Kalyanaraman, Venkatakrishnan

Ramakrishna, P.

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine International Specialty Conference on Cold-Formed Steel Structures (1984) - 7th International Specialty Conference on Cold-Formed Steel Structures Nov 13th, 12:00 AM Non-uniformly Compressed Stiffened Elements Venkatakrishnan Kalyanaraman P. Ramakrishna Follow this and additional works at: https://scholarsmine.mst.edu/isccss  Part of the Structural Engineering Commons Recommended Citation Kalyanaraman, Venkatakrishnan and Ramakrishna, P., "Non-uniformly Compressed Stiffened Elements" (1984). International Specialty Conference on Cold-Formed Steel Structures. 3. https://scholarsmine.mst.edu/isccss/7iccfss/7iccfss-session2/3 This Article - Conference proceedings is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in International Specialty Conference on Cold-Formed Steel Structures by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. Seventh International Specialty Conference on Cold-Formed Steel Structures St. Louis, Missouri, U.S.A., November 13-14, 1984 INTRODUCTION NON-UNIFORMLY COMPRESSED STIFFENED ELEMENTS by 1 V. Ka1yanaraman and p. Ramakrishna2 The local buckling of non-uni form1y compressed sti ffened elements in light-gage cold-formed steel members does not necessarily lead to innnediate fai lure, but instead considerab Ie postbuckling strength may be available depending upon the slenderness of the element and the stress gradient. This paper presents a procedure for calculating the buckling and postbuckling strength of such elements. Considerable attention has been devoted to study the local and postbuckling behaviour of uniformly compressed sti ffened and unsti ffened elements which are frequently encountered in practice (5,12) and the current design codes already have presented effective width desi~proce­dures accounting for the postbuckling strength (1). On the other hand no widely accepted design procedure is available for the thin elements subjected to non-uniform compression which is the state of stress in beam webs and eccentrically compressed column webs. Recently a few research publications have appeared, dealing with post-buckling effective wi dth of non-uni formly compressed sti ffened elements 0,4,6,10). Host of these equations have been derived empirically and have been compared wi th test results in whi ch the stress gradient is ei ther only compressive or only uni form bending but not both cases. An effective width equation is proposed in this paper which is similar in format to that given by Winter for uniformly compressed stiffened elements, but is applicable for the entire range of stress ------------------,--------------.------------------------------------------lAssoc:iate Professor, Dept. of Civil Engineering, Indian Institute of Technology, Madras, India. 2Former Research Scholar, Department of Civil Engineering, Indian Institute of Technology, Madras, India. 75 76 SEVENTH SPECIALTY CONFERENCE gradient from uniform compression to uniform bending. The effective width equation is compared with results of non-uniformly compressed stiffened element webs in beams and eccentri cally loaded colUllll).s. The proposed equation is also compared with the other equations avai lable in the literature for limited range of compressive stress grad:i.e1;lt. It is shown that the proposed equation satisfactorily models the post-buckling behaviour of stiffened elements over the entire range of compressive stress gradie1;lt. EXPERIMENTAL PROGRAM Four beams with two point load causing uniform moment region and four columns under eccentric compression were tested (7) in order to obtain stress gradient in the range of uniform compression to uniform bending. The cross section of the beam and column specimen are shown in Fig.l. The dimension of the specimen are given in Table 1. The specimens were designed to have the sti ffened webs undergo elastic local buckling