About the study of the long rectangle plate by Transfer-Matrix Method.

Suciu, Mihaela ; Paunescu, Daniela ; Balc, Gavril 等

1. INTRODUCTION

The rectangular plate with long lengths can be calculated by the Transfer-Matrix Method--the plate is discredited in parts of its length. One part is equal at the unit and it is as a beam, with a Transfer-Matrix associate.

2. EQUATIONS FOR A LONG RECTANGLE PLATE

In Fig. 1. a) we have a long rectangle plate. The approach consist in discrediting the plate into parts, each part has the width equal to the unit. In general, we consider the load on a line parallel with the length and due to bending the plate is now a curved surface (Fig. 1. b). The mathematical expression of the load on the unit's length is q (x) (Gery,1973).

[FIGURE 1 OMITTED]

The thickness of each part is h and the beam bending's rigidity is D:

D = [Eh.sup.3]/12(1 - [v.sup.2]) (1)

where: E is the Young's Modulus and v is the Poisson's coefficient. In the point x, we have: T(x)-cutting force, M(x)- total flexion moment, v(x)-deformation, [alpha](x)-angle deformation or rotation of the middle fiber, m(x)-bending moment due to a single exterior load. F is the extension reaction in the boundary unit (function of v(x)). For the total bending moment M(x), we can write:

M (x) = m(x) + Fv(x) (2)

with:

[d.sup.2]m(x)d[x.sup.2] = q(x) (3)

For the deformation, the differential equation is:

[d.sup.4]v(x)/[dx.sup.4] - F/D [d.sup.2]v(x)/[dx.sup.2] = q(x)/D (4)

F/D = [[alpha].sup.2] (5)

and we have the associate equation, of the equation (4), without the second member:

v(x) = Ach[alpha]x + Bsh[alpha]x + [C.sub.1] x + [C.sub.2] (6)

With Dirac's function and Heaviside's function and operators, the mathematical approach gives the particular solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and the conditions:

[v.sup.*](0) = [v'.sup.*] (0) = [v".sup.*](0) = [v"'.sup.*] (0) = 0 (8)

In the origin, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and we can write :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The integration constants are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The general solution is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

(12) is the calculus formula for the deformation.

3. THE TRANSFER-MATRIX FOR A LONG RECTANGULAR PLATE

At the origin, in the point 0, the state vector is:

[{U}.sub.0] = [{[v.sub.0], [[omega].sub.0], [M.sub.0], [T.sub.0],}.sup.-1] (13)

and in the point x:

[{U(x)} = {v(x), [omega](x), M(x), T(x)}.sup.-1] (14)

The Transfer-Matrix for the plate, with the Heaviside's function and operators, linking the two state vectors, is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

We can simply write:

{U(x)} = [T] x [{U}.sub.0] + 1/D [{U}.sub.c] (16)

where: {U}(x)} is the state vector in current section x, [[T].sub.x] is the Transfer-Matrix between the side 0 x, [{U}.sub.0] is the state vector for the origin, [{U}.sub.c] is vector of the coefficients.

4. BENDING OF A LONG RECTANGLE PLATE, EMBEDDED AT ITS LENGTHS BOUNDARIES AND WITH A COMPLEX LOAD

We have the scheme (the section) of a long rectangle plate, embedded at its lengths boundaries, with a concentrated load in the middle and a uniformly load on the surface of the plate (Fig. 2.).

[FIGURE 2 OMITTED]

For this load, the function of the distribution, with the Heaviside's function Y, is:

q(x) = -q[Y (x) - Y (x - L)] (15)

We can simply write:

{U(x)} = [T] x [{U}.sub.0] + 1/D [{U}.sub.c] (16)

The coefficients vector [{U}.sub.c] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The conditions for the embedded flank and foe the free length are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

We can write:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

(18) with 19 gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

We obtained the relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

with the solutions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

These values give the expression of the deformation (12), for a long rectangle plate with a uniform load:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

5. CONCLUSIONS

This calculus is very important for a lot of industry domains. With formulas (18) and (22) we can calculate the state vector of the right side of the plate, in function of the state vector of the first side 0 and after, we can obtain with (23) all the state vectors for all the sections of the plate. Using a numerical method, the Transfer-Matrix Method, we can to program this calculus and we can to optimize the form of the plates.

6. REFERENCES

Gery, M., Calgaro, J.-A. (1973). The MATRIX -- TRANSFER in the calculation of structures, Editions Eyrolles, Paris

Tripa, M. (1967). Strengh of Materials, EDP Bucuresti

Warren, C. Y. (1989). ROARK'S Formulas for Stress & Strain, 6-th edition, McGrawHill Book Company