Bending Anaysis of Continuous Casteated Beams * Sahar Eaiwi 1), Boksun Kim ) and Long-yuan Li 3) 1), ), 3) Schoo of Engineering, Pymouth University, Drake Circus, Pymouth, UK PL4 8AA 1) sahar.eaiwi@pymouth.ac.uk ABSTRACT Casteated beams have ess materia but have equa performance as the I-beam of the same size, Therefore engineers and researchers have encouraged to use it as the stee members. The casteated beam is fabricated from a standard universa I- beam or H-coumn by cutting the web on a haf hexagona ine down the center of the beam. The two haves are moved across by a haf unit of spacing and then re-joined by weding. The fabrication process of casteated beams ed to increase the depth of the beam and thence the bending strength and stiffness around the major axis without adding additiona materias. Existing studies have shown that the resistance of casteated beam is infuenced by shear stresses particuary those around web openings and under the T-section, which coud cause the beam to have different faiure modes. However, most of design guidance does not take into account the shear effect. As far as the bending strength is concerned, the negecting the shear effect may not cause probems. However, for the defection cacuation for serviceabiity design, the shear weakness due to web openings in casteated beams coud affect the performance of the beams and thus need to be carefuy considered. The aim of the present paper is to investigate the effect of web openings on the transverse defection of casteated beams by using both anaytica and numerica methods. The purpose of deveoping anaytica soutions, which adopted the cassica principe of minimum potentia energy is for the design and practica use; whie the numerica soutions obtained using ANSYS software are for the vaidation of the anaytica soutions. In addition, this study is aso used to evauate the shear-induced transverse defection of casteated beams subjected to uniformy distributed transverse oad. The anaytica and numerica soutions have been empoyed for a wide spectrum of geometric dimensions of I-shaped casteated beams subjected to a uniformy distributed transverse oad. Keywords: casteated beams; shear effect, defection; the finite eement software; Energy methods. 1) PhD student ) Lecturer 3) Professor 1
1. INTRODUCTION Engineers and researchers have tried various methods to reduce the materia and construction costs to hep optimize the use of the stee structura members. The casteated beam is one of the stee members which uses ess materia, but has comparabe performance as the I-beam of the same size (Harper, 1991). An exampe is shown in Fig. 1 (a). The casteated beam is fabricated from a standard universa I- beam or H-coumn by cutting the web on a haf hexagona ine down the center of the beam. The two haves are moved across by a haf unit of spacing and then re-joined by weding (Harper, 1991). This process increases the depth of the beam and hence the bending strength and stiffness about the major axis without adding additiona materias. This aows casteated beams to be used in ong span appications with ight or moderate oading conditions for supporting foors and roofs. In addition, the fabrication process creates openings on the web, which can be used to accommodate services. As a resut, the designer does not need to increase the finished foor eve. Thus, despite the increase in the beam depth the overa buiding height may actuay be reduced. When compared with a soid web soution where services are provided beneath the beam, the use of casteated beams coud ead to savings in the cadding costs. Moreover, because of its ightweight the casteated beam is more convenient in transportation and instaation than the norma I-beam. The web openings in the casteated beam, however, may reduce the shear resistance of the beam. The saved evidence, that the method of anaysis and design for the soid beam may not be suitabe for the casteated beam (Boyer, 1964), (Kerda & Nethercot, 1984) and. (Demirdjian, 1999). Design guidance on the strength and stiffness for casteated beams is avaiabe in some countries. However, again, most of them do not take into account the shear effect. As far as the bending strength is concerned, negecting the shear effect may not cause probems. However, for the bucking and the cacuation of serviceabiity, the shear weakness due to web openings in casteated beams coud affect the performance of the beams and thus needs to be carefuy considered. Experimenta investigations (Aminian et a., 1; Maaek, 4; Yuan et a., 14; Yuan et a., 16; Zaarour & Redwood, 1996 ) were carried out and finite eements methods (Hosain et a., 1974; Sherbourne & Van Oostrom, 197; Sotani et a., 1; Sonck et