This confirms that the thesis entitled "VIBRATION AND STABILITY OF LAMINATED COMPOSITE DOUBLY CURVED SHELLS WITH HIGHER ORDER SHEAR DEFORMATION THEORY" submitted by Mr. The minimum value of the frequency parameter and buckling loads are calculated for a laminated composite double curved shell. The frequency also increases as the number of shell layers increases for a symmetrical cross layout.
English
A symbol is used for different meanings depending on the context and defined in the text as they occur. Q ij (k) Matrix with reduced stiffness of the constituent layer R Radius of the reference surface of the spherical shell.
INTRODUCTION
INTRODUCTION
- APPLICATION OF SHELLS
- VIBRATION OF COMPOSITE SHELLS
The latter assumption is equivalent to neglecting the extension of the normal to the middle surface of the shell. This displacement field leads to a parabolic distribution of transverse shear stresses (and zero transverse normal strain), so no shear correction factors are used. Many classical theories developed for thin elastic shells are based on the Love-Kirchhoff assumptions in which the
LITERATURE REVIEW
OBJECTIVE AND SCOPE OF PRESENT INVESTIGATION
In this project, analytical solution of frequency characteristics and buckling loads of laminated composite doubly curved shells will be presented. These equations are then reduced to the equations of motion for double-curved panels and the Navier solution is obtained for cross-layer laminated composite double-curved panels. The Eigenvalue problem is then solved to obtain the free vibration frequencies and buckling loads.
THEORY AND FORMULATION
CHAPTER-3
INTRODUCTION
The governing equations including the effect of shear deformation are presented in orthogonal curvilinear coordinates for laminated composite shells. These equations are then reduced to the governing equations for free vibration of doubly composite curved shells interlaced. The Navier solution is used and the generalized eigenvalue problem obtained in the matrix formulation is solved to obtain the eigenvalues which are the natural frequencies and buckling loads.
BASIC ASSUMPTIONS
GEOMETRY OF SHELL
The position vector of a point on the mid-surface is denoted by r and the position of a point at a distance, z, from the mid-surface is denoted by R. The following derivation is restricted to orthogonal curvilinear coordinates that coincide with the principal lines of curvature of the neutral surface. This equation is called the fundamental shape, and A1 and A2 are the fundamental shape parameters, Lame parameters, or surface metrics.
The strain-displacement equations of a shell are, within the assumptions made earlier, an approximation of the strain-displacement relations referring to orthogonal curvilinear coordinates.
STRAIN DISPLACEMENT RELATIONS
- FIRST ORDER SHEAR DEFORMATION THEORY
- HIGHER ORDER SHEAR DEFORMATION THEORY
1 and 2 are the rotations of the normals to the reference surface, z = 0, about the α and β coordinate axes, respectively. The displacement field in equation [10] can be used to derive the general theory of laminated shells. Now substituting equation [12] in the stress-displacement relations referring to an orthogonal curvilinear coordinate system, we get.
Where 1 and 2 are the rotations of the normals to the reference surface, z = 0, about the α and β coordinate axes, respectively. For a particular orthotropic material, in which the direction of the principal axis coincides with the direction axis of the material, For the generalized state of plane stress, the above elastic moduli are related to the engineering constants as follows:
GOVERNING EQUATIONS
- STRAIN ENERGY
If U is the displacement vector, the kinetic energy of the shell element is given by, dV. To eliminate these terms, by integrating equation (26) by parts (the boundary changes t = t1 and t = t2 must disappear) and neglecting the terms. If the shell is subjected to both body and surface forces and if q1,q2 and qn are the components of the body and surface forces along the parametric lines, then the change in work done by the external loads is,.
The equilibrium equations are derived by applying the dynamic version of the principle of virtual work, which is Hamilton's principle. It states that under the set of all admissible configurations of system, the actual motion makes the quantityt Ldt. Here L is called Lagrangian and is equal to L= T - (U –V) (29) Where, T = Kinetic energy, U= Deformation energy, V= potential of all applied loads, = Mathematical operation called variation.
It is clear from the equation [29] that the Lagrangian consists of the kinetic, deformation energy and the potential of the applied loads.
STRESS RESULTANTS AND STRESS COUPLES
In which n = the number of layers in the shell; hk and hk-1 are the top and bottom z-coordinates of the kthlamina. Substitution of equations [11] and [15] into equation [35] leads to the following expressions for stress resultants and stress couples in first-order shear deformation theory:. In which n = the number of layers in the shell; hk and hk-1 are the top and bottom z-coordinates of the kth layer.
Substituting equations [13] and [15] into equation [38] leads to the following expression for the voltage resultants and voltage torques. where Aij, Bij , etc. are the laminate stiffnesses, expressed as. The expression for the stress resultants and stress torques thus obtained are then replaced by the equation of motion (30). The equation of motion in terms of the displacements for doubly curved shells thus reduces to.
For simplicity, only simply supported boundary conditions along all edges of the shell are considered. The boundary conditions for a simply supported doubly curved shell are obtained as given below. For convenience, the elements of the above matrices are suitably non-dimensionalized as follows. line) at the top indicates non-dimensional quantities).
