252
Views
6
CrossRef citations to date
0
Altmetric
Original Articles

Efficient free vibration analysis of FGM sandwich flat panels with conventional shell elements

ORCID Icon, ORCID Icon & ORCID Icon
Pages 3709-3726 | Received 01 Feb 2021, Accepted 23 Mar 2021, Published online: 15 Apr 2021

References

  • H. Altenbach, J. Altenbach, and W. Kissing, Mechanics of Composite Structural Elements, 2nd Edition, Springer, Singapore, 2018. DOI: 10.1007/978-981-10-8935-0.
  • V. N. Burlayenko, and T. Sadowski, Dynamic analysis of debonded sandwich plates with flexible core - Numerical aspects and simulation. In: H. Altenbach, V. A. Eremeyev (Eds.), Shell-like Structures, Vol. 15 of Advanced Structured Materials, Springer, Heidelberg, pp. 415–440, 2011. DOI: 10.1007/978-3-642-21855-2_27.
  • B. O. Baba, Free vibration analysis of curved sandwich beams with face/core debond using theory and experiment, Mech. Adv. Mater. Struct., vol. 19, no. 5, pp. 350–359, 2012. DOI: 10.1080/15376494.2010.528163.
  • V. N. Burlayenko, and T. Sadowski, Linear and nonlinear dynamic analyses of sandwich panels with face sheet-to-core debonding, Shock Vib., vol. 2018, pp. 1–26, 2018. DOI: 10.1155/2018/5715863.
  • A. Vattré, E. Pan, and V. Chiaruttini, Free vibration of fully coupled thermoelastic multilayered composites with imperfect interfaces, Compos. Struct., vol. 259, pp. 113203, 2021. DOI: 10.1016/j.compstruct.2020.113203.
  • F. Avilés, and L. A. Carlsson, Face sheet buckling of debonded sandwich panels using a two-dimensional elastic foundation approach, Mech. Adv. Mater. Struct., vol. 12, no. 5, pp. 349–361, 2005. DOI: 10.1080/15376490591008110.
  • V. N. Burlayenko, and T. Sadowski, Simulations of post-impact skin/core debond growth in sandwich plates under impulsive loading, J. Appl. Nonlinear Dyn., vol. 3, no. 4, pp. 369–379, 2014. DOI: 10.5890/JAND.2014.12.008.
  • V. N. Burlayenko, H. Altenbach, and T. Sadowski, Dynamic fracture analysis of sandwich composites with face sheet/core debond by the finite element method. In: H. Altenbach, A. Belyaev, V. Eremeyev, A. Krivtsov, A. Porubov (Eds.), Dynamical Processes in Generalized Continua and Structures, Vol. 103 of Advanced Structured Materials, Springer, Singapore. pp. 163–194, 2019. DOI: 10.1007/978-3-030-11665-1_9.
  • A. Szekrényes, Analytical solution of some delamination scenarios in thick structural sandwich plates, J. Sandwich Struct. Mater., vol. 21, no. 4, pp. 1271–1315, 2019. DOI: 10.1177/1099636217714182.
  • A. Garg, M.-O. Belarbi, H. Chalak, and A. Chakrabarti, A review of the analysis of sandwich FGM structures, Compos. Struct., vol. 258, pp. 113427, 2021. DOI: 10.1016/j.compstruct.2020.113427.
  • A. M. Zenkour, and N. A. Alghamdi, Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads, Mech. Adv. Mater. Struct., vol. 17, no. 6, pp. 419–432, 2010. DOI: 10.1080/15376494.2010.483323.
  • V. N. Burlayenko, Modelling thermal shock in functionally graded plates with finite element method, Adv. Mater. Sci. Eng., vol. 2016, pp. 1–12, 2016. DOI: 10.1155/2016/7514638.
  • D. Li, Z. Deng, H. Xiao, and L. Zhu, Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores, Mech. Adv. Mater. Struct., vol. 25, no. 3, pp. 179–191, 2018. DOI: 10.1080/15376494.2016.1255814.
  • F. Moleiro, V. Franco Correia, A. Ferreira, and J. Reddy, Fully coupled thermo-mechanical analysis of multilayered plates with embedded FGM skins or core layers using a layerwise mixed model, Compos. Struct., vol. 210, pp. 971–996, 2019. DOI: 10.1016/j.compstruct.2018.11.073.
  • M. Farrokh, M. Afzali, and E. Carrera, Mechanical and thermal buckling loads of rectangular FG plates by using higher-order unified formulation, Mech. Adv. Mater. Struct., vol. 28, no. 6, pp. 608–610, 2021. DOI: 10.1080/15376494.2019.1578014.
  • W. Qu, C.-M. Fan, and Y. Zhang, Analysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical method, Int. J. Heat Mass Transf., vol. 145, pp. 118771, 2019. DOI: 10.1016/j.ijheatmasstransfer.2019.118771.
  • H. Zhang, S. Liu, S. Han, and L. Fan, The numerical manifold method for crack modeling of two-dimensional functionally graded materials under thermal shocks, Eng. Fract. Mech., vol. 208, pp. 90–106, 2019. DOI: 10.1016/j.engfracmech.2019.01.002.
  • K. Z. Uddin, and B. Koohbor, Gradient optimization of transversely graded Ti-TiB structures for enhanced fracture resistance, Int. J. Mech. Sci., vol. 187, pp. 105917, 2020. DOI: 10.1016/j.ijmecsci.2020.105917.
  • J. Tian, H. Zhang, and J. Sun, Study on heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source, J. Therm. Stresses., vol. 43, no. 6, pp. 724–738, 2020. DOI: 10.1080/01495739.2020.1742623.
  • K. Swaminathan, D. Naveenkumar, A. Zenkour, and E. Carrera, Stress, vibration and buckling analyses of FGM plates—A state-of-the-art review, Compos. Struct., vol. 120, pp. 10–31, 2015. DOI: 10.1016/j.compstruct.2014.09.070.
  • A. S. Sayyad, and Y. M. Ghugal, Modeling and analysis of functionally graded sandwich beams: A review, Mech. Adv. Mater. Struct., vol. 26, no. 21, pp. 1776–1795, 2019. DOI: 10.1080/15376494.2018.1447178.
  • S. Brischetto, An exact 3D solution for free vibrations of multilayered cross-ply composite and sandwich plates and shells, Int. J. Appl. Mech., vol. 6, no. 6, pp. 1450076, 2014. DOI: 10.1142/S1758825114500768.
  • V. Burlayenko, H. Altenbach, and T. Sadowski, An evaluation of displacement-based finite element models used for free vibration analysis of homogeneous and composite plates, J. Sound Vib., vol. 358, pp. 152–175, 2015. DOI: 10.1016/j.jsv.2015.08.010.
  • U. Icardi, and A. Urraci, Free and forced vibration of laminated and sandwich plates by zig-zag theories differently accounting for transverse shear and normal deformability, Aerospace., vol. 5, no. 4, pp. 108, 2018. DOI: 10.3390/aerospace5040108.
  • A. Ghaznavi, and M. Shariyat, Higher-order global-local theory with novel 3D-equilibrium-based corrections for static, frequency, and dynamic analysis of sandwich plates with flexible auxetic cores, Mech. Adv. Mater. Struct., vol. 26, no. 7, pp. 559–578, 2019. DOI: 10.1080/15376494.2017.1410900.
  • M. Belarbi, A. Tati, and A. Khechai, Effect of thickness stretching on the natural frequencies of laminate-faced sandwich plates using a new layerwise model, J. Build. Mater. Struct., vol. 6, no. 2, pp. 88–96, 2019. DOI: 10.5281/zenodo.3352310.
  • E. Carrera, Theories and finite elements for multilayered, anisotropic, composite plates and shells, ARCO., vol. 9, no. 2, pp. 87–140, 2002. DOI: 10.1115/1.3013824.
  • M. Filippi, M. Petrolo, S. Valvano, and E. Carrera, Analysis of laminated composites and sandwich structures by trigonometric, exponential and miscellaneous polynomials and a MITC9 plate element, Compos. Struct., vol. 150, pp. 103–114, 2016. DOI: 10.1016/j.compstruct.2015.12.038.
  • F. A. Fazzolari, Sandwich structures. In: H. Abramovich (Ed.), Stability and Vibrations of Thin Walled Composite Structures, Woodhead Publishing, United Kingdom, pp. 49–90, 2017. DOI: 10.1016/B978-0-08-100410-4.00002-8.
  • E. Carrera, S. Valvano, and M. Filippi, Classical, higher-order, zig-zag and variable kinematic shell elements for the analysis of composite multilayered structures, Eur. J. Mech. A Solids., vol. 72, pp. 97–110, 2018. DOI: 10.1016/j.euromechsol.2018.04.015.
  • J. Monge, and J. Mantari, Best non-polynomial shear deformation theories for cross-ply single skin and sandwich shells, Eng. Struct., vol. 203, pp. 109678, 2020. DOI: 10.1016/j.engstruct.2019.109678.
  • J. N. Reddy, Analysis of functionally graded plates, Int. J. Numer. Meth. Engng., vol. 47, no. 1–3, pp. 663–684, 2000. DOI: 10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO;2-8.
  • E. Carrera, S. Brischetto, M. Cinefra, and M. Soave, Refined and advanced models for multilayered plates and shells embedding functionally graded material layers, Mech. Adv. Mater. Struct., vol. 17, no. 8, pp. 603–621, 2010. DOI: 10.1080/15376494.2010.517730.
  • M. Sobhy, Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions, Compos. Struct., vol. 99, pp. 76–87, 2013. DOI: 10.1016/j.compstruct.2012.11.018.
  • L. Iurlaro, M. Gherlone, and M. Di. Sciuva, Bending and free vibration analysis of functionally graded sandwich plates using the refined zigzag theory, J. Sandwich Struct. Mater., vol. 16, no. 6, pp. 669–699, 2014. DOI: 10.1177/1099636214548618.
  • F. A. Fazzolari, and E. Carrera, Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with fgm core, J. Sound Vib., vol. 333, no. 5, pp. 1485–1508, 2014. DOI: 10.1016/j.jsv.2013.10.030.
  • M. Bennoun, M. S. A. Houari, and A. Tounsi, A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates, Mech. Adv. Mater. Struct., vol. 23, no. 4, pp. 423–431, 2016. DOI: 10.1080/15376494.2014.984088.
  • M. Liu, J. Liu, and Y. Cheng, High-order free vibration analysis of FGM sandwich plates with non-monotonically graded flexible core, J. Sandwich Struct. Mater., vol. 20, no. 6, pp. 759–780, 2018. DOI: 10.1177/1099636216680397.
  • M. Di. Sciuva, and M. Sorrenti, Bending and free vibration analysis of functionally graded sandwich plates: An assessment of the refined zigzag theory, J. Sandwich Struct. Mater., pp. 1–43, 2019. in press. DOI: 10.1177/1099636219843970.
  • A. M. Zenkour, and Z. S. Hafed, Bending analysis of functionally graded piezoelectric plates via quasi-3D trigonometric theory, Mech. Adv. Mater. Struct., vol. 27, no. 18, pp. 1551–1562, 2020. DOI: 10.1080/15376494.2018.1516325.
  • R. Ye, N. Zhao, D. Yang, J. Cui, O. Gaidai, and P. Ren, Bending and free vibration analysis of sandwich plates with functionally graded soft core, using the new refined higher-order analysis model, J. Sandwich Struct. Mater., vol. 23, no. 2, pp. 680–731, 2021. in press. DOI: 10.1177/1099636220909763.
  • J. Liu, C. Hao, W. Ye, F. Yang, and G. Lin, Free vibration and transient dynamic response of functionally graded sandwich plates with power-law nonhomogeneity by the scaled boundary finite element method, Computer Methods Appl Mech Eng., vol. 376, pp. 113665, 2021. DOI: 10.1016/j.cma.2021.113665.
  • S. Brischetto, Classical and mixed advanced models for sandwich plates embedding functionally graded cores, J. Mech. Mater. Struct., vol. 4, no. 1, pp. 13–33, 2009. DOI: 10.2140/jomms.2009.4.13.
  • F. A. Fazzolari, Stability analysis of FGM sandwich plates by using variable-kinematics Ritz models, Mech. Adv. Mater. Struct., vol. 23, no. 9, pp. 1104–1113, 2016. DOI: 10.1080/15376494.2015.1121559.
  • A. Gorgeri, R. Vescovini, and L. Dozio, Sublaminate variable kinematics shell models for functionally graded sandwich panels: Bending and free vibration response, Mech. Adv. Mater. Struct., pp. 1–18, 2020. in press. DOI: 10.1080/15376494.2020.1749738.
  • Z. Nouri, S. Sarrami-Foroushani, F. Azhari, and M. Azhari, Application of Carrera unified formulation in conjunction with finite strip method in static and stability analysis of functionally graded plates, Mech. Adv. Mater. Struct., pp. 1–17, 2020. in press. DOI: 10.1080/15376494.2020.1762265.
  • E. Carrera, S. Brischetto, M. Cinefra, and M. Soave, Effects of thickness stretching in functionally graded plates and shells, Compos. Part B: Eng., vol. 42, no. 2, pp. 123–133, 2011. DOI: 10.1016/j.compositesb.2010.10.005.
  • A. M. A. Neves, et al., Influence of zig-zag and warping effects on buckling of functionally graded sandwich plates according to sinusoidal shear deformation theories, Mech. Adv. Mater. Struct., vol. 24, no. 5, pp. 360–376, 2017. DOI: 10.1080/15376494.2016.1191095.
  • D. Ghazaryan, V. N. Burlayenko, A. Avetisyan, and A. Bhaskar, Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method, J. Eng. Math., vol. 110, no. 1, pp. 97–121, 2018. DOI: 10.1007/s10665-017-9937-3.
  • X. Zhang, Z. Ye, and Y. Zhou, A Jacobi polynomial based approximation for free vibration analysis of axially functionally graded material beams, Compos. Struct., vol. 225, pp. 111070, 2019. DOI: 10.1016/j.compstruct.2019.111070.
  • C.-S. Huang, and S. H. Huang, Analytical solutions based on Fourier cosine series for the free vibrations of functionally graded material rectangular Mindlin plates, Materials., vol. 13, no. 17, pp. 3820, 2020. DOI: 10.3390/ma13173820.
  • Z. Li, Y. Xu, and D. Huang, Analytical solution for vibration of functionally graded beams with variable cross-sections resting on Pasternak elastic foundations, Int. J. Mech. Sci., vol. 191, pp. 106084, 2021. DOI: 10.1016/j.ijmecsci.2020.106084.
  • Q. Li, V. Iu, and K. Kou, Three-dimensional vibration analysis of functionally graded material sandwich plates, J. Sound Vib., vol. 311, no. 1–2, pp. 498–515, 2008. DOI: 10.1016/j.jsv.2007.09.018.
  • B. Woodward, and M. Kashtalyan, Bending response of sandwich panels with graded core: 3D elasticity analysis, Mech. Adv. Mater. Struct., vol. 17, no. 8, pp. 586–594, 2010. DOI: 10.1080/15376494.2010.517728.
  • S. Brischetto, Exact elasticity solution for natural frequencies of functionally graded simply-supported structures, Computer Model. Eng. Sci., vol. 95, no. 5, pp. 391–430, 2013. DOI: 10.3970/cmes.2013.095.391.
  • S. Chakraverty, and K. Pradhan, Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions, Aerosp. Sci. Technol., vol. 36, pp. 132–156, 2014. DOI: 10.1016/j.ast.2014.04.005.
  • W.-Y. Jung, S.-C. Han, and W.-T. Park, Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation, Int. J. Mech. Sci., vol. 111–112, pp. 73–87, 2016. DOI: 10.1016/j.ijmecsci.2016.03.001.
  • S. J. Singh, and S. P. Harsha, Exact solution for free vibration and buckling of sandwich s-fgm plates on pasternak elastic foundation with various boundary conditions, Int. J. Str. Stab. Dyn., vol. 19, no. 3, pp. 1950028, 2019. DOI: 10.1142/S0219455419500287.
  • M. Roshanbakhsh, S. Tavakkoli, and BNavayi Neya, Free vibration of functionally graded thick circular plates: An exact and three-dimensional solution, Int. J. Mech. Sci., vol. 188, pp. 105967, 2020. DOI: 10.1016/j.ijmecsci.2020.105967.
