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Automatic design of sloped spread footings
I
Build. Sci. Vol. 7, pp. 53-59. Pergamon Press 1972. Printed in Great Britain
1(16"I)1
I (A3f~
Automatic Design of Sloped Spread Footings A. PISANTY* M. GELLERTt The design of sloped concrete spread footings, conforming to the proposed requirements of the new AC1 Building Code 318-71 is presented. ,4 computer program for automatic design is explained. Results are compared with flat concrete spread footings, proving sloped .footings to be considerably more economical for low soil pressures.
INTRODUCTION
require special formwork. Furthermore, a minimum slope is recommended as a good practice for drainage of casual ground water from the top of the footing. Efficient design may therefore provide sloped spread footings, meeting the requirements of the codes while saving substantial amounts of concrete and reinforcement (at the cost of a small increase of footings' depth at the vicinity of the column only) (figure 1). The purpose of this paper is to outline a design procedure for the proportioning of sloped concrete footings, accounting for slopes for which no special formwork is required, conforming to provisions of the ACI Code. The ACI Building Code treats the subject of footings very extensively, therefore this work follows its provisions. Regretfully, however, even in this code provisions for sloped footings are given in a rather general and noncommittal way. While the 318-63 code made clear distinction
THE DESIGN of single spread footings is a very frequently encountered, trial and error procedure generally involving the checking of several selected values of footing dimensions and reinforcement before satisfactory compliance with codes is achieved. Attempts to facilitate design in the form of homographs[8], design aids[7, 9, 11] and tables[10] have been made. McCormac's tables[10] are by no means the most complete and detailed. Sloped spread footings are mentioned occasionally only and extensive treatment is rare[6] if at all. For a variety of single spread footings (mainly on low bearing capacity soil) their action as a structural element is dominated by the resistance to shear failure, in close analogy to fiat concrete slabs. Accordingly, sufficient structural depth at the center of the footing providing for shear, diagonal tension and bending moment, critical at the vicinity of the column, will provide efficient design. Along
Y
L
Max. strUctural depth
Fig. 1. Structural profiles of sloped footing vs. flat footing.
(and allowance for design) between working stress and ultimate stress design, the new code is based on ultimate strength only. Option is left, however, to the designer to apply the "straight line theory", very similar to the previously outlined WSD. It is felt that foundations, more than other structural elements, should be designed employing the more conservative trend. It should be mentioned also that ultimate strength design is generally backed by wide experimental programs, evidence of which for sloped structural elements in bending and shear is
the periphery minimum dimensions will suffice, provided that any section of the footing meets the requirements of the code. From the point of view of construction, casting and vibrating concrete of 2500 psi quality and above, at slopes 1 : 2.5-1 : 3 do not present any difficulties and certainly do not *Lecturer, Faculty of Civil Engineering, Technion Israel Institute of Technology. tSenior Lecturer, Faculty of Civil Engineering, Technion Israel Institute of Technology. 53
54
A. Pisanty and M. Gellert
scarce, As a result, the straight line theory is adapted, along with other provisions of the new AC! Code.* Results obtained through the developed computer program clearly show the tendency of considerable savings of concrete and reinforcement vs. the quantities obtained for flat footings.
and Q = ( b + 6 in.) 2 0.30,L'+0.0862(f~-./~,'l
(31
f o r b _ > 22 in. The footing is assumed to acquire depth d (to be checked), a minimum edge thickness of 6 in. and a flat area of 3 in. around the column for its future formwork (figure 2).
S U M M A R Y OF THE PROVISIONS FOR FOOTINGS IN THE ACI CODE
The code requires the following to be checked or established: 1, Shear as a diagonal tension--sections 11.2.2 and 11.4.1 2. Wide beam shear--sections 11.2.2 and 8.1.2.3 3. Bending moment--section 8.1.2 4. Development of reinforcement--section 12.5.1 5. Bearing on the top of the footing--sections 10.14 and 15.6.6 6. Minimum edge thickness--section 15.9 7. Maximum tensile stress in plain concrete footings--section 15.7.2 8. Angle of slope--section 15.3.1 9. Effective cross-section in compression--section 8.1.2. I.
~gm~6in~d i
. . . . .
o ..........
~ - ~ - 3,n(reinf. cover)
Fig. 2. Footing section (initial contour).
2. Diagonal shear (punching) is checked at a distance ½dr from the face of the column where depth equals dz t (figure 3) by: q[a 2 - (b + d~ )z]
c~ =
4(b+d~)d,
2\/(.11.').
(4)
DESIGN P R O C E D U R E FOR S L O P E D SPREAD FOOTINGS
In the following a design procedure, based on the preceding provisions and some additional assumptions, for the proportioning of sloped concrete square footings is presented. The initial data assumed are: Column size-b (in.): soil pressure--q (psf); column l o a d - - Q (kips): maximum allowable slope of footings-~b~,~; allowable stress in reinforcement--J]. (psi) and concrete strength in compression--J~' (psi). The footing size is given by:
t
a
(1)
I. Provision for bearing is made by establishing the minimum column size considered for every footing as to comply with one of the following relations (maximum 8 per cent reinforcement is assumed) : Q = 0-92. 1.5b 2 0"30J;' + 0.08bay;
"1
Fig. 3. Section .[or shear (diagonal tension) checking.
Equation (4) provides for dl. The slope angle is obtained and checked: dl - 6 in. tg q5
a =
"
½[a-(b+dO]
< tg qS.,~x
(5)
g is to be increased if the acquired slope exceeds the allowable: g = d~ - ½ [ a - ( b + d O ]
tg ~b.~x
(6)
d--the depth of the footing at the face of the column : d = g + ½ [ a - ( b + 6 in.)] tg q5 (7)
(2)
for b < 20 in.
At this stage the initial profile of the footing is accomplished and left to be checked for bending moment and shear (figure 3).
* This study was originally performed according to the 318-63 Code and later converted to conform to the new Code.
t Diagonal shear (and later shear) is checked for an angle steeper than 45°, which proves realistic for deep beam effects and is on the safe side in general.
Automatic Design of Sloped Spread Footings
55
3. Bending moment at the face of the column: Mm~x = ½qa
(8)
The maximum bending capacity of the section at the face of the column is checked assuming f~ = 0.45f~' for width of (b + 6 in.) at the top. At the neutral line: B = b + 6 in. + 2kd cot q~
(9)
For B < a the bending moment capacity of the section (figure 4a) is obtained by: M = ½f~kd2[(b+ 6 in.)(1 - ½k) + ~kd cot q~(1-~k)]
(io)
Fig. 5. Section for shear (wide beam) checking.
