THIN CIRCULAR PLATE UNIFORMLY LOADED OVER A CONCENTRIC ELLIPTIC PATH AND SUPPORTED ON

Within the limitations of the classical thin plate theory expressions are obtained for the small deflections of a thin isotropic circular plate uniformly load- ed over a c,ncentric ellipse and supported by four columns at the vertices of a recta,g(e whose sides are parallel to the axes of the ellipse. Formulae are given for the mo,aents and shears at the centre of the plate and on the edge. Limiting cases are invest igated.

tests reported in [7].ost of the results in [5-I0,12] are special cases of [B].In qarieq f papers [I-15] complex variable methods were applied to study th,, bending of an ela,;tically restrained circular plate subject to uniform, linearly vac/ing and parabolic loadings over .concentric ellipse.Frishbeir and Lucht [16] used the //L// method ,f complex potentials to derive the solution for a clamped circular plate which is transv_.rselyand uniformly loaded over the area of a polygon.In this paper, expressions are obtained for the deflection at any point of a thin circular plate which i ,niformly load,'.dover a concentric elliptic patch and supported by four equal concentrated forces loc,ted at the corners of a rectangle whose sides are parallel to the axes of the ellipse.
Formulae are given for the boundary and central values of the m,,ents and shears.The limiting cases in which the radius of the plate oo, the eccentricity of the ellipse 0 or its minor axis 0 are investigated.
2. BASIC EQUATIONS AND BOUNDARY CONDITIONS.
Let C denote the boundary of a thin circular plate of centre 0 an radius c.
If 2h is the constant thickness of the plate, then its flexural rigidity D is given by [2] 2 D 2Eh3/3(I- (2.1) where E is the modulus of elasticity and is Poisson's ratio for the material of the plate.
The mid-plane of the plate is chosen as the plane Z 0 of a rectangular Cartesian frame O(x,y,Z) and the notation used is that of [2].According to the classical small bending theory of thin plates the deflection w of the mid- (2.3) w z(z) + zi(i) + ,(z) + o(i) + W(z,z), (2.4) where (z), (z) are functions of z which are regular in the region occupied by the plate and W is a particular integral of (2.2).
The moments and shears at any point (r,v) of the mid-plane of the plate are give by [2] - -Dd("2w)' Qo -Dr-ld' (2w)' ('2.6) and p(z,z) is the normal load intensity at the point z.
The general solution of (2.2) may be written as where 0/.Jr, d' 0/3' In terms of tLe complex potentials .(z), (z) and particular integral W(z,z) we have [3,p.730]51 + Mj -4(;+ )D Qr- where accents denote differentiation with respect to z The conditionfor the circ,lar edge C to be free are [2] ( O, (V r) (2.9) Substitution from (2.5a,c) and (2.6) in (2.9) From (2.)a) and the first equation of (2.9) we see that v($2w) Equation (2.5b) and the second equation of (2.6)   (2.13)This relation and the second equation of (2.9) serve to determine the periphery values of the shears in terms of the moment values.
3. STATEMENT OF THE PROBLEM.
The problem to be solved consists of determining the deflection surface of a thin circular plate of centre 0 and radius c subjected to the following condi- tions: (1) The boundary C of the plate is free. ( 3 The plate is supported by four eqval concentrated forces, each of magnitude L/4, and located at tle points P(z% se ,% ;,2,3,4), where 0 s c, lie in tIe loaded or unloaded region according as s cos y/ + s-sin y/b is less or greater than I.
For y 0 we have two supports at (is,0) while for /2 we have two supports at (0,_+s).See Figure I. Symmetry with respect to both axes show tlmt it is sufficient to find the deflection w at any point z in the positive quadrant.
Deflections, moments and shears at the four points +/-z, +/are the same.

METHOD AND SOLUTION.
Iet F denote the boundary of the ellipse (3..b) and let the indices and 2 ruler to the loaded region inside F and the unloaded region between F and C, respectively.
The particular integrals W and W 2 of (2.2) corresponding to the uniform intensities of normal loading Pl P0 and P2 0 may be taken as W (z,) pOz22/64D, W2(E,) 0. (/.l) The continuity requirements for the deflections, slopes, moments and shears at any (,.l) Introducing (.io), ./;and .18) in (2.4)we get, using (3.14a)where k 8D/L (.21) if @ is the eccentric angle of any point z on F then z a cos + ib sin , Z b cos + ia sin and it is checked that the expression between the square brackets in (4.20) vanishes It is to be noted that Z (z -f is not uniform in region while it is uniform in region 2. In fact, the two branches of Z inter- change when z traces a closed path to, any of the two loci (if,0) of the ellipse.
Thus, the terms containin Z in (4.20) should appear in w 2 and not in w I.
It is also known that the singular part of the deflection w at any point P near a downward concentrated force where R is the distance between P aa the point of application of the force Guided by tllese remarks and using (-.20), we assume that (a F ab) and A ,C (n 0, I,2,...) are real constants to be determined.
It is now easily n n seen that the expressions ( 423) and (4.24) for w and w 2 satisfy the biharmonic equation (Z.Z) corresponding to the load intensities Pl PO and P2 O, satisfy the required transition conditions along F and exhibit the appropriate singular (n 1,2 and behaviour at the four points of support The unkown constants A n C (n 0, 2 ...) will now be determined from the boundary conditions (2.10).
The n condition of zero deflection at any point of support serves to find A O.
To achieve this goal, all the terms in (4.24) will be explicitly expressed in terms of biharmo- nic functions of r and of the types wlere, with the usual notation _l_ n 2-2n /2n (n ) Integrating (4.27a), using (4.27b) and noting that (z,0) in(2z) we get

