INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volue, No 1, 11 Copyright 1 All rights reserved Integrated Publishing services Research article ISSN Coparison of various shear deforation theories for the free vibration of thick isotropic beas Assistant Professor, Departent of Civil Engineering, SRES s College of Engineering, Kopargaon-461, (Maharashtra) India attu_sayyad@yahoo.co.in ABSTRACT In this paper, a coparative study of refined bea theories has been done for the free vibration analysis of thick beas, taking into account transverse shear deforation effect. The theories involves parabolic, sinusoidal, hyperbolic and exponential functions inters of thickness coordinates to include transverse shear deforation effect. The nubers of unknowns are sae as that of first order shear deforation theory. The governing differential equations and boundary conditions are obtained by using the principle of virtual work. The results of bending and thickness shear ode frequencies for siply supported bea are presented and discussed critically with those of other theories. The results are found to agree well with the exact elasticity results wherever applicable. Coparison of dynaic shear correction factor is carried out using various shear deforation theories. Key words: Thick bea, shear deforation, principle of virtual work, free vibration, bending frequency, thickness shear frequency, dynaic shear correction factor. 1. Introduction Since the eleentary theory of bea (ETB) bending based on Euler-Bernoulli hypothesis neglects the transverse shear deforation, it underestiates deflections and overestiates the natural frequencies in case of thick beas where shear deforation effects are significant. Tioshenko (Tioshenko, 191) was the first to include refined effects such as rotatory inertia and shear deforation in the bea theory. This theory is now widely referred to as Tioshenko bea theory or first order shear deforation theory (FSDTs). In this theory transverse shear strain distribution is assued to be constant through the bea thickness and thus requires proble dependent shear correction factor. The accuracy of Tioshenko bea theory for transverse vibrations of siply supported bea in respect of the fundaental frequency is verified by Cowper (Cowper G. R., 1966) with a plane stress exact elasticity solution. The liitations of ETB and FSDTs led to the developent of higher order shear deforation theories. Many higher order shear deforation theories are available in the literature for static and vibration analysis of beas (Hildebrand F. B et al., 194, Bhiaraddi A et al., 199). The trigonoetric shear deforation theories are presented by Vlasov and Leont ev (Vlasov V. Z. et al., 1996) and Stein (Stein M., 1989) for thick beas. However, with these theories shear stress free boundary conditions are not Received on July 11 published on Septeber 11 85

2 satisfied at top and botto surfaces of the bea. Recently Ghugal and Shara (Ghugal Y. M et al., 9) presented hyperbolic shear deforation theory for thick beas. A study of literature by Ghugal and Shipi (Ghugal Y. M et al., ) indicates that the research work dealing with flexural analysis of thick beas using refined trigonoetric, hyperbolic and exponential shear deforation theories is very scant and is still in infancy. In this paper, assessent of various shear deforation theories (Ghugal Y. M., 6, Karaa M et al., ) is carries out for free vibration analysis of thick isotropic beas. The results obtained are copared with those of eleentary, refined and exact bea theories available in literature. 1.1 Bea under Consideration The bea under consideration occupies the region: b b h h x L ; y ; z (1) where x, y, z are Cartesian co-ordinates, L is length, b is width and h is the total depth of bea. The bea is subjected to transverse load of intensity q(x) per unit length of the bea. The bea can have any boundary and loading conditions. 1. Assuptions Made in Theoretical Forulation 1. The in-plane displaceent u in x direction consists of two parts: a) A displaceent coponent analogous to displaceent in eleentary bea theory of bending; b) Displaceent coponent due to shear deforation which is assued to be parabolic, sinusoidal, hyperbolic and exponential in nature with respect to thickness coordinate.. The transverse displaceent w in z direction is assued to be a function of x coordinate. Volue Issue

