THEORY AND METHODS FOR EVALUATION OF nbsp;· Theory and Methods for Evaluation of Elastic Critical Buckling Load 76 use of either the eigenvalue analysis or the second order elastic analysis.

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    3.1 Introduction

    The codes of practice tackle the stability of steel structures by determining the effective

    buckling length of the structural members. Therefore, the problem of stability is very

    important. Underestimation of this effect may lead to disastrous results or unjustified

    factors of safety. Consequently, the Steel Construction Institute (SCI) suggested the

    previous approach for estimating the effective length factor presented BS 449: part 2:

    (1969) should be modified by recommending three methods for evaluating this factor.

    The first, termed extended simple design, also described in the previous chapter, starts

    by evaluating the relative stiffness coefficients of the surrounding columns and beams of

    the column under consideration. Then, using these stiffnesses, the effective length factor

    can be estimated from charts based on the study carried out by Wood (1974a). The

    second method, termed the amplified sway method, states that the bending moments due

    to horizontal loading should be amplified by a factor, as discussed in chapter 2. The

    third, a more accurate method, is to determine the elastic critical load factor fcr from


  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    which the effective length ratios of individual members may be determined. The critical

    load factor fcr is defined as the ratio by which each of the factored loads would have to

    be proportionally increased to cause elastic instability. If this parameter is known, the

    axial load in every compression member Pi at instability is known as well. Then, the

    value of i = )(2 2

    iii LEIP can be computed where iI and iL are the second

    moment of area and length of a column under consideration respectively. Consequently,

    the effective length ratio is evaluated as iii

    LL 1eff = , see SCI (1988).

    Several attempts were suggested in order to overcome some shortcomings of the

    design chart procedure. Several methods, among them Hashemi (1993), Lokkas (1996),

    MacLeod and Zalka (1996) and Lokkas and Croll (1998), were suggested for the

    modification of the design procedure recommended by the British code of practice, but

    this may lead to a design procedure which is not accepted by a practising engineer.

    As well as the British code of practice, the American code of practice also suffers

    from the difficulty of evaluating the effective buckling length accurately enough. This is

    indicated in the studies by Duan and Chen (1988, 1989), Chen and Lui (1991), Kishi et

    al. (1997), White and Clarke (1997) and Essa (1997) who proposed modifications to the

    alignment charts recommended by the American Institute of Steel Construction (AISC).

    Virtually all methods of analysis that have been developed to improve the limit

    strength of structures are based upon a geometrically linear model of the structural

    response. In these methods, the stability concept, addressed in the following section, is

    used. The available methods of calculating the elastic critical load factor are

    subsequently described in chronological order in the section on historical background.

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    3.2 Stability concept

    The question of the stability of various forms of equilibrium of a compressed bar can be

    investigated by using the same theory as used in investigating the stability of

    equilibrium configurations of rigid-body systems (Timoshenko and Gere, 1963).

    Consider three cases of equilibrium of the ball shown in Figure 3.1. It can be concluded

    that the ball on the concave spherical surface (a) is in a state of stable equilibrium, while

    the ball on the horizontal plane (b) is in indifferent or neutral equilibrium. The ball on

    the convex spherical surface (c) is said to be in unstable equilibrium.

    The compressed bar shown in Figure 3.2 can be similarly considered. In the state

    of stable equilibrium, if the column is given any small displacement by some external

    influence, which is then removed, it will return back to the undeflected shape. Here, the

    value of the applied load P is smaller than the value of the critical load Pcr. By

    definition, the state of neutral equilibrium is the one at which the limit of elastic stability

    is reached. In this state, if the column is given any small displacement by some external

    influence, which is then removed, it will maintain that deflected shape. Otherwise, the

    column is in a the state of unstable equilibrium.

