(601-603) DESIGN OF TRAPEZOIDAL COMBINED FOOTINGS

GIVEN: b1 (mm) b2 (mm) D1 (kN) L1 (kN) D2 (kN) L2 (kN) S (m)

(from P501) 480 530 625 475 685 585 4.90

601. For the column dimensions and axial loading given in Problem 501, design a trapezoidal comb
columns. The footing cannot extend beyond both the exterior and interior columns. The colum
center. Use f c ' = 28 MPa. f y = 414 MPa, q a = 150 kPa.

Step 1. Proportion footing dimensions

a) Column loads
P1 1270
P2 1100
R 2370
b) Weight of footing = 10% of column loads
W 237 use 237
c) Required footing area
A 17.38
d) From area of trapezoid,
a+b 6.431
e) Location of resultant from exterior column:
x 2.274
xc 2.539
f) From centroid of trapezoid
a 2.632
b 3.799

(Values may be reversed to maintain consistency with lab manual)

Step 2. Compute factored loads

a) Column loads
P1 1758 Factors: 1.2 1.6
P2 1510
b) Ultimate upward pressure
qu 188.0
c) Equivalent linear pressure
qa 495
qb 714
d) Slope of pressure line
 s 40.52

Step 3. Draw shear and moment diagrams

a) Shear equation
V 714x - 20.26x^2 + C1
b) Shear ordinates
x(m) C1(kN) V(kN)
0.00 0 0.00
0.265 0 187.79
0.265 + dx -1758 -1570.21
0.53 -1758 -1385.27
4.925 -1758 1267.03
5.165 -1758 1389.33
5.165 + dx -3268 -120.67
5.405 -3268 0.00
c) Locate point of zero shear
x1 2.663
d) Moment equation
M 357x^2 - 6.753x^3 + C1x' + C2

x(m) C1(kN) x'(m) M(kN-m)

0.00 0 0.00 0.00
0.265 0 0.00 24.94
0.530 -1758 0.265 -366.59
2.663 -1758 2.398 -1811.52
4.925 -1758 4.660 -339.73
5.165 -1758 4.900 -20.91
-1758 5.140
5.405 -35.42
-1510 0.24

Step 4. Solve "d" for beam shear at small end

a) Equation of width of footing

z 1.167 - 0.216x
b' 3.799 - 0.216x
b) Width at critical section for beam shear
x 4.925 - d
b' 2.735 + 0.216d
c) Shear force at x = 4.925 - d
Vu 714x - 20.26x^2 - 1758
Vu 1267.03-514.44d-20.26d^2
d) Shear capacity at critical section
φVc 2050.24d + 161.92d^2
e) Equate factored shear to shear capacity
Vu ≤ φVc
1267.03-514.44d-20.26d^2 = 2050.24d + 161.92d^2
 d 0.478 500
Try d = 500 mm

Step 5. Check wide beam shear at large end

a) Width of footing at critical section

x 1.030
b' 3.577
b) Shear force at x = 1.03 m
Vu 1044.07
c) Shear capacity at critical section
φVc 1340.71 OK

Step 6. Check punching shear at exterior column

a) Perimeter of critical section

e 0.780
f 1.030
b0 2.590
b) Shear force
Vu 1606.96
c) Shear capacity at critical section
φVc 1941.54 OK

Step 7. Check punching shear at interior column

a) Perimeter of critical section

g 0.730
h 0.980
b0 2.440
b) Shear force
Vu 1375.50
c) Shear capacity at critical section
φVc 1829.10 OK

Step 8. Design negative steel in long direction

a) Factored moment at x = 2.663 m

Mu 1811.52 (from M-diagram)
b' 3.224
b) Moment resistance coefficient
Ru 2.497
c) Strength ratio
m 17.395
d) Required steel ratio
ρ 0.00639
 e) Minimum steel ratio
ρmin 0.00320 ≥ 0.00338
4/3 ρ N/A
Therefore use ρ = 0.00639
f) Steel area
As 10301
db 25 Ab 491
n 21.0
use 21 - 25 mm dia. bars
g) Check spacing of bars
s 98 OK!
s 156 OK!
h) Check development length
Ld,furn 2588 OK!

Step 9. Design short direction steel at exterior column

a) Band width
b 0.905
b2 3.604
bave 3.702
Lave 1.586
b) Equivalent upward soil pressure
qu 525
c) Design moment
Mu 597.56
d) Moment resistance coefficient
Ru 3.252
e) Required steel ratio
ρ 0.00848
f) Minimum steel ratio
ρmin 0.00320 ≥ 0.00338
4/3 ρ N/A
Therefore use ρ = 0.00848
g) Steel area
As 3645
db 28 Ab 616
n 5.9
use 6 - 28 mm dia. bars
h) Check spacing of bars
s 132 OK!
i) Check development length
Ld,furn 1511 OK!

