08. Continuous rectangular plates

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CONTINUOUS RECTANGULAR PLATES

- 52 Simply Supported Continuous Plates Floor slabs used in build- ings, besides being supported by exterior walls, often have intermediate supports in the form of beams and partitions or in the form of columns In the first case we have to deal with proper continuous plates; in the case of columns without intermediate beams we have to deal with flat slabs The floor slab is usually subdivided by its supports into several — S + I + o ! 0 | 0 nà y XIỊ y x21 v Xã ole 1° pt_ a Q, a a a a ee (a) | +‡‡ tỉ} +i‡‡‡tYY “NN ~- VA\ PAŠ (c) Fic 110

panels Only continuous plates with panels of rectangular shape will be

considered in this chapter |

We begin with a case allowing a rigorous solution by methods already

used in the foregoing chapter A rectangular plate of width b and length

a; + a2 + a3, supported along the edges and also along the intermediate

lines ss and ¢t, as shown in Fig 110, forms a simply supported continuous plate over three spans We suppose that the intermediate supports neither yield to the pressure in the transverse direction nor offer any resistance to the rotation of the plate with respect to the axes ss and tt With these assumptions, the bending of each span of the plate can be readily investigated by combining the known solutions for laterally loaded, simply supported rectangular plates with those for rectangular plates bent by moments distributed along the edges

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Let us begin with the symmetrical case in which

a1 = Q@2 = a3 =

and the middle span is uniformly loaded while the side spans are without load (Fig 110b) Considering the middle span as a simply supported rectangular plate and using expression (b) of Art 44 (see page 198), we conclude that the slope of the deflection surface along the edge x2 = a/2 is & dw \ 0x2) za¿=a/2 _ 2q)' (—1)0%-Đ!?2 im ( Ổm _ — ) = 7D » — mà — COS b cosh? Bm tanh Bm (a) m=1,3,5,

where B, = mra/2b Owing to the continuity of the plate, bending moments M, are distributed along the edges x2 = +a/2 From symmetry it is seen that these moments can be represented by the following series: 2 (M2) a—t0/2 = » (—1) "DPE, Cos = 0) m=1,3,5,

The deflections w; produced by these moments can be obtained from Eq (173), and the corresponding slope along the edge xz = a/2 [see

Eq (e), page 198] is

(28 _ b » E (—1)®=0/2

Ore) amaj2 i 2mD „ m

m=1,3,5,

mary Bm

COS > (ton Bum + cosh? 8, «) (c)

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The corresponding slope along the edge x3 = —a/2 is

Owe = " Em J \(m—19)2 () 7 An D » m ( 1)

m= 1,3,5,

——— Bin Bin

cosh — b FY (tanh Bm + coth Bn + cosh? 8, ata) (e) sinh? By The equation for calculating the coefficients E,, 1s

ow -+ đu) J;

OX Lo=a/2 0X2 zz=a/3 Ôz; r3=—a/2

Since this equation holds for any value of y, we obtain for each value of m the following equation: 2qb? 1 Bun _ b Ein Bm —_ x*D m4 (ee Bm tanh Bn) — 22D m <2 (tanh Bm -Ƒ cosh? x) _ b En Bm Bm — 4x D m == (ta nh Bm + coth Bm + cosh? B, sinh? x) J) from which 8qb? | Bm — tanh B, cosh? Bm am 3 tanh Bn cosh? By» + coth Bm cosh? Bn + 38m — Bu coth? Bm - (9) It is seen that E,, decreases rapidly as m increases and approaches the value —2gb?/x?m Having the coefficients H,, calculated from (g), we obtain the values of the bending moments M, along the line tt from

expression (b) The value of this moment at y = 0, that is, at the middle of the width of the plate, is En = (Mz)z;—La/2,y=0 = Lim ( — 1)@œ=1/2 m=1,3,5, Taking, as an example, b = a, we have 8,, = m7/2, and the formula (g) gives 2 By = — "42° 0.1555 By = 0.0092 , = — `: 0.0020 (M,) Zo=ta/2,y=0 — TT 0.038 lqa’ _

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concrete, we get for the first of these moments the values of

(M.,)a = CM,)s = 0.0479 Xx = ga? = 0.0442qa?

(see Table 8, page 120) and for the second moments the values (M,.); = —0.0067qa? and (M,)› = —0.0125qa? Therefore

(M,)2,—0,y=0 = 0.0375qa? (M,);,—o.„-o = 0.0317ga?

If a side span is uniformly loaded, as shown in Fig 110c, the deflection surface is no longer symmetrical with respect to the vertical axis of symmetry of the plate, and the bending moment distributions along the lines ss and tt are not identical Let

(Mz) a=a/2 = » (—1)0-Đ/2F„ cos mary b

| m= 11810, wes | (h)

(Mera), (=1)%-DF cos MA

m=1,3,5,

To calculate the coefficients E,, and F,, we derive two systems of equa-

tions from the conditions of continuity of the deflection surface of the

plate along the lines ss and dt Considering the loaded span and using expressions (a) and (e), we find that the slope of the deflection surface

at the points of the support ss, for a1 = de = a3 = Œ, 1S eo ow — 2qgb° (—1)®=ĐI2 mary ( Bn — (PP) — 4] ) mé cos 6 \cosh? Bn tanh Bn m=1,3,5, — b 4z En (—1)-Đ2 m cosh TP” (tanh Bin may m=1,3,5, Bm Bm |

+ coth đ„ + eosh? đ„ sinh? x) (2) Considering now the middle span as a rectangular plate bent by the moments M, distributed along the lines ss and # and given by the series

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-_ From expressions (2) and (7) we obtain the following system of equations for calculating coefficients „ and „:

An oe + En(Bm + Cn) = —Bn(Em + Fm) — Cn(Em — Fm) (Œ)

where the following notations are used:

—_ Pm_ _ tanh Bn Bn, = — tua + tanh Pn

"` cosh? Bn cosh? By

Bm _ |

Cn = sinh? Ba, coth Bn (2)

The slope of the deflection surface of the middle span at the supporting line it, by using expression (7), 1s

ở _ —Ð _ (—1)@-v2 my

) 7 Ar D » t COS —r— | (Hm + Fn)

m=1,3,5,

Bn fe)

(in + tanh B„) + (Fn — Em) (con Bm — sata.)

