A Simple Higher Order Theory for Bending Analysis of Steel Beams

Transcription

1 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 A Simple Higher Order Theory for Bending Analyi of Steel Beam T.K. Meghare 1, P.D. Jadhao 1 Department of Civil Engineering, K.K. Wagh Intitute of Engineering Education and Reearch, Naik- (Univerity of Pune), Maharahtra, India Prof & Head of civil Engineering Dept., K.K. Wagh Intitute of Engineering Education and Reearch, Naik- (Univerity of Pune), Maharahtra, India. ABSTRACT : A new hear deformation theory for the bending analyi of thick iotropic beam made up of teel i preented in thi paper. The theory preented herein i built upon the elementary theory of beam. The tranvere hear tre can be obtained directly from the contitutive relation atifying the hear tre free urface condition on the top and bottom urface of the beam, hence the theory doe not require hear correction factor. Governing equation and boundary condition of the theory are obtained uing the principle of virtual work. The imply upported thick iotropic beam are conidered for the detailed numerical tudie. Reult of diplacement and tree are compared with thoe of other refined theorie and eact theory to how the efficiency of propoed theory. Keyword - Shear deformation, refined theory,, hear correction factor, Four variable. 1. INTRODUCTION It i well-known that elementary theory of bending of beam baed on Euler-Bernoulli hypothei diregard the effect of hear deformation and tre concentration. The theory i uitable for lender beam and i baed on the aumption that the tranvere normal to neutral ai remain o during bending and after bending, which mean tranvere hear train i zero. Thu thi theory underetimate the deflection in cae of thick beam where hear train i ignificant. The firt-order hear deformation theory (FSDT) i an improvement over the elementary theory of beam. It i baed on the hypothei that the normal to the mid-urface before deformation remain traight but not necearily normal to the mid-urface after deformation. Thi i known a firt order hear deformation theory becaue the thickne wie diplacement field for the in aial diplacement i linear or of the firt order. Timohenko developed the FSDT for the diplacement and tre variation for the thick primatic bar. But the deficiencie in ETB and FSDT till eit. Therefore, higher order hear deformation theorie are developed to obtain the improved reult for the thick beam. In thee theorie the diplacement field i epanded up to the third power of thickne coordinate of beam to have the parabolic variation of tranvere hear tree. The number of diplacement variable are more in thee theorie. The higher order theory i developed by Reddy to get the parabolic hear tre ditribution through the thickne of beam and to atify the hear tre free urface condition on the top and bottom urface of the beam to avoid the need of hear correction factor. Timohenko [1] propoed a hypothei for the development of firt order hear deformation theory which tate that the plane ection which i perpendicular to the neutral ai before bending remain plane but not necearily perpendicular to the neutral ai after bending. In thi theory the tranvere hear train ditribution over the croection of the beam i aumed to be contant through the thickne and thu require hear correction factor. evinon [] preented parabolic hear deformation theorie auming a higher order variation of in-plane diplacement in term of thickne coordinate. Sayyad and Ghugal [] propoed a new hyperbolic hear deformation theory i developed for the tatic fleure of thick iotropic beam, conidering hyperbolic function in term of thickne co-ordinate aociated with tranvere hear deformation effect. Rotation of normal i taken a combined effect of hear lope and bending lope at the neutral ai. The mot important feature of the theory i that the tranvere hear tre can be obtained directly from the contitutive relation atifying the hear tre free urface condition on the top and bottom of the beam. Hence the theory obviate the need of hear correction factor. Comprehenive review of higher order hear deformation theorie have been given by Ghugal and Shimpi []. The theorie are reviewed for the both iotropic and aniotropic laminated beam. Ghugal and Sharma [5] have developed a variationally conitent refined hyperbolic hear deformation theory for fleure and free vibration of thick iotropic beam. Recently Ghugal and Nakhate [6] ha developed trigonometric hear deformation theory for the tatic fleure of thick iotropic beam and obtained the general olution of thick iotropic beam with variou upport and loading condition. Sayyad, Ghugal and Borkar [8] analyzed the ingle layer fibrou compoite beam uing everal diplacement baed hear deformation theorie. A ISSN: Page 9

