Ultimate Capacity of Steel Angles Subjected to Eccentric Compression

ULTIMATE CAPACITY OF STEEL ANGLES SUBJECTED TO ECCENTRIC COMPRESSION BY NONLINEAR GEOMETRIC FINITE ELEMENT ANALYSES

A Thesis by ADIL MD. KAMRUL HASSAN FOISAL

(Student ID: 200304030) submitted in partial fulfillment of requirement for degree of

Bachelor of Science in Civil Engineering

Department of Civil Engineering Bangladesh University of Engineering and Technology (BUET)

Dhaka, Bangladesh

March 2009

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

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

the supervision of Dr. Khan Mahmud Amanat, Professor, Department of Civil Engineering, BUET.

Neither the thesis nor any part of it has been submitted to or is being submitted elsewhere for

any other purposes.

Adil Md. Kamrul Hassan Foisal (Author)

dedicated to

the sweet memory of

AHMED AL FAISAL #200304025

I

ACKNOWLEDGEMENT

At first I would like to express my heartiest gratitude to almighty Allah who is the designer of

every single moment and achievement of my life.

It is a great pleasure to say that I got the opportunity to work and study under the supervision of

Professor Dr. Khan Mahmud Amanat, Department of Civil Engineering, BUET. His motivation and

guidance, continuous encouragement, invaluable support, devotion to work, creative outlook

and dynamic leadership led me the way to complete my thesis work confidently and

successfully. Without his assistance this thesis work would never be possible. His contribution

can’t be expressed in words and I am really very much grateful to him.

I would also like to thank my friend Saad, Sanjoy, Tanim, Aomy, Rich, Rohomot, Tareq and Siam

and to thank Ishrat Apu for their invaluable help and suggestions to complete my thesis.

As for ever, I am expressing my deepest gratitude to relatives, friends and teachers and

especially to my parents and all of my family members whose help, encouragement, support

and cooperation has always inspired me throughout my past, present and future life.

II

ABSTRACT

Lattice steel structures, such as microwave towers and transmission towers, are generally made

of steel angle sections with bolt connection directly or through gussets. Such structures are

normally analyzed to obtain design forces using the linear static methods, assuming that the

members are subjected to only axial loads with negligible eccentricities. But in reality, many of

the load carrying members are connected with considerable eccentricity. In such a structure, the

angles are subjected to both tension and compression. The ultimate capacity of steel angles,

subjected to eccentrically applied axial load, is investigated in this thesis. Apparently, there are

no appropriate analytical methods to determine the ultimate capacity of eccentrically loaded

steel angles. In most of conventional methods, no attempts are made to account for member

imperfections and bending effect due to eccentricity of the applied axial load. In this study, non-

linear analysis of angle members as in typical lattice structures is carried out using both

analytical approach and finite element method. Account is taken for member eccentricity, local

deformation as well as material non-linearity. Results are then compared with experimental

records. The comparative parametric study shows that analytical methods tend to overestimate

the ultimate capacity of eccentrically loaded angle sections under axial compression, at lower

eccentricity and vice versa.

Contents Page

Acknowledgement I

Abstract II

Chapter 1 Introduction

1.1 General 1

1.2 Objective and Methodology 2

1.3 Scopes and Limitations 2

1.4 Organization of the Study 3

Chapter 2 Literature Review

2.1 General 4

2.2 Steel Angles 4

2.2.1 Use of Steel Angles 4

2.2.2 Common Types of Steel Angles 8

2.3

End Connection

9

2.4 Steel Properties (material for angle section) 9

2.5 Linear and Nonlinear Structural Behavior 9

2.6 Compression Members 9

2.7 Failure Modes for Compression Member When Buckling Governs 13

2.7.1 Flexural Buckling 14

2.7.2 Local Buckling 14

2.7.3 Torsional Buckling 14

2.8

Buckling Theories for Compression Members

14

2.8.1 Euler’s Formula for Elastic Buckling 14

2.8.2 Theories for Inelastic Buckling 16

2.8.2.1 Tangent Modulus Theory 16

2.8.2.2 Reduced-Modulus Theory 18

2.8.2.3 Shanley’ Concept on True Column Behavior 18

2.8.2.4 Johnson parabola Approach 20

2.8.2.5 Gordon- Rankine Formula 21

2.8.3

Buckling Theories for Eccentrically Loaded Columns

22

2.8.3.1 Jezek’s Approach 23

2.8.3.2 Secant Formula 24

2.8.3.3 Modified Rankine’s Formula 25

2.8.4

Conventional Engineering Formulae to Determine Ultimate Load Capacity under Buckling Consideration of Structural Compression Members

26

2.8.4.1 ASCE Manual 52 (1988) Formula 26

2.8.4.2 AISC Formula 28

2.8.4.3 AASHTO Formula 28

2.8.4.4 IS Code Formula 29

2.8.4.5 CRC Formula 29

2.9

Previous Research

30

2.9.1 Bathon’s Test 30

Chapter 3 Finite Element Method Of Analysis

3.1 General 32

3.2 A Brief History of Finite Element Method (FEM) 32

3.3 General Description of the FEM 33

3.4 Finite Element Packages 34

3.5 ANSYS: the FEM Package for Present Analysis 35

3.6 Finite Element Modeling of Structure in ANSYS10.0 36

3.6.1 Modeling of Steel Angle with End Plates 36

3.6.2 Element Modeling: SHELL181 36

3.6.2.1 SHELL181 Input Data 37

3.6.2.2 SHELL181 Assumption and Restriction 38

3.6.3 Modeling of Material Properties: BKIN 39

3.7

Buckling Analysis by FEM (Using ANSYS10.0)

39

3.7.1 Linear or Eigen Value Buckling Analysis 39

3.7.2 Nonlinear Buckling Analysis 40

3.7.2.1 Newton-Raphson Method 41

3.7.2.2 The Arc Length Method 42

3.8

Illustrative Model Generation for FEA (in ANSYS)

44

3.8.1 Input Parameters 44

3.8.2 meshing 45

3.8.3 Boundary Conditions 46

3.8.4 Loading 46

3.8.5 Analysis type 47

Chapter 4 Parametric Study Of FEA With Discussions

4.1 General 49

4.2 Basic Features of the Present Parametric Study 49

4.3 Main Features of Bathon’s Test 50

4.4 Presentation of Test Results by FEM 51

4.4.1 Parametric Study of Equal-leg Steel Angles 52

4.4.2 Parametric Study of Unequal-leg Steel Angles 55

4.5

Discussion on Results of the FEA and Comparison with Bathon’s test Results

58

4.5.1 Main Observations 58

4.5.2 Effect of Eccentricity 59

4.5.3 Effect of Slenderness Ratio 59

Chapter 5 Conclusion

5.1 General 60

5.2 Outcomes of the Study 60

5.3 Future Recommendations 60

Appendix

References 62

ANSYS Script 64

Introduction

1

Chapter 1

INTRODUCTION

1.1 GENERAL

Steel angles are used for many purposes in various types of structures. One particular type of these structures is lattice structure like electrical transmission lines, which are supported by a variety of structural configurations, and are most frequently made of steel angles. These lattice structures are normally analyzed and designed assuming that each member is a truss member (i. e. a two force member) and it resist load by tension and/or compression.

Therefore under compression, angles function as column member in a lattice structure. But they are usually a poor choice for column from structural consideration. Yet they are preferred in lattice structure because their simple shape facilitates easier connection with relatively light weight than other type of members. These angles are usually assembled in a structure by bolt connections or by welding. In real structure, these connections are often not provided concentrically along with the center of gravity of cross section of the angle. Therefore application of load on such member induces eccentric axial effect. This imposed situation under compression on the angle is quite similar with the behavior of simple column subjected to eccentric compression. Performance of the angle under such loading depends on the various parameters like the eccentricity, material strength, material imperfections, member geometry, connection type at end and load characteristics. In addition to axial effect, the ultimate capacity of the angles subjected to eccentric compression is significantly affected by special type of bending referred as buckling. However fundamental theories for design of safe lattice structure like electrical transmission tower is based on mostly concentric application of loads which often does not match with real situations where axial effect and buckling simultaneously act. Again these theories often consider linear member behavior. But real structures frequently show nonlinear behavior. Therefore the predicted behavior of member from analytical approach is not always true. In order to ensure proper stability and to approximate actual behavior, eccentricity and geometric nonlinearity must be considered in calculation. For this reason there is a significant scope to investigate this matter.

