1-s2.0-s0141029606004998-main

Engineering Structures 29 (2007) 2344–2352www.elsevier.com/locate/engstruct

Simplified procedure for determining buckling loads of three-dimensionalframed structures

Konuralp Girgin∗, Gunay Ozmen

Faculty of Civil Engineering, Istanbul Technical University, Maslak 34469, Istanbul, Turkey

Received 1 May 2006; received in revised form 27 November 2006; accepted 28 November 2006Available online 17 January 2007

Abstract

A simplified procedure for determining approximate values for the buckling loads of three-dimensional framed structures is developed. Theprocedure utilises lateral load analysis of structures and yields errors less than 10%, which may be considered suitable for design purposes.The structures with or without rigid floor diaphragms may be considered readily. Buckling loads of both regular and irregular structures may beobtained. The proposed procedure is applied to several numerical examples and it is shown that all the errors are in the acceptable range andgenerally on the safe side. Determining the buckling loads of structures using SAP2000 is also discussed.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Buckling load; Buckling modes; Isolated subassembly; Rigid floor diaphragms; Irregular structures; SAP2000 applications

1. Introduction

The stability analysis of framed structures is of paramountimportance in design procedures. However, such an analysisrequires either the usage of eigenvalue computer algorithmsor complex second-order matrix formulations. In spite of theavailability of algorithms based on the finite element methodand powerful computer programs, a stability analysis is stillconsidered a cumbersome and impractical task, particularly forthree-dimensional (3D) structures. Instead, such an analysisis commonly carried out in practice using simplified methodsbased on 2D analyses, i.e., by breaking up the structure intoorthogonal plane frames [1].

Further simplifications are incorporated in contemporary de-sign codes, whereby practical charts, diagrams or formulae aregiven for determining the “effective lengths” of columns [2–6].The so-called “isolated subassembly approach” used in codesand specifications, was originally developed by Galambos [7].A major limitation of the isolated subassembly approach is thatit does not consider the interaction effects of structural elementsother than those in the immediate neighbourhood of the joints.

∗ Corresponding author. Tel.: +90 212 2856556; fax: +90 212 2856587.E-mail address: [email protected] (K. Girgin).

0141-0296/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2006.11.026

Erroneous results corresponding to this fact have been recog-nised by several authors and numerous publications have beenmade to improve the applicability of the subassembly approach.Most of these publications use the so-called “storey-bucklingapproach” which accounts for the horizontal interaction be-tween the columns in a storey of the unbraced frame. Amongthe papers, which use storey-buckling approach, the publica-tions of Lui, Aristizabal-Ochoa and Cheong-Siat-Moy may behighlighted [8–10]. A reasonably comprehensive list of theseimprovement studies is given by Ozmen and Girgin [11].

Apart from the above mentioned improvement studies,certain independent methods for determining an approximatevalue for the overall buckling load of plane frames are alsodeveloped, whereby the displacements due to a fictitious lateralloading is utilised [12–14].

Recently, in AISC (1999), the isolated subassemblyapproach has been abandoned and it has been stated that“. . . the effective length factor K of compression members shallbe determined by structural analysis.” [15]. However in severalwidely used codes (such as ACI [3] and Eurocode 3 [4]) thesubassembly approach and related charts and formulae are stillbeing used.

As for the simplified buckling analysis of 3D structures, veryfew publications exist. Only Aristizabal-Ochoa has extendedhis studies to cover regular 3D structures [1,16]. In this paper,

http://www.elsevier.com/locate/engstruct

mailto:[email protected]

http://dx.doi.org/10.1016/j.engstruct.2006.11.026

K. Girgin, G. Ozmen / Engineering Structures 29 (2007) 2344–2352 2345

(a) System and loading. (b) Buckling mode.

Fig. 1. Multi-storey framed structure and buckling mode.

a practical method for determining the buckling load of 3Dframed structures, will be explained and applied to numericalexamples. The method, which is developed by using theprocedure given by Cakiroglu [12], is performed by applyinga simple quotient, based on the results of a fictitious lateralload analysis. The method is also applicable to irregular framedstructures, whereby the beams of at least one level are curtailed.

2. System buckling load of unbraced 3D structures

An unbraced multi-storey framed structure, which iscomposed of beams and columns made of linear elasticmaterial, is under the effect of vertical loads as shown inFig. 1(a).

