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On the deformation of laminated composite and sandwich curved beams

On the deformation of laminated composite and sandwich curved beams Curved and Layer. Struct. 2022; 9:1–12 Research Article Pravin V. Avhad and Atteshamuddin S. Sayyad* On the deformation of laminated composite and sandwich curved beams https://doi.org/10.1515/cls-2022-0001 stiffness-to-weight ratios as well as the high strength-to- Received Jan 13, 2021; accepted Jun 09, 2021 weight ratio. Therefore, it is significantly important to in- vestigate the accurate static behaviour of curved beams Abstract: Plenty of research articles are available on the subjected to static loading. Various laminated beam theo- static deformation analysis of laminated straight beams us- ries are available in the literature which can be extended for ing refined shear deformation theories. However, research the analysis of curved beams such as classical beam theory on the deformation of laminated curved beams with sim- [1], Timoshenko beam theory [2], Higher order beam theo- ply supported boundary conditions is limited and needs ries [3, 4, 5], etc. In this section literature on various beam more attention nowadays. With this objective, the present theories available in the literature and their applications study deals with the static analysis of laminated compos- in various problems are reviewed. ite and sandwich beams curved in elevation using a new Reddy [6] has developed a well-known third order shear quasi-3D polynomial type beam theory. The theory consid- deformation theory for the analysis of laminated compos- ers the effects of both transverse shear and normal strains, ite beams which is further extended by Khdeir and Reddy i.e. thickness stretching effects. In the present theory, axial [7, 8] for the analysis of cross-ply laminated beams/arches. displacement has expanded up to the fifth-order polyno- Kant and Manjunath [9] have presented a static analysis mial in terms of thickness coordinates to effectively account of symmetric and unsymmetric laminated composite and for the effects of curvature and deformations. The present sandwich beams based on a C continuity finite element theory satisfies the zero traction boundary condition on using various refine higher order beam theories. Kant et the top and bottom surfaces of the beam. Governing differ- al. [10] have developed semi-analytical elasticity bending ential equations and associated boundary conditions are solutions for laminated composite beams. Li et al. [11] have established by using the Principal of virtual work. Navier’s studied the free vibration analysis of laminated compos- solution technique is used to obtain displacements and ite beams of different boundary conditions using various stresses for simply supported beams curved in elevation higher-order beam theories and spectral finite element and subjected to uniformly distributed load. The present method which is also used by Nanda and Kapuria [12] for the results can be benefited to the upcoming researchers. wave propagation analysis of laminated composite curved Keywords: fifth-order polynomial, laminated and sand- beams. Luu et al. [13] investigated non-dimensional deflec- wich, curved beams, static analysis tion and critical buckling loads of shear deformable lam- inated composite curved beams using the NURBS-based isogeometric method. Ye et al. [14] and Mohamad et al. [15] studied vibration analysis of laminated composite curved 1 Introduction beams of various boundary conditions. Jianghua et al. [16] have presented the static and vibration analysis of lami- Laminated composite curved beams/arches are widely nated composite curved beams using a domain decomposi- used in aerospace, automobile, ships, civil and mechani- tion approach. Zenkour [17] has developed a new shear and cal industries due to their superior properties such as high normal deformation theory for the static analysis of cross- ply laminated composite and sandwich beams. Karama et al. [18] have developed an exponential shear deforma- tion theory for the bending, buckling, and free vibration Pravin V. Avhad: Research Scholar, Department of Civil Engineer- ing, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune analysis of multi-layered laminated composite beams. Pio- University, Kopargaon-423603, Maharashtra, India van et al. [19] have analyzed composite thin-walled curved *Corresponding Author: Atteshamuddin S. Sayyad: Department beams with open and closed cross-sections. Hajianmaleki of Civil Engineering, SRES’s Sanjivani College of Engineering, Savit- and Qatu [20] have applied Timoshenko beam theory for ribai Phule Pune University, Kopargaon-423603, Maharashtra, India, the static and free vibration analysis of generally laminated E-mail: attu_sayyad@yahoo.co.in Open Access. © 2022 Avhad and Sayyad, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 2 | Avhad and Sayyad deep curved beams with boundary conditions. Tessler et curved panels using various single layer equivalent and al. [21] have presented the analysis of laminated compos- layerwise theories based on Carrera’s unified formulation. ites and sandwich beams using a zig-zag theory. Carrera Tornabene et al. [44] presented natural frequencies of sev- et al. [22] have investigated the static response of lami- eral doubly-curved shells with variable thickness using var- nated composite beams by using various polynomial and ious higher-order equivalent single layer theories including non-polynomial beam theories. Dogruoglu and Komurcu the Murakami’s function to capture the zig-zag effect. Pa- [23] have presented the static analysis of planar curved gani et al. [45] applied various classical and refined plate beams by using a finite element method. Bhimaraddi et theories for the analysis of laminates and sandwich struc- al. [24] have investigated the static and dynamic analysis tures using finite element method. Carrera et al. [46, 47] of isoparametric thick laminated curved beams by using a have developed a new finite element for the analysis of finite element method. Bouclier et al. [25] have developed metallic and composite plates and shells. shear and membrane locking free isogeometric formula- tion for the analysis of curved beams. Fraternali and Bilotti [26] have presented the stress analysis of composite curved 1.1 Scientific soundness of the topic beams using one-dimensional theory and finite element method. Kurtaran [27] applied a differential quadrature 1) Beams curved in elevation are widely used in many method for the static and transient analysis of functionally engineering industries such as arch type bridges, graded curved beams using first order shear deformation chain links, crane hooks, pipe bends, and curved theory. Matsunaga [28] has investigated the displacements segments of machine tool frames. Curved beam seg- and stresses of the laminated and sandwich curved beams ments used in these structures are often subjected to under mechanical/thermal loading by using higher order the static forces where it needs to analyze and design theory. Qian et al. [29] have studied the displacements and accurately. This forced the authors to consider this stresses of the laminated arches, under thermal load by topic as an area of research. using two-dimensional thermo-elastic theory. Malekzadeh 2) Based on the aforementioned literature review, it is [30] has presented the static analysis of thick laminated observed that a lot of research has been carried out by deep circular arches with various boundary conditions by researchers on static analysis of straight beams using using the two-dimensional theory of elasticity. Casari and higher-order shear deformation theories. However, Gornet [31] have presented the static analysis of composite research on static analysis of curved beams is limited curved sandwich beams under thermo-mechanical load. in the literature. Kress et al. [32] have investigated the stress distributions of 3) From the literature review, it is also observed that, in thick