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THEORY AND METHODS FOR EVALUATION OF ELASTIC

CRITICAL BUCKLING LOAD

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 fcrλ from

I I I

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 fcrλ 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 iρ = )( 2 2iii LEIP π 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 ρ1eff = , 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.


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


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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


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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

(a) Elastic critical loadEigenvalue Analysis

(b) First order elastic analysis

BA

(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)

Pcr

Displacement (∆)


77

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).

Pi

αPi

αPi

Figure 3.4. Deformed shape of braced frame

∆Pi

αPi

αPi

Figure 3.5. Deformed shape of unbraced frame


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3.4 Histor ical 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

Euler’s formula, it was widely criticised when it was established. At first

engineers did not accept the results of Musschenbroek’s experiments and

Euler’s 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 Euler’s 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


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which Euler’s 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


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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


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 program also calculates

the critical load factor of individual struts of varying cross-section, by defining a node at

each change of cross-section. The effective length of each element is also computed

while the actual critical load of frame is estimated. An elastic stability analysis was

carried out by Simitses and Vlahinos (1986) for single-bay multi-storey frames with

support of some rotational stiffness. The computer code implementing the analysis was

applied to a two-storey single-bay in a parametric study, to investigate the effect of: (a)

increasing number of stories, (b) proportional load, (c) the length and stiffness of beam

variation, (d) the support rotational stiffness, and (e) the variation of the column

stiffness of the second floor. Goto and Chen (1987) proposed a second-order elastic

analysis that can be applied to any shape of structural frame. It takes into account the

effect of axial deformation of a structural element. Since the stiffness matrices used

were non-linear, iteration was necessary to arrive at the correct solution. Williams and

Sharp (1990) used a substitute frame technique to obtain the critical load of multi-storey

rigid jointed sway frames.

Duan and Chen (1988) started their study by proposing a simple modification of

the alignment charts in order to take into account the effect of the boundary conditions

at the far ends of columns above and below the column being investigated in braced


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frames. As reported by these authors, these far end conditions have a significant effect

on the K-factor of the column under consideration. As an extension to their research on

braced frames, Duan and Chen (1989) and Chen and Lui (1991) suggested another

modification to the alignment charts to include the effect of far-end conditions of

columns in unbraced frames. Essa (1997) derived expressions for the elastic effective

length factors for columns in unbraced multi-storey frames. The model takes into

account the effects of boundary conditions at the far ends of the columns above and

below the column under consideration. He concluded that using the alignment charts to

estimate the effective length factor for columns may be either overly conservative, or

even unconservative, depending on the boundary conditions and the relative stiffness

ratio of columns. Hashemi (1993) proposed a design methodology for beam-column.

The methodology is based upon the following steps. First, an elastic critical load

analysis is performed on an idealised model, this takes into account the stiffness

interaction with the surrounding frame. Second, a total equivalent imperfection

parameter is defined which accounts for the effects of both adopted geometric tolerances

and all the loading based imperfections. Third, the non-linear elastic response is used to

define the loads at which plastic failure is initiated. Lokkas (1996) extended the work

done by Hashemi to circumstances where more than one mode contributed to the non-

linear elastic behaviour and consequently elastic-plastic failure. In 1998, the author

continued the study by experimental work to investigate the simultaneous action

between the sway and non-sway modes of rigid jointed frames. The experiments show

the importance of taking care of the sway and non-sway critical modes exhibiting

simultaneous or nearly simultaneous critical loads.


83

3.5 Methods for evaluation of elastic cr itical load

Many methods can be used to determine the elastic critical load of structural

frameworks, and these can be summarised in the following sections.

3.5.1 Differential equation method

The basic equations for analysis of beam-columns can be derived by considering the

beam in Figure 3.6. The beam is subjected to an axial load P. The expression for

curvature can be obtained from the following second order differential equation (3.1).

X2

2

Mx

yEI −=

∂∂

. (3.1)

The quantity EI represents the flexural rigidity of the beam in the plane of

bending, that is, in the X-Y plane, which is assumed to be a plane of symmetry. The

general solution of equation (3.1) is

( ) ( ) DxCxBxAy +++= µµ cossin (3.2)

where EI

P=µ .

