element is shown below in details:

SHELL181 Element Description

SHELL181 is suitable for analyzing thin to moderately-thick shell structures. It is a

4-node element with six degrees of freedom at each node: translations in the x, y, and

z directions, and rotations about the x, y, and z-axes. (If the membrane option is

used, the element has translational degrees of freedom only). The degenerate

triangular option should only be used as filler elements in mesh generation.

SHELL181 is well-suited for linear, large rotation, and/or large strain nonlinear

applications. Change in shell thickness is accounted for in nonlinear analyses. In the

element domain, both full and reduced integration schemes are supported.

SHELL181 accounts for follower (load stiffness) effects of distributed pressures.

Figure 3.2 SHELL181 Geometry

xo = Element x-axis if ESYS is not provided.

x = Element x-axis if ESYS is provided.


SHELL181 Input Data

The geometry, node locations, and the coordinate system for this element are shown

in "SHELL181 ". The element is defined by four nodes: I, J, K, and L. The element

formulation is based on logarithmic strain and true stress measures. The element

kinematics allows for finite membrane strains (stretching).The thickness of the shell

may be defined at each of its nodes. The thickness is assumed to vary smoothly over

the area of the element. If the element has a constant thickness, only TK(I) needs to

be input. If the thickness is not constant, all four thicknesses must be input. A

summary of the element input is given in below (Table 3.1).

Table 3.1 SHELL181 Input Summary

Element name SHELL181

Nodes

I, J, K, L

Degrees of Freedom

UX, UY, UZ, ROTX, ROTY, ROTZ if

KEYOPT (1) = 0

UX, UY, UZ if KEYOPT (1) = 1

Real Constants

TK(I), TK(J), TK(K), TK(L), THETA,

ADMSUA, E11, E22, DRILL,

MEMBRANE, BENDING

Material Properties

EX, EY, EZ, (PRXY, PRYZ, PRXZ, or

NUXY, NUYZ, NUXZ),

ALPX, ALPY, ALPZ (or CTEX, CTEY,

CTEZ or THSX, THSY, THSZ),

DENS, GXY, GYZ, GXZ

Modeling Methodology of Target Angle

The finite element analysis has considered both the single bolted and double bolted

target angle specimens as tested by Elgaaly et al (1991). For single bolted target

angles, the connected leg has been divided into four portions and the outstanding leg

has been divided into two areas. The attached area of the connected leg either to the

top or bottom chord has only one portion for single bolted angles (figure 3.3). For

double bolted target angles, the connected leg has been divided into six portions and

mk:@MSITStore:C:\Program%20Files\Ansys%20Inc\v100\CommonFiles\HELP\en-us\ansyshelp.chm::/Hlp_E_SHELL181.html#ai9bXn58ctg


the outstanding leg has been divided into four areas (figure 3.4). The attached area of

the connected leg either to the top or bottom chord has two equal portions of areas

for double bolted angles (figure 3.5). Figure 3.5 shows how a target angle is

connected to the bottom chord.

Figure 3.3 Area formation of target angle for single bolted specimen

Calculation of Member Forces

Experiment of Elgaaly et al(1991)

The method used to calculate member forces from strain readings involves direct

integration of the stress over the cross-sectional area and was developed to handle the

inelastic failures encountered for the specimens tested. Figure 3.8 shows a typical

strain diagram. The member force is computed as follows using numerical

integration:

where AE = portion of the cross section where Eε < FY; AP = portion of the cross

section where Eε > FY; and FY = the actual yield stress of the specimen. An

advantage of this method is that it easily allows the inclusion of residual stresses in

the analysis.

(3.1)


Figure 3.4 Area formation of target angle for double bolted specimen

Figure 3.5 Typical view of junction of a double bolted target angle with bottom

Chord


(a) Single bolted angle (b) Double bolted angle

Figure 3.6 Area formation of target angle (half of the specimen)

Figure 3.7(a) Typical view of calculating member force from element stresses along


mid cross-section (double bolted target angle)

Figure 3.7(b) Typical element divisions along the mid cross-section of target angle

for calculating element stresses to obtain the member force of the

corresponding angle

This is accomplished by combining the residual stress diagram with the measured

stress distribution, and using an elastic-plastic material model. The member force is

then obtained as follows:

where, AE = portion of the cross section where (Eε R )< FY; AP =portion of the

cross section where (Eε R ) > FY; and R = the residual stress.

Finite Element Analysis

In the present study, the member force of target angles have been calculated using

element stresses. Then using the universal equation:

AreaStressForce

the member force has been evaluated. For this purpose, at first, the angle member has

been divided into two equal divisions (figure 3.6 (a) and figure 3.6(b)). Then, taking

either the lower half or the upper half portion, an infinitesimal strip of a number of

elements have been chosen as target whose stresses are to be obtained (figure 3.7(a)).

(3.2)

(3.3)


Fig

ure

3.8

Typic

al s

trai

n d

istr

ibuti

on (

test

of

(Elg

aaly

et

al

(1991))


Finally, the member force has been calculated by directly integrating the

multiplication of individual element stress and corresponding element area as shown

in figure 3.7(b).

3.2.2 Material Properties

The materials for the elements have been taken as bilinear kinematic hardening

(BKIN). The option assumes that the total stress range is equal to twice the yield

stress, which is recommended for general small-strain use for materials that obey von

Mises yield criteria (which includes most metals).

Figure 3.9 Bilinear kinematic hardening (BKIN)

In the figure 3.9,

y = yield stress

y = strain corresponding to yield stress

E1 = modulus of elasticity up to yield point

E2 = modulus of elasticity after exceeding yield point

The Poisson’s ratio is taken as 0.25. The modulus of elasticity of the angle members

has been assumed 200 kN/mm2 (the modulus of elasticity of steel).


3.2.3 Loading and Boundary Conditions

3.2.3.1 Restraints

In case of the bottom horizontal truss member, both the starting leftmost node (where

the angle legs of the bottom horizontal truss member meet) is kept restrained in

horizontal (X) and vertical (Y)-directions (axes notation is mentioned in figure 3.1).

And the corresponding rightmost node is kept restrained in Y and Z- directions. But,

in case of the the top horizontal truss member, the corresponding nodes at the same

location, are restrained in Z direction only to keep the truss frame at its plane. The

other junction nodes of vertical angles with the horizontal members are kept

restrained in the Z-direction only. It has been done so as to facilitate or take the

advantage of the symmetry of the original truss box frame as in the test of Elgaaly et

al. The junction nodes of the leftmost and rightmost vertical angles with the bottom

horizontal truss member are kept restrained in the vertical (Y) direction only. In all

cases, the whole model is kept unrestrained against rotation. These options are

allowed to facilitate the non-linear static analysis of the system. The boundary

conditions for the present problem has been revealed in figure 3.10.