This slope must be equal to the slope in the adjacent unloaded span which is obtained from expression (c) by substituting F,, for E, In this way we find the second system of equations which, using notations (1), can be written in the following form: By (Em + F'n) + Cn (Fm — Em) = — (Bn + Cm) Fm (m) From this equation we obtain Cm — Bn Fn — Em 2(B„ + Cm) (n) Substituting in Eq (k), we find Sạa? -9(B„ + Œ„)

Em = Am Timi (Cy — Bu)? — 4(Bm + Cn)? (0)

Substituting in each particular case for An, Bn», and C,, their numerical values, obtained from Eqs (J), we find the coefficients En and F,,; and then, from expressions (h), we obtain the bending moments along the lines ss and # Take, as an example, b = a Then 8, = mr/2, and we find from Eas (0)

A; = —0.6677 B, = —1.1667 Ci — 0.7936 — 0.9987 For m larger than 3 we can take with sufficient accuracy

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Substituting these values in Eq (0), we obtain E, = — “ae 0.1720 ạ = — ae 0.2496 E; = — ae 0.2500 The moment at the middle of the support ss is (Mz)2,-a/2,y-0 = Hy — E3 + Hs — + + + = —0.0424¢a? Yor the middle of the support t# we obtain (Mz)2,—a/2y-0 = Fy — F3 + F ý — - - - = 0.00412gad?

Having the bending moments along the lines of support, the deflections of the plate in each span can readily be obtained by superposing on the deflections produced by the lateral load the deflections due to the

moments at the supports

The bending moments in the panels of the continuous plate can be obtained in a similar manner Calculating, for example, the moments at the center of the middle span and taking » = 0.2, we arrive at the values a Ft ot FHL, | — : E + T¬—x (M,)z,—o„—=o = —0.0039qa? I |-aIet (M,):z„—o,„—o = — 0.0051qa? 1 Oj; ! Địn LƑ +

Xi xin ele The equations obtained for three ! yo S ý 1| Ý spans can readily be generalized and le at ~ phe as 4) 94 expanded for the case of any number of

+t (a) ' spans In this way an equation similar

atti) | addy aa to the threemoment equation of con-

+ (b) aN tinuous beams will be obtained.! Let

Fic 111 us consider two adjacent spans 7 and 4+ 1 of the length a; and a;.1, respectively (Fig 111) The corresponding values of the functions (/) are denoted by At, B‘,, C*, and At+!, Bit}, Ctt!, The bending moments along

the three consecutive lines of support can be represented by the series oo mir Mi-) = (—1)@-YP Et cos ¬ ` m=1,3,5, 3 " mmr Mi = (—1)-D2Et cos ~ m=1,3,5 Mit! = (—1) D2 E+ cos —” m==1,3,5,

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Considering the span 7 + 1 and using expressions (a) and (7), we find oO ow _ 2qi410° (T1)0-192 mry i+] (ar) oy 7 wD m=1,3,5, m* cos pS Am | — (m—1)/2 =D Tv 2 cos MEH (ins, + ESP) BE =1,3,5 — (EP + L)CH) (re) In the same manner, considering the span 7, we obtain Ow — 2@Ù3 (—1)0®=19⁄2 cos my Ai Ox; z.=a J3 w*D m‡ b m m=1,3,5, œ ij ) (—1) —Í (m—1)/2 cos may [1 + kj)Bÿ, m=1,3,5, + dư, — 1n 1)Cm| (g) Irom the conditlon of continuity we conclude that aw —_ {dow |

Ở:P¡+1 +Z:;i=—(đ:¿)/2 OX; z¡=a¡[2

Substituting expressions (p) and (q) in this equation and observing that it must be satisfied for any value of y, we obtain the following equation for calculating 7-1, E7, and Hit:

E(B, Ct) + kh: ? (Bi, + Ct, +- Bị! + C?+1)

oo 7 8b?

+ Bip (Bip — Cy) = — 5 Gnd + GA) (177)

Equations (k) and (m), which we obtained previously, are particular cases of this equation We can write as many liqs (177) as there are inter-

mediate supports, and there is no difficulty in calculating the moments at the intermediate supports if the ends of the plate are simply supported The left-hand side of Eq (177) holds not only for uniform load but also for any type of loading that is symmetrical in each span with respect to the x and y axes The right-hand side of Eq (177), however, has a different value for each type of loading, as in the three-moment equation for beams

The problem of continuous plates carrying single loads can be treated in a similar manner In the particular case of an infinite number of

equal spans with a single load applied at any point of only one span, the

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finite differences for the unknown coefficient Ei, as functions of the index ¿.!

If the intermediate supports are elastic, the magnitude of the coefh- cients E‘, is governed by the five-term equations, similar to the five- moment equations of the theory of continuous beams.” The torsional rigidity of supporting beams, tending to reduce the rotations of the plate along the support, can also be taken into account in considering the

bending of continuous plates.’ |

As the simplest example of a continuous plate carrying a concentrated load, let us consider an infinitely long plate simply supported along the sides x = 0, x = a, continuous over the support y = 0, and submitted to a concentrated load P at some point x = t,y = (Fig 112a) The load and boundary conditions under consideration can be readily satisfied by superposition of cases shown in Fig 112b and c In the case of Fig 112b each panel of the plate is simply supported along the line y = 0, and the elastic surface is given by the expression +w,/2, in which the sign must be chosen according to whether y is greater or less than zero, w; denotes the deflections (a) of

Art 51, and |y| < || In the case shown in

Fig 112c, each panel is clamped along the edge y = 0, and the corresponding deflections are w/2, w being given by expression

(h) in Art 51 We have therefore Ị TL nh | | m , I Ị ev + rol-o w=wt > forn 2 y > 0 Ya) (b) —(c)

FiG 112 to|§ for <0

and the moments along the edge y = 0 become equal to one-half of the clamping moments of a semi-infinite plate with one edge built in, these latter moments being given by expression (7) of Art 51

