On the deformation of layered composite arches using exponential shear and normal deformation theory

. In the present study, the stresses and displacements are analyzed for layered composite arches of various lamination schemes subjected to uniformly loading. The present work is majorly highlighted the effects of transverse normal stress and transverse normal strain using exponential shear and normal deformation theory (ESNDT). Governing equations are derived using Hamilton’s principle with application of Navier’s method subjected to simply supported end conditions. Present theory is free from use of any shear correction factor and it satisfies the zero traction free end boundary condition at the top and bottom surfaces of the layered composite arches. In the present work symmetric and antisymmetric lamination scheme have been studied to obtain the numerical results for four layered composite arches and is validated through results available in prior literature.


Introduction
Composite arches are widely used across the world due to their exceptional qualities such as excellent strength, an admirable stiffness-to-weight ratio, and outstanding fatigue-resistance.Now a day, there is enormous demand in architectural view of the structures.Composite arches are most suitable for such innovative constructions.Hence, design of layered composite arches is very useful because of their superior properties compared to available materials.Subsequently, these composite materials are mostly used in mechanical and civil engineering, marine, automobile private sector, aerospace engineering, aircraft, ships, bridges and spacecraft.Curved and arched surfaces are commonly used in advanced architectural structures and military engineering for fighter jet, rocket launchers, antiballistic missiles, aircraft carriers, antitank-mines etc.The vast range of literature is available on straight beams/surfaces subjected to several loadings.Bernoulli-Euler [1,2] established beam theory universally named as classical beam theory (CBT).But authors ignored the effect of thickness stretching.Further study has modified and improved by Timoshenko [3] with accounted the effect of shear deformation but it requires shear correction factor.These theories well-known as Timoshenko beam theory (TBT) or first order shear deformation (FSDT) theory in 1921.Classical and Timoshenko beam theory doesn't capture the transverse normal strain and shear deformation effects.In few decades, it is found that a very limited research is completed on arches or curved beams.
Reddy [4] has developed higher-order theory for composite-laminated plates and obtained results are validated through first-order-shear-deformation theory and 3-D elasticity solutions.Carrera [5] studied thermal-stress analysis for layered and isotropic (homogeneous) plates by adopting thickness stretching effect.Carrera et al. [6] addressed a model for static responses of FGM plates known as variable kinematic model subjected to mechanical loads.Zenkour [7] established the 3-D elasticity solution for sandwich and cross-ply laminates exposed to sinusoidally distributed (SDL) and uniformly distributed (UDL) loads.Kant and Shiyekar [8] presented model on cylindrical bending for piezoelectric-laminates plates using higher order theory.Tornabene [9] and his co-authors investigates the responses of static behavior for curved panels using DQM, differential geometry and Carrera-unified-formulation approach and also investigates recovery of repossession of stresses, shear-strains and transverse-normal through-thethickness variations for functionally graded sandwich panels [10].Sayyad and Ghugal [11] studied cylindrical bending for multilayered composite laminate plates using theory of higher-order.Present exponential shear and normal deformation theory captures excellent structural behavior of layered composite arches due to consideration of effect of transverse normal strain and transverse shear deformation.Present theory will be valuable asset in the research field of aerospace, civil and mechanical structures.

Mathematical-formulation for layered arches
Considered layered composite arches with one end is roller supported and another is hinged support with radius of curvature (R), for a length (L), total thickness (h) and unit width of the arch (b) {0 ≤ x ≤ L; -b/2 ≤ y ≤ b/2; -h/2 ≤ z ≤ h/2} as shown in Fig. 1.

Strain displacement relationship
where, 2 2 ( ) where, 0 0 , , and u w φ ψ be the four unknown functions at mid-plane for composite arches.f(z) and f '(z) is shear and normal deformations.In present theory transverse normal strain is not equal to zero i.e. εz ≠ 0. Shear deformation considered at any point on the arch as stated in Eq. (3).

Hooke's law
Two dimensional Hooke's law is applied layerwise to obtained equations for axial bending stresses and shear stresses with reference axes (x,z) from Eq. ( 5).

Navier's method
This technique is applied for simply supported (SS) boundary condition for layered composite arch under the action of transverse uniformly distributed loading (UDL).
In the form of trigonometric, unknown variables are listed below, 4 sin( ) where, m=Positive integer-variables from odd numbers to the infinity (∞).Substituting the values of unit loading from Eq. ( 22) and, ( ) , , , u w φ ψ unknown variables of Eq. (20) and Eq. ( 21) by putting in the governing equations.Bending stresses for layered composite arches are presented in matrix form of Eq. ( 23) is given below.

