Static Examination of Simply Supported Laminated Composite Beam with Varying Load Using Trigonometric Shear Deformation Theory

Objectives: To study the trigonometric shear deformation theory for the evolution of displacements and stresses of crossply simply supported laminated beam subjected to varying load. Methodology: A trigonometric shear deformation is used. The inplain displacement field uses a sinusoidal function in terms of the thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of a shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Finding: Stresses and displacements for orthotropic, single-layer, three-layer symmetric square cross-ply laminated beam subjected to varying load. Novelty: The numerical results of the present theory for displacement and stresses are compared with those of classical (ETB), first-order (FSDT), and higher-order shear deformation beam theories.


Introduction
Nowadays, composite materials are widely used in worldwide. Composite material is a material made up with the help of two or more different properties of materials. The composite material gives more strength to structural components. The composite material having advantage in its life span. Generally, composite material having more uses in aircraft and ships. In previous works, we studied the isotropic beam with different loading condition as well as different supports. Thick beams and plates, either isotropic or anisotropic, basically form two-and three-dimensional problems of elasticity theory. Reduction of these problems to the corresponding one-and two-dimensional approximate problems for their analysis has always been the main objective. The shear deformations in beams and plates with the three-dimensional nature of these problems further intensified the research interest in their accurate analysis. The shear deformation effects are more pronounced in the thick beams when subjected to transverse loads than in the thin beams under similar loading. The shear deformation effects are more significant in the thick beams. These effects are neglected in Elementary Theory of Beam (ETB). In order to describe the correct bending behavior of thick beams including shear deformation effects and the associated cross sectional warping, shear deformation theories are required. This can be accomplished by selection of proper kinematic and constitutes models. The worked on a higher request discrete-layer hypothesis and stated that a limited component is displayed for foreseeing the damping of overlaid composite sandwich beams wherein quadratic and cubic terms for guess of the in-plane relocation in each discrete layer considered. Model frequencies and damping of sandwich composite beam are estimated and associated with anticipated qualities [1]. The displayed another reverse digression shears twisting hypothesis for the statics and free vibration and clasping investigation of covered composite and sandwich plates. Shear stresses are evaporated at top and base surfaces of the plate and shear amendment factors are never again required [2]. The bar assembles into three arrangements, which are Euler-Bernoulli column theory, first solicitation shear deformation speculation and higher solicitation shear twisting hypotheses [3]. The equations governing the dynamic response of laminated structures are derived by using Hamilton's principle. However, equations of equilibrium for buckling problems are given by employing the principle of virtual displacements. Moreover, using Navier's technique and solving the eigenvalue equations, analytical solutions based on the global-local higher-order theory used in this article are first presented in present study. At the same time, the effect of the order number of higher-order shear deformation as well as interlaminar continuity of transverse on the global response of both laminated beams and soft-core sandwiches has been also studied [4]. Previous studies [5] examined extended reason limits and higher solicitation shear distortion theories in the examination of secured composite beams and plates. The built-up a general definition for nothing and transient vibration investigations of composite overlaid beams with subjective lay-ups and limit conditions [6]. The shear cure factor displayed is difficult to decisively for overlaid composite beams, as it dependants on layer bearing, geometric parameters and point of confinement conditions [7]. The structure articulation to decide the compelling flexural modulus of a covered shaft is created and this powerful flexural modulus is applied to the bowing and free vibration reaction of by and large overlaid composite beams with different limit bolsters [8]. The blended limited component conditions which depend on a useful are gotten by utilizing Gateaux differential for overlaid beams [9]. The variationally relentless refined hyperbolic shear distortion theory for flexure and free vibration of thick isotropic beam. This theory considers transverse shear deformations impacts [10]. The refined beam theories are required to describe the correct thermal response of plates as well as shear deformation effects. Three variables are used in displacement field, to represent the effect of shear deformation [11]. It proposes a hyperbolic shear deformation for thick isotropic cantilever beam. A higher-order beam theory which takes into account shear curvature, transverse stresses and rotatory inertia is presented. The displacement field of the present theory was based on a two variable, in which the transverse displacement is partitioned into the bending and shear parts [12].

