1、NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE No. 1851 CRITICAL SHEAR STRESS OF INFINITELY LONG, SIMPLY SUPPORTED PLATE WITH TRANSVERSE STIFFENERS By Manuel Stein and Robert W. Fralich Langley Aeronautical Laboratory Langley Air Force Base, Va. i - Washington April 1949 Provided by IHSN
2、ot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY CWTEE FOR A;EROpsATpTICS TECHNICAL NO323 NO. 1851 CRrrICAL SHEAR STRESS a? m=Y LOE, SIMPLY By Manuel Stein ad Robsrt W. mich Athearetical solution is given far the critical shear stress of an infinite
3、ly long, simply supparted, flat plate with identical, equally spaced, transverse stiffenars of zero torsional stiffness. Result6 are obtained by mans of the Lagrangian multiplier method and are presented in the fm of design charts. are found to be in good agreement wlth the theoretical results. Xxpe
4、rimental results are included and TPJTRODUCTION The design of ehear web beams and nomrinlding skin surfaces requires a knowledge of the critical she.= Stress of stiffened plates. The purpose of the present pqmr ie to give the theoretical critical shear stress of an infinitely long, simply supported,
5、 flat plate rei+ forced with identical, equally spsced, trmmarse stiffeners. The results are found by means of the Lagrangian multiplier method. The stiffeners are assumed to have bending stiffness but no torsional stiffness and are assumed to be concentrated along transverse lines in the middle pla
6、ne of the plate. no torsional stiffness. applies with little error in the case of many open section stiffeners. The asawnption that the stiffeners are con- centrated alwg transverse lines in the midiUe plane of the plate is parison with the stiffener spacing. The assumption that the stiffeners have
7、Y applicable whenever the width of the attached flange is small in com- The theoretical analysis of the problem is given in the appendixes. For completeness, an energy solution for the plate with relatively weak stiffeners is given in appendix A. The solution for a plate with stiffeners of intermedi
8、ate or higher bending stiffness is given in appendix B. curves which cover the complete range of stiffener stiffness and various stiffener spacings and in a table giving values from which the curves The results am presented in the form of nondimnsional Provided by IHSNot for ResaleNo reproduction or
9、 networking permitted without license from IHS-,-,-2 NACA TN No. 1851 . were drawn (table I). Comparison of these results with the present theory indicates good agreement between theory and experiment. merimental results are :resented for 20 panels. I 7 critical shear stress t thickness of the plate
10、 b width of plate I d stiffener spacing %Id panel aspect ratio I I I I D flexural stiffness of the plate E Youngs modulus for plate P E Youngs modulus for stiffener I effective moment of inertia of stiffener L CI Poissons ratio for materiai E w ratio of stiffener atiffness to plate stiffness I x hal
11、f wave length of buckles W deflection of the plate (wSi deflection of the ith stiffener X, Y reference axes 7 1 II Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN NO. 1851 3 coefficients of deflectian function 7n undetermined Iagrangian multi
12、pliers v internal energy of bending of the plate internal energy of bending of stiffeners vS T external work of the stresses The problem of the buckling of stiffened plates in shear has been treated by many authors by the use of both thearetical and semi- empirical methods. In 1930 Schmieden (refere
13、nce 1) solved the differ- ential equation for an infinitely long plate stiffened by closely spaced transverse stiffeners (equivalent to orthotropic plate) and found exact stability criterions for shear buckling of plates with simply supported ems and with clamped edges. simplifying modifications of
14、the stability criterions, Schmieden obtained approximate values of the critical shear stresses. Later in 1930 Seydel (reference 2) obtained exact solutions for infinitely long orthotropic plates with simply supported or clamped edges. With the use of the proper parameters Seydel*s results can be rea
15、dily applied to plate-stiffener combinations. The values of the stresses obtained .from Scbmiedens theory lie slightly below the exact values of Seydel. In 1947 T. K. Wang (reference 3) used the energy method to obtain an appraximate solution for plate-stiffener combinations with simply supported ed
16、ges. All the foregoing solutions me applicable only to the case of weak stiffeners, where the stiffening effect of the stiffeners can be considered to be uniformly distributed over the plate. By making certain Wangs results lie above the exact values of Seydel. Solutions am also available for plates
