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Figure 2.7 Geometry and coordinate system of shells of revolution

Figure 2 7 Geometry and coordinate system of shells of revolution

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Table 1.1 Advantages and disadvantages of composite materials over metals [3]
Table 1.1 Advantages and disadvantages of composite materials over metals [3]
Figure 1.1 Specific strength and specific stiffness of advanced composite materials [1]
Figure 1.1 Specific strength and specific stiffness of advanced composite materials [1]
Figure 1.2 A unidirectional fiber-reinforced lamina with the principal material directions (1, 2, and 3)
Figure 1.2 A unidirectional fiber-reinforced lamina with the principal material directions (1, 2, and 3)
Figure 1.3 Deformation of isotropic, orthotropic and anisotropic rectangular block under uniaxial
Figure 1.3 Deformation of isotropic, orthotropic and anisotropic rectangular block under uniaxial
Figure 2.1 The position vector of a space curve [12].
Figure 2.1 The position vector of a space curve [12].
Figure 2.2 Tangent vector description [12].
Figure 2.2 Tangent vector description [12].
Figure 2.3 Parametric curves of a surface and differential change of a position vector on the surface
Figure 2.3 Parametric curves of a surface and differential change of a position vector on the surface
Figure 2.4 Some examples for surfaces of revolution
Figure 2.4 Some examples for surfaces of revolution
Figure 2.6 Ruled Surfaces (a) Hyperboloid of revolution of one sheet, (b) Conoid [15].
Figure 2.6 Ruled Surfaces (a) Hyperboloid of revolution of one sheet, (b) Conoid [15].
We have the second fundamental magnitudes, Land N for the shells of revolution bi We get the derivatives of the unit normal vector 71 with respect to ¢ and G as follows: inserting Equations (2.74), (2.75), (2.80), and (2.81) into Equation (2.36).
We have the second fundamental magnitudes, Land N for the shells of revolution bi We get the derivatives of the unit normal vector 71 with respect to ¢ and G as follows: inserting Equations (2.74), (2.75), (2.80), and (2.81) into Equation (2.36).
Recalling Equation (2.21), and inserting Equations (2.76) and (2.77) into it, we get
Recalling Equation (2.21), and inserting Equations (2.76) and (2.77) into it, we get
The infinitesimal distance P,P, between two arbitrary points on the reference surface of the other words, the Gauss-Codazzi conditions are clearly satisfied for shells of revolution. expressions for A, A, JR, R, obtained for shells of revolution into Gauss-Codazzi conditions. Ir where A, and A, are sometimes called Lamé parameters which are related with the first fundamental
The infinitesimal distance P,P, between two arbitrary points on the reference surface of the other words, the Gauss-Codazzi conditions are clearly satisfied for shells of revolution. expressions for A, A, JR, R, obtained for shells of revolution into Gauss-Codazzi conditions. Ir where A, and A, are sometimes called Lamé parameters which are related with the first fundamental
Figure 2.8 Differential element and coordinate system of a doubly curved thin shell [12]. Incorporating the orthogonality of curvilinear coordinate system, Equations (2.94), (2.48),
Figure 2.8 Differential element and coordinate system of a doubly curved thin shell [12]. Incorporating the orthogonality of curvilinear coordinate system, Equations (2.94), (2.48),
From the Figure 2.8 and the Equation (2.106), the length of the shaded surface whose unit and for the one whose unit normal vector is in é, is given by:
From the Figure 2.8 and the Equation (2.106), the length of the shaded surface whose unit and for the one whose unit normal vector is in é, is given by:
T, and T, can be expressed finally as follows:
T, and T, can be expressed finally as follows:
manipulations, we obtain From Equation (2.122), we obtain two relations which will be used afterwards.
manipulations, we obtain From Equation (2.122), we obtain two relations which will be used afterwards.
To get the derivatives of the normal vector 7 with respect to é, and és using the position
To get the derivatives of the normal vector 7 with respect to é, and és using the position
Substituting, Equation (2.126) and Equation (2.133) into Equation (2.139), C, become Also, substituting Equation (2.123) into Equation (2.140), C, is found to be
Substituting, Equation (2.126) and Equation (2.133) into Equation (2.139), C, become Also, substituting Equation (2.123) into Equation (2.140), C, is found to be
Substituting Equations (2.146) and (2.153) into Equation (2.159), we get Inserting Equations (2.125) and (2.126) into Equation (2.66) and expanding that
Substituting Equations (2.146) and (2.153) into Equation (2.159), we get Inserting Equations (2.125) and (2.126) into Equation (2.66) and expanding that
Collecting the coefficients of TJ, andT,, we can write a similar expression like Equation
Collecting the coefficients of TJ, andT,, we can write a similar expression like Equation
The equality in Equation (2.161) holds when the coefficients of T, and a are both zero.
