Behavior and Design of Angle Column

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1 Behavior and Design o Angle Column Ricardo Junqueira Justiniano Instituo Superior Técnico, Universidade de Lisboa Abstract This thesis presents a study on the behavior and design o short to medium single angle equal leg column. It shows the state o art o concentric and eccentrically loaded single angle equal leg column, which are related by the major roll that initial imperection/ eccentricity have in the ultimate load capacity. The ocus o this thesis is on the stability behavior, post-critical behavior and ultimate load capacity on angle column under ix, pin and spherical support condition. It is revised the design methodology o angle column presented by some authors and it s taken a particular attention on a design methodology based on Direct Strength Method with good results in ixed and pin slender angle column (b/t>25). Finally, the design methodology is adapted to design o: (i) ixed and pin compact angle column (b/t<25) (ii) spherical angle column with a large b/t spectrum. The evaluation o the design method is made by the comparison o the ultimate load capacity indicated by the method and the numerical results obtained by ABACUS (analysis with inite elements S4) or the experimental results obtained in literature. Whenever the quality or the reliability o the results isn t good enough (generally, conservative), a new design curve is presented. 1. Introduction Thin-walled angle columns are known to possess no primary warping resistance (the crosssection warping constant stems exclusively rom secondary warping), which implies an extremely low torsional stiness and, thereore, a high susceptibility to instability phenomena involving torsion, namely lexural-torsional buckling (equal-leg angles are singly symmetric cross-sections). Since the lexural-torsional deormations exhibited by equal-leg angle columns with short-to-intermediate lengths are very similar to local deormations, these members have been said to ail in local-global interactive modes, which explains why most o the existing methods or their design are based on local strength concepts/curves. However, Dinis e Camotim [1] developed a new design method base on Direct Strength Method (DSM) or ix and pin angle column that is clearly more rational, in the sense that it relects closely the angle column structural behaviour.

2 2. Design Methodology The methodology or a more rational DSM based design approaches or ixed-ended and pin ended angle column must exhibit the ollowing characteristics: (i) Since the intermediate angle columns ail mostly in interactive modes combining lexural (minor-axis) and torsional or lexural-torsional (major-axis) deormations, the strength curves involved must be (i 1 ) the current DSM global strength curve and (i 2 ) lexuraltorsional strength curves. (ii) Several lexural-torsional curves must be developed, in order to enable capturing the progressive (length-dependent) erosion o the column post-critical strength as its length increases within the P cr (L) curve plateau. (iii) The eective centroid shit eects, strongly inluencing the PC (Pin) angle column ailure loads must be incorporated into the design approach through a procedure or parameter that only comes into play or the PC columns exactly the idea put orward by Rasmussen [2]. However, such procedure/parameter must relect, as closely as possible, the length-dependence o the column lexural-torsional behaviour. The proposed DSM-based approaches can be cast in a uniied orm, by means o the expressions: nte β ne crt ne β ne a 1 b crt ne a i i te te b 0.25 b 1 2a 1 2a (2.1) where t y crt (2.2) β ( c) te d 1 or F columns or PCcolumns (2.3) where (i) crt is the lexural-torsional critical buckling, (ii) a and b are variables depending on an adimensional parameter to translate the erosion o the column postcritical strength as its length increases within the P cr (L) curve, (iii) ne is the global lexural strength, (iv) β is a parameter that translate the eective centroid shit eect depending on the slenderness and the variables c and d, which depend rom. 3. Fixed Angle Column Using the methodology rom Dinis and Camotim [1] (to design slender angles), numerical tests were perormed with ixed-ended single angle column L110x5, L180x5, L160x10, L125x8 e L90x7 in order evaluate the design method or ultimate capacity when adapted to b/t<25 (see igure 1 and table 1).

3 u/nte u/nte u/nte vs λt 1,60 1,40 1,20 0,80 0,60 0,000 0, ,500 2,000 2,500 3,000 λ t Fig 1: u/nte vs λt. Table 1: Max, Min, Average, S. Deviation in Fixed-ended angle column. Section b/t Nº Tests Max Min Average S. Deviation L110x5 22, ,27 0,78 0,14 L180x10 18, ,42 0,91 1,12 0,15 L160x10 16, ,44 0,91 1,11 0,15 L125x8 15, ,46 0,91 1,11 0,15 L90x7 12, ,42 0,94 1,13 0,13 Total 252 1,46 0,78 1,08 0,15 The results obtained weren t good enough. This could be related to the inluence o local instability on angle with b/t<25. In this thesis, a new design curve was developed by recalibrate the "a" and "b" variable rom Dinis and Camotim method. New numeric tests were perormed in angles (b/t<25) with corner displacements block in major axe. From the tests results, news expressions are presented: Using these expressions and the methodology rom Dinis and Camotim, the result or ix angle column is presented in table 2 and igure 2: u/nte vs λt 1,40 1,20 0,80 0,60 0,000 0, ,500 2,000 2,500 3,000 λ t Fig 2: u/nte vs λt.

