12. Bending of anisotropic plates
CHAPTER BENDING OF 11 ANISOTROPIC PLATES 85 Differential Equation of the Bent Plate In our previous discussions we have assumed that the elastic properties of the material of the plate are the same in all directions There are, however, cases in which an anisotropic material must be assumed if we wish to bring the theory of plates into agreement with experiments.! Let us assume that the material of the plate has three planes of symmetry with respect to its elastic properties.2 Taking these planes as the coordinate planes, the relations between the stress and strain components for the case of plane stress in the zy plane can be represented by the following equations: o, = Eve, + Be, oy = Eye + Ee, Tey = | (a) GY cy It is seen that in the case of plane stress, four constants, E72, HE}, H’’, and G, are needed to characterize the elastic properties of a material Considering the bending of a plate made of such a material, we assume, as before, that linear elements perpendicular to the middle plane (ry plane) of the plate before bending remain straight and normal to the deflection surface of the plate after bending.* Hence we can use our previous expressions for the components of strain: s„= —# 0? 6, = —# 0° 0? Yeu = —22z dx dy (b) The case of a plate of anisotropic material was discussed by J Boussinesq, J math., ser 3, vol 5, 1879 See also Saint Venant’s translation of ‘‘Théorie de ]’élasticité des corps solides,’’ by A Clebsch, note 73, p 693 | Such plates sometimes are called ‘‘orthotropic.’’ The bending of plates with more general elastic properties has been considered by G Lechnitzky in his book “‘ Anisotropic Plates,’’ 2d ed., Moscow, 1957 The effect of transverse shear in the case of anisotropy has been considered by K Girkmann and R Beer, Osterr Ingr.-Arch., vol 12, p 101, 1958 364 : 365 PLATES ANISOTROPIC OF BENDING “he corresponding stress components, from Eqs (a), are nT —* Big + Phận) / ” o, = =s =: (2,55 Tay — — g”+ (c) 0se) P” + 0*w 2G2z ax ay Vith these expressions for stress components the bending and twisting are aoments M, = h/2 —h/2 sat de = — (D5 oe + D5) oy? h/2 M, = M oy = | = Tye đz , Dy, ay va y ay (0.5 — h/2 2? đz TxuZ Ly 2D» = n which p, = Ev 1/1038 p, = Lx 12 (212) =) D.=*?? 12 Dey =“ 12 (đ) (212) in the differential equation of equilibrium Substituting expressions 100), we obtain the following equation for anisotropic plates: Di O*w =a 2D„) + 2(Di + Pv + nạ ö? aya | ntroducing the notation = H ve obtain = Dị + (e) Dey 0*w St + 2H aaa t Du 0*w = (213) Che corresponding expressions for the shearing forces are readily obtained ‘rom the conditions of equilibrium of an element of the plate (Fig 48) Thus, we have und the previous expressions for the moments Qy 0*w - %(> e+ He) = — 0? (p,2 (214) 0? 2) [n the particular case of isotropy we have it = Bt = E — p? b EY = _ 1—? G= E 2( +) Hence D; = D, = 12(1 Eh’ — SHELLS AND PLATES OF THEORY 366 py?) "hà vi E Eh’ H= Di + 25 = 35 (7a +745) “pa-wA |( | and Eq (213) reduces to our previous Eq (103) Equation (213) can be used in the investigation of the bending of plates of nonisotropic and even nonhomogeneous material, such as reinforced concrete slabs,! which has different flexural rigidities in two mutually perpendicular directions 86 Determination of Rigidities The Cases in Various Specific employed In particular, expressions (d) given for the rigidities in the preceding article are subject to slight modifications according to the nature of the material all values of torsional rigidity D,, based on purely theoretical considerations should be regarded as a first approximation, and a direct test as shown in Fig 25c must be recommended in order Usual values of the rigidities in some to obtain more reliable values of the modulus G | | below given are interest cases of practical Let E, be Young’s modulus of steel, HE that of the conReinforced Concrete Slabs In terms of the elastic concrete, ve Poisson’s ratio for concrete, and n = E./E, For a slab stants introduced in Art 85 we have approximately », = E’’/ a/ E_E, assume can we y and x directions the in nt reinforceme with two-way D, = D, = _ te [lex l1 —v, E say — Đ, + (n — 1)Taz] + (n — 1)Isy] (a) D; = ve V D:D, Dey= — vw / D:D, In these equations, I.2 is the moment of inertia of the slab material, 7, that of the reinforcement taken about the neutral axis in the section z = constant, and Jy and I,„ are the respective values for the section y = constant With the expression given for Dz, (also recommended by Huber) O4w D, ¬T (b) H = V/D.D, and the differential equation VD:D, D,D 04w | | we obtain O4w a the —— —= 79 (c) The application of the theory of anisotropic plates to reinforced concrete slabs is due to M T Huber, who published a series of papers on this subject; see Z Osterr The principal results are collected in his books: Ing u Architektur Ver., 1914, p 557 “Teorya Plyt,” Lvov, 1922, and ‘‘Probleme der Statik technisch wichtiger orthotroper Abstracts of his papers are given in Compt rend., vol 170, Platten,” Warsaw, 1929 pp 511 and 1305, 1920; and vol 180, p 1243, 1925 BENDING OF ANISOTROPIC 367 PLATES hich can readily be reduced to the form (103) by introducing y: = y X⁄ D./D, as a ew variable It is obvious that the values of the state of the concrete (a) are not independent or instance, any difference of the reinforcement in the directions z and y will affect ve ratio D,/Dy much more after cracking of the concrete than before For a plate glued together of three or five plies, the z axis supposed to be Plywood arallel to the face grain, we may use the constants given in Table 79 TaBLE 79 CoNSTANTS ELastic Unit = 10° psi Material E, 1aple,* 5ð-pÌy - 1.87 1.28 2.00 taboon* (Okoumé), 3-pÌy .- - | ireh,† 3- and ð-pÌy -. G 0.60 0.073 0.159 0.11 0.167 0.014 0.077 0.085 0.17 0.165 0.85 1.70 ircht with bakelite membranes Ek” E y 1.96 eee eee 20 e ee fara,* 3-ply PLYWOOD FOR 0.043 0.061 0.110 0.10 * By R F S Hearmon and E H Adams, Brit J Appl Phys., vol 3, p 155, 1952† By G Lechnitzky, Anisotropic ZO ™~ Plates,” p 40, Moscow, ZN NL † fy Ft Po ott P ot | | | | | a Fic Corrugated Sheet — 1947 186 Let E and » be the elastic constants of the material of the sheet, its thickness, , TT = ƒsin he form of the corrugation, and s the length of the arc of one-half a wave (Fig 186) “hen we have! l Eh3 mm 12(1 D, = EI D, H ~ — z?) 90 § kbh3 =2D,„,=-—— 192 +) 1See E Seydel, Ber deut Versuchsanstalt Luftfahrt, 1931, 368 THEORY OF PLATES in which, approximately, §$ = Ấn AND SHELLS wf? +) h , fel, 0881 (f +28(2) Plate Reinforced by Equidistant Stiffeners in One Direction For a plate reinforced symmetrically with respect to its middle plane, as shown in Fig 187, we may take! D, = D , y- 12(1 — vy?) Eh’ g1 ——————12 —_ yp?) + Qi in which £ and » are the elastic constants of the material of the plating, E’ the Young modulus, and J the moment of inertia of a stiffener, taken with respect to the middle axis of the cross section of the plate L fT TT_ ; | k=-—- 0ì — | | Fic Ạ oa —- Ơi ~> | y yt Py | | | | | Ho y 187 Ly Ly I | a _ malt} | | Th 1| Fic 188 Plate Cross-stiffened by Two Sets of Equidistant Stiffeners ment is still symmetrical about the plating we have Provided the reinforce- Eh’ E’'l, D;y =—————— + — 12(1 — y?) + bì D, = Eh 12(1 — vy?) , F'I: ay Eh? HA = ——_ 12(1 — pv?) I, being the moment of inertia of one stiffener and 6; the spacing of the stiffeners in direction z, and J2 and a, being the respective values for the stiffening in direction y Slab Reinforced by a Set of Equidistant Ribs In the case shown in Fig 188 the theory established in Art 85 can give only a rough idea of the actual state of stress and Recommended by Lechnitzky, op cit For more exact values see N J Huffington, J Appl Mechanics, vol 23, p 15, 1956 An experimental determination of the rigidities of stiffened and grooved plates was carried out by W H Hoppmann, N J Huffington, and L 8S Magness, J Appl Mechanics, vol 23, p 343, 1956 ANISOTROPIC OF BENDING 269 PLATES Let E be the modulus of the material (for instance, concrete), rain of the slab Then we may the moment of inertia of a T section