Combined Bending with Induced or Applied Torsion of FRP I-Section Beams

Transcription

1 Combined Bending wih Induced or Applied Torsion of FRP I-Secion Beams MOJTABA B. SIRJANI School of Science and Technology Norfolk Sae Universiy Norfolk, Virginia USA STEA B. BONDI Civil Engineering Technology Old Dominion Universiy Norfolk, Virginia 359 USA hp:// ZIA RAZZAQ Civil and Environmenal Engineering Old Dominion Universiy Norfolk, Virginia 359 USA hp:// Absrac: - Presened herein is he oucome of an experimenal and heoreical sudy on FRP beams wih an I- shaped cross secion subjeced o four-poin loading revealing he significance of laeral bending and warping srains due o pracical imperfecions. The paper also addresses he problem of combined bending and applied orsion. The resuls show ha, for he case of combined bending and induced orsion, he sum of laeral and warping srains in FRP beams is no negligible even in he presence of only he in-plane or verical loads. Based on measured srains, enaive srain-slenderness relaionships are generaed which accoun for he presence of laeral and warping srains in pracical FRP beams. The effecs of boh induced and applied orsion combined wih bending are explained wih he help of numerical examples. Key-Words: - Fiber reinforced plasic beams, laeral bending srain, orsion, FRP beam warping, beam buckling, experimenal sudy, heoreical analysis, load and resisance facor design 1 Inroducion A Fiber-Reinforced Plasic (FRP) beam subjeced o in-plane bending momens abou is crosssecional srong axis can develop laeral-orsional buckling. In heory, such a beam will iniially deflec normal o he srong axis unil he criical value of he bending momen is reached where afer laeral and orsional deflecions develop. In real FRP beams, however, he verical, laeral, and orsional displacemens develop righ from he sar of he loading process, namely, as soon as he inplane bending momens are applied, owing o even inies geomerical, maerial or loading imperfecions. Thus, he acual beam also develops boh laeral bending and warping normal srains. These srains are unaccouned-for in rouine analysis and design procedures. In he presen paper, he magniude of hese srains for such beams based on experimens is firs summarized and discussed. The resuls are hen used o develop enaive srain versus minor-axis (y-axis) slenderness raio relaionships for possible use in he analysis and design of FRP beams. The use of he proposed expressions for laeral bending and ISBN:

2 warping srain versus slenderness raio expressions is demonsraed wih a numerical example. The problem of combined bending and applied orsion is also addressed in his paper and he soluion explained wih he help of anoher numerical example. Problem Formulaion Figure 1 shows a FRP beam of lengh and wih an I-shaped cross secion and subjeced o a pair of gradually increasing applied loads each of magniude P. Figure 1. Beam and loading In an ideal or perfec beam, only verical deflecions (v) would develop iniially unil he beam eiher cracks, or develops laeral displacemen (u) coupled wih an angle of wis (β ) corresponding o he laeral-orsional buckling load, P cr. Experimens conduced by he auhors on real FRP beams, however, showed ha all hree displacemens (u, v, and β ) developed as soon as he loads are applied, unil he peak value of P is reached. The presence of laeral and orsional displacemens of he ype developed resul in an increase in he oal normal sress in he beam beyond ha owing o jus he in-plane bending effec. The problem is o develop a enaive pracical analysis approach which may accoun for he presence of sresses associaed wih he laeral and orsional displacemens in pracical FRP beams. The problem of combined bending and applied orsion deal wih in his paper is schemaically shown in Figure. 3 Experimenal Resuls Table 1 summarizes he experimenal resuls based on five FRP beams wih an I-shaped cross secion ( 4x x0. 5 in.) and having lenghs of 108, 96, 84, 7, and 60 inches. The able presens he applied load values and measured srains. P x 10-6 v (in.) (lbs.) ` Table 1. Summary of experimenal resuls Figure. Combined Bending and Applied Torsion The erm represens he maximum sum of he laeral and warping flange ip srain, v represens he maximum srain due o he in- ISBN:

3 plane bending effec, and is he oal srain. Figure 3 presens maximum srain versus he beam lengh plos. Figure 3. Maxium srains versus lengh Figure 4 exhibis he relaionship beween he sum of he laeral and warping srains versus he beam minor-axis slenderness raio. engh (in.) Sum of laeral and warping srain (in./in.) Figure 4. engh versus sum of laeral and warping srains In his figure, he daa poin for he 96 in. long beam has been excluded in he curve-fiing process o arrive a a conservaive relaionship beween he wo variables. 4 aeral Bending and Warping Srains Based on he larges values of and corresponding o he beam peak loads given in u β Table 1, i is found ha he average of he raio is approximaely 9.69 percen for he five beams. This raio may become even much larger for some beams as demonsraed laer in he paper by means of an example. Thus, in calculaing he value of he maximum axial normal sress, he effec of he sum of he laeral bending and warping srains should no be negleced. 5 Experimenal Srain versus Slenderness Relaionships Figure 3 demonsraes he relaionship beween and he minor-axis slenderness raio, u β, where r y By is he minor-axis radius of gyraion. approximaing he curve in Figure 3 as a bilinear relaionship, he following srain versus minor-axis slenderness raio relaionships are obained corresponding o he beam maximum load, P max : 6 6 ( ) 10 = + x 08.75x10 (1) ry 6 6 ( ) 10 = + x 08.75x10 () ry Equaing he srain expressions from Equaions 1 and, and solving for resuls in defining he criical beam minor-axis slenderness raio, 0 = , which provides he demarcaion basis beween he wo equaions. Thus, he following analysis rules are generaed: If 0 hen Equaion 1 is applicable. If 0 hen Equaion is applicable. The srain expressions presened herein are for he case of four-poin loading and can enaively be used o assess he srain values due o he induced laeral bending and orsional effecs produced by pracical imperfecions in a FRP beam. Fuure research needs o be conduced o develop such expressions for various oher ypes of loading and boundary condiions. The resuling expressions can provide a pracical mehod for he analysis of FRP beams wih a oad and Resisance Facor Design (RFD) approach [1, 5-8]. The absolue value of he oal maximum flange ip normal srain can be obained by using he following expression: = + (3) v ub ISBN:

4 The maximum normal sress can be compued using he following expression: = E 11 cr (4) in which E 11 is he Young s modulus, and cr is he FRP maerial cracking sress. 6 Combined Bending and Induced Torsion: Example 1 Deermine he oal srain of a 6 x 3 x 0.5 in. I- secion FRP beam wih = 144 in., end disance a * = 0 in., load heigh above he cross secion y o = in., E 11 =.53 x 10 6 psi, G 1 = 0.4 x 10 6 psi, I x = in 4., I y = 1.13 in 4., I w = in 6., K = in 4. Soluion: Referring o Figure 1, he beam buckling load can be found using he following formula [1] which is also applicable o I-secion beams: 0.5[ f + f + 4 f1 f 3 ] P cr = (5) f1 in which: 1 π a πa f1 = f ( a) g( a) (6) 16 l π 4 E11I y * f = y 3 0 sin 4 πa 6 π E11I y π E11I w = + G K 4 T 16 f3 1 (7) (8) πa πa πa f ( a) = sin sin (9) 1 a a g( a) = π 1 sinπ 1 (10) Using he numerical values based on Equaions 6 hrough 10 in Equaion 5 gives he following buckling load: P cr = lbs. For he service live load condiion, he RFD-based load facor is 1.6 []. Thus, he service load value of P is given by: P P = cr 1.6 Therefore: P = lbs. The maximum in-plane bending sress is given by: Mc Pac b = = I x I x which leads o: b = psi Since r y = in., he minor-axis slenderness raio becomes: = Therefore Equaion applies and gives: 6 in. = x10 in. which resuls in he following normal sress due o his srain: = psi I is ineresing o noe ha for his example, he normal sress due o induced laeral bending and warping is psi which far exceeds he primary bending sress of psi. The absolue value of he oal maximum flange ip normal sress equals: T = b + = psi For his specific beam, he prediced sress due o combined laeral bending and warping normal sress represens 6 percen of he oal normal sress. 7 Bending and Applied Torsion For he member shown in Figure, he angle of wis, β, a any locaion Z along he member lengh due o a single concenraed orsional momen, M z, can be shown o be equal o [3]: ( ) Z a Z β = Q1 1 α + Q sinh (11) a ISBN:

5 in which: Z = 0.5 and Z / 1 Z β " = Q1 Q sinh (1) a a in which: M Q 1 = (13) GJ α sinh a α Q = cosh (14) anh a a The warping normal sress, w a he flange ips is given by [4]: = E w β " (15) w n in which, E is he modulus of Elasiciy, w n is he normalized uni warping, and β is he second derivaive of β. The analysis example given below demonsraes he procedure for finding he combined bending and normal sress for a FRP beam. 8 Combined Bending and Applied Torsion: Example The beam in Example 1 is subjeced o a pair of bending loads (P, P) as shown in Figure 1 as well as an applied midspan orsional momen of 300 lb-in. Deermine he oal maximum normal sress in he beam including he effec of he warping normal sress due o he applied orsional momen. The maximum value of he normalized uni warping for he flange ip is in. Soluion: Using Equaions 13 and 14, we ge: Q 1 = Q = x 10-4 Using Equaion 1: β" z = 0.5 = x 10-4 rads./in. Equaion 15 gives he following warping normal flange ip sress he beam midspan as: w = psi The maximum bending sress for P = lbs was found earlier in his paper as b = psi. The oal normal sress is he sum of he bending sress plus he warping sress as given by: oal = b + w = 1, psi Thus, he warping sress is abou 75% of he bending sress a he beam midspan. For hese resuls o be valid, he maximum value of he midspan angle of wis, β, should be less han 6 degrees, which is a commonly acceped upper limi in srucural engineering pracice. For his example, β a midspan is found o be abou 1.9 degrees. Therefore, he angle of wis is wihin he small deflecion range. 9 Conclusion Experimenal resuls show ha he sum of laeral and warping srains in FRP beams is no negligible even in he presence of only he in-plane loads. Tenaive srain-slenderness relaionships are presened which accoun for he presence of laeral and warping srains in such beams due o real-life imperfecions. I is also found ha he magniude of he warping normal sress due o an applied orsional momen is of he same order of magniude as ha due o he primary bending loads. References: [1] Razzaq, Z., Prabhakaran, R., and Sirjani, M.M., oad and Resisance Facor Design (RFD) Approach for Reinforced Plasic Channel Beam Buckling, Srucural Composies in Infrasrucures Volume 7, Issues 3-4, 1996, Pages [] oad and Resisance Design Specificaion (RFD), AISC, Chicago, I, 007. [3] Torsional Analysis of Seel Members, AISC, Chicago, I, [4] Galambos, T.V., Srucural Members and Frames, Prenice-Hall, Inc., Englewood Cliffs, New Jersey, [5] Razzaq, Z., Khan, A. N., and Ali, S. M., Effec of Warping Resrains on aeral Torsional Buckling of FRP Beams, Proceedings, 1h Inernaional Conference ISBN:

6 on Composies/Nano Engineering (ICCE/1), Tenerife, Spain, Augus 1-6, 005. [6] Sirjani, M. B., and Razzaq, Z., Sabiliy of FRP Beams under Three-poin oading and RFD Approach, Journal of Reinforced Plasics and Composies, Vol. 4, No. 18, 005. [7] Sirjani, M. B., and Razzaq, Z., Combined Biaxial Bending and Torsion of FRP I-Secion Members a Cracking, Proceedings, 14h Inernaional Conference on Composies/Nano Engineering (ICCE/14), Boulder, Colorado, July -8, 006. ISBN: