Flexure of Thick Simply Supported Beam Using Trigonometric Shear Deformation Theory

Transcription

1 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 1 ISSN 5-15 Flere of Thick Simply Spported Beam Using Trigonometric Shear Deformation Theory Ajay G. Dahake *, Dr. Ywaraj M. Ghgal ** * Research Scholar, Applied Mechanics Department, Govt. College of Engineering, Arangabad (MS), India, 41 5 ** Professor, Applied Mechanics Department, Govt. College of Engineering, Karad (MS), India, Abstract- A trigonometric shear deformation theory for flere of thick beams, taking into accont transverse shear deformation effects, is developed. The nmber of variables in the present theory is same as that in the first order shear deformation theory. The sinsoidal fnction is sed in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy featre of this theory is that the transverse shear stresses can be obtained directly from the se of constittive relations with ecellent accracy, satisfying the shear stress free conditions on the top and bottom srfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential eqations and bondary conditions are obtained by sing the principle of virtal work. The thick simply spported isotropic beams are considered for the nmerical stdies to demonstrate the efficiency of the. Reslts obtained are discssed critically with those of other theories. Inde Terms- Eqilibrim eqations flere, principle of virtal work, trigonometric shear deformation, thick beam. I I. INTRODUCTION t is well-known that elementary theory of bending of beam based on Eler-Bernolli hypothesis disregards the effects of the shear deformation and stress concentration. The theory is sitable for slender beams and is not sitable for thick or deep beams since it is based on the assmption that the transverse normal to netral ais remains so dring bending and after bending, implying that the transverse shear strain is zero. Since theory neglects the transverse shear deformation, it nderestimates deflections in case of thick beams where shear deformation effects are significant. Bresse [1], Rayleigh [] and Timoshenko [] were the pioneer investigators to inclde refined effects sch as rotatory inertia and shear deformation in the beam theory. Timoshenko showed that the effect of transverse vibration of prismatic bars. This theory is now widely referred to as Timoshenko beam theory or first order shear deformation theory () in the literatre. In this theory transverse shear strain distribtion is assmed to be constant throgh the beam thickness and ths reqires shear correction factor to appropriately represent the strain energy of deformation. Cowper [4] has given refined epression for the shear correction factor for different crosssections of beam. The accracy of Timoshenko beam theory for transverse vibrations of simply spported beam in respect of the fndamental freqency is verified by Cowper [5] with a plane stress eact elasticity soltion. To remove the discrepancies in classical and first order shear deformation theories, higher order or refined shear deformation theories were developed and are available in the open literatre for static and vibration analysis of beam. evinson [6], Bickford [7], Rehfield and Mrty [8], Krishna Mrty [9], Balch, Azad and Khidir [1], Bhimaraddi and Chandrashekhara [11] presented parabolic shear deformation theories assming a higher variation of aial displacement in terms of thickness coordinate. These theories satisfy shear stress free bondary conditions on top and bottom srfaces of beam and ths obviate the need of shear correction factor. Irretier [1] stdied the refined dynamical effects in linear, homogenos beam according to theories, which eceed the limits of the Eler- Bernolli beam theory. These effects are rotary inertia, shear deformation, rotary inertia and shear deformation, aial prestress, twist and copling between bending and torsion. Kant and Gpta [1], Heyliger and Reddy [14] presented finite element models based on higher order shear deformation niform rectanglar beams. However, these displacement based finite element models are not free from phenomenon of shear locking (Averill and Reddy [15], Reddy [16]). There is another class of refined theories, which incldes trigonometric fnctions to represent the shear deformation effects throgh the thickness. Vlasov and eont ev [17], Stein [18] developed refined shear deformation theories for thick beams inclding sinsoidal fnction in terms of thickness coordinate in displacement field. However, with these theories shear stress free bondary conditions are not satisfied at top and bottom srfaces of the beam. A stdy of literatre by Ghgal and Shimpi [19] indicates that the research work dealing with fleral analysis of thick beams sing refined trigonometric and hyperbolic shear deformation theories is very scarce and is still in infancy. In this paper development of theory and its application to thick simply spported beam is presented. II. DEVEOPMENT OF THEORY The beam nder consideration as shown in Fig. 1 occpies in y z Cartesian coordinate system the region: h h ; y b ; z where, y, z are Cartesian coordinates, and b are the length and width of beam in the and y directions respectively, and h is