of width a, and a = h/H ssume D, Ea,h? = 12(øn — Í + œ3£) EI D, = — Qi Dị = ‘he effect of the transverse contraction is neglected in the foregoing formulas on orsional rigidity, finally, may be calculated by means of the expressi Dry sy = D., + The c 2a) C the torsional which Dy is the torsional rigidity of the slab without the ribs and igidity of one rib.? 87 Equa- Application of the Theory to the Calculation of Gridworks poe 189 ion (213) can also be applied to the gridwork system shown in Fig (b) (a) Fic This consists of two systems 189 of parallel beams spaced supported at the ends, equal distances points of apart in the x and y directions and rigidly connected at their mtersection ‘The beams are and the load is If the distances a, and b, between the applied normal to the zy plane b of the grid, beams are small in comparison with the dimensions a and the z axis 1s and if the flexural rigidity of each of the beams parallel to is equal to Be, equal to B, and that of each of the beams parallel to y axis we can substitute in Eq (213) D, For a more exact theory Bi op _ B: bạ [An concerning Ụ slabs with ribs in one or two (a) directions and deflection see K Trenks, leading to a differential equation of the eighth order for the vol 16, p 111, 1947 rch., Ingr.-A , Pfliger Bauingenieur, vol 29, p 372, 1954; see also A SHELLS AND PLATES OF THEORY 3/40 The quantity D, in this case is zero, and the quantity D., can be expressed in terms of the torsional rigidities C' and C2 of the beams parallel to the For this purpose we consider the twist of an x and y axes, respectively element as shown in Fig 189b and obtain the following relations between the twisting moments and the twist 0°w/dz dy: _ OF 0?w _ C; (0)| 0*w = ~ ay andy Ma =F dn ay Substituting these expressions in the equation of equilibrium (e) on page 81, we find that in the case of the system represented in Fig 189a the differential equation of the deflection surface is Bị g0 bi ôxzt + (f+ _ g3 ax? ay? + 2) saa B› g° _ - q a, oy? = (215) which is of the same form as Eq (213) In order to obtain the final expressions for the flexural and torsional moments of a rib we still have to multiply the moments, such as given by Eqs (212) and valid for The variation of the moments, the unit width of the grid, by the spacing of the ribs the points (m — 1) and between parabolic assumed be may M,,, say M, and be assigned to the rib may 190) (Fig diagram the of area (m +1) and the shaded F —————— b, _————= > 1Í — Mm Mm-t M H m-† m =-— ~— by rear +1 m m+ >< — Fic bạ > 190 Then, observing the expressions (212), we obtain the (m) running in the direction z following approximate formulas for both moments of the rib (m): M.,= M = 3w _ đì ~ Cìạ 24 Ox? 9? 4+ 22 m—-1l ỏ Ox? 22 ð? m (ate Ì Ox? ư?u m+1 ( C For ribs of the direction y we have to interchange z and y in the foregoing expressions and replace B, by Bz and Ci by C2; (m — 1), (m), and (m + 1) then denote three successive joints on a rib having the direction z ) BENDING OF ANISOTROPIC PLATES 371 Two parameters largely defining the elastic properties of a grid and often used in calculation are n= Be lồi cúi Bia, HỘ @ The parameter \ multiplied by the side ratio a/b (Fig 189) yields the relative carrying capacity of a rectangular plate in the directions y and x, whereas the parameter p characterizes the torsional rigidity of a grid as compared with its flexural rigidity Equation (215) has been extensively used in investigating the distribution of an arbitrarily located single load between the main girders of a bridge stiffened in the transverse direction by continuous floor beams.? 