THEORY AND METHODS FOR EVALUATION OF ELASTIC CRITICAL BUCKLING LOAD

Transcription

1 III TEORY AND METODS FOR EVAUATION OF EASTIC CRITICA BUCING OAD 3. Introduction The codes of practice tackle the staility of steel structures y determining the effective uckling length of the structural memers. Therefore, the prolem of staility 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 : (969) should e modified y recommending three methods for evaluating this factor. The first, termed extended simple design, also descried in the previous chapter, starts y evaluating the relative stiffness coefficients of the surrounding columns and eams of the column under consideration. Then, using these stiffnesses, the effective length factor can e estimated from charts ased on the study carried out y Wood (974a). The second method, termed the amplified sway method, states that the ending moments due to horizontal loading should e amplified y a factor, as discussed in chapter. The f third, a more accurate method, is to determine the elastic critical load factor λ cr from

2 Theory and Methods for Evaluation of Elastic Critical Buckling oad 73 which the effective length ratios of individual memers may e determined. The critical f load factor λ cr is defined as the ratio y which each of the factored loads would have to e proportionally increased to cause elastic instaility. If this parameter is known, the axial load in every compression memer i at instaility is known as well. Then, the value of ρ i = i ( π EI i i ) can e computed where I i and i are the second moment of area and length of a column under consideration respectively. Consequently, the effective length ratio is evaluated as eff i i = ρ, see SCI (988). i Several attempts were suggested in order to overcome some shortcomings of the design chart procedure. Several methods, among them ashemi (993), okkas (996), Maceod and Zalka (996) and okkas and Croll (998), were suggested for the modification of the design procedure recommended y the British code of practice, ut this may lead to a design procedure which is not accepted y 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 uckling length accurately enough. This is indicated in the studies y Duan and Chen (988, 989), Chen and ui (99), ishi et al. (997), White and Clarke (997) and Essa (997) who proposed modifications to the alignment charts recommended y the American Institute of Steel Construction (AISC). Virtually all methods of analysis that have een developed to improve the limit strength of structures are ased upon a geometrically linear model of the structural response. In these methods, the staility concept, addressed in the following section, is used. The availale methods of calculating the elastic critical load factor are susequently descried in chronological order in the section on historical ackground.

3 Theory and Methods for Evaluation of Elastic Critical Buckling oad Staility concept The question of the staility of various forms of equilirium of a compressed ar can e investigated y using the same theory as used in investigating the staility of equilirium configurations of rigid-ody systems (Timoshenko and Gere, 963). Consider three cases of equilirium of the all shown in Figure 3.. It can e concluded that the all on the concave spherical surface (a) is in a state of stale equilirium, while the all on the horizontal plane () is in indifferent or neutral equilirium. The all on the convex spherical surface (c) is said to e in unstale equilirium. (a) Stale equilirium () Neutral equilirium (c) Unstale equilirium Figure 3.. States of equilirium The compressed ar shown in Figure 3. can e similarly considered. In the state of stale equilirium, if the column is given any small displacement y some external influence, which is then removed, it will return ack to the undeflected shape. ere, the value of the applied load is smaller than the value of the critical load cr. By definition, the state of neutral equilirium is the one at which the limit of elastic staility is reached. In this state, if the column is given any small displacement y some external influence, which is then removed, it will maintain that deflected shape. Otherwise, the column is in a the state of unstale equilirium.

4 Theory and Methods for Evaluation of Elastic Critical Buckling oad 75 < cr = cr > cr (a) Stale equilirium () Neutral equilirium (c) Unstale equilirium Figure 3.. Different cases of equilirium for compressed ar 3.3 The concept of uckling in idealised framework models The majority of uilding structures have een designed y the elastic theory y simply choosing allowale stress values for the materials and y imposing limiting ratios such as serviceaility requirements. All structures deflect under loading, ut in general, the effect of this upon the overall geometry can e ignored. In the case of high-rise uilding, the lateral deflections may e such as to add a significant additional moment. This is know as effect. Therefore, the governing equilirium equations of a structure must e written with respect to the deformed geometry; the analysis is referred to as secondorder analysis. On the other hand, when the lateral deflections can e ignored and the equilirium equations are written with respect to the undeformed geometry, the analysis is referred to the first order analysis. The load deflection ehaviours of a structure analysed y first and second order elastic methods are illustrated in Figure 3.3. This is discussed y many authors among them Galamos (968), Allen and Bulson (980) and Chen et al. (996). From this figure, it can e understood that the critical uckling load, needed for the evaluation of the effective length of memers, may e determined y the

