OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION

U. S. FOREST SERVICE RESEARCH PAPER FPL 50 DECEMBER U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION

SUMMARY Theretical buckling cefficients fr simply supprted, flat plywd plates under cmbined edgewise bending and cmpressin are derived by treating plywd as an rthtrpic plate. Curves f buckling cefficients are presented alng with tw examples f their use. Cmputatinal errr in the buckling cefficients was held t less than 1 percent.

BUCKLING OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION By JOHN J. ZAHN, Engineer and KARL M. ROMSTAD, Engineer FOREST PRODUCTS LABORATORY FOREST SERVICE U. S. DEPARTMENT OF AGRICULTURE INTRODUCTION When a flat plate is subjected t a cmpressive edge frce lying in the plane f the plate. failure can ccur by buckling at a stress belw the strength f the material. This Research Paper cnsiders the prblem f a flat, rectangular plywd plate with all edges simply supprted and with edge lad in the plane f the plate n tw ppsite edges. The intensity f this cmpressive edge frce varies linearly alng the laded edges: hence, the resultant f this distributin is a cmpressive edgewise frce P at the center f a laded edge and an edgewise bending mment M. The relative magnitude f P and M are specified by parameter which varies frm 0 <α <2, the extreme values crrespnding t pure cmpressin and pure bending, respectively, This prblem was slved fr a hmgeneus plate by Timshenk 2, but the crrespnding slutin fr an rthtrpic plate appears t be lacking in the literature. Accrdingly, a mathematical analysis fllwing Timshenk was undertaken. The natural rthtrpic axes f the plies were taken t be parallel t the crdinate axes. The energy methd f slutin was used, with the expressin fr strain energy f bending taken frm March s thery f bending f plywd plates. 3 A duble sine series was used t represent-the deflected surface f the buckled plate and fur terms were retained in the cmputatins, althugh nearly always three wuld have been sufficient fr less than 1 percent errr, Curves were cmputed with the aid f an IBM 1620 cmputer and are presented as a family f curves f a nndimensinal buckling cefficient versus a reduced aspect rati. The family is dependent upn tw parameters, α and C, the latter cntaining the elastic prperties f the plies. 1 Maintained at Madisn, Wis., in cperatin with the University f Wiscnsin. 2 Timshenk. S., and Gere, J. Thery f elastic stability. 2nd Ed. McGraw-HiII, New Yrk. 1961. 3 March, H.W. Flat plates f plywd under unifrm r cncentrated lads. Frest Prducts Lab. Rpt. 1312. 1942

a, b A mn C D E, E 1 2 E, E x y E, E L T f F c G 12 G xy h H ni I, I 1 2 K K, K m, n, i M N x N, N p P r U V V w W e x, y, z α ß mn λ NOTATION Length and width f plate (fig. 1). Furier cefficients f lateral deflectin w. Elastic prperties parameter. See equatin (15). Mean bending stiffness f plate. See equatins (7) and (15). Mean elastic mduli in x and y directins. Assciated with bending in equatin (7). Als see equatin (27). Elastic mduli f a ply in x and y directins. Elastic mduli f wd in lngitudinal and tangential directins, See discussin fllwing equatin (23). Factr f safety. N x Allwable cmpressive stress, h, fr plywd plate. Depends n species and cnstructin. shear mdulus f plate. Since we tk rthtrpic axes parallel t edges f plate, we have G = G 12 xy. Shear mdulus f a ply. Thickness f plate. A functin f integers n and i. See equatin (15). Mments f inertia per unit width. See equatins (27) and (7), Lad parameter. See equatin (15). Buckling cefficients. See equatins (20) and (24). Integers. Edgewise bending mment. See equatin (3). Intensity f cmpressive edge frce in x directin, punds per inch. Maximum value f N and its itical magnitude. X Edgewise cmpressive frce. See equatin (2). E T = re L. Rati dependent n species. Strain energy f bending. See equatin (6). Ptential energy f system; strain energy f plate minus wrk dne by lads. Value f V at buckling. Displacement f middle surface f plate in z directin Wrk dne by lads during buckling. Crdinates. variatin parameter. See equatin see equatin (17) since rthtrpic axes are parallel t edges f plate. Reduced aspect rati. Mean Pissn s ratis f plate. See equatin (7). FPL 50 2

