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Nguyenngoc Tuyen
NMSU
LOAD RATING OF A RIVETED STEEL ARCH BRIDGE
BY
NGUYENNGOC TUYEN, B.S.
A thesis submitted to the Graduate School
in partial fulfillment of the requirements
for the degree
Master of Science in Civil Engineering
New Mexico State University
Las Cruces, New Mexico
May 2005
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“Load Rating of a Riveted Steel Arch Bridge,” a thesis prepared by Nguyenngoc
Tuyen in partial fulfillment of the requirements for the degree, Master of Science in
Civil Engineering, has been approved and accepted by the following:
Linda Lacey Dean of the Graduate School
David V. Jáuregui Chair of the Examining Committee
Date
Committee in charge:
Dr. David V. Jáuregui, Chair
Dr. Gabe V. Garcia
Dr. John McNamara
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ACKNOWLEDGMENTS
This research project was funded by the Utilities and Infrastructure Division
(NWIS-UI) of the Los Alamos National Laboratory (LANL). I wish to thank LANL
and New Mexico State University (NMSU) for the financial support that made this
research possible. I would also like to acknowledge Doug Volkman (Technical Staff
Member, LANL NWIS-UI) for his outstanding assistance in this research.
I am deeply indebted to my academic advisor, Dr. David V. Jáuregui, for the
many hours of discussions that enriched my graduate study in the many different
areas of this thesis. His time and effort will forever be appreciated. I also wish to
acknowledge the valuable suggestions of Dr. Kenneth R. White and my committee
members, Dr. Gabe V. Garcia and Dr. John McNamara. For the valuable structural
engineering courses I had the pleasure of taking, I would like to thank the faculty
members, Dr. Craig Newtson and Dr. Clinton Woodward.
I would also like to thank my office mates (Scott Burns, Daniel Lamb, and
Kelly Silliman) for their friendship and assistance, especially in English. I would like
to thank the Vietnamese Government for supporting me throughout the duration of
my graduate studies at NMSU. Finally, special thanks are extended to my parents,
Canh N. Nguyen and Hang T. Pham, and my younger sister, Thanh T. Nguyen, for
their love and moral support.
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VITA September 22, 1977 Born in Thaibinh Town, Thaibinh, Vietnam May 1995 Graduated from Thaibinh Talented High School, Thaibinh, Vietnam May 2000 Graduated from Hanoi University of Civil
Engineering, Bachelor of Science in Civil Engineering, Hanoi, Vietnam January 2001 Lecturer Assistant in Hanoi University of Civil
Engineering Hanoi, Vietnam
Field of Study Major Field: Civil Engineering
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ABSTRACT
LOAD RATING OF A RIVETED STEEL ARCH BRIDGE
BY
NGUYENNGOC TUYEN, B.S.
Master of Science in Civil Engineering
New Mexico State University
Las Cruces, New Mexico, 2005
Dr. David V. Jáuregui, Chair
The Omega Bridge is a riveted steel arch bridge that connects the town of Los
Alamos, New Mexico to the Los Alamos National Laboratory (LANL) over the Los
Alamos Canyon. The bridge was designed and constructed in 1951 based on the ASD
(Allowable Stress Design) method specified in the 1944 AASHO (American
Association of State Highway Officials) Specifications. In 1992, the bridge was
rehabilitated based on the LFD (Load Factor Design) method to satisfy traffic
requirements. However, the present capacity rating was based on the original ASD
criteria and did not incorporate the rehabilitation. Due to new traffic demands, the
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bridge need to be rated based on the current rating method, LFR or Load Factor
Rating, which is widely applied in many states in the U.S. Thus, the major objective
of this study was to determine the capacity of the Omega Bridge according to the
current AASHTO (American Association of State Highway and Transportation
Officials) LFD Specification.
In this study, the structural analysis of the individual members of the Omega
Bridge was carried out using the RISA (Rapid Interface Structural Analysis) program.
Rating calculations were performed for the stringers, floor beams, spandrel beams,
columns (pier, skewback, and arch columns), and the arch rib using MATHCAD
2000 and Excel programs. The results of this study provided important information
and recommendations concerning the capacity level of the Omega Bridge. The major
conclusion is that the Omega Bridge, in general, is in satisfactory condition and no
load posting is necessary but there are some concerns for the floor beams and the arch
columns. In particular, the study pointed out that the floor beam dimensions do not
satisfy the AASHTO compactness requirements and thus, the floor beams are
recommended to be inspected thoroughly for signs of instability. In addition, leaning
of the columns may also reduce their capacity; therefore, monitoring of the column
out-of-plumb (both magnitude and direction) is suggested in future capacity rating of
the bridge. Finally, load testing along with 3-D finite element analysis is also
recommended to refine the calculation of the rating factors.
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TABLE OF CONTENTS
TABLE OF CONTENTS.................................................................................. vii
LIST OF TABLES............................................................................................ x
LIST OF FIGURES .......................................................................................... xii
DATA ON COMPACT DISC (CD) ................................................................ xv
Chapter
1. BRIDGE BACKGROUND.................................................................... 1
1.1. Introduction ............................................................................ 1
1.2. Past Inspection and Evaluation Studies.................................. 4
2. BRIDGE DESCRIPTION ...................................................................... 9
2.1. Floor System .......................................................................... 9
2.1.1. Stringers ................................................................................. 13
2.1.1.1. Exterior stringers .................................................................... 16
2.1.1.2. Interior stringers ..................................................................... 17
2.1.2. Floor Beams ........................................................................... 19
2.1.3. Spandrel Beams...................................................................... 21
2.2. Columns ................................................................................. 25
2.2.1. Pier and Arch Columns .......................................................... 26
2.2.2. Skewback columns................................................................. 27
2.3. Arch ribs................................................................................. 27
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3. AASHTO RATING ANALYSIS........................................................... 30
3.1. General information ............................................................... 30
3.2. Bridge load rating methods .................................................... 31
3.2.1. Introduction ............................................................................ 31
3.2.2. Allowable Stress and Load Factor Rating (ASR and LFR) ... 32
3.2.3. Load and Resistance Factor Rating (LRFR) .......................... 34
3.3. Design, Legal, and Permit Load Rating ................................. 37
4. LOAD RATING FACTOR OF FLOOR SYSTEM ............................... 41
4.1. Stringers ................................................................................. 42
4.1.1. Description of Rating Model for Stringers............................. 42
4.1.2. Load Factor Rating (LFR) Analysis....................................... 45
4.2. Floor beams ............................................................................ 49
4.2.1. Description of rating model.................................................... 49
4.2.2. Load Factor Rating (LFR) Analysis....................................... 56
4.3. Spandrel beams ...................................................................... 58
4.3.1. Description of Rating Model for Spandrel Beams ................. 58
4.3.2. Load Factor Rating Analysis.................................................. 63
5. LOAD RATING OF COLUMNS .......................................................... 68
5.1. Description of Rating Model.................................................. 68
5.2. Load Factor Rating Analysis.................................................. 70
5.2.1. BEAM-COLUMN Model: Combined Axial Load and Bending .................................................................................. 70
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5.2.2. COLUMN Model: Axial Loading.......................................... 85
5.3. Discussion of BEAM-COLUMN and COLUMN Rating Factors .................................................................................... 87
6. LOAD RATING OF ARCH RIB........................................................... 99
6.1. Description of Rating Model.................................................. 99
6.2. Load Factor Rating Analysis.................................................. 101
6.3. Discussion of Rating Factors.................................................. 106
7. SUMMARY AND CONCLUSIONS..................................................... 110
7.1. Summary ................................................................................ 110
7.1.1. Floor System .......................................................................... 110
7.1.2. Columns ................................................................................. 112
7.1.3. Arch rib .................................................................................. 113
7.2. Conclusions ............................................................................ 114
APPENDIX....................................................................................................... 116
REFERENCES ................................................................................................. 273
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LIST OF TABLES
Table Page
2.1 Weight estimate of bridge railing, fencing and utilities ..................... 10
4.1 Moment values and rating factors for interior stringer. ..................... 47
4.2 Moment values and rating factors for exterior stringers. ................... 47
4.3 Rating factors at negative moment region (Section #2) of stringers.. 48
4.4 Moment values and rating factors for floor beams. ........................... 57
4.5 Critical rating factors for floor beams. ............................................... 58
4.6 Live load distribution for spandrel beam ........................................... 62
4.7 Moment values and rating factors of spandrel beam of BEAM model. ................................................................................................ 65
4.8 Moment values and rating factors of spandrel beam of FRAME
model. ................................................................................................ 65
4.9 Rating factors at Section #2 and Section #3 of spandrel beam. ......... 66
5.1 Sample calculation of sidesway moment amplification factor, B2..... 76
5.2 Interaction ratios and rating factors for bridge columns under HS-20 design truck loading. .............................................................. 80
5.3 Interaction ratios and rating factors for bridge columns under
TYPE 3 legal truck loading. .............................................................. 81
5.4 Interaction ratios and rating factors for bridge columns under TYPE 3S2 legal truck loading. .......................................................... 82
5.5 Interaction ratios and rating factors for bridge columns under
TYPE 3-3 legal truck loading............................................................ 83
5.6 Interaction ratios and rating factors for bridge columns under FIRE special truck loading. ............................................................... 84
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5.7 Rating factors for bridge columns based on axial loading only. ........ 87
5.8 Inventory rating factors for bridge columns based on beam-column (RFi,b-c) and column (RFi,c) behavior. ................................................ 88
5.9 Column alignments on east side of Omega Bridge. .......................... 93
5.10 Column alignments on west side of Omega Bridge. ......................... 94
5.11 Load rating of arch column #10 in vertical and inclined position .... 96
6.1 Effective length factor (K) values for arch rib (AASHTO, 2002). ... 103
6.2 Interaction ratio and rating factors for arch rib based on AASHTO Equation (10-47)............................................................................... 107
6.3 Interaction ratio and rating factors for arch rib based on AASHTO
Equation (10-47) for Case 3. ............................................................ 108
7.1 Controlling rating factors of the floor system. .................................. 111
7.2 Controlling rating factors of the columns based on BEAM-COLUMN model. ................................................................ 113
7.3 Controlling rating factors of the arch rib........................................... 114
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LIST OF FIGURES
Figure Page
1.1 Location of Omega Bridge and West Road detour. ........................... 2
1.2 Longitudinal elevation view of the Omega Bridge. ........................... 2
1.3 Cross-section of floor system before rehabilitation in 1992. ............. 4
1.4 Cross-section of floor system after rehabilitation in 1992. ................ 7
2.1 Cross-section of the floor system. ...................................................... 11
2.2 Overall plan view of the bridge floor system..................................... 12
2.3 Exterior stringer layout. ..................................................................... 13
2.4 Interior stringer layout........................................................................ 15
2.5 Positive moment region of the exterior stringer................................. 16
2.6 Positive moment region of interior stringers (in 1st span).................. 17
2.7 Positive moment region of interior stringers (in 6th span).................. 18
2.8 Negative moment region of interior stringers (at pier columns)........ 19
2.9 Floor beam elevation view. ................................................................ 19
2.10 Cross sections of floor beams. ........................................................... 20
2.11 Spandrel beam layout. ........................................................................ 22
2.12 Positive moment region of the spandrel beam. .................................. 23
2.13 Negative moment region of the spandrel beam.................................. 24
2.14 Column layout. ................................................................................... 25
2.15 Cross section of pier and arch columns.............................................. 26
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2.16 Cross-section of skewback columns. ................................................. 27
2.17 Arch rib span and rise. ....................................................................... 28
2.18 Cross-section of the arch rib. ............................................................. 29
3.1 AASHTO design load for ASD and LFD. ......................................... 38
3.2 AASHTO legal loads. ........................................................................ 38
3.3 “Emergency-One Titan” fire truck..................................................... 40
4.1 Rating models of stringers (with critical sections)............................. 42
4.2 Distribution of dead load on floor beam FB#2 of approach span and moment diagram. ............................................................................... 50
4.3 Distribution of dead load on floor beam FB#6 of arch span and
moment diagram. ............................................................................... 50
4.4 Distribution of HS20 live load on floor beam FB#2.......................... 51
4.5 Distribution of live loads on floor beam FB#2 and the corresponding moment diagrams....................................................... 54
4.6 Distribution of live loads on floor beam FB#6 and the
corresponding moment diagrams....................................................... 55
4.7 BEAM rating model of spandrel beam and critical sections.............. 59
4.8 FRAME rating model of spandrel beam. ........................................... 59
5.1 BEAM-COLUMN rating model of pier, skewback, and arch columns.............................................................................................. 68
5.2 COLUMN rating model of pier, skewback, and arch columns.......... 69
6.1 PINNED rating model of arch rib. ..................................................... 99
6.2 RIGID rating model of arch rib.......................................................... 100
6.3 Critical locations of axial force and bending moment of arch rib...... 106
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7.1 Critical locations of the floor system: (a) approach spans and (b) arch spans .......................................................................................... 111
7.2 Critical locations of the columns........................................................ 113
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DATA ON COMPACT DISC (CD)
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Chapter 1
BRIDGE BACKGROUND
1.1. Introduction
The Los Alamos Canyon Bridge (also called the Omega Bridge) is a riveted,
steel arch bridge that carries north and south bound traffic on Diamond Drive (NM
501) over the Los Alamos Canyon between the town of Los Alamos, New Mexico
and technical areas of the Los Alamos National Laboratory (LANL). As shown in
Figure 1.1, the alternate route runs through the canyon on West Road, which entails
approximately 3.1 km (1.9 miles) of additional travel on a steep grade. For emergency
vehicles such as fire trucks, the West Road detour is not a suitable option for obvious
reasons. Consequently, the primary objective of the study reported herein was to
determine the current capacity level of the Omega Bridge, so that more reliable
decisions could be made by the LANL regarding the safety of the bridge under
modern traffic loads. To achieve this objective, a conventional rating analysis was
performed according to the Load Factor Rating (LFR) Method specified in the
American Association of State Highway and Transportation Officials (AASHTO)
Manual for Condition Evaluation of Bridges (1994).
The Omega Bridge was designed by Finney and Turnispeed, fabricated by the
American Bridge Company, and erected by the Vinson Construction Company in
1951. As shown in Figure 1.2, the bridge is 820 ft. long with a 442.5-ft. arch span and
six 62-ft. approach spans (there are three approach spans at each end of the bridge).
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Figure 1.1 Location of Omega Bridge and West Road detour.
15 spans (29.5ft each)
422.5ft
106.6ft
62ft 62ft62ft 62ft 62ft62ft
SOUTH NORTH
Figure 1.2 Longitudinal elevation view of the Omega Bridge.
Omega Bridge
Detour
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The bridge was originally designed for H-20 vehicular live load based on the
ASD (Allowable Stress Design) Method specified in the 1944 AASHO (American
Association of State Highway Officials) Specifications. Normal weight concrete with
a compressive strength of 3000 psi and Grade 40 reinforcement was used for the
deck; for the superstructure, ASTM A7 (Fy = 33 ksi) steel was used. Composite
action, by means of mechanical shear connectors, was not provided between the deck
and the superstructure in the original design. The cross-section of the bridge floor
system before its major repair in 1992 had an overall width of 51’–3 ½”, which
included a 39’–9” wide roadway and a 7’–6” wide pedestrian walkway (see Figure
1.3). The roadway had no shoulders and four lanes, each having a width of 9’–11 ¼”;
the narrow lanes caused significant delays to traffic flow over the bridge, especially
during peak traffic hours. The walkway was separate from the steel superstructure
and consisted of a pre-cast concrete double tee supported by a steel bracket secured to
the west spandrel beam. Figure 1.3 shows the walkway after it was repaired in 1983;
in the original cross-section, the reinforced concrete deck simply extended past the
west spandrel beam to carry pedestrian traffic. For reasons discussed later, the
original cantilever deck overhang was replaced with the walkway configuration
shown in Figure 1.3. Starting from 1983, the floor system remained as shown in the
figure until it was rehabilitated in 1992.
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7' - 412"
35' - 0"
6' - 9"6' - 9"6' - 9" 7' - 412"
39' - 9"7' - 6"
51' - 312"
9' - 1114" 9' - 111
4"9' - 1114"9' - 111
4"Lane 2Lane 1 Lane 3 Lane 4
Figure 1.3 Cross-section of floor system before rehabilitation in 1992.
1.2. Past Inspection and Evaluation Studies
Since the early 1970s, several engineering studies have been performed by
various consultants related to the physical condition and structural integrity of the
Omega Bridge (Merrick & Company, 1989). The first significant study of the bridge
was carried out by HNTB Corporation in 1973, which included an in-depth bridge
inspection and a structural analysis of the deck and steel superstructure. The major
observations made from the inspection were (1) the overall structure was in good
condition; (2) the number of missing rivets was minimal; (3) the test strength of the
steel was more characteristic of ASTM A36 steel (Fy = 36 ksi) rather than ASTM A7
steel (Fy = 33 ksi) as specified in the design; and (4) the use of de-icing salts coupled
with the poor concrete casting techniques used in the original construction was
deteriorating the deck. From the structural analysis, HNTB Corporation found that (5)
the deck was overstressed by 29% under the existing dead loads and H-20 live
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loading; (6) the steel members were also overstressed but to a lesser degree than the
deck; (7) the H-20 vehicular live load used in the original design was consistent with
the type of truck loads currently (i.e., 1973) traveling over the bridge; and (8) the
member stresses would increase under the HS-20 vehicular live load specified for
new bridge designs.
Approximately 10 years after the investigation by HNTB Corporation, two
studies were performed by Holmes and Narver in 1983 with assistance from New
Mexico State University (NMSU) which focused on assessing the structural condition
of the original deck and pedestrian walkway. The first major deficiency identified for
the study was that the deck was structurally adequate only for H-15 vehicular live
load, although the records showed that the original design had been based on H-20
vehicular live load. As a result, significant repair or total replacement of the deck was
recommended. The second major deficiency found was that the overhanging portion
of the deck which served as the walkway was improperly constructed, causing
excessive sag and concern for public safety. Consequently, construction plans were
drawn up by Holmes and Narver to replace the walkway, which was completed in
1983 (see Figure 1.3).
As noted above, the previous investigations of the Omega Bridge concluded
that the deck was deteriorated and overstressed. Accordingly, a study was performed
in 1988 by Merrick & Company to come up with various alternatives along with
construction cost estimates for rehabilitating the bridge. Based on information
provided in that rehab study, the LANL opted to replace the entire deck and to retrofit
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the remaining components of the floor system to meet the current AASHTO and
NMSHTD (New Mexico State Highway and Transportation) standards. The
following year, Merrick & Company continued the rehabilitation project starting with
a feasibility study of three deck replacement alternatives including a normal-weight
concrete deck; a light-weight concrete deck; and a light-weight, concrete filled steel
grid deck. Using a three-dimensional structural analysis program, the level of stress in
the bridge members under dead load and HS-20 vehicular live load (plus impact) was
evaluated for the three deck replacement alternatives. The analysis showed that the
light-weight concrete deck alternative resulted in the lowest member stresses and
thus, would require the least work to retrofit. Ultimately, Merrick and Company
decided on a light-weight, reinforced concrete deck with stay-in-place metal decking.
In 1992, the floor system of the Omega Bridge was rehabilitated, resulting in the
cross-section shown in Figure 1.4. The rehabilitation increased the width of the cross-
section from 51’–3 ½” to 55’-6” and the roadway from 39’–9” to 44’–0” in order to
provide four 11’ – 0” wide traffic lanes (the original lanes had a width of 9’–11 ¼”).
Other major rehabilitation work done on the bridge included: (1) light-weight
concrete with a 28-day compressive strength of 4.5 ksi was used for the deck; (2)
shear studs were installed on the interior stringers and spandrel beams to provide
composite action with the deck; (3) cover plates were added to the interior stringers
and spandrel beams for additional moment capacity; and (4) exterior stringers
supported by outrigger beams were added on both sides of the bridge width. A more
detailed description of the Omega Bridge is provided in Chapter 2.
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35' - 0"
6' - 9"7' - 412"3' - 6" 6' - 9" 6' - 9" 6' - 9" 6' - 9"7' - 41
2 " 3' - 6"
11' - 0"11' - 0" 11' - 0"11' - 0"Lane 1 Lane 3Lane 2 Lane 4
44' - 0"
55' - 6"
8' - 0"
Figure 1.4 Cross-section of floor system after rehabilitation in 1992.
Since the early 1980s, NMSU has conducted regular in-depth inspections of
the Omega Bridge every 2 or 3 years in accordance with NBIS (National Bridge
Inspection System) Standards. The most recent inspection was completed in the
summer of 2003; both the superstructure and substructure were rated as “fair” during
that inspection. No major deficiencies were found with the superstructure, only
isolated areas of corrosion on the arch ribs, spandrel beams, and bracing members.
Cleaning and painting of these rusted areas was recommended within five years.
During the substructure inspection, minor cracking, scaling, and spalling (with
evidence of leaching) was discovered in the concrete abutments and piers; the most
significant deterioration was found at the skewback concrete columns and footings,
which had cracks up to ¼” wide with moderate leaching and spalling. At the time of
the inspection, major repairs were being made to seal the cracks in the substructure.
Overall, the inspection found no major deficiencies which would influence the load
rating of the Omega Bridge. Ultimately, the physical condition of the Omega Bridge
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observed from the inspection was documented in virtual reality format. This
inspection record was referenced frequently throughout the AASHTO load rating
analysis of the bridge and proved to be an extremely helpful aid, particularly for
interpretation of the as-built construction plans.
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Chapter 2
BRIDGE DESCRIPTION
In this chapter, members of the bridge are described in detail. The bridge is
divided into three main components: floor system; columns; and arch ribs. In the first
part of the chapter (2.1 Floor System), details of the floor system are described. Next,
details of the columns are described in the second part (2.2 Columns) and finally,
details of the arch ribs are described in the last part (2.3. Arch Ribs).
2.1. Floor System
The floor system includes a reinforced concrete slab, six stringers, 28 floor
beams and two spandrel beams. Figure 2.1 shows the cross-section while Figure 2.2
shows the overall plan view of the bridge floor system. As shown in Figure 2.1, the
total width of the bridge deck is 55’–6” (out-to-out) and includes a 44’–0” roadway
with four traffic lanes (each lane has a width of 11’–0”) and an 8’–0” sidewalk on the
west side. The slab concrete is light-weight with a density of wc = 120 pcf and a 28-
day compressive strength of fc’ = 4500 psi. The thickness of the slab is ts = 7.25”
which includes a 0.5-in. integral wearing surface. The transverse reinforcement
consists of top and bottom mats of #5 bars placed at a spacing of 6.5”. The
longitudinal reinforcement consists of a top mat of #3 bars spaced at 9” and a bottom
mat of #4 bars spaced at 6” or 9” as shown in Figure 2.1.
Bridge appurtenances include a sidewalk railing; west and east guardrails;
fencing and light poles; and electric and steam utilities. The dead load estimates for
these accessories given in Table 2.1 were furnished by the LANL based on the
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original design and rehabilitation drawings and subsequently field verified. These
dead weights were increased by 4% to account for miscellaneous details. With the
exception of the fencing, the weights of the accessories were distributed over the
entire length of the bridge. The fencing is located on the 150-ft. center portion of the
bridge length on each side of the bridge width and was thus, distributed only over that
region of the bridge. Details of the stringers, floor beams, and spandrel beams are
discussed in subsequent sections.
Table 2.1 Weight estimate of bridge railing, fencing and utilities
Sidewalk Railing 4-L4”X3”X5/16”=4(7.2plf)=28.8plf 1-5WF16 @3.54’/9.83’=5.76plf 1-5C6.7=6.7plf 1-Plate 9.1875”X0.3125”=9.74plf 1-Base Plate 10”X1”X10.5”X1/9.83’=3.03plf 1-Base Plate 10”X3/4”X10.5”X1/9.83’=2.27plf 1-Base Plate 8”X5/8”X8”X1/9.83’=1.15plf 2-Conn Plate 2X5”X3/8”X3.875”@1/9.83’=0.42plf 6-Conn Plate 6X3.25”X3/8”X3.875”@1/9.83’=0.82plf 4-Anchors 4X1”DiaX 8.25”X1/9.83’=0.75plf
Subtotal=59.44plf West Guardrail 1-Pipe 4” Dia=10.79plf 1-Plate ½”X10”X4.25”@1/8.33’=0.72plf 1-Plate ½”X5.5”X8.5”@1/8.33’=0.80plf 1-Bent Plate 18.25”X1/4”X12”@1/8.33’=1.86plf 1-Anch Bolt ½” DiaX 8.5”@1/8.33’=0.06plf
Subtotal=14.23plf East Guardrail 2-Pipe 3.5”Dia=18.22plf 1-Plate ½”X1.83’X4”@1/8.33’=1.50plf 1-Plate ½”X5.5”X9”@1/8.33’=0.84plf 1-Bent Plate 18.25”X1/4”X12”@1/8.33’=1.86plf 1-Anch Bolt ½” DiaX 8.5”X1/8.33’=0.06plf 1-Splash Plate ¼”X9”=7.66plf
Subtotal=30.14plf Fencing 4-Pipe 2”Dia @ 150’=2190lbs 1-Pipe 3” Dia @ 12/10’ X 150’=1364.4lbs 1-Fencing 0.1483”X6/1’X12’X150’=635.35lbs 2-Conn Plate 2X0.375”X8”X8”@1/10’X150’=204.00lbs 2-Bent Plate 2X0.25”X13”X6”@1/10’X150’=165.75lbs Subtotal=4559.5lbs (one side – distributed on center 150’ of bridge) Subtotal=9119.0lbs (both sides – distributed on center 150’ of bridge)
Light Pole 6-Poles 5” Ave Dia X26.5’X1/814.5’=2.85plf 6-Poles 5” Ave Dia X25.25’X1/814.5’=2.72plf 12-Light Arms 3” DiaX 8’X1/814.5’=0.89plf 12-Lamps (Assume 15lbs each)X1/814.5’=0.22plf 2-Conduit 2” Dia=7.30plf
Subtotal=13.98plf Electric Utility 3-Conduit 2” Dia=10.95plf 2-Conduit 5” Dia=29.24plf 1-Conduit 1.25” Dia=2.27plf
Subtotal=42.46plf Steam Utility 1-Steam Pipe 10”Dia=40.48plf 1-CondensatePipe 4”Dia + π(1.913”X1.913”)/144in2X62.4pcf =14.98plf+4.98plf=19.96plf 1-Asbestos Insulation π {[(8.375”)2 - (5.375”)2]/144}153pcf=137.69plf 1-Asbestos Insulation π {[(4”)2 - (2.25”)2]/144}153pcf =36.51plf Hangars—Assume 10% of pipe=4.05plf
Subtotal=238.69plf
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11'-0" traffic lane 11'-0" traffic lane11'-0" traffic lane 11'-0" traffic lane
1'-3"
8'-0" sidewalk9"
CL Bridge
1.5% 1.5%4' - 6"
55'-6" out-to-out
3' - 6" 6' - 9" 7' - 412" 6' - 9" 6' - 9" 6' - 9" 7' - 41
2" 6' - 9" 3' - 6"
1'-3"
3"
714" #5 trans. reinf. bars @61
2" (top and bottom mat)
#3 long. reinf. bars @9"(top mat).
9" 9" 12" 17" 9"6 spa.@6"
= 3'-0"
412"
9" 9"
6 spa.@6"
= 3'-0"9" 9"
412"
412"
9" 9"6 spa.@6"
= 3'-0"9" 9"
412"
412"
9" 9"6 spa.@6"
= 3'-0"
1018"
#4 long. reinf.bars @ 6" or 9"
(bottom mat)30"
62" 9"
10" 6"
3 spa.@9"
= 2'-3"6"
3"
West East
West Outrigger Beam
InteriorStringer
ExteriorStringer
East Outrigger Beam (not shown)Floor Beam (not shown)
Figure 2.1 Cross-section of the floor system.
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31' 31' 31' 31' 31' 31' 29' - 6" 29' - 6" 29' - 6" 29' - 6" 29' - 6" 29' - 6" 29' - 6" 14' - 9"
407' - 3"
FB#1
FB#2
FB#2
FB#2
FB#2
FB#3
FB#4
FB#5
FB#5
FB#5
FB#6
FB#6
FB#6
FB#6
Wind BracingSpandrel Beams
Outrigger BeamColumn
InteriorStringersSkewback Col #1LCPier Col #2CLPier Col #1LC
7' - 412"
6' - 9"
6' - 9"
6' - 9"
7' - 412"
6' - 9"
6' - 9"
35'
Bearing Abutment #1CL
ExteriorStringers
Floor Beam
Arch CL
North
South
407' - 3"
29' - 6"
Skewback Col #2
6' - 9"
35'
6' - 9"
7' - 412"
6' - 9"
7' - 412"
6' - 9"
6' - 9"
ExteriorStringers CL Pier Col #4LC Pier Col #3CL
31' 31'
Bearing Abutment #1
FB#1
LC
FB#2
31'
FB#2
31'
FB#2
FB#2
31' 31'
FB#3
FB#4
InteriorStringersColumn
Outrigger Beam
29' - 6" 29' - 6"
Floor Beam
29' - 6"
FB#5
FB#5
29' - 6"
FB#5
Spandrel Beams
29' - 6"
FB#6
FB#6
29' - 6"
FB#6
Wind Bracing
14' - 9"
FB#6
ArchLC
Figure 2.2 Overall plan view of the bridge floor system.
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2.1.1. Stringers
Each stringer is a continuous beam supported at the locations of the floor
beams over a total of 27 spans as shown in Figure 2.2. The 12 spans on the approach
to the arch (six on both the north and south ends) each have a length of 31’–0” while
the remaining 15 spans over the arch have a length of 29’–6”.
The two exterior stringers are W21x62 sections (ASTM A36 steel) with no
cover plates, which were installed during the 1992 retrofit. Shear studs are distributed
only in the first span on the north and south ends of the stringers as shown in Figure
2.3. Therefore, only the positive moment regions in the end spans are composite; the
remaining length of the stringers is non-composite. The stud spacing is 9” over a
distance of 10’–6” from each end and changes to 11” over the remaining distance of
19’–3”. The studs terminate 3” from the centerline of the outrigger beams.
Symm. about Arch CL
1' 14 spaces@9" = 10'-6"
21 spaces@11" = 19'-3"
3"
31'-0" 376'-3"
Bearing Abutment #1LC
9"
Exterior Stringer
Outrigger Beam
Figure 2.3 Exterior stringer layout.
The four interior stringers are W21x62 sections (ASTM A7 steel), which were
installed when the bridge was originally built in 1951. Shear studs are provided only
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in the positive moment regions of the first and sixth spans and in the negative
moment regions at the floor beam locations above the pier and skewback columns on
the approach to the arch (three on both the north and south ends) as shown in Figure
2.4. The spacing of the shear studs is most dense (i.e., @ 7”) in the negative moment
regions over the floor beams having column supports. In the positive moment regions,
the studs are spaced similar to that of the exterior stringer with the exception of the 7”
spacing close to the first interior floor beam. Cover plates of ASTM A33 steel with
dimensions of 3/8”x7”x14’–0” were provided in the original design in the end spans
(both top and bottom flanges) starting at a distance of 6 ft. from the centerline of the
abutment bearings. During the 1992 retrofit, new cover plates of ASTM A36 steel
with dimensions of 3/8”x9”x8’–0” were provided at the location of the floor beams
having column support (on the bottom flange only).
According to Article 10.38.3 in the AASHTO Standard Specifications (2002),
the effective flange width of the concrete deck acting composite with the steel
stringers shall be the smaller of the following quantities: (1) one-forth the span length
of the girder; (2) the distance center-to-center of the girders; and (3) twelve times the
least thickness of the slab. For both the interior and exterior stringers, criterion (2)
controlled; therefore, the effective flange width for the stringers was taken as 81”.
Ignoring the thickness of the haunch, the section properties of the exterior and interior
stringers were computed.