In this thesis, considering above features and criteria, ultimate capacity of steel angle sections subjected to eccentric compression is to be analyzed by nonlinear geometric finite element analyses following some standard specifications. This investigation is to be carried out for a few numbers of steel angles. Then the results of the analyses would be compared with experimental results of previous research project done by L. Bathon (1993) both numerically and graphically. It is expected that this study will provide some definite guidelines and recommendations to the structural engineer for designing and analyzing steel angles under compression with adequate safety and serviceability.

Introduction

2

1.2 OBJECTIVE AND METHODOLOGY

In this study, steel angles with different geometric and material properties, slenderness ratios and specified edge restraint condition and under specified axial loading would be analyzed by finite element method (FEM) to find the values of critical stress; that occurs before failure. Here following ASCE Manual 52 (1988) specifications for modeling the geometry of the member, analytical formulae and the FEM would be applied to obtain the results. Again to compare the FEM results with real test data, results of previous research regarding this feature done by Bathon in 1993 would be used. For buckling analysis of compression members, Euler’s formula and the ASCE Manual 52 (1988) formulae would be used here. Finally results from various approaches would be compared numerically and graphically with necessary discussions, suggestions and recommendations for further investigation.

1.3 SCOPES AND LIMITATIONS

The whole analyses performed in this research have certain scopes, assumptions and limitations like

• The investigation is based on nonlinear geometric analysis (P-Δ analysis) by finite element method.

• Only a few numbers of angles are to be analyzed here. All of them are single angle with “L” shape.

• Simplified geometric parameters are considered in calculation of angle properties. • End restrained conditions are simplified in FEM than that presents at actual case, i.e.

bolt connections at end are replaced by introducing rectangular end plates with higher stiffness.

• It is assumed that end plates do not influence on angle behavior. • Material imperfections are not considered in FEM. • Static and nonlinear 3D analysis will be performed. • Nonlinear material and element properties will be assigned. • No shear effect would be considered in the analysis. Only the axial and bending effect

would be analyzed. • Lateral deflection is very small.

Introduction

3

1.4 ORGANIZATION OF THE STUDY

The report of the analysis is organized in this paper to represent and discuss the problem and findings that come out from the studies performed by FEM in most convenient way.

Chapter 1 introduces the topic, 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, and is 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 core of this analysis, which elaborately describes the computational investigation made throughout the study in details with presentation by many tables and figures followed by some discussions and recommendations.

Chapter 5, the concluding chapter, summarizes the whole study as well as points out its outcome with some further scope and directions.

Literature Review

4

Chapter 2

LITERATURE REVIEW 2.1 GENERAL

Steel angle, as a structural element, is subjected to both tension and compression while using at electrical transmission towers or at girder plates. Under compression it acts like column member. While axially loaded with tension, the ultimate capacity of a column may be found by multiplying stress developed and the cross sectional area of the column and the relationship is linear. But in case of compression, this is not always true perfectly. Under compression the ultimate capacity and failure mode of an axially loaded member depends on various criteria like buckling effect, intensity of eccentricity of loading, restraint conditions, slenderness ratio, and structural nonlinearities. Hence the design and analysis of such member becomes considerably complex. For this reason, quite a large number of experiments and analysis have been performed to predict the structural behavior of such member and lots of theories have been developed. As our main concern in this thesis is steel angles under compression, important features for analyzing such member with theories for analyzing nonlinear buckling behavior would be discussed on ongoing chapter.

2.2 STEEL ANGLES

2.2.1 Use of Steel Angles • Electrical transmission towers are structurally 3D truss member in most cases. Its

structural form may be H-frame structure, guyed structure or lattice structure. Steel angles are frequently used to make lattice structure where they act as two force member i.e. axially loaded. Figure 2.1 shows such a tower composed of steel angles.

• For large span systems, to facilitate stable frame with light weight, trusses are used as roofs, commonly referred as roof trusses. Steel angle sections are used to make such elements (figure 2.2).

• Steel angles are used as bracing member in bridge girders to provide stability. • Lintels are also made of steel angles where they act like beam element and subjected

to biaxial bending mainly.

Literature Review

5

Figure 2.1: Pictures of Steel Angle Sections Used in Electrical Transmission Tower

Literature Review

6

Figure 2.2: Common Types of Roof Trusses

Literature Review

7

Figure 2.2: Common Types of Roof Trusses (contd.)

Literature Review

8

2.2.2 Common Types of Steel Angles

Steel angles may be of two types such as single angled and double angled depending on the arrangement of the angle. Here we will discuss only single angle sections. The cross section of a single angle is usually “L” shaped with uniform thickness at both legs. The two legs may be equal to each other or not and have some standard dimensions according to proper specifications. For example, an angle of dimensions (in mm) 102×102×6 means a single angle having equal legs of 102mm with uniform thickness of 6mm. Similarly an angle of dimensions (in mm) 89×76×5 means a single angle having one 89mm and one 76mm leg with uniform thickness of 5mm. In figure 2.3 some single angle sections are shown.

Figure 2.3: Steel Angle Sections

Literature Review

9

2.3 END CONNECTION

Usually steel angles are connected at the end by means of bolts or welding with the gusset plates in a structure (figure 2.1). The bolts are made of steel also and have some standard geometric parameters. The connections are provided in such way that the combination of angle, bolts and gusset plate acts as an integrated part of the whole structure to provide necessary stability.

2.4 STEEL PROPERTIES (material for angle section)

To meet the ASTM (American Society for Testing and Materials) specifications, A36 steel is mostly used for angle sections. A36 is a low carbon steel with yield strength of 248MPa and tensile strength of 400MPa-550MPa for a thickness up to 200mm.

2.5 LINEAR AND NONLINEAR STRUCTURAL BEHAVIOUR

Structural behavior is mainly referred to the stress-strain relationship or load-deflection characteristics of a member. In elastic theories it’s usually considered linear. However in practical, almost all structural elements show different types nonlinearity under loading. These nonlinearities are generated from loading, geometric and material considerations. So to approximate the real behavior of angle section accurately, we must consider these nonlinearities with proper assumptions. Therefore we would perform here nonlinear geometric analysis of buckling behavior of steel angle sections under compression.

2.6 COMPRESSION MEMBERS

Compression members, such as columns, are mainly subjected to axial forces. The principal stress in a compression member is therefore the normal stress

ΑFσ = (2.1)

The failure of a short compression member resulting from the compression axial force looks like as shown in figure 2.4.

Literature Review

10

Figure 2.4: Failure Mode of Short Compression Member

However, when a compression member becomes longer, the role of the geometry and stiffness becomes more and more important. For a long (slender) column, buckling occurs before the normal stress reaches the strength of the column material. For example, pushing on the ends of a business card or bookmark can easily reproduce the buckling as shown in figure 2.5.

Figure 2.5: Failure Mode of Slender (Long) Compression Member

For an intermediate length compression member, kneeling occurs when some areas yield before buckling, as shown in the figure 2.6 below.

Literature Review

11

Figure 2.6: Failure Mode of Intermediate Compression Member

Whether a compression member is short, long or intermediate, it depends on the stiffness and

geometry (slenderness ratio, rLe ). The short, intermediate or long classification of columns depending on both the geometry (slenderness ratio) and the material properties (Young's modulus and yield strength) is given in TABLE 2.1.

TABLE: 2.1: Slenderness Ratio (SR) Limits for Different Types of Columns

Material Short Column

(strength limit)

Intermediate Column

(inelastic stability limit)

Long Column

(elastic stability limit)

Structural Steel SR<40 40<SR<150 SR>150

Aluminum Alloy (AA 6061 - T6)

SR<9.5 9.5<SR<66 SR>66

Aluminum Alloy (AA 2014 - T6)

SR<12 12<SR<55 SR>55

Wood SR<11 11<SR<30 30<SR<50

Literature Review

12

The slenderness ratio is mathematically defined by

rLSR e= (2.2)

In equation 2.2

SR = slenderness ratio

Le = effective length of column (TABLE 2.2)

r = minimum radius of gyration, AIr min=

Imin = minimum moment of inertia of area about centroidal axis

A = cross sectional area of column

The effective length or the buckling length (Le) of a column depends on its physical (actual) length (L) and its end conditions. Euler discovered that if a column is hinged at both ends, it will buckle in the form of a sine curve with the inflection points at the hinges. This would be the case in which the buckling length of a column is identical to its length. This is not the case if the ends of the column are other than both hinged.

In following, figure 2.7 and TABLE 2.2 list the effective lengths for columns terminating with a variety of boundary condition combinations. Also a mathematical representation of the buckled mode shape is given in TABLE 2.2.