The total shear force: V = ½qa[a-(b+2d2)]
(14)
Width of the section where shear is to be checked: B = b+2d2+2k2d2 cot ~
(15)
k2 is obtained considering the existing bending moment at the section and its flexural capacity, thus providing for the actual dimensions of the section to be checked for shear.
Id
B = aifk2d2+g >dz
(a)
The shear stress is given by:
Fig. 4(a). Effective cross-section in compression (B <=a).
For B > a (figure 4b) M is given by: M = ½f~kdZ[(b+6 i n . ) ( 1 - k k ) + ~ k d cot 4~(1-½k)] I f COt ~ . . . .
-*Jc ~ t g - a t L - k ) ] 3 [ g + d ( l - k ) ]
(11)
V v~ = Bd2 < 1 "0 ~/(f'e)
(16)
If dz proves unsatisfactory g should be increased. When g reaches d, then d is increased. 5. Development length is given by: L = 0"04as
L
V(D,
-
(17)
The maximum available development length: Lm,~ = ½ [ a - ( b + 6 in.)]
(18)
It is advisable to initiate with bars # 11 and to decrease the diameter if necessary for compliance with Lmax. THE COMPUTER PROGRAM
(b) Fig. (4b). Effective cross-section in compression (B > a).
For the case Mm,x exceeding M, d is increased until satisfactory flexural capacity is obtained. The reinforcement area As: As
Mm,x jdfs
(12)
4. Wide beam shear is checked for a section at a distance dz from face of column where depth is dz (figure 5): d2 = tg ~p(a-b)+2g 2(1 + t g qb)
(13)
The developed computer program is aimed to produce tables for square sloped spread footings, supporting columns having square sections. Soil pressure distribution is assumed uniform. All parameters mentioned previously (q, q~. . . . f~, f'~ etc.) may be varied. For each footing size a series of ten (or more at request) values of column sizes are provided, while the lowest size is established from maximum bearing stresses considerations (but not less than 6 in.). Concerning reinforcement, it is assumed that for economical design greatest possible diameter should be used. Therefore the program's initial check is for # 11. Minimum bars' space of 4 in. and maximum of 14 in. is assumed.
A. Pisanty and M. Gellert
56
Table I. Square Jbotings table. Sloped footings vs. flat footings (fc' = 2500 psi f~ = 20 000 psi
soil press. -- 5000 psf) i
Column Footing load size (Kips) fit-in.)
tgq~ = ] Pier Footing Edge size depth thickn, (in.) (in.)
Footing reinf,
Footing volume (ft 3)
i
Column Footing load size (Kips) (It-in.)
tg0~ = 0 Pier Footing size depth (in.) (in.)
Footing reinf,
Footing volume (ft 3)
172"0 172"4 172"7 172-7 173"1 173"5 173-4 173'6 173'8 173.9
6-0 6-0 6-0 6~ 6-0 6-0 6-0 6-0 6-0 6-0
8 10 12 14 16 18 20 22 24 26
22 22 20 19 18 17 16 15 15 14
13 13 13 13 12 12 12 12 11 12
7 7 7 7 7 7 7 7 6 7
# # # # # # # # # #
7 7 7 7 7 7 7 7 7 7
50'1 50-4 48"2 47"1 44'4 43-2 42-0 40'6 39"4 39"3
170"5 170"9 171'4 171"8 172"3 172'3 172"7 173'2 173"6 173-6
6--0 6-4) 6-0 6~ 6-0 6-0 6-0 6-0 6-0 6-0
8 10 12 14 16 18 20 22 24 26
20 19 18 18 17 16 15 15 14 14
6 6 6 6 7 7 7 7 7 6
# # # # # # # # # #
8 8 8 8 7 7 7 7 7 7
60.0 57"0 54"0 54'0 51-0 48"0 45"0 45.0 42'0 42"0
232"5 233"0 233.4 233"6 233"9 234'4 234'4 234"9 235"2 235.4
7-0 7-0 7-0 7-0 7-0 74) 7-0 7-0 7-0 7-0
8 10 12 14 16 18 20 22 24 26
26 25 24 23 22 21 20 19 18 17
15 15 14 14 14 14 14 13 13 13
7 7 7 7 7 7 7 7 7 7
# # # # # # # # # #
8 8 8 8 8 8 8 8 8 8
79"1 77'9 74'3 73'0 71.7 70-2 68'6 64.9 63-2 61.4
230'2 230"9 231-5 232.1 232.7 232.7 233.3 233'9 234"5 234'5
7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0
8 10 12 14 16 18 20 22 24 26
23 22 22 21 20 19 18 18 17 16
8 8 8 8 8 8 8 7 7 7
# # # # # # # # # #
8 8 8 8 8 8 8 8 8 8
93"9 89"8 89"8 85'7 81"7 77"6 73-5 73"5 69"4 65'3
302-2 303-2 303"5 303"4 304"0 304-3 304'6 305"3 305-3 306"0
8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0
10 12 14 16 18 20 22 24 26 28
29 27 27 26 24 24 23 22 21 20
16 16 16 15 16 15 15 15 14 14
9 8 9 9 9 9 9 9 9 9
# # # # # # # # # #
8 9 8 8 8 8 8 8 8 8
112-9 109.2 109"8 105'0 103"9 101-5 99'6 97"5 92"6 90'4
299-1 299'9 299-9 300-7 301-5 302'3 303"1 303-1 303'9 304"7
8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0
I0 12 14 16 18 20 22 24 26 28
26 25 24 23 22 21 21 20 19 19
8 8 8 8 8 8 8 8 8 9
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 8
138"6 133"3 127"9 122'6 117'3 111'9 111.9 106-6 101.3 101'3
381-1 381'4 381"7 382"6 382"9 382'8 383.6 384"0 384.5 385.3
9-0 9-0 9-0 9-0 9-0 9-0 9--0 9-0 9-0 9-0
12 14 16 18 20 22 24 26 28 30
31 30 30 28 27 27 25 25 24 23
18 17 17 17 17 16 16 16 16 16
9 9 9 9 9 9 9 9 9 9
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 9
156"4 150'4 151"1 146"2 143'9 140-8 135'4 136-0 133"4 130'7
375"6 376-6 377'6 378"6 379"6 379"6 380-6 381"7 382'7 382'7
9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0
12 14 16 18 20 22 24 26 28 30
28 27 26 25 25 24 23 22 22 21
10 10 10 10 10 10 10 10 9 9
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 9
188'9 182"2 175-4 168'7 168.7 161"9 155'2 148"4 148"4 141.7
467.1 467"5 468"6 469.0 469'4 469"8 470-9 471.3 471.2 472.2
10-0 10-0 10-0 10-0 10-0 10~ 10-0 10-0 10~ 10-0
12 14 16 18 20 22 24 26 28 30
35 34 33 32 31 30 29 28 28 26
19 19 19 18 18 18 18 18 17 17
11 11 11 11 11 11 11 I1 II 11
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 9
210.4 208.1 205"6 198.2 195"6 192-9 190-1 187"1 183.3 176-4
459.9 461"2 462.4 463"7 463-7 464'9 466"2 467.4 468"7 468.7
10-0 10-0 10-0 10-0 10-0 10-0 10-0 lifo 10-0 10-0
12 14 16 18 20 22 24 26 28 30
31 30 29 28 28 28 26 25 25 24
10 10 10 10 I0 10 I0 10 10 10
# # # # # # # # # #
10 10 10 10 10 10 10 10 10 10
258'3 250"0 241"6 233"3 233'3 225-0 216"6 208-3 208"3 200-0
Automatic Design o f Sloped Spread Footings
57
Table 2. Square footings table. Sloped footings vs. fiat footings It" = 2500 l~Si f~ = 20 000 psi tg~6 = ½ Column Footing Pier Footing Edge load size size depth thickn, (in.) (in.) (Kips) fit-in.) (in.)