2 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 ISSN 5-15 the thickness of the beam in the z-direction. The beam is made p of homogeneos, linearly elastic isotropic material. beam. The governing differential eqations obtained are as follows: h q(), b z z y 4 d w 4 d EI EI q 4 (6) d d 4 d w 6 d EI EI GA (7) d d The associated consistent natral bondary conditions obtained are of following form: z, w Fig. 1 Beam nder bending in -z plane At the ends = and = A. The displacement field The displacement field of the present beam theory is of the form: dw h z (, z) z sin ( ) d h w(, z) w( ) where is the aial displacement in direction and w is the transverse displacement in z direction of the beam. The sinsoidal fnction is assigned according to the shear stress distribtion throgh the thickness of the beam. The fnction represents rotation of the beam at netral ais, which is an nknown fnction to be determined. The normal and shear strains obtained within the framework of linear theory of elasticity sing displacement field given by (1) are as follows. d w h z d Normal strain: = z sin d h d Shear strain: z dw z cos z d h The stress-strain relationships sed are as follows: E, G (4) z z B. Governing Eqations and Bondary Conditions Using () throgh (4) and sing the principle of virtal work, variationally consistent governing differential eqations and bondary conditions for the beam nder consideration can be obtained. The principle of virtal work when applied to the beam leads to: b zh/ zh/ q( ) wd z. z d dz where the symbol denotes the variational operator. Employing Green s theorem to (4) sccessively, we obtain the copled Eler-agrange eqations which are the governing differential eqations and associated bondary conditions of the (1) () () (5) d w 4 d V EI EI or w is prescribed (8) d d d w 4 d M EI EI d or d 4 d w 6 d M a EI EI d dw d is prescribed (9) d or is prescribed (1) Ths the bondary vale problem of the beam bending is given by the above variationally consistent governing differential eqations and bondary conditions. C. The General Soltion of Governing Eqilibrim Eqations of the Beam The general soltion for transverse displacement w() and warping fnction () is obtained sing (6) and (7) sing method of soltion of linear differential eqations with constant coefficients. Integrating and rearranging (6), we obtain the following epression d w 4 d Q (11) d d EI where Q() is the generalized shear force for beam and it is given by Q qd C1. Now (7) is rearranged in the following form: d w d (1) d 4 d A single eqation in terms of is now obtained sing (11) and (1) as: d d Q( ) EI (1) where constants, and in (1) and (1) are as follows

3 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 ISSN GA, and 4 48 EI The general soltion of (1) is as follows: Q ( ) ( ) Ccosh Csinh (14) EI The eqation of transverse displacement w() is obtained by sbstitting the epression of in (1) and then integrating it thrice with respect to. The general soltion for w() is obtained as follows C 1 EI w( ) qdddd 6 4 EI Csinh Ccosh C4 C5 C6 C, C, C, C, C and C are arbitrary constants and where can be obtained by imposing bondary conditions of beam. III. IUSTRATIVE EXAMPES (15) In order to prove the efficacy of the present theory, the following nmerical eamples are considered. The following material properties for beam are sed E = 1 GPa, μ =. and = 78 kg/m, where E is the Yong s modls, is the density, and μ is the Poisson s Units A. Simply spported beam sbjected to varying load, The simply spported beam is having its origin at left spport and is simply spported at = and =. The beam is sbjected to varying load, on srface z = +h/ acting in the downward z direction with maimm intensity of load q. q q q, General epressions obtained for w and are as follows: E h 4 5 q G w 1EI sinh cosh 1 q 1 1 cosh EI 6 sinh The aial displacement and stresses obtained based on above soltions are as follows 4 7 E h z G 1 h h Eb 48 E z 1 1 sin 4 G h h 6 qh 1 cosh sinh 1 z 1 E h q 1 h h G b 48 E z sin 4 G h z 48 q z 1 1 cosh cos b h h 6 sinh EE q z 1 E h z bh h G 48 zeqh 5 cos h G b (16) (17) (18) (19) () (1) IV. RESUTS In this paper, the reslts for inplane displacement, transverse displacement, inplane and transverse stresses are presented in the following non dimensional form for the prpose of presenting the reslts in this work. z, w Figre : Simply spported beam with varying load For beams sbjected to parabolic load, q() Eb 1Ebh w, w, 4 qh q b, q b z z q

4 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 4 ISSN 5-15 Table I: Non-Dimensional Aial Displacement ( ) at ( =.75, z = h/), Transverse Deflection ( w ) at ( =.75, z =.) Aial Stress ( ) at ( =.75, z = h/) Maimm Transverse EE Shear Stresses z and z ( =, z =.) of the Simply Spported Beam Sbjected to Varying oad for Aspect Ratio 4. Model w z EE z TSDT w w Table II: Non-Dimensional Aial Displacement ( ) at ( =.75, z = h/), Transverse Deflection ( w ) at ( =.75, z =.) Aial Stress ( ) at ( =.75, z = h/) Maimm EE Transverse Shear Stresses z and z ( =, z =.) of the Simply Spported Beam Sbjected to Varying oad for Aspect Ratio 1. Model w z EE z TSDT S Figre 4: Variation of maimm transverse displacement ( w ) of simply spported beam at ( =.75, z = ) when sbjected to varying load with aspect ratio S Figre 5: Variation of aial displacement ( ) throgh the thickness of simply spported beam at ( =.75, z) when sbjected to varying load for aspect ratio 1. Figre : Variation of aial displacement ( ) throgh the thickness of simply spported beam at ( =.75, z) when sbjected to varying load for aspect ratio 4.

