19442847 methods of analysis of steel structures

Course Contents

ELEMENTS

Lecture 7.1 : Methods of Analysis of Steel Structures

Lecture 7.2 : Cross-Section Classification

Lecture 7.3 : Local Buckling

Lecture 7.4.1 : Tension Members I

Lecture 7.4.2 : Tension Members II

Lecture 7.5.1 : Columns I

Lecture 7.5.2 : Columns II

Lecture 7.6 : Built-up Columns

Lecture 7.7 : Buckling Lengths

Lecture 7.8.1 : Restrained Beams I

Lecture 7.8.2 : Restrained Beams II

Lecture 7.9.1 : Unrestrained Beams I

Lecture 7.9.2 : Unrestrained Beams II

Lecture 7.10.1 : Beam Columns I

Lecture 7.10.2 : Beam Columns II

Lecture 7.10.3 : Beam Columns III

Lecture 7.11 : Frames

Lecture 7.12 : Trusses and Lattice Girders

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ESDEP WG 7

ELEMENTS

Lecture 7.1: Methods of Analysis

of Steel Structures OBJECTIVE/SCOPE

To introduce methods of global analysis and to relate them to the assumptions made on the material behaviour and on the effects of deformations.

PREREQUISITES

Elementary mechanics of materials.

Elementary structural analysis.

Elements of plastic design.

Elastic and elastic-plastic behaviour of materials.

RELATED LECTURES

Lecture 7.2: Cross-Section Classification

Lecture 7.8.1: Restrained Beams I

Lecture 7.11: Frames

Lectures 14: Structural Systems: Buildings

SUMMARY

Internal force distributions in structures may be determined using an elastic or a plastic global analysis. Either a first or second-order theory can be used, depending on the type of structure. These concepts are briefly reviewed and comments are made in general terms regarding design practice.

1. INTRODUCTION Checking the strength of cross-sections, the stability of structural members or section components and possibly fatigue requires that the internal force distribution within the structure, is known beforehand; from this, the stress distribution within any cross-section may be deduced as required. The words "internal forces" (also termed "member forces") are used generally and refer to axial forces, shear forces, bending moments, torque moments etc.

The internal forces in a statically determinate structure can be obtained using statics only. In a statically indeterminate structure, they cannot be found from the equations of static equilibrium alone; a knowledge of some geometric conditions under load is additionally required. It is important, at this stage, to recognise this fundamental difference between statically determinate and indeterminate (hyperstatic) structures. The internal forces in a structure may be determined using either an elastic or a plastic global analysis. While elastic global analysis may be used in all cases, plastic global analysis can be used only where both the member cross-sections and the steel material satisfy special requirements.

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The internal forces may be determined using different approaches depending on whether the effects of the deformations in the structure can or cannot be disregarded. In first order theory, the computations are carried out by referring only to the initial geometry of the structure; in this case the deformations are so small that the resulting displacements do not significantly affect the geometry of the structure and hence do not significantly change the forces in the members. Second order theory takes into account the influence of the deformation of the structure and, therefore, reference must be made to the deflected geometry under load. First order theory may, for instance, be used for the global analysis in cases where the structure is appropriately braced, is prevented from sway, or when the design methods make indirect allowances for second-order effects. Second order theory may be used for the global analysis in all cases, without any restrictions.

When first order theory can be used, the behaviour of a structure made with a material obeying Hooke's law is itself linear; the displacements - translation or rotation of any section - vary linearly with the applied forces; that is, any increment in displacement is proportional to the force causing it. Under such conditions, stresses, strains, member forces and displacements due to different actions can be added using the principle of superposition. This principle indeed states that the displacements (internal forces) due to a number of loads acting simultaneously is equal to the sum of the displacements (internal forces) due to each load acting separately. This does not apply if the stress-strain relationship of the material is nonlinear or if the structure (even if it is made of a material obeying Hooke's law) behaves non-linearly because of changes in the geometry caused by the applied loads.

The principle of superposition, when it can be used, is especially useful when determining the most severe condition in each individual member of a statically indeterminate structure; the interaction between different parts of the structure makes it difficult to identify the exact loading which produces the critical condition for design.

In practice, elastic global analysis is generally used to study the serviceability performance of a structure, i.e. limit states beyond which specified service criteria are no longer met. Plastic global analysis is particularly useful when investigating states associated with an actual collapse of the structure and to assess the actual ultimate resistance, i.e. ultimate limit states.

2. ELASTIC GLOBAL ANALYSIS Elastic global analysis presumes elastic behaviour of the structure, and consequently, of the material itself. It is based on the assumption that the load-deformation behaviour of the material is linear, whatever the stress level; the strain is thus assumed proportional to the stress, i.e. the material is obeying Hooke's law in the whole range of loading (Figure 1a). Obviously, actual properties of the material, especially regarding yield stress and possibly strength, shall be considered when checking whether the member forces do or do not exceed the strength resistance of cross-sections and members.



It has already been stated that in the elastic global analysis of statically determinate structures, the internal forces are found from the equations of static equilibrium alone. In continuous construction (statically indeterminate structures) the member forces must satisfy the conditions of equilibrium and produce deformations compatible with the elastic continuity of the structure and with the support conditions. The equilibrium equations are not sufficient to determine the unknown forces and have to be supplemented by simple geometrical relationships between the deformations of the structure. These relationships are termed compatibility conditions because they ensure the compatibility of the deformations in the geometry of the deformed structure.

It is also required that the types of joints employed are able to maintain, virtually unchanged, the original angles between adjacent members, i.e. rigid connections are assumed.

Where first order theory can be used, equilibrium and compatibility conditions are expressed with reference to the initial (non deflected) configuration of the structure.

Two general methods of approach can be used to determine the force resultants and reaction components. The first of these is the flexibility method, in which releases are provided to render the structure statically



determinate; the unknowns are the forces. These are determined by saying that the released structure undergoes inconsistent deformations, which are corrected by the application of appropriate additional forces (Figure 2a).

The second approach is the stiffness method, in which displacement restraints are added to prevent movement of the joints, and the forces required to produce the restraint are determined; the displacements are then allowed to take place at the joints until the fictitious restraining forces have vanished. Once the joint displacements are



known, the forces in the structure are obtained by superposition of the effects of the separate displacements (Figure 2b).

Either the force or the displacement method can be used to analyse any structure. In the force method, the solution is carried out for the forces necessary to restore consistency in geometry; the analysis involves the solution of a number of simultaneous equations equal to the number of unknown forces, that is the number of releases required to render the structure statically determinate. In the displacement method, the unknowns are the possible joint displacements and rotations. The number of the restraining forces to be added to the structure equals the number of possible joint displacements and the analysis similarly involves the solution of a set of equations.

When it is necessary to account for second order effects (geometric non-linearity), second order theory must be



used which involves iterative procedures. Because the principle of superposition is not allowed in this case, reference must be made to a specified reference load distribution. This is increased, in steps using a load multiplier (Figure 3). Each step is chosen sufficiently small so that the behaviour may be assumed linear during this load increment. The deflected configuration reached at the end of a specified loading step is used as the reference geometry for the following step; elastic second order theory thus consists in solving a succession of first order analyses of a structure, the geometry of which is changed at each step on the basis of its former history. Such computations become rapidly unmanageable by hand and appropriate computer programs are needed. These are usually based on the stiffness method - also termed displacement method - because it is easier to define the kinematically determinate structure, which is used as the reference geometry.

Most codes and standards permit member forces, in regular geometric non-linear structures, to be obtained using linear elastic analysis and then amplified, where necessary, to allow for instability effects. Because the principle of superposition is not applicable, this approach would appear to be inconsistent with rigorous theory. Nonetheless, it gives the designer the opportunity to use standard, i.e. linear elastic, frame analysis programs for a wide range of structures, at least for preliminary design.

In certain circumstances, codes and standards permit a limited redistribution of moments. That means that the elastic moment diagram may be modified by up to 5 to 15% of the peak elastic moment, provided that the resulting computed moments and shears remain in equilibrium with the applied external loads (Figure 4). Therefore, although equilibrium is indeed maintained, the elastic compatibility of the structure is somewhat violated. This concept of moment redistribution may be thought of as a very limited recognition of the potential which exists, within statically indeterminate structures, to withstand loads in excess of those that require full elastic member bending strength only at the most critical location. Attention is drawn on the fact that this is possible only if unloading does not follow the attainment of the local maximum strength; some ductility of the cross-sectional behaviour is, therefore, required, which explains the reason for limiting the process to compact sections (see Lecture 7.2).



It should be stressed that the assumption of linear load-deformation behaviour of the material may be maintained for both first order and second order elastic analysis, even where the resistance of a cross-section is based on the plastic resistance (see Lecture 7.2).

3. PLASTIC GLOBAL ANALYSIS The load-deformation behaviour of steel is not infinitely linear. The strain-stress relationship for an ideal elastic-perfectly plastic material is represented in Figure 1b; it follows that Hooke's law is restricted to a stress range σ ≤ fy, fy being the yield stress of the material. Beyond this range, the material yields plastically at constant stress σ = fy. If the stress is reduced at any point in the plastic range, the return path is a straight line parallel to Hooke's law, the slope of which is the elastic modulus E. Both E and fy, and indeed the whole stress-strain relationship are assumed the same for tension and for compression.



The idealised stress-strain relationship, although only a mathematical model, is a close approximation of the behaviour of structural mild steel as well as a reasonable first approximation to many continuously strain-hardening materials used in structural engineering. The assumption of perfect plasticity, after the yield stress is reached, amounts to ignoring the effects of strain hardening and is on the safe side.

Consider a cross-section of area A, having an axis of symmetry and experiencing bending in the plane of symmetry (Figure 5). If the bending moment is small, the stress and the strain vary linearly across the depth. When the moment is increased, yield stress is first attained in one of the top fibres, and with a further increase the yield stress is reached in the bottom fibre as well. If the bending moment continues to increase, yield will spread from the outer fibres inward until the two zones of yield meet; the cross-section in this state is said to be fully plastic. The value of the ultimate moment, termed plastic moment, is deduced from equilibrium conditions. Since there is no axial force, the neutral axis of the fully yielded cross-section divides the latter into two equal areas A/2; the resultant tension and compression are each equal and form a couple equal to the ultimate moment:



Mpl = 0,5 A fy ( ) (1)

where and are respectively the distance of the centroid of the compression and tension area from the



neutral axis, in the fully plastic condition. For a doubly symmetrical cross-section, the distances and are equal, so that 0,5A is the first moment of area S (about the bending axis) of half the cross-section, and the ultimate moment is:

Mpl = 2 S fy (2a)

= Wpl fy (2b)

where Wpl = 2 S is the plastic section modulus for bending about the relevant axis.

The maximum bending moment that the cross-section could carry without exceeding the yield stress at any point is:

Mel = Wel fy (3)

where Wel is the elastic section modulus for bending about the same axis; the relative gain in strength which is achieved by allowing for full yielding of the cross-section is measured by the shape factor:

α = Mpl/Mel = Wpl/Wel (4)

which, for example, equals 1,5 for a rectangular section, 1,7 for a solid circular section, while varying between 1,12 to 1,18 for I and H beams and channels bent about their strong "yy" axis.

When the load on a structure increases, yielding occurs at some locations and the structure undergoes elasto-plastic deformations. On further increase a fully plastic condition will be reached at which a sufficient number of full plastic sections are formed to transform the structure into a plastic mechanism (Figure 6); this mechanism will collapse under any additional loading. A study of the failure mechanism and the knowledge of the associated magnitude of the collapse load are necessary to determine the load factor in analysis. Alternatively, if the load factor is specified, the structure can be designed so that its collapse load is equal to, or higher than, the product of the load factor and the reference service loading. Plastic analysis implies, therefore, not only plastic stress distribution within the cross-section (plastic hinge formation), but also sufficient bending moment redistribution in order to develop all the plastic hinges that are required to give rise to a plastic mechanism.



When yielding develops within a cross-section, the cross-section's effective value of flexural stiffness, EI, decreases progressively (Figure 7); indeed, the effective modulus of the yielded material is zero when assuming the perfectly plastic behaviour beyond yield, hence the term "plastic hinge". Once this hinge is produced, the structure will behave, under additional loading, as if a real hinge was introduced at the yielded section. The onset of the first plastic hinge in a structure results in a reduction of the original redundancy by one; any additional plastic hinge will have a similar effect.



Collapse occurs after sufficient plastic hinges have formed to convert the original redundant structure into a progressively less redundant structure, and finally into a mechanism.

In a statically determinate structure, the gain in strength due to plasticity depends on the value of the shape factor. In a statically indeterminate structure, it is affected by the process of moment redistribution.

The ability of a structure to redistribute stress within the cross-section, and between cross-sections, requires that no other form of failure occurs before the plastic collapse mechanism, so that the ultimate load can be reached. The following requirements must be met for plastic analysis to be allowed:

1. Steel material shall have adequate ductility, so that the plastic resistance of the sections can be developed (Figure 1 b - e);

2. Once formed, a plastic hinge shall be able to rotate at a sensibly constant moment Mp (Figure 7); 3. A plastic hinge shall have sufficient rotation capacity, without local buckling or lateral buckling, so as to

allow for the formation of a collapse mechanism and the corresponding moment redistribution (Figure 7); 4. The structure is subject to predominantly static loading so as to prevent failure from low cycle fatigue

(shake down).

To comply with these requirements, limits must be placed on the type of steel and the proportions of the members and cross-sections. Currently, plastic design is permissible for the usual grades of mild steel, while for other grades a minimum length of the yield plateau and a minimum ratio between the ultimate tensile strength and the yield stress (strain hardening) are required. Members containing plastic hinges must satisfy limitations on flange and web proportions; these are more restrictive for higher steel grades. Because yielding results in a large reduction in stiffness, members where plastic hinges occur are especially prone to failure by member instability; therefore, there are severe limits on the slenderness of such structural elements resulting in a need for appropriate lateral bracing, especially at the plastic hinge locations.

The above implies that the ultimate bending resistance of a section is defined solely by its plastic moment; axial load and shear force, however, will have an effect as discussed in Lecture 7.8.1.



In a structure subject to a specified loading, whose magnitude is increased up to collapse, the sequence of hinge formation is fixed. However, factors such as settlement, variability of material strength between members, residual stresses, thermal effects, etc., can change the sequence while not significantly affecting the plastic collapse load; the latter, indeed, is statically determinate and does not depend on structural imperfections of any kind.

Plastic analysis is based on non linear material behaviour, even where geometric second order effects are negligible. Hand analysis methods are based on the fundamental theorems of the plastic design, which usually neglect elastic curvatures, compared to plastic ones, and concentrate plastic deformations at plastic hinge locations; they use, therefore, rigid-plastic methods (Figure 1c). Information regarding this subject is not within the scope of this lecture and the reader is referred to the literature quoted in Section 6 for further discussion on this matter. Computer methods are less dependent on idealisations and may therefore be based on more realistic material stress-strain relationships accounting for elastic curvatures and deformations. These are termed elastic-plastic and can be distinguished from the perfectly plastic approach, characterised by an infinite yield plateau (Figure 1c), by a slight slope in the yield region (Figure 1d), or by a strain-hardening range following a yield plateau of limited length (Figure 1e). Alternatively, even more precise relationships may be adopted; nowadays, refined finite element programs allow for the spread of yielding and the concept of plastic zones is used instead of plastic hinges.

Second order plastic analysis generally requires the use of computer programs; the collapse load of multi-storey sway frames may however be determined using the Merchant-Rankine formulae, which take into account, in a very simple manner, the interaction between elastic buckling and yielding.

It is worthwhile emphasising that because plastic analysis is essentially nonlinear, the principle of superposition is not, therefore, applicable.

4. ADDITIONAL COMMENTS It should be noted that assumptions made in the global analysis of the structure should be consistent with the anticipated behaviour of the connections. The assumptions made in the design of the members should also be consistent with (or conservative in relation to) the method used for the global analysis and the anticipated behaviour of the connections. More detailed information in this respect will be provided in the lectures devoted to the design of connections.

Current codes and standards require that appropriate allowances shall be incorporated in the global analysis to cover the effects of residual stresses and geometric imperfections, such as lack of verticality, lack of straightness, lack of fit, and the unavoidable minor eccentricities present in practical connections. Suitable equivalent geometric imperfections may be used, with values which reflect the possible effects of all types of imperfections (Figure 8).



5. CONCLUDING SUMMARY The determination of the internal forces in a structure can be made according to either an elastic or a plastic global analysis. The global analysis is usually made using first order theory, where reference is made to the initial geometry of the structure. Second order theory, where equilibrium and compatibility are expressed with respect to the deflected geometry of the structure, is required when sway effects are not negligible. Elastic global analysis implies that the material obeys Hooke's law in the whole range of loading; therefore, the strength resistance of a section is governed by the onset of first yielding. Plastic global analysis makes allowance for a redistribution of the direct stresses within the cross-section(s) and between different cross-sections, resulting in the formation of plastic hinges until a plastic mechanism occurs. Plastic global analysis is allowed provided that the material properties and the proportions of the members and cross-sections comply with appropriate limitations and requirements. Elastic global analysis is generally used where the performance of the structure depends on serviceability criteria; for the ultimate limit states, plastic global analysis is generally appropriate. Whatever the type of global analysis, it shall be consistent with the anticipated behaviour of the connections and shall incorporate the structural and geometric imperfections specified by the appropriate codes and standards.

6. ADDITIONAL READING 1. Baker, J.F., Horne, M.R. and Heyman, J; "The Steel Skeleton: 2.Plastic Behaviour and Design",

Cambridge University Press, 1956, 408pp. 2. Baker, J.F. and Heyman, J; "Plastic Design of Frames: 1.Fundamentals", Cambridge University Press,

1969, 228pp.



3. Lescouarch, Y.; "Calcul en Plasticité des Structures", Edit. COTECO, Paris, 1983. 4. Roik, K.; "Vorlesungen über Stahlbau", W. Ernst und Sohn, Berlin, 1978. 5. Petersen, Chr.; "Statik und Stabilität der Baukoustuktionen", Vieweg Verlag, Braunschweig, 1981. 6. Brohn, D.; "Understanding Structural Analysis", Blackwells Publications Limited, Oxford, 2nd Ed. 1990. 7. Dowling, P. J., Knowles, P. R. and Owens, G. W.; Structural Steel Design, Butterworths, 1988. 8. Coates, R. C., Coutie, M. G. and Kong, F. K.; Structural Analysis, Thomas Nelson & Sons, London,

1972.





ESDEP WG 7

ELEMENTS

Lecture 7.2: Cross-Section Classification OBJECTIVE

To describe the classification of cross-sections and explain how this controls the application of the methods of analysis given in Eurocode 3 [1].

PREREQUISITES

Lecture 7.1: Methods of Analysis of Steel Structures

RELATED LECTURES

Lecture 7.3: Local Buckling

Lectures 7.5.1 & Lecture 7.5.2: Columns

Lectures 7.8: Restrained Beams

Lectures 7.9: Unrestrained Beams

Lectures 7.10: Beam Columns


Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings

RELATED WORKED EXAMPLES

Worked Example 7.1: Cross-Section Classification

SUMMARY

The analysis methods used are primarily dependent upon the geometry of the cross-section and especially on the width-thickness ratio of the elements which make it up.

The lecture describes how sections are classified as plastic, compact or semi-compact and gives the limiting proportions of the elements by which these classifications are made.

1. INTRODUCTION When designing a structure and its components, the designer must decide on an appropriate structural model. The choice of model effects:

the analysis of the structure, which is aimed at the determination of the stress resultants (internal forces and moments), and the calculation of the cross-section resistance.

Thus a model implies the use of a method of analysis combined with a method of cross-section resistance calculation.


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There are several possible combinations of methods of analysis and methods of cross-section calculation, for the ultimate limit state, involving either an elastic or plastic design approach; the possible combinations are listed in Table 1.

Table 1 Ultimate Limit State Design - Definition of Design Models

Model I is related to plastic design of structures. Full plasticity may be developed within cross-sections, i.e. the stress distribution corresponds to a fully rectangular block, so that plastic hinges can form. These have suitable moment rotation characteristics giving sufficient rotation capacity for the formation of a plastic mechanism, as the result of moment redistribution in the structure.

For a structure composed of sections which can achieve their plastic resistance, but have not sufficient rotation capacity to allow for a plastic mechanism in the structure, the ultimate limit state must refer to the onset of the first plastic hinge. Thus, in Model II, the internal forces are determined using an elastic analysis and are compared to the plastic capacities of the corresponding cross-sections. For statically determinate systems, the onset of the first plastic hinge produces a plastic mechanism; both methods I and II should thus give the same result. For statically indeterminate structures, Model II, in contrast to Model I, does not allow moment redistribution.

When the cross-sections of a structure cannot achieve their plastic capacity, both analysis and verification of cross-sections must be conducted elastically. The ultimate limit state, according to Model III, is achieved when yielding occurs at the most stressed fibre. Sometimes yielding in the extreme fibre cannot even be attained because of premature plate buckling of one component of the cross-section; in such cases, the above ultimate limit state should apply only to effective cross-sections (Model IV).

It is obviously not possible to have a model where a plastic method of analysis is combined with an elastic cross-section verification. Indeed, the moment redistribution which is required by the plastic analysis cannot take place without some cross-sections being fully yielded.

2. REQUIREMENTS FOR CROSS-SECTION CLASSIFICATION In the previous section, the models are defined in terms of structural design criteria; these are actually governed by conditions related to stability problems. Plastic redistribution between cross-sections and/or within cross-sections can take place provided that no premature local buckling occurs, as this would cause a drop-off in load carrying capacity.

It must be guaranteed that no local instability can occur before either the elastic (Model III), or the plastic (Model II), bending resistance of the cross-section, or the formation of a complete plastic mechanism (Model I), is achieved.

Such a mechanism, as envisaged by Model I, can occur provided that the plastic hinge, once formed, has the rotational capacity required for the formation of a plastic mechanism.

To ensure sufficient rotation capacity, the extreme fibres must be able to sustain very large strains without any drop-off in resistance. In tension, the usual steel grades have sufficient ductility to allow for the desired amount of tensile strains; in addition, no drop-off is to be feared before the ultimate tensile strength is reached. With

Model Method of Global Analysis (Calculation of internal forces and moments)

Calculation of Member Cross-Section Resistance

I

II

III

IV

Plastic

Elastic

Elastic

Elastic

Plastic

Plastic

Elastic

Elastic Plate Buckling



compressive stresses, however, it is not so much a question of material ductility, as of ability to sustain these stresses without instability occurring.

Table 2 gives a summary of the requirements for cross-sections in terms of behaviour, moment capacity and rotational capacity. As can be seen from this table, the limits are referred to cross-section classes, according to Eurocode 3 [1], each corresponding to a different performance requirement:

Class 1 Plastic cross-sections: those which can develop a plastic hinge with sufficient rotation capacity to allow redistribution of bending moments in the structure.

Class 2 Compact cross-sections: those which can develop the plastic moment resistance of the section but where local buckling prevents rotation at constant moment in the structure.

Class 3 Semi-compact cross-sections: those in which the stress in the extreme fibres should be limited to yield because local buckling would prevent development of the plastic moment resistance of the section.

Class 4 Slender cross-sections: those in which yield in the extreme fibres cannot be attained because of premature local buckling.

Table 2 Cross-section requirements and classification

The moment resistances for the four classes defined above are:



for Classes 1 and 2: the plastic moment (Mpl = Wpl . fy)

for Class 3: the elastic moment (Mel = Wel . fy)

for Class 4: the local buckling moment (Mo < Mel).

The response of the different classes of cross-sections, when subject to bending, is usefully represented by dimensionless moment-rotation curves.

The four classes given above are recognised for beam sections in bending. For struts loaded in axial compression, Classes 1, 2 and 3 become one, and, in the absence of overall buckling are referred to as "compact"; in this case Class 4 is referred to as "slender".

3. CRITERIA FOR CROSS-SECTION CLASSIFICATION The classification of a specific cross-section depends on the width-to-thickness ratio, b/t, of each of its compression elements. Compression elements include any component plate which is either totally or partially in compression, due to axial force and/or bending moment resulting from the load combination considered; the class to which a specified cross-section belongs, therefore, partly depends on the type of loading this section is experiencing.

a. Components of cross-section

A cross-section is composed of different plate elements, such as web and flanges; most of these elements, if in compression, can be separated into two categories:

internal or stiffened elements: these elements are considered to be simply supported along two edges parallel to the direction of compressive stress. outstand or unstiffened elements; these elements are considered to be simply supported along one edge and free on the other edge parallel to the direction of compressive stress.

These cases correspond respectively to the webs of I-sections (or the webs and flanges of box sections) and to flange outstands (Figure 1).



b. Behaviour of plate elements in compression

For a plate element with an aspect ratio, α = a/b (length-to-width), greater than about 0,8, the elastic critical buckling stress (Euler buckling stress) is given by:

σcr = kσ (1)

where kσ is the plate buckling factor (see below),

υ Poisson's coefficient,



E Young's modulus.

The critical buckling stress is proportional to (t/b)2 and, therefore, is inversely proportional to (b/t)2. The plate slenderness, or width-to-thickness ratio (b/t), thus plays a similar role to the slenderness ratio (L/i) for column buckling.

In accordance with the definition of Class 3 sections, the proportions of the plate element, represented by the b/t ratio, must be such that σcr would exceed the material yield strength fy so that yielding occurs before the plate element buckles. The ideal elastic-plastic behaviour of a perfect plate element subject to uniform compression may be represented by a normalised load-slenderness diagram, where the normalised ultimate load:

= σu/fy

and the normalised plate slenderness:

p =

are plotted as ordinates and abscissae respectively (Figure 2).

For p < 1, = 1 which means that the plate element can develop its squash load σu = fy. For p > 1, decreases as the plate slenderness increases, σu being equal to σcr. Substituting the Equation (1) value for σcr into the above and taking υ = 0,3 gives:

p = (2)

This expression is quite general as loading, boundary conditions and aspect ratio all influence the value of the buckling factor kσ .



The factor kσ is a dimensional elastic buckling coefficient, depending on edge support conditions, on type of stress and on the ratio of length to width (a/b), aspect ratio, of the plated element.

In general, the plated elements of a section have an aspect ratio much larger than unity and most of them are submitted to uniform compression. For such cases, Table 3 gives buckling factors for plated elements having various long edge conditions.

Table 3 Elastic buckling factor kσ

When plated elements of sections are submitted to any kind of direct stress, other than uniform compression (e.g. webs of a girder in bending), the buckling factor kσ has to be modified to take account of the stress gradient, given by the stress ratio, ψ.

Table 4 gives the buckling factors for different stress ratios ψ , for internal or outstand elements. In the latter case a distinction is made for elements with tip in compression or in tension.

Table 4 Buckling factors and stress distribution



c. Limit plate element slendernesses

The actual behaviour is somewhat different from the ideal elastic-plastic behaviour represented in Figure 2 because of:

i. initial geometrical and material imperfections,

ii. strain-hardening of the material,

iii. the postbuckling behaviour.

Initial imperfections result in premature plate buckling, which occurs for p < 1. The corresponding limit plate

slenderness p3, for Class 3 sections, may differ substantially from country to country because of statistical variations in imperfections and in material properties which are not sufficiently well known to be quantified accurately; a review of the main national codes shows that it varies from 0,5 to 0,9 approximately. Eurocode 3 [1,2] has adopted p3 = 0,74 as the limit plate slenderness of Class 3 compression elements and p3 = 0,9 for elements in bending where the yield strength may be reached in the extreme fibre of the cross-section. For plate elements for which p < p3, no plate buckling can occur before the maximum compressive strength reaches the yield strength.

A Class 1 section must develop a resistance moment equal to the plastic capacity of the section and must maintain this resistance through relatively large inelastic deformations. In order to fulfil these conditions without buckling, the entire plate element must be yielded and the material must be strained in the strain-hardening region (see Table 2); this is only possible for elements with low reference slendernesses ( p < p1), see Figure 2.

On the basis of certain theoretical approaches [3, 4, 5] values of p1 between 0,46 and 0,6, are proposed in



various standards. The difference can be explained in the choice of the amount of necessary rotation capacity. A value of p1 = 0,6 corresponds to a limited rotation capacity which is estimated to be sufficient for usual plastic design (continuous beams, non-sway frames, etc.). In Eurocode 3 [1], the proposed value is:

p1 = 0,5

A Class 2 (or compact section) is one which can just reach its plastic moment resistance but has a rapid drop-off in resistances at that point (Table 2). The plate element is yielded and the material strained in the plastic range; it occurs for elements with medium reference slendernesses p2 where:

p1 < p2 < p3

In Eurocode 3 [1], the proposed value is p1 = 0,6.

Using formula (2), and the appropriate values of p and kσ , the limiting b/t ratios can be calculated. Table 5 gives some limiting value of b/t for the elements of the cross-section of a rolled I-profile in compression or bending.

Table 5 Maximum slenderness ratios for the elements of a rolled section in compression or in bending

The most important limiting proportions of the elements of a cross-section, which enable the appropriate classifications to be made, are specified in Eurocode 3 [1]. Appendix 1 gives the limiting proportions for compression elements of Class 1 to 3.

The limiting values of the width-to-thickness ratio (b/t) of the plate elements of sections apply to members in

Element Class 1

cross- section

Class 2

cross- section

Class 3 cross-section

Formula kσ b*/t or d/tw

Flange (1)

(b*/t)

9ε 10ε 21ε

0,43 14ε (1)

Web in compression d/tw

33ε 38ε 21

1,0 42ε

Web in pure bending d/tw

72ε 83ε 25,4ε

23,9 124ε

fy 235 275 355

ε =

ε 1,0 0,92 0,81

(1) In practice, b, the half-width of the flange is considered instead of b*. For this reason, the values given in the "Essentials of Eurocode 3" is b = 15 ε > b*.



steel of a specific yield strength. In order to cover all grades of steel, Eurocode 3 presents local buckling data

non-dimensionally, in terms of a reduction factor ε = , where 235 represents the yield stress of mild steel and fy that of the steel considered.

The various compression elements in a cross-section (such as a web or a flange) can, in general, be in different classes and a cross-section is normally classified by quoting the least favourable (highest) class of its compression elements.

It is important, particularly in plastic design, that the sections selected for various members should be, in all cases, appropriate for the assumed mode of behaviour.

