COLUMNAS ACI
w, ¿Xe C.,.\V oV Á^Vs'Sp u . t $ S> *\ .A iO^v\$ ' ^ \V. ,«• , 'í AV vbX'. ryro^ü, & s$^Ofi}aÁ? v*>rTrT5 *A. a\ ts'í
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w, ¿Xe C.,.\V oV Á^Vs'Sp
u .
t $ S> *\ .A iO^v\$ ' ^ \V. ,«• , 'í AV vbX'.
ryro^ü, &
s$^Ofi}aÁ? v*>rTrT5 *A. a\ ts'ír('o\jL o ¿A \ For a 23- in. column, I = 2 3 ,3 2 0 in . Nv
1
ifiA
(h n
a ,
■
K
—
íl tiif-i
T /f
col unan x / * ~ 23,32.0 / (18 x 1 2 ) i ni n )- fe , , D o m t is assumed id é n t ic a ]) A? Ui'.»'. 1 'xCi At top or column
= l o« i n 3 t • > l n - (colunm above ¡;j,
(¿I, I t lií 0 > f r ‘ = -^iS^^oljuinns _ 108 + 10 fi •ÍU
'o'joe
& ?
í '
-70
At bottom fixe d e n d V r * Sec.
i 7n
P
í:
= 1 .0
915 (d) f 3 rp ? ° 4 3 )in ~ ^
R - 0 .8 . Th is valué a p p lie s must be m odified as fo llo w s :
:
( 1 , 7 + 1 • 0) = 1; 3 5 (ACI 318-63,
C » ? I a p p lie s j and „ / t onlv wh c alc U iated above, read ° nly WhSn comPr e s s ion coA trols, and A ccording to 318-63,
S e c . 9 1 6 (b)
w h e n ^ : ' 1^ ^ W ith « i a l ¡« a d tensión Controls, or R * r + ^
~
~ Kc/ Kjj) where
=
? / fc fc2 at balanced conditions K ? = k for column under consideraio n . i s th e R valué computed
.
for compression control,
and R
is
íín R , value for the column under tro ls on when tensión conRb ^
Rc ^
Si nce R^ = o . 8 ,
1•
assume a t r i a l valué of R c » 0 . 9 .
■u
~ P u/
M u
V
íeze M jn ijie
Rc ~ 2 9 8 / 0 . 9
= 331 kips
R C ~ 6 0 0 /0 .9
R c (U n ia x ia l chart No.
'u K = f c ' t¿ E n ter w
p o ttio n
= 667
ft-kips
107),
331
0 .1 5 6
4 x 529 « /t
__ ^
En ter upper p o c tio n w ith |P(.m = 0 . 6 3 ;
^
R
^
obtain e j / t
^
.
0 .5 1
Enter lower p o rtio n w ith efe/ t
= ¡ 0 .5 1
and p fcm = 0 . 6 3 ;
o b tain 1^= 0 .3 1 5 R c = 0 . 8 0 + 0 .2 Be."gxamine P
1 .0
- ( 0 .1 5 6 /0 .3 1 5 )
= 0 .9
' s Further check not r e q u ir e d ,
as assumed.
since R
c
is c alc ulated
to be the assumed v a lu é . D e sig n column: ' P. ’ ^ “ T— ^2 m
= 3 3 1 /(4
23
X
= f y / 0 . 8 5 f c 1 = 6 0 /3 .4
X
23)
* 1 7 .6 5
Pt-ra p £ = _£ m—
=
0 .6 3 /1 7 .6 5
A s t = P ^ Ag
=
0 .0 3 5 7 (23)^' = 1 8 .9 8
Use 19 # 9 b a r s , A
= 0 .1 5 6
= 0 .0 3 5 7 sq in
= 1 9 .0 sq i n .
From Reinforcem ent 4 . 2 . 1 , this is the máximum number of # 9 bars that can be placed in a 23 x 23 column w ith # 4 s p ir a l w ith butt ^ P**i ce (Hen c e , O R . w i th the 3 s p ir a l ass umed) .
PROBLEM 2 : DESIGN OF A LONG SQUARE POLUMN WHERE TENSION CONTROLS USE OF TABLES FOR eb / t AND . G_iven:
A l l data are id e n t ic a l w ith s p i r a l .
So lu tio n;
R efer to T able 4-1, f
= 60 k s i,
to Problem 1.
Assuroe square column
square column,
£ ' = 4 k si,
g = 0 .8 .
II: p fcm = 1 . 0 ,
e ^ /t
1 .0 5 > 0 .6 5 5 4 ,
tensión Controls the d e s ig n .
For p fcm = 0 .6 3 o b ta in
= 0 .6 5 5 4
(from T able 4- 1 ).
(obtained as in Problem 1 ) ,
and e ^ / t . P tm
*b
V
fc
0 .6 0
0 .3 0 6 8
0 .5 1 1 7
0 .6 3
0 .3 0 7 7
0 .5 1 3 8
0 .7 0
0 .3 0 9 8
0 .5 4 8 7
S in c e e / t =
in te rp ó late to
T
^ ! 1? 88-0^ ^ * * 3 r Íng, U n ia x ia l Chart N o - 107 ^ Problem 0 .3 1 5 and e^/t - 0 . 5 1 as compared with the valúes
obtained by in te r p o la tio n using the ta b le s, eb / t
K,
= 0 .3 0 7 7
and
= 0 .5 1 3 8 .
Once Kb has been obtained by in te rp o la tio n using the d esiqn t a b le s , the s o lu t io n proceeds as in Problem 1. 1
PROBLEM 3 : DESIGN OF COLUMN FOR MINIMUM ECCENTRICXTY BY np TABLES 1 THROUGH 4 AND REINFORGEMENT 3 THROUGH 5 BY US~ °* G:iven:
Pu = 1190 k ip s ,
fy ~ 6 0 -° k s i -
Pu ! (0 . 05t) ,
fc • = 4 .0
De sign as a s p i r a l l y rein forced
k si,
square column, |
w ith # 1 4 bars as lo n g it u d in a l reinforcem ent and # 4 as s p ir a l w e ld ab le s te e l and end-bearing s p l i c e s . So lutions
Use
Assume column s i z e , "t = 19 in . Ag = t 2 = 19 X 19 * ¡ 3 6 1 sq in!