The equality in Equation (2.161) holds when the coefficients of T, and a are both zero.
where @. ior jy? 4ivor j) NA icoy j) ALE, Tespectively, the curvilinear coordinates of the geometry, components of the displacement vector and geometrical scale factor quantities. When the Equations The strain-displacement relations by the theory of elasticity in the orthogonal curvilinear
where @. ior jy? 4ivor j) NA icoy j) ALE, Tespectively, the curvilinear coordinates of the geometry, components of the displacement vector and geometrical scale factor quantities. When the Equations The strain-displacement relations by the theory of elasticity in the orthogonal curvilinear
The normal strain components are: obtained in the shell space.
The normal strain components are: obtained in the shell space.
Naghdi-Reissner Shell Theory, they are neglected in the Equations (2.175), (2.176), and (2.178). It and, the shearing strain components are:
Naghdi-Reissner Shell Theory, they are neglected in the Equations (2.175), (2.176), and (2.178). It and, the shearing strain components are:
Equations (2.164), (2.165), (2.166), (2.167) and (2.168) are substituted properly into respectively, are neglected. The strain displacement equations turn into the following Equations from (2.181) to (2.186), then the strain displacement relations can be represented as the sun
Equations (2.164), (2.165), (2.166), (2.167) and (2.168) are substituted properly into respectively, are neglected. The strain displacement equations turn into the following Equations from (2.181) to (2.186), then the strain displacement relations can be represented as the sun
where C ;, are the elastic constants, and are obtained from C aa DY the following change of subscripts:
where C ;, are the elastic constants, and are obtained from C aa DY the following change of subscripts:
where EE sdliys Eo are Young’s moduli in 1, 2, and 3 directions, respectively, Vi is Poisson’s ratio defined as ratio of transverse strain in the a direction to the axial strain in the ith direction, when The elastic coefficients in the stiffness matrixC,.in Equation (2.205) in terms of the stressed in the i-direction —€ j ‘i €, (for O, =O and all other stresses are zero) (i, j=1,2,3), and
where EE sdliys Eo are Young’s moduli in 1, 2, and 3 directions, respectively, Vi is Poisson’s ratio defined as ratio of transverse strain in the a direction to the axial strain in the ith direction, when The elastic coefficients in the stiffness matrixC,.in Equation (2.205) in terms of the stressed in the i-direction —€ j ‘i €, (for O, =O and all other stresses are zero) (i, j=1,2,3), and
Figure 2.9 (a) Specially Orthotropic Lamina, (b) Generally Orthotropic Lamina. and (2.208) are transformed into ones for Generally Orthotropic Lamina.
Figure 2.9 (a) Specially Orthotropic Lamina, (b) Generally Orthotropic Lamina. and (2.208) are transformed into ones for Generally Orthotropic Lamina.
Figure 2.10 Internal Stresses on a Doubly Curved Shell Element The normal and shear force resultants acting on the face perpendicular to the é, coordinat
Figure 2.10 Internal Stresses on a Doubly Curved Shell Element The normal and shear force resultants acting on the face perpendicular to the é, coordinat
Figure 2.11 Force resultants in the shell coordinates.
Figure 2.11 Force resultants in the shell coordinates.
Figure 2.12 Moment resultants in the shell coordinates. A laminate of doubly curved shell element is shown in Figure 2.13. For a laminated shell, the stress components can be integrated across each lamina, then must be added together. In the lamination, laminae are assumed to be bonded perfectly to each other so that the displacements are stresses will be discontinuous across the interface of two laminae.
Figure 2.12 Moment resultants in the shell coordinates. A laminate of doubly curved shell element is shown in Figure 2.13. For a laminated shell, the stress components can be integrated across each lamina, then must be added together. In the lamination, laminae are assumed to be bonded perfectly to each other so that the displacements are stresses will be discontinuous across the interface of two laminae.
Figure 2.13 N-layered laminate of a doubly curved shell element.
Figure 2.13 N-layered laminate of a doubly curved shell element.