5 Theory and Methods for Evaluation of Elastic Critical Buckling oad 76 use of either the eigenvalue analysis or the second order elastic analysis. Unlike a first order analysis in which solutions can e otained in a rather simple and direct manner, a second order analysis often entails an iterative type procedure to otain solutions. Thus, the use of eigenvalue analysis to otain the critical uckling load is the simplest way. oad ( i ) C () First order elastic analysis cr A (a) Elastic critical load Eigenvalue Analysis B (c) Second order elastic analysis (d) First order rigid-plastic analysis Mechanism load (e) First order elastic-plastic analysis (g) Second-order plastic zone (f) Second order elastic-plastic analysis Displacement () Figure 3.3. oad displacement curve (Chen et al., 996) In order to study the uckling response on several possile 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 possiility of sidesway. Both frameworks have initially geometrically perfect memers, which are sujected to a set of point loads i at

6 Theory and Methods for Evaluation of Elastic Critical Buckling oad 77 their joints. If the memers remain elastic as loads are increased, there will e no flexural deformation until a particular level of loading is achieved. This load is known as elastic critical load, corresponding to which a ifurcation of equilirium is possile (see ashemi, 993, Mahfouz, 993 and okkas, 996). i α i α i Figure 3.4. Deformed shape of raced frame i α i α i Figure 3.5. Deformed shape of unraced frame

7 Theory and Methods for Evaluation of Elastic Critical Buckling oad istorical ackground In this section the historical ackground of the staility prolem and methods of staility analysis is presented. Timoshenko and Gere (963) gave the following description of early research in this important field of structural mechanics. The first experiments with uckling of centrally compressed prismatic ars were made y Musschenroek (79). As a result of his tests, he concluded that the uckling load was inversely proportional to the square of the length of the column, a result which was otained y Euler 30 years later from mathematical analysis. Euler (759) investigated the elastic staility of a centrally loaded isolated strut. e 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, elow this critical value the column would suffer no deflection. Although the more recent developments have een ased on Euler s formula, it was widely criticised when it was estalished. At first engineers did not accept the results of Musschenroek s experiments and Euler s theory. Almost 90 years later, amarle (846) was the first to give a satisfactory explanation of the discrepancy etween theoretical and experimental results. e 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. e clarified the fact that when an ideal strut ends, the most stressed fires in the strut may immediately pass the elastic limit of the material. This condition determined the value of the slenderness ratio, elow

8 Theory and Methods for Evaluation of Elastic Critical Buckling oad 79 which Euler s formula is inapplicale, and up to this value of slenderness ratio the strut fails, is due to direct compression rather than to instaility. From that time, the elastic staility prolems of raced and unraced structural frameworks have een addressed y many researchers and a great wealth of literature exists in this field. A considerale amount of the literature is directed towards staility of plane frames within the plane of the frame. The elastic critical load can e evaluated for any symmetrical single-ay multi-storey rigid frame using the relaxation method with no-shear staility function as proposed y Smith and Merchant (956). The analysis was extended to take account of axial deformation. Bowles and Merchant (956) applied a more accurate method ased on the same technique to the staility analysis of a five-storey two-ay steel frame. The results otained were in good agreement with those previously otained using a simpler version of the method. Susequently, Bowles and Merchant (958) proposed the conversion of a multi-storey multi-ay rigid plane frame, to an equivalent single ay frame so that it could e analysed y the method proposed earlier. Timoshenko and Gere (963) treated the uckling ehaviour and the uckling load of single-ay single-storey hinged ase rectangular frame as well as closed frames. Waters (964a, 964) 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 sustitute frame, according to Bolton (955), Bowles and Merchant (956) and McMinn (96). Golderg (968) was the first one to tackle the prolem of lateral uckling load of raced frames. e did not consider the staility of the frame as a whole ut he otained the elastic critical load equations for a typical intermediate column in a multi-storey frame. e considered the effect of girder stiffness at the top and ottom of that column as well as the average