4 u/nte u/nte Table 2: Max, Min, Average, S. Deviation in Fixed-ended angle column Section b/t Nº Tests Max Min Avarege S.Desviation L110x5 22, ,10 0,81 0,95 0,07 L180x10 18, ,20 0,92 1,02 0,07 L160x10 16, ,19 0,94 1,05 0,06 L125x8 15, ,20 0,94 1,05 0,07 L90x7 12, ,26 0,98 1,09 0,07 Total 252 1,26 0,78 1,02 0,09 4. Pin Angle Column In the case o pin-ended angle column, the results (table 3 and igure 3) rom adapting the method rom Dinis and Camotim method rom slender angle to b/t<25: u/nte vs λt 2,00 1,50 0,50 0,00 0,000 0, ,500 2,000 λ t Fig 3: u/nte vs λt. Table 3: Max, Min, Average, S. Deviation in Pin-ended angle column. Section b/t Nº Tests Max Min Avarege S.Desviation L180x10 18,00 24,00 1,55 1,12 1,41 0,12 L160x10 16,00 24,00 1,55 1,04 1,39 0,15 L125x8 15,63 16,00 1,51 1,05 1,37 0,14 L200x14 14,29 16,00 1,48 1,03 1,31 0,15 L90x7 12,86 16,00 1,45 1,02 1,24 0,15 Total 96,00 1,55 1,02 1,35 0,15 Using the methodology proposed by Dinis and Camotim and the new "a" and "b" expression developed to the ultimate capacity o angle with b/t<25 (table 4 and igure 4): u/nte vs λt 2,00 1,50 0,50 0,00 0,000 0, ,500 2,000 λ t Fig 4: u/nte vs λt.

5 Table 4: Max, Min, Average, S. Deviation in Pin-ended angle column. Section b/t Nº Tests Max Min Avarege S. Desviation L180x10 18, ,54 1,16 1,33 0,14 L160x10 16, ,57 1,05 1,33 0,16 L125x8 15, ,50 1,05 1,30 0,14 L200x14 14, ,46 1,03 1,25 0,15 L90x7 12, ,41 1,02 1,20 0,13 Total 96 1,57 1,02 1,29 0,15 The results obtained weren t good enough. This could be the result o 2 aspects: (i) The design o ixed-ended angle column is done with an initial imperection o L/750. The design o pin-ended angle column is done with an initial imperection o L/1000. This dierence must be absolved by the variable "a", "b", "c" and "d". Since only the variable "a" and "b" were recalibrated, that could lead to conservative results. (ii) The design method rom Dinis and Camotim, were developed or slender angle column, with slenderness rom 0,5 to 4. In the case o angle column with b/t<25 the slenderness goes rom 0,5 to 1,5. This means that i the design method is conservative in a small slenderness area (around 1), that could have major impact in the result or the case o angles with b/t< Spherical Angle Column 5.1 Slender Angle (b/t>25) When adapting the design method rom Dinis and Camotim rom the case o pin-ended angle column or the case o spherical-ended angle column, maintaining the expressions, and switching the critical buckling, the results were quite wrong, as show table 5. Table 5: Max, Min, Average, S. Deviation in adapted Dinis and Camotim to spherical-ended Dinis e Camotim [1] Average 2,24 S. Deviation 0,41 Max 2,73 Min 1,04 Taking in account the results it was necessary to look at the stability behavior, pos-critical behavior and ultimate behavior, in order to develop a new design or spherical angle column, using Dinis and Camotim s methodology. This implied: (i) new numerical tests in spherical angle column with the corner block in major axe in order to develop new design curve or "a" and "b". (ii) new elastic non-linear tests in spherical angle column in order to understand the pos-critical behavior. (iii) new elastic non-linear tests in ixed-ended angle column in order to obtain the equilibrium path rom a angle column with no centroide shit eect. Based on the elastic tests or ixed-ended and new spherical-ended angle columns

