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Lecture 7.9.1: Unrestrained Beams IOBJECTIVE/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
Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium
Lectures 6.6: Buckling of Real Structural Elements
Lectures 7.8: Restrained Beams
RELATED LECTURES:
Lecture 7.9.2: Unrestrained Beams II
Lectures 7.10: Beam Columns
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
fy material yield strength
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
LT parameter in design formula, see Equation (2)
LT reduction factor for lateral-torsional buckling
LT beam slenderness
LT basic slenderness
1 parameter used to determine LT, see Equation (4)
twist
LT parameter used to determine LT, see Equation (2)
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 (), 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 ) 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 = LT MRd (1)
where MRd is the moment resistance of the cross-section
LT 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 LT
depends on the beam's slenderness LT and is given by:
LT = 1/ {LT + [LT2 - LT
2]1/2} (2)
where LT = 0,5 [1 + LT( LT - 0,20) + LT2]
and LT = 0,21 for rolled sections
LT = 0,49 for welded beams
Figure 11 illustrates the relationship between LT 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 LT 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 = [LT / 1] (4)
where 1 = [E/fy]1/2
=
and LT =
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 LT 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 LT or Mcr.
For example, if there is a moment gradient between points of lateral restraint, LT is calculated as follows:
LT = (5)
where C1 = 1,75 - 1,05 + 0,3 2 2,35
and is the end moment ratio defined in Figure 13.
Taking as an example the end span of a continuous beam for which = 0 gives C1=1,75 and thus LT will be reduced to 0,76 (= 1/ 1,75) of the value for uniform moment, leading to an increase in LT 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 LT 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 LT or the basic elastic critical moment Mcr.
9. REFERENCES
[1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992.
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.
Lecture 7.9.2: Unrestrained Beams IIOBJECTIVE/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
Lectures 7.5: Columns
Lecture 7.9.1: Unrestrained Beams I
RELATED LECTURES
Lectures 7.10: Beam Columns
RELATED WORKED EXAMPLES
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 2EIw/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 2EIw/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 = (/L) EIxGIt) [1+ (EIw/LGIt)] (17)
Central load: Mcr = (4,24/L) EIxGIt) [1+ (EIw/LGIt)] (21)
The ratio of the two constants /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 K , is:
Mcr =
Rearranging this shows that the beam behaves as if its torsional rigidity GIt were increased to (GIt+ K L2/ 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.
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. 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 BUCKLING
Derivation of Governing Equations
The deformed state of the beam is shown in Figure 3, which identifies the deflections (u and v) and the twist (). A new co-ordinate system , which deflects with the beam, is also illustrated.
Bending in the and planes and twisting about the axis are governed by:
EIy (1)
EIz (2)
GIt (3)
In Equations (1) and (2) the flexural rigidities and curvatures in the and the planes have been replaced by the values for the yx and zx planes, on the basis that 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 and planes and about the axis to be obtained as:
M = Mcos , M = Msin , M = Msin (4)
Since is small, cos 1 and sin , whilst Figure 5 shows that sin may be approximated by
- . Thus Equations (1) - (3) may be written as:
EIy = M (5)
EIz = M (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 , 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 2 = enables the solution to be written as:
= Acos x = Bsin x (10)
Noting the boundary conditions at both ends gives
When x = 0, = 0; then A = 0 (11)
When x = L, = 0; then Bsin L = 0
and either B = 0, or (12)
sin L = 0 (13)
The first possibility gives the unbuckled position whereas the second gives:
L = 0, , 2 (14)
and the first non-trivial solution is:
L = (15)
which gives:
Mcr = (/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 = (/L) EIxGIt) [1+ (EIw/LGIt)] (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:
M =
M = (18)
M =
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 plane causes the end B of the beam to rotate in the 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 o) = Wa o2/2
in which o is the value of at the load point. This must be added to Equation (24).