When any of the compression elements of a cross-section fail to satisfy the limits given in Table 5 for Class 3, the section is classified as "slender" and local buckling shall be taken into account in the design. This may be done by means of the effective cross-section method which is discussed in detail in Lecture 7.3.

4. CONCLUDING SUMMARY The methods of analysis used are influenced by the geometry of the cross-sections and, more particularly, by the width to thickness ratios of the plate elements in compression. It must be guaranteed that no local instability can occur before a complete mechanism is achieved or before the plastic or elastic moment can be reached. Four cross-sectional classes are identified, each corresponding to a different performance requirement: plastic, compact, semi-compact and slender cross-sections. Limiting proportions for the elements of a cross-section, which enable the appropriate classifications to be made, are given in the lecture. When any of the compression elements of a cross-section fail to satisfy the limiting proportions for Class 3 (semi-compact), local buckling shall be taken into account in the design.

5. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992.

[2] Bureau, A. et Galea, Y., "Application de l'Eurocode 3: Classement des sections transversales en I".

Construction métallique, no. 1, 1991.

[3] Salmon, C.G., Johnson, J.E., "Steel Structures. Design and Behaviour", Harper et Row, Publishers, New York.

[4] Dubas, P., Gehri, E., "Behaviour and Design of Steel Plated Structures", Publication no. 44, ECCS, TC8, 1986.

[5] Commentaire de la norme SIA161 "Constructions Métalliques", Publication A5, Centre Suisse de la Construction Métallique, Zurich, 1979.

6. ADDITIONAL READING 1. Bulson, P.S., "The Stability of Flat Plates"

Chatto and Windus, London 1970.

APPENDIX I Table 1 Maximum width-to-thickness ratios for compression elements



Table 2 Maximum width-to-thickness ratios for compression elements



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ESDEP WG 7

ELEMENTS

Lecture 7.3: Local Buckling OBJECTIVE/SCOPE

To describe the design rules in Eurocode 3 [1-3] for members with Class 4 sections.

PREREQUISITES

Lectures 7: Elements

Lectures 8: Plates and Shells

RELATED LECTURES

Lectures 9: Thin-Walled Construction

SUMMARY

For members with Class 4 sections [1] the effect of local plate buckling on the overall member behaviour has to be taken into account. The buckling is allowed for by using effective cross-sections which assume parts of the gross cross-section are inactive. The rules for the determination of the effective cross-section and the design checks required are given.

1. INTRODUCTION For members with Class 4 sections the effect of local buckling on global behaviour at the ultimate limit state is such that the elastic resistance, calculated on the assumption of yielding of the extreme fibres of the gross section (criteria for Class 3 sections), cannot be achieved.

Figure 1 gives the moment deflection curve for a point loaded beam (Class 4). The reason for the reduction in strength is that local buckling occurs at an early stage in parts of the compression elements of the member; the stiffness of these parts in compression is thereby reduced and the stresses are distributed to the stiffer edges, see Figure 2.



To allow for the reduction in strength the actual non linear distribution of stress is taken into account by a linear distribution of stress acting on a reduced "effective plate width" leaving an "effective hole" where the buckle occurs, Figure 2.

By applying this model an "effective cross-section" is defined for which the resistance is then calculated as for Class 3 sections (by limiting the stresses in the extreme fibres to the yield strength).

2. DEFINITION OF THE "EFFECTIVE WIDTHS" The effective widths beff are calculated on the basis of the Winter formula:

beff = ρ .b

where the reduction coefficient ρ is dependent on the plate slenderness p defined by linear plate bucking theory, as shown in Figure 3.



Figure 4 gives some examples of effective cross-sections for members in compression.



For members in bending test results have shown that the effective widths may be determined on the basis of stress distributions calculated using the gross section modulus, Wel , even though the formation of "effective holes" in the compression parts will shift the neutral axis of the effective cross-section; an interactive process is not, therefore, necessary.

Figure 5 gives some examples of effective cross-sections for members in bending.



3. DESIGN OF MEMBERS

3.1 Columns in Compression

As shown by the effective cross-sections 1 and 2, in Figure 4, the neutral axes of doubly-symmetrical cross-sections will not change with the formation of effective holes. Hence the compression load NSd that is central to the gross cross-sections will also be central to the effective cross-section.

The column buckling design check, therefore, is based on the non-dimensional slenderness = , where Ncr is calculated on the basis of the gross cross-section and Npl is calculated using the effective cross-sectional area Aeff (Npl = Aeff . fy ).

The design buckling resistance is given by:

NbRd = χ . Npl /γM

where χ is the reduction factor for the relevant buckling curve.

For singly symmetrical cross-sections - type 3 in Figure 4 - or unsymmetrical cross-sections, the formation of effective holes may lead to a shift, eN, in the neutral axes position. The compression load, NSd, that is central to the gross cross-section will, therefore, be eccentric to the effective cross-section and hence will cause an additional bending moment M=NSd.eN. The member is now a beam-column and must



be checked in accordance with Section 3.3.

3.2 Beams in Bending

Beams must be checked using the section modulus determined for the effective cross-sections, as given in Figure 5. In general the attainment of the yield strength at the compression face will limit the design bending resistance of effective cross-sections:

Mo,Rd = Weff . fy /γM1

For cross-sections similar to type 3 of Figure 5, Weff will be determined for the actual edge (e):

Weff =

and not the edge of the effective hole.

The lateral-torsional buckling check for beams is analogous to that for columns. The non-dimensional

slenderness LT = , is calculated with Mu=Weff.fy for the effective cross-section, and with Mcr calculated for the gross cross-sectional values. The design lateral-torsional buckling resistance is then given by:

Mb,Rd = χLT . Mu /γM1

where χLT is the reduction factor for the relevant buckling curve.

3.3 Beam-Columns

In the case of members that are subject to compression and monoaxial or biaxial bending (e.g. in the case of columns with monosymmetrical or unsymmetrical cross-sections) the design check is carried out using an interaction formula in which the checks for a centrally compressed column, for a beam with bending about the y-axis only, and for a beam with bending about the z-axis only, are combined.

If lateral-torsional bucking is prevented the interaction formula is as follows:

If lateral-torsional bucking can occur:

where eNy or eNz are the eccentricities due to the shift of the neutral axis for compression only. The resistances NbRd and Nbz.Rd are related to the case of central compression; Moy.Rd and Mby.Rd are related to bending about the y-axis only; and Moz.Rd is related to bending about the z-axis (weak axis) only.

4. CONCLUDING SUMMARY The design of members with Class 4 sections is carried out as for members with Class 3 sections (elastic analysis, elastic cross-sectional resistance limited by yielding in the extreme fibres) except that an effective cross-section (derived from gross sections with "effective holes", where buckles may occur) is



used. The buckling checks for columns and the lateral-torsional buckling check for beams, requires the critical values, Ncr and Mcr, to be calculated using the gross cross-sectional data without considering "effective holes". In the case of columns with non-doubly symmetric cross-sections, the formation of "effective holes" may cause a shift in the neutral axis position resulting in eccentric compression and hence a beam-column problem. Beam-columns (compression and biaxial bending) are verified by using an interaction formula where the checks for the column in compression only, for the beam with bending about the y-axis only, and for the beam with bending about the z-axis only, are combined.

5. REFERENCES [1] "Eurocode 3: "Design of steel structures" ENV 1993-1-1: Part 1.1, General rules and rules for buildings, CEN, 1992.

6. ADDITIONAL READING 1. "Eurocode 3: Part 1.3: "Cold formed thin gauge members and sheeting", CEN, (in preparation). 2. Eurocode 3: Background Document 5.5. (Justification of the design resistances, for buckling

verifications)





ESDEP WG 7

ELEMENTS

Lecture 7.4.1: Tension Members I OBJECTIVE

To describe the typical uses of tension members and to explain the derivation of the rules in Eurocode 3 [1].

PREREQUISITES

Basic strength of materials.

General appreciation of material behaviour and limit-state design.

RELATED LECTURES

Lecture 7.12: Trusses and Lattice Girders

Lecture 11.1.2: Introduction to Connection Design


Worked Example 7.3: Tension Members I

SUMMARY

The lecture introduces the use of steel tension members in construction. The modes of failure of these members, especially at holes in connection zones, are discussed; the design formulae, as proposed by Eurocode 3 [1], are presented.

1. INTRODUCTION Structural stability depends on a balance between elements sustaining either tensile or compressive stresses. Because natural materials are more suited to resist compression, the traditional objective of the designer has been to avoid tensile stresses using ingenious systems such as arches, vaults, domes, etc. Special treatment of natural materials , however, allowed the development of structures, usually temporary, where the tension members played a fundamental role (Figure 1). Even in situations of fundamentally massive construction tension elements can be found helping to stabilise the system (Figure 2). The Industrial Revolution, during which time ferrous materials were developed, brought great advances in the use of tensile elements as pure tension could now be safely transmitted without the previous durability problems associated with natural materials.

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Some simple forms of structures with tension members are given in Figure 3: these are an inclined roof with a tension beam, and a truss whose bottom chord and several diagonals are in tension. More recent developments use cables as tension members in such structures as roofs, bridges, masts, cranes etc. The present lecture deals with conventional tension members; cables are discussed in Lecture 7.4.2.



2. BEHAVIOUR OF CROSS-SECTIONS IN TENSION MEMBERS

2.1 General

Generally tension members are designed using rolled sections, bars or flats. When more area is needed or



connection design requires it, it is possible to combine profiles or to build up a specific section using plates (Figure 4). Flats are generally not used because of their high flexibility; for good practice the slenderness should be limited to 300 for principal members or 400 for secondary members (obviously this rule does not apply to round bars). In general, rolled sections are preferred and the use of compound sections is reserved for larger loads or to resist bending moments in addition to tension.

It is generally assumed that the distribution of stresses in cross-sections of members subjected to axial tensile forces is uniform. However, there are some parameters which result in a non-uniform distribution of stresses; these are:

residual stresses connections.

The influence of these parameters on the behaviour of the cross-section is discussed in Sections 2.2 and 2.3.

2.2 Residual Stresses

Residual stresses develop when the member is formed and are due to the production process. Their origin can be



thermal, either developed during the solidification of the steel or during welding parts of the member; or they can be mechanically induced when trying to produce counter-deflection or when straightening the member. The induced stresses are self equilibrated and although they do not affect the ultimate resistance of the member they induce non-linearities in the strain-stress behaviour as well as greater deformability.

Consider, for example, a rectangular section with residual stresses subjected to an axial force (Figure 5); although the distribution of stresses due to this force is uniform, the corresponding distribution of the total stresses is not. When the combined stresses (Figure 5c) reach the yield strength fy, yielding of the relevant fibres commences and the elastic part of the section is continuously reduced as the external force increases (Figure 5d); the ultimate limit state is reached when the entire section has yielded. Although the behaviour of the section is non-linear (Figure 5e), the ultimate limit state is identical for both cases with and without residual stresses.

2.3 Connections

Connections are generally made either by bolting or welding.

When several members have to be connected, additional plates must be used which introduce secondary effects due to the moments developed. Sometimes it is possible to reduce these local eccentricities by varying the weld lengths or the bolt distribution.

In addition, the holes that are needed to fix the bolt significantly distort the ideal behaviour of the cross-section.

Firstly, there is an area reduction that has to be taken into account and also a distortion in the stress distribution that induces a non-uniformity in the strain; the effect of the holes is to increase the stresses locally around them



(Figure 6). For a plate of infinite width the distribution is given by:

σo = σ (1)

for x ≥ R

where R is the hole radius

x is the distance to the point under consideration

when x = R

σo = max σo = 3σ

The above suggests the use of net cross-sections at holes in order to compensate for the weakening effects.



It should be mentioned that, although not reflected in the Codes, the net area might allow for a variety of effects influencing the connection efficiency; these include the ductility of the metal, the care taken when forming the hole (local cracks reduce the ductility), and the relative proportion between hole-diameters and hole distances (inducing a confining effect).

3. ANALYSIS In tension elements the computations are generally related to resistance assessment; however, because of the high slenderness allowed by the optimum use of the material, it is also very important to check their stiffness.

3.1 Stiffness Requirements

Tension elements are generally subjected to bending induced either by their own weight, dynamic effects like wind or passing loads, or even by unavoidable eccentricities. Good practice rules usually allow for this and a rigorous check is not usually required; as mentioned in Section 2.1 the limits are 300 or 400 for principal or secondary members respectively; some American Codes, however, are more restrictive, see Table 1.

TABLE 1

3.2 Resistance of the Cross-Section

The resistance evaluations should check the:

a) Behaviour under static loads, and

b) Behaviour under alternating loads.

This lecture is limited to case a; for tension elements under fatigue conditions reference is made to the Lectures 12.

If the element has no holes, the design axial force resistance is given by:

Npl.Rd = A.fyk / γM1 (2)

AISC AASHTO

Main structure

Secondary structure

Alternating loads

240

300

-

200

240

140

Limit of slenderness λ for tension elements

λ = L/i i2 = I/A

where:

L is the length of the element.

i is the minimum radius of gyration.

I is the minimum 2nd moment area of the section.

A is the area of the cross-section.



where:

A is the gross area of the cross-section.

fyk is the characteristic value of the yield strength.

γM1 is the partial safety factor for the gross section (γM1 ≈ 1,1).

When the member is connected by bolting the section is weakened, suffering a reduction of around 10 to 20% of the gross area. This gives rise to two problems: firstly a reduction in the net area; and secondly, the holes induce stress concentrations that, according to Equation (1) and Figure 6, can reach values three times higher than the uniform distribution. Nevertheless, it is assumed that at the ultimate limit state, due to the ductility of the steel, the stress distribution across the net section is uniform.

The hole diameter should be increased to take account of damaged material when the hole is punched without special precautions; where holes are countersunk the overall diameter should be deducted.

3.2.1 Net area

According to Eurocode 3: "the net area of a cross-section or element section shall be taken as its gross area less appropriate deductions for all holes and other openings. Provided that the fastener holes are not staggered the total area to be deducted shall be the maximum sum of the sectional areas of the holes in any cross-section perpendicular to the member axis" [1].

When the holes are staggered it is necessary to use special formulae to calculate the deduction (Figure 7a).



The stress distribution described in Section 2 is more complicated in this case and the Cochrane rule is applied to add the length S2/4g. This is why the Eurocode suggests the following: "When the fastener holes are staggered the total area to be deducted for fastener holes shall be the greater of:

a) the deduction for non-staggered holes.

b) the sum of the sectional areas of all holes in any diagonal or zig-zag line extending progressively across the member or part of the member less S2t/4g for each gauge space in the chain of holes (Figure 7a).



S is the staggered pitch, the spacing of the centres of two consecutive holes in the chain measured parallel to the member axis.

g is the gauge, the spacing of the centres of the same two holes measured perpendicularly to the member axis.

t is the thickness.

In an angle, or other member, with holes in more than one plane, the gauge shall be measured along the centre of thickness of the material (Figure 7b).

3.2.2 Resistance of the net sections

In principle, the net area check would be as follows:

NR = Anet.fy / γM2 (3)

where

Anet is the net area

γM2 is the partial factor of safety for the net area (γM2 ≈ 1,25)

However, the global behaviour of the tension member has to be taken into account. Imagine, for example, that the length affected by the connection is about 5% of the total member length; then assume that the strain at the ultimate load of the connection is 10 times the yield strain (Figure 8).



When the member reaches the yield condition, and the connection the failure condition, the increases in length would be:

Connection zone:

Member zone:

That is ≈ 0,5 (4)

which means that the elongation within the connection zone is much smaller than that of the entire bar.



This is why Eurocode 3 [1] allows the exceedance of the yield strength in the connection zone up to the ultimate tensile strength, fuk; that is, it is implicitly assumed that the failure of an member can be described by its deformation.

The same philosophy is applied by other codes which, like Eurocode 3, also include a reduction coefficient to take account of the unavoidable eccentricities, stress concentrations, etc. The Eurocode 3 reduction is taken as 10%, so that the recommended formula is:

NnetRd = 0,9Anet.fuk / γM2 (5)

where γM2 is the partial safety factor for resistance with a proposed value of 1,25.

3.2.3 Verification

The verification formula is:

NSd ≤ NRd (6)

where NSd is the design tensile force, and

NRd the design resistance force which is the lesser of the values given by Equations (2) and (5).

If ductile behaviour is desired, it can be seen from Figure 8 that the member has to yield before the connection fails, that is:

Npl.Rd ≤ Nnet.Rd (7)

or

0,9Anet/A ≥ γM2 fy / γM1 fu (8)

Finally, Eurocode 3 [1] considers the verification for Category C connections; these are slip-resistant connections with preloaded high strength bolts, where slip should not occur at the ultimate limit state.

In this case the failure criteria for the net section is a yield criterion, unlike Equation (5), and so the design resistance is given by:

Nnet.Rd = Anet fy / γM1 (9)

4. CONCLUDING SUMMARY The availability of tension members greatly increases the number of structural forms possible. A variety of cross-sections can be used for tension members; to employ their full capacity it is necessary to consider residual stress effects and connection design. The strength of a member in tension is calculated on the assumption that the entire section has yielded. The failure modes of a tension member may be defined by either yielding of the gross section or by rupture of the net section. The design resistances are determined by applying adequate partial safety factors to the relevant strengths. If ductile behaviour is envisaged, e.g. for cyclic loading, yielding of the gross section should precede rupture of the net section. The slenderness of tension members has to be limited in order to avoid excessive deflections during transportation, erection, maintenance, etc.



5. REFERENCES [1] Eurocode 3: " Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992.

6. ADDITIONAL READING 1. Dowling, P. J., Knowles, P., Owens, G. W., "Structural Steel Design", The Steel Construction Institute &

Butterworths, 1988. 2. Worgan, W., "The Elements of Structure", Pitman 1964. 3. Moutklanov, K., "Constructions Métalliques", MIR 1978. 4. Salmon, C. G. and Johnson, J. E., "Steel Structures, 2nd Edition", Harper & Row 1980. 5. Torroja, E., "Razón y Ser de los Tipos Estructurales", IETCC, 1960. 6. Zignoli, V., "Construzioni Metalliche", UTET, 1978. 7. Chr. Petersen, Stahlbauten, Vieweg Verlag, Braunschweig, 1988.





ESDEP WG 7

ELEMENTS

Lecture 7.4.2: Tension Members II OBJECTIVE/SCOPE

To explain the application and behaviour of cables.

PREREQUISITES

Lecture 7.4.1: Tension Members I

RELATED LECTURES

Lecture 14.4: Crane Runway Girders

Lecture 14.6: Special Single-Storey Structures

Lecture 15B.12: Introduction to Bridge Construction

Lecture 16.2: Transformation and Repair


Worked Example 7.4: Tension Members II

SUMMARY

Cable structures (that is those in which the principal element is a cable which transmits only tensile forces) have many and varied applications. This lecture explains how the cables themselves are made and derives the design equations used which take into account their non-linear behaviour.

1. INTRODUCTION The use of cables, as tension members, has been on the increase over the last few decades. These cables are composed of high strength steel wires which are bundled together in order to obtain the required tensile resistances. Cable structures provide most interesting and spectacular solutions to modern architectural and engineering problems; they are used in structures such as roofs, hangars, cranes, guyed masts, suspension bridges, cable stayed bridges, transmission towers and ski-lift facilities (see Slides). Compared to conventional structures these require careful consideration as follows:

they are prone to wind vibrations as they are very light and often span large distances; special care must be taken to protect the cables against corrosion; the overall behaviour of the structure is non-linear as will be explained later.

2. COMPOSITION OF ROPES AND CABLES A cable is a highly flexible member that is primarily capable of transmitting axial forces. It is composed of high strength steel wires which are bundled together; the hierarchy of elements is as follows:

WIRE: This is produced from high-strength steel bars by rolling or cold drawing (Figure 1) thereby reducing the initial area. The cold-forming process results in an increase in the tensile and yield stress and a decrease in



the ductility of the steel (Figure 2); it also effects the residual stresses through the wire thickness (Figure 1) since the velocity of the steel running through the area reducing device is different: for cold drawn wires the velocity in the core is larger than at the surface; for rolled wires the opposite is the case.



STRAND: This is produced from a series of wires that are normally wound together in a helical fashion (Figure 3).

ROPE: This is produced from a series of strands that are also wound together in a helical fashion. If the winding orientation of the strands is, for example, to the left then the strands that produce the rope should be wound together to the right in order to avoid twist of the rope when it is subjected to axial forces (Figure 4).



CABLE: Using the elements mentioned above, it is possible to produce cables; the more usual types of cables are:

Parallel wire cables (Figure 5a) which are composed of a series of parallel wires; Strand cables (Figure 5b) which are composed of parallel or helically combined strands; Locked-coil cables (Figure 5c) which were invented for better corrosion protection. In these the exterior layers, composed of S-shaped wires, surround the inner core of wires. These cables are less flexible than the open types.



3. MECHANICAL PROPERTIES Due to their chemical composition, and the cold forming process, the tensile strength of the wires is high; it is also normally inversely proportional to the wire diameter. The usual tensile strength is about 1600-1800 N/mm2.

The σ -ε diagram for this steel has no yield plateau and so the yield strength is conventionally defined as the stress at which the plastic deformation is 0,2%. The value of this strength typically varies between 80 to 90% of the tensile strength.

The modulus of elasticity of the strands, ropes and cables is smaller than that of the steel material (wire) of



which they are composed (Figure 6). After unloading there is a residual deformation which is not due to plasticity but due to the winding and transportation process, and which stabilises after several loading cycles; this is the reason why, in real structures the cables are post-tensioned after a certain period; after this has been done the modulus of elasticity has a larger value than the initial one.

Mean values of this modulus are normally as follows:

For parallel wire cables E = 200 N/mm2

For locked-coil cables E = 160 N/mm2

For strand cables E = 150 N/mm2

The coefficient of linear thermal expansion has the value of α = 1,2 x 10-5 per degree centrigrade.

4. DESIGN VALUES The metallic area of a rope is equal to:

(1)

where d is the diameter of the rope, and f is a coefficient equal to:

f = 0,55 for multiple strand ropes

f = 0,75 - 0,77 for open spiral ropes

f = 0,81 - 0,86 for locked coil ropes.



The tensile resistance of a rope is equal to:

F = ks Am . fu (2)

where

Am is the metallic area

fu is the tensile strength of the wires

ks is a coefficient equal to:

ks = 0,76 - 0,85 for multiple strand ropes

ks = 0,82 - 0,90 for open spiral ropes

ks = 0,87 - 0,92 for locked coil ropes

ks = 0,93 - 1,0 for parallel wire ropes

The tensile resistance of a complete rope including the anchoring device is equal to:

Fu = ka . ks . Am . fu (3)

where the coefficient ka has values between 0,80 and 1,0 depending on the anchorage system.

The design tensile resistance is then given by the expression:

FRd = Fu /γM (4)

where γM is the partial safety factor.

The design value of the modulus of elasticity is dependent on the stress state under consideration. The σ-ε diagram of a rope, as presented in Figure 7, is non-linear for the first loading; the rope is subsequently subjected to repeated loading cycles, σq (due to traffic loading) over an initial stress σg due to dead loading and prestressing.



From this Figure three values of the modulus of elasticity can be observed:

For the first loading Eg = σg / εg

For the traffic loading Eq = σq / εq

For the total behaviour EA = σg / εA

It is clear that while the value of Eq remains almost constant, the value of Eg decreases as σg increases towards the overall stress σg + σq, while the opposite happens for EA.

5. CONNECTIONS Coupling, saddling and anchorage of cables are the three most important types of connections that occur in these structures (Figure 8); the requirements for these are as follows:



they should be able to safely transmit the loads acting on them; they should not allow the cable to slide; they should be able to sustain alternative loads; they should be easily accessible to allow maintenance.

Examples of connections are given in Figure 9.



6. BEHAVIOUR OF A CABLE Figure 10 shows a cable which is suspended over two points, A and B, at different heights.



In order to derive the equations for this cable a small element CC′ is cut off, and the forces at its ends together with the load acting on it are substituted in order to restore equilibrium; the two loading conditions considered here are:

g = a uniform load related to the inclined length.

q = a uniform load related to the horizontal projection.

It is further supposed that the cable has no bending stiffness, and can therefore only transmit axial loads.

The equilibria for the two loading cases are as follows:



(5), (6), (7) → Hy″ = g (8a) Hy″ = q (8b)

Solution:

The coefficients C1 and C2 may be determined from the boundary conditions. For the special case of a cable suspended from two equally high points (Figure 11) the following expressions may be obtained:

Condition g q Σ H = 0 H - (H + dH) = 0 → H = const (5) Σ M = 0 Vdx - Hdy = 0 → V = Hy′ → V′ = Hy″

(6)

Σ V = 0 V - (V + dV) + g ds = 0 V - (V + dV) + q dx = 0 → V′ = g.ds / dx V′ = q (7)

y = + C2

y = (9)

Catenary Parabola

g q

(10)

Tensile force Ft = H cosh

(11)

Cable length

= = L + L (12)

Sag (13)



A comparison of the values given for sag and cable length by the two approaches is given in Figure 12; it can be seen that the catenary can be reasonably substituted by a parabola when the sag is small relative to the span; this is normally the case in guyed masts, cable stayed bridges, cranes etc. For other cases (e.g. transmission towers) the catenary should be used. Figure 12 shows that the approximation for the cable length is not as good as for sag.



For transmission towers two design situations (Figure 13) should normally be verified:



Winter situation: The cable is loaded by its self weight and the weight of the surrounding ice and the temperature is low, (t=-5° C)

Summer situation: The cable is loaded by its self weight and the temperature is high (t = 40° C).

The first situation yields the highest stress in the cable, whereas the second may be decisive for sag control. The governing equation for the verification is:

(σ / E α) + t + (L2 γ2 / 24σ2) = const (14)

where:

σ is the stress in the cable.

L is the span.

γ is the specific weight of the cable possibly including ice.

t is the temperature.

E and α are the modulus of elasticity and the coefficient of thermal expansion of the cable.

The first term of Equation (14) expresses the strain of the cable due to the tensile force, the second the strain due to temperature and the third the strain due to sag.

7. MODULUS OF ELASTICITY DUE TO SAGGING

When considering a cable whose end B moves from an initial position B to B′ the following expressions are valid:



(15)

The cable length, according to Equation (12), is equal to:

(16)

which gives the displacement of point B as equal to:

(17)

The longitudinal modulus of elasticity, due to sagging, is then equal to (Figure 14):

(18)

The equivalent modulus of elasticity is then derived regarding both the elastic and the sagging deformations:

(19)

The value of Ee is dependent on the stress levels presented in Figure 15. The figure shows that short cables, subjected to high stresses, behave like the steel parent material, whereas long cables, subjected to small stresses, are much less stiff; this is one of the reasons why cable structures must be prestressed as otherwise they would be subjected to very large deformations.



8. CONCLUDING SUMMARY Cables are widely used, as tension members, in various types of structures including roofs, bridges, masts, etc. Cables are composed of high strength steel wires bundled together in various ways. The design values for cable resistance, depend on its type and the anchorage system used. The behaviour of a cable is highly non-linear due to sagging effects. The flexibility of a cable is dependent on its length and the stress conditions. Prestressing of cables is necessary where stiffness is required.

9. ADDITIONAL READING 1. Merkblatt. 496, "Ebene Seiltrogwerke, Benatungsstelle für stahlverwendung", 1980. 2. Petersen, Chr., "Stahlbau", Vieweg Verlag, Braunschweig, 1988. 3. Belenya, E; "Prestressed Load-bearing Metal Structures", MIP, 1977. 4. Podolny, W., Scolzi, J. B.; "Construction and Design of Cable-stayed Bridges", Wiley, 1986. 5. Walther, R., Houriet, B., Isler, W., "Moia, P., "Ponts Houbanés", Presses Polytecniques Romandes,

Lausanne, 1985.





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ELEMENTS

Lecture 7.5.1: Columns I OBJECTIVE

To describe the different kinds of steel columns and to explain the procedures involved in the design of compression members.

PREREQUISITES

Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium


RELATED LECTURES


Lecture 7.6: Built-up Columns


Lecture 7.12: Trusses and Lattice Girders


Worked Example 7.5: Column Design

SUMMARY

Different kinds of compression members (uniform and non-uniform cross-sections and built-up columns) are described. The differences between stocky and slender columns are explained and the basis for design, using the European buckling curves, is outlined.

1. INTRODUCTION Columns are vertical members used to carry axial compression loads. Such structural elements are found, for example, in buildings supporting floors, roofs or cranes. If they are subjected to significant bending moments in addition to the axial loads, they are called beam-columns.

The term 'compression' member is generally used to describe structural components subjected only to axial compression loads; this can describe columns (under special loading conditions) but generally refers to compressed pin-ended struts found in trusses, lattice girders or bracing members.

This lecture deals with compression members and, therefore, concerns very few real columns because eccentricities of the axial loads and, above all, transverse forces, are generally not negligible. Nevertheless, compression members represent an elementary case which leads to the understanding of the term of compression in the study of beam-columns (Lectures 7.10.1 and 7.10.2), frames (Lecture 7.11), and trusses and lattice girders (Lecture 7.12).

Because most steel compression members are rather slender, buckling can occur; this adds an extra bending moment to the axial load and must be carefully checked.



The lecture briefly describes the different kinds of compression members and explains the behaviour of both stocky and slender columns; the buckling curves used to design slender columns are also given.

2. MAIN KINDS OF COMPRESSION MEMBERS

2.1 Simple Members with Uniform Cross-Section

For optimum performance compression members need to have a high radius of gyration, i, in the direction where buckling can occur; circular hollow sections should, therefore, be most suitable in this respect as they maximise this parameter in all directions. The connections to these sections are, however, expensive and difficult to design.

It is also possible to use square or rectangular hollow sections whose geometrical properties are good (the square hollow sections being the better); the connections are easier to design than those of the previous shape, but again rather expensive.

Hot-rolled sections are, in fact, the most common cross-sections used for compression members. Most of them have large flanges designed to be suitable for compression loads. Their general square shape gives a relatively high transverse radius of gyration iz and the thickness of their flanges avoids the effect of local buckling (care has to be taken in the case of light H-sections and high strength steel). The open shape, produced by traditional rolling techniques, facilitates beam-to-column and other connections.

Welded box or welded H-sections are suitable if care is taken to avoid local flange buckling. They can be designed for the required load and are easy to connect to other members; it is also possible to reinforce these shapes with welded cover plates.

Figure 1 illustrates all the shapes mentioned above.



