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NASA / TM--1999-209631 Deflections of a Uniformly Loaded Circular Plate With Multiple Support Points L.D. Craig Marshall Space Flight Center, Marshall Space Flight Center, Alabama J.A.M. Boulet University of Tennessee, Knoxville, Tennessee September 1999

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NASA / TM--1999-209631 Deflections of a Uniformly Loaded Circular Plate With Multiple Support Points L.D. Craig Marshall Space Flight Center, Marshall Space Flight Center, Alabama J.A.M. Boulet University of Tennessee, Knoxville, Tennessee National Aeronautics and Space Administration Marshall Space Flight Center • MSFC, Alabama 35812 September 1999

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Available from: NASA Center for AeroSpace Information 800 Elkridge Landing Road Linthicum Heights, MD 21090-2934 (301) 621-0390 ................................................... National Technical Information Service 5285 Port Royal Road i Springfield, VA 22161 (703) 487-4650 " t

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TABLE OF CONTENTS , INTRODUCTION .................................................................................................................. 1 2. S1NGLERING OF MUL_PLESUPPORTPOINTS ........................................................... 2 3. MULTIPLE RINGSOFEQUALLYSPACEDSUPPORTPOINTS ..................................... 5 4. CONCLUSIONS .................................................................................................................... 14 111

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LIST OF FIGURES 2 , Three-point support ................................................................................................................ 3 2. Four-point support .................................................................................................................. 3 3. Five-point support .................................................................................................................. 4 4. Six-point support .................................................................................................................... 5. Deflection versus r at three azimuthal locations ............................................................................... 12 6. Mathcad surface plot of Mathcad results .................................................................... II 12 7. PATRAN surface plot of NASTRAN results ......................................................................... 13 8. PATRAN fringe plot of NASTRAN results ........................................................................... I3 9. Fringe plot of Mathcad results ............................................................................................... LIST OF TABLES support (v=-0.25) ................................................... 2 , Normalized deflections for a three-point 2. Normalized deflections for a four-point support 3. Normalized deflections for a five-point support 4. Normalized deflections for a six-point support 5. Deflection constants and reactions for various (v=-0.25) ..................................................... 3 (v=0.25) ..................................................... 3 4 (v=-0.25) ....................................................... multipoint support II configurations (v=-0.25) .......................................................................................................... [[ iv

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TECHNICAL MEMORANDUM DEFLECTIONS OF A UNIFORMLY LOADED CIRCULAR PLATE WITH MULTIPLE SUPPORT POINTS 1. INTRODUCTION This technical memorandum (TM) describes methods for determining the transverse deflections of a uniformly loaded, thin circular plate of constant thickness supported by single or multiple rings of equally spaced discreet points. The rotations are assumed free at each point. These methods could have application in the design of telescope primary mirror supports that must minimize structural gravitational deformations. They could also be of general use to the structural analyst. Tables and graphs are presented in section 2 for a variable radius ring of three, four, five, or six equally spaced support points. These contain constants for calculation of the transverse deflection at three locations of interest. Section 3 contains results for multiple rings of various support point configurations. These results include constants for the calculation of root mean square (RMS) and peak-to-valley deflections and the fraction of load supported by each ring. Results obtained from three different methods are summarized and compared. Also presented are equations suitable for programming into a mathematical solver computer program. Once programmed, results configuration. may be obtained for practically any support point