K for minimum e c c e n t r ic it v = K
1
r gt = 19- 2 [ 1 . 5
+ 0 .5
g = g t /t = 1 3 .3 /1 9 .0
+
m
= P / f 't^ u c
!. (1 .6 9 /2 )] =
1 1 9 0 /(4 x 3 6 1 ) = 0 .8 2 5 0 1 3 .3 in .
= 0 .7
Find req uired p^m from T able 4-1: p tm
Km for g = 0 .7
0 .5
0 .8 1 3 0
0 .6
0 .8 6 1 3
For Km “ c -8250 fin d p fcm by in te rp o la tio n
P*-m = 0 .5
1 0 . 8 2 5 0 - Q .8 1 3 0 ) y 0 .1 ( 0 .8 6 1 3 - 0 .8 1 3 0 )
0 .5 2 5 0
D esiqn colum n; m
= V
° *85 V
= 6 0 /(0 .8 5
Pt = P tm/m = 0 . 5 2 5 0 / 1 7 . 6 5
x 4 .0 )
= 1 7 .6 5
= 0 .0 2 9 6
A s t — Ag p^. = 361 x 0 .0 2 9 6 = 1 0 .7 0
sq in .
7
ms
.... ..........
—
A-*- --
f e s - ;.
k
Note from Reinforcernent d ? i Kkaf n a i n sau r ^ T of # 14 is 9 ™ ? ° lumn Wlth r e i n ~ A 0123 = 0 ^ 4Qg Henee -
V
0.0409 x 10 x 17.5 = 7.17 sq in. 0.0123 x 10 x 17.5 = 2.15 sq in.
:¿9
PROBLEM 2 8 : FIND TENSION STEEL FOR COLUMN WITH NO COMPRES 9 0.5998 0.2539 •tí. 3-924 0.6052 u , 692k 0.4457 0.6387 0.2584 O. 43O8 0.6451 0.7403 0.4847 0.6777 0.2628 0 .4680 0.6852 O.7S83 0.5230 0.7170 0.2672 0.5039 0-7257 0.8362 ,5604 O .7561 0.2717 0.5387 0.7660 0.8842 •5972 0.7956 0.2761 O .5723 0.8067 o. 9321 .6332 0.8351 0.2805 0.6049 0.8482 0.9801
0.2202 0.2379 0.2226 O .2812 0.2250 0-3257
0.2274 0.2297
0.7078
0.3584 0 . j 894 0.4192 0.4477
0.2753 0.2809 0.2866 0.4751 0.2923 Or5Sl4 0.2979 0.5267
0.5568
075944 O .6321
0.3261
0.6063
Ik
0 .‘)450
0.4836
0.2362
°- 0 5
g
1
TABLE 2-2—ROUND COLUMNS WITH SPIRALS; REINFORCEMENT AKRANGED IN A C1RCLE
f'
y
ksi
ksi
0.0 0.1
5.0
40.0
0.2151
0.2190
0.3
0.2230
0.4 0.5
0.7
0.8
0.9 1.0 0.0
0.1 0.2
0.3 0.4
60.0
0.2111
0.2
0.6
5.0
lk
Ki
0.5
0.6
0.2269 0.2308 0.2346 0.2387 0.2426 0.2466 0,2505 0.1738 0.1684 O.I 630 0.1577 0.1523 0.1469 0.1415
0.7
0.1362 0.1308
0.9
0.1254
0.8 1.0
0.1201
D
0.2307 0.2620 0.2915 0.3196 0.3462 0.6321 0.2605 0.3715 O .6698 O .2669 0.3956
0.2458 o .2763 0.3056 0.3339 0.3613 0.3877 0.4132 0.4379 0.4618 0.W 4 9 0.5073
0.4^52 0.4829 0.5207 0.5585 0.59^5
0.8205
0.2925
0.2709
0.4452 0.4805 0.5159 0.5515
O .1885 0.1875 0.1864 0.1854 0.1843 0.1833
0.3270
0.3782 0.4329 0.4914 0.5542 0.6218 0.6947 0.7736 0.8592 0.9525
0.7075 0.7451
0.7828
0.5871 0.6230 0.$589 0.6943 0.7308 0.7671 0.8033
0.2285 0.2349 0.2413 0.2477 0.2541
0-2733 0.2797
0.2861
0.1822 0.1812 0.1800 0.1791
0.1780
0.4186 o. 1+405 0.46l4 0.4815
:= 0 .?
= 0.8
i = 0.7
i = 0 .6
Km'
Kfe
0.4452 0.4830
.2460 0 ,.2542 0 ,.2624 i c .2706 0 ,.2788 0 ,.2871 0 ..2953 ó .3035
0.5210 0.5589 0.5971
0.6351
0.6731 0.7112 0.7494 O .7876 0.3259
0.2658 0.1*452 0.3123 0 . 1*816 0.3591* 0.5183 0.4070 0.5551 0.4551 0.5920 0.5038 0.6290 0.5530 0.6663 o .6028 0.7036 0.6532 0.7408 o .7042 0.7781 0.7558 0.8157
"I* D
0.2158 0.2483 0.2787 0.3073 0.3342 0.3596
vm 0.4452 0.4837 0.5224
0.5612 0.5999
0.6388 0.3836 0.6774 0 . 1*062 o .7162
0.2634 0.2731 0.2827 0.2923
0-3020 0.3116 0.3212 O .3308
0 .3117 0.4277 0 .3199 0.4480 .3282 0,4674
y.7550 0.7938
0.8328
0.3405 0.3501 0.3597
0.2526 0 .2052 0.2993 0 .2070 0.3452 0 .2088 0.3903 c -2106 0.4346 0 .2124 0.4782 c .2142 0.5211 .2160 0,563? 0 .2178 o.6o46 0 .2196 0.6454 0 .2214 0.6855
0.4452 0.4830 0.5208
0.2183 0,2222 0.2260
c .2034
0.2298 0.5969 0.2336 0 .63-50 a.2374 0.5538
0.6735 0.7117
0.7502 0.7886 0.8272
0.2412 0.2451 0.2489 0.2527 O .2565
D
K ¡Ti
0.2011
0.4452 0.4848
0.2350
0.2666 0.2961 o .3238
0.3497 0.3741
0.3271
0 .1)187
0.4392 O .4586 0.2395
0.2871
-p - v n -p- 0 O mí
ro vn o\ v
O
o
o
0
fo —'■3 Q os
0
0
6 os G\ G \ es
0
0
0
^ ^ ON H ( J \ ■P' C \ V Q h vo
o
o
o
t o Co l o l o l o
o
ro
o
:
0
0
0
0
V * ;?0
CO £ “ H
V 5JÍ2
^
0
r
p
o
0
0
0
0
fe f e c o Cj fo ~^3 L o v o v n h * ~ h
* ^ j v > V O — 3 ~*s| 0 r
0
c
c\ v
Lo
00
o
0
p
0
H
O
0
0
0
O; 0
p
0
0
0
H
O
O
p
O
O
O
S f e ’f i fe 5 7 ^ b\ C x ^ t v n F ^ ^ ro ^ ro —0 4~ \ / i C \ ~- s C O V O o H ro L o V O : C O \ J \ & S -prpLo l o v n —j c o
0
0
p
O
O
0
0
0
Vi H M o>Mv i 0 en lu H C o - ^ v n v . P 5 S f8 O N 't o 0 ^ 5 4 - H S P
O
O
O
O
p
p
p
p
o
o
-^“ L o l o t 'iQ e n c \ 4^ r o VO' \ r ) rO CTv L o f\]j
0
O
^ VQ —5 Q ~n e n C \V n
o
o
l* j l o ro ro -f^ 1 0 v p o \ vn 0 -p " — 3 s—• > }—¡ q
C
ó?