3 . One diensional constitutive law is used. 4. The bea is subjected to lateral load only. 1. The Displaceent Field Based on the before entioned assuptions, the displaceent field of the present unified refined bea theory is given as below: w u( x, z, t) = z + f ( z) φ( x, t) () x w x, z, t = w x, t () ( ) ( ) Here u and w are the axial and transverse displaceents of the bea center line in x and z -directions respectively and t is the tie. The φ represents the rotation of the crosssection of the bea at neutral axis which is an unknown function to be deterined. The f z assigned according to the shearing stress distribution through the functions ( ) thickness of the bea are given below. Model Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski f ( z) E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) ( ) Touratier Model (Touratier M., Function z h z f ( z) = 4 5z 4z = 1 4 h 1991) ( ) sin Soldatos Model (Soldatos K. P., 199) Karaa et al. Model (Karaa M et al., ) ( ) Akavci Model (Akavci S. S 4 z f z = z 1 h h π z f z = π h 1 z f ( z) = z cosh hsinh h f z z exp z = h π z f z htanh zsec h 1 = h ( ) 7) Noral strain and transverse shear strain for bea are given by: ε x = φ u w = z + f ( z) x x x (4) Volue Issue

4 γ zx u w = + = z x f ' ( z) φ (5) According to one diensional constitutive law, the axial stress / noral bending stress and transverse shear stress are given by: w φ σ x = Eε x = E z + f ( z) x x (6) τ = Gγ = G f ' z φ (7) zx zx ( ). Governing equations and boundary conditions Using the expressions for strains and stresses (4) through (7) and using the principle of virtual work, variationally consistent governing differential equations and boundary conditions for the bea under consideration can be obtained. The principle of virtual work when applied to the bea leads to: x L z h/ x L z h/ u w x L b = =+ = =+ = ( σ xδε x+ τ zxδγ zx) dz dx + ρ b δ u+ δ w dz dx qδ wdx= x= L z= h/ x= L z= h/ t t x= L (8) where the sybol δ denotes the variational operator. Integrating the preceding equations by parts, and collecting the coefficients of δ w andδφ, the governing equations in ters of displaceent variables are obtained as follows: 4 4 w φ ρ A d w ρb d φ d w A B ρh = q x x E dx dt E dxdt dt (9) d w d w ρb d w ρc d φ B C + D φ = dx dx E dxdt E dt (1) and the associated boundary conditions obtained are of following for: d w dφ ρa w ρb dφ A + + = dx dx E x t E dt A B d w dφ B dx or w is prescribed = dx or dw is prescribed dx (1) (11) w dφ B + = x dx C or φ is prescribed (1) Volue Issue

5 where A, B, C and D are the stiffness coefficients given as follows: h / h / h / h / A = E z dz ; B E z f ( z ' ) dz ; C E f ( z) dz ; D G f ( z) = = = dz h/ h/ h/ h/ (14) The stiffness coefficients for various odels discussed are given as follows: Model A B C D Abartsuyan Model Eh.8 Eh.84 Eh.16 Gh (Abartsuian S. A., 1958).8 Eh.8 Eh.874 Eh.8Gh (Kruszewski E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) Touratier Model (Touratier M., 1991) Soldatos Model (Soldatos K. P., 199) Karaa et al. Model (Karaa M et al., ) Akavci Model (Akavci S. S 7). Illustrative exaples.8 Eh.8 Eh.8 Eh.8 Eh.8 Eh.6666 Eh.645 Eh.85 Eh.65 Eh.6 Eh.596 Eh.5 Gh.566 Eh.5 Gh.88 Eh.87 Gh.591 Eh.5156 Gh.5 Eh.51Gh A siply supported bea of rectangular cross-section is considered. The governing equations for free flexural vibration of siply supported bea can be obtained by setting the applied transverse load equal to zero in Eqns.(9) and (1). A solution to resulting governing equations, which satisfies the associated initial conditions, is of the for: π x w= w sin sinωt L (15) π x φ = φ cos sinωt L (16) where w and φ are the aplitudes of translation and rotation respectively, and 5 ω is the natural frequency of the th ode of vibration. Substitution of this solution for into the governing equations of free vibration of bea results in following algebraic equations π π ρ A π ρb π A w B h w L L E L E L φ ω + ρ φ = Volue Issue