P P X

Y

y

x

Figure 3.6. Compressed bar


84

The constants A, B, C, D as well as the elastic critical load Pcr can be evaluated

from applying the end boundary conditions of the member.

Similarly, the elastic critical load can be obtained using the fourth order

differential equation

02

2

4

4

=∂

∂+

∂

∂

x

yP

x

yEI . (3.3)

The use of either the second order differential equation (3.1) or the fourth order

differential equation (3.3) is not a simple task when dealing with the problem of elastic

stability of either two or three-dimensional structural frameworks. That is due to the

large number of boundary and compatibility conditions inherent in structural

frameworks.

3.5.2 Energy method

The energy method can also be used to obtain the elastic critical load of a structural

system assuming a small lateral deflection of a system such as that shown in Figure 3.7.

ω

P

P P

P

L

θ

Figure 3.7. Structural system


85

This deflection leads to an increase in the strain energy, known as ∆U, of the

system. At the same time, the applied load will move through a small distance θL and

does work equal to ∆T. The system becomes stable in its undeflected form if

∆U > ∆T (3.4)

and unstable if

∆U < ∆T (3.5)

where ∆U = 0.5 ω (θL)2, ∆T = 0.5 PLθ 2 and ω denotes the spring constant.

The critical load Pcr is obtained from equating the strain energy of the structural

system due to a virtual lateral deflection with the work done by the loading pattern on

that system. This can be expressed by

∆U =∆T. (3.6)

The theoretical basis of the energy approach is described by Timoshenko and Gere

(1963). At loads lower than the elastic critical load, the gain of strain energy in the

elements is less than the potential energy of the loads. A condition of instability is

defined, as the stage when the change of the above two energies is zero, that is, the

stiffness of the structure is zero. Then the structure will not resist any random

disturbance. Appeltauer and Barta (1964) applied an approximate energy method to

obtain direct formulae for the elastic critical load depending on all the parameters of the

problem. The point of contraflexure was assumed to be at the centre of all elements of

the frame, so that an approximation to the deflected shape at neutral equilibrium could

be obtained. It has been observed from the previous discussion that it is too difficult to

use this method when dealing with the problem of elastic stability of a structural

framework. The reason for this difficulty is as the number of framework elements


86

increases, the complications in formulae of the strain energy and work done increase

too.

3.5.3 Modified slope deflection method

It has been stated by Galambos (1968) that the deformation effects in the equilibrium

equations of any structural framework have been included in the first order elastic

analysis (slope deflection method) to obtain the modified slope deflection method for

the second order elastic analysis. The modified slope method is based on two

assumptions, the first is a relatively small axial force in the beams whilst the second is

nearly identical forces in the columns. Accordingly, the geometrical changes due to

axial shortening can be neglected. This method can be summarised as follows:

• constructing the bending moments at each member end including the stability

functions (see Galambos, 1968),

• constructing the joint and shear equilibrium conditions from which the equilibrium

equations are obtained, and

• eliminating the unknowns from the equilibrium equations and obtaining the

determinantal form of the critical load pattern and finally solving the determinantal

form by a trial and error method.

In order to explain the difficulty of using this method, Mahfouz (1993) studied the

framework shown in Figure 3.8 using two methods of analysis. One of them is the

modified slope deflection method where the framework is subjected to the loading

pattern given in Figure 3.8. It was also assumed that the distorted configuration of the

framework is anti-symmetric as shown in Figure 3.8.


87

In this example, ten preliminary equations must be formulated. These equations

are for MAB, MBA, MCD, MDC, MBC, MBC, RA, RD, HA and HD. These equations are then

substituted into the three basic equilibrium equations to obtain their new form. It can be

concluded that as the number of bays and stories increases, the number of preliminary

equations increases too. This technique therefore cannot be used when dealing with

more highly indeterminate frameworks such as multi-bay or multi-storey frameworks. In

addition to, the technique mainly depends on the trial and error method which makes it

difficult to link with optimization techniques.

