beam columns - weebly
TRANSCRIPT
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Prof. Dr. Zahid Ahmad Siddiqi
BEAM COLUMNS
Beam columns are structural members that are
subjected to a combination of bending and axial
stresses.
The structural behaviour resembles
simultaneously to that of a beam and a column.
Majority of the steel building frames have
columns that carry sizable bending moments in
addition to the usual compressive loads.
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Prof. Dr. Zahid Ahmad Siddiqi
FLOW CHART FOR DESIGN OF
BEAM-COLUMNS
Known Data: Pu, M
ntx, Mltx , M
nty, Mlty, Kx
Lx, K
yL
y
Mr
= Mu
= Mnt
+ Mlt for the first trial
Calculate Mr
both in the x and y directions
Peq
= Pr
+ Mrx
(mx) + M
ry(m
y)
Assume an approximate magnification of
15% for the moments only.
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Prof. Dr. Zahid Ahmad Siddiqi
Select section as a simple column depending
upon the following criteria:
1. Asel
≈ Areq
2. Minimum weight
3. Connecting leg width b > bmin
4. Depth of W-section ≤ 360 mm
mx (for first trial) = 8.5 − 0.7K1xLx
my (for first trial) = 17 − 1.4K1yLy
mx = 10 − 14(d / 1000)2 − 0.7K1xLx
my = 20 − 28(d / 1000)2 − 1.4K1yLy
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Prof. Dr. Zahid Ahmad Siddiqi
The column selection tables may also be
employed to select the section using the values of
Peq
and KyL
y.
See rx/r
yfrom column selection table for selected
section
Calculate (KyL
y)
eq=
yx
xx
rr
LK
Re-enter the table for greater of KyL
yand (K
yL
y)
eq
and revise to obtain suitable section for the load
Peq
.
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Prof. Dr. Zahid Ahmad Siddiqi
Find new values of m for subsequent trials.
Select a new section and repeat until values
of load capacities, Peq, and m are stabilized.
Peq
= Pr
+ Mrx
(mx) + M
ry(m
y)
Select a new section and repeat until values of
load capacities, Peq
and m are stabilized.
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Prof. Dr. Zahid Ahmad Siddiqi
Calculate Cmx
and Cmy
for no sway conditions
Calculate , , and x
xx
r
LK1
y
yy
r
LK1
R = maximum of the above values
Check for maximum slenderness ratio: R ≤ 200
x
xx
r
LK2
y
yy
r
LK2
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Prof. Dr. Zahid Ahmad Siddiqi
Axial strength of trial section:
Calculate φcF
crcorresponding to the R-value or
directly read it from the table in Reference-1 and
evaluate the compression capacity by multiplying
with the area of cross-section.
Pc = φcP
n= φcFcr
Ag
/ 1000
Calculate Euler buckling strength (Pe1
)x, (P
e1)y
, (Pe2
)x
and (Pe2
)y
for both lt and nt cases.
Pe1 = π2 EI / (K1 L)2 / 1000 (kN)
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Prof. Dr. Zahid Ahmad Siddiqi
Calculate no-sway moment magnification factors
B1x
= ≥ 1.0 : B1y
= ≥ 1.0
Note: Pr
in the above formulas is the actual
factored axial load and not Peq
.
Calculate B2x
and B2y
.
B2
=
( )xe
r
mx
P
P
C
1
1 α− ( )ye
r
my
P
P
C
1
1 α−where α = 1.0 for LRFD procedure.
2
1
1
e
nt
P
P
∑∑− α
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Prof. Dr. Zahid Ahmad Siddiqi
Calculate design moments
Mrx
= Mux
= B1x
Mntx
+ B2x
Mltx
Mry
= Muy
= B1y
Mnty
+ B2y
Mlty
Bending strength of the trial section:φbM
ny= φ
bF
yZ
y/ 106 (kN-m)
There are no chances of lateral buckling because
the lateral direction for y-axis bending is the
stronger direction.
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Prof. Dr. Zahid Ahmad Siddiqi
Check conditions of compact section as a beam.
Find Lp
and Lr
from column table and check
against Lbx
.
Calculate φbM
nxas for a beam using L
bx, L
p, L
r
and beam selection tables. Use Cb
= 1.0 in the
expressions.
Calculate to see which interaction
equation is applicable.c
r
P
P
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Prof. Dr. Zahid Ahmad Siddiqi
Check interaction equations:
≤ 1.0For ≥ 0.2
≤ 1.0For < 0.2
Get the value of Left Side of equation (LS) up to
2nd decimal place, truncating the 3rd decimal digit,
which should not be more than 1.00.