3.2.3.2 Loads

The load has been applied on the middle vertical angle member at its junction nodes

with the top chord rather than applying the load at only one node to allow the whole

structure systematically deform. In the present analysis, the load is applied

considering the ultimate load bearing capacity of the specific target angle member

for each case. And the load has been augmented and then subdivided equally into the

junction nodes to be applied on the truss structure (figure 3.11) which helps to

achieve the accuracy of results.

3.2.4 Meshing

Fine meshing will lead to better results at the expense of greater solution time.

Coarse meshing will result in lesser solution times but result accuracy may be

compromised. The balance is therefore to apply the mesh density for which the

solution accuracy is not lost but the computation time is not also that great.


Figure 3.10 Finte element model with loads and boundary conditions

Figure 3.11 Close-up mesh with loads and boundary conditions


An optimal solution is to use a fine mesh in areas of high stress gradient and a

coarser mesh in the remaining areas. Thus, in the finite element model of the present

study the target angle is discretized into finer mesh sizes considering the cross-

sectional dimensions of the target angle rather than the dimensions of other angle

members. The meshing of the remaining truss members has been done in such a

manner so that the overall mesh size for each member remains uniform.

3.2.5 Solution Method

A number of solution tools are available for the solution of nonlinear structural

problems. Complete investigation of the nonlinear behavior of structures must follow

the equilibrium path; identify and compute the singular points like limit or

bifurcation points, whose secondary branches in the equilibrium path must be

examined and followed. Several techniques to achieve the solution pattern on the

equilibrium path were presented in literature. Load controlled Newton-Raphson

method was the earliest method in this regard; but it fails near the limit point. To

overcome difficulties with limit points, displacement control techniques were

introduced. However for structural systems exhibiting snap-through or snap-back

Figure 3.12 Arc-Length Methodology


behavior these techniques lead to error. One way to overcome the errors is the arc-

length method which was first developed by Crisfield in 1991. It has become a

powerful tool to use with finite element formulation for complete analysis of the

load-deflection path. The method uses the explicit spherical iterations to maintain the

orthogonality between the arc-length radius and orthogonal directions. It is assumed

that all load magnitudes are controlled by a single scalar parameter (i.e., the total

load factor). As the displacement vectors and the scalar load factor are treated as

unknowns, the arc-length method itself is an automatic load step method. For

problems with sharp turns in the load-displacement curve or path dependent

materials, it is necessary to limit the arc-length radius using the initial arc-length

radius. During the solution, the arc-length method will vary the arc-length radius at

each arc-length substep according to the degree of nonlinearities that is involved. The

convergence of the arc-length method at a particular substep is shown in figure 3.12.

3.2.6 Mesh Sensitivity

Mesh Density is usually an important factor influencing both the accuracy and cost

of the numerical solution. Analyses to assess the effect of mesh density were

performed on a typical test angle having cross sectional dimensions: L

50.7550.754.83 (test 42 according to the test data of Elgaaly). It has been observed

that both for the target angle and for the other angles in the truss, 4 divisions across

the width of each angle is good enough to obtain an optimal solution with excepted

time limit. The same mesh density was used for each analysis.

3.3 TYPICAL ANALYSIS RESULTS

3.3.1 Typical Load-Deflection Behavior

The load-deflection relationship has been signed out as the best characterization of

the load carrying behavior of single steel angles subjected to eccentric axial loads.

During the finite element analysis of the truss, a load was imposed on the structure

subdiving it on each of the junction nodes of the middle vertical angle with the top

chord. Due to the nodal loads, each time target angle has undergone an axial

compressive force along with some axial shortening (figure 3.13). At different stages


of applying load, corresponding axial forces and axial shortenings have been

obtained. Typical axial load (P) versus lateral displacement ( ) curves obtained

from non-linear finite element analysis using this methodology are shown in figures

3.14 and 3.15, the first one for a double bolted angle (test 34) and the next one for a

single bolted angle (test 26) according to the test data of Elgaaly (1991).

Figure 3.13 General two-dimensional figure of the model

In the figure 3.13,

F = applied axial compressive load

R = support reactions

P = axial compressive load capacity of the target angle

Δ = axial displacement (shortening) due to applied compressive load

3.3.2 Deflected Shapes

Typical deflected shape of the finite element model has been shown in Figure 3.16.

The deflected shape has been exaggerated in Figure 3.17 for illustration process.

Figure 3.18 depicts the deflected model from top view.


Figure 3.14 Typical Load-Deflection Graph for Elgaaly test no.34

angle size L 45.57×45.575.00, double bolted

Figure 3.15 Typical Load-Deflection Graph for Elgaaly test no.26

angle size L 50.42×50.423.63, single bolted


Figure 3.16 Typical deflected shape of the model

Figure 3.17 Typical deflected shape of the model (close-up view)

Figure 3.18 Typical deflected shape of the model (top view)


3.4 VALIDATION OF FINITE ELEMENT MODEL

In the present study it has been assumed that finite element modeling can reasonably

simulate the behavior of the single steel angles subjected to eccentric compressive

load as a part of a truss structure. In his paper, Elgaaly described the load-deflection

characteristics of two single bolted angles (test 42 (angle size: L 50.7550.754.83),

test 26 (angle size: L 50.4250.423.63)) and two double bolted angles (test 34

(angle size: L 50.0650.063.38), test 9 (angle size: L 45.5745.575.00)). The

author has used finite element methods described earlier to recreate the load-

deflection characteristics of the same angle members in order to validate the finite

element analysis adopted in the present study. Both the results obtained from the test

of Elgaaly (1991) and from finite element analysis for each angle member have been

graphically represented in chapter 4 (figure 4.8 to figure 4.11) and then compared.

Figure 3.19 shows typical comparative load-deflection graph for test 34. It has been

observed that, the load-deflection curve obtained from test data of Elgaaly and from

data of the present study follow the same trend and the peak loads are also very close

to each other.

Figure 3.19 Load-deflection graph of test 34: angle size L 45.57×45.575.00

(double bolted)


The main distinguishing point is that in the present study, more data has been taken

to accurately obtain the actual load-deflection behavior of target angle members.


57

Chapter 4

RESULTS AND DISCUSSIONS

4.1 INTRODUCTION

The prime objective of this thesis is to simulate the truss system and the actual

conditions of the tests conducted by Elgaaly et al (1991) on non-slender single-

angle-compression members. These members (denoted as target angles) were tested

to failure as part of a three-dimensional truss. The experimental program of Elgaaly;

including the test specimens, test apparatus, instrumentation, and test procedure has

been described in detail in article no. 2.5. The truss system used for the test of

Elgaaly has been tried to model and analyze using finite element method in the

present study. The necessary input data, loads, boundary conditions, slight

modifications for the advantage of finite element modeling of the actual test set-up

etc. have been discussed concisely in chapter-3. In the present section, the results of

finite element analysis along with the ultimate compressive load capacity using

different code formulas and the test results of Elgaaly has been presented. All the

results are compared in values and graphically represented for the ease to justify the

results from various aspects and to make comments and suggestions for further

recommendations.

