High strength steel: local buckling and residual stresses

(1)

L I C E N T I AT E T H E S I S

Luleå University of Technology

Department of Civil and Environmental Engineering Division of Structural Engineering - Steel Structures

:|: -|: - -- ⁄ -- 

:

High Strength Steel

Local Buckling and Residual Stresses

Mattias Clarin

:

Universitetstryckeriet, Luleå

Mattias Clarin High Strength Steel LICENTIATE THESIS

(2)

High Strength Steel

- Local Buckling and Residual Stresses -

Mattias Clarin

Luleå University of Technology

Dept. of Civil and Environmental Engineering Division of Structural Engineering - Steel Structures

Luleå, November 2004

(3)

(4)

Preface

Not long ago, I met a man I thought I knew fairly well, you know, one of these acquaintances not in your inner circle, but still someone you know. This man works with steel and has done so for a while. During our conversation he exclaimed: “It’s strange, you know about my past, but do you know that working with steel has changed my life?”. I knew that this man had lost everything he held valuable in his life, and he further explained to me that through doing an effort concerning his work, doing these hard things threatening to break you, he found the way back to his life and new valuable things to embrace. From this we may learn that the things we are aiming for may bring other, more valuable, consequences. We know that by using steel we can change the behaviour of a structure, but obviously it may have other effects too. I am not sure if the change in this man’s life was material dependent, but nevertheless it is a nice thought:

That working with steel just may change your life, isn’t it?

Considering valuable, the support I have acquired in the task of steering my sailing ship to the Licentiate island in the ocean of knowledge is invaluable. My supervisors Ove Lagerqvist and Eva Hedman-Pètursson has contributed greatly to this thesis. Thank you for sharing of your energy and technical expertise. You are the hull of my ship, protecting from the waves and supporting me from shore to shore.

The personnel at TESTLAB has also contributed very much to this thesis. Especially Lars Åström, Georg Danielsson, Hans-Olov Johansson and Claes Fahleson who has helped me with the experimental work. The work you have helped me with is the rig and sails that have brought the ship forward.

The people at the division, steel structures in particular, you are the deck of the ship. A solid foundation to support all activities on board. Thanks!

Every ship of dignity has an orchestra bringing joy to the people on board. The orchestra on this ship has been the members in “The Band of Brodders”. Arvid, Jimmy, Karin and Tobias, may the KP live for ever and thank you for the music!

(5)

Another member in the band, as well as vice captain of the ship, that has helped me through is Jonas Gozzi. Ready to help and support when needed, both at sea and in land. The journey continues!

To the people waiting at the destination: family and friends. Now this is done and I promise to improve!

That was it and all.

Luleå, 9^thof November, 2004

Mattias Clarin

“And following our will and wind we may just go where no one's been We'll ride the spiral to the end and may just go where no one's been Spiral out. Keep going.”

/Maynard James Keenan

(6)

Abstract

High strength steel provide designers with the possibility of creating more slender and weight efficient structures than would be possible if using steels with lower strength. To be able to do this, a structural designer needs updated and validated codes as aid in their work. This thesis addresses the behaviour of high strength steel with respect to local buckling and residual stresses. The thesis was aiming to determine if there exists any significant differences in the resistance to local plate buckling of high strength steel (f_y> 460 MPa) compared to steels with lower strength. Furthermore, longitudinal residual stresses induced by welding were also considered on a basis of material strength. Experimental work considering these two issues was conducted concerning the three steel grades Domex 420, Weldox 700 and Weldox 1100.

The investigation concerning the local buckling resistance comprises experiments on 48 welded box section specimens made of the three grades. Nominal plate slenderness values were altered between 0,7 and 1,5. Moreover, the experimental work was founded on plate theory with respect to local buckling and a survey of other conducted comparable experiments. The results from the tests and the literature survey were evaluated with respect to Eurocode 3. The gathered test results from literature and experiments showed that no significant difference between the local buckling resistance of different steel strengths could be concluded if compared to the Winter function. However, the Winter function was concluded to overestimate the resistance for more slender simply supported plates (O_p> 0,9) with residual stresses (in as-welded condition).

The residual stress state present in three box sectioned specimens made of the three grades was measured with the blind hole technique. Evaluation of the test results was made with respect to the steel strength and complemented with test results collected from a literature survey. The study showed that the tensile residual stresses induced by welding could not be directly correlated to the material strength. Results from measurements on high strength steel specimens showed that the longitudinal residual stresses was lower if made dimensionless with respect to the strength of the steel.