9 Theory and Methods for Evaluation of Elastic Critical Buckling oad 80 racing stiffness of that storey. In the same year, Salem (968) studied the prolem of lateral uckling 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 y Salem (973), considering the effect of axial deformation variation in columns. Wood (974a, 974, 974c) adapted an approximate manual technique to e applied in conjunction with effective length and critical load factor charts. The method, which accounts for column continuity, is similar to moment distriution, and called stiffness distriution, involving no-shear staility functions. The elastic critical load factor for a particular storey can e 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: art to e used in the design procedure. orne (975) recommended that a horizontal point load equal to % of the vertical load at that storey should e added at each storey level, and a linear elastic staility analysis e performed. Bolton (976) proposed a single horizontal unit point load to e applied at the top of the frame, and the deflection at each storey to e calculated using an elastic analysis. Then, this deflection was multiplied y the total vertical applied load at that storey level, which was finally divided y 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 (979) adopted a computing method for predicting the lowest elastic critical load factor of sway and non-sway frames applying modified slope deflection equations ased on no-shear staility functions. Anderson (980) derived formulae, from slopedeflection equations which were used for yielding the storey sways ased on su-frames, assuming the point of contraflexure at the mid span of the elements. Then, sway angles

10 Theory and Methods for Evaluation of Elastic Critical Buckling oad 8 were computed from the storey sway, and the expression for the critical load factor y orne (975) was used. A direct calculation of elastic critical loads ased upon the structural system concept involving no staility functions was also presented y Awadalla (983). The computer aspect of this method was discussed and it was shown that the efficiency of the numerical solution can e improved y considering each column as a sustructure. The results from this method consistently exceed those produced y the solution otained y using the staility functions. Carr (985) developed a computer program for the staility prolem. The program also calculates the critical load factor of individual struts of varying cross-section, y 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 staility analysis was carried out y Simitses and Vlahinos (986) for single-ay multi-storey frames with support of some rotational stiffness. The computer code implementing the analysis was applied to a two-storey single-ay in a parametric study, to investigate the effect of: (a) increasing numer of stories, () proportional load, (c) the length and stiffness of eam variation, (d) the support rotational stiffness, and (e) the variation of the column stiffness of the second floor. Goto and Chen (987) proposed a second-order elastic analysis that can e 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 (990) used a sustitute frame technique to otain the critical load of multi-storey rigid jointed sway frames. Duan and Chen (988) started their study y proposing a simple modification of the alignment charts in order to take into account the effect of the oundary conditions at the far ends of columns aove and elow the column eing investigated in raced

11 Theory and Methods for Evaluation of Elastic Critical Buckling oad 8 frames. As reported y these authors, these far end conditions have a significant effect on the -factor of the column under consideration. As an extension to their research on raced frames, Duan and Chen (989) and Chen and ui (99) suggested another modification to the alignment charts to include the effect of far-end conditions of columns in unraced frames. Essa (997) derived expressions for the elastic effective length factors for columns in unraced multi-storey frames. The model takes into account the effects of oundary conditions at the far ends of the columns aove and elow the column under consideration. e concluded that using the alignment charts to estimate the effective length factor for columns may e either overly conservative, or even unconservative, depending on the oundary conditions and the relative stiffness ratio of columns. ashemi (993) proposed a design methodology for eam-column. The methodology is ased 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 oth adopted geometric tolerances and all the loading ased imperfections. Third, the non-linear elastic response is used to define the loads at which plastic failure is initiated. okkas (996) extended the work done y ashemi to circumstances where more than one mode contriuted to the nonlinear elastic ehaviour and consequently elastic-plastic failure. In 998, the author continued the study y experimental work to investigate the simultaneous action etween 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 exhiiting simultaneous or nearly simultaneous critical loads.

