CIVIL ENGINEERING FORMULAS

BUILDING AND STRUCTURES FORMULAS 211

When, however, the compression flange is solid and nearly rectangular in
cross section, and its area is not less than that of the tension flange, the allow-
able stress may be taken as

(9.12)

When Eq. (9.12) applies (except for channels), Fbshould be taken as the larger
of the values computed from Eqs. (9.12) and (9.10) or (9.11), but not more than
0.60Fy.
The moment-gradient factor Cbin Eqs. (9.8) to (9.12) may be computed from

(9.13)

whereM 1 smaller beam end moment and M 2 larger beam end moment.
The algebraic sign of M 1 /M 2 is positive for double-curvature bending and
negative for single-curvature bending.When the bending moment at any point
within an unbraced length is larger than that at both ends, the value of Cb
should be taken as unity. For braced frames, Cbshould be taken as unity for
computation of FbxandFby.
Equations (9.11) and (9.12) can be simplified by introducing a new term:

(9.14)

Now, for 0.2 Q1,

(9.15)

ForQ 1:

(9.16)

As for the preceding equations, when Eq. (9.8) applies (except for chan-
nels),Fbshould be taken as the largest of the values given by Eqs. (9.8) and
(9.15) or (9.16), but not more than 0.60Fy.

LOAD-AND-RESISTANCE FACTOR DESIGN
FOR BUILDING BEAMS

For a compact section bent about the major axis, the unbraced length Lbof the
compression flange, where plastic hinges may form at failure, may not exceed
Lpd,given by Eqs. (9.17) and (9.18) that follow. For beams bent about the
minor axis and square and circular beams, Lbis not restrictedfor plastic analysis.

Fb

Fy

3 Q

Fb

(2Q)Fy

3

Q

(l/rT)^2 Fy

510,000Cb

Cb1.751.05

M 1

M 2

0.3

M 1

M 2 

2

2.3

Fb

12,000Cb

ld /Af