05. Small deflections of laterally loaded plates.PDF
CHAPTER SMALL 21 LOADED LATERALLY OF DEFLECTIONS The Differential Equation of the Deflection Surface PLATES We assume that the load acting on a plate is normal to its surface and that the deflections are small in comparison with the thickness of the plate (see Art 13) At the boundary we assume that the edges of the plate are free to move of the plate; thus the reactive forces at the in the plane With these assumptions we can neglect edges are normal to the plate Taking, as any strain in the middle plane of the plate during bending a put My Myx My Tay + Myx dx " -> i dy~- dy== Oy a T f= Mx —— + aMx Ox tMxv+ 21v L⁄ Qyt By dx Ox onto AQ Ax —— —} — J > ~ Rx t a 38x: dx Fia 47 hefore (see Art 10), the coordinate axes x and y in the middle plane of the plate and the z axis perpendicular to that plane, let us consider an clement cut out of the plate by two pairs of planes parallel to the zz and yz planes, as shown in Fig 47 In addition to the bending moments A, and AM, and the twisting moments J7,, which were considered in the pure lending of a plate (see Art 10), there are vertical shearing forces! acting The magnitudes of these shearing forces on the sides of the element per unit length parallel to the y and x axes we denote by Q, and Qy, respectively, so that Q: = J h/2 —h/2 h/2 Tzz dz Q„ — —h/2 r yz dz (a) %inee the moments and the shearing forces are functions of the coordinates and y, we must, in discussing the conditions of equilibrium of the clement, take into consideration the small changes of these quantities when the coordinates x and y change by the small quantities dz and dy ! Phere will be no horizontal shearing forces and no forces normal to the sides of the clement, since the strain of the middle plane of the plate is assumed negligible z9 80 THEORY OF PLATES AND SHELLS The middle plane of the element is represented in Fig 48a and b, and the directions in which the moments and forces are taken as positive are indicated We must also consider the load distributed over the upper surface of the plate The intensity of this load we denote by g, so that the load acting on the element! is q dz dy M,+ “Ox Mxy My + ðM x2 ar dy r "† \ _ (a) N ð My Mxy My+~gy” 4y | t+ ¿May ox ` a, Q, | ` | y Zz x Q, + oe dx aQy (b) Fic 48 Projecting all the forces acting on the element onto the z axis we obtain the following equation of equilibrium: dQ: dQ, dy dz + gq da dy 3x dx dy + cai from which OQ: , IQ, Dx + Oy —=0 _ +q=0 | (99) Taking moments of all the forces acting on the element with respect to the x axis, we obtain the equation of equilibrium aMa, sy 0# đụ — aM, ây _ dụ dx + Qy dx dụ = | (b) Since the stress component o, is neglected, we actually are not able to apply the load on the upper or on the lower surface of the plate Thus, every transverse single load considered in the thin-plate theory is merely a discontinuity in the magnitude of the shearing forces, which vary according to the parabolic law through the thickness of the plate Likewise, the weight of the plate can be included in the load g without affecting the accuracy of the result If the effect of the surface load becomes of special interest, thick-plate theory has to be used (see Art 19) SMALL DEFLECTIONS LOADED LATERALLY OF PLATES $1 The moment of the load g and the moment due to change in the force &, are neglected in this equation, since they are small quantities of a higher After simplification, Eq (b) becomes order than those retained dM., dM, +9, _=0 “Ta (c) In the same manner, by taking moments with respect to the y axis, we obtain OM,c , OM, (ở) ay Since there are no forces in the x and y directions and no moments with respect to the z axis, the three equations (99), (c), and (d) comLet us eliminate the pletely define the equilibrium of the element shearing forces Q, and Q, from these equations by determining them from In this manner we obtain Eqs (c) and (d) and substituting into Eq (99) eM, 0?M,., , AM, dx? * Oxdy ' oy OMe, () ¢ dxoy Observing that M„¿ = — ăz„, by virtue of 72, = 7„z, we finally represent the equation of equilibrium (e) in the following