Methodology
Consider simply supported laminated composite beam with varying load as shown in Figure 1. The beam is made of many unidirectional plies stacked up in different orientations with respect to the x-axis. In the right-handed Cartesian coordinate system, the x-axis is coincident with the beam axis and the origin is on the mid-plane of the beam. The length, breadth, and height of the beam are represented by L, b, and h, respectively.
The displacement field for overlaid composite beam dependent on the trigonometric shear deformation theory (TSDT), higher-order shear deformation theory (HSDT), and hyperbolic shear deformation theory (HYSDT) can be given as follows (1) (2) where u (1) is the removal along x headings. u (2) is the uprooting along y headings, u (3) is the relocation along z bearings of a point in the beam. u is the removal in the x course and w is transverse dislodging in the y heading of a point on the bar in midplane. The strain-displacement relations between strain-displacement corresponding to the displacement field are given by (2) FIGURE 1. Geometry of simply supported laminated composite beam with varying load.
Using the trigonometric shear deformation theory, the constitution equations of the laminates are 11 12 16  11  12  16  11  12  16   y  12  22  26  12  22  26  12  22  26   xy  16  26  66  16  26  66  16  26  66   11  12  16  11  12  16  11  12  16   12  22  22  12  22  26  12  22  26   16  26  26 where N x , N y , and N xy are the in-plane forces, M x , M y , and M xy the bending and twisting moments, P x , P y , and P xy the refine bending and twisting moments, 0  where   22  26  22  26  22  26   26  66  26  66  26  66  11  11  11  11  11  11  12  16  12  16  12  16  22  26  22  26  22  26  11  11  11  11  11  11  12  16  12  16  12  16  26  11  11  11  12  16  12  16  12  16  11 11 11 The laminated stiffness coefficients ij ij ij ij ij ij A ,B ,D ,E ,F ,H (i,j=1,2,6) and the transverse shear stiffness F 55 , which are capacity of overlay handle direction, material property, and stack succession, are given by All laminates made same orthotropic material, which properties are assumed Where N and Q are the force resultant; M and P are the moment's resultants. The principle of virtual work is used to obtain the governing equations and boundary conditions associated with the present theory. The principle of virtual work is given as ( ) where δ is the variational administrator. Coordination by parts and gathering the coefficients of δu, δw, and δϕ, one can acquire the administering conditions and limit states of the pillar related with the present hypothesis utilizing major lemma of analytics of varieties. where 1

Numerical Result and Discussion
The removals and stresses are determined for essentially bolstered covered composite bar for fluctuating stacking. The relationship of the maximum transverse deflection ( ) w to aspect ratio (AS) is shown for different angle ply in Table 1a and 1b. As would be expected Euler-Bernoulli (ETB) underestimates the beam deflections and gives poor estimates for relatively low value of aspect ratio (AS). The relationship of the maximum axial deflection ( ) u to aspect ratio (AS) is shown for different angle ply in Table 1c and 1d. The axial deflection increases when the aspect ratio (AS) increases. The distribution bending stress ( ) σ of by the five theories is represented in Table 1e and 1f for aspect ratio 4 and 10. The stresses are maximum at top and zero at centre. The distribution of transverse shear stresses according to constitutional relationship and equilibrium equations are presented in Table  1g, 1h, 1i, and 1j. The angle proportion aspect ratio (AS) are considered as 4 and 10. Figure  2a-d shows the transverse deflection, bending stresses by means of condition of harmony and constitutive relationship. As the aspect ratio (AS) increases the transverse deflection becomes constant as shown in Figure 2a. Figure 2b and c shows the through thickness distribution of axial displacement which is maximum at the top and bottom surface of the beam. From Figure 2d, it is seen that as viewpoint proportion expands the estimations of transverse removal got steady, in-plane dislodging and in-plane ordinary burdens are most extreme at top and base surface of the bar and those are zero at nonpartisan hub. The transverse shear pressure is zero at top and base surfaces of the beam while most extreme at nonpartisan pivot. Figure 2 represents non-dimensional (a) transverse deflection, (b) axial deflection for AS 4 for three-layer symmetric (0°/90°/0°), (c) axial deflection for AS 10 for three-layer symmetric (0°/90°/0°), (d) bending stresses for AS 4 for three-layer symmetric (0°/90°/0°).