e n c h .v n ro ^ 3 ro - 3 ro O'v - p ' v n 4-~ \ r , l o O C o 0 3 4 ^ -Lo 0 0
-3
0
O
>
or» m r~ sr
f
0
ü
^ .1 ® la r
O i>
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c:
>
ai ni
0
vn U)
0
H!
3
o
0
0 : 0
O
O
O
O
C
0
°
j
H» H» 0
0
i 3 9 ^ i —-3 r o —q ¡ pD r o r o j 0 D --3 L o
?
p
0
0
O
O
:
p
0
p
0
0
o
0
0
0
0
H
O
O
O
Ó
0
^-o Lo h Lo
0
6
^ ¡} y j y j - f r . V - t ú V I S^ ^ p co 0 h V O L o —'3 H í - ' V l n
M
O
- 9 f f - i CT' c r ^ ro —w ro ro — 3 r ■ > —0 ro ro ro 7 o ro ro ü ÍJ v n ro H '- p - 'O 's C o - p - C D
^ ^ ^ ^ ^ CD CTS Q v V i - r L o t i £T *7Í p ro v n c d CO O S r o i | 0 ~ i3 V 1
0
0
C
0
o
0
0
o
O
O
O
O
O
O
O
0
0
ÍT
O
n O f*
3 •f
0
0
0
0
0
0
0
O
O
O
^
O
O
O
^
O
O
O
O
O
o
O
O
O
O
O
O
O
í- 4 M
O
o
O
O
O
O
V Q v n OO O D “ >3 —j r o r o —-3 n v p C o *3 O S --3 O L O L o P
o
o
o
o
o
O
O
(Q
O
co to ro o s
ro ro covn — 3 o ro lo vn
o
O
-^3 ro OS o
b\ —g V i O
Q \vn r\:> — 3 p -Lo U l CD
c
>
o
o
O
O
O
O
O
O
cr
CO II O OT
H
H
O
O
O
O
V?
O
O
O
->3 vq ro
o s v\yj\ f o — 5 ro ~ > 3 ~-5 o s v i l o v o r o hj c o
O
O
P o o o o o o o o o £ ^ 5^f e ^ ^ L O L o ¿a L o 2 >“ ^ ^ K t í v ^ 2 >o^ osv e n ro ro o s vn vo - * ro
3
4*
p
h
p
o slo o o lo
o-
vq
m
O
p
p
Í k £
o
o
o
p
Í T
o
o
LO LO
o
o
o
fo
fo fo
S
h
o
o
II o * 1 »
0
3 05
3
H
j
v9
P
v
p
p
vk i > o
p
p
o
o
c o a > -^ ^
o
^
H |;t
- P 'V O
^ O
- P 'V O
H
0
O \o
p ~ \ó
O
O
C D CD. í^ v o
o
O
ro
O
-Fv o
co vo vo
O
vn
h
o s o s a
v2
00
Ov en
ro
r ,H P P P P p p >o ó ° o°~^
onc n C n
vALo \.n«N? O v H l\) - P ^ v o H
o
c o o
!
H
H
ro Co
H
Lo
H
cd
e n -p - r o v o v o
H
O
O
O
o
o
kn
12 ;5 > riñ :v m
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3
-f
o
H o v o v o CD C D — 3 t j\ O S M v n v o (_o ~-3 VJI v n l o P * P * v n v n v n cy>K e n O s - ^ O SV O L o e n v o ro v n c o h v n
•tí O w n m 2 IT» Z ■H > a» > Z O m O Z > n 5 n r*
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o
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o
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^ c o ro v» vn en en M - 3 L O VO
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"P ^ Lo L O L O L O L O L O L O L O t i S , ° > ~ íL ^ fo '^ O V f O V O - P O V n
O
-p* lo lo ro O vn h o vo o \ o O sH -P-
o
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\n
0
C o u .) f o f o o \ po \ q \ n 0 “--4 r o Q ro v o 0
O
■% -í t -F" 7^ vn ^ oÑ vc o Ó íH 'v n
o o p o L o . ^ O v vn o s
Lo Lo Lo fO H O 4^ O n v o h v q QSy
» md h Í -
O
s4 f Lo, ^ C ^ -> L O L o ¿ o Q , , S ? ÑQ C O “-3 CTv v n 4 T -L O r o M S 2 . H ^ fO L O - p ~ . p v n OS Lo O OS L o o - 3 -P* h ct» v n
m ox h
0
0
O
O
vi h|
O
O
9 P ‘ ro fí> V j To n
^ \2 ÍV 3 i r i t t c o ¿ 0 ¿ J ro 7 t , +A °'N U J O L^; O -¿1 p > v b H L o -p- - p r L o h Co vn ° m v o - p - r o H T o ^ - S r o u i
0
¿3 cn
CO ro v > v o
0
Cn u > LO -jp*
0
O
H
00 Lo co Lo c o o s v n l o ro o ~^í c d r o c o l ó o v
0
O
ro
£ > 9
0
O
4^ Lo
;•£■ SQ “i
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0
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C O -"3 a \ v n
P
i ,» fs'i p ^
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O
O V O
v n w í (^ ^ ¡ fL ° ¿ ^ J LO O J fo vn vn L/j L o ro ro h O 0 vn Jr* V ¡ V p L o -< I 1—1 v n v o L a j Q \ OÑ O v v n -P" L O L o K ) h -1 H O V O
0
C
H
^ J Ca ) .v n h en ^ W ,v o
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& v f o S l S 8 ^ ^ p g v o i c o 0 0 cfo C D *-'J —nÍ — 3 c ñ j o s c n
