FRV Dynamic Structural Analysis

CLIENT TPZP MADE BY CHECKED BY PROJECT NO.

PROJECT CAHSTP – Fresno River Viaduct BDB 4017026

SUBJECT Rail‐Structure Interaction DATE DATE REVISION SHEET NO.

MIDAS Model Description 11/25/14 1 / 12

Objective

This document describes the analyses performed to evaluate the dynamic effects of trains crossing the

Fresno River Viaduct. These analyses are required per Section 12.6.6 of the Project Design Criteria to

“determine resonancy dynamic impact (ILLV) effects, and limit vertical deck accelerations.”

Structure Description

The analyses for the Fresno River Viaduct are based on the 60% plans prepared by Tutor Perining –

Zachary – Parsons (TPZP) for Contract No. HSR13‐06. The bridge consists of 17 spans with a total length

of over 1580 feet. The superstructure is a cast‐in‐place concrete box girder with a depth of 8’‐6” at the

centerline of bridge. Typical spans are simply supported, though there are two continuous units. Spans

1 and 2 are continuous over Bent 2. Spans 3 through 6 are joined into a four‐span‐continuous unit.

Piers consist of a single column with a flared top to accommodate the bearings. Typically, piers are

founded on a single drilled shaft with an oversized segment near the ground line. This system acts like a

Type II shaft as defined in Caltrans Seismic Design Criteria in that the plastic hinge will form in the pier

just above the oversized segment of the drilled shaft. Bents 4 through 6, which are in the deepest

portions of the Fresno River will each be supported by four drilled shafts and a footing. The approaches

are retained fill.

The bridge carries two tracks of high‐speed rail traffic. Approach track is supported on ties on ballast.

On the aerial structure, the tracks are mounted using direct fixity. Rail is continuously welded without

expansion joints on the bridge or its approaches.

Geometry

The bridge is on a horizontal curve with a radius of 28,000 feet, measured at the centerline of Track S2.

For the seismic and rail‐structure interaction analyses, the bridge will be idealized as being on a tangent

alignment.

AASHTO LRFD BDS gives criteria for considering a curved bridge as straight for dynamic analyses in

Article 4.7.4.3. The FRV does satisfy all of the criteria associated with assuming the bridge is straight.

Article 4.6.1.2.3 of AASHTO LRFD BDS also allows the superstructure to be idealized as straight within a

span for horizontal loads since the subtended angle within a span is less than 12 degrees. The Article

does call for the angle to change at the piers. For FRV, the change in angle is about 1 degree for the 80’

spans.

To facilitate modeling of the dynamic train loadings, as will be described later, the superstructure

geometry was further simplified to have a uniform spacing of nodes. A standard node spacing of 4 feet

was used. Therefore, some span lengths changed slightly to accommodate the uniform node spacing.

Spans 4 and 5 were lengthened by 2’ from 130’ to 132’. Spans 12 and 13 were lengthened by 2’ from





90’ to 92’. Spans 14 through 17 were shortened by 1’ from 85’ to 84’. All other span lengths were

unchanged.

To represent the continuous welded rail beyond the ends of the bridge, the model included 360 feet of

track beyond each abutment as required by Table 12‐24 of the Project Design Criteria.

Section Properties

The typical section, shown on Drawing Number ST‐J3460‐FRV, has a web thickness of 1’‐0”. The typical

bottom slab thickness is 1’‐2”. This section is assigned to Spans 1 – 11 and Spans 14 – 17. Spans 12 and

13 which have a bottom slab thickness of 1’‐4”, which is represented in the section properties used on

those spans.

At all bearing locations, the web thickens to 2’‐0”. In addition, portions of the continuous spans have

thicker webs in the vicinity of intermediate anchorages. These localized areas of thicker webs are not

included in the model. Likewise, the model will not include access openings in the bottom slab or

cutouts for the shear blocks. Stiffness contribution of diaphragms at bearing locations are not expected

to affect global response and are, therefore, not included in the model.

For the “upper bound” stiffness model, the properties of the parapet and crash wall are assumed

composite with the box girder. Dimensions and locations of these walls are taken from the 7/15/2014

plan revision of Drawing Number ST‐J3460‐FRV.

Gross section properties are used to represent the pier members.

For each track, a single element was used to represent both rails on the track. The section properties for

this element were input as two times the individual section properties of an individual 141 RE rail.

Material Properties

Material properties of the structural elements were based on the “upper bound” stiffness as defined in

Section 12.6.8.2 of the project design criteria. Concrete elastic modulus was computed using the

expected material strengths, which are 30% higher than nominal strengths. Materials used the “lower

bound” condition for mass per Section 12.6.8.2 which is 95% of the nominal weight. The superstructure

concrete properties were based on a nominal strength of 7 ksi. Pier concrete properties were based on

a nominal strength of 5 ksi.

