13. Bending of plates under the combined action of lateral loads and forces in the middel plane of the plate

13. Bending of plates under the combined action of lateral loads and forces in the middel plane of the plate

BENDING OF OF PLATES LATERAL LOADS PLANE CHAPTER 12 UNDER THE COMBINED AND FORCES IN THE OF THE MIDDLE PLATE 90 Differential Equation of the Deflection Surface discussion it has always been assumed ACTION In our previous that the plate is bent by lateral loads only If in addition to lateral loads there are forces acting in the middle plane of the plate, these latter forces may have a considerable effect on the bending of the plate and must be considered in deriving the corresponding differential equation of the deflection surface Proceed- ing as in the case of lateral loading (see Art 21, page 79), we consider the equilibrium of a small element cut from the plate by two pairs of planes parallel to the zz and yz coordinate planes (Fig 191) In addi- tion to the forces discussed in Art 21 we now have forces acting in the middle plane of the plate We denote the | magnitude of these forces per unit length by mS N., N,, and Nz, = Ny,z, aS shown in the figure | | | Ny ỳ Net lì on ° , —+>> 1` "SN y ONz , 9Ne _ g pene = Nx | I Projecting these forces on the x and y axes and assuming that there are no body forces or tangential forces acting in those directions at the faces of the plate, we obtain the following equations of equilibrium: ò \ NOT Nyx + a Oy OX + ôNxy dx ON zy Ox dy ` Ny + Fay (b) Fic 191 + ONy oy (216) =0 These equations are entirely independent of the three equations of equilibrium considered in Art 21 and can be treated separately, as will be shown in Art 92 In considering the projection of the forces shown in Fig 191 on the z axis, we must take into account the bending of the plate and the resulting small angles between the forces Nz and N, that act on the opposite sides of the element As a result of this bending the projection 378 PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 379 of the normal forces N, on the z axis gives N —N: dụ 2+ ( dz) dw , ð? & + —, 5x5 is) After simplification, if the small quantities order are neglected, this projection becomes n, ® ae dy oN dy of higher than the second oe (a) In the same way the projection of the normal forces N, on the z axis gives nN, aedy + ae ae dy (b) oat Regarding the projection of the shearing forces N,, on the z axis, we observe that the slope of the deflection surface in the y direction on the two opposite sides of the element is dw/dy and dw/dy + (d?w/dx dy) dz Hence the projection of the shearing forces on the z axis is equal to Ow, 4, ON xy OW Nev ae ay YF ox ay OY An analogous expression can be obtained for the projection of the shearing forces N, = Nz, on the z axis The final expression for the projec- tion of all the shearing forces on the z axis then can be written as 2N “3s iy dx dy + oN Ow ¡ vđy + ane oy72 ow ox dx dy (c) Adding expressions (a), (b), and (c) to the load g dx dy acting on the ele- ment and using Eqs (216), we obtain, instead of Eq (100) (page 81), the following equation of equilibrium: OM, dx? (07M, , OA, mm ^ ôU? ~ - (4 + NSS +N Ge + 2N 0?w san ) Substituting expressions (101) and (102) for A/,, AL,, and ăz„, we obtain aw dtw dw oat T ” Ga? ay? * ays mm | q+ N.S WIN tăm Woy, '” Jw Ox Oy (217) This equation should be used instead of Eq (103) in determining the deflection of a plate if in addition to lateral loads there are forces in the middle plane of the plate ‘880 THEORY OF PLATES AND SHELLS If there are body forces! acting in the middle plane of the plate or tangential forces distributed over the surfaces of the plate, the differential equations of equilibrium of the element shown in Fig 191 become ON Ox + ONey —— oY + ONey ON, Ox oy Xx = (218) =0 — Here X and Y denote the two components of the body forces or of the tangential forces per unit area of the middle plane of the plate Using Eqs (218), instead of Eqs (216), we obtain the following differential equation? for the deflection surface: 04w 0° 04w ð? g?tu ö? “— + N,— +2N.,—— + vaya T ” ax day —-X—ow ox Ow ay (219) Equation (217) or Eq (219) together with the conditions at the boundary (see Art 22, page 83) defines the deflection of a plate loaded ° *T—_x laterally and submitted to the action of forces in the f— ime middle plane of the plate -* va f€ ~-~~@ -==== >| y „ P _Ÿ 91 Rectangular Plate with Simply Supported Edges under the Combined