■ STABILITY Or 4 FEW CURVED PANELS SUBJECTED TO SHEAR May 1947 OBY £ " 2 1948 -tr ATLANTA ATLANTA, .^•ANCH GfeORGIA This Report is One of a Series Issued In Cooperation with the ARMY-NAVy-CIVIL COMMITTEE on AIRCRAFT DESIGN CRITERIA Under the Supervision of the AERONAUTICAL BOARD No. 1571 UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY Madison, Wisconsin In Cooperation with the University of Wisconsin STABILITY OF A FSY, r CUPV5D PAN3LS SUBJECTED TO SKSAR- By EDWARD W. KUENZI, Engineer Summary The major portion of the experimental work of this study was on curved plywood plates subjected to shearing leads. One large specimen of sandwich construction was tested as a preliminary check on the extension of the theory for plywood to the sandwich type of construction. The results of the tests on plywood plates indicated that the "buckling stres; of a curved plate is approximately equal to the sum of the "buckling stress of a complete cylinder subjected to torsion and the buckling stress of a flat plate, of the same size as the curved plate, subjected to shear. The test on the single plate of sandwich type of construction indicated that the theory for plywood can be extended to sandwich constructions. Curves are presented showing the theoretical "buckling coefficients for sandwich material. Introduction A sandwich construction comprised of thin facings bonded to a light-weight core has several advantages when considered for use in aircraft structures. The construction is light in weight, is stiff, and therefore will retain its proper shape uncer load, has a smooth surface that reduces drag, and can be of monocoque design thus eliminating ribs and longerons. The main objective of this study was to investigate the elastic behavior of curved plates of sandwich material subjected to shearing loads such as those imposed upon the wings and fuselage of an airplane. Methods of testing and of confirmation of design criteria were determined by testing k7 curved plates of plywood. A preliminary check on the buck- ling of a curved plate of sandwich construction was obtained by testing one specimen consisting of two curved sheets about k feet square glued to load- ing rails as shown in figure 1. A more complete investigation, including — _ ~ This report is one of a series of progress reports prepared by the Forest Products Laboratory. Results here reported are preliminary a.nd may "be revised as additional data "become available. Rept. No. 1571 -1- such variables as size, curvature, and different materials, is desirable but was not justifiable at the tine. Probably tests on specimens of the same size as those used in a structure or tests on the structure itself would best provide the necessary information. Design Criteria In order that buckling stresses of curved plates in shear can be predicted, it is necessary to obtain design criteria that will represent the experi- mental data. The study of curved sandwich plates in axial compression, presented in Forest Products Laboratory Report No. 1558. shows that the buckling stress of a curved plate is equal to the sum of the buckling stress of a complete cylinder and the buckling stress of a flat plate the same size as the curved plate. It is logical to apply a similar analysis to a curved plate in shear*, for example, to assume that the buckling stress of the curved plate in shear is equal to the sum of the torsional buckling stress of a complete cylinder of the same length and curvature as the curved plate and the shear buckling stress of a flat plate of the same size as the curved plate. Cylinder in Torsion The buckling of plywood cylinders in torsion is discussed in Forest Products Laboratory Report No. 1529- The theoretical formula for the buckling stress of a thin-walled plywood cylinder, as given in that report, is r = kE T $ (1) cr Lr where k is a coefficient depending upon the elastic properties and the para- meter J. Values of k for plywood of three veneers with the faces half as thick as the core are shown in figures 2, 3i ^> aJ1< ^ 5 for different ratios f — — , where ^^m is the modulus of rigidity of the plywood SL K 2 hr b = length of cylinder h = thickness of plywood r = radius of curvature of cylinder E^= modulus of elasticity of the veneer, measured parallel to the direction of the grain The corresponding formula for thin-walled cylinders of sandwich construction is 'cr = k' V1A$ (2) Rept. No. 1571 -2- if the facings and core are assumed to be isotropic (derivation in appendix 1). The value of k' depends upon the parameter 2 /E~ j i - JL_\/_§: (Values of k' are shown in fig. 6) hr " Et_ where