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4'-0" 4'-0"4'-0"4'-0"
4'-0"4'-0"
LC Pier Col #2 C Skewback Col #1L
Pier Col #1CLLC Bearing Abutment #1
6'-0"9"
38" x 7" x 14'-0"
cover plates(top and bottom flange)
Floor Beam
W 21x 62Interior
Stringer
14 spaces @9" = 10'-6"
12 spaces @11" = 11'-0"
14 spaces @7" = 8'-2"
4" 15 spaces @7" = 8'-9"
15 spaces @7" = 8'-9"
1' 31' - 0"
15 spaces @7" = 8'-9"
15 spaces @7" = 8'-9"
15 spaces @9" = 11'-3"
12 spaces @11" = 11'-0"
15 spaces @7" = 8'-9"
15 spaces @7" = 8'-9"
31' - 0"
38" x 9" x 8'-0"
cover plate,(bottom flange only).
38" x 9" x 8'-0"
cover plate, (bottom flange only).
31' - 0" 31' - 0" 221' - 3"
22'-3" 22'-3"
31' - 0"
22'-3"
31' - 0"
22'-3"
Symm. about Arch CL
Figure 2.4 Interior stringer layout.
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2.1.1.1. Exterior stringers
As shown in Figure 2.3, the exterior stringers are composite with the deck
only for positive moment in the first 31’–0” span at the bridge ends; an 81” effective
deck width acts as the concrete compression flange of the composite section. Non-
composite and composite section properties for the exterior stringers (ignoring the
steel reinforcement) are given in Figure 2.5. The figure also shows the section
dimensions and the neutral axis location (labeled N.A.).
6.75"
Composite SectionNon-composite Section
81"
10.5"0.4"
21"
8.24"
N.A.
0.615"N.A.
8.24"
0.4"
20.905"
Non-composite Properties Composite Properties
A I St , Sb I Sct Scb (in2) (in4) (in3) (in4) (in3) (in3)
18.3 1330 126.67 4180 43780 200
Figure 2.5 Positive moment region of the exterior stringer.
In the exterior stringers, the shear studs are provided only within the end spans
as shown in Figure 2.3. Therefore, the negative moment region at the first interior
floor beam support is a non-composite section; the positive and negative moment
regions over the remaining length of the exterior girders are also non-composite.
Non-composite section properties for those regions are given in Figure 2.5.
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2.1.1.2. Interior stringers
In the positive moment regions in the first and sixth spans, an 81” effective
deck width acts as the concrete compression flange of the composite section; the
compression steel reinforcement in the deck slab is ignored. In the negative moment
regions above the pier columns, above the approach columns, the concrete slab is
subject to tension which leads to cracking. Therefore, the concrete slab was assumed
to not carry tension force (according to Article 10.50.2 of the AASHTO Standard
Specifications 2002) with only the tension steel reinforcement contributing to the
stiffness and strength of the composite section. Within the effective width of the slab,
there are nine #3 bars in the top mat and ten #4 bars in the bottom mat. Non-
composite and composite section properties for the interior stringers are given in
Figure 2.6 through 2.9.
81"
10.875"
21.75"21"
8.25"
7"
Non-composite Section
7"x3/8" cover plate
10.5"
N.A.
0.41"0.62"
20.82"
10.875"
21.75"
7"
8.25"
Composite Section
0.41"
N.A.
6.75"
Non-composite Properties Composite Properties
A I St , Sb I Sct Scb (in2) (in4) (in3) (in4) (in3) (in3)
23.77 1943 178.69 5519 5918 265
Figure 2.6 Positive moment region of interior stringers (in 1st span).
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In the positive moment region of the first span, 7”x3/8” cover plates are
provided on both the top and bottom flanges as shown in Figure 2.6. In the positive
moment region of the sixth span, cover plates are not provided as in the first span (see
Figure 2.7).
81"
0.62"
10.5"
21.375"
8.25"
Non-composite Section
0.41"
N.A.
6.75"
Composite Section
0.41"
8.25"
N.A.
20.873" 21"
Non-composite Properties Composite Properties
A I St , Sb I Sct Scb (in2) (in4) (in3) (in4) (in3) (in3) 18.52 1344 128 4217 33290 202
Figure 2.7 Positive moment region of interior stringers (in 6th span).
In the negative moment region at floor beam locations over the pier columns,
a cover plate is provided on the bottom flange only (see Figure 2.8). In the remaining
positive and negative moment regions of the interior girders, neither shear studs nor
cover plates are provided. Section properties of these non-composite regions are
given in Figure 2.7.
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6.75"
81"
21.375"
8.25"9"
9.228"
N.A.
9"x3/8" cover plate
0.41"0.62" 2"
25.625" 23.375" N.A. 21.375"
11.017"
9"8.25"
4.25" 0.41"
Non-composite Section Composite Section
Non-composite Properties Composite Properties
A I St Sb I Sct Scb (in2) (in4) (in3) (in3) (in4) (in3) (in3) 21.9 1670 137.5 180.9 2256 217.85 204.82
Figure 2.8 Negative moment region of interior stringers (at pier columns).
2.1.2. Floor Beams
There are two built-up sections used for the floor beams; one section
corresponds to the floor beams labeled FB#1 and FB#6 while the other section
corresponds to the floor beams labeled FB#2 through FB#5 (see Figure 2.2 for floor
beam labels).
2Ls 8" x 6" x 916" x 32'-9" (Floor Beams FB#1 and FB#6)2Ls 8" x 6" x 5 8" x 32'-9" (Floor Beams FB#2 through FB#5)
Web Plate48" x 3 8" x 32'-9"
2Ls 8" x 6" x 916" x 31'-5" (Floor Beams FB#1 and FB#6)2Ls 8" x 6" x 5 8" x 31'-5" (Floor Beams FB#2 through FB#5)
StringerWF21x62
STIFFENERS2Ls 6" x 4" x 916" x 3'-111
2" (Floor Beams FB#1 and FB#6)2Ls 6" x 4" x 5 8" x 3'-111
2" (Floor Beams FB#2 through FB#5)2 Fills 4" x 916" x 3'-0" (Floor Beams FB#1 and FB#6)2 Fills 4" x 5 8" x 3'-0" (Floor Beams FB#2 through FB#5)
StringerWF21x62
StringerWF21x62
StringerWF21x62
6' - 9" 6' - 9" 6' - 9" 6' - 3"6' - 3"
32' - 9"
Figure 2.9 Floor beam elevation view.
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As shown in Figure 2.9, there are four web stiffeners at the stringer locations.
Each stiffener consists of two angles (6”x4”x3’–111/2”) and two fill plates (4”x3’–0”)
arranged symmetrically about the web plate. The cross-section of the floor beams
consists of two angles (8”x6”x32’–9”) at the top; two angles (8”x6”x31’–5”) at the
bottom; and a web plate (48”x3/8”x32’–9”). The only difference between the two
floor beam sections is the thickness of the angles and fill plates; the thickness is 9/16”
for the floor beams labeled FB#1 and FB#6 and 5/8” for the floor beams labeled FB#2
through FB#5. The span length of a the floor beam is 35 ft. (center-to-center of
spandrel beams). Dimensions and properties for the two floor beams sections are
given in Figure 2.10.
48.5"
L 8" x 6" x 916"
PL 48" x 3 8" x 32'-9" PL 48" x 3 8" x 32'-9"
L 8" x 6" x 5 8"
48.5"
Floor Beams FB#1 and FB#6 Floor Beams FB#2 through FB#5
Floor beam FB#1 and FB#6 Floor beam FB#2 through FB#5
A I St, Sb A I St, Sb (in2) (in4) (in3) (in2) (in4) (in3) 48.4 19320 796.7 51.6 20960 864.2
Figure 2.10 Cross sections of floor beams.
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2.1.3. Spandrel Beams
Each spandrel beam (located on the west and east side of the bridge width) is
a continuous beam supported at the abutments and the column locations over a total
of 21 spans. The three approach spans on the north and south end of the bridge length
are 62 ft each and the remaining 15 spans over the arch are 29.5 ft each. As shown in
Figure 2.11, shear studs were installed over half the length of the three approach
spans (i.e., 93 ft. on both ends of the bridge). The stud spacing is 1’–3” over the first
31’–0” and 12.5” over the remaining 62’–0”. Bottom flange cover plates are provided
at the location of the first interior floor beam from the abutments (see Figure 2.11).
According to Article 10.38.3 of the AASHTO Standard Specifications (2002),
the effective flange width of the concrete deck acting composite with the steel
spandrel beam shall not exceed the following quantities: (1) one-forth the span length
of the girder; (2) the distance center-to-center of the girders; and (3) twelve times the
least thickness of the slab. Hence, the effective flange width of the deck acting
composite with the spandrel beam was controlled by criterion (3), which amounted to
81”.
The cross-section of the spandrel beam consists of two angles (8”x6”x3/4”) on
the bottom; two angles (4”x4”x3/8”) on the top; two web plates (66”x3/8” each); and a
top flange plate (25”x3/8”). The thickness of the haunch (2.87 in.) and the steel
reinforcement was included in the calculation of the composite section properties in
both the positive and negative moment regions. The dimensions and properties of the
spandrel beam sections are given in Figures 2.12 and 2.13.
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BEARING AT ABUTMENTCL
SYMMETRICAL ABOUTARCH SPANLC
24 spaces@1'-3" = 30'-0"
6" 912" 29 spaces
@1212" = 30'-21
2"29 spaces
@1212" = 30'-21
2"6"
31'-0" 31'-0" 31'-0"91
2"
96 72
LC PIER COL #1
9"
New 5 8" x 8" x 14'-0"cover plate, 2 required- 1 each bottom flange
angle of spandrel
314'-3"
Figure 2.11 Spandrel beam layout.
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As shown in Figure 2.11, shear studs were installed in the positive moment
region of the end spans. Cover plates were also provided on the bottom flange angles
of the section during the 1992 retrofit. However, analysis showed that the critical
section for bending moment occurred at the end of the cover plates and thus, the
cover plates were ignored. Within the 81” effective width of the slab, there are nine
#3 bars in the top mat and eight #4 bars in the bottom mat. Section properties for the
spandrel beam in the positive moment region are given in Figure 2.12.
4" x 4" x 3 8" L
8" x 6" x 3 4" L
66.875"
31.789"
49.049"
66.875"
81"
69.75"
6.75"
2.875"
25"
73.125"68.313"
Non-composite Section Composite Section
74"71.75"
N.A.
N.A.
25" x 3 8" top plate
66" x 3 8" web plate
Non-composite Properties Composite Properties
A I St Sb I Sct Scb (in2) (in4) (in3) (in3) (in4) (in3) (in3)
84.545 54300 1548 1708 114200 6406 2328
Figure 2.12 Positive moment region of the spandrel beam.
In the negative moment region at the pier columns closest to the abutments,
shear studs were installed on the spandrel beams to provide composite action. In
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negative flexure, the concrete slab is subject to tension which leads to cracking; thus,
only the reinforcement in the concrete slab contributes to the stiffness and strength of
the cross-section. Section properties for the spandrel beam in the negative moment
region are given in Figure 2.13.
81"
69.75" 66.875"
33.002"
6.75"
73.125"71.75" 74"N.A.
25" x 3 8" top plate
4" x 4" x 3 8" L
66.875" N.A.
31.789"
66" x 3 8" web plate
Non-composite Section
25"
8" x 6" x 3 4" L
Composite Section
Non-composite Properties Composite Properties
A I St Sb I Sct Scb (in2) (in4) (in3) (in3) (in4) (in3) (in3)
84.545 54300 1548 1708 58490 1727 1772
Figure 2.13 Negative moment region of the spandrel beam.
Since shear studs were not installed in the positive moment regions of the
remaining spans and in the remaining negative moment regions at the column
locations, the spandrel beam sections in these regions are non-composite. Non-
composite section properties of the spandrel beam can be found in Figures 2.12 and
2.13.
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2.2. Columns
Each spandrel beam lies in the arch rib plane and is supported by four pier
columns, 14 arch columns and two skewback columns as shown in Figure 2.14. All
pier columns have a riveted connection to the spandrel beam and a pinned support at
the base. The top ends of the skewback and arch columns also are riveted to the
spandrel beam. The base of the skewback columns are fixed to a concrete foundation
while the bottom ends of the arch columns are riveted to the arch rib.
Pier column #1
Skewback column #1 Skewback column #2
Pier column #2
Pier column #3
Pier column #4
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Arch column #
Column Label Length (ft) Column Label Length (ft)
Pier column #1 N2 18.4 Arch column #8 N12 5.8
Pier column #2 N3 41.2 Arch column #9 N13 8.5
Skewback column #1 N4 103.1 Arch column #10 N14 15.1
Arch column #1 N5 99.1 Arch column #11 N15 26.7
Arch column #2 N6 73.1 Arch column #12 N16 41.7
Arch column #3 N7 51.4 Arch column #13 N17 61.2
Arch column #4 N8 34.2 Arch column #14 N18 85
Arch column #5 N9 20.5 Skewback column #2 N19 86.8
Arch column #6 N10 11.7 Pier column #3 N20 47.4
Arch column #7 N11 6.9 Pier column #4 N21 22.1
Figure 2.14 Column layout.
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Figure 2.14 also shows the labels and lengths of all the columns used in the
analysis. The lengths of the columns were taken as the distances between the centers
of the connections at the two ends of the columns. The dimensions as well as the
section properties of all the column sections used in the analysis are given in Figures
2.15 and 2.16.
2.2.1. Pier and Arch Columns
The cross-section of the pier and arch columns are identical which consists of
four 4”x4”x1/2" angles and four 24”x1/2" plates as shown in Figure 2.15.
24"
24.5" back-to-back
4" x 4" x 12"
24" x 12"
24"
Pier/Arch Column Properties
A I r (in2) (in4) (in)
63 6762 10.36
Figure 2.15 Cross section of pier and arch columns.
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2.2.2. Skewback columns
The cross-section of the skewback columns consists of eight 4”x4”x1/2"
angles, three short plates (24”x1/2”), and two long plates (48”x1/2”) as shown in
Figure 2.16. The moment of inertia and radius of gyration are given for the in-plane
and out-of-plane axes.
48"
4" x 4" x 12"
24" x 12"
48" x 12"
48.5" back-to-back
24"
Skewback Column Properties
A Io ro Ii ri
(in2) (in4) (in) (in4) (in)
114 12950 10.66 31680 16.67
Figure 2.16 Cross-section of skewback columns.
2.3. Arch ribs
Each arch rib, which was original built in 1951, is a two-hinge parabolic arch
with a span of 422.5 ft. and a rise of 106.6 ft. as shown in Figure 2.17. The steel used
for the arch ribs is ASTM A33. The transverse distance between the two arch ribs is
equal to 25 ft. and the support locations of the east and west arch are at the same
elevation. Furthermore, each arch rib is symmetrical about its centerline.
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422.5ft
106.6ft
LC Arch rib
Figure 2.17 Arch rib span and rise.
The dimensions and properties of the arch section are given in Figure 2.18. As
shown in the figure, the built-up cross section consists of eight flange angles
(8”x8”x3/4"); two exterior web plate stiffener angles (6”x4”x3/4"); two interior web
plate stiffener angles (4”x4”x3/8"); two web plates (711/2”x1/2”) and two flange plates
(48”x3/4”). The center-to-center distance between the two web plates is 26” while the
center-to-center distance between the top and bottom flange plates is 72.75”. The
section properties given in Figure 2.18 are for the in-plane bending axis.
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46" x 3 4"
4" x 4" x 3 8" L6" x 4" x 3 4" L
71 12" x 12"
8" x 8" x 3 4" L
25.5"
72"
inside to inside of PLs
Arch Rib Properties
A I r St, Sb (in2) (in4) (in) (in3)
252 227100 30.02 6179
Figure 2.18 Cross-section of the arch rib.
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Chapter 3
AASHTO RATING ANALYSIS
3.1. General information
Load rating will provide a basis for determining the safe load capacity that is
expressed in terms of a Rating Factor (RF). In general, a Rating Factor may be
defined as the magnitude of load that a bridge can safely sustain or in other words, it
is the ratio of the actual to the required live-load capacity. Load rating calculations
should be based on information in the bridge file including the existing dimensions
and properties of the bridge of the most recent inspection. Therefore, as part of every
inspection cycle, bridge load ratings should be reviewed and updated to reflect any
relevant changes in condition or dead load noted during the inspection. Types of
events that may occur during the service life of a bridge that can ultimately influence
its load rating (i.e., safe load capacity) include: installation of a new deck wearing
surface; replacement of a bridge deck; section loss of a bridge member due to
deterioration and/or corrosion; retrofit of a bridge member; changes in vehicular live
loads and/or traffic demand; and widening of a bridge roadway. In the case of the
Omega Bridge, rehabilitation work was completed by Merrick & Company in the
early 1990s (to comply with the AASHTO standards current at the time) which
resulted in several changes to the original structure; the reader is referred back to
Chapter 1 for a discussion of the rehab. The study reported in this thesis provides the
load rating of the Omega Bridge as affected by this rehabilitation and according to the
latest AASHTO standards.
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For a typical load rating, all the components of the bridge (i.e. deck;
superstructure; substructure...) should be rated and thus, the component with the
smallest rating factor will control the structure as a whole. However, in this thesis,
attention will be given to the load rating of the superstructure (i.e. floor system,
columns and arch rib). Also, note that rating calculations may be obtained by either
an experimental or analytical means. However, the scope of this thesis research will
be based on analytical ratings only.
3.2. Bridge load rating methods
3.2.1. Introduction
There are three AASHTO methods available for the load rating of highway
bridges including, Allowable Stress Rating (ASR); Load Factor Rating (LFR); and
Load and Resistance Factor Rating (LRFR). The AASHTO Manual for Condition
Evaluation of Bridge (1994), which is consistent with the AASHTO Standard
Specification for Highway Bridges (2002), provides the basis for ASR and LFR while
the AASHTO Manual for Condition Evaluation and Load and Resistance Factor
Rating (LRFR) of Highway Bridges (2003) together with the AASHTO LRFD Bridge
Design Specifications (2004) offers the rating procedures pertinent to the LRFR
method. In addition, the new LRFR Manual (AASHTO, 2003) also includes the ASR
and LFR procedures in the appendix so that all three rating methods are available in a
single source. In this thesis, the Omega Bridge is rated based on the LFR method that
is the customary rating approach used in many states in the U.S. Descriptions of the
three methods are provided in the next sections.
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3.2.2. Allowable Stress and Load Factor Rating (ASR and LFR)
The ASR and LFR methods use the following basic equation to determine the
rating factor for a bridge component or connection subjected to a single load effect
(i.e., axial force, flexure, or shear):
I) L(1A
DA C FR
2
1LFR ASR, +
−=
where RFASR, LFR = ASR or LFR rating factor; C = nominal member capacity; A1 =
dead load factor; D = nominal dead load effect; A2 = live load factor; L = nominal
live load effect caused by rating vehicle; and I = live load impact factor. The rating
factor is the ratio of the available to required live-load capacity and thus, is a direct
measure of the safe live-load capacity of a bridge. If less than one, then the live load
effects caused by the rating vehicle exceed the capacity minus the dead load effects.
Separate rating factors are computed for the different bridge components (i.e., slab,
superstructure, and substructure) and different load effects (i.e., moment, shear, axial
force, etc.). The individual member with the smallest rating factor is the weak link
and thus, controls the load rating of the bridge as a whole.
Bridge members or connections under combined loading such as axial-
bending or shear-bending should be evaluated taking into account the interaction of
load effects. In such cases, the load rating should be based on the appropriate
interaction equation rather than the basic equation given above which applies only to
members subjected to an individual load effect as mentioned earlier (Minervino et al.,
2004).
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Load ratings are computed at an inventory and operating level compliant with
the ASR and LFR methods. The inventory rating represents the magnitude of load
that a bridge can safely carry for an indefinite period of time whereas the operating
rating denotes the absolute maximum load that may be permitted on a bridge but with
appropriate restrictions (AASHTO, 1994; AASHTO, 2003). In the ASR method, the
dead load and live load factors (i.e., A1 and A2, respectively) are taken as unity while
the nominal capacity is determined based on an allowable stress which depends on
the rating level. For steel bridge members in tension or flexure, for example, the
allowable stresses are 55% of the yield stress for inventory and 75% for operating. In
the LFR method, A1 is taken as 1.3 regardless of the rating level whereas A2 is taken
as 1.3 and 2.17 for inventory and operating, respectively. The nominal capacity is
independent of the rating level and is computed according to the AASHTO Standard
Specifications.
The inventory and operating ratings represent the multiple of the load effects
caused by the rating vehicle that a highway bridge can safely carry. For example, an
inventory rating of 1.0 for an HS-20 truck indicates that the bridge safely carry
unlimited passes of a vehicular load that causes load effects equal to those caused by
the HS-20 truck. An operating rating of 1.67, on the other hand, indicates that the
bridge can safely carry a load that causes live load effects equal to 1.67 times that of
an HS-20 truck but on a periodic not continual basis. Safe load capacities are
typically given in terms of the design loading which is the customary reporting format
specified for the National Bridge Inspection Standards (NBIS) and Bridge
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Management Systems (BMS). Hence, inventory and operating ratings of 1.0 and 1.67,
respectively, for an HS-20 truck would be reported as HS-20 and HS-32.
Alternatively, load ratings may be reported in terms of the weight (in tons) of the
rating vehicle. For the example given above, this results in an inventory rating of 36
tons and an operating rating of 60 tons.
3.2.3. Load and Resistance Factor Rating (LRFR)
The same limit states design philosophy used to develop the LRFD Bridge
Design Specifications was extended to the evaluation of existing bridges in the new
LRFR manual (AASHTO, 2003). The load rating equation used in LRFR for
members under discrete loading is given as:
IM) LL(1
DW DC R FR
L
DWDCnscLRFR +γ
γ−γ−φφφ=
where RFLRFR = LRFR rating factor; φc, φs = condition and system factor,
respectively; γDC, γDW, γL = load factors for structural components and attachments
(DC), wearing surfaces and utilities (DW), and live load (L), respectively; DC, DW,
LL = nominal dead load effect due to structural components and attachments, dead
load effect due to wearing surfaces and utilities, and live load effects, respectively;
and IM = dynamic load allowance. Dead loads for the LRFR method are separated
into two categories: component / attachment loads (DC) and wearing surface / utility
loads (DW). The two loads have different dead load factors (i.e., γDC = 1.25 and γDW
= 1.50) in recognition of the lower degree of variability of the component dead loads
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compared to that of the wearing surface. The condition factor, φc, accounts for the
larger uncertainty in the resistance of deteriorated members (and the possibility for
future deterioration) which ranges from 0.85 (poor condition) to 1.0 (satisfactory
condition). It is important to note that this factor does not account for any observed
changes in the physical dimensions of the member (i.e., section loss). The system
factor, φs, relates to the degree of redundancy inherent in a bridge system; that is, the
capability of the bridge to redistribute load in the event of damage or failure to one or
multiple members. Like the condition factor, the system factor ranges from 0.85 to
1.0 with the higher value corresponding to redundant structures (e.g., multiple girder
bridges). Thus, the new LRFR approach accounts for the physical state of a bridge in
a more explicit manner than the LFR approach.
The factored live load effect of the LRFR rating equation represents the
largest discrepancy with LFR. First of all, the LRFR dynamic load allowance (IM) is
a fixed value, whereas the LFR impact factor (I) varies with span length. Also
affecting the live load effects are the distribution factors for moment and shear; the
AASHTO LRFD Bridge Design Specifications introduced new empirical equations
that yield more accurate estimates of live-load distribution. The LRFD-based
distribution factors consider span length, girder spacing, girder stiffness, and slab
thickness, whereas the LFD-based distribution factors consider only the girder
spacing. For interior girders, distribution factors for moment and shear are considered
separately in LRFD. This is not the case for LFD where the same distribution factor is
used for both moment and shear. For exterior beams, the LRFD distribution factors
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are determined by either modifying the distribution factors for the interior beam or by
employing the lever rule; only the lever rule is applied in computing exterior beam
distribution factors in LFD. The LRFD-based distribution factors also account for
support skew and rigid intermediate diaphragms.
A major difference in the live load effect lies in the type of vehicle used in the
analysis; LRFR employs the HL-93 design load while LFR employs the HS-20. The
HL-93 consists of an HS-20 design truck (or design tandem) combined with the
design lane load of 0.64 klf. Conversely, the LFR method considers the HS-20 truck
and lane load separately which yields smaller live load forces (undistributed and
unfactored) compared to LRFR. One last parameter that influences the live load
effects is the live load factor. In LRFR, the live load factor (γL) is 1.75 for the
Strength I check and 1.35 for the Strength II check. Note that Strength I and II in
LRFR is the same as inventory and operating rating in LFR. Under legal loads, the
live load factor ranges from 1.4 to 1.8 depending on the Average Daily Truck Traffic
(ADTT). Separate live load factors are also specified for permit loads depending on
the permit type, frequency of crossing, loading condition, ADTT, and permit weight.
In LFR, the live load factors for legal and permit loads are usually taken to be equal
to those at inventory and operating levels, respectively, under design loads; that is, A2
= 2.17 for legal loads and 1.3 for permit loads. Further discussion of capacity rating
under design, legal, and permit loads are given in the next section.
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3.3. Design, Legal, and Permit Load Rating
The live loads applied in the rating process can be broken down into three
types: design, legal, and permit loads. The new LRFR manual (AASHTO, 2003)
explicitly defines a tiered approach to load rating which starts with a design load
rating, followed by a legal load rating, and ending with a permit load rating. The
design load rating provides a measure of the safe load capacity of existing bridges
according to new bridge design standards. The design load ratings of highway bridges
are customarily reported at both the inventory and operating levels. As mentioned in
Chapter 1, the original design of the Omega Bridge was done for H-20 truck loading
by ASD and based on the 1944 AASHO Specifications. The structure was later
rehabilitated in 1992 to conform to HS-20 vehicular loading based on the AASHTO
Standard Specifications. The live load model adopted by the AASHTO ASD is
designated as H-20 truck while the LFD, which is specified in the AASHTO Standard
Specifications, uses the HS-20 truck or the 0.640 klf lane load, whichever produces
the maximum effect. Figure 3.1 compares the difference between live loads of ASD
and LFD used for design purposes. In the new LRFR manual (AASHTO, 2003), the
live load model adopted by the AASHTO LRFD Bridge Design Specifications is
designated as HL-93 which is combination of the HS-20 truck and the 0.640 klf lane
load.
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8 k
36 k 36 k
14ft 14 to 30 ft 6ft
AXLE NO.1 2 3
(b) HS-20 or LANE LOAD
LANE LOAD = 0.64 klf0.64 klf
14ft
AXLE NO.1
8 k
2
6ft
(a) H-20: Unit weight = 40 kips
20 k 20 k
32 k32 k
32 k
HS-20: Unit weight = 72 kip
Figure 3.1 AASHTO design load for ASD and LFD.
16 k 17 k17 k
15 ft 4 ft
10 k
12 k12 k
15 ft 4 ft
12 k 14 k
16 ft
16 k
4 ft
14 k
15 ft
15.5 k 15.5 k 15.5 k 15.5 k
4 ft 22 ft 4 ft11 ft
AXLE NO.1
AXLE NO.1
AXLE NO.1
2 3
2 3 4 5
2 3 4 5 6
(a) TYPE 3: Unit Weight = 50 kips
(b) TYPE 3S2: Unit Weight = 72 kips
(c) TYPE 3-3: Unit Weight = 80 kips
6ft
25 k 25 k
6ft
36 k 36 k
6ft
40 k 40 k
Figure 3.2 AASHTO legal loads.
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In the second stage of the rating process, load ratings are computed under
legal loads (hereafter designated as legal load rating). Legal load ratings provide
information necessary to the posting of loads or the rehabilitation of the structure.
Therefore, under legal loads, the bridge is rated at the inventory level only. The
AASHTO legal loads shown in Figure 3.2 (i.e., Type 3, Type 3S2, and Type 3-3) are
suitable for posting purposes. In addition, legal load ratings should be evaluated for
legal vehicles particular to the state, particularly when the state legal vehicle is
significantly different than the axle loads and spacings of the AASHTO legal loads.
Figure 3.2 shows the axle weights and longitudinal spacing for the AASHTO legal
trucks; like the design trucks, the legal trucks have the transverse distance between
wheel lines that is equal to 6 ft.
The major difference between LFR and LRFR is the use of these design load
ratings as a screening check for legal loads. In LRFR, an inventory rating greater than
one (computed based on HL-93 design loading) indicates that the bridge has adequate
capacity to carry all AASHTO legal loads. Conversely, in LFR, bridges that have an
inventory rating greater than one must still be checked for legal loads because the
design load is lighter than that of the LRFR (i.e., HS-20 < HL-93).
If the rating factor of the second stage is larger than one, the third stage of the
rating process may be considered. In the third stage, load ratings are computed for
permit loads or loads that exceed legal loads (hereafter referred to as permit load
rating). This type of rating is used for the issuance of overload permits. Permit load
rating is an important activity since granting an overload permit avoids any
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unnecessary detours or restrictive traffic loadings while maintaining a safe and
serviceable structure. Since permit loads are not regular loads like the legal loads,
they are rated only at the operating level. Again, inventory rating factor greater than
one indicates that legal loads can continuously cross over the bridge while operating
rating factor greater than one shows that permit loads can occasionally cross over the
bridge. Highway bridges should only be evaluated under permit loads if shown that
they can safely carry the legal loads.
In the case of the Omega Bridge presented in this thesis study, the LFR
method was used to come up with inventory and operating ratings for five different
vehicular loads including: the AASHTO HS-20 design truck; AASHTO legal load
Type 3, Type 3S2 and Type 3-3; and “Emergency-One Titan” fire truck. The
“Emergency-One Titan” fire truck was included at the request of the Los Alamos
National Laboratory to assess the structural capacity of the Omega Bridge under
emergency response vehicles. Figure 3.3 shows the axle weight and longitudinal
spacing as well as the transverse distance between wheel lines. Note that, unlike the
AASHTO design and legal loads, the wheel line spacing is 7.2 ft.
38.87 k18.98 k
5 ft 12.7 ft 5 ft
18.98 k 19.89 k 19.89 k 38.87 k
7.2 ft
AXLE NO.1 2 3 4
Unit weight = 77.74 kips
Figure 3.3 “Emergency-One Titan” fire truck.
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Chapter 4
LOAD RATING FACTOR OF FLOOR SYSTEM
This chapter covers the load rating of the three floor system components; the
stringers (section 4.1), floor beams (section 4.2), and spandrel beams (section 4.3). A
description of the rating model for each element is first provided followed by a
discussion of the capacity evaluation using the Load Factor Rating (LRF) method.
Rating factors were determined for the bridge members based on flexure using the
equation
L2
D1R
MA
MA M FR
−=
where RF = rating factor (RFi for inventory or RFo for operating); MR = flexural
capacity of the member; A1 = dead load factor = 1.3; MD = bending moment due to
dead load; A2 = live load factor = 2.17 (for inventory rating) or 1.3 (for operating
rating); and ML = bending moment due to live load (equal to the bending moment
caused by a wheel line of the live load vehicle times the distribution, multiple
presence, and impact factors). As mentioned in Chapter 3, the live loads used to rate
the bridge components include the traditional AASHTO HS-20 truck loading;
standard AASHTO rating vehicles (Type 3, Type 3-3, and Type 3S2); and
Emergency-One Titan Fire Truck. The rating analysis complies with the 2nd Edition
of the AASHTO Manual for Condition Evaluation of Bridges (2000) including the
2003 Interim Revisions and the 17th Edition of the AASHTO Standard Specifications
for Highway Bridges (2002). A full listing of the rating calculations for AASHTO
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HS-20 truck loading is provided in the appendix for the stringers (Appendix A1),
floor beams (Appendix A2), and the spandrel beams (Appendix A3).