form: 2M, 0x? 21 0M, dy” 29 Mu Ox OY _ (100) —q To represent this equation in terms of the deflections w of the plate, we make the assumption here that expressions (41) and (43), developed for the case of pure bending, can be used also in the case of laterally This assumption is equivalent to neglecting the effect on loaded plates bending of the shearing forces Q, and Q, and the compressive stress ơ; produced We have already used such an assumption in by the load g the previous chapter and have seen that the errors in deflections obtained in this way are small provided the thickness of the plate is small in com- An approximate parison with the dimensions of the plate in its plane theory of bending of thin elastic plates, taking into account the effect of shearing forces on the deformation, will be given in Art 39, and several examples of exact cussed in Art 26 solutions of bending problems of plates will be dis- Using xz and y directions instead of n and t, which were used in Eqs (41) and (43), we obtain oe Me + 07w SF = ou = Dl =») ở? (102) AND PLATES OF THEORY 82 SHELLS Substituting these expressions in Eq (100), we obtain? 0% 04w ự — (105) art + * ataye * ay* ~ D This latter equation can also be written in the symbolic form (104) AAw = D + ay? Aw = It is reduces tion of ries of from | Aw , Ow where (105) scen that the problem of bending of plates by a lateral load q If, for a particular case, a soluto the integration of Eq (103) this equation is found that satisfies the conditions at the boundathe plate, the bending and twisting moments can be calculated Eqs (101) and normal corresponding The (102) stresses are found from Eq (44) and the expression (T 2) max = and shearing 6M „„ h2 Equations (c) and (d) are used to determine the shearing forces Q, and Q,, from which Q: = 0M, , 9M Oy ôy _ OM, OM ge Qu = Oy —- 7,90 [ow , Ow (32 Dạy & T st) ö” (106) aydy (= + a) (107) Q, = -D " (Am) (108) or, using the symbolic om Q = —D = ; (Aw) The shearing stresses rz, and r,, can now be determined by assuming that they are distributed across the thickness of the plate according to the parabolic law.? Then Qs h (7„z) max — 6, h bo] = bO| Go (722) mux This equation was obtained by Lagrange in 1811, when he was examining the The memoir presented to the French Academy of Science by Sophie Germain history of the development of this equation is given in I Todhunter and K Pearson, ‘History of the Theory of Elasticity,” vol 1, pp 147, 247, 348, and vol 2, part 1, p See also the note by Saint Venant to Art 73 on page 689 of the French transla263 tion of ‘Théorie de ]’élasticité des corps solides,”’ by Clebsch, Paris, 1883 It will be shown in Art 26 that in certain cases this assumption is In agreement with the exact theory of bending of plates SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 83 It is seen that the stresses in a plate can be calculated provided the deflection surface for a given load distribution and for given boundary conditions 1s determined by integration of Eq (103) 22 Boundary Conditions We begin the discussion of boundary conditions with the case of a rectangular plate and assume that the x and y axes are taken paralle! to the sides of the plate Built-in Edge If the edge of a plate is built in, the deflection along this edge is zero, and the tangent plane to the deflected middle surface tong this edge coincides with the initial position of the middle plane of the plate Assuming the built-in edge to be given by # = a, the bound- ary conditions are (w) ren = (5) =0 (109) Simply Supported Edge If the edge x = a of the plate is simply supported, the deflection w along this edge must be zero At the same time this edge can rotate freely with respect to the edge line; z.e., there are no moments M, along this edge This kind of support is represented in Fig 49 The analytical expressions for the boundary condi- - 2⁄2 LIN: tions in this case are — đ”t0 la + V gˆ+0 a) —= (110) 2/8 Ñ (U)z—a ⁄YELLE Ñ bending Fia 49 Observing that 0?w/dy? must vanish together with w along the rectilinear edge « = a, we find that the second of the conditions (110) can be rewritten as 0?w/dx? = or also Aw fore equivalent to the equations (W)raa which not involve = = Equations (110) (AW) = O are there(111) Poisson’s ratio v I'ree Isdge If an edge of a plate, say the edge + = a (Fig 50), is entirely free, it is natural to assume that along this edge there are no bending and twisting moments that (AT 2) cma = and also no vertical shearing lorces, 1.