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1 Send Orders for Reprints to 564 The Open Ciil Engineering Jornal, 16, 1, The Open Ciil Engineering Jornal Content list aailable at: DOI: 1.174/ RESEARCH ARTICE The Elastic Critical oad of Single Angle Colmns Ailin Zhang 1,, Qingfang Zhang 1,3,* and Chao Sn 1 1 College of Architectre and Ciil Engineering, Beijing Uniersity of Technology, Beijing 114, China Beijing Engineering Research Center of High-Rise and arge-span restressed Steel Strctre, Beijing 114, China 3 School of Ciil Engineering, Shijiazhang Tiedao Uniersity, Shijiazhang 543, China Receied: December, 15 Reised: March, 16 Accepted: Jne 3, 16 Abstract: The design of single angle colmns reqires the ability to predict their flexral-torsional bckling strengths and flexral bckling strengths abot the minor principal axis. resent design code proisions are qite different. This paper presents close approximations for the elastic critical load of eqal angle section colmns. It is fond that the member capacities are less than the minor axis capacity at low slenderness and almost remain constant. A formla of the balance point is proposed which is simpler and better nderstanding. Keywords: Balance point, Elastic critical load, Single angle, Simplified eqation. 1. INTRODUCTION Single angles in compression are widely sed in steel strctral members. Single angle section may be eqal-leg or neqal-leg, as shown in Fig. (1). The geometric axes parallel to each leg are x-axis and y-axis separately. The principal axes are -axis and -axis (the -axis is also called the minor principal axis). It shold be noted that, in the following, we will focs or research on the case of eqal-leg angles. α (a) (b) Fig. (1). Single angle section and its axes. For a centrally compressed member, oerall bckling or local bckling may occr. The oerall bckling has three modes: flexral bckling, torsional bckling and flexral-torsional bckling. For a centrally compressed member with an eqal-leg angle, not only flexral bckling abot axis may occr, bt also flexral-torsional bckling abot axis * Address correspondence to this athor at the College of Architectre and Ciil Engineering, Beijing Uniersity of Technology, Beijing 114, China; Tel: ; zqfok@16.com /16 16 Bentham Open

2 The Elastic Critical oad of Angle Colmns The Open Ciil Engineering Jornal, 16, Volme may occr. Howeer, seeral design codes hae different treatments. According to AISC code oad and Resistance Factor Design Specification for Single-Angle Members, it is nnecessary to consider flexral-torsional bckling nder the sitation of calclating the member capacity of single angle colmns [1]. Generally speaking, flexral bckling abot -axis is enogh. It is interesting that similar conclsion has also been drawn by ANSI/AISC36-5 []. In ANSI/AISC36-1 this proision has been changed that flexral-torsional bckling shold be considered for single angles with b / t > [3]. In fact, this applies to only fabricated angles becase all hot rolled angles crrently prodced hae b / t <. In Eropean code EN :5, for angle sections, acconting shold be taken of the possibility that the critical load of flexral-torsional bckling cold be less than flexral bckling [4]. In other words, flexral-torsional bckling force shold be sed to determine the non-dimensional slenderness. In China, rior to Code for design of steel strctres (GB517-3), the design capacity of axially compressed single angles is based on flexral bckling of minor principal axis, and bckling cre b is adopted. Howeer, in GB517-3, the eqialent slenderness ratio is reqired if single angle colmns bend abot -axis, which is deried from flexral-torsional bckling capacity [5]. In this paper, based on elastic stability theory and cross-section characteristic of single angles, a simplified formla abot the balance point of flexral-torsional bckling capacity and flexral bckling capacity was proposed. When the length of colmn is less than the balance point, the elastic critical load can be calclated conseratiely by flexral bckling formla at this point.. EASTIC FEXURA-TORSIONA BUCKING CRITICA OAD OF SINGE ANGE COUMNS For an axial compression member with pinned ends, in the case of single angle, the elastic critical load is the lowest root of the eqation as follows [6]: 1 ( )( )( ) [( ) ( ) ] i z (1) EI () EI (3) z GJ i (4) I i I A (5) where, = elastic critical loads for flexral bckling abot principal axes z = elastic critical load for torsional bckling abot z-axis i = polar radis of gyration abot the shear center, = coordinates of the shear center with respect to the centroid E, G = elastic modls and shear modls of steel J = torsional constant I, I = moment of inertia abot the principal axes = length of the compression member A = cross section area If the section is an eqal leg angle, =, the Eq. (1) redces to