It should be noted that:

the type of connection is important in the design of simple compression members because it defines the effective length to be taken into account in the evaluation of buckling. Circular sections do not represent the optimum solution if the effective length is not the same in the two principal directions; in this case, non symmetrical shapes are preferable. members are frequently subjected to bending moments in addition to axial load; in these conditions I-sections can be preferable to H-sections.

2.2 Simple Members with Non-Uniform Cross-Sections

Members with changes of cross-section within their length, are called non-uniform members; tapered and stepped members are considered in this category.

In tapered members (Figure 2) the cross-section geometry changes continuously along the length; these can be either open or box shapes formed by welding together elements, including tapered webs, or flanges, or both. A classical example is that of a hot-rolled H or I-section whose web is cut along its diagonal; a tapered member is obtained by reversing and welding the two halves together.



Stepped columns (Figures 3) vary the cross-section in steps. A typical example of their use is in industrial buildings with overhead travelling cranes; the reduced cross section is adequate to support the roof structure but must be increased at crane level to cater for the additional loads. Stepped columns can also be used in multi-storey buildings to resist the loads in the columns at the lower levels.



2.3 Built-up Columns

Built-up columns are fabricated from various different elements; they consist of two or more main components, connected together at intervals to form a single compound member (Figure 4). Channel sections and angles are often used as the main components but it is also possible to use I or H-sections; they are laced or battened together with simple elements (bars or angles or smaller channel sections) and it is possible to find columns where both methods are combined (Figure 5). Columns with perforated plates (Figure 6) are also considered in this category.



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The advantage of this system is that it gives relatively light members which, because most of the steel is placed quite far from the centre of gravity of the cross-section have a relatively high radius of gyration. The most important disadvantage comes from the high fabrication costs involved. These members are generally designed for large structures where the compression members are long and subjected to heavy loads. It is to be noted that buckling of each individual element must be checked very carefully.

Figure 7 shows some closely spaced built-up members and gives details of star-battened angle members. These are not so efficient as the previous ones because of their smaller radius of gyration; however, the ease with which these can be connected to other members may make their use desirable. They behave in compression in a similar way to those described above, hence their inclusion in this section.



Built-up columns may have uniform or non-uniform cross-sections; it is possible, for example, to find stepped or tapered built-up columns (Figure 8).



The design of built-up columns will be examined in Lecture 7.6.

3. PURE COMPRESSION WITHOUT BUCKLING

3.1 Stub Columns

Stub (or stocky) columns are characterised by very low slenderness, are not effected by buckling and can be designed to the yield stress fy.

If local buckling does not affect the compression resistance (as can be assumed for Class 1, 2 and 3 cross-sections), the mode of failure of such members corresponds to perfect plastic behaviour of the whole cross-section, which theoretically occurs when each fibre of the cross-section reaches fy. It is to be noted that residual stresses and geometric imperfections are practically without influence on the ultimate strength of this kind of column and that most experimental stub columns fail above the yield stress because of strain-hardening.

The maximum compression resistance Nmax is, therefore, equal to the plastic resistance of the cross-section:

Nmax = Npl = Aeff fy (1)



Aeff is the effective area of the cross-section, see Section 3.2.

Eurocode 3 [1] considers that columns are stocky when their reference slenderness is such that ≤ 0,2 ( is defined in Section 5.1). They are represented by a plateau on the ECCS curves [2], see Section 5.2.

3.2 Effective Area

The effective area of the cross-section used for design of compression members with Class 1, 2 or 3 cross-sections, is calculated on the basis of the gross cross-section using the specified dimensions. Holes, if they are used with fasteners in connections, need not be deducted.

4. STABILITY OF SLENDER STEEL COLUMNS Depending on their slenderness, columns exhibit two different types of behaviour: those with high slenderness present a quasi elastic buckling behaviour whereas those of medium slenderness are very sensitive to the effects of imperfections.

4.1 Euler Critical Stress

If lcr is the critical length, the Euler critical load Ncr is equal to:

Ncr = (2)

and it is possible to define the Euler critical stress σcr as:

σcr = (3)

By introducing the radius of gyration, i = , and the slenderness, λ = lcr/i, for the relevant buckling mode, Equation (3) becomes:

σcr = (4)

Plotting the curve σcr as a function of λ on a graph (Figure 9), with the line representing perfect plasticity, σ = fy, shown, it is interesting to note the idealised zones representing failure by buckling, failure by yielding and safety.



The intersection point P, of the two curves represents the maximum theoretical value of slenderness of a column compressed to the yield strength. This maximum slenderness (sometimes called Euler slenderness), called λ1 in Eurocode 3, is equal to:

λ1 = Π [E/fy]1/2 = 93,9ε (5)

where: ε = [235/fy]1/2 (6)

λ1 is equal to 93,9 for steel grade Fe 235 and to 76,4 for steel grade Fe 355.

A non-dimensional representation of this diagram is obtained by plotting σ/fy as a function of λ/λ 1 (Figure 10); this is the form used for the ECCS curves (see Section5.2). The coordinates of the point P are, therefore, (1,1).



4.2 Buckling of Real Columns

The real behaviour of steel columns is rather different from that described in the previous section and columns generally fail by inelastic buckling before reaching the Euler buckling load. The difference in real and theoretical behaviour is due to various imperfections in the "real" element: initial out-of-straightness, residual stresses, eccentricity of axial applied loads and strain-hardening. The imperfections all affect buckling and will, therefore, all influence the ultimate strength of the column. Experimental studies of real columns give results as shown in Figure 11. Compared to the theoretical curves, the real behaviour shows greater differences in the range of medium slenderness than in the range of large slenderness. In the zone of the medium values of λ (representing most practical columns), the effect of structural imperfections is significant and must be carefully considered. The greatest reduction in the theoretical value is in the region of Euler slenderness λ1. The lower bound curve is obtained from a statistical analysis and represents the safe limit for loading.



a. Slender columns

A column can be considered slender if its slenderness is larger than that corresponding to the point of inflexion of the lower bound curve, shown in Figure 11.

The ultimate failure load for slender columns is close to the Euler critical load Ncr. As this is independent of the

yield stress, slender columns are often designed using the ratio: λ2 = (A lcr2)/I, a geometrical characteristic

independent of the mechanical strength.

Individual or star-battened angles or small flat plates have generally a poor second moment of area about the minor axis, relative to their cross-section area. This can result in large slenderness with high sensitivity to buckling and explains why classical cross-bracing systems, using these shapes, are only designed in tension.

b. Columns of medium slenderness

Columns of medium slenderness are those whose behaviour deviates most from Euler's theory. When buckling occurs, some fibres have already reached the yield strength and the ultimate load is not simply a function of slenderness; the more numerous the imperfections, the larger the difference between the actual and theoretical behaviour. Out-of-straightness and residual stresses are the imperfections which have the most significant effect on the behaviour of this kind of column.

Residual stresses can be distributed in various ways across the section. They are produced by welding, hot-rolling, flame-cutting or cold-forming; Figure 12a shows some of the stress patterns that can occur.



Residual stresses combined with axial stresses are shown in Figure 12b. If the maximum stress σn reaches the yield stress fy, yielding begins to occur in the cross-section. The effective area able to resist the axial load is, therefore, reduced.

Alternatively, an initial out-of-straightness eo, produces a bending moment giving a maximum bending stress σB (see Figure 13a), which when added to the residual stress, σR gives the stress distribution shown in Figure 13b. If σmax is greater than the yield stress the final distribution will be part plastic and part of the member will



have yielded in compression, as shown in Figure 13c.

5. THE EUROPEAN BUCKLING CURVES

5.1 Reference Slenderness



The reference slenderness is defined as the following non-dimensional parameter for Class 1, 2 or 3 cross-sections:

= λ /λ1 (7)

where λ and λ1 are defined in Section 4.1.

can also be written in the following form:

2 = λ2fy/π2E = fy/σcr (8)

or = (9)

5.2 Basis of the ECCS Buckling Curves

From 1960 onwards, an international experimental programme was carried out by the ECCS to study the behaviour of standard columns [2]. More than 1000 buckling tests, on various types of members (I, H, T, U, circular and square hollow sections), with different values of slenderness (between 55 and 160) were studied. A probabilistic approach, using the experimental strength, associated with a theoretical analysis, showed that it was possible to draw some curves describing column strength as a function of the reference slenderness. The imperfections which have been taken into account are: a half sine-wave geometric imperfection of magnitude equal to 1/1000 of the length of the column; and the effect of residual stresses relative to each kind of cross-section.

The European buckling curves (a, b, c or d) are shown in Figure 14. These give the value for the reduction factor χ of the resistance of the column as a function of the reference slenderness for different kinds of cross-sections (referred to different values of the imperfection factor α).



The mathematical expression for χ is:

χ = 1/ {φ + [φ2 − 2]1/2} ≤ 1 (10)

where: φ = 0,5 [1 + α ( - 0,2) + 2] (11)

Table 1 gives values of the reduction factor χ as a function of the reference slenderness .

The imperfection factor α depends on the shape of the column cross-section considered, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed); values for α, which increase with the imperfections, are given in Table 2.

Curve a represents quasi perfect shapes: hot-rolled I-sections (h/b > 1,2) with thin flanges (tf ≤ 40mm) if buckling is perpendicular to the major axis; it also represents hot-rolled hollow sections.

Curve b represents shapes with medium imperfections: it defines the behaviour of most welded box-sections; of hot-rolled I-sections buckling about the minor axis; of welded I-sections with thin flanges (tf ≤ 40mm) and of the rolled I-sections with medium flanges (40 < tf ≤ 100mm) if buckling is about the major axis; it also concerns cold-formed hollow sections where the average strength of the member after forming is used.

Curve c represents shapes with a lot of imperfections: U, L, and T shaped sections are in this category as are thick welded box-sections; cold-formed hollow sections designed to the yield strength of the original sheet; hot-rolled H-sections (h/b ≤ 1,2 and tf ≤ 100mm) buckling about the minor axis; and some welded I-sections (tf ≤ 40mm buckling about the minor axis and tf > 40mm buckling about the major axis).

Curve d represents shapes with maximum imperfections: it is to be used for hot-rolled I-sections with very thick flanges (tf > 100mm) and thick welded I-sections (tf > 40mm), if buckling occurs in the minor axis.



Table 4 helps the selection of the appropriate buckling curve as a function of the type of cross-section, of its dimensional limits and of the axis about which buckling can occur. For cold-formed hollow sections, fyb is the tensile yield strength and fya is the average yield strength. If the cross-section in question is not one of those described, it must be classified analogously.

It is important to note that the buckling curves are established for a pin-ended, end loaded member; it is necessary carefully to evaluate the buckling lengths if the boundary conditions are different, see Lecture 7.7.

5.3 Equivalent Initial Bow Imperfection

To study a column using second order theory, it is necessary to choose geometrical imperfections (initial out-of-straightness and eccentricities of loading) and mechanical imperfections (residual stresses and variations of the yield stress). Eurocode 3 proposes values for a bow imperfection, eo, whose effect is equivalent to a combination of the two previous kinds of imperfections [1].

If the column is designed using elastic analysis, eo is as follows:

eo = α ( - 0,2) Wpl/A for plastic design of cross-sections

or,

eo = α ( - 0,2) Wel/A for elastic design of cross-sections (12)

If it is designed with an elastic-plastic analysis (elasto-plastic or elastic-perfectly plastic), the values of eo are functions of the buckling length L, and are given in Table 3.

5.4 Design Steps for Compression Members

To design a simple compression member it is first necessary to evaluate its two effective lengths, in relation to the two principal axes, bearing in mind the expected connections at its ends. Secondly, the required second moment of area to resist the Euler critical loads should be calculated to give an idea of the minimum cross-section necessary. The verification procedure should then proceed as follows:

the geometric characteristics of the shape, and its yield strength give the reference slenderness (Equation (9)). χ is calculated, taking into account the forming process and the shape thickness, using one of the buckling curves and (Equations 10 and 11).

Buckling resistance of a compression member is then taken as:

Nb.Rd = χ A fy/γM1 (13)

the plastic resistance as:

Npl.Rd = A fy/γM0 (14)

and the local buckling resistance as:

No.Rd = Aeff fy/γM1 (15)

If these are higher than the design axial load, the column is acceptable; if not, another larger cross-section must be chosen and checked.



In addition, torsional or flexural-torsional buckling of the column must be prevented.

6. CONCLUDING SUMMARY Many different kinds of cross-sections are used as compression members; these include simple members, built-up, tapered and stepped columns. A stub column (with ≤ 0,2) can achieve the full plastic resistance of the cross-section and buckling does not need to be checked. If > 0,2, reduction of the load resistance must be considered because of buckling. Columns with medium slenderness fail by inelastic buckling and slender columns by elastic buckling. European buckling curves give the reduction factor for the relevant buckling mode depending on the shape of the cross-section, the forming-process, the reference slenderness and the axis about which buckling can occur. They take into account experimental and theoretical approaches and give reliable results.


8. ADDITIONAL READING 1. Dowling P.J., Knowles, P. and Owens G.W., "Structural Steel Design", The Steel Construction Institute,

Butterworths, 1988. 2. European Convention for Constructional Steelwork, "Manual on Stability of Steel Structures", June 1976. 3. Structural Stability Research Council, "Guide to Stability Design Criteria for Metal Structures", Edited by

B.G. Johnson, John Wiley & Sons, 1976. 4. Trahair, N.S. and Bradford, M.A., "The Behaviour and Design of Steel Structures", 2nd Edition,

Chapman & Hall, 1988. 5. MacGinley T.J. and Ang T.C., "Structural Steelwork: Design to Limit State Theory", Butterworths, 1987.

Table 1 Reduction factors

Reduction factor χ

Curve a Curve b Curve c Curve d

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,0000

0,9775

0,9528

0,9243

0,8900

0,8477

0,7957

0,7339

0,6656

1,0000

0,9641

0,9261

0,8842

0,8371

0,7837

0,7245

0,6612

0,5970

1,0000

0,9491

0,8973

0,8430

0,7854

0,7247

0,6622

0,5998

0,5399

1,0000

0,9235

0,8504

0,7793

0,7100

0,6431

0,5797

0,5208

0,4671



Table 2 Imperfection factors

Table 3 Equivalent initial bow imperfections

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2,0

2,1

2,2

2,3

2,4

2,5

2,6

2,7

2,8

2,9

3,0

0,5960

0,5300

0,4703

0,4179

0,3724

0,3332

0,2994

0,2702

0,2449

0,2229

0,2036

0,1867

0,1717

0,1585

0,1467

0,1362

0,1267

0,1182

0,1105

0,1036

0,5352

0,4781

0,4269

0,3817

0,3422

0,3079

0,2781

0,2521

0,2294

0,2095

0,1920

0,1765

0,1628

0,1506

0,1397

0,1299

0,1211

0,1132

0,1060

0,0994

0,4842

0,4338

0,3888

0,3492

0,3145

0,2842

0,2577

0,2345

0,2141

0,1962

0,1803

0,1662

0,1537

0,1425

0,1325

0,1234

0,1153

0,1079

0,1012

0,0951

0,4189

0,3762

0,3385

0,3055

0,2766

0,2512

0,2289

0,2093

0,1920

0,1766

0,1630

0,1508

0,1399

0,1302

0,1214

0,1134

0,1062

0,0997

0,0937

0,0882

Buckling curve a b c d

Imperfection factor α 0,21 0,34 0,49 0,76

Buckling curve Elasto-plastic Elastic-perfectly plastic

a

b

L/600

L/380

L/400

L/250



Table 4: Selection of buckling curve for a cross-section

c

d

L/270

L/180

L/200

L/150



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ESDEP WG 7

ELEMENTS

Lecture 7.5.2: Columns II OBJECTIVE

To explain the use of the European buckling curves and to introduce the concept of torsional and flexural-torsional buckling.

PREREQUISITES


Lecture 6.6.1: Buckling of Real Structural Elements I

Lecture 7.5.1: Columns I

RELATED LECTURES


Lecture 7.10.1: Beam Columns I


Worked Example 7.5: Column Design

SUMMARY

The analysis of imperfections, leading to the derivation of the Ayrton-Perry formula and the European buckling curves, is explained and justified. The concepts of torsional and flexural-torsional buckling are introduced for the case of simple compression members.

1. INTRODUCTION The behaviour of real steel structures is always different from that predicted theoretically; the main reasons for this discrepancy are:

geometrical imperfections, due to defects causing lack of straightness, unparallel flanges, asymmetry of cross-section etc; material imperfections, due to residual stresses (caused by the rolling or fabrication process) or material inelasticity; deviation of applied load from idealised position due to imperfect connections, erection tolerances or lack of verticality of the member.

Of the above, some are important in the buckling of slender columns (geometrical imperfections), others in the compression of stub columns (material inelasticity) and others in the buckling of columns of medium slenderness (geometric imperfections and residual stresses). The behaviour of these three types of columns is described in Lecture 7.5.1.

In reality, all the imperfections act together simultaneously and their effect depends on their individual intensity and on the slenderness of the column. An experimental study of many columns with various characteristics gives the results shown in Figure 1. The results of the tests should be below the Euler buckling curve because



initial out-of-straightness, eccentricity of applied loads and residual stresses all decrease the allowable buckling load; for small slenderness (stub columns), however, it is possible to find some results above the yield stress line because of possible strain-hardening. A safety curve obtained through a statistical analysis is always situated under the minimum experimental values and has the form shown in Figure 1; the plateau is necessary to limit the allowable stress to the yield value. This is the general form of the European buckling curves [1, 2].

2. ANALYTICAL FORMULATION OF THE EUROPEAN BUCKLING CURVES

2.1 Initial Deflection

Assuming that the initial deflection of a pin-ended column of length l, has a half sine-wave form with magnitude eo (Figure 2), the initial deformation along the column can be written as:

(1)



The differential equation for the deformation of such a pin-ended column loaded by an axial force N is:

(2)

Combining this with the expression for yo, and taking into account the boundary conditions, the solution of this equation is:

(3)

The maximum total deflection, e, of the column is then:

(4)



and the ratio 1/(1 - N/Ncr) is generally called the "amplification factor".

Taking into account the maximum bending moment, Ne, due to buckling, the equilibrium of the column requires that:

(5)

where fy is the yield stress.

If N is the maximum axial load, limited by buckling, and σb the maximum normal stress (σb = N/A), this becomes:

(6)

or, introducing σcr, the Euler critical stress (σcr = π2E/λ2) and including the value of e:

(7)

which can be written as:

(σcr - σb) (fy - σb) = σb σcr eo A/W (8)

This equation is the basic form of the Ayrton-Perry formula.

2.2 Eccentricity of the Applied Load

If the axial compression load is applied with an eccentricity ec on an initially straight pin-ended column (Figure 3), a bending moment (N ec) is introduced which increases the buckling effect. This effect obviously increases along with axial load.



It is possible to show that the total maximum deflection e of the column is equal to:

e = ec - ec/{cos[l/2 (N/EI)1/2]} (9)

and the "amplification factor" to: 1/cos [π/2 (N/Ncr)1/2]

Now, if the combined effect of the initial deflection and of the eccentricity of loading is considered, the stress is approximately equal to:

(10)

This relationship is correct within a few percent for all values of σb from 0 to σcr.

2.3 Ayrton-Perry Formula

The classical form of the Ayrton-Perry formula is:



(σcr - σb) (fy - σb) = η σcr σb (11)

This is the form of Equation (8) if η = (eo A) / W

The coefficient η represents the initial out-of-straightness imperfection of the column but it can also include other defects such as residual stresses in which case it is called the "generalized imperfection factor".

It is possible to write the Ayrton-Perry formula under another form:

(σcr / fy - ) ( 1 - ) = η σcr / fy (12)

where: = σb / fy

If 2 = fy / σcr then, dividing by σcr / fy, gives:

(1 - 2) (1 - ) = η (13)

or: 2 2 - ( 2 + η + 1) + 1 = 0 (14)

This form leads to the European formulation [1].

2.4 Generalized Imperfection Factor

The generalized imperfection factor takes into account all the relevant defects in a real column when considering buckling: geometric imperfections, eccentricity of applied loads and residual stresses; inelastic properties are not considered because they only influence stub columns. The generalized imperfection factor can be expressed through the coefficient η representing the effect of deflections:

(15)

where γ = l / eo, represents the equivalent geometrical imperfection (which is the ratio of the length over the equivalent initial curvature of the column).

Then using L = l.i, W = I / v and i2 = I / A, η can be written as:

η = λ / γ (i/v) (16)

where (i/v) is the relative diameter of the inertia ellipse in the axis where buckling occurs.

As λ = η (E/fy)1/2, introducing the plateau = 1 when ≤ o, the previous relationship can be written as:

(17)

because all the European buckling curves were established with fy = 255 MPa (the real value of the yield stress having a very small influence).

2.3 European Formulation



Using η expressed as:

η = α( - o) (18)

the smallest solution of the Equation (14) is:

= {1 + α( - o) + 2 - [1 + α ( - o)+ 2]2 - 4 2}1/2 / 2 2 (19)

Multiplying by the conjugated term and choosing o = 0,2, this relationship gives the European formulation:

χ = 1 / {φ + [φ2 - 2]}1/2 ≤ 1 (20)

where:

φ = 0,5 [1 + α ( - 0,2) + 2] (21)

χ is the reduction factor considered in Eurocode 3 [1].

The different shapes of cross-sections used to design steel columns have the coefficient α varying from 0,21 to 0,76 and it is possible to represent the real behaviour of all classical columns using the four curves (a, b, c and d) shown in Figure 4, α increasing with the imperfections.

α takes into account two kinds of imperfections (geometrical and mechanical). It can be written as α = α1 + α2, where α1 represents the mechanical and α2 the geometrical imperfections. Considering only the geometrical imperfections, the European buckling curves were established with an initial curvature equal to L/1000 (Lecture



7.5.1); this gives

α2 = 90,15/[1000 (i/v)].

Considering now the equivalent initial deflection: eo = L/γ , linked to the generalized imperfection factor η (Equation (15)) and using Equation (18), gives:

eo = α( - 0,2) W / A (22)

which represents the equivalent initial bow imperfection of a pin-ended column including the initial crookedness and the effect of residual stresses; this has to be taken into account in a second order analysis. The design values relative to each European buckling curve are given in Table 1.

3. TORSIONAL AND FLEXURAL-TORSIONAL BUCKLING For hot-rolled steel members, with the type of cross-sections commonly used for compression members, the relevant buckling mode is generally flexural buckling; however, in some cases, torsional or flexural-torsional modes may govern and these must be investigated for all sections with small torsional resistance.

3.1 Cross-section Subjected to Torsional or Flexural-torsional Buckling

Concentrically loaded columns can buckle by flexure about one of the principal axes (classical buckling), twisting about the shear centre (torsional bucking) or a combination of both flexural and twisting (flexural-torsional buckling).

Torsional buckling can only occur if the shear centre and centroid coincide and the cross-section can rotate; this leads to a twisting of the member. Z-sections and I-sections with broad flanges can be subject to torsional buckling; pylons, fabricated from angle sections, must also be checked for this kind of instability.

Symmetrical sections with axial load not in the plane of symmetry, and non-symmetrical sections such as C-sections, hats, equal-leg angles, T-sections and singly symmetrical I-sections, i.e. sections where the shear centre and the centroid do not coincide, must be checked for flexural-torsional buckling.

Figure 5 gives examples of sections which must be checked for torsional or flexural-torsional buckling.



3.2 Torsional Buckling

The analysis of torsional bucking is quite complex and is too long to be included here. The critical stress depends on the boundary conditions and it is very important to evaluate precisely the possibilities of rotation at the ends. The critical stress depends on the torsional stiffness of the member and on the resistance to warping deformations provided by the member itself and by the restraints at its ends.

The differential equation for torsional buckling is:

(23)

and the critical load for pure torsional buckling, Ncrθ , is:



(24)

where ro is the polar radius of gyration, G the shear modulus of elasticity, N the axial load, θ the twist angle, ID the torsion constant, and Iw the warping constant. Lecture 7.9.2 gives more details about the physical meaning and the computation of the warping constant.

To check a compression member with torsional buckling, a new reference slenderness must be evaluated:

= √(fy/σcrθ) (25)

where σcrθ is the elastic critical stress for torsional buckling obtained with the critical load Ncrθ (Equation (24)).

Generally flexural buckling occurs at a lower critical stress than torsional buckling.

Figure 6 illustrates this phenomenon for the case of a cruciform strut.

3.3 Flexural-torsional Buckling



This is the combination of flexural and torsional buckling and its analysis is too complex to be covered in detail here.

The three basic equilibrium equations governing this sort of buckling are:

(26)

(27)

(28)

where, yo and zo are the coordinates of the shear centre and v and w are the deflections, as shown in Figure 7.



The critical load for pure torsional buckling is obtained from the lowest root of the following equation:

ro2 (Ncr - Ncrz) (Ncr - Ncry) (Ncr - Ncrθ ) - ...

... Ncr2 zo

2 (Ncr - Ncry) - Ncr2 yo

2 (Ncr - Ncrz) = 0 (29)

where, Ncry and Ncrz are respectively the critical loads for pure flexural buckling about the axes y and z, and Ncrθ is defined by Equation (24).

Cross-sections with one (or two) axis of symmetry give yo (or zo) = 0 leading to a simplification of the previous equation; for example, a section with two axes of symmetry gives:



(Ncr - Ncrz) (Ncr - Ncry) (Ncr - Ncrθ ) = 0 (30)

and the members buckle at the lowest of the critical loads without interaction of modes.

This lecture only considered the effects of imperfections on the behaviour of compressed steel columns and, therefore, no end moments are considered. The flexural-torsional buckling, in this case, will be due to the effects such as eccentricity of loading or cross-sectional defects.

To check a compression member with flexural-torsional buckling, a new reference slenderness must be evaluated in a similar way as for torsional buckling (Equation (25)).

In this case σcrθ is the elastic stress for flexural buckling obtained with the critical load relative to flexural-torsional buckling.

4. CONCLUDING SUMMARY The effects of imperfections on the phenomenon of buckling are discussed. Initial out-of-straightness, eccentricity of loading and residual stresses have an important influence on buckling of slender columns and columns of medium slenderness. The Ayrton-Perry formula describes the behaviour of real columns. It is the basis of the European buckling curves. European buckling curves are explained; these include a generalized imperfection factor. Torsional buckling and flexural-torsional buckling are introduced.


6. ADDITIONAL READING 1. Dowling, P.J., Knowles, P. and Owens, G.W., "Structural Steel Design", The Steel Construction Institute,

Butterworths, 1988. 2. European Convention for Construction Steelwork, "Manual on Stability of Steel Structures", June 1976. 3. Structural Stability Research Council, "Guide to Stability Design Criteria for Metal Structures", Edited by

B. G. Johnson, John Wiley & Sons, 1976. 4. Maquoi, R. and Rondal, J., "Mise en Equation des Nouvelles Courbes Européennes de Flambement",

Revue Construction Méttalique, no. 1, 1978. 5. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability",

2nd Edition, McGraw-Hill, 1961.

Table 1 Design values of equivalent initial bow imperfection eo,d

(from Figure 5.5.1, Eurocode 3) [1]



Cross-section Method of global analysis

Method used to verify resistance

Section type and axis Elastic, or Rigid - Plastic, or Elastic - Perfectly Plastic

Elasto-Plasti

(plastic zone

Elastic

[5.4.8.2]

Any α( - 0,2)kγ Wel/A -

Linear

[5.4.8.1(12)]

Any α( - 0,2)kγ Wpl/A -

Plastic

[5.4.8.1(1) to (11)]

I-section

yy-axis

1,33α( - 0,2)kγWpl/A

α( -0,2)kγW

I-section

zz-axis

2,0 kγ eeff/ε kγ eeff/ε

Rectangular hollow section 1,33α( - 0,2)kγWpl/A

α( - 0,2)kγ

Circular hollow section

1,5 kγ eeff/ε kγ eeff/ε

kγ = (1 - kδ) + 2 kδ but ≥ 1,0

Buckling curve α eeff kδ

γM1 =1,05 γM1 =1,10 γM1 =1,15




a 0,21 l/600 0,12 0,23 0,33

b 0,34 l/380 0,08 0,15 0,22

c 0,49 l/270 0,06 0,11 0,16

d 0,76 l/180 0,04 0,08 0,11

Non-uniform members:

Use value of Wel/A or Wpl/A at centre of buckling length l




ESDEP WG 7

ELEMENTS

Lecture 7.6: Built-up Columns OBJECTIVE/SCOPE

To derive the equations for the buckling loads of built-up columns and to present the design methods used in Eurocode 3 [1].

PREREQUISITES

Lectures 6: Applied Stability

Lectures 7.5: Columns

RELATE WORKED EXAMPLES

Worked Example 7.6: Built-Up Columns

SUMMARY

This lecture is divided into two main parts; the first part concentrates on the influence of shear deformations on the elastic critical loads and on the slenderness of columns (this effect is crucial for built-up columns and secondary for solid columns - rolled or welded shapes); the second part deals with the design approach adopted in Eurocode 3 [1] which is related to experimental behaviour.

1. INTRODUCTION Built-up columns are widely used in steel construction especially when the effective lengths are great and the compression forces light. They are composed of two or more parallel main components interconnected by lacing or batten plates (Figures 1 and 2). The greater the distance between the chord axes, the greater is the moment of inertia of the built-up cross section; the increase in stiffness, however, is counterbalanced by the increased weight and cost of the connection of members.



of 22ESDEP WG 7


It should be noted that built-up columns (especially battened built-up columns) are more flexible than solid columns with the same moment of inertia; this must be taken into account in the design.

In order to derive the carrying capacity of steel built-up columns, the following must be studied:

the elastic buckling load and the global behaviour;



the local behaviour of the chords; the internal forces in the connecting members.

2. THE EFFECT OF SHEAR DEFORMATIONS ON THE ELASTIC CRITICAL COLUMN LOAD This section discusses the effect of shear deformation on the elastic critical column load.

The simple case of a pin-ended column, shown in Figure 3, is considered; for M, N, V, x and y, as defined in this Figure, the following relationships hold:

M = N y, (2.1)

The total lateral deflection y of the centreline is the result of two components:

y = y1 + y2 (2.2)

the bending moment M gives rise to the deflection y1, and the shearing force V to the additional deflection y2.

According to elastic theory the curvature due to the bending moment M is as follows:

(2.3)



where

E is the modulus of elasticity or Young's modulus.

I is the moment of inertia of the cross-section.