Action of Uniform Lateral Load and Uniform Tension Assume that the plate is under uniform tension in the tra, 192 z direction, as shown in Fig 192 The uniform lateral load g can be represented by the trigonometric series (see page 109) oy > 9m Mar nny —— sin b » — (a) Equation (217) thus becomes O4w 04w 04w ant | 3gt ay? * ay! N, 0?w D oz? = a » sin — sin =e (b) This equation and the boundary conditions at the simply supported edges An example of a body force acting in the middle plane of the plate is the gravity force in the case of a vertical position of a plate This differential equation has been derived by Saint Venant (see final note 73) in his translation of Clebsch, ‘‘Théorie de ]’élasticité des corps solides,”’ p 704, 1883 UNDER PLATES LOADS LATERAL PLANE THEIR IN FORCES AND will be satisfied if we take the deflection w in the form 381 of the series Mrz ney Omn Sin —— sin |= w= (b), we find the following values for the Substituting this series in Eq coefficients Amn: - 16g Onn = (c) 6n a? 6? (d) - Damn a’ Da? in which m and n are odd numbers 1, 3, 5, , and dm, = O1f m or n Hence the deflection surface of the plate is or both are even numbers “ m (2 n + is) Nm >] T a’ Da? _ | mrn Sin —— a nry sin —- b (e) Comparing this result with solution (131) (page 110), we conclude from in the brackets of the denominator the presence of the term N,m?/x?Da? that the deflection of the plate is somewhat diminished by the action of This is as would be expected the tensile forces N, By using M Lévy’s method (see Art 30) a solution in simple series may be obtained which is equivalent to expression (e) but more conThe maximum values of deflection venient for numerical calculation in this way! for v = 0.3 can be represented and bending moments obtained in the form Wmax — qb" a Eh3 (TM) max — 8q? (M,) max —= 8g)? (f) The constants a, 8, and 8: depend upon the ratio a/b and a parameter _ N,b? Y ™ 42D and are plotted in Figs 193, 194, and 195 If, instead of tension, we have compression, the force N, becomes 1H D Conway, J Appl Mechanics, vol 16, p 301, 1949, where graphs in the case of compression are also given; the case N, = N, has been discussed by R F Morse and H D Conway, J Appl Mechanics, vol 18, p 209, 1951, and the case of a plate clamped all around by C C Chang and H D Conway, J Appl Mechanics, vol 19, For combined bending and compression, see also J Lockwood Taylor, p 179, 1952 The Shipbuilder and Marine Engine Builder, no 494, p 15, 1950 382 negative, THEORY and OF the deflections bent by lateral load only PLATES AND (e) become SHELLS larger than those of the plate It may be seen also in this case that at cer- tain values of the compressive terms in series (e) may vanish force Nz the denominator of one of the This indicates that at such values of N, the plate may buckle laterally without any lateral loading 92 Application of the Energy Method The energy method, which was previously used in discussing bending of plates by lateral loading (see Art 80, page 342), can be applied also to the cases in which the 0.14 _ _——— 0.10 / ` 0.02 25 Wok Z ` ™ 1/1 „ 0.04 \AA AL O12 ® g W - me — — b Fia 193 lateral load is combined with forces acting in the middle plane of the plate To establish the expression for the strain energy corresponding to the latter forces let us assume that these forces are applied first to the unbent plate In this way we obtain a two-dimensional problem which can be treated by the methods of the theory of elasticity.1 Assuming that this problem is solved and that the forces N., N,, and N., are known at each point of the plate, the components of strain of the middle plane of the plate are obtained from the known formulas representing Hooke’s See, for example, ed., p 11, 1951 S Timoshenko and J N Goodier, ‘‘Theory of Elasticity,’’ 2d PLATES UNDER LATERAL law, 012., es = hE (N, — LOADS AND vN,) Cụ Ne, Yzu — FORCES — hi IN (Ny THEIR — PLANE „N,) 383 (a) hG The strain energy, due to stretching of the middle plane of the plate, 1s then | V¥i= AS J (Nez | = | + Nyey + N xyz) du dy [N2 + N2 — 2vN,N, + 2(1 + r)N?,| dà dụ (220) where the integration is extended over the entire plate Let us now apply the lateral load This load will bend the plate and produce additional strain of the middle plane In our previous discussion of bending of plates, this latter strain was always neglected Here, 0.14 \ | 0.12 0.10 ky ⁄ > “ Z5 Ze ⁄Z Fic 194 + Z0 EZ ea CM ss x — a 0.06 / PO 0.08 384 THEORY OF PLATES AND SHELLS 0.06 