  • E. Orakdöğen, S. Küçükarslan, A. Sofiyev, and M. H. Omurtag, Finite element analysis of functionally graded plates for coupling effect of extension and bending, Meccanica., vol. 45, no. 1, pp. 63–72, 2010. DOI: 10.1007/s11012-009-9225-z.
  • J.-H. Kim, and G. H. Paulino, Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials, J. Appl. Mech., vol. 69, no. 4, pp. 502–514, 2002. DOI: 10.1115/1.1467094.
  • M. H. Santare, and J. Lambros, Use of graded finite elements to model the behavior of nonhomogeneous materials, J. Appl. Mech., vol. 67, no. 4, pp. 819–822, 2000. DOI: 10.1115/1.1328089.
  • F. Tornabene, N. Fantuzzi, M. Bacciocchi, and J. N. Reddy, An equivalent layer-wise approach for the free vibration analysis of thick and thin laminated and sandwich shells, Appl. Sci., vol. 7, no. 1, pp. 17, 2016. DOI: 10.3390/app7010017.
  • ABAQUS, User’s Manual, Ver. 2016, Dassault Systémes Simulia Corp., Providence, RI, 2016.
  • W. G. Buttlar, G. H. Paulino, and S. H. Song, Application of graded finite elements for asphalt pavements, J. Eng. Mech., vol. 132, no. 3, pp. 240–249, 2006. DOI: 10.1061/(ASCE)0733-9399(2006)132:3(240).
  • V. Burlayenko, H. Altenbach, T. Sadowski, S. Dimitrova, and A. Bhaskar, Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements, Appl. Math. Modell., vol. 45, pp. 422–438, 2017. DOI: 10.1016/j.apm.2017.01.005.
  • S. Dhital, A. Rokaya, M. R. Kaizer, Y. Zhang, and J. Kim, Accurate and efficient thermal stress analyses of functionally graded solids using incompatible graded finite elements, Compos. Struct., vol. 222, pp. 110909, 2019. DOI: 10.1016/j.compstruct.2019.110909.
  • L. Huang, M. Yang, X. Zhou, Q. Yao, and L. Wang, Material parameter identification in functionally graded structures using isoparametric graded finite element model, Sci. Eng. Compos. Mater., vol. 23, no. 6, pp. 685–698, 2016. DOI: 10.1515/secm-2014-0289.
  • W. Gilewski, and J. Pełczyński, Material-oriented shape functions for fgm plate finite element formulation, Materials., vol. 13, no. 3, pp. 803, 2020. DOI: 10.3390/ma13030803.
  • J. Reinoso, M. Paggi, P. Areias, and A. Blázquez, Surface-based and solid shell formulations of the 7-parameter shell model for layered CFRP and functionally graded power-based composite structures, Mech. Adv. Mater. Struct., vol. 26, no. 15, pp. 1271–1289, 2019. DOI: 10.1080/15376494.2018.1432802.
  • V. N. Burlayenko, T. Sadowski, and S. Dimitrova, Three-dimensional free vibration analysis of thermally loaded FGM sandwich plates, Materials., vol. 12, no. 15, pp. 2377, 2019. DOI: 10.3390/ma12152377.
  • V. N. Burlayenko, T. Sadowski, H. Altenbach, and S. Dimitrova, Three-dimensional finite element modelling of free vibrations of functionally graded sandwich panels. In: H. Altenbach, J. Chróścielewski, V. Eremeyev, K. Wiśniewski (Eds.) Recent Developments in the Theory of Shells. Advanced Structured Materials, vol 110. Springer, pp. 157–177, 2019. DOI: 10.1007/978-3-030-17747-8_10.
  • V. N. Burlayenko, and T. Sadowski, Free vibrations and static analysis of functionally graded sandwich plates with three-dimensional finite elements, Meccanica., vol. 55, no. 4, pp. 815–832, 2020. DOI: 10.1007/s11012-019-01001-7.
  • H. Ashrafi, and M. Shariyat, A three-dimensional comparative study of the isoparametric graded boundary and finite element methods for nonhomogeneous fgm plates with eccentric cutouts, Int. J. Comput. Methods., vol. 14, no. 1, pp. 1750006, 2017. DOI: 10.1142/S0219876217500062.