12 Theory and Methods for Evaluation of Elastic Critical Buckling oad Methods for evaluation of elastic critical load Many methods can e used to determine the elastic critical load of structural frameworks, and these can e summarised in the following sections Differential equation method The asic equations for analysis of eam-columns can e derived y considering the eam in Figure 3.6. The eam is sujected to an axial load. The expression for curvature can e otained from the following second order differential equation (3.). y EI x = M X. (3.) The quantity EI represents the flexural rigidity of the eam in the plane of ending, that is, in the X-Y plane, which is assumed to e a plane of symmetry. The general solution of equation (3.) is ( µ x) B cos( x) C x D y = A sin µ (3.) where µ =. EI Y y X x Figure 3.6. Compressed ar

13 Theory and Methods for Evaluation of Elastic Critical Buckling oad 84 The constants A, B, C, D as well as the elastic critical load cr can e evaluated from applying the end oundary conditions of the memer. Similarly, the elastic critical load can e otained using the fourth order differential equation 4 y y EI = 0. (3.3) 4 x x The use of either the second order differential equation (3.) or the fourth order differential equation (3.3) is not a simple task when dealing with the prolem of elastic staility of either two or three-dimensional structural frameworks. That is due to the large numer of oundary and compatiility conditions inherent in structural frameworks Energy method The energy method can also e used to otain 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 θ Figure 3.7. Structural system

14 Theory and Methods for Evaluation of Elastic Critical Buckling oad 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 θ and does work equal to T. The system ecomes stale in its undeflected form if U > T (3.4) and unstale if U < T (3.5) where U = 0.5 ω (θ), T = 0.5 θ and ω denotes the spring constant. The critical load cr is otained from equating the strain energy of the structural system due to a virtual lateral deflection with the work done y the loading pattern on that system. This can e expressed y U =T. (3.6) The theoretical asis of the energy approach is descried y Timoshenko and Gere (963). 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 instaility is defined, as the stage when the change of the aove two energies is zero, that is, the stiffness of the structure is zero. Then the structure will not resist any random disturance. Appeltauer and Barta (964) applied an approximate energy method to otain direct formulae for the elastic critical load depending on all the parameters of the prolem. The point of contraflexure was assumed to e at the centre of all elements of the frame, so that an approximation to the deflected shape at neutral equilirium could e otained. It has een oserved from the previous discussion that it is too difficult to use this method when dealing with the prolem of elastic staility of a structural framework. The reason for this difficulty is as the numer of framework elements

15 Theory and Methods for Evaluation of Elastic Critical Buckling oad 86 increases, the complications in formulae of the strain energy and work done increase too Modified slope deflection method It has een stated y Galamos (968) that the deformation effects in the equilirium equations of any structural framework have een included in the first order elastic analysis (slope deflection method) to otain the modified slope deflection method for the second order elastic analysis. The modified slope method is ased on two assumptions, the first is a relatively small axial force in the eams whilst the second is nearly identical forces in the columns. Accordingly, the geometrical changes due to axial shortening can e neglected. This method can e summarised as follows: constructing the ending moments at each memer end including the staility functions (see Galamos, 968), constructing the joint and shear equilirium conditions from which the equilirium equations are otained, and eliminating the unknowns from the equilirium equations and otaining the determinantal form of the critical load pattern and finally solving the determinantal form y a trial and error method. In order to explain the difficulty of using this method, Mahfouz (993) 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 sujected 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.

16 Theory and Methods for Evaluation of Elastic Critical Buckling oad 87 α B M BC M BA M CB C M CD A A D D MAB M DC R A R D Figure 3.8. Single-ay single-storey framework: loading pattern and deflected shape In this example, ten preliminary equations must e formulated. These equations are for M AB, M BA, M CD, M DC, M BC, M BC, R A, R D, A and D. These equations are then sustituted into the three asic equilirium equations to otain their new form. It can e concluded that as the numer of ays and stories increases, the numer of preliminary equations increases too. This technique therefore cannot e used when dealing with more highly indeterminate frameworks such as multi-ay 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 Direct method This method is ased on two main steps, see Salem (968). The first step is the ready prepared operations of rotations and the sway of axially compressed memers which are ased 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 possile uckling modes of the given framework. Then the operations of sway and rotations for every memer of this framework are uiltup separately corresponding to its distorted configuration. Since