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1. SINGLE RING OF MULTIPLE SUPPORT POINTS The series solution for this case is lengthy and will not be shown here, but it can be found in reference 1. The number of support points was varied from three to six and the results presented in tables I-4. Each table contains the applicable constant used to determine the transverse deflection at a specific location on the plate. The support ring radius is also varied. These data are displayed graphically to better illustrate the results (figs. 1-4). Note that for a support ring radius equal to zero, the result is identical to a uniformly loaded circular plate supported by one point at the center. Note also that for a support ring radius equal to the outer edge radius (b/a = 1), the normalized deflection at a support (r = a, 0 = 0 °) is zero, as it should be. As shown in reference 2, there is no significant difference in the results when the number of support points is increased beyond six. The variables below are defined as follows: transverse deflection w, uniform load q in force per unit area, radius a of the plate, radius b of the support Table 1. Normalized deflections for a three-point support (v=0.25). wi(qa41D) b/a Onedge On edoe Atcenler r=a, 0=0 ° r=a, o=60 ° r=O, 0=0 ° 0.0 0.096875 0.096875 0 0.05 0.09473 0.094743 --0.00079592 0.1 0.090002 0.09011 -0.0023156 0.15 0.083638 0.084002 -0.0040632 0.2 0.076131 0.076988 -0.0057678 0.25 0.067821 0.069488 --0.0072322 O.3 0.058976 0.061836 -0.0082937 0.35 0.049818 0.054326 -0.0088073 0.4 0.04055 0.047218 -0.0086351 0.45 0.031358 0.040755 -0.0076399 0.5 0.022429 0.03517 -0.0056792 0.55 0.013948 0.030692 -0.0026016 0.6 0.0061139 0.027552 -0.0017582 0.65 -0.00086453 0,025985 -0.0075835 0.7 -0.0067566 0.026242 -0.015081 0.75 -0.011304 0.028595 -0,024489 0.8 -0.014209 0,03335 -0.036089 0.85 -0.015115 0.040869 -0.050224 0.9 -0.013569 0.051611 -0.067343 0.95 -0.0089236 0.066226 -0.088087 1.0 0.0 0.085882 -0.11362 ring, Poisson's ratio v, and flexural rigidity D. On edge, at support --e- On edge, 60° from support •.-a- At center 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 bla. Ratio of supportring radii Io outer radius Figure 1. Three-point support.

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Table 2. Normalized deflections for a four-point support (v=0.25). w/(qa4/_ b/a Onedge Onedge Atcenter r=a, e=O° r=a, e=45 ° r=O, 0=0 ° 0.0 0.096875 0.096875 0 0.05 0.094718 0.094718 -0.0008143 0.1 0.08998 0.089985 -0.0023891 0.15 0.083642 0.083666 -0.004229 0.2 0.076226 0.076301 -0.0060644 0.25 O.068O97 0.068277 -0.0077013 0.3 0.059536 0.059906 -0.0089838 0.35 0.050776 O.O51453 -0.009778 0.4 0.042014 0.043154 -0.009963 0.45 O.033426 0.035226 -0.0094251 0.5 0.025173 0.027872 -0.0080536 0.55 0.017411 0.021288 -0.0057361 0.6 0.01029 0.01567 -0.0023553 0.65 0.003966 0.011209 0.0022154 0.7 -0.0013977 0.0081074 0.0081168 0.75 -0.0056214 0.0065752 0.01551 0.8 -0.0085001 0.0068455 0.024585 0.85 -0.0097884 0.0091873 0.035578 0.9 -0.0091717 0.013937 0.0488 0.95 -0.0061952 0.02157 0.064707 1.0 0.0 0.032954 0.084153 Table 3. Normalized deflections for a five-point support (v=0.25). wi(qa41D) b/a Onedge Onedge Atcenter r=a, e=O° r=a, e=36° r=O, e=O° 0.0 0.096875 0.096875 0 0.05 0.094712 0.O94712 -0.00082045 E 0.1 0.089958 0.089958 -0.0024137 w 0.15 0.083598 0.0836 -0.0042844 O 0.2 0.076161 O.O76169 -0.0061629 em 0.25 0.068021 0.068045 -0.0078555 0.3 0.059469 0.059529 -0.0092066 0.35 0.050746 0.050874 -0.010083 0.4 0.042058 0.042304 -0.010368 0.45 0.033589 0.034023 -0.0099499 0.5 0.0255 0.026222 -0.OO87263 0.55 0.017946 0.01908 -0.0065955 0.6 0.011068 0.012775 -0.0034555 0.65 0.0050073 0.0074816 0.00079924 0.7 -9.4832x10-5 0.0033756 0.006282 -0.01 0.75 -0.0040905 0.00063968 0.013118 0.8 -0.0068171 -0.00053068 0.021454 -0.02 0.85 -0.0080849 8.4254×10-5 0.031467 0.9 -0.0076547 0.0027521 0.043391 0.95 -0.0051828 O.O078453 0.057573 1.0 0.0 0.016058 0.074692 0.11 -.o-On edge, atsupport 0.1 --o-0n edge, 45°from support •-,_-At center 0.01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 b/a. Ratio of supportring radii to outer radius Figure 2. Four-point support. 0.11 .-o- On edge, at support 0.1 0.09 -o-- Onedge, 36° from support 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0. i z ! z z ! ] I I I I I I I I I I ! i 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 bla. Ratio of support ring radii to outer radius Figure 3. Five-point support. 3