  • J. Mars, S. Koubaa, M. Wali, and F. Dammak, Numerical analysis of geometrically non-linear behavior of functionally graded shells, Lat. Am. J. solids Struct., vol. 14, no. 11, pp. 1952–1978, 2017. DOI: 10.1590/1679-78253914.
  • S. Koubaa, J. Mars, and F. Dammak, Efficient numerical modelling of functionally graded shell mechanical behavior, Appl. Comput. Sci., vol. 15, no. 1, pp. 84–94, 2019. DOI: 10.23743/acs-2019-07.
  • H. Jrad, J. Mars, M. Wali, and F. Dammak, Geometrically nonlinear analysis of elastoplastic behavior of functionally graded shells, Eng. Computers., vol. 35, no. 3, pp. 833–847, 2019. DOI: 10.1007/s00366-018-0633-3.
  • H.-S. Shen, and Z.-X. Wang, Assessment of voigt and mori–tanaka models for vibration analysis of functionally graded plates, Compos. Struct., vol. 94, no. 7, pp. 2197–2208, 2012. DOI: 10.1016/j.compstruct.2012.02.018.
  • A. R. Nemati, and M. J. Mahmoodabadi, Effect of micromechanical models on stability of functionally graded conical panels resting on Winkler–Pasternak foundation in various thermal environments, Arch. Appl. Mech., vol. 90, no. 5, pp. 883–915, 2020. DOI: 10.1007/s00419-019-01646-6.
  • M. Aßmus, K. Naumenko, and H. Altenbach, A multiscale projection approach for the coupled global–local structural analysis of photovoltaic modules, Compos. Struct., vol. 158, pp. 340–358, 2016. DOI: 10.1016/j.compstruct.2016.09.036.
  • J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: With Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics, 2nd Edition, Oxford University Press, United Kingdom, 2014. DOI: 10.1093/acprof:oso/9780199641758.001.0001.
  • H. Altenbach, and J. Meenen, Single layer modelling and effective stiffness estimations of laminated plates. In: H. Altenbach, W. Becker (Eds.) Modern Trends in Composite Laminates Mechanics. International Centre for Mechanical Sciences (Courses and Lectures), vol. 448, Springer, Vienna, pp. 1–68, 2003. DOI: 10.1007/978-3-7091-2544-1_1.
  • P. E. Tovstik, and T. P. Tovstik, Generalized timoshenko-reissner models for beams and plates, strongly heterogeneous in the thickness direction, Z Angew. Math. Mech., vol. 97, no. 3, pp. 296–308, 2017. DOI: 10.1002/zamm.201600052.
  • H. Altenbach, An alternative determination of transverse shear stiffnesses for sandwich and laminated plates, Int. J. Solids Struct., vol. 37, no. 25, pp. 3503–3520, 2000. DOI: 10.1016/S0020-7683(99)00057-8.
  • V. Birman, and C. W. Bert, On the choice of shear correction factor in sandwich structures, J. Sandwich Struct. Mater., vol. 4, no. 1, pp. 83–95, 2002. DOI: 10.1177/1099636202004001180.
  • T.-K. Nguyen, K. Sab, and G. Bonnet, Shear correction factors for functionally graded plates, Mech. Adv. Mater. Struct., vol. 14, no. 8, pp. 567–575, 2007. DOI: 10.1080/15376490701672575.
  • R. Menaa, A. Tounsi, F. Mouaici, I. Mechab, M. Zidi, and E. A. A. Bedia, Analytical solutions for static shear correction factor of functionally graded rectangular beams, Mech. Adv. Mater. Struct., vol. 19, no. 8, pp. 641–652, 2012. DOI: 10.1080/15376494.2011.581409.
  • K. Magnucki, D. Witkowski, and J. Lewiński, Bending and free vibrations of beams with symmetrically varying mechanical properties—Shear effect, Mech. Adv. Mater. Struct., vol. 27, no. 4, pp. 325–332, 2020. DOI: 10.1080/15376494.2018.1472350.
  • S. Gagnon, and C. Pirvu, CLT Handbook: Cross-Laminated Timber, Canadian Edition, Special Publication (FPInnovations (Institute)), FPInnovations, Québec, 2011.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.