(a) (b) (c) (d)

Figure 2.7: Buckling Behavior of a Simple Column for Different End Restraint Conditions (a) hinged-hinged (b) fixed-hinged (c) fixed-fixed (d) fixed-free end

Literature Review

13

TABLE 2.2: Effective Length (Le) and Shape of Buckling Mode of Simple Column for Different End Restraint Conditions

Boundary Conditions

Theoretical Effective Length

LeT

Engineering Effective Length

LeE

Buckling Mode Shape

Free-Free L 1.2L Lxπsin

Hinged-Free L 1.2L Lxπsin

Hinged-Hinged (Simply-Supported)

L L Lxπsin

Guided-Free 2L 2.1L Lx

2sin π

Guided-Hinged 2L 2L Lx

2cos π

Guided-Guided L 1.2L Lxπcos

Clamped-Free (Cantilever)

2L 2.1L Lx

2cos1 π

−

Clamped-Hinged 0.7L 0.8L

−+−

LxkLkxkLkx 1cossin

Here L

k π4318.1=

Clamped-Guided L 1.2L Lxπcos1−

Clamped-Clamped 0.5L 0.65L L

xπ2cos1−

L= Physical (actual) length

2.7 FAILURE MODES FOR COMPRESSION MEMBER WHEN BUCKLING GOVERNS

In case of short compression members, buckling may not be critical under working load; and when ultimate capacity is overcome by loading, the member collapses inform of shear failure or ideal compressive failure. But when buckling governs, the failure occurs depending on the type of buckling and the member stiffness characteristics as well as on geometric and material imperfections. Generally there are three types of buckling failure modes for columns (as well as for other compression members) as follows

Literature Review

14

(a) Flexural Buckling (b) Local buckling (c) Torsional Buckling

2.7.1 Flexural Buckling

It’s the primary type of buckling and also referred as Euler Buckling. Members are subjected to flexure or bending when they become unstable.

2.7.2 Local Buckling

This type of failure occurs when some part or parts of the cross section of the column are so thin that they buckle locally in compression before other modes of buckling can occur. The susceptibility of a column to local buckling is measured by the width-thickness ratio of the parts of its cross section.

2.7.3 Torsional Buckling

Columns with some certain cross section fail by buckling resulting from twist or torsion or from combination of flexure and torsion.

2.8 BUCKLING THEORIES FOR COMPRESSION MEMBERS

Buckling refers to the special phenomenon characterized by a sudden failure of a structural member subjected to high compressive stresses, where the actual compressive stress at the point of failure is less than the ultimate compressive stresses that the material is capable of withstanding. This mode of failure is also described as failure due to elastic instability. To analyze buckling characteristics, a lot of elastic as well as inelastic theories have been developed based on different eccentricities of loading and on different failure modes such as local buckling failure, flexural buckling failure, overall buckling failure etc. In following, some of the basic theories are described briefly for compression members.

2.8.1 Euler’s Formula for Elastic Buckling

In 1757, mathematician Leonhard Euler first derived a formula that gives the maximum concentric axial load that a long, slender and ideal column can carry without buckling. Euler defined an ideal column is one that is perfectly straight, homogeneous and free from initial stress. The maximum load, sometimes called the critical load, causes the column to be in a state of unstable equilibrium; that is, any increase in the load, or the introduction of the slightest lateral force, will cause the column to fail by buckling. The formula derived by Euler for columns with no consideration for lateral forces is

Literature Review

15

2

2

ecr L

EIP π= (2.3)

In above equation 2.3

Pcr = critical load (applied concentrically along line of cg of cross section of column)

E = elastic modulus (Young’s modulus) of column material

I = the least moment of inertia of cross sectional area of column

L = actual length of column


Figure 2.8: Euler Ideal Column Subjected to Concentric Axial Compression (F)

Euler first derived the formula for elastic buckling and then the concept was further extended by J. L. Lagrange and Thomas Young. Euler derived the formula only for perfect slender column (figure 2.8) and at later further investigation of the formula brought out following special facts and considerations.

• Boundary conditions other than simply-supported will result in different critical loads and mode shapes (TABLE 2.2).

• The buckling mode shape is valid only for small deflections, where the material is still within its elastic limit.

• The critical load will cause buckling for slender or long columns. In contrast, failure will occur in short columns when the strength of material is exceeded. Between the long and short column limits, there is a region where buckling occurs after the stress exceeds the proportional limit but is still below the ultimate strength. These columns are classified as intermediate columns and their failure is called inelastic buckling.

Literature Review

16

• Elasticity and not compressive strength of the materials of the slender column determines the critical load.

• The critical load is directly proportional to the second moment of area of the cross section.

• Euler formula is applicable while the material behavior remains linearly elastic and loading is concentric. Euler’s approach was generally ignored for design, because test results did not agree with it for nonlinear and inelastic behavior of columns and for eccentricity of loading on columns.

2.8.2 Theories for Inelastic Buckling

As stated on previous article, 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 (figure 2.6). This kind of situation is called inelastic buckling and Euler’s formula is not applicable for such columns to get the value of critical load. In following some basic theories for inelastic buckling are given.

2.8.2.1 Tangent Modulus Theory

Analyzing inelastic behavior, Engesser developed buckling formula to find critical load for intermediate columns based on modified stiffness of the material of column. He considered proportional limit instead of modulus of elasticity to analyze the inelastic behavior.

Figure 2.9: Stress-strain Curve Obtained from Engesser’s Tangent Modulus Theory

Literature Review

17

Suppose that the critical stress σ t in an intermediate column exceeds the proportional Limit of the material σpl. The Young's modulus at that particular stress-strain point is no longer E. Instead, the Young's modulus decreases to the local tangent value, Et. Replacing the Young's modulus E in the Euler's formula with the tangent modulus Et, the critical load becomes

te

tcr P

LIE

P == 2

2π (2.4)

And corresponding critical stress is

te

tcr

rL

Eσ

πσ =

= 2

2

(2.5)

In above equation 2.4 and 2.5

Pcr = critical buckling load for column

Pt = tangent modulus load

Et = tangent modulus of elasticity

I = minimum moment of inertia of cross sectional area of column


σcr = critical stress at buckling

σt = tangent modulus stress

σu = ultimate stress

r = minimum radius of gyration

Some of special features of tangent modulus theory are listed below.

• The proportional limit σpl, rather than yield stress σy, is used in the formula (figure 2.9). Although these two are often arbitrarily interchangeable, the yield stress is about equal to or slightly larger than the proportional limit for common engineering materials. However, when the forming process is taken into account, the residual stresses caused by processing can not be neglected and the proportional limit may drop up to 50% with respect to the yield stress in some wide-flange sections.

• The tangent-modulus theory tends to underestimate the strength of the column, since it uses the tangent modulus once the stress on the concave side exceeds the proportional limit while the convex side is still below the elastic limit (figure 2.9).

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18

• The tangent-modulus theory oversimplifies the inelastic buckling by using only one tangent modulus. In reality, the tangent modulus depends on the stress, which is a function of the bending moment that varies with the displacement w.

2.8.2.2 Reduced-Modulus Theory

The reduced-modulus theory is the modification of tangent modulus theory developed by Engesser and was first described by Salmon and Johnson. 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+

= (2.6)

And the critical load becomes

2

2

e

rcr L

IEP π= (2.7)

In equation 2.6

E = Young’s modulus below proportional limit

2.8.2.3 Shanley’s Concept on True Column Behavior

For the columns with same slenderness ratio, reduced modulus theory always gives slight higher buckling capacity of column than the tangent modulus theory. Both the theories were accepted in analyzing inelastic buckling behavior although the reason for discrepancy in their values was unknown. Later Shanley explained the phenomena based on his true column behavior concept. According to Shanley’s concept, the critical load of inelastic buckling is in fact a function of transverse displacement w. the critical load is located between the critical load predicted by tangent modulus theory (lower bound) and reduced modulus theory (upper bound).

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19

Figure 2.10: Shanley’s Idealized Column

Shanley’s idealized column consists of two rigid legs AC and BC connected at C by an elastic plastic hinge (figure 2.10). 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 tangent modulus theory, increases asymptotically to that given by reduced modulus theory.

He shows relation between applied load P (>Pt) and lateral deflection w (figure 2.10) is given by

−+

++=

ττ

11

2

11

wb

PP t (2.8)

EEt=τ , assumed constant

P = critical load

b = constant

The variation of critical load with variation in lateral deflection in inelastic buckling obtained from tangent modulus theory, reduced modulus theory and from Shanley’s concept is illustrated in figure 2.11.