6 ρb ρc B w + C + D w + = L L E L E π π π φ ω φ The Equations (17) and (18) can be written in the following atrix for: K11 K1 M11 M1 w ω K1 K = M1 M φ (17) (18) (19) Above equation (19) can be written in following ore copact for: where { } ([ K] ω [ M] ){ } T denotes the vector,{ } = { W, φ } The eleents of the coefficient atrix [K] are given by: = (). The [K] and [M] are syetric atrices. 4 4 π π π 11=, 4 1 = 1=, = + K A K K B K C D L L L The eleents of the coefficient atrix [M] are given by ρ A π ρb π ρc M h, M M, M E L E L E 11= + ρ 1 = 1= = For nontrivial solution of Eqn (),{ }, the condition expressed by ([ K] ω [ M] ) = (1) yields the eigen-frequenciesω. Fro this solution natural frequencies of bea for various odes of vibration can be obtained. The following aterial properties for bea are used. E = 1GPa, µ =. and ρ = 78 Kg/ where E is the Young s odulus, ρ is the density, and µ is the Poisson s ratio of bea aterial. 4. Nuerical Results The results for fundaental frequency ω are presented in the following nondiensional for in this paper and discussed. Volue Issue

7 ( ) ω = ω / / L h ρ E The percentage error in results obtained by a theory/odel of various researchers with respect to the corresponding results obtained by theory of elasticity is calculated as follows: value by a particular odel value by exact elasticity solution given by Cowper [] % error = x1 value by exact elasticity solutiongiven by Cowper [] The results obtained for the exaples solved in this paper are presented in Tables 1 through Table 1: Coparison of non-diensional fundaental ( = 1) flexural and thickness shear ode frequencies of the isotropic bea S = 4 S = 1 Model ω w % Error ω φ ω w % Error Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) Touratier Model (Touratier M., 1991) Soldatos Model (Soldatos K. P., 199) Karaa et al. Model (Karaa M et al., ) 1.9 Akavci Model (Akavci S S 7) Bernoulli-Euler Tioshenko (Tioshenko S. P., 191) Ghugal (Ghugal Y. M., ) Heyliger and Reddy (Heyliger P. R et al., 1988) Cowper (Cowper G. R., ) --- ω φ Volue Issue

8 Table : Coparison of non-diensional flexural frequency ( ωw) of the isotropic bea 4 S 1 for various odes of vibration. Model Modes of vibration =1 = = =4 =5 Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) Touratier Model (Touratier M., 1991) Soldatos Model (Soldatos K. P., 199) Karaa et al. Model (Karaa M et al., ) Akavci Model (Akavci S. S 7) Cowper (Cowper G. R., 1968) Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) Touratier Model (Touratier M., 1991) Soldatos Model (Soldatos K. P., 199) Karaa et al. Model (Karaa M et al., ) Akavci Model (Akavci S. S 7) Cowper (Cowper G. R., 1968) Table : Coparison of non-diensional fundaental frequency of thickness shear 4 S Model ode ( ωφ) of the isotropic bea for various odes of vibrations. Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) Touratier Model (Touratier M., 1991) Modes of vibration =1 = = =4 = Soldatos Model (Soldatos K Volue Issue

