Pca Circular Concrete Tanks

Section 9. Wall with Moment Applied at Top

FIG. 17

Whcn the top of the wall and the roof slab are inade continuous, as indicated in Fig. 17, the deflection of the roof slab tends to rotate the top joint and intro- duces a moment at the top of the wall. In this section, the wall is investigated for a momenr of M = 6,700 ft.lb. per ft., the origin of which is discussed later in this section.

The data in Tables VI and XIIwill be used although they are prepared for moment applied at one end of the wall when the other is free. However, these tables may be used with good degree of accuracy also when the far end is hinged or fixed. For ring tension, multiply coefficients from Table VI by MR/H2 = 6,700 X 27/202 = 450 lb. per ft., and for moments, multiply coefficients from Table XI by M = 6,700 ft.lb. per ft. Select coefficients for w/Dt = 6.

Ring tensiori Mornent

FIG. 18

- Mornent added ,

0 .4 at top----

Ring tension Moment .

FIG. 19

It should be noted that ri,ng tension and moment plotted in Fig. 18 are for moment applied at top when base is free. But the relatively small values near the base in Fig. 18 indicate that the results near the top will be practically the same whether the base is hinged or fixed. The fact that ring tension and wall moment created by the moment applied a t top diminish so rapidly is due to the ring elements which exert a strong dampening effect.

The ring tension and the moments determined in this section are now added to those in Section 6.

The effect of adding a moment of M = 6,700 at the top is shown in Fig. 19. The ring tension is in- creased near the top. This increase may in some in- stances become so large that i t affects the design mate- rially. The moments are, of course, large a t the top and are not likely to be ignored, but the more important increase in ring tension may accidentally be overlooked.

In Section 12 i t is shown that the moment at the fixed edge of a roof slab with center support, R = 27 ft., and a total design load of 650 - 432 = 218 lb. per sq.ft.* equals -7,800 it.lb. per ft. of periphery.

*Wcight of thc roof slab and c a n h covcr minus thc sur- prcssurc o n thc storcd liquid.

Point 1 Ringten.,Sec.9

P A G E 1 0

0.OII 1 0.1H ( 0.2H 1 0.3H 1 0.4H 1 0 . 5 9 1 0.6H 0.7H 1 O.BH ( 0.9H 1 1.OH

O +5 ,100 +5 ,900 + - , 6 0 0 +2 ,900 +1 ,500 + 500 300 - 400 - 500 - R - ~ ~ 1 3 , 8 0 0 1 +28,000 1 +31,500 1 +33,700 1 +3:700 1 *17,200 1 +l6,000 1 O

Total ring ten. ~+11,100 1 +20,300 (+25,300 1 +28,400 (+30,900 1 +33,000 1 +34,200 1 +32,700 1 +26,900 1 +15,600 ( - 500

Mom., Sec. 9 + 6,700 + 3,800 + 1,700 + 400 - 200 - 400 - 400 - 300 - Mom., Sec. 6 O 1 O 1 O + O 500 + 1 ,30 , )+ 2,6001 + 4,2001+ 5 ,%/+ P600 1 Total mom. 1 ( + 6,700 1 + 3,800 1 + 1,700 1 + 500 1 + 300 ( + 900 1 + 2,200 / + 3,900 + 5,100 1 + 4,600 1 O

T h ~ s vaiue i s used f o r dzrc:.nn;r.airoi: o/ ::~:I::!C+:T r r anc - Section i C Wall with Moment kpplied at Base mitted frorn the slab thi-oc,yii thc :n.n: inro :kit- t q u; the wall.

The roced dure is so mucn Iikc moment distribu- tion appl~ed t o contlnuous frames that the explanation may be brief. The data in Tables XVIII and XIX are stiffnesses which denote moments required t o impart a unit rotation at the edge of the wall and the slab. Only relative values of stiffness are required in this ~ D P ~ I C ~ ~ I O ~

1 2

The moirirnr required t o rotate the tangent at the edgc through a given angle is proportional t o the following relati ve stiffness factors.

