Research Journal of Engineering Sciences ___________________________________________ ISSN 2278 – 9472Vol. 2(4), 40-42, April (2013) Res. J. Engineering Sci. International Science Congress Association 40 Analysis of Flexural Members using an Alternative approach Patel Rakesh, Dubey S.K and Pathak K.K.2 Department of Civil Engineering, Maulana Azad National Institute of Technology, Bhopal,MP, INDIA Department of Civil and Environmental Engineering, NITTTR, Bhopal, MP, INDIA Available online at: www.isca.in Received 20th November 2012, revised 14th January 2013, accepted 12th February 2013 Abstract An alternative approach used for the analysis of flexural members is method of initial functions (MIF). The equations of two dimensional elasticity have been used for deriving the governing equations. Numerical solutions of the governing equations have been presented for simply supported orthotropic beam. The method of initial function (MIF) is an analytical method of elasticity theory. The method makes it possible to obtain exact solutions of different types of problems, i.e., solutions without the use of hypotheses about the character of stress and strain. This method has applications in various fields of structural engineering such as plates, shells and beams. It is very useful in case of thick, sandwich, and layered beams. Keywords: Flexural members, method of initial functions, stress, strain, flexural Member. Introduction An alternative approach used in this paper for the analysis of flexural members is MIF. The method of initialfunction (MIF) is an analytical method of elasticity theory. The method makes it possible to obtain exact solutions of different types of problems, i.e., solutions without the use of hypotheses about the character of stress and strain. According to this method, the basic desired functions are the displacements and stresses, the system of differential equations which are obtained from equations of Hook’s law and equilibrium equations by replacing stresses by the displacements according to elasticity relations. The order of the derived equations depends on the stage at which the series representing the stresses and displacements are truncated. However, the physical significance of each term in the series is difficult to visualize, especially for the higher order terms. Method of Initial Functions is used for the analysis of beams under symmetric central loading and uniform loading .The method of initial functions is used for the analysis of free vibration of rectangular beams of arbitrary depth. The frequency values are calculated using the Timoshenko beam theory and present the analysis for different values of Poisson's ratio. The method of initial function is applied, to the flexural theory of circular plate subjected to antisymmetric lateral loads. The results are compared with solutions from classical theory. They have used three-dimensional elasticity solutions for some static and dynamic problems of bending multi-layered anisotropic rectangular plates. They are derived by the method of initial functions. Governing equations are developed for composite laminated deep beams by using method of initial functions. The beam theory developed can be used for beam sections of large depth. Applied method of initial functions for the analysis of orthotropic deep beams and compared the results with the available theory. Developed Hyperbolic Shear Deformation Theory for transverse shear deformation effects. It is used for the static flexure analysis of thick isotropic beams. Used method of initial functions for the study of composite beams having two layers of orthotropic material and developed governing equation. Method of initial functions is used for the analysis of composite laminated beams. Formulation of the Problem The equations of equilibrium for solids ignoring the body forces for two-dimensional case are: xyxy+=¶¶ (1) xyyxyts¶¶+=¶¶ (2) The stress-strain relations for isotropic material are: ''1112xxyCCsee=+ (3) ''1222yxyCCsee=+ (4) 33xyxytg (5) The values of the coefficients C’11 to C’33 for isotropic materials are: ''1122CC== (6) 12 2 1 E C m m (7) 33CG (8) Research Journal of Engineering Sciences________________________________________________________ ISSN 2278 – 9472 Vol. 2(4), 40-42, April (2013) Res. J. Engineering Sci. International Science Congress Association 41 The strain displacement relations for small displacements are: x u x e ¶ = ¶ (9) y v y e ¶ = ¶ (10) xy vu xy g ¶¶ =+ ¶¶ (11) Eliminating x s between equations (1) and (2) the following equations are obtained, which can be written in matrix form as 12310010000000 u CC v Y CGC X aa            (12) Where, xy t C= , ''1222 yxy YCC see ==+ 1212 12311 222222;;aa CCCa aGaa - ===- and ''' 111222 111222,, CCC aaa GGG ===The equation (12) can be expressed as: {}[]{} SDS y ¶ (13) The solution of equation (13) is {}[]{} 0 Dy SeS  (14) Where {S} is the vector of initial functions, being the value of the state vector {S} on the initial plane. If , v , Y and 0 are values of u ,v ,Y and respectively, on the initial plane, then {} [ ] 00000,,, T SuvYX (15) Where [ ] [ ] D y Le (16) Expending (16) in the form of a series [][][] [] ....... 