3 566 The Open Ciil Engineering Jornal, 16, Volme 1 Zhang et al. ( )[( )( z ) ] i (6) The three roots for this eqation are gien as follows. z1 (7a) z z z z (1 / i) ( ) ( ) 4 (1 / i ) (7b) z z z z3 (1 / i) ( ) ( ) 4 (1 / i ) (7c) De to z3 z, the critical load is the minimm ale of z1 and z, or, written as min (, z ). In other words, we can say that the compression member wold ndergo flexral bckling abot -axis or flexral-torsional bckling. A serial hot rolled eqal angles was gien by Steel Constrction Manal (14th edition) [7], in which the section 4 4 3/4 has the minimal width-to-thickness ratio(eqals to 19.) and 6 6 5/16 has the maximal(eqals to 5.3). Using the section properties from this book and adopting E=9ksi and G=11ksi, we can plot the elastic critical load er in Fig. (). If the horizontal coordinates of point A in Fig. () is defined as, then we can say that flexraltorsional bckling will happen when the member length is less than. Moreoer, the critical load decreases slowly within this range. As for section / 16, the critical load drops abot only 9.% when the colmn length changes from 5 in. to 114 in. 1 8 cr (kip) 6 4 flexral bckling flexral-torsional bckling (in.) (a) cr (kip) 4 3 flexral bckling flexral-torsional bckling d (in.) (b) 6 6 Fig. (). Critical load-length cres of single angle colmns. Ths, an approximate and conseratie expression can be obtained as follows.

4 The Elastic Critical oad of Angle Colmns The Open Ciil Engineering Jornal, 16, Volme For EI cr = d (8a) For > EI cr = 3. THE BAANCE OINT OF FEXURA BUCKING AND FEXURA- TORSIONA BUCKING When Eq.(7b) eqals to Eq.(7a), we obtain Sbstitting Eq.() to Eq.(4) into Eq.(9) leads to 1 i z z EI ( ) / 1 i GJ I I So the balance point of flexral-torsional bckling and flexral bckling is (8b) (9) (1) EI ( ) / 1 d i GJ I I In fact, we can derie a clearer expression of with a simple model. Using two rectanglar areas represent two legs of single angle, as shown in Fig. (3), we can obtain the cross-section characteristic as follows. (11) Fig. (3). Simplified single angle section model. I = tb 3 /3 I = tb 3 /1 J bt /3 3 b 4 (1) (13) (14) (15)

5 568 The Open Ciil Engineering Jornal, 16, Volme 1 Zhang et al. 3 3 tb /3 tb /1 b i = ( b ) 4 bt 3 (16) in which b is the center-to-center length of a leg segment, t is the thickness of a leg. Sbstitting Eq.(1) to Eq.(16) into Eq.(11), meanwhile E=9ksi and G=11ksi, leads to 1.9 b t (17) 4. COMARISION TO THE EXACT SOUTION To check the alidity of Eq.(17), total 51 eqal angles are selected from Steel Constrction Manal(14th edition). The ales of are obtained by Eq.(11) and Eq.(17) separately and the reslts are shown in Fig. (4). It is worth noting that the horizontal coordinates are the index nmber of 51 eqal angles arranged as the order in that book. Data indicate that the error range is -5.3% to 9.6% Eq.(11) Eq.(16) 8 (in.) No. Fig. (4). Checking the ales of. Table 1 shows all the errors aboe 5% in the approximate expression. The notation E and A are the balance point obtained by exact soltion and Eq.(17) respectiely. λ, d = E / i is the slenderness ratio abot -axis and the colmn length is E. i is the radis of gyration abot the -axis. Table 1. Data of error more than 5%. Angle size E (in.) / / / / 1 / 1 / / 1 / 3 / / / Since the colmns whose slenderness ratios are less than 3 or more than 1 are seldom sed in practice, the error in A is no longer important for them. Then it can be conclded that Eq.(17) can be sed with good accracy to distingish two bckling mode in engineering, althogh this expression is deried from a simplified model. When Eq.(8) is sed to predict the elastic critical load and is obtained by Eq.(17), the reslts are shown in Fig. (5). The section is / 16. The horizontal coordinates are λ, the slenderness ratio abot -axis. Data indicate that the A (in.) Error (%) λ, d