For wall (Table XVIII for H2/Dr = 6): 0.78jt3/H = 0.783 X 153/20 = 132

For slab (Table XIX for C/D = 0.15): 0.332$/R = 0.332 X 123/27 = 21

The distribution factors are

" 132 For wall:

132 + 21 = 0 3 6

L 1 For slab: = 0.14

132 + 21 ,

The dimensions used for the slab are the saine as in Section 12.

Wal I (a) Fixed end momcnts (b) Final moments

FIG. 20

The moment of -7,800 ft.lb. tends t o rotate the fixed joint as shown in Fig. 20(a). When the artificial restraint is removed, the rotation of the joint will in- duce new moments in wall and slab. The sums of the induced moments and the original fixed end moments are the final moments. They must be equal but oppo- site in direction as indicated in Fig. 20(b). The calcu- lations may be arranged in accordance wi th the usual moment distribution procedure.

Wall 1 Slab

The induced moments equal -7,800 times the distribution factors and are recorded wi th signs oppo- site to that of the fixed end moment (unbalanced rnoment). Note that the wall sriffness is more than six times that of the slab.

Distribution factor 0.86 -- Fixed end moment O Induced moment (distributed moment) + 6,700 --

FIG. 21

0.14 -7,800 + 1,100

l n Sections 4 throueh 9, the wall has been as-

Final moment + 6,701: - 6,700

c.

sumed to rest on a footing nor continuous wi th the bottom slab. The condition t o be investigated in this section is illustrated in Fig. 21. in which the wall is " made continuous wi th a reinfbrced bottom slab de- signed for uplift.

The desien of the slab is discussed in Section 13 V

in which i t is shown that the moment at the fixed edge is -27,100 ft.lb. per ft. N o surpressure on the liquid is considered in computing this moment and, there- fore, i t must also be disregarded in the design of the wall. Accordingly in this section, only triangular load is considered, but if the slab had been designed for

u

surpressure, trapezoidal load should be used for the wall design.

he moment at the base of rhe wall is first com- puted on the assumption that the edge is fixed, and a correction is then made for rotation of the edge. The fixed end moment a t base of wall is determined for the triangular loading in Section 4 wi th coeficients selected from Table VI1 for p / D t = 6 Its value is

Mom. = -0.0187 X wH3 = -0.0187 X 62.5 X 203 = -9,350,say, -9,300ft.lb. perft.

As long as the base is artificially fixed ngainst any rotation, i t is subject t o two moments both of which tend t o rotate the joint in the same direction as shown in Fig. 22(a). One moment is due to the out- ward pressure of the Jiquid, the other due to the upward reaction from the subgrade. The base joint is not in equilibrium and when the artificial restraint is removed, i t will rotate. The rotation induces moments in wall and slab, and the induced moments added to

Wall I

(a) Fixed en3 morna-~t+ (O) Final rnomentz

FIG. 22

P A G E 1 1

~ h c c)r jg~n;~, f i x ~ r . e::( rnoments mus; íi; o. -i;-l. ,:

mapnitude thar the combined moments zre cq:ial hn: of opposite direction as indicated in Fig. 223,) . Caicu- lation of the final momr-irs mAy be arranged in accord- anie with the usual mcrnenr distribution procedure.

U 7 1 I Slab Distribution factor

(same as in Section 9) 0.86 0.14 Fixed end moment - 9,300 - 27,100 Induced moment

(distributed moment] + 31,300 + 5,100 Final moment + 22,000 - 22,000

The induced moments, often denoted as distrib- uted moments, are computed by multiplying the "un- balanced momenr", 9,300 + 27,100 = 36,400, by the distribution factors The fixed end moments are re- corded with the same sign, negative, since they have the same direction. The induced moments both have positive signs. .