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20

Figure 2.11: Inelastic Buckling Phenomena, Critical Load vs. Lateral Deflection Characteristics

In figure 2.11

σoc = elastic critical stress

σm = maximum stress which defines ultimate strength of member

2.8.2.4 Johnson Parabola Approach

Johnson derived critical stress as a parabolic function of slenderness ratio (figure 2.12). The equation of Johnson’s parabola to determine critical buckling load is given by

(2.9)

−=Er

L

AP

ey

ycr

2

2

41

π

σσ

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21

Figure 2.12: Column Stress as a Function of Slenderness Ratio

In equation 2.9

Pcr = critical buckling load for the column

σy = 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 = Young’s modulus of elasticity of the material of column

2.8.2.5 Gordon-Rankine Formula

This formula is used for columns with medium length. The critical buckling load is given by the equation

2

1

+

=

rla

AfP c (2.10)

Efa c2π

= (2.11)

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22

Here in equation 2.11

P = buckling load

fc = crushing stress as short column, fc=3200kg/cm2 for mild steel

a = constant, a=1/7500 for mild steel

In addition to above theories, further observations regarding inelastic buckling were made by Perry-Robertson, Tetmajer-Baucschinger, Lin, Duberg and Wilder. Most of their analysis was based on a lot of experiments on inelastic behavior of column member. These researches also included elastic-plastic nature of materials. It should be kept in mind that all the theories mentioned above are applicable for concentric loading.

2.8.3 Buckling Theories for Eccentrically Loaded Columns

In practical, columns are often subjected to eccentric loading which causes development of bending stress as well as generates possible buckling behavior. In following figure 2.13, illustration of the stress that a column experiences under increasing eccentric loading has been shown.

Figure 2.13: Form of the Stress Prism Changes from an Even Distribution to a Very Uneven Distribution Due to Increasing Eccentricity of Loading

From figure 2.13, we find that with increasing eccentricity of loading, the stress becomes bending in nature. This bending stress introduces bending moment, which a column section must resist in addition to compressive stress under equilibrium condition.

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23

Buckling theory for eccentrically loaded column was first originated from the research of Ostenfeld who took the attempt to derive the formula for such case. Later this concern was further developed by Karman, Westergard, Osgood and Chwalla. Regarding this problem, perhaps most valuable contributions were made by Jezek, Young, Pippard-Pritchard and Moncrieff. In following some theories for buckling analysis of eccentrically loaded column are given.

2.8.3.1 Jezek’s Approach

Based on simplified stress-strain curve (figure 2.14) consisting two straight lines, Jezek presented his analytical solution for steel columns. The result agrees reasonably well with that found from actual stress-strain relation. The underlying concept of Jezek involves following formulae.

Figure 2.14: Jezek’s Simplified Stress Distribution for Column

For stress distribution (figure 2.14) case a,

32

2

])1(3

1[)/(

−−=

PArL

EAP

yσκπ

when 0)3(9

)(32

2 >−

−κσ

κπ

y

ErL

(2.12)

Literature Review

24

For stress distribution case b,

])

32(

)/([

324

34

κσ

σπ

σ

−−=

AP

PA

E

rLAP

y

y

y when 0)3(9

)(32

2 <−

−κσ

κπ

y

ErL

(2.13)

In equation 2.12 and 2.13

P = axially applied ultimate column load

A = cross sectional area of column



σy = yield stress of column

E = Young’s modulus of elasticity

e = equivalent eccentricity

s = core radius, s=h/6 for rectangular section where h is depth of section

2.8.3.2 Secant Formula

Secant formula was developed by Young in which he assumed that the failure load is the load which generates beginning of yielding at highest stressed fiber. He developed column stress curves for different eccentricity ratios as shown in figure 2.15.

Figure 2.15: Column Stress Curves for Different Eccentricity

Literature Review

25

Young’s secant formula for critical load is given by

)4

sec1( 2max EAP

rL

rec

AP

+=σ (2.14)

In equation 2.14

σmax = maximum stress not exceeding elastic limit

P = eccentrically applied axial column force

e = eccentricity of loading

A = cross sectional area



E = Young’s modulus of elasticity for column

c = distance of centroidal axis to most stressed fiber in compression

ec/r2 = eccentricity ratio

2.8.3.3 Modified Rankine’ s Formula

Pippard and Pritchard have adjusted Rankine’s formula for axially loaded struts to apply it on eccentrically loaded columns. For axially loaded short columns, according to Rankine maximum compressive stress developed is given by equation

+

=2

1rla

fAP c (2.15)

In equation 2.15

P = axial load applied

A = cross sectional area

fc = allowable unit stress

a = constant

l/r = slenderness ratio

Literature Review

26

Pippard and Pritchard made necessary adjustment of formula given in equation 2.15 for eccentrically loaded columns as follows.

We know that for an eccentrically loaded compression member, the compressive stress developed is given by equation

IPey

APf c

c += from which we get

+

=

21reyf

AP

c

c (2.16)

Pippard and Pritchard combined equation 2.15 and 2.16 as

+⋅

+

=

2

2

11rey

rla

fAP

c

c (2.17)

In equation 2.17

E = eccentricity of loading

yc = distance of stressed fiber from centroidal axis


Equation 2.17 gives the formula to find critical load for eccentrically loaded columns.

2.8.4 Conventional Engineering Formulae to Determine Ultimate Load Capacity under Buckling Consideration of Structural Compression Members

2.8.4.1 ASCE Manual 52 (1988) Formula

According to ASCE Manual 52 (1988), for compression members with normal framing eccentricities at both ends of the unsupported panel

rLrLe 75.030 += for 1200 ≤≤ rLe (2.18)

rLrLe = for 200120 ≤≤ rLe

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27

ycr FF = for yFt

w 80< (2.19)

( )y

y

cr F

F

tw

F

−=

80677.0677.1 for

yy Ftw

F14480

≤≤

2

9500

=

tw

Fcr for yFt

w 144>

crc

e

a FCrL

F

−=

2

211 for ce CrL ≤ (2.20)

2

286000

=

rL

Fa for ce CrL ≥

crc F

EC 2π= (2.21)

In above equations

L/r = actual slenderness ratio

Le/r = effective slenderness ratio

Cc = critical slenderness ratio

Fcr = critical stress

Fy = yield stress

w = effective width of cross section

t = thickness of cross section

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28

2.8.4.2 AISC Formula

AISC specification gives the following requirement for using design of compression members. The allowable stress should not exceed the following values

FSCrL yce

a

σσ

]2/)/(1[ 22−= for ce CrL < (2.22)

And 2

2

)/(2312

rLE

ea

πσ = for ce CrL ≥

Here

yc

EC σπ 22= ; for A36 steel Cc=126.1 (2.23)

The equation of factor of safety is

3

3

8)/(

8)/(3

35

c

e

c

e

CrL

CrL

FS −+= (2.24)

In above equations

Cc = critical slenderness ratio

σy = yield stress

σa = allowable stress

E = Young’s modulus of elasticity

2.8.4.3 AASHTO Formula

AASHTO formula is a simple adoption of the AISC formula in which AASHTO uses a different factor of safety. The AASHTO formulae for allowable stress (Fa) are

−=E

FrL

SFF

Fy

e

ya 3

2

41

.. π for ce CrL ≤ (2.25)

Literature Review

29

And 22

2 135000000

..

=

=

rL

rL

SF

EFee

aπ

for ce CrL ≥

Here y

c FEC

22π= and 12.2.. =SF

2.8.4.4 IS Code Formula

The direct compressive stress on axially loaded column must not be greater than Pc. The value of Pc is given by equation

′+

=

EcmP

rl

mf

P

y

c

4sec2.01

for 160~0=rLe (2.26)

−

′+

=r

l

EcmP

rl

mf

P

y

c 80012

4sec2.01

for 160≥rLe

Here

Pc = allowable average axial compressive stress

m = 1.68, factor of safety

2.8.4.5 CRC Formula

The basic column-strength curve adopted by the Column Research Council (CRC) is based on parabolic equation proposed by Bleich as

])(4

1[ 22 r

LE

eyycr π

σσσ −= (2.27)

In above equation 2.27

σcr = critical stress for column

Literature Review

30

2.9 PREVIOS RESEARCH

To analyze and design steel angles as the compression member, a vast of laboratory and field tests has been performed. To approximate buckling behavior of steel angles under compression, research projects are performed at past by Mueller and Erzurumlu (1981); Prickett and Meller (1983); Callaway (1984); Mueller and Wagner (1985); and Mueller, Prickett and Kempner (1988). Results of these research projects are collected in a data base, which consists of performance of steel angles in elastic, inelastic and post buckling region.