53 Approximate Design of Continuous Plates with Equal Spans.* The layout of a floor slab usually involves continuity not only in one direction, as assumed in Art 52, but rather in two perpendicular direc- tions A continuous slab of this kind is shown in Fig 113 The spans and the thickness of the plate are equal for all rectangular panels Each

1See S Woinowsky-Krieger, Ingr.-Arch., vol 9, p 396, 1938

2 Continuous plates on elastic beams were considered by V P Jensen, Univ Illinois Bull 81, 1938, and by N M Newmark, Univ Illinois Bull 84, 1938

3 See K Girkmann, ‘‘Flachentragwerke,”’ 4th ed., p 274, Vienna, 1956

4The method given below is substantially due to H Marcus; see his book ‘“‘ Die

vereinfachte Berechnung biegsamer Platten,’ Berlin, 1929 The coefficients of Tables

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panel may carry a dead load qo and, r S

possibly, a live load ø, both distri- (4) _ (2

buted uniformly over the area of : [7 † a r7

the panel, the largest intensity of hoy x

the load being g = qo + p

Let us begin with the computation y 4 of bending moments at the inter- R mediate supports of the floor plate ` ©) L Caleulations show that these mo- " @ | LỘ @

ments depend principally on the loading of the two adjacent panels, and the effect of loading panels

farther on is negligible It is justi- X

fiable, therefore, to calculate the I

moments on supports by assuming

the load q uniformly distributed

over the entire floor slab (Fig 114a) Neglecting, at first, the rotations

of the plate along the intermediate supports, each panel in lig 114¢@ will

have the same conditions as a rectangular plate clamped along the inter-

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The maximum bending moments for plates with such boundary conditions have been tabulated (see Tables 51 to 56) Six possible combi- © nations of simply supported and built-in edges of a rectangular plate are shown at the head of these tables The direction of the + and y axes in

each panel of the slab (Fig 113) must be chosen in accordance with I'igs

116 to 121; span a must be measured in the direction of the x axis and span b in the direction of the y axis of the respective panel The six cases shown in Figs 116 to 121 may be numbered 1 to 6, and the corresponding indices are attached to the coefficients of Tables 51 to 56

To illustrate the application of the tables, let us calculate the bending moment at the middle of the support tw (Fig 118) We calculate for this purpose the clamping moment of both panels adjacent to the support For panel 2 we have to use the formula

M>, = Sol? (a)

and Table 52, / being the smaller of spans a and b of the panel In a similar manner we obtain the clamping moment of panel 6 from the expression | M 6x — ysgi? (b) by making use of Table 56 The moment in question now is given with sufficient accuracy by - _ Mwy = 3CM:„ + Max) (¢) and the moments on other intermediate supports are obtainable in a similar manner

It should be noted that Eq (c) expresses nothing else than a moment-

distribution procedure in its simplest form, 7.e., a procedure in which the ‘“carried-over’’ moments from other supports, as well as any difference

in the stiffness values of both adjacent panels, are neglected Such a

simplified procedure is far more justified in the case of a continuous plate than in the case of a continuous beam

Next, let us consider the bending moments at the center of panel 6 (Fig 113) as an example The load distribution most unfavorable for these moments can be obtained by superposition of loads shown in Fig

114b and c

The contribution of the uniformly distributed load qo + p/2 to the values of the moments is obtained by use of Table 56, which gives

i = a (a + ?) P — Mạ, = 6 (a + 5) 1? (d)

1 denoting the smaller of both spans of panel 6

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those of a simply supported plate, and the moments at the center are

readily computed by means of Table 51 for case 1 The load +p/2

acting in panel 6 yields : Mẹ; = SP Mẹ = “PP —— ) and the largest moments at the center of panel 6 are Me = Mị, + MU (Ð $ / vt fk

In order to calculate the largest nega- | of | ole tive moments at the same point we Loot 0.010 La * have only to alter the sign of the load c2 2 Bear ara

in Fig 114c Still using results (đ) and (b) TT] rT] LTTT Tp

(e), we then have Tete yp try Path 40

Me = Mio— MY (ø)

, — , _—_— age mT: !

Mey, = AIG, Mg, © Peery ta,

As a second example of the application of

the approximate method, let us compute TTTT

the bending moments of the continuous | | -‡P ~ fy plate shown in Fig 115, which was treated dì EE ee +9 rigorously in Art 52

First we choose the direction of the z and

y axes in accordance with Figs 117 and 118 (e) Hit itt +? + it} 2P Assuming next a load gq = qo + p uniformly “ À distributed over the entire surface of the

plate (Fig 115b) and using the coefficients | i tp given in Tables 52 and 53 for cases 2 and 3, (f) CTTLT } tit if TT mi with b/a = 1, we obtain at the center of the | r 2P

support ss the moment Fie 115

0.0840 + 0.0697

M ss —= — 2 (go + p)a? = — 0.0769 (go + p)a? (h)

the procedure being the same as in the foregoing example [Eq (c)]._ Using the rigorous solution, the numerically largest moment at ss is produced by the load distribution shown in Fig 115c Superposing the bending moment obtained on page 231 upon those calculated on page 234, the exact minimum value of the moment M,, proves to be

Mss = —[0.0381(qo + p) + 0.0424(qo + p) — 0.0042q]a?

Or OM ss — (0.0805q5 + 0.0763p)a? (2)

Putting, for instanee, go = g/3,p = 24/3, the result (2) yields —0.0777qa? as compared with the value —0.0769qa? obtained by the approximate method

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[Ƒ—~ 0 -> œ,B I t i b l | Y_ y Fe 116

TABLE ðl ĐENDING MOMENTS FOR UNIFORMLY LOADED PLATES IN CASP 1 y = 0.2, 1 = the smaller of spans a and b Center of plate b/a Factor M, = agl?| M, = Bgl? a By 0 0.0250* 0.1250 0.5 0.0367 0.0999 0.6 0.0406 0.0868 0.7 0.0436 0.0742 qb? 0.8 0.0446 0.0627 0.9 0.0449 0.0526 1.0 0.0442 0.0442 ———— 1.1 0.0517 0.0449 1.2 0.0592 0.0449 1.3 0.0660 0.0444 1.4 0.0723 0.0439 1.5 0.0784 0.0426 ga? 1.6 0.0836 0.0414 1.7 0.0885 0.0402 1.8 0.0927 0.0391 1.9 0.0966 0.0378 2.0 0.0999 0.0367 œ 0.1250 0.0250†