9 1 P., 199) Karaa et al. Model (Karaa M et al., ) Akavci Model (Akavci S. S 7) Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski E. T., 1949) Krishna Murty Model (Krishna Murty A. V., 1984) Touratier Model (Touratier M., 1991) Soldatos Model (Soldatos K. P., 199) Karaa et al. Model (Karaa M et al., ) Akavci Model (Akavci S. S 7) Discussion of Results The results obtained fro the present theory are copared with the eleentary theory of bea (ETB), first order shear deforation theory (FSDT) of Tioshenko (Tioshenko S. P., 191), higher order shear deforation theories of Heyliger and Reddy (Heyliger P. R et al., 1988), Ghugal (Ghugal Y. M., 6) and exact elasticity solutions given by Cowper (Cowper G. R., 1968). The value of dynaic shear correction is copared with its exact value given by Lab (H. Lab 1917). a. Fundaental Flexural ode frequency ( ω ): The coparison of lowest natural frequency in flexural ode is shown in Table 1. Observation of Table 1 shows that, Abartsuyan Model (Abartsuian S. A., 1958) overestiates the lowest natural frequencies, in flexural ode by.884 % and.14 % for aspect ratios 4 and 1 respectively. The fundaental frequencies, in flexural ode predicted by Krishna Murty (Krishna Murty A. V., 1984), Touratier (Touratier M., 1991), Soldatos (Soldatos K. P., 199), Karaa et al. (Karaa M et al., ) and Kaczkawski Model (Kruszewski E. T., 1949) odels is identical and in excellent agreeent with the exact solution given by Cowper (Cowper G. R., 1968). Ghugal (Ghugal Y. M., 6) yields the exact value of lowest natural frequencies, in flexural ode for aspect ratios 4 and 1. FSDT of Tioshenko overestiates the flexural ode frequency by.845 % and.14 % for aspect ratios 4 and 1 respectively whereas ETB overestiates the sae by 6.8 % and 1.1 % due to neglect of shear deforation in the theory. The coparison of flexural frequency for various odes of vibration is shown in Table. The exaination of Table reveals that, the flexural frequencies obtained by various odels are in excellent agreeent with each other. Volue Issue

10 b. Fundaental frequency ( ω φ ): Table 1 shows coparison of lowest natural frequency in thickness shear ode. Exact solution for the lowest natural frequency in thickness shear ode is not available in the literature. Fro the Table 1 it is observed that, thickness shear ode frequencies predicted by Kaczkawski (Kruszewski E. T., 1949), Krishna Murty (Krishna Murty A. V., 1984), Touratier (Touratier M., 1991), Soldatos (Soldatos K. P., 199) and Karaa et al. (Karaa M et al., ) odels are in excellent agreeent with each other whereas Abartsuyan Model (Abartsuian S. A., 1958) overestiates the sae. Table shows coparison of thickness shear ode frequencies for various odes of vibration and found in good agreeent with each other. The solution for the circular frequency of thickness shear ode ( = ) for thin rectangular bea is given by K GA ω φ = = Kd () M ρi where K d is dynaic shear correction factor. Model K d % Error Abartsuyan Model (Abartsuian S. A., 1958) (Kruszewski E. T., 1949) Krishna Murty Model.84.4 (Krishna Murty A. V., 1984) Touratier Model (Touratier.8. M., 1991) Soldatos Model (Soldatos.84.4 K. P., 199) Karaa et al. Model (Karaa M et al., ) Akavci Model (Akavci S. S ) Lab (H. Lab 1917) Dynaic shear correction predicted by Touratier Model (Touratier M., 1991) is sae as the exact solution given by Lab (H. Lab 1917). The corresponding values of shear factor for = according to Krishna Murty (Krishna Murty A. V., 1984) and Soldatos (Soldatos K. P., 199) Models is identical. Abartsuyan Model (Abartsuian S. A., 1958) yields higher value of dynaic shear correction factor whereas Akavci Model (Akavci S. S 7) shows lower value for the sae. Volue Issue

