REG NACA-TR-737-1942 Bending of Rectangular Plates with Large Deflections.pdf

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1、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT No. 737BENDING OF RECTANGULAR PLATESWITH LARGE DEFLECTIONSBy SAMUEL LEVYNational Bureau of Standa

2、rds476720-42 IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSHEADQUARTERS, 1500 NEW HAMPSHIRE AVENUE NW., WASHINGTON. D. C.Created by act of Congress approved March 3, 1915, for the supervision and directio

3、n of the scientific study of the problemsof flight (U. S. Code, title 50, sec. 151). Its membership was increased to 15 by act approved March 2, 1929. The members areappointed by the President, and serve as such without compensation.JEROME C. HUNSAKER, Sc. D., Chairman, O.P. ECHOLS, Major General, U

4、nited States Army, Col_-Cambridge, Mass. manding General, The Matdriel Command, Army AirGEORGE J. MEAD, Sc. D., Vice Chairman, Forces, War Department.Washington, D.C. SYDNEY M. KRAUS, Captain, United States Navy, Bureau ofCHARLES G. ABBOT, SC. D., Aeronautics, Navy Department.Secretary, Smithsonian

5、Institution. FRANCIS W. REICHELDERFER, Sc. D.,HENRY H. ARNOLD, Lieut. General, United States Army, Chief, United States Weather Bureau.Commanding General, Army Air Forces, War Depart- JOHN H. TOWERS, Rear Admiral, United States Navy,ment. Chief, Bureau of Aeronautics, Navy Department.LYMAN J. BRIGGS

6、, Ph.D., EDWARD WARNER, Sc. D.,Director, National Bureau of Standards. Civil Aeronautics Board,W. A. M. BURDEN, Washington, D. C.Special Assistant to the Secretary of Commerce. ORVILLE WRIGHT, Sc. D.,VANNEVARBUSH, Sc. D., Director, Dayton, Ohio.Office Scientific Research and Development, THEODORE P.

7、 WRIGHT, So. D.,Washington, D.C. Assistant Chief, Aircraft Branch,WILLIAMF. DURAND, Ph.D., War Production Board.Stanford University, Calif.GEORGE W. LEWIS, Director of Aeronautical Research JOHN F. VICTORY, SecretaryHENRY J. E. REID, Engineer-in-Charge, Langley Memorial Aeronautical Laboratory, Lang

8、ley Field, Va.SMITH J. DEFRANCE, Engineer-in-Charge, Ames Aeronautical Laboratory, Moffett Field, Calif.EDWARD R. SnARP, Administrative O._cer, Aircraft Engine Rese a number of approximate solutions (refer- -3_-t-Z_x-x_by 2-_ by _ Dences 2 to 7) have been developed for the case of a , h O2F b2w 52F

9、52w 2 _2F b2W “_rectangular plate. This paper presents a solution of -t-D_ _f 2 _)x2 bx2bY _ _-_ 5xby (2)yon Krmns equations in terms “of trigonometric where the median-fiber stresses areseries. 52F 52F b2F (3)Acknowledgment is due to the National Advisory _ z=-_-2 _ _= _x-_r _.v=-bxbyCommittee for

10、Aeronautics and the Bureau of Aero-nautics, Navy Department, whose research projects on and the median-fiber strains aresheet-stringer panels have provided the impetus and , lb2F 52F_the necessary financial support for the work presented _ _= _:_-_2._-_-2)in this paper. The author takes this opportu

11、nity to , 1152 F 52F. _acknowledge also the assistance of members of the _ _=_-_-_-_-_) (4)Engineering Mechanics Section of the National Bureauof Standards, particularly Dr. Walter Ramberg, Mr. 2(1-_) b2FPhillip K_rupen, and Mr. Samuel Greenman. “Y*_=- E 5x_y 1Provided by IHSNot for ResaleNo reprodu