(7)

(8)

Notations & Symbols

The notations and symbols used in this thesis are described below in alphabetical order, disregarding being roman or greek letters.

a - Weld size, numerical coefficient or length

A - Area or constant

A₅ - Elongation measurement, 5 %

D - Angle

b , b_w - Plate width b_eff - Effective width

B , C - Constant

'L - Elongation

G - Deformation

D - Flexural plate rigidity

H - Strain or Material depentent parameter

H_r - Radial strain

H_T - Tangential strain

E - Modulus of elasticity, Youngs modulus f_u - Ultimate strength

f_ue - Ultimate strength, electrode

f_y - Yield strength

f - Yield strength, electrode

(9)

f_yk - Characteristic yield strength

F - Force

F_c - Shrinkage force

h - Height

O_p - Plate slenderness

k - Weld factor

k_cr - Buckling load coefficient

L - Plate length

m , n - Number of half waves over plate N_cr - Critical load

n - Number of passes in weld

N_el - Buckling load

N_x , N_y - Normal forces per unit distance N_xy - Shearing force per unit distance q - Distributed variable load

Q - Heat input (circuit voltage x current) R - Radius from drill centre

R_o - Drill radius

R_p0.2 - 0,2 % Proof stress R_m - Ultimate resistance

V - Stress

V_c , V_rc - Compressive residual stress V_cr _- Critical stress

V_r - Residual stress

V’_r - Initial radial stress V_t,rs - Tensile residual stress V_rs - Residual stress

(10)

V´_T - Initial tangential stress V_u - Ultimate strength

V_x - Normal stress

t - Thickness

W'_r_T - Initial shear stress

v - Welding speed

w - Amplitude of plate deflection w₀ - Initial amplitude of plate deflection x, y, z - Cartesian coordinates

X - Position

\ - Stress ratio

Throughout the thesis mean values are marked overlined, e.g_. f_y represents the mean yield strength.

(11)

(12)

Chapter1:

Introduction

The world changes around you! Space travelling is privatized, researchers all around find ways to cure diseases thought not possible to treat, even the very foundation beneath your feet changes. Innovations, or maybe novelties, are also introduced in the field of materials.

Advanced fibre composites, cheramics or steels with strengths and quality only dreamt of 50 years ago are now available on the market for designers.

These new materials provides product designers working in areas ranging from floorball sticks, fighter planes and bullet proof armour to vehicles, bridges and buildings, with new possibilities to develop and construct better merchandises. Nevertheless, the outcome of a designers new creation, has to be carefully investigated before constructed. A floorball stick may be replaced, but regarding the bullet proof armour, one needs to be completely certain of the capabilities and limitations of the new product. This is why researchers bend, pull and twist new innovations in the purpose of establishing rules concerning the behaviour of the new product or material. Does new calculation models have to be installed or may “old” be used regarding the new issue? A malfunctioning product with a possibly lethal outcome (as the bullet proof armour) is a structure, made to work with or around people. This thesis will hopefully be a step towards providing the structural designers with some tools regarding an old material, nevertheless under never ending development, which is named: Steel.

The tree of steel development has many branches; increased toughness, better weldability and enhanced formability are examples. Another very thick branch is the one containing the research and development work put into increasing the strength of the steel. Today, steels with yield strengths of 1100 MPa and above are available on the market. Even though steel has been used as a structural material over decades, most design models used today are based on materials with essentially lower yield strengths. This fact once again raises the question as stated above: Does new calculation models have to be installed or may “old” be used regarding the new issue?

With this in mind the project LiftHigh - “Efficient Lifting Equipment with Extra High Strength Steel” was initiated in 2002. The project, partially funded by RFCS - The Research

(17)

Fund for Coal and Steel, was launched with the purpose of investigating how high strength steel can be used to produce more efficient lifting equipment. The analogy is simple even for the layman: stronger material - higher capacity, in this case lifting capacity. Nevertheless, the calculation rules still needs to be verified for the high strength steel, which in this thesis is defines as steel with a yield strength > 460 MPa.

Today, one of the ruling design codes concerning plated steel structures in Europe, the Eurocode 3, is only validated to comprise steels with strength up to 460 MPa.

