1、,?!REPORT NO. 740SQUARE PLATE WITH CLAMPED EDGES UNDER NORMAL PRESSURE PRODUCINGLARGE DEFLECTIONSBy SAMUELLEVYSUMMARYA theoretical analy_-is is given for the stresses anddeflections of a square plate with damped edges undernormal pressure producing large deflections. Valu6_ ofthe bending s_.ress and
2、 membrane stress at the center of theplate and at the midpoint of the edge are given for centerdeflections up to 1.9 times the plate thiclcness. The shapeof the deflected surface is given for low pressures and forthe highest pressure considered. Convergence of thesolution is considered and it is est
3、imated that the possibleerror is less than _ percent. The results are comparedwith the only previous approximate analysis known tothe author and agree _yithin 5 percent. They are alsoshown go compare favorab!y with the known exact solutionsfor the long rectangular plate and the circular plate.INTROD
4、UCTIONAn exact solution for the small deflections of a platewith clamped edges was given by Hencky in reference 1and an approximate solution for large deflections waspresented by Way in reference 2. In a previous paper(reference 3) there is presented a solution of thefundamental yon K_rm_n large-def
5、lection equations fora simply supported rectangular plate under combinededge compression and lateral loading.In the present paper a theoretical analysis is givenfor the stresses and deflections of a square plate undernormal pressure producing large deflections. The edgesupports are assumed to clamp
6、the plate rigidly againstrotations and displacements normal to the edge but topermit displacements parallel to the edge The analysisreplaces the edge bending moments by an equivalentpressure distribution and then applies the generalsolution for the simply supported rectangldar plate.The results for
7、small deflections obtained by theanalysis agree exactly with those of Hencky and forlarge deflections differ by less than 5 percent from theapproximate solution of Way.The work was carried on with the financial assistanceof the National Advisory Committee for Aeronautics.Acknowledgement is made to t
8、he Bureau of Aeronautics,Navy Department, for its cooperation in a program oftests of rectangular plates under normal pressure thatfurnished the background for the preparation of thispaper. The author is grateful for the assistance ofmembers of the Engineering Mechazfics Section of theNational Burea
9、u of Standards, particularly that ofDr. Walter Ramberg and Mr. Samuel Greenman.,yFUNDAMENTAL EQUATIONSSYMBOLST 7 7 , J T i # IPa tFmul_m I,-Unifcem normal pr“-udure on s _mped square pl_.e.Consider an initiMly flat square plate of uniformthickness (fig. 1) and leta length of sides.h thickness.p norm
10、al pressure, assumed uniform.w normal displacement of points of middlesurface.E Youngs modulus._, Poissons ratio.Eh*D- flexural rigidity of plate.12(1 -_-)x, y coordinate axes lying along edges of plate withtheir origin at one corner.209-_ I I I II II II IIIIIProvided by IHSNot for ResaleNo reproduc
11、tion or networking permitted without license from IHS-,-,-210 REPORT NO. 740-NATIONAL ADVII_,ORY COMMITTEE FOR AERONAUTICS_qxl ;*iCyff7“7(_z, 0“;ffg_ fly1Urn,.FOzl fillp,(x, y)edge bending moments per unit length aboutx and y axes, respectively.normal stress.shearing stress.tensile strain, unit elon
12、gation.shearing strata.extreme-fiber stresses in directions of axes.median-fiber stresses in directions of axes.extreme-fiber hending stresses in directions ofaxes.deflection coefficients.stress function.stress coefficients.average median-fiber stresses in x and ydirections, respectively.auxiliary p
13、ressure replacing edge moments.-_c -_!-Zi-4c -Ica “_YFl(;,Val 2.-Auxiliary pressure distribution for applying ed_ moments alon_ theedges.( and,for the pressure coefficients p are substituted therevalues as given by equation (10). The resultingequations are.0 - 2.835 4- 7.66k_ 0.324k3 + 0.0800ks 0.03
14、03k_4-0.0145k_4- . 4-K_0 = 0.0523 5- 0.324kt 4-1.713k3 4- 0.1405/%4-0.0675kTW0.0360k_4- . 4-K30 = 0.00680 + 0.0800k_ + 0.1405k_ 4- o.956k,4-0.0690kr+0.0433k0+ . . . +Ks (21)0 = 0.001767 -+-0.0303kt 4- O 0675k34- 0.0690ks+0.660kr+0.0402k_4- . 4-Kr0= 0000648 +0.0145kt + 0 0360k3 + 0.0433ks+0.0402k_+0.
