4 Loads and Forces on Steel Railway Bridges
4.4 OTHER STEEL RAILWAY BRIDGE DESIGN LOADS
4.4.3 F ORCES F ROM THE CWR ON S TEEL R AILWAY B RIDGES
4.4.3.3 Acceptable Relative Displacement between Rail-to-Deck and Deck-to-Spanand Deck-to-Span
Assuming a multiple span bridge with n spans of equal length, L, and alternating fixed and expansion bearings on substructures, Equation 4.40 with the boundary conditions outlined inSection 4.4.3.2may be solved to yield
Δx = x2(n+1)(L)− xn+1(L)= α0ΔT
2λ (1+ λL − Cn) (4.48) whereΔT is the bridge temperature change with respect to construction temperature, λ =√
k/EAr, and k2is the equivalent horizontal spring constant for the rail-to-deck-to-superstructure system.
Loads and Forces on Steel Railway Bridges 133
Fixed bearing force at center pier (n = 2 spans)
F = 0.75 F = 1.00
F = (α0 ΔT)/(αΔt)
FIGURE 4.24 An example relationship between fixed bearing force, length of the span, deck and track stiffness, and rail size for two expansion/contraction ratios.
F E F E
45' 45'
5 6
kt
kd kd
kt
1 2 3 4
Track Deck Track
Span Span
Deck
FIGURE E4.10
134 Design of Modern Steel Railway Bridges
Example 4.17
The double track open deck steel multibeam railway bridge shown inFigure E4.1comprises two 45 ft simple spans (Figure E4.10).The CWR with elastic rail fastenings is used on the friction-bolt fastened timber deck. Determine the maximum stress in the CWR, relative displacement between the rail and superstructure, rail separation, and longitudinal bearing force at the pier. The following are characteristics of the bridge:
ΔTc= Δtc= −100◦F kd= 400 lb/in. (normal strain rate)
kt= 100 lb/in. (normal strain rate) Maximum stress in the CWR:
λd=
Substitution into Equation 4.45 with n= 2 yields σcwr= −9425
Force in each rail= 9758(13)/1000 = 127 kips compression, OK.
Rail separation:
kd= 200 lb/in. (rapid strain rate) kt= 50 lb/in. (rapid strain rate)
Loads and Forces on Steel Railway Bridges 135
Substitution into Equation 4.43 yields Δxs= −6.50 × 10−4[(1/(5.16 × 10−4))+ (1/(2.58 × 10−4))] = −3.78 in., which may be excessive, requiring a longitudinally stiffer deck or track. For example, if a frozen ballast with kt= 100 lb/in. (rapid strain rate) is considered, the rail gap is reduced to 3.0 in.
Relative displacement:
Substitution into Equation 4.48 with n= 2 yields Δx = 0.34
1+ λL − (λL + e−λL)e−λL
Δx = 0.34 [1 + 0.39 − (0.39 + 0.67) 0.67] = 0.23 in., which is likely OK and will not cause deck fastener damage.
Fixed bearing force at the pier:
Substitution into Equation 4.47 yields XF= N4(0)− N3(0)= −EArαΔt
For bridges with short spans, the amount of thermal movement per span is small and generally easily accommodated by the normal tolerances of railroad track on bridges. It should be noted that the longitudinal stiffness values assumed for the rail-to-deck and deck-to-superstructure interfaces may not reflect values at a particular railroad location.
Example 4.18
An open deck steel deck truss bridge comprises a single 225 ft span. The CWR with elastic rail fastenings is used on the friction-bolt fastened timber deck.
Determine the maximum stress in the CWR, relative displacement between the rail and superstructure, rail separation, and longitudinal bearing force at the abutment. The following are characteristics of the bridge:
ΔTc= Δtc= −70◦F kd= 400 lb/in. (normal strain rate)
kt= 100 lb/in. (normal strain rate)
136 Design of Modern Steel Railway Bridges
Maximum stress in the CWR:
λd=
Substitution into Equation 4.45 with n= 1 yields σcwr= −9425
1+ 0.31(λdL− 1 + e−λdL)
,
σcwr= −9425 [1 + 0.31(1.97 − 1 + 0.14)] = −12,688 psi for both rails.
Force in each rail= 12.688(13)/1000 = 165 kips compression; there is risk of rail buckling.