whereas flanges do not experience local buckling before yield stress is reached. The beams having 2400 rom simple span were subjected to two equal loads at 800 rom from the supports. The schematic sketch of eccentric compression test frame is shown in Fig,2, The mean yield strength of the hot rolled steel sheets from which the specimens were fabri catedwas 220,3 Nlrml and an ultimate strength of 323.1 ·N/rnro2• The specimens were instrumented wi th di al gauges and strain gauges in order to determine the stress at the web flange junction at each loading stage and local buckling stress of the stiffened web, which are necessary for comparison with analytical method proposed. The stress gradi.ent, ();, «); = 1 - f . If ) and ultimate strength of all these tests are also Dlln max given in Table. 1. Th.e stress at which the webs buckle locally were determined by the stress deviation method en. The effecti.ve cross sec-tion of the specimen in the postbuckling range has been assumed to be as sh.own in Fig.3. The effect:ive depth of the web in the posthuckling range erle' Y'1.e) may be determined from experimental res.ults using moment and axial force eCl,uHibrium eCl,uatio1;ls. 'Since the equilibrium equations are non-linear tn terms of the unknown values of effective depths NON-UNIFORMLY COMPRESSED ELEMENTS 77 (Yle' Y2e)' Sequentiill Unconstrained Minimisation Techniques (SUMT) has been used, in which the RMS error between internal stress result-ant and external forces/moments has been minimised to obtain the experimental values of effective depths at various load stages (7). These experimental values have been used for comparison with analytical method proposed later in this paper. ANALYTXCAL EQUATIONS Local Buckling Stress: The elastic local buckling stress, fcr' of a plate element may be calculated from the following equation (1) whe~e k = local buckling coefficient, E is Young's modulus, ~ = Poisson's ratio, hIt ~ flat height to thickness ratio of the element. DIN 4114 (3) suggests the followi.ng equation for the local buckling coeffi ci ent of non-uni formly compressed sti ffened· element having si mply supported longi tudinal edges. k = .8.4 for Q ~ ~ ~ 1,0 2.1-~ where ~ = 1 - f "If " tIDl"n = minimum edge stress IDln max' 2(a) 2(b) (positi.ve if compressive); f max = ma:s:i.mum edge compressive stre$s. Yll (13) has presented the following equation for the localbuckHng coefficient of non-uniforJjlly compressed simply !!upPol"tedplate elements. (3) Thl"ough a regression analysis the following eqUation!! have been dedved for the local buckling coefficient of nonuniformly compl"e!!s.ed stiffened elements with pinned and dgidly fhed longitudinal boundades en. 78 SEVENTH SPECIALTY CONFERENCE (4a) k = 6.97 + 2.702a + 0.968a2 + 2.92a3 fixed (4b) All these equations compare reasonably well with theoretical values (9) the maximum di fference being less than 3%. PcistbtickHng Effective Depth: Dewolf (2) has presented an empirical equation for the effective depth at failure of the web subjected to non-uni form compression (Fig.4). whi ch may be wri tten as follows: Yle -h 0.7 rr-rf'1CJ. . ,/Lcr/Ly Y2e ~ = (a - l)/a where Y Ie = effective depth of web in compressi on zone; y 2e = effective depth of web in tension zone; h = flat depth of the web plate and f • f cry are local buckling and yield stress (N/rorl) ,respectively. This equation has been compared with tests in which sti ffened elements experience bend-ing stress over the depth (compressive and tensile stress). Labouhe and Yu (6) have presented equations for the effective dep,th of beain webs at ultimate load which were derived by a regression analysis of large number of test results. These equati ons may be wri tten in the following form (Sa) (a - 1)/a ~ 0 (Sb) where (a) for webs wi th adj acent stiffened flanges: Yl = 1.037 -0,000476 (h/t)-If; (Sc) when (w/t)l . . .;:;: (w/t)~ 2(w!t)1· 1m 1m NON-UNIFORMLY COMPRESSED ELEMENTS 79 (wIt) Y2 = 1.074 - 0~073s ( I ) (sd) Vlt liIll when (Vl/t) ~ (w/t)l-1I1l Y2 = 1.0 (b) for webs with adjacent unstiffene4 flanges y 1 = 0.800 when (wIt) ?