€., (M ry) 2—a = (Qz)z—a = The boundary conditions for a free edge were expressed by Poisson! in this form But later on, Kirchhoff? proved that three boundary con- ditions are too many and that two conditions are sufficient for the com- plete determination of the deflections w satisfying Eq (103) He showed ' See the discussion of this subject in Todhunter and Pearson, op cit., vol 1, p 250, wud in Saint Venant, loc cit “See J Crelle, vol 40, p 51, 1850 84 THEORY OF PLATES AND SHELLS also that the two requirements of Poisson dealing with the twisting moment M,, and with the shearing force Qz must be replaced by one boundary condition The physical significance of this reduction in the number of boundary conditions has been explained by Kelvin and Tait.! These authors point out that the bending of a plate will not be changed if the horizontal forces giving the twisting couple M,, dy acting on an element of the length dy of the edge x = a are replaced by two vertical forces of magnitude M,, and dy apart, as shown in Fig 50 Such a replacement does not change the magnitude of twisting moments and produces only local changes in the stress distribution at the edge of the plate, leaving the stress condition of the rest of the plate unchanged We have already discussed a par- pe eae ý aMxy “Mxy ray om = ĐỖ ⁄ ⁄⁄⁄⁄—— / „ | ⁄ A23 - “7 -Mxy mm Mxy+ rd Fic 50 find that the distribution ticular case of such a transforma- tion of the boundary force system in considering pure bending of a ; ° ° plate to an anticlastic surface (see Art 11) Proceeding with the foregoing’replacement of twisting couples along the edge of the plate and considering two adjacent elements of the edge (Fig 50), we “Y dy of twisting moments M,, is statically equiva- lent to a distribution of shearing forces of the intensity Qy ,—.= — ( (ðM„y ay ) Menee the joint requirement regarding twisting moment M„„ and shearing force Q, along the free edge x = a becomes (ø) =0 Vz = (0 — Tái 2) Substituting for Q, and M,, their expressions (106) and (102), we finally obtain for a free edge x = a: 03 03 lợi + — ») Ox mm | =0 _ (112) The condition that bending moments along the free edge are zero requires (Gate a =0 013) Independ1 See ‘Treatise of Natural Philosophy,” vol 1, part 2, p 188, 1883 16, pp vol 2, ser Math., J ently the same question was explained by Boussinesq, 125-274, 1871; ser 3, vol 5, pp 329-344, Paris, 1879 SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 85 Equations (112) and (113) represent the two necessary boundary conditions along the free edge x = a of the plate Transforming the twisting couples as explained in the foregoing discussion and as shown in Fig 50, we obtain not only shearing forces Q; distributed along the edge x = a but also two concentrated forces at the b ends of that edge, as indicated in Fig.51 f The magnitudes of these orces are equal rs ° Ø ~- to the magnitudes corresponding corners of the plate ⁄~~(Myx) | (Myx), _ veh (M xy), y of the twisting couple’ M,, at the (Mxy)„- O;y=O ỳ Te 11G 51 x x=A;Y= b ol;y=b Making the analogous transformation of twisting couples M,, along the edge y = b, we shall find that in this case again, in addition to the distributed shearing forces Q), there will be concentrated forces M,z at the corners This indicates that a rectangular plate supported in some way along the edges and loaded laterally will usually produce not only reactions distributed along the boundary but also concentrated reactions at the corners Regarding the directions of these concentrated reactions, a conclusion can be drawn Take, if the general for example, shape of the deflection surface is known a uniformly loaded square plate simply supported along the edges The general shape of the deflection surface is indicated in Fig 52a by dashed lines