Step 10. Design short direction steel at interior column

a) Band width
 b 0.855
x3 4.550
b3 2.816
bave 2.724
Lave 1.122
b) Equivalent upward soil pressure
qu 648
c) Design moment
Mu 348.74
d) Moment resistance coefficient
Ru 2.009
e) Required steel ratio
ρ 0.00508
f) Minimum steel ratio
ρmin 0.00320 ≥ 0.00338
4/3 ρ N/A
Therefore use ρ = 0.00508
g) Steel area
As 2063
db 20 Ab 314
n 6.6
use 7 - 20 mm dia. bars
h) Check spacing of bars
s 107 OK!
i) Check development length
Ld,furn 1047 OK!

Step 11. Design steel for remaining zones

a) Steel area
ρmin 0.00320 ≥ 0.00338
As 1606
db 20 Ab 314
n 5.1
use 6 - 20 mm dia. bars/meter

Step 12. Compute development lengths

a) For 25 mm diameter tension bars

α 1.0 β 1.0 γ 1.0

= 3.00 ≤ 2.5

= 28.17

simplified formula:

=
 = 46.94 (controls) fraction used 3/5

Ld 1174 (controls)
min Ld 300
b) For 28 mm diameter tension bars
α 1.0 β 1.0 γ 1.0

= 2.68 ≤ 2.5

= 28.17

simplified formula:

= 46.94 (controls) fraction used 3/5

Ld 1314 (controls)
min Ld 300
c) For 20 mm diameter tension bars
α 1.0 β 1.0 γ 1.0

= 3.75 ≤ 2.5

= 28.17

simplified formula:

= 37.55 (controls) fraction used 12/25

Ld 751 (controls)
min Ld 300

Step 13. Check weight of footing

a) thickness of footing
t 600
b) volume of footing
Vf 10.43
c) weight of footing
Wf 246.15 < 237.00 NOT OK!!!

Wf,assumed 237

Step 14. Draw the details

21-25 mm φ bars
 21-25 mm φ bars

a = 2.632 m 0.905 m t = 0.6 m

b = 3.799 m 3.645 3.645 m
6-16 mm φ bars
t = 0.6 m 0.855 m

0.905 m 6 28 φ 3.6456-28
m φ 0.855 m
6-28 φ 22 20 φ 22-20 φ φ
22-20 7-20 φ
7 20 φ 7-20 φ

21 25 mm φ bars
21-25 mm φ bars

6 16 mm φ bars
6-16 mm φ bars

a = 2.632 m
b = 3.799 m

(DRAWING NOT TO SCALE)

(DRAWING NOT TO SCALE)

OOTINGS
fc' (MPa) fy (MPa) qa (kPa) γconc (kN/m3)
28 414 150 23.6

501, design a trapezoidal combined footing to support the 2

and interior columns. The columns are spaces S m center to

685 585 530

625 475 480 0.24
s = 4.9 m
0.265
0.48 m
0.53 m
qa 150 L = 5.405 m

L 5.405

2370 1100 4.90

0.265

685 585
DL LL
625 475

A 17.38

2.632
3.799
 714 20.26
P1 = 1758 kN
P2 = 1510 kN
4.9 m
qb = 714 kN/m
qa = 495 kN/m

188 25
-1570 -367
-1385 -1812
1267 -340
1389 -21
-20.26 714 -1758 -121 -35

357.0 6.753

188

-1570

-1385

1.167 0.216 2.632

2.632 3.799 1.167
5.405 - x
5.405 4.925
2.735

714 20.26 1758

1267.03 -514.44 -20.26

749.63 2050.24 161.92

182.18 2564.68 -1267.03

2.632 m
0.530 0.50 3.799 m
3.799 0.216

714 20.26 1758

0.530 0.50

1758 188.0

0.480 0.50

1510 188.0

2.663

3.799 0.216

1811.52 500
188
fy 414 f c' 28
-1570
 -1385

b 3224 d 500

b 2632 c 75
b 3799

2663 1174

0.530 0.500
3.799 0.216

1758

597.56 905 475

17.395 414

b 905 d 475

b 905 c 75

1586 1314
0.480 0.500
5.405
3.799 0.216
2.632

1510

348.74 855 475

17.395 414

b 855 d 475

b 855 c 75

1122 751

fy 414 fc' 28
0.00338

λ 1.0

db 25
λ 1.0

db 28

λ 1.0

db 20

d 500 c 100

a 2.632 b 3.799 L 5.405

db 16 Ab 201 (bottom bars)
 1389
1267
 1389
1267