3.5.4 Direct method

This method is based on two main steps, see Salem (1968). The first step is the ready

prepared operations of rotations and the sway of axially compressed members which are

based on the decomposition of the general state of sway into the states of no-shear sway

and pure-shear sway. The second step is the pre-study of the possible buckling modes of

the given framework. Then the operations of sway and rotations for every member of

this framework are builtup separately corresponding to its distorted configuration. Since

αP

P PMCB

MCD

MBC

CBMBA

A DHDHA

MDCMAB

RDRA

Figure 3.8. Single-bay single-storey framework: loading pattern and deflected shape


88

at the critical load, there are no external moments or forces at the framework joints to

keep it in its distorted configuration, the sum of moments at each joint of that

framework should be equal to zero. This procedure will give many equations which are

equal to the number of the framework joints. In rectangular frameworks other than

symmetrical ones, another set of equations has to be obtained by equating the relative

displacements of the framework columns. Finally, by eliminating the unknowns from

these equations, a determinantal equation is obtained for the elastic critical load. This

determinantal equation has a number of solutions from which the least is called the first

buckling load. The solution of such determinantal form can only be done by the method

of trial-and-error using a computer program.

Figure 3.9 shows the basic simple operations of rotation and sway of an axially

compressed isolated member for both cases of fixed and pin-ended bases. The principle

of supperposition of any number of states of sway and rotation of an isolated axially

compressed member is applicable so long as the axial compression is kept constant

through all these states. Furthermore, the principle of resolution of any state of an

isolated axially compressed member into any number of states of sway or rotation is also

applied under the same condition. The no-shear stability functions m, n and O were

introduced by Merchant (1955) to deal with the case of a member with fixed ends while

Salem (1968) treated the hinged end case by introducing the stability function n″ for the

no-shear sway for such members. Salem also decomposed the general state of sway into

two components, which are the states of pure-shear sway and no-shear sway. These two

states of sway are shown in Figure 3.10 for members with fixed and pin-ended bases.

The non-dimensional stability functions S, C, S″, m, n, O, and n″ (Appendix A)

indicated in Figures 3.9 and 3.10 are all functions of the ratio ρ of the axial load to


89

22 LEIπ . These stability functions are tabulated by Livesley and Chandler (1956).

A FORTRAN program is developed for the stability analysis of steel frame

structures. The program is based on the direct method. The program evaluates the value

of iρ of each column of the investigated framework at the critical buckling load, then

the effective length factor of each column are computed.

Figure 3.11 illustrates the developed program.


90

M= nKθ M= n″Kθ

∆ ∆

M= HL2

m− =

LKCS

∆+− )(1

PP

∆ ∆

PPM = S″KθM = SKθ

L

P

θ

L

θ

H

H Ha) Rotation

L

KSH −= θ

H H

PP

L

KCSH )(1+−= θ

M = CSKθ

L

Hb) Pure-shear sway

M=n

HL=

LSK

∆−

H H

M= HL2

m−

PP

c) No-shear swayP

P P

θ θ

n

1

L=

∆θ

2

m

L=

∆θ

M = -OKθ

Figure 3.9. Basic simple operation of rotation and sway


91

"1

'11 MMM +=

PPP

P P"2

'22 MMM +=

2

HLmM

'2 −=

"2M = nKθ2-OKθ1

2

m

L=

∆"

(θ1+θ2)

θ1

2

HLmM

'1 −=

"1M = nKθ1-OKθ2

∆"∆' PP P

HH∆

n

HLM

'1 −=

1

2"1 θ

ρπ

n

KM =

θ1θ1

∆"∆'∆

"1

'11 MMM +=

LLL

"' ∆+

∆=

∆

∆"∆'∆

2

m

L=

∆"θ1

P

1nKM θ="1

2

HLmM

'1 −=

PP

θ1θ1

H H

P

K)C(1S2

mHL

L +=

∆'

+

+

+

=

H H

1KOM θ−="

2

2

HLmM

'2 −="

2'22 MMM +=

P PH H

"1

'11 MMM +=

1

1"θ

nL=

∆

nKS

HL

L "