0
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C
b \C n ro H H CO O j 03
b
p
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p r o
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G 0 —-3 0 , v n : 4 r- L O
n Ñ 5 S N£ S c ? ^ ^ ^ U > — J — 3 --g - 4 j G \V J1
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p
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C \ G\ G\ Q \ Q s \.,n vp , \_n v_n v_r¡ v j i r ^ L O ro h 0 0 3 * - 3 O 'x V / 5 -pr r o
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o Q
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n íi lí C/i O Oí o b 7T
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(/) -y
j-* -
j o rtrr
í
“ :¡5\¡¿•ffír^r■ ití~t1tit
]
■ 't' itrfiT?!;2§ I i tttt gkdSg5h:£l
ik ílS ii 3 ü :::::::j:::::::¿ v i :
: f :::
m
#
ir r t r
^■0
"Hlintj'SÍi);;;ilni^-iír.:^ J i :-::J irírH..4rr^j ¿ptjdTTTT^T^" 1.|
ÍT 7
.r
.y T :k
:‘
o
1 . . .. 1..1.Tv - i ia ^ u 4 i+ p ü
jSB m
rti
1 3
m i:i:;i;; mi"*! ;. U^jjt^TTTBW-::: si^•^1::t:;irK r:: 1 : ?T¡T:\< ::;:\:T > p : .:: :: ,-rV i ::) V r. V :: ■.p v - ■t r l!: 13 *
“m UPiMa íííMe K444+l44XIXn^fI^
Éü 2
§ í
M M ir
•1
-u
I!
¡T'Tx
¿ H r;:
teu-
ifítkii ,, Í4T+3K
Cl,
S H lilia
i
V
0> n i
1
a
iv._>
VX-\-•
n
j1 ^ 6
^ %i*
lST-Oí 4inlijC4-4l■ l i i ii'^ l j l it-t*tin^-ri*?++■
n
m i
‘J ¡ AXI AL B-.vnfe± ' +í, 1 o
o m
TntsT-
trP l
4P M
^trricn tlnt:l fíTI :;;kru-rrJ 'íyj¿:—;¿r v>fcMiiiiíá H ^ FSJ f tWirrnt t|Tn»*^U4W^ .;::: Lv W fe -‘
I llffi
r a s f p T O M iF iw f
•sMll
:■ :fj¡.;:¡:::j:|ij|{pp||i||¡ •tHL
JilMjírx 7*íÍlR? + ”
|o
1 Í F !!
lililí;
trríitt
a p iif:
ff
—t"* —tu n
----
n *t H
j (
Lf •''! r— n \ M 1 . ¡i
I B
U n i a x i a l C h a r t No » 88
I..5 TTrrrT-TTiLO
05
! :. ;tí . i.¡.1i... 11ijj5. 1¡1*3*1i-ii-~ i. ¡;!:í j *:;7 •:r^H!r •••L¡
— "T ‘
.• i-pi' •’;
y
. '
■ ;!
- n r. ■ . ;;;};; i : pÍK n*‘
_j ; I. r*’-t •j rr-K..í«j» sprr-r-^r—f r■ ■ . :rnrti¿
0.2
03
í r*; ■ |. ■ í
' M
’ "{¿i-r ;l : j; t:: ¡, ;■ h-;-; Ht■r | 1; ■ 7- Í ! i- - -
TTTT!7ttT
0.4
f¿ -
¿i,iíj .L-i
ilil¡Ü -0.1
-
:í •H -H-íf ’J»
0.5
0-6
07
O.B
0.9
.
4 .0 k s i
f / = ó 0 .0 k s i g =o.s
LO
Pt m
LJ gt
0.04
0.08
OJS
0.12
Py e }24
fr b í2
íi
\ r
\ ....
0.20
T
1
024
1
Un i a x i a l
Ch a r t
Ho , 89
rrr*rr
f¿,s fy
5 . 0 ksi
= 40.0
g = 0.6 II 0.9
,
1.0
ksi
¡Un i a x i a l C h a r t No » 91
fy =
5 .0
ksi
4 0 .0
ksi
g - 0.8 0.1
0 .2
0.3
0.4
0 5 P t
: m
0.6
0:7
0.8
OS
10
Ué
a
M
al
C h a r t No , 92
f¿ = 5.0 ksi fy = 40-0 ksi 9 = 0.9
0.2
0.3
0.4
0-5 P f
0.8 O JQ cC - u
i 1 0.6 v:
m
0.6
"
Uniaxial
•5
—
.. :T t ■
1 .......1 'i_
'—
1 : ,1 i
LO
Ife t
n
i
No ,__9H
I
5.0 ksi f y = ó 0 .0 k s i
íL
M tlh-tH-' Vi:
g = o.7
*
| ~
0.04
130
C h a rt
,
j J rÁ«.
0.08
iÜ [ ilÜ nft 0.7
0.8
0.12 e _ Pu e K- 1 e bt2
0.9
0.16
1.0
O
0.1
0.2
0.3
0.4
0.5
06
07
012 —
Pu e
0
8
0.9
1.0
pf m
0
0 04 .J :
1
0.08
y ®.
0.16
0.20
0.24
L^
X7
U n i a x i a l CHAp Tj o, 97 ;U- i. i; 1 l|i
j 1;
"
i
i : : r ; : |j ; :i
.
TTf
n
P ....
~
f y
=
g
W
Bit 0.4
0 .!
0.6
0,7
0.8
4
4 . 0
k $ i
0
k s i
. 0
JíUI '
|! i
fj.
03
1.0
»
0 . 6
" "' ' ■■*'....... ~.... .
"
... ~S --T'
U n iaxial
Ch a r t
No, 98
^111" 1 ^
i l i l f¿ -
4.0 ksi
íy = 4 0 .0 ksi 9
“
0 . 7
U n i a x i a l C h a r t ;No , 99
1.5
1.0
-
j’r .,;
r if
f*c 1 í í y
05
f:K; ai. i l i l ü
tttít
.iH .
i4ii i: ■W-
U'Pít? “í'I't'lH !!!'■!:; Mían:
-XXif:
' II:í
=
4
4 . 0
k s ¡
0 - 0
k s :
g r. 0.8
•—
—
............. ............ ........—
1"'■ . ..'"l
--- ....... .....