The structure is in the state of “Stable Equilibrium” and, ifthe axial deformations are neglected, all the displacements anddeformations are zero. Internal forces of the columns consist ofonly axial forces N while all the internal forces of the beamsare zero. The axial force of a column may be expressed as

N = n P, (1)

where n is a dimensionless coefficient and P is an arbitrarilychosen load parameter. When the load parameter reaches toa critical Pcr value, another (buckling) state of “UnstableEquilibrium” may exist. The lateral displacement diagramcorresponding to this new state, which is shown schematicallyin Fig. 1(b), is called the “Buckling Mode” of the structure [17].

2.1. Buckling mode shapes

In general, the buckling mode shape of a 3D structure iseither “translational” or “rotational” in character. In Figs. 2 and3, the plan view of translational mode shapes in directions Xand Y are shown, respectively.

For the mode in direction X , although slight displacementsin direction Y and minor rotations occur at each storey,

Fig. 2. Buckling mode in direction X .

Fig. 3. Buckling mode in direction Y .

Fig. 4. Rotational buckling mode.

dominant displacements are in direction X . Similarly, dominantdisplacements are in direction Y for the mode in direction Y .However, it must be remarked that, in certain cases translationalmode displacements may occur in arbitrary directions. Inthe case of the rotational buckling mode, the dominantdisplacements are rotational as shown in Fig. 4.

Investigations on several numerical examples have shownthat, in most of the cases, rotational modes occur for higherbuckling modes only, hence are not important from thedesigner’s point of view. In cases where the first buckling modeis rotational, the next mode appears to be translational with aquite close buckling load value to the value corresponding tothe rotational mode.

2346 K. Girgin, G. Ozmen / Engineering Structures 29 (2007) 2344–2352

3. A simplified procedure for determining the buckling loadof 3D framed structures

In the following, a practical method will be explainedand applied to numerical examples. The method, which isdeveloped by using the procedure given by Cakiroglu [12], isapplied by using a simple quotient based on the results obtainedby standard frame analysis software.

Consider the fictitious lateral loading shown in Fig. 5 appliedto the structure shown in Fig. 1. It is assumed that this loadingprovides displacements identical to (or proportional to) thosecorresponding to the buckling mode.

The buckling load parameter can be determined by usingBetti’s Reciprocal Theorem applied to the states shown inFigs. 1 and 5. According to this theorem, it may be written that

W1 = W2, (2)

(a) Lateral loading.

(b) Displacements.

Fig. 5. Fictitious lateral loading and displacements of 3D structure.

Fig. 6. Column relative displacement diagram for plane structures.

Fig. 7. Column relative displacement diagram for 3D structures.

where W1 is the virtual work of the force system in Fig. 1(a) inconjunction with the displacements in Fig. 5(b), and W2 is thevirtual work of the force system in Fig. 5(a) in conjunction withthe displacements in Fig. 1(b) [18]. Since the displacementsof Figs. 1(b) and 5(b) are assumed to be the same, thedisplacements and deformations corresponding to the lateralfictitious loading will be used in the following.

3.1. Determination of W1

According to the Principle of Virtual Work, W1 can becomputed as the work done by the internal forces of the loadingshown in Fig. 1, in conjunction with the deformations inducedby the fictitious lateral loading. Ozmen and Girgin [11] havegiven an approximate expression for W1 as

W1 = 1.2P∑

nδ2

hc(3)

for two-dimensional (plane) structures. The summation iscarried out for all the columns. It must be noted that, columnindices are omitted for the sake of simplicity. Here P and n are,respectively, the load parameter and a dimensionless coefficientas defined above. As shown in Fig. 6, δ and hc represent therelative displacement and the height of a particular column.

In the case of 3D framed structures, δ relative columndisplacements are in arbitrary directions, and can berepresented by the components δx and δy , as shown in Fig. 7.

In this case, one can transform Eq. (3) into

W1 = 1.2P∑ n

hc(δ2

x + δ2y), (4)


which can be used in Eq. (2).

3.2. Determination of W2

The virtual work of the force system in Fig. 5(a) inconjunction with the displacements in Fig. 1(b) (Fig. 5(b)) cansimply be written as

W2 =

∑Hx dx + Hydy, (5)

where Hx (Hy) and dx (dy) represent the lateral storey loadsand storey displacements, respectively, as shown in Fig. 5. Thesummation is carried out for all joints. Here again joint indicesare omitted for the sake of simplicity.