    (c) Unstable equilibrium

    Figure 3.1. States of equilibrium

    (b) Neutral equilibrium(a) Stable equilibrium

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    3.3 The concept of buckling in idealised framework models

    The majority of building structures have been designed by the elastic theory by simply

    choosing allowable stress values for the materials and by imposing limiting ratios such

    as serviceability requirements. All structures deflect under loading, but in general, the

    effect of this upon the overall geometry can be ignored. In the case of high-rise building,

    the lateral deflections may be such as to add a significant additional moment. This is

    know as P effect. Therefore, the governing equilibrium equations of a structure must

    be written with respect to the deformed geometry; the analysis is referred to as second-

    order analysis. On the other hand, when the lateral deflections can be ignored and the

    equilibrium equations are written with respect to the undeformed geometry, the analysis

    is referred to the first order analysis. The load deflection behaviours of a structure

    analysed by first and second order elastic methods are illustrated in Figure 3.3. This is

    discussed by many authors among them Galambos (1968), Allen and Bulson (1980) and

    Chen et al. (1996). From this figure, it can be understood that the critical buckling load,

    needed for the evaluation of the effective length of members, may be determined by the

    P P

    P < Pcr

    P P P


    P > Pcr P = Pcr

    (c) Unstable equilibrium (b) Neutral equilibrium (a) Stable equilibrium

    Figure 3.2. Different cases of equilibrium for compressed bar

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    use of either the eigenvalue analysis or the second order elastic analysis. Unlike a first

    order analysis in which solutions can be obtained in a rather simple and direct manner, a

    second order analysis often entails an iterative type procedure to obtain solutions. Thus,

    the use of eigenvalue analysis to obtain the critical buckling load is the simplest way.

    In order to study the buckling response on several possible idealised models,

    restricted or not against sidesway, let us consider the two structures in Figures 3.4 and

    3.5. The framework, shown in Figure 3.4, is prevented from sidesway whereas in the

    framework given in Figure 3.5 there is a possibility of sidesway. Both frameworks have

    initially geometrically perfect members, which are subjected to a set of point loads Pi at

    (f) Second order elastic-plastic analysis

    (g) Second-orderplastic zone


    (d) First order rigid-plastic analysis

    (a) Elastic critical loadEigenvalue Analysis

    (b) First order elastic analysis


    (c) Second order elastic analysis

    Mechanism load

    (e) First order elastic-plastic analysis

    Figure 3.3. Load displacement curve (Chen et al., 1996)

    Load (Pi)


    Displacement ()

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    their joints. If the members remain elastic as loads are increased, there will be no

    flexural deformation until a particular level of loading is achieved. This load is known

    as elastic critical load, corresponding to which a bifurcation of equilibrium is possible

    (see Hashemi, 1993, Mahfouz, 1993 and Lokkas, 1996).




    Figure 3.4. Deformed shape of braced frame




    Figure 3.5. Deformed shape of unbraced frame

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    3.4 Historical background

    In this section the historical background of the stability problem and methods of stability

    analysis is presented. Timoshenko and Gere (1963) gave the following description of

    early research in this important field of structural mechanics.

    The first experiments with buckling of centrally compressed prismatic bars

    were made by Musschenbroek (1729). As a result of his tests, he concluded

    that the buckling load was inversely proportional to the square of the length

    of the column, a result which was obtained by Euler 30 years later from

    mathematical analysis. Euler (1759) investigated the elastic stability of a

    centrally loaded isolated strut. He assumed that a column which is originally

    straight (perfect column), remains straight from the onset of loading and in

    order to produce a small deflection of the column, the load should reach a

    critical value, below this critical value the column would suffer no

    deflection. Although the more recent developments have been based on

    Eulers formula, it was widely criticised when it was established. At first

    engineers did not accept the results of Musschenbroeks experiments and

    Eulers theory. Almost 90 years later, Lamarle (1846) was the first to give a

    satisfactory explanation of the discrepancy between theoretical and

    experimental results. He showed that Eulers theory is in agreement with

    experiments provided the fundamental assumptions of the theory regarding

    perfect elasticity of the material and ideal conditions at the ends were

    fulfilled. He clarified the fact that when an ideal strut bends, the most

    stressed fibres in the strut may immediately pass the elastic limit of the

    material. This condition determined the value of the slenderness ratio, below

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    which Eulers formula is inapplicable, and up to this value of slenderness

    ratio the strut fails, is due to direct compression rather than to instability.