1.1. Purpose

This thesis was focused on two aims. First, to investigate if the Winter function, in Eurocode 3 used for estimating the local buckling resistance, is adequate especially concerning plates made of high strength steel. The two main questions to answer was:

• If plates made of steels with higher strength behaves differently than “ordinary”

steel grades, with respect to local buckling.

• If, by using a reference grade in the experimental work (Domex 420), the whole Winter function concept, established mainly trough tests on cold formed profiles, can be improved in general terms.

Residual stresses induced by welding is also of great interest, since these may limit the resistance of a welded member. The second aim of the investigation was to evaluate:

• If a correlation between residual stresses induced by welding can be put in correlation with the strength of the steel.

1.2. Limitations

This thesis is limited to comprise experimental work of three different steel grades; the hot- rolled Domex 420, quenched and tempered Weldox 700 and quenched Weldox 1100.

Furthermore, the measurements of residual stresses was limited to the longitudinal direction (along the welds) and the stress state post-welding.

The evaluation of the experiments and the literature survey is only made with respect to Eurocode 3 and the evaluation of the residual stresses is limited to consider tensile stresses only.

1.3. Method

An experimental investigation comprising welded specimens of the three different steel grades was chosen to evaluate the local buckling behaviour of the high strength steel. The Domex 420 grade was used as a reference enclosed in and validated for Eurocode 3. Uniaxial tests of welded box specimens were done to investigate the local buckling behaviour of simply

(18)

supported plates in as-welded condition. Moreover, a comprehensive survey of the literature was done to support the test results.

The longitudinal residual stresses was chosen to be measured in as-welded condition with the blind hole method. Three different steel grades were considered in the experimental work and the specimens were of the same type as used for the local buckling tests. The steel grades Weldox 700 and Weldox 1100 were chosen to represent the high strength steel and Domex 420 as a steel with “ordinary” strength. With measurements within this strength range,

complemented with evaluated results from a literature survey, an eventual correlation between material strength and tensile residual stresses could be determined.

1.4. Disposition of the Thesis

In chapter 2 the theory behind the plate buckling phenomena is reviewed. Structural stability with focus on local buckling is regarded. The establishment of equations describing local buckling of simply supported plates are presented. Furthermore “the effective width approach”, nowadays widely spread as a theoretical interpretation of the phenomena, is introduced along with the Winter function used in the Eurocode 3 of today.

Chapter 3 comprises a survey of previously conducted experimental work concerning local buckling of plates. Test results concerning range of steel grades from “ordinary” grades with yield strength of approx. 250 MPa to 1100 MPa high strength grades, are presented and re- evaluated.

The experimental work conducted at LTU with respect to local buckling is presented in chapter 4. The test method, used equipment and the measurement of mechanical properties of the three grades are enclosed, as well as the results from the buckling tests.

How residual stresses are formed in steel and some possible consequences of these are presented in chapter 5. Different available measurement techniques, as well as how to avoid or reduce residual stresses is also presented.

In chapter 6 the results of a literature survey concerning measurements of residual stresses in welded members is presented. The chapter was focused on as-welded members and some models of prediction are also described. Moreover, the results from 47 individual measurements of a multitude of steel grades are also presented in this chapter.

Measurements of residual stresses with the blind hole method may be studied in chapter 7.

The results from the experimental work regarding three box shaped specimens are presented and evaluated. Furthermore, the equipment used for the experiments are described.

(19)

The acquired test data concerning local buckling and residual stresses are discussed and concluded in chapter 8. All re-evaluated test data gathered from the two different literature surveys are put together with the test data acquired from experiments at LTU.

Chapter 9 contains the references used herein.

In Appendix A all of the test data concerning the buckling tests are enclosed. Furthermore, the measuring equipment concerning these tests are described more thoroughly.

Appendix B comprises the evaluation model used for the data from the residual stress measurements. Moreover, the relevant test data from these experiments is presented.

In Appendix C data sheets containing the 47 specimens used along with the experimental work conducted with respect to residual stresses. These are extracts from the test data acquired from the literature survey.

(20)

Chapter 2:

Plate Buckling - Theory

The words “stable” or “instable” are used by people in various contexts. Almost everyone have a relation or thought concerning the two words describing the state of something. The terms are used in the wide range from psychology and politics to nuclear and chemical applications. The term “stable” is often connected to something positive and rigid when

“instable” is closely linked to the possibility of an abrupt loss of something. One of the most known and used context of the two words, which almost all people have a relation to, is when used in medical surroundings; a stable or instable health state.

