The Behavior of Structures Composed of Composite Materials Part 5 pps

109 3.9 Quasi-Isotropic Composite Panels Subjected to a Uniform Lateral Load When a composite laminate has a stacking sequence in which it is referred to as quasi-isotropic. In that case it behaves as an isotropic plate in the determination of lateral deformations, w ( x, y ), and stress couples, and For isotropic plate monocoque plates several textbooks such as Timoshenko and Woinowsky-Krieger [4] and Vinson [7] have provided expressions for the maximum deflection, and the maximum stress couple, M , a plate attains when subjected to a constant laterally distributed load such as, where a is the smaller plate dimension; b is the longer plate dimension, E is the modulus of elasticity of the plate; h is the plate thickness; and The dimensionless constants and are given in tabular form for various boundary conditions, and these are repeated herein for completeness in Tables 3.1 through 3.4. Table 3.5 also provides information for the case wherein the plate is subjected to a hydraulic head. These tables and procedures are well known and well used. 110 Of course, for the isotropic plate, the flexural stiffness is given by and the maximum bending stress, which occurs on the top and bottom surfaces of the plate, is Also for Tables 3.1 through 3.5, the numerical coefficients correspond to a Poisson’s ratio of wherein Therefore, for materials with other Poisson ratios, ,, Equation (3.71) must be changed to 111 With Equations (3.75) and (3.72), the well-used Tables 3.1 through 3.4 for monocoque plates can be used to analyze quasi-isotropic composite plates as well. It must be remembered that for these classical theory solutions no transverse-shear deformation effects are included. Table 3.5 can be used to analyze and design composite material plates subjected to a hydraulic head. It is seen that for a monocoque quasi-isotropic plate design, The plate must not be overstressed, i.e., the maximum stress is determined from the use of Equation (3.72) to determine the maximum stress couple, M, and the procedures described earlier to determine the stresses in each lamina. The determined maximum stress cannot exceed some allowable stress, defined by the material’s ultimate stress or yield stress divided by a factor of safety on ultimate stress or yield stress, whichever is smaller. This requires a certain value of plate thickness, h. The monocoque plate must not be over deflected determined by Equation (3.75). This is sometimes specified, but in other cases the plate deflection cannot exceed the plate thickness or some fraction thereof. If the maximum plate deflection reaches a value of the plate thickness, h, the equations discussed herein become inapplicable because the plate behavior becomes increasingly nonlinear which requires that other equations be used. Again, to prevent over deflection, a plate thickness, h, is required as determined by Equation (3.75). (1) (2) Therefore, in monocoque plate design, the plate thickness, h, is determined either from a strength or stiffness requirement, whichever requires the larger thickness. 3.10 A Static Analysis of Composite Material Panels Including Transverse Shearr Deformation Effects The previous derivations have involved "classical" plate theory, i.e., they have neglected transverse shear deformation effects. Because in many composite material laminated plate constructions, transverse-shear deformation effects are important, a more refined theory will now be developed. However, because of its simplicity, and the number of solutions available, classical theory is still useful for preliminary design and analysis to size the structure required in minimum time and effort. In the simpler classical theory, the neglect of transverse shear deformation effects means that To include transverse shear deformation effects, one uses 112 Now substituting the admissible forms of the displacement for a plate or panel, Equation (2.49) into Equations (3.76) and (3.77), shows that No longer are the rotations and explicit functions of the derivatives of the lateral deflection w, as shown by Equation (3.23) for classical plate theory. The result is that for this refined theory there are five geometric unknowns, and instead of just the first three in classical theory. Now one needs to look again at the equilibrium equations, the constitutive equations (stress-strain relations), the strain-displacement relations and the compatibility equations. For the plate, the equilibrium equations are given by Equations (3.9) through (3.15), because they do not change from classical theory. The constitutive equations for a composite material laminated plate and sandwich panel are given by Equations (2.58) through (2.66). The new cogent strain-displacement (kinematic) relations are given above in Equation (3.78) and (3.79). Because the resulting governing equations are in terms of displacements and rotations, any single valued, continuous solution will by definition satisfy the compatibility equations. As an example consider a plate that is mid-plane symmetric and has no coupling terms the constitutive equations for this orthotropic plate can be written as follows, where is a transverse shear coefficient to be discussed later. 