This means that LS can be as high as 1.0099 but
not 1.01.
c
r
P
P
c
r
P
P
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
2
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Prof. Dr. Zahid Ahmad Siddiqi
Values of LS between 0.9 and 1.0 → Very economical design
Values of LS between 0.8 and 0.9 → Economical design
Values of LS between 0.7 and 0.8 → May be acceptable,
but better to try
an economical section
Values of LS lesser than 0.7 → Revise by selecting
a lighter section
Values of LS greater than 1.0099 → Select a stronger section
Check shear strength, which may usually be
omitted in hot rolled W sections because of very
high available strength.
Write the final solution using standard designation.
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Prof. Dr. Zahid Ahmad Siddiqi
Example 5.1: Design the columns in a single-
bay multi-storey unbraced frame shown in Figure
5.6, where P is the load from the top stories.
Ratio of moment of inertia of beams with respect
to columns may be assumed as shown in the
figure. Approximate analyses results are also
provided in Figures 5.7 and 5.8. Assume that
sway is not allowed in the y-direction.
Solution:
Total Factored Loads
1. Load Combination 1, Gravity Load
Combination (1.2 D + 1.6 L)
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Prof. Dr. Zahid Ahmad Siddiqi
8.5 m
II
PP
wI I
H
6.0 m
6.0 m
1.4 I
Figure 5.6. Frame And Loading For Example 5.1.
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Prof. Dr. Zahid Ahmad Siddiqi
Pu
= 1.2(1025) + 1.6(410) = 1886 kN
wu
= 1.2(7.3) + 1.6(22.0) = 43.96 kN/m
P = 1025 kN dead load
= 410 kN live load
w = 7.3 kN/m dead load
= 22.0 kN/m live load
H = 345 kN wind load
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Prof. Dr. Zahid Ahmad Siddiqi
1886kN1886kN
43.96kN/m
75.8 kN-m
37.9 kN-m
227 kN-m
Figure 5.7. Partial Gravity Load Analysis Results.
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Prof. Dr. Zahid Ahmad Siddiqi
2. Load Combination 2, Wind Load
Combination (1.2D+0.5L+1.3W)
Pu
= 1.2(1025) + 0.5(410) = 1435 kN
Hu
= 1.3(345) = 448.5 kN
wu
= 1.2(7.3)+0.5(22.0) = 19.76 kN/m
Value of Kx
Gtop
= = = 2.02( )( ) beamsforLI
columnsforLI
∑∑ ( )
5.84.1
62
I
I
Gbotton
= 1.0 for sway columns
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Prof. Dr. Zahid Ahmad Siddiqi
1435kN1435kN
19.76kN/m
51.2 kN-m
25.6 kN-m
( No Sway Part )
93.7 kN-m
138 kN138 kN
448.5kN
586 kN-m
759 kN-m
∆oh
( Sway Part )
586 kN-m
759 kN-m
Figure 5.8. Partial Lateral Load Analysis Results.
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Prof. Dr. Zahid Ahmad Siddiqi
Kx
= 1.0 for braced frame
Kx
= 1.45 for unbraced conditions
Value of Ky:
No data of connected elements is given for y-
direction and hence the approximate value may
conservatively be assumed for no sway in this
direction.
Ky
= 1.0
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Prof. Dr. Zahid Ahmad Siddiqi
Here, design is made for the wind combination and
check is then made for the gravity combination.
Design for Combination 2:
Pr
= 1435 + 138 + 19.76 × 8.5/2
= 1656.7 kN
According to AISC, max. moments for different
types of loading (nt or lt case), acting at different
locations or of different signs, are to be added
magnitude-wise in any combination.
The Right column is critical for the axial load.
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Prof. Dr. Zahid Ahmad Siddiqi
Mntx
= 51.2 kN-m
Mltx = 759 kN-m
Mnty
= Mlty = 0
K2x
Lx= 1.45 × 6 = 8.7 m for lt-case
K1x
Lx= 1.00 × 6 = 6.00 m for nt-case
K1y
Ly= 1.00 × 6 = 6.00 m
Peq
= Pu
+ 1.15 Mux
(m)
= 1656.7 + 1.15 (51.2 + 759.0) (4.3)
= 5663 kN
mx (for first trial) = 8.5 − 0.7K1xLx
= 8.5 − 0.7 × 6 = 4.3
Assume 15% magnification of moments.