U n i a x i a l C h a r t fio. 100
U n i a x i a l C h a r t No , 102
U n !i a x i a l
Ch a r t
No .
103
í¿ í 4.0 ks| fy = 5 0 . 0 ksi 9 = 0 .8
P
l p
f p
_i"i.1;::: >íii:Júlf'ft.
M
lg g f
í
l
i
m
s 0.20
Uniaxial Chart No. 104
Un i a x i a l
Ch a r t
No ,
105
4,0 ksi |fy = 60.0 ksi
0.1
0.2
0.0¿.
04
0.5
p} m
0.08
06
¡07
0.12
K
e „
0.8
0.16
0.9
0.20
0.24
fu e fr
141
U n i a x i a l C h a r t No , 106
0.5
06
P* m
R p jg ffl
it
S jS?
m t m tM
W
m M m m
Í
*
S
i
i
S Ií
P| m
P *
m
1.90
i,00
1.70
l¿0
C?¡ - o li
O.'.K)
0 .JO
OJO
i« Mwtf.»■ ■ '■ ’;¡1*—
,70pb*
1 .2 0
D
CL
fSst^"
U II
i,0 0
0,»0 0.60
0,70
0.50
o.yo
O JO ^ v
V4B
o
í
£
4 .0 ksi
■= 6 0 ,0 ksi
g = 0.6
Biaxial Charí No
tjtt+'•»
o „^ o
0,00
o.ro
o.so
o.-«o
0,20
r- r ;___ í-¿ * • -/ i [v . — *•*»■ / \ ♦ * 1* \
“T*1it 1: it;'TTvT tlim ilt
m tm ín tttltTl Et
iij
™
ttrSOOTtt xtrnTOT rLo Ttfm i rUv DPi ftI l 5IJilC1i¿
□X-8U4-W',fl44444hH-'HH-rrri
Í T m m i r t i t H 4 4 4ÍÍ1- i w: u n 11 iix ii i »,t í4Oi-L a —¡ 111 n i o x u X ljj rftty,¡nitrn.iL no .00 ¿ m i l n í líltcti I M il
n }
óü
50
t.*c
(¿ *
4 ,0 fes»
fy - 6Q»0 ksi 9 s 0,7
1S
BARS
Biaxial Chart No. 14
o .s o
— — — -— —
-T0S
,1 7 - t í.^ 1»
7
t
i fc -
160
— ;
x
. , ." 7 / . X
* i .x
4 ,0 ksi
~~f «$.
THICKNESS
3. 4.
.24 .20 .17 .15 .13
.42 .35 .30 .25 .22
.11 .09 .08 .07 .06
.19 .16 . 14 .13 .11
.06 .05 .05 .04 .04
.10 .09 .03 .07 .06
.03 .03 .03 .03 .02
.06 .05 .05 .04 .04
.02
.04
31 .04 20.30 14.61 10. 65
40.30 27.00 18 . 96 13. 32
.01
8.00 6,16
1 0 . 39 8.00 6.29 5 .04
.53 .64 !. 92
.37 .95 ..63 . .3? .17
.00
4.85 3.38 3.15
20.29
1 5. 63 1 2 . 29
q s84 6.50
4.10
8.00 6.59 5 . 50 4 .63 5 .94 3 .38
2.60
3.38
>.29
2.17 1 .83
2. 31 2.37 2 .02 1i »~T '
3 . 58 3 . 01 2 . 56 2 . 20
5 .36 1 4.47 3.76 3 . 20 2 „ ~4
l .49 1. 20
2,3~ *36-...... 1 .80 j 1 .39 1.40“
2 .92 2 . 54 2 .22 1. 95 1 .73
1 .25 1. 12 i . 00
1 .54 1 .37
1.55 1 .33
¡
.88
1, U
.5*
. 77 .68
1 .00 .63
1 . 90 1 .cT 1 .44 1.27 1. 12
,46 .41 .36 .33 . 30
.61 .54 .49 .44 .3$
79 . t :• .63
1 .00 . 69 .80
.72
.90
en
.65
.81
1.23 .11 1 .00
.27 .24 .22 .20 .19
.36 .32 .30 .27 .25
.45 .42 .38 .35 .32
. 59 .54 .49 *4 5 .41
.74 .67 .61 .56 ,51
.91 .82 .75 „69 .63
*17
.23
.30
.47
.58
.86 .75 . 66 .58
1. 15 1 .00
;
O b t a in C t f r o m S t i f f n e s s For T- o r L - b e a m s identleal. id column w id T a b l s p r e p a r e d for beara w id t n an ^ th / ? } f o r /■ I f a pan lengths d t f f s r . s u b s t i t u t e / 2/ ( ¿ i * + h-) for h . I f c o l u m n h c ig h t s d i f í e r , s u b s t i t u t e h x 1^ / ( 1^ 23
2.
-
íI
1
l
¡oíni. Rectangular coiumns;
.
