1、r_ _ It R 52848TECH_I CAL NOTES_TATI0_TAL ADVISORY C0_I_IITTEE FOR AER0_T_D-T.ICS _i,_,_o._ OmFmBENDI_G. 0_ R_CT_NGUL_R PLATES fIT?I Lf_RGE DNFLECTI0_TSBy Samuel Levy_Jaoional Burea_l of Stancls_rdsProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-LPro
2、vided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATI0!AL ADVISORY COMI_IT_ -_ -TECHNICAL TOTE N0. 848% . ,. BE_DI_,TG OF R.ECTANGULAR PLATES WIN LARGE D3FL_CTIONS_y Samuel LevySU M_r_u_RY ,The solution of yon KArmAns fundamental equationsfor large defle
3、ctions of plates is presented, for the caseof a simply supported rectangular plate under combinededge compression and lateral loading. Eumerical solutionsare given for square plates and for rectangular plates witha width-span ratio of S:I. The effectiv,e widths under edgecompression are compared wit
4、h effective widths accordingto yon _Iarman,“ _engston, _,arguer_e,_. and Cox and with ex-perimental results by P.e,mberg, c?herson, and Levy. Thedeflections for a square pl,s_te un:.er iater,s_! pressure arecompareC with experimental and theoretical results byKaiser. It is found that the effectiv_ _
5、idths agree close-ly with !4argv_erres formula and with thG experimentally ob-served values and that the deflections agree with the ex-perimental results and with Kaiser|s work.in the design of thin plates that bend under lateraland eEge loading, formulas based on the Kirchhoff theorywhich neglects
6、stretching anE shearing in the middle sur-face are quite satisfactory provided that the de+_lectionsare small compared with the thickness. If deflections areof the same orEer as the thickness, the Xirchhoff theorymay yield results that are considerably in error and a morerigorous theory that takes a
7、ccount of deformations in themiddle surface should therefore be applied. The fu_idamen-tal equations for the more exact theory have been derivedby vo_i I_:_arman“ (reference I); a number o_ approximate solu-tions (references 2 to 7) have been developed for the caseof a rectangular plate. This pa_er
8、presents a solution ofYon _ar Iman s equations in terms of trigonometric serles.Acknowledgment is due to the N _ uiona! _.dvi scryCommittee for ._eronautics and the _ureau of /keronautics,mProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 IqACA Techn
9、ical ITote ITo. 846_Tavy Department, whose research projects on sheet-stringerpanels have provided the impetus and the necessary finan-cial support for the work presented in this paper. Theauthor takes this opportunity to acknowledge also the as-sistance of members of the _ngineering fechanics Secti
10、onof the ITational _ure_ of Standards, particularly Dr.Walter Ramberg, Hr. Phillip Krupen, and Mr: Samuel _reenman.,nl./%TT _. TFUiTDA this difference is prob-ably due to the fact that Yaiser allows distortion of _heedges of the plate. The membrane stresses calculated byKaiser are about one-fifth_s
11、large as those given in thepresent paper. This fact, as well as a comparison of fig-ures 8 and 12, indicates the l_rge influence of edge con-ditions on the membrane stresses.C0:PARISON _ 7,which appeared to be large enough to reduce the effect ofthe torsional stiffness of the stringers an a factor i