a., 15; Srimani & Das, 1978; Wang, Ma, & Wang, 14) were aso used to predict the defection of casteated beams and/or to compare the predictions with the resuts from the experiments. Experiments (Zaarour & Redwood, 1996) demonstrated the possibiity of the occurrence of the bucking of the web posts between web openings. The shear defection of the straight-sided tapering cantiever of the rectanguar cross section (Maaek, 4) were cacuated by using a theoretica method based on Timoshenko s beam theory and virtua work method. Linear genetic programming and integrated search agorithms (Aminian et a., 1) show that the use of the machine earning system is an active method to vaidate the faiure oad of casteated beams. A numerica computer program (Sherbourne & Van Oostrom, 197) was deveoped for the anaysis of casteated beams considering both eastic and pastic deformations by using practica ower imit reationships for shear, moment and
axia force interaction of pasticity. An anaysis on five experimenta groups of casteated beams (Srimani & Das, 1978) was conducted to determine the defection. It (Hosain et a., 1974) is demonstrated that finite eements method is a suitabe method for cacuating the defection of symmetrica section casteated beams. The effect of noninearity in materia and/or geometry on the faiure mode prediction of casteated beams (Sotani et a., 1) was done by using MSC/NASTRAN software to find out bending moments and shear oad capacity, which are compared with those pubished in iterature. The axia compression bucking behavior of casteated coumns was investigated (Yuan et a., 14), in which an anaytica soution for critica oad is derived based on stationary potentia energy and considering the effect of the web shear deformations on the fexura bucking of simpy supported casteated coumn. A parametric study on the arge defection anaysis of casteated beams at high temperatures (Wang, et a., 14) was conducted by using finite eement method to cacuate the growth of the end reaction force, the midde span defection, and the bending moments at susceptibe sections of casteated beams. Recenty, a comprehensive comparison between the defection resuts of ceuar and casteated beams obtained from numerica anaysis (Sonck et a., 15) was presented, which was obtained from different simpified design codes. The comparison showed that the design codes are not accurate for short span beams and conservative for ong span beams. The principe of minimum potentia energy was adopted (Yuan et a., 16) to derive an anaytica method to cacuate the defection of casteated/ceuar beams with hexagona/circuar web openings, subjected to a uniformy distributed transverse oad. The previous research efforts show that there were a few of studies that deat with the defection anaysis of casteated beams. Due to the geometric particuars of the beam, however, it was remarkabe to note that most of the theoretica approximate methods are interested in cacuating the defection of the casteated beams for ong span beams where the shear effect is negigibe. However, the casteated beams/coumns are used not ony for ong span beams/coumns but aso for short beams/coumns. Owing to the compex of section profie of the casteated beams, the shear-effect caused by the web opening on the defection cacuation is not fuy understood. There are no accurate cacuation methods avaiabe in iterature to perform these anayses. Thus, it is important to know how the shear affects the defection of the beam and on what kind of spans the shear effect can be ignored. European buiding standards do not give any guidance on the effects of shear deformation on the deformation on the defection cacuations for casteated beams, which incude shear deformations. In addition, researchers have adopted the finite eements method to predict the defection of casteated beams by using different software programs such as MSC/NASTRAN, ABAQUS, and ANSYS. However, these programs need efficiency in use because any error coud ead to significant distortions in resuts. In spite of the potentia programs, ony the case of simpy supported casteated beams has been treated in most of the studies. This paper presents the deveopment of the anaytica method to cacuate the defection of continuous casteated beams. The defection equation is to be deveoped based on the principe of minimum potentia energy. In order to improve the accuracy and efficiency of this method, shear rigidity factor is determined by using suitabe 3