6 Fu/Fnte elastic tests, it was possible to develop new design curve or "c" and "d" or the spherical design. As results, the new expressions or "a", "b", "c", "d" is presented: In order to evaluate the design proposed, 643 numerical tests were perormed with angle column with slenderness rom 0,5 to 4. The resume o the result are presented in table 6 and the total results in igure 5. 1,20 b/t >25 1,10 0,90 b/t >25 0,80 0,000 0, ,500 2,000 2,500 3,000 3,500 4,000 λ FT Fig 5: u/nte vs λt. Table 6: Max, Min, Average, S. Deviation in spherical-ended angle column. Section b/t nº tests Max Min Average S.Deviation L70x70x1,2 58, ,14 0,98 1,05 0,04 L75x75x1, ,12 0,99 1,04 0,03 L50x50x1,2 41, ,11 0,99 1,05 0,03 L60x60x1, ,10 0,97 1,06 0,04 L90x90x2, ,09 0,96 1,04 0,04 L70x70x ,09 0,97 1,04 0,04 Total 643 1,14 0,96 1,05 0,04 Evaluating the results obtain, and considering the database o 643 numerical tests, the design method propose obtain good results with the numerical tests. As can be seen in igure 5, every results were obtained between 0,95 < u/nte < 1,15.

7 Fu/Fnte 5.2 Angle (b/t<25) In order to considering the geometric eect observed in angle with b/t<25, new numerical tests were perormed to the case o spherical angle column with b/t<25. From the tests results, new expression or "a" and "b" were developed, as presented: Taking in account the news expression and 260 new numerical tests o ultimate capacity o spherical-ended angle column to evaluate the proposed expression, the results are presented in table 7 and igure 6. b/t<25 1,50 0,50 0,00 0,00 0,50 1,50 2,00 2,50 λ t b/t<25 Fig 6: u/nte vs λt. Table 7: Max, Min, Average, S. Deviation in spherical-ended angle column (b/t<25). Secção b/t nº de Média D.Padrao Máximo Mínimo L125x125x ,10 0,07 1,22 0,89 L160x160x ,09 0,08 1,23 0,88 L180x180x10 15, ,08 0,07 1,20 0,83 L200x200x14 14, ,11 0,07 1,26 0,93 L90x90x7 12, ,12 0,06 1,26 0,97 L45x45x3,5 12, ,11 0,06 1,26 0,97 total 260 1,10 0,07 1,26 0,83 From the evaluation o the result, it was obtain good match. It's also worth to point out that in order to have slenderness bigger than 2, it was needed a yield stress around 2000 MPa, which is theoretical steel capacity. 5.3 Experimental results From research in literature, 81 experimental tests on spherical angle column were ound and compared with the design proposed. The result are presented in table 8 and igure 7.

8 Fu/Fnte Experimental Tests 1,60 1,40 1,20 0,80 0,60 0,500 0,600 0,700 0,800 0, ,100 1,200 λ FT Fig 7: u/nte vs λt. Table 8: Max, Min, Average, S. Deviation in spherical-ended angle column (b/t<25). Author nº Tests Average S.Deviation Max Min Kennedy e Murty (1972)[3] 4 1,11 0,17 1,36 0,98 Kitipornchai e Lee (1986) [4] 7 1,05 0,03 1,09 1,04 Adluri e Madugula (1996) [5] 3 1,15 0,12 1,28 1,04 Fan (2009) [6] 9 1,06 0,15 1,39 0,92 Ban, Shi, Shi, Wang (2013) [7] 57 1,14 0,09 1,35 0,98 Total 80 1,12 0,10 1,39 0,92 It s worth to point out that only lexural-torsional buckling mode angle column were considerate or evaluation. Due to a extended database o the initial imperection ound in Ban et al. [7] experimental results, it was possible to evaluate i, on the cases where the design method propose results were more conservative, the initial imperections considered in the design method were bigger than the experimental tests initial imperection. The initial imperection o the experimental tests and the design method or the more conservative results is presented in table 9. Table 9: Experimental test and proposed method design initial imperection. Section Fexp/ v01 v02 v03 v04 Exp Num Fn,te (mm) (mm) (mm) (mm) (mm) (mm) Num/Exp 160x10 1,35 2,68 1,66 1,82 1,88 2,50 2,56 1,02 160x10 1,32 3,44 1,64 3,54 1,9 2,50 2,56 1,02 160x10 1,29 2,7 1,18 2,54 0,92 1,48 1,59 1,07 160x10 1,27 2,08 1,94 3 1,52 2,45 2,56 1,05 180x12 1,27 1,64 0,9 1,76 1,64 1,80 2,87 1,60 160x10 1,25 2,48 1,34 8,82 1,76 2,19 1,92 0,88 125x8 1,24 0,98 0,78 1,06 0,68 1,03 2,00 1,94 125x8 1,24 1,34 1,02 1,42 0,5 1,07 1,25 1,16 125x8 1,24 1,38 0,8 1,3 1,2 1,41 2,00 1,41 160x10 1,24 2,18 1,4 1,92 2,06 2,45 1,92 0,79 180x12 1,23 2 0,82 1,66 1,1 1,36 2,15 1,58 180x12 1,23 1,76 1,96 1,58 1,84 2,69 2,86 1,07 180x12 1,20 4,3 1 4,52 0,64 1,16 2,14 1,85

