International Journal of Engineering & Applie Sciences (IJEAS) Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale Vol.8, Issue 3(016) 67-73 Static analysis of beams on elastic founation by the metho of iscrete singular convolution Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale * Aeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics Antalya-TURKIYE * e-mail: civale@yahoo.com Receive ate: September 016 Accepte ate: October 016 Abstract A iscrete singular convolution metho is presente for computation of the eflection analysis of beams resting on elastic founation.. In the metho of iscrete singular convolution partial space erivatives of a function appearing in a ifferential equation are approimate by means of some ernels. Results are compare with eisting solutions available from other analytical an numerical methos. The metho presente gives accurate results an is computationally efficient. Keywors: iscrete singular convolution, beams on elastic founation. 1. Introuction Real physical systems or engineering problems are often escribe by partial ifferential equations, either linear or nonlinear an in most cases, their close form solutions are etremely ifficult to establish. As a result, approimate numerical methos have been wiely use to solve partial ifferential equations which arise in almost all engineering isciplines. The most commonly use numerical methos for such applications are the finite element, finite ifference an bounary element metho an nowaays, most engineering problems can be solve by these methos to satisfactory accuracy if a proper an sufficient number of gri points are use. The analysis of structures on elastic founations is of consierable interest an wiely use in several fiels such as founation engineering, pavement an railroa engineers, pipelines application, an some aero-space structures. Many problems in the engineering applications relate to soil-structure interaction can be moele by means of a beam or a beam-column on elastic founation. Although few type founation moel are eist, the Winler founation moel is etensively use by engineers an researchers because of its simplicity. Generally, the founation is consiere as an array of springs uniformly istribute along the length of the beam. A etaile eplaine of founation moels can be foun in relate references [1-15]. In this paper, iscrete singular convolution metho technique is presente for computation of the static analysis of beams on elastic founation. The accuracy of the solutions is inferre by comparison with analytical an other numerical solutions. 67
Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale. Discrete singular convolution (DSC) Discrete singular convolutions (DSC) algorithm introuce by Wei [16]. Wei an his coworers [17-6] first applie the DSC algorithm to solve soli an flui mechanics problem. These stuies inicates that the DSC algorithm wor very well for the vibration analysis of plates, especially for vibration an bucling analysis of micro an macro beams, plates an shells [7-66]. Furthermore, it is also conclue that the DSC algorithm has global methos accuracy an local methos fleibility for solving ifferential equations in applie mechanics. In a general efinition, numerical solutions of ifferential equations are formulate by some singular ernels. The mathematical founation of the DSC algorithm is the theory of istributions an wavelet analysis. Consier a istribution, T an η(t) as an element of the space of the test function. A singular convolution can be efine by [16] F ( t) ( T η)( t) T( t ) η( ), (1) where T ( t ) is a singular ernel. For eample, singular ernels of elta type [17] T ( ) ) δ ( n ( ) ; (n =0,1,,...,). () Kernel T ( ) δ( ) is important for interpolation of surfaces an curves, an T ( ) ) δ ( n ( ) for n>1 are essential for numerically solving ifferential equations. With a sufficiently smooth approimation, it is more effective to consier a iscrete singular convolution [18] F ( t) T ( t ) f ( ), (3) α α where F (t) is an approimation to F(t) an { }is an appropriate set of iscrete points on which the DSC (3) is well efine. Note that, the original test function () has been replace by f(). This new iscrete epression is suitable for computer realization. The mathematical property or requirement of f() is etermine by the approimate ernel T α. Recently, the use of some new ernels an regularizer such as elta regularize [19] was propose to solve applie mechanics problem. The Shannon s ernel is regularize as [0] δ sin[( π / )( )] ( ) ( ) ep ; >0. (4) ( π/ )( ) σ,σ where is the gri spacing. It is also nown that the truncation error is very small ue to the use of the Gaussian regularizer, the above formulation given by Eq. (1) is practically an has an essentially compact support for numerical interpolation. Equation (1) can also be use to provie iscrete approimations to the singular convolution ernels of the elta type [18] 68
Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale f ( n) ( ) M M δ ( ) f ( ), (5) In the DSC metho, the function f () an its erivatives with respect to the coorinate at a gri point i are approimate by a linear sum of iscrete values f ( ) in a narrow banwith [- M, + M ]. This can be epresse as n f ( ) M ( n) ( n ) f ( ) δ,σ ( i n M i ) f ( ) ; (n=0,1,,...,). (6) where superscript n enotes the nth-orer erivative with respect to.. For eample the secon orer erivative at = i of the DSC ernels for irectly given () δ,σ ( j) δ,σ ( j), (7) i The iscretize forms of Eq. (7) can then be epresse as f () 3. Beams on elastic founation f M () ( ) δ,σ( N) f. (8) i, j M i Numerous researchers have treate the linear an nonlinear analysis of beams on elastic founation having various support conitions. Close form solutions of the governing ifferential equations have been propose in the literature [1-8]. In recent years, the problem of stability an vibration analysis of beam or beam-columns has been stuie [9-15]. Consier a linear elastic beam of stiffness EI on a winler elastic founation of moulus of. The governing ifferential equation for the eflection of the beam resting on elastic founation in Fig.1 is 4 v EI v q() 4 where v efines the eflection of the beam, E is the moulus of the elasticity of the beam material, I is the moment of inertia of the cross-section, q() is the istribute lateral loa, an is the founation moulus. By using DSC iscretization the Eq. (9) taes the form EI N j1 (4) ( ) v( ) v q( ) (10) /, i i (9) 69
Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale y,v q() A B L Fig.1. a. Beam on elastic founation 4. Results an conclusions A partially loae beam on elastic founation with 5 m is consiere [67]. The uniformly istribute loa (q=1 N/m) is partially effecte to the beam (from the 1 m insie of B support uring 1 m). The results are liste in Table 1. The results are liste for two points (at =3m an at =4m). The beam have EI=45*10 3 N /m. The founation stiffness is =10 6 N/m. It is shown that the reasonable accurate results have been obtaine for DSC metho using 7 points. Table 1. Deflections for beams for two points (m) Ref. 67 DSC (N=7) DSC (N=9) (N=11) 3 0.497*10-6 0.495*10-6 0.497*10-6 0.497*10-6 4 0.518*10-6 0.517*10-6 0.518*10-6 0.518*10-6 References 1. Hetenyi, M., Beams on elastic founation, The University of Michigan Press, 1946.. Hetenyi, M., Beams an plates on elastic founations an relate problems, Appl. Mech. Rev. 19, 95-10, 1966. 3. Kameswara, N.S.V., Das, Y.C., Ananarishnan M., Dynamic response of beams on generalize elastic founation. Int. J. Solis Struc., 11, 55-73, 1975. 4. Kerr, A.D., Elastic an viscoelastic founation moels. J. Appl. Mech., 31, 491-498, 1964. 5. Mirana, C., Nair, K., Finite beams on elastic founation. J. St. Div. ASCE, 9, 131-14, 1966. 6. Ting, B.Y., Finite beams on elastic founation with restraints. J. St. Div. ASCE, 108, 611-61, 198. 7. Coo, R.D., Concept an applications of finite element analysis, Wiley, New Yor, 1981. 8. Lentini, M., Numerical solution of the beam equation with nonuniform founation coefficient, J. App. Mech. ASME, 46, 901-904, 1979. 9. Lai, Y.C., Ting, B.Y., Lee, W.S., Becer, W.R., Dynamic response of beams on elastic founation, J. Struct. Eng. ASCE, 118, 853-58, 199. 10. Wang, J., Vibration of steppe beams on elastic founations, J. Soun Vib., 149, 315-3, 1991. 70
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Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale 33. Civale, Ö., Three-imensional vibration, bucling an bening analyses of thic rectangular plates base on iscrete singular convolution metho, International Journal of Mechanical Sciences, 49, 75 765, 007. 34. Demir, Ç., Mercan, K., Civale, Ö., Determination of critical bucling loas of isotropic, FGM an laminate truncate conical panel, Composites Part B: Engineering, 94, 1-10, 016. 35. Civale, Ö., Funamental frequency of isotropic an orthotropic rectangular plates with linearly varying thicness by iscrete singular convolution metho. Appl Math Moel, 33(10), 385-3835, 009. 36. Civale, Ö., Free vibration analysis of symmetrically laminate composite plates with first-orer shear eformation theory (FSDT) by iscrete singular convolution metho, Finite Elements in Analysis an Design, 44(1-13), 75-731, 008. 37. Civale, Ö., Vibration analysis of conical panels using the metho of iscrete singular convolution, Communications in Numerical Methos in Engineering, 4, 169-181, 008. 38. Civale, Ö., Kormaz, A.K., Demir, Ç., Discrete Singular Convolution Approach for Bucling Analysis of Rectangular Kirchhoff Plates Subjecte to Compressive Loas on Two Opposite Eges, Avance in Engineering Software, 41, 557-560, 010. 39. Civale, Ö., Analysis of thic rectangular plates with symmetric cross-ply laminates base on first-orer shear eformation theory, Journal of Composite Materials, 4(6), 853-867, 008. 