113 Because the plate is mid-plane symmetric there is no bending-stretching coupling, hence the in-plane loads and deflections are uncoupled (separate) from the lateral loads, deflections and rotations. Hence, for the lateral distributed static loading, p ( x, y ) , Equations (3.16) through (3.18) and Equations (3.83) through (3.87) are utilized: 8 equations and 8 unknowns. Substituting Equations (3.83) through (3.87) into Equations (3.16) through (3.18) and using Equation (2.66) results in the following set of governing differential equations for a laminated composite plate subjected to a lateral load, with and no applied surface shear stresses (for simplicity) The inclusion of transverse shear deformation effects results in three coupled partial differential equations with three unknowns, and w, contrasted to having one partial differential equation with one unknown, w, in classical plate (panel) theory; see Equation (3.29). Incidentally if one specified that and substituting that into Equations (3.88) through (3.90) reduces the three equations to Equation (3.29), the classical theory equation. The symbol with no subscript in (3.88) through (3.90) is a transverse shear deformation shape factor which varies from 1 to 2 depending upon the geometry. The classical plate theory governing partial differential equation is fourth order in both x and y, and therefore requires two and only two boundary conditions on each of the four edges, as discussed in Section 3.4. This refined theory, including transverse shear 114 deformation, is really sixth order in both x and y, and therefore requires three boundary conditions on each edge as discussed in Section 3.11 below. If the laminated plate is orthotropic but not mid-plane symmetric, i.e., the governing equations are more complicated than Equations (3.88) through (3.90) and are given by Whitney [8], Vinson [9] and are discussed briefly in Section 3.23 below. 3.11 Boundary Conditions for a Plate Using the Refined Plate Theory Which Includes Transverse Shear Deformation 3.11.1. SIMPLY-SUPPORTED EDGE Again Equation (3.30) holds, but now a third boundary condition is required for the plate bending because it can be shown that Equations (3.88) through (3.90) are sixth order in w with respect to x and y. In addition, since the in-plane and lateral behavior are coupled, a fourth boundary condition enters the picture as well. This has resulted in the use of two different simply supported boundary conditions, both of which are mathematically admissible as natural boundary conditions (to be discussed later) and are practical structural boundary conditions. By convention the simply supported boundary conditions are given as follows: where is the mid-surface displacement in the x-direction and is the mid-surface displacement in the y -direction. Whether one uses S1 or S2 boundary conditions is determined by the physical aspects of the plate problem being studied. 3.11.2. CLAMPED EDGE Similarly, for a clamped edge the lateral deflection w and the rotation or (for an x = constant edge or a y = constant edge, respectively) are zero (note: the slope is not zero) and the other boundary conditions are analogous to Equation (3.31). 115 3.11.3. FREE EDGE The free edge requires three boundary conditions on each edge; therefore, it is no longer necessary to resort to the difficulties of the Kirchhoff boundary conditions for the bending of the plate needed for classical plates. The boundary conditions for the bending of the plate are simply: where n and t are directions normal to and tangential with the edge. Again, the in-plane boundary conditions for the free edge are 3.11.4. OTHER BOUNDARY CONDITIONS In addition to the above boundary conditions, which are widely used to approximate the actual structural boundary conditions, sometimes it is desirable to consider an edge whose lateral deflection is restrained, whose rotation is restrained or both. The means by which to describe these boundary conditions is given for example in [7, pp. 20-21]. 3.12 Composite Plates on An Elastic Foundation Consider a composite material plate that is supported on an elastic foundation. In most cases an elastic foundation is modeled as an elastic medium with a constant foundation modulus, i.e., a spring constant per unit planform area, of k in units such as Therefore, the elastic foundation acts on the plate as a force in the negative direction proportional to the local lateral deflection w ( x,y ) . The force per unit area is -kw, because when w is positive the foundation modulus is acting in a negative direction, and vice versa. In order to incorporate the effect of the elastic foundation modeled as above one simply adds another force to the p ( x,y ) load term. The results are, that for classical theory, Equation (3.29) is modified to (3.94), and for the refined theory, Equation (3.90) is modified to Equation (3.95): 116 3.13 Solutions for Plates of Composite Materials Including Transverse-Shear Deformation Effects, Simply Supported on All Four Edges Some solutions are now presented for the equations in Section 3.10 and 3.11, using the governing differential equations (3.88) through (3.90). In the following with no subscript is a transverse shear factor, often give as or 5/6. Dobyns [10] has employed the Navier approach to solving these equations for a composite plate simply supported on all four edges subjected to a lateral load, using the following functions: It is seen that Equations (3.96) through (3.98) satisfy the simply supported boundary conditions on all edges given in Equation (3.91). Substituting these functions into the governing differential equations (3.88) through (3.90) results in the following: if and is the lateral load coefficient of (3.99) above, then the operators are given by the following: 117 Solving Equation (3.100), one obtains where det is the determinant of the [ L ] matrix in Equation (3.100). Having solved the problem to obtain and w, the curvatures and may be obtained. These then can be substituted back into Equations (3.20) through (3.22) to