The interest in stability / instability is also a central concern regarding mechanical systems, e.g. structural or civil engineering, see Figure 2.1. In this field the stability or instability of a structure is often confined to regard the elastic part of the phenomena. However, as will be shown later herein, a structural engineer may also have to consider the inelastic state of stability / instability. As an example of structural instability one can consider the columns in a building made with a steel frame. These columns have not only to withstand the vertical loads of the dead weight and e.g. snow, but also lateral loads caused by the wind. This well known instability phenomenon is usually referred to as column or flexural buckling.

Figure 2.1: Maybe an up-coming example of global structural instability?

(21)

The buckling may be of global nature, as described above, but may also be of localized (local) type. Buckling of local sort are regional located buckling, e.g. a flange of a beam or at a certain level of a silo, see Figure 2.2. Local buckling occur due to compressive stresses and may in a further perspective cause global buckling because of the loss of resistance of the cross section in question.

Figure 2.2: Different examples of local buckling. Local buckling in a silo, Farshad (1994) (left) and box shaped profile (right).

A structure or a member in an equilibrium state under e.g. compressive load may become unstable and the structure acquires a new equilibrium state or a new trend of behaviour. When considering classical buckling theory the critical stress level is situated where the equilibrium of the load - deformation path diverge. This point is called the bifurcation point or bifurcation load. Usually two more types of elastic instabilities are distinguished. These are limit

equilibrium instability (snap-through buckling) and dynamic or flutter instability, Farshad (1994).

2.1. Bifurcation instability

Considering the load - displacement behaviour of a column or a plate subjected to

compressive stresses, a load level lower than the bifurcation point corresponds to a state where eventual buckles are of elastic type. Hence, the secondary path in Figure 2.3 represents the post buckling stadium.

(22)

Figure 2.3: Schematic description of the bifurcation of equilibrium.

The bifurcation load or critical load has under the years been thoroughly investigated. As mentioned above, the critical load is determined with respect to elastic analysis and have been examined theoretically by many different researchers, e.g. Timoshenko and Gere (1963).

2.2. Plate theory

A thin plate is, by definition, a two-dimensional flexural element of which the thickness is much smaller than its other two dimensions. A plane passing through the middle of the plate is called the middle plane.

Thin plate elements are used in various structures; they may be elements in a complex structure or may themselves constitute the major part of a structure. Examples of plate elements are walls of containers, silos, and reservoirs, flat roofs, flat elements of vehicles and aircrafts, and sheet piles. Examples of plates in civil engineering applications are the flanges and the web of a beam. Plate elements may be homogeneous and isotropic or they may be stiffened and / or have a composite construction.

Depending on the mode of application, a plate can be subjected to various lateral as well as in-plane forces. Under certain circumstances, applied in-plane loading may cause buckling which can be global or in some cases, have a localized nature; delamination buckling of composite plates or buckling of a web in a steel beam are examples of local buckling. For thin plates, buckling is a phenomenon which may influence the load-bearing capacity of plate elements. Therefor, this must be taken into consideration in the design of plate elements.

Load

Deformation Bifurcation point Critical load

Secondary path Primary path

(23)

2.2.1. Elastic analysis / Calculation of critical load

The theory behind the behaviour of a thin plate under compressive forces is usually divided into two parts; firstly the calculation of the critical load and secondly the determination of the ultimate load level. The critical load level is by definition the point were the structure, or member, in question loose its stability.

Analytical calculation of the bifurcation or critical load on the basis of the classical theory of elasticity may be done either through solving the differential plate equation or via the energy method. The differential equation describing the buckling of a plate loaded in its plane was established by Saint-Venant in 1870, Dubas and Gehri (1986), and states

(2.1)

where the flexural rigidity of the plate is given by

(2.2)

This plate equation was derived under the assumptions that the material is behaving in a ideally elasto-plastic way, the plate is without initial imperfections such as initial curvature or residual stresses. Furthermore, the plate deformations are assumed to be small. Under these assumptions the plate shows no lateral deformations until the critical stress level is reached. At this point, the deflection can either be negative or positive regarding the coordinate system of the plate, Figure 2.4.

Figure 2.4: System bifurcation in point A. The plate buckles in either a positive or negative direction, w.

The plate equation may be convenient to use when a rigorous solution of (2.1) is possible.