A non-trivial solution for the column matrix X will give the necessary eigenvalues, which are the values of the square of the frequency parameter in the present case.
NUMERICAL RESULTS AND DISCUSSIONS 4.1 INTRODUCTION
The validation of the formulation and comparison of results
The validation of the buckling load formulation is done in a similar manner by comparing the results with those of Librescu et al [17].
NUMERICAL RESULTS
The first study is conducted to investigate the variation in R/a ratio on the non-dimensional frequency parameter. In the case of an elliptical paraboloidal shell, the non-dimensional frequency decreases as the R/a ratio increases. With the increase of the thickness parameter (a/h), i.e. as the thickness decreases, the frequency parameter also increases.
With the increasing modular ratio, it is seen that the non-dimensional frequency parameter also increases. But hyperbolic parabolic shell has less non-dimensional frequency; therefore it is more stable than the other two double curved shells. Similar study is subsequently done for the laminated composite elliptical paraboloid shell in Table 4.9 and the same variation is observed.
It is found that the dimensionless buckling load increases with increasing modular ratio for all shell geometries. The table above shows that as the number of layers increases, the dimensionless buckling load generally increases.
CONCLUSION
CONCLUSION
The governing equations, including the effect of shear strains, were presented in rectangular curvilinear coordinates for laminated orthotropic doubly curved shells. The theory is based on the displacement field as proposed by Reddy and Liu [24] in which the mid-surface displacements are expanded as cubic functions of the thickness coordinate and the transverse displacement is assumed to be constant through the thickness. This displacement field leads to a parabolic distribution of the transverse shear strain and thus the shear stresses, so no shear correction factors are applied.
These equations are solved for simply supported doubly covered shells and the associated eigenvalue problem is solved by a computer program. The results of the present theory are compared with the previous results available by Reddy and Liu [24]. The non-dimensional buckling load for unidirectional axial compression has also been studied for different shell geometries and variations in other parameters.
But hyperbolic paraboloid scale has less non-dimensional frequency compared to elliptic paraboloid scale and spherical scale. Compared between the spherical hyperbolic paraboloid shell and the elliptical paraboloid shell, the hyperbolic paraboloid shell has the least non-dimensional buckling load. In the case of both shell geometries, the buckling load decreases with the increase of the curvature ratio (R/a).
Thus, the above research shows that by suitably changing the orientation of the layers or the number of layers, the properties of the laminated double-curved composite shells can be tailored to specific needs.
SCOPE FOR FUTURE WORK
- Maurice Touratier (1992): ‘A Generalization of shear deformation theories for axisymmetric multilayered shells’, International Journal of Solids and structures, 29(11),
- Messina Arcangelo (2003): ‘Free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise-smooth function,’ International
- Qatu, S.M(1999).’Accurate equations for laminated composite deep thick shells,’
- Topal U, ‘Mode-Frequency Analysis of Laminated Spherical Shell’ Session ENG pp 501- 001
1989): 'Moderately thick, angular cylindrical shells under internal pressure,' ASME Journal of Applied Mechanics, 56: pp Moderately large vibrations of double-curved shallow open shells composed of thick layers,' Journal of Sound and Vibration, 299, pp 'A new nonlinear higher order shear deformation theory for large amplitude vibrations of laminated double-curved shells', International Journal of Non-Linear Mechanics, 45:pp. 1991): 'Free vibration analysis of double-curved shallow shells on rectangular planform using three-dimensional elasticity theory', International Journal of Solids and Structures, 27(7), pp.897-913. “A higher order theory for free vibration analysis of circular cylindrical shells,” International Journal of Solids and Structures, 20(7): pp.623-630.
1986): 'Arbitrarily laminated anisotropic cylindrical shells under uniform pressure,' AIAA Journal, 24, pp Closed solutions for arbitrarily laminated anisotropic cylindrical shells (tubes) including shear deformation,' AIAA Journal, 27, pp.6059. Maurice Touratier (1992): 'A Generalization of shear deformation theories for axisymmetric multilayered shells', International Journal of Solids and structures, 29(11), axisymmetric multilayered shells', International Journal of Solids and structures, 29(11), pp Influence of rotational inertia and shear in bending motions of isotropic elastic plates', ASME Journal of Applied mechanics, 18(2):pp.31-38. Messina Arcangelo (2003): 'Free Vibrations of Multilayer Doubly Curved Shells Based on a Mixed Variational Approach and Global Piecewise-Smooth Function,' International on a Mixed Variational Approach and Global Piecewise-Smooth Function,' International Journal of Solids and Structures, 40 : p. 3069-3088.
Fourier analysis of thick cross-layer Levy-type clamped double-curved panels', Composite Structures, 80: pp.489-503. The Effect of Shear Deformation on the Bending of Elastic Plates,” ASME Journal of Applied Mechanicals, 12(2): pp.69-77. Bajoria (2003). “Finite element modeling of smart plates/shells using higher order shear deformation theory,” Composite Structures, 62: pp.
2009): 'Analytical studies for buckling and vibration of weld-bonded beam shells of rectangular cross-section,' International Journal of weld-bonded beam shells of rectangular cross-section,' International Journal of Mechanical Sciences, 51, pp.77– 88.
APPENDIX
