CIVIL ENGINEERING FORMULAS

BEAM FORMULAS 71

whereLunbraced length of the member
Emodulus of elasticity
Iymoment of inertial about minor axis
Gshear modulus of elasticity
Jtorsional constant

The critical moment is proportional to both the lateral bending stiffness EIy/L
and the torsional stiffness of the member GJ/L.
For the case of an open section, such as a wide-flange or I-beam section,
warping rigidity can provide additional torsional stiffness. Buckling of a simply
supported beam of opencross section subjected to uniform bending occurs at the
critical bending moment, given by

(2.27)

whereCwis the warping constant, a function of cross-sectional shape and
dimensions (Fig. 2.25).
In the preceding equations, the distribution of bending moment is assumed to
be uniform. For the case of a nonuniform bending-moment gradient, buckling
often occurs at a larger critical moment. Approximation of this critical bending

Mcr

LB

EIyGJECw

2

L^2 

FIGURE 2.25 Torsion-bending constants for torsional buckling. Across-sectional

area;Ixmoment of inertia about x–xaxis;Iymoment of inertia about y–yaxis. (After

McGraw-Hill, New York). Bleich, F., Buckling Strength of Metal Structures.

AreaA

y

y

x–

x x

h

144

(a) Equal-leg angle (b) Un-equal-leg angle (c) T Section

b 2

bt

t

t

h

b^3 t^3

t

b

w

Cw =

A^3

Cw=

4

Iy+x^2 A

(d) Channel (e) Symmetrical I

C h^2

w= I–^4

h^2 Iy

Cw=

4 Ix

h^2 A

(^36144)
h^3 w^3
36
(b (^1) +
(^3) +b
2
C t^33 )
w=
y
y
h x x