numerica techniques. The anaytica resuts were compared by using the numerica resuts obtained from finite eement anaysis using ANSYS software.. ANALYTICAL PHILOSOPHY OF DEFLECTION ANALYSIS OF CASTELLATED BEAMS An approximate method of defection anaysis of casteated beams under distributed transverse oad is presented herein. The method is derived based on the principe of minimum potentia energy. Because of the presence of web openings, the cross-section of the casteated beam is now decomposed into three parts to cacuate the defection and bending stresses, two of which represent the top and bottom T- sections, and the third one represents the mid-part of the web. The anaysis mode is iustrated in Fig. 1 (a). in which the fange width and thickness are b f and t f, the web depth and thickness are h w and t w, and the haf depth of hexagons is a. The haf of the distance between the centroids of the two T-sections is e. In this study, the crosssection of the casteated beam is assumed to be douby symmetric. Under the action of a uniformy distributed transverse oad, the beam section wi have axia and transverse dispacement as shown in Fig. 1 (b), where x is the ongitudina coordinate of the beam, z is the cross-sectiona coordinate of the beam, (u 1, w) and (u, w) are the axia dispacements and the transverse dispacements of the centroids of the upper and ower T-sections. A points on the section have the same transverse dispacement because of the beam assumption used in the present approach (Yuan et a., 14). The corresponding axia strains ɛ 1x in the upper T-section and ɛ x in the ower T- section can be determined by using the strain-dispacement reation as foows: In the upper T-section:-(h w /+t f ) z -a ε 1x x, z = du 1 (z + e) d w (1) In the ower T-section:a z (hw/+tf) ε x x, z = du (z e) d w () The shear strain γ xz in the midde part between the two T-sections can aso be determined using the shear strain-dispacement reation as foows: For the midde part between the two T-sections: -a z a γ xz x, z = du dz + dw = u 1 u + e dw a a e = b f t f h w+t f +t w h w a h w + a 4 b f t f +t w h w a (3) (4) Because the upper and ower T-sections behave according to Bernoui's theory, the strain energy of the upper T-section U 1 and the ower T-section U caused by a transverse oad can be expressed as foows: 4
h w U 1 = Eb f ε 1x Et w dz + ε 1x dz (t f + h w ) ( h w ) = 1 EA tee du 1 a ( h w ) + EI tee d w U = Et w ε x Eb f dz + ε x dz a = 1 EA tee du (t f + h w ) h w + EI tee d w Where E is the Young's moduus of the two T-sections, G is the shear moduus; A tee and I tee are the area and the second moment of area of the T- section, which are determined in their own coordinate systems as foows: A tee = b f t f + t w h w a 3 b f t f I tee = 1 + b ft h w +t f f e + t w 1 h w a 3 + t w h w a h w + a e 4 (5) (6) (7) (8) The mid-part of the web of the casteated beam, which is iustrated in Fig. (a), is assumed to behave according to Timoshenko s theory (Yuan et a., 14). Therefore, its strain energy due to the bending and shear can be expressed as foows: U b = 1 K b (9) where is the reative dispacement of the upper and ower T-sections due to a pair of shear force and can be expressed as ( =aɣ xz ). Whie K b is the combined stiffness of the mid part of the web caused by the bending and shear is determined in terms of Timoshenko beam theory: 1 K b = 3 b GA b + 3 b 1EI b (1) Where A b = 3at w is the equivaent cross-sectiona area of the mid part of the web, I b = ( 3a) 3 t w /1 is the second moment of area and b =a is the ength of the beam. When the Young's moduus of the two T-sections is E=(1+ν)G and Poisson s ratio is 5
ν=.3, the vaue of the combined stiffness of the mid part of the web caused by the bending and shear can be determined as faow: As a resut, for casteated beam the tota shear strain energy of the mid-part of the web U sh due to the shear strain Ɣ xy can be cacuated as foows: K b = 3Gt w 4 U sh = 3 Gt wa γ xz n k=1 3Gt wa 6a γ Gt w a xz = 4 γ xz 3 (11) (1) Let the shear rigidity factor K sh = 1 4 Substituting Eqs (3) into (1) gives the tota shear strain energy of the mid-part of the web: U sh = Gt we k sh a dw u β e (13) Fig. 1 (a) Notations used in casteated beams, (b) dispacement and (c) interna forces. However, the vaue of the combined stiffness of the mid part of the web of the casteated beam caused by the bending and shear by using two-dimensiona inear finite eement anayses (Yuan et a., 16) was found to be 6