4.2 PRINCIPAL FEATURES OF PRESENT ANALYSIS

A total of 50 single angle specimens were tested as a part of a three-dimensional

truss by Elgaaly et al (1991) of which the failure loads have been obtained for 47

specimens. These are the angle members which have been reinvestigated by the

author using finite element analysis. The load capacities of same angles have also

been calculated using the AISC 2005 Specification, ASCE Standard 10-97:

Design of Latticed Steel Transmission Structures, British Standard 8100 (Lattice

towers and masts-Part 3), AASHTO Formula, CRC (Column Research Council)

Formula and IS-800-2007(Indian Standard).

Results and Discussions 63

There are 22 single-bolted and 25 double-bolted angle members having different

width-thickness ratio, slenderness ratio and separate distinguished yield stresses.

In calculating the member force of separate target angles using finite element

analysis, the slenderness ratio and yield stress have been kept as input as

mentioned in the test of Elgaaly et al (1991).

The failure load is determined from axial load versus displacement data that were

obtained from non-linear analysis using finite element method. The failure load is

taken as the maximum axial load, which occurs just prior to buckling of the

specimen. Large increases in displacement coupled with decreasing in axial load

are common for global flexural failures, thus giving a well-defined failure load.

For the purpose of comparison with experimental strength, the factors of safety

employed in different code provisions are taken as unity.

4.3 PRESENTATION OF RESULTS

4.3.1 Axial Force vs Lateral Displacement Response

Different failure modes with distinguished failure loads have been found for both

single bolted and double bolted specimens.. For the ease of discussion, 8 specimens

(4 single bolted and 4 double bolted) are chosen by the author as the representative of

47 specimens to describe the salient features of the buckling analysis of the target

angles. The single bolted target angles are designated by the test number: 53, 35, 31,

and 42 and the double bolted target angles are designated by test number: 1, 34, 20

and 18 according to the test of Elgaaly (1991).

The geometric properties as well as loading conditions etc of the reference specimens

aforementioned are listed in the table 4.1(a) and table 4.1(b). The figures 4.1 and 4.2

show the typical pattern of the load vs displacement graphs, where response is

observed to be linear until failure. As expected, all samples failed due to buckling of

the connected leg of the target angle. The failure mode was global flexural torsional

(FT) mode without local buckling of the angle leg which is similar to the failure

mode of specimen 24 as described by Elgaaly (figure 4.3). As observed, all the load-


deflection graphs show the same trend. Once the pick load reaches, after that point

force eventually diminishes with further increase of deflection.

Table 4.1(a): Properties of Reference Specimens (single bolted)

test

no.

width,

w

(mm)

thickness,

t

(mm)

w/t slenderness

ratio, l/r

yield

stress,

FY

(kN/mm2)

Failure load (kN)

Elgaaly

Test

Present

Analysis

42 50.75 4.83 10.52 81 317.9 80.51 83

35 44.93 5.13 8.76 93 339.9 75.44 77.71

31 50.39 5.08 9.92 81 339.2 85.98 91.41

53 44.42 3.38 13.15 92 353 48.04 47.42

Figure 4.1 Typical axial force vs lateral displacement graph for single bolted angles

based on Finite Element Analysis

From figure 4.1, it has been demonstrated that the peak loads are different for the

target angles. The reason of the variations in results may be due to different width-

thickness ratio, slenderness ratio, yield stress etc of the target angles. For example,

let us consider test 53 and test 31. The specimen of test 31 has larger cross-sectional

width and thickness but lower w/t ratio (9.92) and lesser value of yield stress (339.2

kN/mm2) as compared to the specimen of test 53. It has been observed that in case of

both Finite Element Study and test of Elgaaly, with the increase of w/t ratio, l/r ratio


and yield stress, the ultimate load capacity of angle sections decrease. As expected,

for all the target angles, the load capacities of angles increase with the decrease of l/r

ratio. With the increase of w/t ratio, load capacity increases for the specimens having

same yield stress (test 31 and test 35). For the specimens having same l/r ratio, load

capacity decrease with the increase of w/t ratio and decrease of yield stress (test 42

and 31).

Table 4.1(b): Properties of Reference Specimens (double bolted)

test

no.

width,

w

(mm)

thickness,

t

(mm)

w/t slenderness

ratio, l/r

yield

stress, FY

(kN/mm2)

Failure load (kN)

Elgaaly

Test

Present

Analysis

18 63.17 5.05 12.50 67 315.2 112.7 114.2

20 50.6 5.08 9.96 86 326.9 97.5 92.1

34 45.57 5 9.11 99 342.8 80.2 80.9

1 43.97 3.53 12.45 98 344.1 49.2 53.7

Figure 4.2 Typical axial force vs lateral displacement graph for double bolted angles

based on Finite Element Analysis

From the illustration of figure 4.2, different peak loads for the reference specimens

are seen like single bolted specimens. In all cases, with the increase of slenderness

ratio, load capacity of angle sections decrease. Let us consider, test 18 and test 20. In


this case, test 18 has higher width-thickness ratio and smaller slenderness ratio as

well as yield stress as compared to test 20. It has been observed that in case of both

Finite Element Study and test of Elgaaly, the load capacities are higher for test 18

(with the increase of w/t ratio and decrease of l/r ratio as well as yield stress, the

ultimate load capacity of angle sections increase). The behavior is same for most of

the cases when any two specimens are compared. With the decrease of w/t ratio, l/r

ratio and Fy, failure load capacity increases (test 20 and test1).

Figure 4.3: FT Failure mode of test no: 24 (Elgaaly et al 1991)

4.3.2 Deformation Characteristics of the Target Angle

4.3.2.1 General

The peak load is the indicator which shows that from this point buckling of the

structure initiates especially of target angle, as the other truss members except the

target angle is designed in such a way so that the buckling starts within the target

angle at first and eventually the failure of the target angle occurs without any

significant deformation in the rest of the truss.


Let us consider the case of test 53 of Elgaaly (1991).The specimen has width-

thickness ratio is equal to 13.15 with slenderness ratio of 92.0 (the highest ratio of all

the groups of single bolted target angles). From finite element analysis the obtained

failure load is 47.42 kN whereas compressive load carrying capacity from the test of

Elgaaly is 48.04 kN and from ASCE Manual the calculated load capacity is 50.66

kN. So, it has been observed that finite element analysis reasonably estimates the test

load whereas ASCE Manual overestimates the load capacity to some extent. The

finite element analysis gives the failure patterns represented below:

4.3.2.2 Initial stage

Just when the load reaches the pick, no significant deformation is observed initially.