Pissn s ratis f a ply. Pissn s ratis f wd. MATHEMATICAL ANALYSIS Figure 1.--A simply supprted plywd plate under cmbined edgewise bending and cmpressin. M 129 843 Cnsider a flat plywd panel whse planes f elastic symmetry are parallel t the crdinate planes f a rectangular crdinate system (fig. I). Alng the sides x = 0 and x = a, a cmpressive frce is applied in the x directin whse magnitude varies with y: where N X is cmpressive frce per inch f edge and α is a parameter in the range 0 < α < 2. The resultant edgewise cmpressive frce is and the edgewise bending mment is if P is assumed t act at the midpint f the side. Thus, fr pure cmpressin we take a = 0 s that M vanishes and fr pure bending we take α = 2 s that P vanishes. When P and M are presibed, equatins (2) and (3) are slved fr N and α : (1) (2) (3) We shall prefer t presibe N and α. Since the pel is simply supprted, its buckled surface can be represented by a sine series expansin (4) 3

The wrk dne by external frces during buckling is (5) where n + i is always dd. strain energy f a plywd panel due t bending is fund by March 3 t be: (6) where (7) in which h dentes panel thickness, E dentes elastic mdulus, G dentes shear mdulus, and dentes Pissn s rati. Since the elastic axes f the plies are parallel t the crdinate axes. Maxwell's relatins can be written = cnstant independent f z (8) s that (9) FPL 50 4

f any ply in this case; als, G 12 = G xy f any ply and λ is the same in every ply. Substitituting equatin (4) int (6) we btain (10) Nw let us write the ttal energy f the system, taking the unladed state f Then where V is the ptential energy f the system just prir t buckling, This term is assciated entirely with membrane strains f the middle surface f the plate. If the buckling deflectins given by (4) are infinitesimal, these membrane strains are cnstant during buckling; thus, expressin (5) is justified and V is seen t be independent f the A. Fr equilibrium f the system. V is a minimum; hence mn (11) (12) frm which Using equatins (10) and (5), (13) becmes (13) (14) The nly determinate slutin f system (14) is A = 0, all m, n. This crrespnds t the flat frm mn f equilibrium We nw imagine that α is presibed and N ineased until it reaches the value N at which pint the panel buckles and A # 0, fr sme m. Here the determinant f system (14) mn vanishes. Since there is an infinite set f systems, ne fr each m, there are infinitely many values f N, ne fr each The value f m must be chsen by trial t make N a minimum 5

Befre slving fr N, we nndimensinalize system (14) by intrducing the fllwing ntatin (15) Then, system (14) can be written (16) (17) Then system (16) becmes (18) Nte that the cefficient array is diagnally symmetric since n and i are interchangeable. and that every ther term is zer since every ther H ni. is zer. There is an infinity f systems (18), ne fr each m. The value f m which yields the least K is dependent upn the reduced aspect rati,.. NUMERICAL ANALYSIS System (18) is the cnditin f equilibrium at buckling if nt all A = 0; hence the vanishing f the mn determinant f the cefficients defines a itical value f K which we dente as K. Given, C, and α, K was chsen by trial t make the determinant vanish, with m chsen s as t minimize K. Fr small, m = An IBM 1620 cmputer was prgrammed t start with small and m equal t ne FPL 50 6

