Determination of the rotational restraint coefficient and local buckling of orthotropic box columns

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1 etermination of the rotational restraint oeffiient and loal ukling of orthotropi o olumns A. A. Maghsoudi, H. Saffari & H. Rezaeian epartment of Civil Engineering, Shahid Bahonar University, Iran Astrat This paper presents a method in hih, the eat and simple alulation for finding the plates ukling load of orthotropi o olumns ated on y aial load is possile. First, the differential governing equation for the defletion of orthotropi plates having the semi-rigid onnetions at to parallel edges and to simply supported on the loaded edges ere solved parametrially and afterard the harateristi equation of ukling load as defined. No, having used this equation, the diagrams ere plotted to determine the plates ukling load hile onsidering the rigidity of the semi-rigid onnetion. Sine the proedure for finding the loal ukling load of o olumns is very ompliated, a method as produed to equilirium the ritial load of the o setion ith the ritial load of the plate having semi-rigid onnetions. For this purpose, the e effet on the flange of the setion has een equilirated as a spring ith the rotational rigidity (S) and the nondimensional restraint oeffiient R (oeffiient of rotational restraint), hih is a funtion of (S) and shos the effetive manner of the e on the flange, as introdued and the relations offered for their alulation as ell. To hek the otained results, they are ompared ith the numerial method and it as onluded that the proposed results are very lose to those of the numerial method. Keyords: loal ukling, o olumns, theory of plates, finite element method. Introdution In general, a plenty of the steel setion are thin-alled and therefore, eause of their small ross setions, it is essential to use the high ompressive strength materials. Considering the ne tehnology, noadays, it is possile to produe High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

2 90 High Performane Strutures and Materials II the high strength onrete (HSC) and therefore, the design and onstrution of relatively thin-alled HSC o setions eame popular. Relatively, a more omprehensive researh is presented for the o steel olumn setions [-]. Whereas the researh ork on the thin-alled onrete o olumns are rare. By negleting the loal ukling, undar [] introdued a method to analyse the o onrete setions under iaial ending. A method of analysing and designing of the thin-alled HSC o olumns as offered y Maghsoudi and Rezaeian [5], they also plotted their interations diagrams. Relatively, omprehensive eperimental studies are reported y Taylor et al, [6,7]. The report is ontained the study of the loal ukling effet on thinalled o onrete piers and they are onluded that the flange of the setion ill ukle if the all slenderness ratio is 5 or greater. In most of the researhes, the ompressive flange of a setion is simulated y a plate having simple or rigid support. In this paper first, y use of the lassial theory of orthotropi plates and solving the governing differential equation for the defletion of orthotropi plates having the semi-rigid onnetions (rotational restraint) at to parallel edges and to simply supported on the loaded edges, the harateristi transendental equation for the ukling load of SRSR plate (fig. ) as defined. This paper presents a method to equilirium the ukling load of the o setion ith the ukling load of the plate having semi-rigid onnetions, for this purpose, the e effet on the flange has een equilirated as a spring ith the rotational rigidity (S) and the nondimensional restraint oeffiient (R) hih is a funtion of (S) and shos the effetive manner of e on the flange as introdued and the relations offered for their alulations as ell. (S) is a funtion of length-to-idth ratio of a setion and ratio of orthotropi plate rigidity in to diretions. The otained results are ompared to the numerial method and it as onluded that proposed results are very lose to the numerial method. The governing defletion equation of the orthotropi plates The alls of thin-alled olumns are the isotropi (steel setions) or orthotropi plates (Conrete and polymers). Considering the plates staility theory, Sartz et al [8,9] eperimentally studied the ukling of the onrete plates and y use of their proposed equation, it is possile to predit the ukling load of suh memers. A numerial method analyzing the orthotropi plates as introdued y Harik and Ekamaram [0]. The important onsideration of the previous ork is that, the plates oundary onditions are assumed either hinge or lamp. Suh assumptions are pronouned espeially ith the eperimental studies. Almost in reality, the onditions of semi-rigidity ourred, hih is onsidered in this report. The governing differential equation for the staility analysis of a homogeneous orthotropi plate loaded y in-plane fores in the X-diretion (fig. ) is [], N 0 () y y High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