But gradually when the load tends to decrease and reaches a small but considerable

percentage of the pick load value, some extent of deformation occurs. In this stage,

the deflection initiates with the bending of the connected leg of the target angle. The

rest of the truss members are in the position where they were (figures 4.4, 4.5, and

4.6).

Figure 4.4 Deflection pattern of the truss frame at the early stage of buckling (front

view)

Figure 4.5 Deflection pattern of the truss frame at the early stage of buckling (top

view (close-up))


Figure 4.6 Deflection pattern of the truss frame at the early stage of buckling (front

view (close-up))

4.3.2.3 Final stage

When the load value eventually diminishes and comes to the final diminishing point,

the deflection is associated with the bending of the connected leg along with the

twisting of the unconnected leg of the target angle. Additionally the unconnected leg

of the top horizontal member also faces twisting.

Figure 4.7 Deflection pattern of the truss frame at the final stage of buckling (front

view)

Figure 4.8 Deflection pattern of the truss frame at the final stage of buckling (top

view (closeup))


The lower middle half portion of the target angle faces severe bending stress

specially the lowermost connected region of the target angle. The middle vertical

angle and the corresponding junction have displaced downwards from their original

position. The deflected shapes of the target angle can be easily realized from figure

4.7 and figure 4.8.

4.3.3 Comparative Results

4.3.3.1 Typical comparative load vs deflection graphs

In his paper, Elgaaly described the member force vs displacement graphs of 2 single

bolted (test 42 and test 26) and 2 double bolted (test 34 and test 9) target angles. The

same specimens are analyzed by means of finite element method. It has been

observed from the comparative figures 4.9 to 4.12 that results from test of Elgaaly

and from Finite Element Method are relatively close for all the specimens except for

specimen of test 26 (the reason may be the higher w/t ratio of the specimen, which is

13.88 and it is noteworthy that test 26 is included in group 8, which have single

bolted specimens having higher w/t ratios as compared to the other groups as

mentioned by Elgaaly). The observed deviations may be due to the fact that during

modeling the truss system, bolted connection is simply replaced by modeling the

connecting portions as the integral parts as the component angle members. So in the

finite element model considered here, no stress concentration has occurred, moreover

in modeling, the entire truss system has same stiffness and the restraints are almost

fixed in the majority of the junction points. But these are not the actual conditions

while the test was performed by Elgaaly (1991) on the same angle members. So,

some overestimation occurs for some of the angles. Overall, finite element analysis

gives more logical results for double bolted angles than for single bolted specimens

as the prevailing methodology of modeling the truss frame in finite element is more

compatible to the restraint conditions of double bolted conditions (more fixity in

double bolted than single bolted connections).


Figure 4.9 Load-deflection graph of angle size L 45.5745.575.00, double bolted

(test 34)

Figure 4.10 Load-deflection graph of angle size L 50.7550.754.83, single bolted

(test 42)


Figure 4.11 Load-deflection graph of angle size L 50.0650.063.38, double bolted

(test 9)

Figure 4.12 Load-deflection graph of angle size L 50.4250.423.63, single bolted

(test 26)


4.3.3.2 Correlation between test and present study

For both double bolted and single bolted target angles, the failure loads obtained

from finite element analysis as well as test of Elgaaly and geometric properties of the

specimens are represented in table 4.2 (a) and table 4.2 (b). Correlation between test

data of Elgaaly and present analysis are studied for single and double bolted angles

(shown in figure 4.13 and figure 4.14 respectively).

Table 4.2(a): Results for single bolted angles

Test

no.

w

(mm)

t

(mm)

w/t l/r yield

stress, FY

(kN/mm2)

Failure load (kN)

Elgaaly

test

Present

analysis

P

factor

P

factor

(%)

53 44.42 3.38 13.15 92 353 48.04 47.42 0.99 99

54 44.37 3.35 13.23 92 341.3 44.3 46.17 1.04 104

55 44.55 3.45 12.90 92 353 44.79 48.93 1.09 109

56 44.42 3.43 12.96 92 351.6 46.35 48.31 1.04 104

57 44.48 3.45 12.87 92 359.9 43.99 49.28 1.12 112

24 45.42 4.95 9.17 93 339.2 67.43 75.26 1.12 112

35 44.93 5.13 8.76 93 339.9 75.44 77.71 1.03 103

36 45.16 5.26 8.59 93 343.4 77.8 80.73 1.04 104

37 45.11 4.83 9.35 93 347.5 60 73.66 1.23 123

26 50.42 3.63 13.88 80 342 45.64 61.12 1.34 134

27 50.19 3.51 14.32 80 331.6 38.88 57.16 1.47 147

28 50.83 3.68 13.80 80 356.5 42.39 64.05 1.51 151

38 49.96 3.53 14.15 80 350.3 49.86 59.25 1.19 119

31 50.39 5.08 9.92 81 339.2 85.98 91.41 1.06 106

40 50.75 4.98 10.19 81 322.7 71.08 86.91 1.22 122

41 50.67 5 10.13 81 317.2 81.04 86.25 1.06 106

42 50.75 4.83 10.52 81 317.9 80.51 83 1.03 103

45 62.99 5.13 12.28 65 326.8 87.36 117.03 1.34 134

46 63.53 5.16 12.32 65 331.6 86.69 119.74 1.38 138

47 63.63 5 12.72 65 331.6 89.81 115.64 1.29 129

48 63.45 4.93 12.88 65 328.9 93.72 112.53 1.20 120

49 63.07 5.03 12.54 65 343.4 88.69 118.54 1.34 134

Mean=78.19kN; Standard deviation=25.52kN (present analysis)

Correlation coefficient

Linear regression (trendline) which have been plotted in figure 4.13 and figure 4.14

indicate that:

Co-efficient of determination, R2=0.856 (for single bolted specimens)

0.879 (for double bolted specimens)


Figure 4.13 Correlation between test and present study for single bolted target

angles

Figure 4.14 Correlation between test and present study for double bolted target

angles

And correlation co-efficient, R= 925.0856.0 (single bolted)

= 0.938(double bolted)


Generally, the correlation coefficient, R, ranges from -1 to +1. R equal to 1.0

indicates a perfect correlation. Therefore, the R values obtained for both single and

double bolted angles indicate nearly perfect correlations.