88 Bending of Rectangular Plates ported on all sides Eq (213) When the plate is simply sup- can be solved by the methods used in the case of an isotropic plate Let us apply the Navier method (see Art 28) and assume that the plate is uniformly loaded Taking the coordinate axes as shown in Fig 59 and representing the load in the form of a double trigonometric series, the differential equation ~2 yu + D, 4x4 a Y, = | (9 The roots of the corresponding characteristic equation are T1,3,3,4 “ + mx a Using, in accordance with Eq r - |H RE + LH? D? D, D, (9) (d), Art 87, the notation *LDy D._ Jo H = ————_— "“=⁄7P,p, (k)k we have to consider the following three cases: Case 1,u > 1: Case 2, yu = 1: Case 3, u < Ì: H? > D,D, H? = D,D, @) H? < D,D, In the first case all the roots of Eq (7) are real Considering the part of the plate with positive y and observing that the deflection w and its derivatives must vanish at large distances from the load, we can retain 3/4 THEORY only the negative roots OF PLATES AND SHELLS Using the notation œ — B= ^ + ar ~ vụ X⁄ụ? — ] (m) — wV pe — — the integral of Ey (2) becomes | and expression Yn = Anemv!® + Bele (h) can be represented in the form w= (Anewm/* + B,,e-Ơ!đ) sin = From symmetry we conclude that along the zx axis Ow °* (5; ) and we find Brn and = w= — An (em — B e Am — e cm) sin Ta | (n) The coefficient A,, is obtained from the condition relating to the shearing force Q, along the x axis, which gives -%(v 0” "2y? 9*\ + Hy) _ du max ø Sm —— Substituting for w its expression (n), we obtain An= goa? B? aga 2m°D,(œ? — B®) 2mtm3D,(œ° — 83 and the final expression (n) for the deflection becomes a1 _ = Salm Dela? — BR (2 — Bernt) sin Mrz G) In the second of the three cases (l) the characteristic equation has two double roots, and the function Y,, has the same form as in the case of an isotropic plate (Art 36) In the third of the cases (1) we use the notation ; _ GÀ l —u av | —2 oe Nitu (p) BENDING and thus obtain W _ OF ANISOTROPIC PLATES 315 the solution qoa* 4n?m? ~»/ D,D, my sìn —7 my + ổ , cos — (« , sm a —Max Ìe —mzlB' sin — (q) We can also shift from case to case by using the complex relations 1_1_ „l " a h ’ a’ | eB ta’ (r) Having the deflection surface for the sinusoidal load (f), the deflection for any other kind of load along the x axis can be obtained by expanding the load in the series qd = Am sin mre rN m=] and using the solution series obtained for the load (f) for each term of this The following expressions hold when, for instance, a load P is concentrated at a point x = Case 1,u > w= 1: Pa — œ* m%l)„ —-; w= THTỆ l2 — B Be~mv!B) ) gin ——a mrx sin —— a $ (s) =] Pa? => 21° DX m3 ( — (œe@~ ” Case 2, u = 1: Case ổ, &, y = of the infinite strip (lig 72): ] — m=] m3 mr Y {1 + —Je-™!* T anr mr& Mrz sin —— sin —— a t a ứ) < 1: oan Ha wn DD, v== 2? fal ‘sin sin OY TtÙ2% ) e—mv/#’ sin ` mang sin a “2 + 8! cos m> a’ a’ a g m= (u) Expressions in closed form! can be obtained for bending moments due to a, single load in a manner Art 35 similar to that used for the isotropic plate in Having this solution, the deflection of the plate by a load distributed 1See W Nowacki, Acta Tech Acad Sci Hung., vol 8, p 109, 1954; S WoinowskyKrieger, Ingr.-Arch., vol 25, p 90, 1957 Numerical results regarding influence surfaces of orthotropic rectangular plates may be found in H Olsen and F Reinitzhuber, ‘Die zweiseitig gelagerte Platte,’’ Berlin, 1950, and in H Homberg and J Weinmeister, ‘“Einflussflichen fir Kreuzwerke,’’ 2d ed., Berlin, 1956 376 THEORY OF PLATES AND SHELLS over a circular area can be obtained by integration, as was shown in the case of an isotropic plate (see Art 35) By applying the method of images the solutions obtained for an infinitely long plate can be used in the investigation of the bending of plates of finite dimensions.? 89 Bending of Circular and Elliptic Plates A simple solution of Eq (213) can be obtained in the case of an elliptic plate clamped? on the boundary and carrying uniform load of intensity g Provided the principal directions z and y of the orthotropic material are parallel to the principal axes of the ellipse (Fig 157) the expression +? y? v=w (1-5-4) (a) in which = 9U” 24D, a‘ q 16H a?b? 24D, + b 6) b1 satisfies Eq (213) and the required conditions on the boundary The bending moments of the plate are readily obtained by means of expressions (212) In the particular case of a clamped circular plate (a = b) we have the following results: _ g(a? — 7°)? _— 64D M = Top (Dz + Di)(a? — r*) — 2(Dzx? + Diy?) M, = ED [(Dy + D:)(a? — 12) — 2(D,y? + Diz®)] g May = AD’ (c) Dayxy Qe = — sọ, (3D, + H) Q = — an; BD, + H) in which : r= Vat + y? and D' = $(3Dz + 2H + 3D,) Since the twist is zero along the edge, the reactions of the support are given by a linear combination of the boundary values of the shearing forces Q, and Q, (see page 87) A straightforward solution can also be obtained in the case of pure bending or pure twist of an orthotropic plate Let such a plate be subjected to uniform couples M, = M,, M, = M2, and Mz, = M; By taking the deflection in the form w= Az? + Bay + Cy? (d) Several examples of this kind are worked out in the books by M T Huber: ‘“Teorya Plyt,’’ Lvov, 1922, and ‘‘ Probleme der Statik technisch wichtiger orthotroper Platten,’’? Warsaw, 1929 For bending of a simply supported elliptical plate, see Y Ohasi, Z angew Math u Phys., vol 3, p 212, 1952 BENDING OF ANISOTROPIC PLATES we obviously satisfy the differential equation (213) are given by the linear equations 377 The constants A, B, and C then D,A + DiC = —‡M, D,A + D,C = D,,B = —?M: (e) $M; which ensue from the expressions (212) The bending of a circular plate with cylindrical aeolotropy has been discussed too.} If, in addition to the elastic symmetry, the given load distribution is also symmetrical about the center of the plate, then the ordinary differential equation of the bent plate contains only two flexural rigidities, the radial and the tangential Formal solutions of this equation for any boundary conditions are simple to obtain; the choice of the elastic constants of the material, however, requires special consideration since certain assumptions regarding these constants lead to infinite bending moments at the center of the plate even in the case of a continuously distributed loading Most of the special methods used in solving the problems of bending of an isotropic plate (Chap 10) can be applied with some modifications to the case of an anisotropic plate as well If we take the complex variable method,? for example, the form of the solution proves to be different from that considered in Art 79 As can be shown, it depends upon the roots p1, pz, —p1, and —øp; of the characteristic equation D,p* + 2H p? + D,z = which are either imaginary or complex These roots being determined, the solution of the homogeneous equation D, 04w,/dx! + 2H d*w,/dz? dy? + D, d*wi/dy* = can be represented either in the form = Rlei(z1) + g2(Z2)] if p1 * pe, or else in the form Wi if p:1 = pe = Rlgilez:) + Ziv¿(2¡)] In these expressions ¢; and ¢2 are arbitrary analytic functions of the com- plex variables z; = z + piy and 22 = x + poy In using the Ritz method, expression (6) of Art 80 for the strain energy has to be replaced by the expression 2u veoh I, L (ia) TO 2D; 0?w — 0?w — oat ay? * D, 2? (| — (i) \? b 4D„„ 3?w | —— \? dx d (Gea) | ad while the rest of the procedure remains the same as in the case of the isotropic plate 1G F Carrier, J Appl Mechanics, vo] 11, p A-129, 1944, and Lechnitzky, op cit See S G Lechnitzky, Priklad Mat Mekhan., vol 2, p 181, 1938, and V Morcovin, Quart Appl Math., vol 1, p 116, 1948 For application of the method to the problem of stress concentration, see also G N Savin, ‘‘Stress Concentration around Holes,”’ Moscow, 1951, and S G Lechnitzky, Inzhenernyi Sbornik, vol 17, p 3, 1953 Stress concentration in isotropic and anisotropic plates was also discussed by Holgate, Proc Roy Soc London, vol 185A, pp 35, 50, 1946 ... in the investigation of the bending of plates of finite dimensions.? 89 Bending of Circular and Elliptic Plates A simple solution of Eq (213) can be obtained in the case of an elliptic plate... elastic constants of the material of the plating, E’ the Young modulus, and J the moment of inertia of a stiffener, taken with respect to the middle axis of the cross section of the plate L fT... the rigidities of stiffened and grooved plates was carried out by W H Hoppmann, N J Huffington, and L 8S Magness, J Appl Mechanics, vol 23, p 343, 1956 ANISOTROPIC OF BENDING 269 PLATES Let E