E, = modulus of elasticity of the sandwich plate in bending. E a = modulus of elasticity of the sandwich plate in compression. Values of Et and E a can be computed by the following equations, if the core is assumed to contribute no stiffness to the sandwich. El = Bf(i -4) 1 " h3 E a = E f (l-£) 1 h where E_ = modulus of elasticity of the facing material c - core thickness h = total thickness of the sandwich Flat Plate in Shear The elastic behavior of flat plywood plates subjected to shear is discussed in Forest Products Laboratory Reports 1316 and I316-H. The formula for the buckling stress is q = k Et i5 (3) i-cr ~ -s "L a 2 where k^ is a coefficient depending upon the elastic properties of the ply- wood and the ratio b/a. Values of k_ are determined oy the methods given in Forest Products Laboratory Reports Nos. I3I6 and I316-H, as shown in appendix 3 • a = width of flat plate. For curved plates this width was taken to be the circumferential dimension. If the flat plate is of isotropic sandwich material, formula (3) becomes q = k B. ■£- cr si? a^ Rept. No. 1571 -3- c where k = — -, c„ is obtained from the curve for a. = 1 of figure 8, Report ?k o . 1.316. K = (l - C~) where cr is Poisson's ratio. * = 0.91 (approxi- mately) for isotropic materials. The results of a few tests on flat plates of sandwich material are given in Forest Products Laboratory Report No. I56O. Curved Plato in Shear The buckling stress of the curved plate subjected to shear is assumed to be equal to T 4- q . Confirmation of this assumption will be given in the ''Discussion of Results." Experimental 'fork The curved-plate specimens were cons true ted as shown in the sketch of figure 7« Each specimen was made of two curved plates. This arrangement allowed the use of the same loading apparatus that was used to test flat plates in shear. Materials The plywood specimens were made of four plies of yellow birch veneers l/64 inch thick. The grain directions of the two center plies were parallel; thus the four-ply plywood was equivalent to three-ply plywood having face plies half as thick as the core ply. The veneers were bended together with Tego film glue, set in a hot press. The sheets of plywood "ere made large enough to furnish several coupons which were tested to obtain the elastic properties of the plywood. The sandwich specimen comprised 0. 012-inch 2^+ST ale lad-aluminum facings bonded to a 1 /8-inch end-grain balsa core. The curved plates of this specimen were bag molded as described in Forest Products Laboratory Report No. 157^. Manufacturing the Specimens The curved plywood test specimens, as shown in figure 7. were made by gluing 1- by U— inch maple loading strips (3) to the edges of the plywood while the plywood was flat. Then the plywood was bent and glued to maple end blocks (A) . The maple strips (5) ---ere screwed to a pine filler (F) • Finally holes were drilled for the loading pins (IP). The in~ide dimensions of the loading frame '-ere 10 by 20 inches. t. No. 1571 -*J~ The curved plate of sandwich construction was made in a similar manner, except that the plates were molded to the proper curvature prior to fitting on the 1 loading frame. The sandwich plate was 5° inches wide and 50 inches long, curved to a mean radius of 93 inches. Bending and tension tests were made on coupons of plywood. No coupons of the sandwich construction were tested. Testing Procedures The curved plywood specimens were tested in a hydraulic testing machine. The sandwich specimen was tested in a mechanically operated testing machine. The method of test is shown in figure 1. The "bottom loading block rested on an adjustable spherical "base. The up-oer loading "block was fastened to the upper head of the testing machine . After the specimen was placed in a vertical position, the spherical "base was adjusted so that a good "bearing was assured on all of the loading rollers. The load was applied at a slow uniform rate until "buckling occurred. The buckles usually appeared suddenly, with an accompanying decrease in load; but if the plates were nearly flat, they appeared sloi^ly and it was necessary to determine the critical load by observation of the lateral deflection. The buckling load was picked at the point of greatest curvature of the load-deflection curve. The plywood speci- mens (curved plates and coupons) were conditioned and tested in an atmos- phere having a relative humidity of 65 percent ?nd a temperature of 7^-° F • The methods of testing the coupons are given in appendix 2. C omputation of Results The test results are shown in table 1. The experimental buckling stress was computed by