A Study on Different Methods of Determining the Ultimate Capacity of Steel Angles Subjected to Eccentric Compression Load

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




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.


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


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:


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:


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


52

Table 4.4: Critical stress for different l/r ratio for angle: 51x 51x 3

l/r

ratio



Present

analysis

Test of

Bathon et

al (1993)

ASCE



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


Crit

ical

stre

ss, M

Pa




ASCE formula

Present analysis



53

Table 4.5: Critical stress for different l/r ratio for angle: 89x 89x 6

l/r

ratio



Present

analysis

Test of

Bathon et

al (1993)

ASCE



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


Crit

ical

stre

ss, M

Pa




ASCE formula

Present analysis



54

Table 4.6: Critical stress for different l/r ratio for angle: 102x102x6

l/r

ratio



Present

analysis

Test of

Bathon et

al (1993)

ASCE



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


Crit

ical

stre

ss, M

Pa




ASCE formula

Present analysis



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).


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


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.


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


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.


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.


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)


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.


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.


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)


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.


- 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


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


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


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).


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


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


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.


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


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


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


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

A Study on Different Methods of Determining the Ultimate Capacity of Steel Angles Subjected to Eccentric Compression Load

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