FINITE DIFFERENCE ANALYSIS OF CURVED DEEP BEAMS ON WINKLER FOUNDATION
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1 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. FINITE DIFFERENCE ANALYSIS OF CURVED DEEP BEAMS ON WINKLER FOUNDATION Adel A. Al-Azzaw and Al S. Shaker Department of Cvl Engneerng, Nahran Unversty, Baghdad, Iraq E-Mal: dr_adel_azzaw@yahoo.com ABSTRACT Ths research deals wth the lnear elastc behavor of curved deep beams restng on elastc foundatons wth both compressonal and frctonal resstances. Tmoshenko s deep beam theory s extended to nclude the effect of curvature and the externally dstrbuted moments under statc condtons. As an applcaton to the dstrbuted moment generatons, the problems of deep beams restng on elastc foundatons wth both compressonal and frctonal restrants have been nvestgated n detal. The fnte dfference method was used to represent curved deep beams and the results were compared wth other methods to check the accuracy of the developed analyss. Several mportant parameters are ncorporated n the analyss, namely, the vertcal subgrade reacton, horzontal subgrade reacton, beam wdth, and also the effect of beam thckness to radus rato on the deflectons, bendng moments and shear forces. The computer program (CDBFDA) (Curved Deep Beam Fnte Dfference Analyss Program) coded n fortran-77 for the analyss of curved deep beams on elastc foundatons was formed. The results from ths method are compared wth other methods exact and numercal and check the accuracy of the solutons. Good agreements are found, the average percentages of dfference for deflectons and moments are (5.3%), and (7.3%), respectvely whch ndcate the effcency of the adopted method for analyss. Keywords: curved deep beam, fnte dfferences, elastc foundatons. INTRODUCION The object of the paper s to analyze curved deep beam usng fnte dfference method. The beam s restng on elastc foundaton wth Wnkler frctonal and compressonal resstances, and loaded generally (both transverse dstrbuted load and dstrbuted moment), and nclude the effect of transverse shearng deformatons. The lnear elastc behavor of curved deep beams on elastc foundatons s studed. The governng dfferental equatons of curved deep beams (n terms of w andψ) are developed and converted nto fnte dfferences. A computer program n (FORTRAN language) s developed. Ths program assemblng the fnte dfference equatons to obtan a system of smultaneous algebrac equatons and than solved by usng Gauss elmnaton method. The deflectons and rotatons for each node are obtaned. The shear and moment are obtaned by smply substtutons of the deflectons and rotatons nto the fnte dfference equatons of moment and shear. The obtaned soluton compared wth avalable results to check the accuracy of ths method. Curved beams are one dmensonal structural element that can sustan transverse loads by the development of bendng, twstng and shearng resstances n the transverse sectons of the beam. It s extensvely used n engneerng and other felds snce such beams have many practcal applcatons. The curved beam elements on elastc foundaton would be helpful for the analyss of rng foundaton of structures such as antennas, water towers structures, transmsson towers and varous other possble structures and superstructures. Deep beam model s based on the Tmoshenko theory. Ths theory consders the effect of transverse shearng deformatons. Thus, the cross sectons of the beam reman plane but not normal to the axs of bendng. These are revew of early studes on curved beams: Volterra (95) analyzed the deflectons of crcular beams restng on elastc foundatons. The beam was loaded by symmetrc concentrated forces actng n a plane perpendcular to the plane of the orgnal curvature of the beam. The foundaton s supposed to react followng the classcal Wnkler and Zmmermann hypotheses,.e., the reacton forces due to the foundaton are proportonal at every pont to the deflecton