1. 30 1. 00 .79 .63 .51
.32 .63 .97 . 00
interior
2
.75 .58 .46 .36 .30
INCHES
co
t , INCHES
I 12 10 ¡ ________ l 5.042. 92 3 . 38 1.95 2.37 1. 37 1. 73 1. 00
t
Stiffness 1.1. Cr' factors, ty pical
COLiUMN THICKNESS
IlsC u
CQLUMN D IA M E T E *
00
BEAM
THICKNESS
t,
INCHES
““1 ” 10 .88 .59 .41 .30
1 .72 1. 15 .81 .59
2.97 1 .99 1 . 40
4 .71 3 .16
1 .02
1 .62
.25 .17 .14 .11
.44 .34 .27 .21 17
.76 .59 .46 .37 , 30
.07 .06 ,05 .04 .04
.14 .12 .10 .09 .07
.25 .17 .15 .13
.03
.06
.11
}06
.10
.05
.02 .02
.04 .04
.08 .07 .07
.02
.03
2. 3,
2 .58 1. 99 1. 56 1. 25
.88
.39 .33 .28 . 24
59 49 41
.20
30
.17 .15 .13 .12
26 23
.02 .02 .02
.03
.05 .05 .04 .04 .04
08 07 07 06 06
.05
.02 ,02
.03 .03 .02
.01
.01
.02
.01
.03
1 /C t
Table
and
If
prepared
span
lengths
for beam differ,
coiumn heighti
d iffa r
width
6.46 4 . 71
8.60 6.27
3 .54 2 . 73 2 .14 1 .72 1 .40
4. 71 3 * 68 2.85 2 . 29
.27 .24 .21 .19 .17
.17 .16
.14 .13
,12 10
.09 .08 .07
Obtain
eubstitute . «suúbs iz i t u t e
Coeff ro
46 . 37 3] .06
6. 12 4 . 71 3.71 2 . 97 2 .41
7.78 5.99 4. 71 3.77 3.07
9.72 7 .48 5.89 4. 71 3. 85
11 .95
5.80 -4.71
3~ 1 6 2 .63 2 . 22 1.89 1.62
3 . 88 3. 24 2. 73 2.32 1 .99
1 .99 1 .66 1 .40 1 .19 T . 02
2 . 53 2. 11 1 .78 1 * 51 1 . 29
36 .32 .29 .26 .23
,46 .41 .37 .33 .30
.59 .53 .47 ,42 .38
.66
j[ 1 1
2 i . 82 1 5. 90
K3
n
9.20
i
7 .2 4
—h D n o
1 ,4 0 1 .21 . 1 06 94 í .63
1
.
72
”2 . K o_
1 .49
5'
1 -5 1
9
1 . 15 1
.74 ¡
c/*
.66 :5-Q .'53 ' .¿a
.02
O*
.90 .81 . 72 .65 ^ r- j7 3 . 59
o 5' #
.32 o r* g 4. >
.43 .39 X&
.49
LU
0 C r? Cl r»
.19
.26 .24
.33 .50
,40 j .37
c
.17
.22
.28
.12 .11 .10
.13
/ 2>
!
.45 .40
\ f
.21
Stiffness
diameter ^i d e n t i c a l .
/ 2 / w' l
h3
| 37 . 70 i 25 ,26 I r ; 74 12.93
. 27 .25 .23
from
3
30. 18 20.22 1 4 . 20 1 0 . 35
.21 . 19 . \7 . 16 .15
Ct
30
28
i
23 . 74 1 5. 90 11 .17 8 . 14
1. 12 .97 ,85 .75
.
.14 .13
,10
26
24
.88 .76 .67 . 59 .52
.68 ,59
.44 .39 .34 .30
.20
coiumn
.53 9R
k J? i
. ?2
.
1.86
1 .08
.37 .32 .28 .25
05
For T- or L~beams m u l t ip l y by
9.20
1.
.84 .70 .59 .50 .43
20
22 18.29 12 .25
| 1 3. 74
1 .02
"t C.
14 12 11 10 09
i
6.71 4. 71 3. 44
1.10
.09 .08 .07 .07 .06
.03
.01
1 .81 1 .40
.06 .05 .05 .04 , 04
.01
. 01
10.02
17 15
.03
.01 .01 .01 .01
.94 .74 .59 Ae
7 . 03 4 .7] 3. 31 2 .41
.10
.02 .01
1.21
.21
.02
.01
36
2 .22
20
I
18
16
12
2.
e; *
I
• *
o_ 3 D iA
iliiiiifiiriíiihiiiiii iií .to*iüi»íKtíi»>»■ »■««*■
Stiffness 2 . ^ o e f ficienís C , fo r g ro ss m ornent of ® (flcm ge on o ne o r fw o sü*es)
® 2.0
Ratio of A s
in e rtia of T-$ections
mm H p llp N H i
Stiffness 3. Moments of inerlíc Si of rectangular sections 1 in. wide (¡n.4)
'1
i oo 25 50 75
00 25 50 75 5.00 5.25 5.50 5.75
6 .00
6.25 6.50 6.75 7.00 7.25 7.50 7.75
8. 00 8.25 8.50 8.75 9.00 9.25 9.50 9.75
10.00 10.25 10.50 10.75
11.00 11. 25 11.5 0 11. 75
12.00
12.25 12.50 12.75 13.00 13.25 13.50 13.75
i
CP
r^ b
10
12
7 7
8 9
10
10
14
12
11 12
16
13 14 15
26 53 40
20 21 22 23 24
25
17
22 25 32 39 49
29 37 46 57
18
8 13 1 9~ 28 38
8 14
17 24 34
7 12 18 26 36
9 “ 14— 22 31 43
42
45
48
51
ó¿
ir 85
80 99 122
106 131
113
139
147
81
74 93 114
99 119 141
109 171 155
149 174
166
194
213
129 155 184 216 252
139 167 198 233 272
149 179 212 2 49 291
159 190 2 26 266 31 0
169 202 24 0 283 33 0
179 214
183
119 143 170 19 9 233
202 232 266
224
247 28¿ 3 25 369 417
269
29 2
310 354
337 38 8
359 413
58 2 43 9
413
443
472
502
403
336 3 84 436
314 362
537
57 0
455
493
470 531
503 569
607
645
404 4 65 532 604 683
512 573 639 711
555 621
597
640
683
725
669 746
717
693 770
82 9
765 853 948
812 90 6 1007
768 86 0 959 1066
787
852
910
1049
1115
1 180
42
53
59 73
66
89
134
129 147 16 7
155 177
206 236 268
189
227
181 206 235 265
303
341
258 295 335 3 79
541
384
426
469
39 3
459
430 479 535 590