YY Ree 0.04 ——— _ † B; 0.03 yal —————— YO mm _— we„z3 TT | L7 —-—~ 0.02 F—— _ WwW TT” Nh ` _—— —— + 0.05 b tg 195 however, we have to take it into consideration, since this small strain in combination with the finite forces Nz, N,, Nz, may add to the expression for strain energy some terms of the same order as the strain energy of bending The x, y, and z components of the small displacement that a point in the middle plane of the plate experiences during bending will be denoted by u, 2,, and w, respectively ak 4x15 Considering a linear element AB of that A x plane in the z direction, it may be seen from | Y Fig 196 that the elongation of the element —9; = 9W dx U Fh~ K Ox due to \ the displacement u is equal to (du/dx) dx The elongation of the same element due to the displacement w is Fig 196 4(dw/dx)? dx, as may be seen from the comparison of the length of the element A,B, in Fig 196 with the length of Thus the total unit elongation in the z direcits projection on the z axis tion of an element taken in the middle plane of the plate is z ut gu dx € = Ou fow\? ¬+- &) (221) PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 385 Similarly the strain in the y direction is ,_ v , (aw)? fy | = » (222) (0y) + Considering now the shearing strain in the middle plane due to bending, we conclude as before (see Fig 23) that the shearing strain due to To determine the shearthe displacements u and v is du/dy + dv/dx ing strain due to the displacement w we take two infinitely small linear elements OA and OB in the x and y directions, asshown in Fig 197 The difference between the angle 7/2 and the positions O1A; and O,B; is the shearing strain corresponding to the displacement w bring the plane the with coincidence y B,0,Ai into tionC Thedisplacement B.C'is equal 2, plane B,O,A;* and the point B: to posi- ox ow i wy Ol Ox be >< > = Ow gs A cá + to (dw/dy) dy and is inclined to the verHence tical B.B, by the angle dw/dz BiC is equal to (dw/dx)(dw/dy) dy, ; Fic 197 and the angle CO,B,, which represents the shearing strain corresponding (dw/dx)(dw/dy) % 0, >< we ——> a = dw/dy, a, angle | { ds the dx \ B.O, is parallel to BO Rotating the plane B.O,A, about the axis 0,A, by \ sider the right angle B;O+4;, in which ——>ke-~—=— To determine this difference we con- D> / angle A,0,B, to the come in the z direction these elements of displacements a Because the to displacement w, is this shearing strain to the strain produced by Adding the displacements wu and v, we obtain Vey int hp + Ow Ow dx by (223) Formulas (221), (222), and (223) represent the components of the additional strain in the middle plane of the plate due to small deflections Considering them as very small in comparison with the components «:, €,, and yz, used in the derivation of expression (220), we can assume that With this the forces N., Ny, Nz, remain unchanged during bending assumption the additional strain energy of the plate, due to the strain produced in the middle plane by bending, is Vo = Sf(Nae, + Nye, + Nevyi,) dx dy Substituting expressions * The angles dw/dy and (221), (222), and (223) for «¿, «,, and y;,, we dw/dzx correspond to small deflections of the plate and are regarded as small quantities SHELLS AND PLATES OF THEORY 386 finally obtain -LIxŒ}+w(3) ceEB]zx m | đe dy ve= ff] N + vy 28 + Na (%Oy + án) Ox It can be shown, by integration by parts, that the first integral on the right-hand side of expression (224) is equal to the work done during bendTaking, for ing by the forces acting in the middle plane of the plate example, a rectangular plate with the coordinate axes directed, as shown in Fig 192, we obtain for the first term of the integral b Ou a EM: = | b Nu a du b a ON — [fous dxd Proceeding in the same manner with the other terms of the first integral in expression (224), we finally find Jo Lr [NS + esp tM (Gy + az) Jee Ov =Í (val a + Ou [Nay )+ b fa „+ ƠN vt) u (SE + “LÍ, Ov av ay — Nay + (|! [ I ( b ) as aN 0N, su) ae dy The first integral on the right-hand side of this expression is evidently equal to the work done during bending by the forces applied at the edges Similarly, the second integral is equal to = Qand z = a of the plate the work done by the forces applied at the edges y = Oandy = b The last two integrals, by virtue of Eqs (218), are equal to the work done These during bending by the body forces acting in the middle plane integrals each vanish in the absence of such corresponding forces Adding expressions (220) and (224) to