'=

∆=

LLL

"' ∆+

∆=

∆

H H

P P P

θ2

θ1

θ2

HH

LLL

"' ∆+

∆=

∆

K)C(1S2

mHL

L +=

∆'=

P

State of no-shear swayState of pure-shear swayState of sway

Figure 3.10. Decomposition of the general state of sway

= +


92

3.5.5 Finite element method

The finite element method can be applied to the evaluation of the elastic critical load for

structural frameworks (see Allen and Bulson, 1980). The finite element method is based

upon the use of local functions (i.e. these defined over sub-regions or finite elements of

the structural system). The other methods, such as the modified slope-deflection

Figure 3.11. Flowchart for computer program based on the direct method

Yes

No

Choose the starting value of ρ = 0.0001

Calculate stability functions for each member

Substitute the stability functions into thedeterminantal form

Calculate the determinant value

Does the determinantchange its sign?

Increase ρ

Start

Stop

Input: Structuralgeometry, etc


93

method, are usually thought of as being based upon overall functions (i.e. those defined

over the entire region of the structural system). In the finite element method, each

member of the structure is subdivided into a series of fairly short elements. The

deformation over each element may be defined by a simple polynomial function. The

coefficients of these polynomial functions may be determined if the displacements of

each node are known. As a result, the individual displacements of the entire structure

may be calculated and consequently the behaviour of the structure may be fully

described in terms of the displacements of the nodes. For equilibrium the increment in

total potential energy must be stationary with respect to these nodal displacements. This

leads to a set of linear homogeneous equations, where the dependent variables of these

equations are the nodal displacements Ψ, i.e. the following eigenvalue problem:

[ ]{ } [ ]{ }ΨΨλ CECGf KK = (3.7)

where fλ is the load factor, CEK is the global elastic stiffness matrix corresponding to

the connecting joints (nodes), CGK is the geometric stiffness matrix.

The first eigenvalue, i.e. the smallest value of fλ at which the structure becomes

unstable is termed the critical load factor fcrλ .

This classical eigenvalue approach discussed by many authors among them

Prezemieniecki (1968), Allen and Bulson (1980), Graves Smith (1983), Brebbia and

Ferrante (1986), Coates and Kong, (1988), Galambos (1988) and Bathe (1996). The

eigenvalues and eigenvectors can be obtained by applying several techniques, among

them vector iteration methods i.e inverse iteration, forward iteration and Rayleigh

quotient iteration, transformation methods such as Jacobi method and generalised Jacobi


94

method, and the subspace iteration method. Subroutines, written in FORTRAN 77, are

available in Bathe (1996).

3.6 Ver ification of the developed code for stability analysis

The developed program based on the direct method has been developed for the elastic

critical buckling analysis of 2-D frameworks, Section 3.5.4. In order to verify the

developed program, the established theoretical results presented by Timoshenko and

Gere (1963), Chajes (1974) and Renton (1967) are used. Here, various framework

models have been analysed.

The first example used is the fixed base framework ABCD shown in Figure 3.12a.

The framework is prevented from sidesway. It is subjected to two equal vertical loads P

at corners B and C.

When the vertical loads P reach their critical value crP , the distorted configuration

of the framework will be as shown in Figure 3.12b. The operations of rotations are built

up for every member of the framework separately as given in Figure 3.12c.

Since at the critical load, there are no external moments set-up at corners B and C

to keep the framework in its distorted configuration, thus:

0BCBAB =+=�

MMM ,

024

21

b1

1

b1 =��

��−�� +∴ θθ

K

K

K

KS , and (3.8)

0CDCBC =+=�

MMM ,

042

21

b

1

221

1

b =�� +−��

��∴ θθK

K

K

KS

K

K. (3.9)


95

Eliminating the unknowns ( 21 andθθ ) from (3.8) and (3.9), the elastic critical load

equation becomes:

042

24

det

1

b

1

22

1

b

1

b

1

b1

=��

�

�

��

�

�

��

�

�

�

+−��

�

�

�

��

�

�

�

−��

�

�

�

+

K

K

K

KS

K

K

K

K

K

KS

. (3.10)