4.1. Stringers
4.1.1. Description of Rating Model for Stringers
For purpose of analysis, the stringers were modeled as continuous beams
supported at the floor beam locations. There are a total of 27 spans over the length of
the stringers; the span length is 31 ft in the six approach spans at each end of the
bridge and 29.5 ft in the 15 interior spans over the arch. Figure 4.1 shows the rating
models for the approach spans of the interior and exterior stringers starting from the
south end of the bridge; the floor beam support locations are labeled N1, N2, N3, etc.
Section #1: Positive moment, composite section (top and bottom cover plates)
Section #2: Negative moment, non-composite section (no cover plates)
Section #3: Positive moment, non-composite section (no cover plates)
Section #1: Positive moment, composite section (no cover plates)
Section #2: Negative moment, non-composite section (no cover plates)
Section #3: Positive moment, non-composite section (no cover plates)
Exterior Stringer
Interior Stringer
N1 N2 N3 N4 N5 N6 N7
N3 N4 N5 N6 N7N2N1
(Abutment) (Pier Col #1) (Pier Col #2) (Skewback Col #1)
(Skewback Col #1)(Pier Col #2)(Pier Col #1)(Abutment)
Figure 4.1 Rating models of stringers (with critical sections).
As shown in the figure, the floor beams are assumed to provide unyielding
support to the stringers in the vertical direction. This model assumes that the vertical
stiffness provided by the floor beams (and spandrel beams) to the stringer is
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completely rigid (i.e., the stringers do not deform vertically at the floor beam
locations). In actuality, the support stiffness has a finite spring constant which
depends not only on the position of the stringer above the floor beam but also on the
position of the floor beam relative to the length of the spandrel beam. In the end, a
spring constant of infinity was chosen to simplify the analytical model which results
in smaller positive moments within the spans and larger negative moments at the
floor beam locations. Furthermore, the torsional stiffness of the stringer to floor
beam connections and that of the floor beams themselves was ignored. The section
properties were also assumed constant (i.e., prismatic) along the entire length of the
stringer based on non-composite action. This assumption ignores the change in
stiffness between composite and non-composite regions as well as between areas with
and without cover plates. Incidentally, structural analysis showed no significant
difference in the results between the non-prismatic and prismatic models.
For the interior stringer, the total dead load was wDi = 556 plf which includes
the self-weight of the stringer (66.1 plf); the tributary weight of the slab (456 plf); and
the tributary weight of the integral wearing surface (33.8 plf). The total dead load that
acts on the east exterior stringer was wDe = 609 plf, which includes the self-weight of
the stringer (65.1 plf); the tributary weight of the slab (465 plf); the tributary weight
of the integral wearing surface (33.8 plf); and the weights of the barrier and utilities
(45.9 plf). According to Article 3.23.2.2 of the AASHTO Standard Specification
(2002), the moment distribution factor for an interior stringer (DFint) in bridges
having two or more traffic lanes is
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1.23 5
6.75 5S FD
int===
where S equals the stringer spacing in feet. For an exterior stringer, AASHTO Article
3.23.2.3 states that the moment distribution factor “shall be determined by applying to
the stringer or beam the reaction of the wheel load obtained by assuming the flooring
to act as a simple span between stringers or beams” (AASHTO, 2002). This approach,
also referred to as the lever rule, resulted in a moment distribution factor equal to
unity (i.e., DFext = 1.00) assuming the deck spanned from the top edge of the spandrel
beam. When the deck was assumed to span from the centerline of the spandrel beam,
a value of DFext = 1.19 was computed. AASHTO (2002) also specifies a lower bound
for DFext of
1.19 5(6.75)2.0 4
6.75 5S2.0 4
S FDext
=+
=+
=
where S (the distance between the exterior and adjacent interior stringer) is between
six and 14 feet; thus, the distribution factor for the exterior stringer was taken as DFext
= 1.19. AASHTO Article 3.8.2.1 requires that vehicular live loads be increased a
maximum of 30% to account for dynamic, vibratory, and impact effects using the
equation
( ) 0.30 30.0 ,32.0min 30.0 ,125 29.5or 31
50min 30.0 ,512 L
50min I ==⎟⎠⎞
⎜⎝⎛
+=⎟
⎠⎞
⎜⎝⎛
+=
where I is the impact factor and L is the loaded span length in feet (i.e., 31 ft for
approach spans and 29.5 ft for arch spans). The same moment distribution and impact
factors for the AASHTO HS-20 truck loading were also used to evaluate the bridge
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members under the standard AASHTO rating vehicles and the Emergency-One Titan
Fire Truck.
Taking into account the moment envelopes for dead load and vehicular live
load as well as the varying flexural capacity of the stringers, three critical sections
were identified along the length of the interior and exterior stringers (labeled Section
#1, Section #2, and Section #3 in Figure 4.1). As discussed in Chapter 2, the stringers
are composite in some areas and non-composite in others. Furthermore, ASTM A36
steel (Fy = 36 ksi) was used for the exterior stringers while ASTM A7 steel (Fy = 33
ksi) was used for the interior stringers. Cover plates were also provided in some
regions of the interior stringers. The first critical section (Section #1) is located in the
positive moment region of the end span as shown in Figure 4.1 where the interior and
exterior stringers are both composite. However, cover plates were provided on the top
and bottom flanges of the interior stringers only. The second (Section #2) and third
(Section #3) critical sections are located in the negative moment region at the floor
beam support location N2 and the positive moment region of the third span,
respectively, where both the interior and exterior stringers have no cover plates and
are non-composite. The load rating analysis at these critical sections of the stringers is
discussed next.
4.1.2. Load Factor Rating (LFR) Analysis
The magnitudes of the dead load moment (MD), live load moment (ML),
flexural capacity (MR) and rating factors for the interior and exterior girders are
reported in Tables 4.1 and 4.2, respectively. The moment values given for dead load
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and live load are unfactored; furthermore, the distribution and impact factors are
included in the live load moments as mentioned previously. Note that the dead load
moments for the exterior stringer are larger than those for the interior stringer mainly
due to the added weight of the barrier and utilities. The live load moments, on the
other hand, are slightly larger for the interior stringer since it had a larger distribution
factor than the exterior stringer (i.e., DFint = 1.23 > DFext = 1.19). The flexural
capacity of the stringers was governed by the plastic moment capacities of the
composite section (AASHTO Article 10.50) at Section #1 and the non-composite
section (AASHTO Article 10.48) at Section #2 and Section #3.
The plastic capacity at Section #1 of the interior stringer is about 20% larger
than that of the exterior stringer largely because cover plates were provided for the
interior stringer. Conversely, the plastic capacity of the non-composite section (i.e.,
MR = ZFy where Z is the plastic section modulus of the W-section) at Sections #2 and
#3 of the interior stringer is smaller than that of the exterior stringer due to the
difference in the grade of steel. Both stringers satisfied the compact section
requirements at Sections #1 through #3 needed to develop the plastic moment
capacity.
Based on the rating factors given in Tables 4.1 and 4.2, the negative moment
region (i.e., Section #2) of the stringers controlled the load rating. The rating values at
this location are repeated in Table 4.3. As shown in the table, all the rating factors
(with the exception of the interior stringer inventory rating for the fire truck)
exceeded one.
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Table 4.1 Moment values and rating factors for interior stringer.
Critical Section Section #1 Section #2 Section #3 Dead Load Moment (MD), kip-ft 41.6 56.3 23.4
HS-20 190 139 147 TYPE 3 153 107 121
TYPE 3S2 147 146 94.6 TYPE 3-3 122 121 84.2
Live Load Moment (ML), kip-ft
FIRE 184 156 139 Flexural Capacity (MR), kip-ft 1070 401 401
HS-20 2.46 1.09 1.16 TYPE 3 3.05 1.41 1.42
TYPE 3S2 3.19 1.03 1.80 TYPE 3-3 3.84 1.25 2.03
Inventory Rating Factor (RFi)
FIRE 2.55 0.97 1.23 HS-20 4.11 1.81 1.94
TYPE 3 5.10 2.36 2.36 TYPE 3S2 5.33 1.72 3.01 TYPE 3-3 6.41 2.09 3.38
Operating Rating Factor (RFo)
FIRE 4.25 1.62 2.05
Table 4.2 Moment values and rating factors for exterior stringers.
Critical Section Section #1 Section #2 Section #3 Dead Load Moment (MD), kip-ft 45.6 61.6 25.6
HS-20 184 135 143 TYPE 3 149 104 117
TYPE 3S2 142 142 91.7 TYPE 3-3 118 119 81.6
Live Load Moment (ML), kip-ft
FIRE 178 151 135 Flexural Capacity (MR), kip-ft 888 433 433
HS-20 2.07 1.21 1.29 TYPE 3 2.57 1.57 1.58
TYPE 3S2 2.69 1.15 2.01 TYPE 3-3 3.23 1.39 2.26
Inventory Rating Factor (RFi)
FIRE 2.14 1.08 1.37 HS-20 3.46 2.02 2.15
TYPE 3 4.29 2.62 2.63 TYPE 3S2 4.49 1.92 3.35 TYPE 3-3 5.40 2.33 3.77
Operating Rating Factor (RFo)
FIRE 3.58 1.80 2.28
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Table 4.3 Rating factors at negative moment region (Section #2) of stringers.
Stringer Interior Exterior Rating Factor RFi RFo RFi RFo
HS-20 1.09 1.81 1.21 2.02 TYPE 3 1.41 2.36 1.57 2.62
TYPE 3S2 1.03 1.72 1.15 1.92 TYPE 3-3 1.25 2.09 1.39 2.33
Live Load
FIRE 0.97 1.62 1.08 1.80
Recall that the rating model assumed that the stringer was a continuous beam
rigidly supported by the floor beams (i.e., no vertical deformation occurs at the
support locations). In actuality, however, the floor beams deform vertically as a
function of the flexural spring constant which will reduce the magnitude of the
support reactions. This, in turn, will decrease the negative moment in the stringers at
the floor beam locations and increase the positive moment between floor beams.
Hence, the rigid support assumption results in conservative rating factors for the
negative moment region (i.e., Section #2) which controlled the capacity of the
stringers. However, the same assumption has the opposite effect on the rating factors
(i.e., overestimates) in the positive moment regions (i.e., Sections #1 and #3). This is
not so much a concern at Section #1 (composite section) which had much larger
rating factors compared to Section #3 (non-composite section).
The stiffness effect of the floor beam / spandrel beam system on the stringer
moments could be better evaluated by three-dimensional finite element analysis and
experimental field testing of the Omega Bridge. It was also assumed that the W-
section in the non-composite regions completely resisted the bending moment with no
contribution from the concrete deck. Although there is no definite connection (i.e.,
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shear studs) in these areas, research has shown that some level of interaction
somewhere between a true non-composite and composite section may exist due to
friction and mechanical interlock which will aid the negative moment section
(Section #2) and positive moment section (Section #3) to carry more load than
estimated based on AASHTO provisions. The participation of the deck in regions
without shear connectors, however, must be evaluated by field testing.
4.2. Floor beams
4.2.1. Description of rating model
As shown in Figures 4.2 and 4.3, the floor beams were modeled as simple-
supported spans with an overhang on either side (representing the outrigger beam).
Figure 4.2 corresponds to the floor beams located in the approach spans and Figure
4.3 corresponds to those located in the arch spans (labeled FB#2 and FB#6,
respectively, in Figure 2.2).
The floor beams were assumed to be rigidly supported by the spandrel beams
and dead load was applied as six concentrated loads coincident with the stringer
locations. The two forces at the ends of the overhang include the weight of the
exterior stringers, the deck slab (based on the exterior tributary flange width), the
barriers and utilities. The four interior forces include the weight of the interior
stringers and the deck slab (based on the interior tributary flange width). All forces
were calculated by summing up the total weight over the distance between floor
beams (i.e., 31 ft for the approach spans or floor beam FB#2 and 29.5 ft for the arch
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spans or floor beam FB#6). The distribution of dead load on floor beam FB#6 is
different from that of FB#2 for two reasons. First, the spacing between the floor
beams in the arch spans is 1.5 ft shorter than the floor beam spacing in the approach
spans which explains the larger dead loads at the interior stringer locations of FB#2.
Second, there is a fencing load that acts only on the arch span floor beams which
explains the slightly larger dead loads at the exterior stringer locations of FB#6. The
dead load moment diagrams are shown in Figures 4.2 and 4.3; the analysis also
included the self-weight of the floor beams (equal to a uniform load of about 0.180
k/ft).
-22.1 k-17.2 k -19.5 k-17.2 k -17.2 k -17.2 k 136 k-ft
153 k-ft
255 k-ft252 k-ft
134 k-ft124 k-ft
Figure 4.2 Distribution of dead load on floor beam FB#2 of approach span and moment diagram.
109 k-ft
155 k-ft-22.3 k-16.4 k
235 k-ft232 k-ft
119 k-ft
-16.4 k-16.4 k -16.4 k -19.8k138 k-ft
Figure 4.3 Distribution of dead load on floor beam FB#6 of arch span and moment diagram.
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Figure 4.4 illustrates the analytical model used to determine the live load
forces from an HS-20 truck on floor beam FB#2. The top sketch of the figure shows
the cross-section of the floor system loaded with three HS-20 trucks (centered about
the bridge centerline); the loading of three lanes controlled the capacity rating of the
floor beams. The distance between wheel lines for each HS-20 truck is 6 feet and the
distance between adjacent trucks is 4 feet as specified by AASHTO Article 3.7. In
addition, AASHTO Article 3.12 specifies a reduction in live load of 10% for three
loaded traffic lanes to account for the improbability that the maximum loading in the
three lanes occurs simultaneously. Note that there is no reduction for one or two
loaded traffic lanes and a 25% reduction is permitted for four loaded traffic lanes due
to the higher improbability of having each lane under maximum truck loading.
6'-9"7'-4.5"
Floor beam42.6 k
7'-4.5"6'-9"6'-9"
42.8 k42.8 k 42.6 k
Spandrel Beam
Interior Stringer
0.872 x (36 kips) per force.
4'-0"
6'-9"35'-0"
3'-6" 6'-9" 7'-4.5"
Slab
4'-9" 6'-0"
7'-4.5"6'-9" 6'-9" 3'-6"6'-9"
Exterior Stringer
4'-9"4'-0"6'-0" 6'-0"
Figure 4.4 Distribution of HS20 live load on floor beam FB#2.
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The critical floor beam FB#2 in the approach spans was located one bay from
the abutment (i.e., at support location N2 in Figure 4.1). The live load forces acting
on this critical floor beam were determined by first calculating the fraction of the HS-
20 truck that acts on the deck slab directly over the floor beam. This percentage or
longitudinal distribution factor equaled 0.872 for an HS-20 truck (see Figure 4.4)
which is the ratio of the maximum reaction at floor beam support N2 (see Figure 4.1)
determined from the stringer analysis to the total weight of a single line of HS-20
truck wheels or 36 kips.
Next, the distributed wheel line loads of (0.872 x 36 kips) were applied to the
deck slab which was modeled as a continuous beam rigidly supported by the stringers
and the spandrel beams as shown in Figure 4.4. Deck section properties were
computed assuming a rectangular section with a height equal to the slab thickness and
width equal to the floor beam spacing. With this model, the reactions at the four
interior stringers were computed (equal to 42.6 kips and 42.8 kips) which represented
the HS-20 live load forces applied to floor beam FB#2. The sum of the interior
stringer reactions is smaller than the total load applied to the floor beam (i.e., 2 x 42.6
kips + 2 x 42.8 kips = 170.8 kips < 6 x 0.872 x 36 kips = 188.4 kips) since load is also
distributed transversely to the spandrel beams. It is important to note that this
modeling approach deviates from AASHTO Article 3.23.3 which ignores the
transverse distribution of wheel loads and thus, results in larger bending moments.
However, the AASHTO-based model assumes that live load is entirely distributed to
the spandrel beams by the floor beams rather than through the deck slab and the floor
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beams, an unrealistic and conservative assumption. Furthermore, the torsional
stiffness of the spandrel beam was ignored so that under live-load forces (transferred
from the interior stringers), the floor beam model is a simple-supported beam with
overhangs and rigidly supported by the spandrel beams. Consideration of the torsional
stiffness would decrease the bending moment carried by the floor beam, which could
be evaluated by three-dimensional finite element analysis and field testing.
A similar approach was taken to determine the live load forces on the floor
beams under the AASHTO standard rating vehicles and the fire truck. In Figures 4.5
and 4.6, the live load forces and corresponding moment diagrams for floor beams
FB#2 and FB#6, respectively, under the different vehicular loads are given. As shown
in the figures, floor beam FB#2 had larger live load forces than floor beam FB#6
mainly due to the larger longitudinal distribution factor.
Similar to the stringer analysis, the same dynamic impact factor was applied
to each truck load using the span length of the floor beam from center-to-center of the
spandrel beams (L = 35 feet), which amounted to
( ) 0.30 30.0 ,31.0min 30.0 ,125 35
50min 30.0 ,512 L
50min I ==⎟⎠⎞
⎜⎝⎛
+=⎟
⎠⎞
⎜⎝⎛
+=
as specified in AASHTO Article 3.8.2.1. The use of three lanes of trucks, which has a
multiple presence factor of 0.9, also controlled for the various trucks. As mentioned
previously, the first floor beam from the abutment was critical in the approach spans
while the eight floor beams located on the center of the bridge were equally as critical
in the arch spans (see Figure 2.2). For each floor beam, the location under the third
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-42.8 k
919 k-ft630 k-ft
-42.6 k
919 k-ft
-42.8 k -42.6 k
630 k-ft
659 k-ft451 k-ft
659 k-ft451 k-ft
-30.7 k-30.5 k -30.7 k -30.5 k
-33.2 k
712 k-ft
-33 k
488 k-ft
-33 k-33.2 k
712 k-ft488 k-ft
664 k-ft
-30.9 k
455 k-ft
-30.8 k
664 k-ft455 k-ft
-30.9 k -30.8 k
889 k-ft
-41.3 k
610 k-ft
-41.4 k
889 k-ft610 k-ft
-41.3 k -41.4 k
HS-20
TYPE 3
TYPE 3S2
TYPE 3-3
FIRE
Figure 4.5 Distribution of live loads on floor beam FB#2 and the corresponding moment diagrams.
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846 k-ft580 k-ft
846 k-ft580 k-ft
634 k-ft634 k-ft
-39.3 k
578 k-ft
-39.4 k
396 k-ft
-39.3 k -39.4 k
578 k-ft396 k-ft
-26.9 k
640 k-ft
-29.8 k
439 k-ft
-26.8 k
-29.7 k
-26.9 k
640 k-ft
-26.8 k
439 k-ft
-29.8 k -29.7 k
-29.5 k
878 k-ft
434 k-ft
-29.4 k
609 k-ft
-29.5 k
434 k-ft
-29.4 k
878 k-ft609 k-ft
-40.9 k-40.7 k -40.9 k -40.7 k
TYPE 3S2
FIRE
TYPE 3-3
TYPE 3
HS-20
Figure 4.6 Distribution of live loads on floor beam FB#6 and the corresponding moment diagrams.
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stringer from the east spandrel beam controlled the load rating; specific details of the
load rating analysis are given next.
4.2.2. Load Factor Rating (LFR) Analysis
The dead load moment, live load moment, flexural capacity, and rating factors
for floor beams FB#2 and FB#6 are reported in Table 4.4. The dead and live load
moments are unfactored; in addition, the live load moment has not been reduced to
account for multiple lane loading. As shown in Appendix A2, the compression flange
of the floor beams failed the size requirements for a compact and non-compact
section specified in the 17th Edition of the AASHTO Standard Specifications (2002)
which classified the flange as a slender element. For a section having this type of
structural element, the flexural capacity is less than the yield moment since the stress
is elastic when buckling occurs; the computation of moment strength for sections with
slender elements requires an elastic buckling analysis. When the compression flange
was evaluated based on the 16th Edition of the AASHTO Standard Specification
(1996), however, floor beam FB#2 failed the compact section requirements but
satisfied those for a non-compact section at both the inventory and operating rating
level. Floor beam FB#6, which had a flange thickness 1/16” smaller than floor beam
FB#2, also satisfied the non-compact section requirements but only at the operating
rating level. Based on these findings, the floor beams were considered as non-
compact sections for purposes of computing the flexural capacity; floor beam FB#2
had a slightly larger flexural capacity because of its 1/16” larger flange thickness.
Cases where the compression flange of floor beam FB#6 did not quite satisfy the
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requirements in the 16th AASHTO Standard Specification (1996) for a non-compact
section are designated by an asterisk (*) in Table 4.4.
Table 4.4 Moment values and rating factors for floor beams.
Floor Beam FB#2 FB#6 Dead Load Moment (MD), kip-ft 255 235
HS-20 1080 1030 TYPE 3 771 741
TYPE 3S2 833 749 TYPE 3-3 776 675
Live Load Moment (ML), kip-ft
FIRE 1040 990 Flexural Capacity (MR), kip-ft 2380 2190
HS-20 0.88 0.85*
TYPE 3 1.22 1.17*
TYPE 3S2 1.13 1.16*
TYPE 3-3 1.21 1.29
Inventory Rating Factor (RFi)
FIRE 0.91 0.88*
HS-20 1.46 1.41 TYPE 3 2.04 1.96
TYPE 3S2 1.89 1.94 TYPE 3-3 2.03 2.15
Operating Rating Factor (RFo)
FIRE 1.51 1.47 * Non-compact section requirements for compression flange not satisfied.
Table 4.4 shows comparable rating factors between the two floor beams; floor
beam FB#2 controlled the capacity for two AASHTO rating vehicles (Type 3S2 and
Type 3-3) while floor beam FB#6 controlled for the remaining live loads. The critical
rating values, which were calculated based on the 16th AASHTO Standard
Specification (1996), are repeated in Table 4.5.
It is important to note that the rating factors would have been approximately
10% smaller than those given in Tables 4.4 and 4.5 had no transverse distribution of
live load been assumed as specified in AASHTO Article 3.23.3. As mentioned
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previously, it is more realistic that some of the live load will transfer directly to the
spandrel through the slab and the remainder will transfer through the floor beam; this
approach results in smaller live load moments and thus, larger rating factors. In
addition, the torsional stiffness of the spandrel beam was ignored, which if accounted
for would also possibly decrease the bending moment in the floor beam and further
increase the rating factor; finite element analysis and field testing is recommended to
better evaluate this behavior. Another aspect of the floor beam evaluation of interest
is the new compression flange requirements which are more stringent that those
specified in the original design of the Omega Bridge.
Table 4.5 Critical rating factors for floor beams.
Rating Factor RFi RFo HS-20a 0.85 1.14
TYPE 3a 1.17 1.96 TYPE 3S2b 1.13 1.89 TYPE 3-3b 1.21 2.03
Live Load
FIREa 0.88 1.47
aControlled by floor beam FB#6. bControlled by floor beam FB#2.
4.3. Spandrel beams
4.3.1. Description of Rating Model for Spandrel Beams
The spandrel beam has a total of 21 spans; there are three 62-ft approach
spans at each end of the bridge and fifteen 29.5-ft intermediate spans above the arch.
Two separate models were developed to evaluate the capacity of the spandrel beam.
In the first model (designated BEAM model), the spandrel beam was idealized as a
continuous beam supported at the abutments and the pier, skewback, and arch
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columns as shown in Figure 4.7. This model assumes that the columns provide rigid
support stiffness to the spandrel beam in the vertical direction and no flexural
stiffness.
Section #1: Positive, composite section.
Section #2: Negative, composite section.
Section #3: Positive, non-composite section.
Abutment Pier Col #1 Pier Col #2 Skewback Col Arch Col #1
Section #4: Negative, non-composite section.HS-20 TRUCKSLEGAL LOAD TYPE 3 TRUCKSFIRE TRUCKS
Section #4: Negative, non-composite section.
LEGAL LOAD TYPE 3S2LEGAL LOAD TYPE 3-3
Section #1: Positive, composite section.
Abutment
Section #3: Positive, non-composite section.
Section #2: Negative, composite section.
Pier Col #1 Pier Col #2 Skewback Col Arch Col #1
Figure 4.7 BEAM rating model of spandrel beam and critical sections.
In the second model (designated FRAME model), the entire structure in the
arch rib plane was modeled including the spandrel beam, columns, and arch as shown
in Figure 4.8.
15 spans (29.5ft each)62ft 62ft62ft 62ft 62ft62ft
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Figure 4.8 FRAME rating model of spandrel beam.
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Spandrel-to-column and column-to-arch connections were modeled as rigid
and both the axial and flexural stiffness of the columns and arch were considered. For
the most part, the FRAME model shown in Figure 4.8 resulted in larger negative
moments in the spandrel beam at the column locations and smaller positive moments
between the columns compared to the BEAM model (see Figure 4.7). Furthermore,
axial forces in the spandrel were below 15% of the yield force as required by
AASHTO Article 10.48.1.1(d) for flexural members; otherwise, beam-column
analysis would be necessary.
Similar to the stringer analysis, the section properties of the spandrel beam in
both models were assumed prismatic (based on non-composite action) along the
entire length of the bridge. Again, this assumption ignores the change in stiffness
between composite and non-composite regions as well as between areas with and
without cover plates, but does not significantly influence the bending moments of the
BEAM model. The FRAME model, on the other hand, is affected since the moments
are more dependent on the relative flexural stiffness of the spandrel beam compared
with the columns and arch. However, the use of non-composite instead of composite
section properties for the spandrel will result in larger negative moments at the
column locations, which is the more critical scenario.
The major portion of the dead load that acts on the spandrel beam amounts to
wD1 = 2770 plf which includes the self-weight of the spandrel beam (302 plf) and the
tributary weights of the slab (1910 plf), integral wearing surface (139 plf), floor
beams (162 plf), stringers (198 plf), and wind bracing (55.7 plf). Although the load is
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mostly distributed to the spandrel beam as concentrated forces coming from the floor
beams, the discrete point loads at the floor beam locations were assumed to be
uniformly distributed over the spandrel’s length. This approach simplifies the analysis
but does not significantly affect the dead load moments. As shown in Figure 2.1, the
vehicle traffic lanes are not symmetrical about the bridge centerline but are situated
on the eastern side of the bridge width; there is a sidewalk for pedestrians on the west
side of the traffic lanes. With this layout, the spandrel beam located on the east side of
the bridge width is the most heavily loaded and thus, controls the capacity. In addition
to dead load wD1, the eastern spandrel beam is also subjected to an additional dead
load of wD2 (equal to 307 plf) for barriers and utilities, and wF (equal to 40.4 plf) for
fencing. While the dead loads wD1 and wD2 act over the entire length of the spandrel,
the fencing load wF acts only on the center 150 ft.
Based on Article 3.8.2.1 of the 17th AASHTO Standard Specification (2002),
the live load impact factor amounted to
( ) 0.27 30.0 ,27.0min 30.0 ,125 62
50min 30.0 ,512 L
50min I ==⎟⎠⎞
⎜⎝⎛
+=⎟
⎠⎞
⎜⎝⎛
+=
for the spandrel beam (L = 62 ft); this dynamic amplification factor was applied to all
the trucks. The distribution factors for bending moment were determined using the
lever rule as specified in AASHTO Article 3.23.2.3 for exterior beams and live load
reduction as specified in AASHTO Article 3.12.1 for multiple loaded traffic lanes; the
results for the different trucks are tabulated in Table 4.6. As shown in the table, the
fire truck had unique moment distribution and multiple presence factors since the
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wheel line spacing is 7.2 ft rather than the 6 ft specified for the AASHTO vehicles. In
addition, a maximum of four traffic lanes could be loaded with the AASHTO vehicles
compared to only three lanes for the fire truck because of the wheel line spacing. In
the end, when the moment distribution and multiple presence factors were combined,
the critical case turned out to be three traffic lanes for all the trucks.
Table 4.6 Live load distribution for spandrel beam
AASHTO Vehicles Fire Truck Number of
Loaded Traffic Lanes
Moment Distribution
Factor (DF)
Multiple Presence Factor
(m)
DF x m
Moment Distribution
Factor (DF)
Multiple Presence Factor
(m)
DF x m
1 1.11 1 1.11 1.09 1 1.09 2 1.93 1 1.93 1.86 1 1.86 3 2.46 0.9 2.22 2.31 0.9 2.08 4 2.71 0.75 2.04 N/A 0.75 N/A
Figure 4.7 shows the four critical sections (labeled Section #1 through Section
#4) that were identified by analysis based on the moment envelopes for dead load and
live load for the different trucks. The first (Section #1) and second (Section #2)
critical sections are located in the positive moment region in the first approach span
and the negative moment region at the first pier column, respectively. For these two
sections, the spandrel beam is composite with the deck. The third critical section
(Section #3) is located in the positive moment region of the third approach span
where the spandrel is non-composite. The fourth critical section (Section #4) is
located in a negative moment, non-composite region of the spandrel beam. However,
the critical section is at the skewback column for the HS-20, Type 3, and Fire trucks
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and at the second pier column for the Type 3S2 and Type 3-3 trucks. The load rating
analysis at these critical sections is discussed next.
4.3.2. Load Factor Rating Analysis
Table 4.7 and Table 4.8 compared the rating factors of the critical sections
(shown in Figure 4.7) for the spandrel beam in the BEAM model and FRAME model,
respectively. The dead load effect, the live load effects (with the impact factor,
distribution factor, and multiple presence factor included), flexural capacities and the
resulting rating factors for the spandrel beam were also reported in Table 4.7 and
Table 4.8. The reader is referred to the appendix for more details of the rating
calculations.
In Table 4.7 and Table 4.8, the flexural capacity of the spandrel beam was
governed by the yield moment capacities of the composite section (AASHTO Article
10.50) at Sections #1 and #2 and the non-composite section (AASHTO Article 10.48)
at Sections #3 and #4. Although the spandrel beam met the compact requirements for
a composite section in the positive moment region (i.e., Section #1), the yield
moment controlled the capacity as indicated by AASHTO Article 10.50.1.1 since the
spandrel beam was non-compact (due to the web slenderness) in the negative moment
pier region (i.e., Section #2). For the non-composite sections in the positive and
negative moment regions (i.e., Sections #3 and #4) the spandrel beam is also non-
compact for the web and thus, governed by the yield moment capacity. The web
slenderness actually failed the unstiffened web requirement but satisfied the stiffened
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web requirement for a non-compact section. Since the webs of the spandrel beam
were interconnected by diaphragm plates, the non-compact requirements were
considered to have been met which base the capacity on the yield moment.
In accordance with AASHTO Article 10.50(c), the yield moment capacities of
the positive and negative composite sections (i.e., Sections #1 and #2, respectively)
were computed considering non-composite action (i.e., the steel spandrel beam acting
alone) under dead load and composite action under live load. Thus, for these two
sections, the rating factors are computed as follows
L2
cnc
D1cy
L2
cnc
D1R
MA
S S
MA SF
MA
S S
MA M
FR⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
=
where Snc and Sc are the section moduli at the extreme tension fiber of the steel
spandrel for the non-composite and composite sections, respectively. By replacing Sc
by Snc in the above equation, the rating factors for the positive and negative non-
composite sections (i.e., Sections #3 and #4, respectively) are determined as shown
below
L2
D1ncy
L2
ncnc
D1R
MA
MA SF
MA
SS
MA M
FR−
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
=
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Table 4.7 Moment values and rating factors of spandrel beam of BEAM model.
Critical Section Section #1 Section #2 Section #3 Section #4 Dead Load Moment (MD), kip-ft 930 1223 613 750
HS-20 1867 1154 1382 1252 TYPE 3 1397 822 1045 896
TYPE 3S2 1438 952 1031 894 TYPE 3-3 1269 1026 892 952
Live Load Moment (ML), kip-ft
FIRE 1823 1166 1322 1263 Flexural Capacity (MR), kip-ft 6403 4749 4256 4256
HS-20 1.17 1.19 1.15 (*) 1.21 TYPE 3 1.57 1.67 1.53 (*) 1.69
TYPE 3S2 1.52 1.44 (*) 1.55 1.53 TYPE 3-3 1.73 1.34 (*) 1.79 1.44
Inventory Rating Factor (RFi)
FIRE 1.20 1.18 (*) 1.21 1.20 HS-20 1.96 1.98 1.93 (*) 2.02
TYPE 3 2.62 2.78 2.55 (*) 2.82 TYPE 3S2 2.54 2.40 (*) 2.58 2.56 TYPE 3-3 2.88 2.23 (*) 2.99 2.40
Operating Rating Factor (RFo)
FIRE 2.01 1.96 (*) 2.01 2.00 Table 4.8 Moment values and rating factors of spandrel beam of FRAME model.