3.3. Simplified buckling load formula

Substituting the expressions for W1 and W2 givenrespectively by Eqs. (4) and (5) into Eq. (2) and solving forP(Pcr), the buckling load is obtained as

Pcr =

∑Joints

Hx dx + Hydy

1.2∑

Columns

nhc

(δ2x + δ2

y). (6)

It must be noted that this formula is approximate, since thefictitious lateral loading corresponding to the buckling modedisplacements are not known initially. However, application onseveral numerical examples has shown that the value of Pcr isnot strongly dependent on the initial choice of lateral loads. Itmay be recommended that the lateral load at each joint shouldbe selected as proportional to the vertical load Pi existing at thejoint.

3.4. Analysis procedure

Buckling loads of 3D framed structures can be determinedas follows:

• apply lateral forces proportional to the vertical loads at eachjoint,

• compute relative storey displacements using any existingsoftware, and

• compute the critical load Pcr by using Eq. (6).

In practice, lateral loads should be applied first in directionX , then in direction Y and two separate values for Pcr shouldbe obtained. The smaller of the two values should be selectedas the approximate buckling load. In practice, rigid floordiaphragms may or may not exist in storeys. The proposedmethod is general, i.e., is applicable to both cases without anymodification. In certain cases, beams of a certain storey (orstoreys) may be curtailed. In this case, the structure is called“irregular” and storey-buckling approaches are not applicable.However, the proposed method is applicable to both regular andirregular 3D structures just as easily.

4. Numerical examples

In the following, the procedure outlined above will beapplied to three examples and the results will be discussed.

(a) System and loading.

(b) Plan.

Fig. 8. Dimensions and loading of Example 1.

Fig. 9. Fictitious lateral loading of Example 1.

4.1. Example 1

Dimensions and loading of a one-storey 3D structure areshown in Fig. 8.

This example is first introduced by Razzaq and Naim [19]and then used by Aristizabal-Ochoa [1]. It is assumed thatthe floor acts as a rigid diaphragm with flexural restraintsprovided by the girders connecting the columns. A W 10 × 33section is utilised for all beams and columns with the followingproperties: Iy = 173.498 in.4 (72,215,320 mm4), Ix =

34.182 in.4 (14,227,260 mm4), A = 10.08 in.2 (6503.2 mm2),elastic moduli E = 29,000 ksi (200.1 × 106 kN/m2) andG = 11,600 ksi (80 × 106 kN/m2). The minor axis of the crosssection of the girders is parallel to the global vertical axis Zwhile the columns are oriented as shown in Fig. 8(b) with theirmajor axes along the global X axis.

It can easily be deduced from the characteristics of thestructure that the first buckling mode is in direction X . Hence,the fictitious lateral loading is chosen as shown in Fig. 9.


Table 1Buckling load calculations for Example 1

Joint (column) no. Hx dx (δx ) × 102 dy(δy) × 102 Hx dx n hcnhc

(δ2x + δ2

y)

1 0.50 82.53 −4.40 41.27 0.50 250 13.662 1.00 82.53 4.40 82.53 1.00 250 27.323 0.50 75.49 4.40 37.75 0.50 250 11.444 0.25 75.49 −4.40 18.87 0.25 250 5.72Sum 180.42 2.25 58.14


After carrying out the lateral load analysis for the fictitiousloading, joint displacements are obtained. The terms used forthe application of Eq. (6) are shown in Table 1.

Applying Eq. (6) yields

Pcr =180.42 × 10−2

1.2 × 58.14 × 10−4 = 258.60 Kips (1150 kN).

The total critical load for the structure is found as

Pcr = 2.25 × 258.60 = 581.85 Kips (2588 kN).

This value of the total critical load compares well with theresults reported by Razzaq and Naim [19] (590.0 Kips) andAristizabal-Ochoa [1] (587.9 Kips). The exact value of the totalbuckling load for this structure is found to be

Pcr = 583.27 Kips (2594 kN).

Thus, the value found by the proposed method has an error of−0.24%. Determination of the exact values will be discussed ina separate chapter in the following.