3 High Performane Strutures and Materials II 9 here is the transverse displaement and N is the ompressive fore per unit idth of the plate. It is lear that, for applying the isotropi onditions, it is enough to sustitute / in eqn. (). / R.R S E E S X R.R a Figure : Plate ith rotationally restrained unloaded edges and simply supported loaded edges (S: simple, R.R: rotational restraint (Semirigid)). Timoshenko and Gere [] proved that for the ompressive retangular plates ith the high length-to-idth ratios (suh as those found in olumns and piers), the degree of rotational restraint along the short edges has a negligile influene on the ukling plate load. Therefore, The oundary onditions on the short edges are assumed as 0, a 0 0, a M 0 0 () here M is the moment on the unit idth of the plate (y-diretion). The oundary onditions on the parallel edges of the olumn ais (-diretion) y 0, y 0 y 0, y ( M y S ) 0 S( ) ( ) 0 (5) y y y here S is onsidered as the unit length stiffness of a semi-rigid onnetion. In other ords, the e effet on preventing the flange rotation is assumed as a spring has rotational rigidity ith a unit length of S. The nondimensional oeffiient of rotational restraint (R) is defined as, S R (6) here is the plate idth and therefore the eqn. (6) an e ritten as, High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

4 9 High Performane Strutures and Materials II R y 0, y ( + ) 0 (7) y the nondimensional variales, X and, are introdued as, y X, (8) a In terms of these variales, hih have limits of 0 and, eqn. () an e ritten as, ( ) + ( ) + + π ( ) K 0 (9) a X a X a X here K is the ukling oeffiient defined as, K N (0) π the oundary onditions given y eqns. (), (), (5) and (6) are ritten in terms of nondimensional variales X 0, 0 X 0, X 0, 0 () 0, ( + R ) 0 Solution method and harateristi equation The levy method as used to solve the equation. Hene, the solution to eqn. (9) is hosen in a form that satisfies eqns. () and (). f ( ) sin mπx (5) sustituting eqn. (5) into eqn. (9) gives the governing equation as, d f d f ( ) ( mπ ) d a d (6) + { ( ) ( mπ) K( ) m π } f 0 a a the solution to eqn. (6) is in the form, f ( ) C sinh λ + C osh λ + C sin λ + C os λ (7) High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

5 High Performane Strutures and Materials II 9 K λ πm( ) ( ) + ( ) + (8) a m λ πm( ) ( ) + ( ) (9) a m The oundary onditions for eqn. (6) given on the edges y eqns. () and () are used to otain a set of four equations in terms of the oeffiients C-C. For a nontrivial solution to this set of equations the determinant of the system is set to zero and the harateristi transendental equation for the ukling of the SRSR plate is otained as, + λ ) sinh λ sin λ + R (λλ + ( λ λ )sin λ sinh λλ osλ osh λ ) + R[ ( λ + λ )( λ sin λ osh λ, λ sinh λ osλ ) + λ λ ( λ sin λ osh λ λ os λ sinh λ), + sin λ osh λ λ sinh λ osλ ] 0 ( λ λ K a a, λ (0) Figure : Bukling urves for 0 /. / The harateristi equation is solved numerially and the diagrams of ukling oeffiient K ), hih is a funtion of the plate aspet ratio ( ( /, / ( φ a / ), ratios of ), the degree of the edge restraint oeffiient (R) and the mode numer (m) are plotted for SRSR onditions. The High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