Rail separation:
kd= 200 lb/in. (rapid strain rate) kt= 50 lb/in. (rapid strain rate) Substitution into Equation 4.44 yields
Δxs= −4.55 × 10−4
Substitution into Equation 4.48 with n= 1 yields Δx = 0.24(1 + λL − e−λL),
Δx = 0.24 (1 + 1.97 − 0.14) = 0.68 in., which may be excessive requiring a longitudinally stiffer deck.
Fixed bearing force at the abutment:
Substitution into Equation 4.47 yields
XF= N3(0)− N2(0)= −EArαΔt
XF= 131,250(0.30 − 0.14) = 20,386 lb for both bearings.
Loads and Forces on Steel Railway Bridges 137
Example 4.19
In order to reduce the relative displacements at the rail-to-deck-to-superstructure system in Example 4.18, a fastening system on the bridge with greater horizontal elastic spring stiffness is proposed. Determine the maximum stress in the CWR, relative displacement between the rail and superstructure and rail separation.
kd= 850 lb/in. (normal strain rate) kt= 100 lb/in. (normal strain rate) Relative displacement:
λd=
kd
EA= 1.06 × 10−3in.−1, ldL= 2.87.
Substitution into Equation 4.48 with n= 1 yields Δx = 0.16(1 + λL − e−λL),
Δx = 0.16(1 + 2.87 − 0.056) = 0.63 in.; relative displacement remains quite large.
Maximum stress in the CWR:
Substitution into Equation 4.45 with n= 1 yields σcwr= −9425
1+ 0.31(λdL− 1 + e−λdL)
σcwr= −9425 [1 + 0.31(2.87 − 1 + 0.056)] = −15,052 psi for both rails.
Force in each rail= 15,052(13)/1000 = 196 kips compression; the rail may buckle.
Rail separation:
kd= 425 lb/in. (rapid strain rate) kt= 50 lb/in. (rapid strain rate) Substitution into Equation 4.44 yields
Δxs= −4.55 × 10−4
138 Design of Modern Steel Railway Bridges
Examples 4.18 and 4.19 illustrate that, for long open deck spans, there are conflict-ing design requirements that the rail-to-deck-to-superstructure connection be flexible enough to avoid excessive compressive stress in the CWR (that could precipitate buckling), and rigid enough to reduce rail separation and relative displacements at the rail-to-deck-to-superstructure interfaces.
Therefore, in order to allow for movement between the rail and superstructure while providing sufficient rail anchoring to preclude excessive relative displacements, the CWR may be anchored to only a portion of the span length. The portion of the span length to which the CWR is anchored should be adjacent to the fixed bearings to allow the necessary movement between the rail and superstructure (anchored CWR over the expansion bearing areas will resist the thermal movements of the span). The effect of this is illustrated in Example 4.20.
Example 4.20
In order to reduce the relative displacement between the rail and super-structure in Example 4.18, anchoring the CWR to only a portion of the rail is proposed. If only 1/3 of the span length (from fixed bearings) has the CWR anchored to the deck, determine the maximum stress in the CWR, relative displacement between the rail and superstructure and rail separation, Maximum stress in the CWR:
λd=
into Equation 4.45 with n= 1 yields σcwr= −9425
1+ 0.31(λdL− 1 + e−λdL)
,
σcwr= −9425 [1 + 0.31(0.66 − 1 + 0.52)] = −9937 psi for both rails.
Force in each rail= 9937(13)/1000 = 129 kips compression, OK.
Relative displacement:
Substitution into Equation 4.48 with n= 1 yields Δx = 0.24
1+ λL − e−λL ,
Δx = 0.24 (1 + 0.66 − 0.52) = 0.27 in., OK.
Rail separation:
Substitution into Equation 4.44 yields Δxs= −4.55 × 10−4
Loads and Forces on Steel Railway Bridges 139
Fixed bearing force at the abutment:
Substitution into Equation 4.47 yields XF= N3(0)− N2(0)= −EAαΔt
α0ΔT
2αΔt(C2− C1)
= −131,250(C2− C1),
C1= e−λL= 0.52, C2=
λdL+ C1
e−λL= 0.61 = 0.61,
XF= 131,250(0.61 − 0.52) = 11,585 lb for both bearings.