IRJET- Parabolic Loading in Fixed Deep Beam using 5th Order Shear Deformation Theory
0 likes21 views
This document presents a study analyzing parabolic loading in fixed deep beams using a 5th order shear deformation theory. The authors develop a variationally consistent 5th order refined shear deformation theory for fixed-fixed beams. They obtain the governing differential equations and boundary conditions using the principle of virtual work. Numerical results are presented comparing displacements, stresses, and transverse shear stresses using the proposed 5th order theory versus higher-order plate and shear deformation theories and classical beam theory. The results show excellent agreement of the flexural stresses and shear stress distributions through the beam thickness between the proposed theory and other refined shear deformation theories. This validates the usefulness of the 5th order shear deformation theory.
IRJET- Parabolic Loading in Fixed Deep Beam using 5th Order Shear Deformation Theory
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1843 Parabolic Loading in Fixed Deep Beam using 5th Order Shear Deformation Theory Ajinkya Patil1, Sandeep Mahajan2, Sagar Gaikwad3, Ajay Yadav4, Sagar Gawade5 1,2,3,4,5PG Student, G.H.Raisoni College of Engineering and Management, Wagholi, Pune, India. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract - In this paper, a variationally consistent 5th order refined shear deformation theory for deep fixed- fixed beams is developed. The governing differential equations and boundary conditions are obtained using principle of virtual work. The general solutiontechnique is developed to solve the governing differential equations. The theory is applied to static flexural analysis of fixed-fixed beams of homogeneous and isotropic material with uniform solid rectangular cross- section. The general solutions for field variables w (x) and x are obtained for beams under consideration using appropriate boundary conditions. The general expressionsfordisplacementsandstressesarepresented. Key Words: Fixed, Deep beam, 5th order 1. INTRODUCTION Beams and plates are widely used in civil and mechanicalindustries.Asthethicknessismuchsmaller than the length, it can be converted from 3D to 1D also it is conceivable to work-out the variation of the stress inthethicknesscoordinate.Bernoulli-Eulerestablished the most commonly used classical or elementary theory of beam. Galileo in 1638 have made first attempt till 1856 mentioned by Saint Venant Barre de is also presented by Love . The classical theoryofbeam bending (ETB) is founded on the hypothesis that the plane sections remain plane and normal to the axis after bending, implyingthatthe transverseshearstrain is zero. Due to negligence of the transverse shear deformation, it is acceptable for the analysis of thin beams. Due to underestimationofdeflectionsincaseof thick beams where shear deformation effectsaremore pronounced. 1.1 Theoretical Formulation The beam is made up of isotropic material and occupiesin 0 x y z Cartesiancoordinatesystemthe region: 0 ; ; 2 2 2 2 b b h h x L y z where, x, y, z = Cartesian coordinates, L = Length of beam in x direction b = breadth of beam in y direction, and h = thickness of the beam in the z-direction. The beam is up of homogeneous, linearly elastic isotropic material. 1.2 Equilibrium Equations Using the expressions for strains and stresses (2) through (4)and the principle ofvirtualwork,following equilibrium equations canbeobtained.Theexpression obtained by using principle of virtual work as follows: . /22 20 /2 0 ( ) 0x x zx zx x L z h x Ly b y bx z h x dxdydz q x wdx (5) Integrating Eqn. (5) successively, we obtain the coupled equilibrium equations of the beam. 4 3 3 2 0 0 0 04 3 3 2 , 0 d w d d w d EI A EI q x A EI B EI C GA dx dx dx dx where the constants, 0 0 0 12 , 2.96, 2.4635 7 A B C A fixed beam with parabolic load 2 0 2 ( ) x q x q L A fixed-fixed beam has its origin at left hand side support and is fixed at x = 0 and L. The beam is subjected to parabolic load, 2 0 2 ( ) x q x q L on surface z = +h/2 acting in the downward z direction with maximum intensity of load 0q . parabolic load is considered as shown in Fig. 1. The material assumed for beam are as modulus of elasticity (E) = 2.1 x105 MPa, Poisson’s ratio (μ) = 0.3 and density ( )=7800kg/m3.
