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A Study on Different Methods of Determining the Ultimate Capacity of Steel Angles Subjected To
Eccentric Compression Load
Submitted by
Ishrat Jabin Student no- 0204074
Course: CE 400 (Project and Thesis )
Thesis
Submitted in Partial Fulfillment of the Requirements for the Degree of
BACHELOR OF SCIENCE IN CIVIL ENGINEERING
Department of Civil Engineering
BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY, DHAKA
January, 2008
A Study on Different Methods of Determining the Ultimate Capacity of Steel Angles Subjected To
Eccentric Compression Load
Submitted by
Ishrat Jabin Student no- 0204074
Course: CE 400 (Project and Thesis )
Thesis
Submitted in Partial Fulfillment of the Requirements for the Degree of
BACHELOR OF SCIENCE IN CIVIL ENGINEERING
Department of Civil Engineering BANGLADESH UNIVERSITY OF
ENGINEERING AND TECHNOLOGY, DHAKA
January, 2008
DECLARATION
Declared that except where specified by reference to other works, the studies
embodied in thesis is the result of investigation carried by the author. Neither the
thesis nor any part has been submitted to or is being submitted elsewhere for any
other purposes.
Signature of the student
( Ishrat Jabin)
ACKNOWLEDGMENTS
First of all the author would like to express her heartiest gratitude to the
Almighty Allah, The sovereign lord, who is the absolute king of entire universe and
who regulates each and every achievement of every individual life.
The author has the great pleasure to express her deepest sense of gratitude to
Prof. Dr. Khan Mahmud Amanat, Department of Civil Engineering, Bangladesh
University of Engineering and Technology (BUET), Dhaka for his guidance,
continuous encouragement and invaluable support throughout the course of this
work. His tireless devotion, unfailing support and dynamic leadership in pursuit of
excellence have earned the authors highest respect. The author will remain ever
grateful to him for his supervision and inspiration to work hard in writing this thesis.
The author wishes to convey sincere thanks to all the teachers of Department
of Civil Engineering, BUET for their entire effort to teach her the subject of civil
engineering.
The author is also grateful to Iftesham Bashar senior student of BUET who
helped her a lot.
Last but not the least, the author would like to express her deep sense of
gratitude to her parents, sisters and brother, their underlying love, encouragement
and support throughout her life and education. Without their blessings, achieving the
goal would have been impossible.
ABSTRACT
Lattice microwave towers and transmission towers are frequently made of
angles bolted together directly or through gussets. Such towers are normally
analyzed to obtain design forces using the linear static methods, assuming the
members to be subjected to only axial loads and the deformations to be small. In
such a tower, the angles are subjected to both tension and compression. The ultimate
compressive load carrying capacity of single steel angles subjected to eccentrically
applied axial load is investigated in this project. Apparently, there is no suitable
analytical method for the analysis of ultimate compressive load carrying capacity of
single steel angles. In conventional methods, no attempt is made to account for
member imperfection and bending effect due to eccentricity of the applied axial
load. In this study, non-linear analysis of angle compression members as in typical
lattice towers, are carried out using both analytical approach and finite element
software. Account is taken of member eccentricity, local deformation as well as
material non-linearity. Results are then compared with experimental records. The
comparative study shows that analytical methods tend to overestimate the axial
compression capacity of eccentrically loaded angle sections at lower eccentricity
ratio and vice versa.
Table of contents
Page No.
Chapter 1 Introduction 1.1 General 1 1.2 Objective and Significance of the Study 1 1.3 Scopes and Limitations 2 1.4 Organization of the study 2
Chapter 2 Literature Review
2.1 Introduction 3 2.2 Steel Angles 6 2.2.1 Types of Steel Angles (available) 6 2.2.2 Designation of the Steel Angles 6 2.2.3 Materials Used for Producing Steel Angles 8 2.3 End Plates 8 2.4 Ultimate Load Capacity of Structural Steel Members 8 2.5 Development of Column Buckling Theory 9 2.5.1 Elastic Buckling 9 2.5.2 Inelastic Buckling 12 2.6 Eccentrically Loaded Column Theory-Historical Development 21 2.7 Conventional Formulas to Determine Ultimate Load Capacity of
Structural Steel Angles 28
2.8 Previous Research 30 Chapter 3 Methodology for Finite Element Analysis
3.1 Introduction 32 3.2 The Finite Element Packages 33 3.3 Finite Element Modeling of the Structure 35 3.3.1 Modeling of Steel Angle and End Plates 35 3.3.2 Material Properties 38 3.4 Types of Buckling Analyses 39 3.4.1 Non-linear Buckling Analysis 39 3.4.2 Eigenvalue Buckling Analysis 39 3.5 Finite Element Model Parameters 43 3.6 Meshing 43 3.6.1 Meshing of the End Plate 43 3.6.2 Meshing of the Steel Angle 43 3.7 Boundary Conditions 44 3.7.1 Restraints 44 3.7.2 Load 44
Chapter 4 Results and Discussions 4.1 Introduction 47 4.2 Test of Bathon et al (1993) 47 4.3 Major Features of Present Analysis 49 4.4 Presentation of Results 50 4.5 Discussion on Results 55 Chapter 5 Conclusion 5.1 General 56 5.2 Outcomes of the Study 56 5.3 Future Scopes and Recommendations 57 References 58 Appendix
ANSYS Script Used in this Analysis
List of tables
Page
No.