of the beam at that pont. Rodrguez, (959), solved the three dmensonal bendng of a rng (curved beam n the form of a complete crcle) of unform cross sectonal area and supported on a transverse elastc foundaton. Close, (964), presented a mathematcal analyss for determnng the vertcal deflecton at the free end of a crcular cantlever I-beam (curved n the horzontal plane). Chaudhur and Shore, (977), worked on thn walled curved beam fnte elements, they presented a procedure for the consstent matrx formulaton of a thn walled crcularly curved beam element. Yoo, (979), presented matrx formulaton for the statc analyss of the spatally curved beams of thn walled members. Ths formulaton was qute general and consstent. Fukumoto and Nshda, (98), derved a fundamental equatons of a sngle curved I-beam subjected to the acton of bendng and torsonal moments. They nvestgated the behavor of curved flexural members under large torsonal deflecton. Dasgupta and Sengupta, (988), suggested a formula for the analyss of a horzontal curved beam by usng three node soparametrc fnte elements. The formulaton presented was general and the method, therefore, may be utlzed for straght beams as well. The 4
2 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. beam was wth or wthout an elastc base throughout ts length. FORMULATION Accordng to the small deflecton theory and lnear stress-stran relatonshps, a formulaton for the bendng of curved deep beams s presented heren, and based on the followng assumptons: a) Plane cross-secton before bendng remans plane after bendng. b) The cross-secton wll have addtonal rotaton due to transverse shear warpng of the cross-secton by transverse shear wll be taken nto consderaton by ntroducng a shear correcton factor (c ). Knematcs consderatons A cross secton n θz plane s consdered. The deflecton w and the rotaton of transverse secton ψ n θ drecton are shown n Fgure-. Fgure-. Deformatons of a beam secton. A normal lne to the neutral plane has two degrees of freedom (deflecton w and rotatonψ). The dsplacement n θ-drecton (v) at a pont at dstance z above the neutral plane wll be: v = z. ψ () where v = v(θ) s the dsplacement at neutral plane. w = w(θ) (ndependent of z), ψ = ψ(θ) (postve when n clockwse drecton) u = 0 () The mathematcal expressons of strans are: ε θ = ( R dv ) + dθ u R ε θ = z dψ ( ) (3) Also, the engneerng shearng strans are: γ θz dv dw = ( ) + ( ) dz R d θ γ θ = ψ + α z (4) α = dw ( ) (5) where α s the slope of the deflecton curve. The non-zero stresses are : z dψ σ θ = E εθ = E ( ) (6) dw τθ z = Gγ θ z = G ) R d θ ( ψ + α ) = G ψ + ( Fgure-. Arbtrary cross secton of a curved deep beam. (7) 43
3 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. From Fgure-, the bendng moment s: M = A σ z da (8) θ Usng equaton (6): E dψ M = z zda A EI dψ M = ( ) (9) Where I = Α z da s the second moment of area of the cross-secton. Also, usng equaton (7), the transverse shearng force s: dw Q = Gc A ψ + (0) R dθ Where c s numercal factor representng the restrant of the cross secton aganst warpng, commonly assumed to be (5/6) for rectangular sectons. Statc consderatons Fgure-3 shows the vertcal and moment equlbrum. Fgure-3. Curved beam under appled loadngs. By equlbrum of forces n z-drecton: Q + dq - Q + q. R. dθ = 0 or dq = q.r () dθ Equlbrum of moments n θζ- plane, gves: M + dm - M - Q.R. dθ + µ.r. dθ + q. (Rd θ) = 0 The last term wll be gnored (very small), thus dm = Q.R µ.r dθ () where µ = µ(θ) s the dstrbuted moment (per unt length). By substtutng equaton (0) nto equaton (), and both equatons (9) and (0) nto equaton (), then : dψ d w Gc A[( ) + ( )] = q.r (3) d θ R d θ EI d ψ dw ( ) = Gc A.R[ψ + ( )] -µ.r (4) Equatons (3) and (4) are the governng dfferental equatons of curved deep beams n terms of two deformaton functons (w andψ). These equatons are coupled through the deformaton functons. Governng equatons of curved deep beams on elastc foundatons The governng equatons of curved deep beams on elastc foundatons characterzed by Wnkler model for both compressonal and frctonal resstances are gven: Gc dψ d w A[ + ] + qr K zw.r = 0 (5) dθ EI d ψ dw ( ) Gc A.R[ψ + ] R d θ R d θ h K θ ( ) ψ.r = 0 (6) FINITE DIFFERENCE ANALYSIS The governng dfferental equatons for curved deep beams on elastc foundatons represented by a Wnkler model for frctonal restrants can be rewrtten usng fnte dfferences equatons for an nteror node (): ψ Gc A Gc A[ + ψ ] + [w w + θ R θ Kz wr θ + w ] + q R = 0 Gc A (7) 44