When the plate in question is for example reinforced with stiffeners, the problem gets more x⁴

4

w

w w 2 w⁴w wx²wy² ---

y⁴

4

w

+ +w w 1

D---- q N_x x²

2

w – w w N_y

y²

2

w

– w w 2N_xyw²w w yxw --- +

=

D Et³

12 1 –Q² ---

=

V

w

V_cr

A

(24)

advanced. These more advanced applications led to the development of other models, better describing the actual behaviour of plates.

The solution to this problem was delivered by Bryan in 1891 through the establishment of an energy based approach. The approach of this method is to study the plate energy in the

bifurcation point, where the plate cease to be in its assumed perfectly flat state and instead follow its secondary equilibrium path (see Figure 2.3) in a laterally deformed state. The energy based solution is built on the classical correlation between the internal energy of bending and the external work done by the forces acting in the middle plane of the plate. The expression for describing the strain energy stored in the deformed plate is

(2.3)

Furthermore the equation describing the work conducted by the externally applied forces is

(2.4)

The equations (2.3) and (2.4) are only valid for small deformations, which is assumed to be the case up to the bifurcation point. With Figure 2.3 in mind, the comparison between the internal energy and external work gives, according to Timoshenko and Gere (1963), the following information concerning the stability of the plate in question at the bifurcation point:

• If U > T, the flat form of equilibrium of the plate is stable (primary path)

• If U < T, the plate is unstable and buckling occurs (secondary path) However, the critical load amplitude may be found by setting

(2.5) which can be solved under the condition that the change in energy potential must have a minimum value for a stable equilibrium. This may be used for the derivation of a differential equation form of the equilibrium. Another way to solve the problem is to apply an expression for the lateral deformation of the plate.

U 1

2---D w²w x² G

--- w²w y² G ---

¨ ¸

§ ·²

2 1 –Q w²w x² G --- w

2w y² G

--- w²w G yxG ---

© ¹

§ ·²

¨ ¸

§ ·

– dx yd

³

=

T 1

2--- N_x x²

2

w

w w N_y y²

2

w

w w 2N_xyw²w w yxw ---

+ + dx yd

³

–

=

T = UU–T = 0

(25)

2.2.2. Simply supported plates under uniform compression

Figure 2.5: Simply supported plate under uniform compressive load. Dubas and Gehri (1986).

If considering a plate subjected to evenly distributed forces along two of the edges, according to Figure 2.5, the determination of the critical load level of the plate in question is dramatically simplified. Since the only load applied on the plate, in the form of a uniform distributed compressive force, acting along the edges x = a and x = -a, the rest of the external applied loads according to equation (2.1) equals zero:

(2.6)

The edge constraints of the plate leads to the following boundary conditions:

Along the edges x = a and x = -a

(2.7)

and along the edges y = 0 and y = b

(2.8) q = N_y = N_xy = 0

w x²

2

w

w w 0

= =

w y²

2

w

w w 0

= =

(26)

The boundary conditions implies that the deformed shape of the simply supported plate may be described by a double trigonometric Fourier series on the form

(2.9)

By substituting the proposed solution according to equation (2.9) into (2.3) and (2.4) under the above described conditions in (2.6), (2.7) and (2.8), and by using the relation between the external work done by the applied load and the strain energy according to equation (2.5), the following relation may be evolved:

(2.10)

To satisfy the equation (2.10) for all positions on the plate, i.e. all values of x and y, the following relation has to be true:

(2.11)

or in another form

. (2.12)

The combination of the two integer parameters now have to be chosen in such a way that the applied load, N_x, reach a minimum value, i.e. the sought critical load value, N_cr. It can be shown that the lowest critical load is reached when the plate buckles in a shape such that one half sinus wave is formed over the width of the plate (y-direction), hence the integer parameter n = 1, Timoshenko and Gere (1963). With this, the equation (2.12) may be evaluated to

(2.13)

in which the integer parameter m describes the number of half sinus waves over the length of the plate (x-direction). The equation (2.13) are more often formed as