representing the section of the middle surface a apne a >| k——” Y ! frre of the plate by planes parallel to the xz Considering and yz coordinate planes 77x thes lines, it may be seen that near the 1A decreases numerically with increasing y Hence 02w/dx dy is positive at the cor- corner A the derivative dw/dx, representing the slope of the deflection surface in the x direction, is negative and (a) * ner A From Eq (102) we conclude that M,, is positive and M,, is negative /⁄ From this and from at that corner the directions of M,, and M,, in Fig 48a it follows that both concentrated forces, indicated at the point x = a, —A ¥R (b) Fic 52 From symmetry we conclude y = bin Fig 51, have a downward direction also that the forces have the same magnitude and direction at all corners Hence the conditions are as indicated in Fig 52b, in which of the plate = 2(M cy) c=a, y=a = TT 2D(1 _— v) ö” (75 ray) +=a,1/=G The couple Mz, is a moment per unit length and has the dimension of a force 86 THEORY OF PLATES AND SHELLS It can be seen that, when a square plate is uniformly loaded, the eorners in general have a tendency to rise, and this is prevented by the concentrated reactions at the corners, as indicated in the figure ee ¬ Elastically Supported and Elast- | mz cally Built-in Edge x Iftheedgex = a ofarectangular plate is rigidly joined to a supporting beam (Fig 53), the ZZZLL LLL LLL, deflection along this edge is not zero and is equal to the deflection of the beam Also, rotation of the edge is Fig 53 equal to the twisting of the beam Let B be the flexural and C the torsional rigidity of the beam The pressure in the z direction transmitted from the plate to the supporting beam, from Eq (a), is y z -om) 7, 7= -(9, (2 oy Vs =p? \ DxOx lœ T (2 — Đ ô®uu |, and the differential equation of the deflection curve of the beam is B d4w (57) - D—ax lœ This equation represents one of the along the edge x = a To obtain the second condition, The angle of rotation! considered — (dw/0dx)2-a, and the rate of change gle along the edge is _ ( ÔZ ÔW ¬+- (2 — LÀ) 3” | (114) two boundary conditions of the plate the twisting of the beam should be of any cross section of the beam is of this an- /z—a Hence the twisting moment in the beam 1s —C(0?w/dx dy)z—a This moment varies along the edge, since the plate, rigidly connected with y the beam, transmits continuously distributed twisting moments to the beam The magni(b) tude of these applied moments per unit length Fic, 54 is equal and opposite to the bending moments M, in the plate Hence, from a consideration of the rotational equilibrium of an element of the beam, we obtain — Y C (/ — 0” — ay (5: a) — — | (Mz) z= The right-hand-screw rulc is used for the sign of the angle DEFLECTIONS SMALL PLATES LOADED LATERALLY OF 87 or, substituting for M, its expression (101), ö ( =2 0” (sa) ^ D ð? 0° & $e) (115) This is the second boundary condition at the edge x = a of the plate In the case of a plate with a curvilinear boundary (Fig 54), we take at a point A of the edge the coordinate axes in the direction of the tangent ¢ and the normal ø as shown in the figure The bending and twisting moments at that point are (b) M, = [ j,„zơudz — Mạ = — [og em de h/2 h/2 Using for the stress components o, and r,, the known expressions! 272, SIN a COS a o, Sin? a + = 0, COS? a + gn Tnt = Tzy(Cos? a — sin? a) + (o, — oz) SM a COS a (b) in the following form: we can represent expressions M,cos? M, a+ = Maz = M,,(cos? a — sin? a) + The shearing force Q, at point equation of equilibrium from which 1/ 2M,, sin? a — M, UY sin a cos a (M, — M,) A of the boundary of the plate On — = QQ: dy shown Qy dx in Fig 54D, (d) Q„ = Q, cos a + Q, sin a or () will be found from the of an element ds sin a cosa Having expressions (c) and (d), the boundary condition in each particular case can be written without difficulty If the curvilinear edge of the plate is built in, we have for such an edge Ow w= on = (e) In the case of a simply supported edge we have (f) =0 M, w=0 Substituting for M, its expression from the first of equations (c) and using Eqs (101) and (102), we can represent the boundary conditions (f) in terms of w and its derivatives If the edge of a plate is free, the boundary conditions are cm M„ =0 Vạ = Q„ — aM, =0 | (g) The x and y directions are not the principal directions as in the case of pure bending; hence the expressions for AZ, and Af,, will be different from those given by Lqs (39) and (40) 88 THEORY OF PLATES AND SHELLS where the term —0M,,/0s is obtained in the manner shown in Fig 50 and represents the portion of the edge reaction which is due to the distribution along the edge of the twisting moment M,,; Substituting expressions (c) and (d) for Mn, Mn, and Q, and using Eqs (101), (102), (106), and (107), we can represent boundary conditions (g) in the following form: 0*w ö? yAw + (1 —— ) (cos s„U a a3 + sin? a 2y? COS a Aw + sin a ay Aw + (1 — ») 2a |eo 0° si) — 0? 2œ ax oy aw + 5sin 2a ($9 — (116) 0° 5) | = where, as before, Aw Another method of derivation = 0*w 0*w of these conditions _ 6ø? ° Oy? will be shown in the next article 23 Alternative Method of Derivation of the Boundary Conditions equation The differential (104) of the deflection surface of a plate and the boundary conditions can be obtained by using the principle of virtual displacements together with the expression for the strain energy of a bent plate.! Since the effect of shearing stress on the deflections was entirely neglected in the derivation of Eq (104), the corresponding expression for the strain energy will contain only terms depending on the action of bending and twisting moments as in the case of pure bending discussed in Art 12 Using Eq (48) we obtain for the strain energy in an infinitesimal element dV =-D HT? {fat , aw lZ+:+ ~~) — 2(1 — 20 Sw aw —) E ay? dw \? (2 5) | dz sấyd 6) votoff (+8) -0-[ BE -(Sey nw om The total strain energy of the plate is then obtained by integration as follows: dw dw a%y \’ where the integration is extended over the entire surface of the plate Applying the principle of virtual displacements, we assume that an infinitely small variation dw of the deflections w of the plate is produced Then the corresponding change in the strain energy of the plate must be equal to the work done by the external forces during the assumed virtual displacement In calculating this work we must consider not only the lateral load q distributed over the surface of the plate but also the bending moments M, and transverse forces Qn— (8Mn:z/ds) distributed along the boundary of the plate Hence the general equation, given by the principle of virtual displacements, i 18 This is the method by which the boundary conditions were satisfactorily established for the first time; see G Kirchhoff in J Crelle, vol 40, 1850, and also his Vorlesungen tiber Mathematische Physik, Mechantk, p 450, 1877 Lord Kelvin took an interest in Kirchhoff’s derivations and spoke with Helmholtz about them; see the biography of Kelvin by Sylvanus Thompson, vol 1, p 432 90 THEORY OF PLATES AND SHELLS The first term on the right-hand side of this expression is zero, since we are integrating along the closed boundary of the plate Thus we obtain [ Bo sin «c08 «2B as = — | Ow (Be & sin a c08 «) oo a Substituting this result in Eq (f), we finally obtain the variation of the first term in the expression for the strain energy in the following form: ff (“*) Ox? dx dy = —-.—- +2] |29s Noa? win | có2x3 a COS @ cos œ | sw ôu ds (0) (g Transforming in a similar manner the variations of the other terms of expression (117), we obtain s/f J) d2w \? (7) \ey? dx dy ff sw aeay v2 | oe oY an dy = Ox? dy? Ox = + Ox? ö?1⁄ +=>=-dq 0? 050, 0% _ _— si M dx? © ay? „áo Both these equations are of the same kind as that obtained for a uniformly stretched and laterally loaded membrane.? The solution of these equations is very much simplified in the case of a simply supported plate of polygonal shape, in which case along each rectilinear portion of the boundary we have 0?w/ds*? = since w = at the boundary Observing that M, = at a simply supported edge, we conclude also that 0?w/dn*? = at the boundary Hence we have [see Eq (34)] vu | 0U, 0° ð?u , 9?” ast + on? ~ an? Taye ~~ D9 ©) at the boundary in accordance with the second of the equations (111) It is seen that the solution of the plate problem reduces in this case to the integration of the two equations (120) in succession We begin with 1This method of investigating the bending of plates was introduced by H Marcus in his book “‘ Die Theorie elastischer Gewebe,’’ 2d ed., p 12, Berlin, 1932 See Timoshenko and J N Goodier, ‘‘Theory of Elasticity,?? 