Footing reinf,
Footing volume (ft a)
soil press. = 10 000 psf)
Column Footing Pier load size size (Kips) fit-in.) (in.)
18 20 22 24 26 28
8#8 8#8 8#8 10# 10# 10# 9#7 9#7 9#7 11#
7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0
12 14 16 18 20 22 24 26 28 30
30 29 28 27 26 25 24 23 22 21
8#9 8#9 10# 10# 10# 10# 10# 9#8 12# 12#
612"7 613"5 614"3 615-1 615-9 616'7 617'5 618'3 619"I 619-9
8-0 8-0 8-0 8-0 8--0 8-0 8-0 8-0 8-0 8-0
14 16 18 20 22 24 26 28 30 32
33 32 31 30 29 28 27 27 26 25
221"4 219"4 213"4 207"5 198"5 199"3 193.4 184'2 185"0 179.0
772-5 773"5 774"5 775"5 776'5 777.5 778.6 779.6 779"6 780'6
9--0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0
16 18 20 22 24 26 28 30 32 34
301"6 294"3 286"9 276"0 277"0 269"6
948"7 949"9 951"2 952"4 953"7 954"9 956"2 957"4 958'7 959'9
10-0 10-0 10-0 10-0 10-0 10-4) 104) 104) 10-0 10-0
18 20 22 24 26 28 30 32 34 36
6-0 6--0 6-0 6-0 6--0 6-0 6-0 6-0 6-0 6-0
10 12 14 16 18 20 22 24 26 28
30 28 27 26 24 24 23 21 21 20
21 20 19 18 17 17 16 15 15 14
7 7 7 9 9 8 8 8 8 10
# # # # # # # # # #
8 8 8 7 7 7 7 7 7 6
74'4 70"5 67"8 65'2 61-1 61.4 58.8 54-5 54"9 52"2
347.8 348'2 348.7 349.1 349.6 350.0 350"5 350"9 351"4 351'8
6-0 6-0 6-0
10 12 14
6-0
16
6-0 6-0 6-0 6-0 6--0 6-0
473'3 473"2 473'8 474"6 475"1 475-7 475.9 476"4 477-0 477"2
7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0
12 14 16 18 20 22 24 26 28 30
33 32 31 29 29 28 26 26 25 23
23 22 21 20 20 19 18 18 17 17
8 8 9 I0 9 9 9 8 11 11
# # # # # # # # # #
9 9 8 8 8 8 8 8 7 7
111"0 •07"4 103"8 98"4 98"9 95-3 89"7 90"1 86"5 82-6
471"6 472'2 472"8 473-4 474"0 474-6 475'2 475"9 475"9 476"5
615'1 615'8 616.9 617'6 617"5 618-6 619'3 620'0 620"3 621"0
8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0
14 16 18 20 22 24 26 28 30 32
37 36 35 34 33 31 31 30 28 28
25 24 24 23 22 21 21 20 19 19
I0 # 10 # 10# 10# 9 # 10 # 9 # 11 # 11 # I1 #
9 9 9 9 9 9 9 8 8 8
160-0 155"3 153'6 148-9 144'2 137"0 137"6 132'9 125"5 126"1
775"7 777'0 776'9 777"8 779-1 780"0 779"9 781'3 782"2 783"1
9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0
16 18 20 22 24 26 28 30 32 34
41 40 39 38 36 36 35 33 33 32
27 27 26 25 24 24 23 22 22 21
IOB 10# 10# 10# 12 # 12 # 11 # 12 # 11 # 11 #
10 10 10 10 9 9 9 9 9 9
953-4 954"6 955.7 957"3 957"1 958-2 959'9 961.0 960.9
10-0 10-0 104) 10-0 10-0 10-0 10-4) 10-0 104) 104)
18 20 22 24 26 28 30 32 34 36
45 44 43 41 41 40 38 38 37 35
30 29 28 27 27 26 25 25 24 23
10 12 12 12 12 12 12 11 11 14
11 10 10 10 10 10 10 10 10 9
962"6
Footing reinfi
26 25 24 23 22 21 21 20 19 18
348"8 349'4 349"8 350"2 350'3 350"7 351"1 351'3 351"7 352-1
# # # # # # # # # #
tg~b = 0 Footing depth (in.)