Table 2.1 Coefficient of Reduced (Effective) Length 11
Table 3.1: SHELL181 Input Summary 37
Table 3.2: Various input parameters 43
Table 4.1: Typical representative test results for various single equal-
leg angles
49
Table 4.2: The position of applied load g for the angle sections used
in present analysis
50
Table 4.3: Critical stress for different l/r ratio for angle: 44 x 44 x 3 51
Table 4.4: Critical stress for different l/r ratio for angle: 51x 51x 3 52
Table 4.5: Critical stress for different l/r ratio for angle: 89x 89x 6 53
Table 4.6: Critical stress for different l/r ratio for angle: 102x102x6 54
List of figures
Page No.
Figure 2.1: Four-legged electrical transmission tower (pylon) with
single steel angle
4
Figure 2.2: Typical images of roof trusses 5
Figure 2.3: Image of single equal leg angle members 6
Figure 2.4: Image of single unequal leg members 7
Figure 2.5: A single steel angle cross section of designation
L A x B x C
7
Figure 2.6: An ideally pinned column 10
Figure 2.7: Different end conditions of axially loaded column 11
Figure 2.8: Inelastic buckling of a column with intermediate length 12
Figure 2.9:
Column stress as function of slenderness 13
Figure 2.10: Enjessers Basic Tangent-Modulus Theory
(in terms of a typical stress vs. strain curve)
14
Figure 2.11: Shanleys idealized column 16
Figure 2.12: Shanleys Theory of Inelastic Buckling 17
Figure 2.13: Critical buckling load vs. transverse deflection (w) 18
Figure 2.14: The form of the stress prism changes from an even distribution to a very uneven distribution due to eccentricity of loading
21
Figure 2.15: Two possible stress distributions for columns according to
Jezeks Approach
23
Figure 2.16: Column models for which secant formula is applicable 25
Figure 2.17: Column curves for various values of the eccentricity 25
Figure 2.18: Eccentrically loaded column 27
Figure 2.19: Details of end connections (a) Two bolt configuration
(b) Three-bolt configuration (c) Five-bolt configuration
31
Figure 3.1: Collection of nodes and finite element 32
Figure 3.2: General sketch of a single steel angle with end plates at its
both ends subjected to eccentric load
35
Figure 3.3: SHELL181 Geometry 36
Figure 3.4: Bilenear kinematic hardening 38
Figure 3.5: Buckling Curves 39
Figure 3.6: Nonlinear vs. Eigenvalue Buckling Behavior 40
Figure 3.7: Snap Through Buckling 40
Figure 3.8: Newton - Raphson Method 41
Figure 3.9: Arc-Length Methodology 42
Figure 3.10: Arc-Length Convergence Behavior 42
Figure3.11: Preliminary model of a single steel angle connected to end
plates at its both ends (prior to meshing)
44
Figure 3.12: Finite elements mesh of the steel angle with end plates at its
both ends
45
Figure 3.13: Finite elements mesh with loads and boundary conditions 45
Figure 3.14: Typical deflected shape of the model 46
Figure 3.15: Typical deflection versus load curve obtained from non-
linear buckling analysis of the steel angle with end plates at
its both ends.
46
Figure 4.1: Cross-section dimensions for test performed by Bathon et al
(1993)
48
Figure 4.2: Experimental results (buckling curve) for 102x102x6 angle 48
Figure 4.3: Critical Stress vs l /r ratio for 44x44x3 angle 51
Figure 4.4: Critical Stress vs l /r ratio for 51x51x3 angle 52
Figure 4.5: Critical Stress vs l /r ratio for 89x89x6 angle 53
Figure 4.6: Critical Stress vs l /r ratio for 102x102x6 angle 54
Chapter 1 : Introduction
1
Chapter 1
INTRODUCTION 1.1 GENERAL
The simplest type of compression member is a single steel angle. This is very commonly used as primary compression member in electrical transmission towers. 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 addition, steel angles are subjected to bending due to the eccentricity of the applied load, which has a pronounced effect on the performance of the steel angles. 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 OBJECTIVE AND SIGNIFICANCE OF THE STUDY The performance of steel angle in carrying eccentrically applied compression
loads is of great importance in the design of electrical transmission towers. As, these angles are an integral part of the tower structures (which are often subjected to tremendous wind forces and may be subjected to other kinds of forces), it becomes therefore obvious to make a formulation for predicting ultimate compression load carrying capacity of the single steel angles. In previous researches, there had been some attempts for analyzing ultimate load capacity of the steel angles. The AISC also has its own formulas for determining the ultimate capacity of angles. But unexpectedly, nobody including the AISC considered the effects of load eccentricity in their formulation. So, it becomes necessary to consider the eccentricity effects in all cases to predict ultimate load capacity of steel angles which is carefully done in this thesis project. A comparative of results obtained using both analytical approach and non-linear finite element analysis versus calculated capacity using the procedure of ASCE manual 52 (1988) and previous test results will be made. The purpose is to compare all these results to the design requirements and to make observations and recommendations.