2d ed., p 269, 1951 © SMALL LOADED LATERALLY OF DEFLECTIONS PLATES 93 the first of these equations and find a solution satisfying the condition Substituting this solution in the second equaM = Oatthe boundary.! tion and integrating it, we find the deflections w Both problems are of the same kind as the problem of the deflection of a uniformly stretched and laterally loaded membrane having zero deflection at the boundary This latter problem is much simpler than the plate problem, and it can always be solved with sufficient accuracy by using an approximate method of integration such as Ritz’s or the method of finite differences Some examples of the application of these latter methods will be disSeveral applications of Ritz’s method cussed later (see Arts 80 and 83) are given in discussing torsional problems.? A simply supported plate of polygonal shape, bent by moments M, uniformly distributed along the boundary, is another simple case of the (120) application of Eqs (120) in such a case become Equations aM , eM _ 9+? ay? 9? 03w, - (121) M an? * oy? ~D Hence = Along a rectilinear edge we have again d°w/ds? M, = —-p?” on? and we have at the boundary 3? g?› dx? ' ay? —_— 0?w ôn — owe Mu D ——— — — — M D This boundary condition and the first of the equations (121) will be satisfied if we take for the quantity M the constant value M = Ma at all points of the plate, which means that the sum of the bending moments M, and M, remains constant over the entire surface of the The deflections of the plate will then be found from the second plate of the equations (121),* which becomes ð?w , 9°? aa? | By? M, dD ) It may be concluded from this that, in the case of bending of a simply supported polygonal plate by moments M,, uniformly distributed along the boundary, the deflection surface of the plate is the same as that of Note that if the the boundary when See Timoshenko This was shown plate is not of a polygonal shape, M generally does not vanish at M, = and Goodier, op cit., p 280 first by Woinowsky-Krieger, Ingr.-Arch., vol 4, p 254, 1933 94 THEORY OF PLATES AND SHELLS ol — “4 RS Q kl a uniformly stretched membrane with a uniformly distributed load There are many cases for which the solutions of the membrane problem are known These can be immediately applied in discussing the corresponding plate problems Take, for example, a simply supWenn nnn m= q - > ported equilateral triangular plate (Fig 55) bent by moments MM, uniformly distributed along the The deflection surface boundary C of the plate is the same as that of a a uniformly stretched and uni| formly loaded membrane The LZ (a) latter can be easily obtained ex- a R y Fig 55 (b) perimentally by stretching a soap py film on the triangular boundary and loading it uniformly by air pressure.! The analytical expression of the deflection surface is also comparatively simple in this case We take the product of the left-hand sides of the equations of the three sides of the triangle: (= +5) (Sg t¥-5 3/\xv⁄3 3) (Sg -4- 5) v⁄3/\x⁄3 V3 —dự+,_ da +) 4a? 3-27 This expression evidently becomes zero at the boundary Hence the boundary condition w = O for the membrane is satisfied if we take for deflections the expression w= ụ | ae — 3y2x = a(x? + yy?) 3 4d) T8.37 te) where N is a constant factor the magnitude of which we choose in such a manner as to satisfy Eq (d) In this way we obtain the required solution: W _ Mi, E _ đaD — 31/?*x = a(x? + y?) + ne | Substituting = y = 0in this expression, we obtain the centroid of the triangle | (f) deflection at the M,a? 1Such experiments are used Goodier, op cit., p 289 in solving torsional problems; see Timoshenko and SMALL DEFLECTIONS LOADED OF LATERALLY PLATES 95 The expressions for the bending and twisting moments, from Eas (101) and (102), are M, OX 0-93] pi+y- Mz = Ey My = Betty tba 3x À1, r 3(1 Mu, = _ Shearing forces, from Eqs (h) | — - ?)