Recently, the ultimate compressive load carrying capacity of steel angles of equal and unequal leg has been investigated through a large number of tests by Leander Bathon (1993). These tests involve simply supported steel angles (at end) under eccentric compressive axial forces. Following the ASCE Manual 52 (1988) specifications for eccentricity and design equations for capacity, test results were compared for further investigations, recommendations, and analyses and design requirements. Following article describes basic features of Bathon’s test briefly.

2.9.1 Bathon’s Test

Bathon’s test involves determination of ultimate capacity of steel angle with its ends unrestrained against rotation and eccentricities within the limits specified in the ASCE Manual 52 (1988). These conditions define the design equations that are used. A comparison of test results vs. calculated capacity using the procedure of ASCE Manual 52 (1988) is made then. Thirty one single angle equal-leg and forty four single angle unequal-leg specimens are tested. To perform the test, except geometric properties within limits of ASCE Manual 52 (1988), not any other type of member imperfection is accounted for.

Figure 2.16: Schematic Test Apparatus of Bathon’s Test

Literature Review

31

Figure 2.16 shows a schematic diagram of the apparatus of Bathon’s test. Bathon used three types of end connections with bolts for his tests. These are the two-bolt, three-bolt and the five-bolt configuration. The details of end connections are shown in figure 2.17. Three-bolt and five-bolt configurations showed bolt slip while in case of two-bolt configuration the test member actually bears on the end plate. In the tests, a member was grinded down at unconnected leg of angle to ensure no contact with end plate and to ensure that the axial load only travels through the connected leg.

Figure 2.17: End Connections of Angle in Bathon’s Test

(a) Two-bolt (b) Three-bolt and (c) Five-bolt Configuration

The ball end connection was fabricated to load the angle through the centroid of the bolt pattern and along the connected face of the angle. The test member was loaded by a hydraulic actuator that was controlled by the material test system. A load cell was placed in series with the actuator providing the direct reading of the applied load that was displacement stroke controlled. The displacement was measured by a displacement transducer.

To verify the accuracy of test setup and to correlate the experimental result with the theory, a concentrically member was first tested. Then the test was performed for all the selected angles for eccentric loading. All of the specimens were failed due to overall member buckling. During the test, data were collected in an organized way and finally the results were interpreted. At last Bathon compared his test results with predicted behavior of the angles found from ASCE Manual formulae and theories.

Finite Element Method of Analysis

32

Chapter 3

FINITE ELEMENT METHOD OF ANALYSIS

3.1 GENERAL

On this thesis paper, the ultimate capacity of steel angles subjected to eccentric loading would be determined using finite element method. Nonlinear buckling analysis under compression, using suitable finite element package software (computer based program) would be performed for this purpose. Finite element method (FEM) is a numerical technique. In this method, all the complexities of the problems, like varying shape, boundary conditions and loads are maintained as they are but the solution obtained is approximate. Because of its diversity and flexibility as an analysis tool, it is receiving much attention in engineering. In this chapter, finite element modeling technique of steel angles will be described clearly and concisely. Also related features regarding finite element analysis (FEA) would be described briefly.

3.2 A BRIEF HISTORY OF FINITE ELEMENT METHOD (FEM)

The FEM was originated as a method of stress analysis in the design of aircrafts. But the engineers, physicists and mathematicians developed the method independently. In 1943 Courant made the first effort to use piecewise (finite elements) continuous functions. Later method of analyses based on this concept was developed by Weinberger, Polya, Argyris and Kelsey. In 1960 Clough first introduced the word ‘Finite Element Method’.

During the era of 60’s, 70’s and 80’s, considerable progress was made in the field of FEA. The improvement in speed and memory capacity of computers largely accelerated the advancement of this method. Continuous development induced FEM to handle three dimensional problem, stability and vibration problems, nonlinear analysis etc quite successfully.

Today the FEM is used not only for the analysis of solid mechanics but even in the analysis of fluid flow, heat transfer, electric and magnetic fields and many others. Civil engineers use this method for analysis of beams, space frames, columns, plates, shells, folded plates, foundations, rock mechanics etc. both static and dynamic problems can be handled by FEA.


33

3.3 GENERAL DESCRIPTION OF THE FEM

In engineering problems, there are some basic unknowns. If they are found, the behavior of the entire structure can be predicted. The basic unknowns or field variables which are encountered in engineering problems are for example, displacements in solid mechanics, velocities in fluid mechanics and temperature in heat problems.

In a continuum, these unknowns are infinite. In FEM, such unknowns are reduced to finite number by dividing (meshing) the solution region into small parts called elements and by expressing the field variables in terms of assumed interpolating functions or shape functions. The finite elements are connected by certain points called nodes or nodal points (figure3.1). The shape functions are defined in terms of nodal field variables and the field variable at any point within the element can be found using appropriate shape functions.

Figure 3.1: A Typical Finite Element Model


34

After selecting elements and nodal unknowns, the element properties are assembled for each element mathematically and then global properties for whole structure is developed using the element properties. Next the boundary conditions are imposed and solution of the mathematical relationship is carried out to find nodal variables. The whole process of FEM may be mentioned in following steps.

Figure 3.2: Discretization of Continuum

• Select suitable field variables and the elements. • Discretize the continuum considering various discontinuities (figure 3.2). • Select interpolating functions. • Find the element properties for each element. • Assemble element properties to global properties. • Impose boundary conditions. • Solve the system equations to get nodal unknowns. • Find field variables required using nodal variables.

3.4 FINITE ELEMENT PACKAGES

The fast improvement of computer hardware technology and slashing of cost of computers have boosted the use of FEM, since computer is the basic need for the application of this method on complex structures. A large number of FEA packages are available at present for analyzing


35

different type of structures. They vary in degree of complexity and versatility. Some of popular packages are

• ANSYS • STAAD-PRO • ABAQUS • GT-STRUDEL • NASTRAN • DYNA-3D • SDRC • ALGOR • PATRAN • NISA

3.5 ANSYS: THE FEM PACKAGE FOR PRESENT ANALYSIS

ANSYS is very widely used for FEA analysis of several types of structures for its versatility, reliability and easy application technique. Moreover ANSYS can be used in several fields of physics, mathematics and engineering. ANSYS enables users to

• Build computer models or transfer CAD models of structures, products, components, or systems.

• Apply operating loads or other design performance conditions.

• Study the physical responses, such as stress levels, temperature distributions, or the impact of electromagnetic fields.

• Optimize a design early in the development process to reduce production costs.

• Do prototype testing in environments where it otherwise would be undesirable or impossible.

• Both static and dynamic analysis considering linear or nonlinear behavior of element and material can be performed without complexities.

The ANSYS program has a comprehensive graphical user interface and quick access commands that help user to study and function the program quite comfortably. For analysis of buckling behavior of steel angles, we will perform finite element modeling of the member using ANSYS version 10.0 (i.e. ANSYS10.0).


36

3.6 FINITE ELEMENT MODELING OF STRUCTURE IN ANSYS10.0

Figure 3.3: Qualitative Sketch of Steel Angle with End Plates at Its Both Ends and Subjected to Axial Compressive Load (Isometric View)

3.6.1 Modeling of Steel Angle with End Plates

For simplification, bolt connections at the end of steel angle are to be replaced by introducing end plates (figure 3.3) to facilitate nonlinear 3D analysis by ANSYS. But since our main concern is the behavior of angle, modeling of end plates should be such that they don’t interfere to the angles behavior under load. Therefore it would be modeled with sufficient stiffness so that applied load is fully imposed on steel angles and doesn’t affect considerably to the end plate.

3.6.2 Element Modeling: SHELL181

In ANSYS shell elements are special type of elements that are designed to efficiently model thin structures. Therefore regarding our concern, shell elements may be used to model angles.

SHELL181 is a 4-noded finite strain element with six degrees of freedom at each node, translation in x, y and z direction; and rotation about the x, y and z axes. SHELL181 is well suited


37

for linear, large rotation and/or large strain nonlinear applications. Therefore incase of nonlinear and inelastic problem, the element can also be used for modeling.

The steel angles in this analysis would be modeled with SHELL181 element. The basic geometric parameters of SHELL181 element are given in figure 3.4.