Table 4.2(b): Results for double bolted angles

Test

no.

w

(mm)

t

(mm)

w/t l/r yield

stress, FY

(kN/mm2)

Failure load (kN)

Elgaal

y test

Present

analysis

P

factor

P

factor

(%)

1 43.97 3.53 12.45 98 344.1 49.2 53.7 1.09 109

2 43.69 3.58 12.20 98 363.4 66.1 56 0.85 85

3 43.99 3.45 12.74 98 341.4 63.6 52.1 0.82 82

4 44.02 3.56 12.38 98 344.8 61.7 54.3 0.88 88

5 44.07 3.43 12.85 98 340.7 58.2 51.6 0.89 89

6 44.75 5.05 8.85 99 328.3 98.2 78.7 0.8 80

7 44.88 4.93 9.11 99 335.9 97 77.9 0.8 80

8 44.88 5.03 8.92 99 329 93.5 78.7 0.84 84

33 44.91 4.98 9.02 99 353.1 85.5 81.1 0.95 95

34 45.57 5 9.11 99 342.8 80.2 80.9 1.01 101

9 50.06 3.38 14.82 85 324.1 51.4 56.6 1.1 110

10 50.11 3.33 15.06 85 320 48.8 55.2 1.13 113

11 50.14 3.43 14.62 85 324.8 64.5 57.9 0.9 90

12 51.21 3.33 15.39 85 342.8 52.8 57.7 1.09 109

13 49.96 3.38 14.79 85 329.7 64.5 57.1 0.89 89

20 50.6 5.08 9.96 86 326.9 97.5 92.1 0.94 94

21 50.47 5.13 9.84 86 315.9 88.3 90.8 1.03 103

22 50.42 4.95 10.18 86 328.3 94.6 89.7 0.95 95

43 50.83 5.08 10.00 86 315.9 84.9 90.3 1.06 106

44 51 5.03 10.17 86 313.8 82.1 89.1 1.09 109

18 63.17 5.05 12.50 67 315.2 112.7 114.2 1.01 101

19 63.07 5.05 12.48 67 327.6 110.1 117.3 1.07 107

50 63.6 5.16 12.33 67 336.6 117.9 122.9 1.04 104

51 63.7 5.08 12.54 67 327.6 111 118.7 1.07 107

52 63.8 5.31 12.02 67 329.7 116.1 124.8 1.07 107

Mean=79.98kN; Standard deviation=24.82kN (present analysis)

Capacity factor (P factor)

The capacity factor (P-factor) is the ratio of angle capacity obtained from present

analysis to the actual test capacity. Evaluating the P-factor is a way to show how

close the actual angle capacity is to the angle capacity obtained from finite element

analysis. It has been observed that for single bolted specimens, all values of P-factors

except for test 53 is greater than 1.00 which indicates that finite element analysis


gives reasonably close results to the test with a slight overestimation. In case of

double bolted angles, P-factors for some of the angles are less than 1.00 indicating

underestimation of corresponding test results.

Standard deviation

Standard deviation is a widely used measure of variability or diversity which shows

how much variation or dispersion exists from the average (mean, or expected value).

The standard deviation is the root mean square (RMS) deviation of the values from

their arithmetic mean. Generally, a low standard deviation indicates that the data

points tend to be very close to the mean, whereas high standard deviation indicates

that the data points are spread out over a large range of values. In case of single

bolted angles, the arithmetic mean of the results of finite element study is equal to

78.19 with a standard deviation of 25.57. In case of double bolted angles, the

arithmetic mean is 79.98 and the corresponding standard deviation is 24.82. Both the

values of standard deviations are much lower than corresponding means indicating

that most of the data points are very close to the mean values. Moreover, it also

means that most of the data (about 68%, assuming a normal distribution) have a

failure load within one standard deviation of the mean and almost all the data (about

95%) have a failure load within two standard deviations of the mean.

4.3.3.3 Comparison with codes and discussions

Comparative results of single bolted angles

The bar charts are presented in figure 4.15 to 4.22 for the ease of discussion of

results of buckling loads of single and double bolted angles using various renowned

code provisions, test performed by Elgaaly et al and finite element analysis. In case

of single and double bolted target angles, the test result, finite element output and

result evaluated from ASCE code show relatively close values. The figures for single

bolted angles depict that finite element outputs slightly overestimates the load

carrying capacity of angles found from test.

86

87

http://en.wikipedia.org/wiki/Statistical_dispersion

http://en.wikipedia.org/wiki/Mean

http://en.wikipedia.org/wiki/Mean

http://en.wikipedia.org/wiki/Normal_distribution


Figure 4.15 Comparison of buckling load for Elgaaly test no.53

angle size L 44.4244.423.38, single bolted




Figure. 4.17 Comparison of buckling load for Elgaaly test no.31





The figures for double bolted angles also show the same trend for the majority of

cases. Of the other codes, Indian standard code gives relatively close results to test,

FEM and ASCE in case of double bolted angles, whereas underestimates the load

capacities of single bolted angles. The AASHTO and CRC Formula give the same

result. The British standard code and the AISC Formula underestimate the test results

at a large extent. The difference in results obtained using code formulas may be due

to some reasons- different codes have different formulas for effective slenderness

ratio (KL/r). Secondly, there are some codes (BS Code, IS Code, ASCE standard 10-

97 etc) having built in factor of safeties which are the integral part of the formulas

and very difficult to identify them and to exclude them in evaluating the ultimate

load capacity of angle sections.

Comparative results of double bolted angles


angle size L 43.9743.973.53, double bolted



angle size L 63.17 63.175.05, double bolted

4.4 INTERPRETATION AND EXPLANATION OF

OBSERVATIONS For convenience and also for comparing the buckling loads a variable n, which is

defined as the ratio of the failure load divided by the uniform yield capacity of the

section (yield stress multiplied by the cross-sectional area), has been calculated. This

facilitates the accounting for the effect of the variations in area and yield stress

among the test specimens. Moreover % difference in n values of both single bolted

and double bolted angle specimens have been calculated. Table 4.4 pairs groups by

size and lists the percent difference in n values between the corresponding groups.

For better comparison, the average n values for both single and double bolted target

angles have been summarized in figure 4.20 and figure 4.21 respectively.