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Table4. Normalized deflections for a sixpoint support (v=0.25). w/(qa4/D) On edge On edge At center b/a r=a, e=O ° r=a, 8=30 ° r=O, e=O ° 0.0 0.096875 0.096875 0 0.05 0.094709 0.094709 -0.0008231 0.1 0.089947 0.089947 -0.0024243 0.15 0.083575 0.083575 -0.0043082 0.2 0.076122 0.076123 -0.0062054 0.25 0.067965 0.067968 -0.0079218 0.3 0.059398 0.059409 -0.0093022 0.35 0.050666 0.050693 -0.010214 0.4 0.04198 0.042041 -0.010538 0.45 0.033529 0.033649 -0.010168 0.5 0.025479 0.0257 -0.0089995 0.55 0.017987 0.018367 -0.0069353 0.6 0.011196 0.011818 -0.0038785 0.65 0.0052455 0.0062157 0.00026869 0.7 0.00026902 0.0017249 0.0056073 0.75 -0.0035977 -0.0014877 0.012246 0.8 -0.0062118 -0.0032473 0.020307 0.85 -0.0074135 -0.0033622 0.029934 0.9 -0.007007 -0.0016062 0.041316 0.95 -0.0047165 0.0023282 0.054729 1.0 0.0 0.0090156 0.070726 = 2 4 0.11 -.o- On edge, at support --o- On edge, 30 ° from support At center 0.01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 b/a. Ratio of support ring radii 1o outer radius Figure 4. Six-point support.

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1. MULTIPLE RINGS OF EQUALLY SPACED SUPPORT POINTS The solution to the multiple ring problem is described in a paper by Nelson, Lubliner, and Mast. 2 A solution for a single ring of discreet support points multiple ring solution 2 is expressed as the summation is derived in appendix B of reference 2. The of single ring solutions with each ring weighted by its portion of the total load reacted. This summation of weighted single ring solutions is not easily obtained for the analyst with only a basic understanding of plate theory and the method of superposition. The solution is complicated and lengthy, but results may be obtained quickly with the aid of a computer. In 1998, a summer faculty fellow, Dr. Toby Boulet of the University of Tennessee, attempted to develop a true, closed-form solution. In the process, he developed a Mathcad® (a registered trademark of MathSoft, Inc.) document using the solution in reference 2. This document can be used to determine transverse deflections of a uniformly loaded circular plate resting on multiple rings of equally spaced support points, multiple rings of equally spaced support points with a center support point, or a single ring of equally spaced support points with or without a center support. The number of rings and support points must be two or greater to obtain results. Deflections for a single ring of points may be found by specifying different azimuthal positions of two support rings located at the same radius. To create a center support point, the radius of one ring must he set equal to zero. Dr. Boulet programmed the following equations in Mathcad: v:= 0.25 Poisson's ratio j] (,) := (f12 + 2). ln(fi) + [3+v-(1-v). 2 i+v l 2 f2(_,/3) := ([32 + 2). In() + 2] 3 + v - (1 - v)./3 2] 2 l+v J:=20 (number of terms in Fourier expansion) J:= 1,2 .... J + A(j, fl, N). = flN'J37V [ (l-v). ( N.j-I1 N-7.j]_2 ) (N.j) 2 (N.j-1).(1-v) ] -1 8 . l+v /

