appendix xx column interaction diagrams of the steel
TRANSCRIPT
176
Appendix XX
Column Interaction Diagrams of the Steel Girder Bridge
West Bound Bridge
Figure XX-1. The cross section of the West Bound bridge columns [Brown, 1993]. The ¾” = 1’-0” scale is
no longer correct.
( )
2
10
2
9
2
8
2
7
2
6
2
5
2
4
2
3
2
2
2
1
22
00.2
00.2
00.2
00.2
00.2
00.2
00.2
00.2
00.2
0.1
4.1385.424
1
60
3'
inA
inA
inA
inA
inA
inA
inA
inA
inA
inA
ininA
ksif
ksif
s
s
s
s
s
s
s
s
s
s
g
y
c
=
=
=
=
=
=
=
=
=
=
==
=
=
π
177
00207.0000,29
60
0144.04.1385
20
20
00.1
2
2
2
2
11
===
===
==
=
∑
ksi
ksi
E
f
in
in
A
A
inAA
inA
s
y
y
g
st
t
sist
s
ε
ρ
Nominal Concentric Axial Load Capacity
( )( )( )( )( ) ( )( )
( )( )
( )
( )
kP
kPP
AkP
kkP
PP
kinksiAfP
k
inksiininksi
AfAAfP
n
n
a
gca
stystgc
985,2
511,385.085.0
'3511
511,3682,475.0
75.0
552.4.13853133.0'133.0
682,4
.2060.20.4.1385385.0
'85.0
(max)
0(max)
0
0
0
2
222
0
=
==
→=
==
=→>
===
=
+−=
+−=
φ
φφ
φ
φ
φ
Capacity in Axial Tension
( )
( )( )
( )
'080,1
080,1200,19.0
1200
.20602
1
1
EkP
kkP
kP
inksiP
AfP
AfP
nt
nt
nt
nt
n
i
siynt
n
i
siynt
→−=
−=−=
−=
−=
−=
−=
∑
∑
=
=
φ
φ
178
d1 d2d3
d4d5
d6
d7d8
d9d10
d11
( )( )( )( )
( )( )( )( ).5.3
.36.472sin5.1721
.84.654sin5.1721
.7.1036sin5.1721
.6.1518sin5.1721
.21
.4.2672cos5.1721
.3.3154cos5.1721
.2.3536cos5.1721
.6.3718cos5.1721
.5.38
11
10
9
8
7
6
5
4
3
2
1
ind
ind
ind
ind
ind
ind
ind
ind
ind
ind
ind
=
=−=
=−=
=−=
=−=
=
=+=
=+=
=+=
=+=
=
o
o
o
o
o
o
o
o
179
φPn and φMn for the balanced failure (εs1 = -εy)
( )( )
( )
( )
( )
( )
( )
( )
( ) 00159.0003.0.8.22
.7.10.8.22
000947.0003.0.8.22
.6.15.8.22
000237.0003.0.8.22
.21.8.22
000388.000207.0.2.19
.8.22.4.26
000916.000207.0.2.19
.8.22.3.31
00134.000207.0.2.19
.8.22.2.35
00160.000207.0.2.19
.8.22.6.37
00207.0
.8.22.5.3800207.01003.0
003.0
8
7
6
5
4
3
2
1
=
−=
=
−=
=
−=
−=−
−=
−=−
−=
−=−
−=
−=−
−=
−=−=
=×−−
=
in
inin
in
inin
in
inin
in
inin
in
inin
in
inin
in
inin
ininc
s
s
s
s
s
s
s
ys
ε
ε
ε
ε
ε
ε
ε
εε
c = 22.8 in.
19.2 in.