(2.14)

w a_mn mSx

---a nSy

---b m n sin

sin

n 1=

f

¦

m 1=

f

¦

^{1 2 3}}

= =

D mS

---a

© ¹

§ ·² nS

---b

© ¹

§ ·² +

2

N_x mS ---a

© ¹

§ ·² +

¯ ¿

® ¾

½

a_mn mSx ---a nSy

---b sin

sin = 0

D mS

---a

© ¹

§ ·² nS ---b

© ¹

§ ·² +

2

N_x mS ---a

© ¹

§ ·²

+ = 0

N_x

D mS

---a

© ¹

§ ·² nS ---b

© ¹

§ ·² +

2

mS ---a

© ¹

§ ·² ---

=

N_cr a²S²D m² --- m²

a² --- 1

b² ---

¨ ¸

§ ·

=

2

m = 1 2 3}

Ncr k

cr

S²D b² ---

=

(27)

where the dimensionless parameter k_cr is the buckling load coefficient and is given by

. (2.15)

Furthermore, with the expression for the flexural rigidity of the plate given in (2.2), inserted in (2.14) the well known expression for the critical, or bifurcation, stress may be expressed as

(2.16)

with the insight of that

(2.17)

The buckling load coefficient, k_cr, is, as can be seen in (2.15), a function of the plate width b, the length a and the number of sinus half waves over the length, m. For different values of the plate width and length ratio a / b, the lowest critical stress level will be found for different numbers of half waves according to Figure 2.6 below.

Figure 2.6: The buckling load coefficient for a simply supported thin plate.

Timoshenko and Gere (1963).

2.2.3. Initial plate imperfections

In section 2.2.1 above, a quite straight forward method for calculating the critical stress level is presented. However, as always concerning theoretical models describing nature, it is

important to remember the assumptions made for the theory in question. Emphasizing the assumptions made of a initially perfect flat plate and a perfectly isotropic behaviour in a homogenous material the understanding of the limitations in the presented theory are obvious.

All materials have different levels inherent imperfections, also steel. A plate delivered from the steel fabricator has an initial curvature and probably also residual stresses from uneven cooling

kcr

mb ---a a

mb---

§ ·²m 1 2 3}

= =

V_cr k

cr

S²E 12 1 –Q² --- t

b--- ²

=

V_cr N

cret

=

(28)

of the material. These facts makes the assumptions made above somewhat untrue, which also has been proven experimentally and may be found in chapter 3.

Now when the assumptions are found to be a quite utopical description of the real behaviour of the considered plates, the question arises how these initial imperfections affect the plate behaviour before, as well as after, the bifurcation point. Figure 2.7 below shows the difference in the plate behaviour when plate imperfections are considered.

Figure 2.7: The influence of initial plate imperfections in relation to perfect plates. Farshad (1994).

Considering Figure 2.7 above two conclusions concerning how the imperfection influence the plate behaviour may be drawn. Firstly, buckling of a plate with inherent imperfections is gradual and the exact critical load may be difficult to determine. Hence, difficulties arises when a comparison between theoretically and experimentally determined critical loads are to be conducted. Secondly, as mentioned before, the plate may accept continued loading after the bifurcation load is reached. Thus the critical load is shown to be a non-representative measure on the ultimate resistance of the plate in question, Brush and Almroth (1975).

2.2.4. Geometric imperfections

When considering the initial out-of-plane imperfections, i.e. initial buckles, the influence of these on the maximal out-of-plane deformation / load correlation are shown in Figure 2.8.

The graph and the calculations behind was made by H. Nylander in 1951 and shows how an applied initial deformed shape with the amplitude w_o (in the same shape as the deformed plate) affects the magnitude of lateral deformations under applied load. Furthermore, when the material is assumed to be ideal elastic, the model gives no information concerning the ultimate load, Johansson (2005). Concluded, the initial geometric imperfections primarily influences the plate stiffness and becomes more obvious with an increased plate slenderness.

(29)

Figure 2.8: The effect of initial geometric imperfections. Relation between the lateral deformation, w, plate thickness, d, and load, N, concerning different amplitudes of initial imperfections w_o. StBK-K2 (1973).

2.2.5. Residual stresses

How residual, or initial, stresses are formed, distributed and under which magnitudes these may occur is more thoroughly described in chapter 5. However, knowing that residual stresses are present in all materials, it is evident that this must affect also the elastic plate buckling theory. Geometrical imperfections and residual stresses in a plate under compression mainly affects the initial stiffness of the plate. In Figure 2.9 below, a schematical distribution of residual stresses caused by edge welding a plate is shown.

Figure 2.9: Schematic distribution of residual stresses in an edge welded plate.