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Prof. Dr. Zahid Ahmad Siddiqi
Using column tables of Reference 1 for this Peq
and KyL
y= 6.0 m;
Trial Section = W360 × 262
Peq
= Pu
+ 1.15 Mux
(m)
= 1656.7 + 1.15 (51.2 + 759.0) (3.99)
= 5374 kN
Revised mx = 10 − 14(d / 1000)2 − 0.7K1xLx
= 10 − 14 × 0.362 − 0.7 × 6
= 3.99
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Prof. Dr. Zahid Ahmad Siddiqi
Trial Section-1: W360 × 237
A = 30,100 mm2
rx
= 162 mm, ry
= 102 mm
rx/r
y= 1.60
Ix
= 79,100 × 104 mm4
M1
/ M2
is positive because of reverse curvature
Cmx
= 0.6 – 0.4 2
1
M
M
= 0.6 – 0.4 = 0.4
2.51
6.25
= = 37.04 ( for nt case)x
xx
r
LK1
162
10000.6 ×
yx
xx
rr
LK
/
2
60.1
7.8(K
yL
y)
eq= = = 5.44 m (not critical)
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Prof. Dr. Zahid Ahmad Siddiqi
= = 53.70 (for lt-case)x
xx
r
LK2
162
10007.8 ×
= = 58.82y
yy
r
LK1
102
10000.6 ×
R ≈ 59 < 200 OK
φcF
cr= 187.09 MPa
Pc
= φcF
crA
g= = 5,631 kN
1000
100,3009.187 ×
for nt-case
Pe1x
= π2 EI / (K1x
L)2
= 10006000
10100,79000,2002
42
××××π
= 43,371 kN
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Prof. Dr. Zahid Ahmad Siddiqi
B1x
= =
= 0.42 ∴ B1x
= 1.0
Pe2x
= π2 EI / (K2x
L)2
= 10008700
10100,79000,2002
42
××××π
= 20,628 kN for lt-case
xer
mx
PP
C
,11 α− 371,437.165611
4.0
×−
ΣPnt
= 1435 × 2 + 19.75 × 8.5 = 3038 kN
ΣPe2,x
= 2 × 20,628 = 41,256 kN
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Prof. Dr. Zahid Ahmad Siddiqi
B2x
= = = 1.08
Mrx
= B1x
Mntx
+ B2x
Mltx
= 1.0 (51.20) + 1.08(759.00)
= 870.9 kN-m
From column selection table:
Lp
= 5.06 m, Lr
= 25.43 m
xe
nt
P
P
,2
1
1
∑∑− α
256,41
30380.11
1
×−
Pr
= Pnt
+ B2
Plt
= 1518.98 + 1.08(138)
= 1668.02kN
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Prof. Dr. Zahid Ahmad Siddiqi
Check conditions of compact section:
= 6.5 < λp
= 10.8 OKf
f
t
b
2
= = 0.245yb
u
P
P
φ ( ) 100,301000/2509.0
7.1656
××For web, λ
p= 3.4233.28.31 ≥
−
yb
u
P
P
φ= 66.3 for A36 steel
= 15.3 < λp
OKwt
h
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Prof. Dr. Zahid Ahmad Siddiqi
Lb
= 6.00m > Lp, bending strength is to be
calculated using the inelastic LTB formula.