Critical Section Section #1 Section #2 Section #3 Section #4 Dead Load Moment (MD), kip-ft 927 1231 560 881
HS-20 1872 1145 1386 1097 TYPE 3 1399 816 1049 785
TYPE 3S2 1439 941 1036 1010 TYPE 3-3 1271 1009 900 930
Live Load Moment (ML), kip-ft
FIRE 1824 1157 1327 1105 Flexural Capacity (MR), kip-ft 6403 4749 4256 4256
HS-20 1.17 (*) 1.19 1.17 1.31 TYPE 3 1.57 1.67 1.55 (*) 1.83
TYPE 3S2 1.52 1.45 (*) 1.57 1.46 TYPE 3-3 1.72 1.35 (*) 1.81 1.58
Inventory Rating Factor (RFi)
FIRE 1.20 1.18 (*) 1.23 1.30 HS-20 1.96 (*) 1.99 1.96 2.18
TYPE 3 2.62 2.79 2.59 (*) 3.05 TYPE 3S2 2.54 2.42 (*) 2.62 2.43 TYPE 3-3 2.88 2.26 (*) 3.02 2.64
Operating Rating Factor (RFo)
FIRE 2.01 1.97 (*) 2.05 2.16
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Based on the rating factors given in Table 4.7 and Table 4.8, Section #2
(where the spandrel beam was composite and subjected to negative moment)
controlled the capacity under TYPE 3S2, TYPE 3-3 and FIRE truck loads while
Section #3 (where the spandrel beam was non-composite and subjected to positive
moment) controlled the capacity under HS-20 and TYPE 3 truck loads. The rating
factors for this section are repeated in Table 4.9 which shows all values larger than 1;
the smallest rating factors are those for the HS-20 and Fire trucks.
Table 4.9 Rating factors at Section #2 and Section #3 of spandrel beam.
Rating Factor RFi RFo HS-20 a 1.15 1.93
TYPE 3 a 1.52 2.55 TYPE 3S2 b 1.44 2.40 TYPE 3-3 b 1.34 2.23
Live Load
FIRE b 1.18 1.96 a Controlled by Section #3. b Controlled by Section #2.
It is recommended that a three-dimensional finite element model be developed
and field testing be performed to further evaluate the spandrel. One particular
improvement that could be made is with the moment distribution factor. Recall that
the AASHTO distribution factor was determined based on the lever rule. In many
cases, this approach overestimates the load carried by exterior beams and thus, results
in low rating factors. A better estimate of the load distribution to the spandrel beam
could help improve the capacity rating for the spandrel beam. In addition, the actual
stiffness of the spandrel-to-column connections as well as that of the column supports
in the approach spans could be better evaluated. In the FRAME model, the base
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support was assumed to be pinned for the pier columns and fixed for the skewback
column. Another parameter that could be evaluated is the spandrel beam stiffness
properties. Both the BEAM and FRAME models assumed prismatic section
properties based on non-composite action. As mentioned previously, the bending
moments in the spandrel depend on its stiffness relative to the columns and arch rib.
Finally, finite element analysis and field testing allows the overall three-dimension
behavior of the bridge to be evaluated opposed to the two-dimensional models used in
this study that are common in design.
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Chapter 5
LOAD RATING OF COLUMNS
5.1. Description of Rating Model
As discussed earlier in Chapter 2, there are four pier columns, 14 arch
columns and two skewback columns in each arch rib plane of the Omega Bridge as
shown in Figure 5.1. All the columns have a riveted connection to the spandrel beam
at their top end; however, the columns have different support conditions at their
bottom end (see Figure 5.1). Pier columns #1 and #4, which are located closest to the
abutments, are supported by rocker bearings (i.e., rollers) while pier columns #3 and
#4 are pin supported at their bases. The two skewback columns, #1 and #2, are both
fixed at their bottom ends. For the 14 arch columns, labeled #1 through #14, the base
connections with the arch rib are riveted.
Skewback
Pier column #
Pinned
Roller
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Arch column #
SOUTH NORTH1 2
Pier column #
43
column #2column #1Skewback
Fixed
Roller
Pinned
Fixed
Figure 5.1 BEAM-COLUMN rating model of pier, skewback, and arch columns
As mentioned above, the column-spandrel beam connections as well as the
connections between the arch columns and the arch rib are all riveted; however, two
separate design details were used to make these connections. Based on a review of
these details, it is anticipated that the connections will provide semi-rigid stiffness
somewhere between a true hinge (i.e., no moment transfer and independent rotation
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of connecting members) and a fully rigid connection (i.e., full moment transfer and
upholding original angle between connecting members). Due to the uncertainty of the
rotational connection stiffness at the column ends, two separate models were
developed to evaluate the capacity of the columns. In the first model (designated
BEAM-COLUMN model), all the riveted column connections with the spandrel beam
and the arch rib as well as the base connections of the skewback columns were
idealized as rigid connections as shown in Figure 5.1. This modeling approach
simulates frame behavior and therefore, produces both bending moment and axial
compression in the vertical members. Under combined axial load and bending, the
vertical members are evaluated as beam-columns according to AASHTO Article
10.54.2. In the second model (designated COLUMN model), all the riveted
connections as well as the base support of the skewback columns were discretized as
hinged connections as shown in Figure 5.2. Also, the boundary conditions at the south
abutment and pier columns #1 and #4 needed to be changed from rollers to pinned
supports in order to provide stability to the model. With the model given in Figure
5.2, the columns are subjected to pure axial loading (with no bending moment) and
thus, are evaluated according to AASHTO Article 10.54.1.
53 4 6 7 98 10 1311 12 141 2
Arch column #
column #2Skewback NORTH
Pier column #
3 4
SOUTH column #1Skewback
Pier column #
1 2
Pinned
Pinned
Pinned Pinned
Pinned
Pinned
Roller
Figure 5.2 COLUMN rating model of pier, skewback, and arch columns
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In addition to the self-weight of the columns, the dead load applied to both the
BEAM-COLUMN and COLUMN rating models included the weights of the floor
system; wind bracing; barriers and utilities; and fencing. A detailed discussion of the
dead loads can be found in Section 4.3 which covers the rating of the spandrel beam.
Like the spandrel beam, the columns in the eastern arch rib plane control the capacity
due to the position of the traffic lanes. Live-load distribution factors for the columns
were also determined in the same manner as that of the spandrel beam; that is, using
the lever rule for exterior beams (as specified in AASHTO Article 3.23.2.3) and live
load reduction for multiple loaded traffic lanes (as specified in AASHTO Article
3.12.1). Hence, the live-load distribution factors for the columns in both rating
models are the same as those tabulated in Table 4.6 for the eastern spandrel beam (see
Section 4.3).
5.2. Load Factor Rating Analysis
5.2.1. BEAM-COLUMN Model: Combined Axial Load and Bending
Since a two-dimensional model was used to evaluate the columns of the
Omega Bridge, there is no bending moment generated in the out-of-plane direction.
As a result, the columns were assumed to act as beam-column members in the plane
of the arch rib (i.e., in-plane direction) and as pure column members in the out-of-
plane direction. In reality, the columns are subjected to flexure perpendicular to the
arch rib plane; however, the out-of-plane bending moments applied to the columns
are generally smaller than those in the in-plane direction. Thus, the out-of-plane
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71
direction will not control the column capacity based on beam-column behavior.
Furthermore, a three-dimensional analysis is required to properly evaluate the
columns as beam-column members in the out-of-plane direction which was outside
the scope of this study. In consequence, the evaluation of the columns under axial
loading and bending moment is based on in-plane behavior. In the out-of-plane
direction, the columns are treated as pure compression members (covered in Section
5.2.2).
Recall from AASHTO Article 3.8.2.1, the impact factor for a bridge member
is a function of the loaded span length. For pier columns #2 and #3 and skewback
columns #1 and #2, the loaded length equaled the span length of the spandrel beam in
the approach spans as given below in the impact factor computation
( ) 0.27 30.0 ,27.0min 30.0 ,125 62
50min 30.0 ,512 L
50min I ==⎟⎠⎞
⎜⎝⎛
+=⎟
⎠⎞
⎜⎝⎛
+=
Pier columns #1 and #4, closest to the abutments, are not subject to flexure since they
are leaner columns and thus, were not evaluated as beam-column members. The
capacity evaluation of these two columns was done based on pure column behavior
which is discussed in Section 5.2.2. For the arch columns, numbered #1 through #14,
the loaded span length was taken as two times the arch column spacing (i.e., 59 ft).
This length was determined by influence line analysis of both axial compression and
bending moment in the arch columns. In both analysis cases, the minimum distance
between inflection points (i.e., locations of zero axial compression or bending
moment on the influence line) was about two times the arch column spacing. Thus,
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using a loaded span length of 59 ft, the dynamic impact factor of the arch columns
amounted to
( ) 0.27 30.0 ,27.0min 30.0 ,125 29.5 x 2
50min 30.0 ,512 L
50min I ==⎟⎠⎞
⎜⎝⎛
+=⎟
⎠⎞
⎜⎝⎛
+=
With no bracing between the columns in the arch rib plane (see Figure 5.1), the
portion of the Omega Bridge above the arch line (i.e., columns and spandrel beam)
was considered an unbraced or sway frame for analysis purposes. The arch rib is
laterally supported on both ends with pin connections and thus, is not subject to
sidesway; hence, the arch rib provided vertical and lateral support at the bottom of the
arch columns. In an unbraced frame, lateral resistance is generally provided by rigid
beam-column connections. For the Omega Bridge, lateral stiffness is provided by the
connections between the columns and the spandrel beam as well as the fixed base
connections of the skewback columns and the connections between the arch columns
and arch rib.
As specified in AASHTO Article 10.54.2.1, the capacity of bridge
components under combined axial loading and bending shall satisfy interaction
equations (10-155) and (10-156) given below:
1
FAP1
CM
M F0.85A
P
es
ucrs
≤
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−+
AASHTO Equation (10-155) and
1M
M F0.85A
P
pys
≤+ AASHTO Equation (10-156)
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where P = maximum axial compression (due to dead load and live load plus impact);
As = cross-sectional area of column; Fcr = critical buckling stress; M = maximum
bending moment (due to dead load and live load plus impact); Mu = maximum
flexural strength (amounted to the yield moment or FyS for all the columns); C =
equivalent moment factor; Fe = Euler Buckling stress; Fy = yield stress; and Mp = full
plastic moment of the section or FyZ. Note that Mu, Fe, and Mp are computed in the
plane of bending (i.e., arch rib plane).
In beam-column analysis for unbraced frames, the maximum moment applied
to a bridge member is generally expressed as
lt2nt1MBMB M +=
where Mnt is the first order moment assuming no lateral translation of the member
ends (i.e., non-sway case) and Mlt is the first order moment due to lateral end
translation (i.e., sway case). The moment amplification factors, B1 and B2, account for
second order effects due to the displacement between member ends (i.e., P-δ effects)
and due to lateral end translation (i.e., P-Δ effects), respectively, which are computed
as follows:
1
FAP1
CB
e1s
1≥
−=
and
1
FAP1
1B
e2s
2≥
∑∑−
=
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where Fe1 is the Euler Buckling stress for the non-sway case and Fe2 is the Euler
Buckling stress for the sway case. Note that separate B1 factors are computed for all
the columns of the frame while a single B2 value applies to all the columns of the
frame since second order effects in frames subject to lateral translation is a story
phenomenon. It is therefore observed that AASHTO Equation (10-155) is meant to be
applied directly to braced frames. For unbraced frames such as the Omega Bridge, the
equation may be rewritten as follows to account for non-sway and sway effects
1M
MBMB
F0.85AP
u
lt2nt1
crs
≤+
+
The greatest difficulty in the evaluation of beam-columns is determining the
maximum bending moment, M = B1Mnt + B2Mlt. To calculate B1, the column ends are
assumed to be restrained from lateral end translation. According to AASHTO Article
10.54.1.2, the effective length factor under braced conditions used to compute Fe1 is
specified as 0.75 for members with riveted-end conditions (i.e., arch and skewback
columns) and 0.875 for members with pinned-end conditions (i.e., pier columns). The
equivalent load factor, C, is computed according to AASHTO Article 10.54.2.2 using
the equation C = 0.6 + 0.4a where “a” is the ratio of the smaller to larger moment at
the column ends; the value of “a” is positive for members in single curvature and
negative for members in double curvature. Thus, the equivalent load factor for the
pier columns is C = 0.6 since the moment at the pinned base is zero. The arch and
skewback columns, on the other hand, have bending moments on both ends which are
approximately equal in magnitude and act in the same direction (causing double
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curvature). The “a” ratio for the arch and skewback columns was conservatively
taken as -0.5 (i.e., the smaller end moment is half in magnitude of the larger end
moment) and thus, C = 0.4. Because of the small C values, the moment amplification
factor for the unbraced condition (i.e., B1) was equal to unity for all the columns
under the different vehicular loads.
Contrary to B1, B2 is computed based on unrestrained lateral end translation of
the column ends. As shown above, the denominator of the B2 equation contains the
term ∑AsFe2 which represents the total Euler Buckling resistance (of all the columns)
under unbraced conditions. Since the spandrel beam and arch rib were found to be
significantly stiffer in flexure than the columns, the end conditions were taken to be
fixed-pinned for the pier columns and fixed-fixed for the arch and skewback columns
to compute Fe2; the top end of each of the columns was free to translate relative to the
bottom end. These end conditions resulted in effective length factors equal to 2.0 for
the pier columns and 1.2 for the arch and skewback columns as specified in
AASHTO Appendix C. The two pier columns closest to the abutment (i.e., pier
column #1 and #2 in Figure 5.1) are leaner columns and theoretically have an
effective length factor equal to infinity and no axial capacity under unbraced
conditions. As a result, these columns do not contribute to the overall sway buckling
strength of the Omega Bridge.
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Table 5.1 Sample calculation of sidesway moment amplification factor, B2
Column L (ft) L (in) K Pe2 (kips)
Pier col #2 41.2 494.4 2 1980
Pier col #3 47.4 568.8 2 1496
Skewback col #1 103.1 1237.2 1.2 4114
Skewback col #2 86.8 1041.6 1.2 5804
Arch col #1 99.1 1189.2 1.2 950
Arch col #2 73.1 877.2 1.2 1747
Arch col #3 51.4 616.8 1.2 3533
Arch col #4 34.2 410.4 1.2 7980
Arch col #5 20.5 246 1.2 22210
Arch col #6 11.7 140.4 1.2 68183
Arch col #7 6.9 82.8 1.2 196042
Arch col #8 5.8 69.6 1.2 277455
Arch col #9 8.5 102 1.2 129184
Arch col #10 15.1 181.2 1.2 40935
Arch col #11 26.7 320.4 1.2 13093
Arch col #12 41.7 500.4 1.2 5368
Arch col #13 61.2 734.4 1.2 2492
Arch col #14 85 1020 1.2 1292
Frame strength (kips) 783855
2510203
3704 Frame force at inventory level (kips)
3. Sidesway moment amplification factor calculation at inventory level
1. Frame strength calculation
2. Total frame force calculation
Total dead load (kips) HS20 live load plus impact (kips)
2eP =∑
1.3 2510 2.17 203P = × + × =∑
2
2
1 1 1.005370411 783855e
BP
P
= = =−− ∑
∑
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Table 5.1 shows a sample computation of B2 under HS-20 vehicular loading at
the inventory rating level. Note the pier columns #1 and #2 were not included in the
calculation for the reasons discussed above. As shown in the table, the value of B2 is
very close to unity since the total load acting on the columns under dead load and live
load plus impact (i.e., ∑P) is small compared to the total elastic sidesway buckling
strength (i.e., ∑AsFe2). This is true under all the vehicular loading cases at both the
inventory and operating rating level.
With both the moment amplification factors equal to unity (i.e., B1 = 1 and B2
= 1), AASHTO Equation (10-155) becomes
1M
MM
F0.85AP
u
ltnt
crs
≤+
+
where P is the maximum compressive load and Mnt + Mlt is the maximum first order
moment acting on the column under dead load and live load plus impact. In general,
the longitudinal position of the vehicular load that causes the maximum axial force in
a given column is not the same as that producing the maximum bending moment. For
maximum axial compression, the truck is generally positioned straddling the column
with axles on both of the adjacent spans. To maximize the bending moment, the truck
is generally positioned in only one of the spans adjacent to the column. In order to
simplify the analysis of the Omega Bridge columns, the largest magnitudes for axial
force and bending moment from the design envelopes were used. This approach is
conservative but reasonable since the influence line analysis showed that the critical
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truck locations for axial force and bending moment were in close proximity (i.e.,
within about 60 ft for the pier columns and 30 ft for the arch and skewback columns).
The final variable needed in the interaction equation is the critical buckling
stress, Fcr, of the column which is specified in AASHTO Article 10.54.1.1 for
inelastic and elastic buckling as
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
r
KL
E24
F1FF cy
ycr π Inelastic Buckling
2c
2
rKL
EFcr
⎟⎠⎞
⎜⎝⎛
=π Elastic Buckling
where E = modulus of elasticity (29000 ksi); K = effective length factor in the plane
of buckling; Lc = length of the member between points of support; and r = radius of
gyration in the plane of buckling. In basic frame analysis, the maximum slenderness
ratio, (KLc/r)max, for in-plane or out-of-plane buckling is used to compute Fcr. For the
Omega Bridge, however, in-plane buckling was not considered likely for three
reasons. First, for sway buckling to occur, the columns would all have to buckle
simultaneously as a story. This type of behavior is likely for rectangular framing
where the columns all have about the same stiffness and applied loads. The Omega
Bridge, on the other hand, is composed of columns having different end conditions,
unsupported lengths, and axial loads. Under truck loading, only a single column is
subjected to maximum loading; the remaining columns are subjected to axial loads
below their largest magnitude (since the truck load is positioned to maximize the
force in a discrete column) and thus, act to brace the most heavily loaded column.
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Second, there is only a limited amount of movement that can take place at the
expansion joints (at the abutment ends) of the bridge. Once the expansion joint gap is
exhausted, sway will be restrained to some degree by the abutment. Third, in the
computation of B2 (see Table 5.1), the total elastic sidesway buckling strength of the
bridge columns (i.e., ∑AsFe2) was found to be large compared to the total applied
compressive loads (i.e., ∑P). For these three major reasons, the columns were all
assumed to be braced in the in-plane direction and the critical buckling stress was
determined based on nonsway in-plane buckling.
With support from the discussion given above, Equations (10-155) and (10-
156) of the AASHTO Standard Specifications may now be written in terms of the
factored dead load and live load (plus impact) as follows
1SF
MAMA
F0.85A
PAPA
M M
PP
y
L2D1
crs
L2D1
ucr
≤+
++
=+ AASHTO Eq (10-155)
1ZF
MAMA
F0.85A
PAPA
M M
PP
y
L2D1
ys
L2D1
py
≤+
++
=+ AASHTO Eq (10-156)
where PD and PL are the unfactored axial forces; MD and ML are the
unfactored bending moments; and A1 and A2 are the load factors under dead load and
live load (plus impact), respectively. Tables 5.2 through 5.6 gives the final interaction
ratios for the pier columns (excluding the ones closest to the abutments), arch
columns, and skewback columns computed based on AASHTO Equations (10-155)
and (10-156) at the inventory and operating rating levels; the five tables correspond to
the five different live loads. The interaction ratios are designated as IRi at inventory
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Table 5.2 Interaction ratios and rating factors for bridge columns under HS-20 design truck loading.
AASHTO Eq. (10-155) AASHTO Eq. (10-156) Column
IRi IRo RFi RFo IRi IRo RFi RFo
Pier Col #2 0.87 0.59 1.18 1.97 0.79 0.53 1.33 2.21
Pier Col #3 0.76 0.53 1.42 2.36 0.69 0.48 1.60 2.68
Arch Col #1 0.80 0.54 1.30 2.16 0.67 0.45 1.59 2.66
Arch Col #2 0.93 0.62 1.10 1.83 0.81 0.54 1.29 2.15
Arch Col #3 1.02 0.68 0.98 1.64 0.90 0.60 1.13 1.89
Arch Col #4 1.07 0.72 0.93 1.55 0.95 0.63 1.06 1.77
Arch Col #5 1.17 0.77 0.83 1.39 1.04 0.69 0.95 1.59
Arch Col #6 1.21 0.80 0.80 1.34 1.08 0.71 0.92 1.53
Arch Col #7 1.08 0.70 0.91 1.52 0.97 0.63 1.04 1.73
Arch Col #8 0.92 0.59 1.10 1.83 0.83 0.54 1.23 2.06
Arch Col #9 1.19 0.78 0.82 1.36 1.06 0.70 0.93 1.56
Arch Col #10 1.26 0.83 0.76 1.27 1.13 0.74 0.87 1.45
Arch Col #11 1.17 0.76 0.84 1.40 1.04 0.68 0.96 1.59
Arch Col #12 1.10 0.72 0.90 1.50 0.98 0.64 1.03 1.71
Arch Col #13 0.98 0.65 1.02 1.71 0.86 0.57 1.19 1.98
Arch Col #14 0.79 0.52 1.31 2.19 0.68 0.45 1.55 2.59
Skewback Col #1 0.44 0.31 2.76 4.60 0.38 0.27 3.21 5.36
Skewback Col #2 0.45 0.32 2.63 4.39 0.40 0.28 3.03 5.06
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Table 5.3 Interaction ratios and rating factors for bridge columns under TYPE 3 legal truck loading.
AASHTO Eq. (10-155) AASHTO Eq. (10-156) Column
IRi IRo RFi RFo IRi IRo RFi RFo
Pier Col #2 0.66 0.46 1.69 2.82 0.60 0.42 1.90 3.17
Pier Col #3 0.59 0.42 2.03 3.38 0.53 0.39 2.30 3.83
Arch Col #1 0.61 0.42 1.85 3.09 0.51 0.35 2.26 3.78
Arch Col #2 0.70 0.48 1.56 2.61 0.61 0.42 1.83 3.06
Arch Col #3 0.76 0.52 1.39 2.32 0.68 0.47 1.60 2.68
Arch Col #4 0.80 0.54 1.32 2.20 0.71 0.49 1.50 2.51
Arch Col #5 0.88 0.60 1.18 1.96 0.79 0.54 1.35 2.24
Arch Col #6 0.91 0.62 1.13 1.89 0.81 0.55 1.30 2.16
Arch Col #7 0.80 0.54 1.29 2.16 0.72 0.48 1.47 2.45
Arch Col #8 0.68 0.45 1.56 2.61 0.61 0.40 1.76 2.93
Arch Col #9 0.89 0.61 1.15 1.92 0.80 0.54 1.32 2.20
Arch Col #10 0.94 0.64 1.07 1.79 0.84 0.57 1.23 2.05
Arch Col #11 0.87 0.58 1.19 1.98 0.78 0.52 1.35 2.26
Arch Col #12 0.82 0.55 1.27 2.13 0.73 0.49 1.46 2.43
Arch Col #13 0.73 0.50 1.46 2.43 0.65 0.44 1.69 2.82
Arch Col #14 0.59 0.40 1.87 3.12 0.51 0.35 2.21 3.69
Skewback Col #1 0.35 0.25 3.87 6.47 0.30 0.22 4.52 7.55
Skewback Col #2 0.35 0.26 3.71 6.19 0.31 0.23 4.27 7.13
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Table 5.4 Interaction ratios and rating factors for bridge columns under TYPE 3S2 legal truck loading.
AASHTO Eq. (10-155) AASHTO Eq. (10-156) Column
IRi IRo RFi RFo IRi IRo RFi RFo
Pier Col #2 0.85 0.57 1.22 2.04 0.77 0.52 1.37 2.29
Pier Col #3 0.74 0.51 1.48 2.46 0.67 0.47 1.67 2.79
Arch Col #1 0.74 0.50 1.43 2.38 0.63 0.42 1.73 2.90
Arch Col #2 0.87 0.58 1.19 1.99 0.76 0.51 1.40 2.34
Arch Col #3 0.95 0.63 1.07 1.79 0.84 0.56 1.24 2.06
Arch Col #4 0.98 0.65 1.02 1.71 0.87 0.58 1.17 1.96
Arch Col #5 1.07 0.71 0.92 1.54 0.95 0.64 1.06 1.77
Arch Col #6 1.10 0.73 0.89 1.48 0.98 0.66 1.02 1.71
Arch Col #7 1.02 0.66 0.98 1.64 0.91 0.59 1.12 1.87
Arch Col #8 0.87 0.56 1.17 1.96 0.78 0.50 1.33 2.22
Arch Col #9 1.09 0.72 0.90 1.51 0.97 0.65 1.04 1.73
Arch Col #10 1.15 0.76 0.84 1.41 1.03 0.68 0.97 1.62
Arch Col #11 1.07 0.70 0.92 1.54 0.95 0.63 1.06 1.77
Arch Col #12 1.02 0.67 0.98 1.63 0.91 0.60 1.12 1.87
Arch Col #13 0.92 0.61 1.11 1.85 0.81 0.54 1.28 2.14
Arch Col #14 0.74 0.49 1.43 2.38 0.64 0.42 1.69 2.82
Skewback Col #1 0.39 0.28 3.24 5.41 0.34 0.25 3.77 6.30
Skewback Col #2 0.40 0.29 3.10 5.18 0.36 0.25 3.57 5.95
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Table 5.5 Interaction ratios and rating factors for bridge columns under TYPE 3-3 legal truck loading.
AASHTO Eq. (10-155) AASHTO Eq. (10-156) Column
IRi IRo RFi RFo IRi IRo RFi RFo
Pier Col #2 0.90 0.60 1.13 1.89 0.82 0.55 1.28 2.13
Pier Col #3 0.78 0.54 1.37 2.29 0.70 0.49 1.56 2.60
Arch Col #1 0.77 0.52 1.36 2.27 0.65 0.44 1.65 2.75
Arch Col #2 0.90 0.61 1.13 1.89 0.79 0.53 1.33 2.22
Arch Col #3 0.98 0.66 1.02 1.70 0.87 0.58 1.18 1.97
Arch Col #4 1.02 0.67 0.98 1.64 0.90 0.60 1.13 1.88
Arch Col #5 1.11 0.73 0.89 1.48 0.98 0.66 1.02 1.70
Arch Col #6 1.14 0.76 0.86 1.43 1.01 0.67 0.99 1.65
Arch Col #7 1.06 0.69 0.94 1.56 0.94 0.62 1.07 1.79
Arch Col #8 0.91 0.59 1.11 1.85 0.82 0.53 1.26 2.10
Arch Col #9 1.12 0.74 0.87 1.45 1.00 0.66 1.00 1.67
Arch Col #10 1.19 0.78 0.81 1.36 1.06 0.70 0.94 1.56
Arch Col #11 1.11 0.73 0.89 1.48 0.99 0.65 1.02 1.70
Arch Col #12 1.06 0.70 0.93 1.56 0.94 0.62 1.07 1.79
Arch Col #13 0.96 0.63 1.05 1.75 0.84 0.56 1.22 2.03
Arch Col #14 0.77 0.51 1.36 2.26 0.66 0.44 1.60 2.67
Skewback Col #1 0.38 0.28 3.31 5.53 0.34 0.25 3.85 6.43
Skewback Col #2 0.39 0.28 3.18 5.31 0.35 0.25 3.65 6.09
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Table 5.6 Interaction ratios and rating factors for bridge columns under FIRE special truck loading.
AASHTO Eq. (10-155) AASHTO Eq. (10-156) Column
IRi IRo RFi RFo IRi IRo RFi RFo
Pier Col #2 0.88 0.59 1.16 1.94 0.80 0.54 1.31 2.19
Pier Col #3 0.77 0.53 1.40 2.34 0.69 0.48 1.59 2.65
Arch Col #1 0.81 0.54 1.29 2.15 0.68 0.45 1.57 2.63
Arch Col #2 0.93 0.62 1.09 1.82 0.81 0.55 1.28 2.13
Arch Col #3 1.03 0.68 0.97 1.62 0.91 0.60 1.12 1.87
Arch Col #4 1.07 0.71 0.92 1.54 0.96 0.63 1.05 1.76
Arch Col #5 1.17 0.78 0.83 1.38 1.05 0.69 0.94 1.58
Arch Col #6 1.21 0.80 0.80 1.33 1.08 0.71 0.91 1.52
Arch Col #7 1.09 0.71 0.90 1.51 0.98 0.64 1.02 1.71
Arch Col #8 0.93 0.60 1.09 1.82 0.84 0.54 1.22 2.04
Arch Col #9 1.19 0.78 0.81 1.36 1.06 0.70 0.93 1.56
Arch Col #10 1.27 0.83 0.76 1.26 1.13 0.74 0.86 1.44
Arch Col #11 1.17 0.76 0.83 1.39 1.05 0.68 0.95 1.58
Arch Col #12 1.11 0.72 0.89 1.48 0.98 0.64 1.02 1.70
Arch Col #13 0.99 0.65 1.01 1.69 0.87 0.57 1.17 1.96
Arch Col #14 0.80 0.53 1.30 2.17 0.69 0.45 1.54 2.57
Skewback Col #1 0.44 0.31 2.73 4.56 0.39 0.27 3.18 5.32
Skewback Col #2 0.45 0.32 2.61 4.35 0.40 0.28 3.00 5.01
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and IRo at operating; values less than one indicate satisfactory passing performance.
Also reported in Tables 5.2 through 5.6 are the inventory (RFi) and operating (RFo)
rating factors. These factors were determined by solving for the multiple of the live
load effects (axial force and bending moment) that set the left part of the AASHTO
equations equal to one as shown below. Contrary to the interaction ratios, the capacity
check is confirmed when the rating factors exceed one.
1SF
RF)(MAMA
F0.85A
RF)(PAPA
y
L2D1
crs
L2D1 =+
++
1ZF
RF)(MAMA
F0.85A
RF)(PAPA
y
L2D1
ys
L2D1 =+
++
5.2.2. COLUMN Model: Axial Loading
Using the COLUMN rating model, the second term of AASHTO Equation
(10-155) that accounts for the bending moment effects in beam-column behavior goes
away leaving the following equation
1 F0.85A
RF)(PAPA
crs
L2D1 =+
This equation can now be solved for the rating factor, RF, which gives
L2
D1R
PA
PAPRF
−=
where PR = 0.85AsFcr as specified by AASHTO Equation (10-150) for concentrically
loaded compression members. It is important to note that the dead and live load axial
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forces, PD and PL, did not differ significantly between the BEAM-COLUMN and
COLUMN rating models.
Table 5.7 lists the rating factors computed for the columns with the equation
given above. No significant difference were found in the rating factors for the arch
columns symmetrical about the centerline of the arch (i.e., arch columns #1 and #14,
#2 and #13, etc.) so only one value is reported for each arch column pair. The same
distribution factors and impact factors used in the beam-column analysis (based on
braced conditions) were also used for the pure compression analysis. However, the
effective length factors were conservatively taken as K = 2 (for in-plane buckling) for
the pier columns located closest to the abutments which assumes fixed-free end
conditions. For these two columns, the strength was controlled by in-plane buckling
since the effective slenderness ratio was larger compared to out-of-plane buckling.
This is reasonable considering the large stiffness of the spandrel beam and the rocker
bearing at the base support. In the out-of-plane direction, the buckling strength of all
the columns was computed assuming an effective length factor of K = 1. The columns
were also assumed to be unsupported over their full length with bracing only at the
top and bottom ends. This was true also for the skewback columns which had wind
bracing at intermittent distances along their length; an effective length factor of unity
for these columns assumes the pair buckles together. A discussion of the rating
factors based on beam-column and column behavior is provided in the following
section.