2 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 tatic fleural analyi i carried out for imply upported fibrou compoite beam ubjected to different mechanical loading. The reult obtained uing all the theorie are compared with thoe obtained by eact elaticity olution for a inuoidal load and then their validity i checked for other loading condition. Ghugal and Dahake [9] introduced a trigonometric hear deformation theory for fleure of thick or deep beam, taking into account tranvere hear deformation effect. The number of variable in the preent theory i ame a that in the firt order hear deformation theory. The inuoidal function i ued in diplacement field in term of thickne coordinate to repreent the hear deformation effect. The fied iotropic beam ubjected to parabolic load are eamined uing the preent theory. Sayyad [1] preented, a refined hear deformation theory i developed for the tatic fleure and free vibration analyi of thick iotropic beam, conidering inuoidal, hyperbolic and eponential function in term of thickne coordinate aociated with tranvere hear deformation effect. Rotation of normal i taken a combined effect of hear lope and bending lope at the neutral ai. Vinay, Raju, Adil Dar and Manzoor[11]focue on a numerical model developed for Hollow tube and concrete filled teel tube (CFST) column under monotonic loading and. The tudy wa conducted uing MSC NASTRAN.PATRAN finite element oftware. Three dimenional nonlinear finite element model developed to tudy the force tranfer between teel tube and concrete core. 8 Noded Heagonal element are conidered for finite element analyi. Analyi are done for different rate of concrete M, M & M. 1%, %, % Epoy ued a infill in M, M, M.For different length of pecimen that i 5mm, 5mm, 5mm with the different thickne.mm,mm and.mm and different Diameter that i.mm,8.mm,6.mm are to be analyed. Analyi wa run for both Hollow tube and Concrete filled teel tube (CFST). Reult of Analytical olution wa compared with Eperimental reult and Deign code uch a EUROCODE, ACI CODE, BS5 CODE. Kumar and Ravih[1] introduced the ue of fiber reinforced polymer (FRP) reinforcement in concrete tructure ha increaed rapidly in the lat 1 year due to their ecellent corroion reitance, high tenile trength, and good nonmagnetization propertie. Fiber-reinforced polymer (FRP) application i a very advanced method for the purpoe of repair and trengthen tructure which are weak during their life pan. FRP repair ytem provide an economical and effective method for repairing repair ytem and a a material. In thi paper a diplacement baed higher order hear deformation theory (HOSDT) i ued for the bending analyi of thick iotropic beam which include effect of tranvere hear deformation and rotary inertia. The diplacement field of the theory contain one variable of beam. The theory i hown to be imple and more effective for the bending analyi of iotropic beam.. Theoretical formulation.1. Iotropic Beam under Conideration Conider a beam made up of iotropic material. The plate occupie a region a, h/ z h/, where, a i length and h i total thickne of beam... Aumption Made In the Preent Theory Theoretical formulation of preent theory i baed on the following aumption. 1. The diplacement are mall in comparion with the beam thickne and therefore train involved are infiniteimal.. The diplacement u in -direction conit of etenion, bending, and hear component. The tranvere diplacement include two component i.e. bending ( w ) and hear ( w ). The beam i ubjected to tranvere load only. 5. The body force are neglected... The Diplacement Field Baed upon the before mentioned aumption, the diplacement field of the propoed beam theory i given a below: dwb( ) dw( ) u( ) u( ) z f ( z) d d (1) w( ) w ( ) w ( ) () ISSN: Page b.. Strain-Diplacement Relationhip Normal train and hear train b are obtained within the framework of linear theory of elaticity uing the diplacement field given by Eq. (1). du du d wb d w z f () z () d d d d du dw d dwb( ) dw( ) d u( ) z f ( z) wb ( ) w ( ) d d d d z dw 1. h d.5. Stre-Strain Relationhip du d wb d w E. E. z f ( z) d d d () (5)