17、 reinforced by rigid In 1936 Timoshenko (reference 4) treated the case of By mean8 of ths energy method Timoshenko found the stiffeners. simply supported rectangular plates reinforced with one or two stiffeners. stiffener flexural rigidity necessary to prevent bwAcling across stiffeners with the con
18、servative assumption that the stiffeners act as - simple supports. In 1948, Budiansky, Comer, and Stein (reference 5) found the critical shear stress for an infinitely long, cla.m?ed plate divided into square panels by nondeflecting intermediate supports which Provided by IHSNot for ResaleNo reprodu
19、ction or networking permitted without license from IHS-,-,-4 correspond to rigid stiffeners. They also considered the case of a NACA TN No. 1831 plate of infinite length and width having nondeflecting intermediate supports that form an array of square panels. Kuhn has written a number of papers on r
20、elated subjects in which he presents semiempirical results for the critical shear stress of stiffened plates. (See, for example, reference 6.) The available theoretical solutions treat the relatively unim- portant case of weak or closely spaced stiffeners and the case of rigid stiffeners that divide
21、 a plate into square panels. None of the theoretical solutions presents results for the practical range of intermediate stiffener stiffness and very little theory is presented for the practical range of spacing of rigid stiffeners. Also, it is felt that the semiempirical results for transverse stiff
22、ened plates cannot. be extended to all stiffener spacings and stiffnesses without a sound theoretical basis. The theoretical results of the present paper cover the complete range of stiffener stiffness and the practical range of s tiff ener spacing . RESULTS AND DISCUSSION The critical shear stress
23、for a plate-stiffener combination is given by the formula f12D b2t T=k - Curves are presented in figure 1 giving corresponding values of kS and the stiffness parameter - for simply supported, transversely stiffened plates with panel aspect ratios of 1, 2, and 5. These results are replotted in logari
24、thmic fo,m in figure 2 for comparison with experiment,al results. Dd The points of discontinuity of the slopes in the curves of figure 1 The present results for an ortho- represent chaises in buckle patterns. lropic plate agree with the exact results of reference 2. vation of the buckling criterion
25、for an orthotropic plate (a plate sLiffened by stiffeners of low bending stiffness) is given in appendix A. The derivation of the buckling criterion for plates stiffened by si,it“feilern of higher bending stiffness is given in appendix B. The deri- Provided by IHSNot for ResaleNo reproduction or net
26、working permitted without license from IHS-,-,-NACA TN No. 1851 5 ?i- previous solutions, values of k, were found by using the orthotropic-plate curve and a cut-off at the value of ks for simply supported panels. (See fig. 1.) These figures show that the present solution yields values of k, by the o
27、rthotropic-plate curve in the intermediate range of stiffener stiffness. Also, the present solution for more rigid stiffeners yields a curve that is higher than the cu-f, which is obtained by assuming the stiffeners to have the effect of aimple supparts. Since the conti- nuity of the plate across th
28、e stiffeners of higher bending stiffness certainly adds a constraint to the plate, a higher buckling stress than that carresponding to a simply supported eQe is obtained. that am considerably belar those given In figure 2, experimental results axe cnmpared with the theoretical curves. These results
29、am fram two sources. The first set of erperi- mental data is taken from NACA tests on shear webs of 24%” aluminum alloy attached to torsion boxes. box and the method of loading are given in reference 7. were obtained fromthe stiffener load-deflection curves which were taken from the original data. p
30、resent paper is the average load at which the stiffeners start to deflect. given in table 11. Ikawings of a shear web and torsion Buckling loads Each of the buckling loads given in the The properties of the specimens and the buckling data are The second set of experimental data is taken from NACA te
31、sts on The beams were made of thick web beams described in reference 8. 24s-T aluminum alloy with heavy flanges and with joggled stiffeners riveted to the flanges. The open spaces in the joggles were filled with soft metal. The load was applied at the center and the reactions were at the ends of the
32、 beams. supports. was taken as the buckling load. The properties of the specimens and the buckling data are given in table 111. A picture of a failed beam is sham in figure 3. Lateral deflections were prevented by lateral The load, when strain was first observed in the stiffeners, The stiffener spac