and t inement after each slutin fr K, thereby generating a plt f K versus. Whenever had ineased sufficiently, m was ineased by unity. The t inease m was prgrammed as fllws. Let MM dente the current value f m. Figure 2A shws a plt f the value f the determinant versus K fr m = MM. The cmputer pltted this curve and chse K = K by trial s that the determinant was equal t zer, as shwn in the figure. With this value f K, the cmputer then calculated ne pint n the curve fr m = MM + 1. If Figure 2.--Value f determinant versus K, illustrating lgic used by when inementing t decide whether m is yielding the lwest pssible value f K. M 129 844 A, cmputer will inement and retain value f m; in B, cmputer will inement m and deease value f. the determinant was psitive, as figure 2A, the rt crrespnding t m = MM + 1 was knwn t be greater than the current K, s was inemented and m = MM was retained. If, hwever, the determinant was negative as in figure 2B, the rt crrespnding t m = MM + 1 was knwn t be less than the Current K s MM was inemented by unity and was nt ineased. In fact, it was decided t deease at this pint s that the new branch f the curve f K versus fr larger m wuld clearly intersect the ld branch fr smaller m. This prvided gd definitin f the cusps in the f K versus. f K versus were platted fr m = 1, 2, 3, and 4. The envelpe f the branches is a hrizntal straight line, and since the cusps d nt lie much abve the envelpe fr m >4 it was decided t stp cmputing at m = 4 and shw nly the straight line envelpe fr larger. A furth rder determinant was used thrughut. Results using a third rder determinant were btained fr α = 2 (pure bending), C = 0.2 (the smallest C value and >0.4, which shwed a difference f 1.02 percent at = 0.4 and less than 0.3 percent fr larger. Hence in by trial, the inement in K was nt refined beynd K = 0.01. The accuracy f the curves is thus mre by the scale f the drawings than by cmputing errr. A flw chart f the cmputing methd and its Frtran cding are given in the Appendix In

CURVES OF BUCKLING COEFFICIENTS The intensity f cmpressive edge lad is (19) where N is the maximum intensity α is the slpe f the intensity distributin. Parameters N and α are related t ttal edge frce P and ttal edge mment M by equatins (2) (3). The itical value f N is given by where (20) and E 1, E 2, and λ are defined by equatins (7). A reduced aspect rati is defined as (21) (22) and five curves f K versus are presented in figures 3-7, fr α = 0, 0.5, 1.5, and 2.0. Each f these is a family dependent upn an elastic parameter C defined as (23) where the subsipts L and T refer t principal directins in the plies-assumedhere t be flat grain. The directin L is parallel t the grain and T is tangential t the annual rings. Curves fr values f C = 0.2, 0.4, 0.6, 0.8, and 1.0 are shwn, the latter value crrespnding t an istrpic plate. The curves fr C = 1 check the values btained by Timshenk 2. The curves indicate that linear interplatin with respect t C is extremely accurate. Nte that all curves apprach a hrizntal asymptte fr > 1; hence ne culd design cnservatively n the basis f K, the asympttic value f K. Let (24) Then (25) Since K is independent f, it is presented in figure 8 as a ne-parameter family f curves f K versus α with C as parameter. Again, linear interplatin with respect t C is seen t be highly accurate. These curves are prbably the mst useful f all, since the influence f negligible fr >1. With <1, the panel wuld have a small length t width rati r a small value f FPL 50 8

Figure 3.--Buckling cefficient, K, versus reduced aspect rati,, fr α=0 (pure cmpressin). M 129 846 either such cnstructin wuld be uncmmn. The curves have tw basic uses: (A) select a plywd panel, knwing a, b, P, M, and (B) t check the stability f a given panel. These will be discussed separately. A. T Design a Panel 1. Given P, M, and b, find α: (26) (27) 9

M 129 847 Figure 4.--Buckling cefficient, K, versus- reduced aspect rati,, fr α =0.5 (cmbined bending and cmpressin). where I 1 = mment f inertia per unit width f parallel plies nly; that is, plies whse grain is parallel t the directin f lading, r x directin. This will rdinarily be the directin f the face grain I = mment f inertia per unit width f perpendicular plies nly. 2 Let us intrduce a fractin p, which will depend nly n the number f plies and their relative thicknesses, as (28) and the rati (29) Fr mst wds 0.01<r<0.07. and r = 0.05 is a gd average value. With r intrduced, equatins (27) becme (30) FPL 50 10