478 533
rtze;
298 334 373 414
526 586 651 721
426 4 74 524
7
54
53
112
256 286 319
6
6 10 16 23 31
69 86 106
48
179
213 239 266 29.6 327
36
58
157
201
34
58 75 96 119
32 41
155
32
38 50 64 79 98
79 95 113 153
70 83 97
30
35 46 56 73 90
11 15 21
69 83 99 116 136
39
28
5 9 13 19 26
7
59 71 85 99 116
49
26
5 8 12 17 24
4
29 37
65
22
20
25 33
K\
27 28 29 30
20
8 12
c* co
26
4 6 9 14 19
3 5
6
16 17 18 19
16
14
24
107 127
302
592
655
9 14
21 29
n
f' For w e i g h t
other
than
145
p c f , m u l t i p .y
^
J
=
799 88 8
983
4000
PSI
21 - 30 41 54
■ 71 90
255 299 349
w
Round columns, dia. = t
KJ O 2 . 00+ ! 2 . 2 5 * 2 . 25 2 5 ¡ 2 . 50 i2 -
2.00
8
REINFORCEMENT 2.2 i PITCH OF SPIRALS (IÑCHES) FOR SQUARE COLÜMNS lH INCH CONCRETE PROTECTION f
y f
O
o
4 .0
#5
#5
21
26 28
23 25
2.50 *2.25
2.75
V
3.0 1 #4
#5
S T o c h ! 1 Í2 . 0 Q] 2 . 00 + 2 . 00 + !| 2.50} 2.50 |2.00+, |,2 . 50| 2.75 [2^00+i 112 . 5 0 j 3.00 ~ II?¿5PJ
2.25 2.25
3.50
#5
[2 .oó; 2 . 00 + .2 .OOl 2.50 2.75 *3.00 ¡ 2.75 J
5 .0
6.0
#5
#5
12. 00 + 2.50 2.50 2.25 2.25
Q M
g U
¡2 00+1 12 |2. 00+i 14 00+1 16 00+! 18
i
20 22
2.75 2.50 2.50 2.50
24 26
27
^2. 25
2.25
¡3.50¡
12.25 s2 . 00 +
2.00
I *2-*l
36
33
38 40 42
35 37
¡2 . 00 + ¡2 . 00 + ¡2^ 00 +
[3 ¿25\
¿2 -00 ] 3 . 0 0 t2 ._00 j 3.00
=60
k si
£q ', k si
i 2 . 00 + i 2 .00+ | 2 . 00+ 0+ | |2
3 .0
3. 0 #4
#5
12. 00 +' ' ¡2 . 00 Í 2.25 1 ¡ 2 . 2 5 ! 2.25 ! ¡2-25* 2.25 1 12 ,_00 ¡ 2 . 00+1 2 . 0 0 I 2 . 00 + r27doi ¡2 . 00+1 |2 -_P0f [2 J>0 +¡
28 30
i
2.25 |
2.25 | 2 .25 | \2*_00t) J-
T ab la Preparad for 1 ” Máximum S i z e A g g re g ste + Valuee R e q u ir e 3/ 4 11 Máximum Si s a A g g re g ate Notes. The dashed l in e a in d íc ate that a p i r a l epacing cannot bm p ro v id e d w i t h in lim ítatío n s of ACX 318- 63" for those com binatíons of colman s i z e and m a te ria l str«ngths*
32
136 I38 Note:
Most
are
5 .0
#5
#5
2 . 00 +
¡2 . 00+1 12. 25 1 't2 . 5 0 i >2.5-0 1 ¡2.25 ¡ í ¡2.25 | 12. 25 | ^2.25 , i2 . 00 + 12 . 00 +* i |
2.25 2.50 3 .00 [3^25] 3.00 3 .0 0 firo 0 ! {2 .751 '2 7 5 1 r * /3¡ 1 ..~ ¡ ! i 2 .7 5 , 2*50, Í2t_50j
‘ 2 . 00+1 ¡2 . 00 +| (2 ^ 00 +j
--
Economical
indicated
#5 ¡2 . 00 +i ¡2 . 00 +| |2 . 00 +| 12,-00 +j —— — -
-
---
Selections
thus?
_
COLUMNS
¿Ja
y
FOR SQUARE
2.75 j 2.25
29 31
f
H
4.0 #4
f - 40 y ksil f c 1, ksif
ksi
*3.50
30
39
'
oí w
ksi
p r 7 5¡
32 34
2.00
=60
OF SPIRALS
24
2.50 2.50
20
y
2.2; PITCH
2 . 00 ^ 2.50
22
18
f
c * , ka i
3 .0
13 15 17 19
16
= 40 k s i
2 INCH CONCRETE PROTECTION
REINFORCEMENT
W N H W W 26 £y
oí w
R E IN F O R C E M E N T
3 . 1 . 1 .
H A X IH U H AND
NUMBER
S P L .IC E
TVPE
OF
3ARS
P E R M IT T E D
i l l u s t r a t e í
)
m. 4*-
Pt
w “ l s"-‘ “
FOR ( 1 / 2
CO LU *. lH C K
TYPE
T1£S
s»»*.«.
The s t e & l r a t i o , As t/A g.
- r r ^ r / r - J / i
1 ia¿
»-
Reinforcement 3.1.2. Máximum AND
Ag
number
of
bars
permítted
for
cclumn
(1/2
SPLICE TYPE 1LLUSTRATED
INCH T!£S)
BAR S1ZE DESIGNA!ION
7
eq
10
11
2 ¡S* 0 0 . 0416
¡25 ►34 . 0440
24,98 . 0434
9 28 21. 99 ¿3 82
24
20
U>
24
í>76
32 9.82 . 01 70
32 14. 14 0245
28 16. 84
25
625
36 1 1. 04 . 0177
32 14.14 0226
28 16. 84 . 0269
21.99 . 0352
24 24.00 . 0384
.20 25. 34 . 0405
26
676
36 1 1 . 04 . 016 3
32 14. 14 0209
32 19. 24 . 02 85
28 21.99 G 325
24 2 4 . 00 . 03 55
25, 34 .0375
20 31. 23 . 04 62
27
729
1 2 .2 ? . 0163
36 15* SO .0218
32 1 9. 24 . 0264
32 25. 13 . 0345
28 23.00 . 0384
24 30.40 . 0417
20 31.23 . 0428
28 28. 00 . 0357
24 30 . 4 0 . 03 88
20 31. 23 . 0 39 8
28
24 37. 47 . 0446
40
2¡8 [ r
|:
20
16
24.98 . 0400
28
784
40 12.27 . 01 57
36 1 5 . 90 * 0203
36 2.1. 65 . 02 76
32 25.13 . 0321
29
841
44 13. 50 . 0161
40 17.07 02 10
36 2 1 . 65 . 0257
28. 27 . 03 36
28.00 . 0333
24 30. 40 . 0362
32 3 2 . 00 . 03 55
28 ,35. 47 . 0394
24 37 . 47 . C 4 16
.