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B(j, fl, N) := flN.j ._ 3+v1-v .( N.j1 N.j+I -1. fiN.j+2 C(j,fl, N) := N.j.(N.j+I) flN.j D(j,,N) := l. j.(N. j-1) E(j, fl, N) := A(j, fl, N) -_ N.j.(N.j-I) -(N.j) F(j,,N):=B(j, fl, N) W.j.(N.j+l) rl(j,,fl, N):= A(j,,N)._ N'j + B(j,fl, N)._ N'j+2 +C(j, fl, N)._ -(N'j) + D(j, fl, N)._ -(N'j)+2 Z(j,,/3, N) := E(j, fl, N)' N'j + F(j, fl, N)'_ N'j+2 wj(j,,fl, N) "= if( < fl,)t(j,,fl, N),o(j,,fl, N)) F(,fl, U,O):= fu(,fl)- 2 wj(j,_,fl, N)c°s(N" j'O))j N R :=? (integer number of rings) NI N2 N: = : N i is the number of supports in each ring (>1). NNR 6

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/3i is the ring radius divided by the outer radius of the plate• _NR _ 41 _i is the azimuthal (clocking angle) location (radians) of supports in each ring. 0 R] Note that the supports within each ring are equally spaced; that is, if NI=3, the support points in ring 1 are 120 ° apart or, if Nj =4, they are 90 ° apart and so on. s:= 1,2 .... N R k:=l,2 .... N R bk,s := F( flk ,fls, Ns, (Pk - dPs) t := 2,3...N R Ki,k:= 1 Kt,k := bt,k - bl,k h := XX.XX" in. p:= YY.YY. lb__f_f,h in. 3

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Cl:= P c t:= 0- psi q:= K -1 • c Emodulus:= ZZ.ZZ. psi A: = Em°dulus " h3 q = a: = RR.RR. in. w(,0) := __--__a4•qk.(F(,k,Nk,O-(k)-F(fll,flk,Nk,-Ok)) 1 where N n is the number of circumferential points at which the deflection is calculated and N: is the number of radial points. n "= 1,2 .... N. k'=l,2 .... N,a • = Ic.-- N- 2.,7' 0,,:= -zr + (n - 1).-- N n

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rx transverse displacement at r, 0 PV:=max(d) - min(d) PV= max(d)= min(d)= The following equations calculate the Zernike coefficients for bias, x tilt, and y tilt: -. w(¢O).¢d dO If CO= Cl= C2:: /:a4 ..x_lrSO1w('O)'¢2d_'sin(O)dO C2= The following equations remove the bias from the transverse deflections and calculate the residual RMS deflection: 6(¢o):: w(¢O)-Co _RMS = Calculate 7for comparison with reference 2. 2 SA: = ;,r . a NS:= _., N (sum of the support points) YN:= 6RMS p 1_SA ) reference 2, equation YN = 4 solved for YN-

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This concludestheMathcadinput.The casesin table 1of reference2 weresolvedwith this input andtheresultsconvertedto a form for comparison.The samecasesweresolvedvia the finite elementmethod(FEM) with NASTRAN for further validation,andtheseresultsarealsoshownin table5. No attemptwasmadeto explainthe discrepanciesbetweenthe resultsfrom reference2 and thosefrom NASTRAN or the equationsabove.Todeterminethe RMS deflection(with biasremoved; thatis, the first Zernikecoefficient),useequation(4) from reference2 shownbelow. aRMS = _'N D _, N S ) " Mathcad or NASTRAN deflections may be illustrated graphically with various software packages. Once the results are generated, they can be plotted internally or exported to a spreadsheet and plotted externally. Results for the 4-ring, 12-point support in table 5 are shown in figures 5"9. Note that deflection downward is positive with the exception of the PATRAN fringe plot of NASTRAN results where the downward direction is negative. The Mathcad-generated deadweight deflections shown in figure 5 are for a 0.1-i n. thick, 20-in. diameter aluminum plate. Note the zero deflection at the inner and intermediate ring support points. Figures 6-9 illustrate different ways of displaying the deadweight deflections of the aluminum plate. 10

