Straight and curved steel I-girder bridges - Computational Analysis and Design of Bridge Structures - page 199

required. For curved structure, EC w is often the dominant contributor to
the individual girder torsional stiffness. Without consideration of EC w ,
the local twisting responses of the girders cannot be modeled accurately.
On the other hand, a full three-dimensional (3D) finite element analysis
(FEA), in which the thin-wall sections are modeled by plane shell elements,
bypasses the need for the modeling of warping stiffness within the single
beam element used to model the girder in two-dimensional (2D) grid anal-
ysis approaches.
A rigorous solution of grid analysis to take care of the warping prob-
lem of a thin-wall beam requires the warping deflection as an additional
DOF. Several researchers (e.g., Hsu and Fu 1990; Fu and Hsu 1995) have
included the warping deflection as the seventh DOF, in addition to the
regular six DOFs, at each node for the curved beam analysis to consider
the warping effect. For the case of partial warping restrained, an effec-
tive torsional constant, K eff , was proposed by Fu and Hsu (1994) and later
improved by Elhelbawey and Fu (1998) to consider warping effects in a
regular six DOFs analysis. A simple, easy-to-apply effective torsional con-
stant for the rotational stiffness of a restrained open section was developed
to take both the pure torsion and the warping torsion into account. This
effective (equivalent) torsional constant, K te , can be easily calculated and
used for any generic finite element structural analysis program.
The original torsional constant for most common structural shapes, J ,
can be approximated by Equation 7.2.

= ∑ 3 /

J bt

(7.2)

where b and t are the width and thickness of the thin-wall elements, respec-
tively. The effective (equivalent) torsional constant, K te , developed by Fu
and Hsu (1994), can be expressed as

λ λ

K

te =

J cosh / cosh − 1 0 . C (7.3)

2 2

where:
λ 2 is the GJ / EC w , (λ = l / a , where a is used in AISC documents)
C is the correction factor that equals {1.0/[1.0 + 2.95 ( b/l ) 2 ]}
l is the unbraced length
b is the flange width

Once the effective torsional constant is determined, the stiffness matrix
for a grid structure can be derived by using the traditional straight beam
method with three DOFs (torsional rotation, bending rotation, and deflection)