Figure 3.4: SHELL181 Geometry

3.6.2.1 SHELL181 Input Data

The geometry, node locations, and the coordinate system for this element are shown in figure 3.4. The element is defined by four nodes: I, J, K, and L. The element formulation is based on logarithmic strain and true stress measures. The element kinematics allows for finite membrane strains (stretching).The thickness of the shell may be defined at each of its nodes. The thickness is assumed to vary smoothly over the area of the element. If the element has a constant thickness, only TK (I) needs to be input. If the thickness is not constant, all four thicknesses must be input. A summary of the element input is given in Table 3.1.


38

TABLE 3.1: SHELL181 Input Data Summary

Element Name SHELL181 Input Data Nodes

I, J, K, L

Degrees of Freedom

UX, UY, UZ, ROTX, ROTY, ROTZ if KEYOPT (1)=0 UX, UY, UZ if KEYOPT (1) = 1

Real Constants

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

E11, E22, E12, DRILL, MEMBRANE, BENDING Material Properties

EX, EY, EZ, (PRXY, PRYZ, PRXZ, or NUXY, NUYZ, NUXZ),

ALPX, ALPY, ALPZ (or CTEX, CTEY, CTEZ or THSX, THSY, THSZ),

DENS, GXY, GYZ, GXZ

3.6.2.2 SHELL181 Assumption and Restriction

• Zero area elements are not allowed (this occurs most often whenever the elements are not numbered properly).

• Zero thickness elements or elements tapering down to a zero thickness at any corner are not allowed (but zero thickness layers are allowed).

• In a nonlinear analysis, the solution is terminated if the thickness at any integration point that was defined with a nonzero thickness vanishes (within a small numerical tolerance).

• This element works best with full Newton-Raphson solution scheme.

• The through-thickness stress, SZ, is always zero.

• Using this element in triangular form is not recommended.

• If reduced integration is used (KEYOPT (3) = 0) SHELL181 will ignore rotary inertia effects when an unbalanced laminate construction is used.

• If multiple load steps are used, the number of layers may not change between load steps.


39

3.6.3 Modeling of Material Properties: BKIN

The Bilinear Kinematic Hardening (BKIN) option assumes that the total stress range is equal to twice the yield stress; this option is recommended for general small-strain use for materials that obey von Mises yield criteria (which includes most metals). It is not recommended for large strain applications.

Figure 3.5: Bilinear Kinematic Hardening

In figure 3.5, denotation σ, ε, E and T represent stress, strain, modulus of elasticity and the temperature of the material respectively.

3.7 BUCKLING ANALYSIS BY FEM (USING ANSYS10.0)

There are two techniques available for performing buckling analysis in FEM as linear or Eigen value buckling analysis and nonlinear buckling analysis. These two techniques frequently yield quite different results. The basic features of these two approaches are described below.

3.7.1 Linear or Eigen Value Buckling Analysis

The first method, Eigen value buckling analysis, predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure. This method corresponds to the textbook approach to elastic buckling analysis; for instance, an Eigen value buckling analysis of a column will match the classical Euler solution. However, imperfections and nonlinearities prevent most real-world structures from achieving their theoretical elastic buckling strength. Thus, Eigen value buckling analysis often not yields conservative results, and is not generally used in actual day-to-day engineering analysis (figure 3.6 (b)).


40

Figure 3.6: Load vs. Deflection Curves for (a) Nonlinear Buckling (b) Linear (Eigen Value) Buckling

3.7.2 Nonlinear Buckling Analysis

Nonlinear buckling analysis is usually the more accurate approach. It is recommended for design or evaluation of actual structures. This technique employs a nonlinear static analysis with gradually increasing loads to seek the load level at which a structure becomes unstable. Using the nonlinear technique, models can include features such as initial imperfections, plastic behavior, gaps, and large-deflection response. In addition, using deflection-controlled loading, the post-buckled performance of structures (which can be useful in cases where the structure buckles into a stable configuration, such as "snap-through" buckling of a shallow dome)can be obtained (figure 3.6 (a)).

In contrast to Eigen value buckling, nonlinear buckling phenomenon includes a region of instability in the post-buckling region whereas Eigen value buckling only involves linear, pre-buckling behavior up to the bifurcation (critical loading) point (figure 3.7).


41

Figure 3.7: Nonlinear vs. Eigen Value Buckling Behavior

The unstable region above (figure 3.7) is also known as the “snap through” region, where the structure “snaps through” from one stable region to another. To illustrate, the loaded shallow arch (figure 3.8) may be considered.

Figure 3.8: ‘Snap Through’ Buckling Behavior

Generally the FEM involves two approaches for nonlinear buckling analysis as (a) Newton-Raphson method and (b) arc length method. They are briefly described below.

3.7.2.1 Newton-Raphson Method

In FEM, for most nonlinear analyses, the Newton-Raphson method is used for convergence of the solution at each time step along the force deflection curve. The Newton-Raphson method works by iterating the equation


42

[KT]{u} = {Fa}-{Fnr} (3.1)

Where {Fa} is the applied load vector and {Fnr} is the internal load vector, until the residual, {Fa} - {Fnr}, falls within a certain convergence criterion. The Newton-Raphson method increments the load to a finite amount at each sub step and keeps that load fixed throughout the equilibrium iterations. Because of this, it cannot converge if the tangent stiffness (the slope of the force-deflection curve at any point) is zero (figure 3.9).

Figure 3.9: Newton-Raphson Method

3.7.2.2 The Arc Length Method

The arc-length method causes the Newton-Raphson equilibrium iterations to converge along an arc, thereby often preventing divergence, even when the slope of the load vs. deflection curve becomes zero or negative. This iteration method is represented schematically in figure 3.10. Therefore the arc-length method should be used for solving nonlinear post-buckling where Newton-Raphson method fails to converge. To handle zero and negative tangent stiffness, the arc-length multiplies the incremental load by a load factor, λ, where λ is between -1 and +1. This addition introduces an extra unknown, altering the equilibrium equation slightly to

[KT]{u} = λ{Fa}-{Fnr} (3.2)

To deal with this, the arc-length method imposes another constraint, stating that

=+∆ 22 λnu (3.3)


43

Equation 3.3 is valid throughout a given time step, where is ℓ the arc-length radius. Figure 3.10 illustrates this process.

Figure 3.10: The Arc Length Method of Nonlinear Solution

The arc-length method therefore allows the load and displacement to vary throughout the time step as shown in figure 3.11.

Figure 3.11: The Arc Length Convergence Behavior


44

3.8 ILLUSTRATIVE MODEL GENERATION FOR FEA IN (ANSYS)

Based on above FEM concepts mentioned, now considering finite strain element (SHELL181), small deflections, plastic material properties (BKIN) and nonlinear approach (the arc length method), an illustrative model of steel angle will be generated with following features and also necessary sketches obtained from modeling in ANSYS will be given.

3.8.1 Input Parameters

Figure 3.12: Angle Parameters and Coordinate Direction for Present Modeling

Various angle and end plate input parameters are listed in following.

Parameter Reference value

Angle dimension in X-direction, aa 102 mm

Angle dimension in Y-direction, bb 102 mm

Thickness of angle, t1 6 mm

Thickness of end plate, t2 25 mm

Corner dimension of end plate same as for angle in both direction

Eccentricity (location of cg of bolt pattern in x-direction), gg

38 mm

Young’s modulus of elasticity, E 200 KN/mm2

Yields stress for the angle, σy 0.3259 KN/mm2

Poison’s ratio, ν 0.30

Applied load Slightly greater than critical buckling load obtained from Euler’s formula

For end plates, E=2000 KN/mm2, ν=0 are assigned to obtain greater stiffness.


45

3.8.2 Meshing

Performing rectangular meshing, the end plates (in xy plane) are modeled in such way that element aspect ratio closes to value one. Angles are meshed in z direction (in xz and in yz plane) also with same concept (figure 3.13).

(a)

(b) (c)

Figure 3.13: Finite Element Meshing of Present Model [Angle L 102×102×6; l/r= 60] (a) Isometric View (b) Front (c) Side Elevation


46

3.8.3 Boundary Conditions

One end plate is kept restrained in all the x, y and z direction while the other is kept restrained in x and y direction only to allow free deflection along z direction or along length of the angle. Load is applied on the later one. In all cases, the whole model is kept unrestrained against rotation (figure 3.14).

Figure 3.14: Boundary Conditions for Present Model [Angle L 102×102×6; l/r= 60]

3.8.4 Loading

While modeling, options are provided for both the concentric and eccentric loading. Eccentricity of loading is applied following Bathon’s test parameters specified by ASCE Manual 52 (1988). Load on the angle, is applied slightly higher than the value obtained from Euler’ formula along z direction (figure 3.15).