4 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. EI ψ + [ R θ w+ w ( R θ ψ + ψ APPLICATIONS ] Gc AR[ ψ + h ) - Kθ ( ) ψ.r = 0 (8) Rng beam on Wnkler foundaton under four concentrated loads A rng foundaton of (E = 0.7 kn/mm, ν = 0.5), havng a radus of (R = mm), wdth (b = 76 mm), thckness (h = 76mm). The rng beam carres four equal column loads (perpendcular to the rng), each column load (P = kn). The rng beam s restng on an elastc foundaton whch s represented by Wnkler model for compressonal restrant wth the coeffcent (KZ = 0.35*0-4 kn/mm 3 ), as shown n Fgure-4. Ths problem was solved by Dasgupta and Sengupta, (988). In the present study, the same problem s solved by usng the fnte dfference method. The results of deflectons and bendng moments are plotted wth the results of Dasgupta and Sengupta, (988) as shown n Fgures 5 and 6. The fgures show acceptable agreement. The percentage of the dfference between the maxmum deflectons and bendng moments for Dasgupta and Sengupta,(988) soluton and the present study s equal to (4.3%), (.48%), respectvely. Fgure-4. Rng beam on an elastc foundaton under four concentrated loads. Fgure-5. Deflecton curves for free curved beam restng on an elastc foundaton. Fgure-6. Bendng moment dagrams for free curved beam restng on an elastc foundaton. 45
5 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Fxed ends curved deep beam restng on an elastc foundaton under unform loadng A fxed curved deep beam wth Young s modulus of (E = 5 kn/mm ), Posson s rato (ν = 0.5), radus (R = 000 mm), wdth (b = 00mm), and thckness (h = 400 mm), and loaded by a unform load (q = 0.05 kn/mm) s consdered. The load s actng n a plane perpendcular to the plane of the orgnal curvature of the beam and at a total angular dstance (θ = 80 ). The beam s restng on an elastc foundaton whch s represented by Wnkler model for compressonal restrant wth coeffcent (K Z = 0.*0-4 kn/mm 3 ) and by Wnkler frcton model wth sprng coeffcent (K θ = 0.*0-4 kn/mm 3 ), as shown n Fgure-7. Ths problem was solved by usng Plate Foundaton Analyss Program (PFAP) [Al-Allaf, (005)]. In the present study, the problem s solved by usng the fnte dfference method. The results of deflectons obtaned from present study are plotted wth the results of (PFAP) as shown n Fgure-8. The Fgure shows acceptable agreement. The percentage of the dfference between the maxmum deflectons for (PFAP) soluton and the present study s (9.4%). Fgure-7. Fxed ends curved deep beam restng on an elastc foundaton Under unform dstrbuted load (q). Fgure-8. Deflecton curves for fxed ends curved deep beam restng on an elastc foundaton. PARAMETRIC STUDY A parametrc study s performed to show the nfluence of several mportant parameters on the behavor of free curved beam. The values of the vertcal and horzontal subgrade reactons (0.3577*0-3 kn/mm 3 and 0.