obtain the stress couples and to determine the location where they are maximum, to help in determining where the stresses are maximum. For a laminated composite plate, to find the bending stresses in each lamina one must use the above equations to find the values for and in Equation (3.23). Finally, for each lamina the bending stresses can be found using: The stresses in each lamina in each direction must be compared to the strength of the lamina material in that direction. Keep in mind that quite often the failure occurs in the weaker direction in a composite material. Looking at the load p ( x,y ) in Equation (3.99), if the lateral load p ( x,y ) is distributed over the entire lateral surface, then the Euler coefficient, is found to be 118 If that load is uniform then, For a concentrated load located at and where P is the total load. For loads over a rectangular area of side lengths u and v whose center is at and as shown in Figure 3.8, is given as follows: [...]... considered as a homogeneous material, then the lower face can be considered as lamina 1, the core as lamina 2, and the upper face as lamina 3 If the face were, for example made of a four ply cross-ply, then the lower face can be considered as laminae 1 through 4, the core is lamina 5 and the top face is laminae 5 through 9 Either classical theory of the inclusion of transverse shear deformation can be... torsional stiffness of the construction of Figure 3.10 is 138 The above merely illustrates what one can and must do to develop the basic mechanics of materials gross formulation for the extension, bending or twisting of a rectangular section, perhaps composed of very esoteric composite materials but used for a water ski, windmill blade or other shapes for many other purposes 3.21 Methods of Analysis for... which value of m and n result in the lowest critical buckling load All values of n appear in the numerator, so n = 1 is the necessary 132 value for this case of all four edges simply supported But m appears several places, and depending upon the value of the flexural stiffness and and the length to width ratio of the plate, a/b, it is not clear which value of m will provide the lowest value of However,... of four things happens: (1) The amplitude of vibration grows until the ultimate strength of a brittle material is exceeded and the structure fails (2) Portions of the structure exceed the yield strength, plastically deform and the behavior changes drastically (3) The amplitude grows until nonlinear effects become significant, and there is no natural frequency (4) Due to damping or other mechanism the. .. other hence the compartmentalization of the energy) The normal modes comprise the solutions to the homogeneous governing differential equations, which are now zero only at the eigenvalues for those equations and boundary conditions If there is a forcing function, then the particular solution for the specific forcing function (which can be cyclical or a one time dynamic impact load) is added onto the. .. 3.20 Some Remarks On Composite Structures So far in this chapter, plates made of composite materials have been discussed However, there are complicated constructions, which are made either from composite materials or from isotropic materials, which can be referred to as composite structures One such structure is a box beam shown below in Figure 3.10, which could be the crosssection of a windmill blade,... compared to the other ones, so Similar expressions can easily be constructed for the torsional stiffness Consider the construction of Figure 3.10 subjected to a torsional load T in inch-lbs about the xaxis Then it is clear that Now from Equation (2.66), for both elements, If is the angle of twist caused by the torque T over the length L, then for element 1 and 2 It is also seen that So the GJ, the torsional... for the natural frequencies of vibration where the unprimed L quantities were defined below Equation (3.100) and 124 Three eigenvalues (natural frequencies) result from solving Equation (3.1 25) for each value of m and n However, two of the frequencies are significantly higher than the other because they are associated with the rotatory inertia terms, which are the last terms on the left-hand sides of. .. symmetry no other couplings but includes transverse shear deformation, then one sets p(x,y) = 0 in Equation (3.90), but adds to the right-hand side In addition, because and are both dependent variables that are independent of w, there will be an oscillatory motion of the lineal element across the plate thickness about the mid-surface of the plate This results in the last term on the left-hand side of Equations... will be included The constitutive equations are given in (2.66), where the curvatures are defined by (3.23) For the buckling of the plate due to in-plane loads, the equilibrium equations for a buckled plate are [17]: 133 The five equilibrium equations can be written in terms of the five unknowns, the three displacements, u, v and w, and the two rotations, and Solving the five coupled partial differential . necessary to resort to the difficulties of the Kirchhoff boundary conditions for the bending of the plate needed for classical plates. The boundary conditions for the bending of the plate are simply: where. when n/b = 1/v, m/a = 1/u, then Of course, any other lateral load can be characterized by the use of Equation (3.1 05) . 3.14 Dynamic Effects on Panels of Composite Materials Seldom in real life. until one of four things happens: The amplitude of vibration grows until the ultimate strength of a brittle material is exceeded and the structure fails. Portions of the structure exceed the yield