The slope due to the shearing force V is as follows:

(2.4)

where

A is the cross-sectional area.

G is the modulus of rigidity or shear modulus.

β is the shape factor of the column cross-section (β =1,11 for solid circular cross-sections; β = 1,2 for rectangular cross-sections).

The curvature due to the effect of the shearing force V is as follows:

(2.5)

The total curvature of the buckling curve is due both to the bending moment, Equation (2.3), and to the shearing force, Equation (2.5):

(2.6)

It is possible to rearrange Equation (2.6) in the form:

(2.7)

Adopting the same procedure as in the Euler case, the critical load is defined by the equation:

(2.8)

Solving for N, the following expression for the elastic critical load Ncr,id is obtained:

(2.9)

where:

Ncr = is the Euler buckling load obtained disregarding the deformations due to shearing force



Sv = is the shear stiffness of the column.

Obviously Ncr,id < Ncr ; the greater the ratio Ncr / Sv, the smaller the ratio Ncr,id/ Ncr < 1. The ratio Ncr,id / Ncr obtained from Equation (2.9) is plotted, in Figure 4, as a function of the ratio Ncr / Sv.

For solid rolled cross-sections the shear stiffness Sv is much greater than N. The difference between Ncr,id and Ncr is very small therefore, and can be disregarded for design purposes.

However, as will be shown below, the shear stiffness Sv, of built-up columns, is much smaller than it is for solid shapes; in this case, therefore, the influence of the shearing forces on the reduction of the critical load is very significant.

In order to compare the shear stiffness Sv to the Euler buckling loads of solid columns Ncr, consider as an example a HE200A column buckling in the plane of the web.

The shear stiffness Sv is as follows:

Sv =

E = 200 kNmm-2



ν = 0,3

Aw = the area of the web = 6,5 x 170 = 1105 mm2

The Euler buckling load Ncr is:

Ncr =

where:

A is the cross-sectional area = 5380 mm2.

λ is the slenderness of the column.

In Table 1 the critical buckling loads Ncr,id, the Euler buckling loads Ncr, and the ratios Ncr / Sv are given as functions of the slenderness; it clearly shows that in the case of solid cross-sections, Ncr is always far smaller than Sv; therefore, for technical purposes, it is possible to disregard the influence of the shear deformations on the elastic buckling loads Ncr,id.

3. EVALUATION OF THE SHEAR STIFFNESS OF LACED AND BATTENED COLUMNS

3.1 Laced Columns

In laced columns the elastic extension of the diagonals and the horizontals must be considered in order to derive the shear stiffness Sv; the elongation of the chords (the main components) should not be taken into account, because they are already considered in the global flexural stiffness E I of the built-up column.

With an N-shape arrangement of lacing, as shown in Figure 5, the elongations of one diagonal and of one horizontal are taken into account in order to derive Sv:



1/Sv = δ/a = γ (3.1)

where δ is the lateral displacement due to the unit shearing force.

The total displacement δ is the result of two components: δ1 is the contribution from the elongation of the diagonal; δ2 is the contribution from the shortening of the horizontal. From virtual work theory:

Thus, for one plane of lacing:

(3.2)

With an A shape arrangement of lacing, as shown in Figure 6:



Therefore:

(3.3)

The cross-bracings, shown in Figure 7, have the same shear stiffness, because the horizontals do not take part in the transmission of the shearing force:



and the shear stiffness is:



(3.4)

3.2 Battened Built-up Columns

For battened built-up columns, as shown in Figure 8, the flexural deformations both of the chords and of the battens must be considered in order to derive the shear stiffness Sv; as is the case for laced built-up columns, the extensions of the main chords are not considered because their contributions appear in the global flexural stiffness E I.

Adopting the virtual work method, the displacement δ due to unit shearing force is obtained:

and the shear stiffness is:

(3.5)

where Ic is the in-plane second moment of area of one chord.

Ib is the in-plane second moment of area of one batten.

The above formula may be refined by taking into account the deformations due to the shearing force in the



battens.

3.3 Quantitative Comparison

Table 2 gives a comparison of the shear stiffness Sv for a solid column and three different built-up columns; the global dimensions of the cross-sections and the weights of steel per unit length are kept constant.

The solid column (a) is made up of two HE400A shapes each welded to a common plate 8mm thick x 1000mm wide.

Built-up columns (b) and (c) are laced: the chords are HE400A shapes and there are two planes of lacing with 100 × 10 equal angles.

Built-up column (d) is with batten plates: the same shape HE400A as above is adopted for the chords; the batten plates, which have rectangular cross-section 400 × 20mm, are on two planes.

Whereas the steel employed in the web and in the connecting members has almost the same weight in the four columns, Table 2 shows that the shear stiffness Sv has a wide range of variation.

4. THE ELASTIC CRITICAL LOADS OF BUILT-UP COLUMNS For design purposes the elastic critical loads of built-up columns can be obtained from Equation (2.9): that is, it is possible to assume a continuous distribution of the shear stiffness Sv if the number of panels is greater than or equal to six; if not, more complex analysis, with methods suitable for frames, has to be performed (for further reading on this topic see (1, 2, and 3).

By introducing formulae for the shear stiffness Sv (3.2, 3.3, 3.4 and 3.5) in Equation (2.9), it is easy to derive, for the various types of lacings and battens, specific formulae for the elastic critical loads Ncr,id and, as a consequence, for the effective length of built-up columns; formulae of this type are widely adopted in the European codes and standards for steel construction.

The details of the analytical procedure are now outlined.

When the lacing is N shaped (as shown in Figure 5), the expression for the shear stiffness Sv (3.2) is substituted in Equation (2.9), where:

Ncr = π2 E I / l2 is the Euler critical load, as for a solid column.

I = 2 Ic + Ac h2, is the moment of inertia of the built-up cross-section.

Ac is the cross-sectional area of the chords.

Ic is the relevant moment of inertia of the chords.

Substitution gives the following:

(4.1)

By introducing the slenderness λ for the column without shear deformations, such that:



λ2 = 1/ ρ2 = 2 Ac l2 /I

the elastic critical stress σcr,id, for the built-up column becomes:

(4.2)

where:

(4.3)

is the equivalent slenderness of the built-up column.

Following the same procedure, when simple A bracing (Figure 6) is adopted, the equivalent slenderness of the column is:

(4.4)

For cross-bracing (Figure 7):

(4.5)

Finally, for battened built-up columns (Figure 8), the equivalent slenderness is:

(4.6)

If the batten plates are very stiff, their flexural deformations may be disregarded and it is possible to put:

, in Equation (4.6).

In this case the equivalent slenderness of battened built-up columns becomes:

λeq = √{λ2 + π2Aca2 /(12Ic)} = √{λ2 + π2λ12 /12} (4.7)

where λ1 is the local slenderness of the chords between the centrelines of the batten plates.

5. THE BEARING CAPACITY OF STEEL BUILT-UP COLUMNS AND THE DESIGN PHILOSOPHY OF EUROCODE 3 The experimental behaviour of a steel built-up column (with batten plates) [2] in a compression test up to collapse, is summarised in Figure 9.



in Figure 9(a) the lateral displacement v at midspan is plotted as a function of the applied external compression load Ni. in Figure 9(b) the axial forces N1 and N2 in the chords at midspan are plotted as a function of the compression load Ni. finally in Figure 9(c) the shearing forces in the batten plates are plotted, and in Figure 9(d) the distribution of the shearing forces in the battens is shown.



Due to the initial geometrical imperfections and residual stresses, the lateral displacements increase with the applied load more and more rapidly up to the bearing capacity Ncr of the column; the presence of lateral displacements explains why one chord is more compressed than the other, which tends to unload after it has reached maximum compression.

In the batten plate at the end, the framing effect is smaller than in the next internal one, because of the presence of only half a field.

The maximum carrying capacity of the built-up column is reached when one of the following possibilities occurs:

at mid-span the more compressed chord buckles. at the end a chord fails by compression and flexure. a batten-plate and/or its connection to the chords fails by shear and flexure.

The main features of the experimental behaviour of a built-up compression member summarised above may be



represented by a simple elastic column with equivalent initial geometrical imperfections and shear flexibility (Figure 10).

The design philosophy of Eurocode 3 [1] is based on this simple model: it is assumed that in the initial unloaded state the column is not perfectly straight and that the initial deflection y (x) is given by a sine curve for a pin-ended member:

yo (x) = wo sin (π x / l) (5.1)

where

wo is the equivalent geometrical out of straightness at mid-span (wo = 0,002l = l / 500);

l is the length of the pin-ended column.

When the design axial load N is applied to the built-up column the initial geometrical imperfection is then amplified elastically; the lateral displacements in the equivalent state are:

sin (π x / l) (5.2)

where, in Equation (5.2), Ncr,id is the elastic critical load of the built-up column as given in Section 4.

At mid-span of the built-up column, the axial force is N and the bending moment M equals:



(5.3)

The axial force Nf in the most loaded chord is:

(5.4)

The buckling resistance of the chords NRd must be greater than Nf:

NRd ≥ Nf (5.5)

In laced built-up columns the effective length of the chords is taken equal to the system length between lacing connections; in battened built-up columns (for the sake of simplicity and disregarding any possible end restraint) the effective length of the chords is taken equal to the distance between the centre lines of the battens.

The shearing force V at the ends of a built-up column is given by:

(5.6)

The forces in the lacing members and in the chords adjacent to the ends are derived from the shearing force V and from the axial force N.

The battens, their connections to the chords, and the chords themselves are checked for the moments and forces due to the shearing force V and the axial force N as shown in Figure 11.



The design philosophy of Eurocode 3 [1] may be summarized in the following six steps:

1. derive the design axial load. 2. derive the flexural stiffness and the shear stiffness of the built-up columns. 3. derive the elastic critical load of the built-up column. 4. compute the design bending moment at midspan and the design shear force at the ends of the built-up-

column. 5. check the buckling strength of the chords at midspan. 6. check the strength of the web members and of their connections to the chords for the most loaded panel at

the ends.

6. CONCLUDING SUMMARY The bearing capacity of built-up columns is largely affected by the shear deformations. It is possible to study the behaviour of built-up columns using a simple elastic model. Because of shear deformations the initial lack of straightness of the column is strongly amplified. The design method proposed in Eurocode 3 is based on the above approach.


[2] Ballio, G. & Mazzolani, F. M., "Theory and Design of Steel Structures", Chapmann and Hall, New York, 1983.



8. ADDITIONAL READING 1. Bleich, F., "Buckling Strength of Metal Structures", McGraw Hill, New York, 1952. 2. Timoshenko, S., "Theory of Elastic Stability", McGraw Hill, New York, 1936. 3. Galambos, Th. V., "Guide to Stability Design Criteria for Metal Structures - Fourth Edition", J. Wiley &

Sons, New York, 1988.

Table 1 Example of the influence of shear flexibility on stability

Table 2 Examples of shear stiffnesses for solid, laced and battened columns

λ 80 90 100 110 120 130 140

Ncr,id (MN) 1,63 1,29 1,05 0,869 0,731 0,624 0,538

Ncr (MN) 1,66 1,31 1,06 0,878 0,738 0,628 0,542

Ncr / Sv 0,020 0,015 0,12 0,010 0,009 0,007 0,006

a)



Table 2 - Continued

Shear stiffness

Sv =

E = 20 MN cm-2 υ = 0,3

Aw = 100 x 0,8 = 80 cm2

Sv =

Volume and mass of steel

V = 0,8 x 100 x 100 = 8000 cm3 m-1

W = Ys V = 0,00785 x 8000 = 63 kg m-1

b)

Shear Stiffness

Sv =

L 100 × 10 Ad = 19,2 cm2

h = 100 cm a = 115,5 / 2 = 57,75 cm

d = 2 a = 115,5 cm

Sv =


V = 19,2 x 115,5 x 4 / 1,155 = 7680 cm3 m-1

W = 0,00785 x 7680 = 60 kg m-1

c)

Shear Stiffness



Table 2 - Continued

Sv =

Ad = Ao = 19,2 cm2

h = 100 cm a = 115,5 cm

d = (a2 + h2)0,5 = 152,8 cm

Sv =


V = 19,2 x (100 + 152,8) x 2 / 1,155 = 8405 cm3 m-1

W = 0,00785 x 8405 = 66 kg m-1

d)

Shear Stiffness

Sv =

Ic = 8564 cm4 H E 400 A

Ib = 403 x 2 x 2 / 12 = 21333 cm4

a = 200 cm h = 100 cm

Sv =


V = 2 x 40 x 100 x 2/2 = 8000 cm3

W = 0,00785 * 8000 = 63 kg m-1

a) Web 8 × 1000

Shear Stiffness

Sv

Mass of Steel

W




b) Lacing L 100 × 10

c) Lacing L 100 × 10

d) Batten plates 400 × 20

(MN)

615

288

194

73

(kg m-1)

63

60

66

63




ESDEP WG 7

ELEMENTS

Lecture 7.7: Buckling Lengths OBJECTIVE/SCOPE

To introduce the concept of effective length and to describe its application in the design of practical columns.

PREREQUISITES

Lecture 6.3: Elastic Instability Modes


Lecture 7.6: Built-up Columns

RELATED LECTURES


Lectures 7.12: Trusses and Lattice Girders


Worked Example 7.7: Effective Lengths

SUMMARY

For pin-ended columns the buckling length equals the actual length; such columns are, however, relatively rare in practice. Predicting strength under other than pin-ended conditions can be achieved by using the notion of effective length (LE).

LE is the length of a similar pin-ended column (of the same section) which has the same buckling load as the column being considered. Approximate values for effective length, which can be used in design, are given for a wide range of end-restraint conditions.

1. INTRODUCTION For the determination of the elastic Euler critical buckling load

(1)

it is assumed (Lectures 6.1. and 7.5.1) that both ends of the column are pinned (Figure 1); practical end connections on real columns, however, will often not behave in this manner and this will, therefore, significantly affect the buckling load. Two aspects of the end condition must be considered:



the rotation restraints which can vary between 0 and ∞ (i.e. a hinge without friction or fully restrained); the translational restraints (sway or no sway).

The usual design approach consists of reducing the practical case under consideration to an equivalent pin-ended case by means of an effective length factor K.

2. EFFECTIVE LENGTH OF COLUMNS The effective length, LE, of a member hinged at its ends is the distance between the axes of the hinges. For general end restraints, the effective length LE, is the length of an end-hinged member which has the same load bearing resistance as the member under consideration.

The application of the above definition is not easy in practice. Numerical studies have shown that the notion of effective column length can be derived from elastic stability theory. In this case the effective length factor, K, is the ratio of the length (LE) of the equivalent column to the actual length (L); and the length of the equivalent column is the distance between two consecutive points of contraflexure (points of zero moment) in the actual column (Figure 2).



For the pin-ended column (fundamental case of buckling of a prismatic bar, see Figure 1) the effective length factor is equal to 1, and the distance between the points of zero moment is equal to the actual column length.

More generally, let us consider, for example, the columns of the frame illustrated in Figure 3a; if it is assumed that the flexural rigidity of the beam is much higher than that of the columns, no rotation of the upper ends of the columns occurs when the frame moves laterally. This situation is shown in Figure 3b.

The bending moment at a point along the column is given by



M = Nv + Hz (Figure 3c).

The differential equation becomes:

(2)

Using the notation K2 = N/EI:

(2b)

The solution of Equation (2b) is given by:

(3)

To find the constants A and B, the boundary conditions are used:

for z = 0, v = 0 and for z = L, = 0, therefore A = 0 and

B K cos K L = 0 (4)

From (4), it follows that either B or cos K L = 0.

If B = 0, v = -Hz/N and d2v/dz2 = 0; in this case, the bending moment M should be zero at any point along the column.

The other possibility is that cos KL = 0 and this condition requires that K = nπ/2L where n = 1,3,5, .... (5)

To obtain the smallest value of N for which Equation (5) is satisfied, using n=1 gives KL = π/2 from which K = π/2L and K2 = N/EI,

Ncr = K2 EI = π2 EI/4L2 = π2 EI/(2L)2 (6)

The comparison of Equations (6) and (1) shows that the effective length factor K is equal to 2 and therefore, that the effective length of the column is twice the actual length. In other words, the critical load for the column of length L, shown in Figure 3, is the same as the critical load of a pin-ended column of length 2L. The situation is shown geometrically in Figure 3a.

The use of an effective column length is basically a device to relate the behaviour of columns with any form of support to the behaviour of the basic pin-ended case. The design procedure for columns with particular end conditions is the same as for pin-ended columns (see Lecture 7.5.1) but in establishing the design strength from the column design curve, the slenderness (LE/ry) would be used instead of L/ry.

Table 1 gives theoretical K-values for idealized conditions in which the rotational and/or translational restraints at the ends of the column are either fully realized or non-existent. Each of these values is obtained as for the previous example.

Table 1 also recommends K-values which are equal or slightly higher than the equivalent theoretical values derived from elastic stability theory. When higher values are specified it is usually in recognition of the practical difficulties of providing complete restraint against rotation or translation.



The comparison of cases (b) and (e) in Table 1, shows the influence of the translational restraints on the buckling load. Case (e) represents the situation of the column of Figure 3a, with lateral displacement, while in case (b) no translation is permitted; the buckling load is multiplied by a factor 8 ((2,0/0,7)2) when translation is prevented. For this reason, it is absolutely necessary that the designer knows the difference between sway and non-sway frames.

According to Eurocode 3 [1], a frame may be classified as non-sway if its response to in-plane horizontal loads is sufficiently stiff for it to be acceptably accurate to neglect any additional internal forces or moments arising from horizontal displacement of its nodes. Any other frame shall be treated as a sway frame and the effects of the horizontal displacements of its nodes taken into account in its design.

More details concerning the distinction between sway and non-sway frames are given in the Lectures 14.

A column in a non-sway frame would have no sideways movement at the top relative to the bottom. The buckling of a non-sway frame would result in a buckled column shape having at least one point of contraflexure between the ends of the member, such as cases (a), (b), and (c) of Table 1 (see Figure 4). The effective length factor K is always less than or equal to 1 (0,5 ≤ K ≤ 1).

In a sway frame, the top of the column moves relative to the bottom. Cases (d), (e) and (f) of Table 1 are sidesways buckling cases which are illustrated in Figure 5. The effective length factor K is always greater than or equal to 1 and is unlimited (1 ≤ K ≤ ∞).



The above considerations concerning single storey frames can be generalized, so as to extend to frames of more than one storey.

The fully rigid end-restraints (shown in Figures 4b, 4d, 5b and 5d) can rarely be achieved in practice and partial end-restraints are much more common.

In the case of partial end-restraint the effective length factor K can be determined either by a generalized second order rotation method or by using stability functions [2].

The solution to the problem is expressed in the form:

K = f(ηt,ηb) (7)

where ηt and ηb, are elastic restraint coefficients at the top and bottom of the column considered.

Simplified approaches are available for evaluating the effective length factor K, [3-7].

Using Donnell's approximate formula [3] (see Figure 6)



K = 1/√n (8)

where n = (9)

and (10)

In the case of a compressed bar of a truss

(11)

where Ri = Σj 3EIj/lj (12)

characterizes the restraint of the adjacent bars, "j".

Wood [4] and Johnston [5] have given other simplified approaches differing only in their presentation.

In Eurocode 3 [1], the approach suggested by Wood has been adopted and two cases are considered: non-sway and sway frames.

In some cases, a compressed bar can be supported elastically at several intermediate points along the length. Figure 7 shows, for example, the compressed chord of a truss girder; the intermediate supports of the chord are represented by the framed cross girder.



In such a case the effective length is greater than the distance "a" between the cross girders [8] and is given by:

(13)

where f = 1/Kr, the displacement of a spring (intermediate support) due to a unit force.

3. COLUMNS OF NON-SWAY FRAMES Wood [4], considers a sub-element of a non-sway frame as illustrated in Figure 8b (part AB of the frame represented in Figure 8a.).



The two elastic restraint coefficients ηt and ηb (which are closely analogous to the cross-distribution coefficients at the upper and lower ends of the column) are calculated using the following formulae:

ηt = KC/(KC + ΣKb,t) (14)

ηb = KC/(KC + ΣKb,b) (15)

where KC is the column stiffness I/L

ΣKb is the sum of effective beam stiffness at a joint and the suffices b and t indicate the bottom or top end of the column.

Where the beams are not subject to axial forces, their effective stiffnesses can be determined by reference to Table 2, provided that they remain elastic under the design moments.

Where, for the same load case, the design moment in any of the beams exceeds the elastic moment, the beam should be assumed to be pinned at the point or points concerned.

Where a beam has semi-rigid connections its effective stiffness should be reduced accordingly.

Where the beams are subject to axial forces, their effective stiffnesses should be adjusted accordingly; stability functions can be used in this case. As a simple alternative, the increased stiffness due to axial tension can be neglected and the effects of axial compression can be allowed for by using the conservative approximations given in Table 3.

Considering the sub-element illustrated in Figure 8b, and the distribution coefficients given above yields results that can be graphically illustrated [4] by the curves of Figure 9. These can also be represented by the following expression:

(16)



The model can be adapted for the design of continuous columns, by assuming that each length of column is loaded to the same value of the ratio (N/Ncr). In the general case, where (N/Ncr) varies, this leads to a conservative value of K for the most critical length of column.

For each length of a continuous column this assumption can be introduced using the model shown in Figure 10 and obtaining the distribution coefficients ηt and ηb as follows:



(17)

(18)

4. COLUMNS OF SWAY FRAMES In unbraced frames (and some braced frames) sway is permitted; the effective length factor, K, is therefore greater than unity and can tend towards infinity if the horizontal beams are very flexible.

K can be calculated using the same approach as that adopted for frames in which sway is prevented; it should, however, be pointed out that the results for sway frames must be considered as even more approximate than those given for non-sway frames.



Wood's method can be considered as acceptable when sway is permitted only if the frames are regular, i.e. heights, moments of inertia and axial forces in the columns do not differ considerably.

The effective length factor of a column in a sway frame may be obtained from Figure 11 or Equation (19):

(19)



The elastic restraint coefficients ηt and ηb are calculated as for the case of non-sway frames.

Introducing the effective length concept in the elastic design of sway columns requires that second order or load destabilizing effects (due to the displacement of the top of the columns) are approximately taken into account by an effective length factor K, greater than 1. The advantage of this approach is its simplicity but it should be recognised that it is limited and may, in some cases, be inaccurate.

A design considering the whole structure, based on approximate methods for elastic critical load analysis, is recognized as more reliable. These methods consider the effects of horizontal sway forces on the structure which subject the column to bending moments as well as axial loads. More information on this topic is given in Lecture 7.11.

5. CONCLUDING SUMMARY Effective lengths enable column design curves, for pin-ended columns, to be used for the design of practical columns which have a wide range of end-restraint conditions. Simplified methods are available for evaluating the effective length of a compressed bar. For sway columns, effective lengths are greater than the actual lengths. For non-sway columns, effective lengths are less than the actual lengths.

6. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules ad rules for buildings, CEN, 1992.

[2] Livelsey, R.K. and Chandler, P.B., "Stability Functions for Structural Frameworks", Manchester University Press, 1956.

[3] Massonnet, Ch., "Flambement des Constructions Formées de Barres Droites". Note technique B10.52, CRIF, Bruxelles, 1955.

[4] Wood, R.H., "Effective Lengths of Columns in Multistorey Buildings". The Structural Engineer, vol. 52, 1974 (pp. 235-244; 295-302; 341-346).

[5] Johnston, G., "Design Criteria for Metal Compression Members". John Wiley and Sons, Inc., New York, 1960.

[6] Djalaly, H., "Longueur de Flambement des Eléments de Structures". Construction métallique, no. 4, 1975.

[7] Kamal Hassan., "Zur Bestimmung der Knicklänge of Rahmenstreben", IVBH Abhandlungen 28-I-1968.

[8] SIA 161, Constructions Metalliques 1979.

Table 1 Effective length factor for centrally loaded columns with various end conditions

With lateral restraint

(a) (b) (c)

Without lateral restraint

(a) (b) (c)

Ideal buckling conditions

See Fig T1-1

See Fig T1-2



Fig T1-1 (a) (b) (c)

Fig T1-2 (a) (b) (c)

Table 2 Effective stiffness of a beam

Theoretical K-values 1,0 0,7 0,5 2,0 2,0 1,0

Recommended K-values when ideal conditions are approximated

1,0 0,8 0,65 2,0 2,0 1,2

Conditions of rotational restraint at far end of beam

Effective beam stiffness (provided beam remains elastic)



Table 3 Approximate formulae for reduced stiffness due to axial compression


Fixed at far end 1,0 I/L

Pinned at far end 0,75 I/L

Rotation as at near end (double curvature) 1,5 I/L

Rotation equal and opposite to that at near end (single curvature)

0,5 I/L

General case. Rotation ΘA at near end and ΘB at far end

(1 + 0,5 ΘB/ΘA) I/L

Far end condition Effective beam stiffness

Fixed 1,0 I/L (1 - 0,4 N/Ncr)

Pinned 0,75 I/L (1 - 1,0 N/Ncr)

Double curvature 1,5 I/L (1 - 0,2 N/Ncr)

Single curvature 0,5 I/L (1 - 1,0 N/Ncr)

Where Ncr = π 2 EI/L2




ESDEP WG 7

ELEMENTS

Lecture 7.8.1: Restrained Beams I OBJECTIVE/SCOPE

To derive and discuss the procedures used to design restrained compact beams for bending, shear and deflection, according to the principles of Eurocode 3 [1].

PREREQUISITES

Elastic theory for uniaxial bending

Simple torsion theory


RELATED LECTURES

Lecture 7.8.2: Restrained Beams II



Worked Example 7.8: Laterally Restrained Beams

SUMMARY

This lecture is restricted to beams whose design may be based on simple strength of materials considerations. Behaviour in simple bending is discussed, leading to the concept of section modulus as the basis for strength design. Subsidiary considerations of shear strength, resistance to local loads and adequate stiffness against deflection are also mentioned. Behaviour under complex loading, producing bending about both principal axes, or combined bending and torsion is introduced.

ADDITIONAL NOTATION

A area

Av shear area

d depth of section

F applied load

fd limiting stress in material

fy material yield strength

fyd material design strength

I second moment of area



M moment

Mpl plastic moment

t thickness

tf thickness of flange

tw thickness of web

Vs shear due to applied load

W elastic section modulus

Wmax maximum value of W

Wmin minimum value of W

Wpl plastic section modulus

α,β coefficients, see Equation (7)

σ normal stress

1. INTRODUCTION Probably the most basic structural component is the beam, spanning between two supports, and transmitting loads principally by bending action. Steel beams, which may be drawn from a wide variety of structural types and shapes, can often be designed using little more than the simple theory of bending. However, situations will arise in which the beam's response to its loading will be more complex, with the result that other forms of behaviour must also be considered. The main purpose of this lecture is to concentrate on the design of that class of steel beam for which strength of materials forms the basis of the design approach. These are termed "restrained compact beams"; in order to come within this category the beam must not be susceptible to either local instability (see Lecture 7.2 for Class 4 beams) or lateral-torsional instability (see Lecture 7.9.1 and 7.9.2 for unrestrained beams). A further limitation is that the beams are assumed to be statically determinate or, if statically indeterminate, that the distribution of internal bending moments has been obtained on a simple linear elastic basis.

The first requirement will be met if the width of the individual plate elements of the cross-section are limited, relative to their thickness. Sections for which the ratios of flange width/flange thickness, web depth/web thickness etc, have been limited, so that their full plastic moment capacity may be achieved, correspond to either Class 1 or Class 2 according to Eurocode 3 [1]. The majority of hot rolled sections, e.g. UB's and IPE's, meet these requirements. However, care is necessary when using fabricated sections and suitable rules are provided in codes of practice. This lecture assumes that the beam cross-section will at worst correspond to Class 2 or Class 3 and that, if it meets the Class 1 limits, then the global analysis of the structure will be conducted using plastic methods.

Lateral-torsional instability will not occur if any of the following conditions apply:

the section is bent about its minor axis. full lateral restraint is provided, e.g. by positive attachment of the top flange of a simply supported beam to a concrete slab. closely spaced, discrete bracing is provided so that the weak axis slenderness (L/iz) of the beams is low. adequate torsional restraint of the compression flange is provided, e.g. by profiled sheeting. the section has high torsional and lateral bending stiffness; for example, rectangular box sections bent



about their major axis are unlikely to fail in this way.

For the special case of continuous beams supporting a roof or floor, care must be taken to ensure adequate stability of those regions in which the bottom flange is in compression, e.g.the support region under gravity loading, the mid-span region under wind uplift, etc... For the purpose of this lecture, beams within any of these categories will be classed as "restrained".

2. BEAM TYPES Several factors, some of them tending to conflict with one another, influence the designer's choice of beam type for a given application. Clearly the beam must possess adequate strength, but it should also not deflect too much. It must be capable of being connected to adjacent parts of the structure; this will often involve the use of site connections which should be simple and quick to erect, with minimum requirements for skilled labour or special equipment. Particular features, such as the passage of services beneath the floor of a building, may dictate the use of a section with openings in the web, whilst architectural requirements may require the profile to be varied, to improve the line. Table 1, summarising the main types of steel beam, indicates the range of spans for which each is most appropriate and gives some idea of any special features.

3. DESIGN OF BEAMS FOR SIMPLE BENDING For a doubly-symmetrical section or a singly-symmetrical section bent about the axis of symmetry, the basic theory of bending, assuming elastic behaviour, gives the distribution of bending stress shown in Figure 1.



Since the maximum stress σmax is given by:

σmax = (1)



where M is the moment at cross-section under consideration.

d is the overall depth of section.

I is the second moment of area about the neutral axis (line of zero strain).

it follows that limiting this to a fraction of the material yield stress gives a design condition of the form:

W ≥ M/fd (2)

where W =

fd is the limiting normal bending stress.

When M is taken as the moment produced by the working loads, this approach to design is termed 'elastic' or 'permissible stress' design and is the method traditionally used in many existing codes of practice. In more modern Limit States codes fd is taken as the material strength fy possibly divided by a suitable material factor γM and M is taken as the moment due to the factored loads i.e. the working loads suitably increased so as to provide a margin of safety in the design. Equation (2), in this case, represents the condition of first yield. Values of W for the standard range of sections are available in tables of section properties.