Combining Equations (2.187) to (2.191), (2.210), (2.211), (2.213) to (2.216) with Equations
Combining Equations (2.187) to (2.191), (2.210), (2.211), (2.213) to (2.216) with Equations
As a result, the constitutive equations for the anisotropic laminated composite doubly curvec the force and moments resultants as
As a result, the constitutive equations for the anisotropic laminated composite doubly curvec the force and moments resultants as
where A, , the extensional stiffness matrix which relates the in-plane stress resultants to the reference surface strains, are given by
where A, , the extensional stiffness matrix which relates the in-plane stress resultants to the reference surface strains, are given by
Substituting the strain-displacement relations, (2.187) to (2.191) into Equation (2.268) yields Thus, the expression for the variation of the strain energy takes the form
Substituting the strain-displacement relations, (2.187) to (2.191) into Equation (2.268) yields Thus, the expression for the variation of the strain energy takes the form
can only be satisfied if each of the triple and double integral parts is equal to zero. Furthermore, since
can only be satisfied if each of the triple and double integral parts is equal to zero. Furthermore, since
It should be noted that each of the double integrals given in Equation (2.270) is equal to zero only if the coefficients of the variational displacements, variational displacements or one of the two for each term are zero. Since variational displacements are only zero at all times when the
It should be noted that each of the double integrals given in Equation (2.270) is equal to zero only if the coefficients of the variational displacements, variational displacements or one of the two for each term are zero. Since variational displacements are only zero at all times when the
Writing the each element of the left hand side of Equation (2.299) in terms of strain
Writing the each element of the left hand side of Equation (2.299) in terms of strain
The equations (3.9) to (3.12) can be written in compact form as:
The equations (3.9) to (3.12) can be written in compact form as:
Up to this point, the displacements and rotations terms of the system of partial differentia!
Up to this point, the displacements and rotations terms of the system of partial differentia!
The elements of the coefficient matrix K are given in the Appendix F. 3.4 NONDIMENSIONALIZATION OF THE SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS The Equation (3.37) can be put into nondimensional form by using the following scheme
The elements of the coefficient matrix K are given in the Appendix F. 3.4 NONDIMENSIONALIZATION OF THE SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS The Equation (3.37) can be put into nondimensional form by using the following scheme
Figure 3.1 Division of meridional length of the shell of revolution into segments [71]. Equations (3.71) and (3.72) constitute a system of 2M linear homogeneous matrix equations T'(9., w,(¢)+T? (9, w.(¢)-w,¢.,)=0 T (9, W,(@)4+7;' (@., W.(9)-w,(9.,)=0
Figure 3.1 Division of meridional length of the shell of revolution into segments [71]. Equations (3.71) and (3.72) constitute a system of 2M linear homogeneous matrix equations T'(9., w,(¢)+T? (9, w.(¢)-w,¢.,)=0 T (9, W,(@)4+7;' (@., W.(9)-w,(9.,)=0
frequency of the example problem with traditional Fourier Decomposition procedure.
frequency of the example problem with traditional Fourier Decomposition procedure.
Figure 3.3 Finding first natural frequency of the example problem with (a)traditional Fourier Decomposition Method, and (b) Method of Finite Exponential Fourier Transform.
Figure 3.3 Finding first natural frequency of the example problem with (a)traditional Fourier Decomposition Method, and (b) Method of Finite Exponential Fourier Transform.
frequency of the example problem with method of Finite Exponential Fourier Transform. Based on Figure 3.3, it can be said that in order to determine the root by the application of Table 3.2 Trial natural frequencies and their computed determinants in order to find first natural
frequency of the example problem with method of Finite Exponential Fourier Transform. Based on Figure 3.3, it can be said that in order to determine the root by the application of Table 3.2 Trial natural frequencies and their computed determinants in order to find first natural
Figure 4.1 General Configuration and Coordinate System of Laminated Composite Circular Cylindrical Shell and its Laminate. where R is the mean radius of laminated composite circular cylindrical shell.
Figure 4.1 General Configuration and Coordinate System of Laminated Composite Circular Cylindrical Shell and its Laminate. where R is the mean radius of laminated composite circular cylindrical shell.
Figure 4.2 Circumferential Nodal Patterns and Axial Nodal Patterns for a Simply Supported Circular
Figure 4.2 Circumferential Nodal Patterns and Axial Nodal Patterns for a Simply Supported Circular
Figure 4.3 shows the comparison of natural frequencies calculated by the exact method of cylindrical shell used in comparison study of the method of solution.