9 160x10 1,20 2,84 1,42 1,8 1,4 1,99 1,92 0,96 180x12 1,20 1,86 1 1,94 1,18 1,54 2,86 1,86 160x10 1,20 3,24 1,04 2,46 0,82 1,32 1,60 1,22 180x12 1,19 2,6 0,68 2,06 1,16 1,30 2,15 1,65 125x8 1,19 2,1 0,78 2,3 0,8 1,12 1,50 1,34 Analyzing the tests results, in 15 o 18 cases, the initial imperection considerate in the design method were bigger than the experimental results. This result conirms the major role o initial imperection in the behavior and ultimate capacity o single angle equal leg column. 6. Global Design In order to present a more simple and practical version o the design method, it was propose to aggregate all the expressions developed in this thesis,, that would still respect the mechanic behavior o single angle columns. These expressions are presented as: Fixed-ended and Pin-ended angle column: Spherical-ended angle column ( c and d expression may be considered (6.3) and (6.4)). Base on the proposed expression, the reliability o the design method was evaluated by LRFD rom AISI The results are presented in table 10. Table 10: LRFD resistance actors ϕ calculated according to AISI2007. Final Table LRFD Numerical LRFD Experimental Total typo Fixed Pin Spherical Fixed Pin Spherical Pm Average) 1,08 1,07 1,08 0,99 1,06 1,13 1,08 Vp (S. Deviation) 0,12 0,14 0,06 0,10 0,23 0,12 0,10 φ (Reliability index) 0,84 0,81 0,89 0,78 0,69 0,88 0,85

10 From the analysis o the result it is proved that it is possible to develop a rational design method or ixed-end, pin-ended and spherical-ended angle column or. 7. Conclusion This thesis dealt with the (i) analysis o stability, post-critical behaviour and ultimate capacity o spherical-ended angle column and (ii) development and assessment o novel procedures or the design o ixed-ended (F), pin-ended (P) and spherical-ended (PS) equalleg angle columns with short-to-intermediate lengths, (those buckling in lexural-torsional modes). As remarkable conclusion is worth to highlight: (i) the major role o initial imperection on the behaviour and ultimate capacity o angle column, (ii) Several lexural-torsional curves must be developed, in order to enable capturing the progressive erosion o the column postcritical strength (iii) Fixed-ended angle column don t have eective centroid eects, but Pinended and Spherical-ended do (iv) in angles with b/t<25 local buckling mode have more inluence than in slender angles, 8. Bibliography [1] Dinis, P. B. e Camotim, D. (2014) A novel DSM-based approach or the rational design o ixed-ended and pin-ended short-to-intermediate thin-walled angle columns, Thin-Walled Structures, submitted to publication. [2] Rasmussen, K. J. R. (2003) Design o angle columns with locally unstable legs. The University o Sydney, Department o Civil Engineering, Research Report No R830. [3] Kennedy, J.B. e Murty, M.K.S. (1972) Buckling o steel angle and tee stucts, Journal o Structural Division, ASCE, Vol. 98, No. 11, pp [4] Kitipornchai, S. e Lee, H.W. (1986) Inelastic experiments on angle and tee struts, Journal o Constructional Steel Research,Vol. 6, No. 3, pp [5] Adluri, S.M.R. e Madugula, M.K.S. (1996) Flexural buckling o steel angles: Experimental investigation, Journal o Structural Engineering, ASCE, Vol. 122, No. 3, pp [6] Fan, J.K. (2009) "Theoretical and Experimental Study on Q460 Single Equal-Leg Angle under Axial Compression", Master Thesis, Xi an University o Architecture and Technology, Xi an, China. [7] Ban, H. Y., Shi, G., Shi, Y.J. e Wang, Y. Q. (2013) " Column buckling tests o 420 Mpa high strength steel single equal angles", International Journal o Structural Stability and Dynamics, Vol. 13, No 2.

to introduce the principles of stability and elastic buckling in relation to overall buckling, local buckling

to introduce the principles of stability and elastic buckling in relation to overall buckling, local buckling In the case of elements subjected to compressive forces, secondary bending effects caused by,