Selection of a suitable beam therefore comes down to:

1. determination of the maximum moment in the beam, 2. extraction of the appropriate value of fd from a suitable code, 3. selection of a section with an adequate value of W subject to considerations of minimum weight, depth of

section, rationalisation of sizes throughout the structure, etc...

Clearly, sections for which the majority of the material is located as far away as possible from the neutral axis will tend to be the most efficient in elastic bending. Figure 2 gives some quantitative idea of this for some of the more common structural shapes. I-sections are most often chosen for beams because of their structural efficiency; being open sections they can also be connected to adjacent parts of the structure without undue difficulty. Figure 3 gives some typical examples of beam-to-column connections.



Utilisation of the plastic part of the stress-strain curve for steel enables moments in excess of those which just cause yield to be carried. At full plasticity the distribution of bending stress in a doubly-symmetrical section will be as illustrated in Figure 4, with half the section yielding in compression and half in tension.



The corresponding moment is termed the fully plastic moment Mpl. It may be calculated by taking moments of the stress diagram about the neutral axis to give:

Mpl = fy Wpl (3)

where fy is the material yield stress (assumed identical in tension and compression).

Wpl is the plastic section modulus.

Basing design on Equation (3) means that the full strength of the cross-section in bending is now being used, with the design condition being given by:

Wpl ≥ M/fyd (4)

where M is the moment at cross-section under consideration.

fyd is the design strength (material yield strength divided by a suitable material factor).

When M is due to the factored loads Equation (4) represents the design condition of ultimate bending strength used in Eurocode 3 for beams whose cross-sections meet at least the Class 2 limits [1]. It is usual in codes such as Eurocode 3 for the value of fyd to be taken as the material yield strength, reduced slightly so as to cover possible variations from the expected value.

For continuous (statically indeterminate) structures, attainment of Mpl at the point of maximum moment will not normally imply collapse. Providing nothing triggers unloading at this point, e.g. local buckling does not occur, then the local rotational stiffness will virtually disappear, i.e. the cross-section will behave as if it were a hinge, and the pattern of moments within the structure will alter from the original elastic distribution as successive plastic hinges form. This redistribution of moments will enable the structure to withstand loads beyond that load



which produces the first plastic hinge, until eventually collapse will occur when sufficient hinges have formed to convert the structure into a mechanism, as shown in Figure 5. Utilisation of this property of redistribution is termed "plastic design". It can only be used for continuous structures and then only when certain restrictions on cross-sectional geometry, member slenderness, etc.., are observed. The topic is covered in detail in Lecture 7.8.2 in the context of beams and in Lecture 7.11 in the context of frames.

4. DESIGN OF BEAMS FOR SHEAR Although bending will govern the design of most steel beams, situations will arise, e.g. short beams carrying heavy concentrated loads, in which shear forces are sufficiently high for them to be the controlling factor.

Figure 6 illustrates the pattern of shear stress found in a rectangular section and in an I-section assuming elastic behaviour. In both cases shear stress varies parabolically with depth, with the maximum value occurring at the neutral axis. However, for the I-section, the difference between maximum and minimum values for the web, which carries virtually the whole of the vertical shear force, is sufficiently small for design to be simplified by working with the average shear stress, i.e. total shear force/web area. Since the shear yield stress of steel is approximately 1/√3 of its tensile yield stress, a suitable value for "permissible" shear stress when using elastic design is 1/√3 of the permissible tensile stress.



The ultimate shear resistance (based on plastic principles) used in Eurocode 3 [1] is fyd/√3; this is used in conjunction with a shear area Av, examples of which are:

rolled I-section, load parallel to web Av = A-2btf + (tw+2r)tf

plates and solid bars Av = A

circular hollow sections Av = 2A/π

In cases where high shear and high moment coexist, e.g. the internal support of a continuous beam, it may



sometimes be necessary to allow for interaction effects. However, since the full shear capacity may be developed in the presence of quite large moments and vice versa (Eurocode 3 only requires reductions in moment resistance when the applied shear exceeds 50% of the ultimate shear resistance [1]) this will not often be required.

5. DEFLECTIONS Excessive deflections, whilst not normally leading to a structural failure, may, nonetheless, impair the serviceability of the structure.

Deflection might, for example, cause:

cracking of plaster ceilings. misalignment of crane rails. difficulty in opening large doors.

Since these affect the performance of the structure in its working conditions, it is usual to conduct this type of check at service load levels. Eurocode 3 suggests a limit of span/300 for the maximum deflection of beams supporting floors under service live load [1]. Such limits should, however, be regarded as advisory only, and smaller or larger values can be more appropriate in any given situation. Wherever possible, specialist advice should be sought; crane manufacturers, for example, should be able to give detailed guidelines regarding permissable deflection for gantry beams.

6. BENDING OF UNSYMMETRICAL SECTIONS For sections with a single axis of symmetry bent about the perpendicular axis, the elastic distribution of bending stresses will be as shown in Figure 7. Due to its lack of symmetry about the neutral axis, the stress on the top and bottom flanges will not be equal; for example, the stress on the smaller flange, in the case of an unequal flanged I-beam, will exceed that on the bigger. Therefore, in elastic design, it is necessary to use the smaller of the two elastic section moduli Wmin.



At full plasticity, the line of zero stress must be positioned so that it divides the area of the section into two equal parts, as shown in Figure 7c. Since the plastic section modulus Wpl is defined from Equation (3) as the ratio of Mpl/fy, there will be a single value for both faces of the section. Thus, when using ultimate moment resistance, the design condition will remain as Equation (4).

7. BIAXIAL BENDING Doubly or singly-symmetrical sections subject to bending moments My, Mz about both principal axes may be treated as the sum of two uniaxial problems. Thus, in elastic design, it is necessary to satisfy the linear interaction equation:



(6)

in which Wy and Wz are the elastic section moduli of the cross-section with respect to the axes y and z and

fd is the limiting normal bending stress.

This is based on the idea of limiting the maximum combined stress in the section to the design value. Care is necessary when dealing with sections such as angles for which the principal axes are not rectangular.

When using the plastic strength of the cross-section, analysis (which entails the location of the plastic neutral axis) shows that the shape of the interaction between moments will depend upon the geometry of the cross-section. Safe approximations to these interaction diagrams have therefore been provided in Eurocode 3 [1]. These adopt the form:

(7)

in which Mply and Mplz are moment capacities about the y and z axis respectively and the values of α and β depend on the particular section under consideration; safe values are α = β = 1,0.

8. BENDING AND TORSION Loads which do not act through the shear centre of the section (see Figure 8) will also cause twisting (the shear centre coincides with the centroid for doubly-symmetrical sections, and lies on the axis of symmetry for singly-symmetrical sections.) This will induce shear stresses due to torsion and, in the case of open sections, may also produce significant additional longitudinal stresses due to the structural effect known as warping. Proper consideration of this complex topic requires an appreciation of the theory of torsion which is outside the scope of this lecture. In many cases torsional effects can be minimised by careful detailing such that load is transferred into members in a way that avoids twisting.



9. CONCLUDING SUMMARY The main design requirement for beams is the provision of adequate bending strength. The types of member normally used to achieve this have been indicated. Means of recognising beams whose design can be based on relatively simple structural principles have been presented. Section modulus, either elastic or plastic depending upon the design philosophy adopted, is the most appropriate property for use in selecting a suitable section. Shear capacity and deflection must also be checked in simple beam design. Situations in which a more complex structural response occurs, resulting in the need to consider biaxial bending, torsion, etc.., have been discussed.




11. ADDITIONAL READING 1. Dowling, P.J., Owens, G.W. and Knowles, P., "Structural Steel Design", Butterworths, 1988. 2. Petersen C., "Stahlbauten", Vieweg Verlog Braunshweig, 1988. 3. Galambos, T.V., "Structural Members and Frames", Prentice-Hall, Englewood Cliffs, 1968. 4. Narayanan, R., "Beams and Beam Columns - Stability and Strength", Applied Science, London, 1983. 5. Salmon, C. G. and Johnson, J. E., "Steel Structures - Design and Behaviour", Harper and Row, New

York, 1980.

TABLE 1


Beam Type Span Range (m)

Notes

0. Angles 3 - 6 used for roof purlins, sheeting rails, etc., where only light loads have to be carried.

1. Cold-formed sections 4 - 8 used for roof purlins, sheeting rails, etc., where only light loads have to be carried.

2. Rolled sections UB, IPE, UPN, HE

1 - 30 most frequently used type of section; proportions selected to eliminate several possible types of failure.

3. Open web joists 4 - 40 prefabricated using angles or tubes as chords and round bar for web diagonals; used in place of rolled sections.

4. Castellated beams 6 - 60 used for long spans and/or light loads, depth of UB increased by 50%, web openings may be used for services, etc.

5. Compound sections e.g. IPE+UPN

5 - 15 used when a single rolled section would not provide sufficient capacity; often arranged to provide enhanced horizontal bending strength as well.

6. Plate girders 10 - 100 made by welding together 3 plates, sometimes automatically; web depths up to 3-4m sometimes need stiffening.

7. Box girders 15 - 200 fabricated from plate, usually stiffened; used for OHT cranes and bridges due to good torsional and transverse stiffness properties.




ESDEP WG 7

ELEMENTS

Lecture 7.8.2: Restrained Beams II OBJECTIVE/SCOPE

To present and discuss procedures for the design of restrained beams including consideration of moment redistribution for Class 1 sections [1].

PREREQUISITES

Elastic theory for uniaxial and biaxial bending combined with shear forces.

Lectures 2.3: Engineering Properties of Steels

RELATED LECTURES




Worked Example 7.8: Laterally Restrained Beams

SUMMARY

The mechanism of redistribution of bending stress in a steel beam after the attainment of first yield is explained. For statically determinate beams, the idea of failure by the development of full plasticity at a cross-section is introduced, leading to the concepts of plastic section modulus and shape factor. Statically indeterminate beams are shown to be capable of developing several such regions - termed plastic hinges - before collapsing as a mechanism. The influence of shear forces is discussed from a design point of view, and behaviour in bending about both principal axes is briefly considered.

ADDITIONAL NOTATION

b width

ε strain

ε1 & ε2 strains on extreme fibres

d overall depth of section

Est strain hardening modulus

εy yield strain

φ curvature

F applied load



fd limiting stress in material

fly lower yield stress

fp limit of proportionality

fuy upper yield stress


L span

M moment

Mplr plastic moment resistance in presence of an axial load

Mplv plastic moment resistance in presence of shear load

My moment at first yield (fyW)

Npl squash load (fyA)

τy yield shear stress

s shape factor (Wpl/W)

fmax maximum stress on extreme fibre

tw web thickness

Vpw web shear resistance, at shear yield

Vs applied shear force

W elastic section modulus

Wpl plastic section modulus

σ normal stress

τ shear stress

1. INTRODUCTION When a restrained steel beam of "compact" proportions (see Lecture 7.8.1) is subjected to loads producing vertical bending, its response will consist of a number of stages. Initially it will behave elastically, with vertical deflections being related linearly to the applied load. As the loading is increased, the most highly stressed regions will develop strains in excess of yield, resulting in a local loss of stiffness. For the beam as a whole, deflections will now start to increase rather more rapidly. Additional load will cause this process to continue until complete plasticity is reached at one cross-section. For a simply supported beam, this point will correspond to the maximum load that can be carried without strain hardening and will also be the point at which deflections become very large. On the other hand, for continuous structures, further increases of load are possible as



redistribution of moments takes place.

It is the purpose of this lecture to discuss the concept of plasticity as it influences the design of beams, showing how the basic procedures of Lecture 7.8.1 may be modified to allow for this behaviour.

2. BEHAVIOUR OF STEEL BEAMS IN BENDING

2.1 Statically Determinate Beams

Figure 1 presents the relationship between applied load and central deflection that would be obtained from a test on a simply supported steel beam of Class 2 proportions or better. Three distinct phases may be identified:

OA elastic, linear relation between load and deflection. AB elastic-plastic, deflections increase at a progressively faster rate. BC plastic, growth of large deflections at reasonably constant load.

Because phases 2 and 3 involve the development of plasticity in regions of high stress, full understanding of the beam's response first requires that the behaviour of the material itself be considered.

Figure 2 presents a stress-strain curve of the type that would be obtained from a tensile test on a small piece of steel, called a 'coupon'. Rather than work directly with this, calculations may be considerably simplified if the idealised, bilinear response of Figure 3 is substituted. This comprises an initial elastic portion, followed by a horizontal, perfectly plastic portion. It therefore neglects such features as the upper yield point, strain hardening



etc., the inclusion of which would have only a very small effect on the resulting analysis for a substantial increase in complexity.



Using Figure 3, the relationship between moment (M) and curvature (φ) for a cross-section can be derived by considering the distributions of strain and stress at various stages, shown in Figure 4. Providing the maximum strain at the extreme fibre (ε1) remains below the yield strain (εy), then the corresponding stress distribution will be linear, as shown in Figure 4a.



Assuming a beam of depth d, curvature will be given by:

φ = (ε1 - ε2)/ d (1)

The corresponding moment of resistance may be determined by taking moments of the stress diagram about the neutral axis (yy). Assuming a rectangular section of width b gives:

M = σ (bd2/6) (2)



in which (bd2/6) is termed the elastic section modulus (W).

Once ε1 exceeds εy, since σ cannot exceed fy, the shape of the stress distribution will change, as shown in Figure 4b, with the result that φ will now increase more rapidly, i.e. the M-φ relationship will become non-linear, as shown in Figure 5. φ may still be obtained from Equation (1) whilst M may be calculated as before, using Figure 4b.

Eventually the condition shown in Figure 4c will be approached (replacing the actual stress distribution with the two rectangular stress blocks is, of course, an approximation that greatly simplifies calculation for negligible loss in accuracy) for which M is given by:

M = fy (bd2/4) (3)

This is termed the fully plastic moment (Mpl) of the cross-section; it represents the theoretical upper limit on

moment resistance based on the stress-strain behaviour of Figure 3. The quantity Wpl = bd2/4 is termed the plastic section modulus and the ratio Wpl/W, which is a measure of the additional moment that can be carried beyond the first yield, is the shape factor. For a rectangular section:

s = Wpl/W = (bd2/4)/bd2/6) (4)



Therefore s = 1,50

Values of W, Wpl and s for several structural shapes are given in Table 1.

Since the slope of the M-φ curve decreases effectively to zero at Mpl, the cross-section is said to form a plastic hinge, because bending stiffness locally in this region will be zero, i.e. the beam now acts as if it contains a real hinge, with the difference that the moment at this hinge remains at Mpl. For a simply supported beam, the formation of a plastic hinge, rather than the attainment of first yield, provides a close estimate of its maximum load carrying resistance. A simple method of determining the load at which this occurs, consists of treating the elastic portions of the beam as rigid and equating the work done by the external loads to the energy dissipated by the plastic hinge. For the example of Figure 1 this gives:

FLθ/2 = Mpl.2θ (5)

F = 4Mpl/L

This approach uses the concept of the structure being transformed into a mechanism at collapse.

2.2 Statically Indeterminate Beams

If the steel beam is continuous and Class 1, then the formation of the first plastic hinge at the point of maximum moment, previously obtained from an elastic analysis, will not mark the limit of its load-carrying resistance (see Figure 6). Rather, it signifies a change in the way in which the beam responds to further loads. For the two-span beam of Figure 6 the insertion of a real hinge at the central support (B) would cause each span to behave as if it were simply supported. Thus both would be capable of sustaining load, and would not collapse until this load caused a plastic hinge to form at mid-span. The formation of a plastic hinge at B produces qualitatively similar behaviour. Thus continuous structures do not collapse until sufficient plastic hinges have formed to convert them into a mechanism. At collapse, the beam will appear as shown in Figure 6.



Equating external and internal work as before for this collapse mechanism gives:

2FLθ/2 = Mpl (2θ + 2θ + 2θ) (6)

F = 6Mpl/L

Figure 6 shows the load-deflection curve for this two-span beam obtained from an elastic-plastic analysis. Each change in slope corresponds to the formation of a plastic hinge which produces progressive "softening" of the structure. Collapse occurs at an applied load given by Equation (6) when sufficient hinges have formed to transform the beam into a mechanism with no inherent stiffness, corresponding to the final horizontal segment of the curve. It is usual, when conducting this type of analysis, to assume that plastic hinges form suddenly; this corresponds to regarding the cross-section as having a unit shape factor, i.e. Wpl = W. Comparing the resulting load-deflection curve with the more rounded curve obtained from an actual test shows how neglecting the spread of plasticity has comparatively little effect on the overall behaviour.

Although the elastic-plastic analysis of statically indeterminate beams is quite complex, the determination of the collapse load from considerations of the collapse mechanism for most types of continuous beam is relatively straightforward. A similar approach is also possible for plastically designed, portal frame structures.



It should be noted that, in passing from the elastic state corresponding to loads up to Fyield to the plastic state corresponding to the load Fcollapse, the beam of Figure 6 has undergone a redistribution of moment. Thus the shape of the plastic moment diagram will differ from that of the elastic moment diagram. In this case the former will correspond to Mpl at all three plastic hinges, whilst the latter will have a maximum at the central support.

2.3 Bending of I-Sections

Applying the stress-strain curve of Figure 3 to the idealised I-section of Figure 7a bending about the yy axis, in which half of the area is assumed to be concentrated at the mid-depth of each flange, leads to the bilinear M-φ curve of Figure 8. Thus, for this hypothetical cross-section, full plasticity and first yield coincide and the shape factor is unity.



Taking the more realistically proportioned beam with shape as shown in Figure 7b, produces an M-φ curve with a rounded 'knee' as indicated in Figure 8. Since this 'knee' corresponds to the spread of plasticity, it starts at a moment equal to Mpl/s. A typical value of s for an I-beam section would be 1,15 for strong axis bending. For weak axis bending of I-sections s ≈ 1,5.

An elastic-plastic analysis for any cross-sectional shape, bent about its axis of symmetry, may be performed using the approach described previously for a rectangular section. If necessary a more precise representation of the material stress-strain curve may be employed but this does, of course, complicate the arithmetic.

2.4 Bending of Singly-symmetrical Sections

An elastic-plastic analysis for a cross-section that is not bent about an axis of symmetry, is complicated by the shift in the position of the neutral axis that occurs as plasticity develops. Initially the elastic neutral axis will pass through the centroid as shown in Figure 9a. However, because of the unequal strains on the top and bottom surfaces of the section, yield will occur first at one edge only as shown in Figure 9b. Plasticity will thus spread inwards in an unsymmetrical fashion as shown by Figure 9c. Equilibrium of axial forces at all such intermediate stages requires a balance of compressive and tensile forces. For this requirement to be met, the line of zero



stress must move away from the yielded material as the force in this part of the cross-section will be increasing less rapidly. Eventually, upon attaining full plasticity as shown in Figure 9d, the section's neutral axis must coincide with the equal area axis, i.e. the line dividing the section into two halves. Calculation of Mpl for an unsymmetrical section therefore requires a knowledge of the location of the equal area axis, rather than the use of the centroidal axis used when dealing with elastic behaviour.

Singly-symmetrical sections have two different values of the elastic section modulus for the two faces, because of the different distances of the latter from the elastic neutral axis. However, such sections possess a single



value of plastic section modulus equal to Mpl/fy. They will therefore have two values of shape factor depending on which of the faces is under consideration. For extreme sections such as tees, these may differ quite widely with one value actually being less than unity. Although this does not, of course, affect the basic relationship between My and Mpl for the section, it can be a source of confusion in situations which require the specific consideration of yield in compression (or tension).

3. EFFECT OF SHEAR FORCE Consideration of the effect of shear on plastic moment resistance first requires an assumption about the way in which steel yields under the action of direct and shear stress. This is represented by:

(σ/fy)2 + (τ/τy)2 = 1 (7)

in which τy = yield shear strength (taken as fy/√3 according to the Von Mises criterion) and rounded to 0,6fy.

Applying this to assumed plastic patterns of bending and shear stress for a rectangular section, leads to the following approximate relationship between the applied shear force (Vs), reduced plastic moment resistance (Mplv), the shear force required for yield in shear (Vpw), and Mpl :

Mplv = for (8)

Applying Equation (7) to a fully plastic I section (see Figure 10), assuming that the shear stresses in the web are uniform and that Vpw = 0,6dtwfy (τy = 0,6 fy), gives:



Mplv = for 0,5Vpw ≤ Vs ≤ Vpw (9)

Mplv = Mpl for Vs ≤ 0,5Vpw

Thus for I-sections bent in the plane of the web, shear forces of less than 50% of the plastic resistance of the web in shear have no significant effect on Mpl. For bending about the minor axis, behaviour is similar to that of a rectangular section, as described by Equation (8).

4. PLASTIC BEHAVIOUR UNDER GENERAL COMBINED LOADING Analyses for cross-sectional resistance under combinations of moment and shear have been performed for various structural shapes using a number of variants of plasticity theory. For design purposes, results are normally expressed either graphically or as approximate interaction formulae. In the case of an I-section bent about its major axis, no reduction in Mpl for shears of less than one half of Vpw is necessary.

When bending takes place about both axes of the cross-section, the plastic neutral axis will be inclined to the rectangular axes by an amount which depends on the ratio of the applied moments and the exact shape of the section. By assuming stress distributions and calculating corresponding moments of resistance, a relationship between the two reduced plastic moment resistances (Mplry and Mplrz) and the basic plastic moment resistances (Mply and Mplz) may be obtained. If required, such calculations may also include the effects of axial load (N). Although tedious, these have been performed for a variety of structural shapes (see Figures 11 and 12), including unsymmetrical sections such as angles. For an I or H shape under biaxial bending, a suitable



expression relating Mplry and Mplrz is: Page 15 of 18Previous | Next | Contents


(10)

Providing the section has at least one axis of symmetry and the problem is worked in terms of its principal axes, it is always safe to use the linear equivalent of Equation (10), viz:

(11)

5. CONCLUDING SUMMARY



Providing instability effects do not occur, steel beams can withstand moments in excess of those that just cause yield. Stress redistribution over the cross-section enables the fully plastic pattern of bending stresses to develop, for which the moment of resistance is the full plastic moment, Mpl. Statically indeterminate beams only collapse when sufficient regions of full local plasticity, termed "plastic hinges" have developed to convert the "structure" into a mechanism. The value of Mpl is given by the product of yield stress (fy) and a geometrical property of the cross-section termed the "plastic section modulus" - Wpl; the ratio Wpl/W is called the "shape factor". Mpl will be reduced if shear is also present, although in most practical situations reductions due to shear will be negligible.


7. ADDITIONAL READING 1. Dowling, P.J., Owens, G.W. and Knowles, P., "Structural Steel Design", Butterworths, 1988. 2. Ballio, G., and Mazzolani, F., "Theory and Design of Steel Structures", Chapman and Hall, New York,

1983. 3. Salmon, C.G. and Johnson, J.E., "Steel Structures - Design and Behaviour", Harper and Row, New York,

1980. 4. Trahair, N.S. and Bradford, M.A., "The Behaviour and Design of Steel Structures", Chapman and Hall,

1988. 5. McGinley, T.J. and Ang, T.C., "Structural Steelwork Design to Limit State Theory", Butterworths, 1987.

Table 1 Values of W, Wpl and s for various shapes

Section Elastic section modulus W Plastic section modulus Wpl Shape factor

s

Rectangle bd2/6 bd2/4 1,5

I: major axis [bd3-(b-tw)h3]/6d btf(d-tf)

+ tw(d-2tf)2/4

approx. 1,15

I: minor axis

b2tf/2 +

(d-2tf)tw2/4

approx. 1,67

Solid circular π d3/32 d3/6 1,7

Circular hollow section π [d4-(d-2t)4]/32d

If t << d; td2

for t = d/10;

1,4

for t << d; 1,27

Channel [bd3-(b-tw)h3]/bd btf(d-tf)

+ tw(d-2tf)2/4

approx. 1,15



Previous | Next | ContentsPage 18 of 18Previous | Next | Contents



ESDEP WG 7

ELEMENTS

Lecture 7.9.1: Unrestrained Beams I OBJECTIVE/SCOPE

To develop an understanding of the phenomenon of lateral-torsional instability; to identify the controlling parameters and to show how theory, experiment and judgement are combined to produce a practical design method. The design procedure given in Eurocode 3 [1] is used as an illustration of such a method.

PREREQUISITES


Lectures 6.6: Buckling of Real Structural Elements


RELATED LECTURES:

Lecture 7.9.2: Unrestrained Beams II


SUMMARY

This lecture begins with a non-mathematical introduction to the phenomenon of lateral torsional buckling. It presents a simple analogy between the behaviour of the compression flange and the flexural buckling of a strut. It summarises the principal factors influencing lateral stability and briefly describes the role of bracing in improving this.

A brief explanation is given of the reasons why the elastic theory, discussed in Lecture 7.9.2, requires modification before being used as a basis for the design rules for unrestrained beams. A summary of the background to Eurocode 3 [1] is also presented.

NOTATION

C coefficient to account for type of loading

d overall depth

EIz flexural rigidity about the minor axis

fd design strength of material


iz minor axis radius of gyration

k coefficient to account for conditions of lateral support

L span



Mb.Rd buckling resistance moment

Mcr elastic critical buckling moment

Mpl plastic moment of cross-section

MRd moment resistance of cross-section

tf flange thickness

u lateral deflection

αLT parameter in design formula, see Equation (2)

χLT reduction factor for lateral-torsional buckling

LT beam slenderness

λLT basic slenderness

λ1 parameter used to determine LT, see Equation (4)

φ twist

φLT parameter used to determine χLT, see Equation (2)

ψ moment ratio, see Equation (5)

1. STRUCTURAL PROPERTIES OF SECTIONS USED AS BEAMS When designing a steel beam it is usual to think first of the need to provide adequate strength and stiffness against vertical bending. This leads naturally to a cross-sectional shape in which the stiffness in the vertical plane is much greater than that in the horizontal plane. Sections normally used as beams have the majority of their material concentrated in the flanges, which are relatively narrow so as to prevent local buckling. The need to connect beams to adjacent members with ease normally suggests the use of an open section, for which the torsional stiffness will be comparatively low. Figure 1, which compares section properties for four different shapes of equal area, shows that the high vertical bending stiffness of typical beam sections is obtained at the expense of both horizontal bending and torsional stiffness.



2. RESPONSE OF SLENDER BEAMS TO VERTICAL LOADING It is known from our understanding of the behaviour of struts that, whenever a slender structural element is loaded in its stiff plane (axially in the case of the strut), there exists a tendency for it to fail by buckling in a more flexible plane (by deflecting sideways in the case of the strut). Figure 2 illustrates the response of a slender cantilever beam to a vertical end load; this phenomenon is termed lateral-torsional buckling. Although it involves both a lateral deflection (u) and twisting about a vertical axis through the web (φ), as shown in Figure 3, this type of instability is quite similar to the simpler flexural buckling of an axially loaded strut. Loading the beam in its stiffer plane (the plane of the web) has induced a failure by buckling in a less stiff-direction (by deflecting sideways and twisting).



Of course, many types of construction effectively prevent this form of buckling, thereby enabling the beam to be designed with greater efficiency as fully restrained (see Lecture 7.8.1). In this context it is important to realise that during erection of the structure certain beams may well receive far less lateral support than will be the case when floors, decks, bracings, etc., are present, so that stability checks, at this stage, are also necessary.

Lateral-torsional instability influences the design of laterally unrestrained beams in much the same way that flexural buckling influences the design of columns. Thus the bending strength will now be a function of the beam's slenderness, as indicated in Figure 4, requiring the use in design of an iterative procedure similar to the use of column curves in strut design. However, because of the type of structural actions involved, the analysis of lateral-torsional buckling is considerably more complex. This is reflected in a design approach which requires a rather greater degree of calculation.



3. SIMPLE PHYSICAL MODEL Before considering the analysis of the problem, it is useful to attempt to gain an insight into the physical behaviour by considering a simplified model. Since bending of an I-section beam is resisted principally by the tensile and compressive forces developed in two flanges, as shown in Figure 5, the compression flange may be regarded as a strut. Compression members exhibit a tendency to buckle and in this case the weaker direction would be for the flange to buckle downwards. However, this is prevented by the presence of the web. Therefore the flange is forced to buckle sideways, which will induce some degree of twisting in the section as the web too is required to deform. Whilst this approach neglects the real influence of torsion and the role of the tension flange, it does, nevertheless, approximate the behaviour of very deep girders with very thin webs or of trusses or open web joists. Indeed, early attempts at analysing lateral-torsional buckling started with this approach.



4. FACTORS INFLUENCING LATERAL STABILITY The compression flange/strut analogy, discussed in the previous section, is also helpful in understanding the following:

1. The buckling load of the beam is likely to be dependent on its unbraced span, i.e.the distance between points at which lateral deflection is prevented, and on its lateral bending stiffness (ELz) because strut

resistance ∝ ELz/L2.

2. The shape of the cross-section may be expected to have some influence, with the web and the tension flange being more important for relatively shallow sections, than for deep slender sections. In the former case the proximity of the stable tension flange to the unstable compression flange increases stability and also produces a greater twisting of the cross-section. Thus torsional behaviour becomes more important.

3. For beams under non-uniform moment, the force in the compression flange will no longer be constant, as shown in Figure 6. Therefore such members might reasonably be expected to be more stable than similar



members under a more uniform pattern of moment. 4. End restraint which inhibits development of the buckled shape, shown in Figure 3, is likely to increase the

stability of the beam. Consideration of the buckling deformations (u and φ) should make it clear that this refers to rotational restraint in plan, i.e.about the z-axis (refer back to Figure 5 and 3). Rotational restraint in the vertical plane affects the pattern of moments in the beam (and may thus also lead to increased stability) but does not directly alter the buckled shape, as shown in Figure 7.



A more rigorous analysis, substantiating the above four points, is presented in Lecture 7.9.2. This lecture also deals with warping of the cross-section and the influence of level of application of load on stability; factors not illustrated by the simplified model presented here.

5. BRACING AS A MEANS OF IMPROVING PERFORMANCE Bracing may be used to improve the strength of a beam that is liable to lateral-torsional instability. Two requirements are necessary:

1. The bracing must be sufficiently stiff to hold the braced point effectively against lateral movement (this can normally be achieved without difficulty).

2. The bracing must be sufficiently strong to withstand the forces transmitted to it by the main member (these forces are normally a percentage of the force in the compression flange of the braced member).