Figure 4.3 shows the comparison of natural frequencies calculated by the exact method of cylindrical shell used in comparison study of the method of solution.
for m=1-3 and n=0-10.
for m=1-3 and n=0-10.
Figure 4.3 The comparison of natural frequencies calculated by the exact method of solution [13] with those calculated by the present method of solution from n=0 to n=10
Figure 4.3 The comparison of natural frequencies calculated by the exact method of solution [13] with those calculated by the present method of solution from n=0 to n=10
laminated composite circular cylindrical shell used for the case study on the fiber orientation angle. When ply stacking sequence, material, and geometry (ply thickness) are symmetric about the Table 4.3 Geometrical, material properties, and laminate properties of the simply supported
laminated composite circular cylindrical shell used for the case study on the fiber orientation angle. When ply stacking sequence, material, and geometry (ply thickness) are symmetric about the Table 4.3 Geometrical, material properties, and laminate properties of the simply supported
variation of stretching stiffness matrix elements with respect to alpha Figure 4.4 Effect of the fiber orientation angle on the elements of stretching stiffness matrix.
variation of stretching stiffness matrix elements with respect to alpha Figure 4.4 Effect of the fiber orientation angle on the elements of stretching stiffness matrix.
Figure 4.5 Effect of the fiber orientation angle on the elements of transverse shear stiffness matrix.
Figure 4.5 Effect of the fiber orientation angle on the elements of transverse shear stiffness matrix.
the increment of 10 degrees for n=0, and m=1,2, and 3. Table 4.4 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=0, and m=1,2, and 3. Table 4.4 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
variation of flexural stiffness matrix elements with respect to alpha Figure 4.6 Effect of the fiber orientation angle on the elements of flexural stiffness matrix.
variation of flexural stiffness matrix elements with respect to alpha Figure 4.6 Effect of the fiber orientation angle on the elements of flexural stiffness matrix.
Table 4.5 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with Table 4.6 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.5 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with Table 4.6 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=2, and m=1,2, and 3.
the increment of 10 degrees for n=2, and m=1,2, and 3.
the increment of 10 degrees for n=3, and m=1,2, and 3. Table 4.7 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=3, and m=1,2, and 3. Table 4.7 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.8 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.8 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.9 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.9 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=6, and m=1,2, and 3. Table 4.10 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=6, and m=1,2, and 3. Table 4.10 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=7, and m=1,2, and 3. Table 4.11 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=7, and m=1,2, and 3. Table 4.11 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=8, and m=1,2, and 3. Table 4.12 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=8, and m=1,2, and 3. Table 4.12 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=9, and m=1,2, and 3. Table 4.13 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=9, and m=1,2, and 3. Table 4.13 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=10, and m=1,2, and 3.
the increment of 10 degrees for n=10, and m=1,2, and 3.
Figure 4.7 Natural Frequency versus circumferential wave number changing from 0 to 10 for fiber orientation angle of 0° to 90° with the increment of 20 degrees when m=1.
Figure 4.7 Natural Frequency versus circumferential wave number changing from 0 to 10 for fiber orientation angle of 0° to 90° with the increment of 20 degrees when m=1.
Figure 4.8 Natural Frequency versus circumferential wave number changing from 0 to 10 for fiber orientation angle of 0° to 90° with the increment of 20 degrees when m=2.
Figure 4.8 Natural Frequency versus circumferential wave number changing from 0 to 10 for fiber orientation angle of 0° to 90° with the increment of 20 degrees when m=2.
Figure 4.9 Natural Frequency versus circumferential wave number changing from 0 to 10 for fibe: orientation angle of 0° to 90° with the increment of 20 degrees when m=3.
Figure 4.9 Natural Frequency versus circumferential wave number changing from 0 to 10 for fibe: orientation angle of 0° to 90° with the increment of 20 degrees when m=3.
Figure 4.10 Variation in strain energy in a particular cylindrical shell with increasing number of
Figure 4.10 Variation in strain energy in a particular cylindrical shell with increasing number of
Figure 4.11 Natural Frequency versus fiber orientation angle for the first lowest fundamental mode
Figure 4.11 Natural Frequency versus fiber orientation angle for the first lowest fundamental mode
Figure 4.12 Natural Frequency versus fiber orientation angle for the second lowest fundamental mode
Figure 4.12 Natural Frequency versus fiber orientation angle for the second lowest fundamental mode
Figure 4.13 Natural Frequency versus fiber orientation angle for the third lowest fundamental mode
Figure 4.13 Natural Frequency versus fiber orientation angle for the third lowest fundamental mode
Figure 4.14 Mode shapes for the first fundamental transverse mode when Q changes from 0° to 90° with the increment of 10° (n=1).