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Table 5. Deflection constants and reactions configurations (v=-0.25). for various multipoint support Ns ,6 {o yNxll_ "/Nxl03 YNXl_ P-I/]RMS P-V/RMS P-V/RMS e e e deg. (NLM) (FEM) (8oulet) (NLM) (FEM) (8oulet) (NLM) (FEM) (Boulet) 3 0,645 " " 5.76 " 5.73" 5,76 4.2 4,19 4,19 1.0 1.0 1.0 6-, 0.6,._ 81 2.93 2.91 2.90 4.3 4.3t 4.31 1.0 1.0 1.0 7 0.0 2.36 3.00 2.93 4.9 4.81 4.88 0.1183 0.1301 0.1301 0.737 0.8817 0.8686 0.87 9 0.2825 0.0 3.76 4.83 4.75 5.0 4,42 4.44 0.2309 0.2365 0.2450 0.7936 0.0 0.770 60.0 0.3637 0,3573 0.3582 0.4054 0.3962 0.3968 12 0.3151 0.0 1.94 2.13 207 5.1 5.01 5,0 0.2783 0.2781 0.2786 0.7662 60.0 0.8257 20.0 0.8257 -20.0 0.2843 0.2804 0,2805 0.2187 0.2201 0.2204 0,2187 0.2201 0.2204 15 0.3192 0.0 2.32 3.00 2.97 5.4 4.18 4,16 0.2833 0.2810 0.2810 0.7765 44.88 0.8412 150 0.7765 -44.88 0.8412 -15.0 0.2046 0.2030 0.2037 0.1538 0.1565 0.1558 0.2046 0,2030 0.2037 0,1538 0.1565 0.1558 18 0,4741 0.0 1,89 2.09 2.02 5.5 4.89 5.01 0.1625 0.1689 0.1704 0.3195 60.0 0.8171 44.8 0.8536 15.26 0.8171 -44.8 0.8536 -15.26 6.0 5.27 5.38 0.1671 0.1687 0.1674 36 0.2569 0.0 1.63 1.71 1.65 0.5771 15.18 0.5771 44.82 0.8830 9.76 0.8834 30.0 0.8830 50.24 ,8 is the ratio of the ring radius to the outer radius 0.2071 0.2018 0.2008 0.1731 0,1731 0,1730 0.1421 0.1416 0.1414 0.1731 0.1731 0.1730 0,1421 0.1416 0.1414 0.1812 0.1791 0.1810 0.1812 0.1791 0.1810 0.1549 0.1571 0.1552 0.1607 0,1577 0.1602 0.1549 0.1571 0.1552 is clocking or the azimuthal location (in degrees) of suppods in one ring relative to one of the other rings " is the fraction of load carried by one ring of supports, equal to qlp above NLM is Nelson, Lubliner, and Mast.2 9.0E-05 --.o- at -75 ° 8.0E-05 .-o- at 0 ° 7.0E-05 .w = 6.0E-05 o 5.0E-05 (13 4.0E-05 = 3.0E-05 2.0E-05 1.0E-05 O.OE+O0 i i _ i 1 2 3 4 r(in.) i 1 j.. zuq ! 5 6 7 8 9 10 Figure 5. Deflection versus r at three azimuthal locations. 11

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Figures 6 and 7 are three-dimensional surface the deformed shape. Figures 8 and 9 show the fringe 8 6 4 2 0 -10 -2 0 2 4 6 8 Figure 6. Mathcad surface plots of the transverse deflections and display plots of the NASTRAN and Mathcad results. 0 2 4 6 8 10 10 plot of Mathcad results. MSC/PATRANVersion 8.022-Feb-99 13:15:49 Deform:-1GZ on 12 PTS.SC1.Static Subcase; Displacements, Translational t; Figure 7. PATRAN surface 12 Default Deformation: Max 8.05-005 @Nd 1863 plot of NASTRAN results.