47

Figure 3.15: Boundary Conditions with Loading for Present Model [Angle L 102×102×6; l/r= 60]

3.8.5 Analysis Type

After the model has been generated, type of analyses is assigned. For present problem, the analysis is static and nonlinear. Nonlinear geometry is also assigned for the model. To satisfy the convergence criteria, arc length method is introduced for the solution. Adequate numbers of load steps are provided to analyze load-deflection characteristics of the model (figure 3.16 and 3.17).


48

Figure 3.16: Typical Deflected Shape of the Model [Angle L 102×102×6; l/r= 60]

Figure 3.17: Typical Load vs. Deflection Curve for Concentric Loading from Nonlinear Buckling Analysis Using FEA for Different Slenderness Ratios, (l/r) [Angle L 102×102×6].

0

50

100

150

200

250

300

350

400

450

0 0.5 1 1.5 2 2.5 3 3.5

load

(kn)

deflection(mm)

(130)

(120) (110)

(90)

(60) (40)

(20)

(10)

l/r

Parametric Study of FEA with Discussions

49

Chapter 4

PARAMETRIC STUDY OF FEA WITH DISCUSSIONS

4.1 GENERAL

Based on analytical theories and FEA concept as mentioned in chapter 2 and in chapter 3, present chapter includes parametric study of nonlinear buckling analysis of steel angle compression members. The specifications of ASCE Manual 52 (1988) would be followed here for this purpose. Test results will be compared with most recent research results performed by Bathon in 1993 under same specifications. Also the observations will be compared with fundamental buckling theory of Euler and the ASCE Manual 52 (1988) formulae for compression members. The comparison of the results will be presented both numerically and graphically with appropriate discussion on the outcome of the study; and with recommendation for further investigations regarding this analysis.

4.2 BASIC FEATURES OF THE PRESENT PARAMETRIC STUDY

• The present analysis performed similarly to that performed by Bathon 1993 i.e. various geometric, strength and loading parameters with boundary conditions of the present model is taken as same as Bathon with ASCE Manual 52 (1988) specifications.

• Using Euler’s formula, axial displacement of the model under buckling is obtained at first. Then based on that displacement value, nonlinear buckling analyses through FEM in ANSYS is carried out to find the critical load causing that displacement.

• The application of load is eccentric. In this analysis, the value of the eccentricity is equal to the value of distance between the center of gravity (cg) location of bolt pattern connected with one side of the angle and the cg line of other side (figure 3.13). The bolt is always connected with the longer side of the angle cross section (ASCE Manual 52 (1988)).

• During analysis, the displacement values, number of loading sub steps, arc length radius limits and solution convergence criteria are adjusted with a number of trials to get the optimum results.

• A total six number of steel angle sections are analyzed among which, three are equal leg angles and the other three are unequal leg angles. The geometric and strength parameters follow ASCE Manual 52 (1988) specifications. These parameters are listed in TABLE 4.1 and in TABLE 4.2.

• The analyses are performed for the slenderness ratio of 10, 20, 40, 60, 90, 120, 150 and 180 respectively.


50

4.3 MAIN FEATURES OF BATHON’S TEST

• Bathon’s test is basically an ongoing research program on analyses of nonlinear buckling behavior of steel angles. Certain tests were selected by Bathon concerning this analysis, in order to add missing information to prior research projects of Callaway, Mueller, Erzurumlu, Wagner, Prickett and Kempner.

• He tested a total number of seventy five steel angles among which, thirty one are equal leg and forty four are unequal leg angles. Eccentricity was included in the tests for applying axial load on angles.

• Bathon followed ASCE Manual 52 (1988), for his tests on nonlinear buckling analyses. He also compared his results with values obtained from ASCE Manual 52 (1988) formulae.

• The tests include determination of critical load and stress within presence of buckling phenomenon. Results of his research consist of performance of single angle members in the elastic, inelastic and post buckling region. For example, figure 4.1 shows the overall buckling behavior of L 102×102×6 steel angles from Bathon’s test results.

Figure 4.1: Load-Deflection Curve from Bathon’s Test Results for Steel Angle L 102×102×6

• All the test specimens were failed due to overall member buckling. • No attempts were made to account for member imperfections in the test results.


51

Among the test numbers of Bathon’s analyses, the following tests as listed in TABLE 4.1 and in TABLE 4.2; are selected for present analysis by FEM using ANSYS10.0.

TABLE 4.1: Test Results of Bathon for Various Equal-Leg Steel Angles.

Test Number

Angle Size Area

(mm2)

Eccentricity of Loading

(mm)

Slenderness Ratio, l/r

Yield Stress,

fy (MPa)

Actual Capacity

(MPa)

Capacity by ASCE

Formulae (MPa)

14 44×44×3 272 25 60 378.3 57.9 97.1 12 44×44×3 272 25 90 378.3 59.9 75.4 11 44×44×3 272 25 120 378.3 53.1 57.7 8 76×76×6 928 37 60 347.9 212.2 321.5 6 76×76×6 928 32 90 347.9 209.5 256.9 3 76×76×6 928 32 120 347.9 177.8 196.7 7 102×102×6 1250 38 60 325.9 304.5 394.8 4 102×102×6 1250 37 90 325.9 226.7 334.3 5 102×102×6 1250 38 120 325.9 164.0 265.1

TABLE 4.2: Test Results of Bathon for Various Unequal-Leg Steel Angles.

Test Number

Angle Size Area

(mm2)

Eccentricity of Loading

(mm)


Yield Stress, fy

(MPa)

Actual Capacity

(MPa)

Capacity by ASCE

Formulae (MPa)

72 63×51×5 522 25 60 344.5 121.3 179.8 68 63×51×5 522 25 90 344.5 89.6 144.0 66 63×51×5 522 25 120 351.4 72.3 110.9 19 89×76×6 1006 35 60 340.4 229.4 345.9 26 89×76×6 1006 35 90 340.4 192.2 278.8 31 89×76×6 1006 35 120 340.4 124.7 213.8 44 102×89×6 1168 38 90 349.3 206.7 317.6 53 102×89×6 1168 38 120 332.1 147.4 247.4 58 102×89×6 1168 38 150 349.3 111.6 158.5

4.4 PRESENTATION OF TEST RESULTS BY FEM

This article consists of results of FEA of above tests mentioned in TABLE 4.1 and TABLE 4.2, their comparison with Bathon’s test results, Euler’s formula and with ASCE Manual 52 (1988) formula; and finally graphical representation of obtained results from different approaches.


52

4.4.1 Parametric Study of Equal-Leg Steel Angles

TABLE 4.3: Critical Stress for Angle L 44×44×3; Eccentricity= 25mm; fy= 378.3MPa


Critical Stress (MPa) Euler’s Formula ASCE Formula Present Analysis Bathon’ Test

10 376.5 167.3 20 371.1 158.5 40 349.3 150.5 60 313.0 97.1 137.8 57.9 90 231.5 75.4 116.2 59.9

120 137.7 57.7 91.5 53.1 150 88.0 68.2 180 61.1 51.1

Figure 4.2: Critical Stress vs. Slenderness Ratio (l/r) Curve [Angle L 44×44×3]

0

50

100

150

200

250

300

350

400

0 50 100 150 200

criti

cal s

tres

s (M

pa)

l/r

present analysisEuler's formulaBathon's testASCE formula

Angle: L 44×44×3


53

TABLE 4.4: Critical Stress for Angle L 76×76×6; Eccentricity= 32mm; fy= 347.9MPa Slenderness

Ratio, l/r Critical Stress (MPa)

Euler’s Formula ASCE Formula Present Analysis Bathon’ Test 10 346.4 226.6 20 341.8 208.4 40 323.4 203.7 60 292.7 321.5 195.3 212.2* 90 223.7 256.9 143.6 209.5

120 136.4 196.7 100.7 177.8 150 87.4 72.1 180 60.7 53.1

*Eccentricity=37mm


0

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100

150

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300

350

400

0 50 100 150 200

criti

cal s

tres

s (M

pa)

l/r


Angle: L 76×76×6


54



Euler’s Formula ASCE Formula Present Analysis Bathon’ Test 10 324.6 217.8 20 320.5 210.6 40 304.4 189.6 60 277.5 394.8 164.9 304.5 90 216.9 334.3 122.8 226.7*