Rail Springs

Properties of the rail fasteners are presented in Section 12.6.8.5 of the Project Design Criteria. For

direct fixity, which applies on the aerial structures, the slip threshold is computed assuming a pair of

fasteners with 1.35 kip unloaded longitudinal restraint at 27” spacing. Then, the unloaded threshold

occurs at a 1.2 kips per foot of track uniformly distributed. The loaded threshold occurs at double that

value, 2.4 kips per foot of track uniformly distributed. The slip displacement for both conditions is 0.02





inches. For the analyses described herein, the trains traverse the bridge on Track 1. Therefore, springs

for Track 1 are loaded properties and springs for Track 2 are for unloaded properties.

For ballasted track, which applies on the embankments, the value from Table 12‐24 will be used. The

yield load is 1.37 kips per foot of track uniformly distributed. The slip displacement is 0.08 inches.

These properties apply for both loaded and unloaded conditions.

The spring stiffness for each spring was computed by multiplying the stiffness and yield force times the

tributary length for the spring. The typical node spacing used is 4 feet. Nodes at expansion joints,

therefore, have a tributary length of 2 feet.

Track Loading

Typical Spacing At Expansion Joints

Movement Yield Force Stiffness Movement Yield Force Stiffness

(ft) (kip) (kip/ft) (ft) (kip) (kip/ft)

Direct Unloaded 0.0017 4.80 2880 0.0017 2.40 1440

Direct Loaded 0.0017 9.60 5760 0.0017 4.80 2880

Ballasted Both 0.0067 5.48 822 0.0067 2.74 411

Boundary springs were included at the ends of the model as described in Table 12‐24 in the Project

Design Criteria.

Track Loading

Boundary Spring Properties

Stiffness Movement Yield

(kip/ft) (ft) (kip)

Direct Unloaded 24,200 0.0017 40.3

Direct Loaded 34,200 0.0017 57.0

Ballasted Both 12,900 0.0067 86.0

As was previously described in the discussion of Section Properties, a single element was used to

represent both rails on a track. Accordingly, the stiffness and strength of spring elements used to

represent the slip of the rail clips is set to represent both rails.

Bearing Springs

Typical bearings on the bridge are steel‐reinforced elastomeric bearings. Bearings do not have a

masonry plate or a sole plate and depend on friction for the connection to the superstructure and

substructure. Analyses assume that friction will not be overcome for the small shear loads associated

with train passage. In addition, each pier has a shear key that resists transverse motion of the

superstructure relative to the pier.

Although each bearing line contains two bearings, the link between the superstructure and substructure

was modeled using a single element. To accurately represent the rotational degrees of freedom,

rotation about the bridge longitudinal axis was restrained while the remaining rotational degrees of





freedom were left free. Vertical and transverse translation was also restrained at each bearing line.

Longitudinal stiffness was based on the bearing dimensions and the applicable connection to the shear

key at the bearing line.

Foundation Springs

The stiffness of the foundation in response to service loads was computed at each pier using FB‐

MultiPier. Soil properties used for input into those models were based on data presented in the AMEC

Technical Memorandum prepared for TPZP dated April 16, 2014.

The foundation stiffness was represented as a 6x6 stiffness matrix. For the Type C bents, the spring was

applied at the top of the enlarged drilled shaft. For Bents 4 through 6, the foundation springs were

applied at the bottom of the footing.

Mass Modeling

The weight of elements will be based on Table 12‐1 in the Project Design Criteria. Masses will be

introduced by the MIDAS feature converting loads to masses. Per Section 12.6.8.2 of the project

criteria, the “lower bound” mass condition, 95% of nominal, is applied.

The mass of the superstructure, including the parapet and crash walls, was computed directly from the

section properties of the structural elements in the model. The remaining mass on the superstructure

was input as uniformly‐distributed line loads which were subsequently converted into masses. These

additional masses included OCS poles and support, cable trough and walkway surface, rails and

fasteners, sound walls, and systems in the cable troughs. These applied masses were located both sides

of the bridge centerline with appropriate eccentricity to provide an accurate estimate of the second

mass moment of inertia of the cross section about the centerline of the bridge section. For each track,

the additional masses have a nominal weight of about 5.6 kips per foot acting at an eccentricity of 12.6

feet from the centerline of the bridge.

Eigenvalue Analysis Results

An eigenvalue analysis was performed to determine the first 50 vibration modes of the structure. These

results were used to verify the properties of the model as well as to determine the frequency at which

the mass participation was at least 90% in each of the three principal directions of translation.

The figures below show the cumulative mass participation in terms of the number of modes as well as in

terms of the frequency.





