Laminated plates stiffnesses

From the pre determined elastic and strength properties of the plies determine the laminate plate stiffness and strength properties from 4.10.3 and 4.10.5 respectively. 

Classical laminated plate tlieoiy is used to determine the stiffness of laminated composites. Details of the Kitchoff-Love hypothesis on which the theory is based can be found in standard texts (1,7,51). Essentially, the strains in each ply of the laminate ate represented as middle surface strains plus... 

A simply supported rectangular plate is used consistently in all sections to illustrate the kinds of results that can be obtained, i.e., the influence of the various stiffnesses on laminated plate behavior. In addition, only the simplest types of loading will be studied in order to avoid the solution difficulties inherent to complex loadings. Accordingly, in the interest of simplicity, just the bare thread of laminated plate results will be displayed. 

Note the presence of the bend-twist coupling stiffnesses in the boundary conditions as well as in the differential et uation. As with the specially orthotropic laminated plate, the simply supported edge boundary condition cannot be further distinguished by the character of the in-plane boundary conditions on u and v because the latter do not appear in any plate problem for a symmetric laminate. 

We will see that the unsymmetric laminate has more bending stiffness in the y-direction than the all-0°-layer laminate and almost as much bending stiffness in the x-direction. Thus, the center deflection of the unsymmetrically laminated plate should exceed that of the all-0°-layer laminated plate. However, we are already aware that bending-extension coupling increases deflections, so the center deflection of the unsymmetric laminate should exceed that of the orthotropic laminated plate. 

In their pioneering paper on laminated plates, Reissner and Stavsky investigated an approximate approach (in addition to their exact approach) to calculate deflections and stresses for antisymmetric angie-ply laminated plates [5-27]. Much later, Ashton extended their approach to structural response of more general unsymmetrically laminated plates and called it the reduced stiffness matrix method [5-28]. The attraction of what is now called the Reduced Bending Stiffness (RBS) method is that an unsymmetrically laminated plate can be treated as an orthotropic plate using only a modified D matrix in the solution, i.e.,... 

Alternatively, tests can be used to obtain the basic stiffness properties of the material form and their corresponding range measured by some statistical property such as the standard deviation. In two-dimensional cases where there are no significant loads in the out-of-plane direction, the basic orthotropic stiffness properties in Eqn (6.1) can be measured experimentally. Then, the classical laminated plate theory described in previous sections for determination of stiffness can be used effectively to model these sttuctures. Alternatively, the four basic stiffiiesses for 3-D woven composites can be... 

However, especially for three-dimensional stmcture, or, even, in two-dimensional stmcture with significant out-of-plane loads, the stiffness averaging of the classical laminated plate theory is not sufficient. In such cases, a specialized finite element model such as the binary model by Cox, Carter, and Fleck [25] can be used. In this model, Monte Carlo simulation allows the user to randomly orient tows and to randomly assign strength properties to the different components. 

Once the new matrix properties accounting for porosity are known, the ply-level stiffnesses can be determined using micromechanics, and then the classical laminated plate theory outlined in previous sections can be used. 

Engineering Sciences Data Unit (ESDU), Transverse (through-thethickness) shear stiffnesses of fibre reinforced composite laminated plates (computer program). Item No. 89013, S14, 1989. 

Laminate plate and shell stiffness classical lamination theory (CLT)... 

ESDU (1994) Stiffnesses of laminated plates. No. 94003, Engineering Sciences Data Unit, London. 

As stated by Remark 6.4, the wall thickness of the considered beams is small in comparison to the cross-sectional dimensions, being reflected in r (s) and rs(s), and to the radius of curvature R s). These geometric orders of magnitude also enter the constitutive description of the laminated beam wall in Eq. (6.4b). The plate stiffness coefficients 613(5) and 633(5) and coupling stiffnesses 831(5) and 633(5) essentially depend upon the difference of cubed, respectively squared, laminae positions in the thickness direction, while the membrane stiffness kss s) is a function of the laminae thicknesses. To comply with Remark 6.4, it is necessary to revise the formulation of the warping... 

Stiffnesses for single-layered configurations are treated first to provide a baseline for subsequent discussion. Such stiffnesses should be recognizable in terms of concepts previously encountered by the reader in his study of plates and shells. Next, laminates that are symmetric about their middle surface are discussed and classified. Then, laminates with laminae that are antisymmetrically arranged about their middle surface are described. Finally, laminates with complete lack of middle-surface symmetry, i.e., unsymmetric laminates, are discussed. For all laminates, the question of laminae thicknesses arises. Regular laminates have equal-thickness laminae, and irregular laminates have non-equal-thickness laminae. 

Often, because specially orthotropic laminates are virtually as easy to analyze as isotropic plates, other laminates are regarded as, or approximated with, specially orthotropic laminates. This approximation will be studied by comparison of results for each type of laminate with and without the various stiffnesses that distinguish it from a specially orthotropic laminate. Specifically, the importance of the bend-twist coupling terms D,g and D26 will be examined for symmetric angle-ply laminates. Then, bending-extension coupling will be analj ed for antisym-... 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

M. S. Ewing, R. J. Hinger, and A. W. Leissa, On the Validity of the Reduced Bending Stiffness Method for Laminated Composite Plate Analysis, Composite Structures, Volume 9, 1988, pp. 301-317. 

Using the unidirectional stiffness properties of the composite material (HTA/6376) in Table 11.1, the laminates were modelled with one orthotropic solid element per ply in the thickness direction, leading to 16 elements through-thickness for both the splice plate and skin plate. As before, the titanium bolts were modelled with isotropic material properties, with material constants Eb = 110 GPa, Vb = 0.29. Linear 8-node hexa-hedral brick elements with a reduced integration scheme were used for the laminates and bolts. This element formulation was used to reduce the cost of the analysis and size of the output files, which were very large. 

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