120 136.6 265.1 89.3 164.0 150 87.5 66.1 180 60.7 49.9

*Eccentricity=37mm


0

50

100

150

200

250

300

350

400

450

0 50 100 150 200

criti

cal s

tres

s (M

pa)

l/r


Angle: L 102×102×6


55

4.4.2 Parametric Study of Unequal-Leg Steel Angles



Euler’s Formula ASCE Formula Present Analysis Bathon’ Test 10 343.0 411.6 20 338.5 266.4 40 320.5 242.2 60 290.4 179.8 238.1 121.3 90 222.8 144.0 223.9 89.6

120 136.9 110.9 164.3 72.3* 150 87.6 105.1 180 60.8 72.9

*fy= 351.4MPa


0

50

100

150

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250

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450

0 50 100 150 200

criti

cal s

tres

s (M

Pa)

l/r


Angle: L 63×51×5


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TABLE 4.7: Critical Stress for Angle L 89×76×6; Eccentricity=35mm; fy= 340.4MPa Slenderness


Euler’s Formula ASCE Formula Present Analysis Bathon’ Test 10 338.9 251.2 20 334.5 233.2 40 316.9 224.0 60 287.6 345.9 204.9 229.4 90 221.5 278.8 169.7 192.2

120 136.7 213.8 136.4 124.7 150 87.5 105.0 180 60.7 72.9


0

50

100

150

200

250

300

350

400

0 50 100 150 200

criti

cal s

tres

s (M

pa)

l/r


Angle: L 89×76×6


57

TABLE 4.8: Critical Stress for Angle L 102×89×6; Eccentricity=38mm; fy= 349.3MPa Slenderness


Euler’s Formula ASCE Formula Present Analysis Bathon’ Test 10 347.8 254.2 20 343.1 235.3 40 324.6 215.3 60 293.7 191.5 90 224.1 317.6 153.9 206.7

120 137.5 247.4 118.5 147.4* 150 88.0 158.5 91.9 111.6 180 61.2 72.2

*fy= 332.1MPa


0

50

100

150

200

250

300

350

400

0 50 100 150 200

criti

cal s

tres

s (M

Pa)

l/r

present analysis

Euler's formula

Bathon's test

ASCE formula

Angle: L 102×89×6


58

4.5 DICUSSION ON RESULTS OF THE FEA AND COMPARIZON WITH BATHON’S TEST RESULTS

Based on above parametric study of angle sections, following main observations are made from numerical and graphical representation of test results.

4.5.1 Main Observations

• Capacity of the angle section under compressive load is a function of its strength and buckling characteristics, slenderness ratio, geometric and material properties, eccentricity of loading and the type of the analysis.

• Theoretical results vary from both test and FEA results to a significant amount. Because theoretical result from the ASCE approach is based on the uniform cross section with eccentricity along the axes of symmetry; again Euler’s formula is based on concentric loading. But for present analysis, loading is eccentric and eccentricity is not applied along axis of symmetry. Yet the results are shown with Euler and ASCE formulae to compare the effect of unsymmetrical eccentricity in loading.

• For both the equal and unequal-leg angles, it’s found that the capacity from actual tests increases also with increase in angle dimensions of cross section and may exceed the capacity found by theoretical and FEM analyses. From figure 4.2 to figure 4.7 we see that for smaller cross section angles, actual critical stress may be less than that found from FEA; again for large cross sections the actual capacity is more than obtained value by FEM. This may be due to improper integrity of geometric and strength behavior of smaller cross sections that is often present in actual structures. So for smaller cross sections, elaborate investigation of member behavior should be undertaken and results of FEM should be properly checked with actual conditions. For intermediate cross sections, actual test results closely matches with FEA.

• One of the most important concerns is that the individual member behavior often varies from the same member assembled with a real structure as an integrated part in real cases to considerable degrees. For this reason, although single member analyses matches with analytical approaches, adequate measurements should be taken for real situation. Again in Bathon’s test, the members were connected by bolts at the end while in present analysis; members are connected at ends by means of high stiff rectangular steel plates. The obtained result may vary from actual situations for this reason also.

• Material imperfections are often present in actual structures at some specific or at whole regions which take account for member collapse by local failure at those regions. But for present analysis, except only the elastic and plastic nonlinear behavior, any other material imperfections are not considered.

• Comparing analytical and FEA approach with Bathon’s test results, we also find that analytical approaches tend to overestimate the axial compression capacity of eccentrically loaded angle sections at lower eccentricity ratios and vice versa.


59

4.5.2 Effect of Eccentricity

• Eccentricity reduces the load bearing capacity of angles from that of under concentric loading consideration by analytical approach of Euler. This effect is quite significant for short and intermediate members subjected to compression (slenderness ratio less than 90). This occurs due to the fact that, for such members both material strength characteristics and bending characteristics influence on member capacity; but the eccentricity induces bending to a certain degree that failure occurs before the material strength is fully utilized.

• For slender members (slenderness ratio above 120), the effect of eccentricity on capacity becomes negligible. Because for slender members, always bending governs on failure of member whether eccentricity present or not.

4.5.3 Effect of Slenderness Ratio

• The ultimate compressive load carrying capacity or the critical stress decreases with slenderness ratio (approximately in quadratic manner) because with increasing slenderness ratio, member becomes more susceptible to bending or buckling. When slenderness ratio exceeds value 150, the variation of critical stress diminishes.

• For slenderness ratio<10, test results show relatively higher stiffness against buckling of member; and for slenderness ratio>120, results closely match with the capacity obtained from Euler’ formula. For slenderness ratios of intermediate range between 10 and 120, eccentricity, angle dimensions and strength characteristics have significant effect on the capacity of the member. These phenomena show the behavior of short, intermediate and slender columns.

• For slender members or with slenderness ratios above 120, test results show greater capacity than analytic approach for unequal-leg angles. This may be due to the fact that although bending governs the capacity for slender columns, through FEA member stiffness and orientation may be fully utilized; which also may have some influence on strength of member.

• In case of short members, greater stiffness is obtained from material strength characteristics in addition to overall member behavior under loading.

Since only six angles are analyzed in this thesis, complete nature of buckling behavior of steel angle sections with various geometric, material, loading and other structural parameters under axial compression can not be predicted accurately based on only the results and information of this research. Analyzed information of present research can be used for further investigations regarding steel angle behaviors.

Conclusion

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Chapter 5

CONCLUSION

5.1 GENERAL

This study is emanated with an aim to perform the 3-D finite element analysis of steel angles, subjected to eccentrically applied, axial compressive load with nonlinear geometric and buckling consideration; and to compare the results with the previous research project of Bathon and with the fundamental theories of buckling for compression members. For this purpose, certain loading and restraint conditions were maintained throughout the whole analysis following ASCE Manual 52 (1988). In previous chapters, the whole study was carried out in a systematic and organized way. Then the computational investigation with numerical and graphical comparison was analyzed with necessary recommendations. Also suggestions are given for further research on related topics.

5.2 OUTCOMES OF THE STUDY

The whole research on ultimate capacity of steel angles by FEM concludes with following outcomes.

• Analytical approach does not show reliable results under practical situations often. Therefore in order to design and analyze steel angles, these theories should be developed by simulating behavior of angle sections through adequate and appropriate study.

• Under practical consideration, FEA gives more reliable and conservative results than analytical theories if appropriate approximation of geometric, material, loading and structural conditions is made.

• With proper and logical approximation, FEA may be used for analyzing such members with reasonable degree of accuracy without performing complex laboratory and field tests.

5.3 FUTURE RECOMMENDATIONS

Analytical approach requires adequate modification and development in the study of steel angles. Test results and FEM approach closely match with Euler’s analytical approach for large slenderness ratios i.e. for angles with larger lengths. But in case of short and intermediate length angles, test results vary from those approaches considerably. Therefore more study on such angles is recommended to achieve accurate results.

Conclusion

61

For short members (l/r<10), material strength assigned to present analyses overestimates test results to a considerable amount due to the fact that, present method fully utilizes material strength which is rarely available in rare structures. Therefore more study on the effect of material strength (for transient and long term loading) on angle behavior is recommended.

FEM approach and test results also show that the capacity is also a function of angle cross section. Therefore appropriate study on the effect of cross section on the capacity is recommended and suitable theory should be developed on this concern for real situations.

The present analysis does not consider any lateral effect or any member imperfections. Therefore there is a scope to analyze same research considering these considerations.

The present FEA is performed only for single angle members. So it may probably not approximate the capacity of angles under combined or group actions in real structures accurately; therefore further study is recommended to get behavior of angles under such cases.

References

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References

1. ASCE - ASCE Manuals and Reports on Engineering Practice No. 52 - Guide for Design

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Ultimate Capacity of Steel Angles Subjected to Eccentric Compression