K b =.78 3 Gt w 4 (14) Hence the reason for this is due to the smear mode used for the cacuation of the shear strain energy for the mid-part of the web in Eq. (1). However, the factor of.78 in Eq. (14) was obtained for ony one specific section of a casteated beam. It is not known whether this factor can aso be appied to other dimensions of the beams. A finite eement anaysis mode for determining the shear rigidity factor K sh is therefore deveoped herein (see Fig. (c)), in which the unit of ength and depth are (4a/ 3) and (a+a/), respectivey. In the unit mode the reative dispacement can be cacuated numericay when a unit oad F is appied (see Fig. (c)). Hence, the combined rigidity K b =1/ is obtained. Note that in the unit mode a dispacements and rotations of the bottom ine are assumed to be zero, whereas the ine where the unit oad is appied is assumed to have zero vertica dispacement. The caibration of the shear rigidity for beams of different section sizes shows that the use of the expression beow gives the best resuts and therefore Eq. (15) is used in the present anaytica soutions. K sh =.76 b f 1 4 (15) Thus the tota strain energy of the casteated beam U T is expressed as foows, U T = U 1 + U + U sh (16) For simpicity of presentation, the foowing two new functions are used: u u u 1 u u 1 u ( 17) (18) In summary, by using Eqs. (17) and (18), the tota potentia energy of the casteated beam subjected to a uniformy distributed oad can be expressed as foows: = EA tee du β +EI tee d w + Gt w e k sh a dw u β e W (19) Where W is the potentia of the uniformy distributed oad q max due to the transverse dispacement, which can be expressed as foows: W = q max w () 7
Where q max is the uniformy distributed oad, which can be expressed in terms of design stress σ y, as foows q max = 16 σ yi reduced (h w + t f ) (1) I reduced = b f h w + t f 3 1 t wa 3 1 h w 3 b f t w 1 () Fig. Shear strain energy cacuation mode: (a) unit considered, (b) shear deformation cacuation mode and (c) finite eement mode of 4a/ 3 ength unit 3. DEFLECTION OF CONTINUOUS CASTELLATED BEAMS SUBJECT TO UNIFORMLY DISTRIBUTED LOADS For a continuous casteated beam, as shown in Fig. 3: u β (x) and w(x) can be assumed as foows: u β (x) = w(x) = A k k 1 sin k=1,,.. B k sin k=1,,.. k + 1 sin kπ x k 1 kπ x k + 1 π x sin π x Where A k, B k, (k = 1,, ) are the constants to be determined. It is obvious that the dispacement functions assumed in Eqs. (3)- (4) satisfy the continuous support w end boundary conditions that are w, and d du (3) (4) at x = zero and the camped end dw boundary conditions, that are w u at x =, Substituting Eqs. (3), (4) into (19), according to the principe of minimum 8
potentia energy: Fig. 3.Continous beam δu T W = (5) The variation of Eq. (5) with respect to A k and B k resuts in the foowing three agebraic equations: A k = π 4 (16k 4 + 4k + 1)(4k 1) 18qk 5 Gk sh t w e coskπ 1 + 4k 16EA tee Gk sh t w k e + 16EA tee EI tee π ak 4 +4EA tee Gk sh t w e + 16EI tee Gk sh t w k + 4EA tee EI tee π ak + 4Gk sh t w EI tee + EA tee π aei tee (6) B k = 18qk4 coskπ 16EA tee π ak 4 + 16Gk sh t w k + 4EA tee π ak + 4Gk sh t w + EA tee π a 16EA tee Gk sh t w k e + 16EA tee EI tee π ak 4 π 5 (16k 4 + 4k + 1)(4k 1) +4EA tee Gk sh t w e + 16EI tee Gk sh t w k + 4EA tee EI tee π ak + 4Gk sh t w EI tee + EA tee π aei tee Therefore, the defection of continuous casteated beam can be expressed as: (7) wx = k=1,,3 18qk 4 kπ x coskπ Z sin π 5 16k 4 + 4k + 1 4k 1 π x sin (8) Where: Z = 16EA teeπ ak 4 + 16Gk sh t w k + 4EA tee π ak + 4Gk sh t w + EA tee π a 16EA tee Gk sh t w k e + 16EA tee EI tee π ak 4 +4EA tee Gk sh t w e + 16EI tee Gk sh t w k + (9) 4EA tee EI tee π ak + 4Gk sh t w EI tee + EA tee π aei tee Defection of a continuous casteated beam can be determined by using Eq. (8). 9
Ceary, the denominator of Eq. (8) is a very compicated situation, and the position of maximum defection point is at (x=.39). The Eq. (8) is not easy to simpify but it is regarded as Bernoui-Euer beam with modifications. To determine the approximate defection, the cacuation does not consider the shear effect of the web, but rather can be empoyed Eq. (3) w max = q 4 185EI reduced (3) 4. NUMERICL STUDY Casteated beams of various spans were anayzed using the ANSYS Programming Design Language (APDL). The FEA modeing of the casteated beams is carried out by using 3D inear 4-Node thin she eements (SHELL181). This eement presents four nodes with six DOF per node, i.e., transations and rotations on the X, Y, and Z axes, respectivey. The fu ength of the casteated beams is used. The dispacement boundary conditions are appied to a nodes at the two ends. The atera and transverse defections and rotation are restrained (u y =, u z = and θ x =) at the simpy supported end, whie continuous support boundary condition is appied at the other end by restricting the axia dispacement, transverse defections and rotations around the three axes within the cross-section (u x =, u y =, u z =, θ x =, θ y =, and θ z =). The materia properties of the casteated beam are assumed to be inear eastic materia with Young s moduus E =.1 1 5 MPa and Poisson s ratio v =.3. 