and singly curved laminates by using finite element the higher order theories available in the literature, method. Marur and Kant [33] have analyzed the static be- thickness coordinates are expanded up to third order haviour of laminated arches using transverse shear and nor- only. However, for the accurate prediction of static mal deformation theory. Sayyad and his co-authors’ [34-38] behaviour laminated curved beams, it is necessary to have presented various polynomial, and non-polynomial expand thickness coordinates up to minimum fifth type beam theories for the static, vibration, and buckling order. Therefore, in this paper, a fifth-order shear analysis of isotropic, functionally graded, laminated, and deformation theory is developed for the curved beam. sandwich beams. Recently, Sayyad and Ghugal [39] pre- 4) It is recommended by Carrera et al. [22] that the trans- sented static analysis sandwich curved beams using a si- verse normal strain plays an important role in predict- nusoidal shear deformation theory considering the effects ing the accurate bending behavior of thick composite of thickness stretching. Fazzolari et al. [40] presented a beams. Therefore, the present theory also considers comparative study between two computational techniques the effects of transverse normal strain i.e. thickness (hierarchical ritz formulation and generalized differential stretching effects for the modeling and analysis of quadrature) to evaluate the natural frequencies of com- curved laminated beams. posite beams and shells. Dimitri et al. [41] also presented 5) In this paper, the static analysis of laminated com- the analysis of various curved segments using generalized posites and sandwich beams curved in elevation is differential quadrature method. Fantuzzi and Tornabene analyzed under uniform load. [42] presented analysis of arbitrarily shaped plates using 6) The present theory and the approach can be extended strong finite element formation and differential quadrature to analyze the curved beams subjected to dynamic method. Tornabene et al. [43] have presented the through- loading, thermal loading, nonlinear problems, and the-thickness distribution of strains and stresses for doubly- On the deformation of laminated composite and sandwich curved beams | 3 the curved beams accounting for variable radii of where (u, u ) and (w, w ) are the x- and z- directional 0 0 curvatures and thicknesses. displacements, respectively; ϕ , ψ , ϕ , ψ are the un- x x z z known rotations to be determined. In the case of plates, the displacement model of the theory is modeled by con- sidering displacement components in all three directions 2 Mathematical formulation of the of the plate. But, in the case of beam, rotation in the y- present theory for curved beam direction is assumed as zero and the width of the beam is taken as unity in the y-direction. Hence, the displacement Consider a laminated curved beam with radius of curvature in the y-direction is neglected. For the curved beam, the R in the Cartesian coordinate system as shown in Figure non-zero strain components are determined using the fol- 1. The beam has curved length L, and rectangular cross- lowing strain-displacement relationship for the theory of section b×h where b is the width (b=1) and h is the total elasticity. thickness (0≤ x ≤ L; -b/2≤ y ≤ b/2; -h/2≤ z ≤ h/2). The beam is composed of N number of layers perfectly bonded together ∂u w ∂w ∂u ∂w u ε = + , ε = , 𝛾 = + − (2) x xz and made up of fibrous composite materials. The bond ∂x R ∂z ∂z ∂x R between the two adjacent layers is of zero thickness. The Using displacements from Eq. (1), the following are the beam is subjected to uniform load (q). expressions for normal strains and transverse shear strain at any point of the beam. ∂u ∂ w ∂ϕ 0 0 ε = − z + f (z) + x 1 ∂x ∂x ∂x ∂ψ w ϕ ψ x z z 0 ′ ′ + f (z) + + f (z) + f (z) 1 2 ∂x R R R (3) ′′ ′′ ε = f (z)ϕ + f (z)ψ z z z 1 2 ∂ϕ ∂ψ z z ′ ′ ′ ′ Figure 1: Geometry ad coordinates system of laminated curved 𝛾 = f (z)ϕ + f (z)ψ + f (z) + f (z) xz x x 1 2 1 2 ∂x ∂x beam. where Following are the kinematic assumptions made for the (︂ )︂ (︂ )︂ 3 5 development of the displacement field of the present theory. 4z 16z f (z) = z − , f (z) = z − 1 2 2 4 3h 5h 1) The x-directional displacement contains the extension, (4) (︂ )︂ (︂ )︂ 2 4 bending, and shear components. Theories that consider the 4z 16z ′ ′ f (z) = 1 − , f (z) = 1 − 1 2 2 4 effects of bending deformation, shear deformation, and nor- h h mal deformation (thickness-stretching) are called quasi-3D The prime ( ) indicates the derivative with respect to theories. 2) The extension component is presented interms the z-coordinate. A generalized Hooke’s law is used to de- of the radius of curvature. 3) The z-directional displacement th termine the stresses in the k layer of laminated curved considered the effects of transverse normal deformations beams. i.e. ε ≠ 0. 4) The present theory consists of fifth order poly- nomial to account for the traction-free boundary conditions ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎪ ⎪ ⎪ ⎪ σ C C 0 ε x x at the top and bottom surfaces of the beam. Based on the ⎨ ⎬ 11 13 ⎨ ⎬ ⎢ ⎥ σ = C C 0 ε (5) z ⎣ ⎦ z before mentioned assumptions, the displacement field of 13 33 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ τ 0 0 C 𝛾 xz xz the present theory is written as follows: 55 where C are the reduced stiffness coefficients. ij (︁ )︁ z ∂w u(x, z) = 1 + u (x) − z + R ∂x (︂ )︂ (︂ )︂ (︂ )︂ (︂ )︂ E E 3 5 1 3 4z 16z C = , C = , 11 33 + z − ϕ (x) + z − ψ (x) x x 1 − µ µ 1 − µ µ 2 4 13 31 13 31 3h 5h (︂ )︂ (6) (︂ )︂ (1) µ E 2 13 1 4z C = , C = G 13 55 13 w(x, z) = w (x) + 1 − ϕ (x)+ 0 z 1 − µ µ 13 31 (︂ )︂ where E and E represent elastic moduli, G represent 16z 1 3 13 + 1 − ψ (x) h shear modulus and µ , µ represent Poisson’s ratios. Ax- 13 31 4 | Avhad and Sayyad ial force, bending moment and shear force resultants as- (︂ )︂ sociated with the present theory can be derived using the 2 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z Q = AG + − AN + AT + AY + BC + 13 13 13 13 13 following relations. 2 ∂x R ∂x ∂x ∂x R + BF + BH ϕ + BI ψ 13 33 z 33 z h/2 h/2 (︂ )︂ ∫︁ ∫︁ ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z b Q = AH + − AO + AU + AZ + BD + 13 13 13 13 13 N = σ dz, M = σ zdz, 2 x x x x ∂x R ∂x ∂x ∂x R −h/2 −h/2 z + BG + BI ϕ + BJ ψ 13 33 z 33 z h/2 h/2 (︂ )︂ (︂ )︂ ∫︁ ∫︁ ∂ϕ ∂ψ z z 1 2 s1 s2 Q = BK ϕ + + BL ψ + and Q = 55 x 55 x M = σ f (z)dz, M = σ f (z)dz xz xz x x 1 x x 2 ∂x ∂x (︂ )︂ (︂ )︂ −h/2 −h/2 ∂ϕ ∂ψ z z = BL ϕ + + BM ψ + 55 x 55 x h/2 h/2 ∂x ∂x ∫︁ ∫︁ (8) 1 ′ 2 ′ V = σ f (z)dz, V = σ f (z)dz, x x x x 1 2 Integration constants appeared in Eq. (8) are defined −h/2 −h/2 (7) as follows: h/2 ∫︁ 1 ′′ Q = σ f (z)dz z 1 AA , AB , AC , AD , AE , AF = ( ) 11 11 11 11 11 11 −h/2 h/2 h/2 h/2 ∫︁ ∫︁ ∫︁ [︀ ]︀ ′ ′ 2 ′′ 1 ′ C 1, z, f (z), f (z), f (z), f (z) dz 11 1 2 1 2 Q = σ f (z)dz, Q = τ f (z)dz, z z 2 xz xz 1 −h/2 −h/2 −h/2 h/2 (AI , AJ , AK , AL , AM ) = ∫︁ 11 11 11 11 11 2 ′ Q = τ f (z)dz h/2 xz xz 2 ∫︁ [︀ ]︀ ′ ′ −h/2 C z, f (z), f (z), f (z), f (z) zdz 11 1 2 1 2 −h/2 Substitution of stresses from Eq. (5) into Eq. (7) one can AG , AH = ( ) 13 13 get the final expressions for stress resultants as follows: h/2 ∫︁ [︀ ]︀ (︂ )︂ n ′′ C f (z), f (z) dz AN , AO = ∂u w ∂ w ∂ϕ ∂ψ ϕ ( ) 0 0 0 x x z 13 1 2 13 13 N = AA + − AB + AC + AD + AE + 11 11 11 11 11 ∂x R ∂x ∂x ∂x R −h/2 h/2 + AF + AG ϕ + AH ψ 11 13 z 13 z ∫︁ R [︀ ]︀ n n (︂ )︂ C f (z), f (z) zdz 2 (9) 13 1 2 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z M = AB + − AI + AJ + AK + AL + 11 11 11 11 11 ∂x R ∂x ∂x ∂x R −h/2 (AP , AQ , AR , AS ) = 11 11 11 11 + AM + AN ϕ + AO ψ 11 13 z 13 z (︂ )︂ h/2 2 ∫︁ ∂u w ∂ w ∂ϕ ∂ψ ϕ x x z 0 0 0 s1 [︀ ]︀ M = AC + − AJ + AP + AQ + AR + 11 11 11 11 11 ′ ′ C f (z), f (z), f (z), f (z) f (z)dz ∂x R ∂x ∂x ∂x R 11 1 2 1 1 2 −h/2 + AS + AT ϕ + AU ψ 11 13 z 13 z (︂ )︂ AT , AU = ( ) 2 13 13 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z s2 M = AD + − AK + AQ + AV + AW + 11 11 11 11 11 2 h/2 ∂x R ∂x ∂x ∂x R ∫︁ [︀ ]︀ ′′ ′′ z C