Transverse shear

The Kirchhoff hypothesis of negligible transverse shear strains, Yxz and tutes... 

Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

HOLES IN LAMINATES TRANSVERSE SHEAR EFFECTS POST-CURING LAMINATE SHAPE ... 

Composite materials typically have a low matrix Young s modulus in comparison to the fiber modulus and even in comparison to the overall laminae moduli. Because the matrix material is the bonding agent between laminae, the shearing effect on the entire laminate is built up by summation of the contributions of each interlaminar zone of matrix material. This summation effect cannot be ignored because laminates can have 100 or more layersi The point is that the composite material shear moduli and G are much lower relative to the direct modulus than for isotropic materials. Thus, the effect of transverse shearing stresses. 

Study of transverse shearing stress effects is divided in two parts. First, some exact elasticity solutions for composite laminates in cylindrical bending are examined. These solutions are limited in their applicability to practical problems but are extremely useful as checl oints for more broadly applicable approximate theories. Second, various approximations for treatment of transverse shearing stresses in plate theory are discussed. 

Obviously, the classical lamination theory stresses in Pagano s example converge to the exact solution much more rapidly than do the displacements as the span-to-thickness ratio increases. The stress errors are on the order of 10% or less for S as low as 20. The displacements are severely underestimated for S between 4 and 30, which are common values for laboratory characterization specimens. Thus, a practical means of accounting for transverse shearing deformations is required. That objective is attacked in the next section. 

The preceding subsection was devoted to a comparison of a special exact elasticity solution with classical lamination theory results. The importance of transverse shear effects was clearly demonstrated. However, that demonstration was for a special problem of rather narrow interest. The objective of this subsection is to display approaches and results for the approximate consideration of transverse shear effects for general laminated plates. 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

The transverse shearing stress distribution is then approximated... 

Figure 6-23 Transverse Shear Stress Distribution along...

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

Eric Reissner, The Effect of Transverse Shear Deformation on the Bending of Elastic Plates, Journal of Applied Mechanics, June 1945, pp. A-69-77. 

E. Reissner, A Consistent Treatment of Transverse Shear Deformations in Laminated Anisotropic Piates, AIAA Journal, May 1972, pp. 716-718. 

Another issue that turns out to be very important for the sandwich-blade stiffener, but not at all important for the hat-shaped stiffener, is shear in the vertical web. Not shear in the plane of the web, but shear in the plane perpendicular to the web. This transverse shear stiffness turns out to dominate the behavior or be very important in the behavior of the sandwich blade, but simply is not addressed at all in the hatshaped stiffener. You can imagine that the transverse shearing stiffness would be more important in the sandwich blade when you consider the observation that the sandwich blade is a thick element and the hatshaped stiffener is a thin element. That is, bending and in-plane shear would dominate this response, whereas transverse shear, because the sandwich blade is thick, can very easily be an important factor in the sandwich blade. For both stiffeners, appropriate analyses and design rationale have been developed to be able to make an optimally shaped stiffener. 

The next problem area is transverse shearing effects. There are some distinct characteristics of composite materials that bear very strongly on this situation because for a composite material the transverse shearing stiffness, i.e., perpendicular to the plane of the fibers, is considerably less than the shear stiffness in the plane of the fibers. There is a shear stiffness for a composite material in a plane that involves one fiber direction. Shear involves two directions always, and one of the directions in the plane is a fiber direction. That shear stiffness is quite a bit bigger than the shear stiffness in a plane which is perpendicular to the axis of the fibers. The shear stiffness in a plane which is perpendicular to the axis of the fibers is matrix-dominated and hardly fiber-influenced. Therefore, that shear stiffness is much closer to that of the matrix material itself (a low value compared to the in-plane shear stiffness). 

Now recognize an apparent contradiction in classical plate theory. First, from force equilibrium in the z-direction, we saw transverse shear forces and Qy must exist to equilibrate the lateral pressure, p. However, these shear forces can only be the resultant of certain transverse shearing stresses, i.e.. 

However, these transverse shearing stresses were neglected implicitly when we adopted the Kirchhoff hypothesis of lines that were normal to the undeformed middle surface remaining normal after deformation in Section 4.2.2 on classical lamination theory. That hypothesis is interpreted to mean that transverse shearing strains are zero, and, hence, by the stress-strain relations, the transverse shearing stresses are zero. The Kirchhoff hypothesis was also adopted as part of classical plate theory in Section 5.2.1. 

We can reexamine the beam problem to determine the distribution of the transverse shearing stress -c z- know that the resultant of -c z is V which we obtain from Equation (D.7), i.e.,... 

Accordingly, we find it difficult to determine the distribution of the transverse shearing stress in a beam, much less in a plate. Procedures for determining the approximate transverse shear stress distribution in plates are described in Section 6.5.2. 

The failure mechanisms of interest in reinforced masonry wall elements include flexural, transverse shear, in-plane shear and in some cases, combined axial compression and flexure. Buckling failure modes of compression elements and connection failures are to be avoided. 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

Figure 4. Theoretical trends for —(storage) and dissipation as the inner slip is varied between no slip (0) and strong slip (1) for a coated transverse shear acoustic wave device in water. The thickness of the film is 5 nm. The solid line displays the decrease in storage, and the dashed line shows the change in dissipation.

The Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

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