* Mmax = 0.0364gÒ? at 0.48b from the short edge † Muax = 0.0364ga? at 0.48ø from the short edge

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———O ——- ó°Š°ˆŠBẾBÊB⁄⁄⁄4 ă ồ b| % x 1 Y_ y Fic 117

TABLE 52 BENDING MoMENTS FOR UNIFORMLY LOADED PLATES IN CASE Z

y = 0.2;1 = the smaller of spans a and b Middle of Center of plate fixed edge b/a Factor M, = agl?| M, = Bogl? | M, = d2q/? ae B» 62 0 0.0125 0.0625 —0.1250 0.5 0.0177 0.0595 —0.1210 0.6 0.0214 0.0562 —0.1156 0.7 0.0249 0.0514 —0.1086 qb? 0.8 0.0272 0.0465 —0.1009 0.9 0.0294 0.0415 —0.0922 1.0 0.0307 0.0367 —0.0840 1.1 0.0378 0.0391 —0.0916 1.2 0.0451 0.0404 —0.0983 1.3 0.0525 0.0415 —0.1040 1.4 0.0594 0.0418 —0.1084 1.5 0.0661 0.0418 —0.1121 ga? 1.6 0.0722 0.0414 —0.1148 1.7 0.0780 0.0408 —0.1172 1.8 0.0831 0.0399 —0.1189 1.9 0.0879 0.0390 —0.1204 2.0 0.0921 0.0382 —0.1216 œ 0.1250 0.0250* —0.1250

* Max = 0.0387qa2 at 0.80a from the built-in edge

It is of interest to verify the foregoing approximate values by use of the results obtained on pages 232 and 234 Distributing the load again as shown in Fig 115d and interchanging the indices z and y in the results mentioned above, we have

M, = 0.0317(q¢o + p)a? — (0.0051 + 0.0051)qoa? = (0.0215q) + 0.0317p)a?

M, = 0.0375(qo + p)a? — (0.0039 + 0.0039)qoa?

= (0.0297qo + 0.0375p)a?

Setting again qo = q/3 and p = 2q/3, we obtain for the moments the exact values of 0.0283qa? and 0.0349qa2, respectively Eqs (7) yield for the same moments the approximate values of 0.0291qa? and 0.0358qa’

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~—~=—=08——- V72 0, 8 — 1/⁄///////// y Fic 118

TAnILE 53 BENDING Moments FOR UNIFORMLY LOADED PLATES IN CASE 3 y = 0.2, 1 = the smaller of spans a and b Center of plate fixed edee b/a †- Factor M, = azql? | M, = Bsql?| M, = 63ql? a3 8: 63 7 0 0.0083* 0.0417 —0.0833 0.5 0.0100 0.0418 —0.0842 0.6 0.0121 0.0410 —0.0834 0.7 0.0152 0.0393 —0.0814 qb? 0.8 0.0173 0.0371 —0.0783 0.9 0.0196 0.0344 —0.0743 1.0 0.0216 0.0316 —0.0697 1.1 0.0276 0.0349 —0.0787 1.2 0.0344 0.0372 —0.0868 1.3 0.0414 0.0391 —0.0938 1.4 0.0482 0.0405 —0.0998 1.5 0.0554 0.0411 —0.1049 | ga? 1.6 0.0620 0.0413 —0.1090 1.7 0.0683 0.0412 —0.1122 1.8 0.0741 0.0408 —0.1152 1.9 0.0795 0.0401 —0.1174 2.0 0.0846 0.0394 —0.1191 00 0.1250 0.02507 —0.1250

* Max = 0.0174qb2 at 0.300 from the supported edge Tt Mmax = 0.0387qa? at 0.80a from the built-in edge

The largest error of the approximate method ensues from the fact that the largest positive moments do not always occur at the center of the panel This is especially far from being true in the case of distinctly oblong rectangular panels If b, for example, is much larger than a, the largest moment M, occurs near the short side of the rectangular plate Some values of these largest moments are given in footnotes to the tables, and they should be considered as the least possible values of the corresponding columns, regardless of the actual ratio b/a

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-Q - VLLLLLM LLL LL ồ —K R w —Ý——=—Œ———~- ¬N ‘ 4 Fic 119

TABLE 54 Benning MomENTS FOR UNIFORMLY LOADED PLATES IN CasE 4* y = 0.2, 1 = the smaller of spans a and b Center of plate Middle of fixed edge „ cọ "ã b/a Factor M, = œaq!2 M, = Bgl? M, = +4qÌ? My = 54ql? M max = €,ql? œ4 8a “Y4 Ồa E4 0.5 0.0191 0.0574 —0.0787 —0.1180 0.0662 0.6 0.0228 0.0522 —0.0781 —0.1093 0.0570 0.7 0.0257 0.0460 —0.0767 —0.0991 0.0501 qb? 0.8 0.0275 0.0396 —0.0746 —0.0882 0.0430 0.9 0.0282 0.0336 —0.0715 —0.0775 0.0363 1.0 0.0281 0.0281 —0.0678 —0.0678 0.0305 ————— 1.1 0.0330 0.0283 —0.0766 —0.0709 0.0358 1.2 0.0376 0.0279 —0.0845 —0.0736 0.0407 1.3 0.0416 0.0270 —0.0915 —0.0754 0.0452 1.4 0.0451 0.0260 —0.0975 —0.0765 0.0491 1.5 0.0481 0.0248 —0.1028 —0.0772 0.0524 ga? 1.6 0.0507 -0.0236 —0.1068 —0.0778 0.0553 1.7 0.0529 0.0224 —0.1104 —0.0782 0.0586 1.8 0.0546 0.0213 —0.1134 —0.0785 0.0608 1.9 0.0561 0.0202 —0.1159 —0.0786 0.0636: 2.0 0.0574 0.0191 —0.1180 —0.0787 0.0662

* The authors are indebted to the National Research Council of Canada for a grant which greatly facilitated the computation of the table

is the largest bending moment at the center of the plate Table 54 shows, however, that the difference between Max and the largest of the values of M, and A/, does not exceed 10 per cent of the latter values and that the general procedure described on page 238 is justified in case 4 as well |