258"5 259'5
252"1 240'7
7 7 7
6
Foo-ing volume (ft 3) 78'0 75.0 72-0 69"0 66"0 63"0 63"0 60.0 57.0 54.0
7 7
122"4 118-4 114'3 110'2 106"1 102'0 98.0 93.9 89-8 85-7
11# 11# 11# 11# 10# 10# 10# 12# 12# 12#
9 9 9 9 9 9 9 8 8 8
175.9 170-6 165'3 159"9 154"6 149"3 143"9 143-9 138.6 133"3
36 35 34 34 33 32 31 30 29 28
11 # 11 # 11 # 11 # 13# 13# 13# 12# 12# 12#
10 10 10 I0 9 9 9 9 9 9
242.9 236"2 229'4 229.4 222-7 215.9 209-2 202-4 195.7 188.9
40 39 38 37 36 35 34 33 32 32
11 # 13 # 13 # 13 # 13 # 13 # 13 # 13 # 13 # 15#
11 10 10 10 10 10 10 10 10 9
333'3 325'0 316-6 308'3 300-0 291 "6 283 "3 275.0 266"6 266"6
8 8 8 8 8
58
A. Pisanty and M. Gellert
From the initial design load the footing's weight is reduced and the footing redesigned. The process is repeated until a tolerance of I in. is achieved, A minimum concrete cover of 3 in. is provided for the reinforcement (to the bar centre). Additional I in. is given for bars larger than # 6. In the final stage the design of each footing is completed by checking tensile stresses of concrete in flexure. For values lower than 1-6 ~,/(.f'~) reinforcement is omitted. All parameters in the program are easily varied and different basic data nested. Computer time is extremely low--more than 60 footings are obtained in one minute.
pressure
Soft
20001b/ft
,,~ Concrete ,i, R e m f o r c e m e n l
25
#
.
L=
£
'
'
20
(a)
.... :'
', 5
10
/a
Footing's size, Soil
EVALUATION AND COMPARISON OF RESULTS
ft
pressure=5000
ib/ft 2
25
Concrele Reinforcemenl
~'~
20
Concrete and reinforcement quantities for sloped footings having a maximum slope of 1:3 (approx. 18.5 ° ) are compared with flat footings. The comparison indicates sharp reductions for low soil pressures which tend, however, to diminish for higher soil pressures. For soil pressure of 2000 psf. the concrete quantities approach average values of 30 per cent less than those in fiat footings, while the corresponding reduction of reinforcement approaches 20 per cent. For increased soil pressures, concrete reduction tends to be less significant, until it vanishes for 20 000 psf, while reinforcement is approximately 10 per cent less, mainly due to the increased footing depth at the centre of the footing, Tables I and 2 provide a small range of the values obtained for soil pressures of 5000 and 10 000 psf. respectively compared with values for fiat footings produced by the same computer program. Figure 6(a-e) provides a graphical representation of the average reductions of concrete and reinforcement quantities for five different values of soil
2
30
z, , . . ' ' +~
2~
" ' ,. : '
' '
"
L m
L `%
(b) "6 "8
:\
2~
4
~
8
L! Eootrng'S
15t
~O
,'.
Concrete
,3
Reinforcement
(C)
~
Soil
pressure =lOOO0
0
'
!4
¢
iB
ft
size.
Ib/ft 2
"
DE
2
4
~
5
IC
FoOtlng's
12
size,
4
Ig
,B
4
t6
~8
ft
Soi} pressure= 1 5 0 0 0 I b / f t 2
Concrete Reinforcement
A
pressures. (d)
'~
(
~ ,s
~ ,
:
:'
::~:'
CONCLUDING REMARKS A procedure for the design of sloped spread footings is outlined following the provisions of the proposed ACI Code 318-71. A computer program performing the automatical design is explained. The tabulated values presented in this paper serve as an example and for comparison purposes. The authors intend to produce tables covering a wide range of footing sizes. Tables appear to be in many ways preferable to nomographs and design aids of other kinds. Finally it is of interest to point out that comparison between results obtained employing the new code and those using the 318-63 Code indicate
4
6
8
i,C*
Footing's
size,
t2
ft
SoJI pressure= 2 © O 0 0 ~b/f t z L5
COncrete Reinforcemen!
L c
}
(e) c
0 LO 2
'
, ,
£' 4
£' 6
~
"~
~" 8
'
FOOting'S
I0
size,
[2
14
J6
J5
ft
Fig. 6(a), (b), (c), (d), (e). Average reduction o f concrete and reinforcement quantities for various soll pressures.
Automatic Design of Sloped Spread Footings insignificant differences in structural depth averaging 1 in. and almost no difference in reinforcement. The only real difference lies in the fact that due to more severe regulations for minimum embedment length in the new code, most o f the very small footings are unreinforced. Some increase of structural depth is obtained as well due to reduced allowable shear (wide beam) stresses.
d2 g M V j
L vc
NOMENCLATURE a b Q q ~b d
d~
side of the footing (square), ft. side of the column (square), ft. load carried by footing, kips. soil pressure, psf. angle of slope. distance from extreme compression fibre to centroid of tension reinforcement
As as f~ fc f~ fy
59
footing depth at section diagonal tension to be checked. footing depth at section shear to be checked. edge thickness. bending moment. total shear, kips. ratio of distance between centroid of compression and centroid of tension reinforcement to the depth d. embedment length of reinforcing bars. shear stress carried by concrete (also diagonal), psi. reinforcement area, in 2. area of reinforcing bar, in 2. specified compressive strength of concrete, psi. allowable concrete stress, psi. allowable stress in reinforcing steel, psi. yield stress of reinforcing steel, psi.
REFERENCES 1. ACI Committee 318, Proposed Revision of AC1 318-63, "Building Code Requirements
for Reinforced Concrete", ACIJ. Proc. 67, No. 2 (1970). 2. Discussion 67-8--"Discussion of a Report by ACI Committee 318--Discussion and Committee Closure", ACIJ. Proc. 67, No. 9 (1970). 3. AC1 Committee 318, Building Code Requirements for Reinforced Concrete (ACI 318-63), American Concrete Institute, Detroit, June 1963, 144 p. 4. ACI Committee 317, Reinforced Concrete Design Handbook SP-3, American Concrete Institute, Detroit, 1965, 271 p. 5. Commentary on Building Code Requirements for Reinforced Concrete (ACI 318-63), ACI Publication SP 10. 6. C. W. DUNHAM,Foundations of Structures, McGraw-Hill, New York (1962). 7. R.W. FtJaLONO, Design aids for square footings, ACIJ. Proc. 62, 363 (1965). 8. E. HENYE,Nomographs for design of rectangular spread footings, ACIJ. Proc. 66, 545 (1969). 9. J.P. KOnLV, Optimum design of concrete spread footing by computer, ACIJ. Proc. 65, 384 (1968). 10. M. S. McCORgAC, Square Footing Tables, Stress Publications., 208 p. Rockville, Maryland, (1968). 11. C. RODRIOt~eZ, Design of isolated square column footing, ACIJ. Proc. 61, 889 (1964).