12、nthe edge conditions.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technic_l Note _o. 846 19Deflection under Lateral PressureKaiser (reference 5) has conducted a carefully con-trolled experiment on one simply supported plate. Inthis experiment
13、, as in Kai.serls theoretical work, the edgeconditions are such that the membrane stresses at the edge “are zero. The initial deflections obtained by Kaiser arein agreement with the results in this paper. At large de-flections, however, the fact that the membrane stress at ,the edge of the plate was
14、 zero in the experiment causesthe measured _ofloctions to exceed by appreciable amountsthe deflections calculated in this paper.National Bureau of Standards,Washington, D. C., May 27, 1941.i Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20 NACA Tec
15、hnical )ote I_o. 848I_.EFERE NC_ Sp ,*I. von I2 W5l -4.00 0 0 0 18.3121.1875hWl3 W3l Wl5 1,685 _.00 -9.25 16,790 0 10.25h h hProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-l_CATbchnio_Note No. 846 _4T_LE _ (ContinuedO= O= O= O= O= O= oWl3w3z w51 1.6
16、85 -9._5 -8.00 16.790 1_.25 0h h hWi, 3 w3 3 Wl5 -2.34 0 16.790 0 -46.16 -19.485h h hWI,3 W3,3 W5,1 -4.68 0 16.790 0 -19.485 Uh h hW3l W33 WI5 -4.68 16,790 0 0 0 -19.485h h hoW_p_ W5I -2_34 16.790 0 0 -19.485 -46.16h h h_2 Wl,5 2.385 0 0 0 38.27_ 20._5h/ h(,_2 “5,1 2.385 0 0 0 _0.25 38._72h0 18.313
17、0 -B3.08 0 0Wl,3 _w_._2-2.825 13.451 0 0 -13.625 0hw3,1 -2.825 0 13.45 0 0 -13.625h2w31(51/- - 0 0 18.31_ -_3.08 0 0h h/W13 Wl5 W5l 0 0 10.25 -19.485 0 -27.25h h hW3l Wl5 wSl 0 10.25 0 -19.485 -_.25 0h h h0 “23o08 0 38.272 0 0hw33_51_ 2 0 0 -23,08 38.272 0 0h h/W33 Wl5 W5l 3.645 -19.485 -19.485 _.5
18、0 0h h h_/3 0 0 0 0 39.125 0Io o o o a !Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technical Note No. 846 25TABLE III - VALUES OF COEFFICIENTS IN DEFLECTIONFUI_CTi0N OF EQUATIO_ (6) FOR SQUARE PLATE UIXD_REDC-E C0_,_SSIO_ _ = 0.31_3w w w w
19、Wcenterpxa wi,1 w _z_3 3-_-x-_ 3_s l_l_s _h e h h h h h h h3.66 o o 0 0 o 0 o.ooo3.72 .25 .ooo1o9 ,ooo164 .oooooo .oooooo .oo0ooo .25o396 .5o .ooos4s .oo13o8 .oooo12 .OOOO01 .0oooo1 .4984.34 .75 .o0275 .00434 .oooo_6 .000005 .o00009 .7434.S7 1.oo .oo615 .OlO43 .ooo36o .oooo17 .oooo36 .9S45.51 1.25 .
20、o1127 .0203 .OO1063 .o00044 .00o104 1.2206.3o 1.5o .o181 .0350 .o0257 .00oo92I .ooo241 145o!722 1.75 .0267 .0561 .0o541 .oo0166 .000484 1.673S.24 2.oo I .0370 .O846 .01040 .0o0284 .0oo879 1.8899.38 2.25 .0493 .12o8 .0184 .ooo467 .00143 2.101!10.61 2.50 .o635 I .1670 .0307 .00082 .00215 2.303i1199 2.
21、75 .0790 .226_ .04S8 .00145 .00313 2.49813.4S 3.0o .095 .299 .0743 .00273 .0041 2.687I14.97 325 .112 .384 .1o7 .oo483 .oo51o 2.S7116.79 3.5o .129 .493 .151 .00S93 ! .00565 3.0_4I18.77 3.75 138 .626 .206 0161 i .00392 3.212“ “ i I21.45 4.oo .124 _ ._OS .2S7 .o3o3 -.oo21 3.376Provided by IHSNot for Re
22、saleNo reproduction or networking permitted without license from IHS-,-,-NACA Teohn.l.oa3. Note No. 846 26TABLE IVVALUES OF COEFFICIENTS IN STRESS FUNCTION OF EQUATION (8) FOR SQU_RE PLATE UNDER EDGE COHPRESSIONo3163,66 3.7_“3.96 4.34 4.87 5.51 6.30 7.82 8.84 10,61 11.99 13.48 14.97 16.79 18.77 81_.
