Plates - solution methods
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The linear thin plate problem is solved by finding a displacement function that satisfies the differential equation obtained and the boundary conditions for the plate. First, however, we must take a quick glance at boundary conditions for a plate.
- M
_{n}- Bending moment - M
_{nt}- Torsion moment - Q
_{n}- Shear force
_{nt}, and the shear force, Q_{n},
into one "edge shear force", V_{n}.
In addition, we have the two geometrical boundary conditions, expressed by: - w - displacement at edge
- w,
_{n}- slope at edge
Boundary conditions on a plates edge can therefore be determined by two of the four entities
w, w,
- (1) - Clamped edge
w = 0 w,_{n}= 0 - (2) - Free edge
M_{n}= 0 V_{n}= 0 - (3) - Pinned edge
M_{n}= 0 w = 0
The requirement of bending moment, M
We now want to solve the plates differential equation, that is: Very limited solution method, and analogous to the inverse method for solution of shell problems. You start by assuming a displacement pattern, e.g. sinusoidal form, and determine what load and boundary conditions that gives this displacement, where the loads also comes out as a sinusoidal distribution.**Direct solution on closed form****Solution of the homogeneous part of the equation, and then superposition of the particular solution****Navier solution**- Use of double Fourier seriesThis solution metod can be used for a plate that is pinned along all edges. The method gives the displacement function from an arbitrary distributed load. Click to see the theory of Naviers solution here. Click to experiment with Navier's plate solution in this Java applet. When the displacement w(x,y) is known, inner moment, shear forces and stresses can be calculated. This calculation method has quick convergence for load distributions that covers the entire plate, and the method is well suited for solution on computers. **Levy's solution**- Use of single Fouries seriesThis method is used for plates with at least two pinned edges. The other two can have arbitrary boundary conditions. Click to see the theory of the Levy solution here. The method has quick convergence, and few parts are needed to establish a good expression for w(x,y). If moment and stresses are to be determined with a minimal error, more parts must be taken into consideration.
In ship or ocean structures you often have stiffened plate fields with hydrostatic pressure. To look at a single plate here, the appropriated boundary condition is more closer to that of a full fixed edge. This type of boundary condition means that Navier or Levy solution methods cannot be applied. A solution of this kind of problem requires a more sophisticated approach, but this theory is to deep to be presented here. |