The fictitious lateral loading shown in Fig. 9 may beconsidered as representing the earthquake loading, since all thehorizontal loads are proportional to the vertical loads acting atthe same joints. The same procedure is applied by using windloads as the fictitious loading and the critical load is obtained as

Pcr = 593.26 Kips (2639 kN),

which has an error of 1.71%.In the case when rigid floor diaphragms do not exist, the

procedure can be applied in just the same manner. The onlydifference is not to take into account the diaphragm constraintsin the lateral load analysis. The total critical load for this caseis found as

Pcr = 575.67 Kips (2561 kN),

which has an error of −0.15% compared with the exact valueof 576.54 Kips (2564 kN).

4.2. Example 2

Dimensions and loading of a one-storey L-shaped structureare shown in Fig. 10.

Moment of inertia for all the beams and for bothprincipal directions of the columns is I . Both axial and sheardeformations will be neglected in the analysis. It is againassumed that the floor acts as a rigid diaphragm. Due tothe diagonal symmetry of the system, fictitious loading indirections X and Y will yield identical results. The loading ischosen in direction X as shown in Fig. 11.

Lateral load analysis is carried out by taking the moment ofinertia I , modulus of elasticity E , and storey height h equal tounity. Thus the buckling load parameter will have a multiplierof E I

h2 . After performing the lateral load analysis, the terms usedfor the application of Eq. (6) are obtained as shown in Table 2.Column numbers are the same as their upper joint numbers,which are shown in Fig. 10.


Pcr =250.286 × 10−2

1.2 × 5222.27 × 10−4E Ih2 = 3.994

E Ih2 .

The exact value for the buckling load is found to be

Pcr = 4.184E Ih2 .

Thus, the value computed by the proposed method has an errorof −4.54%. Here again, the lateral loading shown in Fig. 11may be considered as representing the earthquake loading. Ifthe same analysis is performed by using wind loading Pcr =

4.036 E Ih2 is obtained, which has an error of −3.54%.


Fig. 11. Fictitious lateral loading of Example 2.

(a) Real mode shape. (b) Mode shape according to fictitiousloading.

Fig. 12. Real and assumed modal shapes of Example 2.


Joint (column) no. Hx dx (δx ) × 102 dy(δy) × 102 Hx dx n hcnhc

(δ2x + δ2

y)

1 1.00 21.209 0.360 21.209 1.00 1.00 449.952 2.00 21.209 −0.060 42.418 2.00 1.00 899.653 1.00 21.209 −0.480 21.209 1.00 1.00 450.054 2.00 20.787 0.360 41.574 2.00 1.00 864.465 3.00 20.787 −0.060 62.361 3.00 1.00 1296.316 1.00 20.787 −0.480 20.787 1.00 1.00 432.337 1.00 20.364 0.360 20.364 1.00 1.00 414.828 1.00 20.364 −0.060 20.364 1.00 1.00 414.70Sum 250.286 5222.27

The real buckling mode shape of the structure is shown inFig. 12(a) whereby joint displacements are in a direction whichmakes an angle of 45◦ to the horizontal plane. On the otherhand, the modal displacements due to the fictitious loading aremainly in direction X .

It is interesting to note that the rather great discrepancybetween the real and assumed modal shapes does not affect theresulting buckling load to a great extent.

4.3. Example 3

Dimensions and loading of a two-storey 3D reinforcedconcrete structure are shown in Fig. 13.

The beams at the lower right corner of the 1st storey arecurtailed, hence, the structure is irregular, i.e., the storey-buckling approaches are not applicable. However, it will beshown that the proposed method can be applied to this kind ofstructure just as readily. The floors consist of 12 cm thick plates,which act as rigid diaphragms. Beam and column cross sectionsare 25 × 50 cm2 and 35 × 35 cm2, respectively. Beam sectionsare considered as tee-sections with flange widths as shown inFig. 14.

The elastic modulus for concrete is 3 × 107 kN/m2. Inorder to account for the effect of cracking and reinforcementon relative stiffness, the rigidities of beams and columns aremultiplied by 0.35 and 0.50, respectively [3].



Fig. 14. Flange widths of tee-beams of Example 3, (m).

Fig. 15. Fictitious lateral loading of Example 3, (kN).

It can be deduced that the first buckling mode is in directionX . Hence the fictitious lateral loading is chosen as shown inFig. 15.