Thus we have
In a + ---z + Z In a+b2r 2 1 Z !
It is easily seen that such points exist only if f b, i.e., if the eccentricity of the ellipse e 2/2.
In any case, w 2 is given by (4.24).
The constant A 0 can always be determined from the condition that the deflection vanishes at any of the four points of support.
If all these points lie in the loaded region then (4.44) gives If all the su,oort ,oints lie in the unloaded region, then either (4.45) or (4.30b)   can be used to determine A 0 according as s f or s f.
If s -> f we have In any case, the deflection at the centre of the plate is where the appropriate value of A 0 is taken.

BOUNDARY ANt) CENTRAL VALUES OF MOMENTS AND SHEARS
It can be easily shown that the deflections (4.23) and ( 424) may be written in  (5.52a) (5.52b) Z% z z% and the real constants An, C n have been determined in the previous section.
The moments and shears at any point of the plate can be obtained either by substitution from (4.15), (5.Sla,b) and (5.52a,b) in (2.7a,b) and (2.8) or by introd- ucing (4.44), (4.45), (4.30b) in (2.5a,b,c) and (2.6), noting that S is defined by (4.25), (4.26) and its expansion is (4.31b) if r s and interchanging r,s in (4.3b) if r -< s.After extensive algebraic manipulation, it is found that (Mr)r=c=O as expected and (+)L v -u + E 2-2u +--u2ncos (5.54b) It is easily seen that (5.53b), (5.54a) satisfy the second boundary condition in (2.9)   and (5.53a), (5.54b) satisfy (2.13) All the infinite series appearing in this section and in section 4 are convergent in the intervals mentioned and some of them will be summed in section 8.
The following formulae are obtained for the moments and shears at the centre: (Mr) 0 (0)0 .a-b +u 2 f2 2 a+--' 3) cos 20 (5 55a)  A 0 ab + s s in sin y +cos y in cos y-+ in a+ < s 2 ) 2 { a_b2 a-b s + +b)cos 2y-+ k7 cos 4y (6.59) if the supports lie in te loaded region; if the supports lie in the unloaded region and s e f 2 ; " When the minor axis of the loaded elliptic patch 0 we have the case of a variable line loading extending along tL,e x-axis from x -a to x a.If b 0 and PO such [hat 2bP0 p! then the intensity of this line loading at a 2/a2 distance x from the centre equals p| /(|-x ).
Deflections, moments and shears corresponding to this case can be dedcued from those for region 2 in sections 4, 5 by setting b O, d f a, L apl and noting that separate expressions are orel, =ro,pvpqnt (,r,.O aqcord,ing as r is greater or less than a and tn columns lie outside or inside the circle r a 8. THIN CIRCULAR PLATE UNIFORMLY LOADED OVER A CONCENTRIC CIRCULAR PATCH AND The last function is the dilogaritnm studied in the last three references of [7].,/_l(U ,2) + u a l_j + (K-I) td,u ,0) + d(u 2,2)} It is verified that (8.71a,b) agree with (3.14) and (3.6a) of [3], noting the difference in notation.
There are misprints in equation (3.5a), p.738 of [3] and cos a which appears twice in this equation must be replaced by It is easily verified that (8.73a), (.74b) satisfy (2.13) and (8.73b),(8.74a)satisfy the second equation in (2.9)   Formulae (5.55a,b) and (5.56) for the moments at the centre reduce to

Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities.This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics.It is possible to make a division in two main categories: astronomy and astrodynamics.By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth.Many important topics of research nowadays are related to those subjects.By astrodynamics, we mean topics related to spaceflight dynamics.
It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects.Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts.Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines.The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research.All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/mpe/guidelines.html.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/according to the following timetable:

Manuscript Due
July 1, 2009 First Round of Reviews October 1, 2009
kw Letting c in (4.44), (4.45) and (4.30b) leads to' in(4s siny) + s cos y ln(4s cosy) CIRCULAR PLATE UNDER A VARIABLE LINE LOADING ALONG A DIAMETER AND SUPPORTED ON COLUMNS.