Build. Sci. Vol. 7, pp. 53-59. Pergamon Press 1972. Printed in Great Britain
1(16"I)1
I (A3f~
Automatic Design of Sloped Spread Footings A. PISANTY* M. GELLERTt The design of sloped concrete spread footings, conforming to the proposed requirements of the new AC1 Building Code 318-71 is presented. ,4 computer program for automatic design is explained. Results are compared with flat concrete spread footings, proving sloped .footings to be considerably more economical for low soil pressures.
INTRODUCTION
require special formwork. Furthermore, a minimum slope is recommended as a good practice for drainage of casual ground water from the top of the footing. Efficient design may therefore provide sloped spread footings, meeting the requirements of the codes while saving substantial amounts of concrete and reinforcement (at the cost of a small increase of footings' depth at the vicinity of the column only) (figure 1). The purpose of this paper is to outline a design procedure for the proportioning of sloped concrete footings, accounting for slopes for which no special formwork is required, conforming to provisions of the ACI Code. The ACI Building Code treats the subject of footings very extensively, therefore this work follows its provisions. Regretfully, however, even in this code provisions for sloped footings are given in a rather general and noncommittal way. While the 318-63 code made clear distinction
THE DESIGN of single spread footings is a very frequently encountered, trial and error procedure generally involving the checking of several selected values of footing dimensions and reinforcement before satisfactory compliance with codes is achieved. Attempts to facilitate design in the form of homographs[8], design aids[7, 9, 11] and tables[10] have been made. McCormac's tables[10] are by no means the most complete and detailed. Sloped spread footings are mentioned occasionally only and extensive treatment is rare[6] if at all. For a variety of single spread footings (mainly on low bearing capacity soil) their action as a structural element is dominated by the resistance to shear failure, in close analogy to fiat concrete slabs. Accordingly, sufficient structural depth at the center of the footing providing for shear, diagonal tension and bending moment, critical at the vicinity of the column, will provide efficient design. Along
Y
L
Max. strUctural depth
Fig. 1. Structural profiles of sloped footing vs. flat footing.
(and allowance for design) between working stress and ultimate stress design, the new code is based on ultimate strength only. Option is left, however, to the designer to apply the "straight line theory", very similar to the previously outlined WSD. It is felt that foundations, more than other structural elements, should be designed employing the more conservative trend. It should be mentioned also that ultimate strength design is generally backed by wide experimental programs, evidence of which for sloped structural elements in bending and shear is
the periphery minimum dimensions will suffice, provided that any section of the footing meets the requirements of the code. From the point of view of construction, casting and vibrating concrete of 2500 psi quality and above, at slopes 1 : 2.5-1 : 3 do not present any difficulties and certainly do not *Lecturer, Faculty of Civil Engineering, Technion Israel Institute of Technology. tSenior Lecturer, Faculty of Civil Engineering, Technion Israel Institute of Technology. 53
54
A. Pisanty and M. Gellert
scarce, As a result, the straight line theory is adapted, along with other provisions of the new AC! Code.* Results obtained through the developed computer program clearly show the tendency of considerable savings of concrete and reinforcement vs. the quantities obtained for flat footings.
and Q = ( b + 6 in.) 2 0.30,L'+0.0862(f~-./~,'l
(31
f o r b _ > 22 in. The footing is assumed to acquire depth d (to be checked), a minimum edge thickness of 6 in. and a flat area of 3 in. around the column for its future formwork (figure 2).
S U M M A R Y OF THE PROVISIONS FOR FOOTINGS IN THE ACI CODE
The code requires the following to be checked or established: 1, Shear as a diagonal tension--sections 11.2.2 and 11.4.1 2. Wide beam shear--sections 11.2.2 and 8.1.2.3 3. Bending moment--section 8.1.2 4. Development of reinforcement--section 12.5.1 5. Bearing on the top of the footing--sections 10.14 and 15.6.6 6. Minimum edge thickness--section 15.9 7. Maximum tensile stress in plain concrete footings--section 15.7.2 8. Angle of slope--section 15.3.1 9. Effective cross-section in compression--section 8.1.2. I.
~gm~6in~d i
. . . . .
o ..........
~ - ~ - 3,n(reinf. cover)
Fig. 2. Footing section (initial contour).
2. Diagonal shear (punching) is checked at a distance ½dr from the face of the column where depth equals dz t (figure 3) by: q[a 2 - (b + d~ )z]
c~ =
4(b+d~)d,
2\/(.11.').
(4)
DESIGN P R O C E D U R E FOR S L O P E D SPREAD FOOTINGS
In the following a design procedure, based on the preceding provisions and some additional assumptions, for the proportioning of sloped concrete square footings is presented. The initial data assumed are: Column size-b (in.): soil pressure--q (psf); column l o a d - - Q (kips): maximum allowable slope of footings-~b~,~; allowable stress in reinforcement--J]. (psi) and concrete strength in compression--J~' (psi). The footing size is given by:
t
a
(1)
I. Provision for bearing is made by establishing the minimum column size considered for every footing as to comply with one of the following relations (maximum 8 per cent reinforcement is assumed) : Q = 0-92. 1.5b 2 0"30J;' + 0.08bay;
"1
Fig. 3. Section .[or shear (diagonal tension) checking.
Equation (4) provides for dl. The slope angle is obtained and checked: dl - 6 in. tg q5
a =
"
½[a-(b+dO]
< tg qS.,~x
(5)
g is to be increased if the acquired slope exceeds the allowable: g = d~ - ½ [ a - ( b + d O ]
tg ~b.~x
(6)
d--the depth of the footing at the face of the column : d = g + ½ [ a - ( b + 6 in.)] tg q5 (7)
(2)
for b < 20 in.
At this stage the initial profile of the footing is accomplished and left to be checked for bending moment and shear (figure 3).
* This study was originally performed according to the 318-63 Code and later converted to conform to the new Code.
t Diagonal shear (and later shear) is checked for an angle steeper than 45°, which proves realistic for deep beam effects and is on the safe side in general.
Automatic Design of Sloped Spread Footings
55
3. Bending moment at the face of the column: Mm~x = ½qa
(8)
The maximum bending capacity of the section at the face of the column is checked assuming f~ = 0.45f~' for width of (b + 6 in.) at the top. At the neutral line: B = b + 6 in. + 2kd cot q~
(9)
For B < a the bending moment capacity of the section (figure 4a) is obtained by: M = ½f~kd2[(b+ 6 in.)(1 - ½k) + ~kd cot q~(1-~k)]
(io)
Fig. 5. Section for shear (wide beam) checking.