L

IK b

b =

-2 bK θ2 -4 bK θ2

K2=L

I 2K1=

L

I1I1 I2 K2=L

I 2K1=L

I 1

B

DA

L

L

P

Rigid bracing

a) Loading pattern

P P P

bI

I1 I2

L

IK b

b =

2 bK 1θ4 bK θ1P P

θ2θ1111 θKS

-S2K2θ2

C1S1K1θ1 - C2S2K2θ2

c) Operation of rotationb) Distorted configuration

Figure 3.12. Single storey single-bay fixed base framework prevented from sway

bI

C


96

Following the same previous procedure, the elastic critical load equation (3.11) of

a single-bay single-storey fixed base framework permitted to sway (Figures 3.13), is

obtained:

0

)(12)(1222

2

42

2

24

det

2

1

22

2

11

121

2

1

b

1

22

1

b

1

1

b

1

b1

=

��

�

�

��

�

�

��

�

�

�

++

+−��

�

�

�

−��

�

�

��

�

�

−��

�

�

�

+��

�

�

�

��

�

��

�

�

��

�

�

�

+

K

K

CS

m

CS

mmm

m

KK

KnK

K

m

K

K

K

Kn

K (3.11)

where the third equation is obtained by equating the sidesways at joints B and C. In this

equation, the unknowns are 121 and KHL, θθ .

For a single-bay single-storey hinged base framework permitted to sway (Figures

3.14), the elastic critical load equation is

0

1111

142

124

det

2

1

22"

11"

21

21

b

1

22"

1

b

11

b

1

b

1"

=

��

�

��

�

�

��

�

��

�

�

+−��

�

��

�

�

−��

�

��

�

�

��

�

��

�

�

−��

�

��

�

�

+��

�

��

�

�

��

�

��

�

�

��

�

��

�

��

�

��

�

�

+

K

K

nSnSnn

nK

K

K

KnK

K

nK

K

K

Kn

. (3.12)

In order to verify the developed program, the previous described framework

models have been analysed and the results obtained are compared with established

theoretical results are given in Table 3.1.


97

Figure 3.13.Single-storey single-bay fixed base framework permitted to sway

H H

-O1K1θ1

+ HLm

2

1

-O2K2θ2

- HLm

2

2

θ2 222 θKn

- HLm

2

2

111 θKn

+ HLm

2

1

(b) Operation of rotation(a) Loading pattern and distorted configuration

2 1bθK4 1bθK

2 2bθK 4 2bθK

P P

L

bK

K1 K2

L

θ1

bKθ2

P P

Figure 3.14.Single-storey single-bay hinged base framework permitted to sway

(a) Loading pattern and distorted configuration (b) Operation of rotation

H H

θ1 22"2 θKn

- HLn2

111

"1 θKn

+ HLn1

1

2 1bθK4 1bθK

2 2bθK 4 2bθK

L

LK1 K2


98

Table 3.1. Comparison of the theoretical and developed code in plane buckling loads.

Model Theoretical Results Obtained results

Chajes (1974) and Renton (1967)

2cr 2.25L

EIP =

553292

2

2cr .

L

EI

P==

πρ

ρ = 2.554

Chajes (1974) and Renton (1967)

2cr 34.7L

EIP =

74370

2

2cr .

L

EI

P==

πρ

ρ = 0.7475

Timoshenko and Gere (1963)

2cr 82.1L

EIP =

18440

2

2cr .

L

EI

P==

πρ

ρ = 0.1843

Timoshenko and Gere (1963)

2

2

cr4L

EIP

π=

250

2

2cr .

L

EI

P==

πρ

ρ = 0.2499

P P

L

L

I I

I

P P

L

L

I I

I

P P

L

L

I I

I

P P

L

L

I I

Ib=0


99

3.7 Concluding remarks

In this chapter, the stability concept of idealised framework model has been presented.

The methods of evaluating the elastic critical load as well as literature on the stability

analysis are also reviewed. Finally, verification of the developed program for the

stability analysis of frameworks has been carried out.

In Section 3.5 it has been concluded that the methods of analysis based on trial

and error are difficult to use in the design optimization process. Consequently, the

eigenvalue approach is more suitable for the design optimization process.

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