11 6. Conclusions Fro the study of coparison of various shear deforation theories for the free vibration of isotropic beas following conclusions are drawn. 1. Results of lowest natural frequencies for flexural ode predicted by Krishna Murty (Krishna Murty A. V., 1984), Touratier (Touratier M., 1991) and Soldatos (Soldatos K. P., 199) Models are identical and are in excellent agreeent with the exact solution. Abartsuyan Model (Abartsuian S. A., 1958) overestiates the flexural ode frequency with that of exact solution. Flexural ode frequencies predicted by Kaczkawski (Kruszewski E. T., 1949) and Akavci (Akavci S. S 7) Models are in tune with the exact solution.. The results of thickness shear ode frequencies are in excellent agreeent with each other for all odes of vibration.. Touratier Model (Touratier M., 1991) yields the exact value of dynaic shear correction factor and it is in excellent agreeent when predicted by Krishna Murty (Krishna Murty A. V., 1984) and Soldatos (Soldatos K. P., 199) odels. 7. References 1. Tioshenko S. P., (191), on the correction for shear of the differential equation for transverse vibrations of prisatic bars, philosophical agazine, series 6, 41, pp Cowper G. R., (1966), the shear coefficients in Tioshenko bea theory, ASME Journal of Applied Mechanics,, pp Cowper G. R., (1968), On the accuracy of Tioshenko s bea theory, ASCE Journal of Engineering Mechanics Division, 94(6), pp Hildebrand F. B., and Reissner E. C., (194), Distribution of stress in built-in bea of narrow rectangular cross section ASME Journal of Applied Mechanics, 64, pp Levinson M., (1981), A new rectangular bea theory, Journal of Sound and vibration, 74, pp Bickford W. B., (198), A consistent higher order bea theory, Developent in Theoretical Applied Mechanics, SECTAM, 11, pp Rehfield L. W., and Murthy P. L. N., (198), Toward a new engineering theory of bending: fundaentals, AIAA Journal,, pp Krishna Murty A. V., (1984), Toward a consistent bea theory, AIAA Journal,, pp Volue Issue

12 9. Baluch M. H., Azad A. K., and Khidir M. A., (1984), Technical theory of beas with noral strain, Journal of Engineering Mechanics Proceeding ASCE, 11, pp Heyliger P. R., and Reddy J. N., (1988), A higher order bea finite eleent for bending and vibration probles, Journal of Sound and vibration, 16(), pp Bhiaraddi A., and Chandrashekhara K., (199), Observations on higher-order bea theory, Journal of Aerospace Engineering Proceeding of ASCE, Technical Note., 6, pp Vlasov V. Z., and Leont ev U. N., (1996), Beas, plates and shells on elastic foundation. Chapter 1, pp 1-8. (Translated fro Russian) Israel progra for scientific translation ltd., Jerusale. 1. Stein M., (1989), Vibration of beas and plate strips with three-diensional flexibility, Transaction ASME Journal of Applied Mechanics, 56(1), pp Ghugal Y. M., and Shara R., (9), Hyperbolic shear deforation theory for flexure and vibration of thick isotropic beas, International Journal of Coputational Methods, 6(4), pp Ghugal Y. M., and Shipi R. P., (), A review of refined shear deforation theories for isotropic and anisotropic lainated beas, Journal of Reinforced Plastics and Coposites, 1, pp Ghugal Y. M., (6), A siple higher order theory for bea with transverse shear and transverse noral effect, Departental Report 4, Applied echanics Departent, Governent college of Engineering, Aurangabad, India, pp Abartsuian S. A., (1958), On the theory of bending plates, Izv otd Tech Nauk an Sssr, 5, pp Kruszewski E. T., (1949), Effect of transverse shear and rotatory inertia on the natural frequency of a unifor bea, NACA TN, Touratier M., (1991), An efficient standard plate theory, International Journal of Engineering Science, 9(8), pp Soldatos K. P., (199), A transverse shear deforation theory for hoogeneous onoclinic plates, Acta Mechanica, 94, pp Karaa M., Afaq K. S., and Mistou S., (), Mechanical behavior of lainated coposite bea by new ulti-layered lainated coposite structures odel with transverse shear stress continuity, International Journal of Solids and Structures, 4, pp Volue Issue

13 . Akavci S. S., (7), Buckling and free vibration analysis of syetric and antisyetric lainated coposite plates on an elastic foundation, Journal of Reinforced Plastics and Coposites, 6(18), pp H. Lab, (1917), On waves in an elastic plates, Proceeding of Royal society, London, series a. 9, pp Volue Issue