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Flexural analysis of thick beams using single Flexural analysis of thick beams using single Document Transcript

  • INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME TECHNOLOGY (IJCIET)ISSN 0976 – 6308 (Print)ISSN 0976 – 6316(Online)Volume 3, Issue 2, July- December (2012), pp. 292-304 IJCIET© IAEME: www.iaeme.com/ijciet.aspJournal Impact Factor (2012): 3.1861 (Calculated by GISI) IAEMEwww.jifactor.com FLEXURAL ANALYSIS OF THICK BEAMS USING SINGLE VARIABLE SHEAR DEFORMATION THEORY Ansari Fatima-uz-Zehra1 S.B. Shinde2 1 2 P.G. Student Dept. of Civil Associate Professor Dept. of Civil Engineering, J.N.E.C., Aurangabad Engineering, J.N.E.C., Aurangabad (M.S.) India. (M.S.) India. E-mail : fatima.ansari19@gmail.com E-mail : sb_shinde@yahoo.co.in ABSTRACT In this paper, unified shear deformation theory is used to analyse simply supported thick isotropic beams for the transverse displacement, axial bending stress and transverse shear stress. A single variable shear deformation theory for flexural analysis of isotropic beams taking into account transverse shear deformation effects is developed. The number of variables in the present theory is same as that in the elementary theory of beam (ETB). The polynomial function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that thee transverse shear stress can be obtained directly from the use of constitutive relation with excellent accuracy, satisfying the shear stress free condition on the top and bottom surfaces of the beam. Hence the theory obviates the need of shear correction factor. The governing differential equation and boundary conditions are obtained by using the principle of virtual work. The results of displacement, stresses and natural bending for simply supported thick isotropic beams subjected to various loading cases are presented and discussed critically with those of exact solution and other higher order theories. Keywords: Thick beam, Shear deformation, Principle of virtual work, Bending, transverse shear stress , Various loading cases,. 1. INTRODUCTION Since the elementary theory of beam (ETB) bending based on Euler-Bernojulli hypothesis neglects the transverse shear deformation, it underestimates deflections and overestimates the natural frequencies in case of thick beams where shear deformation effects are significant. Timoshenko [1] was the first to include refined effects such as rotatory inertia and shear deformation in the beam theory. This theory is now widely referred to as Timoshenko beam 292
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEtheory or first order shear deformation theory (FSDTs). In this theory transverse shear straindistribution is assumed to be constant through the beam thickness and thus requires problemdependent shear correction factor. The accuracy of Timoshenko beam theory for transversevibrations of simply supported beam in respect of the fundamental frequency is verified byCowper [2, 3] with a plane stress exact elasticity solution. The limitations of ETB and FSDTs led to the development of higher order sheardeformation theories. Many higher order shear deformation theories are available in theliterature for static and dynamic analysis of beams [4-11]. The trigonometric sheardeformation theories are presented by Vlasov and Leont’ev [12] and Stein [13] for thickbeams. However, with these theories shear stress free boundary conditions are not satisfied attop and bottom surfaces of the beam. Recently Ghugal and Sharma [14] presented hyperbolicshear deformation theory for the static and dynamic analysis of thick isotropic beams. Astudy of literature [15-23] indicates that the research work dealing with flexural analysis ofthick beams using refined trigonometric, hyperbolic and exponential shear deformationtheories is very scant and is still in infancy. In the present study, Single Variable shear deformation theory is applied for the bendingAnalysis of simply supported thick isotropic beams considering transverse shear andtransverse normal strain effect.2. BEAM UNDER CONSIDERATIONThe beam under consideration occupies the region given by Eqn. (1). 