f (z), f (z) f (z)dz 13 1 2 1 + AX + AY ϕ + AZ ψ 11 13 z 13 z (︂ )︂ −h/2 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z V = AE + − AL + AR + AW + BA + 11 11 11 11 11 2 AV , AW , AX = ( ) 11 11 11 ∂x R ∂x ∂x ∂x R z h/2 ∫︁ + BB + BC ϕ + BD ψ 11 13 z 13 z [︀ ]︀ ′ ′ C f (z), f (z), f (z) f (z)dz (︂ )︂ 11 2 1 2 2 ∂u w ∂ w ∂ϕ ∂ψ ϕ x x z 0 0 0 V = AF + − AM + AS + AX + BB + 11 11 11 11 11 −h/2 ∂x R ∂x ∂x ∂x R + BE + BF ϕ + BG ψ 11 13 z 13 z R On the deformation of laminated composite and sandwich curved beams | 5 AY , AZ = ( ) 13 13 h/2 ∫︁ L L (︂ )︂ (︂ )︂ ∫︁ ∫︁ [︀ ]︀ 2 ′′ ′′ ∂δu δw ∂ δw 0 0 b 0 C f (z), f (z) f (z)dz 13 1 2 2 N dx + N − M dx+ x x x ∂x R ∂x −h/2 0 0 (︂ )︂ BA , BB = ∫︁ ( ) 11 11 ∂δϕ s x 1 M + Q δϕ dx x x= x h/2 ∫︁ ∂x [︀ ]︀ ′ ′ ′ C f (z), f (z) f (z)dz 11 1 2 1 (11) (︂ )︂ ∫︁ −h/2 ∂δψ s2 x 2 + M + Q δψ dx x xz x ∂x (BC , BD ) = 13 13 h/2 ∫︁ (︂ )︂ ∫︁ [︀ ]︀ ′′ ′′ 1 δϕ ∂δϕ 1 z 1 1 z C f (z), f (z) f (z)dz 1 2 1 + V + Q δϕ + Q dx x z z x= R ∂x −h/2 BF , BG = ( ) 13 13 ∫︁ (︂ )︂ h/2 δψ ∂δψ ∫︁ z z 2 2 2 + V + Q δψ + Q − [︀ ]︀ z x z xz ′′ ′′ ′ R ∂x C f (z), f (z) f (z)dz, BE = 13 1 2 2 11 −h/2 ∫︁ (︀ )︀ h/2 ′ ′ ∫︁ q δw + f (z)δϕ + f (z)δψ = 0 z z 0 1 2 [︀ ]︀ C f (z) dz 11 2 −h/2 Integrating Eq. (11) by parts, collecting the coeffi- BH , BI = ( 33 33) cients of δu , δw , δϕ , δψ , δϕ , δψ and setting 0 0 x x z z h/2 them equal to zero, the following six governing differential ∫︁ [︀ ]︀ n n ′′ equations are obtained. C f (z), f (z) f (z) · dz BJ = 33 33 1 2 1 −h/2 ∂N δu : = 0 h/2 0 ∫︁ ∂x [︀ ]︀ ′′ 2 b C f (z) dz 33 2 ∂ M N x x δw : − + q = 0 −h/2 ∂x R s1 ∂M x 1 (BK , BL ) = 55 55 δϕ : − Q = 0 xz ∂x (12) h/2 ∫︁ s2 [︀ ]︀ ∂M x 2 ′ ′ ′ δψ : − Q = 0 x xz C f (z), f (z) f (z)dzBM = 55 1 2 1 55 ∂x 1 1 −h/2 ∂Q V xz x 1 ′ δϕ : − − Q + qf (z) = 0 z 1 h/2 ∂x R ∫︁ [︀ ]︀ 2 2 ∂Q V C f (z) dz xz x 2 ′ δψ : − − Q + qf (z) = 0 z z ∂x R −h/2 The boundary conditions at the supports (x=0 and x=L) The principle of virtual work is employed to derive the are presented in Table 1. governing equation and boundary condition associated Using Eq. (8), in terms of unknown variables, Eq. (12) with the present theory. can be written as L h/2 L ∫︁ ∫︁ ∫︁ (︂ )︂ 2 3 ∂ u 1 ∂w ∂ w 0 0 0 (σ δε + σ δε + τ δ𝛾 ) dzdx − qδwdx = 0 δu : −AA + + AB − x x z z xz xz 0 11 11 2 3 ∂x R ∂x ∂x 0 0 −h/2 2 2 ∂ ϕ ∂ ψ AE ∂ϕ x x 11 z (13) (10) AC − AD − 11 11 2 2 R ∂x ∂x ∂x where δ is the variational operator. Eq. (10) can be writ- AF ∂ψ ∂ϕ ∂ψ 11 z z z − − AG − AH = 0 ten as follows using strains from Eqs. (3) and stress resul- 13 13 R ∂x ∂x ∂x tants from Eq. (8). 6 | Avhad and Sayyad Table 1: Natural and essential boundary conditions associated with the present theory. (︂ )︂ (︂ )︂ 1 ∂u w ∂u w 0 0 0 0 δϕ : AE + + AG + − z 11 13 R ∂x R ∂x R Natural Essential 2 2 2 AL ∂ w ∂ w AR ∂ ϕ ∂ϕ x x 11 0 0 11 − AN + + AT given N |x = 0, L or u |x = 0, L x 0 13 13 2 2 2 R ∂x ∂x R ∂x ∂x b ∂w given M |x = 0, L or |x = 0, L ∂x ∂ϕ AW ∂ψ ∂ψ ∂ψ x x x x − BK + + AY − BL ∂M 55 13 55 given |x = 0, L or w |x = 0, L ∂x R ∂x ∂x ∂x ∂x (17) s1 BA BC BB given M |x = 0, L or ϕ |x = 0, L 11 13 11 + ϕ + 2 ϕ + BH ϕ + ψ − z 33 Z Z Z s2 R R R given M |x = 0, L or ψ |x = 0, L 1 ∂ ϕ BD BF Z 13 13 given Q |x = 0, L or ϕ |x = 0, L xz BK + ψ + ψ + BI ψ − 55 Z 33 Z R R 2 ∂x given Q |x = 0, L or ψ |x = 0, L xz ∂ ψ Z ′ BL = qf (z) ∂x (︂ )︂ (︂ )︂ (︂ )︂ (︂ )︂ 1 ∂u w ∂u w 3 2 0 0 0 0 1 ∂u w ∂ u 2 ∂ w δψ : AF + + AH + − 0 0 0 0 z 11 13 δw : AA + − AB + R ∂x R ∂x R 0 11 11 2 3 2 R ∂x R ∂x R ∂x 2 2 2 AM ∂ w ∂ w AS ∂ ϕ ∂ϕ 4 3 11 0 0 11 x x ∂ w ∂ ϕ − AO + + AU 0 13 13 2 2 2 + AI − AJ + R R ∂x ∂x ∂x ∂x 11 11 4 3 ∂x ∂x ∂ϕ AX ∂ψ ∂ψ ∂ψ x 11 x x x 3 2 − BL + + AZ − BM AD ∂ψ ∂ ψ AE AL ∂ ϕ 55 13 55 11 x x 11 11 z ∂x R ∂x ∂x ∂x − AK + ϕ + − 11 z 3 2 R ∂x ∂x R R ∂x BB BF BD 11 13 13 2 + ϕ + ϕ + ϕ + BI ϕ − z z Z Z 33 ∂ ϕ AF AG z 11 13 R R R AN + ψ + ϕ z z 2 2 (18) ∂x R R 2 2 AH AM ∂ ψ ∂ ψ z z 13 11 + ψ − − AO = q 2 2 R R ∂x ∂x ∂ ϕ BE BG Z 11 13 (14) − BL + ψ + 2 ψ + BJ ψ − 55 Z Z 33 Z 2 2 ∂x R ∂ ψ (︂ )︂ Z ′ 2 3 BM = qf (z) ∂ u 1 ∂w ∂ w 2 0 0 0 ∂x δϕ : −AC + + AJ − x 11 11 2 3 ∂ x R ∂x ∂x 2 2 ∂ ϕ ∂ ψ x z AP + BK ϕ − AQ + BL ψ 11 55 x 11 55 x 2 2 ∂x ∂x (15) 3 The Navier solution AR ∂ϕ ∂ϕ ∂ϕ z z 11 Z − − AT + BK − 13 55 R ∂x ∂x ∂x Analytical solution of above mentioned six governing equa- AS ∂ψ ∂ψ ∂ψ z z z − AU + BL = 0 13 55 tions is obtained using Navier’s solution technique. The R ∂x ∂x ∂x solution is obtained to investigate the static behaviour (︂ )︂ of simply-supported laminated composites and sandwich 2 3 ∂ u 1 ∂w ∂ w 0 0 0 δψ : −AD + + AK − x 11 11 beams curved in elevation. Following are the boundary 2 3 ∂ x R ∂x ∂x conditions at the simple supports of the curved beams. 2 2 ∂ ϕ ∂ ψ x z AQ + BL ϕ − AV + BM ψ 11 55 x 11 55 x 2 2 ∂x ∂x (16) AW ∂ϕ ∂ϕ ∂ϕ b s1 s2 z z z N = 0, w = 0, M = 0, M = 0, M = 0ϕ = 0, ψ = 0 x z z − − AY + BL − 0 x x x 13 55 R ∂x ∂x ∂x at x = 0 and x = L AX ∂ψ ∂ψ ∂ψ z z 11 Z − AZ + BM = 0 13 55 (19) R ∂x ∂x ∂x According to Navier’s technique, unknown variables are expanded in the single trigonometric series satisfying simply-supported boundary conditions stated in Eq. (19). The following trigonometric form is assumed for the un- known variables. On the deformation of laminated composite and sandwich curved beams | 7 (︂ )︂ ∞ ∞ AD (︁ )︁ (︁ )︁ 11 3 ∑︁ ∑︁ mπx mπx K = K = − α − AK α , 24 42 11 u = u cos , w = w sin R 0 m 0 m L L (︂ )︂ m=1 m=1 AE AG AL 11 13 11 2 2 ∞ ∞ K = K = + + α + AN α , (︁ )︁ (︁ )︁ 25 52 13 ∑︁ ∑︁ mπx mπx R R ϕ = ϕ cos , ϕ = ϕ sin x xm z zm (︂ )︂ L L AF AH AM m=1 m=1 11 13 11 2 2 K = K 2 = + + α + AO α , 26 62 13 ∞ ∞ R R R (︁ )︁ (︁ )︁ ∑︁ ∑︁ mπx mπx (︁ )︁ (︁ )︁ ψ = ψ cos , ψ = ψ sin x xm z zm 2 2 L L K = AP α + BK , K = K = AQ α + BL , 33 11 55 34 43 11 55 m=1 m=1 (︂ )︂ (20) AR K = K = − α − AT α + BK α , 35 53 13 55 where u , w , ϕ , ψ , ϕ , ψ are the un- m m xm xm zm zm (︂ )︂ known coefficients to be determined; the transverse uni- AS K = K = − α − AU α + BL α , 13 55 form load (q) acting on the beam is also expanded in the 36 63 (︁ )︁ single trigonometric form: K = AV α + BM , 44 11 55 (︂ )︂ (︁ )︁ (︂ )︂ ∑︁ 4q mπx AW q = sin (21) K = K = − α − AY α + BL α , 45 54 13 55 mπ L m=1 (23) where q is the maximum intensity of uniform load. By substituting unknown variables from Eq. (20) and trans- (︂ )︂ AX verse load from Eq. (21) into governing equations (13)-(18), K = K = − α − AZ α + BM α , 46 64 13 55 the following equation is derived. (︂ )︂ BA BC 11 13 2 K = + 2 + BH + BK α 55 33 55 R R ⎡ ⎤ ⎧ ⎫ K K K K K K u 11 12 13 14 15 16 ⎪ 0 ⎪ ⎪ ⎪ K = K = 65 56 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ (︂ )︂ K K K K K K w 21 22 23 24 25 26 0 ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ BB BD BF ⎢ ⎥ ⎨ ⎬ 11 13 13 2 K K K K K K ϕ = + + + BI + BL α ⎢ 31 32 33 34 35 36 ⎥ x 33 55 ⎢ ⎥ × = R R R ⎢ ⎥ ⎪ ⎪ K K K K K K ψ (︂ )︂ 41 42 43 44 45 46 x ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ BE BG 11 13 2 ⎣ ⎦ ⎪ ⎪ K K K K K K ⎪ ϕ ⎪ K = + 2 + BJ + BM α 51 52 53 54 55 56 z 66 33 55 ⎪ ⎪ ⎩ ⎭ R R K K K K K K ψ 61 62 63 64 65 66 z (22) ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ 4 Illustrative problems ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 4q 0 In this section, static analysis of simply supported lami- mπ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ nated composites and sandwich beams curved in elevation ⎪ ⎪ ⎪ ⎪ f (z) ⎪ ⎪ ⎪ ⎪ is presented to prove the efficiency and accuracy of the ⎩ ⎭ f (z) present theory. The numerical results for the static analysis where, of laminated curved beams are not available in the litera- ture, therefore, the present theory is validated with straight (︂ )︂ beam results available in the literature. Three types of lami- AA 2 3 0 0 0 0 0 0 0 K = AA α , K = K = − α − AB α , 11 11 12 21 11 nation schemes (0 /90 , 0 /90 /0 , 0 /core/0 ) have been (︁ )︁ solved in the present study. The following material prop- 2 2 K = K = AC α , K = K = AD α , 13 31 11 14 41 11 erties have been used to present the numerical results for (︂ )︂ laminated curved beams. AE K = K = − α − AG α , 15 51 13 M1:E = 172.4 GPa, E = 6.89 GPa, G = 3.45 GPa, G = 1.378 1 3 13 31 AF K = K = − α − H α, 16 