Figure 4.14 Mode shapes for the first fundamental transverse mode when Q changes from 0° to 90° with the increment of 10° (n=1).
Figure 4.15 Mode shapes for the first fundamental transverse mode when Q@ changes from 0° to 90°
Figure 4.15 Mode shapes for the first fundamental transverse mode when Q@ changes from 0° to 90°
Figure 4.16 Mode shapes for the first fundamental transverse mode when Q@ changes from 0° to 90°
Figure 4.16 Mode shapes for the first fundamental transverse mode when Q@ changes from 0° to 90°
Table 4.15 Six symmetric layups used in the case study on the stacking sequence. The geometrical, mechanical and laminate properties of the clamped laminated composite
Table 4.15 Six symmetric layups used in the case study on the stacking sequence. The geometrical, mechanical and laminate properties of the clamped laminated composite
laminated composite circular cylindrical shell used for the case study on stacking sequence. The results of the six different stacking sequence for the lowest three natural frequencies
laminated composite circular cylindrical shell used for the case study on stacking sequence. The results of the six different stacking sequence for the lowest three natural frequencies
Figure 4.17 Effect of the stacking sequence on the elements of stretching stiffness matrix (h/R=0.01). variation of stretching stiffness matrix elements
Figure 4.17 Effect of the stacking sequence on the elements of stretching stiffness matrix (h/R=0.01). variation of stretching stiffness matrix elements
Figure 4.18 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements
Figure 4.18 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements
Table 4.17 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CC (h/R=0.01). Table 4.18 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CC (h/R=0.01).
Table 4.17 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CC (h/R=0.01). Table 4.18 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CC (h/R=0.01).
Figure 4.19 Effect of the stacking sequence on the elements of flexural stiffness matrix (h/R=0.01) variation of flexural stiffness matrix elements
Figure 4.19 Effect of the stacking sequence on the elements of flexural stiffness matrix (h/R=0.01) variation of flexural stiffness matrix elements
Table 4.19 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CC (h/R=0.01). Table 4.20 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CC (h/R=0.01).
Table 4.19 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CC (h/R=0.01). Table 4.20 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CC (h/R=0.01).
Table 4.21 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CC (h/R=0.01)
Table 4.21 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CC (h/R=0.01)
Table 4.22 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CC (h/R=0.01).
Table 4.22 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CC (h/R=0.01).
Figure 4.20 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=1 and h/R=0.01).
Figure 4.20 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=1 and h/R=0.01).
Figure 4.21 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=2 and h/R=0.01).
Figure 4.21 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=2 and h/R=0.01).
Figure 4.22 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=3 and h/R=0.01).
Figure 4.22 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=3 and h/R=0.01).
Table 4.23 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CF (h/R=0.01) Table 4.24 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CF (h/R=0.01).
Table 4.23 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CF (h/R=0.01) Table 4.24 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CF (h/R=0.01).
Table 4.25 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CF (h/R=0.01). Table 4.26 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CF (h/R=0.01).
Table 4.25 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CF (h/R=0.01). Table 4.26 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CF (h/R=0.01).
Table 4.27 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CF (h/R=0.01)
Table 4.27 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CF (h/R=0.01)
Table 4.28 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CF (h/R=0.01).
Table 4.28 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CF (h/R=0.01).
Figure 4.23 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.23 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.24 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.24 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.25 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.25 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.26 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=1 and
Figure 4.26 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=1 and
Figure 4.27 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=2 and
Figure 4.27 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=2 and
Figure 4.28 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=3 and
Figure 4.28 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=3 and
Figure 4.29 Effect of the stacking sequence on the elements of stretching stiffness matrix (h/R=0. 1).
Figure 4.29 Effect of the stacking sequence on the elements of stretching stiffness matrix (h/R=0. 1).