Considering Figure 2.9 above, the influence of the initial load due to the present residual stresses is clear. Since the middle region of the plate before external loads are applied, already is under compressive stresses, it is obvious that yielding of the plate in question will occur at a lower external load level compared to a residual stress free plate, see Figure 2.10.

The effect of inherent residual stresses is more marked for stockier or intermediate slender plates, for which the yielding process of the plate is the governing cause of failure. Concerning more slender plates, the initial geometric imperfection tend to surpass the influence of residual

(30)

stresses, Dubas and Gehri (1986). Hence, the influence of residual stresses decreases with increasing plate slenderness.

Figure 2.10: Schematic influence on the behaviour of a plate with (S) and without (A) residual stresses.

2.3. Non linear theory / Post buckling behaviour

As shown above, the estimation of the critical load may be done by a straight forward method. However, the elastic analysis assumes, as described in previous sections, that the plate in question is perfectly flat and that no initial stresses are present. Because of the presence of these imperfections non-linear models were evolved. Furthermore, the initial plate

imperfections were not solely the reason to why non-linear theories had to be evolved. The assumption concerning the constitutive relations, in this case ideal elastic material, is not suitable to use when the ultimate resistance is sought for.

Another reason why non-linear models were established was that many researchers showed that the ultimate load of a plate under compression may significantly surpass the critical load level. This was especially evident concerning more slender plates. Regarding stockier plates the resistance is often limited by yielding in the material and the ultimate load may be lower than the critical.

In linear elastic analysis, the distribution of the load is assumed to remain uniform until the plate buckles. However, when the plate starts to buckle, the stresses are re-distributed in the plate. The plate behaviour under these large deformations, or post critical behaviour, is a complicated area to describe. Some differential equations describing the phenomenon were derived by von Kármán in 1910 but the methods for solving these are complex, Dubas and Gehri (1986). The finite difference method, fourier series or different perturbation methods are possible tools for this work.

V

'^{L / L} V_cr

(31)

Other methods may also be used for studying the post critical plate behaviour. One example is the numerical methods, e.g. the finite element method, FEM, which probably is the most powerful tool available today. However, other methods have been used during the years of research. Analytical methods such as the Ritz energy method or a method based on a theory by Skaloud and Kristek called the “Folded plate theory method” are both excellent examples.

As described above, the theory behind plate buckling is rather complicated due to the combination between the membrane stresses from the applied load and bending stresses in the deformed plate, as well as shear stresses due to rotation of the corners of the plate. For design purposes the above described methods may be too advanced to use. This is why the “Effective width approach” by von Kármán et al. (1932), is widely spread as the model for determining the ultimate resistance of plates under compression.

2.3.1. The von Kármán effective-width formula

The starting point for the effective width approach is that the ultimate resistance is reached when the largest edge stress reaches the yield stress level. Since the formed buckle in the middle of the plate reduces the plates ability to carry the load, the stresses are re-distributed as shown in Figure 2.11 below. The real stress distribution in the plate is approximated, or substituted, with two strips which describes the load carrying effective width of the plate.

Figure 2.11: Stress distribution in a plate before (a) and after buckling (b).The von Kármán assumption concerning the effective width is presented in (c).

Brush and Almroth (1975).

(32)

von Kármán’s hypothesis was that the “new” plate with the width of b_effwould have the critical stress equal to the yield stress, i.e.

(2.18)

Furthermore, the critical stress according to (2.16) under the condition that the plate is under uniform compression and simply supported (k_cr = 4) the following expression may describe the relation between effective width and yield stress level:

(2.19)

or with the original plate width equal to b

(2.20)

which is usually referred to as the von Kármán effective-width formula. Furthermore, the relation

(2.21)

was made as a generalization of the corresponding well known parameter for column buckling and was called the reference slenderness of the plate. In modern design rules, when design is done with respect to the ultimate load level, this expression is the only one considering the critical load. And as expressed in von Kármán et al. (1932) the following may be stated

(2.22)

or

, for (2.23)

under the circumstances that the plate is simply supported and under uniform compressive load.

Although, von Kármán’s theories gained reputation as a good method to use for the

determination of the ultimate load of the plate in question, the method was a strictly theoretical method based on plates without initial imperfections and when compared to test results it was found to be true only for large b / t ratios. However, von Kármán still stands as the first researcher proposing a reduction factor function.