61 13 GPa, =0.25. (︂ )︂ M2:E = 0.276 GPa, E = 3.45 GPa, G = 3.45 GPa, G = 1.378 AA AB 1 3 13 31 11 4 11 2 K = + AI α + 2 α , 22 11 R R GPa, =0.25. (︂ )︂ AC 11 3 K = K = − α − AJ α , 23 32 11 The thickness of the beam is assumed as a unity (h=1.0) and other dimensions depend on L/h and R/h ratios. The nu- 8 | Avhad and Sayyad merical results are expressed in the following normalized axial displacement does not change its sign and remains form in Tables 2 through 5. positive throughout the thickness of the beam. Table 4 presents the normalized values of bending (︂ )︂ (︂ )︂ stresses in laminated and sandwich curved beams sub- 3 3 E h L E h h 3 3 w ¯ = 100 w , u ¯ = 100 u 0,− , jected to uniform load. Bending stresses of the top fiber q L 2 q L 2 0 0 (︂ )︂ i.e (z=-h/2) are summarized in Table 4. From the compar- h h h (24) σ ¯ = σ 0,− , τ ¯ = τ (0, 0) ison of results, it is observed that the present results are x x xz xz q L 2 q L 0 0 closely matched with other theories. Bending stresses are where E = 6.89GPa (24) increasing with respect to an increase in radius of curva- ture i.e. bending stress is maximum at R=∞. Figure 4 shows Tables 2 through 5 show the normalized displacements through-the-thickness distributions of bending stresses in and stresses of laminated composite and sandwich curved laminated and sandwich curved beams. It is observed that beams subjected to uniform load. In the case of laminated bending stresses are maximum in 0 layer and minimum composite beams, all layers are of equal thickness. How- in 90 layer. ever, in the case of sandwich beams, each face sheet is of Numerical values of normalized transverse shear thickness 0.1h and the core is of thickness 0.8h where h is stresses in laminated and sandwich curved beams sub- the overall thickness of the beam. Laminated composite jected to uniform load are compared in Table 5 and through- beams are made up of material M1. In the case of sand- the-thickness distributions are plotted in Figure 5. When wich beams, each face sheet is made up of material M1 and transverse shear stresses are obtained using the constitu- the core is made up of material M2. The numerical results tive relations (CR), it shows a discontinuity at the layer of straight beams are compared with those presented by interfaces which is practically not acceptable. Therefore, Reddy [6], Kant et al. [10], and Sayyad and Ghugal [37]. The transverse shear stresses are recovered using direct integra- normalized displacement and stresses are obtained for L/h tion (DI) of equilibrium equations of the theory of elasticity = 4, 10 and R/h=5, 10, 20. The straight beam (R=∞) results to achieve continuity at the layer interfaces. are compared with previously published results. Table 1 shows the comparison of non-dimensional h/2 (︂ )︂ ∫︁ ∂σ vertical displacement for laminated composite and sand- k x τ = − dz + C xz ∂x wich beams curved in elevation. Examination of Table −h/2 1 reveals that the present results are in good agreement where integration constants are determined after im- with those presented by Reddy [6], Kant et al. [10], and posing boundary conditions of top, bottom surfaces and Sayyad and Ghugal [37] when R=∞. The minimum value continuity at the layer interface. Numerical results are in of non-dimensional vertical displacement is observed for 0 0 0 good agreement with previously published results of the 0 /90 /0 due to the absence of extension-bending cou- straight beam (R=∞). It is observed that the transverse pling stiffness. Also, it is observed that the non-dimensional shear stresses are more or less the same for all curvature vertical displacement is maximum for deep curvature and values. Figure 5 shows that traction-free conditions are sat- minimum for shallow curvature i.e. vertical displacement isfied along with continuity of shear stresses at the layer decreases with respect to an increase in radius of curvature. interface. Transverse shear stresses are found maximum in Figure 2 shows the through-the-thickness distribution of 0 layer. vertical displacement. Due to the consideration of thick- ness stretching, i.e. the effect of transverse normal strain, vertical displacement is not constant through the thickness. In the well-known theory of Reddy [6] also this effect is 5 Conclusions neglected. Table 3 shows a comparison of normalized axial dis- In the present study, a higher order shear and normal de- placement of laminated and sandwich curved beams sub- formation theory is developed and applied to investigate jected to uniform load. The present results are in good agree- the static analysis of laminated composite and sandwich ment with other theories for R=∞. Through-the-thickness beams curved in elevation subjected to uniform load. The distributions for all lamination schemes are plotted in Fig- present theory considers the effects of both transverse shear ure 3. From the g fi ures is observed that the axial displace- and normal deformations. A simply-supported boundary 0 0 ment is zero at z=+0.369h for 0 /90 scheme, however, in condition is analyzed using Navier’s solution technique. A 0 0 0 0 0 the case of 0 /90 /0 and 0 /Core/0 lamination schemes, close agreement with other theories for straight beams is ob- On the deformation of laminated composite and sandwich curved beams | 9 Table 2: Normalized vertical displacement (w ¯ ) of simply-supported laminated composite and sandwich curved beams. R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 5.8578 3.5166 12.0776 3.7370 1.1345 3.0862 10 Present 5.8565 3.5151 12.0777 3.7381 1.1346 3.0863 20 Present 5.8571 3.5151 12.0777 3.7381 1.1346 3.0863 ∞ Present 5.8578 3.5151 12.0777 3.7390 1.1346 3.0863 Reddy [6] 5.5900 3.3680 12.4550 3.6970 1.0980 3.0920 Kant et al. [10] 5.9000 3.6050 13.7500 3.7440 1.1710 3.3300 Sayyad and Ghugal [37] 5.5230 3.3940 12.4630 3.6830 1.1060 3.1000 Figure 2: Through-the-thickness variations of vertical displacement for laminated and sandwich beams curved in elevation (L/h=4, R/h=5). Table 3: Normalized axial displacement (u ¯) of simply-supported laminated composite and sandwich curved beams. R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 5.7738 3.5159 9.8338 248.0907 78.7989 205.205 10 Present 4.0330 2.5155 6.3123 144.7630 47.4498 119.291 20 Present 3.0898 1.9712 4.3962 88.5963 30.3921 72.5438 ∞ Present 2.0979 1.3976 2.3764 29.4110 12.4121 23.2686 Reddy [6] 2.2580 1.1620 2.3650 29.8050 11.7340 23.2400 Sayyad and Ghugal [37] 2.2680 1.1950 2.3910 29.7990 11.8910 23.0300 Figure 3: Through-the-thickness variations of axial displacement for laminated and sandwich beams curved in elevation (L/h=4, R/h=5). 10 | Avhad and Sayyad Table 4: Normalized bending stresses (σ ¯ ) of simply-supported laminated composite and sandwich curved beams. R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 40.9198 22.2994 40.1524 219.8111 88.1728 169.2986 10 Present 40.9294 22.3754 40.1892 219.8409 88.1083 169.1772 20 Present 40.9363 22.4434 40.2073 219.8508 88.0762 169.1176 ∞ Present 40.9352 22.5115 40.2254 219.8585 88.0442 169.0576 Reddy [6] 40.2390 19.6710 39.1610 221.0170 85.0300 168.1300 Kant et al. [10] 36.6780 21.5680 43.4880 217.3300 89.1200 172.6000 Sayyad and Ghugal [37] 40.4970 20.2880 39.5690 221.4020 85.6640 168.7600 Table 5: Normalized shear stress (τ ¯ ) of simply-supported laminated composite and sandwich curved beams. xz R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 3.6812 1.9855 2.0462 8.9175 5.6263 5.8645 10 Present 3.6832 1.9852 2.0462 8.9175 5.6255 5.8645 20 Present 3.6851 1.9850 2.0462 8.9178 5.6249 5.8645 ∞ Present 3.6860 1.9848 2.0462 8.9181 5.6243 5.8645 Reddy [6] 5.0240 1.8310 2.6620 11.5440 6.0690 5.2870 Kant et al. [10] 3.8480 2.4880 2.2800 10.7380 6.1500 5.2400 Sayyad and Ghugal [37] 5.0780 1.7610 2.7970 11.5860 6.0160 5.2650 Figure 4: Through-the-thickness variations of bending stresses for laminated and sandwich beams curved in elevation (L/h=4, R/h=5)). 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On the deformation of laminated composite and sandwich curved beams