the energy of bending [see ka (117), page 88], we obtain the total strain energy of a bent plate under the combined action of lateral loads and forces acting in the middle plane This strain energy is equal to the work T, done by the of the plate lateral load during bending of the plate plus the work T; done by the Observing that this latter forces acting in the middle plane of the plate work is equal to the strain energy V; plus the strain energy represented by the first integral of expression (224), we conclude that the work pro- - PLATES LATERAL UNDER LOADS AND FORCES IN THEIR PLANE 387 uced by the lateral forces is Ow Ow Ow OW =zJJ|xv-(2) + w, (3) + OMe ox syBy | + JJ Z + am) —?0 95 g?› ? dụ |p mem aự — (say) 0?w 07w 6? \/ (225) pplying the principle of virtual displacement, we now give a variation 6w o the deflection w and obtain, from Eq (225), r= 58 ff [ve GB) +055) + 20 Se ay | eae Ow Ow 0*w 0N, +72 || (Ss +=) aw tw aw atu aw \’ “ag =» [Be ay? - (25,) |} egy (226) ‘he left-hand side in this equation represents the work done during the irtual displacement by the lateral load, and the right-hand side is the The application orresponding change in the strain energy of the plate f this equation will be illustrated by several examples in the next article 93 Simply Supported Rectangular Plates under the Combined Action f Lateral Loads and of Forces in the Middle Plane of the Plate Let us egin with the case of a rectangular plate uniformly stretched in the direction (ig 192) and carrying a concentrated load P at a point with and The general expression for the deflection that satisoordinates es the boundary conditions is co ©Ị w= m=1,2,3, n=1,2,3, _ Marx TW] sin sỶ Amn Sin b (a) ‘o obtain the coefficients ann in this series we use the general equation 226) Since N, = N., = in our case, the first integral on the right- and side of Eq (225), after substitution of series (a) for w, is "mm ©© m=l1n=1 ‘he strain energy of bending 225) is [see Eq (d), page 343] representing the second polSaguy m=1n=1]1 integral in Kq « THEORY 388 SHELLS AND PLATES OF To obtaina virtual deflection dw we give to a coefficient @m,n, increase The corresponding deflection of the plate is 5Qm,n, MyWr ồu = ồđ„,„, SIn Nyw.H sin b y | The work done during this virtual displacement by the lateral load P is (5.83 ‹ố — Tù17TTỊ P6Qm,n, SIN Sn (d) The corresponding change in the strain energy consists of the two terms which are sã I i ) dx dy ~ i (3 Na.” cok oman, aV and 5V = OV sa OOm,n, Í = ~ My mE Minn Ti Na "an bdmin, 2\2 ab Da» (Ga +3) (e) in Eq (226), we obtain QÙ mr? n Tp =7 Ms tmm Cạn Oman + © from which Oem, = 4P Mart sin ame sin 2\2 Dames (7 +1) nin —— — abz*D l + m) + ?a?Ù mir _ 00 4P = Tp mr & NTN sin a sin _b_ me na\? mEN mal n=l (= +4) TH _ 5Qm,n, Ø Substituting these values of the coefficients an,n, in expression find the deflection of the plate to be (Ê) Ôn, Substituting expressions (d) and P6dm,n, S10 2-2 TLTỤ sin — — sin [= (a), we (9) T 722D If, instead of the tensile forees W;, there are compressive forces of the same magnitude, the deflection of the plate is obtained by substituting — WN, in place of N, in expression (g) This substitution gives 60 4P ~ abrtD / , sin mr &é —— mn a NTN sin —— 2? b m2N; mat nai \q? * b) ~ wa®D _ Mmnx sin — a sin nry chb (A) PLATES UNDER IN FORCES AND LOADS LATERAL PLANE THEIR 389 The smallest value of Nat which the denominator of one of the terms (h) becomes in expression It is evident that this critical value is obtained by pressive force N, taking n = equal to zero is the critical value of the com- Hence 2772 m? where m must be chosen Plotting the factor 2D b? a expression so as to make mb a k= (= + =) (227) a minimum \? against the ratio a/b, for various integral values of m, we obtain a system 198 in Fig of curves shown The portions of the curves that must be 10 m=l kề O 4⁄2 q b Fic 198 used in determining k are indicated by heavy lines It is seen that the factor k is equal to for a square plate as well as for any plate that can be subdivided into an integral number of squares with the side It can also be seen that for long plates k remains practically constant at a value Since the value of m in Eq (227) may be other than for oblong of 4.