The total shear force: V = ½qa[a-(b+2d2)]
(14)
Width of the section where shear is to be checked: B = b+2d2+2k2d2 cot ~
(15)
k2 is obtained considering the existing bending moment at the section and its flexural capacity, thus providing for the actual dimensions of the section to be checked for shear.
Id
B = aifk2d2+g >dz
(a)
The shear stress is given by:
Fig. 4(a). Effective cross-section in compression (B <=a).
For B > a (figure 4b) M is given by: M = ½f~kdZ[(b+6 i n . ) ( 1 - k k ) + ~ k d cot 4~(1-½k)] I f COt ~ . . . .
-*Jc ~ t g - a t L - k ) ] 3 [ g + d ( l - k ) ]
(11)
V v~ = Bd2 < 1 "0 ~/(f'e)
(16)
If dz proves unsatisfactory g should be increased. When g reaches d, then d is increased. 5. Development length is given by: L = 0"04as
L
V(D,
-
(17)
The maximum available development length: Lm,~ = ½ [ a - ( b + 6 in.)]
(18)
It is advisable to initiate with bars # 11 and to decrease the diameter if necessary for compliance with Lmax. THE COMPUTER PROGRAM
(b) Fig. (4b). Effective cross-section in compression (B > a).
For the case Mm,x exceeding M, d is increased until satisfactory flexural capacity is obtained. The reinforcement area As: As
Mm,x jdfs
(12)
4. Wide beam shear is checked for a section at a distance dz from face of column where depth is dz (figure 5): d2 = tg ~p(a-b)+2g 2(1 + t g qb)
(13)
The developed computer program is aimed to produce tables for square sloped spread footings, supporting columns having square sections. Soil pressure distribution is assumed uniform. All parameters mentioned previously (q, q~. . . . f~, f'~ etc.) may be varied. For each footing size a series of ten (or more at request) values of column sizes are provided, while the lowest size is established from maximum bearing stresses considerations (but not less than 6 in.). Concerning reinforcement, it is assumed that for economical design greatest possible diameter should be used. Therefore the program's initial check is for # 11. Minimum bars' space of 4 in. and maximum of 14 in. is assumed.
A. Pisanty and M. Gellert
56
Table I. Square Jbotings table. Sloped footings vs. flat footings (fc' = 2500 psi f~ = 20 000 psi
soil press. -- 5000 psf) i
Column Footing load size (Kips) fit-in.)
tgq~ = ] Pier Footing Edge size depth thickn, (in.) (in.)
Footing reinf,
Footing volume (ft 3)
i
Column Footing load size (Kips) (It-in.)
tg0~ = 0 Pier Footing size depth (in.) (in.)
Footing reinf,
Footing volume (ft 3)
172"0 172"4 172"7 172-7 173"1 173"5 173-4 173'6 173'8 173.9
6-0 6-0 6-0 6~ 6-0 6-0 6-0 6-0 6-0 6-0
8 10 12 14 16 18 20 22 24 26
22 22 20 19 18 17 16 15 15 14
13 13 13 13 12 12 12 12 11 12
7 7 7 7 7 7 7 7 6 7
# # # # # # # # # #
7 7 7 7 7 7 7 7 7 7
50'1 50-4 48"2 47"1 44'4 43-2 42-0 40'6 39"4 39"3
170"5 170"9 171'4 171"8 172"3 172'3 172"7 173'2 173"6 173-6
6--0 6-4) 6-0 6~ 6-0 6-0 6-0 6-0 6-0 6-0
8 10 12 14 16 18 20 22 24 26
20 19 18 18 17 16 15 15 14 14
6 6 6 6 7 7 7 7 7 6
# # # # # # # # # #
8 8 8 8 7 7 7 7 7 7
60.0 57"0 54"0 54'0 51-0 48"0 45"0 45.0 42'0 42"0
232"5 233"0 233.4 233"6 233"9 234'4 234'4 234"9 235"2 235.4
7-0 7-0 7-0 7-0 7-0 74) 7-0 7-0 7-0 7-0
8 10 12 14 16 18 20 22 24 26
26 25 24 23 22 21 20 19 18 17
15 15 14 14 14 14 14 13 13 13
7 7 7 7 7 7 7 7 7 7
# # # # # # # # # #
8 8 8 8 8 8 8 8 8 8
79"1 77'9 74'3 73'0 71.7 70-2 68'6 64.9 63-2 61.4
230'2 230"9 231-5 232.1 232.7 232.7 233.3 233'9 234"5 234'5
7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0
8 10 12 14 16 18 20 22 24 26
23 22 22 21 20 19 18 18 17 16
8 8 8 8 8 8 8 7 7 7
# # # # # # # # # #
8 8 8 8 8 8 8 8 8 8
93"9 89"8 89"8 85'7 81"7 77"6 73-5 73"5 69"4 65'3
302-2 303-2 303"5 303"4 304"0 304-3 304'6 305"3 305-3 306"0
8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0
10 12 14 16 18 20 22 24 26 28
29 27 27 26 24 24 23 22 21 20
16 16 16 15 16 15 15 15 14 14
9 8 9 9 9 9 9 9 9 9
# # # # # # # # # #
8 9 8 8 8 8 8 8 8 8
112-9 109.2 109"8 105'0 103"9 101-5 99'6 97"5 92"6 90'4
299-1 299'9 299-9 300-7 301-5 302'3 303"1 303-1 303'9 304"7
8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0
I0 12 14 16 18 20 22 24 26 28
26 25 24 23 22 21 21 20 19 19
8 8 8 8 8 8 8 8 8 9
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 8
138"6 133"3 127"9 122'6 117'3 111'9 111.9 106-6 101.3 101'3
381-1 381'4 381"7 382"6 382"9 382'8 383.6 384"0 384.5 385.3
9-0 9-0 9-0 9-0 9-0 9-0 9--0 9-0 9-0 9-0
12 14 16 18 20 22 24 26 28 30
31 30 30 28 27 27 25 25 24 23
18 17 17 17 17 16 16 16 16 16
9 9 9 9 9 9 9 9 9 9
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 9
156"4 150'4 151"1 146"2 143'9 140-8 135'4 136-0 133"4 130'7
375"6 376-6 377'6 378"6 379"6 379"6 380-6 381"7 382'7 382'7
9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0
12 14 16 18 20 22 24 26 28 30
28 27 26 25 25 24 23 22 22 21
10 10 10 10 10 10 10 10 9 9
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 9
188'9 182"2 175-4 168'7 168.7 161"9 155'2 148"4 148"4 141.7
467.1 467"5 468"6 469.0 469'4 469"8 470-9 471.3 471.2 472.2
10-0 10-0 10-0 10-0 10-0 10~ 10-0 10-0 10~ 10-0
12 14 16 18 20 22 24 26 28 30
35 34 33 32 31 30 29 28 28 26
19 19 19 18 18 18 18 18 17 17
11 11 11 11 11 11 11 I1 II 11
# # # # # # # # # #
9 9 9 9 9 9 9 9 9 9
210.4 208.1 205"6 198.2 195"6 192-9 190-1 187"1 183.3 176-4
459.9 461"2 462.4 463"7 463-7 464'9 466"2 467.4 468"7 468.7
10-0 10-0 10-0 10-0 10-0 10-0 10-0 lifo 10-0 10-0
12 14 16 18 20 22 24 26 28 30
31 30 29 28 28 28 26 25 25 24
10 10 10 10 I0 10 I0 10 10 10
# # # # # # # # # #
10 10 10 10 10 10 10 10 10 10
258'3 250"0 241"6 233"3 233'3 225-0 216"6 208-3 208"3 200-0
Automatic Design o f Sloped Spread Footings
57
Table 2. Square footings table. Sloped footings vs. fiat footings It" = 2500 l~Si f~ = 20 000 psi tg~6 = ½ Column Footing Pier Footing Edge load size size depth thickn, (in.) (in.) (Kips) fit-in.) (in.)
