11. Special and approximate methods intheory of plates

11. Special and approximate methods intheory of plates

CHAPTER 10 SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES The state of stress in a plate 75 Singularities in Bending of Plates s said to have a singularity at a point! (xo0,yo) if any of the stress comFrom expressions (101), yonents at that point becomes infinitely large 102), and (108) for moments and shearing forces we see that a singuarity does not occur as long as the deflection w(z,y) and its derivatives 1p to the order four are continuous functions of z and y Singularities usually occur at points of application of concentrated In certain cases a singularity due to reactive forces forces and couples ean occur at a corner of a plate, irrespective of the distribution of the surface loading In the following discussion, let us take the origin of the coordinates The expressions at the point of the plate where the singularity occurs for the deflection given below yield (after appropriate differentiations) stresses which are large in comparison with the stresses resulting from loading applied elsewhere or from edge forces, provided x and y are small If the distance of the Single Force at an Interior Point of a Plate point under consideration from the boundary and from other concentrated loads is sufficiently large, we have approximately a state of axial Consequently, the radial shearing symmetry around the single load P force at distance r from the load P is Q; = —-sa> 27m? Observing the expression (193) for Q; we can readily verify that the respective deflection is given by Wo _= 87D Po r in which a is an arbitrary length log ar (206) The corresponding term r? log a yields negligible stresses when the ratio r/a remains small Single Couple at an Interior Point of a Plate | More exactly, at a point (20,Y0,2) 325 Let us apply a single 326 THEORY OF PLATES AND SHELLS force —M,/Az at the origin and a single force +M,/Az at the point (—Az,0), assuming that Jf, isa known couple From the previous result [Eq (206)] the deflection due to the combined action of both forces is _ Ms — 8xD (2 + Ax)? + Ax y? log [(z + Ax)? + y?}} a Mi 2+y 8rD Ax (x? + ?)‡ log a (a) As Ax approaches zero, we obtain the case of a couple Af; concentrated at the origin (lig 168a) and the deflection is ‘ WwW; = lim [wlaz—o = Mf, PP OWo 2x where Wp is the deflection given by expression differentiation we obtain wy = ^» 82D (tog ea a `2 42 | (206) 1) Performing the (0) If we omit the second term M,x/8rD, which gives no stresses, and use polar coordinates, this expression becomes WwW, = iD i r log — cos (207) In the case of the couple Af, shown in Fig 168b we have only to replace by + 7/2 in the previous formula to obtain the corresponding deflection Fic (b) 168 Double Couple at an Interior Point of a Plate Next we consider the combined action of two equal and opposite couples acting in two parallel planes Ax apart, as shown in Fig 169 Putting Af, Ax = H, and fixing the value of H, we proceed in essentially the same manner as before and arrive at the deflection — ows _ H) 02g (c) SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 327 jue to a singularity of a higher order than that corresponding to a couple.! Substitution of expression (206), where rectangular coordinates may be ised temporarily, yields the deflection i, Oe Sar D (2 log= "+2 + cos 20) (208) Expressions containing a singularity are also obtainable in the case of a couple acting at the corner of a wedge-shaped plate with both edges free, as well as in the case of a semi-infinite plate sub- mitted to the action of a transverse force or a couple at some point along the free edge.’ Single Load Acting in the Vicinity of a Burlt-in Aq ⁄ 4) TE | sles £ 0⁄ | r A NEP 4T X ⁄1 of ! k T1 | | >i x Pit jo ae ⁄ mm | Fic Edge (Fig 170) 169 Fig 170 The deflection of a semi-infinite cantilever plate carry- ing a single load P at some point (£,7) is given by the expression _T— + ) — „À9 where r? = (a — £)? + (y — »)? We confine ourselves to the consideration of the clamping moment at the origin Due differentiation of expression (d) yields M, = ~~ cos ¢ (209) at x = y = 0, provided £ and y not vanish simultaneously It is seen that in general the clamping moment MM, depends only on the ratio n/£ 1'To make the nature of such a loading clear, let us assume a simply supported beam of a span L and arigidity EI with a rectangular moment diagram Az by M, symmetrical to the center of beam and due to two couples Af applied at a distance Ar from each other Proceeding as before, 7.e., making Az — 0, however fixing the value of H = M Az, we would arrive at a diagram of magnitude H concentrated at the middle of beam Introducing a fictitious central load H/EI and using Mohr’s method, we would also obtain a triangular deflection diagram of the beam with a maximum ordinate HL/4EI A similar deflection diagram would result from a load applied at the center of a perfectly flexibie string 2See A Naddai, ‘‘llastische Platten,” p 203, Berlin, 1925 328 