Numerical results with discussions
Numerical results are presented in tabular form consists of Table 2 to Table 5 and variations of displacements and stresses for layered composite arches through the thickness are plotted using Grapher as shown in Figure 2 to Figure 5. Present theory analyzed the different lamination schemes for arches viz, symmetric and antisymmetric layered composite arches.The material properties for various arches are given below in Table 1.
Table 1 Materials property for layered composite arches.
Table 2 presented the non-dimensional results for four layered symmetric straight beams when subjected to transverse uniformly distributed loadings.Present results are in good-agreement with earlier published results by Sayyad and Ghugal [11] of normalized stresses and displacements for aspect ratio L/h = 4, 10 and 100.Present numerical results are compared and closely matches with well-known theory of Reddy [4] for transverse and axial deformation at aspect ratio L/h = 100.
Normalized axial displacements and transverse deflections are plotted through the thickness of symmetric layered composite arch as shown in Figure 2.With the application of constitutive relation, it is observed that at the interlaminar surfaces of arches shows two values for shear stress and axial bending through the thickness as shown in Figure 3 for symmetric layered and Figure 5 for antisymmetric layered composite arch.Table 3 presented the normalized transverse ( w ) deformation, axial ( u ) deformation, axial bending stress ( x σ ) and shear stress ( x z τ ) deformation for four-layered symmetric composite arch subjected to uniformly loading for aspect ratio (L/h = 4, 10, 100).It is observed that, maximum non-dimensional value of axial deformation and axial bending stress have noted at the top fibre of layered arch i.e. (z/h = -h/2) due to placing of fibers in 0 0 horizontal direction along the length of arches.While minimum non-dimensional bending stress and axial displacement have been reported at bottom surface of the arch i.e. (z/h = +h/2), it means that fibers are laid in 90 0 direction or perpendicular to zero degree layer of the arch.From Table 3 it is observed that transverse deflection and shear stress deformation remains constants with varying radius of curvature.Table 4 presented numerical results of stresses and displacements for four-layered antisymmetric composite arch.It is found that normalized shear stress and transverse deflection are remains constant with varying radius of curvature for aspect ratio L/h = 4, 10, 100 (R/h = 1.0 to ∞).It is observed that the variations of bending stress through the thickness is nearly equal to zero when the fibers are placed in 90 0 direction of 2 nd and 4 th layer of antisymmetric composite arch under the action of transverse uniformly distributed loading as shown in Figure 5.It is also observed that bending stress is increasing parabolically in 1 st layer from maximum nondimensional to zero and 3 rd layer varying from minimum to maximum non-dimensional of the layered arch.But in case of shear stress is linearly increasing in the 1 st layer, parabolic nature in 2 nd and 3 rd layer and nearly equal to zero variations in the 4 th layer of four layered antisymmetric composite arch subjected to uniformly loading through the thickness variations.
Table 5 shows that, present theory have been great-agreement to Sayyad and Ghugal [11] for stresses and displacements at aspect ratio (L/h = 10).Present numerical results of transverse deformation are closely-matches with well-known theory of Reddy [4] at aspect ratio (L/h = 4, 10 and 100).In the present investigation it is found that transverse shear stress is slightly improved for aspect ratio (L/h = 4, 10 and 100).Present theory well captures the effect of normal deformation which is not considered in some prior available literature and numerical results are exceptional matches with Sayyad and Ghugal [11] and Reddy [4].

Conclusions
Present scientific study mainly contributes the precise exponential shear and normal deformation theory for symmetric and antisymmetric four-layered composite arches when subjected to uniform load.The effects of transverse normal-stress and transverse normal-strain have been taken into account by present theory.The present theory meets the zero traction free end boundary condition and does not require any shear correction factors.Present theory is very accurate estimation for displacements and stresses which are rarely found in the literature.The obtained numerical results can be use for accurate design of such complex engineering structures.These innovative results will obviously set the benchmark for upcoming researchers in the area of composite arches.

Table 3
Normalized displacements and stresses for four layered symmetric composite arch.

Table 4
Normalized displacements and stresses for four layered antisymmetric composite arch.