Figure 4.30 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements
Figure 4.30 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements
Figure 4.31 Effect of the stacking sequence on the elements of flexural stiffness matrix (h/R=0.1) variation of flexural stiffness matrix elements
Figure 4.31 Effect of the stacking sequence on the elements of flexural stiffness matrix (h/R=0.1) variation of flexural stiffness matrix elements
Table 4.29 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CC (h/R=0.1). Table 4.30 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CC (h/R=0.1).
Table 4.29 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CC (h/R=0.1). Table 4.30 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CC (h/R=0.1).
Table 4.31 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CC (h/R=0.1)
Table 4.31 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CC (h/R=0.1)
Table 4.32 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CC (h/R=0.1).
Table 4.32 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CC (h/R=0.1).
Table 4.33 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CC (h/R=0.1). Table 4.34 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CC (h/R=0.1).
Table 4.33 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CC (h/R=0.1). Table 4.34 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CC (h/R=0.1).
Figure 4.32 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=1 and h/R=0.1).
Figure 4.32 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=1 and h/R=0.1).
Figure 4.33 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=2 and h/R=0.1).
Figure 4.33 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=2 and h/R=0.1).
Figure 4.34 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.34 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Table 4.35 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CF (h/R=0. 1). Table 4.36 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CF (h/R=0.1).
Table 4.35 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CF (h/R=0. 1). Table 4.36 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CF (h/R=0.1).
Table 4.37 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CF (h/R=0.1).
Table 4.37 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CF (h/R=0.1).
Table 4.38 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CF (h/R=0.1)
Table 4.38 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CF (h/R=0.1)
Table 4.39 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CF (h/R=0. 1). Table 4.40 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CF (h/R=0.1).
Table 4.39 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CF (h/R=0. 1). Table 4.40 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CF (h/R=0.1).
Figure 4.35 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=1 and h/R=0.1).
Figure 4.35 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=1 and h/R=0.1).
Figure 4.36 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=2 and h/R=0.1).
Figure 4.36 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=2 and h/R=0.1).
Figure 4.37 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=3 and h/R=0.1).
Figure 4.37 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=3 and h/R=0.1).
Figure 4.38 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=1 and
Figure 4.38 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=1 and
‘igure 4.39 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=2 and
‘igure 4.39 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=2 and
Figure 4.40 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=3 and
Figure 4.40 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=3 and
Figure 4.41 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.41 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.42 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-clamped (CC) boundary condition (m=2).
Figure 4.42 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-clamped (CC) boundary condition (m=2).
Figure 4.43 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-clamped (CC) boundary condition (m=3).
Figure 4.43 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-clamped (CC) boundary condition (m=3).
Figure 4.44 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.44 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.45 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-free (CF) boundary condition (m=2).
Figure 4.45 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-free (CF) boundary condition (m=2).
Table 4.41 Three different layups used in the case study on effect of coupling terms.
Table 4.41 Three different layups used in the case study on effect of coupling terms.
Figure 4.46 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.46 Comparison of the natural frequencies of the thick and thin shell configurations
Stretching, transverse shear, stretching-flexural coupling, and flexural stiffness coefficients laminated composite circular cylindrical shell used for the case study on effect of coupling terms
Stretching, transverse shear, stretching-flexural coupling, and flexural stiffness coefficients laminated composite circular cylindrical shell used for the case study on effect of coupling terms
Figure 4.47 Effect of the stacking sequence on the elements of stretching stiffness matrix
Figure 4.47 Effect of the stacking sequence on the elements of stretching stiffness matrix
Figure 4.48 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements for three different layups in the case study on effect of coupling terms.
Figure 4.48 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements for three different layups in the case study on effect of coupling terms.
Figure 4.49 Effect of the stacking sequence on the elements of stretching-flexural coupling stiffness matrix for three different layups in the case study on effect of coupling terms.
Figure 4.49 Effect of the stacking sequence on the elements of stretching-flexural coupling stiffness matrix for three different layups in the case study on effect of coupling terms.