dividing the component of force parallel to the straight edge of the specimen by the area of the plates at that edge. Let t 1 = the experi- mental buckling stress, then r' = P cos/3 2hb where ? is the buckling load, /? is the angle between the diagonal and the long side of the specimen, b is the long dimension of the specimen, and h is the thickness of one curved plate. The theoretical buckling stresses were computed by means of the formulas given in the preceding section on "Design Criteria." The values of k used in this study were obtained by extending the work of Forest Products Laboratory Report ITo. 1529 to include values for J < 250, and are given in the curves of figures 2, J, b, and 5 for three-ply plywood having a ratio of s — ~- equal to 0.835* Values of k' for use in E l + E 2 Rept. Ho. 1571 -5- buckling stress of isotropic Lch coi -ions are r. in :" i. This curve of ".:' v • . J ' ■■■ ■ *?.ined analysis in appendix 1. The values of k were obtained from the curves of Forest Products La", on tory Reports Nos. I316-H end 1316. pie computations for olywood and for sandwich construction are given in apoendix 3 • Presentation and Discussion of Results Experimental rnd computed values of the buckling stresses are given in table 1 and comparisons of then are shown in figures 8 and 9 • '■"' 8 shows the exDerimental stress values compared with the computed buckling stresses ( T ) of a complete cylinder in torsion. ~he experimental values are as muck as 70 percent higher, provided each plate is not half of a com- pleted cylinder, for example, | h • In figure 9 the ere ntal ": stress is plotted against r + q . The points fit the theory much I I r than if just T cr is used for a computed value. The experimental values specimens that are complete cylinders are lower than computed values* No attempt has been made to analyse the distribution of stresses along the curved edge of the plate. The difference between experimental and computed values for specimens *f a complete cylinder indicates that the plates are probably not subjected to pure shear. The experimental buckling stress of the one specimen of sandwich 3C c- tion agreed well with the theoretical value of r + . For a cylinder of isotropic material with Young's modulus E, equation (29) of Heport Ho. 1529 (See equations (30), (31) , and (32) of that report and note that g = 0) can he written in the form r = |Ej f + — 2X| h where I " J j r f-,. and fg^ are the values of f-, and f of a.s defined by equation (32) for an isotropic material. In this expression the first term comes from the first term in brackets in (29) together with its coefficient, while t second term comes from the second term in "brackets in (2y) together wit] its coefficient, Now, for isotropic material, the constants c. . are pro- 1 portional to -^, while the constants M, P, S, Q, and N arc- either zero or * 1 proportional to —5. Hence, the whole first bracket expression is -oro-Dor- E^ tional to E. The factor E also appears in the denominator of the expression for i-]±' ^' et E f li = 'li* In the seccn( ^ bracket in (29) the coefficients "b-j_, h£, bo , bi|, "be;, b6 are either zero or proportional to E, and, consequently the whole bracket expression is proportional to E, . Then K li J 2i, r can be written for an isotropic cylinder where ^ . and v^. are each propor- tional to E. Rept. No. 1571 -8- For a sandwich cylinder the first term is to be that for an isotropic cylinder with E replaced by E , and the second term is to be that for an isotropic cylinder with E replaced by E - Hence, for a sandwich cylinder r Jj i* + ]lf lh IE li J E 2i! r a E i:L J E ; r 2 I .* h = ;E a J f.. + -± f 9 . i j a xi j 2i r where f-t^ and f^j are to be calculated from equations (32) for an isotropic cylinder with Young's modulus S. The appearance of this modulus is purely incidental, since it is found in both the numerators and the denominators of the expressions for f , . and f_. and cancels out. li 2i In the expression for t , let J" = J \lrr ' Et Then T =\/sS J,f " + 5H 5 k ' \/ E l E a * where fo, k' = J'f n . + -£f li J' This is the value of k of Eorest Products Laboratory Report No. 1529 for an isotropic cylinder for which rh |fB x Kept. No. 1571 -9- a? : 2 Testing of Plywood Coupons Bending Test: To obtain the bending moduli of elasticity (E-, and ^2) °? ^he plywood, 1- by 6-inch strips having their face grain direction parallel to their length were tested on a i*— inch span and 1- by k-1 /2-inch specimens having their face grain direction perpendicular to their length were tested on a 2-l/2-inch span. The load was applied at the center of