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-—-—(Q - AAXXX ce SOS TK Mss W Y To œ; 8 YÁ s\—V WX ARAN y Fie 120

‘ABLE 55 BENDING Moments FoR UNIFORMLY LOADED PLATES IN CasE 5* y = 0.2, 1 = smaller of spans a and b Center of plate Middle of fixed edge b/a Factor M, = asql? M, = B:ql? M 2ysql? M, = ồ;gˆ œ§ 8s Y6 65 0.5 0.0206 0.0554 —0.0783 —0.114 0.6 0.0245 0.0481 —0.0773 —0.102 0.7 0.0268 0.0409 —0.0749 —0.0907 b3 0.8 0.0277 0.0335 —0.0708 —0.0778 | ! 0.9 0.0274 0.0271 —0.0657 —0.0658 1.0 0.0261 0.0218 —0.0600 —0.0547 1.1 0.0294 0.0204 —0.0659 —0.0566 1.2 0.0323 0.0192 —0.0705 —0.0573 1.3 0.0346 0.0179 —0.0743 —0.0574 1.4 0.0364 0.0166 —0.0770 —0.0576 1.5 0.0378 0.0154 —0.0788 —0.0569 1.6 0.0390 0.0143 —0.0803 —0.0568 qo" 1.7 0.0398 0.0133 —0.0815 —0.0567 1.8 0.0405 _0.0125 —0.0825 —0.0567 1.9 0.0410 0.0118 —0.0831 —0.0566 2.0 0.0414 0.0110 —0.0833 —0.0566 so 0.0417 0.0083 —0.0833 —0.0566

* The data of this table are due substantially to F Czern

p 33, W Ernst & Sohn, Berlin, 1955 y, Bautech.-Arch., vol 11, The method given in this article is still applicable if the spans, the flexural rigidities, or the intensity of the load differs only slightly from panel to panel of the continuous plate Otherwise more exact methods should be used |

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restrain-~ Q -= LLLLLLLLL AL, ề “,B ¥ >< mè TK SN SYS —~ SN N ⁄⁄⁄⁄⁄⁄//// y Fie 121

TABLE 56 BENDING MoMENTS FOR UNIFORMLY LOADED PLATES IN CasE 6 y = 0.2, 1 = the smaller of spans a and b Center of plate Middle of fixed edge b/a _ _ Factor M, = agl?| My, = Beql? | Mz = yegl? | My = ồsgP arg Be +6 ò6 _ 0 0.0083 0.0417 —0.0571 —0.0833 0.5 0.0118 0.0408 —0.0571 —0.0829 0.6 0.0150 0.0381 —0.0571 —0.0793 0.7 0.0178 0.0344 —0.0569 —0.0736 qb? 0.8 0.0198 0.0299 —0.0559 —0.0664 0.9 0.0209 0.0252 —0.0540 —0.0588 1.0 0.0213 0.0213 —0.0513 —0.0513 |_——— 1.1 0.0248 0.0210 —0.0581 —0.0538 1.2 0.0284 0.0203 — 0.0639 —0.0554 1.3 0.0313 0.0193 — 0.0687 —0.0563 1.4 0.0337 0.0181 —0.0726 —0.0568 1.5 0.0358 0.0169 —0.0757 —0.0570 qa? 1.6 0.0372 0.0157 —0.0780 —0.0571 1.7 0.0385 0.0146 —0.0799 —0.0571 1.8 0.0395 0.0136 —0.0812 —0.0571 1.9 0.0402 0.0126 —0.0822 —0.0571 2.0 0.0408 0.0118 —0.0829 —0.0571 2o 0.0417 0.0083 —0.0833 —0.0571

ing effect of the surrounding walls, the anisotropy of the plate itself, and the inaccuracy in estimating the value of such constants as the Poisson ratio »v

However, we can simplify the procedure of calculation by restricting the Fourier series, representing a bending moment in the plate, to its initial term or by replacing the actual values of moments or slopes along some support of the plate by their average values or, finally, by use of a moment distribution procedure.!

54 Bending of Plates Supported by Rows of Equidistant Columns— (Flat Slabs) If the dimensions of the plate are large in comparison with

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the distances a and b between the columns (Fig 122) and the lateral load is uniformly distributed, it can be concluded that the bending in all panels, which are not close to the boundary of the plate, may be assumed to be identical, so that we can limit the problem to the bending of one panel only Taking the coordinate axes parallel to the rows of columns and the origin at the center of a panel, we may consider this panel as a uniformly loaded rectangular plate with sides a and b From symmetry we conclude that the deflection surface of the plate is as shown by the | dashed lines in Fig 122b The maximum deflection is at the center of the plate, and the deflection at the corners is zero To simplify the problem we assume that the cross-sectional dimensions of the columns

are small and can be neglected in so far as deflection and moments at — i ! t Í a ' { vy (a) Yh Fic 122

the center of the plate are concerned.!_ We then have a uniformly loaded rectangular plate supported at the corners, and we conclude from symmetry that the slope of the deflection surface in the direction of the normal to the boundary and the shearing force are zero at all points along the edges of the plate except at the corners.?

Proceeding as in the case of a simply supported plate (Art 30), we take the total deflection w in the form

W = Wi + We (a)

a qb4 _ 4y? 2

where Wi = số (1 | (0)

‘In this simplified form the problem was discussed by several authors; see, for example, A Nddai, Uber die Biegung durchlaufender Platten, Z angew Math Mech., vol 2, p 1, 1922, and B G Galerkin, “‘Collected Papers,” vol 2, p 29, Mos- cow, 1953

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represents the deflection of a uniformly loaded strip clamped at the ends

y = +6/2 and satisfies the differential equation (103) of the plate as well as the boundary conditions

OW) _ _ ở O?0 2U) ¬

(ae) = 0 (z)z-xe2 = —D (se + dy? i = 0 ©)

