Eug

J area I

dx dy

A suitable deflection function w(x, y) satisfying the boundary conditions of the given plate may be chosen. The strain energy U and the work done by the given load, q(x, y),

q(x, y)w(x, y) dx dy can be calculated. The total potential energy is, therefore, given as V = U + W. Minimizing the total potential energy the plate problem can be solved:

The term 18 is known as the Gaussian curvature.

If the function w(x, y) — f(x)f(y) (product of a function of x only and a function of y only) and w — 0 at the boundary are assumed, then the integral of the Gaussian curvature over the entire plate equals zero. Under these conditions

If polar coordinates instead of rectangular coordinates are used and axial symmetry of loading and deformation is assumed, the equation for strain energy, U, takes the form

and the work done, W, is written as

J J area

Detailed treatment of Plate Theory can be found in Timoshenko and Woinowsky-Krieger (1959).

2.5.6 Plates of Various Shapes and Boundary Conditions

2.5.6.1 Simply Supported Isosceles Triangular Plate Subjected to a Concentrated Load

Plates of shapes other than circular and rectangular are used in some situations. A rigorous solution of the deflection for a plate with a more complicated shape is likely to be very difficult. Consider, for example, the bending of an isosceles triangular plate with simply supported edges under concentrated load P acting at an arbitrary point (Figure 2.38). A solution can be obtained for this plate by considering a mirror image of the plate as shown in the figure. The deflection of OBC of the square plate is identical with that of a simply supported triangular plate OBC. The deflection owing to the force P can be written as

Upon substitution of — P for P, (a — y1) for x1 and (a — x1) for y1 in Equation 2.81 we obtain the deflection due to the force — P at A;:

4Pa2 xmin sinmrcxi/a) sin(nrevi /a) mpx npy w2 = —> > (—1)m+n—^-—-\ 717 7 sin-sin— (2 . 82)

The deflection surface of the triangular plate is then w = w1 + w2 (2 . 83)

2.5.6.2 Equilateral Triangular Plates

The deflection surface of a simply supported plate loaded by uniform moment Mo along its boundary and the surface of a uniformly loaded membrane, uniformly stretched over the same triangular boundary, are identical. The deflection surface for such a case can be obtained as

W 4aD

FIGURE 2.38 Isosceles triangular plate.

If the simply supported plate is subjected to uniform load po the deflection surface takes the form x3 — 3xy2 — a(x2 + y2 )+ — a3

For the equilateral triangular plate (Figure 2.39) subjected to uniform load and supported at the corners, approximate solutions based on the assumption that the total bending moment along each side of the triangle vanishes were obtained by Vijakkhna et al. ( 1 973), who derived the equation for deflection surface as qa4

27 (7 + v)(2 — v) — (7 + v)( 1 — v)( x2 + a2 27 a a2

. w ,,'x3 xy2\ 9, ,, fx4 x2y2 y4 —(5 — v)( ^ v)( - —3 Jyr) + 9 ( ^ v2)( - +2 -f- + y a3

a4 a4

The errors introduced by the approximate boundary condition, that is, the total bending moment along each side of the triangle vanishes, are not significant since its influence on the maximum deflection and stress resultants is small for practical design purposes. The value of the twisting moment on the edge at the corner given by this solution is found to be exact. The details of the mathematical treatment may be found in Vijakkhna (1973).

2.5.6.3 Rectangular Plate Supported at the Corners

Approximate solutions for rectangular plates supported at the corners and subjected to uniformly distributed load were obtained by Lee and Ballesteros (1960). The approximate deflection surface is given as

0 0

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