Äl„ụ _ (107), are (106) and Q: —= Oy = Along the boundary, from Eq (d) of Art 22, the shearing force Q, = 0, and the bending twisting moment The to M, is equal moment the side BC (Fig 55) from Eqs (c) of Art 22 1s 3(1 — M,, = 23 a along — (y — V32) The vertical reactions acting on the plate along the side BC (I'ig 55) are = Vn From symmetry Qn — OM nt QO 3(1 — ») eg = Ma we conclude that the same uniformly R v) (2) distributed reac- These forces are tions also act along the two other sides of the plate balanced by the concentrated reactions at the corners of the triangular plate, the magnitude of which can be found as explained on page 85 and is equal to = (AM ni) c=20,y=0 — (1 ~~ V3 Mp (7) The distribution of the reactive forces along the boundary is shown in The maximum bending stresses are at the corners and act on Fig 55b The magnitude of the corresponding the planes bisecting the angles bending moment, from Has (h), is (AL) max = (AT ,) 23a — Ma(3 — ») — — (k) This method of determining the bending of simply supported polygonal plates by moments uniformly distributed along the boundary can be applied to the calculation of the thermal stresses produced in such plates In discussing thermal stresses in clamped plates, by nonuniform heating it was shown in Art 14 [Eq (b)] that nonuniform heating produces uniformly distributed bending moments along the boundary of the plate The magnitude of these which prevent any bending of the plate 96 THEORY momerts is! OF M, PLATES AND = tet SHELLS „) (Ð To obtain thermal stresses in the case of a simply supported plate we need only to superpose on the stresses produced in pure bending by the moments (l) the stresses that are produced in a plate with simply supported edges by the bending moments —atD(1 + v)/h uniformly distributed along the boundary The solution of the latter problem, as was already explained, can be obtained without much difficulty in the case of a plate of polygonal shape.? Oe xiên xtEh | atthe Ty | fateh? NP | + the - œtEhF 24 tr R atEh?/ 8a (a) ate Fie 56 (b) Take again, as an example, the equilateral triangular plate If the edges of the plate are clamped, the bending moments due to nonuniform heating are Mt = Mt = ah + v) (m) To find the bending moments M, and M, for a simply supported plate we must superpose on the moments from Eqs (h) by letting M, obtain M = ale + v) _ al (m) the moments that will be obtained = —atD(1 + v) + v)/h +y—(1—») In this way we finally =| atEh? M, = atu + v) alt — ¢ + =) Y) ji + y+ (1 — ») =| —_ ath? _ latkh’y Moy = a 32x 94 1— 3x | a Jt is assumed that the upper surface of the plate is kept at a higher temperature than the lower one and that the plate thus has the tendency to bend convexly upward See dissertation by J L Maulbetsch, J Appl Mechanics, vol 2, p 141, 1935 SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 97 The reactive forces can now be obtained from qs (z) and (7) by substitution of M, = —atD(1 + v)/h Hence we find Vn = Qn - ƠM„ _ Mh? ds — =, _ 8a V3 ot HD? 12 The results obtained for moments and reactive forces due to nonuniform heating are represented in Fig 56a and 6, respectively 25 Effect of Elastic Constants on the Magnitude of Bending Moments from Eqs (101) and (102) that the magnitude It is seen of the bending and twisting moments in a plate is considerably affected by the numerical value of Poisson’s ratio » On the other hand, it can be easily shown that in the case of a transverse load the magnitude of the quantity Dw is independent of both constants £ and » if the plate is either simply supported at rectilinear edges or clamped along some edges, whether rectilinear or not Assuming such boundary conditions in any combination, let us consider the following problem Some values of the bending moments M, and M, being given numerically for an assumed numerical value of v, these moments must be computed for a new value, say v’, of the same elastic constant Let M, and M ' be the new values of the bending moments Writing Eqs (101) first for », then for