Providing these two conditions are satisfied, then the full in-plane strength of a beam may be developed through braces at sufficiently close spacing. Figure 8, which illustrates buckled shapes for beams with intermediate braces, shows how this buckling involves the whole beam. In theory, bracing should prevent either lateral or torsional displacement from occurring. In practice, consideration of the buckled shape of the beam cross-section shown in Figure 3 suggests that bracing is potentially most effective when used to resist the largest components of deformation, i.e. a lateral brace attached to the top flange is likely to be more effective than a similar brace attached to the bottom flange.



6. DESIGN APPLICATION Direct use of the theory of lateral-torsional instability for design is inappropriate because:

The formulae are too complex for routine use, e.g. Equation (17) of Lecture 7.9.2. Significant differences exist between the assumptions which form the basis of the theory and the characteristics of real beams. Since the theory assumes elastic behaviour, it provides an upper bound on the true strength (this point is discussed in general terms in Lecture 6.6.2).

Figure 9 compares a typical set of lateral-torsional buckling test data obtained using actual hot-rolled sections with the theoretical elastic critical moments given by Equation (17) of Lecture 7.9.2.



In Figure 9a only one set of data for a narrow flanged beam section is shown. The use of the LT non-dimensional format in Figure 9b has the advantage of permitting results from different test series (using different cross-sections and different material strengths) to be compared directly. In both figures three distinct regions of behaviour can be observed:



Stocky beams which are able to attain Mpl, with values of LT below about 0,4 in Figure 9b.

Slender beams which fail at moments close to Mcr, with values of LT above 1,2 in Figure 9b.

Beams of intermediate slenderness which fail to reach either Mpl or Mcr, with 0,4 < LT < 1,2 in Figure 9b.

Only in the case of beams in region 1 does lateral stability not influence design; such beams can be designed using the methods of Lecture 7.8.1. For beams in region 2, which covers much of the practical range of beams without lateral restraint, design must be based on considerations of inelastic buckling suitably modified to allow for geometrical imperfections, residual stresses, etc., (see Lecture 6.1). Thus both theory and tests must play a part, with the inherent complexity of the problem being such that the final design rules are likely to involve some degree of empiricism.

Section 7 outlines the provisions of Eurocode 3 [1] with regard to beam design, assuming typical sections as shown in Figure 10a and 10b. It should be noted that sections of the type illustrated in Figure 10b, with one axis of symmetry, e.g.channels, may only be included if the section is bent about the axis of symmetry, i.e. loads are applied through the shear centre parallel to the web of the channel. Singly-symmetrical sections bent in the other plane, e.g. an unequal flanged I-section bent about its major-axis as shown in Figure 10c, may only be treated by an extended version of the theory of Lecture 7.9.2, principally because the section's shear centre no longer lies on the neutral axis.

7. METHOD OF EUROCODE 3 The buckling resistance moment [1] is given by:

MbRd = χLT MRd (1)

where MRd is the moment resistance of the cross-section

χLT is the reduction factor for lateral-torsional buckling



In determining MRd the section classification should, of course, be noted and the appropriate section modulus used in conjunction with the material design strength fd. The value of χLT depends on the beam's slenderness

LT and is given by:

χLT = 1/ {φLT + [φLT2 - LT

2]1/2} (2)

where φLT = 0,5 [1 + αLT( LT - 0,20) + LT2]

and αLT = 0,21 for rolled sections

αLT = 0,49 for welded beams

Figure 11 illustrates the relationship between χLT and LT, showing how it follows the pattern of behaviour

exhibited by the test data of Figure 9. When LT ≤ 0,4, the value of χLT is sufficiently close to unity that design may be based on the full resistance moment MRd.

The slenderness LT, which is a measure of the extent to which lateral-torsional buckling reduces a beam's load carrying resistance, is a function of MRd and Mcr. Mcr is the elastic critical buckling moment, a quantity similar in concept to the Euler load for a strut since it is derived from a theory (see Lecture 7.9.2) that assumes "perfect" behaviour, i.e. an initially straight member, elastic response, no misalignment of the loading, etc..



Thus LT is taken as:

LT = (3)

For calculation purposes Equation (3) may be rewritten as:

LT = [λLT / λ1] (4)

where λ1 = π[E/fy]1/2

=

and λLT =

This expression represents a conservative approximation for any uniform plain I or H shape with equal flanges -see Annex F2 of EC3.

The above expression for λLT is valid for loading giving uniform moment over a span whose ends are prevented from deflecting laterally and from twisting about a vertical axis passing through the web. This is the basic case for lateral stability (Figure 12) for which a full theoretical treatment is provided in Lecture 7.9.2. Variations in the conditions of loading and/or lateral support may be allowed for by introducing modifying factors into the expressions for λLT or Mcr.

For example, if there is a moment gradient between points of lateral restraint, λLT is calculated as follows:



λLT = (5)

where C1 = 1,75 - 1,05 ψ + 0,3 ψ2 ≤ 2,35

and ψ is the end moment ratio defined in Figure 13.

Taking as an example the end span of a continuous beam for which ψ = 0 gives C1=1,75 and thus λLT will be reduced to 0,76 (= 1/√ 1,75) of the value for uniform moment, leading to an increase in χLT and thus in MbRd.

Variations in the conditions of lateral restraint may be treated by introducing k-coefficients to modify the geometrical length L into kL when determining Mcr. For conditions with more restraint, values of k < 1,0 are

appropriate, leading to an increase in Mcr and thus, via a reduction in LT, to increases in χLT and MbRd.

Similarly additional C-coefficients may be used directly in the determination of Mcr to provide modified values

of LT appropriate for a wide range of load types. In particular, this method should be used to calculate the reduced Mcr appropriate for destabilising loads. These are loads that act above the level of the beam's shear centre and are free to move sideways with the beam as it buckles, as shown in Figure 14.



For cross-sections of the type illustrated in Figure 9c, for which the shear centre and centroid do not lie on the same horizontal axis, evaluation of Mcr becomes more complex and is covered by Annex F of Eurocode 3 [1].

8. CONCLUDING SUMMARY Beams that are not restrained along their length and are bent about their strong axis are subject to lateral torsional buckling. Unbraced span, lateral slenderness (L/iz), cross-sectional shape (torsional and warping rigidities), moment distribution and end restraint are the primary influences on buckling resistance. Bracing of sufficient stiffness and strength, that restrains either torsional or lateral deformations, may be used to increase buckling resistance. Although elastic critical load theory provides a background for understanding the behaviour of laterally unrestrained beams, it requires both simplifications and empirical modification if it is to form a suitable basis for a design approach. In order to check the lateral buckling resistance of a trial section, its effective slenderness LT must first be obtained. Variation in either lateral support conditions or the form of the applied loading may be accommodated in the design process by means of coefficients k and C, used to modify either the basic slenderness λLT or the basic elastic critical moment Mcr.




10. ADDITIONAL READING 1. Narayanan, R., Editor, "Beams and Beam Columns: Stability and Strength", Applied Science Publishers

1983.

Chapters 1 - 3 deal with various aspects of behaviour and design of laterally unrestrained beams.

2. Chen, W. F. and Atsuta, T. "Theory of Beam Columns Volume 2, Space Behaviour and Design", McGraw Hill 1977.

Chapter 3 deals with laterally unrestrained beams.

3. Timoshenko, S. P. and Gere, J. M., "Theory of Elastic Stability" Second Edition, McGraw Hill 1961.

Basic derivations for the elastic critical moment for a variety of beam problems are provided in Chapter 6.

4. Bleich, F., "Buckling Strength of Metal Structures", McGraw Hill 1952.

Chapter 4 presents the basic theory of lateral buckling of beams.

5. Galambos, T. V., "Structural Members and Frames", Prentiss Hall 1968.

Chapter 2 deals with the fundamentals of elastic behaviour, whilst Chapter 3 covers elastic and inelastic behaviour and design of laterally unrestrained beams.

6. Trahair, N. S. and Bradford, M. A., "The Behaviour and Design of Steel Structures", Chapman and Hall, Second Edition, 1988.

Laterally unrestrained beams are dealt with in Chapter 6.





ESDEP WG 7

ELEMENTS

Lecture 7.9.2: Unrestrained Beams II OBJECTIVE/SCOPE

To derive the basic theory of elastic lateral-torsional buckling and to discuss the physical significance of the resulting expressions.

PREREQUISITES

Simple bending theory

Simple torsion theory

Lecture 6.4: General Methods for Assessing Critical Loads


Lecture 7.9.1: Unrestrained Beams I

RELATED LECTURES



Worked Example 7.9: Laterally Unrestrained Beams

SUMMARY

This lecture presents the basic elastic theory for lateral-torsional buckling of beams, commencing with the case of a simply supported beam under uniform moment. Variations in load pattern, load level and degree of end restraint are discussed. The theoretical derivations are separated into two Appendices, the main text being limited to a discussion of the underlying assumptions and the physical significance of the derived expressions.

NOTATION

b flange width

C1 coefficient to allow for type of loading

hf distance between flange centroids

EIw warping rigidity

EIz minor axis flexural rigidity

E Young's modulus

F applied load



Fcr elastic critical buckling load

GIt torsional rigidity

d overall depth of section

L span

M moment

Mcr applied elastic critical buckling moment

tf flange thickness

tw web thickness

1. INTRODUCTION The basic model used to illustrate the theory of lateral-torsional buckling is shown in Figure 1. It assumes the following:

beam is initially straight elastic behaviour uniform equal flanged I-section ends simply supported in the lateral plane (twist and lateral deflection prevented, no rotational restraint in plan) loaded by equal and opposite end moments in the plane of the web.

This problem may be regarded as being analogous to the basic pin-ended Euler strut.



The beam is placed in its buckled position, as in Figure 2, and the magnitude of the applied load necessary to hold it there determined by equating the disturbing effect of the end moments, acting through the buckling deformations, to the internal (bending and torsional) resistance of the section.

The derivation and solution to the equations leading to the critical value of applied end moments (Mcr) at which the beam of Figure 1 just becomes unstable is provided in Appendix 1. The physical significance of the solution and its application in cases where the assumptions listed above do not apply are discussed in Sections 2 and 3 that follow.

2. PHYSICAL SIGNIFICANCE OF THE SOLUTION The buckled shape of the beam, Figure 2, is now compared with the expression for the elastic critical moment of Equation (17) in Appendix 1, i.e.

Mcr = (17)

The presence of the flexural (EIz) and torsional (GIt and EIw) stiffnesses of the member in the equation is a direct consequence of the lateral and torsional components of the buckling deformations. The relative importance of the two mechanisms for resisting twisting is reflected in the second square root term. Length is also important, entering both directly and indirectly via the π2EIw/L2GIt term. It is not possible to simplify Equation (17) by omitting terms without imposing limits on the range of application of the resulting approximate solution. Figure 6 shows quantitatively the application of Equation (17) to the different types of beam sections defined in the earlier Lecture 7.8.1. The region of the curves for both I-sections of low length/depth ratios corresponds to the situation in which the value of the second square root term in Equation (17) adopts a value significantly in excess of unity. Since warping effects (see Appendix 1) will be most important for deep sections composed of thin plates, it follows that the π2EIw/L2GIt term will, in general, tend



to be large for short deep girders and small for long shallow beams.

Figure 7 gives some quantitative indication of the effect of shape of cross-section for structural steel I-beams, by comparing values of Mcr for a beam (I) and a column (H) having approximately equal in-plane plastic moment capacities. Clearly, lateral-torsional buckling is a potentially more significant design consideration for the beam section which is much less stiff laterally.



3. EXTENSION TO OTHER CASES

3.1 Load Pattern

The equivalent of Equation (8) (Appendix 1) may be set up and solved for a variety of other load cases. Since the applied moment at any point within the span will now be a function of x, the mathematics will be more complex. As an example, consider the beam subjected to a central load acting at the level of the centroidal axis shown in Figure 8, for which the analysis is outlined in Appendix 2.



The solution for this example may conveniently be compared with the basic case in terms of the critical moments for each, i.e. maximum moment when the beam is on the point of buckling.

Basic case: Mcr = (π/L) √(EIxGIt) √[1+ (π2EIw/L2GIt)] (17)

Central load: Mcr = (4,24/L) √(EIxGIt) √[1+ (π2EIw/L2GIt)] (21)

The ratio of the two constants π/4,24=0,74 is the reciprocal of the coefficient C1 introduced in Lecture 7.9.1. Its value is a direct measure of the severity of a particular pattern of moments relative to the basic case.

Figure 9, which gives C1 factors for various loading patterns, shows how lateral stability generally increases as the moment pattern becomes less uniform.



3.2 Level of Application of Load

For transverse loads free to move sideways with the beam as it buckles, the level of application of load (relative to the centroid) is important. The solution for a point load applied at any level relative to the beam's centroidal axis may conveniently be obtained using an energy approach, as outlined in Appendix 2. When the load is applied to either the top flange or the bottom flange, e.g. by a crane trolley, the solution of Equation (21) may still be used, providing the numerical constant is replaced by a variable, the value of which depends upon the ratio L2GIt/EIw as shown in Figure 10. The reason why top flange loading and bottom flange loading are respectively more or less severe than centroidal loading may be appreciated from the sketches in Figure 10, which show the destabilising and stabilising effects. Clearly this would be expected to become more significant as the depth of the section increases and/or the span reduced, i.e. as L2GIt/EIw becomes smaller.



3.3 Conditions of Lateral Support

It has already been suggested in Lecture 7.9.1, that lateral support arrangements which inhibit the growth of the buckling deformations will improve a beam's lateral stability. Equally, less effective conditions will reduce stability. Providing the appropriate boundary conditions can be incorporated into the analysis methods of Appendices1 and 2, any arrangement can be dealt with.

A convenient way of including the effect of different support conditions is to redefine L in Equation (17) as an effective length l, with the exact value of l/L depending upon the degree of lateral bending and/or warping restraint provided. In Eurocode 3 [1] this approach is split into the use of two factors:

k referring to end rotation on plan.

kw referring to end warping.

It is recommended that kw be taken as unity unless special provision for warping fixity is made; k may vary from 0,5 for full fixity, through 0,7 for one end fixed and one end free, to 1,0 for no rotational fixity.

One case of particular practical interest is the cantilever, for which some results are presented in Figure 11.



These show that:

1. Cantilevers under end moment are less stable than similar, simply supported, beams. 2. Concentrating the moment adjacent to the support, as happens when the applied loading changes from

pure moment to an end load or to a distributed load, improves lateral stability. 3. The effect of load height is even more significant for cantilevers than for simply supported beams.

3.4 Continuous Beams

Continuity may be present in two different forms:

1. In a beam that has a single span vertically but is subdivided, by intermediate lateral supports, so that it exhibits horizontal continuity between adjacent segments, see Figure 12a.

2. In the vertical plane as illustrated in Figure 12b.



For the first case a safe design will result if the most critical segment, treated in isolation, is used as the basis for designing the whole beam. For the second case account should be taken of the actual moment diagram within each span, produced by the continuity, by using the C1 factor. If the top flange can be considered as laterally restrained because of attachment to a concrete slab, particular attention should be paid to the regions in which the lower flange is in compression, e.g. the support regions or regions where uplift loads can occur.

3.5 Beams Other than Doubly-Symmetrical I-sections

The basic theoretical solution of Equation (17) is valid for members that are symmetrical about their horizontal axis, e.g. a channel with the web vertical, providing the moments act through the shear centre (which will not now coincide with the centroid). However, sections symmetrical only about the vertical axis, e.g. an unequal flanged I, require some modification so as to allow for the so-called Wagner effect. This arises as a direct result of the vertical separation of the shear centre and the centroid and leads to either an increase or a decrease in the section's torsional rigidity. Thus lateral stability will be improved when the larger flange is in compression and reduced when the smaller flange is in compression as compared with equal flange sections having comparable properties.

Sections with no axis of symmetry will not actually buckle but will deform by bending about both principal axes and by twisting from the onset of loading. They should therefore be treated in the same way as symmetrical sections under biaxial bending.



3.6 Restrained Beams

The elastic critical moment for a doubly symmetrical I-beam provided with continuous elastic torsional restraint, of stiffness equal to Kφ , is:

Mcr =

Rearranging this shows that the beam behaves as if its torsional rigidity GIt were increased to (GIt+ Kφ L2/π 2), thereby permitting a ready assessment of the effectiveness of the restraint. An important practical example of such a restraint would be that provided by the bending stiffness of profiled steel sheeting (used typically in roof construction) spanning at right angles to the beam.

4. CONCLUDING SUMMARY The elastic critical moment which causes lateral-torsional buckling of a slender beam may be determined from an analysis which has close similarities to that used to study column buckling. Examination of the expression for the elastic critical moment for the basic problem enables the influence of cross-sectional shape, as it affects the beam's resistance to lateral bending (EIz), torsion (It) and warping (Iw), to be identified; it also demonstrates the importance of unbraced span length. Extensions to the basic theory permit the effects of load pattern, end restraint and level of application of destabilising loads to be quantified. Load patterns which produce non-uniform moment may be compared with the basic, uniform moment case using the coefficient C1; since most of these other cases will be less severe, C1 values greater than 1,0 are the norm.


6. ADDITIONAL READING 1. Narayanan, R., Editor, "Beams and Beam Columns: Stability and Strength", Applied Science Publishers

1983.

Chapters 1 - 3 deal with various aspects of behaviour and design of laterally unrestrained beams.

2. Chen, W. F. and Atsuta, T. "Theory of Beam Columns Volume 2, Space Behaviour and Design", McGraw Hill 1977.

Chapter 3 deals with laterally unrestrained beams.

3. Timoshenko, S. P. and Gere, J. M., "Theory of Elastic Stability" Second Edition, McGraw Hill 1961.

Basic derivations for the elastic critical moment for a variety of beam problems are provided in Chapter 6.

4. Bleich, F., "Buckling Strength of Metal Structures", McGraw Hill 1952.

Chapter 4 presents the basic theory of lateral buckling of beams.

5. Galambos, T. V., "Structural Members and Frames", Prentice Hall 1968.

Chapter 2 deals with the fundamentals of elastic behaviour, whilst Chapter 3 covers elastic and inelastic



behaviour and design of laterally unrestrained beams.

6. Trahair, N. S. and Bradford, M. A., "The Behaviour and Design of Steel Structures", Chapman and Hall, Second Edition, 1988.

Laterally unrestrained beams are dealt with in Chapter 6.

APPENDIX 1: ANALYSIS OF LATERAL-TORSIONAL BUCKLINGDerivation of Governing Equations

The deformed state of the beam is shown in Figure 3, which identifies the deflections (u and v) and the twist (φ). A new co-ordinate system ξ η ζ , which deflects with the beam, is also illustrated.

Bending in the ξ ζ and η ζ planes and twisting about the ζ axis are governed by:

EIy (1)



EIz (2)

GIt (3)

In Equations (1) and (2) the flexural rigidities and curvatures in the ξ ζ and the η ζ planes have been replaced by the values for the yx and zx planes, on the basis that φ is a small angle. Equation (3) includes both mechanisms available in a thin-walled section to resist twist; the first term corresponds to that part of the applied torque which is resisted by the development of shear stresses, whilst the second term allows for the influence of restrained warping. This latter phenomenon arises as a direct result of the type of axial flange deformation, illustrated in Figure 4a, that occurs in an I-section subject to equal and opposite end torques. The two flanges tend to bend in opposite senses about a vertical axis through the web, with the result that originally plane sections do not remain plane. On the other hand, for the cantilever of Figure 4b, it is clear that warping deformations must be at least partly inhibited elsewhere along the span, since they cannot occur at the fixed end. This induces additional axial stresses in the flanges; the pair of couples, or bimoment, due to this additional stress system provides part of the section's resistance to twist. In the case of lateral instability, restraint against warping arises as a result of adjacent cross-sections wanting to warp by different amounts.



For an I-section, the relative magnitudes of the warping constant Iw and the torsion constant It are:

Iw = Iz hf2/4 and It =

They will be affected principally by the thickness of the component plates and by the depth of the section. For compact column-type sections the first term in Equation (3) will tend to provide most of the twisting resistance, whilst the second term will tend to become dominant for deeper beam shapes.

Consideration of the buckled shape using Figures 2, 3 and 5 enables the components of the applied moment in the ξ ζ and η ζ planes and about the ζ axis to be obtained as:

Mξ = Mcosφ , Mη = Msinφ , Mζ = Msinα (4)



Since φ is small, cosφ ≈ 1 and sinφ ≈ φ, whilst Figure 5 shows that sinα may be approximated by - . Thus Equations (1) - (3) may be written as:

EIy = M (5)

EIz = Mφ (6)



GIt (7)

Since Equation (5) contains only the vertical deflection (v), it is independent of the other two; it controls the in-plane response of the beam described in Lecture 7.5.1. Equations (6) and (7) are coupled in u and φ , the buckling deformations; their solution gives the value of elastic critical moment (Mcr) at which the beam becomes unstable. Combining them gives:

EIw (8)

Solution

The solution of Equation (8) is made far simpler if the warping stiffness (Iw) is assumed to be zero. The results obtained are then directly applicable to beams of narrow rectangular cross-section but are conservative for the normal range of I-sections. Equation (8) therefore reduces to:

(9)

Putting µ2 = enables the solution to be written as:

φ = Acos µx = Bsin µx (10)

Noting the boundary conditions at both ends gives

When x = 0, φ = 0; then A = 0 (11)

When x = L, φ = 0; then Bsin µL = 0

and either B = 0, or (12)

sin µL = 0 (13)

The first possibility gives the unbuckled position whereas the second gives:

µL = 0, π, 2π (14)

and the first non-trivial solution is:

µL = π (15)

which gives:

Mcr = (π/L) √(EIxGIt) (16)

Since the form of Equation (9) is identical to the form of the basic Euler strut equation all of the same arguments about its solution apply.

Returning to the original Equation (8), this may be solved to give:



Mcr = (π/L) √(EIxGIt) √[1+ (π2EIw/L2GIt)] (17)

The inclusion of warping effects therefore enhances the value of Mcr by an amount which is dependent on the relative values of EIw and GIt.

APPENDIX 2: BUCKLING OF A CENTRALLY LOADED BEAM Using the approach of Appendix 1 and noting from Figure 7 that the vertical load will produce a moment about the x-axis of W(uo - u)/2 when the beam is in its buckled position, enables Equations (4) to be re-written as:

Mξ =

Mη = (18)

Mζ =

Replacing Equations (5) - (7) by their revised forms and eliminating u from the second and third of these gives:

EIw (19)

which may be solved for Wcr to yield approximately:

Wcr = 5,4 (20)

The moment at midspan is then:

Mcr = (21)

The alternative means of obtaining elastic critical loads uses the energy method, in which the work done by the applied load during buckling is equated to the additional strain energy stored as a result of the buckling deformations. Considering an element of the longitudinal axis of the beam of length dx located at C, bending in the ξ ζ plane causes the end B of the beam to rotate in the ξ ζ plane by:

(22)

The vertical component of this is:

(23)

Summing these for all elements between x= 0 and x = L/2 gives the lowering of the load W from which the work is:



(24)

The strain energy stored as a result of lateral bending, twisting and warping is:

(25)

Assuming a buckled shape of the form:

(26)

and equating Equations (24) and (25) enables the critical value of W to be obtained.

Use of this technique permits examination of the case in which the load is applied at a level other than the centroidal axis. Assuming W to act at a vertical distance (a) above the centroid, the additional work will be:

Wa (1 - cos φo) = Wa φo2/2

in which φo is the value of φ at the load point. This must be added to Equation (24).





ESDEP WG 7

ELEMENTS

Lecture 7.10.1: Beam Columns I OBJECTIVE/SCOPE

To introduce the principles of beam-column behaviour and design through the concepts of the interaction of the compressive and bending load components.

PREREQUISITES

Simple bending theory




RELATED LECTURES



Worked Example 7.10: Beam Columns

SUMMARY

This lecture explains the basic concepts of interaction between bending and compression effects by concentrating on uniaxial, in-plane behaviour. This permits topics such as amplification of moments, interaction formulae and the use of compressive resistance and bending resistance to be discussed without the complication of requiring consideration of torsion and out-of-plane response.

1. INTRODUCTION Beam-columns are defined as members subject to combined bending and compression. In principle, all members in frame structures are actually beam-columns, with the particular cases of beams (F = 0) and columns (M = 0) simply being the two extremes. Depending upon the exact way in which the applied loading is transferred into the member, the form of support provided and the member's cross-sectional shape, different forms of response will be possible.

The simplest of these involves bending applied about one principal axis only, with the member responding by bending solely in the plane of the applied moment. Only this case will be considered in this lecture; more complex behaviour is covered in Lecture 7.10.2.

2. CROSS-SECTIONAL BEHAVIOUR Figure 1 shows a point somewhere along the length of an H-shape column where the applied compression and moment about the y-axis produce the uniform and varying stress distributions shown in Figures 1a and 1b.



For elastic behaviour the principle of superposition may be used to simply add the two stress distributions as shown in Figure 1c. First yield will therefore develop at the edge where the maximum compressive bending stress occurs and will correspond to the condition:

fy = σc + σb (1)

where:



fy is the material yield stress

σc = N/A is the stress due to the compressive load N

σb = is the maximum compressive stress due to the moment M, h is the overall depth of section and, I is the second moment of area about the y axis.

Alternatively, if full plasticity is allowed to occur, then the failure condition will be as shown in Figure 2 and the combination of axial load and moment giving this condition will be:

a. For yn ≤ (h-2tf)/2 neutral axis in web

NM = 2fytwyn

MN = fybtf (h-tf) + fy tw

(2)

b. For yn > (h-2tf)/2 neutral axis in flange

NM = fy

MN = tf

(3)



Figure 3 compares Equations (2) and (3) with the approximation used in Eurocode 3 [1] of:

MNy = Mpl.y (1 - n) / (1 - 0,5a) (4)

in which

n = NSd / Npl.Rd is the ratio of axial load to "squash" load (fy A)



a = (A - 2btf)/A ≤ 0,5

Further simplifications and approximations for a range of common cross-sectional shapes are provided in Table 1. In all cases the value of MN should, of course, not exceed that of Mpl.

3. OVERALL STABILITY The treatment of cross-sectional behaviour in the previous section took no account of the exact way in which the moment M at the particular cross-section under consideration was generated. Figure 4 shows a beam-column undergoing lateral deflection as a result of the combination of compression and equal and opposite moments applied at the ends.



The moment at any point within the length may conveniently be regarded as being composed of two parts:

primary moment M

secondary moment Nv

Analysing this problem elastically using strut theory gives the maximum deflection at the centre as:

vmax = (5)

where PEy = is the Euler load for major axis buckling

and the maximum moment is:

Mmax = (6)

In both equations the secant term may be replaced by noting that the first order deflection (due to the end moments M acting alone) and the first order moment M - determined by ordinary beam theory - are approximately amplified by the term:



1/(1 - N/PEy) (7)

as shown in Figure 5.

Thus:

vmax = (ML2/8EIy) {1/[1 - N/PEy]} (8)

Mmax = (9)

Since the maximum elastic stress will be:

σmax = σc + σb M/Mmax (10)

Equation (10) may be rewritten as:

= 1,0 (11)



Equation (11) may be solved for values of σc and σb that just cause yield, taking different values of PEy (which is dependent on slenderness L/ry). This gives a series of curves, as shown in Figure 6, which indicate that as σb → 0, σc tends to the value of material strength fy. Thus Equation (11) does not recognise the possibility of buckling under pure axial load at a stress σEy given by:

σEy = PEy / A (12)

= π2EIy/AL2 = π2Ey/λ2 (13)

Use of both Equation (11) and Equation (12) ensures that both conditions are covered as shown in Figure 7.



4. TREATMENT IN DESIGN CODES Equations (11) and (13) are written in terms of stresses and originate from the concept of "failure" being defined as either the attainment of first yield or elastic buckling of the perfect member. Limit state design codes normally take ultimate load as the design criterion when considering resistance under static loading. Thus these equations must be re-written in terms of forces and moments. In doing this it is also necessary to make some allowance for those effects present in real steel structures that have not so far been explicitly allowed for, e.g. initial lack of straightness, residual stresses, etc.. For consistency in design it is, of course, essential that the interaction equation for combined loading reduces down to the column and beam design procedures as moment and axial load respectively reduce to zero.

The approach taken in Eurocode 3 [1] (assuming bending about the y-axis) is to use:

≤ 1 (14)

in which χy is the reduction factor for column buckling



ky is a coefficient

The value of ky depends in a rather complex way on:

the level of axial load as measured by the ratio NSd/χyAfy.

the member slenderness λy. the margin between the cross-section's plastic and elastic section moduli (Wpl and Wel). the pattern of primary moments.

When all of these combine in the most severe way the safe value of ky is 1,5. The role of ky is to allow for the secondary bending effect described earlier plus the effects of non-uniform moment and spread of yield. Figures 8 - 10 show how, depending upon the particular case selected, the form of interaction may vary from concave to convex. In constructing these figures use has been made of the design formulae of Clause 5.5.4(1) of Eurocode 3 [1].



5. EFFECT OF PATTERN OF PRIMARY MOMENTS Figure 4 showed how, for the particular case of equal and opposite end moments, the primary moments are amplified due to the effect of the axial load N acting through the lateral displacements v. When the pattern of primary moment is different the two effects will not be so directly additive since maximum primary and secondary moments will not necessarily occur at the same location. Figure 11 illustrates the situation for end moments M and ψM, where ψ can adopt values between +1 (uniform single curvature) and -1 (double curvature). The particular case shown corresponds to a ψ value ≈ -0,5.



For the case illustrated the maximum moment still occurs within the member length but the situation is clearly less severe than that of Figure 4 assuming all conditions to be identical apart from the value of ψ. It is customary to recognise this in design by reducing the contribution of the moment term to the interaction relationship. Thus in Eurocode 3 "ky" in Equation (14) depends upon the ratio ψ, as already shown in Figure 10. The exact way in which this should be implemented is explained in Clause 5.5.4 and Figure 5.5.3 of Eurocode 3 [1].

Since the case of uniform single curvature moment is the most severe, it follows that a safe simplification is always to use the procedure for ψ = 1.0.

Returning to Figure 11, it is possible for the point of maximum moment to be at the end at which the larger primary moment is applied. This would usually occur if the axial load was small and/or slenderness was low so that secondary bending effects were relatively slight. In such cases design will be controlled by the need to ensure adequate cross-sectional resistance at this end. The formula from Table 1, for the particular shape of cross-section being used, should therefore be employed. In cases where only the uniform moment (ψ = 1,0) arrangement is being considered, the overall bucking check of Equation (14) will always be more severe than (or in the limit equal to) the cross-sectional check.