Table 4.4 Comparison of single bolted versus double bolted angles

Group

(1)

Angle size

(2)

l/r

(3)

End

condition

s

(4)

Elgaaly test Present analysis

Average

n

(5)

Percent

differen

ce in n

(6)

Average

n

(7)

Percent

differen

ce in n

(8)

1 44.45x44.45x3.18 98 Double 0.582 31 0.521 11

6 44.45x44.45x3.18 92 Single 0.444 0.468

2 44.45x44.45x4.76 99 Double 0.636 34 0.554 6

7 44.45x44.45x4.76 93 Single 0.476 0.522

3 50.8x50.8x3.18 85 Double 0.525 43 0.530 6

8 50.8x50.8x3.18 80 Single 0.368 0.502

4 50.8x50.8x4.76 86 Double 0.575 12 0.581 4

9 50.8x50.8x4.76 81 Single 0.514 0.560

5 63.5x63.5x4.76 67 Double 0.556 27 0.585 2

10 63.5x63.5x4.76 65 Single 0.438 0.572

Single bolted versus double bolted specimens

Figure 4.20 Comparison of buckling load factor, n

for single bolted angles


Figure 4.21 Comparison of buckling load factor, n

for double bolted angles

From Tables 4.4, it is seen that the double-bolted specimens are stronger than the

single-bolted specimens. From test, the average n value for the double-bolted

specimens is 0.575, which is 28% higher than that of the single-bolted specimens and

from finite element analysis, the average n value for the double-bolted specimens is

0.554, which is 5.5% higher than that of the single-bolted specimens. Moreover,

from finite element analysis, % difference in average n values for the double-bolted

specimens is 3.79, which is within the expected accuracy limit. But, for the single

bolted angles, the % difference is 17.14 which is above the accuracy limit. In

general, as observed, failure mechanism and load versus displacement characteristics

vary depending on:

-Effect of b/t ratio

-Effect of end restraints

-Effect of slenderness ratio

Effect of b/t ratio

It has been observed that in case of single bolted specimens, target angles of group-6,

as the width-thickness ratio decreases, the failure load evaluated from finite element

analysis increases proportionally. For higher b/t ratios, out of plane buckling occurs

as in the case of single bolted specimen number 53.


Effect of end restraints

The difference in strength between the corresponding groups with double and single

bolted connections is primarily due to the difference in end restraint conditions. In

the test of Elgaaly, the largest difference in n values occurs between groups 3 and 8

(43%). On the other hand, from the finite element analysis, the largest difference in n

values occurs between groups 1 and 6 (11%). This difference is mostly attributed to

significantly higher stress concentrations in the single-bolted connections as

compared with the double-bolted connections. The smallest difference in strength is

between groups 4 and 9 (12%) from test results, whereas, the smallest difference in n

values occurs between groups 5 and 10 (2%) from finite element study. According to

the test results of Elgaaly, the failure modes for both of these groups are the same,

with the dominant effect being global flexural buckling, which tends to emphasize

the importance of the difference in rotational end restraint as opposed to local leg

crippling.

Effect of slenderness ratio

It is expected that with the increase of slenderness ratio, axial load carrying capacity

of single steel angles decrease. But this is the case with a concentrically loaded

structure. But in case of eccentrically loaded structures, with the increase of

slenderness ratio, failure load does not decrease, rather it increases. This is the case

with the test specimens of Elgaaly. Both the test results and the results from finite

element analysis exhibit same behavior.

Relationship between Slenderness and Strength

Another area that requires explanation is the difference in failure loads between

groups of different sizes and similar end conditions. It is intuitively expected that

column strength increases with decreasing L/r ratios, and n approaches unity as L/r

approaches a limiting value close to zero. However, this is the case only for

concentrically loaded struts, which do not exhibit local failures or torsional effects.

All of the specimens tested were loaded eccentrically, and most exhibited significant

local and torsional deformations. As a result, n does not necessarily increase with

decreasing L/r values. This is true for both the single and double bolted specimens


for the results obatained from test of Elgaaly et al. For example, group 2 has an L/r

ratio of about 99 and double-bolted ends, and group 4 has an L/r ratio of 86 and

double-bolted ends as well, yet the average n value for group 4 is 12% lower than

that of group 2. One reason for the difference is the presence of local-torsional

effects in group 4 (b/t is about 10), which do not occur in group 2 (b/t is about 9).

Further, since all of the angles were fabricated with the bolt holes centered on the

connected legs, the load eccentricity was slightly greater for group 4 (b = 2 in.) than

for group 2 (b = 1.75 in.). This same reasoning applies when comparing any two

groups with similar end conditions that indicate decreasing n values with decreasing

L/r values. But, the same groups (group 2 and group 4) exhibit increase of average n

value for the decrease of L/r ratio observed from finite element analysis. In this case,

n value for group 4 is about 4.9% higher than that for group 2. It is of interest to note

that groups 7 and 9, which are the single-bolted counterparts of groups 2 and 4,

exhibit increasing n values with decreasing L/r values from test results of Elgaaly as

is intuitively expected. This is because the single-bolt connections cause identical

flexural-torsional failure in both groups 7 and 9, whereas the predominant failure

mode in group 2 is different from that of group 4. Finite element study also shows

the same behavior for groups 7 and 9, where n value for group 9 is about 7.3% higher

than that for group 7. The average n values for angles of all the groups exhibit same

behavior as in the test of Elgaaly, except for group 5 and group 10.

4.5 REMARKS

The results of present study are generally in well agreement with those obtained from

test of Elgaaly (1991). Hence, hopefully finite element studies may be satisfactorily

conducted for predicting the ultimate load capacity of steel angles for designing safe

tower structures. Among the codes, only the ASCE Standard 10-97 evaluates the

angle capacities with reasonable accuracy which concurs with test results as well as

with the finite element study. Therefore, design criteria of ASCE Standard 10-97

may be followed for calculating design load capacity of single angles.

Chapter 5

CONCLUSIONS

5.1 GENERAL

The behavior of a single angle compression member is complicated by the details of its

connection to the rest of the structure, and by the behavior of the members connected to

it. Connections are usually made to one leg of the angle so that the angle is loaded

eccentrically in a plane that lies between its principal planes. Different types of end

connection provide different types of restraint both in and out of the plane of the

connected leg. The thesis originated with the aim to validate the eccentric compressive

load carrying capacity of a single steel angle (designated as target angle; either single

bolted or double bolted) as part of a three-dimensional truss tested by Elgaaly et al

(1991). The entire system has been modeled and analyzed using Finite Element Method.

It has been proved that Finite Element Analysis can simulate the truss system and its

connection, load as well as overall practical conditions satisfactorily. And the failure

loads of the target angles obtained from finite element analysis are fairly close to the test

results obtained by Elgaaly. Therefore, the finite element model can accurately predict

the strength and behavior of single steel angles as part of a truss system.

5.2 FINDINGS

Based on the study, the following conclusions can be arrived at:

The response of steel single angles subjected to axial eccentric loading is

investigated by means of numerical modeling based on finite element techniques.

Results show that Finite Element Analysis can simulate the practical test

conditions satisfactorily and analysis results closely match with past experimental

records. So finite element studies may be an easy alternative of physical testing of

Conclusions 86

single angle structures and can be used for routine design of steel angles which will

be helpful to find out better solutions for engineers.

The load carrying capacity of the same angles has been evaluated by means of

internationally adopted standard codes for a better understanding and comparison

of results. Among the codes, The ASCE Standard 10-97 gives relatively

satisfactory results as compared to test of Elgaaly and finite element method. The

rest of the codes either overestimate or underestimate the compressive load

capacity of angle members.