the span by means of the movable head of a mechanical testing machine. The load was measured en a scale that could be read to the nearest gram. Deflections at the center of the span were measured with a dial reading to 0.001 inch. The main spring had been removed from the dial so that the load exerted by the dial stem was negligible. Tension Tests Tension tests were made on 1-inch strips to obtain the moduli of elasticitj for example, E a (face grain direction parallel to direction cf loa . E^ (face grain direction perpendicular to the direction of loadj c- (face grain direction at an angle of k^° to the direction of loading). The specimens were 11 inches long. The ends were held in Templin 3-rips fastened to the testing machine by bolts passed through spherical bearings. The strain over a 2-inch gage length was measured with an extensometer reading to 0.0001 of an inch. Computation of Results Values of Br were computed by the equation r ^L " 5 1 T2 * *h 2(1 + r ) E where — = C.0^-5 f 3r yellov birch plywood. Values of the shearing modulus , p jm, were computed by the equation where r a ^ = O.C^-6 for yellow birch plywood. ' 3 formula wp.s obtained from the equation for Errc given in Forest Products Laboratory Report No. 1317* Rept. No. 1571 -10- APPENDIX 3 Sanvolc Computations of Suckling Stresses Sandwich Construction Specimen data — Curvature r = 93 inches Width a = 5^ inches Length "b - 5° inches Total thickness h = 0.149 inch Facing thickness f = 0.012 inch (24 ST alclad aluminum) Core thickness c = C.125 inch (end-grain balsa) Modulus of elasticity , of facing E^ = 1C x 10 p.s.i. Then E = 10 x 1C 6 (l « ^25) = 1,610,000 p.s.i. a ]_49 6 f, ,\2^ E, =10 x 10 6 Jl -(I=i) 1=4,100,000 p.s.i 1 I 149 J j' = 2500 /22 = 113 0.149 x 93 v4lo From curve of figure 6, k' = 0.245 Therefore, r = 0.245 v'1.61 x 4.10 x 10 6 x °' lZ *9 = 1,010 p.s.i. cr 93 for determining k a = 1 and jS = -^2 = 1 s a 50 from figure 8 of Report No. 1316 c =22.9. Then k = 22 '9 = 8.39 s 3 x 0.91 Therefore, s 2 q cT . = 8.39 x 4.1 x lC b x (%^2) = 305 p.s.i., ox 50 and the "buckling stress of the curved plate = T + q = 1,315 p.s.i. cr w i. Kept. No. 1571 -11- Plywood Construction Specimen data — Curvature r = 3 inches Width a = 10.8 inches Length b = 20 inches Total thickness h = G.^59 inch Plywood is three-ply 1:2:1 yellow "birch £= 0° (face grair. parallel to cylinder axis) . Elastic properties (from coupon tests) Ei - 2.32 x 13° p.s.i. E = C.435 x 10° p.s.i. 6 E a = E^ = 1.3^2 x 10 p.s.i. &L = 2.602 x 10 p.s.i. ^ LT « 2.01 x 10- p.s.i. Then ^M = c.077 and 1 1 — = 0.84-2 El e x e 2 J = ^C = Q h? O.C59 x 8 Then entering the curves of figure 2 and interpolating to the value of 0.077 for -jHt, k = 0.C31. E L Therefore, r = 0.031 x 2.602 x 1C 6 x °'°59 = 59? p.s.i. cr g 3 For determining k g — — Enter curve of figure 130, Forest Products Laboratory Report Bto. I3I6-E, and obtain „ E ^ = 1.65 for ±=- = 0.842. a E]_ + E 2 Enter curve of figure 129 °f Heport No. I3I6-H, and obtain k = 0.92. 3 It should be noted that Q of this report refers to the angle of the face grain to the cylindrical nxis , but 6 of Reports I3I6 and I3I6-H refers to the angle between the short side of th° specimen and the face grain direction. Rept. Ho. 1571 -12- Compute £_ = 1 ; £1 = 1>8 5 _ 1.12. d' a a 1.65 Enter curve of figure 12 of Report No. I316, and obtain ^ — = 1.35^ then k = 1.35 2 * x O.92 = 1.25 K s S (X Therefore, q = 1.25 x 2.602 x 10 6 (^Sl) 2 = 95 p.s.i., cr 10.3 and the "buckling stress of the curved plate = T cr + q = 69^ p.s.i Kept. 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OS C I ^ I • B c - 01 c U O 3 - 8 C5 OS C5 CS ^ CURVED PLATES 14 PIN MAPLE BLOCK iWmaple PINE FILLER 2" ROLLER Z W 72984 F Figure 7. — Sketch of curved-plate specimen. 2fK)0 I S i 1,600 1,200 800 400 / A • A p * * ¥ A A • < » y^ • A • • / • r 1 i TGEND I P/ Vl_ • i 1 L ■A c Xf-i L ¥000 6-0° v/inn a - on o P yT \ PLYWOOD 6= +45° i. pi vwnnn a=-/Lti °? 1 A^ Y SPECIMENS WITH y- z7r p CAMDiA/iru CDCriUCM L «<90 £<%> l % 200 T cr (PSIJ IfiOO 2fiO0 Figure 8. — Experimental buckling stress of a curved plate in shear vs. theoretical buckling stress of a complete cylinder in torsion. ■l * '?985 p 2,400 2,000 i 5 1 1 ki 1,600 l,2O0 800 400 A \* □ ▲ y ¥ ¥ , k . i k >/* • /* • • / \ $ T T LEGEND: mP /v f A PLYWOOD 6=0° • PLYWOOD 6=90° A PLYWOOD 6=+45° a 0/ vuz/wi o--x*r* X° V \ Y SPECIMENS WITH y- =1?~ n cAfunwiru COC^/MJCAJ 400 500 /,£00 /,#X? Icr + qcr (PS.I.) 2000 2,400 Figure 9. — Experimented, vs. theoretical buckling stresses of curved plates in shear. Z M 72986 F UNIVERSITY OF FLORIDA 1 1 iiiii i in 3 1262 08929 0646