Chapter 1 : Introduction
2
1.3 SCOPES AND LIMITATIONS Like many other studies this study has also its limitations :
i) Study shall be carried out for a few equal leg angle sections which have
been tested earlier by others.
ii) Non-linear (both geometric and material) finite element analysis shall be
carried out to determine the axial compression capacity of angles under
eccentric loading.
iii) Capacity of angle sections according to non-linear analytical formulations
shall be evaluated.
iv) Comparison of compression capacity obtained by different methods shall be
made.
1.4 ORGANIZATION OF THE STUDY The report 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 4 : Results and Discussions
47
Chapter 4
RESULTS AND DISCUSSIONS
4.1 INTRODUCTION
In the previous chapter, modeling of the steel angle with end plates has been
discussed in details. In the ongoing chapter, we shall analyze a few angle sections
under axially applied eccentric compression load by means of theoretical approach
(Jezeks formula and Youngs secant formula) and non-linear buckling analysis in
finite element method. The results obtained from both the analysis have been
compared with results obtained from the test of Bathon et al (1993) and ASCE
formula (obtained from ASCE Manual 52 for the Design of Steel Transmission
towers (1988)).
The results are presented in tables and supporting graphs are also provided
for convenience to justify the results from various aspects and for making comments
and suggestions and for further recommendations.
4.2 TEST OF BATHON et al (1993)
Bathon et al (1993) tested a total number of seventy five single steel angles
(thirty one single-angle equal leg and forty four single angle unequal leg
member), for determining ultimate compression load carrying capacity
considering the effect of eccentricity of the applied axial load.
For giving emphasize of eccentricity effect of applied load, the load was
applied through the center of gravity of the bolt pattern (which is, according
to ASCE Manual 52(1988)), located between the centroid of the angle and the center line of the connected leg. Figure 4.1 shows the portion g that
determines the eccentricity of the applied load.
Chapter 4 : Results and Discussions
48
Figure 4.1:Cross-section dimensions for test performed by Bathon et al (1993)
The results of this research project consisted of the performance of single
angle members in the elastic, inelastic and post buckling regions.
All of the test specimens failed due to overall member buckling.
Experimental test results for a 102x102x6 angle obtained from test of
Bathon et al (1993) are shown below (Figure 4.2).
Figure 4.2: Experimental results (buckling curve) for 102x102x6 angle
Chapter 4 : Results and Discussions
49
Table 4.1: Typical representative test results for various single equal-leg angles.
Test
number
(1)
Angle size
(2)
Area
(sq.mm)
(3)
l/r (4) Actual Fy
(MPa) (5)
Actual capacity
test (MPa) (6)
Predicted capacity Fy=actual (MPa)
(7) 14 44x44x3 272 60 378.3 57.9 97.1
12 44x44x3 272 90 378.3 59.9 75.4
11 44x44x3 272 120 378.3 53.1 57.7
13 51x51x3 312 60 403.1 89.6 108.6
10 51x51x3 312 90 403.1 82.7 86.6
9 51x51x3 312 120 403.1 57.2 66.3
67 89x89x6 1089 60 334.9 268.7 370.5
46 89x89x6 1089 90 339.0 207.4 300.8
51 89x89x6 1089 120 339.0 144.7 231.0
7 102x102x6 1250 60 325.9 304.5 394.8
4 102x102x6 1250 90 325.9 226.7 334.3
5 102x102x6 1250 120 325.9 164.0 265.1
Where,
yF is the yield stress of the angle.
4.3 MAJOR FEATURES OF PRESENT ANALYSIS
The present study will compare the findings of similar study carried out by
Bathon et al (1993).In this study, the ultimate compression load carrying capacity
has been analyzed by means of analytical approach (Modified Rankine formula and
Youngs secant formula) and non-linear buckling analysis in finite element method.
A total number of four single equal leg angles have been studied under present
investigation. For each angle, critical stress is determined for slenderness ratios: 12,
20,40,60,90,120,150,180,210 and 240 respectively. The position of applied load has
fixed eccentricity for each angles, these values were used both in non-linear finite
element buckling analysis and in analytical formulas as listed below:
Chapter 4 : Results and Discussions
50
Table 4.2: The position of applied load g for the angle sections used in present analysis
Angle size l/r g (mm)
44x44x3 60 25
44x44x3 90 25
44x44x3 120 25
51x51x3 60 25
51x51x3 90 25
51x51x3 120 25
89x89x6 60 35
89x89x6 90 35
89x89x6 120 35
102x102x6 60 38
102x102x6 90 37
102x102x6 120 38
4.4 PRESENTATION OF RESULTS In this article, the result of the present investigation will be presented and
compared with previous similar research investigations simultaneously in following
tables and figures in the next pages:
Chapter 4 : Results and Discussions
51
Table 4.3: Critical stress for different l/r ratio for angle: 44 x 44 x 3
l/r
ratio
Critical stress, MPa