12、ction or networking permitted without license from IHS-,-,-2 REPORT NO. 737-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSThe extreme-fiber bending and shearing stresses are andb2w_ b% q-1 , Eh “_ B,=_._ p.kt(p_)(_t)_k_(q_t)2lw_.,_eor /hcorx plates due to the addition of lateral load is seen to be_ pa

13、_c E_I appreciable for _-_2.25.0 IO0 200 300paa/Eh 4“ XO , -T r-_ - iA, (_%aVEh9D, (o,%aVEh9J_ ,_:3_,0-%_a21Eh9a, C, 0-“._aZlEh9BD_,(a“.a_/Eh 2)_, (auaVEh2)_ E, (a“.a_lEh9 c, (o“_a2/Eh2)v tDx a z l_x , _ ,plate under uniform normal pressure. Edge displacement=0; _=0.816. Lineartheory from reference

14、9. .8As a measure of the rapidity of eonvergenee, the re-sults obtained by using one, three, and six equations of -q _ pa4/Eh ._: 0 mil,oqu ion ivonin 11convergence of the value of the pressure is both rapid _-_ _ 23.5and monotonic. In the case of the center deflection, _c_the convergence is oscilla

15、tory. For small pressures, _ _ /_this oscillation decreased rapidly (reference 4, p. 316). _ I,_/For larger pressures the decrease in amplitude of oscil- _lation is less rapid, as is indicated by table 11 (b), butan estimate of the asymptotic value may be obtainedby noting that this value, if it exi

16、sts, must lie betweenthe value at any particular maximum (minimum) and 2the average of that maximum (minimum) with thepreceding minimum (maximum). Since the next fourequations for _-_ wT,_ wz.s _ will cause a decrease inWW o / _ _ 4Avez-oge edge ._tr-o/nw_,_%h the correct value of w_,must lie betwee

17、n (o-/co/,_/,-o,)_oO1.827 (average of 1.807 and 1.846) and 1.846 when vm,_ 14.-Effeetofnormalpressureoneffectivewidthorasquareplateloadedbyedge compression.pa _ pa _-_=278.5. At higher values of _ it may be neces- As a measure of tbe convergence, the results obtainedsary to use the first ten equatio

18、ns of the family of equa- by using one, three, four, and six of the equations intion (9) to get a solution accurate to within 1 percent table 2 are given in table 13. The convergence is rapidfor center deflection, and monotonic.Provided by IHSNot for ResaleNo reproduction or networking permitted wit

19、hout license from IHS-,-,-BENDING OF RECTANGULAR PLATES WITH LARGE DEFLECTIONS 9SPECIFIC SOLUTION FOR A RECTANGULAR PLATE by successive approximation from their respectiveWl., and P_b2_(a=3b) WITH NORMAL PRESSURE SYMMETRICAL TO equations the corresponding values of -_AXES OF PLATE /5/b“The first two

20、 equations of the family of equation (9) These calculations have been made for 13 values of pb4Eh 4for the case of a rectangular plate whose length is threetimes its width (a-3b) are, for _=0.316. and w3., with the results given in table 14. Tl_e ratio-h-b4p,.,_0.1142w,l_h _xb 2 Wl., _,b 2 Wl,_ / of

21、 effective width to initial width was computed from7r4Eh4 97r2_L_.h2 h _=_Eh2“h equation (11) and table 14, with the results given inthe last two columns of table 14 and in figure 16. +0.0632 ) “ LWl l 3 Wlt 2_)3 1 Wl 1 qfl31 2_=“) - O“1873(_-) -_= + O“267-_(-_ 14) /.0 “_=_- ,- 4.50 _-0.0625 +0.267

22、_+0.125 _- .8 _.In the previous solutions a close approximation wasobtained with one equation as long as _1-. For _l _this reason in the following problem only the first two _ B _ a -equations, as given by equation (14), will be used and the _-: h_:-7 VP_b : b :_bhdeflections will be limited to valu