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Table 5.7 Rating factors for bridge columns based on axial loading only.
HS-20 TYPE 3 TYPE 3S2 TYPE 3-3 FIRE Column
RFi RFo RFi RFo RFi RFo RFi RFo RFi RFo
Pier Col #1 3.28 5.48 4.68 7.82 3.49 5.83 3.32 5.54 3.24 5.41
Pier Col #2 3.35 5.59 4.77 7.96 3.61 6.03 3.46 5.77 3.31 5.52
Pier Col #3 3.26 5.44 4.64 7.74 3.51 5.86 3.37 5.62 3.22 5.37
Pier Col #4 3.19 5.32 4.55 7.59 3.39 5.66 3.22 5.38 3.15 5.25
Arch Col #1 and #14 2.69 4.48 3.72 6.21 3.65 6.10 4.03 6.72 2.67 4.46
Arch Col #2 and #13 3.41 5.70 4.72 7.89 4.70 7.84 5.19 8.66 3.40 5.67
Arch Col #3 and #12 3.93 6.56 5.45 9.09 5.41 9.04 5.99 9.99 3.92 6.54
Arch Col #4 and #11 4.20 7.01 5.81 9.70 5.78 9.65 6.39 10.66 4.18 6.98
Arch Col #5 and #10 4.33 7.23 6.00 10.01 5.97 9.96 6.59 11.01 4.32 7.20
Arch Col #6 and #9 4.38 7.31 6.07 10.13 6.03 10.07 6.67 11.13 4.37 7.29
Arch Col #7 and #8 4.40 7.34 6.09 10.17 6.06 10.11 6.69 11.17 4.38 7.31
Skewback Col #1 3.78 6.30 5.35 8.92 4.26 7.11 4.25 7.09 3.74 6.23
Skewback Col #2 4.56 7.60 6.45 10.77 5.14 8.58 5.13 8.56 4.51 7.52
5.3. Discussion of BEAM-COLUMN and COLUMN Rating Factors
Recall that in the BEAM-COLUMN model, the riveted connections as well as
the base supports of the skewback columns were discretized as rigid connections; this
approach resulted in the column rating factors shown in Tables 5.2 through 5.6.
Contrary to the BEAM-COLUMN model, the column connections in the COLUMN
model were idealized as pinned connections which significantly increased the rating
factors (see Table 5.7). In Table 5.8, the inventory rating factors from the BEAM-
COLUMN and COLUMN models (designated RFi,b-c and RFi,col, respectively) are
summarized. The rating values reported for the BEAM-COLUMN model correspond
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to AASHTO Equation (10-155) which controlled the capacity for combined axial
load and bending rather than (10-156). No rating values are given for pier columns #1
and #2 for the BEAM-COLUMN model since these two columns are under axial
compression only regardless of the connection stiffness.
Table 5.8 Inventory rating factors for bridge columns based on beam-column (RFi,b-c) and column (RFi,c) behavior.
HS-20 TYPE 3 TYPE 3S2 TYPE 3-3 FIRE Column
RFi,b-c RFi,col RFi,b-c RFi,col RFi,b-c RFi,col RFi,b-c RFi,col RFi,b-c RFi,col
Pier Col #1 N/A 3.28 N/A 4.68 N/A 3.49 N/A 3.32 N/A 3.24
Pier Col #2 1.18 3.35 1.69 4.77 1.22 3.61 1.13 3.46 1.16 3.31
Pier Col #3 1.42 3.26 2.03 4.64 1.48 3.51 1.37 3.37 1.40 3.22
Pier Col #4 N/A 3.19 N/A 4.55 N/A 3.39 N/A 3.22 N/A 3.15
Arch Col #1 1.30 2.69 1.85 3.72 1.43 3.65 1.36 4.03 1.29 2.67
Arch Col #2 1.10 3.41 1.56 4.72 1.19 4.70 1.13 5.19 1.09 3.40
Arch Col #3 0.98 3.93 1.39 5.45 1.07 5.41 1.02 5.99 0.97 3.92
Arch Col #4 0.93 4.20 1.32 5.81 1.02 5.78 0.98 6.39 0.92 4.18
Arch Col #5 0.83 4.33 1.18 6.00 0.92 5.97 0.89 6.59 0.83 4.32
Arch Col #6 0.80 4.38 1.13 6.07 0.89 6.03 0.86 6.67 0.80 4.37
Arch Col #7 0.91 4.40 1.29 6.09 0.98 6.06 0.94 6.69 0.90 4.38
Arch Col #8 1.10 4.40 1.56 6.09 1.17 6.06 1.11 6.69 1.09 4.38
Arch Col #9 0.82 4.38 1.15 6.07 0.90 6.03 0.87 6.67 0.81 4.37
Arch Col #10 0.76 4.33 1.07 6.00 0.84 5.97 0.81 6.59 0.76 4.32
Arch Col #11 0.84 4.20 1.19 5.81 0.92 5.78 0.89 6.39 0.83 4.18
Arch Col #12 0.90 3.93 1.27 5.45 0.98 5.41 0.93 5.99 0.89 3.92
Arch Col #13 1.02 3.41 1.46 4.72 1.11 4.70 1.05 5.19 1.01 3.40
Arch Col #14 1.31 2.69 1.87 3.72 1.43 3.65 1.36 4.03 1.30 2.67
Skewback Col #1 2.76 3.78 3.87 5.35 3.24 4.26 3.31 4.25 2.73 3.74
Skewback Col #2 2.63 4.56 3.71 6.45 3.10 5.14 3.18 5.13 2.61 4.51
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For the BEAM-COLUMN model, the two pier columns (labeled #2 and #3 in
Figure 5.1), the two skewback columns, and four arch columns (labeled #1, #2, #13,
and #14 in Figure 5.1) had inventory ratings greater than 1 for all live loads. These
eight columns are the ones located on the north and south ends of the Omega Bridge
(four on each end) as shown in Figure 5.1. Arch column #8 (located at the apex of the
arch rib) had rating values also exceeding 1 at inventory while those for arch column
#3 were close to or greater than 1. Thus, these 10 columns were not critical to the
bridge capacity. Note also that all the columns had inventory ratings exceeding 1
under TYPE 3 legal truck loading.
The inventory capacity ratings were smallest and equal to 0.80 and 0.76,
respectively, for arch columns #6 and #10 under HS-20 and FIRE truck loading;
hence, these two arch columns were the most critical. The next three arch columns
with the lowest capacity ratings were #5, #9, and #11 which all had inventory ratings
between 0.81 and 0.84, also under HS-20 and FIRE truck loading. The column group
that followed was arch columns #4, #7, and #12 with inventory rating values between
0.89 and 0.93. From the rating analysis of the BEAM-COLUMN model, it was
observed that arch columns #4 through #12 had much smaller axial forces than the
pier and skewback columns. However, the use of rigid connections caused large end
moments in the arch columns which were the dominate effects in the interaction
equation and decreased the rating factors.
Comparing the inventory rating factors between the BEAM-COLUMN and
COLUMN models shows that bending of the columns significantly reduces their
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capacity. The rating factors of pier columns #2 and #3 decreased by a factor of 2 to 3
when treated as beam-columns rather the pure columns. For the arch columns, the
reduction was as much as seven-fold for columns located at the top of the arch rib
(i.e., the shorter columns) and between two-fold and three-fold for those located
closer to the pinned ends (i.e., the longer columns). This behavior is reasonable since
short columns have a high axial compressive strength when treated as a pure column.
Furthermore, the two skewback columns experienced a decrease in rating factor
between 1 and 2 when bending was considered. These observations suggest that
flexure impacts the shorter columns more than it does the longer columns. Thus,
when flexure is considered in the rating, the capacity of the short columns becomes
much more critical; under axial loading only, short column capacity is not as much of
a concern.
An important observation is that the moments applied at the column ends are
directly proportional to the stiffness of the riveted connections and thus, the capacity
ratings are inversely proportional. The largest bending moments and smallest capacity
ratings result when the connections are assumed completely rigid (i.e., BEAM-
COLUMN model). Conversely, the moment magnitudes are equal to zero for pinned
connections and results in the largest capacity ratings (i.e., COLUMN model) which
overestimates the true column capacities. In actuality, the connection stiffness may be
somewhere between fully rigid and pinned behavior which if modeled appropriately
would improve the rating factors compared to the BEAM-COLUMN model which
underestimates the column capacities. Due to the very wide range in the magnitudes
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of the rating factors between the BEAM-COLUMN and COLUMN models, it is very
difficult to judge where the actual rating factors for the columns will fall. Field testing
is recommended to determine a better estimate of the stiffness of the riveted
connections and thus, more realistic rating factors. For safety purposes, it is
recommended that the rating factors produced from the BEAM-COLUMN be used
until a field test can be carried out.
In the summer of 2004, a terrestrial survey of the Omega Bridge was carried
out by Lasergeomatics (a division of Bohannan-Huston, Inc. in Albuquerque, NM)
using laser scanning techniques. As shown in Tables 5.9 and 5.10, results of the
survey showed that a few arch columns were out-of-plumb with angular variations
exceeding the AISC (American Institute of Steel Construction) erection tolerance of
1:500 specified in Section 7.13 of the Code of Standard Practice for Steel Buildings
and Bridges (AISC, 2001). The columns with out-of-plumb ratios greater than the
AISC erection limit were primarily the shorter arch columns located in the central
portion of the arch rib. For these columns, the angular variation was as high as 1:80.
Incidentally, these short arch columns were also found to be the more critical columns
when evaluated based on beam-column behavior. In addition, a high angular variation
(equal to 1:220) was measured at the pier column closest to the south abutment on
both the east and west arch rib plane.
Column misalignment can influence beam-column capacity in two major
ways. First, it can reduce the total unbraced buckling strength of a frame since more
deformation is needed to reach the bifurcation buckling load (i.e., the buckling load
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assuming plumb columns). However, this particular impact was not considered a
concern since the sway buckling resistance of the Omega Bridge was shown to be
quite large for several reasons (see discussion given in Section 5.2.1). In addition, the
columns were not all misaligned in the same direction in the arch rib plane as shown
in Tables 5.9 and 5.10; some lean in the north direction and some lean in the south
direction. Since the tendency of the Omega Bridge is to sway in the north direction
(due to the road alignment), the south-leaning columns would act to brace the north-
leaning columns which would further increase the sway buckling resistance. Note that
a decrease in the sway buckling resistance increases the moment magnification factor
(i.e., B2) for the unbraced condition. However, B2 was left at unity for the Omega
Bridge columns for reasons given above and in Section 5.2.1.
The second key impact of column misalignment is that the first order
moments at a column end may increase or decrease depending on which direction the
column is leaning; recall that the columns lean in opposite directions in the arch rib
plane as shown in Tables 5.9 and 5.10. The axial and shear forces of a column will
also change but by a smaller amount compared to the change in first order moments.
Since column offsets were found in the north-south and east-west direction, the first
order force effects will change in both the in-plane and out-of-plane direction of the
columns. As mentioned previously, bending at the column ends in the out-of-plane
direction was ignored. Although the columns will be subjected to out-of-plane flexure
due to the misalignment, these effects were not considered critical since the in-plane
bending moments are larger in magnitude. With the exception of the skewback
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Table 5.9 Column alignments on east side of Omega Bridge.
N-S Direction* E-W Direction* Column
Column Length
(ft) Offset (ft) Ratio Offset (ft) Ratio
Pier Col #1 15.47 -0.071 1:218 +0.039 1:397
Pier Col #2 38.20 +0.015 1:2547 -0.056 1:682
Skewback Col #1 102.92 +0.050 1:2058 +0.268 1:384
Arch Col #1 100.73 +0.028 1:3598 +0.315 1:320
Arch Col #2 75.01 -0.036 1:2084 +0.192 1:391
Arch Col #3 52.87 -0.042 1:1259 +0.097 1:545
Arch Col #4 35.02 -0.055 1:637 +0.056 1:625
Arch Col #5 21.57 -0.102 1:211 -0.010 1:2157
Arch Col #6 13.19 -0.169 1:78 -0.005 1:2638
Arch Col #7 7.57 +0.050 1:151 -0.020 1:379
Arch Col #8 5.98 -0.002 1:2990 -0.101 1:59
Arch Col #9 8.46 +0.024 1:353 +0.047 1:180
Arch Col #10 15.17 +0.012 1:1264 +0.085 1:178
Arch Col #11 26.21 -0.079 1:332 +0.047 1:558
Arch Col #12 42.28 -0.070 1:604 +0.049 1:863
Arch Col #13 61.42 -0.189 1:325 +0.046 1:1335
Arch Col #14 84.63 -0.164 1:516 -0.003 1:28210
Skewback Col #2 87.22 -0.104 1:839 +0.059 1:1478
Pier Col #3 45.89 +0.080 1:574 +0.038 1:1208
Pier Col #4 23.21 -0.025 1:928 -0.099 1:234
* – positive offset values designate North or East direction
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Table 5.10 Column alignments on west side of Omega Bridge.
N-S Direction* E-W Direction* Column
Column Length
(ft) Offset (ft) Ratio Offset (ft) Ratio
Pier Col #1 15.44 +0.072 1:214 +0.028 1:551
Pier Col #2 38.14 +0.007 1:5448 -0.113 1:337
Skewback Col #1 103.66 +0.044 1:2356 +0.214 1:484
Arch Col #1 97.86 -0.012 1:8155 +0.316 1:310
Arch Col #2 74.04 -0.075 1:987 +0.237 1:312
Arch Col #3 52.90 +0.008 1:6613 +0.116 1:456
Arch Col #4 35.26 -0.106 1:333 -0.039 1:904
Arch Col #5 21.06 +0.043 1:490 -0.007 1:3009
Arch Col #6 12.41 +0.004 1:3103 +0.006 1:2068
Arch Col #7 7.16 +0.029 1:247 +0.080 1:90
Arch Col #8 6.21 +0.062 1:100 +0.136 1:46
Arch Col #9 9.26 +0.045 1:206 +0.005 1:1852
Arch Col #10 16.26 -0.010 1:1626 +0.054 1:301
Arch Col #11 26.29 +0.040 1:657 +0.026 1:1011
Arch Col #12 42.18 -0.040 1:1055 +0.047 1:897
Arch Col #13 61.90 +0.960 1:64 -0.034 1:1821
Arch Col #14 85.02 +0.024 1:3543 +0.040 1:2126
Skewback Col #2 87.46 +0.005 1:17492 +0.064 1:1367
Pier Col #3 47.91 -0.087 1:551 -0.103 1:465
Pier Col #4 22.89 +0.029 1:789 +0.035 1:654
* – positive offset values designate North or East direction
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columns, the bending stiffness of the columns are the same in both the in-plane and
out-of-plane directions; hence, the in-plane direction with the larger moments will
control. Furthermore, the columns are adequately braced in the out-of-plane direction
by cross-bracing between the skewback columns. Based on these observations, the
effects of column misalignment discussed in the following paragraph focuses on in-
plane behavior (i.e., in the arch rib plane).
To illustrate the potential impact of column misalignment on the load rating
capacity, consider arch column #10 located on the east side of the Omega Bridge. As
discussed earlier, this column had the lowest rating factor based on beam-column
behavior. Table 5.9 shows the column to be out-of-plumb in the northern direction a
distance of 0.012 ft or 0.14 in. The column is assumed to lean in the direction that
increases the first order moment effects compared to the vertical column position.
Table 5.11 shows the first order dead load and live load effects and the rating factors
of arch column #10 with the column in a vertical position (i.e., offset distance = 0)
and an inclined position (i.e., offset distance = 0.14 in.). The change in axial force
caused by column misalignment was ignored and thus, the same axial forces
determined beforehand with the columns vertical were also used for the inclined case.
The increase in first order moments due to column misalignment was approximated
by multiplying the axial column load and the offset distance. It is important to note
that this approach provides only an estimate of the column misalignment effects.
Evaluation of the actual impact of the column misalignment would require
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remodeling of the entire structure in the arch rib plane and including the measured
inclination (direction and magnitude) of each individual column.
Table 5.11 Load rating of arch column #10 in vertical and inclined position
AASHTO Eq. (10-155)
AASHTO Eq. (10-156) Offset
Distance (in)
Axial force due to
Dead Load (kips)
Axial force due to
Live Load (kips)
Moment due to
Dead Load (kip-in)
Moment due to
Live Load (kip-in) RFi RFo RFi RFo
0.00 101.3 47.0 115.5 219.3 0.76 1.27 0.87 1.45
0.14 101.3 47.0 130.1 226.1 0.73 1.21 0.84 1.39
As shown in Table 5.11, the rating factors computed by AASHTO Equations
(10-155) and (10-156) for beam-column behavior decreased about 4% with the
column misaligned a distance of 0.14 in. However, this reduction in the load rating
assumes that the column was out-of-plumb under dead and live load axial forces
which may not be the case. If the effects of misalignment under dead load are
neglected, the total change in moment under live load amounts to about 7 kip-ft (i.e.,
226.1 – 219.3 kip-ft = 6.8 kip-ft) which is one-third the change caused by dead and
live load combined (i.e., 130.1 – 115.5 + 6.8 kip-ft = 21.4 kip-ft). Under this scenario,
the reduction in capacity ratings would be less than 4%.
Aside from the arch columns, another notable impact of the column
misalignment has to do with the pier columns. As mentioned previously, these
columns were analyzed as compression members since the bottom ends are rocker
bearings and thus, there is no shear force or bending moment. This is true only if the
columns are perfectly plumb. If the column is out-of-plumb, the axial load acts
through an eccentricity equal to the offset distance which produces both shear force
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and bending moment in the column. Tables 5.9 and 5.10 showed the pier column
closest to the south abutment to be out-of-plumb a distance of about 0.07 ft (0.85 in)
which gave an angular variation of 1:215 exceeding the AISC erection tolerance of
1:500. Shear and flexure can also be produced if the rocker bearing does not permit
free rotation at the bottom end of the column; in this case, shear and flexure result
from the horizontal force that develops at the rocker bearing. In either of these two
scenarios, the pier columns behave as beam-column members which as stated earlier
decreases the capacity rating compared to pure compression. As shown in Table 5.8,
the pier columns closest to the abutment had rating factors comparable to the pier
columns closest to the skewback columns based on column behavior. When beam-
column behavior was used to evaluate the latter two columns, the inventory ratings
exceeded unity. It is therefore anticipated that the pier columns closest to the
abutments will also have inventory ratings greater than one, particularly if the
moment effects are produced primarily from column misalignment rather than
locking of the rocker bearing which will cause larger moments.
As illustrated by the examples given above, column misalignment can reduce
the rating factor should the column lean in the direction which increases the first
order moments; a more significant reduction in the capacity ratings will result with
larger column offsets. Thus, it is recommended that the column misalignment (both
direction and magnitude) continue to be monitored periodically by the LANL,
particularly the arch columns; the procedure given in this section provides a simple
approach to approximate the out-of-plumb effects on the capacity ratings. It is
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important to note that misalignment of the columns will only change the rating factors
of the BEAM-COLUMN model; trivial changes will occur in the column rating
factors of the COLUMN model because the axial force in the columns remain about
the same and also, there is no bending moment in the columns since the ends are
pinned.
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Chapter 6
LOAD RATING OF ARCH RIB
6.1. Description of Rating Model
As discussed earlier in Chapter 2, the arch rib is a two-hinge parabolic arch
with a span of 422.5 ft and a rise of 106.6 ft. Recall that riveted connections were
provided between all the columns and the spandrel beam as well as between the arch
columns and arch rib; however, the stiffness of these connections is not certain.
Therefore, two separate models were developed to load rate the arch rib. Similar to
the column evaluation approach presented in Chapter 5, rigid connections were
applied at the column ends in the first arch rib model (designated RIGID model)
while pinned connections were assumed in the second model (designated PINNED
model). The PINNED and RIGID models are shown in Figures 6.1 and 6.2,
respectively, which are identical to the COLUMN and BEAM-COLUMN models
used to evaluate the columns (see Chapter 5). As shown in these two figures, the
entire structure in the arch rib plane was modeled including the spandr.el beam,
columns and arch rib.
15 spans (29.5ft each)62ft 62ft62ft 62ft 62ft 62ft
106.6ft
422.5ft
SOUTH NORTH
Figure 6.1 PINNED rating model of arch rib.
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15 spans (29.5ft each)
422.5ft
106.6ft
62ft 62ft62ft 62ft 62ft62ft
SOUTH NORTH
Figure 6.2 RIGID rating model of arch rib.
In the RIGID model, it is assumed that no relative rotation occurs between the
connecting members (i.e., the original angle between the members is maintained). On
the contrary, the PINNED model assumes that the columns are free to rotate relative
to the connecting component (i.e., the spandrel beam or arch rib). In order to maintain
structural stability of this model, however, a horizontal restraint was placed at the
south abutment. The major difference between the two models is with regard to the
forces transferred to the arch rib at the base of the arch columns. The RIGID model
results in axial forces, bending moments, and shear forces at these locations while the
PINNED model results in only axial forces. The reader is referred back to Section 5.1
for more discussion regarding these two modeling schemes.
The dead load that acts on the arch rib includes that applied to the arch
columns (see Section 5.1) plus the self-weight of the arch rib and the wind bracing
spanning between the arches. The live load impact factor amounted to
( ) 0.21 30.0 ,21.0min 30.0 ,125 29.5 x 4
50min 30.0 ,512 L
50min I ==⎟⎠⎞
⎜⎝⎛
+=⎟
⎠⎞
⎜⎝⎛
+=
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for the arch rib according to AASHTO Article 3.8.2.1. In this computation, the loaded
length (L) was taken as the distance between the two points on the influence lines
where the ordinates were equal to zero. Influence line analysis showed the smallest
distance to be equal to 4.5 times the distance between adjacent arch columns (i.e., 4.5
x 29.5 ft). In the end, a conservative decision was made to use 4 instead of 4.5 times
the arch column spacing for the loaded length (i.e., 4 x 29.5 ft). As discussed in
Chapter 4, the arch rib situated on the east side of the Omega Bridge is the most
heavily loaded arch due to the location of the traffic lanes and thus, controls the arch
capacity. For the eastern arch rib, the distribution factor (including multiple presence
effects) is equal in magnitude to the one used to evaluate the eastern spandrel beam
(see Table 4.6). Recall that the same distribution factor was also used to evaluate the
columns (see Section 5.1).
6.2. Load Factor Rating Analysis
According to AASHTO Articles 10.37 and 10.55, the capacity of the arch rib
shall satisfy interaction equation (10-47) given below:
1F
f
F
f
b
b
a
a ≤+ AASHTO Equation (10-47)
where fa = the computed axial stress (under dead load and live load plus impact); Fa =
the allowable axial stress; fb = the computed bending stress, including moment
amplification, at the extreme fiber (under dead load and live load plus impact); and Fb
= the allowable bending stress. In terms of axial forces, fa may be expressed as
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A
NANAf L2D1
a
+=
where ND and NL are the unfactored axial forces and A1 and A2 are the load factors
under dead load and live load (plus impact). The variable A represents the cross-
sectional area of the arch rib. Similarly to fa, fb may be rewritten in terms of the
bending moments in the arch rib as follows
S
)A x (MAMAf FL2D1
b
+=
where MD and ML are the unfactored, first-order bending moments under dead load
and live load (plus impact); AF is the amplification factor for the live load plus impact
moment; and S is the section modulus of the arch rib at the extreme fiber. The
amplification factor, AF, is computed by AASHTO Equation (10-159) as shown
below:
e
F
AFT18.11
1A−
= AASHTO Equation (10-159)
where T is the thrust at the quarter point (under dead and live load plus impact) and Fe
is the Euler buckling stress, π2E / (KL/r)2, of the arch rib. In terms of the dead and
live load (plus impact) thrusts, TD and TL, AASHTO Equation (10-159) may be
rewritten as
e
L2D1F
AF)TAT(A 18.1
1
1A+
−=
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The effective slenderness ratio (i.e., KL/r) used to compute Fe employs L
equal to one-half the arch rib length and r equal to the radius of gyration in the plane
of bending; E is equal to the modulus of elasticity in the calculation of Fe. The
effective length factor, K, depends on the rise-to-span ratio and the type of arch (see
K values given in Table 6.1).
Table 6.1 Effective length factor (K) values for arch rib (AASHTO, 2002).
Rise to Span Ratio
3-Hinged Arch
2-Hinged Arch
Fixed Arch
0.1 – 0.2 1.16 1.04 0.70
0.2 – 0.3 1.13 1.10 0.70
0.3 – 0.4 1.16 1.16 0.72
The arch rib of the Omega Bridge is a 2-hinged arch with a rise-to-span ratio
equal to 0.25 (i.e., 106.6 ft divided by 422.5 ft) which gave a K value equal to 1.10.
The same KL/r ratio is also used to determine the allowable axial stress, Fa, which is
computed by AASHTO Equation (10-160) as shown below
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
π
⎟⎠⎞
⎜⎝⎛
−=E4
F r
KL
118.1
FF
2
yy
a AASHTO Equation (10-160)
The allowable bending stress, Fb, is taken to be equal to Fy (i.e., the yield
stress). In order to use AASHTO Equation (10-47) to evaluate the arch rib capacity,
the web plates, stiffener angles, and flange plates of the arch rib must all satisfy the
slenderness checks given in AASHTO Article 10.37. The slenderness limits are
computed based on the axial and bending stress in the arch rib (under dead load and
Nguyenngoc Tuyen
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104
live load plus impact). Appendix A5 shows that all the slenderness requirements were
satisfied for the five different rating vehicles.
Based on the discussion given above, AASHTO Equation (10-47) may be
rewritten in terms of the dead load and live load (plus impact) effects as follows
( )
1SF
AFTT18.11
1MM
AF
NN
b
e
LDLD
a
LD ≤⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−
+
++
For solid rib arches evaluated by the Load Factor Method, AASHTO Article
10.55 specifies the same load factors as the Allowable Stress Method (i.e., A1 = A2 =
1) at the inventory rating level. Similar to the beam-column analysis of the Omega
Bridge columns (see Section 5.2.1), the left side of AASHTO Equation (10-47) given
above represents the interaction ratio at the inventory rating level, IRi. The arch rib is
shown to have adequate capacity when the interaction ratio is less than unity. The
inventory rating factor, RFi, may then be determined by solving the multiple of live
load effects which sets the interaction ratio (at the inventory level) equal to 1 as
shown below.
( )
( ) ( )( )
1SF
AFRFTT18.11
1RFMM
AF
RFNN
b
e
iLDiLD
a
iLD =⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−
+
++
To determine the operating rating factor, RFo, the inventory rating factor is
simply multiplied by a factor 1.67; hence, RFi = RFo / 1.67 and the equation becomes
Nguyenngoc Tuyen
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105
( )
( ) ( )( )
1SF
AF67.1/RFTT18.1
1
167.1/RFMM
AF
67.1/RFNN
b
e
oLDoLD
a
oLD =⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−
+
++
which may be resolved for RFo.
Structural analysis results showed that the maximum axial force occurs at the
supports and the maximum bending moment occurs at the quarter points of the arch
rib. However, the location of the rating vehicle that produces the maximum axial
force in the arch rib does not coincide with the location that produces the maximum
bending moment. Thus, four separate loading cases (designated Case 1 through Case
4) for each rating vehicle were considered in both the RIGID and PINNED rating
models as described below.
Case 1: Nmax @ Point A, Mmax @ Point C2, T @ Point E
Case 2: Nmax @ Point B, Mmax @ Point D2, T @ Point F
Case 3: Mmax @ Point C1, Nmax @ Point A, T @ Point E
Case 4: Mmax @ Point D1, Nmax @ Point B, T @ Point F
Cases 1 and 2 maximized the axial forces in the arch rib (on the south and
north end, respectively) while Cases 3 and 4 maximized the bending moments (on the
south and north half, respectively). The locations of points A, B, C1, C2, D1, D2, E,
and F on the arch rib mentioned above are shown in Figure 6.3.
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106
SOUTH
M25
M22
A
M23
M24C1
M28
M26M27
M29 M30
M33
M36
B
D1 M35
M34
M31
NORTH
C2 D2
M32
C1 : j end of element M24
C2 : j end of element M27
D2 : i end of element M31
D1 : i end of element M34
E : j end of element M25
F : i end of element M33
A : i end of element M22 (the south support of the arch)
B : j end of element M36 (the north support of the arch).
NOTES:
FE
Figure 6.3 Critical locations of axial force and bending moment of arch rib.
Table 6.2 reports the inventory interaction ratio and the rating factors
(inventory and operating) for the arch rib. Values are given for the RIGID and
PINNED rating models of the arch rib under the different rating vehicles.
6.3. Discussion of Rating Factors
As shown in Table 6.2, the rating factors for the RIGID model were larger in
magnitude than those for the PINNED model. These results indicate that the arch rib
has a lower capacity rating when the riveted connections are modeled as pinned rather
than rigid, particularly for Case 3 and 4 which maximized the arch rib bending
moments. This makes sense since the bending moments in the arch rib will decrease
(thus, increasing the capacity rating) if bending moments are also carried by the
bridge columns. Note that this is contrary to the column rating factors which showed
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107
smaller magnitudes when rigid connections were used and beam-column behavior
was considered.
Table 6.2 Interaction ratio and rating factors for arch rib based on AASHTO Equation (10-47).
RIGID Model PINNED Model Rating Vehicle Case
IRi RFi RFo IRi RFi RFo
1 0.56 2.53 4.23 0.57 2.41 4.02
2 0.55 2.60 4.34 0.57 2.36 3.94
3 0.63 2.14 3.57 0.69 1.77 2.96 HS20
4 0.61 2.20 3.67 0.68 1.81 3.02
1 0.48 3.58 5.98 0.48 3.47 5.79
2 0.47 3.73 6.23 0.48 3.48 5.81
3 0.53 3.05 5.10 0.58 2.53 4.23 TYPE 3
4 0.52 3.13 5.23 0.56 2.58 4.31
1 0.53 2.81 4.69 0.56 2.35 3.92
2 0.52 2.88 4.81 0.55 2.36 3.95
3 0.60 2.32 3.87 0.66 1.80 3.01 TYPE 3S2
4 0.59 2.38 3.98 0.65 1.84 3.07
1 0.54 2.68 4.48 0.58 2.54 4.24
2 0.51 2.96 4.95 0.57 2.56 4.27
3 0.62 2.21 3.68 0.69 1.91 3.19 TYPE 3-3
4 0.60 2.26 3.78 0.67 1.95 3.25
1 0.56 2.49 4.17 0.57 2.39 3.99
2 0.55 2.56 4.28 0.57 2.40 4.01
3 0.63 2.12 3.54 0.70 1.76 2.93 FIRE
4 0.62 2.18 3.63 0.68 1.79 2.99
It is observed that the development of end moments in the bridge columns (at
the riveted connections) helps the capacity rating of the arch rib but inhibits the
capacity rating of the bridge columns. No significant difference was observed in the
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108
axial forces in the arch rib between the PINNED and RIGID models. In addition, the
rating factors were about the same on the north and south half of the arch rib due to
symmetry (i.e., Case 1 agreed with Case 2 and Case 3 agreed with Case 4). Only
slight differences occurred due to the incline of the roadway. Of the four cases, Case
3 had the lowest rating factors (and largest interaction ratios) and thus, controlled the
capacity of the arch rib. This was true for the two rating models of the arch rib and
the five different rating vehicles. The Case 3 results for both the RIGID and PINNED
model are repeated in Table 6.3.
Table 6.3 Interaction ratio and rating factors for arch rib based on AASHTO Equation (10-47) for Case 3.