v’, eliminating from them the curvatures 0°w/dz? and 0°w/dy?, and solving the resulting equations for M and M H we obtain M, , = TT a8 [(1 — ` vv’) M, + (øw? — »)M,] (122) M, = T—7zLd — v)Mụ + 0! — v)Ma] — yp Thus #⁄¿ and ụ can be readily calculated if M, If the constant »v is implied in some of the given of a free edge [Eq (112)], Eqs (122) not hold If the plate is elastically supported or elastically and M, are known boundary conditions, as in the case any more clamped, the moments also depend on the flexural rigidity D of the plate with respect to the stiffness of its restraint The thermal stresses, finally, are affected not only by all the above-mentioned factors, but also by the absolute value of the rigidity D of the plate Average values of v for some materials are given in Table The last value of the table varies widely, depending on the age of the concrete, on the type of aggregate, and on other factors.! TABLE AVERAGE VALUES Material Steel 0.30 Aluminum 0.30 Glass German Code Porsson’s (DIN 4227) Ratio v Ụ Concrete 1The OF 0.25 0.15-0.25 gives values of » which approximately can be expressed by »y = N4 f./350, ƒ, being the compressive strength of concrete at 28 days in pounds per square inch See also J C Simmons, Mag of Concrete Research, vol p 39, 1956 98 THEORY OF PLATES AND SHELLS 26 Exact Theory of Plates The differential equation (103), which, together with the boundary conditions, defines the deflections of plates, was derived (see Art 21) by neglecting the effect on bending of normal stresses o, and shearing stresses rz, and ty: This means that in the derivation each thin layer of the plate parallel to the middle plane was considered to be in a state of plane stress in which only the stress components or, Sy, and rzy may be different from zero One of the simplest cases of this kind is that of pure bending The deflection surface in this case is a second-degree function in x and y [see Eq (c), Art 11] that satisfies Eq (103) The stress components oz, oy, and rz, are proportional to z and independent of x and y There are other cases of bending in which a plane stress distribution takes place and Eq (103) holds rigorously Take, for example, a circular plate with a central circular hole bent by moments M, uniformly distributed along the boundary of the hole (Fig 57) Each thin layer of the plate cut out by two adjacent planes parallel to the middle plane is in the same stress condition as a thick-walled cylinder subjected to a uniform internal pressure or tension (Fig 57b) The sum oa, + o of the two principal stresses is constant in such a case,! and it can be concluded that the deformation of the layer in the z direction is also constant and does not interfere with the deformation of adjacent layers Hence we have again a planar stress distribution, and Eq (103) holds Let us discuss now the general question regarding the shape of the deflection surface of a plate when bending results in a planar stress distribution To answer this question it is necessary to consider the three differential equations of equilibrium together with the six compatibility conditions If body forces are neglected, these equations are? Oo x OT zy OT xz =0 Ox OY Oz Oo OT: OT yz — Ox Oo OT zz OT yz 02 Ox OY Aiox = — Ai, “ = — Aw, = — A1Txyw = — A y4—" Oy Txz —= — AiTyz = — 02 — 076 — + y ox? ] 076 — 026 — » 9z? +wôy? + 1 + 076 vp Ox 02 076 l +w9oy9z in which 6=o07, and 0? A, =— +o, b v Ox Oy 1 + ©) 076 | (a) +a: 9? +— 0? +— 1See Timoshenko and Goodier, op cit., p 60 See zbid., pp 229, 232 nà (€ ... accuracy of the result If the effect of the surface load becomes of special interest, thick-plate theory has to be used (see Art 19) SMALL DEFLECTIONS LOADED LATERALLY OF PLATES $1 The moment of the... biography of Kelvin by Sylvanus Thompson, vol 1, p 432 SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 89 The first integral on the right-hand side of this equation represents the work of the lateral... in certain cases this assumption is In agreement with the exact theory of bending of plates SMALL DEFLECTIONS OF LATERALLY LOADED PLATES 83 It is seen that the stresses in a plate can be calculated