After performing the lateral load analysis for the fictitiousloading, joint displacements are obtained and the terms usedfor the application of Eq. (6) are shown in Table 3. Here again,



Joint (column) no. Hx (kN) dx × 105 (m) dy × 105 (m) Hx dx δx × 105 (m) δy × 105 (m) n hc (m) Ca

1 0.15 39.36 −0.74 5.90 17.61 −0.48 0.15 4.00 11.642 0.40 39.36 0.00 15.74 17.61 0.00 0.40 4.00 31.013 0.15 39.36 0.85 5.90 17.61 0.56 0.15 4.00 11.644 0.40 39.89 −0.74 15.96 17.96 −0.48 0.40 4.00 32.285 1.00 39.89 0.00 39.89 17.96 0.00 1.00 4.00 80.646 0.40 39.89 0.85 15.96 17.96 0.56 0.40 4.00 32.297 0.15 40.42 −0.74 6.06 18.30 −0.48 0.15 4.00 12.578 0.40 40.42 0.00 16.17 18.30 0.00 0.40 4.00 33.499 0.15 40.42 0.85 6.06 40.42 0.85 0.15 8.00 30.65

10 0.18 21.75 −0.26 3.92 21.75 −0.26 0.33 4.00 39.0311 0.48 21.75 0.00 10.44 21.75 0.00 0.88 4.00 104.0712 0.18 21.75 0.29 3.92 21.75 0.29 0.33 4.00 39.0313 0.48 21.93 −0.26 10.53 21.93 −0.26 0.88 4.00 105.8214 0.90 21.93 0.00 19.74 21.93 0.00 1.90 4.00 228.4415 0.18 21.93 0.29 3.95 21.93 0.29 0.58 4.00 69.7516 0.18 22.12 −0.26 3.98 22.12 −0.26 0.33 4.00 40.3717 0.18 22.12 0.00 3.98 22.12 0.00 0.58 4.00 70.95Sum 188.08 973.67

a C =nh (δ2

x + δ2y).

c

column numbers are taken as the same as their upper jointnumbers, which are shown in Fig. 13.


Pcr =188.08 × 10−5

1.2 × 973.67 × 10−10 = 16,097 kN.

The exact value for the buckling load is found to be

Pcr = 16,765 kN.

Thus, the value computed by the proposed method has an errorof −3.98%. When the same analysis is carried out by usingearthquake and wind loadings, the buckling load values arefound to have respective errors of −2.23% and −4.95%.

5. Conclusions

In this paper, determining the buckling loads of multi-storey3D framed structures is investigated. The main conclusionsderived may be summarised as follows:

1. A simplified procedure for determining an approximatevalue for system buckling load is developed. The procedureutilises a simple quotient based on the results of a fictitiouslateral load analysis.

2. The structures may or may not have rigid diaphragms at floorlevels. The proposed procedure is applicable to both casesequally.

3. The procedure is applicable to both regular and irregularcases equally easily.

4. The procedure yields errors, which are less than 10% forall the considered examples. This order may be regardedacceptable from the designer’s point of view.

5. The buckling load value is not strongly dependent on thechoice of lateral loading. Hence, any existing lateral loadingon the structure under consideration may be used withoutlosing a significant amount of accuracy.

6. The proposed procedure is applied to several numericalexamples and it is seen that all the errors are in the acceptablerange and for most of the cases on the safe side.

7. Determining the buckling loads of structures usingSAP2000 [20] is discussed and it is shown that thecompression members should be divided into a propernumber of pieces when using SAP2000.

Appendix A. Exact values of the buckling loads

In the numerical examples presented above, all the resultsare compared with the “exact” values of buckling loadsand the corresponding errors are determined. In view of thecharacteristics of existing software, determining the exactvalues seems to be a somewhat delicate matter, which will bediscussed herein.

Since the most widely used contemporary structural programis SAP2000 [20], the buckling loads of the exact values ofthe above examples are determined by a special application ofthis software. The results are then checked by using special-purpose software developed by Girgin [21]. However, standardapplication of SAP2000 causes somewhat extensive errors asshown in Table A.1.

These rather large (and unsafe) errors are due to the fact that,SAP2000 uses the so-called “geometric stiffness” formulae forthe members with P–∆ effects. Formerly, Horne and Merchanthave described P–∆ (or second-order) effects by using so-called “stability functions” [17]. However, the geometric

Table A.1Errors in buckling load values for standard SAP2000 application (%)

Example Error

1 18.762 13.813 12.25


Fig. A.1. Cantilever beam and loading.

