Bending of a circular plate with an eccentric circular patch symmetrically loaded with respect to its centre

Abstract

The complex variable method is applied to obtain solutions for the deflexion of a supported circular plate with uniform line loading along an eccentric circle under a general boundary condition including the clamped boundary , a boundary with zero peripheral couple , a boundary with equal boundary cross-couples , a hinged boundary and a boundary for which , η being Poisson's ratio. These solutions are used to obtain the deflexion at any point of a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre. Expressions for the slope and cross-couples over the boundary and the deflexions at the centres of the plate and the loaded patch are obtained.

Bending of an eccentrically loaded and concentrically supported thin circular plate. I

“…The transverse flexure of thin isotropic circular plates subject to various distributions of normal loadings have been considered by several authors when the boundary of the plate is rigidly clamped, simply supported or free. In a series of papers ((6), (5), (2)) the methods of complex function theory were applied to obtain solutions for a thin circular plate under a general boundary condition including the usual clamped and simply supported boundaries and acted upon by the following types of loadings: (a) the load p o r m . nd over the entire plate or over a concentric sin circle (m,n being positive integers including zero), (b) the loadingsp 0 R m , p 0 R m cos® COS and^o-R" 1 .…”

“…We now proceed to determine £l Q (z) and « 0 (z) from the boundary condition where Let y denote the unit circle in the £-plane and ^ any point inside it. To solve for the regular functions ^(£) and i/r(Q we follow the usual procedure of multiplying (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) and its conjugate throughout by (l/27ri)dcr/(or -Q and integrating around y. The method is the same as that used by Muskhelishvili ((10)) in obtaining the general solution for a circular plate loaded in its plane.…”

“…I 109 P = np 1 a 2m /m is the total load on the patch and T n is given by . To determine w 1 we use equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) Results corresponding to the four limiting cases of section 3 will now be briefly considered.…”

“…2 -a 2 -(-R 2 + a 2 )log|], (3-9c) where R = \Z\ = \z-f\. (3-10)Noting that zz = b 2 , e~i a = b\z, q = -t(d) on T,(3)(4)(5)(6)(7)(8)(9)(10)(11) where t(6) is given by(3)(4), it can be easily shown that the transition conditions (2-3) along r lead to Equation (2-26) now yields / 2 )log r -Applying (3-9), (3-12) and taking into account the fact that Qy(z), o)j{z) are regular functions in region j (j = 1,2, 3) we may put b » a n cos w6> Tr 2 "+ 2 -6 2w + 2 6 2m r 2 -brl . o , " .…”

“…(2)(3)(4), where the functions Q(z), o)(z) are replaced by (3-16a), (3-166), respectively. Putting z = c£ at any point of the plate and £ = cr on 0 and writing we find that(2)(3)(4) gives /c^(cr) + cr^''{a) + ^(o=) = J'(o-),(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) …”