3 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 z G. E. (6) (1 ). Governing Equation and Boundary Condition The variationally conitent governing equation of equilibrium and boundary condition aociated with the preent theory can be derived uing the principle of virtual work. The analytical form of principle of virtual work can be written a- h/ h/ q( ) d.. d. (7) Separating above equation for further calculation, h/ h/ E z dw z d w.. d d. (1 ) h d h/ h/ h d (8) Where be the arbitrary variation. Integrating above equation by part and collecting the coefficient of u, wb and w to obtain governing equation of equilibrium and boundary condition aociated with the preent theory. The governing equation of equilibrium are a follow: u wb w A B C (9a) u wb w (9b) B D E q C u wb w w E F A q m q( ) qm in (11) m1,,5 Uniformly ditributed load q qm For m 1,,5,..., m Sinuoidal load q m = q (1) The following olution form i aumed for unknown diplacement variable, w w b atifying the boundary condition for imply upported beam eactly. m wbm in wb (1) w m wm in Where w and are arbitrary contant, which bm can be calculated by ubtituting the above olution in governing differential equation reulting in to following equation(hown in Matri form).subtitution of thi form of olution and tranvere load q() into governing equation lead to following equation. K11 K1 K1 um K1 K K (1) wbm qm K1 K K w m q m Where element of tiffne matri K are a follow: Where, C A h/ h/ h/ h/ E. E. f ( z). D B h/ E.. z h/ h/ h/ E. z. h/ h/ E E. z. f ( z). F E. f ( z). h/ h/ h/ A G.(1 f '( z)). h/ (9c) (1). Illutrative Eample A imply upported iotropic beam occupying the region given by the Eq. (1) i conidered. The beam i ubjected to uniformly ditributed tranvere load, q() on urface z = h/ acting in the downward z-direction a given below: 5.1 Numerical Reult Reult obtained for diplacement and tree will now be compared and dicued with the correponding reult of higher order hear deformation theory (HSDT) of Reddy, (FSDT) by Timohenko, hyperbolic hear deformation theory m K11 A m K1 K1 C m K D (15) m m K K E K F Obtaining u m, wbm and w m from the equation 1 and 1 one can calculate all diplacement and tree. 5. Numerical Reult and Dicuion Following non dimenional parameter are ued to analyze the iotropic teel beam under variou loading condition. by Sayyad and Ghugal and ETB theory by Bernoulli-Euler. The numerical reult are preented in the following non-dimenional form and following material propertie of iotropic beam are ued. ISSN: Page 1

4 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 bue 1Ewh b Material Propertie: E = 1 GPa u w qh q q µ =. E b (16) G= (1+μ) q Table1. Comparion of non-dimenional inplane diplacement ( u ), Tranvere diplacement ( w ), inplane normal tre ( ) and tranvere hear tre in iotropic teel beam ubjected to inuoidal load. S Theory Method u w Preent Sayyad and Ghugal TSDT Reddy HSDT Timohenko FSDT Bernoulli-Euler ETB _ Ghugal Eact Preent Sayyad and Ghugal TSDT Reddy HSDT Timohenko FSDT Bernoulli-Euler ETB _ Ghugal Eact Table. Comparion of non-dimenional inplane diplacement ( u ), Tranvere diplacement ( w ), inplane normal tre ( ) and tranvere hear tre in iotropic teel beam ubjected to uniformly ditributed load. S Theory Method u w Preent Sayyad and Ghugal TSDT Reddy HSDT Timohenko FSDT Bernoulli-Euler ETB _ Timohenko and Goodier Eact Preent Sayyad and Ghugal TSDT Reddy HSDT Timohenko FSDT Bernoulli-Euler ETB _ Timohenko and Goodier Eact Table. Comparion of non-dimenional inplane diplacement ( u ), Tranvere diplacement ( w ), inplane normal tre ( ) and tranvere hear tre in iotropic teel beam ubjected to inuoidal, uniformly ditributed load and linearly varying load. S Theory Method u w Preent Sayyad and Ghugal TSDT Reddy HSDT Timohenko FSDT Bernoulli-Euler ETB _ Timohenko and Goodier Eact ISSN: Page