Theoretical approach with ec/r2=0.13
Present
analysis
Test of
Bathon et
al (1993)
ASCE
formula Youngs Secant formula
Modified Rankine formula
12 286.4 281.88 82.6 - -
20 284.8 272.74 81.7 - -
40 275.9 236.78 77.6 - -
60 254.5 193.62 70.7 57.9 97.1
90 189.1 138.12 55.3 59.9 75.4
120 123.2 98.44 34.9 53.1 57.7
150 82.7 71.99 22.3 - -
180 58.7 54.00 15.5 - -
210 43.6 41.76 11.4 - -
240 33.6 33.10 8.7 - -
0
50
100
150
200
250
300
350
0 50 100 150 200 250 300
Slendereness ratio ( l/r )
Crit
ical
stre
ss, M
Pa
Test of Bathon et al(1993)
Analytical approach( Secant formula )
Modified Rankineformula
ASCE formula
Present analysis
Figure 4.3: Critical Stress vs. slenderness ratio for 44x44x3 angle
Chapter 4 : Results and Discussions
52
Table 4.4: Critical stress for different l/r ratio for angle: 51x 51x 3
l/r
ratio
Critical stress, MPa
Theoretical approach with ec/r2=0.4
Present
analysis
Test of
Bathon et
al (1993)
ASCE
formula Youngs Secant formula
Modified Rankine formula
12 232.3 229.08 96.2 - -
20 229.5 221.65 95.1 - -
40 217.1 192.42 90.3 - -
60 195.3 157.75 82.4 89.6 108.6
90 149.3 112.25 64.4 82.7 86.6
120 105.7 79.96 40.7 57.2 66.3
150 75.0 58.37 26.0 - -
180 54.9 43.89 18.0 - -
210 41.5 33.94 13.2 - -
240 32.4 33.10 10.1 - -
0
50
100
150
200
250
0 50 100 150 200 250 300
Slendereness ratio ( l/r )
Crit
ical
stre
ss, M
Pa
Test of Bathon et al(1993)
Analytical approach( Secant formula )
Modified Rankineformula
ASCE formula
Present analysis
Figure 4.4: Critical Stress vs. slenderness ratio for 51x51x3 angle
Chapter 4 : Results and Discussions
53
Table 4.5: Critical stress for different l/r ratio for angle: 89x 89x 6
l/r
ratio
Critical stress, MPa
Theoretical approach with ec/r2=1.2
Present
analysis
Test of
Bathon et
al (1993)
ASCE
formula Youngs Secant formula
Modified Rankine formula
12 147.8 105.04 334.3 - -
20 146.0 101.64 330.8 - -
40 137.2 88.24 314.1 - -
60 124.2 72.34 286.3 268.7 370.5 90 100.7 51.47 223.9 207.4 300.8 120 78.1 36.66 141.5 144.7 231 150 60.0 26.76 90.5 - -
180 46.5 20.12 62.9 - -
210 36.6 15.56 46.2 - -
240 29.3 12.33 35.4 - -
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300
Slendereness ratio ( l/r )
Crit
ical
stre
ss, M
Pa
Test of Bathon et al(1993)
Analytical approach( Secant formula )
Modified Rankineformula
ASCE formula
Present analysis
Figure 4.5: Critical Stress vs. slenderness ratio for 89x89x6 angle
Chapter 4 : Results and Discussions
54
Table 4.6: Critical stress for different l/r ratio for angle: 102x102x6
l/r
ratio
Critical stress, MPa
Theoretical approach with ec/r2=1.27
Present
analysis
Test of
Bathon et
al (1993)
ASCE
formula Youngs Secant formula
Modified Rankine formula
12 142.5 99.60 384.9 - -
20 140.8 96.37 380.8 - -
40 132.7 83.66 361.6 - -
60 120.3 68.59 329.6 304.5 394.8
90 97.9 48.80 257.7 226.7 334.3
120 74.8 34.76 162.8 164 265.1
150 58.9 25.38 104.2 - -
180 45.8 19.08 72.4 - -
210 36.2 14.75 53.2 - -
240 29.1 11.70 40.7 - -
0
50
100
150
200
250
300
350
400
450
0 50 100 150 200 250 300
Slendereness ratio ( l/r )
Crit
ical
stre
ss, M
Pa
Test of Bathon et al(1993)
Analytical approach( Secant formula )
Modified Rankineformula
ASCE formula
Present analysis
Figure 4.6: Critical Stress vs. slenderness ratio for 102x102x6 angle
Chapter 4 : Results and Discussions
55
4.5 DISCUSSION ON RESULTS
The numerical results have shown that the ultimate load-carrying capacity
drops with an increase in column slenderness ratio.
For angle section with eccentricity ratio, ec/r2 1, results from finite element analysis give higher stress.
This is due to the fact that angle dimension has large effect on stress than that of
eccentricity ratio in analytical approach.
It has been further observed that with the increase in the dimensions of angle
sections, the curves tend to merge for higher l/r ratios. This leads to the conclusion
that the influence of the eccentricity on the member performance increases from
slenderness ratio 60 to 120 and decreases from 120 to 240.
FEA results closely match with Bathon test results which indicate FEA can
simulate real condition acceptably. Theoretical results differ from both test & FEA
results. Because theoretical formula developed based on-
uniform and symmetric cross-section.
eccentricity along the axis of symmetry.
But angle cross-section is not symmetric.
Individual member analysis, while serving a very useful purpose of verifying
analytical methods, often fail to model the behavior of members assembled as a
structure. The angle analysis reported use ball end connections, thus removing all
beneficial end restraint effect present in a tower. This may the reason for the
significant difference in capacity predicted by the test and code equation (ASCE
Manual)
Chapter 5:Conclusions
56
Chapter 5 CONCLUSIONS 5.1 GENERAL
The thesis emanated with an aim to find out the ultimate load carrying
capacity of single steel angles subjected to eccentrically axial compressive loads.