Moment ~t base / -

í '/t- Hinged

Actual +22,000

Lease FIG. 23

First, assume the base fixed; and second, apply a mo- ment of 9,300 + 22,000 = 31,300 fr.lb. per ft. of rhe base. Finally, combine the resulrs of the two steps. The triangular loading is the same as in Section 4, and the value of H2/Dt = 6 is the same as brfore. For ring tension, multiply coefficients by WHR '= 33,750 lb. per f t . (triangular), and by MRIH2 = 31,300 X 27/202 = 2,110 lb. per ft. (M a t base).

The rotation of the base and the consequent dis- For moments in a vertical strip, 1 ft. wide, mul- tribution of moment reveal a significant fact. The tiply bv wH3 = 500,000 ft.lb. per ft. (triangular), and change in moment is from -27,100 to -22,000 in the by M = 31,300 ft.lb. per ft. (M at base).

slab but from -9,300 t o +22,000 in the wall. For the Ring tension and moments both for fixcd base wall, the effects of three conditions of restraint a t the and for actual base condition are plotted in Fig. 24. base are shown diagram- matically in Fig. 23. The ac tua l c o n d i t i o n is n o t between fixed and hinged but is far beyond the hinged base a s s u m p t i o n . S ince the distance between the straight line and the deflec- tion curves in Fig . 23 repre-

7 sents the magnitude of ring tension, i t is obviously un- safe to base the design on hinged and especially o n fixed-base assumptions.

The wall will now be Ring tension M o m e n t investigated in two steps. FIG. 24

7 P A G E 1 2

The ni;xirnun! 7:r.g rcnr :~; . :: ;-.4311 ií rhc basc is fixcd; bu; actuaIIy i r 1: ~ ; ~ F T o > : ~ I ~ : c L ~ \ . j5,801) lb.. aí. increase of 117 per cent. ~ o m e n r a t ~ i i e base is changed from -9,300 ft.lb. (tension 1rA insideji to 4-22.00 ft.lb. (tension in outside). l c is clcar that continuity between wall and bottom slab materiaily affects both ring tension and moments. It must be considered in the desirrn.

L.

Shear a t base of wall when the base is fixed may be compured as the sum of the products of coefficients takeii from Tatilr XVI multiplied by zuH2 = 67.5 X 202 = 25.00C lh. prr f r . (triangular), and M / H = 31,300/2¿, = 1,505 11:. per f t . (M at base).

When basc is fixed: 0.197 X wH2 = 0.197 X 25,00C = 3- 4,930 lb.

ERect of M at base: - 4.49 M/H = - 4.49 X 1,565 = - 7,020 lb. - .-

Shear with base released: V = - 2,090 lb.

The tensile stress on the transformed scction in Section 6 (A, = 2.34 sq.in.) is .

Section 1 1 . Roof Slab without Center Support

FIG. 25

The applicatioo of data for design of roof slabs without interior support is illustrated for the tank sketched in Fig. 25 which carrics a superimposed load of 500 lb. per sq.ft. of roof area. The diameter of 54 ft. used in other sections is too large for economical design of this roof slab without center support, so the dimen- sions in Fig. 25 have been substituted. The total design load is p = 500 + 125 = 625 lb. per sq.ft.

For the wall, H2/Dt = 16.02/26.0 X 1.0 = 9.8, say, 10. From Table XVIII, for @/Dt = 10, the relative stiffness of the wall is 1.010P/H=1.010~ 123/16= 109. The relative stiffness of a circular plate without interior support (from Table XIX) is 0.104fl/R = 0.104 X 103113 = 8.0. The relative values computed suffice for the cal- culation of distribution factors which are

109 For wall: --- = 0.93 109 + 8

8 For slab: --- -

109 + 8 - 0.07

When the slab is considered fixed at the edge, the edge moment may be computed by multiplying pR2 by

[!le ~t.:z2'i,~:cri: i:ur) Tah:t XIT L: P<I:,.-I: I .Wl<: -9 1:: :., pi.:: = -I.115 j . f.': > 132 = .-1,.2nO ft.lb. prr f t . U! ll:ri!..h::i- insld: L i . ~ I : - : . C , - 4 L O ~ C L 10: al1 calcu- lations k r r : ,ictuall\-. ;i .::-;;. ..\ i . i !..-crr r ~ l u e shoui? be uscd iur >Gne c j i rhc r.~icr.:::!:.>r,s. ttüi 1-roportionin~ of the slab shoulc' he m:ide a r insidr face.of wall.