5 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 1 Preent Sayyad and Ghugal TSDT Reddy HSDT Timohenko FSDT Bernoulli-Euler ETB _ Timohenko and Goodier Eact Table. Non-dimenional inplane diplacement ( u ), Tranvere diplacement ( w ), inplane normal tre ( ) and tranvere hear tre in iotropic teel beam for different apect ratio and ubjected to inuoidal load. S u w Table5. Non-dimenional inplane diplacement ( u ), Tranvere diplacement ( w ), inplane normal tre ( ) and tranvere hear tre in iotropic teel beam for different apect ratio and ubjected to uniformly ditributed load. S u w Table6. Non-dimenional inplane diplacement ( u ), Tranvere diplacement ( w ), inplane normal tre ( ) and tranvere hear tre varying load. in iotropic teel beam for different apect ratio and ubjected to linearly S u w ISSN: Page

6 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April u 1 Ʈz Fig.1. Through thickne variation of in-plane diplacement of iotropic plate ubjected to inuoidal load for apect ratio. W S Fig.. Through thickne variation of tranvere hear tre of iotropic plate ubjected to inuoidal load for apect ratio u -. Fig..Through thickne variation of tranvere diplacement of iotropic plate ubjected to inuoidal load for different apect ratio..5 Fig.5. Through thickne variation of inplane diplacement of iotropic plate ubjected to uniformly ditributed load for apect ratio ơ w Fig..Through thickne variation of in-plane normal tre of iotropic plate ubjected to inuoidal load for apect ratio. Fig. 6.Through thickne variation of tranvere diplacement of iotropic plate ubjected to uniformly ditributed load for different apect ratio. S ISSN: Page

7 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 Fig. 1.Through thickne variation of tranvere diplacement of iotropic plate ubjected to.5 linearly varying load for different apect ratio ỡ ỡ Fig.7.Through thickne variation of in plane normal tre of iotropic plate ubjected to uniformly ditributed load for apect ratio..5 Fig. 11.Through thickne variation of in plane normal tre of iotropic plate ubjected to linearly varying load for apect ratio. 1 τz Fig.8.Through thickne variation of tranvere hear tre of iotropic plate ubjected to uniformly ditributed load for apect ratio τz u Fig.9.Through thickne variation of in-plane diplacement of iotropic plate ubjected to linearly varying load for apect ratio. w S Fig. 1.Through thickne variation of tranvere hear tre of iotropic plate ubjected to linearly varying load for apect ratio. 5. Dicuion of Reult Table 1 how the comparion of maimum diplacement and tree for the iotropic beam (teel) ubjected to inuoidal load. The preent theory overetimate and other higher Order theorie underetimate the reult of in-plane diplacement a compared to thoe of eact olution. Through thickne variation of in-plane diplacement for iotropic beam ubjected to inuoidal load i hown in Fig.. The HSDT and FSDT overetimate the value of maimum tranvere deflection for apect ratio repectively while the ETB underetimate the value of maimum tranvere diplacement for apect ratio and 1 repectively due to neglect of tranvere hear deformation. The value of maimum normal bending tre obtained by preent theory i in tune with the eact olution for all apect ratio. Theory of Reddy overetimate the normal bending tre for apect ratio and 1 repectively compared to thoe of eact value. The value of normal bending tre predicted by FSDT and ETB are identical for all apect ratio. FSDT and ETB underetimate the value of normal bending tre ISSN: Page 5