For attaining the objective, a gradual sequence has been maintained. First of all, the
theoretical background of eccentrically loaded column is enumerated and the
necessary equations adopted in the analytical approach related to our study are also
mentioned for convenience. All these information are included in chapter 2.The
present issue has also been analyzed in finite element method. The details of
modeling procedure, selection of finite element along with boundary conditions have
been explained in chapter 3.The subsequent chapter is the summarization of
computational investigation and comparative study of the obtained results from
analytical approach and finite element analysis to similar studies carried out in the
previous research and those using ASCE formula. The results of the present study
are informative and hopefully would be useful for practical purposes.
5.2 OUTCOMES OF THE STUDY The design of eccentrically loaded single steel angles is a major design
concern to practicing engineers. As there are no specific codal provisions for
determining the ultimate load capacity of single steel angles subjected to eccentric
loads, an attempt has been made in the study to achieve a practical solution.
As finite element analysis gives reliable results analytical formula can be
developed from FEA only and no further laboratory test is necessary.
Analytical approach may not give reliable results in case of angle section.
Chapter 5:Conclusions
57
5.3 FUTURE SCOPES AND RECOMMENDATIONS
In the study of angle section, analytical approach is not realistic. So in order
to design angle section subjected to eccentric compressive load, development of
analytical formula by simulating the behavior of angle section is necessary and more
study in this sector is recommended.
Chapter 2 : Literature Review 3
Chapter 2
LITERATURE REVIEW
2.1 INTRODUCTION Steel angles are one of the most important structural compression members.
In the majority of structural applications single angles are usually loaded in such a
manner that the applied load is eccentric. Eccentrically loaded single-angle struts are
among the most difficult structural members to analyze and design. Single-angles
are used extensively in towers, particularly in transmission towers. They are some
times used in small roof trusses. They can also be used as structural frame elements.
Electrical Transmission Towers:
Electrical transmission lines are supported by a variety of structural
configurations including single poles, H-frame structures, guyed structures and
lattice structures. Of these structures, the most common is the lattice which is
composed of single hot-rolled steel angles. The lattice tower is analyzed and
designed assuming that each member is a two force member or truss. These
members resist load by tension and/or compression. Although an angle would
normally be a poor choice for a column, it is the member of choice in a lattice
structure, because its shape makes it easy to connect one to the other. The
performance of steel angle is of great interest to the field of electrical transmission
tower design (Figure 2.1).
Roof Trusses:
Trussing, or triangulated framing, is a means for developing stability with a
light frame. It is also a means for producing very light two dimensional or three
dimensional structural elements for spanning systems or structures in general. The
double angle member is widely used in roof trusses (Figure 2.2).
Chapter 2 : Literature Review 4
Figure 2.1: Four-legged electrical transmission tower (pylon) with single steel angle
Chapter 2 : Literature Review 5
It may be used for both web and chord members of riveted or bolted trusses with
connections made to gusset plates between the vertical legs.
(a) Bowstring roof truss
(b) Multi piece roof truss
(c) Double-inverted roof truss Figure 2.2: Typical images of roof trusses
Other Uses:
Single angles are also used as bracing members in plate girder bridges and as bracings in large build-up columns. Double angle section are also used in
wind bracing in plate girder bridges.
In the case of larger compression members or when suitable channel sections are not available, the built-up sections using angles can conveniently be used.
Chapter 2 : Literature Review 6
2.2 STEEL ANGLES 2.2.1 Types of Steel Angles (available) According to arrangement:
I. Single angle (Used in towers, lintels, etc.)
They are available in two types of shapes (according to size of their individual legs):
-Single equal leg angles: The two legs of these angles are of same size (Figure 2.3).
-Single unequal leg angles: The two legs of these angles are of different sizes.
(Figure 2.4).
II. Double angles (Used as members of light steel trusses) 2.2.2 Designation of the Steel Angles
Structural angles are rolled section in the shape of the letter L. Both legs
of an angle have same thickness.
- Angles are designated by the alphabetical symbol L, followed by dimensions of
the legs and their thickness.
Figure 2.3: Image of single equal leg angle members
Chapter 2 : Literature Review 7
Figure 2.4: Image of single unequal leg members
Figure 2.5: A single steel angle cross section of designation L Ax B x C
- Thus the designation L 4 x 4 x indicates an equal leg angle with 4-inch legs
and -inch thickness.
Similarly, the designation L 5 x 3 x indicates an unequal leg angle with
one 5-inch and one 3 inch leg, both of -inch thickness.
Chapter 2 : Literature Review 8
2.2.3 Materials Used for Producing Steel Angles Steel that meets the requirements of the (American Society for Testing and
Materials (ASTM)) Specification, A36 is the grade of structural steel commonly
used to produce rolled single angles.
Properties of A36 steel:
In order to understand the variation in the mechanical properties of the
structural steel available that may be grouped by strength grade for ease of
discussion. The structural carbon steel is one of them. These steels depend upon the
amount of the carbon used to develop their steel through the way the range of
thickness.
- A36 steel is a low carbon steel.
