Refined One-Dimensional Models for the Multi-Field Analysis of Layered Smart Structures

  • Enrico ZappinoEmail author
  • Erasmo Carrera
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 81)


The analysis of layered structures requires the use of numerical tools that able to describe the complex behavior that can appear at the interface between two different materials. The use of the Finite Element Method can only lead to accurate results if the kinematic assumptions of the structural models allow complex deformation fields to be evaluated, and as a consequence classical models are often ineffective in the analysis of such structures. The use of the Carrera Unified Formulation provides a general tool that can be used to derive refined one-dimensional models in a compact form. The use of a refined kinematic description over the cross-section of an element leads to accurate results even when multi-field problems are considered, that is when complex stress fields appear. A comprehensive derivation of a class of refined one-dimensional models, which are able to deal with multilayer structures and multi-field problems, is presented in this section. Thermal and piezoelectric effects are considered, and a fully coupled thermo-piezo-elastic model is presented. Finally, some benchmarks are shown in order to verify the accuracy of the presented models.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahmad SN, Upadhyay CS, Venkatesan C (2006) Electro-thermo-elastic formulation for the analysis of smart structures. Smart Materials and Structures 15(2):401Google Scholar
  2. Ambrosini R (2000) A modified Vlasov theory for dynamic analysis of thin-walled and variable open section beams. Engineering Structures 22(8):890–900Google Scholar
  3. Bailey T, Hubbard J (1985) Distributed piezoelectric polymer active vibration control of a cantilever beam. AIAA Journal 8:605–611Google Scholar
  4. Benjeddou A, Trindade M, Ohayon R (1997) A unified beam finite element model for extension and shear piezoelectric actuation mechanisms. Journal of Intelligent Material Systems and Structures 8(12):1012–1025Google Scholar
  5. Berdichevsky VL (1976) Equations of the theory of anisotropic inhomogeneous rods. Dokl Akad Nauk 228:558–561Google Scholar
  6. Biscani F, Nali P, Belouettar S, Carrera E (2012) Coupling of hierarchical piezoelectric plate finite elements via arlequin method. Journal of intelligent materials systems and structures 23:749Google Scholar
  7. Carrera E (1997a) \( C_{z}^{0} \) Requirements – Models for the two dimensional analysis of multilayered structures. Composite Structure 37:373–384Google Scholar
  8. Carrera E (1997b) An improved reissner-mindlin-type model for the electromechanical analysis of multilayered plates including piezo-layers. Journal of Intelligent Material Systems and Structures 8:232–248Google Scholar
  9. Carrera E (2000) An assessment of mixed and classical theories for thermal stress analysis of orthotropic multilayered plates. Journal of Thermal Stresses 23:797–831Google Scholar
  10. Carrera E (2003) Theories and finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering 10:215–297Google Scholar
  11. Carrera E, Boscolo M (2007) Hierarchic multilayered plate elements for coupled multifield problems of piezoelectric adaptive structures: Formulation and numerical assessment. Archives of Computational Methods in Engineering 14(4):383–430Google Scholar
  12. Carrera E, PetroloM(2012) Refined beam elements with only displacement variables and plate/shell capabilities. Meccanica 47:537–556Google Scholar
  13. Carrera E, Robaldo A (2007) Extension of reissner mixed variational principle to thermopiezelasticity. Atti della Accademia delle Scienze di Torino Classe di Scienze Fisiche Matematiche e Naturali 31:27–42Google Scholar
  14. Carrera E, Boscolo M, Robaldo A (2007) Hierarchic multilayered plate elements for coupled multifield problems of piezoelectric adaptive structures: Formulation and numerical assessment. Archives of Computational Methods in Engineering 14(4):383–430Google Scholar
  15. Carrera E, Brischetto S, Nali P (2008) Variational statements and computational models for multifield problems and multilayered structures. Mechanics of Advanced Materials and Structures 15(3-4):182–198Google Scholar
  16. Carrera E, Giunta G, Nali P, Petrolo M (2010) Refined beam elements with arbitrary crpss-section geometries. Computers and Structures 88:283–293Google Scholar
  17. Carrera E, Gaetano G, M P (2011a) Beam Structures, Classical and Advanced Theories. John Wiley & SonsGoogle Scholar
  18. Carrera E, Petrolo M, Nali P (2011b) Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations 18(3):485–502Google Scholar
  19. Carrera E, Petrolo M, Varello A (2012a) Advanced beam formulations for free vibrations analysis of conventional and joined wings. Journal of Aerospace Engineering 25(2):282–293Google Scholar
  20. Carrera E, Zappino E, Petrolo M (2012b) Advanced elements for the static analysis of beams with compact and bridge-like sections. Journal of structural engineering 56:49–61Google Scholar
  21. Carrera E, Cinefra M, Petrolo M, Zappino E (2014a) Comparisons between 1d (beam) and 2d (plate/shell) finite elements to analyze thin walled structures. Aerotecnica Misssili & Spazio The journal of Aerospace Science, Technology and Systems 93(1-2)Google Scholar
  22. Carrera E, Cinefra M, Petrolo M, Zappino E (2014b) Finite Element Analysis of Structures Through Unified Formulation. John Wiley & SonsGoogle Scholar
  23. Carrera E, Filippi M, Zappino E (2014c) Free vibration analysis of laminated beam by polynomial, trigonometric, exponential and zig-zag theories. Journal of Composite Materials 48(19):2299–2316Google Scholar
  24. Caruso G, Galeani S, Menini L (2003) Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators. Simulation modelling prectice and theory 11:403–419Google Scholar
  25. Cowper GR (1966) The shear coefficient in Timoshenko’s Beam Theory. Journal of Applied Mechanics 33(2):335–340Google Scholar
  26. Crawley E, Luis J (1987) Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal 25:1373–1385Google Scholar
  27. Davies JM, Leach P (1994) First-order generalised beam theory. Journal of Constructional Steel Research 31(2-3):187–220Google Scholar
  28. Davies JM, Leach P, Heinz D (1994) Second-order generalised beam theory. Journal of Constructional Steel Research 31(2-3):221–241Google Scholar
  29. Dong SB, Alpdogan C, Taciroglu E (2010) Much ado about shear correction factors in Timoshenko beam theory. International Journal of Solids and Structures 47(13):1651–1665,
  30. Dong XJ, Meng G, Peng JC (2006) Vibration control of piezoelectric actuators smart structures based on system identification technique. Journal of sound and vibration 297:680–693Google Scholar
  31. Euler L (1744) De curvis elasticis. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis iso-perimetrici lattissimo sensu accepti. Bousquet & Socios, Lausanne & GenevaGoogle Scholar
  32. Friberg PO (1985) Beam element matrices derived from Vlasov’s theory of open thin-walled elastic beams. International Journal for Numerical Methods in Engineering 21:1205–1228Google Scholar
  33. Giavotto V, Borri M, Mantegazza P, Ghiringhelli G, Carmaschi V, Maffioli GC, Mussi F (1983) Anisotropic beam theory and applications. Computers & Structures 16(1):403–413Google Scholar
  34. Kim J, Varadan VV, Varadan VK (1997) Finite element modelling of structures including piezoelectric active devices. International journal for numerical methods in engineering 832:817–832Google Scholar
  35. Kim TW, Kim JH (2005) Optimal distribution of an active layer for transient vibration control of an flexible plates. Smart Material and Structures 14:904–916Google Scholar
  36. Kpeky F, Abed-Meraim F, Boudaoud H, Daya EM (2017) Linear and quadratic solid–shell finite elements shb8pse and shb20e for the modeling of piezoelectric sandwich structures. Mechanics of Advanced Materials and Structures pp 1–20Google Scholar
  37. Kumar K, Narayanan S (2007) The optimal location of piezolectric actuators and sensors for vibration controls of plate. Smart Material and Structures 16:2680–2691Google Scholar
  38. Kusculuoglu ZK, Royston TJ (2005) Finite element formulation for composite plates with piezoceramic layers for optimal vibration control applications. Smart Material and Structures 14:1139–1153Google Scholar
  39. Liu G, Dai K, Lim K (2004) Static and vibration control of composite laminates integrated with piezoelectric sensors and actuators using radial point interpolation method. Smart Material and Structures 14:1438–1447Google Scholar
  40. Miglioretti F, Carrera E, Petrolo M (2014) Variable kinematic beam elements for electro-mechanical analysis. Smart Structures and Systems 13(4):517–546Google Scholar
  41. Moita J, Soares C, Soares C (2005) Active control of forced vibration in adaptive structures using a higher order model. Composite Structures 71:349–355Google Scholar
  42. Moitha J, Correira I, Soares C, Soares C (2004) Active control of adaptive laminated structures with bonded piezoelectric sensors and actuators. Computer and Structures 82:1349–1358Google Scholar
  43. Robaldo A, Carrera E, Benjeddou A (2005) Unified formulation for finite element thermoleastic analysis of multilayered anisotropic composite plates. Journal of Thermal Stresses 28:1031–1064Google Scholar
  44. Robaldo A, Carrera E, Benjeddou A (2006) A unified formulation for finite element analysis of piezoelectric adaptive plates. Computers & Structures 84(22):1494–1505Google Scholar
  45. de Saint-Venant A (1856) Mémoire sur la Torsion des Prismes, avec des considérations sur leur flexion, ainsi que sur l’équilibre interieur des solides élastiques en général, et des formules pratiques pour le calcul de leur résistance à divers efforts s’exerçant simultanément. Académie des Sciences de l’Institut Impérial de Frances 14:233–560Google Scholar
  46. Sarvanos D, Heyliger P (1999) Mechanics and computational models for laminated piezoelectric beams, plate, and shells. Applied Mechanic Review 52(10):305–320Google Scholar
  47. Schardt R (1966) Eine Erweiterung der Technischen Biegetheorie zur Berech- nung Prismatischer Faltwerke. Der Stahlbau 35:161–171Google Scholar
  48. Silvestre N, N S, Camotim D (2002) First-Order Generalised Beam Theory for Arbitrary Orthotropic Materials. Thin-Walled Structures 40(9):791–820Google Scholar
  49. Sun C, Zhang X (1995) Use of thickness-shear mode in adaptive sandwich structures. Smart Materials and Structures 4(3):202Google Scholar
  50. Timoshenko SP (1921) On the corrections for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine 41:744–746Google Scholar
  51. Tzou HS, Ye R (1994) Piezothermoelasticity and precision control of piezoelectric systems: Theory and finite element analysis. Journal of Vibration and Acoustics 116(4):489–495Google Scholar
  52. Vasques C, Rodrigues J (2006) Active vibration of smart piezoelectric beams: comparison of classical and optimal feedback control strategies. Computer and Structures 84:1459–1470Google Scholar
  53. Vidal P, D’Ottavio M, Thaier M, Polit O (2011) An efficient finite shell element for the static resposne of piezoelectric laminates. Journal of intelligent materials systems and structures 22:671Google Scholar
  54. Vlasov VZ (1984) Thin Walled Elastic Beams, 2nd edn. National Technical Information Service, JerusalimGoogle Scholar
  55. Volovoi VV (1999) Asymptotic theory for static behavior of elastic anisotropic I-beams. International Journal of Solids and Structures 36(7):1017–1043,
  56. Washizu K (1968) Variational methods in elasticity and plasticity. Oxford: Pergamon PressGoogle Scholar
  57. Xu S, Koko T (2004) Finite element analysis and design of actively acontrolled piezoelectric smart structure. Finite element in Analysis and Design 40:241–262Google Scholar
  58. Yu W, Hodges DH (2004) Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams. Journal of Applied Mechanics 71(1):15Google Scholar
  59. Yu W, Volovoi VV, Hodges DH, Hong X (2002) Validation of the variational asymptotic beam sectional analysis (VABS). AIAA Journal 40:2105–2113Google Scholar
  60. Zappino E, Carrera E, Rowe S, Mangeot C, Marques H (2016) Numerical analyses of piezoceramic actuators for high temperature applications. Composite Structures 151:36 – 46Google Scholar
  61. Zhang X, Sun C (1996a) Formulation of an adaptive sandwich beam. Smart Materials and Structures 5(6):814Google Scholar
  62. Zhang XD, Sun CT (1996b) Formulation of an adaptive sandwich beam. Smart Materials and Structures 5(6):814Google Scholar
  63. Zhou X, Chattopadhyay A, Gu H (2000) Dynamic resposne of smart composites using a coupled thermo-piezoelectric-mechanical model. AIAA Journal 38:1939–1948Google Scholar
  64. Zhou YS, Tiersten HF (1994) An elastic analysis of laminated composite plates in cylindrical bending due to piezoelectric actuators. Smart Materials and Structures 3(3):255Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Mul2TeamPolitecnico di TorinoTorinoItaly

Personalised recommendations