0 ≤ x ≤ L ; - b / 2 ≤ y ≤ b / 2 ; -h / 2 ≤ z ≤ h / 2 (1)where x, y, z are Cartesian co-ordinates, L is length, b is width and h is the total depth of thebeam.The beam can have any boundary and loading conditions.2.1 Assumptions Made in Theoretical Formulation1. The axial displacement consists of two parts: a) Displacement given by elementary theory of beam bending. b) Displacement due to shear deformation, which is assumed to be polynomial in nature with respect to thickness coordinate, such that maximum shear stress occurs at neutral axis as predicted by the elementary theory of bending of beam .2. The axial displacement u is such that the resultant of axial stress σ x , acting over the cross-section should result in only bending moment and should not in force in x direction.3. The transverse displacement w is assumed to be a function of longitudinal (length) co- ordinate ‘x’ direction.4. The displacements are small as compared to beam thickness. 293
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME5. he body forces are ignored in the analysis. (The body forces can be effectively taken into account by adding them to the external forces.)6. One dimensional constitutive laws are used.7. The beam is subjected to lateral load only.2.2 The Displacement Field Based on the before mentioned assumptions, the displacement field of the present unifiedshear deformation theory is given as below: dw (1 + µ) h 2  4  z 2  d 3 w u ( x, z ) = − z − × z 1 −    3 (2) dx 4  3  h   dx   w( x, z ) = w( x) (3)Here u and w are the axial and transverse displacements of the beam center line in x and z -directions respectively.Normal strain and transverse shear strain for beam are given by: d 2 w (1 + µ)h2  4  z 2  d 4 w εx = − z 2 − × z 1 −    4 (4) dx 4  3  h   dx   (1 + µ)h  4 z  d w 2 2 3 γ zx = − 1 − h2  dx3 (5) 4  According to one dimensional constitutive law, the axial stress / normal bending stress andtransverse shear stress are given by:  d 2 w (1 + µ)h 2   4  z 2  d 4 w   σ x = E − z 2 − × z 1 −    4  (6)  dx  4  3  h   dx       (1 + µ)h  4 z  d w  2 2 3  τ zx = G − 1 − h 2  dx 3  (7) (7)   4    3. GOVERNING EQUATIONS AND BOUNDARY CONDITIONSUsing Eqns. (4) through (7) and the principle of virtual work, variationally consistentgoverning differential equations and boundary conditions for the beam under considerationcan be obtained. The principle of virtual work when applied to the beam leads to: h/ 2 L L ∫ ∫ ( σx × δε x + τzx × δγ zx ) dxdz = ∫ q × δw × dx − h /2 0 0 (8)where the symbol δ denotes the variational operator. Integrating the preceding equations byparts, and collecting the coefficients of δw , the governing equation in terms ofdisplacement variables are obtained as follows: d 4w d 6w d 8w A 4 + ( 2B − D ) 6 + C =q (9) dx dx dx8and the associated boundary conditions obtained are of following form: 294
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME d 3w d 5w d 7w−A + ( −2B+D ) 5 − C =0 or w is prescribed (10) dx 3 dx dx 7 d 2w d 4w d 6w dw A 2 + (2B − D) 4 + C 6 = 0 or is prescribed (11) dx dx dx dx d 3w d 5w d 2w (-B+D) −C 5 = 0 or is prescribed (12) dx 3 dx dx 2 d 2w d 4w d 3w B +C 4 = 0 or is prescribed (13) dx 2 dx dx 3where A, B, C and D are the stiffness coefficients given as follows:  h /2  A = E ∫ z dz 2   −h/ 2   2 h /2  2 4  z 4  B = E (1 + µ) h   4 ∫/2  3  h   z −  2   dz   −h   2 2  (14)  E (1 + µ) h2 4 h/ 2  4 z    ∫/2 z 1 − 3  h   dz  2 C =     16 −h      2  D = G (1 + µ) h 2 4 h/ 2  4z 2    16 ∫/2  h  −h  1 − 2  dz  3.1 Illustrative examples In order to prove the efficacy of the present theory, the following numerical examples areconsidered. The following material properties for beam are used. E E = 210GPa , µ = 0.3 , G = 2(1 + µ)Where E is the Young’s modulus and µ is the Poisson’s ratio of beam material.Example 1A simply supported laminated beam subjected to single sine load on surface z = − h / 2 ,acting in the downward z direction. The load is expressed as  πx q ( x ) = q0 sin    L where q0 is the magnitude of the sine load at the centre.Example 2A simply supported laminated beam subjected to uniformly distributed load, q ( x) on surfacez = − h / 2 acting in the z -direction as given below: ∞  mπx q ( x ) = ∑ qm sin   m =1  L where qm are the coefficients of Fourier expansion of load which are given by 295