* plates, such plates, being submitted to a lateral load combined with compression, not generally deflect! in the form of a half wave in the direcIf, for instance, a/b = 2, 4, tion of the longer side of the plate the respective elastic surface becomes markedly unsymmetrical with respect to the middle line x = a/2 (Fig 192), especially so for values of N, close to the critical value (Nz) cr By using the deflection (g) produced by one concentrated load, the * A more detailed discussion of this problem is given in S Timoshenko, ‘‘Theory of Elastic Stability,’’ p 327, 1936 Several examples of such a deformation have been considered by K Girkmann, Stahlbau, vol 15, p 57, 1942 390 THEORY OF PLATES AND SHELLS deflection produced by any lateral load can be obtained by superposition Assuming, for example, that the plate is uniformly loaded by a load of intensity g, we substitute q dé dn for P in expression (g) and integrate the expression over the entire area of the plate In this way we obtain the same expression for the deflection of the plate under uniform load as has already been derived in another manner (see page 381) If the plate laterally loaded by the force P is compressed in the middle plane by uniformly distributed forces N, and N,, proceeding as before we obtain w=— 4P — = sin mene sini nen a b abx4D m2 m=1 The critical condition! n=] n?\° az value b2 + of the có forces m?N, n2N, ra2D x?b?D Nz and m?(Na)er , 22(Ny)cr — aD + wD sin mr — sin > WN, is obtained (m2, ~\art (2) from the n\? 0) where m and n are chosen so as to make N, and N, a minimum for any given value of the ratio N,/N, In the case of a square plate submitted to the action of a uniform pressure p in the middle plane we have a = b and N, = N, = p Equation (7) then gives 2D | Der = “ (m2 + 17) | (È) The critical value of p is obtained by taking m = n = 1, which gives cr — 2m?D a? (228) In the case of a plate in the form of an isosceles right triangle with simply supported edges (lig 161) the deflection surface of the buckled plate which satisfies all the “nes conditions is? w= alsin WX Lí — sin ong +- sin 272~ sin Tử a a a Thus the critical value of the compressive stress is obtained by substituting m = 1,n = or m = 2, n = into expression (k) This gives c® — 5x? q2 '(229) A complete discussion of this problem is given in Timoshenko, “Elastic Stability,” p 333 ? This is the form of natural vibration of a square plate having a diagonal as a nodal line PLATES UNDER LATERAL LOADS AND FORCES IN THEIR PLANE 391 94 Circular Plates under Combined Action of Lateral Load and Tension or Com- pression Consider a circular plate (Fig 199) submitted to the simultaneous action of a symmetrical lateral load and a uniform compression N, = N; = N in the middle plane of the plate Owing to the slope ¢ of the deformed plate (Fig 27) the radial compression N gives a transverse component N dy/dr which we have to add to the shearing force Q (Fig 28) due to the lateral load Hence the differential equation ở? ld k? (54) becomes ] ry i%4(2-4)6dr? r dr a? r? -£D @ in Which k2 _ Na? 6) D In the case of a circular plate without a hole! the solution of Eq (a) is of the form k: CW; (*) a p= + Yo | (c) Fic 199 where J; is the Bessel function of the order one, go a particular solution of Eq (a) depending on Q, and C; a constant defined by the boundary conditions of the plate Let us take as an example a rigidly clamped? plate carrying a uniform load of intensity g Then, as a particular solution, we use _ f6 and therefore | ° It follows, by integration, that qra? 92D dw dự = — Cia = — „ww _ qr 9N kr mỹ Œ) = CJ¡| J,{ — kr (=) qr 2N — Ì —_—— qr? }] —- — d “ C AN TC (e) where Jo 1s the Bessel function of the order zero and C2 a second constant Having calculated Ci from the condition ¿ = onr = a, and C2 from the condition w = on r = a, we obtain the final solution? (5) 20 | —— 48D gear = — 2:57 1(k) D kr nS The deflections (ƒ) beeome infinite for J¡(k) = Denoting Ớ) the zeros of the func- tion J; in order of their magnitude by ji, j2, we see that the condition k = j; In the case of a concentric hole a term proportional to a Bessel function of second kind must be added to expression (c) The inner boundary must be submitted then to the same compression JN, or else the problem becomes more complex because of the inconstancy of stresses N, and N: ? The case of an elastic restraint without transverse load has been discussed by H Reismann, J Appl AMfechanics, vol 19, p 167, 1952 This result may be found in A Nddai, ‘‘Elastische Platten,’ p 255, Berlin, 1925 392 SHELLS AND PLATES OF THEORY defines the lowest critical value No = | 100 =8(1—#) (1-8) Now, for the function Ji(k) we have the expression of the compressive stress N (9) As k