Wal l (a)Fixed end moments (b) ~ i n a i nlornents

FIG. 26

The procedure in determining the final moment at the edge has already been illustrated in Sections 9 and 10. The fixed end moments at the edge of the slab in this secrion are shown in Fig. 26(a), and the final moments in Fig. 26(b) are computed below by the ordinary moment distribution procedure.

Distribution factor 0.93 -- "'11 I E Fixed end moment O - 13,200 Induced moment

(distributed moment) + 12,300 Final moment f 12,300 1 - 12,300

It is seen that a large moment is induced in the top of the wall. It has been shown in Section 9 how to determine ring tensions and moments in a wall caused by a moment at top of the wall. The slab only is dis- cussed in this section.

v Unir shear: v = - =

4,060 0.875bd 0.875 X 12 X 8.5

= 45 p.s.i.

The roof slab in Fig. 25 is first assumed to be fixed and a correction is then added for the effect of a moment applied at the edge. For illustration, consider a tank in which the joint at top of wall is discontinuous so the slab may be assumed to be hinged. Thc moments in the hinged slab may be computed by determining moments in a fixed slab, using coefficients in Table XII, and adding to them the moments in a slab in which an edge moment of 0.125pR2 ft.lb. per ft . is app!ied. The most convenient way to do t h is to add 0.125 to al1 the coefficients in Table XII, both for radial and tangential momcnts, and then to multiply the modified coeñicients by pR2. Note that the coefficient for radial moment at the edge becomes zero by the addition of 0.125, and the tangential moment becomes 0.100. These are the values for a slab hinged at the edge.

P A G E 1 3

. . . , - , i:i r i ? , : r : .-*?itm :h: n,::!:,zr : . : ,. . i ;:,. !;I*<cs! í i l ir : . i>~~ ~f r;lrlial bars fo; p~si i i \ . :

slab equals + 993 ft.1 b. pc: t : T Y-,.. . t i .):.c. í . i L... ..i!i:. !. hii-:.ct!: T'oirir- :!.?l.: and 0.4R wlirre rhc moment cocfficients are thosc fc)r fixcci edge i:: ' I a h i c J5; 1; ,~ik. I : ! ~ C . n:~: l i s ciliximuni value. At Point 0.4R, the t o each of which musr be addrd a quAntir!- equal r ; i7?o!;>cnr is 5,50Ci fr.lb. Fe; fr., and rhr length of th t $- 900/pK2 = $- 900/625 X 13? = + O.OOy. Thc ccl- soncentric c ~ r c l r r t ; rouyh 0.4R is 7s ( 0 . 4R) = 21; >: 0.a

efficirnts and moments are as follows, Point O.OR dcnor- X 13 = 32.7 f[ .

ing the centcr and Point 1.OR theedge of slab. Multiply 3?-7M 32.7 X.5.5 = 13.g sq,in At Point 0 .4R: As = --- = -

c o e ~ c i e n t s by fiR2 = 625 X 132 = 105,600 ft.lb. per fr. ad 1 . 4 4 X 9 . 0

. - . .. . . - - - - Use th i r ty - two Gc,,. ,. i O.0Ii / 0.111 1 0.2R ( 0.3R 1 0.4X 0.5R 1 0.6R / 0.7K j 0.81: ( 0.9R ' !.DI( 1 - -

; a - i n . r o u n d . . _ -- . .- - -...p.-

, . . '. ~0.075 1 -0.073 140.067 1 -0.057 ; +0.043 +o.o~!, ' -0.003 1 -0.023 / -0.053 ; -0.08; - 0.1?5: hars(A,= 14.06). Add - 0.003 / Coef.: T ~ ~ I O X I I T i n p

The dash ! ~ d c + 0.009 l i ne in F ig . 27 - . - .- - - -. pp -

Ra.1 n m,. per it. S ~ O W S t h a t the tan^. nom. per ft. Ra,!. niom per seg:

r ad ia l moment per segment con- v e r a e s t o w a r d "

zcro at chr center. Actually, most of the radial bar5 musr bc extended close to or across the center.