33、inga for the test results are not the same as those for the theoretical results. All the test results fall in the expected regions among the theoretical curves. results for which fall in the range which serves to verify the present theory over previous theory which considered the orthotropic- plate
34、curve to hold up to the cut-off at which the stiffeners are assumed to act as simple supports. agree with the present theory, but they do not cover the range in which an appreciable difference exists between the present theory and previous theory. theory fully. Only the group of test = 2.4 d The oth
35、er groups of test results More experimental results are required to confirm the present Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 mACA TN lo. 1851 CONCLUDING REMARKS Charts are presented from which the theoretical critical shear stresses cm b
36、e obtained for infinitely long, sim$ly supported plates stiffened with identical, equally spaced, transverse stiffeners of zero torsional stiffness. multiplier method. to hold up to a cut-off value corresponding to the stiffener stiffness at which the buckling load was equal to the buckling load of
37、a simply supported panel the size of each bay. and previous theory shows that previous theory gives unconservative results for stiffeners of intermediate stiffness and conservative results for stiffeners of high stiffness. Test results of 20 panels me presented which are in good agreement with the p
38、resent theory. a conclusive check additional test results are required. The theoretical results am based on the Lagrangian Previous theory considered the orthotropic curve Comparison of the present theory For Langley Aeronaut I cal Labarat ory National Advisory Committee for Aeronautics Langley Air
39、Force Base, Va., JaxmrY 28, 1949 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN No. 1851 7 THEORETICAL SOLUTION OF CRITICAL SBEAR SlCliESS OF pw-1TES WIT“ TRANSVERSE SIlFFENERS OF LOW BENDING STIFFNESS If the stiffener bending stiffness is l
40、uw ad the stiffeners are fairly closely spaced, the buckle pattern may be considered independent of the stiffener spacing, and the plate stlffener combination can then be analyzed as a plate with different bending properties in each direction, tht is, an orthotropic plate. shear of an orthotropic pl
41、ate is analyzed by means of the energy method. In this appendix buckling in The buckling configuration of the plate sham in figure 4 is represented by the trigonamstric series nfly nfly + cos - 7 bn sin - h, b sin - b w = sin “- x n=2,4,. . . n=1,3,. . . which satisfies the boundary conditions of si
42、m2le support term by term. The internal bending energy of the plate of the stiffeners are given by the expressions V, the internal bending energy Vs, and the external work of the shear stresses T Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA
43、 TN No. 1851 l Substitution of the expansion for w (equation (Al) into these energy integrals gives v =- EIM4(9 an 2n4 + S 8ab3 n=2,4,. . . n=1,3,. . . n=1,3,. . . 4=2,4,. . Then I- 1 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-9 NACA TN No. 1851
44、 where According to the enera method the -potential energy (V + Vs - T) must be minimized with respect to the unknown coefficients % and bn. By minimizing (V + Vs - T) with respect to the coefficients an and bn, the following set of equations is obtainsd: =o (n=2,4,6, . . . ) The coefficients an can
45、 be found in terms of br from equation (Ab). Substitution of the resulting expression for an in equation (A5) results in the following equations: I i q=2,4,. . . (A4 1 (A5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-nant vanishes: c31 51 C where
46、- cnn - ($ + n2 + 4 n c13 c33 53 C . c15 c35 55 C . 1 NACA TN No. 1851 .e. e e =o 22 na q=2,4, (where n 4 r) A solution including all the a *a and b can be obtained by n 1 setting equal to zero the first appraximatian of the determinant equation (A7) Provided by IHSNot for ResaleNo reproduction or n
47、etworking permitted without license from IHS-,-,-NACA TN No. 1851 11 Similarly the second appraximatian includes all the antsJ bl, 3 and b 2 %1 c33 - c13 = 0 Higher approximations me found in a similar manner. mation was found to give satisfactory results. mation .It is necessary to try values of sp
48、onding values of ks until a minimum value of ks with respect to b/X is found for each - The results am given in table I and in figure 1. A second approxi- For a given approxi- b/X and find the corre- Dd. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACA TN No. 1851 APPENDIX B THEORETICAL SOLUTION OF CRITICAL SHEAR STRESS OF PLATES In appendix A a theoretical solution for a plate stiffened by stiffeners of lar bending stiffness is presented where the buckle p
