Deflection of plates

You will find the formulae for the elastic deflections of plates and beams under their own weight in standard texts on mechanics or structures (one is listed under Further Reading at the end of this chapter). We need only one formula here it is that for the deflection, 8, of the centre of a horizontal disc, simply supported at its... 

Obvious and sometimes drastic simplifications occur when the laminate is symmetric about the middle surface (By = 0), specially orthotropic (all the terms with 16 and 26 subscripts vanisn in addition to the By), homogeneous ( = 0 and Djj = Ayr/12), or isotropic. In all those cases. Equations (5.6) and (5.7) are coupled to each other, but uncoupled from Equation (5.8). That is. Equation (5.8) contains derivatives of the transverse displacement w only, and Equations (5.6) and (5.7) contain both u and V but not w. Accordingly, only Equation (5.8) must be solved to determine the transverse deflections of a plate with the aforementioned... 

A prominent part of many of the techniques is separation of variables. In that method, the deflection variables, or the variation In deflection variables, are arbitrarily separated into functions of plate coordinate x alone times functions of y alone. Wang [5-8] determined that separation of variables leads to exact solutions for some classes of plate problems, but does not for others, I.e., the deflections are not always separable. A specific example of an approximate use of separation of variables due to Ashton [5-9] will be discussed in Section 5.3.2. Other exact uses of the method abound throughout Section 5.3 through 5.5. 

DEFLECTION OF SIMPLY SUPPORTED LAMINATED PLATES UNDER DISTRIBUTED TRANSVERSE LOAD... 

Because exact solutions for skew isotropic plates are readily available, Ashton was able to get some exact solutions for anisotropic rectangular plates by the special identification process outlined in the preceding paragraph. Specifically, values for the center deflection of a uniformly loaded square plate are shown in Table 5-2. There, the exact solution is shown along with the Rayleigh-Ritz solution and the specially orthotropic solution. For the case already discussed where D22/D11 -1, (D.,2-i-2De6)/Di. = 1.5, and D.,g/Di., = D2 Di., =-.5, the exact solution is... 

Figure 5-13 Deflection of an Antisymmetric Cross-Ply Laminated Plate under Sinusoidal Transverse Load...

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. 

Robert M. Jones and Harold S. Morgan, Deflection of Unsymmetrically Laminated Cross-Ply Rectangular Plates, Proceedings ol the 12th Annual Meeting of the Society of Engineering Science, 20-22 October 1975, Austin, Texas, pp. 155-167. 

If now a second, time varying, voltage is applied to the horizontal plates, this will cause a vertical deflection of the beam, and the result is a trace on the screen, which corresponds to the variation of this second voltage with time. The action of the beam on the screen causes a fluorescent trace to appear on the screen as the beam is deflected. The time for which this trace persists is a function of the electron density in the beam and the material with which the screen is coated. 

In optics, one frequently wants to know the deflections of an axially loaded plate. Mirrors are often good approximations to thin plates and it is important to understand the deflections of mirrors when loaded by gravity, wind, or other point loads. Fortunately this is a relatively well studied subject, so many solutions exist in the literature. A particularly valuable general reference book is Theory of Plates and Shells, by Timoshenko and Woinowsky-Krieger. 

To prevent failure due to the disengagement of the pane out of the frame, bite or edge engagement depths are required. They are based upon the assumption that the plate will distort as a spheroid surface. At the maximum design center deflection of 15 pane thicknesses, this conservatively approximates the deflection shape function. To be conservative, a 0.5-inch safety margin is added to all calculations. 

The compressive behavior of a DL is a very important mechanical property. Therefore, to study the mechanical properties of various diffusion materials (carbon cloths, carbon fiber papers, and carbon felts), Escribano et al. [251] used a compression cell. The sample diffusion materials were placed between the two plates of the cell, and the thickness and deflection of each sample were measured as a function of the compression pressure. These researchers... 

Fig. 9.8. Deflection of a bimorph. Two long, thin plates of piezoelectric material are glued together, with a metal film sandwiched in between. Two more metal films cover the outer surfaces. Both piezoelectric plates are poled along the same direction, perpendicular to the large surface, labeled z. (A) By applying a voltage, stress of opposite sign is developed in both plates, which generates a torque. (B) The bimorph flexes to generate a stress to compensate the torque. The neutral plane, where the stress is zero, lies at hi i from the central plane.

For efficient removal, these plates should ideally be placed at the point at which ions attain their tightest temporal and spatial focus prior to detection. Although deflection of this type would be challenging with a swept-beam geometry, the space focus plane of the two-stage acceleration design offers an excellent opportunity to deflect the maximum number of ions over the shortest distance in the shortest time. This placement also ensures the most resolution of mass deflection. For an ICP-TOF-MS, the efficiency of removal for this parallel-plate scheme was promising [24]. 

This action eliminates the need for a costly mechanical roughening process that most other materials require. The depositing of a metal surface on plastic parts can increase environmental resistance of the part, also its mechanical properties and appearance. As an example a plated ABS part (total thickness of plate 0.015 in.) exhibited a 16% increase in tensile strength, a 100% increase in tensile modulus, a 200% increase in flexural modulus, a 30% increase in Izod impact strength, and a 12% increase in deflection temperature. Tests on outdoor aged samples showed complete retention of physical properties after six months. 

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