THEORY OF PLATES AND SHELLS If, however, £ = » = the moment M, vanishes, and thus the function M,(&,n) proves to be discontinuous at the origin Of similar character is the action of a single load near any edge rigidly or elastically clamped, no matter how the plate may be supported elsewhere This leads also to the characteristic shape of influence surfaces plotted for moments on the boundary of plates clamped or continuous along that boundary (see Figs 171 and 173) For the shearing, or reactive, force acting at x = y = in Fig 170 we obtain in similar manner Q„ = = cos? (210) where r? = £2 + 7?, 76 The Use of Influence Surfaces in the Design of Plates In Art 29 we considered an influence function K(z,y,£,n) giving the deflection at some point (x,y) when a unit load is applied at a point (é,7) of a simply supported rectangular plate Similar functions may be constructed for any other boundary conditions and for plates of any shape We may also represent the influence surface K(é,n) for the deflection at some fixed point (z,y) graphically by means of contour lines By applying the principle of superposition to a group of n single loads P; acting at points (&:,n:) we find the total deflection at (x,y) as w= > i=1 P:K (x,y, &i,n:) | (a) In a similar manner, a load of intensity p(é,y) distributed over an area A of the surface of the plate gives the deflection w = Jf p(E,n)K(œ,,E,n) đề dn (b) By Maxwell’s reciprocal law we also have the symmetry relation K(x,y,&,n) = K(é,n,2,y) (c) z.e., the influence surface for the deflection at some point (z,y) may be obtained as the deflection surface w(é,n) due to a unit load acting at (x,y) The surface w(é,n) is given therefore by the differential equation AAw(é,n) = 0, and the solution of this equation not only must fulfill the boundary conditions but also must contain a singularity of the kind represented in Eq (206) at & = x2, = y Of special practical interest are the influence surfaces for stress resultants! given by a combination of partial derivatives of w(z,y) with respect to z and y To take an Such surfaces have been used first by H M Westergaard, Public Roads, vol 11, 1930 See also F M Baron, J Appl Mechanics, vol 8, p A-3, 1941 SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 329 example, let us consider the influence surfaces for the quantity 0? —ÐĐ ö? = —D Ox? - K (x,y, &,n) (d) By result (c) of Art 75 this latter expression yields the ordinates of a deflection surface in coordinates ¢, containing at — = 2,7 = ya singularity due to a ‘‘couple of second order’? H = which acts at that point in accordance with Fig 169 The procedure of the construction and the use of influence surfaces may be illustrated by the following examples.} Influence Surface for the Edge Momeni of a Clamped Circular Plate? (Fig 171) By representing the deflection (197), page 293, in the form w = PK(z,0,é,6), we can consider K as the influence function for the deflection at some point (z,0), the momentary position of the unit load being (£,@) In calculating the edge couple M,atz = r/a = 1, y = we observe that all terms of the respective expressions following one, vanish along the clamped edge z = a — öZ2 /z~: (192), except for the The only remaining term yields ¢2\2 4r ‡? — 2£ cos Ø +1 Eor brevity let us put ‡? — 2‡ cos + = 7»? and, furthermore, introduce the angle ¢ (Fig 171a) Then we have ¢? = M, = — 2n cos ¢ + 7? and — — Ar (2 cos¢ — n)? which, for negligible values of », coincides with the expression (209) The influence surface for the moment M, is represented by the contour map in Fig 171b, with the ordinates multiplied by 47 Influence Surface for the Bending Moment M, at the Center of a Simply Supported Square Plate.* It is convenient to use the influence surfaces for the quantities Mio = —D 0*w/dzx? and Myo = —D d*w/dy? with the purpose of obtaining the final result by means of Eqs (101) The influence surface for Mz»9 may be constructed on the base of Fig 76 The influence of the single load P = acting at point is given by the first of the equations (151) and by Eq (152) This latter expression also contains the required singularity of the type given by Eq (206), located at the point The effect of other loads may be calculated by means of the first of the equations (149), the series being rapidly convergent The influence surface is shown in Fig 172 with ordinates multiplied by 8z Let us calculate the bending moment M, for two single loads P; and P; < P; ata fixed distance of 0.25a from each other, each load being distributed uniformly over For details of the so-called singularity method see A Pucher, Ingr.-Arch., vol 12, p 76, 1941 Several influence surfaces for the clamped circular plate are given by M ElHashimy, ‘‘Ansgewadhlte Plattenprobleme,’’ Zurich, 1956 The most extensive set of influence surfaces for rectangular platcs with various edge conditions is due to A Pucher, ‘‘Einflussfelder elastischer Platten,’’ 2d ed., Vienna, 1958 See also his paper in ““Eederhofer-Girkmann-Festschrit,” p 303, Vienna, 1950 For influence surfaces of continuous plates, see G Hoeland, Ingr.