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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• THEORY AND METHODS FOR EVALUATION OF ELASTIC

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

III

• Theory and Methods for Evaluation of Elastic Critical Buckling Load

73

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

74

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

75

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

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

76

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

C

(d) First order rigid-plastic analysis

(b) First order elastic analysis

BA

(c) Second order elastic analysis

(e) First order elastic-plastic analysis

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

Pcr

Displacement ()

• Theory and Methods for Evaluation of Elastic Critical Buckling Load

77

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

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

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

Pi

Pi

Pi

Figure 3.4. Deformed shape of braced frame

Pi

Pi

Pi

Figure 3.5. Deformed shape of unbraced frame

• Theory and Methods for Evaluation of Elastic Critical Buckling Load

78

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

79

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

80

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

81

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...