(.w/t)l" . lI1l 1.024 - 0.024 -r~(wr./~t:.<.)_ (Vl/t> 1" I.IIl when (w/t)5 (w/t)l" - . lI1l Y2 = 1.0 (w/t)l" = J169'/ f i I,IIl Y (se) (Sf) (sg) (sh) (sj) (sk) Effective depth equation, ~q.s 'has also been caHbrated only wi.th beam web test where the stress gradient is large enough to cause both com-pressive and tensile stresses within the stiffened web (a>1.0). usami (10) has presented the following semi-empirical equati.on for the postbuckli.ng behaviour of non-uniformly compressed sti ffened elements: (1 - O.sd.) tcr I ((1+0.1d.) - (0.22 + o.osa.J!cr ') (6) max max Usami has also proposed the following effective depth equations for the postbuckHng range of non-uni formly compressed elements. Yle.= h 0.5 ref fcro (1 max _ 0.22 pcro~ max (7a) 80 SEVENTH SPECIALTY CONFERENCE YZe 11 (7b.) YZe 11 for a ~1.0 (7c) where Yle ' YZe '" effective widths in compression and tension zone (Fj.g.3); f '" local buckling stress under uniform compression· (a"'O). =0 ..Thes.e equati ons have been empi ri cally deri ved and have been compared with eccentrically loaded rectangular box section test results. It is not very clear as to how the effective depth equation CEq.]) is derived from the postbuckling behaviour equation (Eq.6). ECCS Committee recollllllendation ·(4) for the postbuckHngeffective depth may be written in th~ following form. for a < 1.0 (8a) C8b} fora> 1.0 fIe", 2.!i!: .rr7£' (1 - Q,zzJT7£') h a 'f tr·y '" tr· y (8c) YZe '" Yle a-1 h 1.5 11 + a (8d) The value of f If calculated from experimental valuea discussed av maX earlier when compared with Eq.6 indicated good correlation validating equation for the postbuckling range of both beam and ec·centrically loaded column web". Consequently the effective depth equation for non-uni formly compressed webs can be analytically derived using Eq.6. The effectj ve depth" Cy Ie' y 2e)' were derived for vari ous values of maximum edge streas, f . If by equating the intetnal stress. resultants max cr to the corresl?onding extetnal forces and moments obtained froD). Eq.6 (7 ). NON-UNIFORMLY COMPRESSED ELEMENTS 81 These equi librium equati ons, which are non-linear in terms of the unknown effective depths, were solved using SUMT.On the basis of this exercise, the following effective wi dth equations for non-uni formly compressed stiffened elements have been .derived through regression analysis (7). (9a) The Co and C1 are fUnctions of stress gradient factor, il, as shown in Fi g.5. ·It can be seen that there is a . sudden change in the values of the two variables, Co'Cl , at a = 1.5. This is due to the fact that the non-linear equations of equilibrium, from which they are derived, have two real solutions at a = 1.5, both satisfying the equilibrium equations. The Yle and Y2e are given by the following equations (7). Y2e (Yle + 'l2el/h T;=< (2.0 - O.65a) 'lIe = 11. for a < 1.5 for a ? 1.5 (9b) (9c) (9d) In Fig.6 the equations for ratio of depth of compression zone ¥ Ie' to total depth for non-uni formly compressed sti ffened elements, suggested by different authors (Eq.4,5,7,8) are comparedwith the effective depth equation proposed in this paper (Eq.9) for various valuefl of a and f . If , ·Eqs. 4 and 5 presented in refs .. 