Plate equilibrium 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. 

This review of the foregoing simple derivation will help you to understand the following derivation of the plate equilibrium equations. The major difference between plate and beam problems is that beams are one-dimensional and plates are two-dimensional. Therefore, beams have ordinary differential equations as governing equations whereas plates have partial differential equations. Moreover, in the derivation of the governing differential equations, there will necessarily be more force equilibrium and moment equilibrium equations for plates than for beams. 

We call a plate the shallow shell when k =k2 = 0. This implies that the plate mid-surface coincides with the plane z = 0, and the plate is limited by the two parallel planes z = h, z = —h and a boundary contour. Let us redenote the horizontal and vertical displacements of the plate mid-surface by u = ui, u = U2, w. In this case, the plate horizontal and vertical displacements are not coupled. Indeed, it follows from (1.18), (1.19), that U = (ui,U2) is described by the following equilibrium equations ... 

Substituting the moments into the equilibrium equation, we obtain the equation for an isotropic viscoelastic plate,... 

In this section we derive a nonpenetration condition between crack faces for inclined cracks in plates and discuss the equilibrium problem. As it turns out, the nonpenetration condition for inclined cracks is of nonlocal character. This means that by writing the condition at a fixed point we have to take into account the displacement values both at the point and at the other point chosen at the opposite crack face. As a corollary of this fact, the equilibrium equations hold only in a domain located outside the crack surface projection on the mid-surface of the plate. This section follows the papers (Khludnev, 1997b Kovtunenko et ah, 1998). 

Here i —> i is the convex and continuous function describing a plasticity yield condition. The function w describes vertical displacements of the plate, rriij are bending moments, (5.139) is the equilibrium equation, and equations (5.140) give a decomposition of the curvatures —Wjj as a... 

Here i —> i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij ... 

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

In plate theory, the problem is reduced from the deformation of a solid body to the deformation of a surface by use of the Kirchhoff hypothesis (normals to the undeformed middle surface remain straight and normal after deformation, etc., as discussed in Chapter 4). Then, we attempt to apply boundary conditions to that surface which is usually the middle surface of the plate. There should be no surprise that the boundary conditions for the unapproximated solid body are not the same as those for the solid approximated with a surface. The problem arises when these boundary conditions are applied to an approximate set of equilibrium equations that result when force-strain and moment-curvature... 

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... 

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... 

For a four component mixture, the above equilibrium relations will yield three independent equations thereby providing the three extra equations required to determine all the eight unknowns. The equilibrium equations together with the material balance equations are applied to each plate to calculate the vapor and liquid compositions for each plate as follows. 

The second method is a simplified one used to estimate the number of plates. This equation can only be used if both the equilibrium and operating... 

Now consider the same experimental set-up as above, but with the top plate rotated around its normal by radians so that the fixed director alignments on the boundaries are at right angles to each other as shown in Fig. 3.2(a). The expressions for the energy and equilibrium equation remain as stated in equations (3.4) and (3.6), respectively. The solution for 0 is again given by (3.7) but with the boundary conditions changed to... 

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

We consider a boundary value problem for equations describing an equilibrium of a plate being under the creep law (1.31)-(1.32). The plate is assumed to have a vertical crack. As before, the main peculiarity of the problem is determined by the presence of an inequality imposed on a solution which represents a mutual nonpenetration condition of the crack faces... 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

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