‘The deflection w is taken in the form of the series

co

= Ao+ Ym COS Ta (đ)

m= 2,4,6,

each term of which satisfies the conditions (c) The functions Y,, must

he chosen so as to satisfy the homogeneous equation

AAw, = 0 (e)

und so as tc make w satisfy the boundary conditions at the edges

y = +b/2 Equation (e) and the conditions of symmetry are satisfied hy taking series (d) in the form

= Ae CÀ) (Ancosh MY + Be ME nn cos ứ m= 2,4,6,

where the constants Ao, Am, and B, are to be determined from the

boundary conditions along the edge y = 6/2 Irom the condition concerning the slope, v7z., that Ow OW, OW? — — { _ ˆ _ = () (= i ( oy + oy ) we readily find that tanh am Bm = —Am Qm + tanh ap» (9) in which, as before, mab om = 2a Ú)

Considering now the boundary condition concerning the shearing force,

we see that on a normal section nn (Jig 122b) of the plate infinitely close to the boundary y = 6/2, the shearing force Q, 1s equal to zero at :Ul points except those which are close to the column, and at these points (J), must be infinitely large in order to transmit the finite load 3gab to the

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symmetry, has the form QM = Cot » Cm cos —=* (2) m= 2,4,6, and observing that Vv, = 90 forO<2<5-c a/2 and I Q,dxz = — gab a/2—c 4 we find, by applying the usual method of calculation, that _ gu _ _P (o= — ng 4 [2/2 marx P | = — —— — — — Í — m{Z and Cm a lo Q, Cos ¬ dx 5 (—1)

where P = gab is the total load on one panel of the plate Substituting these values of the coefficients Cy and C,, in series (7), the required boundary condition takes the following form: ew 03 (9/)„3/+ = —D lạm T dx? on _1)ml2 ¢ mre P (—1) os a 2a si ?n = 2,4,6, ,

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The deflection of the plate takes the form — a Marx Thế X eÐ (— 1)”2 cos w= 384D or ( - #) + Aot Xp b? Pa — MỸ sinh a, tanh an = m = 2,4,6, — (œm -+ tanh œ„) cosh THỊ (Ù | tanh Om 4 gs sin nh” a

The constant A» can now be determined from the condition that the

deflection vanishes at the corners of the plate Hence (W)z-a/2,y—v/2 = O ©© _ ga'b 1 _ r3 ]) mề Om m= 2,4,6, and Ao = Qn + tanh se) (m) tanh? a,

The deflection at any point of the plate can be calculated by using expressions (1) and (m) The maximum deflection is s evidently at the center of the plate, at which point we have co (w) z=0,=0 _ _gbt _ ga%d 394 9sS4D - 25 (—1)"? am + tanh am mŠ sinh am tanh am m= 2,4,6, _ ga*b i —— fm + tanh Am 23 ]) mm? Om tanh? an (n) m= 2,4,6,

Values of this deflection calculated for several values of the ratio b/ø are given in Table 57 Values of the bending moments (M,)z—o„„-o and (M,)z-0,-0 calculated by using formulas (101) and expression (/) for deflection are also given It is seen that for b > a the maximum bend-

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ing moment at the center of the plate does not differ much from the moment at the middle of a uniformly loaded strip of length b clamped at the ends

Concentrated reactions are acting at the points of support of the plate, and the moments calculated from expression (/) become infinitely large We can, however, assume the reactive forces to be distributed uniformly over the area of a circle representing the cross section of the column The bending moments arising at the center of the supporting area remain finite in such a case and can be calculated by a pro- | cedure similar to that used in the case of rectangular plates and described on page 147 With reference to Fig 122, the result can be expressed by the formulas! 0 b2? 1 (M x) c—a/2,y—b/2 = Mo — La _ + (1 _— v) re 4143 mab sinh? = a | n=] (0) b2 1 a (My)enoiny-vrs = My +2 | — 5 _ + (1 — z) nrb +(1—»)— arb sinh? —— Le n=] a In these expressions M = -] +») ON Ae 7 10 Š 2„c(1 — q21 — q93 - - - +1

g = e */le, and c denotes the radius of the circle, supposed to be small compared with spans a and b of the panel Carrying out the required calculations, we can reduce Eqs (0) to the form b ¬ (ÄấMÃz)z~—4/2,y—b/2 = _—= ak + v) log - _ (œ + 8y) - (p) (My) c-o/2,ymb/2 = Tan la+ + p) log - — (8 + av) T7

in which œ and Ø are coeffioients given for several values of the ratio b /ain Table 58 _ TABLE 58 VALUES OF COEFFICIENTS a AND Ø IN lQs (p) FOR MOMENTS ON SUPPORT b/a 1 1.1 1.2 1.3 1.4 1.5 2.0 a 0.811 0.822 0.829 0.833 0.835 0.836 0.838 B 0.811 0.698 0.588 0.481 0.374 0.268 —0.256

The bending moments corresponding to the centers of columns of rectangular cross section also can be calculated by assuming that the reactions are uniformly distributed over the rectangles, shown shaded in Fig

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TABLE 59 BENDING MoMENTS AND LARGEST SHEAR FoRCE OF A SQUARE PANEL OF A UNIFORMLY LOADED PLATE (Fig 123) y = 0.2 uja =k (M)z~y-a/+2 = Bga? (M)z-v~o = Biga?|(Mz)z-a/2,y-0 = B2qga? (My)z-0/2,y-0 = Baga? Qinax = ygg j 8 Bi B2 B3 * i 0 — 0 0.0331 —0.0185 0.0512 eo 0.1 —0.196 0.0329 —0.0182 0.0508 2.73 0.2 —0.131 0.0321 —0.0178 0.0489 0.3 —0.0933 0.0308 —0.0170 0.0458 0.842 0.4 —0.0678 0.0289 —0.0158 0.0415 0.5 — 0.0487 0.0265 —0.0140 0.0361 0.419

123, that represent the cross sections of the columns.! In the case of square panels and square columns we have u/a = v/b = k, and the

moments at the centers of the columns and at the centers of the panels

are given by the following formulas: _ (+ »)ga? |S — k)(2 — k) 4 12 (M xz)Ìz=v=al3 = (M y)2=y=a/2 = 1 2 + Tư mr(2 —k) +- ape » mã sinh ma sinh _ cosh 5 sin met | (q) m=1 (M z)z—v=0 — (M )z—=v=0 _ G+ »)qa? 1am 1 » (—1m+¡ Sinh mak sỉn mk | () 4 12 3k? n3 sinh rrr