*0-3 kn/mm 3 ) are consdered. The effect of ncreasng the thckness to radus rato (h/r) s shown n Fgure-9. From ths Fgure the maxmum deflecton wll decrease at decreasng rate as the beam thckness ncreased. It was found that by ncreasng the rato of (h/r) from (0. to ), the maxmum deflecton for the free curved beam under four concentrated loads are decreased by (63.5%). The effect of ncreasng the beam wdth s shown n Fgure-0. From ths Fgure the maxmum deflecton wll decrease at decreasng rate as the beam wdth ncreased. It was found that by ncreasng the wdth of the free curved beam from (76 to 3048 mm), the maxmum deflecton for the curved beam under four concentrated loads are decreased by (75%). The effect of ncreasng the vertcal and horzontal subgrade reactons (K z ) and (K θ ) are shown n Fgures and. From these Fgures, the maxmum deflecton wll decrease at a decreasng rate as the vertcal subgrade reacton s ncreased. Whle the maxmum deflecton wll decrease almost lnearly as the horzontal subgrade reactons s ncreased. It was found that by ncreasng the vertcal and horzontal subgrade reactons for the free curved beam from (0.35*0-3 kn/mm 3 to 0.6*0-3 kn/mm 3 ) for the vertcal, and from (0.*0-3 kn/mm 3 to 0.*0 - kn/mm 3 ) for the horzontal, the maxmum deflecton decreased by (65.3%), (7.84%), respectvely. 46
6 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Fgure-9. Effect of (h/r) on the maxmum deflecton for free curved beam under four concentrated loads. Fgure-0. Effect of beam wdth on the maxmum deflecton for free curved beam under four concentrated loads. Fgure-. Effect of vertcal subgrade reacton on maxmum deflecton for free curved beam under four concentrated loads. Fgure-. Effect of horzontal subgrade reacton on maxmum deflecton for free curved beam under four concentrated loads. 47
7 VOL. 6, NO. 3, MARCH 0 ISSN Asan Research Publshng Network (ARPN). All rghts reserved. CONCLUSIONS a) Orgnal Tmoshenko s deep beam theores are extended to nclude the effects of externally dstrbuted moments along the curved deep beam and modeled by fnte dfference; b) The beneft of ncludng the externally dstrbuted moments to the orgnal theores for beams s to ncrease the doman of applcatons. The dstrbuted moments can be generated from varous problems; externally appled dstrbuted moments, the problems of curved deep beams on elastc foundatons wth frctonal restrants, curved deep beam subjected to temperature gradents, prestressed curved deep beams and the problem of shrnkage n concrete n renforced concrete curved deep beams; c) In the problems of curved deep beams on elastc foundatons, varous models can be suggested to represent both the compressonal and frctonal restrants. The frctonal component can be represented by Wnkler model (proportonal to the horzontal dsplacements) or by Coulomb model (proportonal to the transverse dsplacements) or by a constant value (ndependent of horzontal and transverse dsplacements); d) The numercal technques of the fnte-dfference method are used to solve the problem of curved deep beams restng on elastc foundatons. The formulatons n fnte-dfference method are based on the governng equatons; and e) When the wdth of the beam ncreases, the deflecton wll decrease. Fukumoto Y. and Nshda S. 98. Ultmate Load Behavor of Curved I-Beams. Journal of Mechanc Engneerng Dvson, ASCE. 07(ST): Ross A. Close Deflecton of Crcular Curved I- Beams. Journal of Structural Dvson, ASCE. 93(ST): Volterra E. 95. Bendng of a Crcular Beam Restng on an Elastc Foundaton. Journal of Appled Mechanc, Trans. ASME. 74: -4. Yoo C. H Matrx Formulaton of Curved Grders. Journal of Mechanc Engneerng Dvson, ASCE. 05(ST6): REFERENCES Al-Allaf M.H Three Dmensonal Fnte Element Analyss of Thck Plates on Elastc Foundatons. M. Sc. Thess. Faculty of Engneerng, Nahran Unversty, Iraq. Al-Jubor A. A. 99. Deep Beams and Thck Plates under Generalzed Loadng. M. Sc. Thess. Faculty of Engneerng, Nahran Unversty, Iraq. Al-Musaw A. N Three Dmensonal Fnte Element Analyss of Beams on Elastc Foundaton. M. Sc. Thess. Faculty of Engneerng, Nahran Unversty, Iraq. Bowles J. E. Foundaton, Analyss and Desgn. Chaudhur S. K. and Shore S. 977c. Thn-Walled Curved Beam Fnte Element. Journal of Mechanc Engneerng Dvson, ASCE. 03(ST5): Dasgupta S. and Sengupa D Horzontally Curved Isoparametrc Beam Element Wth or Wth Out Elastc Foundaton Includng Effect of Shear Deformaton. Comp. and Struct. 9(ST (6):
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