2 and 6 max cr respectively are valid only for elements with bending stress (a>l.O) and hence the correflponding curves appear only in such caSeS. For the case of uniform c.ompression (a=O) Eqs.7, 8 and 9 give essentially the same value (Winter's effective width value) and hence only one curve is drawn for a = O. It can be seen that there is considerable variation between the values of effecti ve depth,y l/h, given by the di fferent equati ons. Equations from refs, 2 and 6 (Eqs.4 and .5) generally give a higher value, 82 SEVENTH SPECIALTY CONFERENCE whereas refs.4 and 10 (Eqs.7 and 8) generally give a lower val~e. The proposed eq~ation (Eq.9) is in between the val~e. being closer to that given by Eqs.4 and 5 at lower stresses and closer to that given by Eqs. 7 and 8 at higher stresses. Yle + Y2e In. Fi g. 7 the effective depth to total depth rati 0 ( h ) for vario~s val~es of c;i, and f I f as gi ven by di fferent eq~ati ons are max cr compared. The test values are also plotted in the same figure. It can be seen that the proposed effective depth equation compares best with test results for over a range of a values. The failure moments or axial loads calculated on the basis of effect-ive depth equation proposed in this paper are compared with test values in Tab Ie 2. It is seen that the comparison is generally good except for exceptional case. The mean error is 6 percent and the standard deviation is 10 percent. DESIGN PROCEDURE Step 1: Assume a value of a whi ch may be based on the full cross secti on of beams or full cross section and eccentricity of load in columns. Step 2: Calculate the buckling stress from Eqs.4 and hence f If. cr y Step 3: Calculate the effective depth of webs from Eqs.9. Step 4: Calculate the neutral axis location of the resulting effective section and from this the new value of a.,CLnew' Step 5: Repeat steps 1 through 4 until assumed a converges to a • new Step 6: Calculate the strength of effective section. SUMMARY AND CONCLUSIONS An effective depth equation for non-uniformly compressed stiffened plates has been derived on the basis of regression analysis, which is suit-able for the entire possible range of stress gradient (a). This equation has been compared with other equati ons avai lable in the literature and wi th tes.ts conducted using beams and beam columns by the authors and the other tests rel?orted in the literature. It is seen that the strength predicted by the proposed effective depth method compares well with tests. NON-UNIFORMLY COMPRESSED ELEMENTS APPENDIX I ~ REFERENCES 1. "AISI - Sped fi cad on for Design of Cold-Formed Steel Structural Members", Amedcan Iron and Steel Institute, Washington, D.C., 1980. 2. bewolf, J.T. and Gladding, C., "Web Buckling in Beams", Proceedings of Fourth International Conference on Research and Developments in Cold-Formed Steel Structures, University of Missouri, Rolla, MO, June 1-2, 1978, pp 191-218. 3. bIN 414, Blatt 2, "Stahlbau, Stabi.li tatsfalle (Kni ckung, Kippang, Beulung)", Berechnungsgrundlagen, Ri chtlinien, Berlin, Germany, Feb. 1953. 4. ECCS Commi ttee TC7, "European Recommendations for the Desi,gn of Profiled Sheeting", working Group TWG 7.1, Apr. 1983. 5. Kalyanaraman, V. Pekoz' , T. and Winter, G., "Unstiffened Compression Elements" Journal of Structural Division, ASCE, Vol.103, No.ST9, Sep. 1977, pp 1982-1848. 6. Laboube, R.A~ and Yu Wei-wen, "Bending Strength of Cold-Formed Steel Beams", Journal of Structural Division, ASCE, Vol.108, No.ST7, July 1982, pp 1589-1604. 83 7. J:{amakdshna, p., "Local and Post Buckling Strength of Nonuniformly Compressed Stiffened Elements", M.S. Thesis, Dept. of civi 1 Engineedng, lIT, Madras, 1983. 8. ]:{hodes, ol. and Harvey, J.M., "'Effect of Eccentricity of Load or Com-pression on Buckling and Postbuckling Behaviour of Plates", Inter-national Journal of Mechanical Science, Vol.13, NO.10, Oct. 1971, pp 867-879. 9. Tomoshenko, S.p. and Gere, J.M., "Theory of Elastic Stability", McGraw Hill Book Co. Inc., N.Y., 1961. 10. Usami, J.,"Postbuckling of Plates in Compression and Bending", Journal of Structural Division, ASCE, Vol.108, No.ST3, Mar. 1982, pp 591-609. 11. Walker, A.C., "Flat Rectangular Plates subjected to Linearly Varying Edge Compression Loading", Thin Walled Structures, e4s. A.H. Chiluer, Chatto and Windus, London, 1967, pp 208-247. 12. Winter, G., "Cold-Formed Light Gage Steel Construction", Journal of Structural Division, ASCE, Vol.85, No.ST9, Nov. 1959. 13. Yu, W.W., '''Cold-Formed Steel Structures", McGraw-Hi 11· Book Co., Inc., ' New York, N.Y. 1973. 