B-14

1
LRFD Design of Compact Beams
1 Full section Yielding (Lb Lp) Mn Mp
FyZx (AISC F 2.1) 2 Inelastic Lateral
Torsional Buckling ( Lp lt Lb Lr ) 3
Elastic Lateral Torsional Buckling (Lr lt
Lb) Mn Fcr Sx Mp (AISC F-2.3) where
J Torsional constant (in4). ho Distance
between flanges centroids (in). C 1.0 for w
shapes. rts radius of gyration of the
compression flange plus one-sixth of the web.
B-14
2
Equations (F 2.2) (F2.4) for compact beams
affected by lateral torsional buckling, require
the introduction of the Moment Gradient Factor
(Cb) for non-uniform bending moment values
between the lateral bracing points for (Lb). AISC
provides value for Cb as
The effect of Cb on Nominal Strength is shown
below
B-15
3
Example B - 5
Determine (Cb) for a uniformly loaded, simply
supported beam with lateral supports at its ends
only.
Solution
B-16
4
(Cb)
Some Examples on
For unbraced cantilever beams, AISC recommends
the value of Cb 1.0. A value of Cb 1.0 is
always conservative and represent uniform banding
throughout the unbraced length (Lb), (See Table
3-1) AISC.
B-17
5
Example on Bending of Compact Sections
Example B - 6

• Determine the design strength (?b Mn) for W14 ?
  68 made of A-572-Gr50 steel and
• Continuous lateral support.
• Unbraced length 20 ft, Cb 1.0
• Unbraced length 20 ft, Cb 1.75

Solution
A) Check compactness
web is always compact ! ? Mn Mp FyZ 20 ?
115 5750 ink 479 ftk. ?b Mn 0.9 ? 479
431 ftk.
B)
B-18
6
Continued
Since Lp lt (Lb 20 ft) lt Lr ? Equation F
2.2 controls
?b Mn 0.9 ? 316.25 284.6 ftkip.

• For Cb 1.75, other conditions unchanged
• ? Mn 1.75 ? 316.25 553.4 ftk.
• Since Mn Mp,
• then Mn Mp 479 ftk
• ?bMn 0.9 ? 479 431 ftk.

B-19
7
Design of Un-braced Beams
Example B - 7
A simply supported beam of span 20ft is to
carry static dead load of (1.0 k/ft) and a live
load of (2.0 k/ft) in addition to its own dead
weight. The flange is laterally supported at
support points only. Select the most economical W
shape using A572-Gr50 steel.
Solution
Estimate self weight 0.06 k/ft. Wu 1.2 x 1.06
1.6 x 2 4.47 k/ft
Check your selection From Load Factor
Design Selection Table 3.2 in AISC (page 3-17)
Zx 77.9 in3 , Lp 8.76 ft , Lr 28.2 ft.
Determine (Cb) for UDL 1.14 (see B-16) ?bMn ?
Mu ? ?bMn ? 224 k.ft
Enter Beam Design Moments Chart at AISC for Lb
20 ft, and ?bMn 234, select W12 x 53 (page
3.126 but for Cb 1.0)
B-20
8
Design Problem Contd..
Since Lplt(Lb20 ft) lt Lr then equation ( F-2
2b AISC)
where Cb 1.14 Mp FyZx 50x77.9 3895
k. in 324.6 kft. Sx 70.6 in3
?bMn 0.9 x 292 262.7 ? (Mu 223.6 kft) OK
B-21
9
Bending Strength of Non-compact Sections
As noted earlier, most W,M S shapes are
compact for Fy 36 ksi and Fy 50 ksi , very
few sections are non-compact because of their
flanges, but non are slender. The effect of
non-compact flange is recognized in the AISC as
the smaller value of LTB (AISC F 2.2) and
where
B-22
10
Non Compact Flange Section
Example B - 8
A simple supported beam with span 45 ft is
laterally supported at ends only , and is subject
to the following service loading D.L. 0.4
k/ft ( including self wt.) L.L 0.7
k/ft Is W 14 x 90 made of A572-Gr50 steel
adequate?
Solution
Wu 1.2 x 0.4 1.6 kft
B-23
11
Non Compact Flange Section Contd.
? ?p lt ? lt ?r The shape is non-compact.
Section properties Zx 157 in3 , Sx 143
in3 (properties 1) Lp
15.2 ft , Lr 42.6 ft
(Table 3.2 p. 3.16)

• Now we check the
• capacity due to LTB
• Lr lt ( Lb 45 ft)
• ? Elsatic LTB controls

?bMn 0.9 x 638.2 574.4 ? Mu OK
B-24
12
Non Compact Flange Section Contd.
Mn Fcr Sx ? Mp
(probably this beam is O.K.).
?bMn 398.7 k. ft lt 405 kft
B-25