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1844 Fig. 1: Fixed beam with parabolic load Boundary conditions for the beam used as 0 dw w dx at x = 0, L 2. Numerical Results The results for flexural and transverse shear stresses are mentioned in Table 1 in the followingnon- dimensional form. 3 4 0 0 0 0 10 , , ,x zx x zx b bEbu Ebh w u w q h q L q q The transverse shear stresses ( zx ) are obtained directly by constitutive relation and, alternatively, by integrationof equilibriumequationoftwodimensional elasticity and are denoted by ( CR zx ) and ( EE zx ) respectively. The transverse shear stress satisfies the stress free boundary conditions on the top 2 h z andbottom 2 h z surfaces of the beam when these stresses are obtained by both the above mentioned approaches. Non-Dimensional Axial Displacement (u ) at (x = 0.25L, z = h/2), Transverse Deflection ( w ) at (x = 0.25L, z =0.0) Axial Stress ( x ) at (x = 0.0, z = h/2) Maximum Transverse Shears Stresses CR zx at (x=0.01L, z =0.0) and EE zx at (x=0.01L, z =0.0) of the Fixed Beam Subjected to Parabolic Load for Aspect Ratio 4 V order 0.2890 15.5504 1.4037 -1.832 HPSD 0.289 16.309 1.458 -3.017 HSDT 0.2965 14.6098 1.2744 1.9020 FSDT 0.1943 6.4000 0.1905 2.5400 ETB 0.1340 6.4000 — 2.5400 Table -1: Variation of axial displacement -3 -2 -1 0 1 2 3 u -0.50 -0.25 0.00 0.25 0.50 z/h Present V order HPSDT HSDT FSDT ETB Fig. 2: Variation of axial displacement (u ) through the thickness of fixed-fixed beam at (x = 0.25L, z) when subjected to parabolic load for aspect ratio 4. 0 10 20 30 40 50 S 0.0 0.2 0.4 0.6 0.8 w Present V order HPSDT HSDT FSDT ETB Fig. 3: Variation of maximum transverse displacement ( w ) of fixed-fixed beam at (x=0.25L, z = 0) when subjected to parabolic load with aspect ratio S. 2 0 2 ( ) 1 x q x q L q0 x, u z, w L
- 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1845 -18 -12 -6 0 6 12 18 x -0.50 -0.25 0.00 0.25 0.50 z/h Present V order HPSDT HSDT FSDT ETB Fig. 4: Variation of axial stress ( x ) through the thickness of fixed-fixed beam at (x = 0, z) when subjected to parabolic load for aspect ratio 4. 0.0 0.3 0.6 0.9 1.2 1.5 zx -0.50 -0.25 0.00 0.25 0.50 z/h Present V order HPSDT HSDT FSDT Fig. 5: Variation of transverse shear stress ( zx ) through the thickness of fixed-fixed beam at (x = 0.01L, z) when subjected to parabolic load and obtain using constitutive relation for aspect ratio 4. -4 -2 0 2 4 6 zx -0.50 -0.25 0.00 0.25 0.50 z/h Present V order HPSDT HSDT FSDT ETB Fig. 6: Variation of transverse shear stress ( zx ) through the thickness of fixed-fixed beam at (x = 0.01L, z) when subjected to parabolic load and obtain using equilibrium equation for aspect ratio 4. 3. CONCLUSIONS 1. Theflexuralstressesandtheirdistributionsthrough the thickness of beam given by proposed theoryare in excellent agreement with those of other refined shear deformation theories. 2. The shear stresses and their distributions over the thicknessofbeamfromconstitutiveandequilibrium equations are matching with that of other refined shear refined theories. 3. In general, use of proposed theory gives precise results by the numerical considered. 4. This validates the usefulness of the 5th order shear deformation theory. REFERENCES 1. Gol’denviezer, A. L., “Methods for Justifying and Refining the TheoryShells,”Prikladnaya Matematika I Mekhanika, Vol. 32, No.4, pp. 684-695, Journal of Applied Mathematics and Mechanics, Vol. 32, No.4, pp. 704-718, 1968. 2. Kil’chevskiy, N. A., “Fundamentals of the Analytical Mechanics of Shells,” NASA TT F- 292, Washington, D.C., pp. 80-172, 1965. 3. Donnell, L. H., Beams, Plates and Shells, McGraw-Hill Book Company. New York, pp. 453, 1976. 4. Vlasov, V. Z., and Leont’ev, U. N., Beams, Plates and Shells on Elastic Foundations, Translated from Russian by Barouch, A.,and edited by Pelz, T. Israel program for scientific translations Ltd., Jerusalem, Chapter 1, pp. 1-8, 1960. 5. Sayir, M., and Mitropoulos, C., “On Elementary Theories of Linear Elastic Beams, Plates and Shells,” Zeitschrift f¨ur AngewandteMathematikundPhysik,Vol.31, No.1, pp. 1-55, 1980. 6. Rankine, W. J. M., A Manual of Applied Mechanics, R. Griffin and Company Ltd., London, U. K., pp. 342-344, 1858.