Footing reinf,
Footing volume (ft a)
soil press. = 10 000 psf)
Column Footing Pier load size size (Kips) fit-in.) (in.)
18 20 22 24 26 28
8#8 8#8 8#8 10# 10# 10# 9#7 9#7 9#7 11#
7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0
12 14 16 18 20 22 24 26 28 30
30 29 28 27 26 25 24 23 22 21
8#9 8#9 10# 10# 10# 10# 10# 9#8 12# 12#
612"7 613"5 614"3 615-1 615-9 616'7 617'5 618'3 619"I 619-9
8-0 8-0 8-0 8-0 8--0 8-0 8-0 8-0 8-0 8-0
14 16 18 20 22 24 26 28 30 32
33 32 31 30 29 28 27 27 26 25
221"4 219"4 213"4 207"5 198"5 199"3 193.4 184'2 185"0 179.0
772-5 773"5 774"5 775"5 776'5 777.5 778.6 779.6 779"6 780'6
9--0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0
16 18 20 22 24 26 28 30 32 34
301"6 294"3 286"9 276"0 277"0 269"6
948"7 949"9 951"2 952"4 953"7 954"9 956"2 957"4 958'7 959'9
10-0 10-0 10-0 10-0 10-0 10-4) 104) 104) 10-0 10-0
18 20 22 24 26 28 30 32 34 36
6-0 6--0 6-0 6-0 6--0 6-0 6-0 6-0 6-0 6-0
10 12 14 16 18 20 22 24 26 28
30 28 27 26 24 24 23 21 21 20
21 20 19 18 17 17 16 15 15 14
7 7 7 9 9 8 8 8 8 10
# # # # # # # # # #
8 8 8 7 7 7 7 7 7 6
74'4 70"5 67"8 65'2 61-1 61.4 58.8 54-5 54"9 52"2
347.8 348'2 348.7 349.1 349.6 350.0 350"5 350"9 351"4 351'8
6-0 6-0 6-0
10 12 14
6-0
16
6-0 6-0 6-0 6-0 6--0 6-0
473'3 473"2 473'8 474"6 475"1 475-7 475.9 476"4 477-0 477"2
7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0 7-0
12 14 16 18 20 22 24 26 28 30
33 32 31 29 29 28 26 26 25 23
23 22 21 20 20 19 18 18 17 17
8 8 9 I0 9 9 9 8 11 11
# # # # # # # # # #
9 9 8 8 8 8 8 8 7 7
111"0 •07"4 103"8 98"4 98"9 95-3 89"7 90"1 86"5 82-6
471"6 472'2 472"8 473-4 474"0 474-6 475'2 475"9 475"9 476"5
615'1 615'8 616.9 617'6 617"5 618-6 619'3 620'0 620"3 621"0
8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0 8-0
14 16 18 20 22 24 26 28 30 32
37 36 35 34 33 31 31 30 28 28
25 24 24 23 22 21 21 20 19 19
I0 # 10 # 10# 10# 9 # 10 # 9 # 11 # 11 # I1 #
9 9 9 9 9 9 9 8 8 8
160-0 155"3 153'6 148-9 144'2 137"0 137"6 132'9 125"5 126"1
775"7 777'0 776'9 777"8 779-1 780"0 779"9 781'3 782"2 783"1
9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0 9-0
16 18 20 22 24 26 28 30 32 34
41 40 39 38 36 36 35 33 33 32
27 27 26 25 24 24 23 22 22 21
IOB 10# 10# 10# 12 # 12 # 11 # 12 # 11 # 11 #
10 10 10 10 9 9 9 9 9 9
953-4 954"6 955.7 957"3 957"1 958-2 959'9 961.0 960.9
10-0 10-0 104) 10-0 10-0 10-0 10-4) 10-0 104) 104)
18 20 22 24 26 28 30 32 34 36
45 44 43 41 41 40 38 38 37 35
30 29 28 27 27 26 25 25 24 23
10 12 12 12 12 12 12 11 11 14
11 10 10 10 10 10 10 10 10 9
962"6
Footing reinfi
26 25 24 23 22 21 21 20 19 18
348"8 349'4 349"8 350"2 350'3 350"7 351"1 351'3 351"7 352-1
# # # # # # # # # #
tg~b = 0 Footing depth (in.)