5. DISCUSSION Fig. 3 shows a comparison of the maximum defations between anaytica soutions using different shear rigidity factors incuding one with zero shear factor and FEA numerica soution for four casteated beams of different fange widths. It can be seen from the figure that, the anaytica soution using the proposed shear factor is cosest to the numerica soution, whereas the anaytica soutions using other shear factors is not as good as the present one. This demonstrates that the shear factor is aso affected by the ratio of the fange width to the beam ength. Aso, it can be seen from the figure that, the onger the beam, the coser the anaytica soution to the numerica soution; and the wider the fanges, the coser the anaytica soution to the numerica soution. Fig. 4 shows the reative error of each anaytica soution when it is compared with the finite eement soution. From the figure it is evident that the error of the anaytica soutions using the present shear rigidity factor does not exceed 5.% for a of discussed four sections in a the beam ength range (>3 meter). In contrast, the anaytica soution ignoring the shear effect, or considering the shear effect by using smear mode or by using the ength-independent shear rigidity factor wi have arge error, particuary when the beam is short. 1
Fig. 3 Maximum defections of Continuous casteated beam with uniformy distributed oad between anaytica soutions using different shear rigidity factors obtained by Eq. (9) incuding one with zero shear factor obtained by Eq. (3) and FEA numerica soution for four casteated beams of different fange widths.(a) b f =1 mm, (b) b f =15 mm, (c) b f = mm and (d) b f =5 mm. (h w =3mm, t f =1mm,t w =8mm and a=1mm) 11
Fig. 4 Divergence of maximum defections of Continuous casteated beam with uniformy distributed oad between anaytica soutions using different shear rigidity factors obtained by Eq. (9) incuding one with zero shear factor obtained by Eq. (3) and FEA numerica soution for four casteated beams of different fange widths. (a) b f =1 mm, (b) b f =15 mm, (c) b f = mm and (d) b f =5 mm. (h w =3mm, t f =1mm,t w =8mm and a=1mm) 1
6. CONLUSIONS This study has reported the theoretica and numerica soutions for cacuating the defection of hexagona casteated beams with Continuous support subjected to a uniformy distributed transverse oad. The anaysis is based on the tota potentia energy method, by taking into account the infuence of web shear deformations. The anaytica and numerica soutions are empoyed for a wide spectrum of geometric dimensions of I-shaped casteated beams. In order to evauate the anaytica resuts, ANSYS software is used. From the present study, the main concusions can be summarized as foows: 1. The present anaytica resuts are in exceent agreement with those obtained from the finite eement anaysis, which demonstrates the appropriateness of proposed approach.. Shear effect on the defection of casteated beams is very important, particuary for short and medium ength beams with narrow or wide section. Ignoring the shear effect coud ead to an under-estimation of the defection. 3. Divergence between anaytica cacuation and numerica cacuation does not exceed (1.13%-31.94%) even for short span casteated beam with narrow or wide section. 4. The effect of web shear on the defection reduces when casteated beam ength increases. 5. Despite that the numerica soution based on FEA has been widey used in the anaysis of casteated beams, it is usuay time consuming and imited to specific geometrica dimensions. Thus, a simpified cacuation soution that is abe to deiver reasonabe resuts but requires ess computationa effort woud be hepfu for both researchers and designers. REFERENCES Aminian, P., Niroomand, H., Gandomi, A. H., Aavi, A. H., & Arab Esmaeii, M. (1), "New design equations for assessment of oad carrying capacity of casteated stee beams: a machine earning approach," Neura Computing and Appications, 3(1), 119-131. Boyer, J. P. (1964), "Casteated Beam- New Deveopment," AISC Engineering 1(3), 14-18. Demirdjian, S. (1999), Stabiity of casteated beam webs, PhD thesis, Department of Civi Engineering and Appied Mechanics. McGi University, Montrea, Canada. Harper, C. (1991), Design in stee 4: casteated & ceuar beams, British Stee. Hosain, M., Cheng, W., & Neis, V. (1974), Defection anaysis of expanded open-web stee beams, Computers & Structures, 4(), 37-336. Kerda, D., & Nethercot, D. (1984), Faiure modes for casteated beams, Journa of Constructiona Stee Research, 4(4), 95-315. Maaek, S. (4), "Shear defections of tapered Timoshenko beams, Internationa Journa of Mechanica Sciences, 46(5), 783-85. doi:1.116/j.ijmecsci.4.5.3 13
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