40. Seçin, A., Sarıgül, A.S., Free vibration analysis of symmetrically laminate thin composite plates by using iscrete singular convolution (DSC) approach: algorithm an verification. J Soun Vib, 315, 197-11, 008. 41. Xin, L., Hu, Z., Free vibration of simply supporte an multilayere magnetoelectroelastic plates. Comp Struct, 11, 344-350, 015. 4. Wang, X., Xu, S., Free vibration analysis of beams an rectangular plates with free eges by the iscrete singular convolution. J Soun Vib, 39, 1780-179, 010. 43. Baltacıoğlu AK, Civale Ö, Agöz B, Demir F. Large eflection analysis of laminate composite plates resting on nonlinear elastic founations by the metho of iscrete singular convolution. Int J Pres Vessel Pip 011;88:90-300. 44. Civale Ö, Agöz B. Vibration analysis of micro-scale sector shape graphene surroune by an elastic matri. Comp. Mater. Sci. 013;77:95-303. 45. Gürses M, Civale Ö, Kormaz A, Ersoy H. Free vibration analysis of symmetric laminate sew plates by iscrete singular convolution technique base on first-orer shear eformation theory. Int J Numer Methos Eng 009;79:90-313. 46. Baltacıoglu, A.K., Agöz, B., Civale, Ö., Nonlinear static response of laminate composite plates by iscrete singular convolution metho. Compos Struct, 93, 153-161, 010. 47. Gürses, M., Agöz, B., Civale, Ö., Mathematical moeling of vibration problem of nano-size annular sector plates using the nonlocal continuum theory via eight-noe iscrete singular convolution transformation. Appl Math Comput, 19, 36 340, 01. 48. Mercan, K., Civale, Ö., DSC metho for bucling analysis of boron nitrie nanotube (BNNT) surroune by an elastic matri, Composite Structures, 143, 300-309, 016. 49. Agöz, B., Civale, Ö., A new trigonometric beam moel for bucling of strain graient microbeams, International Journal of Mechanical Sciences, 81, 88-94, 014. 50. Civale, Ö., Agöz, B., Static analysis of single walle carbon nanotubes (SWCNT) base on Eringen s nonlocal elasticity theory, International Journal of Engineering an Applie Sciences, 1, 47-56, 009. 7
Beir Agöz, Kair Mercan, Çiğem Demir, Ömer Civale 51. Demir, Ç., Civale, Ö., Torsional an longituinal frequency an wave response of microtubules base on the nonlocal continuum an nonlocal iscrete moels, Applie Mathematical Moelling, 37, 9355-9367, 013. 5. Agöz, B., Civale, Ö., Shear eformation beam moels for functionally grae microbeams with new shear correction factors, Composite Structures, 11, 14-5, 014. 53. Tornabene, F., Fantuzzi, N., Viola, E., Ferreira, A.J.M., Raial basis function metho applie to oubly-curve laminate composite shells an panels with a General Higher-orer Equivalent Single Layer formulation. Compos Part B: Eng, 55, 64 659, 013. 54. Tornabene, F., Viola, E., Inman, D.J., -D ifferential quarature solution for vibration analysis of functionally grae conical, cylinrical shell an annular plate structures. J Soun Vib, 38, 59 90, 009. 55. Civale, Ö., Application of ifferential quarature (DQ) an harmonic ifferential quarature (HDQ) for bucling analysis of thin isotropic plates an elastic columns, Engineering Structures, 6(), 171-186, 004. 56. Civale, Ö., Geometrically non-linear static an ynamic analysis of plates an shells resting on elastic founation by the metho of polynomial ifferential quarature (PDQ), PhD. Thesis, Fırat University, (in Turish), Elazığ, 004. 57. Civale, Ö., Finite Element analysis of plates an shells, Elazığ, Fırat University,(in Turish), Seminar Manuscript, 1998. 58. Gürses, M., Civale, Ö., Ersoy, H., Kiracioglu, O., Analysis of shear eformable laminate composite trapezoial plates, Materials & Design, 30, 3030-3035, 009. 59. Agöz, B. an Civale, Ö., Bening analysis of embee carbon nanotubes resting on an elastic founation using strain graient theory, Acta Astronautica, 119, 1-1, 016. 60. Agöz, B. an Civale, Ö., A novel microstructure-epenent shear eformable beam moel, International Journal of Mechanical Sciences, 99, 10-0, 015. 61. Agöz, B. an Civale, Ö., Bening analysis of FG microbeams resting on Winler elastic founation via strain graient elasticity, Composite Structures, 134, 94-301, 015. 6. Agöz, B. an Civale, Ö., A Microstructure-Depenent Sinusoial Plate Moel Base on the Strain Graient Elasticity Theory, Acta Mechanica, 6, 77-94, 015. 63. Agöz, B. an Civale, Ö., Bucling analysis of linearly tapere micro-columns base on strain graient elasticity, Structural Engineering an Mechanics,48(), 195-05, 013. 64. Civale, Ö., Demir, Ç., A simple mathematical moel of microtubules surroune by an elastic matri by nonlocal finite element metho. Applie Mathematics an Computation, 89, 335-35, 016. 65. Demir, Ç., Civale, Ö., Nonlocal Deflection of Microtubules Uner Point Loa, International Journal of Engineering an Applie Sciences, 7(3), 33-39, 015. 66. Civale, Ö., Demir, Ç., Elasti zemine oturan iriģlerin ayrı teil onvolüsyon ve harmoni iferansiyel quarature yöntemleriyle analizi. BAÜ FBE Dergisi, 11(1), 56-71, 009. 67. Eisenberger, M., Yanelevsy, D.Z., Eact stiffness matri for beams on elastic founation, Comp. Struct., 1(6), 1355-1359, 1985. 73