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME 4q0qm = for m = 1,3,5....... mπqm = 0 for m = 2,4,6.......Example 3 q xA simply supported laminated beam subjected to linearly varying load,  0  on surface  L z = − h / 2 acting in the z –direction. The coefficient of Fourier expansion of load in theequation of example 2 is given by: 2qqm = − 0 cos(mπ) for m = 1,3,5....... mπqm = 0 for m = 2, 4, 6.......Example 4A simply supported laminated beam subjected to center concentrated load P. The magnitudeof coefficient of Fourier expansion of load in the equation of example 2 is given by:  2 P   mπξ qm =   sin    L   L where ξ represent distance of concentrated load from x axis.4. NUMERICAL RESULTS AND DISCUSSIONSThe results for inplane displacement (u), transverse displacement (w), axial bending stress( σ x ) and transverse shear stress ( τ zx ) are presented in the following non dimensional form. Ebu 10 Ebh 3 w bσ bτ u= , w= 4 , σ x = x , τ zx = zx , S = Aspect ratio = L / h qh qL q qFor centre concentrated load q becomes P  value by a particular model − value by exact elasticity solution  % error =   ×100  value by exact elasticity solution Table 1: Comparison of axial displacement u at (x = 0, z = ± h / 2), transverse displacement w at (x = L / 2, z = 0), axial stress σ x at (x = 0.5L, z = ± h / 2) and transverse shear stress τzx at (x = 0, z = 0) for isotropic beam subjected to single sine load.S Theory Model u w σx τ zx Present Theory SVSDT 12.311 1.414 9.950 2.631 Bernoulli-Euler ETB 12.385 1.232 9.727 -4 Timoshenko FSDT 12.385 1.397 9.727 1.273 Reddy HSDT 12.715 1.429 9.986 1.906 Ghugal Exact 12.297 1.411 9.958 1.900 Present Theory SVSDT 202.142 1.247 55.709 8.711 Bernoulli-Euler ETB 193.509 1.232 60.793 -10 Timoshenko FSDT 193.509 1.258 60.793 3.183 Reddy HSDT 193.337 1.264 61.053 4.779 Ghugal Exact 192.950 1.261 60.917 4.771 296
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Table 1 shows the comparison of displacements and stresses for the simply supportedisotropic beam subjected to single sine load. The maximum axial displacement predicted bypresent theory is in good agreement with that of exact solution (see Figure 1). The maximumcentral displacement predicted by present theory is underestimated for all aspect ratios ascompared to exact solution. Figure 2 shows that, bending stress predicted by present theory isin excellent agreement with that of exact solution for aspect ratio 4. Theory of Reddy yieldsthe higher value of bending stress compared to the exact value for all aspect ratios whereasFSDT and ETB predicts lower value for the same. The transverse shear stress predicted bypresent theory is in excellent agreement for all aspect ratios when obtained using constitutiverelation. Fig.1 : Variation of Inplane Displacement (u ) through the thickness of simply supported beam at (x = 0, z = ± h / 2) when subjected to single sine load for aspect ratio 4. Fig.2 : Variation of Inplane Normal stress (σx ) through the thickness of simply supported beam at (x = 0.5L, z = ± h / 2) when subjected to single sine load for aspect ratio 4. 297