The values of these moments, to-

gether with values of moments at half a distance between columns, ob-

tained from the same solution and calculated for various values of k and

for vy = 0.2, are given in Table 59

It is seen that the moments at the

columns are much larger than the

moments at the panel center and that

their magnitude depends very much

on the cross-sectional dimensions of the columns The moments at the

panel center remain practically con-

stant for ratios up tok = 0.2 Hence the previous solution, obtained on Fig 123

1 This case was investigated by S Woinowsky-Krieger; see Z angew Math Mech., vol 14, p 13, 1934 See also the papers by V Lewe, Bauingenieur, vol 1, p 631,

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the assumption that the reactions are concentrated at the panel corners, is sufficiently accurate for the central portion of the panel

An approximate calculation of moments given by Eq (q) in the form of a series can also be made by means of expressions (p) Using for this purpose Eq (c), Art 37, we substitute u 2 e*i4~1 = 0.57 ®

t.e., the radius of a circle equivalent to the given square area % by œ, in Eqs (yp) In the particular case of square panels numerical results obtained in this manner are but slightly different from those given in the second column of Table 59

The shearing forces have their maximum value at the middle of the sides of the columns, at points m in Fig 123 This value, for the case of Square panels, depends on the value of the ratio k and can be represented

by the formula Q = yqa? Several numerical values of the factor y are |

given in Table 59 It is interesting to note that there is a difference of only about 10 per cent between these values and the average values obtained by dividing the total column load ga?(1 — k?) by the perimeter joan = 0 = 4ka of the cross section of the column

77777 Uniform loading of the entire plate

gives the most unfavorable condition at the columns To get the maximum bending moment at the center of a panel, the load must be distributed as shown by the shaded areas in Fig 124a The solution for this case is readily obtained by combining

bg the uniform load distribution of in-

2 tensity ¢g/2 shown in Fig 124 with the load q/2 alternating in sign in tq consecutive spans shown in Fig 124c | | | -_ ` ‡iiii1tii111111111lđ/] (b)

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From Table 59 we conclude, furthermore, that

(Mz)z—o,—s;› = $q° 0.05120? + ~qa? = 0.0881 qa?

The foregoing results are obtained in assuming that the plate is free to rotate at the points of support Usually the columns are in rigid connection with the plate, and, in the case of the load distribution shown in Fig 124, they produce not only vertical reactions but also couples with a restraining effect of those couples on the bending of the panels A frame analysis extended on the flat slab and the columns as a joint structure

therefore becomes necessary in order to obtain

more accurate values of bending moments under ~ Jf, Z

alternate load.! ( KEY 4 ⁄44⁄⁄

The case in which one panel is uniformly PLL > 2 2⁄2 |

loaded while the four adjacent panels are not — V4 ⁄ loaded is obtained by superposing on a uniform “iy LLLL ZZ, load q/2 the load q/2, the sign of which alter- Ay — OS, nates as shown in Fig 125 In this latter case j

each panel is in the same condition as a simply Fic 125 supported plate, and all necessary information

regarding bending can be taken from Table 8 Taking the case of a square panel, we find for the center of a panel that 4 4 -+24.0.00406 % = 0.00494 2% D 2 D D Ly) 2-y—0 = 5 g: 0.0331a2 + 5 g: 0.04/9- Le a? = 0.0387qa? (U)z—y=o = 5 0.00581 — Œ 1.3 (Mz)z—„—=o =

The case of an infinitely large slab subjected to equal concentrated loads centrally applied in all panels can be handled substantially in the

same manner as in the preceding case, 7.e., by using the double periodicity in the deflections of the plate.’

The problem of bending of a uniformly loaded flat slab with skew panels has also been discussed ?

55 Flat Slab Having Nine Panels and Slab with Two Edges Free So far, an infinite extension of the slab has always been assumed Now

let us consider a plate simply supported by exterior walls, forming the square boundary of the plate, together with four intermediate columns

(Fig 126) From symmetry we conclude that a uniformly distributed

1 The procedure to be used is discussed in several publications; see, for instance, H Marcus, ‘‘Die Theorie elastischer Gewebe,” p 310, Berlin, 1932

2 This problem was discussed by V Lewe in his book ‘‘Pilzdecken und andere tragerlose Eisenbetonplatten,’”’ Berlin, 1926, and also by P Pozzati, Riv math Univ ~ Parma, vol 2, p 123, 1951

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mofo ~~~ 0 ote 0 —~ 0 — Fig 126

load of intensity q produces equa! column reactions R, which we may consider as redundant in the given statically indeterminate structure Removing all columns, we obtain a simply supported square plate carrying merely the given load g The deflections wo produced by this load at the center of the columns can easily be calculated by means of the theory given in Chap 5 Next, removing the load g and distributing a load R = 1 (acting downward) uniformly over each area u by u, we obtain some new deflections w; at the ˆ same points x = +a/2, y = +a/2 as before From the condition that in the actual case these points do not de- flect, we conclude that wo — Rw, = 0, which yields R = wo/w; Now it remains only to combine the effect of the uniform load qg with the effect of four known reactions on the bending moments of the square plate of

the size 3a by 3a

In the case of a partial loading, such as shown in Fig 126) and c, we have to superpose one-half of the

moments previously obtained on the

moments of a simply supported plate with the area a by 3a, carrying a uniformly distributed load +q/2 Calculations of this kind carried out by Marcus! led to the values of bending moments given in Table 60 The reaction of acolumn is R = 1.196qa? in this case The bending of an infinite plate which is supported not only along both its parallel sides