Radial mom. per f t of width t t. w

Y O OS 0;2 O;? 0i4 0;s 0:6

FIG. 27

The solid-line curves in Fig. 27 are for moments per ft. of width. The dash line indicates radial moments for a segment that is 1 fr. wide a t the edge. Values on the dash line are obtained by multiplying the radial moment per f t . by the fraction indicating its distance from the center.'For example, multiply 12,300 by 1.0; 8,200 by 0 .9 ; 4,700 by 0.8; and so forth.

The maximum negative moment is 12,300 ft.lb. per f t .

Use 1-in. round bars spaced 9% in. O.C. ( A J = 1.00) i n top of slab and outside of wall a t corner. Total number required is 2nR,/9.5 = 2 n X 13 X 12/9.5 = 103, say, 104 bars.

From Table 4 (Handbook**), for bd = 12 X 8%: F = 0.072, and K = M/F = 12.3/0.072 = 171. K = 236 is allowed for fJ /n/ fc ' = 20,000/10/3,000.

I t is seen from the dash line in Fig. 27 that one- half of the 104 t op bars may be discontinued a t a dis- tance from the inside of the wall equal t o 0.13R + 12 diameters = 0.13 X 13 + 12 X 1.0/12 = 1.69 + 1.00 = 2.69 ft., say, 2 ft. 9 in. The other 52 top bars may be discontinued a t a distance of 0.37R + 12 diameters =

0.37 X 13 + 12 X 1.0/12 = 4 .8 - k 1.0 = 5.8ft . , say, 5 ft. 10 in. from the inside of the wall. All these bars are placed radial.] y.

- 4 Bars

4 b r s - 4 Barc Total : 16-5/44

18'-3" long

Use S'minimum spacing where bar5 cross at center

FIG. 28

Fig. 28 shows one arrangement wi th eight radial bars in each quadrant. Sixteen bars, 18 ft . 3 in. long, are required for the whole slab and are bent as shown, the minimum spacing at center being approximately 3 in. If desired, some of the bars in Fig. 28 rnay be discon- tinued in accordance wi th the steel requirements repre- sented by the dash line in Fi-g. 27. Note that there are only two layers where the bars cross a t center in Fig. 28 and that onlv four types of bent bars are required.

Ring bars are proportioned so as to fit the tangential moment curve in Fig. 27. The radius of the smallest ring bar may be 1 ft. Maximum area is required

M near the center and equals AJ = - = = 0.73 ad 1 . 4 4 X 8 . 5 sq.in. Use ?.$-in. round bars spaced 10 in. O.C.

Ring bar areas decrease graduallv roward Point 0.9R. Inside this point, the bars are al1 'n the bottom, but outside, thep are in the top. Laps may be spliced in accordance wi th code requirements, or the joints may be welded.

-

'1 f r . widc ar cdgc. **Rrinforcrd Conrrrrr Drsign Handbook o[ rhr Amrrjcaa Coricrrtr Itlrrzturr.

P A G E 1 4

Section 12. Rooc 5lab ,*i:b Cenbe: iup?c.d

FIG. 29

l n chis secrion the original rank dimensions pven in Secrions 4 through 10 wiIl be used. The rop slab 1s as sketched in Fig. 29. It is designed for a super- imposed load of 500 lb. per sq.ft. Its thickness is 12 in., and i r has a drop paneI with 6-in. depth and 12-ft. diameter. The capital of the column has a diameter of r = 8 fr. Slab and wall are assumed to be continuous.

Data are presented in Tables XIR, XIV and XV for slabs with center support for the following ratios of capital to wall diameter: C / D = 0.05,0.10,0.15,0.20, and 0.25. The tables are for fixed and hinged edge as well as for a moment applied ar the edgc.

