Plate buckling equations

The equilibrium equations for a beam are derived to illustrate the derivation process and to serve as a review in preparation for addressing plates. Then, the plate equilibrium equations are derived for use in Chapter 5. Next, the plate buckling equations are discussed. Finally, the plate vibration equations are addressed. In each case, the pertinent boundary conditions are displayed. Nowhere in this appendix is reference needed to laminated beams or plates. All that is derived herein is applicable to any kind of beam or plate because only fundamental equilibrium, buckling, or vibration concepts are used. 

The plate buckling equations inherently cannot be derived from the equilibrium of a differential element. Instead, the buckling problem represents the departure from the equilibrium state when that state becomes unstable because the in-plane load is too high. The departure from the equilibrium state is accompanied by waves or buckles in the surface of the plate. That is, the plate cannot remain flat when the... 

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. 

One of the major complications in the plate buckling solution is the need to investigate the influence of buckle mode shape on the buckling load itself. That is, the plate buckling load in Equation (5.81) is a function... 

However, because we are usualiy interested oniy in the lowest buckling load for a column, m is always one. For plates, both m and n enter the buckling equation as well as the plate aspect ratio, a/b, so the lowest buckling load does not typically occur for m = 1 and n = 1. Thi, we must find the absolute minimum of the values of the buckling load, N, or more generally, X, for a wide range of m and n values. 

Note that if B g and 825 are zero, then and T23 are also zero, so Equation (5.92) reduces to the specially orthotropic plate solution. Equation (5.65). The character of Equation (5.92) is the same as that of Equation (5.81) for antisymmetric cross-ply laminated plates, so the remarks on finding the buckling load in Section 5.4.3 are equally applicable here. 

The solution to this fourth-order partial differential equation and associated homogeneous boundary conditions is just as simple as the analogous deflection problem in Section 5.3.1. The boundary conditions are satisfied by the variation in lateral displacement (for plates, 5w actually is the physical buckle displacement because w = 0 in the membrane prebuckling state however, 5u and 8v are variations from a nontrivial equilibrium state. Hence, we retain the more rigorous variational notation consistently) ... 

The presence of D g 26 governing differential equation and the boundary conditions renders a closed-form solution impossible. That is, in analogy to both bending and buckling of a symmetric angle-ply (or anisotropic) plate, the variation in lateral displacement, 5vy, cannot be separated into a function of x alone times a function of y alone. Again, however, the Rayleigh-Ritz approach is quite useful. The expression... 

GOVERNING EQUATIONS FOR BEAM EQUILIBRIUM AND PLATE EQUILIBRIUM, BUCKLING, AND VIBRATION... 

In the particular problem being discussed in this section, all deformation fields over the midplane are independent of y and are represented in terms of the coordinate x as illustrated in Figure 5.2. As noted by Fung (1965), equilibrium deformation within the buckle can be described by means of the nonlinear von K man equations for thin plates. Such an analysis is predicated upon several assumptions and observations, as listed below. 

As is the case with most plate theories, the governing equations are expressed in terms of fields defined over the midplane surface of the undeformed plate or over the uniformly compressed midplane of the plate prior to buckling. Deformation at material points away from the midplane is described in terms of these fields. 

If the magnitude of the center point deflection of the film wq increases to values on the order of film thickness h, then the potential arises for generation of significant membrane stress in the film due to transverse deflection (in addition to any residual membrane stress which may be present in the film prior to deflection). As in the case of film buckling, the von Karman plate theory provides a useful and effective framework for describing response with center point deflection wq of magnitude equal to several times the film thickness. In the present case, the von Karman equations reduce to the pair of ordinary differential equations... 

Vessels of noncircular cross section may be subjected to external pressure Membrane and bending stresses are considered the same as for internal pressure unless the resulting stresses are compressive where stability may be a possible mode of failure. Interaction equations are used to examine die various plates for stability. Calculated stresses are compared with critical buckling stresses with ii factor of safety applied. This is described in Article 13-14 of the ASME Code,... 

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