Equivalent Static Lateral Force

= −131,250(C2− C1),

C1= e^−λL= 0.52, C2=

λ_dL+ C1

e^−λL= 0.61 = 0.61,

X_F= 131,250(0.61 − 0.52) = 11,585 lb for both bearings.

4.4.3.4 Design for the CWR on Steel Railway Bridges

Based on similar considerations, AREMA (2008) and many railway companies estab-lish standard practice for anchoring CWR to long open deck steel spans. In general, the recommended practice is to use longitudinal rail anchors on approaches, and near fixed ends of spans, allowing some movement near expansion ends of spans.^∗

4.4.4 S^EISMICF^{ORCES ON}S^TEELR^AILWAYB^RIDGES

The level of seismic dynamic analysis required depends on the location and characteristics of the bridge.

An equivalent static analysis is often used in the analysis of ordinary steel rail-way bridges where the response to seismic forces is depicted primarily by the first or fundamental vibration mode. Steel railway bridges that may be analyzed by an equiv-alent static analysis are typically simply supported, not (or only slightly) skewed or curved, and have spans of almost equal length and supporting substructures of almost equal stiffness. Seismic forces in an equivalent static analysis are developed based on a period-dependent coefficient and the weight of the bridge. AREMA (2008) recommends the use of a seismic response coefficient and the uniform load method.^† The seismic forces on complex steel railway bridges are generally determined for use in a dynamic structural analysis.^‡These loads are typically represented by an elastic design seismic response spectrum. AREMA (2008) recommends the use of a normalized response spectrum based on the seismic response coefficient.

4.4.4.1 Equivalent Static Lateral Force

The equivalent static distributed lateral force, p(x), applied to the steel superstructures of a railway bridge is

p(x)= Cnw(x) (4.49)

where Cn= (1.2 ASD/Tn^2/3)≤ 2.5AD is the seismic response coefficient for the nth mode of vibration and 5% damping ratio; w(x) is the distributed weight of the

∗Rail expansion joints are sometimes used on very long bridges or bridges with unusual bearing configuration (i.e., adjacent expansion bearings on long spans).

†For some bridges it may be appropriate to consider the multimode dynamic analysis method.

‡AREMA Chapter 9 indicates that a modal analysis is appropriate for such railway bridges.

140 Design of Modern Steel Railway Bridges

superstructure, A is the base acceleration ratio determined from appropriate geo-logical sources^∗ for the design return period,^†S is the site coefficient between 1.0 and 2.0 depending on foundation soil conditions,^‡D= [1.5/(0.4ξ + 1) + 0.5] is the damping adjustment factor to account for the actual superstructure percentage of crit-ical damping,ξ,^§Tnis the natural period of the nth mode of vibration and is equal to 2π/ωn, andωnis the natural frequency of the nth mode of vibration (seeTables 4.2 and4.3andFigure 4.8).

However, in some cases, the development of the equivalent static distributed lateral force based on the seismic response coefficient is inappropriate and consideration of loading based on site-specific information is required.^∗∗

The equivalent static lateral distributed force, p(x), is calculated in two orthogonal directions (longitudinal and transverse for ordinary bridges). Following a linear elastic analysis^††in each direction, forces are distributed to superstructure members based on load path, support conditions, and stiffness. Since these member loads are orthogonal and uncorrelated, they must be combined^‡‡ for design purposes. AREMA (2008) recommends the method often referred to as the 100%–30% rule (Equations 4.50a and b) to combine the seismic loads for member design.

EQ= 1.00FT+ 0.30FL, (4.50a)

EQ= 0.30FT+ 1.00FL, (4.50b)

where EQ is the combined seismic design force, FTis the absolute value of the seismic force in the transverse direction, and FLis the absolute value of the seismic force in the longitudinal direction.

4.4.4.2 Response Spectrum Analysis of Steel Railway Superstructures The response spectrum used to represent the seismic loading of more complex steel superstructures is a plot of the peak value of the response as a function of the natural period of vibration of the superstructure. These are typically plotted for a particular damping ratio^§§and response (deformation, velocity, or acceleration). AREMA (2008) recommends the use of a normalized spectral response based on the seismic response

∗ For example, the U.S. Department of the Interior Geological Survey maps.