It is therefore recommended that in the design of three-dimensional lattice towers

the provisions of ASCE Standard 10-97 should be followed.

5.3 SCOPE FOR FUTURE INVESTIGATION

The following recommendations may be suggested for future research work:

The present study analyzed the load capacity of some equal leg single steel angles

which were tested by Elgaaly et al (1991). The accuracy of the finite element

model considered in the study may be verified for unequal leg angle sections tested

in similar past experiments.

The angle sections investigated in this project are in the range of slenderness ratio

less than critical value i.e. all the angles are non-slender. Therefore, study can be

extended for the slender angle members.

Column curves may be established for steel angle sections which can be adopted

for routine design or may be used to calibrate special code clauses.

Finite Element study can be conducted for the ultimate capacity of bracing angle

members used in towers and trusses.

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APPENDIX A

Results for single bolted angles

Test

no.

width, w

(mm)

thickness, t

(mm)

slenderness

ratio,

l/r

yield stress,

FY

(kN/mm2)

Failure load (kN)

Elgaaly

test

Present

analysis

AISC ASCE BS AASHTO CRC IS

53 44.42 3.38 92 353 48.04 47.42 23.13 50.66 30.07 63.38 63.34 36.43

54 44.37 3.35 92 341.3 44.3 46.17 22.91 47.64 29.31 61.96 61.92 35.36

55 44.55 3.45 92 353 44.79 48.93 23.66 51.91 30.82 64.9 64.9 37.67

56 44.42 3.43 92 351.6 46.35 48.31 23.44 51.42 30.47 64.14 64.14 37.14

57 44.48 3.45 92 359.9 43.99 49.28 23.62 51.82 31.09 65.25 65.25 38.03

24 45.42 4.95 93 339.2 67.43 75.26 33.45 73.93 42.92 90.61 90.61 59.29

35 44.93 5.13 93 339.9 75.44 77.71 34.21 75.57 43.95 92.7 92.7 61.38

36 45.16 5.26 93 343.4 77.8 80.73 35.18 77.75 45.46 95.72 95.77 63.87

37 45.11 4.83 93 347.5 60 73.66 32.43 71.66 42.12 88.65 88.65 58.09

26 50.42 3.63 80 342 45.64 61.12 35.05 62.18 40.83 87.23 87.23 44.75

27 50.19 3.51 80 331.6 38.88 57.16 33.72 58.45 37.99 82.33 82.33 41.59

28 50.83 3.68 80 356.5 42.39 64.05 35.81 63.3 42.12 91.45 91.41 46.97

38 49.96 3.53 80 350.3 49.86 59.25 33.76 58.58 38.65 85.36 85.31 43.28

31 50.39 5.08 81 339.2 85.98 91.41 47.37 93.27 55.42 118.45 118.41 70.14

40 50.75 4.98 81 322.7 71.08 86.91 46.79 91.01 53.15 113.42 113.42 66.45

41 50.67 5 81 317.2 81.04 86.25 46.97 90.78 52.8 112.58 112.58 66.05

42 50.75 4.83 81 317.9 80.51 83 45.46 87.94 51.15 109.11 109.11 63.25

45 62.99 5.13 65 326.8 87.36 117.03 73.53 130.82 81.26 167.2 167.2 85.76

46 63.53 5.16 65 331.6 86.69 119.74 74.55 133.44 83.18 171.43 171.43 87.63

47 63.63 5 65 331.6 89.81 115.64 72.55 128.5 80.95 166.8 166.84 84.02

48 63.45 4.93 65 328.9 93.72 112.53 71.3 124.99 79.09 162.84 162.84 81.62

49 63.07 5.03 65 343.4 88.69 118.54 72.24 129.66 82.47 170.67 170.67 86.16

91

Results for double bolted angles

Test

no.

width, w

(mm)

thickness, t

(mm)

slenderness

ratio, l/r

yield

stress, FY

(kN/mm2)

Failure load (kN)

Elgaaly test Present

analysis

AISC ASCE BS AASHTO CRC IS

1 43.97 3.53 98 344.1 49.2 53.7 21.6 49.5 36 59.6 59.6 47

2 43.69 3.58 98 363.4 66.1 56 21.7 49.8 37.3 60.8 60.8 49

3 43.99 3.45 98 341.4 63.6 52.1 21.2 48.5 35.2 58.3 58.3 45

4 44.02 3.56 98 344.8 61.7 54.3 21.8 49.9 36.4 60.1 60.1 48

5 44.07 3.43 98 340.7 58.2 51.6 21 48.2 35 57.9 57.9 45

6 44.75 5.05 99 328.3 98.2 78.7 30.4 70.2 49.9 83 83 74

7 44.88 4.93 99 335.9 97 77.9 29.8 68.8 49.5 81.8 81.8 73

8 44.88 5.03 99 329 93.5 78.7 30.4 70.1 49.9 82.9 82.9 74

33 44.91 4.98 99 353.1 85.5 81.1 30.1 69.5 51.2 83.8 83.7 76

34 45.57 5 99 342.8 80.2 80.9 30.7 70.9 51.5 84.8 84.8 76

9 50.06 3.38 85 324.1 51.4 56.6 29.6 52.7 41.9 74.5 74.5 48

10 50.11 3.33 85 320 48.8 55.2 29.2 51.3 40.5 72.9 73 47

11 50.14 3.43 85 324.8 64.5 57.9 30.1 54.2 43.1 75.8 75.8 50

12 51.21 3.33 85 342.8 52.8 57.7 29.8 50.3 40.4 77.5 77.5 49

13 49.96 3.38 85 329.7 64.5 57.1 29.1 52.5 41.9 75.1 75 49

20 50.6 5.08 86 326.9 97.5 92.1 43.4 89.5 64.7 110.7 110.7 88

21 50.47 5.13 86 315.9 88.3 90.8 43.7 89.3 63.9 109.3 109.3 87

22 50.42 4.95 86 328.3 94.6 89.7 42.2 87.1 63.1 107.9 107.9 85

43 50.83 5.08 86 315.9 84.9 90.3 43.6 89.1 63.7 109 109.1 87

44 51 5.03 86 313.8 82.1 89.1 43.3 88.4 63.2 108 108 85

18 63.17 5.05 67 315.2 112.7 114.2 70.9 125.7 96 158.5 158.5 107

19 63.07 5.05 67 327.6 110.1 117.3 70.8 127.7 98.4 163.2 163.1 110

50 63.6 5.16 67 336.6 117.9 122.9 72.8 132.8 103 171.3 171.2 115

51 63.7 5.08 67 327.6 111 118.7 71.9 129.7 99.9 165.6 165.6 111

52 63.8 5.31 67 329.7 116.1 124.8 75.1 135.8 104.8 173.9 173.8 119

92

93

APPENDIX B

Ultimate Capacity of Angles Using Code Formulas

Problem Statement 1: Double bolted specimen (Test no. 20 as per the experiment of