- According to ASTM Code, for thickness up to 8 inch,
Minimum Yield point=36,000psi
Tensile strength=58,000-80,000psi
2.3 END PLATES
End plates are often used with single steel angles. In such cases, they are
either welded to the steel angle or connected to the steel angle by bolts.
2.4 ULTIMATE LOAD CAPACITY OF STRUCTURAL STEEL MEMBERS
It may be defined as the load carrying capacity of the structural steel
members at which it fails by buckling.
For long members:
Failure:
- Elastic
- Fails by buckling
- No yielding.
Chapter 2 : Literature Review 9
For short members:
Failure:
- Inelastic
- Fails by yielding
- No buckling. For many years, theoretical determinations of Ultimate load did not agree with test
results.
Test results included - Effects of initial crookedness of the member
- Accidental eccentricity of load
- End restraint
- Local or lateral buckling
- and residual stress
2.5 DEVELOPMENT OF COLUMN BUCKLING THEORY The behavior of single steel angles subjected to buckling is almost similar to
that of columns.So, it is desired to have a clear and concise idea about the buckling
phenomenon of columns prior to any study and analysis regarding steel angles.
Development of column buckling theory is a gradual process, which is briefly
illustrated in this article.
2.5.1 Elastic Buckling Euler Formula
Column buckling theory originated with Leonhard Euler in 1757. He
considered a column with both end pinned as shown (Figure 2.6).
The underlying assumptions were:
1) Perfectly straight column
2) No eccentric axial load
3) Plane remains plane after deformation
4) Bending deflection only (no shear deformation)
Chapter 2 : Literature Review 10
5) Hook's law
6) Small deflection
Figure 2.6: An ideally pinned column The critical or Euler load, Pcr for such a column is,
2
2
ecr L
EIP = (2.1)
Where, E = Youngs modulus of elasticity of the material of column
I = The least moment of inertia of the constant cross-sectional area of a
column
L = Actual length of column Le = The effective length (usually the unbraced length) of the column ( effective
length Le related to the actual length of the arrangement by a factor k
which reflects the degree of end fixing.
Chapter 2 : Literature Review 11
Figure 2.7: Different end conditions of axially loaded column
Table 2.1: Coefficient of Reduced (Effective) Length
Indication Strut mounting Coef.(theor) Coef.(pract)
A fixed - fixed 0.50 0.65 B fixed - Hinged 0.70 0.80 C fixed - Guided 1.00 1.20 D Hinged - Hinged 1.00 1.00 E fixed - Free end 2.00 2.10 F Hinged - Guided 2.00 2.00
Examination of this formula reveals the following interesting facts with regard to the
load-bearing ability of columns.
1. Elasticity and not compressive strength of the materials of the column
determines the critical load.
2. The critical load is directly proportional to the second moment of area of the
cross section.
3. The boundary conditions have a considerable effect on the critical load of
slender columns. The boundary conditions determine the mode of bending and the
distance between inflection points on the deflected column. The closer together the
inflection points are, the higher the resulting capacity of the column.
Euler formula is applicable while the material behavior remains linearly
elastic. Eulers Approach was generally ignored for design, because test results did
not agree with it; columns of ordinary length used in design were not as strong as
determined from Euler formula.
Chapter 2 : Literature Review 12
2.5.2 Inelastic Buckling The Euler formula describes the critical load for elastic buckling and is valid only
for long columns. The ultimate compression strength of the column material is not
geometry-related and is valid only for short columns.
In between, for a column with intermediate length, buckling occurs after the stress in
the column exceeds the proportional limit of the column material and before the
stress reaches the ultimate strength. This kind of situation is called inelastic
buckling.
This section discusses some commonly used inelastic buckling theories that fill the
gap between short and long columns.
Figure 2.8: Inelastic buckling of a column with intermediate length
I."Johnson Parabola" Approach
Many empirical and semi-empirical methods have been proposed for matching
the experimental data.
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 ( y,0 ) tangent to the Euler curve.
The equation of Johnsons parabola is given by:
])4
/(1[ 2 y
eycr E
rLAP
= (2.2)
Chapter 2 : Literature Review 13
Where,
Pcr = Critical buckling load for the column
y = Yield stress of the column
A = Cross-sectional area of the column
r = Least radius of gyration of column cross-section
Le = Effective length of the column
E = Youngs modulus of elasticity of the material of column
Figure 2.9: Column stress as function of slenderness.
II. Basic Tangent Modulus Theory Considere and Engesser in 1889 independently realized that for columns
those fail subsequent to the onset of inelastic behavior, the constant of
proportionality must be used rather than the modulus of elasticity (E) (Engesser
formula, as described by Bleich (1952)).
- The constant of proportionality (Et) is the slope of the stress-strain diagram
beyond the proportional limit, termed the tangent modulus, where, within the
linearly elastic range, E = Et.
Chapter 2 : Literature Review 14
- The theory assumes that no strain reversal takes place and the tangent modulus
Et applies over the whole section.