RIGID Model PINNED Model Rating Vehicle IRi RFi RFo IRi RFi RFo
HS20 0.63 2.14 3.57 0.69 1.77 2.96
TYPE 3 0.53 3.05 5.10 0.58 2.53 4.23
TYPE 3S2 0.60 2.32 3.87 0.66 1.80 3.01
TYPE 3-3 0.62 2.21 3.68 0.69 1.91 3.19
FIRE 0.63 2.12 3.54 0.70 1.76 2.93
As mentioned before, the PINNED model has the smaller capacity ratings
since the riveted connections were assumed to be pinned; this model underestimates
the true capacity of the arch rib. On the other hand, the RIGID model is based on
rigid connection behavior which results in an overestimate of the actual arch rib
capacity for reasons discussed earlier. Similar to the bridge columns, the actual
capacity ratings of the arch rib will be somewhere between the values for the pinned-
connection and rigid-connection models. A field test could aid in obtaining a better
Nguyenngoc Tuyen
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109
estimate of the arch rib capacity. Nevertheless, the inventory rating values given in
Table 6.3 are all greater than 1.75 (i.e., IRi > 1.75) which shows that the arch rib has
substantial capacity to indefinitely carry the five rating vehicles. These results are
quite comforting considering the arch rib is a fracture critical element.
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Chapter 7
SUMMARY AND CONCLUSIONS
7.1. Summary
7.1.1. Floor System
From the capacity evaluation of the floor system reported in Chapter 4, the
smallest rating factors calculated for the stringers (see Table 4.3), floor beams (see
Table 4.5), and spandrel beams (see Table 4.8) are repeated in Table 7.1. Figure 7.1
shows the critical locations of the floor system components. As shown in the figure,
the critical section for the stringers is at the negative moment region (i.e., above floor
beam FB#2). The two critical floor beams are FB#2 (located one bay from the
abutment on the approach spans) and FB#6 (located above arch column #4 on the
arch spans). There are two critical sections for the east spandrel beam; the negative
moment region located above pier column #1 closest to the abutment and the positive
moment region located at mid-span of the third approach span (i.e., at floor beam
FB#3). Of the three floor system components, the floor beam controlled the capacity
rating. Although the inventory rating of the floor beam at inventory level for the
design load is smaller than one (i.e., RFi = 0.85 for the HS-20 Truck), all the rating
factors under legal loads are larger than one; thus, load posting of the Omega Bridge
is not required based on the floor system capacity. Also note that the operating rating
for the FIRE truck is larger than one and the smallest inventory rating is 0.88. This
indicates that the floor system is safe for passing of the emergency vehicle.
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Table 7.1 Controlling rating factors of the floor system.
Design Load Legal Load Permit Load
HS-20 Truck TYPE 3, TYPE 3S2 or TYPE 3-3 Truck FIRE Truck
Floor System Component
RFi RFo RFi RFo RFi RFo Stringer 1.09 1.81 1.03 1.72 0.97 1.62
Floor Beam 0.85 1.14 1.13 1.89 0.88 1.47
Spandrel Beam 1.15 1.93 1.34 2.23 1.18 1.96
31' 31' 31' 31' 31' 31'
FB#1
FB#2
FB#2
FB#2
FB#2
FB#3
FB#4
Skewback Col #1LCPier Col #2CLPier Col #1LCCL
South
Skewback Col #1
29' - 6"
LC
FB#4
29' - 6"
FB#5
FB#5
Floor Beam
29' - 6"
FB#6
FB#6
FB#5
29' - 6"29' - 6"
Spandrel Beams
North
Critical sections for interior stringers
Critical section for spandrel beams
(under HS-20and TYPE3 Trucks)
Critical section for spandrel beams
(under TYPE 3S2, TYPE 3-3and FIRE Trucks)
North
South
Critical floor beam FB#2
Critical floor beam FB#6
Bearing Abutment #1
Arch Col #4CL
(a)
(b)
Figure 7.1 Critical locations of the floor system: (a) approach spans and (b) arch spans.
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112
7.1.2. Columns
As discussed in Chapter 5, because of the uncertainty of the rotational
connection stiffness at the column ends, two separate models were developed to
obtain the rating factors for the columns. The BEAM-COLUMN model represented
the most conservative case for the columns (i.e., small rating factors) while the
COLUMN model represented the least conservative case (i.e., large rating factors).
The lowest rating factors from Tables 5.2 through 5.6 in Chapter 5 for the BEAM-
COLUMN model are repeated in Table 7.2. Figure 7.2 shows the locations of the
critical columns (i.e., those having rating values smaller than unity at the inventory
level) which included arch columns #3 – #6 and #9 – #12. Of these eight columns,
arch column #10 had the smallest rating capacity rating. Thus, the rating factors for
this critical column controlled the arch column capacity and are the ones reported in
Table 7.2. As mentioned earlier, the rating factors produced from the BEAM-
COLUMN model represent a lower bound of the column capacity; however, in the
interest of safety it is recommended that they be used until the connection stiffness
can be more accurately estimated. Based on the rating factors in Table 7.2, all the pier
and skewback columns are satisfactory since their rating factors are larger than one at
the inventory and operating levels. The rating factors for the arch columns at
inventory level under design, legal and permit loads are all less than one. However,
they are larger than one at the operating level for all the five rating vehicles and thus,
no load posting is required. However, more frequent inspection of the columns than
the two-year interval may be warranted as well as traffic monitoring for overloads.
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Table 7.2 Controlling rating factors of the columns based on BEAM-COLUMN model.
Design Load Legal Load Permit Load
HS-20 Truck TYPE 3, TYPE 3S2 or TYPE 3-3 Truck FIRE Truck Column
RFi RFo RFi RFo RFi RFo
Pier Column 1.18 1.97 1.13 1.89 1.16 1.94
Arch Column 0.76 1.27 0.81 1.36 0.76 1.26
Skewback Column 2.63 4.39 3.1 5.18 2.61 4.35
Pier column #1
Skewback column #1 Skewback column #2
Pier column #2
Pier column #3
Pier column #4
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Arch column #
Controlled
Figure 7.2 Critical locations of the columns.
7.1.3. Arch rib
Table 7.3 shows the lowest rating factors for the arch rib taken from Table
6.3. Unlike the rating factors of the columns, the rating factors of the arch rib were
not much different between the two separate connection models, RIGID and
PINNED. Recall that the RIGID model assumed that the riveted connections at the
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column ends are completely rigid (similar to the BEAM-COLUMN model for the
columns) and thus, provide full moment transfer. The PINNED model, on the other
hand, assumed pinned connections (similar to the COLUMN model for the columns)
and no moment transfer. As shown in Table 7.3, all the rating factors for the arch rib
were found to be larger than one at both the inventory and operating level, and thus
the arch rib capacity is satisfactory.
Table 7.3 Controlling rating factors of the arch rib.
Design Load Legal Load Permit Load
HS-20 Truck TYPE 3, TYPE 3S2 or TYPE 3-3 Truck FIRE Truck Model
RFi RFo RFi RFo RFi RFo
Pinned-connection 1.77 2.96 1.80 3.01 1.75 2.93
Rigid-connection 2.14 3.57 2.21 3.68 2.12 3.54
7.2. Conclusions
In general, the Omega Bridge is in satisfactory condition and no load posting
is necessary but there are some concerns for the floor beams and the arch columns.
Based on the lowest rating factors provided in this report (i.e., RFi = 0.76 and RFo =
1.26 for HS-20 and FIRE trucks), the arch columns are most critical and therefore,
control the capacity of the Omega Bridge. However, if a better estimate of the
rotational stiffness of the riveted connections is obtained, the arch columns may no
longer be the critical components since the rating factors are inversely proportional to
the column moments (i.e., connection stiffness ↓ rating factor ↑). In such a case, the
floor beam capacity may become more critical than that of the columns. Future
evaluation of the Omega Bridge under other vehicles should be based on the column
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115
rating values determined under legal loads (i.e., RFi = 0.81 and RFo = 1.36) until
further study is carried out to improve the rating factors.
Recall that the floor beam dimensions did not satisfy the AASHTO (2002)
compactness requirements and thus, it is recommended that the floor beams be
inspected thoroughly for signs of instability. As discussed in Chapter 5, leaning of the
columns may also reduce their capacity; thus, monitoring of the column out-of-plumb
(both magnitude and direction) is suggested in future capacity rating of the bridge.
Load testing along with 3-D finite element analysis is also recommended to refine the
calculation of the rating factors. From a load test, the rotational stiffness of the
column-end connections and the load distribution to each bridge component could be
determined more accurately and thus, provide a better evaluation of the structure.
Using the actual stiffness of the riveted connections and the load distribution found
from field testing, a 3-D finite element model can be developed and calibrated to the
test data. Recall that the column rating factors, which controlled the bridge capacity,
increase if the connection stiffness at the column ends decreases. Conversely, the
rating factors of the spandrel beams and arch ribs will decrease. However, the
spandrel beam and arch rib rating factors will converge to the rating values for the
case of pinned connections. Hence, a better estimate of the connection stiffness will
not change the final rating factors of the other components (i.e., stringers, floor
beams, spandrel beams, arch ribs) since they were conservatively calculated. Finally,
a calibrated finite element model can also be used to evaluate the effects of other
loads (e.g., temperature, settlement, seismic) on the Omega Bridge capacity.
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APPENDIX
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The appendix shown in this section only represents the calculations for the
most critical load case for the bridge members under HS-20 vehicular live load. The
reader is referred to the CD (included in this thesis) for all the detailed calculations
under the five rating vehicles. The appendix section provided hereafter includes:
Appendix A1 (RATING STRINGERS): rating of interior stringer; Appendix A2
(RATING FLOOR BEAMS): rating of floor beam FB#6; Appendix A3 (RATING
SPANDREL BEAMS): rating of the eastern spandrel beam; Appendix A4 (RATING
COLUMNS): rating of the pier, skewback, and arch columns; and finally Appendix
A5 (RATING ARCH RIB): rating of the eastern arch rib.
117
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APPENDIX A1
RATING STRINGERS
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
(distance between stringers)Ss 6.75
(span length)ftLe 31
ksiEs 29000
ksiFy 33
ksiEc 2.91 103Ec 33000 0.1201.5 f'c
ksif'c 4.5
intws 0.5
ints 6.75
Initial Data
119
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Effective flange width: (AASHTO Article 10.38.3)
0.25 of the span length:
b1 31 12( ) 0.25 or b1 93 in
12 times the thickness of the slab:
b2 12 ts or b2 81 in
the average spacing of adjacent beams:
b3 6 12 9 or b3 81 in
the effective flange width:
beff min b1 b2 b3
beff 81 in( )
1. Rating for Positive Moment (Section #1)
1.1. Non-composite Section
21
0.41 10.5
N.A.
78.25
21.75
10.875
7x3/8" cover plate
0.62
d 21.75 in( )
tw 0.41 in( )
bf 8.25 in( )
tf 0.62 in( )
As 18.52 5.25
As 23.77 in2
Ix 1343.6 5.25 10.68752
Ix 1.943 103 in4
StIx
0.5d
St 178.691 in3
1.2. Composite Section
120
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
in( )
yct d ycb or yct 0.933 in( )
Cross-section Properties
Ic Ix As ycbd2
2Acs d
ts2
ycb
2 bcs ts3
12
Ic 5.519 103 in4
SctIcyct
or Sct 5.918 103 in3
ScbIc
ycbor Scb 265.129 in3
81
21.75
10.875
8.25
7
0.41
6.75
20.82
NA
Modular ratio (n)
nEsEc
or n 9.966
Transformed Flange width and Flange area:
bcsbeff
nor bcs 8.128 in( )
Acs ts bcs or Acs 54.863 in2
Location of the Neutral Axis of the composite section:
ycb
Acs dts2
Asd2
Acs Asor ycb 20.817
121
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
DF 1.227soDFSs5.5
Distribution factor (AASHTO Article 3.23.2.2)
I 0.3soI min I 0.3( )
I50
Le 125
Impact factor (AASHTO Article 3.8.2.1)
Live Load: (rating for HS20)
( lbs / ft )wD 555.525orwD wstri wslab wws
Total:
( lbs / ft )
1.3. Loads and Factors
Dead loads for non-composite section:
Stringer :
wstri 1.05 63( ) or wstri 66.15 ( lbs / ft )
(add 5% to account for connection weight)
Slab :
wslabbeff12
ts12
120( ) or wslab 455.625 ( lbs / ft )
Integral Wearing Surface:
wwsbeff12
tws12
120( ) or wws 33.75
122
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
the plastic neutral axis is located in the slab.C C1Since
kips( )C 784.41soC min C1 C2
kips( )C2 784.41soC2 As Fy
kips( )C1 2.091 103soC1 0.85 f'c beff ts
Determine the compression capacity of the slabStep 1:
(AASHTO Article 10.50.1)1.5. Capacity of Section
ft.kipsML 190.019
ML 1 I( ) DF MHS20
ft.kipsMHS20 119.1
Live load
ft.kipsMD 41.6
Moment diagram due to dead load
Dead load
1.4. Unfactored Moments due to Dead Load and Live Load
123
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
ft-kips
MR Mn (if the cross section is compact)
1.6. Slenderness Checks (AASHTO Article 10.50.1.1.2.)
Checking the web compactness:
cp
w y
2D 19,230t F
Eq. 10-129
Since the plastic neutral axis is located above the web (all of the web is on the tension side),the web is compact and Eq. 10-129 is satisfied.
Checking the distance from the top of the slab to the plastic neutral axis:
pD5
D'Eq. 10-129a
where:
Dpa
0.82or Dp 3.088 in
Step 2: Compression stress block (Eq: 10-125)
aC
0.85 f'c beffso a 2.532 in( )
21.75
8.25
7
10.875
0.41
16.36
81
2.536.75
Step 3: Nominal Capacity
armd2
tsa2
or arm 16.359 in( )
Mn Carm12
or Mn 1.069 103
124
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
RFo 4.11orRFoMR A1 MD
A2 ML
A2 1.3
A1 1.3
Operating Level
RFi 2.462orRFiMR A1 MD
A2 ML
A2 2.17
A1 1.3
Inventory Level
1.7. Load Factor Rating for Section #1
kip-ftMR 1.069 103
as calculated in part 1.5.
MRthe capacity of the cross section isin( )D' 3.42<in( )Dp 3.088Since
=> OK5<DpD'
0.903
inD' 3.42orD' 0.9d ts
7.5
125
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
in4
StIx
0.5dor St 127.962 in3
SbIx
0.5dor Sb 127.962 in3
Z bf tfd2
tf2
twd2
tf
2 12
2 or Z 144.266 in3
2.2. Unfactored Moments due to Dead Load and Live Load
Dead load
Moment diagram due to dead load
MD 56.3 ft.kips
2. Rating for Negative Moment (Section #2)
2.1. Non-composite Section
0.620.41
21
8.25
NA
10.5
d 21 in( )
tw 0.41 in( )
bf 8.25 in( )
tf 0.62 in( )
As 18.52 in2
The location of the neutral axis:
ynad2
yna 10.5 in
Ix 1343.6
Ix 1.344 103
126
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
(Eq 10-93)=> OK4110
1000 Fy22.625
bftf
13.306
Checking flange compactness:
AASHTO Article 10.48.1.12.4. Slenderness Checks
(if the section is compact)MR Mn
ft-kipsMn 400.808orMnAs2
Fyarm12
The nominal capacity of the section
in
Live load
MHS20 87.1 ft.kips
ML 1 I( ) DF MHS20
ML 138.964 ft.kips
2.3. Capacity of Section
The height of the web:
dw d 2.tf or dw 19.76 in
Distance between the center of gravity of tension part and compression part:
arm 2
dw2
twdw4
tf bfdw2
tf2
dw2
tw tf bf
or arm 15.74
127
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Since all of the eqs are satisfied, the section is compact. Use the capacity as caculated in part 2.3.
=> OK3.6( ) 106
1000Fy109.091 <
Lbry
40.862
= 0 (M1 is the smaller moment value, which is equal to zero)M1Mu
(6 ft from the center of the support)Lb 6( ) 12
= the lateral bracing of the compression flange, but due to the sharp drop in negative moment Lb was taken as the distance from the center of the support to the point of zero moment.
Lb
inry 1.762orry57.5As
(Eq 10-96)
Checking web compactness:
D d 2tf or D 19.76 in (Clear distance between flanges)
Dtw
48.19519230
1000 Fy105.858 => OK (Eq 10-94)
Dtw
48.19575100
19230
1000 Fy79.393 => OK, no need to check Eq 10-95
Check the spacing of lateral bracing for compression flange:
613.6 2.2 / 10ub
y y
M MLr F
128
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
2.5. Load Factor Rating for Section #2
Inventory Level
A1 1.3
A2 2.17
RFiMR A1 MD
A2 MLor RFi 1.086
Operating Level
A1 1.3
A2 1.3
RFoMR A1 MD
A2 MLor RFo 1.814
129
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NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
in4
StIx
0.5dor St 127.962 in3
SbIx
0.5dor Sb 127.962 in3
3.2. Unfactored Moments due to Dead Load and Live Load
Dead load
MD 23.4 ft.kips
Live load
MHS20 92.3 ft.kips
3. Rating for Positive Moment (Section #3)
3.1. Non-composite Section
0.620.41
21
8.25
NA
10.5
d 21 in( )
tw 0.41 in( )
bf 8.25 in( )
tf 0.62 in( )
As 18.52 in2
The location of the neutral axis:
ynad2
yna 10.5 in
Ix 1343.6
130
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
MR Mn (if the section is compact)
3.4. Slenderness Checks AASHTO Article 10.48.1.1
No need to check the compression flange compactness (Eq 10-93) since there is a concrete slab above the compression flange and therefore, the compression flange is compact.
Checking web compactness:
D d 2tf or D 19.76 in (Clear distance between flanges)
Dtw
48.19519230
1000 Fy105.858 => OK (Eq 10-94)
No need to check (Eq 10-96) since there is a concrete slab above the compression flange and therefore, the compression flange is fully braced.
Since 10-94 is satisfied, the section is compact. Use the capacity as calculated in part 3.3.
ML 1 I( ) DF MHS20
ML 147.26 ft.kips
3.3. Capacity of Section
The height of the wing:
dw d 2.tf or dw 19.76 in
Distance between the center of gravity of tension part and compression part:
arm 2
dw2
twdw4
tf bfdw2
tf2
dw2
tw tf bf
or arm 15.74 in
The nominal capacity of the section
MnAs2
Fyarm12
or Mn 400.808 ft-kips
131
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING INTERIOR STRINGERS
3.5. Load Factor Rating for Section #3
Inventory Level
A1 1.3
A2 2.17
RFiMR A1 MD
A2 MLor RFi 1.159
Operating Level
A1 1.3
A2 1.3
RFoMR A1 MD
A2 MLor RFo 1.935
132
Nguyenngoc Tuyen
NMSU
APPENDIX A2
RATING FLOOR BEAMS
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
in3S 796.736orSI
0.5 48.5
The section modulus with respect to the top or bottom flange:
in4I 1.932 104orI 4 7.6148.5
21.49
224.1
112
38
483
The moment of inertia:
in2As 48.44orAs 4 7.61 4838
The area of the cross-section:
48.5"
L 8" x 6" x 916"
PL 48" x 3 8" x 32'-9"
ksiFy 33The yield stress of the steel:
Initial Data
Floor Beam FB#6
134
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
Pwheel 4 16 16( ) 0.833 or Pwheel 29.988 kips
where: 0.833 is the longitudinal distribution factor
Therefore:
Concentrated load at the location of the first stringer:
PHS20_1 40.7 kips
Concentrated load at the location of the second stringer:
PHS20_2 40.9 kips
Concentrated load at the location of the third stringer:
PHS20_3 40.9 kips
Concentrated load at the location of the fourth stringer:
PHS20_4 40.7 kips
Impact Factor (AASHTO Article 3.8.2.1)
I50
35 125I min I 0.3( ) so I 0.3
1. Loads and Factors
Dead loads :
Selfweight :
wsw 1.05As144
490 or wsw 173.072 ( lbs / ft )
(add 5% to account for connection weight)
Concentrated loads at the location of the interior stringers:
Pi_st556 29.5( )
1000or Pi_st 16.402 kips
Concentrated loads at the location of the west exterior stringer:
Pe_st_l564 1.04 59.44 42.46 30.4( ) 1.4[ ] 29.5( )
1000or Pe_st_l 22.321 kips
Concentrated loads at the location of the right exterior stringer:
Pe_st_r564 1.04 30.14 13.98 30.4( ) 1.4[ ] 29.5( )
1000or Pe_st_r 19.839 kips
Live Load: (rating for HS20)
The concentrated load on the slab:
135
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
Multiple Presence Factor
For 3 lanes of HS20
m 0.9
2. Unfactored Moments due to Dead Load and Live Load
Dead Load:
109 k-ft
155 k-ft-22.3 k-16.4 k
235 k-ft232 k-ft
119 k-ft
-16.4 k-16.4 k -16.4 k -19.8k138 k-ft
MD 235 ft-kips
Live Load:
6'-9"
Floor beam
6'-9"7'-4.5" 6'-9"
40.7k 40.9k 40.9k
7'-4.5"
Interior Stringer
Spandrel Beam
40.7k
6'-0"
6'-9"35'-0"
6'-9"3'-6"
Slab
6'-9"7'-4.5" 6'-9"
0.833x (36 kips) per force.
4'-0"6'-0"4'-9" 6'-0" 4'-0"
6'-9" 3'-6"7'-4.5"
Exterior Stringer
4'-9"
136
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
Equation 10-100 is not satisfied. This implies that the 2002 AASHTO Standard Specification recognizes the flange as a slender element. Therefore, the section was not evaluated using the 2002 AASHTO Standard Specification.
(Eq 10-100) => NG2 8 0.375 29.1 240.563
f
f
bt
Compression flange:
If the beam meets the requirements in Article 10.48.2. However,
(Eq 10-99)or R cr xcM =F S bR
(Eq 10-98)R y xtM =F S
According to AAHSTO Article 10.48.2, the maximum strength shall be computed as:
2002 AASHTO Article 10.48.2For braced, non-compact section:
=> This is not a compact section.
4110
1000Fy22.625
878 k-ft609 k-ft
878 k-ft609 k-ft
-40.9 k-40.7 k -40.9 k -40.7 k
MHS20 878 kip-ft
ML MHS20 1 I( ) m( ) or ML 1.027 103 kip-ft
3. Capacity of Section
For braced, compact section: 2002 AASHTO Article 10.48.1
R u yM M F Z (Eq 10-92)
Check compression flange compactness:
4110f
f y
bt F
(Eq 10-93)
2 8 0.375( )0.563
29.085 >
137
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
rM is the square root of the ratio between the yielding moment and the maximum moment of the cross section.
For inventory level:
Mm 1.3 MD 2.17 ML or Mm 2.535 103 kip-ft
MyFy S
12or My 2.191 103 kip-ft
rMMyMm
or rM 0.93 use: rM 1
8 0.563( )0.563
13.21 > 2200
1000FyrM 12.111
This section does not satisfy the 1996 AASHTO Standard Specification requirements at the inventory level.
Using the 1996 AASHTO Standard Specification:
For braced, compact section: 1996 AASHTO Article 10.48.1
R u yM M F Z (Eq 10-92)
Check projecting compression flange:
' 2,055
y
bt F
(Eq 10-93)
8 0.563( )0.563
13.21 > 2055
1000Fy11.312
=> This is not a compact section; refer to Article 10.48.2.
For braced, non-compact section: 1996 AASHTO Article 10.48.2
R u yM M F S (Eq 10-98)
Check projecting compression flange:
' 2, 200M
y
b rt F
(Eq 10-99)
where:
138
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
(Eq 10-100)
0.5 48.50.375
64.667 < 15400
1000Fy84.774
Check spacing of lateral bracing for compression flange:
20,000,000 fb
y
AL
F d(Eq 10-101)
Lb 88.5 in < 20000000 16 0.625( )1000 Fy 48.5
124.961 in
=> This is a braced non-compact section. Use Eq (10-98)
MR Fy S112
or MR 2.191 103 kip-ft
For operating level:
Mm 1.3 MD 1.3 ML or Mm 1.641 103 kip-ft
MyFy S
12or My 2.191 103 kip-ft
rMMyMm
or rM 1.156
8 0.563( )0.563
13.21 < 2200
1000FyrM 13.994
This section does satisfy the 1996 AASHTO Standard Specification requirements at the operating level.
Check web thickness:
15400c
w y
Dt F
139
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING FLOOR BEAM (#6)
4. Load Factor Rating (According to 1996 AASHTO Standard Specification)
Inventory Level
A1 1.3
A2 2.17
RFiMR A1 MD
A2 MLor RFi 0.846
Operating Level
A1 1.3
A2 1.3
RFoMR A1 MD
A2 MLor RFo 1.412
140
Nguyenngoc Tuyen
NMSU
APPENDIX A3
RATING SPANDREL BEAMS
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
Section #1: Positive moment, composite section
Section #2: Negative moment, composite section
Section #3: Positive moment, non-composite section
Abutment Pier Col #1 Pier Col #2 Skewback Col #1
Section #4: Negative moment, non-composite section
HS-20, Type 3, and Fire Trucks
Section #4: Negative moment, non-composite section
Type 3S2 and Type 3-3 Trucks
Section #1: Positive moment, composite section
Abutment
Section #3: Positive moment, non-composite section
Section #2: Negative moment, composite section
Pier Col #1 Pier Col #2 Skewback Col #1
Critical Sections:
ksiEs 29000The elastic modulus of steel:
ksiFy 33The yield stress of the steel:
ksi( )Ec 2.91 103Ec 33000 0.1201.5 f'c0.5
ksif'c 4.5The compressive strength of the concrete:
intws 0.5The thickness of the integral wearing surface:
intha 2.875The thickness of the haunch:
ints 6.75The thickness of the deck:
Initial Data
142
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
in2As 84.575orAs Atopplate Atopangle Awebplate Abotangle
in2Abotangle 19.98orAbotangle 2 9.99( )
in2Awebplate 49.5orAwebplate 2 6638
in2Atopangle 5.72orAtopangle 2 2.86( )
in2Atopplate 9.375orAtopplate 25 0.375
25" x 3/8" top plate
4" x 4" x 3 8" L
66" x 3 8" web plate
8" x 6" x 3 4" L
d=66.875
31.789
25
1.1. Non-composite Section
1. Rating for Positive Moment (Section #1)
143
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
in
The moment of inertial:
I1 Atopplate y1 yb2
I2 Atopangle y2 yb2 2 4.32( )
I3 Awebplate y3 yb2 2 0.375 663
12
I4 Abotangle y4 yb2 2 30.8( )
Ix I1 I2 I3 I4 or Ix 5.43 104 in4
The section modulus of the top fiber:
StIx
d ybor St 1.548 103 in3
The section modulus of the bottom fiber:
SbIxyb
or Sb 1.708 103 in3
d 66.875 in
y1 d316
or y1 66.688 in
y2 d38
1.13832 or y2 65.362 in
y3 d38
33 or y3 33.5 in
y4 1.56368 in
The location of the neutral axis:
ybAtopplate y1 Atopangle y2 Awebplate y3 Abotangle y4
As
or yb 31.789
144
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
49.083
d=66.875
81
69.75
6.75
2.875
73.12568.313
7471.75
in( )beff 81
beff min b1 b2 b3
the effective flange width:
inb3 420orb3 88.5 2 81 3
the average spacing of adjacent beams:
inb2 81orb2 12 ts
12 times the thickness of the slab
inb1 186orb1 62 12( ) 0.25
0.25 of the span length:
Effective flange width:
1.2. Composite Section
145
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
Atrbeff ts
nor Atr 54.863 in2
Transformed haunch width area:
Transfromed haunch width: bha25n
or bha 2.509 in
Transformed haunch area: Ahatha 25
nor Aha 7.212 in2
Locate the neutral axis of the composite section:
Distance from the center of gravity of the slab to the bottom of the beam:
ds d thats2
or ds 73.125 in
Distance from the ceter of gravity of the haunch to the bottom of the beam:
dha dtha2
or dha 68.313 in
The total area of the composite section:
Ac As Atr Aha or Ac 146.651 in2
The area of the #3 reinforcement bars in the slab:
Arb3 9 0.11( ) or Arb3 0.99 in2
The location of the #3 reinforcement bars from the bottom of the spandrel beam:
drb3 d tha 4.25 or drb3 74 in
The area of the #4 reinforcement bars in the slab:
Arb4 8 0.2( ) or Arb4 1.6 in2
The location of the #4 reinforcement bars from the bottom of the spandrel beam:
drb4 d tha 2 or drb4 71.75 in
Transformed flange width and area:
Modular ratio: nEsEc
or n 9.966
Transformed flange width: btrbeff
nor btr 8.128 in
Transformed flange area:
146
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
The section modulus of the bottom fiber:
SbcIc
ybcor Sbc 2.328 103 in3
1.3. Loads and Factors
Dead loads for non-composite section:
Selfweight : wspandrel 1.05As144
490 or wspandrel 302.179 lbs / ft
Slab : wslab 1.0255.5
2
ts12
120( ) or wslab 1.911 103 lbs / ft
Integral Wearing Surface:
wws55.5
2
tws12
120( ) or wws 138.75 lbs / ft
The location of the neutral axis from the bottom of the spandrel beam:
ybcAs yb Atr ds Aha dha
Acor ybc 49.049 in
The location of the neutral axis from the top of the spandrel beam:
ytc d ybc or ytc 17.826 in
The moment of inertial of the composite section:
I1 Atr ds ybc2 btr ts
3
12
I2 Aha dha ybc2 bha tha
3
12
I3 As yb ybc2 Ix
Ic I1 I2 I3 or Ic 1.142 105 in4
The section modulus of the top fiber:
StcIc
d ybcor Stc 6.406 103 in3
147
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
Utilities : wutility 1.04 238.69 1.279 13.98( )
Total : wD2 wbarriers wutility or wD2 306.924 lbs / ft
Fence : wfence 1.04 1.279 30.4( ) (distributed on center 150' of bridge)
wfence 40.437 lbs / ft
Note: 1.05 = the weight of steel members was increased by 5% to account for connection weight
1.04 = the weight of barriers, fencing, utilities was increased by 4% as recommended by LANL
1.02 = the weight of the slab was increased by 2% to account for the weight of the haunch
Live Load: (rating for HS20)
Impact factor (AASHTO Article 3.8.2.1)
I50
62 125
I min I 0.3( ) so I 0.267
Floor beams:
wfloorbeam
1.05 0.548.23 55.5
144490( )
29.5or wfloorbeam 162.099 lbs / ft
Stringers: wstringers 1.05 3 63( )[ ] or wstringers 198.45 lbs / ft
Wind Bracing: wbracing
1.05 34.313.4144
490
29.5or wbracing 55.667 lbs / ft
wD1 wspandrel wslab wws wfloorbeam wstringers wbracingTotal :
wD1 2.768 103 lbs / ft
Dead loads for composite section:
Barriers: wbarriers 1.04 1.279 30.14( )
148
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
mDF 2.218ormDF max 0.75 DF4 0.9DF3 DF2 DF1
(AASHTO Article 3.12.1)Distribution factor (accounting for multiple presence factor):
DF1 1.107orDF1 1.10714
Distribution factor for 1 lane:
DF2 1.929orDF2 1.10714 0.82143
Distribution factor for 2 lanes:
DF3 2.464orDF3 1.10714 0.82143 0.53571
Distribution factor for 3 lanes:
DF4 2.714orDF4 1.10714 0.82143 0.53571 0.25000
Distribution factor for 4 lanes:
0.53571
465.0
420.0
345.0
420.0
48.0
81.0
Level rule:
225.0
105.0
0.25
81.042.0 88.5 81.0
48.072.0 72.0
1.107140.82143
1
24.0
42.088.581.0 81.0
48.072.0 72.0
1.27857
72.0
120.0120.0120.0168.0
Lane 4Lane 2 Lane 3Lane 1
Distribution factor ( use level rule spandrel beam AASHTO Article 3.23.2.3)
149
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
C2 As Fy so C2 2.791 103 kips( )
C min C1 C2 so C 2.522 103 kips( )
Since C C2 the plastic neutral axis is located in the steel section.