εcu = 0.003
εs1 = -0.00207
180
( )
( )
( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )
( )( ) kinksiC
radin
inin
h
ah
hfC
OKindininca
ksifksiksiksiEf
ksifksiksiksiEf
ksifksiksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
in
inin
in
inin
in
inin
c
cc
ssss
ssss
ssss
sss
sss
sss
sss
sss
sss
sss
sss
s
s
s
585,14
49.1cos49.1sin49.1.42385.0
49.1.21
.38.19.21cos
2
2cos
4
cossin'85.0
.42.38.19.8.2285.0
606066.73000,2900254.0
606047.70000,2900243.0
60609.60000,2900210.0
11.46000,2900159.0
46.27000,29000947.0
87.6000,29000237.0
25.11000,29000388.0
56.26000,29000916.0
86.38000,2900134.0
4.46000,2900160.0
60000,2900207.0
00254.0003.0.8.22
.5.3.8.22
00243.0003.0.8.22
.36.4.8.22
00210.0003.0.8.22
.84.6.8.22
2
11
2
1
111111
101010
999
88
77
66
55
44
33
22
11
11
10
9
=
−=
=
−=
−
=
−=
→=<===
=→>===
=→>===
=→>===
===
===
===
−=−==
−=−==
−=−==
−=−==
−=−==
=
−=
=
−=
=
−=
−−θ
θθθ
β
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
181
( )( )( )( )( )( )( )( )( )( )( )( )
( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( ) kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiAfF
kinksiAfF
kinksiAfF
kinksiAfF
kinksiAfF
kinksiAfF
sccss
sccss
sccss
sccss
sccss
sss
sss
sss
sss
sss
sss
45.57.00.1385.060'85.0
9.114.00.1385.060'85.0
9.114.00.2385.060'85.0
12.87.00.2385.011.46'85.0
82.49.00.2385.046.27'85.0
74.13.00.287.6
5.22.00.225.11
12.53.00.256.26
72.77.00.286.38
8.92.00.24.46
60.00.160
2
111111
2
101010
2
999
2
888
2
777
2
666
2
555
2
444
2
333
2
222
2
111
=−=−=
=−=−=
=−=−=
=−=−=
=−=−=
===
−=−==
−=−==
−=−==
−=−==
−=−==
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( )
( )
( ) bn
bn
yst
b
i
isicn
bsicn
MftkipsftkipsM
PkkP
M
ftkips
inkips
inink
ininkinink
ininkinink
ininkinink
ininkinink
ininkinink
dh
Fah
CM
PkFCP
φφ
φφ
φεεε
=−=−=
===
=→−==
=
−=
−=
−+
−+−+
−+−+
−−+−−+
−−+−−+
−−+−=
−+
−=
==+=
∑
∑
=
17503.333,275.0
288,1717,175.0
75.0
3.2333
000,28
.5.3.2145.57
.36.4.219.114.7.10.2112.87
.6.15.2182.49.21.2174.13
.4.26.215.22.3.31.2112.53
.2.35.2172.77.6.37.218.92
.5.38.2160.69.9.21585,1
222
717,1
1
9
1
The column interaction diagram is shown in Figure XX-4.
182
-2000
-1000
0
1000
2000
3000
4000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Phi Mn (k-ft)
Ph
i P
n (
k)
Figure XX-2. The column interaction diagram for the West Bound bridge columns. The grey points show
the factored axial loads and moments in the columns.
East Bound Bridge
Figure XX-3. The cross section of the East Bound bridge columns [Brown, 1993]. The ¾” = 1’-0” scale is
no longer correct.
183
( )
∑ ==
=
=
=
=
=
=
=
=
=
=
=
=
==
=
=
2
2
12
2
11
2
10
2
9
2
8
2
7
2
6
2
5
2
4
2
3
2
2
2
1
22
22
00.1
00.2
00.2
00.2
00.2
00.2
00.2
00.2
00.2
00.2
00.2
0.1
4.1385.424
1
60
3'
inAA
inA
inA
inA
inA
inA
inA
inA
inA
inA
inA
inA
inA
ininA
ksif
ksif
sist
s
s
s
s
s
s
s
s
s
s
s
s
g
y
c
π
00207.0000,29
60
0159.04.1385
222
2
===
===
ksi
ksi
E
f
in
in
A
A
s
y
y
g
st
t
ε
ρ
184
Nominal Concentric Axial Load Capacity
( )( )( )( )( ) ( )( )
( )( )
( )
( )
kP
kPP
AkP
kkP
PP
kinksiAfP
k
inksiininksi
AfAAfP
n
n
a
gca
stystgc
058,3
598,385.085.0
'3598
598,3797,475.0
75.0
552.4.13853133.0'133.0
797,4
.2260.22.4.1385385.0
'85.0
(max)
0(max)
0
0
0
2
222
0
=
==
→=
==
=→>
===
=
+−=
+−=
φ
φφ
φ
φ
φ
Capacity in Axial Tension
( )
( )( )
( )
'188,1
188,1320,19.0
1320
.22602
1
1
EkP
kkP
kP
inksiP
AfP
AfP
nt
nt
nt
nt
n
i
siynt
n
i
siynt
→−=
−=−=
−=
−=
−=
−=
∑
∑
=
=
φ
φ
185
d1d2
d3d4
d5
d6
d7
d8
d9d10 d11
d12
( )( )( )( )( )( )( )( )( )( ).5.3
.21.44.16cos5.1721
.28.67.32cos5.1721
.54.91.49cos5.1721
.7.135.65cos5.1721
.5.188.81cos5.1721
.5.238.81cos5.1721
.3.285.65cos5.1721
.5.321.49cos5.1721
.7.357.32cos5.1721
.8.374.16cos5.1721
.5.38
12
11
10
9
8
7
6
5
4
3
2
1
ind
ind
ind
ind
ind
ind
ind
ind
ind
ind
ind
ind
=
=−=
=−=
=−=
=−=
=−=
=+=
=+=
=+=
=+=
=+=
=
o
o
o
o
o
o
o
o
o
o
186
φPn and φMn for the balanced failure (εs1 = -εy)
( )( )
( )
( )
( )
( )
( )
( )
( ) 00120.0003.0.8.22
.7.13.8.22
000566.0003.0.8.22
.5.18.8.22
0000755.000207.0.2.19
.8.22.5.23
000593.000207.0.2.19
.8.22.3.28
00105.000207.0.2.19
.8.22.5.32
00139.000207.0.2.19
.8.22.7.35
00162.000207.0.2.19
.8.22.8.37
00207.0
.8.22.5.3800207.01003.0
003.0
8
7
6
5
4
3
2
1
=
−=
=
−=
−=−
−=
−=−
−=
−=−
−=
−=−
−=
−=−
−=
−=−=
=×−−
=
in
inin
in
inin
in
inin
in
inin
in
inin
in
inin
in
inin
ininc
s
s
s
s
s
s
s
ys
ε
ε
ε
ε
ε
ε
ε
εε
c = 22.8 in.