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TAbLB 6Ú CorrFricIENTS B FOR CALCULATION OF BENDING MoMENTS Äƒ = 8qa? oF A SIMPLY SUPPORTED SQUARE PLATE WITH FourR INTERMEDIATE CoLuMns (Fig 126) u/a = 0.25, v = 0.2 y Load a Load b Load c Point _ — — z M, M, M, M, M, M, 1 0 0 0.021 0.021 | —0.048 | —0.004 0.069 0.025 2 0510 —0.040 0.038 | —0.020 0.019 | —0.020 0.019 a 1.010 0.069 0.025 0.095 0.027 | —0.024 | —0.002 4 0 0.5 0.038 | —0.040 | —0.036 | —0.036 0.074 | —0.004 5 0.5 |0.5 | —0.140 | —0.140 | —0.070 | —0.070 | —0.070 | —0.070 6 1.0 | 0.5 0.074 | —0.004 0.092 0.014 | —0.018 | —0.018 7 0 1.0 0.025 0.069 | —0.028 0.017 0.052 0.052 8 0.5 | 1.0 | —0.004 0.074 | —0.002 0.037 | —0.002 0.037 9 1.0 | 1.0 0.053 0.053 0.066 0.044 ; —0.013 0.009 _Ă~ but also by one or several rows of equidistant columns! can be discussed in a similar manner

The case of bending of a long rectangular plate supported only by the

two parallel rows of equidistant columns (Fig 127) can also be solved without any difficulty for several types of loading We begin with the case

in which the plate is bent by the moments M, represented by the series

0

MTX

(My)y=40/2 = Mo + » Em COS — — (a)

m= 2,4,6,

Since there is no lateral load, the deflection surface of the plate can be

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and from the condition that the deflection vanishes at the columns

Substituting series (b) in Eqs (c), we find that

Mo Ai=- 5D

A = — a? Em (1 + v) sinh an — (1 — v)am cosh am (đ) m mm D (3 + v)(1 — v) sinh am cosh am — Om(1 — v)?

BR = a? Hm | sinh am

_ rmD (3+ y) sinh œ„ cosh œ„ — œ„(Í — v)

Combining this solution with solution (), Art 54, we can 1nvestigate the bending of the plate shown in Fig 127a under the action of a uniformly a a ap ° 0 ° + „— ⁄ /* ob y £ sa R ^ ^” "mm: | 0 } t y⁄ fF” VN NMMMMA Fig 127

distributed load For this purpose we calculate the bending moments M, from expression (/) by using formula (101) and obtain

b2

(My) yatb/2 —= 1

_ gab (—1)”?| 1+ — Wm(Ï — v) max

| 2m m= 2,4,6, » ™m tanh am sinh? a, | °° a (e) Equating this moment to the moment (a) taken with the negative sign, we obtain the values of My and E,, which are to be substituted in Eas (d) for the constants Ai, Am, and B,, in expression (b) Adding expression (b) with these values of the constants to expression (1), Art 54, we obtain the desired solution for the uniformly loaded plate shown in Fig

127a |

Combining this solution with that for a uniformly loaded and simply supported strip of length b which is given by the equation

—~ — 9 (8 _ „z\(ðps _ „a

mẽ tp (a “)(t 4

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56 Effect of a Rigid Connection with Column on Moments of the Flat Slab In discussing the bending of a flat slab it has always been assumed that the column reactions are concentrated at some points or distributed uniformly over some areas corresponding to the cross section of the columns or their capitals Asa rule, however, concrete slabs are rigidly connected with the columns, as shown in Fig 128

In discussing moments at such rigid joints, let us begin with the case of a circular column and let c be the radius of its cross section The calculation of bending x -~ 0.22 0 -> a Tấ

moments using expression (1) in Art 54 shows! that, in the case of a square panel (a = b) and small values of c/a, the bending moments in the radial direction practically vanish along a circle of radius e = 0.22a (Fig 122a) Thus the portion of the plate around the column and inside such a circle is in the state of an annular plate simply supported along the circle r = 0.22a and clamped along the circle r =: ¢, with a transverse displacement of one circle with respect to the other Hence the maximum yo Middle line of panel _y Middle line of ponel ; _ " UL | _ FT.“ - | % = p @ | t © c=O.†q | 5 | O us | \ = é | Glos Blog Z u=0.2q | oles ‘oO ° © c | cS 2 x } — ' Y % — (4 ` Z ‘ Ỷ X <= Ậ a Ậ O r | Ị oO Y 6‹ r | | 7! ' I sO = , OE NI | | = | oley có ^ ty ! |_ - _ | Ý - - _ | # Fic 129 Fic 130

bending stress around the column can be obtained by using formulas (75), previously derived for circular plates (see page 61), and combining cases 3 and 8 in Fig 36

A more elaborate discussion of the same problem is due to F Télke.2 Numerical results obtained by F Télke for a square panel and c/a = 0.1 (Fig 129) are given in Table 61, together with values of bending moments calculated for the same case on the 1 Such calculations were made by A Nddai; see his book ‘‘Elastische Platten,” p 156, Berlin, 1925

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basis of the customary theory It is seen that a rigid connection between slab and column tends to increase numerically the moments on support and to reduce the positive moments of the slab

TApLE 61 CoEFFICIENTS B FOR CALCULATION oF BENDING MomENTS M = ga? oF A UNIFORMLY LOADED SQuaRE PANEL oF A Fuat SLAB y = 0.2 Circular column Square column (Fig 129) (Fig 130)

Bending moment Location Rigid Rigid

connection | Customary | connection | Customary

with theory with theory column column M, = M, | x =a/2,y = a/2 0.0292 0.0323 0.0264 0.0321 M, x =a/2,y =0 0.0399 0.0494 0.0348 0.0487 M, =a/2,y =0 —0.0161 —0.0179 —0.0146 —0.0178 11, = M, + =0,=0 | —0.143 | —0.131 M,z z=u/2,u=0 | | —0.0626 —0.0808 M, + =/2,=uwu/2| | — œ —0.0480 1M r=cC —0.1682 —0.0629

The same table also gives moments for a flat slab rigidly connected with a column of a square cross section! (Fig 130) The infinitely large stresses occurring at the corners of columns in this case are of a highly localized character Practically, they are limited by a cracking of concrete in tension and a local yielding of the steel reinforcement

From this discussion we may conclude that (1) the actual values of bending moments of a flat slab at the columns generally lie between the values given in Table 61 for the rigid connection and those given by the usual theory, and (2) circular columns secure a more uniform distribution of clamping moments than columns with a square-shaped supporting area.?

1 See S Woinowsky-Krieger, J Appl Mechanics, vol 21, p 263, 1954

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