The general procedure in this section is the same as in Secrion 11. First consider the edge fixed and com- pute fixed end moments. Then, distribute moments a t the edge, and finally, make adjustments for the change in edge moment. -

Al1 the table values are based on a uniform slab thickness. Adding the drop panel will have some effect, but it is believed that the'chantre is relativelv small " especially since the ratio of panel area to total slab area is as small as 1 :20.

The relative stiffness factors are 0.86 for the wall and 0.14 for the slab ( ~ c c Scction 9).

The radial fixed end moment equals the coeffi- cient of -0.0490 from Table 'XIII (for C/D = 8/54 = 0.15 a t Point 1.OR) multiplied by pR2. Two values of p will be considered. For the slab, use p = 650, which gives -0.0490 X 650 X 2T2 = -23,200 ft.ib. per f t . Whcn therc is a surpressure on the liquid in the tank of 432 lb. per sq.ft., the combined downward load on the slab is p = ,650 - 432 = 218, and the fixcd cnd mo- ment is -0.0490 X 218 X 272 = -7,800 ft.ib: per ft.

. - , ;ir: ::lri. : :o;.? disrriburioi?. The fiiial edpe irio:!icr,- . .

J' s. : : : - . : V I : , si:ih 1s designed 1s -23,20,3 ', 1 \J.¡ .: = --. ?,:!:0;t ir . i h . per fr.

'Th: yrc)cedure is to design the slab for fixed e d ~ r (-23.200 i ~ . l h . ) . 2 n d thcn add rhe effect of a rnorricn: of 23,200 - 20,000 = 3,200 ft.lb. applied at rhe edgc. but first, shearing stresses are investigared.

The column load is determined by multiplying coeficienrs taken from Table XVII hv pR2. \i7hen edge is fixed :

1.007 pR2 = 1.0C7 X 650 'b. 27: = 478,0001b. Effect of momrnr ar rdse:

9.29 M = 9.29 X 3,200 = 30,000 lb. Total colurnn load = 508,000 lb.

Load on concrete in 30-in. round ried column:

0.225 X 3,000 X 0.8 XAg. = 382,000 lb. Balance: 508,000 - 382,000 = 126,000 lb.

Use ten l-in. square bars.

Radius of critical section for shear around capital is 48 + 18 - 1.5 = 64.5 in. = 5.37 ft. Length of this sec- tion is 2 r X 64.5 = 405 in. Load on area within the section is 650 X r X 5.372 = 59,000 lb. Unit shear equals

v u = - - - - 508,000 - 59,000

= 77 p.s.i. 0.875bd 0.875 X 405 X 16.5

Radius of critical section for shear around drop panel is 72 + 12 - 1.5 = 82.5 in. = 6.88 f t . Length of rhis secrion is 2 r X 82.5 = 518 in. Load on area wirhin the secrion 1s 650 X r X 6.882 = 96.000 lb. Unir shear equals

v 1' = - - 508,000 - 96,000

0.875bd - 0.875 X 518 X 10.5 = 87 p.s.i.

Shear at edge of wall: V = rpR2 - column load = ?r X650 X 272 - 508,000 = 1,489,000 - 508,000 = 981,000 lb. Unit shear is

v u = - -

981,000 0.87561 - 0.875 X ?r X 2 X 27 X 12 X 10.5

= 52 p.s.i.

The radial moments are computed by selecting coefficients for C / D = 0.15 from Tables XIII and XV, and multiplying them by pR2 = 650 X 272 = 474,000 ft.lb. per ft. (for fixed edge), and by M = 3,200 ft.lb. per fr. (for moment at edge).

