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.
The following forces can act on a plate edge:
This gives us 3 conditions that must be fulfilled, but it can be problematic to work
with more than two boundary conditions in the thin plate theory. Kirchhoff and Kelvin
solved this by combining the torsion moment, Mnt, and the shear force, Qn,
into one "edge shear force", Vn.
- Mn - Bending moment
- Mnt - Torsion moment
- Qn - Shear force
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
w, w,n, Mn, Vn, where n is the direction normal to the edge.
Her is an illustration of the three typical boundary conditions of a plate:
- (1) - Clamped edge
w = 0
w,n = 0
- (2) - Free edge
Mn = 0
Vn = 0
- (3) - Pinned edge
Mn = 0
w = 0
The requirement of bending moment, Mn can, however, be replaced by requirement
of zero curving/camber, expressed by w,nn, on the edge.
Soultion of the differential equation
We now want to solve the plates differential equation, that is:
As discussed earlier, this is a linear, non-homogeneous, partial differential equation
and can be solved in several ways. Four main solution methods exists:
Direct solution on closed form
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.
Solution of the homogeneous part of the equation, and then superposition of
the particular solution
Navier solution - Use of double Fourier series
This 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 series
This 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.
Other solution methods
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.