15、50.Sk_+ . . . 4-K_where K_ . . . K, are functions of the pressure p andof the cubes ofthe deflection functions w The first22 terms in the equations for the first five coefficientsK, _re given in table 2. As an example of the use oftable 2,w_ _ _ *_Zh4 _Yl 1 2 q/)1.3“ . pa_O7,“Ehw, t y/ws _)+o.1-_r-T
16、t -Uz- . . . (22)SOLUTION OF EQUATIONSVALUI_ OW DEFLECTION COI_IeFIC1ENTS w AND EDGI_ MOMENTCO I_WlffICII_NTS koThe method of obtaining the required values of thedeflection coeffic|ents w. and the edge momentcoefficients k, consists of assuming values for w_.t-_- andpa _ W_ a Ws .3then solving for _
17、h- i, -_-, -_- kt, k_, k_ bysuccessive approximation from the simultaneous equa-tiorm in t_.ble 1 and equations (10) and (21). Thesecalculations have been made for 10 values of w_._hThe corresponding values of the first 36 deflectioncoefficients -_ and of the first five moment coefficientsk, are giv
18、en in table 3 and table 4, respectively. Theerror arising from the use of only the first 22 terms inthe equations in table I will be considered in a latersection.CINTIgU DI_IeLI_CTIONFrom equation (12) the center deflection is“_(_ = - - I -“i- w,., (23)_-t, 3, 5, . n-l, 3, 5 Provided by IHSNot for R
19、esaleNo reproduction or networking permitted without license from IHS-,-,-:roSQUARE PLATE WITH CLAMPED EDGES UNDER NORMAL PRESSUREThe center deflection was obtained by substituting thevalues of w,., from table 3 in equation (23) with theZO_ - - _ _ _- : . ,-i : i : , ._ .5_ , I, .-_ “ / i I5 / / : -
20、 1 /J f ;, I i j ;_ ,o /oo zoo soo ,ooPressur-e ratio, pa/Eh FIGURt 3.-Centd_r deflection of sqna_ plate with clamped edges. #, 0.318.resultsgiven in table 5 and figure3. Figure 3 showsthat the deflectionpressurecurve deviates increasingly1,0,-/,902.4 tt:; i Il _ .( aO_sto_ce from clamoed edcJeF1GUI
21、tl 4.-Shal_ ol deflected surf_ along center line Z-a/2 for very small deflectionwe,t_fll_/_,_ _ /I II I/ I/_,/ o Colcu/ofed /oo/nPs, 10 /00 COO 300 400Presxure rot/o, paalEh_FIOUII 8,-Variation of deflectio_ It center with pre_ttr_ for sqtuwe lalat_, circ,.llirplate, and long rectsmgu_ar plate.(refe
22、rence 14) of diameter a and those for a clampedlong rectangular plate (referencs 4 and 15) of width a.As would be expected, the ._uare plate is more rigidthan the long rectangular p,ate and more flexible thanthe circular plate./IProvided by IHSNot for ResaleNo reproduction or networking permitted wi
23、thout license from IHS-,-,-JitWt?/iiik216 REPORT NO. 740-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSi ! i L i 1 i !I_ Clomped ion 9 rectongulor plote _rf. 15) _Po,ssons r/o, 0.3/6. I_, Clomped _quore plofe fpresen_ pope/-)Po,_sons roMo, 0.3/5.C, Clomped orcular plo?e (reference 2)Pots$ons rOt O, 0 _
24、. I _ I , , ii i_ ii i i i /i !i !/!z,x . . / _ ( (/_ ieo- II/_ ! ) , _ I 1/_t/ , ! I I ,/ , I ,!0 /00 ZOO 300 400Pressure r“OMO, oa4/Eh*FIGURll 9.-Y_r/alion O( _i_tlUl3a e%trerae-flber stre_l at _dge with prt_sure for$qua_ plate, irc-ahu“ plate, az_d lon_ re_t_n9 -11.0_ .810 0 089.5 -19-,0411 -. O_
25、,.q -.0621 -. 0_11-, 02_7 -. 0_181 -._8 -. 0_8.0133 .00_ .0_1 , XY28-. 0112 -. 0138 -. 0186 -. 0196 _016 . 0017 .0018 . . .018.OO2O .0Cg4 .002_ .00_2-.0_3 -.0_ -. OGOI -.0110.OOO6 .000_ . IXC6 .OOCO._00 .0910 ._113 .0015-. 004 -. 00,_ -.CC_ -.00_.occa .occ_ . oCO_l .oc_.0_04 .0006 . o(lc_ .0007.0c04
26、 .0006 .coco .0007-.00_ -.0(33 -. c_lil -.004./.0002 .oco_ .0002 .olx_.0c02 .0003 .0004 .c_04.0002 .00_ .cool .01104Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-222TABLE 4.-VALUES OF EDGE MOMENT COEFFICIENTSk, k, AS FUNCTIONS OF THE NORMAL PRESSURE p- Pa4; 0 -0. 372 17,8 -.3_:_. 3 - 352, gt 4 - ,T_95.0 - 30_l:bl, 9 - 28617?+aProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