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Fig.3 : Variation of Transverse shear stress (τ zx ) through the thickness of simply supported beam at (x = 0, z = 0) when subjected to single sine load and obtained using constitutive relation for aspect ratio 4.Table 2: Comparison of axial displacement (u ) at (x = 0, z = ± h / 2), transverse displacement w at (x = L / 2, z = 0), axial stress σ x at (x = 0.5L, z = ± h / 2) and transverse shear stress τzx at (x = 0, z = 0) for isotropic beam subjected to uniformly distributed load.S Theory Model u w σx τ zx Present Theory SVSDT 15.753 1.808 12.444 2.980 Bernoulli-Euler ETB 16.000 1.5630 12.000 -4 Timoshenko FSDT 16.000 1.8063 12.000 2.400 Reddy HSDT 16.506 1.8060 12.260 2.917 Timoshenko and Exact 15.800 1.7852 12.200 3.000 Goodier Present Theory SVSDT 250.516 1.6015 75.238 7.4875 Bernoulli-Euler ETB 249.998 1.5630 75.000 -10 Timoshenko FSDT 250.000 1.6015 75.000 6.0000 Reddy HSDT 251.285 1.6010 75.246 7.4160 Timoshenko and Exact 249.500 1.5981 75.200 7.5000 Goodier Table 2 shows comparison of displacements and stresses for the simply supportedisotropic beam subjected to uniformly distributed load. The axial displacement and transversedisplacement obtained by present theory is in good agreement with those of Reddy’s theory.The bending stress σx predicted by present theory is in excellent agreement with Reddy’stheory whereas FSDT and ETB underestimate the bending stress compared with those ofpresent theory and theory of Reddy for all aspect ratios. Variation of axial displacement andbending stress through the thickness of isotropic beam subjected to uniformly distributed loadare shown in Figure 4 and Figure 5 respectively. When beam is subjected to uniformlydistributed load, the present theory underestimates the transverse shear stresses whenobtained using constitutive relations and overestimates the same when obtained usingequations of equilibrium. Variation of transverse shear stress τzx is shown in Figure 6 usingconstitutive relations. 298
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Fig.4 : Variation of Inplane Displacement (u ) through the thickness of simply supported beam at (x = 0, z = ± h / 2) when subjected to uniformly distributed load for aspect ratio 4. Fig.5 : Variation of Inplane Normal stress (σ x ) through the thickness of simply supported beam at (x = 0.5L, z = ± h / 2) when subjected to uniformly distributed load for aspect ratio 4. Fig.6 : Variation of Transverse shear stress (τ zx ) through the thickness of simply supported beam at(x = 0, z = 0) when subjected to uniformly distributed load and obtained using constitutive relation for aspect ratio 4. 299
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMETable 3: Comparison of axial displacement u at (x = 0, z = ± h / 2), transverse displacement w at (x = L / 2, z = 0), axial stress σ x at (x = 0.5L, z = ± h / 2) and transverse shear stress τzx at (x = 0, z = 0) for isotropic beam subjected to linearly varying load.S Theory Model u w σx τ zx Present Theory SVSDT 7.773 0.8923 6.141 1.386 Bernoulli-Euler ETB 8.000 0.7815 6.000 -4 Timoshenko FSDT 8.000 0.9032 6.000 1.200 Reddy HSDT 8.253 0.9030 6.130 1.458 Timoshenko and Exact 7.900 0.8926 6.100 1.500 Goodier Present Theory SVSDT 123.620 0.7903 37.129 3.0769 Bernoulli-Euler ETB 124.999 0.7815 37.500 -10 Timoshenko FSDT 125.000 0.8008 37.500 3.0000 Reddy HSDT 125.643 0.8005 37.623 3.7080 Timoshenko and Exact 124.750 0.7991 37.600 3.7500 Goodier Comparison of displacements and stresses for the simply supported isotropic beamsubjected to linearly varying load are shown in Table 3. The maximum axialdisplacement and transverse displacement predicted by present theory are in close agreementwith Reddy’s theory (see Figure 7). FSDT underestimate the axial displacement andoverestimate the transverse displacement. Figure 8 show that, the bending stress σ x predictedby present theory is in close agreement with Reddy’s theory whereas FSDT and ETBunderestimate the same for all aspect ratios. The maximum transverse shear stress τzx is inexcellent agreement with Reddy’s theory for all aspect ratios when obtained by constitutiverelation (see Figure 9). Fig.7 : Variation of Inplane Displacement (u ) through the thickness of simply supported beam at (x = 0, z = ± h / 2) when subjected to linearly varying load for aspect ratio 4. 300