The critical mode shapes in each of the principal directions are plotted in the figures below. The first

transverse mode is the structure’s critical mode and has a frequency of 1.59 Hz. Acting on Spans 1





through 6, this mode engages over 41% of the total structure mass. The first longitudinal mode is Mode

5 which engages Spans 1 through 6. With a frequency of 2.88 Hz, the first longitudinal mode engages

over 41% of the structure’s mass. Mode 13 is the first vertical mode. At a frequency of 5.17 Hz, the

first vertical mode is reflected in Spans 3 through 6. This mode only engages about 2% of the total

structural mass. The most influential vertical mode is Mode 24 with a frequency of 6.68 Hz. This mode

acts on almost all of the spans and has a modal mass participation of over 27% of the total mass.

First transverse mode. (Mode 1. Frequency = 1.59 Hz)

First longitudinal mode. (Mode 5. Frequency = 2.88 Hz)





First vertical mode. (Mode 13. Frequency = 5.17 Hz)





Rayleigh Damping

Structural damping for the dynamic analyses was accounted for using the Rayleigh damping formulation

in which the damping is proportional to the system’s mass and stiffness. Damping values for the

dynamic model were set at 1.0% per Table 12‐23 of the Project Design Criteria. The target frequencies

used for computing the Rayleigh damping coefficients were computed based on Section 11.7.3.18 of the

Project Design Criteria. One target frequency was 90% of fundamental frequency of the system which is

1.59 Hz, which corresponds to a period of about 0.63 seconds. Then, the lower target frequency is 1.43

Hz corresponding to a period of 0.70 seconds. The other target frequency is from the mode at which

90% of the structural mass is participating. This mode, Mode 37, has a frequency of 8.14 Hz which

corresponds to a period of about 0.12 seconds.

A plot of the modal damping ratios computed for each period is presented in the figure below. Using

these target frequencies will produce conservative damping ratios for the first 37 modes.

Dynamic Train Loading

Section 12.6.6 of the Project Design Criteria presents the requirements for the “Dynamic Structural

Analysis” of a train crossing the bridge. The Criteria call for a fully dynamic analysis for each of five train

sets at a suite of speeds ranging from 90 mph to 250 mph. Given the similarities in the train sets, Train

Set 1 will be the only load pattern considered for evaluation of the 60% design.





MIDAS does not have a built‐in module for dynamic analysis of a train crossing the bridge. Therefore, a

methodology was developed to represent how the applied vertical load at each node on the bridge

changed with time as the train moves across the bridge. The load history at each node, then, depends

on the node spacing, the train axle pattern, the train’s velocity, and the node’s distance from the

beginning abutment.

For example, consider the development of a loading time history for Train Set 1. This train has eight cars

with four axles per car. Then, the load history must capture the arrival of each of these 32 axles. For





Train Set 1, the last axle is about 677 feet behind the first axle. For the case where the train is moving at

100 mph, about 146.7 feet per second, it takes about 4.6 seconds for the train to pass over a fixed point

on the track. The figure below shows a load time history for Train Set 1 at 100 mph.

Once the time history has been developed, it must be adjusted for each point along the track to account

for the time it takes for the first axle to reach that point. For Train Set 1 at 100 mph, the train will take

about 10.8 seconds to go about 1584 feet from the beginning abutment to the end abutment.

Vertical Acceleration Results

Dynamic structural analyses were carried out for Train Set 1 with train velocities from 90 mph to 250

mph in 10 mph increments. Analyses were performed for time steps of 0.010 seconds and 0.005

seconds to test for the sensitivity of the response to analysis time step.

For each train velocity, the maximum vertical acceleration for the entire structure over all time steps

was tabulated. In general, the peak vertical acceleration occurred near the middle of the 92‐foot‐long

spans in the model. A plot of the peak vertical accelerations is plotted in the figure below.

Results show that higher speeds bring higher accelerations. The reduced time step has a small effect on

the results. Smaller time steps than 0.005 seconds are not practical for preliminary analtyical work. All





analyses produced peak accelerations less than the 0.50g threshold stipulated in Section 12.6.6.4 of the

Project Design Criteria.

Impact Factor Results

Impact factor for the single‐track loading (LLV) was computed from the dynamic analyses. Based on the

peak acceleration results in which the peak accelerations occurred in the 92‐foot‐long spans, bending

moments in Span 13 were used as a benchmark for computing the dynamic impact factor.

The maximum static bending moment was computed by performing a static analysis to determine the

maximum moment in the span using the moving load generator feature in MIDAS. For both the static

and dynamic models, the rails were modeled explicitly as well as the nonlinear response of the clips

fastening the rails to the superstructure.

Dynamic analyses were performed for times steps of 0.010 seconds as well as for 0.005 seconds. Peak

bending moments occurred at a velocity of 170 mph for both cases. Peak dynamic response is about

2,680 kip‐ft. For a peak static response of around 1,730 kip‐ft, the effective impact factor is 1.55.

References

AASHTO, LRFD Bridge Design Specifications, 6th Edition, 2012. [AASHTO LRFD BDS]

FRV Dynamic Structural Analysis

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