258"5 259'5
252"1 240'7
7 7 7
6
Foo-ing volume (ft 3) 78'0 75.0 72-0 69"0 66"0 63"0 63"0 60.0 57.0 54.0
7 7
122"4 118-4 114'3 110'2 106"1 102'0 98.0 93.9 89-8 85-7
11# 11# 11# 11# 10# 10# 10# 12# 12# 12#
9 9 9 9 9 9 9 8 8 8
175.9 170-6 165'3 159"9 154"6 149"3 143"9 143-9 138.6 133"3
36 35 34 34 33 32 31 30 29 28
11 # 11 # 11 # 11 # 13# 13# 13# 12# 12# 12#
10 10 10 I0 9 9 9 9 9 9
242.9 236"2 229'4 229.4 222-7 215.9 209-2 202-4 195.7 188.9
40 39 38 37 36 35 34 33 32 32
11 # 13 # 13 # 13 # 13 # 13 # 13 # 13 # 13 # 15#
11 10 10 10 10 10 10 10 10 9
333'3 325'0 316-6 308'3 300-0 291 "6 283 "3 275.0 266"6 266"6
8 8 8 8 8
58
A. Pisanty and M. Gellert
From the initial design load the footing's weight is reduced and the footing redesigned. The process is repeated until a tolerance of I in. is achieved, A minimum concrete cover of 3 in. is provided for the reinforcement (to the bar centre). Additional I in. is given for bars larger than # 6. In the final stage the design of each footing is completed by checking tensile stresses of concrete in flexure. For values lower than 1-6 ~,/(.f'~) reinforcement is omitted. All parameters in the program are easily varied and different basic data nested. Computer time is extremely low--more than 60 footings are obtained in one minute.
pressure
Soft
20001b/ft
,,~ Concrete ,i, R e m f o r c e m e n l
25
#
.
L=
£
'
'
20
(a)
.... :'
', 5
10
/a
Footing's size, Soil
EVALUATION AND COMPARISON OF RESULTS
ft
pressure=5000
ib/ft 2
25
Concrele Reinforcemenl
~'~
20
Concrete and reinforcement quantities for sloped footings having a maximum slope of 1:3 (approx. 18.5 ° ) are compared with flat footings. The comparison indicates sharp reductions for low soil pressures which tend, however, to diminish for higher soil pressures. For soil pressure of 2000 psf. the concrete quantities approach average values of 30 per cent less than those in fiat footings, while the corresponding reduction of reinforcement approaches 20 per cent. For increased soil pressures, concrete reduction tends to be less significant, until it vanishes for 20 000 psf, while reinforcement is approximately 10 per cent less, mainly due to the increased footing depth at the centre of the footing, Tables I and 2 provide a small range of the values obtained for soil pressures of 5000 and 10 000 psf. respectively compared with values for fiat footings produced by the same computer program. Figure 6(a-e) provides a graphical representation of the average reductions of concrete and reinforcement quantities for five different values of soil
2
30
z, , . . ' ' +~
2~
" ' ,. : '
' '
"
L m
L `%
(b) "6 "8
:\
2~
4
~
8
L! Eootrng'S
15t
~O
,'.
Concrete
,3
Reinforcement
(C)
~
Soil
pressure =lOOO0
0
'
!4
¢
iB
ft
size.
Ib/ft 2
"
DE
2
4
~
5
IC
FoOtlng's
12
size,
4
Ig
,B
4
t6
~8
ft
Soi} pressure= 1 5 0 0 0 I b / f t 2
Concrete Reinforcement
A
pressures. (d)
'~
(
~ ,s
~ ,
:
:'
::~:'
CONCLUDING REMARKS A procedure for the design of sloped spread footings is outlined following the provisions of the proposed ACI Code 318-71. A computer program performing the automatical design is explained. The tabulated values presented in this paper serve as an example and for comparison purposes. The authors intend to produce tables covering a wide range of footing sizes. Tables appear to be in many ways preferable to nomographs and design aids of other kinds. Finally it is of interest to point out that comparison between results obtained employing the new code and those using the 318-63 Code indicate
4
6
8
i,C*
Footing's
size,
t2
ft
SoJI pressure= 2 © O 0 0 ~b/f t z L5
COncrete Reinforcemen!
L c
}
(e) c
0 LO 2
'
, ,
£' 4
£' 6
~
"~
~" 8
'
FOOting'S
I0
size,
[2
14
J6
J5
ft
Fig. 6(a), (b), (c), (d), (e). Average reduction o f concrete and reinforcement quantities for various soll pressures.
Automatic Design of Sloped Spread Footings insignificant differences in structural depth averaging 1 in. and almost no difference in reinforcement. The only real difference lies in the fact that due to more severe regulations for minimum embedment length in the new code, most o f the very small footings are unreinforced. Some increase of structural depth is obtained as well due to reduced allowable shear (wide beam) stresses.
d2 g M V j
L vc
NOMENCLATURE a b Q q ~b d
d~
side of the footing (square), ft. side of the column (square), ft. load carried by footing, kips. soil pressure, psf. angle of slope. distance from extreme compression fibre to centroid of tension reinforcement
As as f~ fc f~ fy
59
footing depth at section diagonal tension to be checked. footing depth at section shear to be checked. edge thickness. bending moment. total shear, kips. ratio of distance between centroid of compression and centroid of tension reinforcement to the depth d. embedment length of reinforcing bars. shear stress carried by concrete (also diagonal), psi. reinforcement area, in 2. area of reinforcing bar, in 2. specified compressive strength of concrete, psi. allowable concrete stress, psi. allowable stress in reinforcing steel, psi. yield stress of reinforcing steel, psi.
REFERENCES 1. ACI Committee 318, Proposed Revision of AC1 318-63, "Building Code Requirements
for Reinforced Concrete", ACIJ. Proc. 67, No. 2 (1970). 2. Discussion 67-8--"Discussion of a Report by ACI Committee 318--Discussion and Committee Closure", ACIJ. Proc. 67, No. 9 (1970). 3. AC1 Committee 318, Building Code Requirements for Reinforced Concrete (ACI 318-63), American Concrete Institute, Detroit, June 1963, 144 p. 4. ACI Committee 317, Reinforced Concrete Design Handbook SP-3, American Concrete Institute, Detroit, 1965, 271 p. 5. Commentary on Building Code Requirements for Reinforced Concrete (ACI 318-63), ACI Publication SP 10. 6. C. W. DUNHAM,Foundations of Structures, McGraw-Hill, New York (1962). 7. R.W. FtJaLONO, Design aids for square footings, ACIJ. Proc. 62, 363 (1965). 8. E. HENYE,Nomographs for design of rectangular spread footings, ACIJ. Proc. 66, 545 (1969). 9. J.P. KOnLV, Optimum design of concrete spread footing by computer, ACIJ. Proc. 65, 384 (1968). 10. M. S. McCORgAC, Square Footing Tables, Stress Publications., 208 p. Rockville, Maryland, (1968). 11. C. RODRIOt~eZ, Design of isolated square column footing, ACIJ. Proc. 61, 889 (1964).