Thus, Engesser modified Eulers equation to become formula,
2
2
e
tcr L
IEP = =Pt (2.3)
Where,
Pcr = Critical buckling load for the column
Pt = The tangent modulus load
Et = Tangent modulus of elasticity
I = Moment of inertia (usually the minimum) of the column cross-section
Le = Effective length of the column
This theory, however, still did not agree with test results, giving computed
loads lower than measured ultimate capacity. 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. The relationships between E and Et are shown in the following figure:
Figure 2.10: Enjessers Basic Tangent-Modulus Theory
(in terms of a typical stress vs. strain curve)
In this figure,
u = Ultimate stress of the column
Chapter 2 : Literature Review 15
t = Tangent modulus stress
pl = Proportional limit
III.Double Modulus Theory
In 1895, Engesser changed his theory, reasoning that during bending some
fibers are under going increased strain (lowered tangent modulus) and some fibers
are being unloaded (higher modulus at the reduce strain): therefore, a combined
value should be used for the modulus. This is referred to as either Double Modulus
Theory or the Reduced Modulus Theory, described by Salmon and Johnson (1971).
The Reduced Modulus theory defines a reduced Young's modulus Er to compensate
for the underestimation given by the tangent-modulus theory.
For a column with rectangular cross section, the reduced modulus is defined by,
( )24
t
tr
EE
EEE+
= (2.4)
Where E is the value of Young's modulus below the proportional limit. Replacing E
in Euler's formula with the reduced modulus Er, the critical load becomes,
22
eff
rr L
IEF = (2.5)
The corresponding critical stress is,
( )22
/ rLE
eff
rr
= (2.6)
For the column of same slenderness ratio, this theory always gives a slightly higher
column buckling capacity than the tangent modulus theory. The discrepancy
between the two solutions is not very large. The reason for this discrepancy was
explained by F. R.Shanley. IV. Shanley Concept True column behavior
Chapter 2 : Literature Review 16
Both the Tangent-Modulus Theory and Reduced-Modulus Theory were
accepted theories of inelastic buckling until F. R. Shanley published his logically
correct paper in 1946.According to Shanleys concept, as described by Bleich(1952)
buckling proceeds simultaneously with the increasing axial load.
- Shanley 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.
Figure 2.11: Shanleys idealized column
Shanleys idealized column consists of two rigid legs AC and BC connected at C by
an elastic-plastic hinge.
He shows the relation between the applied load P (>Pt ) and the deflection y is given
by
+
++=
11
2
11
ybPP t
(2.7)
EEt= and is assumed constant
Chapter 2 : Literature Review 17
Pt = critical load and b = constant
Figure 2.12: Shanleys Theory of Inelastic Buckling In this figure,
Foc = Elastic critical stress
Fr = Reduced modulus stress
Ft = Tangent modulus stress
Fm = Maximum stress, which defines the ultimate strength of the member
Both tangent-modulus theory and reduced-modulus theory were accepted theories of
inelastic buckling until F. R. Shanley published his logically correct paper in 1946.
The critical load of inelastic buckling is in fact a function of the transverse
displacement w. According to Shanley's theory, the critical load is located between
the critical load predicted by the tangent-modulus theory (the lower bound) and the
reduced-modulus theory (the upper bound / asymptotic limit).
Chapter 2 : Literature Review 18
Figure 2.13: Critical buckling load vs. transverse deflection (w)
The above figure shows buckling loads according to different theories.
V. Gordon- Rankine Formula
In practice the column is of medium length. Its strength is affected by buckling which causes bending stress. Therefore, in such columns the direct stress as
well as stress caused by buckling is important. This has been taken into account in
Gordon-Rankine formula.
21
+
=
rla
AfP c (2.8)
E
fa c2= (2.9)
Where,
P = buckling load
fc = the crushing stress as a short column
a = constant, which depends on material
l/r = slenderness ratio
Incase of mild steel, fc = 3200 kg / cm2
a = 1 / 7500
Chapter 2 : Literature Review 19
VI. Perry-Robertson Formula The ideal column is that which is initially straight and which is loaded
concentrically. The behaviors of ideally long column are represented by Eulers
formula which is based on stability of column. In practice there will always be initial
curvature i.e. the compression member will not be straight and the load cannot be
applied concentrically, the behavior of such column is absolutely different.
Depending upon the experimental results, Robertson modified the theoretical
formula suggested by Perry. The standard formula is
( ) ( )
++
++
= eyeyey
a fffnffnf
cf2
12
1 (2.10)
Where,
c = load factor taken as 2
fa = permissible average stress
fe = Euler buckling stress = 22
ecr L
EIP =
n = 0.003
The expression is used for values of l/r > 80. For values between 0 to 80, a straight
line formula fa = p (1- 0.00538 l/r) is used where fa is allowable stress on the
column and p is permissible stress as a short column.
VII. IS. Code Formula The direct stress in compression on the gross sectional area of axially loaded compression members shall not exceed the values of Pc as given by formula
for l/r = 0-160 (2.11)
for l/r = 160 and above (2.12)
+
=
+
==
rl
EcmP
rlmf
P
EcmP
rlmf
PP
radian
y
c
radian
y
cc
80012
4sec2.01
4sec2.01
Chapter 2 : Literature Review 20
Where,
Pc = the allowable average axial compressive stress
fy = the guaranteed minimum yield stress
m = factor of safety taken as 1.68
l/r = slenderness ratio
VIII. Straight- line Formula In this formula it is assumed that allowable stress varies linearly with respect to l/r ratio.