    From that time, the elastic stability problems of braced and unbraced structural

    frameworks have been addressed by many researchers and a great wealth of literature

    exists in this field. A considerable amount of the literature is directed towards stability

    of plane frames within the plane of the frame. The elastic critical load can be evaluated

    for any symmetrical single-bay multi-storey rigid frame using the relaxation method

    with no-shear stability function as proposed by Smith and Merchant (1956). The

    analysis was extended to take account of axial deformation. Bowles and Merchant

    (1956) applied a more accurate method based on the same technique to the stability

    analysis of a five-storey two-bay steel frame. The results obtained were in good

    agreement with those previously obtained using a simpler version of the method.

    Subsequently, Bowles and Merchant (1958) proposed the conversion of a multi-storey

    multi-bay rigid plane frame, to an equivalent single bay frame so that it could be

    analysed by the method proposed earlier. Timoshenko and Gere (1963) treated the

    buckling behaviour and the buckling load of single-bay single-storey hinged base

    rectangular frame as well as closed frames. Waters (1964a, 1964b) presented, in two

    parts, direct approximate methods, involving no trial and error, for the elastic critical

    load parameter of plane rigid-jointed rectangular and triangulated frameworks. Two

    approaches were considered: equal rotations and the substitute frame, according to

    Bolton (1955), Bowles and Merchant (1956) and McMinn (1961). Goldberg (1968) was

    the first one to tackle the problem of lateral buckling load of braced frames. He did not

    consider the stability of the frame as a whole but he obtained the elastic critical load

    equations for a typical intermediate column in a multi-storey frame. He considered the

    effect of girder stiffness at the top and bottom of that column as well as the average

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    bracing stiffness of that storey. In the same year, Salem (1968) studied the problem of

    lateral buckling of rectangular multi-storey frames. These frames are loaded at

    intermediate floor levels and the column sections vary according to an arithmetic series.

    An investigation on the sway critical load factor of symmetrical and unsymmetrical

    frames, loaded with unequal and equal axial loads was carried out by Salem (1973),

    considering the effect of axial deformation variation in columns. Wood (1974a, 1974b,

    1974c) adapted an approximate manual technique to be applied in conjunction with

    effective length and critical load factor charts. The method, which accounts for column

    continuity, is similar to moment distribution, and called stiffness distribution, involving

    no-shear stability functions. The elastic critical load factor for a particular storey can be

    estimated. The same procedure is followed for the rest of the stories and the lowest

    critical load is the elastic critical load of the original frame. This technique was

    recommended in BS 5950: Part 1 to be used in the design procedure. Horne (1975)

    recommended that a horizontal point load equal to 1% of the vertical load at that storey

    should be added at each storey level, and a linear elastic stability analysis be performed.

    Bolton (1976) proposed a single horizontal unit point load to be applied at the top of the

    frame, and the deflection at each storey to be calculated using an elastic analysis. Then,

    this deflection was multiplied by the total vertical applied load at that storey level,

    which was finally divided by the height of the storey, to yield the storey critical load

    factor. The lowest of all load factors corresponds to the critical load factor of the frame.

    Al-Sarraf (1979) adopted a computing method for predicting the lowest elastic critical

    load factor of sway and non-sway frames applying modified slope deflection equations

    based on no-shear stability functions. Anderson (1980) derived formulae, from slope-

    deflection equations which were used for yielding the storey sways based on sub-frames,

    assuming the point of contraflexure at the mid span of the elements. Then, sway angles

  • Theory and Methods for Evaluation of Elastic Critical Buckling Load


    were computed from the storey sway, and the expression for the critical load factor by

    Horne (1975) was used. A direct calculation of elastic critical loads based upon the

    structural system concept involving no stability functions was also presented by

    Awadalla (1983). The computer aspect of this method was discussed and it was shown

    that the efficiency of the numerical solution can be improved by considering each

    column as a substructure. The results from this method consistently exceed those

    produced by the solution obtained by using the stability functions. Carr (1985)

    developed a computer program for the stability problem. The progr...


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