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Curved and Layer. Struct. 2022; 9:1–12 Research Article Pravin V. Avhad and Atteshamuddin S. Sayyad* On the deformation of laminated composite and sandwich curved beams https://doi.org/10.1515/cls-2022-0001 stiffness-to-weight ratios as well as the high strength-to- Received Jan 13, 2021; accepted Jun 09, 2021 weight ratio. Therefore, it is significantly important to in- vestigate the accurate static behaviour of curved beams Abstract: Plenty of research articles are available on the subjected to static loading. Various laminated beam theo- static deformation analysis of laminated straight beams us- ries are available in the literature which can be extended for ing refined shear deformation theories. However, research the analysis of curved beams such as classical beam theory on the deformation of laminated curved beams with sim- [1], Timoshenko beam theory [2], Higher order beam theo- ply supported boundary conditions is limited and needs ries [3, 4, 5], etc. In this section literature on various beam more attention nowadays. With this objective, the present theories available in the literature and their applications study deals with the static analysis of laminated compos- in various problems are reviewed. ite and sandwich beams curved in elevation using a new Reddy [6] has developed a well-known third order shear quasi-3D polynomial type beam theory. The theory consid- deformation theory for the analysis of laminated compos- ers the effects of both transverse shear and normal strains, ite beams which is further extended by Khdeir and Reddy i.e. thickness stretching effects. In the present theory, axial [7, 8] for the analysis of cross-ply laminated beams/arches. displacement has expanded up to the fifth-order polyno- Kant and Manjunath [9] have presented a static analysis mial in terms of thickness coordinates to effectively account of symmetric and unsymmetric laminated composite and for the effects of curvature and deformations. The present sandwich beams based on a C continuity finite element theory satisfies the zero traction boundary condition on using various refine higher order beam theories. Kant et the top and bottom surfaces of the beam. Governing differ- al. [10] have developed semi-analytical elasticity bending ential equations and associated boundary conditions are solutions for laminated composite beams. Li et al. [11] have established by using the Principal of virtual work. Navier’s studied the free vibration analysis of laminated compos- solution technique is used to obtain displacements and ite beams of different boundary conditions using various stresses for simply supported beams curved in elevation higher-order beam theories and spectral finite element and subjected to uniformly distributed load. The present method which is also used by Nanda and Kapuria [12] for the results can be benefited to the upcoming researchers. wave propagation analysis of laminated composite curved Keywords: fifth-order polynomial, laminated and sand- beams. Luu et al. [13] investigated non-dimensional deflec- wich, curved beams, static analysis tion and critical buckling loads of shear deformable lam- inated composite curved beams using the NURBS-based isogeometric method. Ye et al. [14] and Mohamad et al. [15] studied vibration analysis of laminated composite curved 1 Introduction beams of various boundary conditions. Jianghua et al. [16] have presented the static and vibration analysis of lami- Laminated composite curved beams/arches are widely nated composite curved beams using a domain decomposi- used in aerospace, automobile, ships, civil and mechani- tion approach. Zenkour [17] has developed a new shear and cal industries due to their superior properties such as high normal deformation theory for the static analysis of cross- ply laminated composite and sandwich beams. Karama et al. [18] have developed an exponential shear deforma- tion theory for the bending, buckling, and free vibration Pravin V. Avhad: Research Scholar, Department of Civil Engineer- ing, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune analysis of multi-layered laminated composite beams. Pio- University, Kopargaon-423603, Maharashtra, India van et al. [19] have analyzed composite thin-walled curved *Corresponding Author: Atteshamuddin S. Sayyad: Department beams with open and closed cross-sections. Hajianmaleki of Civil Engineering, SRES’s Sanjivani College of Engineering, Savit- and Qatu [20] have applied Timoshenko beam theory for ribai Phule Pune University, Kopargaon-423603, Maharashtra, India, the static and free vibration analysis of generally laminated E-mail: attu_sayyad@yahoo.co.in Open Access. © 2022 Avhad and Sayyad, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 2 | Avhad and Sayyad deep curved beams with boundary conditions. Tessler et curved panels using various single layer equivalent and al. [21] have presented the analysis of laminated compos- layerwise theories based on Carrera’s unified formulation. ites and sandwich beams using a zig-zag theory. Carrera Tornabene et al. [44] presented natural frequencies of sev- et al. [22] have investigated the static response of lami- eral doubly-curved shells with variable thickness using var- nated composite beams by using various polynomial and ious higher-order equivalent single layer theories including non-polynomial beam theories. Dogruoglu and Komurcu the Murakami’s function to capture the zig-zag effect. Pa- [23] have presented the static analysis of planar curved gani et al. [45] applied various classical and refined plate beams by using a finite element method. Bhimaraddi et theories for the analysis of laminates and sandwich struc- al. [24] have investigated the static and dynamic analysis tures using finite element method. Carrera et al. [46, 47] of isoparametric thick laminated curved beams by using a have developed a new finite element for the analysis of finite element method. Bouclier et al. [25] have developed metallic and composite plates and shells. shear and membrane locking free isogeometric formula- tion for the analysis of curved beams. Fraternali and Bilotti [26] have presented the stress analysis of composite curved 1.1 Scientific soundness of the topic beams using one-dimensional theory and finite element method. Kurtaran [27] applied a differential quadrature 1) Beams curved in elevation are widely used in many method for the static and transient analysis of functionally engineering industries such as arch type bridges, graded curved beams using first order shear deformation chain links, crane hooks, pipe bends, and curved theory. Matsunaga [28] has investigated the displacements segments of machine tool frames. Curved beam seg- and stresses of the laminated and sandwich curved beams ments used in these structures are often subjected to under mechanical/thermal loading by using higher order the static forces where it needs to analyze and design theory. Qian et al. [29] have studied the displacements and accurately. This forced the authors to consider this stresses of the laminated arches, under thermal load by topic as an area of research. using two-dimensional thermo-elastic theory. Malekzadeh 2) Based on the aforementioned literature review, it is [30] has presented the static analysis of thick laminated observed that a lot of research has been carried out by deep circular arches with various boundary conditions by researchers on static analysis of straight beams using using the two-dimensional theory of elasticity. Casari and higher-order shear deformation theories. However, Gornet [31] have presented the static analysis of composite research on static analysis of curved beams is limited curved sandwich beams under thermo-mechanical load. in the literature. Kress et al. [32] have investigated the stress distributions of 3) From the literature review, it is also observed that, in thick and singly curved laminates by using finite element the higher order theories available in the literature, method. Marur and Kant [33] have analyzed the static