Table A.2Values of coefficient k found by SAP2000 for cantilever beam

Number of pieces k Error (%)

1 3.000 21.592 2.597 5.254 2.499 1.288 2.475 0.31

16 2.469 0.0632 2.468 0.02

stiffness formulae recognise these effects only approximately.These formulae are reasonably accurate when the memberlengths are sufficiently small. Hence, when using SAP2000, itis necessary to divide the compression members into a propernumber of pieces. This fact can easily be demonstrated on asimple example shown in Fig. A.1.

When the shear deformations are neglected, the exact valueof the buckling load of a cantilever beam is given by the well-known expression

Pcr =π2

4E IL2 = k

E Ih2 (k = 2.467).

The value of the coefficient k is found through the use ofSAP2000 by dividing the beam into a varying number of piecesand the corresponding results are shown in Table A.2.

It is seen that the standard application of SAP2000, i.e.,without dividing the beam into pieces, produces 21.59% error,which can be considered rather high. Errors diminish quiteswiftly, when the beam is divided into pieces and the number ofpieces increases. Research on several numerical examples hasrevealed that, in order to achieve reasonably accurate results,i.e., results with an error order of less than 5%, compressionmembers should be divided into at least 4 pieces. It is clearthat this operation increases the number of degrees of freedomconsiderably.

In the above numerical examples, all the exact valuesare determined through SAP2000 solutions by dividing the

compression members into 20 pieces. The results are thenchecked by using special-purpose software developed byGirgin [21], which computes the system-buckling load utilisingthe stability functions.

References

[1] Aristizabal-Ochoa JD. Classic buckling of three-dimensional multi-column systems under gravity loads. Journal Engineering Mechanics,ASCE 2002;128(6):613–24.

[2] AISC. Specification for structural steel buildings. Chicago (IL): AmericanInstitute of Steel Construction; 1988.

[3] ACI 318-02. Building code requirements for structural concrete.Farmington Hills (MI): American Concrete Institute; 2002.

[4] Eurocode 3. Design of steel structures, final draft. Brussels (Belgium):CEN; 2002.

[5] DIN 18800. Part2: Analysis of safety against buckling of linear membersand frames. Berlin: Beuth Verlag GmbH; 1990.

[6] BS 5950-1 British standard. Part 1: Code of practice for design-rolled andwelded sections. 2000.

[7] Galambos TV. Structural members and frames. New York: Prentice-Hall,Inc.; 1968.

[8] Lui EM. A novel approach for K factor determination. EngineeringJournal, AISC 1992;29(4):150–9.

[9] Aristizabal-Ochoa JD. Braced, partially braced and unbraced frames:Classical approach. Journal of Structural Engineering, ASCE 1997;123(6):799–807.

[10] Cheong-Siat-Moy F. An improved K-factor formula. Journal of StructuralEngineering, ASCE 1999;125(2):169–74.

[11] Ozmen G, Girgin K. Buckling lengths of unbraced multi-storey framecolumns. Structural Engineering and Mechanics, An International Journal2005;19(1).

[12] Cakiroglu A. Buckling analysis of multi-storey frames. In: Proc. oftechnical conference of Turkish civil engineers. 1962 [in Turkish].

[13] Stevens LK. Elastic stability of practical multi-storey frames. Proceedingsof the Institute of Civil Engineers 1967;36.

[14] Horne MR. An approximate method for calculating the elastic criticalloads of multi-storey plane frames. The Structural Engineer 1975;53(6).

[15] AISC. Load and resistance factor design specification for structural steelbuildings. Chicago (IL): American Institute of Steel Construction; 1999.

[16] Aristizabal-Ochoa JD. Elastic stability and second-order analysis of three-dimensional frames: Effects of column orientation. Journal of EngineeringMechanics, ASCE 2003;129(11):1254–67.

[17] Horne MR, Merchant W. The stability of frames. London: PergamonPress; 1965.

[18] Neal BG. Structural theorems and their applications. London: PergamonPress; 1964.

[19] Razzaq Z, Naim MM. Elastic instability of unbraced frames. Journal ofStructural Division, ASCE 1980;106(ST7):1389–400.

[20] Wilson EL. Three-dimensional static and dynamic analysis of structures.Berkeley (CA, USA): Computers & Structures, Inc.; 2002.

[21] Girgin K. A method of load increments for determining the second-orderlimit load and collapse safety in R/C structures. Ph.D. thesis. Istanbul:Istanbul Technical University; 1996 [in Turkish].

1-s2.0-s0141029606004998-main

Documents