6 9 High Performane Strutures and Materials II ukling mode is given y the mode numer for hih the smallest K is otained for a given set of parameters (for eample fig. ). No, it is easy to use these diagrams to alulate the ukling load and mode of a speified plate of knon dimensions. It is lear from the ukling urves of fig., the predition of ukling load is depend on the degree of edges restraint oeffiient (R) hih an separately determine. Aording to eqn. (6) R is a funtion of S (rotational rigidity of spring). To aid in the determination of R (and S) the ukling urves are plotted in a modified fashion as shon in fig., here the ukling stress is plotted against the oeffiient of restraint R as a funtion of the modes numer. Fig. shos for a speified dimensional ratio and thikness of a omposite material alled as a graphite epoy (T00/508) []. These urves an e used to determine the value of R from tests of full-setion thin-alled memers. Sine the numer of half-aves and the ukling load an e determined y eperiment, the value of R an therefore e otained from ukling urves of type shon in fig.. Figure : Modified ukling urves for graphite/epoy (T00/508). Bukling ehavior of o setion It is ovious that, setion deformation during the ukling has a diret relation ith the types of loading. The deformation due to pure ending is different from that of pure ompression. This report elongs to the study of olumn under pure aial load. When a ompressed plate ukled it develops out-of-plane ripples, along it s length, suh a ehavior is illustrated if fig.. As an e seen oth flange and e of the setion are ukled and have a symmetrial onditions. The governing differential equations for the defletion of orthotropi plates and in fig. are ritten. The oundary onditions are, () [ ] 0 0, [ ] 0 () 0, High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

7 0 0 0 () + + X a ν 0 + X a ν () (5) (6) Figure : Loally ukled thin-alled o setion under aial load. By plaing the to equations at eight aove oundary onditions, eight equations ill get and to get the nontrivial solution, the determination of matri oeffiient must set to zero. No y sustituting the plate rigidities and plates aspet ratios in the determinant of oeffiients equating it to zero, the value of K is determined. By assuming the same plates rigidity (plate & plate) and also / and different ratios of / (hih is equal to / E E ) equal to 0., 0., 0., 0.5, 0.8,,,, 5, 8 & 0 and the ratios of / equal to,.,.5,.5,,, 5, 8 & 0 diagrams are plotted []. But here, only the ukling urves for / 0, / (eqn. ) and K for speified ratio of / are shon in Fig. 5. It is evident in figure High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN High Performane Strutures and Materials II 95

8 96 High Performane Strutures and Materials II i) By inreasing the ratio of the setion dimension ( / ), the values of K and the oeffiient of rotational restraint ( R ) is inreased ii) The olumn length has no effet on the value of R, for eample for the ratio of / 0, the value of R is equal to 0 (independent of a/). iii) As it as predited, the value of R for a square setion ( / ) is equal to zero, it means, during the loal ukling of suh a setion, the onnetion of flange to e is ated as a hinge. Figure 5: Bukling urves for / 0, / and K for speified ratios of / In Fig. 6, the urves R are plotted against /, for the ratios of / equal to 0.,, 0. As an e seen, y inreasing / E / E, the value of R is dereased. By inreasing the /, the effet of value of / on R ill e dereased. Considering the plotted diagrams it is onluded that, the R value has a nonlinear relation at eginning ith the / ratio ( / < ) and then after, it is linear ( / ). For the otained values of mentioned equations, the folloing proposed relations are offered to determine the R values. It is reminded that the ratio of / assumed to e unity. For isotropi materials, For / <, R 5.λ 6.7λ + 9.5λ 7.9 (7) High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

High Performane Strutures and Materials II 97 For /, R.9λ +.8 (8) For orthotropi materials, For / <, R.68λ 7.5λ + 55.67λ.7λβ + 0.55λ β + 0.08λβ +.9β + 0.6β 0.05β.66. (9) For /, R.8λ 0.β + 0.0λβ +.