5 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 5 ISSN z z -.5 Figre 6: Variation of aial stress ( ) throgh the thickness of simply spported beam at ( =.75, z) when sbjected to varying load for aspect ratio 4. Figre 8: Variation of transverse shear stress ( z ) throgh the thickness of simply spported beam at ( =, z) when sbjected to varying load and obtain sing constittive relation for aspect ratio z z Figre 7: Variation of aial stress ( ) throgh the thickness of simply spported beam at ( =.75, z) when sbjected to varying load for aspect ratio 1. Figre 9: Variation of transverse shear stress ( z ) throgh the thickness of simply spported beam at ( =, z) when sbjected to varying load and obtain sing constittive relation for aspect ratio 1.

6 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 6 ISSN Figre 1: Variation of transverse shear stress ( z ) throgh the thickness of simply spported beam at ( =, z) when sbjected to varying load and obtain sing eqilibrim eqation for aspect ratio z z Figre 11: Variation of transverse shear stress ( z ) throgh the thickness of simply spported beam at ( =, z) when sbjected to varying load and obtain sing eqilibrim eqation for aspect ratio 1. V. DISCUSSION OF RESUTS z z The comparison of reslts of maimm non-dimensional aial displacement ( ) for the aspect ratios of 4 and 1 is presented in Tables I and II 6 for simply spported beams sbjected to linearly varying and parabolic load (see Figre ). The vales of aial displacement given by present theory are in close agreement with the vales of other refined theories for aspect ratio 4 and 1. The throgh thickness distribtion of this displacement obtained by present theory is in close agreement with other refined theories as shown in Figres and 5 for aspect ratio 4 and 1. The comparison of reslts of maimm non-dimensional transverse displacement ( w ) for the aspect ratios of 4 and 1 is presented in Tables I and II for simply spported beams sbjected to linearly varying load. The vales of present theory are in ecellent agreement with the vales of other refined theories for aspect ratio 4 and 1 ecept those of classical beam theory () and of Timoshenko. The variation of w with aspect ratio (S) is shown in Figre 4. The refined theories converge to the vales of classical beam theory for the higher aspect ratios. The reslts of aial stress ( ) are shown in Tables I and II for aspect ratios 4 and 1. The aial stresses given by present theory are compared with other higher order shear deformation theories. It is observed that the reslts by present theory are in ecellent agreement with other refined theories as well as and. The throgh the thickness variation of this stress given by all the theories is linear at =.75. The variations of this stress are shown in Figres 6 and 7. The comparison of maimm non-dimensional transverse shear stress for simple beams with varying load obtained by the present theory and other refined theories is presented in Tables I and II for aspect ratio of 4 and 1 respectively. The maimm transverse shear stress obtained by present theory sing constittive relation is in good agreement with that of higher order theories for aspect ratio 4 and for aspect ratio 1. The throgh thickness variation of this stress obtained via constittive relation are presented graphically in Figres 8 and 9 and those obtained via eqilibrim eqation are presented in Figres 1 and 11. The throgh thickness variation of this stress when obtained by varios theories via eqilibrim eqation shows the ecellent agreement with each other. The maimm vale of this stress occrs at the netral ais. VI. CONCUSION The variationally consistent theoretical formlation of the theory with general soltion techniqe of governing differential eqations is presented. The general soltions for beam with varying load are obtained in case of thick simply spported beam. The displacements and stresses obtained by present theory are in ecellent agreement with those of other eqivalent refined and higher order theories. The present theory yields the realistic variation of aial displacement and stresses throgh the thickness of beam. Ths the validity of the present theory is established. REFERENCES [1] J. A. C. Bresse, Cors de Mechaniqe Appliqe, Mallet- Bachelier, Paris, [] J. W. S. ord Rayleigh, The Theory of Sond, Macmillan Pblishers, ondon, [] S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, Third International Edition, McGraw-Hill, Singapore. 197.