84 SEVENTH SPECIALTY CONFERENCE Af~ENI>IX II - NOTATIONS fav average compressive stress; f(::r local buckli.ng stress; fc:ro local buckling stress under uniform compression; fma~ ma~imum edge strells jn web; yield strength of the material h flat depth of web; k local buckling coeffi dent; t thicknells of element; 2 (N!mm ); wit flat width to thickness ratio of flange elements; Yle effective depth of web adjacent to ma~imum compression edge; YZe El£;fective depth of web adjacent to minimum compression or tensjon edge; Compressive stress gradient factor. TABLE 1 SPEClMENDlMENSIONS* AND TEST RESULTS H B T1 T2 D ___ ~~E~!i~~U~!l_~l~i~!~~ ____ Specimen in i.n in in in (1 Load Moment I1llll I1llll I1llll in in I1llll I1llll kN kN-m ---------------------------------------------COLUMNS C1 150.0 52.0 1,00 0,00 58.9 C2 150.0 52.0 1.00 0.16 63.8 C3 210.0 52.0 1,00 0,77 52.2 C4 210.0 52.0 1,00 0.80 44.5 BEAMS B1 219.0 44.0 0.90 0,0 19.5 2.00 4.32 B2 216.0 42.5 0.95 0.95 21.4 1,92 5.10 B3 220.1 44.0 0.99 1.99 19,7 1.76 5.15 B4 216.0 43.5 0.95 2.99 21.4 1.48 4.59 *See F:ig,l Z 0 ~ C1 Z ...... "".J 0 ~ ~ t'" ><l (") 0 ~ "tI ~ t'=j r:n r:n t'=j t1 t'=j t'" t'=j ~ t'=j Z ~ r:n 00 01 86 (a) Co1ulims, Spec:imen SEVENTH SPECIALTY CONFERENCE TABLE 2 COPWARISON OF TEST WITH STRENGTH BASED ON AUTHORS' EFFECTIVE WIDTH Fully effective section P in kN y P .theory u in kN P • expt. u :in kN l?tl' theory P)l' expt. - - - - - - - - -- - - - - - - - ~ -- ----- -- -- -, --- - - - ----Cl C2 C3 C4 (b) Beams Spec:iIl)en 111.0 102.2 84.6 82.5 ;Fully effective section M y in N-m 69.0 64.8 46.7 45.6 M • theory in )1 N~m 58.9 63.8 52.2 44.5 Mu' test in N-m 1.17 1.02 0.89 1.03 Mtl theory Mu test ------------------------------------Author's tests; Bl 5277 4288 4316 0,99 B2 7408 5313 5100 1.04 B3 8662 6361 5150 1,22 B4 8598 5788 4591 1,26 DeWolf~s Tests; Al 30942 26127 24408 1,07 A2 47223 35164 34578 1.02 A3 66452 44049 44183 1.00 A4 88628 52874 52432 1.01 HI 23309 20331 16950 1.20 B2 45106 30922 32431 0.95 H3 72783 41545 40680 1.02 B,4 107038 47497 43957 1.08 NON-UNIFORMLY COMPRESSED ELEMENTS 87 Q) BEAM SECTION b) COLUMN SECTION Fig.'. lest Specimen Sections 88 L SEVENTH SPECIALTY CONFERENCE i i !--I--i-----I.Centre line locations I' ot column speclmenl ~I) 935 2000 All dimensions are In mm Fig. 2. Non- Uniform Compression Test Frame NON-UNIFORMLY COMPRESSED ELEMENTS f max -I~=+-------,-- ------t---' h I~= '="'"fm-;n.J----cx.so fmin OI~O Fig.3. Effective Depth Under Non-Uniform Compression (ref.l0) t mQ• TII--------+------' ------y liE----+-. ---.--- --------,I 'min g(,~o Fig.4.Effective Depth Under Non-Uniform Compression (ref.2 ) 89 90 SEVENTH SPECIALTY CONFERENCE .", .D·I .0·6 -Co 0·0 ~---------------------1 0·0 -o·z ,-----c;---- .0·2 -0", OL.Z--O ... ,--O ..... :---O-'.:---,-'.O=--..::', '=-:---:, ":-,---:,': .•---:,': .•----:!z.o 0·' "' Fig.S Coeffl~nt. Co.C, in Effective Dltpth Equatlon( Eq·9) NON-UNIFORMLY COMPRESSED ELEMENTS 91 fmax / fer f_x/'er ",.00 £11.4 ~ ~ -- -- --- -~·I - - --. .-. ~ ~i7 \ .. 1-'- q.' 0-J (). I ~./h K ~ ~ ... r-- O· 2 O· 2 o· , o· 4 3 o· ',./h 2 O· 0, , ~ , .., o· J9 3 '2 o· ',./h o o ., \I~, " ,t\, . ~ 1"'-:., 1-"": ........., ~, .... , ~ ~ r-~ \\ \ ' ' " ~~ .... ......... ' .... --I=::.:..:", -....:. ~ r--- 1::0.--n '\.. \ , ~ . .:::.:: '~ -.. ~= ~ ·0 o 00() ',0 2-0 3-0 4'0 5"0 0-0 ',0 Z·O ,-0 4·0 s-o 'max/'.r 'mox/ tor Fig,S Comparison of Effective Depth Equations O· o 0 O· I 2 a. O· o O· 4 0 ., 0 ·z 0 ., 92 SEVENTH SPECIALTY CONFERENCE Yle • Y2e h Yle ·Y2e --h-',e • Y2e --h-1·0 0·8 0·6 0·4 0·2 0·8 0·6 0·4 0·2 0·8 0·6 0·4 0·4 1·0 2'0 3'0 '·0 0·0 1·0 2·0 3·0 4·0 5-0 f mo./ fer 'maxi fer Fig.7 Comparison of Effective Depth Equations with Tests 

Non-uniformly Compressed Stiffened Elements

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Non-uniformly Compressed Stiffened Elements

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