19.2 in.
εcu = 0.003
εs1 = -0.00207
187
( )
( ) 00217.0003.0.8.22
.28.6.8.22
00174.0003.0.8.22
.54.9.8.22
10
9
=
−=
=
−=
in
inin
in
inin
s
s
ε
ε
( )
( ) 00254.0003.0.8.22
.5.3.8.22
00245.0003.0.8.22
.21.4.8.22
12
11
=
−=
=
−=
in
inin
in
inin
s
s
ε
ε
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )
( )( ) kinksiC
radin
inin
h
ah
hfC
OKindininca
ksifksiksiksiEf
ksifksiksiksiEf
ksifksiksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
ksiksiEf
c
cc
ssss
ssss
ssss
sss
sss
sss
sss
sss
sss
sss
sss
sss
585,14
49.1cos49.1sin49.1.42385.0
49.1.21
.38.19.21cos
2
2cos
4
cossin'85.0
.42.38.19.8.2285.0
606066.73000,2900254.0
606005.71000,2900245.0
606093.62000,2900217.0
46.50000,2900174.0
8.34000,2900120.0
41.16000,29000566.0
19.2000,290000755.0
20.17000,29000593.0
45.30000,2900105.0
31.40000,2900139.0
98.46000,2900162.0
60000,2900207.0
2
11
2
1
121212
111111
101010
99
88
77
66
55
44
33
22
11
=
−=
=
−=
−
=
−=
→=<===
=→>===
=→>===
=→>===
===
===
===
−===
−=−==
−=−==
−=−==
−=−==
−=−==
−−θ
θθθ
β
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
188
( )( )( )( )( )( )( )( )( )( )( )( )
( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( ) kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiksiAffF
kinksiAfF
kinksiAfF
kinksiAfF
kinksiAfF
kinksiAfF
kinksiAfF
sccss
sccss
sccss
sccss
sccss
sccss
sss
sss
sss
sss
sss
sss
45.57.00.1385.060'85.0
9.114.00.2385.060'85.0
9.114.00.2385.060'85.0
82.95.00.2385.046.50'85.0
5.64.00.2385.08.34'85.0
72.27.00.2385.041.16'85.0
38.4.00.219.2
4.34.00.220.17
9.60.00.245.30
62.80.00.231.40
96.93.00.298.46
60.00.160
2
121212
2
111111
2
101010
2
999
2
888
2
777
2
666
2
555
2
444
2
333
2
222
2
111
=−=−=
=−=−=
=−=−=
=−=−=
=−=−=
=−=−=
−=−==
−=−==
−=−==
−=−==
−=−==
−=−==
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( )
( )
( ) bn
bn
yst
b
i
isicn
bsicn
MftkipsftkipsM
PkkP
M
ftkips
inkips
inink
ininkinink
ininkinink
ininkinink
ininkinink
ininkinink
ininkinink
dh
Fah
CM
PkFCP
φφ
φφ
φεεε
=−=−=
===
=→−==
=
−=
−=
−+
−+−+
−+−+
−+−−+
−−+−−+
−−+−−+
−−+−=
−+
−=
==+=
∑
∑
=
811,1414,275.0
295,1726,175.0
75.0
414,2
967,28
.5.3.219.114
.21.4.219.114.28.6.219.114
.54.9.2182.95.7.13.215.64
.5.18.2172.27.5.23.2138.4
.3.28.214.34.5.32.219.60
.7.35.2162.80.8.37.2196.93
.5.38.2160.69.9.21585,1
222
726,1
1
9
1
189
-2000
-1000
0
1000
2000
3000
4000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Phi Mn (k-ft)
Ph
i P
n (
k)
Figure XX-4. The column interaction diagram for the East Bound bridge columns. The grey points show
the factored axial loads and moments in the columns.