Loads and Forces on Steel Railway Bridges - Design of Modern Steel Railway Bridges - page 127

The partial differential equation of horizontal motion from force equilibrium on a
simply supported span bridge of constant mass and stiffness is

EA ∂ 2 x(x , t)
∂x 2

m ∂ 2 x(x , t)
∂t 2

∂x(x , t)
∂t

− + + c x = h(x , t) , (4.39)

where x(x , t) is the superstructure horizontal deflection at distance x and time t ,EA
is the axial stiffness of the span, c x is the s up erstructure equivalent longitudinal vis-
cousdampingcoefficient,and h(x , t)

k d x(x , t) isthedistributedlongitudinalforce
due to thermal expansion transferred through an elastic rail-to-deck-to-superstructure
system represented by an equivalent horizontal spring stiffness, k d . ∗ Longitudinal
movement will typically occur primarily at the rail-to-deck or deck-to-superstructure
interface depending on their respective degree of longitudinal restraint. † This model
also oversimplifies the rail-to-deck-to-superstructure system with a single elastic hor-
izontal stiffness. More sophisticated models may be developed ‡ that use different
elastic horizontal stiffnesses at the rail-to-deck and deck-to-superstructure interfaces.
Assuming negligible longitudinal viscous damping and neglecting superstructure
longitudinal inertia effects (acceptable for ordinary steel railway superstructures),
Equation 4.39 may be expressed as the differential equation (Fryba, 1996)

=−

EA d 2 x(x)
d x 2

− + k d x(x) = 0, (4.40)

which may be solved considering various failure criteria, such as:

• Safe rail gap (separation) on a bridge after fracture of the CWR. Rail frac-
ture § may occur due to cold weather contraction. The safe rail gap depends
on individual railroad operating practice but is generally considered to be
between 2 and 6 in.
• Safe stress in the CWR to preclude buckling. ∗∗ Rail buckling, particularly
at the typically weaker †† bridge approach track, may occur when rails on
the bridge are highly longitudinally restrained such that large rail forces are
created during hot weather rail expansion.

∗ A linear elastic spring is assumed for all levels of displacement in this model. Rail-to-deck and deck-
to-superstructure interfaces may be more accurately modeled using bilinear springs, which following
initial elastic behavior then acts perfectly plastic during steady state sliding friction displacement.
† For example, longitudinal movement may occur at the deck-to-superstructure interface for open deck
beams and girders with smooth tops, and at the rail-to-deck interface for girders with substantial longitu-
dinal resistance at the deck-to-superstructure interface (e.g., by rivet and bolt heads, restraint angles, and
bars) positive deck connection or well-tensioned deck anchor bolts. Some modern elastic rail fasteners
allow for longitudinal movement (no “hold-down” forces) at the rail-to-deck interface.
‡ Usually used in conjunction with computer-based FEA.
§ Modern NorthAmerican heavy freight railroad CWR is considered to have a minimum fracture strength
of about 300 kips.
∗∗ Modern North American heavy freight railroad CWR is considered to have a minimum safe buckling
strength of about 150 kips.
†† Weaker lateral restraint behind abutment backwalls and approach track.