17 Theory and Methods for Evaluation of Elastic Critical Buckling oad 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 e equal to zero. This procedure will give many equations which are equal to the numer of the framework joints. In rectangular frameworks other than symmetrical ones, another set of equations has to e otained y equating the relative displacements of the framework columns. Finally, y eliminating the unknowns from these equations, a determinantal equation is otained for the elastic critical load. This determinantal equation has a numer of solutions from which the least is called the first uckling load. The solution of such determinantal form can only e done y the method of trial-and-error using a computer program. Figure 3.9 shows the asic simple operations of rotation and sway of an axially compressed isolated memer for oth cases of fixed and pin-ended ases. The principle of supperposition of any numer of states of sway and rotation of an isolated axially compressed memer is applicale 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 memer into any numer of states of sway or rotation is also applied under the same condition. The no-shear staility functions m, n and O were introduced y Merchant (955) to deal with the case of a memer with fixed ends while Salem (968) treated the hinged end case y introducing the staility function n for the no-shear sway for such memers. 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 3.0 for memers with fixed and pin-ended ases. The non-dimensional staility functions S, C, S, m, n, O, and n (Appendix A) indicated in Figures 3.9 and 3.0 are all functions of the ratio ρ of the axial load to

18 Theory and Methods for Evaluation of Elastic Critical Buckling oad 89 π EI. These staility functions are taulated y ivesley and Chandler (956). A FORTRAN program is developed for the staility analysis of steel frame structures. The program is ased on the direct method. The program evaluates the value of ρ i of each column of the investigated framework at the critical uckling load, then the effective length factor of each column are computed. Figure 3. illustrates the developed program.

19 Theory and Methods for Evaluation of Elastic Critical Buckling oad 90 a) Rotation θ M = Sθ θ M = S θ = S( C) θ = S θ M = CSθ ) ure-shear sway m M= = S( C) M= = S n m M= M= nθ M= n θ c) No-shear sway θ θ m = θ = θ n M = -Oθ Figure 3.9. Basic simple operation of rotation and sway

20 Theory and Methods for Evaluation of Elastic Critical Buckling oad 9 ' " M = M M ' M ' = m " M θ θ " = nθ ' = " = ' m = S( C ) " m = θ = ' " M M M M ' = m " M = Oθ ' " M = M M θ M ' = n ' π ρ " M = θ n " θ ' = " = ' = " S n " = θ n ' " M = M M θ M ' = m ' " M = nθ -Oθ " θ ' = " = ' m = S( C ) " m = (θ θ ) θ θ ' " M = M M State of sway M ' = m = State of pure-shear sway " M = nθ -Oθ State of no-shear sway Figure 3.0. Decomposition of the general state of sway

21 Theory and Methods for Evaluation of Elastic Critical Buckling oad 9 Start Input: Structural geometry, etc Choose the starting value of ρ = Calculate staility functions for each memer Sustitute the staility functions into the determinantal form Calculate the determinant value Increase ρ No Does the determinant change its sign? Yes Stop Figure 3.. Flowchart for computer program ased on the direct method Finite element method The finite element method can e applied to the evaluation of the elastic critical load for structural frameworks (see Allen and Bulson, 980). The finite element method is ased upon the use of local functions (i.e. these defined over su-regions or finite elements of the structural system). The other methods, such as the modified slope-deflection

22 Theory and Methods for Evaluation of Elastic Critical Buckling oad 93 method, are usually thought of as eing ased upon overall functions (i.e. those defined over the entire region of the structural system). In the finite element method, each memer of the structure is sudivided into a series of fairly short elements. The deformation over each element may e defined y a simple polynomial function. The coefficients of these polynomial functions may e determined if the displacements of each node are known. As a result, the individual displacements of the entire structure may e calculated and consequently the ehaviour of the structure may e fully descried in terms of the displacements of the nodes. For equilirium the increment in total potential energy must e stationary with respect to these nodal displacements. This leads to a set of linear homogeneous equations, where the dependent variales of these equations are the nodal displacements Ψ, i.e. the following eigenvalue prolem: λ f [ ]{ Ψ } [ ]{ Ψ } = (3.7) CG CE where f λ is the load factor, the connecting joints (nodes), CE is the gloal elastic stiffness matrix corresponding to CG is the geometric stiffness matrix. The first eigenvalue, i.e. the smallest value of f unstale is termed the critical load factor λ cr. f λ at which the structure ecomes This classical eigenvalue approach discussed y many authors among them rezemieniecki (968), Allen and Bulson (980), Graves Smith (983), Breia and Ferrante (986), Coates and ong, (988), Galamos (988) and Bathe (996). The eigenvalues and eigenvectors can e otained y applying several techniques, among them vector iteration methods i.e inverse iteration, forward iteration and Rayleigh quotient iteration, transformation methods such as Jacoi method and generalised Jacoi