=
rlapfa 1 (2.13)
Where,
fa = allowable stress
p = working stress as a short column
a = constant depends on material
= 0.0053 for mild steel
IX.Tetmajer and Bauschinger Formula This formula was obtained as a result of experiments of Tetmajer and Bauschinger
on structural steel columns with hinged ends. The formula
( )wc =16000 70 (l/r) (2.14)
Where,
( )wc = allowable average compressive stress
For main members 30 < l/r < 120
For secondary members 30 < l/r < 150
For l/r < 30 ( )wc = 14000 psi.
Chapter 2 : Literature Review 21
The experiments suggested for the critical value of the average compressive stress
the formula
cr = 48000 210 (l/r) (2.15)
Tetmajer recommended this formula for l/r < 110.
Further Observations
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.
2.6 ECCENTRICALLY LOADED COLUMN THEORY-
HISTORICAL DEVELOPMENT Eccentrically loaded columns usually fail by buckling. The figure below illustrates the stress that a column experiences as a load, N, is applied with
increasing eccentricity.
Figure 2.14: The form of the stress prism changes from an even distribution to a
very uneven distribution due to eccentricity of loading
Chapter 2 : Literature Review 22
The case at the left illustrates how the axial load creates a compressive stress
which is evenly distributed across the column's section. The load on each column to
the right has an increasing eccentricity. As the load moves away from the centroidal
axis, it introduces a bending moment which the column's cross-section must also
resist.
In a brief account of the development of the theory of eccentrically loaded
columns, Ostenfeld (1898) must be mentioned, who, half 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.
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.
Westergaard and Osgood (1928) presented a paper in which the behavior of
eccentrically loaded columns and initially curved columns were discussed
analytically. The method is based upon the same equations as were used by Karma
but assumes the deflected center line of eccentrically loaded compression members
to be part of a cosine curve, thereby simplifying Karmans method without
impairing the practical accuracy of the results.
Starting from Karmans exact concept, 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. Chwalla 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
Chapter 2 : Literature Review 23
influenced by shape of the column cross section, slenderness ratio, and eccentrically
and that his exact results can serve as a measure for the accuracy of approximate
methods.
In the course of development of the theory of eccentrically loaded columns
another simplified stability theory by assuming 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 was established
in 1928.
A very valuable contribution to the solution of the problem was offered by
Jezek (1934), who gave an analytical solution for steel columns based upon a
simplified stress-strain curve consisting of two straight lines and showed that the
results agree rather well with the values obtained from the real stress-strain relation.
The underlying concept of Jezeks theory proves useful in devising analytical
expressions from which, in a rather simple manner, diagrams, tables, or design
formulas for all kinds of material having sharply defined yield strength can be
derived.
Figure 2.15: Two possible stress distributions for columns according to Jezeks
Approach
Chapter 2 : Literature Review 24
For stress distribution, case (a),
32
2
])1(3
1[)/(
=
PArL
EAP
y (2.16)
valid for, )3(9
)(32
2
y
ErL >0
For stress distribution, case (b),
])
32(
)/([
324
34
=
AP
PA
E
rLAP
y
y
y (2.17)
valid for, 0)3(9
)(32
2 80. For values between 0 to 80, a straight line formula fRaR = p (1- 0.00538 l/r) is used where fRaR is allowable stress on the column and p is permissible stress as a short column.VII. IS. Code FormulaThe direct stress in compression on the gross sectional area of axially loaded compression members shall not exceed the values of PRcR as given by formulafor l/r = 0-160 (2.11)for l/r = 160 and above (2.12)Where,PRcR = the allowable average axial compressive stressfRyR = the guaranteed minimum yield stressm = factor of safety taken as 1.68l/r = slenderness ratioVIII. Straight- line FormulaIn this formula it is assumed that allowable stress varies linearly with respect to l/r ratio.(2.13)Where,fRaR = allowable stressp = working stress as a short columna = constant depends on material= 0.0053 for mild steelIX. Tetmajer and Bauschinger FormulaThis formula was obtained as a result of experiments of Tetmajer and Bauschinger on structural steel columns with hinged ends. The formula=16000 70 (l/r) (2.14)Where,= allowable average compressive stressFor main members 30 < l/r < 120For secondary members 30 < l/r < 150For l/r < 30 = 14000 psi.The experiments suggested for the critical value of the average compressive stress the formula= 48000 210 (l/r) (2.15)Tetmajer recommended this formula for l/r < 110.Further Observations 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).In a brief account of the development of the theory of eccentrically loaded columns, OstenfeldP P(1898) must be mentioned, who, half a century ago, made an attempt to derive design formulas for centrally and eccentrically loaded columns. His ...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...In the course of development of the theory of eccentrically loaded columns another simplified stability theory by assuming that the deflected center line of the column can be represented by the half wave of a sine curve but based the computa...A very valuable contribution to the solution of the problem was offered by Jezek (1934), who gave an analytical solution for steel columns based upon a simplified stress-strain curve consisting of two straight lines and showed that the resu...
chapter-33.1AN INTRODUCTION TO FINITE ELEMENT ANALYSISSHELL181 Element DescriptionSHELL181 Input DataTable 3.1: SHELL181 Input Summary
SHELL181 Assumptions and Restrictions:3.4.1 Nonlinear Buckling Analysis
referencesReferences
appendixAppendixANSYS Script Used in this Analysis