Table 4.43 Natural frequencies (rad/sec) of circular cylindrical shell with symmetric layup 10/30/ 90/60 | ; Table 4.44 Natural frequencies (rad/sec) of circular cylindrical shell with antisymmetric layup
Table 4.43 Natural frequencies (rad/sec) of circular cylindrical shell with symmetric layup 10/30/ 90/60 | ; Table 4.44 Natural frequencies (rad/sec) of circular cylindrical shell with antisymmetric layup
variation of flexural stiffness matrix elements Figure 4.50 Effect of the stacking sequence on the elements of flexural stiffness matrix for three
variation of flexural stiffness matrix elements Figure 4.50 Effect of the stacking sequence on the elements of flexural stiffness matrix for three
Natural frequencies for the three layup cases are plotted with respect to circumferential wave Table 4.45 Natural frequencies (rad/sec) of circular cylindrical shell with unsymmetric layup
Natural frequencies for the three layup cases are plotted with respect to circumferential wave Table 4.45 Natural frequencies (rad/sec) of circular cylindrical shell with unsymmetric layup
‘igure 4.51 Effect of coupling terms on the natural frequencies of the simply supported laminated number in Figures 4.51, 4.52, and 4.53 for m=1,2, and 3, respectively.
‘igure 4.51 Effect of coupling terms on the natural frequencies of the simply supported laminated number in Figures 4.51, 4.52, and 4.53 for m=1,2, and 3, respectively.
Figure 4.52 Effect of coupling terms on the natural frequencies of the simply supported laminated composite circular cylindrical shell (m=2).
Figure 4.52 Effect of coupling terms on the natural frequencies of the simply supported laminated composite circular cylindrical shell (m=2).
Figure 4.53 Effect of coupling terms on the natural frequencies of the simply supported laminated
Figure 4.53 Effect of coupling terms on the natural frequencies of the simply supported laminated
Table 4.46 Dominant mode shapes for three layups for n=0, and n=1. As one can see from Table 4.46 for n=0, axisymmetric case, the dominant modes are all
Table 4.46 Dominant mode shapes for three layups for n=0, and n=1. As one can see from Table 4.46 for n=0, axisymmetric case, the dominant modes are all
variation of stretching stiffness matrix elements Figure 4.54 Elements of stretching stiffness matrix for the layups used in the case study on
variation of stretching stiffness matrix elements Figure 4.54 Elements of stretching stiffness matrix for the layups used in the case study on
Figure 4.55 Elements of transverse shear stiffness matrix for the layups used in the case study on variation of transverse shear stiffness matrix elements the existence of coupling terms.
Figure 4.55 Elements of transverse shear stiffness matrix for the layups used in the case study on variation of transverse shear stiffness matrix elements the existence of coupling terms.
‘igure 4.56 Elements of stretching-flexural coupling stiffness matrix for the layups used variation of stretching-flexural coupling stiffness matrix elements in the case study on the existence of coupling terms.
‘igure 4.56 Elements of stretching-flexural coupling stiffness matrix for the layups used variation of stretching-flexural coupling stiffness matrix elements in the case study on the existence of coupling terms.
The variation of the natural frequencies versus circumferential wave number is given in
The variation of the natural frequencies versus circumferential wave number is given in
Figure 4.57 Elements of flexural stiffness matrix for the layups used variation of flexural stiffness matrix elements in the case study on the existence of coupling terms.
Figure 4.57 Elements of flexural stiffness matrix for the layups used variation of flexural stiffness matrix elements in the case study on the existence of coupling terms.
Figure 4.58 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=1). omitted is seen to be higher.
Figure 4.58 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=1). omitted is seen to be higher.
Figure 4.59 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=2).
Figure 4.59 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=2).
Figure 4.60 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=3).
Figure 4.60 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=3).
layup case and unsymmetric layup with coupling terms omitted for n=0,1, and 2. The results given in Table 4.48 also confirm that at low circumferential modes (low n) the Table 4.48 Comparison of the natural frequencies (rad/sec) between antisymmetric influence of bending stiffness D, is very little. Although these two layups have different bending
layup case and unsymmetric layup with coupling terms omitted for n=0,1, and 2. The results given in Table 4.48 also confirm that at low circumferential modes (low n) the Table 4.48 Comparison of the natural frequencies (rad/sec) between antisymmetric influence of bending stiffness D, is very little. Although these two layups have different bending
Table 4.50 Natural Frequencies for the laminated
Table 4.50 Natural Frequencies for the laminated
composite spherical shell (n=1). As one can see from the mode shape plots, first three modes for n=1 correspond to transverse
composite spherical shell (n=1). As one can see from the mode shape plots, first three modes for n=1 correspond to transverse
Figure 4.61 The side and perspective views of the laminated composite spherical shell.
Figure 4.61 The side and perspective views of the laminated composite spherical shell.