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Fig.8 : Variation of Inplane Normal stress (σ x ) through the thickness of simply supported beam at (x = 0.5L, z = ± h / 2) when subjected to linearly varying load for aspect ratio 4. Fig.9 : Variation of Transverse shear stress (τ zx ) through the thickness of simply supported beam at (x = 0, z = 0) when subjected to linearly varying load and obtained using constitutive relation for aspect ratio 4.Table 4: Comparison of axial displacement u at (x = 0, z = ± h / 2), transverse displacement w at (x = L / 2, z = 0), axial stress σ x at (x = 0.5L, z = ± h / 2) and transverse shear stress τzx at (x = 0, z = 0) for isotropic beam subjected to centre concentrated load.S Theory Model u w σx τ zx Present Theory SVSDT 6.2220 2.8650 4.6972 0.9201 Bernoulli-Euler ETB 6.0000 2.5000 6.0000 -4 Timoshenko FSDT 6.0000 2.9875 6.0000 0.6000 Ghugal and Sharma HPSDT 6.1290 2.9740 5.7340 0.7480 Ghugal and Nakhate TSDT 4.7664 2.9725 8.2582 0.7740 Timoshenko and Exact - 2.9125 5.7340 0.7500 Goodier Present Theory SVSDT 37.619 2.570 13.844 0.7664 Bernoulli-Euler ETB 37.500 2.500 15.000 -10 Timoshenko FSDT 37.500 2.578 15.000 0.6000 Ghugal and Sharma HPSDT 37.629 2.577 14.733 0.7480 Ghugal and Nakhate TSDT 36.624 2.577 17.258 0.7740 Timoshenko and Exact - 2.569 14.772 0.7500 Goodier 301
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Table 4 shows comparison of displacements and stresses for the simply supportedisotropic beam subjected to center concentrated load. The maximum axial displacement andtransverse displacement predicted by present theory are in close agreement with Ghugal’stheory. Variation of axial displacement through the thickness of beam is shown in Figure 10.The bending stress σx predicted by present theory is in close agreement with Ghugal’s theoryand non-linear in nature due to effect of local stress concentration (see Figure11). FSDT andETB underestimate the axial displacement and bending stress for all aspect ratios. Themaximum transverse shear stress τzx is in excellent agreement with Ghugal’s theory for allaspect ratios when obtained by constitutive relation as shown in Figure 12 respectively. Fig.10 : Variation of Inplane Displacement (u ) through the thickness of simply supported beam at (x = 0, z = ± h / 2) when subjected to centre concentrated load for aspect ratio 4. Fig.11 : Variation of Inplane Normal stress (σ x ) through the thickness of simply supported beam at (x = 0.5L, z = ± h / 2) when subjected to centre concentrated load for aspect ratio 4. 302
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Fig.12 : Variation of Transverse shear stress (τ zx ) through the thickness of simply supported beam at(x = 0, z = 0) when subjected to centre concentrated load and obtained using constitutive relation for aspect ratio 4.5. CONCLUSION Following conclusions are drawn from this study. 1. Present theory is variationally consistent and requires no shear correction factor. 2. Present theory gives good result in respect of axial displacements. Results of available higher-order and refined shear deformation theories for the axial displacement are in tune with the results of present theory. 3. The use of present theory gives good result in respect of transverse displacements. Results of available higher-order and refined shear deformation theories for the transverse displacement are in tune with the results of present theory. 4. In this paper, present theory is applied to static flexure of thick isotropic beam and it is observed that, present theory is superior to other existing higher order theories in many cases. 5. Present theory capable to produce excellent results for deflection and bending stress because of effect of transverse normal. 