Mp
= 250 × 4700 × 103 / 106 = 1175.0 kN-m
φbM
p= 0.9 × 1175 = 1057.5 kN-m
Mr
= 0.7 × Fy× S
x/ 106
= 0.7 × 250 × 4160 × 103 / 106
= 728.0 kN-m
BF = = 21.94 kN
−−=−
−06.543.25
7281175
pr
rp
LL
MM
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Prof. Dr. Zahid Ahmad Siddiqi
Mcx = φb[Mp − BF(Lb − Lp)]
= 0.9 [1175 − 21.94(6.0 − 5.06)]
= 1038.9 kN-m
Check Interaction Equation:
c
r
P
P
631,5
02.1668= = 0.296 > 0.2
+cx
rx
c
r
M
M
P
P
9
8= 0.296 +
9.1038
92.870
9
8
= 1.041 > 1.00 NG
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Prof. Dr. Zahid Ahmad Siddiqi
Trial Section-2: W360 × 262
A = 33,400 mm2
rx = 163 mm, ry = 102 mm
rx/ry = 1.60
Ix = 89,100 × 104 mm4
(KyL
y)eq
= yx
xx
rr
LK
/
2
= 60.1
7.8= 5.44 m (not critical)
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Prof. Dr. Zahid Ahmad Siddiqi
Cmx = 0.4 (as before)
x
xx
r
LK1
163
10000.6 ×= = 36.81 ( for nt case)
x
xx
r
LK 2
163
10007.8 ×= = 53.37 ( for lt case)
y
yy
r
LK1
102
10000.6 ×= = 58.82
R ≈ 59 < 200 OK
φcFcr = 187.09 MPa
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Prof. Dr. Zahid Ahmad Siddiqi
Pc
= φcP
n= φ
cF
crA
g
= 1000
400,3309.187 ×= 6,248 kN
Pe1x
= π2 EI / (K1x
L)2
= 10006000
10100,89000,2002
42
××××π
= 48,854 kN for nt-case
Pe2x
= π2 EI / (K2x
L)2
= 10008700
10100,89000,2002
42
××××π
= 23,236 kN for lt-case
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Prof. Dr. Zahid Ahmad Siddiqi
xer
mx
PP
C
,11 α−854,487.165611
4.0
×−
B1x
=
= = 0.41 ∴ B1x
= 1.0
ΣPnt = 1435 × 2 + 19.75 × 8.5
= 3038 kN
ΣPe2,x = 2 × 23,236 = 46,472 kN
B2x
=
xe
nt
P
P
,2
1
1
∑∑− α =
472,46
30380.11
1
×− = 1.07
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Prof. Dr. Zahid Ahmad Siddiqi
Mrx = B1x Mntx + B2x Mltx
= 1.0 (51.20) + 1.07(759.00)
= 863.33 kN-m
Pr = Pnt + B2 Plt
= 1518.98 + 1.07(138)
= 1666.64 kN
From column selection table:
Lp = 5.08 m, Lr = 30.44 m
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Prof. Dr. Zahid Ahmad Siddiqi
Check conditions of compact section:
f
f
t
b
2= 6.0 < λ
p= 10.8 OK
yb
u
P
P
φ ( ) 400,331000/2509.0
7.1656
××= = 0.220
For web,
λp
= 3.4233.28.31 ≥
−
yb
u
P
P
φ for A36 steel
= 67.1
wt
h= 13.7 < λ
pOK
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Prof. Dr. Zahid Ahmad Siddiqi
Lb = 6.00m > Lp, bending strength is to be
calculated using the inelastic LTB formula.
Mp = 250 × 5240 × 103 / 106
= 1310.0 kN-m
φbMp = 0.9 × 1310 = 1179 kN-m
Mr = 0.7 × Fy × Sx / 106
= 0.7 × 250 × 4600 × 103 / 106
= 805.0 kN-m
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Prof. Dr. Zahid Ahmad Siddiqi
BF =
−−=−
−08.544.30
8051310
pr
rp
LL
MM
= 19.91 kN
Mcx
= φb[M
p− BF(L
b− L
p)]
= 0.9 [1310 − 19.91(6.0 − 5.08)]
= 1162.5 kN-m
c
r
P
P
248,6
64.1666= = 0.267 > 0.2
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Prof. Dr. Zahid Ahmad Siddiqi
Check Interaction Equation:
+cx
rx
c
r
M
M
P
P
9
8
5.1162
33.863
9
8= 0.267 +
= 0.927 < 1.00 OK
Section Selected For Wind
Combination: W360 × 262
Check for Combination 1:
Pr = Pu = 1886 + 43.96 × 8.5/2
= 2073 kN
Mntx = 75.8 kN-m
![Page 39: BEAM COLUMNS - Weebly](https://reader030.vdocuments.site/reader030/viewer/2022012018/61686985d394e9041f6f69e4/html5/thumbnails/39.jpg)
Prof. Dr. Zahid Ahmad Siddiqi
Cmx
= 0.4 same as before
xer
mx
PP
C
,11 α− 854,48207311
4.0
×−B1x
= =
= 0.42 ∴ B1x
= 1.0
Mrx = B1x × Mntx = 75.8 kN-m
c
r
P
P
248,6
2073= = 0.332 > 0.2
![Page 40: BEAM COLUMNS - Weebly](https://reader030.vdocuments.site/reader030/viewer/2022012018/61686985d394e9041f6f69e4/html5/thumbnails/40.jpg)
Prof. Dr. Zahid Ahmad Siddiqi
Check Interaction Equation:
+cx
rx
c
r
M
M
P
P
9
8
5.1162
8.75
9
8= 0.332 +
= 0.39 < 1.00 OK
Final Selection: W360 × 262