Abstract
The present study investigates the bending, buckling, and vibration responses of shear deformable laminated composite and sandwich beams using trigonometric shear and normal deformation theory. The most important feature of the present theory is that it includes the effects of transverse shear and normal deformations, i.e., the effect of thickness stretching. Therefore, the theory is also called as a quasi-2D theory. The axial displacement uses sine function in terms of the thickness coordinate to include the effect of transverse shear deformation, and the transverse displacement uses cosine function in terms of the thickness coordinate to include the effect of transverse normal deformation, i.e., the thickness stretching. The present theory satisfies the zero shear stress conditions at top and bottom surfaces of the beam without using shear correction factor. Governing differential equations and associated boundary conditions of the theory are derived by employing the dynamic version of principle of virtual work. Navier-type closed-form solutions are obtained for simply supported boundary conditions. The numerical results are obtained for deflections, stresses, natural frequencies, and critical buckling loads for isotropic, laminated composite, and sandwich beams. Since exact elasticity solutions for laminated composite and sandwich beams are not available in the literature, the results are compared with those obtained by using other higher-order shear deformation theories to demonstrate the accuracy of the proposed theory.
Similar content being viewed by others
References
Sayyad AS, Ghugal YM (2015) On the free vibration analysis of laminated composite and sandwich plates: a review of recent literature with some numerical results. Compos Struct 129:177–201
Bernoulli J (1694) Curvatura laminae elasticae. Acta Eruditorum Lipsiae 262–276 (Also in Bernoulli J (1744) Basileensis Opera 1(LVIII): 576)
Euler L (1744) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Apud Marcum-Michaelem Bousquet and Socio, Lausanne, Geneva, Switzerland, pp 1–322
Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philos Mag 46:744–746
Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross-section. Philos Mag 43:125–131
Carrera E (1999) A study of transverse normal stress effect on vibration of multilayered plates and shells. J Sound Vib 225(5):803–829
Carrera E (1999) Transverse normal stress effects in multilayered plates. ASME J Appl Mech 66(4):1004–1012
Carrera E (2005) Transverse normal strain effects on thermal stress analysis of homogeneous and layered plates. AIAA J 43(10):2232–2242
Levinson M (1981) A new rectangular beam theory. J Sound Vib 74:81–87
Krishna Murty AV (1984) Toward a consistent beam theory. AIAA J 22:811–816
Reddy JN (1984) A simple higher-order theory for laminated composite plates. ASME J Appl Mech 51:745–752
Kant T, Manjunatha BS (1989) Refined theories for composite and sandwich beams with C 0 finite elements. Comput Struct 33:755–764
Ghugal YM, Shimpi RP (2000) A trigonometric shear deformation theory for flexure and free vibration of isotropic thick beams. Structural Engineering Convention (SEC-2000), IIT Bombay, India
Sayyad AS, Ghugal YM (2015) Static flexure of soft core sandwich beams using trigonometric shear deformation theory. Mech Adv Compos Struct 2:45–53
Soldatos K, Elishakoff I (1992) A transverse shear and normal deformable orthotropic beam theory. J Sound Vib 155(3):528–533
Karama M, Afaq KS, Mistou S (2003) Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 40:1525–1546
Sayyad AS, Ghugal YM (2011) Flexure of thick beams using new hyperbolic shear deformation theory. Int J Mech 5(3):113–122
Sayyad AS (2012) Static flexure and free vibration analysis of thick isotropic beams using different higher order shear deformation theories. Int J of Appl Math Mech 8(14):71–87
Benatta MA, Mechab I, Tounsi A, Bedia EAA (2008) Static analysis of functionally graded short beams including warping and shear deformation effects. Comput Mater Sci 44:765–773
Benatta MA, Tounsi A, Mechab I (2009) Bouiadjra mathematical solution for bending of short hybrid composite beams with variable fibers spacing. Appl Math Comput 212:337–348
Aydogdu M (2009) A new shear deformation theory for laminated composite plates. Compos Struct 89:94–101
Mahi A, Bedia EAA, Tounsi A, Mechab I (2010) An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions. Compos Struct 92:1877–1887
Shi G, Voyiadjis GZ (2011) A sixth-order theory of shear deformable beams with variational consistent boundary conditions. J Appl Mech 78:1–11
Sayyad AS, Ghugal YM, Naik NS (2015) Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved Layer Struct 2:279–289
Sayyad AS, Ghugal YM (2016) On the free vibration of angle-ply laminated composite and soft core sandwich plates. J Sandw Struct Mater. https://doi.org/10.1177/1099636216639000 (in press)
Sayyad AS, Ghugal YM, Shinde PN (2015) Stress analysis of laminated composite and soft-core sandwich beams using a simple higher order shear deformation theory. J Serb Soc Comput Mech 9(1):15–35