6. CONCLUDING SUMMARY The main features of the behaviour and design of beam-columns have been presented within the context of members subject to uniaxial bending, whose response is such that deformation takes place only in the



plane of the applied moments. For the cross-section, the interaction of normal force and bending may be treated elastically using the principle of superposition or plastically using equilibrium and the concept of stress blocks. When considering the member as a whole, secondary bending effects must be allowed for. Strut analysis may be used as a basis for examining the role of the main controlling parameters. Design is normally based on the use of an interaction equation, an essential feature of which is the resistance of the component as a beam and as a column.


8. ADDITIONAL READING 1. Chen, W. F. and Atsuta, T., "Theory of Beam-Columns" Vol. 1, McGraw-Hill, 1976.

Comprehensive treatment of the beam-column problem for the in-plane case, with an emphasis on methods of analysis for the determination of the maximum load carrying capacity.

2. Trahair, N. S. and Bradford, M. A., "Behaviour and Design of Steel Structures", 2nd edition, Chapman and Hall, 1988.

Chapter 7 refers to beam-columns, including a comparison of the subject's treatment in three design codes (not including Eurocode 3).

3. Ballio, G. and Mazzolani, F. M., "Theory and Design of Steel Structures", Chapman and Hall, 1983.

Gives basis of original European approaches to the use of interaction formulae, including derivations.

4. Galambos, T. V., "Guide to Stability Deign Criteria for Metal Structures", 4th edition, Wiley Interscience.

Chapter 8 presents a comprehensive review of theoretical, experimental and design-oriented contributions to the topic of beam-column behaviour.

5. Dowling, P. J., Owens, G. W. and Knowles, P., "Structural Steel Design", Butterworths, 1988.

Chapter 24 deals with beam-column behaviour and design, including explanations of the physical significance of the concepts of interaction and slenderness.

6. Nethercot, D. A., "Limit State Design of Structural Steelwork", 2nd edition, Chapman and Hall, 1991.

Chapter 6 deals with beam-column behaviour and design.

Cross-Section

Expression for MN

Rolled I or H MN,y = 1,11 Mpl.y (1-n)

MNz = 1,56Mpl.z (1-n)(n+0,6)



Table 1 Expressions for reduced plastic moment resistance MN


Square hollow section

MN = 1,26Mpl(1-n)

Rectangular hollow section

MNy = 1,33Mpl.y(1-n)

MNz = Mpl.z(1-n)/(0,5 + ht/A)

Circular hollow section

MN = 1,04Mpl(1-n1,7)




ESDEP WG 7

ELEMENTS

Lecture 7.10.2: Beam Columns II OBJECTIVE/SCOPE

To extend the introductory coverage of beam columns given in Lecture 7.10.1 to cover the full three-dimensional case.

PREREQUISITES

Simple bending and torsion theory





Lecture 7.10.1: Beam columns I

RELATED LECTURES




SUMMARY

This lecture expands on the treatment of beam-columns given in Lecture 7.10.1 to include the cases of out-of-plane buckling and biaxial bending. The basis for the Eurocode 3 interaction formulae is discussed and related to physical behaviour [1].

1. INTRODUCTION Lecture 7.10.1 introduced all the main aspects of beam-column behaviour and design within the context of the uniaxial in-plane case. More general forms of response are, however, possible. This lecture broadens the coverage to include all of the main cases.

2. FORMS OF BEHAVIOUR Three separate forms of beam-column behaviour are illustrated in Figure 1.

Page 1 of 12L071002


If the member is bent about its weaker principal axis, or is prevented from deflecting laterally when bent about its stronger principal axis as shown in Figure 1a, then its response will be confined to the plane of bending. This case has been covered in Lecture 7.10.1.

When a laterally unbraced beam-column of open cross-section is bent about its stronger principal axis as shown in Figure 1b, then it may buckle prematurely out of the plane of loading by deflecting laterally and twisting. Such behaviour is conceptually and mathematically very similar to the lateral-torsional buckling of beams described in Lectures 7.9.1 and 7.9.2.

Page 2 of 12L071002


The most general situation is illustrated in Figure 1c. When bending is applied about both principal axes the member's response will be 3-dimensional in nature, involving biaxial bending and twisting.

Page 3 of 12L071002


In Figure 1 the nature of the interaction in each case is listed in the caption. Clearly the behaviour shown as Figure 1c is the most general, with that of Figures 1a and 1b being simpler and more limited cases. For a full treatment of the in-plane case of Figure 1a refer back to Lecture 7.10.1.

3. FLEXURAL-TORSIONAL BUCKLING When a laterally unrestrained I-section beam-column is bent about its major axis, it may buckle by deflecting laterally and twisting at a load which is significantly less than the maximum load predicted by an in-plane analysis. Assuming elastic behaviour and the arrangement of applied loading and support conditions given in Figure 2, the critical combinations of N and M may be obtained from the solution of:

= (1)

in which io = [(Iy + Iz)/A]1/2 is the polar radius of gyration

Page 4 of 12L071002


Nz = π2 EIz/L2 is the minor axis critical load

No = (GJ/io2) (1 + π2 EIw/GItL

2) is the torsional buckling load

Equation (1) reduces to the buckling of a beam when N → 0 and to the buckling of a column in either flexure (Nz) or torsion (No) as M → 0. In the first case the critical value of M will be given by:

Mcr = (2)

in which EIz is the minor axis flexural rigidity

GIt is the torsional rigidity

Page 5 of 12L071002


EIw is the warping rigidity

In deriving Equation (1) no allowance was made for the amplification of the in-plane moments M by the axial load acting through the in-plane deflections. As explained in Lecture 7.10.1 this may be approximated as M/(1-N/Ny). Equation (1) can, therefore, be modified to:

(3)

Noting the relative magnitudes of Ny, Nz and No and re-arranging gives the following approximation:

N/Nz + {1/(1-N/Ny)}{M/i(NzNo)1/2} = 1 (4)

or

N/ Nz + {1/(1-N/ Ny)}M/Mcr = 1 (5)

4. DESIGN For design purposes it is necessary to make suitable allowances for effects such as initial lack of straightness, partial yielding, residual stresses, etc., as has been fully discussed in earlier lectures in the context of columns and beams. Thus some modification to Equation (5) is necessary to make it suitable for design. In particular, the end points (corresponding to the cases of M = 0 and N = 0) must conform to the established procedures for columns (Lectures 7.5.1 and 7.5.2) and beams (Lectures 7.9.1 and 7.9.2).

Eurocode 3 [1] uses the interaction equation:

≤ 1 (6)

In which kLT is a coefficient whose value depends upon:

the level of axial load as measured by the ratio Nsd / χz A fy.

the member slenderness z. the pattern of primary moments.

and χLT is the reduction factor for lateral-torsional beam buckling.

For the most severe combination kLT adopts the value unity, corresponding to a linear combination of the compressive and bending terms. This reflects the reduced scope for amplification effects in this case, since the value of Nsd cannot exceed χz A fy, which will, in turn, be significantly less than the elastic critical load for in-plane bucking Ny.

It is, of course, also necessary to ensure against the possibility of in-plane failure by excessive deflection in the plane of the web at a lower load than that given by Equation (6). This might occur, for example, in situations where different bracing and/or support conditions are provided in the xy and xz planes as illustrated in Figure 3.

Page 6 of 12L071002


Such cases should be treated by checking, in addition to Equation (6), an in-plane equation of the form:

≤ 1 (7)

in which χmin depends on the in-plane conditions. Usually, however, Equation (6) will govern.

5. BIAXIAL BENDING Analysis for the full 3-dimensional case, even for the simple elastic version, is extremely complex and closed-form solutions are not available. Rather than starting analytically it is more convenient to approach the question of a suitable design approach from considerations of behaviour and the use of the methods already derived for the simpler cases of Figures 1a and 1b.

Figure 4 presents a diagrammatic version of the design requirement. The N-Mz and N-My axes correspond to the two uniaxial cases already examined. Interaction between the two moments Mz and My corresponds to the horizontal plane. When all three load components N, Mz and My are present the resulting interaction plots somewhere in the 3-dimensional space represented by the diagram. Any point falling within the boundary

Page 7 of 12L071002


corresponds to a safe combination of loads.

Assuming proportional loading any load combination may be regarded as a straight line starting at the origin, the orientation of which depends upon the relative sizes of the three load components. Increasing the loads extends this line from the origin until it just reaches and then exceeds the boundary. Non-proportional loading would correspond to a series of lines.

In each case the axes have been taken as the ratio of the applied component to the member's resistance under the load component alone, e.g. Nsd / χmin Afy in the case of the compressive loading. Thus Figure 4 actually represents the situation for one particular example with particular values of cross-sectional properties, slenderness and load arrangement. Changing some or all of these will alter the shape of the interaction surface shown, but not the general principle involved.

6. DESIGN FOR BIAXIAL BENDING AND COMPRESSION For design purposes, it is necessary to have a convenient representation of the situation described in Section 5 by using an interaction equation containing the three load components N, Mz and My. Parts of this equation, corresponding to the two 2-dimensional cases represented by the N, Mz and N, My planes, have already been discussed. The full equation must clearly reduce to these in the absence of the third load component.

Eurocode 3 [1] uses the pair of formulae:

Page 8 of 12L071002


≤ 1 (8)

≤ 1 (9)

Two checks are necessary because, under the action of compression plus major axis moment on an I-section with different support conditions in the zx and yx planes, it is not known whether the in-plane or out-of-plane interaction will be the more critical; that is to say whether, in the absence of Mz, failure would occur as shown in Figure 1a or Figure 1b. For the same conditions in both planes and z > y, χmin will correspond to χz and Equation (9) will govern since χLT, the reduction factor for lateral-torsional buckling under pure bending, will be less than or (if LT is small) equal to unity.

For cross-sections not susceptible to lateral-torsional buckling, e.g. tubes, only Equation (8) is required since χLT = 1.

7. TREATMENT OF OTHER THAN CLASS 1 OR 2 SECTIONS The design formulae given as Equations (6) - (9) relate specifically to the case of Class 1 or 2 sections, i.e. those for which the proportions of the plate elements meet the limitations necessary to ensure the development of full cross-sectional plasticity, as explained in Lecture 7.2. When using either Class 3 or Class 4 sections some modifications are necessary.

For Class 3 cross-sections the quantities Wpl.y and Wpl.z should be replaced by the equivalent elastic quantities Wel.y and Wel.z.

When Class 4 sections are being employed the section properties A and W must relate to the effective cross-section; the shift of the neutral axis of the effective cross-section from its original position due to loss of effectiveness of some parts of the cross-section must also be allowed for. Thus Equations (8) and (9) become:

≤ 1 (10)

≤ 1 (11)

in which Aeff, Weff.y and Weff.z are the effective properties in the presence of only uniform compression or moment about the y and z axes respectively and eN is the shift of the neutral axis when the cross-section is subject to uniform compression.

An important point to note from the definition of Aeff and Weff above is that the calculation of cross-sectional properties, and thus also cross-sectional classification, should be undertaken on a separate basis for each of the three load components N, My and Mz. This does, of course, mean that the same member may be classified as (say) Class 1 for major-axis bending, Class 2 for minor-axis bending and Class 3 for compression. In such cases the safe design approach is to conduct all beam-column checks using the procedures for the least favourable class.

8. DETERMINATION OF k-FACTORS The value of kLT for use in Equation (6) is actually given by:

Page 9 of 12L071002


(but kLT ≤ 1) (12)

in which µLT = 0,15 z βM.LT - 0,15 (but µLT ≤ 0,90) (13)

and βM.LT is the equivalent uniform moment factor for lateral-torsional buckling determined from Table 2.

In Equations (7) - (11) the values of ky and kz should be obtained from:

k = but k ≤ 1,15 (14)

µ = (2βM- 4) + (Wpl - Wel)/Wel but µ ≤ 0,9 (15)

in which µ, χ, , βM, Wpl and Wel all relate to the axis under consideration, i.e. y or z, and βM is determined from Table 2.

For Class 3 or 4 cross-sections the second term in Equation (15) should be omitted.

9. CROSS-SECTION CHECKS

If allowance has been made when determining the k-factors (through the use of βM) for the less severe effect of patterns of moment other than uniform single curvature bending, it is necessary further to check that the cross-section is everywhere capable of locally resisting the combination of compression and primary moment(s) present at any point. This location will usually be one or other of the ends as explained in Lecture 7.10.1.

Expressions for checking several types of cross-section under compression plus uniaxial bending were given in Lecture 7.10.1. For biaxial bending Eurocode 3 [1] uses:

≤ 1 (16)

in which the values of α and β depend upon the type of cross-section as indicated in Table 1.

A simpler but conservative alternative is:

≤ 1 (17)

10. CONCLUDING SUMMARY Depending on the form of the applied loading, 3 types of beam-column problem may be identified. The biaxial bending case is the most general and includes the 2 others as simpler and more restricted component cases. Interaction equations are used for design purposes. These make use (as end points) of the design procedures for beams (N = 0) and columns (M = 0). The class of cross-section will affect some of the values used in the interaction equations.

11. REFERENCES

Page 10 of 12L071002


[1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992.

12. ADDITIONAL READING 1. Chen, W. F. and Atsuta, T., "Theory of Beam-Columns" Vol. 2, McGraw-Hill, 1977.

Comprehensive treatment of the beam-column problem for the cases of flexural-torsional bucking and biaxial bending.

2. Trahair, N. S. and Bradford, M. A., "Behaviour and Design of Steel Structures", 2nd edition, Chapman and Hall, 1988.

Chapter 7 refers to beam-columns, including a comparison of the subject's treatment in three design codes (not including Eurocode 3).

3. Ballio, G. and Mazzolani, F. M., "Theory and Design of Steel Structures", Chapman and Hall, 1983.

Gives basis of original European approaches to the use of interaction formulae, including derivations.

4. Galambos, T. V., "Guide to Stability Design Criteria for Metal Structures", 4th edition, Wiley Interscience.

Chapter 8 presents a comprehensive review of theoretical, experimental and design-oriented contributions to the topic of beam-column behaviour.

5. Dowling, P. J., Owens, G. W. and Knowles, P., "Structural Steel Design", Butterworths, 1988.

Chapter 24 deals with beam-column behaviour and design, including explanations of the physical significance of the concepts of interaction and slenderness.

6. Nethercot, D. A., "Limit State Design of Structural Steelwork", 2nd edition, Chapman and Hall, 1991.

Chapter 6 deals with beam-column behaviour and design.

Table 1 Values of α and β for use in Equation (16)

Type of cross-section α β

I and H - sections

Circular tubes

Rectangular hollow sections

Solid rectangles and plates

2

2

but ≤ 6

1,73 + 1,8n3

5n but ≥ 1

2

but ≤ 6

1,73 + 1,8n3

n = Nsd / Npl.Rd

Page 11 of 12L071002


Table 2 Equivalent uniform moment factors βM


Moment diagram Equivalent uniform moment factor β M

End moments

βM,ψ = 1,8 - 0,7 ψ

Moments due to in-plane lateral loads

βM,Q = 1,3

βM,Q = 1,4

Moments due to in-plane lateral loads plus end

moments

βM = βm, ψ + (βM,Q - βM, ψ )

MQ = ⏐ Max M⏐ due to lateral load only

∆M = ⏐ Max M⏐ for moment diagram without change of sign

∆M = ⏐ Max M⏐ + ⏐ Min M⏐ where sign of moment diagram changes

Page 12 of 12L071002



ESDEP WG 7

ELEMENTS

Lecture 7.10.3: Beam Columns III OBJECTIVE/SCOPE

To describe the methods of beam-column analysis, either by verification of a single member or by verification of the whole frame.

PREREQUISITES



RELATED LECTURES

Lecture 6.2: General Criteria for Elastic Stability




SUMMARY

This lecture explains the basic methods of beam - column verification; this is done firstly for an isolated member, taking the end deformations and end forces from a global analysis of the whole frame and calculating the resistance using the rules given in Eurocode 3 [1]. The case where the member cannot be isolated from the frame structure is also covered by giving an alternative procedure, which yields an overall slenderness, , of the frame (including lateral-torsional buckling effects) that allows for frame verification using the European buckling curves.

1. INTRODUCTION Normally, the design of an individual member in a frame is done by separating it from the frame and dealing with it as an isolated substructure. The end conditions of the member should then comply with its deformation conditions, in the spatial frame, in a conservative way, e.g. by assuming a nominally pinned end condition, and the internal action effects, at the ends of the members, should be considered by applying equivalent external end moments and end forces, Figure 1. Methods of verification for these members are given in Section 2.

Page 1 of 11L071003


A more general procedure is given in Section 3, for the case where members cannot be isolated from the frame structure in the way described above.

2. METHODS OF VERIFICATION FOR ISOLATED MEMBERS

2.1 Beam-columns with Mono-axial Bending only

For the design of beam-columns, with mono-axial bending only, two checks must be carried out:

Page 2 of 11L071003


the in-plane buckling check taking into account the in-plane imperfections. the out-of-plane buckling check, including the lateral-torsional buckling verification that takes account of the out-of-plane imperfections (see Figure 2).

It has been found by test calculations that twist imperfections, ρ, of beam-columns that are susceptible to lateral-torsional buckling, can be substituted by flexural imperfections, see Figure 3.

Page 3 of 11L071003


Members with sufficient torsional stiffness, e.g. hollow section members, need not be verified for lateral-torsional bucking.

The in-plane or out-of-plane flexural buckling check is done using the interaction formula:

where χmin is the lesser of χy and χz and the factor ky is given by:

ky = 1 - (µyNSd)/(χy Afy) ≤ 1,5

where:

µy = y{2βMy - ψ + (Wply - Wely)/Wely} ≤ 0.90

βMy takes account of the shape of the moment distribution diagram between the member ends. Values for this are given in Table 1.

In the case where lateral-torsional buckling is possible, the verification must be carried out using the following formula:

where χz refers to the direction of the lateral-torsional buckling.

χLT is the relevant reduction factor for lateral-torsional buckling

kLT =

Page 4 of 11L071003


where:

µLT = 0,15 z βM - 0,15 ≤ 0,90

and βM is given by Table 1.

When the non-dimensional slenderness LT ≤ 0,4, the reduction coefficient χLT need not be taken into account. This rule may be used for spacing the lateral restraints to resist lateral-torsional buckling.

The above verification formulae are valid for members with Class 1 and Class 2 sections; in the case of Class 3 sections, Wpl must be substituted by Wel; in the case of Class 4 sections, reference should be made to Lecture 7.3.

2.2 Beam-columns with Bi-axial Bending

In the case of beam-columns with bi-axial bending, the formulae given in Section 2.1 have to be expanded.

The interaction without lateral-torsional buckling becomes:

The interaction with lateral-torsional buckling reads:

The definition of kz and Wplz are analogous to those for ky and Wply, given in the previous Section.

Figure 4 illustrates these interactive formulae.

Page 5 of 11L071003


3. METHOD OF VERIFICATION OF WHOLE FRAMES

3.1 General

Figure 5 gives an example of a portal frame with tapered columns and beams, the external flanges of which are laterally supported by the purlins which, due to their flexural stiffness, also provide torsional restraint; the beams and columns may, however, be subject to distortion of the cross-section, due to the flexibility of the web.

Page 6 of 11L071003


An accurate verification of this arrangement should be based on a finite element model which takes the above effects into account. The basic assumptions made regarding the imperfections in this model, would be such that the standard verification given in Section 2 would produce equally favourable results since the standard procedure has been calibrated against test results.

A more simplified procedure is, therefore, given here which is related to the verification of columns for flexural buckling, and beams for lateral-torsional buckling.

3.2 Basic Assumption

The basic principles governing the standard verification of columns for flexural buckling, and beams for lateral-torsional bucking, are as follows:

1. The non-dimensional slenderness is defined by:

FB = √{Npl/Ncr); LT = √{Μpl/Mcr)

where Npl, Mpl are the characteristic values of the elastic/plastic resistances of the column or beam neglecting any out-of-plane effects; and Ncrit, Mcrit are the critical bifurcation values for the column resistance, or the beam resistance, when considering out-of-plane deflections and hyperelastic behaviour in the equilibrium state.

2. Using the non-dimensional slenderness, , a reduction factor χ can be determined from the European buckling curves that allows the design value of the resistance of the column or beam to be defined by:

Nbd = χ Npl / γM1 for the column

Mbd = χ Mpl / γM1 for the beam.

Page 7 of 11L071003


In applying this principle to any loaded structure, see Figure 6, the procedure is as follows:

1. As a first step the structure is analysed for a given load case with an elastic or plastic analysis assuming that any out of plane deflections are prevented. By this analysis a multiplier, γpl, of the given loads is found that represents the ultimate resistance of the structure.

2. The structure is then checked assuming hyperelastic material behaviour allowing for lateral and torsional deflections. This leads to a multiplier γcrit of the given loads that represents the critical elastic resistance of the structure to lateral buckling or lateral-torsional bucking.

3. The overall slenderness, , of the structure can then be defined by:

and by using the reduction coefficient χ from the relevant European buckling curve, e.g. curve c, the final safety factor:

γ = χ γpl

can be derived.

This procedure is analogous to the Merchant-Rankine procedure for the non- elastic verification of frames.

Page 8 of 11L071003


3.3 Tools for the Procedure

In general the procedure described in Section 3.2 needs a computer program that performs a planar elastic-plastic analysis of the frame and determines the elastic bifurcation load of the structure for lateral and torsional deflections, including distortion. Such a program, for calculating the elastic bifurcation loads, can either be based on finite elements or on a grid model where the flanges and stiffeners are considered as beams and the web is represented by an equivalent lattice system that allows for second order effects; such programs are available on PC's.

4. CONCLUDING SUMMARY A three dimensional frame may generally be analysed by separating it into plane frames and analysing these on the assumption of no imperfections; the individual members of the frame should then be checked with the imperfection effects taken into account. The isolated members in general represent beam-columns with either inplane or biaxial bending. Beam - columns with mono-axial bending, which are not susceptible to lateral-torsional buckling, e.g. hollow sections, shall be verified for inplane and out of plane buckling by means of an interaction formulae. In the case of beam-columns which are susceptible to lateral-torsional buckling the out of plane flexural buckling of the column has to be combined with the lateral-torsional buckling of the beam using the

Page 9 of 11L071003


relevant interaction formulae. For beam-columns with biaxial bending, the interaction formula is expanded by the addition of an analogous term. In certain cases the standard procedure for the verification of a beam- column is not applicable and more accurate models must be used. As a non-linear spatial analysis including the effect of imperfections is difficult, an alternative procedure is provided by which the overall slenderness of a frame is defined; this allows verification of the frame using the European buckling curves which take account of lateral- torsional buckling. This procedure is analogous to the Merchant-Rankine procedure.

5. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992.

6. ADDITIONAL READING 1. Eurocode 3 - Background Document 5.05 "Design rules for thin-walled plate girders for the ultimate and

serviceability limit state taking account of the buckling phenomena". Part A: Results Eurocode 3 Editorial Group CEC 1991.

2. Braham M. et al: Ein alternatives Verfabren zur Bestimmung der Biegedrillknicksicherheit von Konstruktionen. Der Stahlbau 1992.

Moment diagram Equivalent uniform moment factor β M

End moments

β M,ψ = 1,8 - 0,7 ψ

Moments due to in-plane lateral loads

β M,Q = 1,3

β M,Q = 1,4

Moments due to in-plane lateral loads plus end

moments

β M = β m, ψ + (β M,Q - β M, ψ )

MQ = ⏐ Max M⏐ due to lateral load only

∆ M = ⏐ Max M⏐ for moment diagram without change of sign

Page 10 of 11L071003


Table 1 Equivalent uniform moment factors βM


∆ M = ⏐ Max M⏐ + ⏐ Min M⏐ where sign of moment diagram changes

Page 11 of 11L071003



ESDEP WG 7

ELEMENTS

Lecture 7.11: Frames OBJECTIVE/SCOPE

To introduce the basic ideas of frame behaviour as a prelude to the more detailed description of design in later lectures.

PREREQUISITES

Lectures 1B.7: Introduction to Design of Multi-Storey Buildings

Lectures 6.6: Buckling of Real Structural Elements


RELATED LECTURES

Lecture 14.2: Analysis of Portal Frames: Introduction and Elastic Analysis

Lecture 14.3: Analysis of Portal Frames: Plastic Analysis

Lecture 14.8: Classification of Multi-Storey Frames

Lecture 14.9: Methods of Analysis for Multi-Storey Frames

Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings

Lecture 14.13: Design of Multi-Storey Frames with Partial Strength and Semi-Rigid Connections

Lecture 14.14: Methods of Analysis of Rigid Jointed Frames

SUMMARY

Techniques for the determination of the individual member forces in steel frames are described and discussed. These techniques include first and second order approaches based on both elastic and plastic theory. The different approaches to the design and construction of steel frames permitted by Eurocode 3 [1] are explained and their implementation outlined.

1. INTRODUCTION Frames of varying size and complexity represent one of the most frequent uses of structural steel. Whilst the most obvious application is in buildings, support frames for bridges, offshore platforms, falsework and industrial storage systems also constitute a significant usage.

The main components of a rectangular arrangement are identified in Figure 1. Vertical loads on the roof and floors are transmitted by bending and shear into the columns, which, in turn, transfer load into the foundations by means of compressive, bending and shearing actions. Horizontal loading, e.g. due to wind, must also be transferred into the foundations and may, depending upon the frame geometry and the relative magnitudes of the vertical and lateral loads, induce tension in some columns and therefore uplift on the foundations.



In general, a 3-dimensional building frame may be separated into a set of planar frames with well-defined support or restraint conditions for out-of-plane deformations, see Figure 1. These planer frames should be investigated for two different limit state conditions:

Ultimate limit state. Serviceability limit state.

Details of these investigations are provided in later lectures. In this introductory lecture attention is focused on describing the main aspects of frame behaviour and the ways in which these aspects are linked to various techniques for predicting structural response. The lecture is a preparation for later more detailed treatments.

2. FRAMING SYSTEMS



For the purposes of analysis and design, steel frames have traditionally been regarded as belonging to one of two categories:

pin-jointed (simple construction). rigid-jointed (continuous construction).

Of course, examples exist for which elements of both types are present; the present discussion does not consider these cases in developing the subject.

In Eurocode 3 this simple subdivision has been extended by including consideration of the method of analysis and including joints that behave as semi-rigid. This leads to the more complex classification system that is shown in Table 1.

Type of framing Method of global analysis

Types of connections

Simple Pin joints ⋅ Nominally pinned

⋅ Nominally pinned

Continuous Elastic ⋅ Rigid


Rigid-Plastic ⋅ Full-strength


Elastic-Plastic ⋅ Full-strength - Rigid


Semi-continuous Elastic ⋅ Semi-rigid

⋅ Rigid


Rigid-Plastic ⋅ Partial-strength

⋅ Full-strength


Elastic-Plastic ⋅ Partial-strength - Semi-rigid

⋅ Partial-strength - Rigid

⋅ Full-strength - Semi-rigid

⋅ Full-strength - Rigid




Table 1 Methods of framing and global analysis in Eurocode 3

This introductory lecture does not cover the cases where the joints are semi-rigid. Further information on this type of joint and their influence on frame design is given in Lectures 11.7 and 14.13.

3. SIMPLE CONSTRUCTION Building frames designed and executed according to this principle require comparatively little analysis as the loads may be allocated to individual members on the basis of simple statics. Since the joints are assumed to be incapable of transmitting moments, lateral stability requires the use of bracing because a rectangular bay with pinned beam to column connections, of course, possess no lateral stiffness. The only exception to this condition is when the feet of the columns are rigidly fixed to a solid foundation so that they can function as vertical cantilevers. Figure 2 illustrates these points.



Once the forces in the members have been decided upon, design may be conducted by considering individual beams, columns and joints, using the procedures presented in Lectures 7 and 11 respectively. All horizontal loading is assumed to be resisted by the bracing system and, in the terminology of Eurocode 3 [1], the structure is designed as a braced frame.

4. CONTINUOUS CONSTRUCTION When rigid joints are used, considerable interaction between beams and columns occurs due to transfer of moments around the frame. Various approaches to analysis and design may be employed; the most important consideration is the extent to which the effects of deflections on the response of the frame must be taken into account. Although rigid joints may be used in conjunction with bracing, construction economies make this arrangement an uncommon solution. Rigid joints are normally only used as an alternative to bracing so that lateral stiffness is provided by so-called frame action as illustrated in Figure 3.

5. METHODS OF ANALYSIS

5.1 First-Order Elastic Analysis

In first-order elastic analysis a linear relationship between the applied loading F and the deformations (δ) is assumed. The internal force distribution in the frame is assumed to be unaffected by the displacements in the frame.

Frame analysis can therefore be conducted according to linear elastic principles as outlined in Lecture 7.1. The frame responds according to line 1 in Figure 4.



5.2 First-Order Rigid-Plastic Analysis

Rigid-plastic analysis (or the application of simple plastic theory) neglects the effects of elastic deflections and assumes that all structural deformation takes place in discrete regions, called plastic hinges, where plasticity has developed. When using first-order, rigid-plastic theory only the collapse condition is addressed. This condition occurs when sufficient plastic hinges are assumed to have formed to convert the structure into a mechanism. Thus the path by which this stage is reached, i.e. the sequence of formation of the hinges and any intermediate distributions of internal forces, are not considered. Figure 5 illustrates the concept for 3 cases of vertical, horizontal and combined loading, whilst curve 2 in Figure 4 gives the frame response according to this approach. Due to the form of the analysis, no information is provided on the magnitude of the deflections. The analysis gives only that all stiffness is lost at the collapse load and deflections therefore (in theory) become uncontrolled.



5.3 Elastic Critical Load

Using the methods described in Lectures 6, it is possible to calculate the buckling loads for frames under suitably idealised loading. Depending upon the content and complexity of the frame, several different buckling modes, each with its associated elastic vertical load, may be possible. For the simple portal frame shown in Figure 6, both a symmetrical and an antisymmetrical mode are possible; the latter will usually be avoided with a much lower critical load. Once again the analysis provides no information on the magnitude of the deflections; it simply identifies a particular load level. The curve 3 in Figure 4 gives the representation of the critical load obtained by an elastic buckling analysis.