be- thickness coordinates are expanded up to third order haviour of laminated arches using transverse shear and nor- only. However, for the accurate prediction of static mal deformation theory. Sayyad and his co-authors’ [34-38] behaviour laminated curved beams, it is necessary to have presented various polynomial, and non-polynomial expand thickness coordinates up to minimum fifth type beam theories for the static, vibration, and buckling order. Therefore, in this paper, a fifth-order shear analysis of isotropic, functionally graded, laminated, and deformation theory is developed for the curved beam. sandwich beams. Recently, Sayyad and Ghugal [39] pre- 4) It is recommended by Carrera et al. [22] that the trans- sented static analysis sandwich curved beams using a si- verse normal strain plays an important role in predict- nusoidal shear deformation theory considering the effects ing the accurate bending behavior of thick composite of thickness stretching. Fazzolari et al. [40] presented a beams. Therefore, the present theory also considers comparative study between two computational techniques the effects of transverse normal strain i.e. thickness (hierarchical ritz formulation and generalized differential stretching effects for the modeling and analysis of quadrature) to evaluate the natural frequencies of com- curved laminated beams. posite beams and shells. Dimitri et al. [41] also presented 5) In this paper, the static analysis of laminated com- the analysis of various curved segments using generalized posites and sandwich beams curved in elevation is differential quadrature method. Fantuzzi and Tornabene analyzed under uniform load. [42] presented analysis of arbitrarily shaped plates using 6) The present theory and the approach can be extended strong finite element formation and differential quadrature to analyze the curved beams subjected to dynamic method. Tornabene et al. [43] have presented the through- loading, thermal loading, nonlinear problems, and the-thickness distribution of strains and stresses for doubly- On the deformation of laminated composite and sandwich curved beams | 3 the curved beams accounting for variable radii of where (u, u ) and (w, w ) are the x- and z- directional 0 0 curvatures and thicknesses. displacements, respectively; ϕ , ψ , ϕ , ψ are the un- x x z z known rotations to be determined. In the case of plates, the displacement model of the theory is modeled by con- sidering displacement components in all three directions 2 Mathematical formulation of the of the plate. But, in the case of beam, rotation in the y- present theory for curved beam direction is assumed as zero and the width of the beam is taken as unity in the y-direction. Hence, the displacement Consider a laminated curved beam with radius of curvature in the y-direction is neglected. For the curved beam, the R in the Cartesian coordinate system as shown in Figure non-zero strain components are determined using the fol- 1. The beam has curved length L, and rectangular cross- lowing strain-displacement relationship for the theory of section b×h where b is the width (b=1) and h is the total elasticity. thickness (0≤ x ≤ L; -b/2≤ y ≤ b/2; -h/2≤ z ≤ h/2). The beam is composed of N number of layers perfectly bonded together ∂u w ∂w ∂u ∂w u ε = + , ε = , 𝛾 = + − (2) x xz and made up of fibrous composite materials. The bond ∂x R ∂z ∂z ∂x R between the two adjacent layers is of zero thickness. The Using displacements from Eq. (1), the following are the beam is subjected to uniform load (q). expressions for normal strains and transverse shear strain at any point of the beam. ∂u ∂ w ∂ϕ 0 0 ε = − z + f (z) + x 1 ∂x ∂x ∂x ∂ψ w ϕ ψ x z z 0 ′ ′ + f (z) + + f (z) + f (z) 1 2 ∂x R R R (3) ′′ ′′ ε = f (z)ϕ + f (z)ψ z z z 1 2 ∂ϕ ∂ψ z z ′ ′ ′ ′ Figure 1: Geometry ad coordinates system of laminated curved 𝛾 = f (z)ϕ + f (z)ψ + f (z) + f (z) xz x x 1 2 1 2 ∂x ∂x beam. where Following are the kinematic assumptions made for the (︂ )︂ (︂ )︂ 3 5 development of the displacement field of the present theory. 4z 16z f (z) = z − , f (z) = z − 1 2 2 4 3h 5h 1) The x-directional displacement contains the extension, (4) (︂ )︂ (︂ )︂ 2 4 bending, and shear components. Theories that consider the 4z 16z ′ ′ f (z) = 1 − , f (z) = 1 − 1 2 2 4 effects of bending deformation, shear deformation, and nor- h h mal deformation (thickness-stretching) are called quasi-3D The prime ( ) indicates the derivative with respect to theories. 2) The extension component is presented interms the z-coordinate. A generalized Hooke’s law is used to de- of the radius of curvature. 3) The z-directional displacement th termine the stresses in the k layer of laminated curved considered the effects of transverse normal deformations beams. i.e. ε ≠ 0. 4) The present theory consists of fifth order poly- nomial to account for the traction-free boundary conditions ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎪ ⎪ ⎪ ⎪ σ C C 0 ε x x at the top and bottom surfaces of the beam. Based on the ⎨ ⎬ 11 13 ⎨ ⎬ ⎢ ⎥ σ = C C 0 ε (5) z ⎣ ⎦ z before mentioned assumptions, the displacement field of 13 33 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ τ 0 0 C 𝛾 xz xz the present theory is written as follows: 55 where C are the reduced stiffness coefficients. ij (︁ )︁ z ∂w u(x, z) = 1 + u (x) − z + R ∂x (︂ )︂ (︂ )︂ (︂ )︂ (︂ )︂ E E 3 5 1 3 4z 16z C = , C = , 11 33 + z − ϕ (x) + z − ψ (x) x x 1 − µ µ 1 − µ µ 2 4 13 31 13 31 3h 5h (︂ )︂ (6) (︂ )︂ (1) µ E 2 13 1 4z C = , C = G 13 55 13 w(x, z) = w (x) + 1 − ϕ (x)+ 0 z 1 − µ µ 13 31 (︂ )︂ where E and E represent elastic moduli, G represent 16z 1 3 13 + 1 − ψ (x) h shear modulus and µ , µ represent Poisson’s ratios. Ax- 13 31 4 | Avhad and Sayyad ial force, bending moment and shear force resultants as- (︂ )︂ sociated with the present theory can be derived using the 2 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z Q = AG + − AN + AT + AY + BC + 13 13 13 13 13 following relations. 2 ∂x R ∂x ∂x ∂x R + BF + BH ϕ + BI ψ 13 33 z 33 z h/2 h/2 (︂ )︂ ∫︁ ∫︁ ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z b Q = AH + − AO + AU + AZ + BD + 13 13 13 13 13 N = σ dz, M = σ zdz, 2 x x x x ∂x R ∂x ∂x ∂x R −h/2 −h/2 z + BG + BI ϕ + BJ ψ 13 33 z 33 z h/2 h/2 (︂ )︂ (︂ )︂ ∫︁ ∫︁ ∂ϕ ∂ψ z z 1 2 s1 s2 Q = BK ϕ + + BL ψ + and Q = 55 x 55 x M = σ f (z)dz, M = σ f (z)dz xz xz x x 1 x x 2 ∂x ∂x (︂ )︂ (︂ )︂ −h/2 −h/2 ∂ϕ ∂ψ z z = BL ϕ + + BM ψ + 55 x 55 x h/2 h/2 ∂x ∂x ∫︁ ∫︁ (8) 1 ′ 2 ′ V = σ f (z)dz, V = σ f (z)dz, x x x x 1 2 Integration constants appeared in Eq. (8) are defined −h/2 −h/2 (7) as follows: h/2 ∫︁ 1 ′′ Q = σ f (z)dz z 1 AA , AB , AC , AD , AE , AF = ( ) 11 11 11 11 11 11 −h/2 h/2 h/2 h/2 ∫︁ ∫︁ ∫︁ [︀ ]︀ ′ ′ 2 ′′ 1 ′ C 1, z, f (z), f (z), f (z), f (z) dz 11 1 2 1 2 Q = σ f (z)dz, Q = τ f (z)dz, z z 2 xz xz 1 −h/2 −h/2 −h/2 h/2 (AI , AJ , AK , AL , AM ) = ∫︁ 11 11 11 11 11 2 ′ Q = τ f (z)dz h/2 xz xz 2 ∫︁ [︀ ]︀ ′ ′ −h/2 C z, f (z), f (z), f (z), f (z) zdz 11 1 2 1 2 −h/2 Substitution of stresses from Eq. (5) into Eq. (7) one can AG , AH = ( ) 13 13 get the final expressions for stress resultants as follows: h/2 ∫︁ [︀ ]︀ (︂ )︂ n ′′ C f (z), f (z) dz AN , AO = ∂u w ∂ w ∂ϕ ∂ψ ϕ ( ) 0 0 0 x x z 13 1 2 13 13 N = AA + − AB + AC + AD + AE + 11 11 11 11 11 ∂x R ∂x ∂x ∂x R −h/2 h/2 + AF + AG ϕ + AH ψ 11 13 z 13 z ∫︁ R [︀ ]︀ n n (︂ )︂ C f (z), f (z) zdz 2 (9) 13 1 2 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z M = AB + − AI + AJ + AK + AL + 11 11 11 11 11 ∂x R ∂x ∂x ∂x R −h/2 (AP , AQ , AR , AS ) = 11 11 11 11 + AM + AN ϕ + AO ψ 11 13 z 13 z (︂ )︂ h/2 2 ∫︁ ∂u w ∂ w ∂ϕ ∂ψ ϕ x x z 0 0 0 s1 [︀ ]︀ M = AC + − AJ + AP + AQ + AR + 11 11 11 11 11 ′ ′ C f (z), f (z), f (z), f (z) f (z)dz ∂x R ∂x ∂x ∂x R 11 1 2 1 1 2 −h/2 + AS + AT ϕ + AU ψ 11 13 z 13 z (︂ )︂ AT , AU = ( ) 2 13 13 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z s2 M = AD + − AK + AQ + AV + AW + 11 11 11 11 11 2 h/2 ∂x R ∂x ∂x ∂x R ∫︁ [︀ ]︀ ′′ ′′ z C f (z), f (z) f (z)dz 13 1 2 1 + AX + AY ϕ + AZ ψ 11 13 z 13 z (︂ )︂ −h/2 ∂u w ∂ w ∂ϕ ∂ψ ϕ 0 0 0 x x z V = AE + − AL + AR + AW + BA + 11 11 11 11 11 2 AV , AW , AX = ( ) 11 11 11 ∂x R ∂x ∂x ∂x R z h/2 ∫︁ + BB + BC ϕ + BD ψ 11 13 z 13 z [︀ ]︀ ′ ′ C f (z), f (z), f (z) f (z)dz (︂ )︂ 11 2 1 2 2 ∂u w ∂ w ∂ϕ ∂ψ ϕ x x z 0 0 0 V = AF + − AM + AS + AX + BB + 11 11 11 11 11 −h/2 ∂x R ∂x ∂x ∂x R + BE + BF ϕ + BG ψ 11 13 z 13 z R On the deformation of laminated composite and sandwich curved beams | 5 AY , AZ = ( ) 13 13 h/2 ∫︁ L L (︂ )︂ (︂ )︂ ∫︁ ∫︁ [︀ ]︀ 2 ′′ ′′ ∂δu δw ∂ δw 0 0 b 0 C f (z), f (z) f (z)dz 13 1 2 2 N dx + N − M dx+ x x x ∂x R ∂x −h/2 0 0 (︂ )︂ BA , BB = ∫︁ ( ) 11 11 ∂δϕ s x 1 M + Q δϕ dx x x= x h/2 ∫︁ ∂x [︀ ]︀ ′ ′ ′ C f (z), f (z) f (z)dz 11 1 2 1 (11) (︂ )︂ ∫︁ −h/2 ∂δψ s2 x 2 + M + Q δψ dx x xz x ∂x (BC , BD ) = 13 13 h/2 ∫︁ (︂ )︂ ∫︁ [︀ ]︀ ′′ ′′ 1 δϕ ∂δϕ 1 z 1 1 z C f (z), f (z) f (z)dz 1 2 1 + V + Q δϕ + Q dx x z z x= R ∂x −h/2 BF , BG = ( ) 13 13 ∫︁ (︂ )︂ h/2 δψ ∂δψ ∫︁ z z 2 2 2 + V + Q δψ + Q − [︀ ]︀ z x z xz ′′ ′′ ′ R ∂x C f (z), f (z) f (z)dz, BE = 13 1 2 2 11 −h/2 ∫︁ (︀ )︀ h/2 ′ ′ ∫︁ q δw + f (z)δϕ + f (z)δψ = 0 z z 0 1 2 [︀ ]︀ C f (z) dz 11 2 −h/2 Integrating Eq. (11) by parts, collecting the coeffi- BH , BI = ( 33 33) cients of δu , δw , δϕ , δψ , δϕ , δψ and setting 0 0 x x z z h/2 them equal to zero, the following six governing differential ∫︁ [︀ ]︀ n n ′′ equations are obtained. C f (z), f (z) f (z) · dz BJ = 33 33 1 2 1 −h/2 ∂N δu : = 0 h/2 0 ∫︁ ∂x [︀ ]︀ ′′ 2 b C f (z) dz 33 2 ∂ M N x x δw : − + q = 0 −h/2 ∂x R s1 ∂M x 1 (BK , BL ) = 55 55 δϕ : − Q = 0 xz ∂x (12) h/2 ∫︁ s2 [︀ ]︀ ∂M x 2 ′ ′ ′ δψ : − Q = 0 x xz C f (z), f (z) f (z)dzBM = 55 1 2 1 55 ∂x 1 1 −h/2 ∂Q V xz x 1 ′ δϕ : − − Q + qf (z) = 0 z 1 h/2 ∂x R ∫︁ [︀ ]︀ 2 2 ∂Q V C f (z) dz xz x 2 ′ δψ : − − Q + qf (z) = 0 z z ∂x R −h/2 The boundary conditions at the supports (x=0 and x=L) The principle of virtual work is employed to derive the are presented in Table 1. governing equation and boundary condition associated Using Eq. (8), in terms of unknown variables, Eq. (12) with the present theory. can be written as L h/2 L ∫︁ ∫︁ ∫︁ (︂ )︂ 2 3 ∂ u 1 ∂w ∂ w 0 0 0 (σ δε + σ δε + τ δ𝛾 ) dzdx − qδwdx = 0 δu : −AA + + AB − x x z z xz xz 0 11 11 2 3 ∂x R ∂x ∂x 0 0 −h/2 2 2 ∂ ϕ ∂ ψ AE ∂ϕ x x 11 z (13) (10) AC − AD − 11 11 2 2 R ∂x ∂x ∂x where δ is the variational operator. Eq. (10) can be writ- AF ∂ψ ∂ϕ ∂ψ 11 z z z − − AG − AH = 0 ten as follows using strains from Eqs. (3) and stress resul- 13 13 R ∂x ∂x ∂x tants from Eq. (8). 6 | Avhad and Sayyad Table 1: Natural and essential boundary conditions associated with the present theory. (︂ )︂ (︂ )︂ 1 ∂u w ∂u w 0 0 0 0 δϕ : AE + + AG + − z 11 13 R ∂x R ∂x R Natural Essential 2 2 2 AL ∂ w ∂ w AR ∂ ϕ ∂ϕ x x 11 0 0 11 − AN + + AT given N |x = 0, L or u |x = 0, L x 0 13 13 2 2 2 R ∂x ∂x R ∂x ∂x b ∂w given M |x = 0, L or |x = 0, L ∂x ∂ϕ AW ∂ψ ∂ψ ∂ψ x x x x − BK + + AY − BL ∂M 55 13 55 given |x = 0, L or w |x = 0, L ∂x R ∂x ∂x ∂x ∂x (17) s1 BA BC BB given M |x = 0, L or ϕ |x = 0, L 11 13 11 + ϕ + 2 ϕ + BH ϕ + ψ − z 33 Z Z Z s2 R R R given M |x = 0, L or ψ |x = 0, L 1 ∂ ϕ BD BF Z 13 13 given Q |x = 0, L or ϕ |x = 0, L xz BK + ψ + ψ + BI ψ − 55 Z 33 Z R R 2 ∂x given Q |x = 0, L or ψ |x = 0, L xz ∂ ψ Z ′ BL = qf (z) ∂x (︂ )︂ (︂ )︂ (︂ )︂ (︂ )︂ 1 ∂u w ∂u w 3 2 0 0 0 0 1 ∂u w ∂ u 2 ∂ w δψ : AF + + AH + − 0 0 0 0 z 11 13 δw : AA + − AB + R ∂x R ∂x R 0 11 11 2 3 2 R ∂x R ∂x R ∂x 2 2 2 AM ∂ w ∂ w AS ∂ ϕ ∂ϕ 4 3 11 0 0 11 x x ∂ w ∂ ϕ − AO + + AU 0 13 13 2 2 2 + AI − AJ + R R ∂x ∂x ∂x ∂x 11 11 4 3 ∂x ∂x ∂ϕ AX ∂ψ ∂ψ ∂ψ x 11 x x x 3 2 − BL + + AZ − BM AD ∂ψ ∂ ψ AE AL ∂ ϕ 55 13 55 11 x x 11 11 z ∂x R ∂x ∂x ∂x − AK + ϕ + − 11 z 3 2 R ∂x ∂x R R ∂x BB BF BD 11 13 13 2 + ϕ + ϕ + ϕ + BI ϕ − z z Z Z 33 ∂ ϕ AF AG z 11 13 R R R AN + ψ + ϕ z z 2 2 (18) ∂x R R 2 2 AH AM ∂ ψ ∂ ψ z z 13 11 + ψ − − AO = q 2 2 R R ∂x ∂x ∂ ϕ BE BG Z 11 13 (14) − BL + ψ + 2 ψ + BJ ψ − 55 Z Z 33 Z 2 2 ∂x R ∂ ψ (︂ )︂ Z ′ 2 3 BM = qf (z) ∂ u 1 ∂w ∂ w 2 0 0 0 ∂x δϕ : −AC + + AJ − x 11 11 2 3 ∂ x R ∂x ∂x 2 2 ∂ ϕ ∂ ψ x z AP + BK ϕ − AQ + BL ψ 11 55 x 11 55 x 2 2 ∂x ∂x (15) 3 The Navier solution AR ∂ϕ ∂ϕ ∂ϕ z z 11 Z − − AT + BK − 13 55 R ∂x ∂x ∂x Analytical solution of above mentioned six governing equa- AS ∂ψ ∂ψ ∂ψ z z z − AU + BL = 0 13 55 tions is obtained using Navier’s solution technique. The R ∂x ∂x ∂x solution is obtained to investigate the static behaviour (︂ )︂ of simply-supported laminated composites and sandwich 2 3 ∂ u 1 ∂w ∂ w 0 0 0 δψ : −AD + + AK − x 11 11 beams curved in elevation. Following are the boundary 2 3 ∂ x R ∂x ∂x conditions at the simple supports of the curved beams. 2 2 ∂ ϕ ∂ ψ x z AQ + BL ϕ − AV + BM ψ 11 55 x 11 55 x 2 2 ∂x ∂x (16) AW ∂ϕ ∂ϕ ∂ϕ b s1 s2 z z z N = 0, w = 0, M = 0, M = 0, M = 0ϕ = 0, ψ = 0 x z z − − AY + BL − 0 x x x 13 55 R ∂x ∂x ∂x at x = 0 and x = L AX ∂ψ ∂ψ ∂ψ z z 11 Z − AZ + BM = 0 13 55 (19) R ∂x ∂x ∂x According to Navier’s technique, unknown variables are expanded in the single trigonometric series satisfying simply-supported boundary conditions stated in Eq. (19). The following trigonometric form is assumed for the un- known variables. On the deformation of laminated composite and sandwich curved beams | 7 (︂ )︂ ∞ ∞ AD (︁ )︁ (︁ )︁ 11 3 ∑︁ ∑︁ mπx mπx K = K = − α − AK α , 24 42 11 u = u cos , w = w sin R 0 m 0 m L L (︂ )︂ m=1 m=1 AE AG AL 11 13 11 2 2 ∞ ∞ K = K = + + α + AN α , (︁ )︁ (︁ )︁ 25 52 13 ∑︁ ∑︁ mπx mπx R R ϕ = ϕ cos , ϕ = ϕ sin x xm z zm (︂ )︂ L L AF AH AM m=1 m=1 11 13 11 2 2 K = K 2 = + + α + AO α , 26 62 13 ∞ ∞ R R R (︁ )︁ (︁ )︁ ∑︁ ∑︁ mπx mπx (︁ )︁ (︁ )︁ ψ = ψ cos , ψ = ψ sin x xm z zm 2 2 L L K = AP α + BK , K = K = AQ α + BL , 33 11 55 34 43 11 55 m=1 m=1 (︂ )︂ (20) AR K = K = − α − AT α + BK α , 35 53 13 55 where u , w , ϕ , ψ , ϕ , ψ are the un- m m xm xm zm zm (︂ )︂ known coefficients to be determined; the transverse uni- AS K = K = − α − AU α + BL α , 13 55 form load (q) acting on the beam is also expanded in the 36 63 (︁ )︁ single trigonometric form: K = AV α + BM , 44 11 55 (︂ )︂ (︁ )︁ (︂ )︂ ∑︁ 4q mπx AW q = sin (21) K = K = − α − AY α + BL α , 45 54 13 55 mπ L m=1 (23) where q is the maximum intensity of uniform load. By substituting unknown variables from Eq. (20) and trans- (︂ )︂ AX verse load from Eq. (21) into governing equations (13)-(18), K = K = − α − AZ α + BM α , 46 64 13 55 the following equation is derived. (︂ )︂ BA BC 11 13 2 K = + 2 + BH + BK α 55 33 55 R R ⎡ ⎤ ⎧ ⎫ K K K K K K u 11 12 13 14 15 16 ⎪ 0 ⎪ ⎪ ⎪ K = K = 65 56 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ (︂ )︂ K K K K K K w 21 22 23 24 25 26 0 ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ BB BD BF ⎢ ⎥ ⎨ ⎬ 11 13 13 2 K K K K K K ϕ = + + + BI + BL α ⎢ 31 32 33 34 35 36 ⎥ x 33 55 ⎢ ⎥ × = R R R ⎢ ⎥ ⎪ ⎪ K K K K K K ψ (︂ )︂ 41 42 43 44 45 46 x ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ BE BG 11 13 2 ⎣ ⎦ ⎪ ⎪ K K K K K K ⎪ ϕ ⎪ K = + 2 + BJ + BM α 51 52 53 54 55 56 z 66 33 55 ⎪ ⎪ ⎩ ⎭ R R K K K K K K ψ 61 62 63 64 65 66 z (22) ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ 4 Illustrative problems ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 4q 0 In this section, static analysis of simply supported lami- mπ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ nated composites and sandwich beams curved in elevation ⎪ ⎪ ⎪ ⎪ f (z) ⎪ ⎪ ⎪ ⎪ is presented to prove the efficiency and accuracy of the ⎩ ⎭ f (z) present theory. The numerical results for the static analysis where, of laminated curved beams are not available in the litera- ture, therefore, the present theory is validated with straight (︂ )︂ beam results available in the literature. Three types of lami- AA 2 3 0 0 0 0 0 0 0 K = AA α , K = K = − α − AB α , 11 11 12 21 11 nation schemes (0 /90 , 0 /90 /0 , 0 /core/0 ) have been (︁ )︁ solved in the present study. The following material prop- 2 2 K = K = AC α , K = K = AD α , 13 31 11 14 41 11 erties have been used to present the numerical results for (︂ )︂ laminated curved beams. AE K = K = − α − AG α , 15 51 13 M1:E = 172.4 GPa, E = 6.89 GPa, G = 3.45 GPa, G = 1.378 1 3 13 31 AF K = K = − α − H α, 16 61 13 GPa, =0.25. (︂ )︂ M2:E = 0.276 GPa, E = 3.45 GPa, G = 3.45 GPa, G = 1.378 AA AB 1 3 13 31 11 4 11 2 K = + AI α + 2 α , 22 11 R R GPa, =0.25. (︂ )︂ AC 11 3 K = K = − α − AJ α , 23 32 