A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams RameshchandraP.Shimpi*,YuwarajM.Ghugal Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India Abstract A new layerwise trigonometric shear deformation theory for the analysis of two-layered cross-ply laminated beams is presented. The number of primary variables in this theory is even less than that of first-order shear deformation theory, and moreover, it obviatestheneedforashearcorrectionfactor.Thesinusoidalfunctionintermsofthicknesscoordinateisusedinthedisplacement field to account for shear deformation. The novel feature of the theory is that the transverse shear stress can be obtained directly from the use of constitutive relationships, satisfying the shear-stress-free boundary conditions at top and bottom of the beam and satisfying continuity of shear stress at the interface. The principle of virtual work is used to obtain the governing equations and boundary conditions of the theory. The effectiveness of the theory is demonstrated by applying it to a two-layered cross-ply lami- nated beam. Keywords: Sheardeformation;Laminatedthickbeam;Transverseshearstress;Interfaceshearcontinuity;Cross-plybeam 1. Introduction The use of fiber-reinforced composite laminates has greatly increased in weight sensitive applications such as aerospaceandautomotivestructuresbecauseoftheirhigh specificstrengthandhighspecificstiffness.Theincreased use of laminated beams in various structures has stimu- lated considerable interest in their accurate analysis. On accountoftheirlowratiooftransverseshearmodulusto thein-planemodulus,sheardeformationeffectsaremore pronounced in the composite beams subjected to trans- verseloads. It is well-known that the classical Euler–Bernoulli theoryofbeambending,alsoknownaselementarythe- oryofbending(ETB),disregardstheeffectsoftheshear deformation. The theory is suitable for slender beams butnotforthickordeepbeamssinceitisbasedonthe assumptionthatthetransversenormaltotheneutralaxis remains so during bending and after bending, implying that the transverse shear strain is zero. Since the theory neglectsthetransversesheardeformation,itleadstoless accurateresultsinthecaseofisotropicthickbeamsand moresointhecaseoflaminatedcompositethickbeams, where shear effects are significant. Bresse [1], Rayleigh [2], and Timoshenko [3] were the pioneer investigators who included refined effects such as thesheardeformatonandrotatoryinertiainthebeamthe- ory.Timoshenkoshowedthattheeffectoftransverseshear ismuchgreaterthanthatofrotatoryinertiaontheresponse of transverse vibration of prismatic beams. This theory is nowwidelyreferredtoastheTimoshenkobeamtheoryin the literature. In this theory, transverse shear strain dis- tribution is assumed to be constant through the beam thickness and, thus, requires a shear correction factor to appropriatelyrepresentthestrainenergyofdeformation. Kant and Manjunatha [4], Manjunatha and Kant [5], Maiti and Sinha [6] and Vinayak et al. [7] used the equivalentsinglelayer,displacementbased,higher-order shear deformation theories (HSDT) in the analysis of symmetric and unsymmetric laminated beams and employed the finite-element method as a solution tech- nique.ThesetheoriesarethespecialcasesofLoetal.[8] higher-ordertheory. Levy [9] and Stein [10] developed refined plate the- ories expressing the displacement field in terms of trigonometric functions to represent the thickness effect and approximated the shear stress distribution through the thickness. Recently, Liu and Li [11] presented an overall com- parison of laminate theories based on displacement hypothesis emphasizing the importance of layerwise