23 Theory and Methods for Evaluation of Elastic Critical Buckling oad 94 method, and the suspace iteration method. Suroutines, written in FORTRAN 77, are availale in Bathe (996). 3.6 Verification of the developed code for staility analysis The developed program ased on the direct method has een developed for the elastic critical uckling analysis of -D frameworks, Section In order to verify the developed program, the estalished theoretical results presented y Timoshenko and Gere (963), Chajes (974) and Renton (967) are used. ere, various framework models have een analysed. The first example used is the fixed ase framework ABCD shown in Figure 3.a. The framework is prevented from sidesway. It is sujected to two equal vertical loads at corners B and C. When the vertical loads reach their critical value cr, the distorted configuration of the framework will e as shown in Figure 3.. The operations of rotations are uilt up for every memer of the framework separately as given in Figure 3.c. 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: M = M M 0, B BA BC = 4 S θ θ = 0, and (3.8) M = M M 0, θ C CB CD = S 4 θ = 0. (3.9)

24 Theory and Methods for Evaluation of Elastic Critical Buckling oad 95 Eliminating the unknowns ( θ andθ ) from (3.8) and (3.9), the elastic critical load equation ecomes: det S 4 S 4 = 0. (3.0) I I B I = C I = I I = I I = I = I = I I A D Rigid racing a) oading pattern 4 θ θ - θ -4 θ S θ θ θ -S θ C S θ - C S θ ) Distorted configuration c) Operation of rotation Figure 3.. Single storey single-ay fixed ase framework prevented from sway

25 Theory and Methods for Evaluation of Elastic Critical Buckling oad 96 Following the same previous procedure, the elastic critical load equation (3.) of a single-ay single-storey fixed ase framework permitted to sway (Figures 3.3), is otained: 0 ) ( ) ( 4 4 det = C S m C S m m m m n m n (3.) where the third equation is otained y equating the sidesways at joints B and C. In this equation, the unknowns are and, θ θ. For a single-ay single-storey hinged ase framework permitted to sway (Figures 3.4), the elastic critical load equation is det " " " " = n S n S n n n n n n. (3.) In order to verify the developed program, the previous descried framework models have een analysed and the results otained are compared with estalished theoretical results are given in Tale 3..

26 Theory and Methods for Evaluation of Elastic Critical Buckling oad 97 θ θ n n θ θ m 4 θ θ θ 4 θ m - -O θ m -O θ m - (a) oading pattern and distorted configuration () Operation of rotation Figure 3.3.Single-storey single-ay fixed ase framework permitted to sway 4 θ θ θ 4 θ θ θ " " n θ n θ n - n (a) oading pattern and distorted configuration () Operation of rotation Figure 3.4.Single-storey single-ay hinged ase framework permitted to sway

27 Theory and Methods for Evaluation of Elastic Critical Buckling oad 98 Tale 3.. Comparison of the theoretical and developed code in plane uckling loads. Model Theoretical Results Otained results Chajes (974) and Renton (967) I I I EI cr = 5. cr ρ = = π EI ρ =.554 Chajes (974) and Renton (967) I I I EI cr = cr ρ = = π EI ρ = Timoshenko and Gere (963) I I I EI cr =. 8 cr ρ = = π EI ρ = Timoshenko and Gere (963) I I = 0 I π EI cr = 4 cr ρ = = 0. 5 π EI ρ = 0.499

28 Theory and Methods for Evaluation of Elastic Critical Buckling oad Concluding remarks In this chapter, the staility concept of idealised framework model has een presented. The methods of evaluating the elastic critical load as well as literature on the staility analysis are also reviewed. Finally, verification of the developed program for the staility analysis of frameworks has een carried out. In Section 3.5 it has een concluded that the methods of analysis ased on trial and error are difficult to use in the design optimization process. Consequently, the eigenvalue approach is more suitale for the design optimization process.