Step 2: The depth of the compressive stress block: (Eq: 10-125)
aC 60 Arb3 Arb4
0.85 f'c beffso a 7.637 in( )
a 7.637 > ts 6.75 => T section behavior.
1.4. Unfactored Moments due to Dead Load and Live Load
Dead Load:
MD 929.6 kip-ft
Live Load: (LFD 3.12)
MHS20 664.3 k-ft
ML MHS20 mDF 1 I( ) or ML 1.867 103 kip-ft
1.5. Capacity of Section (AASHTO Article 10.50.1)
Step 1: Determine the compression capacity of the slab and the haunch
C1 0.85 f'c beff ts tha 25 60 Arb3 Arb4 so C1 2.522 103 kips( )
150
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
inytpna 0.163orytpnaC'
Atopplate Fy
38
The location of the plastic neutral axis within the steel section (measured from the top of the steel section) is determined as follows:
kipsAtopplate Fy 309.375C' < Since:
kipsC' 134.667orC'C2 C
2
The top portion of the steel section will be subjected to the compressive force C'
The location of the plastic neutral axis:Step 3:
d=66.875
2.875
66.712
PNA
81
6.75a=9.625
7471.75
in( )a 9.625ora Find a( )
C 0.85 f'c ts beff 25 0.85f'c a 25( ) 60 Arb3 Arb4=
Given
Finding the value of a:
151
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
is the depth of the web in compression at the plastic moment
Since the plastic neutral axis is located in the top flange Dcp < 0
=> all of the web is on the tension side, therefore the web is compact.
Equation 10-129a
5'
PDD
where: DP the distance from the top of the slab to the plastic neutral axis
DP ts tha ytpna or DP 9.788 in
D' 0.9d ts tha
7.5or D' 9.18 in
The location of the plastic neutral axis measured from the bottom of the steel section is:
ybpna d ytpna or ybpna 66.712 in
Step 4: Nominal Capacity
Mn 0.85 f'c ts beff 25 ytpna thats2
0.85 f'c a 25( ) ytpnaa2
Mn Mn 60 Arb3 drb3 d ytpna Arb4 drb4 d ytpna
Mn Mn C'ytpna
212
38
ytpna
225 Fy
Mn Mn ybpna y2 Atopangle ybpna y3 Awebplate ybpna y4 Abotangle Fy
Mn 1.123 105 kip-in
1.6. Slenderness Checks
Checking web compactness according to AASHTO Article 10.50.1.1.2.
Equation 10-129
cp
w y
2D 19,230t F
where : Dcp
152
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
1
2 2
.. .
remainingR D
L L
MM A MRFA M A M
as for negative moment composite region (Section #2).
1.7. Capacity of Section (based on AASHTO Article 10.50.c)
Note: The capacity mentioned in previous parts was calculated based on the assumption that all the deadload and live load acts on the composite section. In reality, some of the dead load acts on the non-composite section and some of the dead load plus the live load acts on the composite section. To simplify the calculations, the non-composite section was assumed to carry all the dead load and thus, the composite section carried all the live load.
The capacity minus 1.3 times the dead load effect:
Mremaining MR 1.3 MD
Since the section modulus with respect to the bottom fiber is much smaller than the section modulus with respect to the top fiber for the composite sections, the bottom fiber will yield first.
Non-composite Section: St 1.548 103 and Sb 1.708 103
Composite Section: Stc 6.406 103 and Sbc 2.328 103
Stress in the top flange due to 1.3 times the dead load effect:
fbot1.3 12 MD
Sbor fbot 8.49 ksi
Since D' < DP 9.788 < 5 D' 45.9 use Eq 10-129c to calculate the bending capacity
Mp Mn or Mp 1.123 105 kip-in
My Fy Sbc or My 7.683 104 kip-in
The bending capacity of the section:
MR112
5Mp 0.85My4
0.85My Mp4
DPD'
or MR 9.29 103 kip-ft
However, Moment capacity is controlled by yield moment since negative moment region is non-compact. Calculate rating factor using the equation:
153
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
RFo 1.959orRFoMremaining
A2 ML
1
2 2
.. .
remainingR Do
L L
MM A MRFA M A M
A2 1.3
A1 1.3
Operating Level
RFi 1.174orRFiMremaining
A2 ML
1
2 2
.. .
remainingR Di
L L
MM A MRFA M A M
A2 2.17
A1 1.3
Inventory Level
1.8. Load Factor Rating for Section #1
k-ftMremaining 4.755 103orMremainingMremaining
12
(k-in)Mremaining Fy fbot Sbc =>
remainingtop y
tc
Mf F
S
The capacity minus 1.3 times the deadload effect will be calculated based on the condition that the bottom fiber has just yielded (i.e., the yield moment).
154
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
ind 66.875
in2As 84.575orAs Atopplate Atopangle Awebplate Abotangle
in2Abotangle 19.98orAbotangle 2 9.99( )
in2Awebplate 49.5orAwebplate 2 6638
in2Atopangle 5.72orAtopangle 2 2.86( )
in2Atopplate 9.375orAtopplate 25 0.375
25" x 3/8" top plate
4" x 4" x 3 8" L
66" x 3 8" web plate
8" x 6" x 3 4" L
d=66.875
31.789
25
2.1. Non-composite Section
2. Rating for Negative Moment (Section #2)
155
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
I4 Abotangle y4 yb2 2 30.8( )
Ix I1 I2 I3 I4 or Ix 5.43 104 in4
The section modulus of the top fiber:
StIx
d ybor St 1.548 103 in3
The section modulus of the bottom fiber:
SbIxyb
or Sb 1.708 103 in3
2.2. Composite Section
Effective flange width:
0.25 of the span length:
b1 62 12( ) 0.25 or b1 186 in
y1 d316
or y1 66.688 in
y2 d38
1.13832 or y2 65.362 in
y3 d38
33 or y3 33.5 in
y4 1.56368 in
The location of the neutral axis:
ybAtopplate y1 Atopangle y2 Awebplate y3 Abotangle y4
As
or yb 31.789 in
The moment of inertia:
I1 Atopplate y1 yb2
I2 Atopangle y2 yb2 2 4.32( )
I3 Awebplate y3 yb2 2 0.375 663
12
156
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
indrb3 74ordrb3 d tha 4.25
The location of the #3 reinforcement bars from the bottom of the spandrel beam:
in2Arb3 0.99orArb3 9 0.11( )
The area of the #3 reinforcement bars in the slab:
81
69.75d=66.875
33.002
6.75
73.12571.75 74NA
in( )beff 81
beff min b1 b2 b3
the effective flange width:
b3 420orb3 2 88.5 3 81
the average spacing of adjacent beams:
inb2 81orb2 12 ts
12 times the thickness of the slab
157
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
ft-kipsMD 1223.2
Dead Load:
2.3. Unfactored Moments due to Dead Load and Live Load
in3Sbc 1.772 103orSbcIc
ycb
in3Stc 1.727 103orStcIc
d ycb
in4Ic 5.849 104
Ic Ix As ycb yb2 Arb4 ycb drb4
2 Arb3 ycb drb32
Cross-section Properties
inycb 33.002orycbAs yb Arb4 drb4 Arb3 drb3
As Arb4 Arb3
The location of the neutral axis from the bottom of the spandrel
indrb4 71.75ordrb4 d tha 2
The location of the #4 reinforcement bars from the bottom of the spandrel beam:
in2Arb4 1.6orArb4 8 0.2( )
The area of the #4 reinforcement bars in the slab:
158
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
Atotal 89.284 in2
The distance from the plastic neutral axis to the bottom of the spandrel beam:
ypna yb
Given
ypna 0.534
AbotangleAtotal
2=
ypna Find ypna or ypna 33.383 in
(Based on the condition that the tension area is equal to the compression area)
Step 2: Locate the centers of gravity of tension and compression parts:
Compression part:
d1
ypna 0.534
ypna 0.5
20.5 Abotangle y4
0.5 Atotal
or d1 10.059 in
Live Load:
MHS20 410.4 k-ft
ML MHS20 mDF 1 I( ) or ML 1.154 103 k-ft
2.4. Capacity of Section
Step 1: Determine the location of the plastic neutral axis
Since the reinforcement bars in the slab have a yield strength of 60 ksi (greater than that of the spandrel beam), the area of the reinforcement bars is multipled by the ratio of yield strengths.
ny60Fy
or ny 1.818
The total area of the steel cross-section:
Atotal As Arb3 Arb4 ny or
159
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
(if the section is compact)MR Mn
ft-kipsMn 5.864 103orMnAtotal
2Fy
arm12
in( )arm 47.766orarm d2 d1
Nominal CapacityStep 3:
d2=57.825
d1=10.059
PNA 71.75 74
33.383
ind2 57.825or
d2
ny Arb3 drb3 ny Arb4 drb4 Atopplate y1 Atopangle y2 yw234
yw22
ypna
0.5 Atotal
yw2 66.5 ypna
Tension part:
160
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
=>use: 2 Dcp instead of D (AASHTO Article 10.50.2.1)
Dcp ypna34
or Dcp 32.633 in (Depth of the web in compression)
tw38
in (thickness of the web)
2 Dcptw
174.04119230
1000 Fy105.858 => NG (Eq 10-94)
Since 10-94 is not satisfied, the section is non-compact. Refer to AASHTO Article 10.50.2.2
Checking equations in AASHTO Article 10.48.2.1:
Compression flange:
bt
10.667 24 => OK (Eq 10-100)
Web thickness (for girder with transverse stiffeners):
Dtw
175.33336500
1000Fy200.926 => OK (Eq 10-104)
2.5. Slenderness Checks AASHTO Article 10.50.2.1
Note: The cross-section of the spandrel beam was divided into two equal part and each part was considered as an independent beam for the compactness checks. The strength of each independent beam is thus half the strength of the whole cross-section.
Checking equations in AASSTO Article 10.48.1.1:
Checking flange compactness:
b 8 in (width of comp flange)
t34
in or t 0.75 in (thickness of comp flange)
bt
10.6674110
1000 Fy22.625 => OK (Eq 10-93)
Checking web compactness:
D d34
38
or D 65.75 in (Clear distance between flanges)
Since:
ypna 33.383 > D2
32.875
161
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
Fcr min1
10004400
tb
2Fy or Fcr 33 ksi
Rb = the flange stress reduction factor:
1 0.002 1.0c w cb
fc w b
D t DRA t f
fb = factored bending stress in the compression flange (not to exceed Fy)
min ,D D L Lb y
M Mf FS
or, to be conservative use: fb Fy
Dc = depth of the web in compression
Dc ycb34
or Dc 32.252 in
15400 since Dc < d 7.125( )2
37 in
Spacing of lateral bracing for compression flange:
Af 834
2 (area of the flange)
Lb 7 ft112
20000000 Af1000Fy d 7.125( )
8.19 => OK (Eq 10-101)
Af = the area of the two flanges since there are diaphragm plates between the two flanges and therefore, the two compression flanges buckle together.
Lb = the lateral bracing of the compression flange, but due to the sharp drop in negative moment Lb was taken as the distance from the center of the support to the point of zero moment.
The maximum strength shall be computed as the lesser of Eq 10-98 and Eq 10-99:
First yield in tension fiber:
MR1 Fy Stc or MR1 5.698 104 kip-in (Eq 10-98)
First yield in compression fiber:
R2 cr bcM =F S bR (Eq 10-99)
where:
162
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
Sb 1.708 103
Composite Section: Stc 1.727 103 and Sbc 1.772 103
Stress in the top flange due to 1.3 times the dead load effect:
ftop1.3 12 MD
Stor ftop 12.329 ksi
The capacity minus 1.3 times the deadload effect will be calculated based on the condition that the top fiber has just yielded (i.e., the yield moment).
remainingtop y
tc
Mf F
S
=> Mremaining Fy ftop Stc (k-in)
MremainingMremaining
12or Mremaining 2.975 103 k-ft
Rb min 1 0.002Dc tw
Af
Dctw 1000fb
1 or Rb 0.998
MR2 Fcr Sbc Rb or MR2 5.834 104 k-in
The capacity of the section:
MR112
min MR1 MR2 or MR 4.749 103 k-ft
2.6. Capacity of Section (based on AASHTO Article 10.50.c)
Note: The capacity mentioned in previous parts was calculated based on the assumption that all the deadload and live load acts on the composite section. In reality, some of the dead load acts on the non-composite section and some of the dead load plus the live load acts on the composite section. To simplify the calculations, the non-composite section was assumed to carry all the dead load and thus, the composite section carried all the live load.
The capacity minus 1.3 times the dead load effect:
Mremaining MR 1.3 MD
Since the section modulus with respect to the top fiber is smaller than the section modulus with respect to the bottom fiber for both the non-composite and composite sections, the top fiber will yield first.
Non-composite Section: St 1.548 103 and
163
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
ksi < Fy 33 ksi => OK
Compression stress in the bottom angles of the spandrel beam:
fbot1.3 12 MD
Sb
2.17 12 MLSbc
or fbot 28.119 ksi
fbot 28.119 ksi < Fy 33 ksi => OK
=> At inventory level, no component of the section will yield (i.e., RF > 1).
Check yield of rebars, top and bottom fibers of the spandrel beam:
Tension stress in the rebars:
frebar2.17 12ML
Icytc 7.125 or frebar 12.814 ksi
frebar 12.814 ksi < Fy_rebar 60 ksi => OK
Tension stress in the top plate of the spandrel beam:
ftop1.3 12 MD
St
2.17 12 MLStc
or ftop 29.725 ksi
ftop 29.725
164
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
2.7. Load Factor Rating for Section #2
Inventory Level
A1 1.3
A2 2.17
1
2 2
.. .
remainingR Di
L L
MM A MRFA M A M
RFiMremaining
A2 MLor RFi 1.188
Operating Level
A1 1.3
A2 1.3
1
2 2
.. .
remainingR Do
L L
MM A MRFA M A M
RFoMremaining
A2 MLor RFo 1.983
165
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
in2As 84.575orAs Atopplate Atopangle Awebplate Abotangle
in2Abotangle 19.98orAbotangle 2 9.99( )
in2Awebplate 49.5orAwebplate 2 6638
in2Atopangle 5.72orAtopangle 2 2.86( )
in2Atopplate 9.375orAtopplate 25 0.375
25" x 3/8" top plate
4" x 4" x 3 8" L
66" x 3 8" web plate
8" x 6" x 3 4" L
d=66.875
31.789
25
3.1. Non-composite Section
3. Rating for Positive Moment (Section #3)
166
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
in
The moment of inertia:
I1 Atopplate y1 yb2
I2 Atopangle y2 yb2 2 4.32( )
I3 Awebplate y3 yb2 2 0.375 663
12
I4 Abotangle y4 yb2 2 30.8( )
Ix I1 I2 I3 I4 or Ix 5.43 104 in4
The section modulus of the top fiber:
StIx
d ybor St 1.548 103 in3
The section modulus of the bottom fiber:
SbIxyb
or Sb 1.708 103 in3
d 66.875 in
y1 d316
or y1 66.688 in
y2 d38
1.13832 or y2 65.362 in
y3 d38
33 or y3 33.5 in
y4 1.56368 in
The location of the neutral axis:
ybAtopplate y1 Atopangle y2 Awebplate y3 Abotangle y4
As
or yb 31.789
167
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
ind1 8.848ord1
ypna 0.534
ypna 0.5
20.5 Abotangle y4
0.5As
Distance from the center of gravity of tension part to the extreme bottom fiber:
(Based on the condition that the tension area equal to the compression area)
inypna 30.243orypna Find ypna
ypna 0.534
AbotangleAs2
=
Given
Distance from the plastic neutral axis to the extreme bottom fiber:
3.3. Capacity of Section
k-ftML 1.382 103orML MHS20 mDF 1 I( )
k-ftMHS20 491.5
Live Load:
ft-kipsMD 612.9
Dead Load:
3.2. Unfactored Moment due to Dead Load and Live Load
168
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NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
d1=8.848
d2=54.73
30.243
PNAarm=45.883 66.5 66.875
(if the section is compact)MR Mn
ft-kipsMn 5.336 103orMnAs2
Fyarm12
The nominal capacity of the section
inarm 45.883orarm d2 d1
Distance between the center of gravity of tension part and compression part:
ind2 54.73ord2
Atopplate y1 Atopangle y2 yw234
yw22
ypna
0.5As
yw2 66.5 ypna
Distance from the center of gravity of compression part to the extreme bottom fiber:
169
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
R y xtM =F S (Eq 10-98)
or
R cr xcM =F S bR (Eq 10-99)
If the beam meets the requirements in AASHTO Article 10.48.2.
Checking equations in AASHTO Article 10.48.2
Compression flange is compact (no need to check Eq 10-100)
Web thickness (for girder with transverse stiffeners):
Dtw
175.33336500
1000Fy200.926 => OK (Eq 10-104)
Spacing of lateral bracing for compression flange:
Af252
38
or Af 4.688 in2 (area of the flange)
Lb 0 ft112
20000000 Af1000Fy d
3.54 => OK (Eq 10-101)
3.4. Slenderness Checks
Note: The cross-section of the spandrel beam was divided into two equal part and each part was considered as an independent beam for the compactness checks. The strength of each independent beam is thus half the strength of the whole cross-section.
Checking equations in AASHTO Article 10.48.1
The compression flange is embedded in the deck and compact (no need to check Eq 10-93).
The thickness of the compression flange: t 0.375 in
The width of the compression flange: b 12.5 in
Checking web compactness:
D d34
38
or D 65.75 in (Clear distance between flanges)
tw38
in (thickness of the web)
Dtw
175.33319230
1000 Fy105.858 => NG (Eq 10-94)
Since 10-94 is not satisfied, the section is non-compact.
According to AAHSTO Article 10.48.2, the maximum strength shall be computed as:
170
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
The capacity of the section:
MR112
min MR1 MR2 or MR 4.256 103 k-ft
3.5. Load Factor Rating for Section #3
Inventory Level
A1 1.3
A2 2.17
RFiMR A1 MD
A2 MLor RFi 1.154
Operating Level
A1 1.3
A2 1.3
RFoMR A1 MD
A2 MLor RFo 1.926
The maximum strength shall be computed as the lesser of Eq 10-98 and Eq 10-99:
First yield in tension fiber:
MR1 Fy Sb or MR1 5.637 104 (Eq 10-98)
First yield in compression fiber:
R2 cr tM =F S bR (Eq 10-99)
Since the compression flange is embedded in the deck is fully braced and compact, we use:
Fcr Fy or Fcr 33
Rb 1 (The flange stress reduction factor)
MR2 Fcr St Rb or MR2 5.107 104 k-in
171
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NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
in2As 84.575orAs Atopplate Atopangle Awebplate Abotangle
in2Abotangle 19.98orAbotangle 2 9.99( )
in2Awebplate 49.5orAwebplate 2 6638
in2Atopangle 5.72orAtopangle 2 2.86( )
in2Atopplate 9.375orAtopplate 25 0.375
25" x 3/8" top plate
4" x 4" x 3 8" L
66" x 3 8" web plate
8" x 6" x 3 4" L
d=66.875
31.789
25
4.1. Non-composite Section
4. Rating for Negative Moment (Section #4)
172
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
in
The moment of inertial:
I1 Atopplate y1 yb2
I2 Atopangle y2 yb2 2 4.32( )
I3 Awebplate y3 yb2 2 0.375 663
12
I4 Abotangle y4 yb2 2 30.8( )
Ix I1 I2 I3 I4 or Ix 5.43 104 in4
The section modulus of the top fiber:
StIx
d ybor St 1.548 103 in3
The section modulus of the bottom fiber:
SbIxyb
or Sb 1.708 103 in3
d 66.875 in
y1 d316
or y1 66.688 in
y2 d38
1.13832 or y2 65.362 in
y3 d38
33 or y3 33.5 in
y4 1.56368 in
The location of the neutral axis:
ybAtopplate y1 Atopangle y2 Awebplate y3 Abotangle y4
As
or yb 31.789
173
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
ind1 8.848ord1
ypna 0.534
ypna 0.5
20.5 Abotangle y4
0.5As
Distance from the center of gravity of tension part to the extreme bottom fiber:
(Based on the condition that the tension area is equal to the compression area)
inypna 30.243orypna Find ypna
ypna 0.534
AbotangleAs2
=
Given
Distance from the plastic neutral axis to the extreme bottom fiber:
4.3. Capacity of Section
k-ftML 1.252 103orML MHS20 mDF 1 I( )
k-ftMHS20 445.5
Live Load:
ft-kipsMD 749.8
Dead Load:
4.2. Unfactored Moment due to Dead Load and Live Load
174
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NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
d1=8.848
d2=54.73
30.243
PNAarm=45.883 66.5 66.875
(if the section is compact)MR Mn
ft-kipsMn 5.336 103orMnAs2
Fyarm12
The nominal capacity of the section
inarm 45.883orarm d2 d1
Distance between the center of gravity of tension part and compression part:
ind2 54.73ord2
Atopplate y1 Atopangle y2 yw234
yw22
ypna
0.5As
yw2 66.5 ypna
Distance from the center of gravity of compression part to the extreme bottom fiber:
175
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
(thickness of the web)
Dtw
175.33319230
1000 Fy105.858 => NG (Eq 10-94)
Since 10-94 is not satisfied, the section is non-compact.
According to AAHSTO Article 10.48.2, the maximum strength shall be computed as:
R y xtM =F S (Eq 10-98)
or
R cr xcM =F S bR (Eq 10-99)
If the beam meets the requirements in AASHTO Article 10.48.2.
Checking equations in AASHTO Article 10.48.2
Compression flange:
bt
10.667 24 => OK (Eq 10-100)
Web thickness (for girder with transverse stiffeners):
Dtw
175.33336500
1000Fy200.926 => OK (Eq 10-104)
4.4. Slenderness Checks ASSHTO Article 10.48
Note: The cross-section of the spandrel beam was divided into two equal part and each part was considered as an independent beam for the compactness checks. The strength of each independent beam is thus half the strength of the whole cross-section.
Checking equations in AASHTO Article 10.48.1
Checking flange compactness:
b 8 in (width of comp flange)
t34
in or t 0.75 in (thickness of comp flange)
bt
10.6674110
1000 Fy22.625 => OK (Eq 10-93)
Checking web compactness:
D d34
38
or D 65.75 in (Clear distance between flanges)
tw38
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NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
where:
Fcr min1
10004400
tb
2Fy or Fcr 33 ksi
Rb = The flange stress reduction factor:
1 0.002 1.0c w cb
fc w b
D t DRA t f
fb = factored bending stress in the compression flange but not to exceed Fy
min ,D D L Lb y
M Mf FS
or, to be conservative use: fb Fy
Dc = depth of the web in compression
Dc yb34
or Dc 31.039 in
15400 since Dc < d/2
Spacing of lateral bracing for compression flange:
Af 834
2 (area of the flange)
Lb 7 ft112
20000000 Af1000Fy d( )
9.063 => OK (Eq 10-101)
Af = the area of the two flanges since there are diaphragm plates between the two flanges and therefore, the two compression flanges buckle together.
Lb = the lateral bracing of the compression flange, but due to the sharp drop in negative moment Lb was taken as the distance from the center of the support to the point of zero moment.