† The design return period depends on the earthquake event frequency and the limit state under consideration (serviceability, ultimate, or survivability).

‡ Rock, soil type, stratigraphy, depth, soil stiffness, and shear wave velocity are considered in the site coefficient.

§ Established from tests or other sources in the literature of structural dynamics. The percentage of critical damping for steel superstructures is often less than 5% and depends on materials, structural system/foundation, deck type and whether the structural response is linear elastic or post yield.

∗∗For example, some bridges on soft-clays and silts where vibration modes greater than the fundamental mode have periods of less than 0.3 s and bridges near faults or in areas of high seismicity. In these cases, alternative equations, available in seismic design standards and guidelines, for Cnmay apply.

††Linear elastic analysis is used for the equivalent lateral force method at the serviceability limit state.

‡‡These combined forces account for the directional uncertainty and simultaneous occurrence of the seismic design forces in members.

§§Often established for a damping ratio (percentage of critical damping) of 5%.

Loads and Forces on Steel Railway Bridges 141

Natural period, T_n (seconds) C_n

FIGURE 4.25 Typical actual response spectrum.

Natural period, T_n (seconds) C_n

T₀ T_s

T_r

FIGURE 4.26 Typical design response spectrum.

coefficient. This is essentially a pseudo-acceleration^∗response spectrum normalized by the natural period of vibration, Tn. The actual pseudo-acceleration response spec-trum for a given earthquake and the design pseudo-acceleration response specspec-trum will typically look like the plots of Figures 4.25 and 4.26, respectively.

AREMA (2008) recommends the following with respect to the normalized design response spectra: T_r is the maximum natural vibration period for essentially rigid response, T₀= 0.096S, and Ts= (0.48S)^3/2.

However, dynamic analyses of railway bridges typically underestimate the actual natural vibration period and, therefore, the response of the bridge for low period structures. AREMA (2008) recommends a design response spectrum without reduced response (or C_n) below T₀(Figure 4.27)unless the effects of foundation flexibility, foundation rotational movement, and lateral span flexibility were included in the dynamic analysis. In some cases, the development of the response spectra from the seismic response coefficient is inappropriate and consideration of loading based on site-specific response spectra is required.^†

The response spectrum must be calculated in each orthogonal and uncorrelated direction (longitudinal and transverse) and, therefore, must be combined for design purposes. AREMA (2008) recommends either the square root sum of squares (SSRC)

∗For steel bridge superstructures with low damping and short vibration periods the pseudo-acceleration response is a close approximation the actual acceleration response.

†For example, where A≥ 0.2 and Tn≥ 0.7 for bridges on very soft clays and silts, and for bridges on soft clays and silts where vibration modes greater than the fundamental mode have periods of less than 0.3s and bridges near faults or in areas of high seismicity. In these cases, alternative equations, available in seismic design standards and guidelines, for Cnmay apply.

142 Design of Modern Steel Railway Bridges

Natural period, T_n (seconds) C_n

T_s

FIGURE 4.27 Typical AREMA design response spectrum used with simple dynamic analyses.

method (Equation 4.51) or the method often referred to as the 100%–30% rule (Equations 4.50a and b) to combine the seismic loads.

F=

F_T²+ F²_L. (4.51)

Example 4.21

The normalized response spectrum is required for a 100 ft long steel girder span with the following properties:

Weight: 3000 lb/ft Ix= 100,000 in.⁴ Iy= 5000 in.⁴ ξ = 3%

The bridge is located where A= 10% and founded on material with S= 1.0 (rock)

Ts= [0.48(1.0)]^3/2= 0.33 s Cn= 0.142

T_n^2/3 ≤ 0.30.

The normalized response spectrum is shown in Figure E4.11.

0 0.35

0.3 0.25 0.2 0.15 0.1 0.05 0

0.1 0.2 0.3 0.4 0.5 Period, T (seconds) Sesmic response coefficient, Cn

Normalized response spectra (ex. 4.21)

0.6 0.7 0.8 0.9 1

FIGURE E4.11

Loads and Forces on Steel Railway Bridges 143

Q 5'

FIGURE 4.28 Derailment load.

4.4.5 LOADSRELATING TOOVERALLSTABILITY OF THESUPERSTRUCTURE