Plyty Wzmocnione Belkami
The establishment of the boundary conditions is based on the assumptions that
1) the edge beam (constituting a whole with the plate) is deformed in the same way as the plate at the contact line,
2) the conditions of equilibrium of the beam and the plate are
The concepts and equations for the statement of the problem are described in Art. 2 to 7. The reasoning is based on the Kirchhoff hypothesis and the hypothesis of plane cross-sections. The condition 1 is expressed by the Eqs. (6.8) and (6.9) and determines the relations between the mechanical quantities in the beam and in the plate. This follows from the equality of deformations of the beam and the plate at the contact line. The equations considered are derived from
the Eqs. (2.18) describing the relations between the coordinates of the strain tensor and the mechanical quantities in the plate, and from the Eqs. (6.1), (6.2) expressing the mechanical quantities in the beem in function of the coordinates of the strain tensor at the edge of the plate. The condition 2 is expressed by the equilibrium equations for the beam (4.6), (4.9) acted on by the plate and the external load. In general, we obtain six equations (6.8) and (6.9), all the mechanical quantities of the beam being thus expressed in terms of those of the plate. For an unequivocal statement of the problem, only four of these relations will be used. These are the relations between the axial force and the moment’s in the beam and the statical quantities in the plate. Such a procedure is explained in Art. 8 and illustrated by an example of a rectangular beam. From the equations of equilibrium of the beam (8.8), and from the Eqs. (8.11); we can obtain the conditions for an edge stiffened with a beam. The mechanical quantities in the plate being described by means of two stress functions 0(x,y) and w (x, y), the conditions for an edge stiffened with a beam (8.14) are expressed in terms of these two functions.
They ccmprise such particular cases as those of:
a) free edge (8.15),
b) clemped edge (8.18) and (8.19),
c) hinged support (8.17),
d) elastic support on a beam, as commonly used hitherto, (8.16).
The boundary conditions for a circular plate (9.11) are obtained in an analogous manner in Art. 9. They contain also the cases of free and damped edge.
Boundary conditions for any other particular case may be established in a similar manner.
The conditions for a stiffened edge are expressed in a differential form and do not depend on the way in which the beam is supported. Therefore, some support conditions should be assumed at the ends of the beam or at the points where the beam axis is broken (if the beam constitutes a broken line). These are established according to the principles of strength of materials for lattice structures. The particular cases in Art. 10 illustrate the method presented, and concern a rectangular plate with two edges stiffened. The plane state of stress is considered in addition to that of bending. The influence of beam prestressing on the mechanical quantities of the plate is also calculated.
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