8 SSRG International Journal of Civil Engineering (SSRG-IJCE) volume Iue April 15 for apect ratio and 1 repectively a compared. Theory avoid the need of hear correction factor to eact value. The tranvere hear tre atifie 5. The reult of tranvere diplacement, in-plane the tre free boundary condition on the top and normal, in-plane hear and tranvere hear bottom urface of the beam when obtained by tree obtained are identical for iotropic contitutive relation. The FSDT yield lower value beam and for all the loading cae. Thi how of tranvere hear tre when obtained uing that tree are independent of modulu of contitutive relation. The maimum tranvere elaticity when material i homogeneou. hear obtained by preent theory uing equation of 7. Acknowledgement equilibrium i in ecellent agreement with that of We would like to epre our gratitude to Prof.A.S.Sayyad, eact olution for the apect ratio and 1. The Department of Civil Engineering, SRES College of preent theory, HSDT, FSDT and ETB give Engineering Kopargaon, for hi valuable guidance related to thi identical value of thi tre for apect ratio 1 topic. when obtained uing equation of equilibrium. Comparion of diplacement and tree REFERENCES for the iotropic beam ubjected to uniformly [1] S. P. Timohenko, On the correction for hear of the ditributed load are hown in Table. Through differential equation for tranvere vibration of primatic bar, Philoophical Magazine, Serie 6, Vol. 1, pp. 7-76, 191. thickne variation of diplacement and tree for the iotropic beam ubjected to uniformly [] M. evinon, A new rectangular beam theory, Journal of ditributed load for apect ratio are hown in Sound and Vibration, Vol. 7, pp , Fig. 5 through 8. [] A. Sayyad and Y. Ghugal, Fleure of Thick Beam uing The comparion of aial diplacement for New Hyperbolic Shear Deformation Theory, International iotropic beam ubjected to linearly varying load i Journal of Mechanic, Iue, Volume 5, 11. hown in Table. The eamination of Table reveal that the aial diplacement predicted by [] Y. Ghugal and R. Shimpi, A review of refined hear deformation theorie for iotropic and aniotropic laminated preent theory i in ecellent agreement with that beam, Journal of Reinforced Platic and Compoite, Vol., of eact olution for apect ratio 1 wherea HSDT No., pp. 557, 1. of Reddy overetimate the ame. The aial diplacement predicted by FSDT and ETB are [5] Y. Ghugal and R. Sharma, Hyperbolic hear deformation theory for fleure and vibration of thick iotropic beam, identical for both the apect ratio. FSDT and ETB International Journal of Computational Method, Vol. 6, No., how the identical value for tranvere pp , 9. diplacement for both the apect ratio. The aial bending tre predicted by preent theory i in [6] Y. Ghugal and V. Nakhate, Fleure of thick beam uing trigonometric hear deformation theory-the Bridge and Struct. ecellent agreement with that of eact olution Engineer, The Quarterly Journal of the Indian National Group wherea FSDT and ETB underetimate the ame of International Aociation for bridge and Structural for both the apect ratio. The through thickne variation of aial bending tre for the iotropic Engineering, Vol. 9, No., pp. 1-17, 1. beam ubjected to linearly varying load for apect [7] Y. Ghugal, A two-dimenional eact elaticity olution ratio i hown in Fig. 11. The tranvere hear of thick beam, Departmental Report-I, Dept. of Applied tre predicted by preent theory i in ecellent Mechanic, Govt. Engineering College Aurangabad, India, pp. agreement when obtained uing contitutive 1-96, 6. relation. The through thickne variation of [8] A. Sayyad,Y. Ghugal and R. Borkar, Fleural analyi tranvere hear tre for iotropic beam ubjected of fibrou compoite beam under variou mechanical loading to linearly varying load i hown in Fig 1. uing refined hear deformation theorie, Compoite: Table, 5 and 6 how the maimum value for Mechanic, Computation, Application. An International Journal 5(1), 1 19 (1). correponding diplacement and tree for inuoidal load, uniformly ditributed load and [9] Y. Ghugal and A. Dahake, Fleural analyi of deep linearly varying load repectively. beam ubjected to parabolic load uing refined hear deformation theory, Applied and Computational Mechanic 6 6. Concluion (1) From the tudy of bending analyi of thick iotropic beam by uing higher order hear deformation theory (HOSDT), following concluion are drawn: 1. The reult of diplacement and tree obtained by preent theory for the all loading cae are in ecellent agreement with thoe of eact olution.. The preent theory atifie the hear tre free urface condition on the top and bottom urface of the plate.. Theory i variationally conitent. [1] A. Sayyad, Static fleure and free vibration analyi of thick iotropic beam uing different higher order hear deformation theorie, International Journal of Applied Mathematic and Mechanic, 8 (1): 71-87, 1. [11] G. Vinay, J. Raju, M. Adil Dar and S.B. Manzoor, A Study on Compoite Steel Tube, SSRG International Journal of Civil Engineering (SSRG-IJCE) EFES April 15, ISSN: [1] D.Kumar and G.Ravih, Ue Of GFRP (Gla Fiber Reinforced Polymer) for Strenthening of Reinforced Concrete Beam, SSRG International Journal of Civil Engineering (SSRG-IJCE) EFES April 15, ISSN: ISSN: Page 6