36
30
900
44 13. 50 . 0150
40 .17. 67 . 0196
40 24.05
.026'/
3 6' 26. 27 . 0314
31
961
44 13. 50 . 0140
44 19. 44
. 0202
40 24. 05 . 0250
36 28. 27 . 0294
32 32.00 . 0333
28 35. 47 . 0 369
24 37. 47 . 03 90
48
32
1024
14.73
44 19. 44 .0.190
40 24, 05 . 0235
40 31. 42 . 0307
32 32.00 «0312
28 35. 47 . 0346
24 37. 47 . 03 66
48
. 01 44
33
1089
48 1 4. 73 . 0135
21 . 21 .0195
44 26. 46 . 0243
40 31.42 .0288
36 3 6 . 00 . 0330
32 40. 54 . 03 72
28 43. 72 . 04 01
28
34
1156
52 1 5 . 95 . 0138
48 21. 21 . 0183
44 26. 46 . 0229
44 34. 56 . 0299
36 3 6 . 00 . 0311
32 4 0 . 54 . 0351
43.72 . 0378
35
1225
52 1 5 . 95 . 0130
48 2 1 . 21 . 01 7 3
48 28. 86 . 02 36
44 34. 56 . 02 82
40 40. 00 . 0326
32 40. 54 . 0331
28 4 3.72 .0357
36
1296
56 17. 18 . 01 33
52 22. 97 . 01 77
48 28. 86 . 0223
44 34 . 56 . 0 2 67
40 40. 00 . 03 08
36 45.60 . 03 52
32 49.97 . 0386
37
1369
56 1 7. 18 .0125
52 2 2. 97 . 0168
52 31. 27 . 0 2 28
48 37. 70 . 02 75
40 40.00 . 0292
36 45.60 .0333
32 49.97 . 03 65
t°=11 c
type
& á>
# t
Si
JU __
Pt
For each bar &ize and dimensión* thre* valuee ar® ehoun.. They repres en t , in o rder: The maximutn n u m b jr of bar a permittad; claaimancat cavar and aplica type oonoidertd. The area of úteel, A9 t probidad by ¡V bars, sq. in. Thit eteel r a t i o , A$%/A 9*
RADIAL SPL ICE Tablea con&ider 1 inch máximum a g g r eg a te. Add 1 i n. for p iles and formed aolumns exponed to weather or earth. Add 2 in. for columna or piere formed again&t ea r t h .
3,2,1.
REINFORCEMENT
MAXIMUM NUMBER OF 3ARS PERM1TTED FOR COLUMN TYPE
(1/2 1HCH TI ES)
AND SPLICf TYPE iLLUSTRATED BAR SIZE DESiGHATION
ng sq. in
in.
10 ! 8
10
11
2 * 45 . 0245
121
12 3,68 ,0304
j 3.53 1.0292
12
12
14
15
12
169
12 3.68 €218
5. 30 . 0314
7 . 22 . 04 27
0400
4 6.25 0625
8 6.28 0519
8 8.00 0661
4 5.07 0419
4 6.35 . 0516
4 9, OOl .0744
8
10. 13 . 0704
4 6.25 . 0434
4 9 . 00 .0625
8 1 2. 49 . 0739
4 9.00 . 0533
8 12.49 . 0637
4 9.00 0459
8 12* 49 . Q55S
4 9.00 . 04 00
S 16 7.07 1.0361
12
196
16 4.91 0250
225
16 4.91 0218
16 7.07 . 03 14
6.14 0240
i
6. 2 8 . 0436
8. 00
0555
18
12 9.42 . 05 58
0473
1 0 . 13 . 0600
7.22 . 0368
12 9.42 . 0481
12 1 2 . 00 . 0612
10. 13 . 0517
16 9.62 .0428
16 12. 57 . 0559
12 12.00 .0533
| 20
16 9.62 . 0376
16 12.57 . 04 91
1 2 . 00 . 0468
15. 20 . 0594
12 18. 74 . 0732
8 18.00 . 0703
>6.00
20
16 12.57 . 04 35
16 16. 00 . 0 5 53
12 15.20 . 05 26
12 18. 74 .0648
8 18. 00 . 0623
lé.OÜ
.0554
20
16 20. 27 . 0626
12 18.74 . 05 78
8 18.00 . 0556
4 10.00 .0494
12
I 8.84 1.0345
8 . 00
12
15. 20 .0676
12
12
4
16-00 .07 n
4 0625 4
289
6.14 . 0212
8.84 | . 03 0 6
20 12. 03 . 0416
324
24 7. 36 .. 0227
20 l. 84 , 0273
20 12.03 .0371
15.71 .0485
16 1 6. 00 . 0 4 93
361
24 7. 36 . 02 04
24 ¡10.60 . 02 94
20 1 2 . 03 . 0333
20 15.71 .0435
16 16. 00 0443
16 2 0. 27 .0561
12 1 8. 74 . 05 19
2 7. 00 . 0748
24 14.43 . 03 61
24 18. 85 . 04 71
20 20 . 05 0 0
.00
16 20.27 . 0507
16 24.98 .0625
27.00 . 0675
16.00
400
28 I 24 8.59 1 0. 60 0215 i . 02 65 28 12.37 I . 02 8 0
24 14. 43 . 0327
24 18.85 .0427
20
441
28 8.59 . 01 95
2 0 00
.
. 0453
16 20.27 . 0460
16 2 4. 93 . 05 67
12 27.00 . 0613
32.00 .0726
484
32 9. 82 . 0203
28 12.37 . 0 2 56
28 16.84 . 0 3 48
24 1 8. 85 .0389
20 2 0 00 . 04 13
.
20 25. 34 . 0523
16 2 4 . 98 . 0516
12 2 7. 00 . 05 58
ft 32.00 .066’l
32 14. 14 . 02 67
28 16.84 .0318
28 2 1 . 99 .0416
24 24.00 . 0453
20
529
32 9.82 . 0186
25.34 . 0479
16 2 4 . 98 . 0472
16 36.00 . 0681
32.00 .0605
20
17
7. 22 , 0501
3.68
256
. 06 28
4 5. 0 7 0507
8
12
5. 30 ¡. 0368
20
16
4. 81 0398
— — r W-
00
6.28
8
. 02 56
13
4 -81 0481
3.53 .0353
100
144
12
a
8
8
(
u
18
19
20 21
22 23
16.00 .0443
12
4
.0400
» •» ¡ " ''• y ü U .
——--
fi
i
A 1
4
12
-di— \Pt i
m _____ f l
The máximum nimbar o f bar* ***•«* aover and » p l i c « t-UP* bare The área o f * . t p r o v e a by » bare. The eteol v a t i o , Át t / * g •
bearing s p l i c e
acolumna % s orr ^ * = ^ 3 « ,r s « , - - í ! í p ier ^ formed agarnet earth.
.
•