Figure 4.62 Dominant mode shapes (transverse displacement) of the corresponding natural
Figure 4.62 Dominant mode shapes (transverse displacement) of the corresponding natural
Figure 5.1 Cross-sections of the circular cylindrical shells with
Figure 5.1 Cross-sections of the circular cylindrical shells with
Let us split the Equation (A.3) and do the variational computations term by term, anc
Let us split the Equation (A.3) and do the variational computations term by term, anc
Carrying out the Finite exponential Fourier transform to Equation (E. 1) (c,, Yu’
Carrying out the Finite exponential Fourier transform to Equation (E. 1) (c,, Yu’
Separating the real and imaginary parts of Equation (E.4)
Separating the real and imaginary parts of Equation (E.4)
We now separate the real and imaginary parts of Equation (E.10). First, writing real parts Then, writing imaginary parts of Equation (E.10) term by term.
We now separate the real and imaginary parts of Equation (E.10). First, writing real parts Then, writing imaginary parts of Equation (E.10) term by term.
We separate the real and imaginary parts of Equation (E. 16)
We separate the real and imaginary parts of Equation (E. 16)
We separate the real and imaginary parts of Equation (E.22). First, writing real parts of Then, writing imaginary parts of Equation (E.22) term by term.
We separate the real and imaginary parts of Equation (E.22). First, writing real parts of Then, writing imaginary parts of Equation (E.22) term by term.
Carrying out the Finite exponential Fourier transform to Equation (E.25). Using Equation (3.28) in Equation (E.26) yields
Carrying out the Finite exponential Fourier transform to Equation (E.25). Using Equation (3.28) in Equation (E.26) yields
We separate the real and imaginary parts of Equation (E.28).
We separate the real and imaginary parts of Equation (E.28).
Separating the real and imaginary parts of Equation (E.34)
Separating the real and imaginary parts of Equation (E.34)
Using Equation (3.28) in Equation (E.38) yields Carrying out the Finite exponential Fourier transform to Equation (E.37)
Using Equation (3.28) in Equation (E.38) yields Carrying out the Finite exponential Fourier transform to Equation (E.37)
We separate the real and imaginary parts of Equation (E.40). First, writing real parts of Equation (E.40) term by term.
We separate the real and imaginary parts of Equation (E.40). First, writing real parts of Equation (E.40) term by term.
Using Equation (3.28) in Equation (E.44) yields
Using Equation (3.28) in Equation (E.44) yields
We separate the real and imaginary parts of Equation (E.46)
We separate the real and imaginary parts of Equation (E.46)
Using Equation (3.28) in (E.50) yields
Using Equation (3.28) in (E.50) yields
Kutta method for system of ODEs is given in [69]. As a result, IT (G lis given by
Kutta method for system of ODEs is given in [69]. As a result, IT (G lis given by
The formal solution of (G.26) in the interval (% G ) can be given by
The formal solution of (G.26) in the interval (% G ) can be given by
(G.30) is expanded to let us carry out single step numerical integration as follows:
(G.30) is expanded to let us carry out single step numerical integration as follows:
IT. (Gg, ) can be obtained from solutions of 16 initial value problems such that
IT. (Gg, ) can be obtained from solutions of 16 initial value problems such that
Kutta Method.
Kutta Method.
Nonzero elements of iT, (G, )}computed by using Matlab Symbolic Toolbox are listed a:
Nonzero elements of iT, (G, )}computed by using Matlab Symbolic Toolbox are listed a:
Figure H.1 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=0) and (m=1);
Figure H.1 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=0) and (m=1);
Figure H.2 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=1) and (m=1);
Figure H.2 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=1) and (m=1);
Figure H.3 Mode Shapes in displacements of (a) W (b) Ug (c) Ug for (n=2) and (m=1);
Figure H.3 Mode Shapes in displacements of (a) W (b) Ug (c) Ug for (n=2) and (m=1);
Figure H.4 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=0) and (m=1);
Figure H.4 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=0) and (m=1);
Figure H.5 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=1) and (m=1);
Figure H.5 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=1) and (m=1);
Figure H.6 Mode Shapes in displacements of (a) W (b) U g(©) Ue for (n=2) and (m=1);
Figure H.6 Mode Shapes in displacements of (a) W (b) U g(©) Ue for (n=2) and (m=1);
Figure H.7 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=0) and (m=1);
Figure H.7 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=0) and (m=1);
Figure H.8 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=1) and (m=1);
Figure H.8 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=1) and (m=1);
Figure H.9 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=2) and (m=1);
Figure H.9 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=2) and (m=1);
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