6. Transverse shear stresses obtained by constitutive relations satisfy shear free condition on the top and bottom surfaces of the beams. The present theory is capable of producing reasonably good transverse shear stresses using constitutive relations.ACKNOWLEDGEMENTS The authors wish to thank the Management, Principal, Head of Civil EngineeringDepartment and staff of Jawaharlal Nehru engineering College, Aurangabad and Authoritiesof Dr. Babasaheb Ambedkar Marathwada University for their support. The authors expresstheir deep and sincere thanks to Prof. A.S. Sayyad (Department of Civil Engineering, SRES’sCollege of Engineering, Kopargaon) for his tremendous support and valuable guidance fromtime to time.REFERENCES [1] Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine 41 (6) (1921) 742-746. [2] Cowper, G. R., The shear coefficients in Timoshenko beam theory, ASME Journal of Applied Mechanics 33 (1966) 335-340 303
  • International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME [3] Cowper, G. R., On the accuracy of Timoshenko’s beam theory, ASCE Journal of Engineering Mechanics Division 94 (6) (1968) 1447-1453. [4] Hildebrand, F. B., Reissner, E. C., Distribution of stress in built-in beam of narrow rectangular cross section, Journal of Applied Mechanics 64 (1942) 109-116. [5] Levinson, M., A new rectangular beam theory, Journal of Sound and Vibration 74 (1981) 81- 87. [6] Bickford, W. B., A consistent higher order beam theory. Development of Theoretical and Applied Mechanics, SECTAM 11 (1982) 137-150. [7] Rehfield, L. W., Murthy, P. L. N., Toward a new engineering theory of bending: fundamentals, AIAA Journal 20 (1982) 693-699. [8] Krishna Murty, A. V., Toward a consistent beam theory, AIAA Journal 22 (1984) 811 816. [9] Baluch, M. H., Azad, A. K., Khidir, M. A., Technical theory of beams with normal strain, Journal of Engineering Mechanics Proceedings ASCE 110 (1984) 1233-1237. [10] Heyliger, P. R., Reddy, J. N., A higher order beam finite element for bending and vibration problems, Journal of Sound and Vibration 126 (2) (1988) 309-326. [11] Bhimaraddi, A., Chandrashekhara, K., Observations on higher-order beam theory, Journal of Aerospace Engineering Proceeding ASCE Technical Note 6 (1993) 408-413. [12] Vlasov, V. Z., Leont’ev, U. N., Beams, Plates and Shells on Elastic foundation, (Translated from Russian) Israel Program for Scientific Translation Ltd. Jerusalem (1996) [13] Stein, M., Vibration of beams and plate strips with three-dimensional flexibility, Transaction ASME Journal of Applied Mechanics 56 (1) (1989) 228- 231. [14] Ghugal, Y. M., Sharma, R., Hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams, International Journal of Computational Methods 6 (4) (2009) 585-604. [15] Akavci, S. S., Buckling and free vibration analysis of symmetric and anti-symmetric laminated composite plates on an elastic foundation, Journal of Reinforced Plastics and Composites 26 (18) (2007) 1907-1919. [16] Ambartsumyan, S. A., On the theory of bending plates, Izv Otd Tech Nauk AN SSSR 5 (1958) 69–77. [17] Ghugal, Y. M., Shimpi, R. P., A review of refined shear deformation theories for isotropic and anisotropic laminated beams, Journal of Reinforced Plastics and Composites 21 (2002) 775-813. [18] Ghugal, Y. M., A simple higher order theory for beam with transverse shear and transverse normal effect, Department Report 4, Applied of Mechanics Department, Government College of Engineering, Aurangabad, India, 2006. [19] Karama, M., Afaq, K. S., Mistou, S., Mechanical behavior of laminated composite beam by new multilayered laminated composite structures model with transverse shear stress continuity, International Journal of Solids and Structures, 40 (2003) 1525–46. [20] Kruszewski, E. T., Effect of transverse shear and rotatory inertia on the natural frequency of a uniform beam, NACATN (1909). [21] Lamb, H., On waves in an elastic plates, Proceeding of Royal Society London Series A. 93 (1917) 114-128. [22] Soldatos, K. P., A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica 94 (1992) 195–200. [23] Touratier, M., An efficient standard plate theory, International Journal of Engineering Science 29 (8) (1991) 901–16. 304