7 International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 7 ISSN 5-15 [4] G. R. Cowper, The shear coefficients in Timoshenko beam theory, ASME Jornal of Applied Mechanic, vol., no., 1966, pp [5] G. R. Cowper, On the accracy of Timoshenko beam theory, ASCE J. of Engineering Mechanics Division. vol. 94, no. EM6, 1968, pp , [6] M. evinson, A new rectanglar beam theory, Jornal of Sond and Vibration, Vol. 74, No.1, 1981, pp [7] W. B. Bickford, A consistent higher order beam theory, International Proceeding of Development in Theoretical and Applied Mechanics (SECTAM), vol. 11, 198, pp , [8]. W. Rehfield, P.. N. Mrthy, Toward a new engineering theory of bending: fndamentals, AIAA Jornal, vol., no. 5, 198, pp [9] A. V. Krishna Mrty, Towards a consistent beam theory, AIAA Jornal, vol., no. 6, 1984, pp [1] M. H. Balch, A. K. Azad, and M. A. Khidir, Technical theory of beams with normal strain, ASCE Jornal of Engineering Mechanics, vol. 11, no. 8, 1984, pp [11] A. Bhimaraddi, K. Chandrashekhara, Observations on higher order beam Theory, ASCE Jornal of Aerospace Engineering, vol. 6, no. 4, 199, pp , [1] H. Irretier, Refined effects in beam theories and their inflence on natral freqencies of beam, International Proceeding of Eromech Colloqim, 19, on Refined Dynamical Theories of Beam, Plates and Shells and Their Applications, Edited by I. Elishak off and H. Irretier, Springer-Verlag, Berlin, 1986, pp [1] T. Kant, A. Gpta, A finite element model for higher order shears deformable beam theory, Jornal of Sond and Vibration, vol. 15, no., 1988, pp [14] P. R. Heyliger, J.N. Reddy, A higher order beam finite element for bending and vibration problems, Jornal of Sond and Vibration, vol. 16, no., pp [15] R. C. Averill, J. N. Reddy, An assessment of for-noded plate finite elements based on a generalized third order theory, International Jornal of Nmerical Methods in Engineering, vol., 199, pp [16] J. N. Reddy, An Introdction to Finite Element Method. nd Ed., McGraw-Hill, Inc., New York, 199. [17] V. Z. Vlasov, U. N. eont ev, Beams, Plates and Shells on Elastic Fondations Moskva, Chapter 1, 1-8. Translated from the Rssian by A. Baroch and T. Plez Israel Program for Scientific Translation td., Jersalem, [18] M. Stein, Vibration of beams and plate strips with three dimensional fleibility, ASME J. of Applied Mechanics, vol. 56, no. 1, 1989, pp [19] Y. M. Ghgal, R. P. Shmipi, A review of refined shear deformation theories for isotropic and anisotropic laminated beams, Jornal of Reinforced Plastics and Composites, vol., no.,, 1, pp [] Y. M. Ghgal, R. Sharma, A hyperbolic shear deformation theory for flere and vibration of thick isotropic beams, International Jornal of Comptational Methods, vol. 6, no. 4, 9, pp [1] F. B. Hildebrand, E. C. Reissner, Distribtion of Stress in Bilt-In Beam of Narrow Rectanglar Cross Section, Jornal of Applied Mechanics, vol. 64, 194, pp [] S.P. Timoshenko, On the correction for shear of the differential eqation for transverse vibrations of prismatic bars, Philosophical Magazine, vol. 41, no. 6, 191, pp APPENDIX Notations A Cross sectional area of beam = bh b Width of beam in y directions E, G, μ Elastic constants of the material h Thickness of beam I Moment of inertia of beam Span of the beam q Intensity of niformly distribted transverse load S Aspect ratio for beam = / h w Transverse displacement in z direction w Non-dimensional transverse displacement Non-dimensional aial displacement, y, z Rectanglar Cartesian coordinates Non-dimensional aial stress in directions Non-dimensional transverse shear stress ZX via constittive relations EE Non-dimensional transverse shear stress ZX via eqilibrim eqation Unknown fnctions associated with the shear slopes ist of abbreviations Elementary Theory of Beam Constittive Relations EE Eqilibrim Eqations TSDT Trigonometric Shear Deformation Theory Hyperbolic Shear Deformation Theory Third Order Shear Deformation Theory First Order Shear Deformation Theory AUTHORS First Athor Ajay G. Dahake, ME (Strctre), Research Scholar, Applied mechanics Department, Govt. College of Engineering, Arangabad (MS). India. ajaydahake@gmail.com Second Athor Ywaraj M. Ghgal, Ph. D. (Strctres), Professor, Govt. College of Engineering, Karad (MS), India. ghgal@rediffmail.com Correspondence Athor Ajay G. Dahake, ajaydahake@gmail.com, d_ajayan@rediffmail.com Contact nmber : ,