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Figure 8. PATRAN fringe plot of NASTRAN results. 10,0 --6.0- -8.0- 10.0 -8,0 -6.0 -4,0 -2.0 0.0 8.00E.5 7.00E.5 6.00E-5 5.00E-5 4.00E-5 3.00E-5 200E-5 I,OOE-5 O,OOE-5 --T [ T-- 2.0 40 6.0 8.0 10.0 Figure 9. Fringe plot of Mathcad results. 13

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1. CONCLUSIONS This TM describes three methods for defining the deflected shape of a uniformly loaded, thin circular fiat plate supported with multiple discreet points. A comparison of these methods for specific examples is shown. These methods can provide a preliminary support system design for thin telescope mirrors. The equations programmed into Mathcad can solve Virtually any system of support points but are limited to thin, circular fiat plates (although contributio_n !:o thg deflection due to shear could be added). The finite element method (NASTRAN) can solve any support system and mirror geometry but requires much more computer time and memory and more of the analyst's time and effort. The tables and graphs contained in section 2 are sUbsets-of the results of-sec_n 3_and are generated v_ith different equations. The graphs may be used to determine (he optimum Support ring radius. . _2 . _ 14

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REFERENCES o Pan, H.H.; and Yu, J.C.L.: "Uniformly Loaded Circular Plate Supported at Discrete Points," blternational Journal of Mechanical Sciences, pp. 333-340, May 1966. ° Nelson, J.E.; Lubliner, J.; and Mast, T.S.: "Telescope Mirror Supports: Plate Deflections on Point Supports," SPIE, Vol. 332, pp. 212-228, 1982. 15

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REPORT DOCUMENTATION PAGE FormApprove_ OMB No. 0704-0188 Public reporting burden for this collection of information sestimated to average 1 hour per response, including the time for reviewing instructions, searching existing data SOurces, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operation and Reports, 1215 Jefferson Davis Highway, Suite 12C,4+ Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503 1, AGENCY USE ONLY (Leave Blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED September 1999 Technical Memorandum 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Deflections of a Uniformly Loaded Circular Plate With Multiple Support Points 8. AUTHORS L.D. Craig and J.A.M. Boulet* 7.PERFORMINGORGANIZATIONNAMES(S)ANDADDRESS(ES) George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama 35812 8. PERFORMING ORGANIZATION REPORT NUMBER M-942 ADDRESS(ES) 10• SPONSORING/MONITORING 9. SPONSORING/MONITORINGAGENCYNAME(S)AND National Aeronautics and Space Administration Washington, DC 20546-0001 11. SUPPLEMENTARY NOTES Structures, Mechanics, and Thermal Department, *University of Tennessee, Knoxville, Tennessee 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified-Unlimited Subject Category 39 Nonstandard Distribution 13, ABSTRACT (Maximum 200 words) AGENCY REPORT NUMBER NASA/TM--1999-209631 Engineering Directorate 12b. DISTRIBUTION CODE This technical memorandum describes a method for determining the transverse deflections of a uniformly loaded, thin circular plate of constant thickness supported by single or multiple rings of equally spaced discreet points. The rotations are assumed free at each point. This could have application in the design of telescope mirror supports that must minimize structural gravitational deformations. It could also be of general use to the structural analyst. 14. SUBJECT TERMS load, telescope, optics, 2O circular plates, multipoint support, uniform mirror supports 17. SECURITY CLASSIFICATION 18, SECURITY CLASSIFICATION OF REPORT OF THIS PAGE Unclassified Unclassified NSN 7540-01-280-5500 15. NUMBER OF PAGES 16. PRICE CODE A03 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF ABSTRACT Unclassified Unlimited Standard Form 298 (Rev 2-89) PrEc,cribed by ANSI SId 239-18 2-102