9 High Performane Strutures and Materials II 97 For /, R.9λ +.8 (8) For orthotropi materials, For / <, R.68λ 7.5λ λ.7λβ λ β λβ +.9β + 0.6β 0.05β.66. (9) For /, R.8λ 0.β + 0.0λβ +.77 (0) here, it is assumed λ / and β /. Figure 6: Curves of R against / for 0.,, 0 / Figure 7: Finite element ukling Analysis (a: vie, : left vie, : right vie). High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

98 High Performane Strutures and Materials II 5 Finite element analysis of loal ukling setion To hek the auray of proposed method, the loal ukling load of olumn as found out y use of the Ansys

10 98 High Performane Strutures and Materials II 5 Finite element analysis of loal ukling setion To hek the auray of proposed method, the loal ukling load of olumn as found out y use of the Ansys Softare. For this shell9 element ith eight nodes ere adopted. The results are in very good agreement for oth methods (Tale ). Tale : Results of proposed solution and finite element analysis (P.S.: proposed solution, F.E.A.: finite element analysis). 6 Conlusion An aurate and simple method as proposed to alulate the ukling load of the o olumns made of either the isotropi or orthotropi materials. In the proposed method, the loal ukling load of the o setion is simulated ith that of the ukling load of the plate onsidering a semi-rigid onnetion. The validity of the method ompared ith the numerial method and it as shon that results of oth methods are very lose to eah other. Referenes [] Shanmugan, N.E., Chie, S.P. & Lee, S.L., Strength of Thin-Walled Square Steel Bo Columns. ASCE Journal of Strutural Engineering, (), pp. 88-8, 987. [] Lee, S.L., Shanmugan, N.E. & Chie, S.P., Thin-Walled Bo Columns under Aritrary End Loads. ASCE Journal of Strutural Engineering, (6), pp. 90-0, 988. High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

11 High Performane Strutures and Materials II 99 [] Rihard Lie, J.., Shanmugan, N.E. & Lee, S.L., Behavior of Thin- Walled Steel Bo under Biaial Loading. ASCE Journal of Strutural Engineering, 5(), pp ,989. [] undar, C., Conrete Bo Setion under Biaial Bending and Aial Load. ASCE Journal of Strutural Engineering, 6(), pp , 990. [5] Maghsoudi, A.A. & Rezaeian, H., Analysis and esign of ifferent Shapes of Thin-Walled Columns having HSC. Pro. Of the st Iranian Thin-Walled Strutures Conferene, Urmia, Iran, pp , 00. [6] Taylor, A.W., Roell, R.B., & Breen, E.B., Behavior of Thin-Walled Conrete Piers. ACI Strutural Journal, 9(), pp. 9-, 995. [7] Taylor, A. W. & Breen, J.E., esign Reommendations for Thin-Walled Bo Piers and Pylons. ACI Conrete International. 6(), 99. [8] Sartz, S.E., Roseraugh, V., & Berman, M., Bukling Tests on Retangular Conrete Panels. ACI Strutural Journal, 7(5), pp. -9, 97. [9] Sartz, S.E., & Roseurgh, V., Bukling of Reinfored Conrete Plates. ASCE Journal of Strutural ivision, 00(ST), pp , 97. [0] Harik I.E., & Ekamaram, R., Elasti Staility of Orthotropi Plates. Journal of Thin-Walled Strutures, 6(988), pp. 05-6, 988. [ ] Timoshenko, S.P. & Krieger, S.W., Theory of Plates & Shells, MGra- Hill, Tokyo, Japan, 970. [] Timoshenko, S.P., & Gere, J.M., Theory of Elasti Staility, Seond Edition, MGra-Hill Book Company, Ne ork, 96. [] Bank. L.C., & in, J., Bukling of Orthotropi Plates ith Free and Rotationally Restrained Unloaded Edges. Journal of Thin-Walled Strutures, (996), pp. 8-96, 996. [] Rezaeian, H., Analys and design of different shapes of thin-alled high strength onrete olumns, MS thesis, Civil Eng. ept., Shahid ahonar university, Kerman, Iran, 00. High Performane Strutures and Materials II, C.A. Breia & W.P. e Wilde (Editors) 00 WIT Press,.itpress.om, ISBN

Buckling loads of columns of regular polygon cross-section with constant volume and clamped ends

Buckling loads of columns of regular polygon cross-section with constant volume and clamped ends 76 Bukling loads of olumns of regular polygon ross-setion with onstant volume and lamped ends Byoung Koo Lee Dept. of Civil Engineering, Wonkwang University, Iksan, Junuk, 7-79, Korea Email: kleest@wonkwang.a.kr