V_cr = f_y

4S²E 12 1 –Q² --- t

beff

---

© ¹

§ ·² = f_y

beff b V_cr f_y ---

=

O_p V_cr f_y ---

= 1 05b

---t f_y kcrE ---

=

b_eff 1 9t E f_y

---

=

beff

---b 1 O_p ---

= O_pt1

(33)

2.3.2. The Winter function

Theodor von Kármáns work was a milestone concerning the simplified design methods concerning plate buckling. Many researchers followed his work (Figure 2.12), aiming for an expression describing a real plate with inherent initial imperfections. One of the more known and widely spread in design codes, are the one proposed by Winter in 1947. Winter conducted numerous experimental tests on cold formed specimens and suggested

, for (2.24)

as a suitable function regarding the effective width, Winter (1947). Winters first suggestion was with the coefficient 0,25 but was later changed to the 0,22 used nowadays. However, it is interesting to notice the small difference between the “original” equation (2.23) and the experimentally based (2.24).

Other researcher proposed different solutions, or modifications, of the initial von Kármán formula. Two reported in Dubas and Gehri (1986) are

, for (2.25)

by Faulkner in 1965 and

(2.26)

suggested by Gerard in 1957.

Figure 2.12: Reduction functions according to Winter, Faulkner, von Kármán and Gerard as described in the text above.

b_eff ---b 1

O_p

--- 1 0 22 O_p ---

§ ·

= O_pt0 673

beff

---b 1 05 O_p

--- 1 0 26 O_p ---

§ ·

= O_pt0 55

b_eff

---b 0 82 O_p^{0 85} ---

=

0 0.5 1 1.5 2 2.5

O_p, Plate slenderness

0 0.4 0.8 1.2

U, Reduction factor

Winter function Faulkner function von Kármán function Gerard function

(34)

Even though a lot of effort has been put into this reseach field, the Winter function, based on the cold formed members survived and is nowadays set as the function used in the present design regulation in Europe, the Eurocode 3.

In Eurocode 3 the plate slenderness, O_p, is calculated according to

(2.27)

and His defined according as

. (2.28)

Furthermore the buckling load coefficient, k_V , for a simply supported plate under uniform compressive load is determined according to Figure 2.6.

As mentioned above, design with respect to local buckling of flat compression elements is made through a reduction of the cross sectional area of the plate in question. Concerning internal compression elements this is, according to Eurocode 3, done through the use of the expression

(2.29)

in which the factor<, represents the actual stress distribution over the plate. Concerning uniform distribution of compressive stress this factor equals 1. Thus, the equation reflects the original Winter function (2.24) used for these kind of plate elements in Eurocode 3.

O_p b te 28 4H k _V ---

=

H 235

f_y ---

=

U O_p–0 055 3 +\ O_p²

---d1 0

=

(35)

(36)

Chapter 3:

Plate Buckling - Survey of Literature

The plate buckling phenomena has, as mentioned in previous chapters, been quite thoroughly investigated. This also on a strictly experimental basis. The research work is forthgoing when new steel grades and design rules enter the field of constructional applications.

However, to acquire all the test data and experimental reports concerning plate buckling are difficult and the author to this theses makes no claims of have accomplished this. Though the work presented below should be sufficient to validate the experiments presented in following chapter 4.

The articles and papers presented in this chapter have been chosen to be comparable to the tests in chapter 4. This with respect to specimen layout, welding conditions, support conditions, steel grades and other comparable similarities. Furthermore, all the test results presented in this chapter are evaluated with respect to the Winter function discussed in chapter 2 and according to the Eurocode 3 specifications concerning plate slenderness values.

3.1. “Experimental Investigation of the Buckling of Plates with Residual Stresses”

An investigation aiming to clarify how residual stresses influence the resistance against local buckling was presented by Nishino et al. (1967). Specimens used in this research work were fabricated of plates welded together to form a square cross section, see Figure 3.1, and tested in as-welded condition.

High strength steel: local buckling and residual stresses

L I C E N T I AT E T H E S I S

High Strength Steel

Local Buckling and Residual Stresses

Mattias Clarin

High Strength Steel

- Local Buckling and Residual Stresses -

Mattias Clarin

Preface

Abstract

Notations & Symbols

Table of Contents

Chapter1:

Introduction

Chapter 2:

Plate Buckling - Theory

³

³

³

³

¦

¦

Chapter 3:

Plate Buckling - Survey of Literature