Elgaaly et al (1991))

Angle size: width, d 992.1 inch, thickness, t 200.0 inch

86r

l,

2mm

kN200E , ksi4.47yF


For, 80xr

L,

5.1398625.13225.132 xr

L

r

KL

Now,

51.116895.64.47

1020071.471.4

3

yF

E


E

r

KL71.4 ,

So,

43.1015.139

102002

32

2

2

r

KL

EFe MPa

Now, 80.143895.64.4744.044.0 yF MPa


95.8843.101877.0 crF MPa

Therefore, kN43.4326.48895.88 gcrn AFP

94

Or, 99.8nP kips


2596.8

200.0

200.0992.1

t

w

62.114.47

18080

lim

yFt

w 1 (for Fy in ksi)

Check: lim

t

w

t

w

907.109895.64.47

1020022 3

y

cF

EC


panel,

103865.0605.060 r

L

r

KL

cCr

KL , for which,

y

c

a FC

r

KL

F

2

5.01

895.64.47907.109

1035.01

2

2mm

kN29.183

So, gan AFP

26.48829.183

Or, 49.89nP kN 12.20 kips

95

BS Standard Code:

m

rAjN

Where,

j=1.0

26.488200.0200.0984.12222 ttdA mm

2

222

1

effΛ

222

2427.14495.14495.1

1

5684.0

Here,

4495.12427.12.02427.134.015.02.015.0 22 effeff ΛΛa

and 2427.1093.1137.1 KΛΛeff

Now,

ΛK


1

58.07.0

158.07.0

=1.23

86

895.64.47

205000

58.07.0

5.0

96

Now,

96.9200.0

992.1

t

B

2.14895.64.47

205000567.0567.0

y

E

Now, for t

B,

823.326895.64.47 yr MPa


Therefore,

0.1

823.326257.4885684.00.1 N

644.90701 N/mm2

39.20 kips

AASHTO Formula:

Allowable stress,

E

Fr

L

SF

FF

ye

y

a 2

2

41

. (for c

e Cr

L )

2050004

895.64.478675.01

0.1

895.64.472

2

89.24 kips

Though, AASHTO Code, factor of safety, F.S=2.12, but in the present study, it has been

assumed 1.0.

IS Code Formula:

875.0895.64.47

2502505.05.0

yf

97

106.1

250

10200875.0

86

250

322

E

r

l

vv

vv

1281.0

250

10200875.008.52

25968.50

250

2322

21

E

t

bb

9779.01281.020106.135.020.0 222

3

2

21 kkk vveq

1687.19779.02.09779.049.015.02.015.0 22 eqeq

553.0

9779.01687.11687.1

115.0225.022

eq

So, 697.180895.64.47553.0 mo

y

cd

ff

MPa

23.88697.18026.488 dP kN 84.19 kips

CRC Formula:

2

32

2

286

102004

895.64.471895.64.47])(

41[

r

L

E

ey

ycr

= 226.668 MPa

kips88.242571.488668.226 xAPgcrn

Problem Statement 2: Single bolted specimen (Test no. 53 as per the experiment of

Elgaaly et al)

Angle size: width, d=1.749 inch, thickness, t=0.133 inch

92r

l,

2mm

kN200E , ksi2.51yF

98


For, 80xr

L,

0.1479225.13225.132 xr

L

r

KL

Now,

11.112895.62.51

1020071.471.4

3

yF

E


E

r

KL71.4 ,

So,

35.910.147

102002

32

2

2

r

KL

EFe MPa

Now, 895.62.5144.044.0 yF MPa 33.155 MPa


11.8035.91877.0 crF MPa

So, 13.2374.28811.80 gcrn AFP kN

Or, 20.5nP kips


2515.12

133.0

133.0749.1

t

w

18.112.51

18080

lim

yFt

w 1 (for Fy in ksi)

13.202.51

1144144

yF

99

Check:

yFt

w

t

w 144

lim

, for which

19.482.5118.11

15.12677.0677.1677.0677.1

lim

ycr F

t

w

t

w

F ksi

00.109895.619.48

1020022 3

cr

cF

EC


panel,

106925.0605.060 r

L

r

KL

cCr

KL , for which,

cr

c

a FC

r

KL

F

2

5.01

895.619.4800.109

1065.01

2

= 2mm

kN15.175

So, 57.507381.28815.175 gan AFP kN

Or, 37.11nP kips

BS Standard Code:

m

rAjN

100

Where,

j=0.8

74.288133.0133.0749.12222 ttdA mm

2

222

1

effΛ

222

43.1733.1733.1

1

3688.0

Here,

733.143.12.043.134.015.02.015.0 22 effeff ΛΛa and

43.1215.1177.1 KΛΛeff

ΛK


1

58.07.0

158.07.0

92

895.62.51

205000

58.07.0

5.0

=1.177

Now,

15.13133.0

749.1

t

B

66.13895.62.51

205000567.0567.0

y

E

101

Now, for t

B,

024.353895.62.51 yr MPa


Therefore,

0.1

024.35374.2883688.08.0 N

86.30073 N

76.6 kips

AASHTO Formula:

Allowable stress,

E

Fr

L

SF

FF

ye

y

a 2

2

41

. (for c

e Cr

L )

2000004

895.62.51921

0.1

895.62.512

2

24.14 kips

Though, in AASHTO Code, factor of safety, F.S=2.12, but in the present study, it has

been assumed 1.0.

IS Code Formula:

842.0895.62.51

2502505.05.0

yf

231.1

250

10200842.0

92

250

322

E

r

l

vv

vv

102

176.0

250

10200842.0378.32

2425.44

250

2322

21

x

E

t

bb

378.1176.020231.135.075.0 222

3

2

21 kkk vveq

738.1378.12.0378.149.015.02.015.0 22 eqeq

357.0

378.1738.1738.1

115.0225.022

eq

So, 172.1261

2.51357.0

xff

mo

y

cd

MPa

43.36172.12674.288 dP kN 19.8 kips

CRC Formula:

2

32

2

292

102004

895.62.511895.62.51])(

41[

r

L

E

ey

ycr

= 219.292 MPa

32.6374.288292.219 g

AP crn kN 24.14 kips

103

APPENDIX C

Comparative results of single bolted angles

Figure 1 Comparison of buckling load for Elgaaly test no.54 angle size L 44.3744.373.35, single bolted.

Figure 2 Comparison of buckling load for Elgaaly test no. 36

angle size L 45.16×45.165.26, single bolted.

104





105

Comparative results of double bolted angles


angle size L 43.69×43.693.58, double bolted.



106





ultimate capacity of steel angles subjected to eccentric

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