COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD

Transcription

1 COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros, H-355 Hungary e-mal: Abstract Ths paper gves an analytcal soluton for deflecton, slp and nternal forces n composte beams wth weak shear connecton. The appled loads are the mechancal and thermal load. The thermal load s caused by unform temperature change and the consdered beam s statcally ndetermnate. The Euler-Bernoull beam hypothess s assumed to hold for both two beam components. The consttutve equaton between the horontal slp and nter-lamnar shear force s lnear. An example llustrates the applcaton of presented method.. INTRODUCTION The paper deals wth the soluton of statc problem of two-layer composte beam wth weak shear connecton. The consdered beam and ts load are shown n Fg.. The beam carres the unform mechancal load at hgh temperature, so that the temperature change T t t, where t s the absolute temperature of the beam and t s the reference temperature at whch no deformaton and the beam s stress free. The presented analytcal soluton s based on the Euler-Bernoull beam theory and the one-dmensonal verson of Duhamel-Neumann s law [,]. The beam component B has a rectangular cross secton A whose dmensons are h and b (,) as presented n Fg.. The modulus of elastcty for beam component B s E and the lnear thermal expanson coeffcent s (,). The length of the beam s denoted by L and the cross secton at s fxed and the cross secton at L s smply supported. The orgn O of the rectangular Cartesan coordnate system Oxy s the E-weghted centre of the left end cross secton, so that the axs s the E-weghted centre lne of the consdered composte beam wth flexble shear connecton. A pont P n B B B s ndcated by the poston vector OP r xex ye y e, where ex, ey, e are the unt vectors of the coordnate system Oxy. It s known the poston of E-weghted cross secton s obtaned from next equaton (Fg. ) A E A E h h c CC c, c CC c, CC c, AE A E A E. AE AE () DOI:.6649/musc.5.66

2 The common boundary of the beam components B and B s determned by y y c.5 h, x.5 b, L.. GOVERNING EQUATIONS Fgure. Two-layer composte beam Accordng to the Euler-Bernoull beam theory, whch s vald for each homogeneous beam components, the deformed confguraton s descrbed by the next dsplacement feld dv u( x, y, ) v( ) e y w ( ) y e, ( x, y, ) B, (,). () d On the common boundary of B and B the axal dsplacement may have jump whch s called the nterlayer slp s( ) w ( ) w ( ). (3) Applcaton of the stran dsplacement relatonshp of the lneared theory of elastcty gves [,] x y xy y x, ( x, y, ) B, (4) dw y, ( x, y, ) B, (,), d d (5) where x, y, are the normal strans, xy, x, denote the shearng strans. The y normal stress s obtaned from the one-dmensonal verson of Duhamel Neumann s law [,] dw E y, (,, ), (,). T x y B d d (6)

3 The temperature of the two-layer composte beam ntally s the reference temperature t. Its temperature s slowly rased to constant unform temperature t t T. Below we defne the secton forces and secton moments [3] (Fg. ) where dw N A A E c T (7) d ( ), (,), d d A dw M y A A E c T E I (8) d ( ), (,), d d A I y d A, (,). (9) A Fgure. Normal forces and bendng moments The nterlayer slp s s assumed to be a lnear functon of shear force transmtted between the two beam components that s we have [3,4] Q ks, () where k s the slp modulus. In the present problem there are no axal forces, so N N N that s, dw dw A E A E AE T, AE A E A E. () d d Combnaton of Eq. (3) wth Eq. () provdes where ds N AE c T, d d ds N AE c T, d d () (3)

4 . (4) AE A E A E The bendng moment on the whole cross secton can be expressed as ds,. M c AE T IE IE I E I E d d (5) The cross sectonal shear force s as follows d d d V ( ) c AE IE. (6) 3 M s v 3 d d d From the equlbrum condton of a small beam element B (Fg. 3) we receve dn d ks. (7) N N In detaled form of Eq. (7) Fgure 3. Equlbrum condton n axal drecton 3 d s d k s c v. (8) d AE d 3 Combnaton of Eq. (6) wth Eq. (8) results where d s d c IE s V, (9) IE IE c AE, () IE k AE IE. ()

5 The cross sectonal rotaton n terms of deflecton s dv ( ) () d accordng to the Euler-Bernoull beam theory. From Eq. (5) we get IE ( ) () M I ( ) c AE s( ) s() T, M I ( ) M d. (3) Integraton of Eq. (3) yelds the expresson of v v( ) IE v( ) v() () M ( ) c AE s d s().5 T, (4) II M ( ) M d. II I Eqs. (), (3) and (4) wth the boundary condtons gve the possblty to obtan the deflecton, slp and cross sectonal rotaton. The applcaton of followed method s llustrated by the soluton of problem depcted n Fg.. 3. EXAMPLE Denote F the unknown reacton at L. By the applcaton of equaton of statcs we gan V ( ) F f L, (5) The boundary condtons n our case are f M ( ) F L L. (6) v(), (), s(), v( L), N ( L). (7) It can be proved that from Eq. (7) 5 and M ( L) t follows that ds T at L. (8) d Eqs. (4), (7) 3 and Eq. (8) can be used to get the soluton of Eq. (9) n terms of F. Substtuton of expresson of s s( ) nto Eq. (6) and usng the boundary condtons (7), (7) and (7) 4 we get the value of the reacton F, whch

6 essentally the soluton of the consdered problem snce V V ( ), M M ( ) wll be known functons. The followng numercal data are used n the example: h.3 m, h.6 m, E. Pa, E 8 Pa, b. m, L.5 m, k 6 Pa,.8 / K,.43 / K, T 5 K, f N. The computatons result for the reacton at L, F N. In Fg. 4 the graph of deflecton functon and n Fg. 5 the graph of slp functon are shown for f, T. The plots of bendng moment M M ( ) obtaned from formula (5) and formula (6) are llustrated n Fg. 6 for f, T. The plot of the axal force N ( ) N s presented n Fg. 7 for f, T. If there s no appled thermal load then f N and T, the plot of v v( ) and s s( ) are shown n Fgs. 8 and 9. In ths case the bendng moment M M ( ) and axal forces can be seen n Fgs. and. Fgure 4. The graph of v v( ) f, T 4. CONCLUSIONS Fgure 5. The graph of s s( ) f, T Ths paper presents an analytcal method to obtan the deflecton, slp and nternal forces for composte beams wth weak shear connecton subjected to mechancal and thermal load. The soluton of ths strength of materals problem can be used to desgn composte beams wth mperfect connecton workng n hgh temperature.

7 Fgure 6. The plot of M M ( ) f, T Fgure 7. The plot of N N ( ) f, T Fgure 8. The plot of v v( ) f, T Fgure 9. The plot of s s( ) f, T

8 Fgure. The graph of M M ( ) f, T ACKNOWLEDGEMENTS Fgure. The graph of N N ( ) f, T Ths research was (partally) carred out n the framework of the Center of Excellence of Innovatve Engneerng Desgn and Technologes at the Unversty of Mskolc REFERENCES [] BOLEY, B. A. and WEINER, I. M.: Theory of Thermal Stresses. Wley, New York, 96. [] HETNARSKI, R. B. and ESLAMI, M. R.: Thermal Stresses Advanced Theory and Applcaton. Sprnger, New York,. [3] ECSEDI, I. and BAKSA, A.: Statc analyss of composte beams wth weak shear connecton. Appled Mathematcal Modellng 35,, [4] GIRHAMMAR, U.: A smplfed analyss method for composte beams wth nterlayer slp. Internatonal Journal of Mechancal Scences 5(7), 9,