-Arch., vol 24, p 124, 1956 330 THEORY OF PLATES oY AND SHELLS © Z O< ' sọ = (a) 0| -01f (b) /8 @\ P=t x “0 y -05[ -10f y -15{ -20 Co x Multiplication factor z' = 0.0796 _ Fig 171 an area 0.la-0.1la Outside those areas the plate may carry a uniformly distributed live load of an intensity g < P./0.01a? The influence surface (Fig 172) holds for M zo, and the distribution of the loading which yields the largest value of Mois given in this figure by fulllines Because of the singularity, the ordinates of the surface are infinitely large at the center of the plate; SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES therefore it is simplest to calculate the effect of the load P; separately, Eqs (163) and (165), in connection with Tables 26 and 27.! For this y = 0, v/u = k = 1, ¢ = 1.5708, y = 0, A = 2.669, and » = O, which and a value of M calculated hereafter As for the effect of the load assumed as proportional to the ordinate 2.30 of the surface at the center 381 by means of case we have yields N = P2, it can be of the loaded ae ì be —_— GH_ oa KD m—E ee ee ees ES — Ï— oy a SS SG —~= — “~~ _ a” a £ _” wa ao ED or ' | ee +“ = — — ——Ể—— | | el an — TỶ — — e=.dmeÐ {sms — S2 come, ~ Multiplication factor = 0.0398 | Fig For uniform lood M, =0,0369 qa* 172 area Introducing only the excesses of both single loads over the respective loads due to g, we have to sum up the following contributions to the value of Afro: Load P: from Eqs (163), (165), with € = ø/2, d = 0.1 V/2a, , M_ ÄMạẹ=—= z9 P;—-0.0lga? Sir = 0.219(P; — 0.01ga?) ( log —————= + 06 O.la 4⁄2 T 2.669 — 1.571 ) ‘The effect of the central load may also be calculated by means of influence lines similar to those used in the next example or by means of Table 20 332 THEORY OF PLATES AND SHELLS Load P2: Ms =— i 8z 2.30(P; — 0.01ga?) = 0.092(P; — 0.01ga3) Uniform load g: from data on Fig 172, Therefore Mo Ma = 0.0369qa? = 0.219P; + 0.092P2 + 0.0338q¢a? Owing to the square shape of the plate and the symmetry of the boundary conditions we are in a position to use the same influence surface to evaluate Myo The location of the load P2 corresponding to the location previously assumed for the surface Mzo is given by dashed lines, and the contribution of the load P: now becomes equal to M v0 = 0.035(P2 — 0.01ga*), while the contributions of P; and ø remain the same as before This yields Myo = 0.219P: + 0.035P2 + 0.0344ga? Now assuming, for example, M, vy = 0.2 we have the final result = Mio + 0.2Myo = 0.263P; + 0.099P; -+ 0.0407qa? Influence Surface for the Moment M, at the Center of Support between Two Interior Square Panels of a Plate Continuous in the Direction x and Simply Supported aty = +b/2 This case is encountered in the design of bridge slabs supported by many floor beams and two main girders Provided the deflection and the torsional rigidity of all supporting beams are negligible, we obtain the influence surface shown! in Fig 173 In the case of a highway bridge each wheel load is distributed uniformly over some rectangular area u by v For loads moving along the center line y = of the slab a set of five influence lines (valid for v/b = 0.05 to 0.40) are plotted in the figure and their largest ordinates are given, which allows us to determine without difficulty the governing position of the loading Both the surface and the lines are plotted with ordinates multiplied by 8r EXAMPLE OF EVALUATION Let us assume a = b = 24 ft in.; furthermore, for the rear tire P, = 16,000 lb, u = 18 in., v = 30 in., and for the front tire Py = 4,000 lb, u=18in.,v =15in The influence of the pavement and the slab thickness on the distribution of the single loads may be included in the values u and v assumed above For the rear tire we have v/b =~ 0.10 and for the front tire v/b ~ 0.05 Assuming the position of the rear tires to be given successively by the abscissas = 0.20a, 0.25a, 0.30a, 0.35a, and 0.40a, the respective position of the front tires is also fixed by the wheel base of 14 ft = 0.583a The evaluation of the influence surface for each particular location of the loading gives a succession of values of the moment plotted in Fig 173 versus the respective values of — by a dashed line The curve proves to have a maximum at about = 0.30a The procedure of evaluation may be shown for this latter position only _ The influence lines marked 0.10 and 0.05, respectively, yield the contribution of both central loads (at y = 0) equal to — (16,000 - 3.24 + 4,000 - 3.82) = —65,100 lb and the influence surface gives the contribution of the remaining six loads as —16,000(1.66 + 2.25 + 0.44) — 4,000(1.59 + 2.25 + 0.41) = For methods of its construction see references given in Art 52 —86,600 Ib „pÐ O/00— =ŸW_ - —_—— —_ a $I8u0đ J0 uị pDOI 020/10A 204 - — = ° ‘| HL BS | fol | | Đ —_ > ° —¬ S nn lŠ A ELT “Ol || „“ se0o0^/0“ ;⁄ i | €0- , ot „/ i „ | | / lls ⁄/ sIR to i == —_ — paiaoddns iđung- ¿0-7 ỷ— = _—~ aa em {| + oom #8 B6£0'0‹^ =—— —— —_— Ot JO} Saul]\ 20120j U011021Iđ111W me aes — = /A 09} GO'O=0 BUAN| JUL §nonu1‡u02~