11 The thickness of the beam is assumed as a unity (h=1.0) and other dimensions depend on L/h and R/h ratios. The nu- 8 | Avhad and Sayyad merical results are expressed in the following normalized axial displacement does not change its sign and remains form in Tables 2 through 5. positive throughout the thickness of the beam. Table 4 presents the normalized values of bending (︂ )︂ (︂ )︂ stresses in laminated and sandwich curved beams sub- 3 3 E h L E h h 3 3 w ¯ = 100 w , u ¯ = 100 u 0,− , jected to uniform load. Bending stresses of the top fiber q L 2 q L 2 0 0 (︂ )︂ i.e (z=-h/2) are summarized in Table 4. From the compar- h h h (24) σ ¯ = σ 0,− , τ ¯ = τ (0, 0) ison of results, it is observed that the present results are x x xz xz q L 2 q L 0 0 closely matched with other theories. Bending stresses are where E = 6.89GPa (24) increasing with respect to an increase in radius of curva- ture i.e. bending stress is maximum at R=∞. Figure 4 shows Tables 2 through 5 show the normalized displacements through-the-thickness distributions of bending stresses in and stresses of laminated composite and sandwich curved laminated and sandwich curved beams. It is observed that beams subjected to uniform load. In the case of laminated bending stresses are maximum in 0 layer and minimum composite beams, all layers are of equal thickness. How- in 90 layer. ever, in the case of sandwich beams, each face sheet is of Numerical values of normalized transverse shear thickness 0.1h and the core is of thickness 0.8h where h is stresses in laminated and sandwich curved beams sub- the overall thickness of the beam. Laminated composite jected to uniform load are compared in Table 5 and through- beams are made up of material M1. In the case of sand- the-thickness distributions are plotted in Figure 5. When wich beams, each face sheet is made up of material M1 and transverse shear stresses are obtained using the constitu- the core is made up of material M2. The numerical results tive relations (CR), it shows a discontinuity at the layer of straight beams are compared with those presented by interfaces which is practically not acceptable. Therefore, Reddy [6], Kant et al. [10], and Sayyad and Ghugal [37]. The transverse shear stresses are recovered using direct integra- normalized displacement and stresses are obtained for L/h tion (DI) of equilibrium equations of the theory of elasticity = 4, 10 and R/h=5, 10, 20. The straight beam (R=∞) results to achieve continuity at the layer interfaces. are compared with previously published results. Table 1 shows the comparison of non-dimensional h/2 (︂ )︂ ∫︁ ∂σ vertical displacement for laminated composite and sand- k x τ = − dz + C xz ∂x wich beams curved in elevation. Examination of Table −h/2 1 reveals that the present results are in good agreement where integration constants are determined after im- with those presented by Reddy [6], Kant et al. [10], and posing boundary conditions of top, bottom surfaces and Sayyad and Ghugal [37] when R=∞. The minimum value continuity at the layer interface. Numerical results are in of non-dimensional vertical displacement is observed for 0 0 0 good agreement with previously published results of the 0 /90 /0 due to the absence of extension-bending cou- straight beam (R=∞). It is observed that the transverse pling stiffness. Also, it is observed that the non-dimensional shear stresses are more or less the same for all curvature vertical displacement is maximum for deep curvature and values. Figure 5 shows that traction-free conditions are sat- minimum for shallow curvature i.e. vertical displacement isfied along with continuity of shear stresses at the layer decreases with respect to an increase in radius of curvature. interface. Transverse shear stresses are found maximum in Figure 2 shows the through-the-thickness distribution of 0 layer. vertical displacement. Due to the consideration of thick- ness stretching, i.e. the effect of transverse normal strain, vertical displacement is not constant through the thickness. In the well-known theory of Reddy [6] also this effect is 5 Conclusions neglected. Table 3 shows a comparison of normalized axial dis- In the present study, a higher order shear and normal de- placement of laminated and sandwich curved beams sub- formation theory is developed and applied to investigate jected to uniform load. The present results are in good agree- the static analysis of laminated composite and sandwich ment with other theories for R=∞. Through-the-thickness beams curved in elevation subjected to uniform load. The distributions for all lamination schemes are plotted in Fig- present theory considers the effects of both transverse shear ure 3. From the g fi ures is observed that the axial displace- and normal deformations. A simply-supported boundary 0 0 ment is zero at z=+0.369h for 0 /90 scheme, however, in condition is analyzed using Navier’s solution technique. A 0 0 0 0 0 the case of 0 /90 /0 and 0 /Core/0 lamination schemes, close agreement with other theories for straight beams is ob- On the deformation of laminated composite and sandwich curved beams | 9 Table 2: Normalized vertical displacement (w ¯ ) of simply-supported laminated composite and sandwich curved beams. R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 5.8578 3.5166 12.0776 3.7370 1.1345 3.0862 10 Present 5.8565 3.5151 12.0777 3.7381 1.1346 3.0863 20 Present 5.8571 3.5151 12.0777 3.7381 1.1346 3.0863 ∞ Present 5.8578 3.5151 12.0777 3.7390 1.1346 3.0863 Reddy [6] 5.5900 3.3680 12.4550 3.6970 1.0980 3.0920 Kant et al. [10] 5.9000 3.6050 13.7500 3.7440 1.1710 3.3300 Sayyad and Ghugal [37] 5.5230 3.3940 12.4630 3.6830 1.1060 3.1000 Figure 2: Through-the-thickness variations of vertical displacement for laminated and sandwich beams curved in elevation (L/h=4, R/h=5). Table 3: Normalized axial displacement (u ¯) of simply-supported laminated composite and sandwich curved beams. R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 5.7738 3.5159 9.8338 248.0907 78.7989 205.205 10 Present 4.0330 2.5155 6.3123 144.7630 47.4498 119.291 20 Present 3.0898 1.9712 4.3962 88.5963 30.3921 72.5438 ∞ Present 2.0979 1.3976 2.3764 29.4110 12.4121 23.2686 Reddy [6] 2.2580 1.1620 2.3650 29.8050 11.7340 23.2400 Sayyad and Ghugal [37] 2.2680 1.1950 2.3910 29.7990 11.8910 23.0300 Figure 3: Through-the-thickness variations of axial displacement for laminated and sandwich beams curved in elevation (L/h=4, R/h=5). 10 | Avhad and Sayyad Table 4: Normalized bending stresses (σ ¯ ) of simply-supported laminated composite and sandwich curved beams. R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 40.9198 22.2994 40.1524 219.8111 88.1728 169.2986 10 Present 40.9294 22.3754 40.1892 219.8409 88.1083 169.1772 20 Present 40.9363 22.4434 40.2073 219.8508 88.0762 169.1176 ∞ Present 40.9352 22.5115 40.2254 219.8585 88.0442 169.0576 Reddy [6] 40.2390 19.6710 39.1610 221.0170 85.0300 168.1300 Kant et al. [10] 36.6780 21.5680 43.4880 217.3300 89.1200 172.6000 Sayyad and Ghugal [37] 40.4970 20.2880 39.5690 221.4020 85.6640 168.7600 Table 5: Normalized shear stress (τ ¯ ) of simply-supported laminated composite and sandwich curved beams. xz R/h Theory L/h=4 L/h=10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] [0 /90 ] [0 /90 /0 ] [0 /Core/0 ] 5 Present 3.6812 1.9855 2.0462 8.9175 5.6263 5.8645 10 Present 3.6832 1.9852 2.0462 8.9175 5.6255 5.8645 20 Present 3.6851 1.9850 2.0462 8.9178 5.6249 5.8645 ∞ Present 3.6860 1.9848 2.0462 8.9181 5.6243 5.8645 Reddy [6] 5.0240 1.8310 2.6620 11.5440 6.0690 5.2870 Kant et al. [10] 3.8480 2.4880 2.2800 10.7380 6.1500 5.2400 Sayyad and Ghugal [37] 5.0780 1.7610 2.7970 11.5860 6.0160 5.2650 Figure 4: Through-the-thickness variations of bending stresses for laminated and sandwich beams curved in elevation (L/h=4, R/h=5)). 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Journal

Curved and Layered Structuresde Gruyter

Published: Jan 1, 2022

Keywords: fifth-order polynomial; laminated and sandwich; curved beams; static analysis

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