Vo TP, Thai HT (2012) Static behavior of composite beams using various refined shear deformation theories. Compos Struct 94:2513–2522
Akavci SS (2010) Two new hyperbolic shear displacement models for orthotropic laminated composite plates. Mech Compos Mater 46:215–226
Ray MC (2003) Zeroth-order shear deformation theory for laminated composite plates. ASME J Appl Mech 70:374–380
Mantari JL, Oktem AS, Soares CG (2011) Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Compos Struct 94:37–49
Mantari JL, Oktem AS, Soares CG (2012) A new higher order shear deformation theory for sandwich and composite laminated plates. Compos B Eng 43:1489–1499
Mantari JL, Oktem AS, Soares CG (2012) A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int J Solids Struct 49:43–53
Meiche NE, Tounsi A, Ziane N, Mechab I, Bedia EAA (2011) New hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int J Mech Sci 53:237–247
Daouadji TH, Henni AH, Tounsi A, Bedia EAA (2013) A new hyperbolic shear deformation theory for bending analysis of functionally graded plates. Model Simulat Eng 2012:1–10
Thai CH, Tran LV, Tran DT, Thoi TN, Xuan HN (2012) Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method. Appl Math Model 36:5657–5677
Kant T, Manjunatha BS (1990) Higher-order theories for symmetric and un-symmetric fiber reinforced composite beams with C 0 finite elements. Finite Elem Anal Des 6:303–320
Zenkour AM (1999) Transverse shear and normal deformation theory for bending analysis of laminated and sandwich elastic beams. Mech Compos Mater Struct 6:267–283
Zenkour AM (1997) Maupertuis-Lagrange mixed variational formula for laminated composite structure with a refined higher order beam theory. Int J Non Linear Mech 32(5):989–1001
Maiti DK, Sinha PK (1994) Bending and free vibration analysis of shear deformable laminated composite beams by finite element method. Compos Struct 29:421–431
Sayyad AS, Ghugal YM (2011) Effect of transverse shear and transverse normal strain on bending analysis of cross-ply laminated beams. Int J of Appl Math Mech 7(12):85–118
Vo TP, Thai HT, Nguyen TK, Inam F, Lee J (2015) Static behaviour of functionally graded sandwich beams using a quasi-3D theory. Compos B Eng 68:59–74
Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN, Soares CMM (2011) Bending of FGM plates by a sinusoidal plate formulation and collocation with radial basis functions. Mech Res Commun 38:368–371
Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN, Soares CMM (2012) A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Struct 94:1814–1825
Kant T, Pendhari SS, Desai YM (2007) On accurate stress analysis of composite and sandwich narrow beams. Int J Comput Methods Eng Sci Mech 8:165–177
Vidal P, Polit O (2010) Vibration of multilayered beams using sinus finite elements with transverse normal stress. Compos Struct 92:1524–1534
Chakrabarti A, Chalak HD, Iqbal MA, Sheikh AH (2011) A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core. Compos Struct 93:271–279
Khdeir AA, Reddy JN (1994) Free vibration of cross-ply laminated beams with arbitrary boundary conditions. Int J Eng Sci 32(12):1971–1980
Aydogdu M (2005) Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Int J Mech Sci 47:1740–1755
Vo TP, Thai HT (2012) Vibration and buckling of composite beams using refined shear deformation theory. Int J Mech Sci 62:67–76
Vo TP, Thai HT, Inam F (2013) Axial- flexural coupled vibration and buckling of composite beams using sinusoidal shear deformation theory. Arch Appl Mech 83(4):605–622
Matsunaga H (2001) Vibration and buckling of multilayered composite beams according to higher order deformation theories. J Sound Vib 246(1):47–62
Chalak HD, Chakrabarti A, Iqbal MA, Sheikh AH (2011) Vibration of laminated sandwich beams having soft core. J Vib Control 18(10):1422–1435
Arya H (2003) A new zig-zag model for laminated composite beams: free vibration analysis. J Sound Vib 264:485–490
Rao MK, Desai YM, Chitnis MR (2001) Free vibration of laminated beams using mixed theory. Compos Struct 52:149–160
Zhen W, Wanji C (2008) An assessment of several displacement based theories for the vibration and stability analysis of laminated composite and sandwich beams. Compos Struct 84:337–349
Kapuria S, Dumir PC, Jain NK (2004) Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Compos Struct 64:317–327
Khdeir AA, Reddy JN (1997) Buckling of cross-ply laminated beams with arbitrary boundary conditions. Compos Struct 37(1):1–3
Chakrabarti A, Chalak HD, Iqbal MA, Sheikh AH (2011) Buckling analysis of laminated sandwich beam with soft core. Lat Am J Solids Struct 9:367–381
Cheng S (1979) Elasticity theory of plates and refined theory. ASME J Appl Mech 46:644–650
Author information
Authors and Affiliations
Corresponding author
Additional information
Technical Editor: André Cavalieri.
Rights and permissions
About this article
Cite this article
Sayyad, A.S., Ghugal, Y.M. Effect of thickness stretching on the static deformations, natural frequencies, and critical buckling loads of laminated composite and sandwich beams. J Braz. Soc. Mech. Sci. Eng. 40, 296 (2018). https://doi.org/10.1007/s40430-018-1222-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40430-018-1222-5