The maximum strength shall be computed as the lesser of Eq 10-98 and Eq 10-99:
First yield in tension fiber:
MR1 Fy St or MR1 5.107 104 k-in (Eq 10-98)
First yield in compression fiber:
R2 cr bM =F S bR (Eq 10-99)
177
Nguyenngoc Tuyen
NMSUAASHTO HS20 TRUCK RATING SPANDREL BEAM (BEAM MODEL)
RFo 2.016orRFoMR A1 MD
A2 ML
A2 1.3
A1 1.3
Operating Level
RFi 1.208orRFiMR A1 MD
A2 ML
A2 2.17
A1 1.3
Inventory Level
4.5. Load Factor Rating for Section #4
k-ftMR 4.256 103orMR112
min MR1 MR2
The capacity of the section:
k-inMR2 5.637 104orMR2 Fcr Sb Rb
Rb 1orRb min 1 0.002Dc tw
Af
Dctw 1000fb
1
178
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NMSU
APPENDIX A4
RATING COLUMNS
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
(an initial value is needed for Madcad Calculations)RF 0
Rating factor is denoted as RF:
ksiFy 33
The yield stress of the steel:
ksiE 29000
The elastic modulus of steel:
inr 10.36
rI
As
The radius of gyration: 24
24.5" back-to-back
4" x 4" x 12"
24" x 12"
24
in3Z 610.05
Pier and Arch Columns
Cross-section of the columns
The area of the cross-section:
As 4 24 0.5 3.75( ) or As 63 in2
The moment of inertia:
I 2 12 12.52 4 3.75 12.25 1.18( )2 4 5.52 20.5 243
12
I 6.762 103 in4
The section modulus:
SI
0.5 25.5or S 530.373 in3
Z 24 0.5( )252
2 12 0.5( )122
2 3.75( ) 12.25 1.18( ) 2
180
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
mDF 2.218ormDF max 0.75 DF4 0.9DF3 DF2 DF1
(AASHTO Article 3.12.1)Distribution Factor (accounting for Multiple Presence Factor):
DF1 1.107orDF1 1.10714
Distribution factor for 1 lane:
DF2 1.929orDF2 1.10714 0.82143
Distribution factor for 2 lanes:
DF3 2.464orDF3 1.10714 0.82143 0.53571
Distribution factor for 3 lanes:
DF4 2.714orDF4 1.10714 0.82143 0.53571 0.25000
Distribution factor for 4 lanes:
0.53571
465.0
420.0
345.0
420.0
48.0
81.0
Level rule:
225.0
105.0
0.25
81.042.0 88.5 81.0
48.072.0 72.0
1.107140.82143
1
24.0
42.088.581.0 81.0
48.072.0 72.0
1.27857
72.0
120.0120.0120.0168.0
Lane 4Lane 2 Lane 3Lane 1
Distribution Factor (use level rule, AASHTO Article 3.23.2.3)
181
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NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 255.45 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 414.771 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 248.48 kips
1.3. Compressive Capacity of the Section
Check if
K Lcr
41.755 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
1. Rating Column N3 (Pier Column # 2)
1.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 41.2 ft or Lc 12 Lc or Lc 494.4 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.875
Impact factor:
I50
62 125or I 0.267
1.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 196.5 kips
Unfactored axial force due to Live Load:
uPL 68.0 kips
Factored axial force due to Dead Load:
182
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NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 8.242 103 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 4.937 103 kip-in
1.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 164.163 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.6
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 31.342 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.678 103 kips
1.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 2.8 kip-ft
Unfactored moment due to Live Load:
uML 112.6 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 43.68 kip-in
Factored moment due to Live Load at Inventory Level:
183
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NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.791
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.585
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.873
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
1.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
184
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 2.211
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.325
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.965
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.177
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
1.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.533
At Operating Level:
185
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 259.61 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 413.551 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 247.749 kips
2.3. Compressive Capacity of the Section
Check if
K Lcr
48.039 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
2. Rating Column N20 (Pier Column #3)
2.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 47.4 ft or Lc 12 Lc or Lc 568.8 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.875
Impact factor:
I min50
62 1250.3 or I 0.267
2.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 199.7 kips
Unfactored axial force due to Live Load:
uPL 67.8 kips
Factored axial force due to Dead Load:
186
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 5.724 103 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 3.429 103 kip-in
2.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 124.026 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.6
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 30.805 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.65 103 kips
2.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 27.8 kip-ft
Unfactored moment due to Live Load:
uML 78.2 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 433.68 kip-in
Factored moment due to Live Load at Inventory Level:
187
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.687
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.528
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.76
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
2.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
188
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 2.678
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.604
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 2.363
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.416
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
2.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.479
At Operating Level:
189
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 131.3 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 277.261 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 166.101 kips
3.3. Compressive Capacity of the Section
Check if
K Lcr
86.088 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
3. Rating Column N5 (Arch Column #1)
3.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 99.1 ft or Lc 12 Lc or Lc 1.189 103 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
3.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 101.0 kips
Unfactored axial force due to Live Load:
uPL 45.3 kips
Factored axial force due to Dead Load:
190
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 8.042 103 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 4.818 103 kip-in
3.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 38.62 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 25.951 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.39 103 kips
3.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 52.3 kip-ft
Unfactored moment due to Live Load:
uML 109.5 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 815.88 kip-in
Factored moment due to Live Load at Inventory Level:
191
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.671
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.536
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.8
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
3.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
192
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 2.656
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.591
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 2.175
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.303
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
3.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.448
At Operating Level:
193
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 138.32 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 225.236 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 134.934 kips
4.3. Compressive Capacity of the Section
Check if
K Lcr
63.502 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
4. Rating Column N6 (Arch Column #2)
4.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 73.1 ft or Lc 12 Lc or Lc 877.2 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
4.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 106.4 kips
Unfactored axial force due to Live Load:
uPL 36.8 kips
Factored axial force due to Dead Load:
194
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.082 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 6.481 103 kip-in
4.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 70.979 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 29.164 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.562 103 kips
4.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 83.0 kip-ft
Unfactored moment due to Live Load:
uML 147.3 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.295 103 kip-in
Factored moment due to Live Load at Inventory Level:
195
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.807
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.619
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.925
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
4.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
196
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 2.153
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.29
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.834
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.099
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
4.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.541
At Operating Level:
197
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 131.69 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 278.485 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 166.834 kips
5.3. Compressive Capacity of the Section
Check if
K Lcr
44.651 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
5. Rating Column N7 (Arch Column #3)
5.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 51.4 ft or Lc 12 Lc or Lc 616.8 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
5.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 101.3 kips
Unfactored axial force due to Live Load:
uPL 45.5 kips
Factored axial force due to Dead Load:
198
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.204 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 7.212 103 kip-in
5.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 143.561 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 31.104 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.666 103 kips
5.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 93.5 kip-ft
Unfactored moment due to Live Load:
uML 163.9 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.459 103 kip-in
Factored moment due to Live Load at Inventory Level:
199
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.903
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.675
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.017
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
5.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
200
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.885
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.129
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.635
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.98
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
5.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.6
At Operating Level:
201
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 128.05 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 279.709 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 167.568 kips
6.3. Compressive Capacity of the Section
Check if
K Lcr
29.709 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
6. Rating Column N8 (Arch Column #4)
6.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 34.2 ft or Lc 12 Lc or Lc 410.4 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
6.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 98.5 kips
Unfactored axial force due to Live Load:
uPL 45.7 kips
Factored axial force due to Dead Load:
202
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.304 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 7.81 103 kip-in
6.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 324.273 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.16 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.722 103 kips
6.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 94.4 kip-ft
Unfactored moment due to Live Load:
uML 177.5 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.473 103 kip-in
Factored moment due to Live Load at Inventory Level:
203
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.951
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.702
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.066
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
6.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
204
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.77
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.06
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.548
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.928
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
6.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.628
At Operating Level:
205
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 131.82 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 283.993 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 170.134 kips
7.3. Compressive Capacity of the Section
Check if
K Lcr
17.808 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
7. Rating Column N9 (Arch Column #5)
7.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 20.5 ft or Lc 12 Lc or Lc 246 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
7.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 101.4 kips
Unfactored axial force due to Live Load:
uPL 46.4 kips
Factored axial force due to Dead Load:
206
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.448 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 8.677 103 kip-in
7.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 902.518 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.698 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.751 103 kips
7.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 115.5 kip-ft
Unfactored moment due to Live Load:
uML 197.2 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.802 103 kip-in
Factored moment due to Live Load at Inventory Level:
207
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
> 1 => NGPD PL_in0.85 As Fy
MD ML_inMP
1.044
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.771
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.168
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
7.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
208
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.585
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 0.95
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.386
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.83
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
7.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.691
At Operating Level:
209
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 119.47 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 277.873 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 166.468 kips
8.3. Compressive Capacity of the Section
Check if
K Lcr
10.164 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
8. Rating Column N10 (Arch Column #6)
8.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 11.7 ft or Lc 12 Lc or Lc 140.4 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
8.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 91.9 kips
Unfactored axial force due to Live Load:
uPL 45.4 kips
Factored axial force due to Dead Load:
210
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.514 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 9.068 103 kip-in
8.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 2.771 103 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.902 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.762 103 kips
8.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 128.2 kip-ft
Unfactored moment due to Live Load:
uML 206.1 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 2 103 kip-in
Factored moment due to Live Load at Inventory Level:
211
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
> 1 => NGPD PL_in0.85 As Fy
MD ML_inMP
1.076
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.795
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.205
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
8.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
212
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.529
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 0.916
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.335
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.8
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
8.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.712
At Operating Level:
213
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 120.38 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 269.916 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 161.701 kips
9.3. Compressive Capacity of the Section
Check if
K Lcr
5.994 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
9. Rating Column N11 (Arch Column #7)
9.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 6.9 ft or Lc 12 Lc or Lc 82.8 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
9.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 92.6 kips
Unfactored axial force due to Live Load:
uPL 44.1 kips
Factored axial force due to Dead Load:
214
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.391 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 8.334 103 kip-in
9.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 7.966 103 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.966 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.765 103 kips
9.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 75.3 kip-ft
Unfactored moment due to Live Load:
uML 189.4 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.175 103 kip-in
Factored moment due to Live Load at Inventory Level:
215
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.97
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.703
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.083
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
9.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
216
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.728
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.035
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.523
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.912
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
9.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.632
At Operating Level:
217
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 125.84 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 283.381 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 169.768 kips
10.3. Compressive Capacity of the Section
Check if
K Lcr
73.839 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
10. Rating Column N18 (Arch Column #14)
10.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 85 ft or Lc 12 Lc or Lc 1.02 103 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2*29.5 1250.3 or I 0.272
10.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 96.8 kips
Unfactored axial force due to Live Load:
uPL 46.3 kips
Factored axial force due to Dead Load:
218
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 8.388 103 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 5.025 103 kip-in
10.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 52.496 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 27.814 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.489 103 kips
10.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 42.0 kip-ft
Unfactored moment due to Live Load:
uML 114.2 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 655.2 kip-in
Factored moment due to Live Load at Inventory Level:
219
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.681
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.523
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.791
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
10.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
220
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 2.593
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.553
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 2.189
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.312
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
10.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.449
At Operating Level:
221
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 138.06 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 228.909 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 137.134 kips
11.3. Compressive Capacity of the Section
Check if
K Lcr
53.164 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
11. Rating Column N17 (Arch Column #13)
11.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 61.2 ft or Lc 12 Lc or Lc 734.4 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2*29.5 1250.3 or I 0.272
11.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 106.2 kips
Unfactored axial force due to Live Load:
uPL 37.4 kips
Factored axial force due to Dead Load:
222
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.217 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 7.291 103 kip-in
11.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 101.265 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 30.312 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.623 103 kips
11.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 66.1 kip-ft
Unfactored moment due to Live Load:
uML 165.7 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.031 103 kip-in
Factored moment due to Live Load at Inventory Level:
223
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.863
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.645
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.98
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
11.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
224
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.98
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.186
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.708
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.024
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
11.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.569
At Operating Level:
225
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 129.35 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 284.605 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 170.501 kips
12.3. Compressive Capacity of the Section
Check if
K Lcr
36.225 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
12. Rating Column N16 (Arch Column #12)
12.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 41.7 ft or Lc 12 Lc or Lc 500.4 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2*29.5 1250.3 or I 0.272
12.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 99.5 kips
Unfactored axial force due to Live Load:
uPL 46.5 kips
Factored axial force due to Dead Load:
226
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.373 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 8.224 103 kip-in
12.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 218.118 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 31.752 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.7 103 kips
12.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 79.4 kip-ft
Unfactored moment due to Live Load:
uML 186.9 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.239 103 kip-in
Factored moment due to Live Load at Inventory Level:
227
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.978
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.717
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.099
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
12.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
228
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.713
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.027
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.496
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.896
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
12.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.64
At Operating Level:
229
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 126.75 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 285.218 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 170.868 kips
13.3. Compressive Capacity of the Section
Check if
K Lcr
23.194 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
13. Rating Column N15 (Arch Column #11)
13.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 26.7 ft or Lc 12 Lc or Lc 320.4 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2*29.5 1250.3 or I 0.272
13.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 97.5 kips
Unfactored axial force due to Live Load:
uPL 46.6 kips
Factored axial force due to Dead Load:
230
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.494 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 8.95 103 kip-in
13.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 532.036 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.488 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.74 103 kips
13.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 84.8 kip-ft
Unfactored moment due to Live Load:
uML 203.4 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.323 103 kip-in
Factored moment due to Live Load at Inventory Level:
231
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
> 1 => NGPD PL_in0.85 As Fy
MD ML_inMP
1.041
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.758
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.166
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
13.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
232
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.594
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 0.955
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.397
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.837
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
13.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.679
At Operating Level:
233
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 131.69 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 287.666 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 172.334 kips
14.3. Compressive Capacity of the Section
Check if
K Lcr
13.117 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
14. Rating Column N14 (Arch Column #10)
14.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 15.1 ft or Lc 12 Lc or Lc 181.2 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2*29.5 1250.3 or I 0.272
14.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 101.3 kips
Unfactored axial force due to Live Load:
uPL 47.0 kips
Factored axial force due to Dead Load:
234
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.611 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 9.649 103 kip-in
14.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 1.663 103 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.836 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.758 103 kips
14.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 115.5 kip-ft
Unfactored moment due to Live Load:
uML 219.3 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.802 103 kip-in
Factored moment due to Live Load at Inventory Level:
235
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
> 1 => NGPD PL_in0.85 As Fy
MD ML_inMP
1.127
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.827
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.262
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
14.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
236
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.449
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 0.868
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.266
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.759
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
14.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.741
At Operating Level:
237
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 119.21 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 278.485 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 166.834 kips
15.3. Compressive Capacity of the Section
Check if
K Lcr
7.384 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
15. Rating Column N13 (Arch Column #9)
15.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 8.5 ft or Lc 12 Lc or Lc 102 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2*29.5 1250.3 or I 0.272
15.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 91.7 kips
Unfactored axial force due to Live Load:
uPL 45.5 kips
Factored axial force due to Dead Load:
238
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.489 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 8.919 103 kip-in
15.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 5.25 103 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.948 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.764 103 kips
15.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 123.4 kip-ft
Unfactored moment due to Live Load:
uML 202.7 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.925 103 kip-in
Factored moment due to Live Load at Inventory Level:
239
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
> 1 => NGPD PL_in0.85 As Fy
MD ML_inMP
1.06
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.782
At Operating Level:
> 1 => NGPD PL_in
PR
MD ML_in AFiMu
1.186
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
15.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
240
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 1.557
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 0.933
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.361
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 0.816
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
15.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.701
At Operating Level:
241
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
PD 1.3uPD or PD 119.99 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 268.692 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 160.968 kips
16.3. Compressive Capacity of the Section
Check if
K Lcr
5.038 < 22
EFy
131.706
According to AASHTO Article 10.54.1.1:
22c
y
KL Er F
For: Eq 10-152
16. Rating Column N12 (Arch Column #8)
16.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 5.8 ft or Lc 12 Lc or Lc 69.6 in
The effective length factor (AASHTO Article 10.54.1.2):
K 0.75
Impact factor:
I min50
2 29.5 1250.3 or I 0.272
16.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 92.3 kips
Unfactored axial force due to Live Load:
uPL 43.9 kips
Factored axial force due to Dead Load:
242
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 1.17 104 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 7.009 103 kip-in
16.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 1.127 104 ksi
Maximum flexural strength:
Mu Fy S or Mu 1.75 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 32.976 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 1.766 103 kips
16.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 35.6 kip-ft
Unfactored moment due to Live Load:
uML 159.3 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 555.36 kip-in
Factored moment due to Live Load at Inventory Level:
243
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.829
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.591
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.92
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
16.6. Check AASHTO Equations
kip-inMP 2.013 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
244
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING PIER AND ARCH COLUMNS
RF 2.059
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 1.234
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 1.831
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 1.097
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
16.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.535
At Operating Level:
245
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
Z 1.532 103 in3
The radius of gyration:
rI
Asor r 16.67 in
48
4" x 4" x 12"
24" x 12"
48" x 12"
48.5" back-to-back
The elastic modulus of steel:
E 29000 ksi
The yield stress of the steel:
Fy 33 ksi
Rating factor is denoted as RF:
RF 0 (an initial value is needed for Madcad Calculations)
Skewback Columns
Section properties of the skewback in the strong direction
The area of the cross-section:
As 3 24 0.5( ) 8 3.75( ) 2 48 0.5( ) or As 114 in2
The moment of inertia:
I 2 12492
24 3.75
48.52
1.182
4 3.75 1.1814
28 5.52 2
0.5 483
12
I 3.168 104 in4
The section modulus:
SI
0.5 49.5or S 1.28 103 in3
Z 24 0.5( )492
2 24 0.5( )242
2 3.75( )48.5
21.18 2 3.75( ) 1.18
14
2
246
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
mDF 2.218ormDF max 0.75 DF4 0.9DF3 DF2 DF1
(AASHTO Article 3.12.1)Distribution Factor (accounting for Multiple Presence Factor):
DF1 1.107orDF1 1.10714
Distribution factor for 1 lane:
DF2 1.929orDF2 1.10714 0.82143
Distribution factor for 2 lanes:
DF3 2.464orDF3 1.10714 0.82143 0.53571
Distribution factor for 3 lanes:
DF4 2.714orDF4 1.10714 0.82143 0.53571 0.25000
Distribution factor for 4 lanes:
0.53571
465.0
420.0
345.0
420.0
48.0
81.0
Level rule:
225.0
105.0
0.25
81.042.0 88.5 81.0
48.072.0 72.0
1.107140.82143
1
24.0
42.088.581.0 81.0
48.072.0 72.0
1.27857
72.0
120.0120.0120.0168.0
Lane 4Lane 2 Lane 3Lane 1
Distribution Factor (use level rule, AASHTO Article 3.23.2.3)
247
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
Factored axial force due to Dead Load:
PD 1.3uPD or PD 273.13 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 414.81 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 248.504 kips
1.3. Compressive Capacity of the Section
Check if
K Lcr
55.608 < 22
EFy
131.706
According to AASHTO (10.54.1.1)
22c
y
KL Er F
For: Eq 10-152
1. Rating Column N4 (Skewback Column # 1)
1.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 103 ft or Lc 12 Lc or Lc 1.236 103 in
The effective length factor:
K 0.75
Impact factor:
I50
29.5 125or I min I 0.3( ) or I 0.3
1.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 210.1 kips
Unfactored axial force due to Live Load:
uPL 66.3 kips
248
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 7.508 103 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 4.498 103 kip-in
1.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 92.56 ksi
Maximum flexural strength:
Mu Fy S or Mu 4.224 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 30.059 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 2.913 103 kips
1.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 65.9 kip-ft
Unfactored moment due to Live Load:
uML 100.0 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.028 103 kip-in
Factored moment due to Live Load at Inventory Level:
249
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.384
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.31
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.438
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
1.6. Check AASHTO Equations
kip-inMP 5.054 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
250
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
RF 5.364
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 3.214
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 4.598
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 2.755
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
1.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.272
At Operating Level:
251
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
Factored axial force due to Dead Load:
PD 1.3uPD or PD 260.65 kips
Factored axial force due to Live Load at Inventory Level:
PL_in mDF 1 I( ) 2.17 uPL or PL_in 417.313 kips
Factored axial force due to Live Load at Operating Level:
PL_op mDF 1 I( ) 1.3 uPL or PL_op 250.003 kips
2.3. Compressive Capacity of the Section
Check if
K Lcr
46.862 < 22
EFy
131.706
According to AASHTO (10.54.1.1)
22c
y
KL Er F
For: Eq 10-152
2. Rating Column N19 (Skewback Column # 2)
2.1. Characteristics of the Beam-ColumnThe unsupported length of the column:
Lc 86.8 ft or Lc 12 Lc or Lc 1.042 103 in
The effective length factor:
K 0.75
Impact factor:
I50
29.5 125or I min I 0.3( ) or I 0.3
2.2. Axial Load due to Dead Load and Live Load
Unfactored axial force due to Dead Load:
uPD 200.5 kips
Unfactored axial force due to Live Load:
uPL 66.7 kips
252
Nguyenngoc Tuyen
NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
ML_in 12mDF 1 I( ) 2.17 uML or ML_in 8.341 103 kip-in
Factored moment due to Live Load at Operating Level:
ML_op 12mDF 1 I( ) 1.3 uML or ML_op 4.997 103 kip-in
2.5. Flexural Capacity of the Section and Moment Amplification
The Euler Buckling stress in the plane of bending:
FeE
2
K Lcr
2or Fe 130.334 ksi
Maximum flexural strength:
Mu Fy S or Mu 4.224 104 kip-in
Equivalent moment factor
Cm 0.4 (to be conservative)
The buckling stress:
Fcr Fy 1Fy
42E
K Lcr
2
or Fcr 30.911 ksi
The maximum Capacity:
PR 0.85 As Fcr or PR 2.995 103 kips
2.4. Moment due to Dead Load and Live Load
Unfactored moment due to Dead Load:
uMD 71.4 kip-ft
Unfactored moment due to Live Load:
uML 111.1 kip-ft
Factored moment due to Dead Load:
MD 12 1.3uMD or MD 1.114 103 kip-in
Factored moment due to Live Load at Inventory Level:
253
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NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
< 1 => OKPD PL_in0.85 As Fy
MD ML_inMP
0.399
At Inventory Level:
1.00.85 s y P
P MA F M
Check Equation 10-156:
< 1 => OKPD PL_op
PR
MD ML_op AFoMu
0.315
At Operating Level:
< 1 => OKPD PL_in
PR
MD ML_in AFiMu
0.45
At Inventory Level:
1.00.85 0.85
1
D Lm D L
s cr s cr uu
s e
M M AFMC P PPA F A F MPM
A F
Check Equation 10-155:
2.6. Check AASHTO Equations
kip-inMP 5.054 104orMP Fy Z
The full plastic Capacity:
AFo 1orAFo maxCm
1PD PL_op
As Fe
1
The amplification factor at Operating Level:
AFi 1orAFi maxCm
1PD PL_in
As Fe
1
The amplification factor at Inventory Level:
254
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NMSUBEAM-COLUNM MODEL AASHTO HS20 TRUCK RATING SKEWBACK COLUMNS
RF 5.063
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_op0.85 As Fy
MD RF ML_opMP
1=
Given
At Operating Level:
RF 3.033
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_in0.85 As Fy
MD RF ML_inMP
1=Given
At Inventory Level:
Equation 10-156
RF 4.394
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_opPR
MD RF ML_op AFoMu
1=
Given
At Operating Level:
RF 2.632
RF Find RF( )Rating Factor is obtained by solving the above equation:
PD RF PL_inPR
MD RF ML_in AFiMu
1=Given
At Inventory Level:
Equation 10-155
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-155 or 10-156 equal to 1.
2.7. Calculate the corresponding Rating Factor
< 1 => OKPD PL_op0.85 As Fy
MD ML_opMP
0.281
At Operating Level:
255
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NMSU
APPENDIX A5
RATING ARCH RIB
Nguyenngoc Tuyen
NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
Analysis 1: Pinned connections at the ends of the columns.
1. Cross-section of the Arch-Rib
The area of the cross-section:
As 2 4634
71.512
6.90 2.86 8 11.5 or As 252.02 in2
The moment of inertia:
I 212
71.53 112
2 4634
3638
28 69.9 11.5 36 2.26( )2 2 8.63 4.32( )
or I 2.271 105 in4
46" x 3 4"
4" x 4" x 3 8" L6" x 4" x 3 4" L
71 12" x 12"
8" x 8" x 3 4" L
25.5"
72"
inside to inside of PLs
257
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
RF 0
Rating factor is denoted as RF:
intw 0.5
The thickness of the web:
inD 70.5orD 72 1.5
The clear distance between flanges:
1. Impact factor will vary depending on the location of the live load;2. Impact factor for axial force is not equal to impact facor for moment;3. Distance between 2 inflexion points of influence lines, which is considered as the loaded length L in equation 3-1 of AASHTO Article 3.8.2.1, is greater than 4.5 times the disstance between columns (4.5 x 29.5ft) in any case.
=> To be conservative, use the loaded length L = 4 x 29.5 ft.
Note:
I 0.206orI min50
4 29.5 1250.3
(AASHTO Article 3.8.2.1)
The radius of gyration:
rI
Asor r 30.017 in
The elastic modulus of steel:
E 29000 ksi
The yield stress of the steel:
Fy 33 ksi
The section modulus:
SI
722
34
or S 6.179 103 in3
Impact factor:
258
Nguyenngoc Tuyen
NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
mDF 2.218ormDF max 0.75 DF4 0.9DF3 DF2 DF1
(AASHTO Article 3.12.1)Distribution Factor (accounting for Multiple Presence Factor):
DF1 1.107orDF1 1.10714
Distribution factor for 1 lane:
DF2 1.929orDF2 1.10714 0.82143
Distribution factor for 2 lanes:
DF3 2.464orDF3 1.10714 0.82143 0.53571
Distribution factor for 3 lanes:
DF4 2.714orDF4 1.10714 0.82143 0.53571 0.25000
Distribution factor for 4 lanes:
0.53571
465.0
420.0
345.0
420.0
48.0
81.0
Level rule:
225.0
105.0
0.25
81.042.0 88.5 81.0
48.072.0 72.0
1.107140.82143
1
24.0
42.088.581.0 81.0
48.072.0 72.0
1.27857
72.0
120.0120.0120.0168.0
Lane 4Lane 2 Lane 3Lane 1
2. Distribution Factor (use level rule, AASHTO Article 3.23.2.3)
259
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
kipsT1L 66.8
Arch rib thrust at the quarter point (point E) due to Live Load:
k-ftM1D 168.74
Moment at point C2 due to Dead Load:
kipsN1D 1338.3
Axial force at point A due to Dead Load:
k-ftM1L 1162.4
Moment at point C2 due to Live Load:
kipsN1L 69.0
Axial force at point A due to Live Load:
Truck location that causes maximum axial force at point A (critical point for axial force) produces a maximum moment in the arch rib at point C2.
Case 1:
The maximum axial force and moment in the arch rib depend on the location of the truck. The location of the truck that causes the maximum axial force will not produce the maximum moment in the arch rib and vice versa (i.e. the location of the truck that causes the maximum moment will not produce the maximum axial force in the arch rib). Consider 4 cases:
D2
SOUTH
C2
C1
A
M22
M23
M24
M25
M28M27M26
M30M29
D1
B
M36
M35
M34
M33
M31M32
NORTH
C1 : j end of element M24
C2 : j end of element M27
D2 : i end of element M31
D1 : i end of element M34
E : j end of element M25
F : i end of element M33
A : i end of element M22 (the south support of the arch)
B : j end of element M36 (the north support of the arch).
NOTES:
E F
The connections between the columns and spandrel beam as well as the connections between the columns and arch rib are considered as pinned connections:
3. Unfactored Moments and Axial Forces
260
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
kips
Arch rib thrust at the quarter point (point F) due to Dead Load:
T2D 1066.1 kips
Case 3:
Truck location that causes maximum moment at point C1 (critical point for moment) produces a maximum axial force in the arch rib at point A.
Axial force at point A due to Live Load:
N3L 57.4 kips
Moment at point C1 due to Live Load:
M3L 1769.4 k-ft
Axial force at point A due to Dead Load:
N3D 1338.3 kips
Moment at point C1 due to Dead Load:
M3D 389.96 k-ft
Arch rib thrust at the quarter point (point E) due to Dead Load:
T1D 1066.1 kips
Case 2:
Truck location that causes maximum axial force at point B (critical point for axial force) produces a maximum moment in the arch rib at point D2.
Axial force at point B due to Live Load:
N2L 69.0 kips
Moment at point D2 due to Live Load:
M2L 1211.8 k-ft
Axial force at point B due to Dead Load:
N2D 1333.7 kips
Moment at point D2 due to Dead Load:
M2D 30.16 k-ft
Arch rib thrust at the quarter point (point F) due to Live Load:
T2L 66.7
261
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
kipsT4D 1066.1
Arch rib thrust at the quarter point (point F) due to Dead Load:
kipsT4L 26.6
Arch rib thrust at the quarter point (point F) due to Live Load:
k-ftM4D 142.9
Moment at point D1 due to Dead Load:
kipsN4D 1333.7
Axial force at point B due to Dead Load:
k-ftM4L 1775.7
Moment at point D1 due to Live Load:
kipsN4L 57.3
Axial force at point B due to Live Load:
Truck location that causes maximum moment at point D1 (critical point for moment) produces a maximum axial force in the arch rib at point B.
Case 4:
kipsT3D 1066.1
Arch rib thrust at the quarter point (point E) due to Dead Load:
kipsT3L 27.2
Arch rib thrust at the quarter point (point E) due to Live Load:
262
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
kips
Arch rib thrust at the quarter point from dead plus live plus impact loading for Case 2:
T2 T2D T2L 1 I( ) mDF( ) or T2 1.244 103 kips
Arch rib thrust at the quarter point from dead plus live plus impact loading for Case 3:
T3 T3D T3L 1 I( ) mDF( ) or T3 1.139 103 kips
Arch rib thrust at the quarter point from dead plus live plus impact loading for Case 4:
T4 T4D T4L 1 I( ) mDF( ) or T4 1.137 103 kips
Amplification factor for Case 1:
A1F1
11.18 T1As Fe
or A1F 1.304 (Eq 10-159)
4. Moment Amplification and Allowable Stress (AASHTO Article 10.55)
Moment Amplification and Allowable Stress are calculated according to AASHTO Article 10.55 and Article 10.37.1.1:
Factor to account for effective length: (AASHTO Article 10.37.1.1)
K 1.1
(For Rise to Span Ratio = 1279.75070
0.252 and a 2-Hinged Arch)
One-half of the length of the arch-rib:
L12
486.93 12( ) or L 2.922 103 in
Euler buckling stress:
Fe
2E
K Lr
2or Fe 24.969 ksi
Arch rib thrust at the quarter point from dead plus live plus impact loading for Case 1:
T1 T1D T1L 1 I( ) mDF( ) or T1 1.245 103
263
Nguyenngoc Tuyen
NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
ksiFb 33orFb Fy
The maximum bending unit stress (Eq. 10-160):
ksiFa 18.726orFaFy
1.181
K Lr
2Fy
42
E
The maximum axial unit stress (Eq. 10-160):
(Eq 10-159)A4F 1.271orA4F1
11.18 T4As Fe
Amplification factor for Case 4:
(Eq 10-159)A3F 1.272orA3F1
11.18 T3As Fe
Amplification factor for Case 3:
(Eq 10-159)A2F 1.304orA2F1
11.18 T2As Fe
Amplification factor for Case 2:
264
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
or Nmax 1.523 103 (kips)
faNmax
Asor fa 6.042 (ksi)
The computed bending stress at the extreme fiber:
Mmax MD ML 1( ) A1F 1 I( ) mDF( )[ ] or Mmax 4.224 103 (ft-kips)
fb12Mmax
Sor fb 8.203 (ksi)
Slenderness checks
Check web plates:
13500
w a
Dt f
Eq 10-163
Dtw
141 < 13500
1000fa173.671 => OK
5. Inventory Rating (AASHTO Table 3.22.1A)
5.1. Case 1
Maximum unfactored axial force at support due to dead load:
ND N1D or ND 1.338 103 (kips)
Maximum unfactored axial force at support due to live load:
NL N1L or NL 69 (kips)
Corresponding unfactored moment within span due to dead load:
MD M1D or MD 168.74 (ft-kips)
Corresponding unfactored moment within span due to live load:
ML M1L or ML 1.162 103 (ft-kips)
The computed axial stress:
Nmax ND NL 1( ) 1 I( ) mDF( )
265
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
90.75
12 < 2200
1000 fa fb18.433 => OK Eq 10-166
Check Equation 10-47
faFa
fbFb
0.571 < 1 => OK
Find the Rating Factor
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-47 equal to 1:
Let: FL 1 I( ) mDF or FL 2.674
Given
ND RF NL 1( ) FL
As
Fa
MD RF ML 1( )1
11.18 T1D RF T1L FL
As Fe
FL 12
S
Fb1=
the rating factor is obtained by solving the above equation:
RF Find RF( )
RF 2.408
Check the ratio for the stiffeners:
60.75
8 < 2200
1000 fafb3
23.483 and 12 => OK Eq 10-164
and
40.375
10.667 < 2200
1000 fafb3
23.483 and 12 => OK
Check flange plate (for width between webs) :
240.75
32 < 5700
1000 fa fb47.757 => OK Eq 10-165
Check flange plate (for overhang width) :
266
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
or Nmax 1.518 103 (kips)
faNmax
Asor fa 6.024 (ksi)
The computed bending stress at the extreme fiber:
Mmax MD ML 1( ) A2F 1 I( ) mDF( )[ ] or Mmax 4.257 103 (ft-kips)
fb12Mmax
Sor fb 8.268 (ksi)
Slenderness checks
Check web plates:
13500
w a
Dt f
Eq 10-163
Dtw
141 < 13500
1000fa173.934
5.2. Case 2
Maximum unfactored axial force at support due to dead load:
ND N2D or ND 1.334 103 (kips)
Maximum unfactored axial force at support due to live load:
NL N2L or NL 69 (kips)
Corresponding unfactored moment within span due to dead load:
MD M2D or MD 30.16 (ft-kips)
Corresponding unfactored moment within span due to live load:
ML M2L or ML 1.212 103 (ft-kips)
The computed axial stress:
Nmax ND NL 1( ) 1 I( ) mDF( )
267
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
90.75
12 < 2200
1000 fa fb18.402 => OK Eq 10-166
Check Equation 10-47
faFa
fbFb
0.572 < 1 => OK
Find the Rating Factor
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-47 equal to 1:
Let: FL 1 I( ) mDF or FL 2.674
Given
ND RF NL 1( ) FL
As
Fa
MD RF ML 1( )1
11.18 T2D RF T2L FL
As Fe
FL 12
S
Fb1=
the rating factor is obtained by solving the above equation:
RF Find RF( )
RF 2.358
Check the ratio for the stiffeners:
60.75
8 < 2200
1000 fafb3
23.479 and 12 => OK Eq 10-164
and
40.375
10.667 < 2200
1000 fafb3
23.479 and 12 => OK
Check flange plate (for width between webs) :
240.75
32 < 5700
1000 fa fb47.679 => OK Eq 10-165
Check flange plate (for overhang width) :
268
Nguyenngoc Tuyen
NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
or Nmax 1.492 103 (kips)
faNmax
Asor fa 5.919 (ksi)
The computed bending stress at the extreme fiber:
Mmax MD ML 1( ) A3F 1 I( ) mDF( )[ ] or Mmax 6.407 103 (ft-kips)
fb12Mmax
Sor fb 12.442 (ksi)
Slenderness checks
Check web plates:
13500
w a
Dt f
Eq 10-163
Dtw
141 < 13500
1000fa175.467
5.3. Case 3
Corresponding unfactored axial force at support due to dead load:
ND N3D or ND 1.338 103 (kips)
Corresponding unfactored axial force at support due to live load:
NL N3L or NL 57.4 (kips)
Maximum unfactored moment within span due to dead load:
MD M3D or MD 389.96 (ft-kips)
Maximum unfactored moment within span due to live load:
ML M3L or ML 1.769 103 (ft-kips)
The computed axial stress:
Nmax ND NL 1( ) 1 I( ) mDF( )
269
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
90.75
12 < 2200
1000 fa fb16.236 => OK Eq 10-166
Check Equation 10-47
faFa
fbFb
0.693 < 1 => OK
Find the Rating Factor
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-47 equal to 1:
Let: FL 1 I( ) mDF or FL 2.674
Given
ND RF NL 1( ) FL
As
Fa
MD RF ML 1( )1
11.18 T3D RF T3L FL
As Fe
FL 12
S
Fb1=
the rating factor is obtained by solving the above equation:
RF Find RF( )
RF 1.772
Check the ratio for the stiffeners:
60.75
8 < 2200
1000 fafb3
21.927 and 12 => OK Eq 10-164
and
40.375
10.667 < 2200
1000 fafb3
21.927 and 12 => OK
Check flange plate (for width between webs) :
240.75
32 < 5700
1000 fa fb42.065 => OK Eq 10-165
Check flange plate (for overhang width) :
270
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
or Nmax 1.487 103 (kips)
faNmax
Asor fa 5.9 (ksi)
The computed bending stress at the extreme fiber:
Mmax MD ML 1( ) A4F 1 I( ) mDF( )[ ] or Mmax 6.179 103 (ft-kips)
fb12Mmax
Sor fb 11.999 (ksi)
Slenderness checks
Check web plates:
13500
w a
Dt f
Eq 10-163
Dtw
141 < 13500
1000fa175.754
5.4. Case 4
Corresponding unfactored axial force at support due to dead load:
ND N4D or ND 1.334 103 (kips)
Corresponding unfactored axial force at support due to live load:
NL N4L or NL 57.3 (kips)
Maximum unfactored moment within span due to dead load:
MD M4D or MD 142.9 (ft-kips)
Maximum unfactored moment within span due to live load:
ML M4L or ML 1.776 103 (ft-kips)
The computed axial stress:
Nmax ND NL 1( ) 1 I( ) mDF( )
271
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NMSUPINNED MODEL AASHTO HS20 TRUCK RATING ARCH-RIB
90.75
12 < 2200
1000 fa fb16.444 => OK Eq 10-166
Check Equation 10-47
faFa
fbFb
0.679 < 1 => OK
Find the Rating Factor
The Rating Factor is definded as a factor times the live load effect that makes the left part of equation 10-47 equal to 1:
Let: FL 1 I( ) mDF or FL 2.674
Given
ND RF NL 1( ) FL
As
Fa
MD RF ML 1( )1
11.18 T4D RF T4L FL
As Fe
FL 12
S
Fb1=
the rating factor is obtained by solving the above equation:
RF Find RF( )
RF 1.806
Check the ratio for the stiffeners:
60.75
8 < 2200
1000 fafb3
22.111 and 12 => OK Eq 10-164
and
40.375
10.667 < 2200
1000 fafb3
22.111 and 12 => OK
Check flange plate (for width between webs) :
240.75
32 < 5700
1000 fa fb42.604 => OK Eq 10-165
Check flange plate (for overhang width) :
272
Nguyenngoc Tuyen
NMSU
REFERENCES
American Association of State Highway and Transportation Officials (AASHTO). (1994). Manual for Condition Evaluation of Bridges, 2nd Edition, Washington, DC.
American Association of State Highway and Transportation Officials (AASHTO).
(1996). Standard Specifications for Highway Bridges, 16th Edition, Washington, DC.
American Association of State Highway and Transportation Officials (AASHTO).
(2002). Standard Specifications for Highway Bridges, 17th Edition, Washington, DC.
American Association of State Highway and Transportation Officials (AASHTO).
(2003). Manual for Condition Evaluation and Load and Resistance Factor Rating (LRFR) of Highway Bridges, Washington, DC.
American Association of State Highway and Transportation Officials (AASHTO).
(2004). LRFD Bridge Design Specifications, 3rd Edition, Washington, DC. American Institute of Steel Construction (AISC). (2001). Manual of Steel
Construction: Load and Resistance Factor Design, 3rd Edition, Chicago, IL. Merrick & Company. (1989). “Feasibility Study: Los Alamos Canyon Bridge Deck
Replacement.” Report prepared for Los Alamos National Laboratory, Los Alamos National Laboratory, Los Alamos, NM.
Minervino, C., Sivakumar, B., Moses, F., Mertz, D., and Edberg, W. (2004). “New
AASHTO Guide Manual for Load and Resistance Factor Rating of Highway Bridge.” J. Bridge Engrg., ASCE, 9(1), 43-54.
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