M 129 848 Figure 5.--Buckling cefficient, K, versus reduced aspect rati,, fr α=1.0 (cmbined bending and cmpressin). Assuming that the face grain is parallel t the directin f lading. we can say Thus ri 2 can be neglected in cmparisn t I 1. We cannt neglect ri 1 in cmparisn t I 2, hwever, since fr certain cnstructins I 2 is cnsiderably less than I 1. Therefre frm equatin (30) 11

Figure 6.--Buckling cefficient, K, versus reduced aspect rati,, bending and cmpressin). s that, using equatin (28) 129 849 fr α=1.5 (cmbined Manufacturers assciatins such as the American Plywd Assciatin prvide tables f I 1 and I 2 fr cnstructins 4, s that p can be estimated by type f cnstructin. Fr example, with three equal plies p = with five equal plies p = 0.44; and with many plies, p appraches 3. Estimate C as (32) 4 American Plywd Assciatin. Fir plywd technical data handbk. Amer. Plywd Assc., Tacma, Wash. 1948. (31) 12

Figure 7.--Buckling cefficient, K, versus reduced aspect rati,, fr bending). r, simply make C = 1 n the =et trial. 4. Use α and C t get K frm figure 8 fr K. M 129 850 α=2.0 (pure 5. Get maximum edge lad intensity frm P, M, and b; (33) 6. Frm N and K, cmpute minimum panel thickness using definitin f K: (34) 7. Select a panel, recmpute p and make a secnd trial if necessary. 13

M 129851 Figure 8.--Buckling cefficient, K, versus lad variatin parameter, a, fr varius values f K is the asymptte f K fr large values f. 8. Nte we have used K fr K. Therefre the step is t check If is less than 1 it may be pssible t imprve the design by using figures 3-7 in lieu f figure 8. 9. This design is based n stability. It shuld als be checked fr strength. Example: Select a 4- by 8-ft Duglas-fir plywd panel t carry an eccentric axial lad f 9,000 punds in the 8-ft directin with an eccentricity f 4 inches and all sides simply supprted. Since eccentricity we have Anticipating a cnstructin with many equal plies, we estimate fr Duglas-fir. Thus frm figure 8, K = 4.0. Intrducing an arbitrary factr f safety f 3, we have P = punds. Then FPL 50 14

Thus h > 0.90 inch; Use h = 0.9375 inch We select a 15/16-inch, seven ply sanded plywd with 1/8-inch face plies, 3/16-inch centers, and 1/8-inch ssbands fr which Therefre, which agrees s well with ur first estimate that a secnd trial is unnecessary. A check f? shws The allwable cmpressive stress fr this cnstructin (Exterir Grade AB) is 1,375 punds per square inch. The maximum cmpressive stress is the allwable stress. punds per square inch which is less than B. T Check a Given Panel fr and Strength 1. Safe lading is determined by (35) fr stability, where f is a factr f safety, (36) fr strength, where F c is the safe allwable cmpressive stress. Cnditin (36) can be written 2. Since K depends n it is cnvenient t represent cnditins (35) and (37) n a plane f versus α as figure 9. Tgether, these tw cnditins define a safe regin f the K-α plane. 3. T every pint (K, α) in the safe regin there a safe pa (P, M) by frmulas (2) and (3). Example: Given a 4-ft by 8-ft by 1-inch Duglas-fir plywd panel fr which (37) 15

Figure 9.--Typical shape f safe regin f K- α plane. t the safe allwable cmpressive stress. M 129 845 K C is the value f K crrespnding find the maximum safe lad with an eccentricity f 6 inches. Apply a factr f safety 3 t the stability iterin. As in the earlier example, we knw α. We als cmpute FPL 50 16

and btain K frm figure 8 as Fr safety, (35) (37) In this case stability gverns and K = thus max It must be emphasized that this thery assumes the eccentricity f P is achieved by means f a unifrmly varying edge lad intensity. Fr any ther bundary cnditin, the thery must be used with cautin. APPENDIX FLOW CHART OF COMPUTING METHOD 17

FORTRAN PROGRAM FOR BUCKLING OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION FPL 50 18

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20 1.2-21