23、es of _1. It _: _/bshould be noted that the two equations of (14) will be “ _ _ adequate only-as long as the normal pressure can be -hdescribed by the first two terms of equation (7): .2_ry “ 3_rx Tryp_=p,., sin vx sin T +P3., sin - sina a -b-For more complicated pressure distributions as well asfor

24、 _1, more equations of the family of equation o. .4 .8 /2 /.6Averoqe edye stro/n(9) should be used. (crZ/ca/s/ro_n)p.oThe following results apply to rectangular plates Fm,_ 16.-Effect of normal pressme on effective width of a rectangular plate(a= 3b) loaded by a uniform lateral pressure p and by (a=

25、38)loadedby edge compression on theshortsides.edge compression acting on the shorter edges as shown The reduction in effective width of rectangular platesin figure 15. (a=3b) due to the addition of lateral load is seen tobe less than in the case of square plates (fig. 14).IY a _ COMPARISON WITH APPR

26、OXIMATE FORMULASl EFFECTIVE WIDTH,Q. b _5_ h Approximat e formulas for effective width have beenderived in references 2, 3, 6, and 7._“ Marguerre (reference 2) expresses the deflection for.t5_b h_t,P, ttl,ll ;5_bh a square plate by a series similar to equation (6), He_ limitshimself, however, to w_.

27、, ego3.1,and w8,3 and in1F_._ _.-Combinednormalpressureand edgecompressionfor a reotangu_ar his numerical work requires that wa.3-=-_ Waa and thatplate (a=Bb). g=0. His stress function corresponds to the firstPoissons ratio _ is taken as 0.316. The edge com- terms of equation (8). He uses the energy

28、 principle topression in the y-direction “_ah is zero. The coeffi- determine the values of w,_ and w8,t instead of thecients p_,_ in the Fourier series for the pressure as differential equation given as equation (2) in the present1(4) _given in equation (7) equal _ p. The method of work. Marguerres

29、approximate solution is given ascurve c in figure 17. It is evident that, even thoughobtaining a solution of equations (14) for this ease con- Marguerre has limited the number of his arbitrary_r P b_ w3,_sists of assuming values ,_ _-_ana _- and determining parameters to two and has taken _=0, his r

30、esults arein excellent agreement with the results obtained in theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-0 REPORT NO. 737-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSpresent paper. Marguerres approxhnate formula be/b The well-known formtfla of

31、 von Krmn (see refer-=3_/e_-_/e is given as curve b. This curve checks within ence 7) b,/b=_/_/_ is plotted as curve a in figure 17.about 7 percent with the exact results. It is in good agreement with the effective widths ob-Bengston (reference 3) assumes a sinusoidal deflection tanned in this paper

32、 for small values of the ratio e/ec_equivalent to the first term in equation (6) in his solu- but is about 20 percent low for e/ec,=-4.tion for a square plate. He then chooses his displace- Cox (reference 6) in his solution for the sin_ply sup-meEts so that the strain at the supported edges is uni-

33、ported square plate uses energy methods together withform but, in order to do so, he violates equation (1). the approximation that the strain is uniform along the/.O_ Exocf-_ _ :-% e d-“ b“ -.2o 2 4 G 8 I0 /2 14A veroge edge _fz-o/27 Cri/co/ _tFa/h J Eo_FmuRE 17.-Effective-width curves for a simply

34、supported square plate according to different sources.Curve Soulco7 b.=_/Ua reference b V_=_/_b approximate formula of reference 2, b Yc approximate solution of reference 2.d solution of reference 3.e formula of reference 6, _-=0.09J-0.80-_ _Exact derived from present paper.Owing to the method of ch

35、oosing the displacements, entire length of narrow element of the panel. Thehowever, the resulting errors should be small. The effective-width curve thus obtained is plotted in figureenergy principle is then used to obtain the solution. 17 as curve e. It gives effective widths 10 to 20 percentIn orde