2! LIyDD=+++ (17) Application of MIF An isotropic beam of length l, depth, H and loaded with sinusoidal normal load p = psin (x/l) in the y- direction. The bottom plane of the beam is taken as the initial plane (p). Due to loading at the top plane of the beam one has 0 = Y0 = 0 On the plane, y = H, the conditions are X = 0, Y = -p Y= -p on y=H, after simplification yields the governing partial differential equation: (..)YuXvYvXu LLLLp f -=- (18) Initial functions are obtained by substituting the value of : Xu vL f =- (19)From the value of initial functions the value of displacements and stresses are obtained. 00 00 00 00 .. .. .. .. uuuvvuvvYuYvXuXv uLuLv vLuLv YLuLv XLuLv =+ =+=+=+ (20)Analysis of Flexural Member The following values of beam dimensions are chosen for the particular problem, =100 cm, = 400 cm The following material properties are taken: =2.10x105 N/mm2 = 0.30, = 0.10x105 N/mmThe boundary conditions of the simply supported edges are: X = Y = v = 0, at x = 0 and x = l The boundary conditions are exactly satisfied by the auxiliary function. = sin (x/l) A sinusoidal normal loading is assumed, on the top surface of the beam: P(x) = -Psin (x/l) Taking P0 = 100N/mm and x = l/2 Initial functions are obtained from expression (19)The values of u and v0 are substituted in expression (20) for obtaining the values of stresses and displacements.u =0.1274 cm v = -0.2791cm Y = -100.00N/mm2 X = 0 Conclusion The deflection obtained is also equal to the deflection obtained by other theories. The normal stress equal to the intensity of , XvuL f = Research Journal of Engineering Sciences________________________________________________________ ISSN 2278 – 9472 Vol. 2(4), 40-42, April (2013) Res. J. Engineering Sci. International Science Congress Association 42 loading and shear stress equal to zero at the top of beam are obtain, this shows that MIF is successfully applied for the analysis of beam. The beam theories based on MIF have advantage over other theories because of their capabilities to converge to an exact linear elasticity solution and so provide a governing equation of desired order according to the requirements of a beam problem. MIF gives accurate results in case of small thickness, large thickness and layered members. The MIF assumes a significant importance in the analysis of thick, composite or sandwich beams. Notation l - Span of beam H - Total thickness of beam E - Young’s modulus of Elasticity G - Shear modulus of Elasticity µ - Poisson’s ratio e - Strain x - Bending stress - Normal stress xy - Shear stress u - Displacements in x directions v - Displacements in y directions - x ¶ ¶ References 1.Iyengar K.T.S., Chandrashekhara K. and Sebastian V.K., Thick Rectangular Beams, Journal of the Engineering Mechanics Division,100(6), 1277-1282 (1974) 2.Iyengar K., Raja, T.S., and Raman P.V., Free vibration of rectangular beams of arbitrary depthActa Mechanica, 32(1), 249-259 (1979) 3.Sargand S.M., Chen H.H. and Das Y.C., Method of initial functions for axially symmetric elastic bodies, International Journal of Solids and Structures, 29(6), 711- 719 (1992) 4.Galileev S.M. and Matrosov A.V., Method of initial functions: stable algorithms in the analysis of thick laminated composite structures, Composite Structures 39(3), 255-262, (1997) 5.Dubey S.K., Analysis of composite laminated deep beams, Proceedings of the third International Conference on Advances in Composites, Bangalore, 30-39, (2000) 6.Dubey S.K., Analysis of homogeneous orthotropic deep beams, Journal of Structural Engineering,32(2), 109-166 (2005)7.Ghugal Y.M.and Sharma Rajneesh, A refined shear deformation theory for flexure of thick beams, Latin American Journal of Solids and Structures, (8), 183–195 (2011)8.Patel Rakesh Dubey S.K. and pathak K.K., Method of initial functions for composite laminated beams, ICBEST-12 Proceedings published by International Journal of Computer applications, 4-7 (2012) 9.Patel Rakesh, Dubey S.K. and Pathak K.K., Analysis of Composite Beams using Method of Initial Functions, International Journal of Advanced Structures and Geotechnical Engineering,1(2), 83-86 (2012)