6 The Elastic Critical oad of Angle Colmns The Open Ciil Engineering Jornal, 16, Volme capacity is hardly nderestimate. The error is -16.6% if λ eqals to 4, althogh the error of is only 4.4%. If the constant 1.9 in Eq.(17) is replaced with 1.5, the error will be within -1% exact soltion Eq.(8) 11 cr (kip) Fig. (5). Checking the accracy of Eq.(8). CONCUSION The stable capacity of single angle colmns may be controlled by flexral-torsional bckling or flexral bckling, bt the bondary is not so clear. According to a simplified model, an approximate expression of diiding point, the specified colmn length, has been proide. The elastic critical load will be calclated by flexral bckling instead of flexral-torsional bckling in the range of. Moreoer, since formla of inoles only two parameters of an angle section, engineers will find it to be easy to nderstand. NOTATIONS The following symbols are sed in this paper: A = cross section area of member b = fll width of a leg b = width of a leg referring to midlines, eqal to b t / E = elastic modls of steel = 9 ksi G = elastic shear modls of steel = 11 ksi I, I = moments of inertia abot -axis and -axis i = polar radis of gyration abot the shear center i = radis of gyration abot -axis J = torsional constant = length of compressie member = balance point of flexral-torsional bckling and flexral bckling E A = exact soltion of = approximate soltion of, = elastic critical loads for flexral bckling abot -axis and -axis z = elastic critical load for torsional bckling abot -axis z1, z, z3 = three roots obtained from cbic eqation t = thickness of leg, = coordinates of the shear center with respect to the centroid λ, d = slenderness ratio abot -axis CONFICT OF INTEREST The athors confirm that this article content has no conflict of interest.

7 57 The Open Ciil Engineering Jornal, 16, Volme 1 Zhang et al. ACKNOWEDGEMENTS The athors deeply appreciate the generos spport of the National Natral Science Fondation of China (No ) and the Beijing Natral Science Fondation (No. 13H189). REFERENCES [1] "oad and Resistance Factor Design Specification for Single-Angle Members", American Institte of Steel Constrction: Chicago, I,. [] "Specification for Strctral Steel Bildings", ANSI/AISC 36-5, American Institte of Steel Constrction: Chicago, I, 5. [3] "Specification for Strctral Steel Bildings", ANSI/AISC 36-1, American Institte of Steel Constrction: Chicago, I, 1. [4] "Erocode 3: Design of Steel Strctre art 1-1: General Rles and Rles for Bildings", EN , Eropean Committee for Standardization: Brssels, 5. [5] "Code for Design of Steel Strctres", GD517-3, Chinese lanning ress: Beijing, 3. (in Chinese) [6] S.. Timoshenko, and J.M. Gere, Theory of Elastic Stability., McGraw-Hill: New York, [7] "Steel Constrction Manal", M35-11D, 14 th ed. American Institte of Steel Constrction: Chicago, I, 11. Zhang et al.; icensee Bentham Open This is an open access article licensed nder the terms of the Creatie Commons Attribtion-Non-Commercial 4. International blic icense (CC BY-NC 4.) ( which permits nrestricted, non-commercial se, distribtion and reprodction in any medim, proided the work is properly cited.