23、45Eh_4_Bb08 .00 .08 .31 .69 1.82 1.89 8.73 3.69 4.80 6.10 7.52 9.15 I0.92 18.88 15.08 17.58 _0.55Eh8.00 .08 .51 .68!1.81 1.86 8.66 3.53:4.58 5.59 6.70 7.88 8.84 9.88ii0.60 11.12 11.83 !Eh816_8b04 .00 .00 .00 .00 .01 .03! .07 ,18 .80 .38 .50 .77 1.18 1,77 8,70 3.99 6.15Eh816_864,0 00 .00 .00 .01 .03
24、.06 .13 .84 .48 .68 1-06 1.60 8.34 3.30 4.64 6.39 879Eh84_b8, _ IQ O0 i00 .00 .01 .04 .09 .19 .34 .57 .89 1.38 1.90 8.64i 3.56 4.74 6.19 8.07Eh_-36r_b06 .00 .00 .00 .00 .00 .00 .00 00 .00 .00 .01 .05 .I0 .17! .35 .64 1,84EhZ56n;_60 .00 .00 .00 .00 .00 .00 .00 .00 .0_ .0_ .06 .ii .S0 .35 .59 .98 1.70
25、,h816_8b8,4 .00 .00 .00 .00 .00 .00 .01 .03 .08 .17 .38 .58 .99 1.56 8.38 3.43 4.86Eh816_b4 8 .00 .00 .00 .00 .00 .00 .00 .01 .04 .Ii .$3 .441 .76 1.84 1.98 8.81 4.10Eh836n_bs6 .00 .00 .00 .00 .00 .00 .00 .00 .00 .01 .08 .03! .06 ,11 .19 .38 .54_h836nSb6, .00 .00 .00 .00 .00 .00 .00 .00 .01 .08 .04
26、.08 .17 .38 .59 1.03 1.87Eh_16w_b44 .00 .00 .00 .00 .00 .00 .00 .00 .00 -.01:-.08 -.04 -.06 -.08 -.II -.ii -.08_h864_8b0 8 .00! .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .01 .01_h864Tri_b8,0 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .01 .01 .00_h836nSb46 .00 .00 .00 .00 .
27、00 ,00 .00 .00 ,00 .00 .00 .00 -.01 -.08 -.05 -.i0 -;80Eh_-36w266,4 .00 .00 .00 .00 .00 .00 .00 .00 .00 ,00 -.01 -.01 -.03 -.05 -.09 -.15 -.88_h864_b8_“ .00 .00 .00 .OOi .00 .00 .00 .00 .00 .00 .00 .00 .00 .01 .01 .01 -.01Eh864rrSb_.,8 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .01 .01 .04
28、.10Eh8Others .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .O0 .00Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technlcal Note No. 846 27TABLE V - CONVERGENCE OF S OLUTION FOR EFFECTIVE WIDTHOF A S_UARE PLATE UNDER EDGE COMPRESSI
29、ON AS THENUMBER OF EQUATIONS OF THE FAMILY OF EQUATION (9)USED IN THE SOLUTION IS INCREASED = o.316Effective _d dthInitial _i dthUsing one Using three Using four Using sixAverage edge strain from from from fromCritical strain equation (9) equation (9) eqD1ation (9) equation (_)i.1.00 1.000 1.000 1.0
30、00 1.0001.67 .797 .797 .797 .7977. Ol .570 .535 .525 , .525l13.50 .538 .480 .434 .434Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technical Note No. 846 28TABLE VI - VALUES OF COEFFICIENTS IN DEFLECTIONFUNCTION, EQUATION (6), FOR SQUARE PLATE
31、 UNDERUNIFORM NORMAL PRESSURE pEdge compression = 0; _ = 0.316pa 4 w _penter,i _ w w w wEh 4 h h h h h h hI0 0 0 0 0 012.1 .500 .00781 .000814 .000644 .48629.4 1.000 .02165 .00254 .00156 .96256.9 1.500 .0447 .00666 ,00303 1.42499.4 2.000 .0776 -0152 .00524 1.870161 2,500 .1195 .0299 .00831 2.307247 3.000 .167 .0516 .0123 2.742358 _.500 .221 .0813 .0175 3.174497 4.0-00 .282 .116 .0236 3.600iProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