5.4 Second-Order Elastic Analysis

In second-order elastic analysis the effect of elastic deformations on the internal force distribution is taken into account. The result is a transition from the linear analysis line 1 at low loads to the elastic critical line 3 at large deflections. For frames the second-order effects may be separated into 2 parts:

reduction in the effective bending stiffnesses of individual members due to compressive loading. a destabilising effect due to the overturning moment produced by the vertical loads acting through the horizontal deflections caused by the lateral loads.

5.5 Second-Order Rigid-Plastic Analysis



If the deformations that may develop as a result of the formation of the plastic collapse mechanism are allowed for when formulating the equilibrium of the frame, then the result is the developing mechanism curve of line 5 in Figure 4. This curve shows that equilibrium can only be maintained with a reduction in the level of the applied loads.

5.6 First-Order, Elastic-Plastic Theory

If a linear elastic analysis is modified to allow for reductions in frame stiffness with the progressive formation of plastic hinges at increasing levels of the applied load, then the response curve of line 6 is obtained. This line exhibits progressive loss of stiffness as each plastic hinge is formed and eventually merges with the rigid-plastic line 2.

5.7 Second-Order, Elastic-Plastic Analysis

When the analysis that traces the formation of plastic hinges also allows for the effects of deformations in setting up the governing equations, then line 6 is modified somewhat into line 7. Line 7 initially follows the first-order elastic line 1 but diverges from this line to follow the second-order elastic line 4 as destabilising effects become more significant. Formation of the first plastic hinge - which occurs at a slightly lower applied load than is the case with the first-order, elasto-plastic analysis due to the larger deformation associated with second-order analysis - further reduces the stiffness, causing line 7 to diverge from line 4. This divergence becomes more pronounced as more plastic hinges form. The peak of this curve corresponds to the failure load predicted by this type of analysis. At large deformations line 7 tends to merge with the curve for the mechanism, line 5.

5.8 Second-Order, Plastic Zone Analysis

If the spread of plasticity both through the cross-section and along the member length is taken into account, instead of assuming that it is concentrated into the desirable regions of the plastic hinges, then the resulting type of analysis is usually termed plastic zone theory. It provides an even closer representation of actual behaviour and leads to a curve similar to line 7.

6. COMMENTS In principle, any of the above approaches to frame analysis may be adopted. In practice, some of the effects may be found to be of little real significance for certain classes of structure, e.g. for many low-rise frames second-order effects are very small and may reasonably be neglected. Certain cases may also arise where particular forms of response should be avoided, e.g. for buildings containing heavy cranes which will cause repeated loading, elastic design is normally employed.

The more complex approaches will almost certainly require the use of suitable computer software to implement the volume of calculation. It is therefore important to select an approach which is compatible with both the accuracy required and the level of importance of the project under consideration.

When calculating deflections at working load levels for the purpose of checking serviceability, it is usual to employ only linear elastic analysis.

7. FRAME CLASSIFICATION In order to provide guidance on the most appropriate type of analysis to use in particular cases, Eurocode 3 has introduced the idea of frame classification [1]. A double condition is used:

braced or unbraced. non-sway or sway.

7.1 Braced Frames

A frame may be classified as braced if its sway resistance is supplied by a bracing system with a response



to in-plane horizontal loads which is sufficiently stiff for it to be acceptably accurate to assume that all horizontal loads are resisted by the bracing system.

This may be further quantified as:

A steel frame may be classified as braced if the bracing system reduces its horizontal displacements by at least 80%.

For such frames first-order elastic or plastic theory should be used.

When the above conditions are not satisfied the frame must be considered as unbraced.

When designing the bracing system:

The effects of the initial sway imperfections, given in Clause 5.2.4.3 of Eurocode 3 [1] and discussed later in these notes, in the braced frame shall be taken into account in the design of the bracing system. The initial sway imperfections plus any horizontal loads applied to a braced frame, may be treated as affecting only the bracing system. The bracing system should be designed to resist:

- any horizontal loads applied to the frames which it braces.

- any horizontal or vertical loads applied directly to the bracing system.

- the effects of the initial sway imperfections (or the equivalent horizontal forces) from the bracing system itself and from all the frames which it braces.

Where the bracing system is a frame or sub-frame, it may itself be either sway or non-sway.

7.2 Unbraced Frames

An unbraced frame may be classified as a non-sway frame according to Clause 5.2.5.2 of Eurocode 3 [1] providing:

Its response to in-plane horizontal forces is sufficiently stiff for it to be acceptably accurate to neglect any additional internal forces or moments arising from horizontal displacements of its nodes.

This classification may be further quantified as:

A frame may be classified as non-sway for a given load case if the elastic critical load ratio VSd /Vcr for that load case satisfies the criterion:

VSd /Vcr ≤ 0,1

where VSd is the design value of the total vertical load.

Vcr is its elastic critical value for failure in a sway mode.

Beam-and-column type plane frames in building structures with beams connecting each column at each storey level may be classified as non-sway for a given load case if the following criterion is satisfied. When first-order theory is used, the horizontal displacements in each storey due to the design loads (both horizontal and vertical), plus the initial sway imperfection applied in the form of equivalent horizontal forces, should satisfy the criterion:



where δ is the horizontal displacement at the top of the storey, relative to the bottom of the storey.

h is the storey height.

H is the total horizontal reaction at the bottom of the storey.

V is the total vertical reaction at the bottom of the storey.

Both requirements follow from the idea that, if satisfied, the load-carrying resistance determined by neglecting sway effects will be only a ten per cent less than that calculated by including such effects. This approach is, in turn, based upon the Merchant-Rankine concept for estimating the true ultimate load of a frame that fails by some from of inelastic instability from a knowledge of its elastic critical load and its first-order, rigid-plastic collapse load. Both loads are relatively straightforward to calculate.

The original Merchant-Rankine formulae for the failure load Vsd is:

where:

Vcr is the elastic critical load.

Vpl is the first-order, rigid-plastic collapse load.

From this it is clear that when Vcr >> Vpl, then Vsd ~ Vpl.

Non-sway frames should be designed using first-order elastic or plastic theory to resist safely the arrangements of loads that lead to the most severe combinations of internal forces and moments in the individual members and connections. The effects of restraint to columns in improving their stability should be taken into account by using the concept of effective buckling length as explained in Lecture 7.7.

Frames that do not meet the above requirements must be designed as sway frames.

7.3 Sway Frames

Sway frames shall be analysed under those arrangements of the variable loads which are critical for failure in a sway mode. In addition, sway frames shall also be analysed for the non-sway mode.

The initial sway imperfections, and member imperfections where necessary, shall be included in the global analysis of all frames.

The allowance for imperfections in the analysis of sway frames is intended to cover effects such as lack of verticality, lack of straightness, residual stresses, etc. It is expressed in Eurocode 3 by means of a set of equivalent geometrical imperfections [1]. These imperfections are not actual construction tolerances but, because they are intended to represent the effect of a number of factors, are likely to be larger than such tolerances. The form specified in Eurocode 3 is:

The effects of imperfections shall be allowed for in frame analysis by means of an equivalent geometric imperfection in the form of an initial sway imperfection φ determined from:

φ = kc ks φo

with φo = 1/200



kc = [0,5 + 1/nc]0,5 but kc ≤ 1,0

and ks = [0,2 + 1/ns]0,5 but ks ≤ 1,0

where nc is the number of columns per plane.

ns is the number of storeys.

Columns which carry a vertical load NSd of less than 50% of the mean value of the vertical load per column in the plane considered, shall not be included in nc. Columns which do not extend through all the storeys included in ns shall not be included in nc. Those floor levels and roof levels which are not connected to all the columns included in nc shall not be included when determining ns. These initial sway imperfections apply in all horizontal directions, but need only be considered in one direction at a time. The possible torsional effects on the structure of anti-symmetric sways, on two opposite faces, shall also be considered. If more convenient, the initial sway imperfection may be replaced by a closed system of equivalent horizontal forces, see Figure 7. In beam-and-column building frames, these equivalent horizontal forces should be applied at each floor and roof level and should be proportionate to the vertical loads applied to the structure at that level, see Figure 8. The horizontal reactions at each support should be determined using the initial sway imperfection and not the equivalent horizontal forces. In the absence of actual horizontal loads, the net horizontal reaction is zero.



First-order or second-order analysis may be used. If the analysis is first-order, second-order effects may be allowed for in an appropriate way when designing the columns by using the results of a first-order analysis and either:

- using amplified sway moments, or

- using the sway-mode buckling lengths.

When second-order elastic global analysis is used, in-plane buckling lengths for the non-sway mode may be used for member design. In the amplified sway moments method, the sway moments found by a first-order elastic analysis should be increased by multiplying them by the ratio:

where VSd is the design value of the total vertical load.

Vcr is its elastic critical value for failure in a sway mode.

The amplified sway moments method should not be used when the elastic critical load ratio VSd/Vcr is more than 0,25. Sway moments are those associated with the horizontal translation of the top of a storey relative to the bottom of that storey. They arise from horizontal loading and may also arise from vertical loading if either the structure or the loading is asymmetrical. As an alternative to determining VSd/Vcr directly, the following approximation may be used in beam-and-column type frames:



where δ, h, H and V are as defined previously.

When the amplified sway moments method is used, in-plane buckling lengths for the non-sway mode may be used for member design. When first-order elastic analysis with sway-mode in-plane buckling lengths is used for column design, the sway moments in the beams and the beam-to-column connections should be amplified by at least 1,2 unless a smaller value is shown by analysis to be adequate.

Rules for the application of plastic analysis procedures to sway frames are given in Clause 5.2.6.3 of Eurocode 3 [1].

8. MEMBER CHECK AND FRAME DESIGN Satisfying the verification rules for resistance and stability of frames has to assure that neither the frame as a whole, nor the isolated members in the frame on their own, will collapse under a load which is smaller than the design load. For the safety verification of the individual members, the members may be separated from the frames to be dealt with as independent isolated sub-structures. The end conditions of the members should then comply with the deformation conditions of the members in the special frame in a conservative way (e.g. by assuming nominally pinned end conditions) and the interaction effects at the ends of the members should be considered by applying equivalent end moments and end forces, see Figure 9.



In the safety verification of these separated members, the member imperfections must be taken into account; these imperfections have normally been included when formulating the member design rules as explained in other Lectures 7. In general, the isolated members by their loading and end-conditions represent simply supported beam-columns with or without restraints between their ends, see Figure 10. Beam-columns are members loaded by normal forces and moments about one or two axes.



If, in a second-order elasto-plastic calculation, the real behaviour of the frame has been approximated and local instability and out-of-plane buckling is prevented, then further verification is not needed. In this case the strength check - also called the cross-sectional check - is implicity satisfied by working with the actual distribution of forces and moments. This is also valid for the stability check. It has to be shown that the equilibrium is stable under the design load. In other words: VSd representing the design load, must be less than



the elasto-plastic collapse load Vk.

If the distribution of forces and moments, as a result of the design load, is calculated with a first-order elastic method, then it is quite possible that the actual elasto-plastic resistance Vk of the frame is exceeded. Verification rules to overcome this problem are therefore needed. On the one hand, cross-sectional checks are needed, to show that each cross-section can offer enough resistance to withstand normal forces, shear forces and bending moments due to the design load. On the other hand, stability checks are necessary to show that every member and the frame as a whole are stable.

In general, for each method of calculating the distribution of forces and moments, additional verification rules are necessary related to the specific method of calculating the distribution of forces and moments. All collapse mechanisms which are relevant for the frame and which have not been taken into account in calculating the distribution of forces and moments, should be checked by using adequate verification rules.

If a frame can deform only in its own plane and plate buckling (of web and/or flange), torsion, torsional buckling and lateral-torsional buckling are not relevant, then there are only two types of verification rules of importance: cross-sectional checks and stability checks. Depending on the method of calculating the distribution of forces and moments, specific verification rules must be taken into account. Table 2 shows that these rules depend on the calculation method used.

Table 2 Relation between global analysis and code check

When the method of calculating the distribution of the forces and moments is relatively simple, the verification rules are complex and the other way around. In general verification rules for members are used in the step after the calculation of the distribution of forces and moments in a frame.

9. CONCLUDING SUMMARY Frame behaviour and thus the approach which should be used in design has been shown to be crucially dependent upon the type of joints used. The 2 main forms of construction are:

i. simple construction - assumed joints act as if pinned.

ii. continuous construction - assumed joints act as if rigid.

Simple statics is usually all that is needed to determine the distribution of internal forces in the individual members of frames designed according to the principles of simple construction. 8 different approaches - varying in precision and complexity - to the analysis of rigid jointed frames are possible. Eurocode 3 classifies frames as braced/unbraced and for the latter as either non-sway/sway. The basis for this classification is an assessment of the extent to which deformations influence the response of the frame. Different design approaches are necessary for the three classes:

i. braced.

Method for calculation the forces and moment distribution

Cross-sectional verification rules

Stability checks

First-order elastic

First-order plastic

Second-order elastic

Second-order plastic

YES

NO

YES

NO

YES

YES

NO

NO



ii. unbraced and non-sway.

iii. unbraced and sway.

Each approach has been outlined, including the treatment of imperfections and the link between the approach adopted to consider overall frame behaviour and that necessary when considering individual members.

10. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1, General rules and rules for buildings, CEN, 1992.

11. ADDITIONAL READING 1. Galambos, T.V. "Guide to Stability Design Criteria for Metal Structures", 4th Edition, John Wiley & Son,

1988. 2. Horne, M. R, and Merchant, W., "The Stability of Frames", Pergamen Press, 1965. 3. Chen, W.F. and Lui, E.M. "Stability Design of Steel Frames", CRC Press, 1991. 4. Ballio, G. and Mazzolani, F. M., "Theory and Design of Steel Structures", Chapman and Hall, 1983.

Chapter on stability.

5. Trahair, N. S. and Bradford, M. A., "Behaviour and Design of Steel Structures", Chapman and Hall, 1983.

Chapter on frames.





ESDEP WG 7

ELEMENTS

Lecture 7.12: Trusses and Lattice Girders OBJECTIVE/SCOPE

To introduce two-dimensional trusses: types, uses and principal design considerations.

PREREQUISITES

None.

RELATED LECTURES

None.

SUMMARY

This lecture presents the types and uses of trusses and lattice girders and indicates the members that are most often used in their construction. A discussion of overall truss design considers primary analysis, secondary stresses, rigorous elastic analysis, cross- braced trusses and truss deflections. The practical design of truss members is discussed.

1. INTRODUCTION The truss or lattice girder is a triangulated framework of members where loads in the plane of the truss or girder are resisted by axial forces in the individual members. The terms are generally applied to the planar truss. A 'space frame' is formed when the members lie in three dimensions.

The main uses are:

in buildings, to support roofs and floors, to span large distances and carry relatively light loads, see Figure 1. in road and rail bridges, for short and intermediate spans and in footbridges, as shown in Figure 2. as bracing in buildings and bridges, to provide stability where the bracing members form a truss with other structural members such as the columns in a building. Examples are shown in Figure 3.



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The principle of a truss is simple. The structure is composed of top and bottom chords triangulated with diagonals in the webs so that each member carries purely axial load. Additional effects do exist but in a well designed truss these will be of a secondary nature.

A global moment on a truss is carried as compression and tension in the chords. A global shear is carried as tension or compression in the diagonal members. In the simplified case, where joints are considered as pinned, and the loads are applied at the panel points, the loading creates no bending moment, shear, or torsion in any single member. Loads applied in such a way as to cause bending, shear, or torsion will usually result in



inefficient use of material.

Trusses and lattice girders are classified in accordance with the overall form and internal member arrangement. Pitched trusses are used for roofs. Parallel chord lattice girders are used to support roofs and floors and for bridges, although in continuous bridges, additional depth is often required at the piers. In the past, proper names were given to the various types of trusses such as the Fink truss, Warren girder, etc.. The most commonly used truss is single span, simply supported and statically determinate with joints assumed to act as pins.

The Vierendeel girder should also be mentioned. It consists of rigid jointed rectangular panels as shown in Figure 1d. This truss is statically indeterminate and will not be further considered in this lecture, although it has a pleasing appearance and is often used in foot-bridges.

The saving over a plate girder is clear when the webs are considered. In a truss the webs are mainly fresh air - hence less weight and less wind force.

A truss can be assembled from small easily handled and transported pieces, and the site connections can all be bolted. Trusses can have a particular advantage for bridges in countries where access to the site is difficult or supply of skilled labour is limited.

2. TYPICAL MEMBERS Truss, lattice girder and bracing members for buildings are selected from:

open sections, primarily angles, channels, tees and joists. compound sections, i.e. double angle and channels. closed sections, in practice structural hollow sections.

For bridges, members are selected from:

rolled sections. compound sections. built-up H, top hat and box sections.

Typical sections are shown in Figure 4.



The selection of members depends on the location, use, span, type of connection and the appearance required. Hollow sections are more expensive than open sections but are cheaper to maintain and have a better appearance. However, in exposed trusses corrosion can occur at the crevices which are formed at gusset positions. Angles are the sections traditionally used for small span truss construction.

3. LOADS ON TRUSSES AND LATTICE GIRDERS The main types of loads on buildings are shown in Figure 5, namely:



1. Dead loads. These are caused by self-weight, sheeting, decking, floor or roof slabs, purlins, beams, insulation, ceilings, services and finishes. Dead loads for the construction to be used in any particular case must be carefully estimated from material weights given in handbooks and manufacturers' literature.

2. Imposed loads. These are given in Eurocode 1 [1] for floors in various types of building and for roofs with or without access. The imposed load may cover the whole or part of the member and should be applied in such a way as to cause the most severe effect.

3. Wind loads. These are given in Eurocode 1 [1] and can be estimated from the location of the building, its



dimensions and the sizes of openings on its faces. Wind generally causes uplift on roofs and this can lead to reversal of load in truss members in light construction. In multi-storey buildings, wind gives rise to horizontal loads that must be resisted by the bracing.

In special cases, trusses resist dynamic, seismic and wave loads. A careful watch should be kept for unusual loads applied during erection. Failures may occur at this stage when the final lateral support system is not fully installed.

For bridges, in addition to the dead loads and the vertical effects of live loads due to highway or railway loading, horizontal effects of live load have to be considered. These include braking and traction effects, centrifugal loads and accidental skidding loads. Temperature effects are significant in some bridges.

4. ANALYSIS OF TRUSSES

4.1 General

Trusses may be single span, statically determinate or indeterminate, or may be continuous over two or more spans, as shown in Figure 6. Only single span, statically determinate, trusses are considered in this section.



A truss is usually statically determinate when:

m = 2j - 3,

where m is the number of members in a truss

j is the number of joints.

However compliance with this formula for the truss as a whole does not preclude the possibility of a local mechanism in part of the truss.

Manual methods of analysis for trusses, where the loads are applied at the nodes, are joint resolution, the method of sections and the force diagram. Joint resolution is the quickest method for analysing parallel chord lattice girders when all the forces are required. The method of sections is useful where the values of the forces in only a few critical members are required. The force diagram is the best general manual method. Computer programs are also available for truss analysis.



4.2 Secondary Stresses in Trusses

In many cases in the design of trusses and lattice girders, it is not necessary to consider secondary stresses. These stresses should, however, be calculated for heavy trusses used in industrial buildings and bridges.

Secondary stresses are caused by:

Eccentricity at connections Loads applied between the truss nodes Moments resulting from rigid joints and truss deflection.

They are discussed in detail below:

1. Eccentricity at connections

Trusses should be detailed so that either the centroidal axes of the members or the bolt gauge lines meet at a point at the nodes. Otherwise, members and connections should be designed to resist the moments due to eccentricity. These moments should be divided between members meeting at joints in proportion to their rotational stiffnesses. Stresses due to small eccentricities are often neglected.

2. Loads applied between the truss nodes

Moments due to these loads must be calculated and the stresses arising combined with those due to primary axial loads; that is the members concerned must be designed as beam-columns. This situation often occurs in roof trusses where the loads are applied to the top chord through purlins which may not be located at the nodes, as shown in Figure 7. The manual method of calculation is first to analyse the truss for the loads applied at the nodes which gives the axial forces in the members. Then a separate analysis is made for bending in the top chord which is considered as a continuous beam. The ridge joint E is fixed because of symmetry, but the eaves joint A should be taken as pinned; otherwise, moment will be transferred into the bottom chord if the joint between the truss and column is assumed to be pinned. The top chord is designed for axial load and bending. Computer analysis is mentioned below.



3. Moments resulting from rigid joints and truss deflection

Stresses resulting from secondary moments are important in trusses with short thick members. Approximate rules specify when such an analysis should be made. Secondary stresses will be insignificant if the slenderness of the chord members in the plane of the truss is greater than 50 and that of most of the web members is greater than 100. In trusses in buildings, the loads are predominantly static and it is not necessary to calculate these stresses. The maximum stresses from secondary moments occur at the ends of members and are not likely to cause collapse. However, where fatigue effects are significant, these secondary stresses have to be considered. The method of analysis for secondary moments is set out below.

4.3 Rigorous Elastic Analysis



Rigid jointed, redundant or continuous trusses or trusses with loads applied between the nodes can be analysed using a plane frame program based on the matrix stiffness method of frame analysis. The truss can also be modelled taking account of joint eccentricity. Member sizes must be determined in advance using a manual analysis. All information required for design is output including joint deflections.

It is important that a consistent approach is adopted for analysis and design. This means that if secondary moments are to be ignored then the primary axial forces to be used in design must be obtained from the simple analysis of the truss as a pin jointed frame. The axial forces obtained from a rigid frame computer analysis may be modified considerably by the joint moments.

5. SECONDARY CONSIDERATIONS

5.1 Cross-Braced Trusses in Buildings

In the bracing provided to stabilise multi-storey buildings, the panels often have cross-diagonals as shown in Figure 8a. It is customary to consider the truss as statically determinate, with only the set of diagonals in tension assumed to be effective. When the wind reverses the other set becomes active.



Another common case is the lattice girder with an odd number of panels. The centre panel is cross-braced as shown in Figure 8b. Under symmetrical loading there are no forces in these diagonals. If imposed load is placed over part of the span, only the diagonal in tension is assumed to be effective.

5.2 Lateral Bracing for Bridges

Stringer bracing, braking girders and chord lateral bracing are needed to transmit the longitudinal live loads and



the wind and/or earthquake loads to the bearings and also to prevent the compression chords from buckling.

For the top laterals, a diamond system with kickers at the panel points halves the transverse effective length of the compression chord as shown in Figure 9.

For railway bridges, Figure 9 illustrates an economic lateral system at deck level which consists of a simple single member which doubles up as part of the braking girder. The lateral is supported by the stringers, so the effective length is only about a third of the panel length.

Wind loading on diagonals and verticals can be split equally between top and bottom lateral systems, remembering that the end portals (either diagonals or verticals) have to carry the load applied to the top chord down to the bottom chord.

Obviously where only one lateral system exists (as in semi-through or underslung trusses) then this single system must carry all of the wind load.

In addition to resisting externally applied transverse loads due to wind, etc., lateral bracing stabilizes the compression chord. Its presence is necessary to ensure that reasonably small effective lengths are obtained for the truss members. Lateral bracing is also required at all kinks in the chords where compressive loads are induced into the web members, irrespective of whether the chord is in tension or compression.



5.3 Deflection of Trusses

The deflection for a pin jointed truss can be calculated using either the strain energy or virtual work method. The deflection using the strain energy method is given by:

δ = Σ FuL/EA

where:

A is the area of a truss member

E is the modulus of elasticity

L is the length of a truss member between nodes

F is the force in a member due to the applied loads

u is the force in a member due to unit load applied at the truss node and in the direction of the required deflection.

The Williot-Mohr graphical method can also be used to determine truss deflections. If a computer analysis is carried out, joint deflections are given as part of the output.

A truss may be cambered during fabrication to offset deflections due to applied loads. The term cambering means that a given upward deflection can be built into, say, a nominally horizontal truss during fabrication by adjusting the member lengths slightly to cause the truss to bow upwards.

6. DESIGN OF TRUSS MEMBERS The truss should be analysed for the separate load cases. These cases are combined to give the most severe conditions for design of each element. Some important aspects of design are set out below.

6.1 Compression Members in Buildings

Maximum slenderness ratios are normally defined in codes, and these often limit the minimum size of the members that can be used in light trusses.

Acceptable maximum slenderness values are:

Members resisting dead and imposed load - 180

Members resisting wind load - 250

Any member normally acting as a tie but subject to reversal of stress due to wind - 350

These limits ensure that reasonably robust members are selected when only light loads are involved. Wind loads are transient and larger slenderness values are permitted than for dead and imposed loads. These rules also reduce the likelihood of damage occurring during transport and erection. In this regard it has been common practice to specify that the minimum sizes for angles should be as follows:

equal angles 50 x 50 x 6L unequal angles 65 x 50 x 6L

For the design of members in trusses where secondary bending stresses are insignificant, the following assumptions are made:

For the purpose of analysis, the joints are taken as pinned.



For calculating effective lengths, the fixity of connections and rigidity of adjacent members may be taken into account. Where the exact position of point loads on the rafter relative to the connection of the web members is not known, the local bending moment may be taken as WL/6. In accordance with Clause 5.8.2 of Part 1.1 of Eurocode 3 [2], the buckling length of chord members may be taken as the distance between connections to web members in the plane and the distance between purlins or ties out of plane of the truss.

For web members the buckling length for in-plane buckling may be taken as 0.9L, where L is the length between truss nodes.

Figure 10 shows roof trusses in place in a building with the purlins providing lateral support to the top chord, and a lower chord bracing system providing lateral support to the bottom chord.



Two common internal truss members are the single angle discontinuous strut connected to a gusset or another member and the double angle discontinuous strut connected to both sides of a gusset or another member. These should be connected by at least two bolts or the equivalent in welding. Eurocode 3: Part 1.1: Clause 5.8.3 states that the end eccentricity may be ignored and the struts designed as axially loaded members in accordance with that clause [1].



6.2 Compression Members in Bridges

Generally the truss members in bridges are much larger than in buildings, and much more attention has to be paid to the detailed design of the member. Eurocode 3: Part 1.1 [2] applies to buildings, and the very conservative buckling length values of L and 0,9L are not very significant for relatively small span trusses [1]. However, for bridges, where absolute economy in steel weight is vital, it is assumed that the matter of effective length will be dealt with fully in Eurocode 3: Part 2 [3].

When making up the section for the compression chord, the ideal disposition of material will be one that produces a section with radii of gyration such that the ratio of effective length to radius of gyration is the same in both planes. In other words, the member is just as likely to buckle horizontally as vertically.

The depth of the member needs to be chosen so that plate dimensions are sensible. If they are too thick, the radius of gyration will be smaller than it would be if the same area of steel was used to form a larger member using thinner plates. The plates should be as thin as possible without losing too much material when the effective section is derived.

6.3 Tension Members for Buildings

Structural hollow sections connected by welding may be fully effective. The 'effective area' is to be used for angles connected through one leg. Theoretically rounds or cables could be used; but these are unsuitable for practical reasons, because they lack stiffness and are easily damaged. The same minimum sections for angle members set out above for compression members should be adopted for tension members.

6.4 Tension Members for Bridges

Tension members should be as compact as possible, but the depths will have to be large enough to provide adequate space for bolts at the gusset positions. The width out-of-plane of the truss should be the same as that of the verticals and diagonals so that simple lapping gussets can be provided without the need for packing.

Allowance has to be made for the nett section when bolt holes are removed. It should be possible to achieve a nett section about 85% of the gross section by careful arrangement of the bolts.

6.5 Members Subject to Reversal of Load

For buildings, Eurocode 3: [2] only requires fatigue assessment for:

a. Members supporting lifting appliances or rolling loads.

b. Members subject to repeated stress cycles from vibrating machinery.

c. Members subject to wind-induced or crowd-induced oscillations.

Even in these cases, assessment is not required if the stress range or number of stress cycles is low.

Otherwise, members subject to reversal of load should be designed for the worst condition.

For bridges fatigue assessment is required for all members subject to reversal of load.

7. PRACTICAL DESIGN a. Buildings

1. It is not always economic to make every member a different size. The designer should rationalise the sizes and use only two or three different sections in small span trusses.

2. Minimum sizes should be adopted to prevent damage during transport and erection. Recommendations are set



out above.

3. Safe load tables are very useful and members subjected to axial load can be selected directly. Members subjected to axial load and moment must be designed by successive trials. Select the initial size by assuming the compression resistance is 60% of full resistance.

4. Large trusses must be sub-divided for transport. Bolted site splices are used to assemble the truss on site.

b. Bridges

1. The optimum value for the span-to-depth ratio depends on the magnitude of the live load that has to be carried. It should be in the region of 10, being greater for road traffic and less for rail traffic. (For twin track rail loading the ratio would drop to about 71/2.) However, one should always make a check on the economic depth for the given bridge.

2. An even number of bays should be chosen to suit the configuration of diagonals. If an odd number is chosen there will be a central bay with crossed diagonals. This is not usually desirable except perhaps at the centre of a swing bridge. The diagonals should be at an angle between 50° and 60° to the horizontal.

3. Grade 50 steel should be used for the main members with Grade 43 used only for members carrying nominal load, unless the truss has to be fabricated in a country where the supply of higher grade steel is a problem. For a truss designed using Grade 50 steel, the amount of Grade 43 steel used would normally be about 7%.

4. The problems that may confront the bridge maintenance team should be fully appreciated. Details which could trap rainwater, dirt and debris should be avoided. All exposed areas should be fully accessible for painting. Box sections make painting easier, but rolled hollow sections leave nasty crevices at gusset positions unless the joints are welded.

8. CONCLUDING SUMMARY Trusses and lattice girders are important elements in building where they are used to support floors and roofs and provide bracing. For bridges, trusses can be economic for spans of 30m to 200m. They can be assembled from small pieces and are particularly advantageous where site access is difficult. Statically determinate trusses are generally used. Keep the configuration simple, using a minimum of members and connections. Avoid eccentricity of loading and connections to reduce secondary stresses. Secondary stresses due to loads applied between the nodes must be calculated. Careful consideration must be given in design to the provision of lateral support. Fatigue effects have to be considered in bridges and in some elements of buildings. Configuration of members and careful design of connections are particularly important. Avoid potential corrosion areas on all exposed steelwork.

9. REFERENCES [1] Eurocode 1: "Basis of Design and Actions on Structures": CEN (in preparation).

[2] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992.

[3] Eurocode 3: "Design of Steel Structures", Part 2: Bridges, CEN (in preparation).




19442847 methods of analysis of steel structures

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