This is used as basis for the moment distribution in Section 9 which results in a final edgc moment of -7,800 ( 1 - 0.14) = -6,700. The wall is designed for this moment with opposite sign combined with a

Radial moments in the last line are for a segment haring an arc 1 ft . long a t the edge (Point 1.OR). Thcy are obtained by multiplying the original momcnt per fr. by the fraction indicating its distance from the

The rnorncTw ir. ::,c ,v !?C. ne, c : .st :.A;,,.

arc plocred in Fig 3ü. Thc ni;!xirnsi~; ,,,,LaL~ve . - - - - - - ~ I C ~ I T I L - ~

ar the cencer occurs ac [he edge of rhe column capii :. Th¿ circumference of the capital is ¿ir fr . , and thc tri:;!; maximum negative momenr around the edge i i 52.73,' X 6r = 1,425,000 fr.lb.

Radial moments ver cegrnent

The theoretical moment across the section around the capital is larger than the moment that actually exists. I t should be remembcred that the m,ment coficients in this section are computed for a slab that is assumed t o be fixed at the edge of the capital. Actually, the edge is not fixed, but it has some rotation and a reduction in the theoretical moment results.

The problem of determining the actual moment at the capital is similar to that which exists in regular flat slab design. As a matter of fact, the region around the centcr column in the tank slab is stressed very much as in ordinary flat slab floor construction, so that the design should be practically identical in the column region of both types of structures.

Westergaard* has worked out momcnts in flat slab in terms of the quantity: 0.125WL (1 - 2 ~ / 3 L ) ~ . In al1 modern codes, however, the coefficient of 0.125 is vcplaced by 0.09, a reduction of 28 per cent. Other adjustments made in such codcs introduce still greater reductions in some of the theoretical moments a t the column capital. Such modified design moments have been thoroughly investigated by numerous test loadings of flat slab floors and are generally accepted for use in design. -

In view of the facts discussed, it seems reasonable and conservative t o allow a 28 psr cent reduction in the theoretical moments around the center column of the

P A G E 1 6

!;.ni.. slat,. Ti:c rt:icc;ini: \ , . , lo he used here for radia! -.,;)rr:c-n:~ ai r h t cnnir;l c,n> *.-i;cnrial monienci ar . ; c ; . , : - . i co~il; prub:l!4! >c r d ~ ~ ~ c . : :i.lc,r,. hut [he\- are !ir.ca;i. ;nrriparariveI!. irn:i:I c:,:: .,~,i;hout the re- . . !.:

, c.- .-'.,, r h c siar in F ig 7 4 t h e r . : i . . . iiiomenc around :ii- r :!:t o! rhe capital \vili thrii ht r,nen.as (1 - 0.28) i: ; .4:j.%X, = 1,026,000 ft.lb. The strel area is

Use twenty-eight 1'4-in. square bars (AJ = 43.68) and arrange the bars in top of the slah as in Fig. 31.

i Edge ofdrop panel

Total: 14. 18'-9" luriu

Use 3Mrninimum spacing 1 \ \ ' where bars croir at center

FIG. 31

Across thr edge of the drop panel the moment is 20,000 ft.lb. per ft. at Point (6/27)R = 0.22R, or

M = 127r X 20,000 X (1 - 0.28)= 543,000 ft.lb.

The twenty-eight 1%-in. square bars are ample. Positive moment per segment is maximuin at

Point 0.6R as indicatcd by the dash-line curve in Fig. 30. The total moment a t this point is

M = 14,400 X 2 r X 0;6 X 27 = 1,465,000'ft.lb.

Use one hundred sixty B-in. round bars(AJ = 96). 27r X 0.6 X 27 X 12

Spacing a t Point 0.6R is 160

=7.6 in.

Positive reinforcement may be discontinued at points 12 diameters beyond sections 0.30 X 27 = 8.1 ft. and 0.83 X 27 = 22.4 ft. from the center as shown by the curves in Fig. 30. The total over-al1 length of posi- tive reinforcement is

22.4 - 8.1 + 2 X = 16.0 ft. If some of these bars are t o be made shorter [han

16 ft . , use the dash-line curvc in Fig. 30 for determining where bars can br discontinued.

'"Morncnrs and Srrcsscs in Slabs", Procctdtng~ oj Amrricrn Co~rcrct, Inrrrrr~rc, 1921, pagcs 41 5-538.