36、r to take account of secondary buckling, it is below those obtained in this paper.assumed that buckling of Y3 and g the original wave DEFLECTIONUNDERLATERALPRESSURElength will occur independently and that the resulting Naviers solution for the simply supported squareeffective width will be the produ

37、ct of each of the separate plate with small deflections (linear theory), given ineffective widths. Finally, an envelope curve to the reference 9, is included in figures 7, 9, 11, and 13. Iteffective widths thus constructed is drawn. This curve is seen that for small deflections the solution given in

38、is given as curve d in figure 17. It differs less than 7 this paper is in agreement with Naviers linear theory.percent from the effective widths obtained in this paper. Kaiser (reference 5) converted yon Kgrmns differ-The fact that Bengstons values are lower indicates that ential equations into diff

39、erence equations and calcu-the increased strength which should result from the con- lated deflections and stresses for a square plate underditions of uniform strain at the edges is lost due to the constant.pressure assuming simple support at the edgesapproximate method of taking account of secondary

40、 w_,._,buckling, with zero membrane stress. He obtained hProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-BENDING OF RECTANGULAR PLATES WITH LARGE DEFLECTIONS .2.47 for Pa4-i18.8. This center deflection is about specimen the agreement is excellent up

41、to stressesEh 4- for which yielding due to the combined bending and25 percent higher than the curve in figure 7; this differ- membrane stresses was probably taking place. In the ence is probably due to tile fact that Kaiser allows dis- case of the 0.025-inch aluminum-alloy specimen thetortion of the

42、 edges of the plate. The membrane observed effective width exceeded the calculated valuesstresses calculated by Kaiser are about one-fifth as fore/ecrU7 but the agreement was excellent for d/ec,_7,large as those given in the present paper. This fact, which appeared to be large enough to reduce the e

43、ffectas well as a comparison of figures 8 and 12, indicates the of the torsional stiffness of the stringers as a factor inlarge influence of edge conditions on the membrane the edge conditions.stresses.DEFLECTION UNDER LATERAL PRESSURECOMPARISON WITH EXPERIMENTAL RESULTS- Kaiser (reference 5) has co

44、nducted a carefully con-EFFECTIVEWIDTH trolled experiment on one simply supported plate. InExtensive experiments on two aluminum-alloy sheet- this experiment, as in Kaisers theoretical work, thetringer panels 16 inches wide, 19 inches long, and 0.070 edge conditions are such that the membrane stress

45、es at/.O_ a Tesf on O.070-/nch 245-T o/clod olurninurn:olloy pone/.Str/zTgera op2rox/moled simple auppor?.o Tesf on O.025-/noh _4_-T o/um/nurn-o/oy cone/.2fringeF_ provided restFo/nf ogO_TSf Fofo/bn.8 a _“Edge _ t/-ess exceeds 25, 000 Ib/_q #7.2O 2 4 6 8 I0 12 14AVeFoTe edge sfrOln ECrifiCo/ .sfroz

46、J EcrFIGURE18.-Comparison of computed effective width and experimental results from reference 8. The critical strain is the e_mputed critical strain for simply supportedsquare plates.and 0.025 inch in thickness are reported in reference 8. the edge are zero. The initial deflections obtained byThe sh

47、eet of the 0.070-inch panel was 24S-T alclad Kaiser are in agreement with the results in this paper.aluminum-alloy and the 0.025-inch panel was 24S-T At large deflections, however, the fact that the mem-aluminum-alloy sheet. The panels were reinforced by brane stress at the edge of the plate was zer

48、o in thestringers (0.13 sq in. in area) spaced 4 inches on centers, experiment causes the measured deflections to exceedDeflection curves measured at the time of the experi- by appreciable amounts the deflections calculated inments indicated that in the panel having 0.025-inch this paper.sheet the torsional stiffness of the stringers was largeenough compared with the stiffness of the sheet toprovide appreciable restraint against rotation at theedges; in the case of the 0.070-inch _lclad aluminum- NATIONAL BUREAU OF STANDARDS